Rewards-Supply Aggregate Planning in the Management of Loyalty Reward Programs - A Stochastic Linear Programming Approach

Transcription

1 Rewards-Supply Aggregate Planning in te Management of Loyalty Reward Programs - A Stocastic Linear Programming Approac YUHENG CAO, B.I.B., M.Sc. A tesis submitted to te Faculty of Graduate and Postdoctoral Affairs in partial fulfillment of te requirements for te degree of Doctor of Pilosopy in Management Sprott Scool of Business Carleton University Ottawa, Ontario, Canada September YuengCao All Rigts Reserved

2 1*1 Library and Arcives Canada Publised Heritage Branc 395 Wellington Street OttawaONK1A0N4 Canada Biblioteque et Arcives Canada Direction du Patrimoine de I'edition 395, rue Wellington Ottawa ON K1A 0N4 Canada Your Tile Votre reference ISBN: Our file Notre reference ISBN: NOTICE: Te autor as granted a nonexclusive license allowing Library and Arcives Canada to reproduce, publis, arcive, preserve, conserve, communicate to te public by telecommunication or on te Internet, loan, distribute and sell teses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any oter formats. Te autor retains copyrigt ownersip and moral rigts in tis tesis. Neiter te tesis nor substantial extracts from it may be printed or oterwise reproduced witout te autor's permission. AVIS: L'auteur a accorde une licence non exclusive permettant a la Biblioteque et Arcives Canada de reproduire, publier, arciver, sauvegarder, conserver, transmettre au public par telecommunication ou par I'lnternet, preter, distribuer et vendre des teses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats. L'auteur conserve la propriete du droit d'auteur et des droits moraux qui protege cette tese. Ni la tese ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. In compliance wit te Canadian Privacy Act some supporting forms may ave been removed from tis tesis. Wile tese forms may be included in te document page count, teir removal does not represent any loss of content from te tesis. Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de cette tese. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. 1*1 Canada

3 Acknowledgements First of all, I would like to tank my supervisor Dr. Aaron L. Nsakanda for is patient guidance, valuable suggestions and enligtening comments during te period of my study in Carleton University. Tis dissertation owes a lot to is gentle, yet effective guidance. Witout is elp, tis dissertation could not ave been finised. His profound knowledge in management science definitely benefited me a lot. Working wit im as always been an enjoyable and rewarding experience. I would like to tank all te members of my tesis committee: Dr. Vinod Kumar, Dr. Micael Armstrong, Dr. Akif A. Bulgak, and Dr. Yiqing Zao wit individually and collectively contributed to te tesis. Secondly, I would like to tank te Sprott Scool of Business, Carleton University, for providing financial support and necessary researc facilities for tis tesis. It would ave been impossible for me to concentrate on my researc witout suc financial support. Many tanks go to members and staff of te Scool of Business, wo elped me during my studies at Carleton, in particular Melissa Doric, Greg Scmidt, and Jason Holtz. I would like to tank all of te great teacers and researcers I ave encountered ere at Carleton University, in particular Dr. Saobo Ji, Dr. Roland Tomas, and Dr. Uma Kumar. Finally, I would like to tank my parents, wose love, support, and sacrifice made me wat I am today. No words can properly express te love, gratitude, and admiration I ave for tem. 1

4 Abstract Loyalty reward programs (LRPs), initially developed as marketing programs to enance customer retention, ave now become an important part of customer-focused business strategies. One of te operational callenges faced by LRP managers is tat of planning for te supply of rewards in a given period of time. We ave developed tree matematical models for solving tis problem under various settings. In eac setting, te problem as been formulated as a two-stage stocastic linear programming model wit recourse. A euristic optimization procedure based on sample average approximation (SAA) is proposed for solving eac of tese models. We carried out extensive computational experiments to demonstrate te viability of te modeling and solution approaces for solving realistically sized (large-scale) problems as well as to evaluate te impacts of canges tat internal dynamics and external uncertainties ave on te performance of a loyalty reward program operating as a profit center. Findings from tese computational studies ave led to a number of managerial insigts. Our results sow tat demand variability as negative impacts on LRP performance. As suc, adopting an option contract provides good means for mitigation, especially wen demand uncertainty is ig. Our results ave also sown tat offering cooperative advertisement troug bonus points is a double-edged sword. It may bring in iger LRP profitability, but it also results in iger liability. Wen demand variability is ig, offering bonus points is not preferred in rewards-supply planning. Finally, our results indicate tat budget tigtness and liability control tigtness ave an impact on LRP performance to an extent tat varies across different system settings. Tis researc contributes to te literature in several ways: it syntesizes and extends te concept of supply cain management in te context of LRPs and it enances te understanding of LRPs and rewards-supply planning problems troug quantitative modeling and stocastic programming. Our findings will elp LRP managers to understand te roles of cooperative advertising troug bonus points and option contract in planning for te supply of rewards as well as to evaluate te impact of canges in te internal dynamics and external uncertainties on te performance of loyalty reward program operations. n

5 Table of Contents Acknowledgements Abstract List of Tables List of Figures List of Appendices i ii vi viii ix Capter 1 Introduction Background and Researc Motivation Researc Objectives Outline of te Tesis 8 Capter 2 Literature Review Loyalty Reward Programs (LRPs) Overview Typology Framework for LRPs Literature Review of LRPs Summary Supply Cain Contracts Overview Option Contracts Summary Cooperative Advertising Overview Cooperative Advertising and Budget Allocation Cooperative Advertising in LRP Operations Summary 51 Capter 3 Researc Framework and Matematical Models Loyalty Reward Programs - "Rewards-Points" Supply Cains LRP Rewards - Supply Aggregate Planning Models 57 in

6 3.2.1 LRP Rewards - Supply Planning Problem witout Bonus Points Modeling Assumptions Problem Description and Model Formulation LRP Rewards - Supply Planning Problem wit Bonus Points Modeling Assumptions Problem Description and Model Formulation LRP Rewards - Supply Planning Problem wit Option Contracts Modeling Assumptions Problem Description and Model Formulation Summary 78 Capter 4 Solution Metodology Stocastic Programming and Its Implementation Model Reformulation for Problem BP Model Reformulation for Problem EP Model Reformulation for Problem EP Solution Procedure Sample Average Approximation (SAA) Reformulation SAA Model for Problem BP-2SLPR SAA Model for Problem EP1-2SLPR SAA Model for Problem EP2-2SLPR SAA-based Heuristic Solution Procedure Implementation Issues in te Solution Procedure Summary 100 Capter 5 Design of Numerical Studies Procedure for Generating Testing Problems Testing te Effectiveness of te Solution Metodology Model Solvability Quality of Stocastic Solutions Testing te Impacts of Demand Variability Testing te Impacts of Budget Tigtness Testing te Impacts of Liability Control Tigtness 116 IV

7 Capter 6 Results and Analysis Testing te Effectiveness of te Solution Metodology Model Solvability Quality of Stocastic Solutions Testing te Impacts of Demand Variability Under BP Setting Under EP1 Setting Under EP2 Setting Comparison across BP, EP1, and EP2 Model Settings Summary Testing te Impacts of Budget Tigtness Under BP Setting Under EP1 Setting Under EP2 Setting Comparison across BP, EP1, and EP2 Model Settings Summary Testing te Impacts of Liability Control Under BP Setting Under EP1 Setting Under EP2 Setting Comparison across BP, EP1, and EP2 Model Settings More on Management Insigts from Liability Control Analysis Summary 200 Capter 7 Conclusions and Future Researc Directions Findings and Implications Limitations and Contributions Future Researc Directions 209 References 213 Appendices 222 v

8 List of Tables Table 2.1 Summary of te main streams of LRP literature 27 Table 2.2 Traditional supply cain versus "rewards-points" supply cain 34 Table 2.3 Summary of te main contributions of option contracts in OM/SC literature 42 Table 4.1 Summary of te models 101 Table 5.1 Problem generation parameters 103 Table 5.2 Problem generation parameters per partner type 104 Table 5.3 Summary caracteristics of te set of random generated problems 109 Table 5.4 Demand variability parameter and its values used for normal distributions Ill Table 5.5 Demand variability parameter and its values used for uniform distributions 112 Table 5.6 Values of LUB used for BP setting 118 Table 5.7 Values of L UB used for EP1 setting 119 Table 5.8 Values of LUB used for EP2 setting 119 Table 6.1 Quality of stocastic solutions of BP model 127 Table 6.2 Quality of stocastic solutions of EP1 model 129 Table 6.3 Quality of stocastic solutions of EP2 model 131 Table 6.4 Summary table of computational results wit different levels of demand variability under BP setting 134 Table 6.5 Summary table of computational results wit different levels of demand variability under EP1 setting 138 Table 6.6 Summary table of computational results wit different levels of demand variability under EP2 setting 142 Table 6.7 Impacts of budget tigtness under BP setting 151 Table 6.8 Impacts of budget tigtness under EP1 setting 156 Table 6.9 Impacts of budget tigtness under EP2 setting 160 Table 6.10 Computational results under BP setting 178 Table 6.11 Comparison of te budget usages under BP setting 180 Table 6.12 Computational results under EP1 setting 182 Table 6.13 Comparison of te budget usages under EP1 setting 184 Table 6.14 Computational results under EP2 setting 186 vi

9 List of Tables (Cont.) Table 6.15 Comparison of te budget usages under EP2 setting 190 Table 6.16 Comparison of te impacts of liability control on LRP profitability across different model settings 193 Table 6.17 Comparison of te impacts of liability control on cost of rewards across different model settings 194 Table 6.18 Comparison of te impacts of liability control on ordering quantity of rewards across different model settings 195 vn

10 List of Figures Figure 2.1 General LRP system 13 Figure 2.2 Organizational structure models: Type A (I, II, III) 14 Figure 2.3 Organizational structure models: Types B (I, II, III, IV) and Type C 15 Figure 2.4 Typology framework for LRPs 21 Figure 3.1 Conceptual model of a rewards-points supply cain 53 Figure 3.2 Illustration example of value creation 56 Figure 3.3 Effect of bonus points on accumulation demands 70 Figure 6.1 Estimated mean gaps wit different sample sizes and sample replications 122 Figure 6.2 Confidence interval upper bounds of te estimated mean gaps wit different sample sizes and sample replications 123 Figure 6.3 CPU times wit different sample sizes and sample replications 124 Figure 6.4 LRP profitability under BP, EP1, and EP2 settings 163 Figure 6.5 Liability ratios under BP, EP1, and EP2 settings 167 Figure 6.6 Ordering quantity of rewards under BP, EP1, and EP2 settings 171 Figure 6.7 Comparison of budget usage ratios across different model settings 195 vin

11 List of Appendices Appendix A.l Sample of LRPs in today's marketplace in Canada 223 Appendix B. 1 Computational outputs for examining BP model solvability and determining sample size and sample replications 228 Appendix B.2 Computational outputs for examining EP1 model solvability and determining sample size and sample replications 229 Appendix B.3 Computational outputs for examining EP2 model solvability and determining sample size and sample replications 231 Appendix C. 1 Summary table of bonus points to offer wit different levels of demand variability under EP1 Setting 234 Appendix C.2 Summary table of options to purcase and to exercise wit different demand variability under EP2 Setting 237 Appendix C.3 LRP profitability comparisons across BP, EP1, and EP2 239 Appendix C.4 Liability ratio comparisons across BP, EP1, and EP2 242 Appendix C.5 Budget usage comparisons across BP, EP1, and EP2 245 Appendix D. 1 Comparison of te impacts of budget tigtness wit a given level of demand variability under BP setting 249 Appendix D.2 Comparison of te impacts of demand variability wit a given level of budget tigtness under BP setting 252 Appendix D.3 Comparison of te impacts of budget tigtness wit a given level of demand variability under EP1 setting 256 Appendix D.4 Comparison of te impacts of demand variability wit a given level of budget tigtness under EP1 setting 259 Appendix D.5 Comparison of te impacts of budget tigtness wit a given level of demand variability under EP2 setting 263 Appendix D.6 Comparison of te impacts of demand variability wit a given level of budget tigtness under EP2 setting 266 Appendix D.7 Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability 270 ix

12 List of Appendices (Cont.) Appendix E. 1 Extended matematical model for Problem BP wit revenue-saring type of contracts 283 Appendix E.2 Extended matematical model for Problem EP2 wit multi-layer of contracts 286 x

13 Capter 1 Introduction Tis capter provides an introduction to te study. It starts by examining te status quo of loyalty reward programs (LRPs) wile introducing te concept and management of LRPs. After tat, te motivation of te study and te researc objectives are discussed. Finally, te organization of te tesis is presented. 1.1 Background and Researc Motivation Since te debut of te American Airlines' AAadvantage program in 1981 (Duffy, 1998), LRPs ave been employed by a wide range of companies in te consumer goods and service industries. LRPs are today being offered in a number of industries suc as airline, retail, otels, financial services, telecommunications, and gaming and entertainment. In te airline industry alone, more tan 130 companies ave an LRP, and 163 million people trougout te world collect loyaltybased air miles 1. LRPs ave been quite popular in te United States, United Kingdom, Canada, and a ost of oter countries. Recent studies (e.g., Berman, 2006) sow tat no less tan 90 percent of te consumers in tose countries ave at least one loyalty card. Many of tem are enrolled in multiple loyalty reward programs. Altoug various LRPs exist in today's business world, te fundamental business logic beind tese programs is te same: to offer consumers incentives or rewards for repeat business. Tese incentives or rewards, in turn, serve as motivation for consumers to continue buying products from te same product provider. Terefore, LRPs ave been widely adopted as an important component of customer relationsip 1 "Funny Money," Economist, December 24, 2005,

14 management strategy tat targets long-term customer profitability (e.g., Jain and Sing, 2002; Liu, 2007; Meyer-Waarden, 2008). Generally speaking, in a typical LRP system, tere are four key players: LRP members, an LRP ost, LRP partner(s), and LRP service provider(s). LRP members refer to te end consumers wo own a member account and/or a membersip card. LRP ost is te firm tat owns or manages te program. LRP partners refer to te business entities oter tan te ost firm, wo join te program to offer accumulation and/or redemption options to LRP members. LRP service providers refer to firms wic provide tecnical support (e.g., maintaining a customer database tat stores all LRP members' purcase istory) for te LRP business. In suc a system, LRP members, based on some specified accumulation sceme, earn loyalty units (e.g., points or miles) along wit teir purcases of products trougout te network of LRP commercial partners. Tese units can later be redeemed based on a "reward cart" pre-establised by te LRP ost. Loyalty units tat are not redeemed by LRP members are saved in a "reserve" account and constitute te LRP outstanding balance (referred to as "liability" 1 ). Loyalty units (e.g., points) earned by LRP members during a given period (e.g., a year) constitute te LRP accumulated points (referred to as "accumulation"), wereas loyalty units redeemed by LRP members for rewards during a given period constitute te LRP redeemed points (referred to as "redemption"). In recent years, tere as been a general recognition in te industry of a need for more sopisticated loyalty-based systems capable of responding to long-term competitive treats suc 1 Liability refers to te value of future redemption obligation of loyalty units (e.g., points or miles) earned by LRP members in an LRP. 2

15 as retail overcapacity, spending on mass advertising, and consumer attrition issues. Two new trends of LRP development ave been noticed. One trend is tat some of te LRP service providers ave replaced te traditional LRP ost firms (e.g., airline companies or retail firms) to become LRP osts temselves and treat LRP as teir core business. For tese loyalty-based service companies, te primary source of teir revenue comes from te sale of loyalty units to commercial partners' (referred to as "accumulation partners"). Tese LRP osts incur teir main costs at te time an LRP member redeems points for rewards, as te LRP ost as to purcase te rewards from its commercial partners (referred to as "redemption partners"). Anoter trend is tat in order to compete effectively and to continue contributing to value growt, many existing LRPs ave been restructured or expanded in scope to partner wit oter non-lrp firms to offer new products and/or services. As suc, te competition between tese LRPs is no longer among individual business entities, but among te networks of business entities involved in tese LRPs. Te growt of LRPs in recent years as led to a considerable increase in teir management and control complexities. For example, Aeroplan, Canada's premier loyalty program, was founded in 1984 by an airline company (Air Canada) as an internal marketing program. Since ten, Aeroplan as experienced organizational restructure and expansion several times. Now te program is owned and operated by Group Aeroplan Inc., a loyalty-service-oriented company. At present time, Aeroplan as an accumulation and redemption network of over 75 commercial partners representing more tan 150 brands. In 2010, more tan two million rewards were issued to members. Te revenue from te sale of Aeroplan 2 Miles was $1,033 million, and te total cost of rewards was $665 million in te year Air 1 ttp:// Reports/ MDA.pdf (page 7, accessed in July, 2011) 2 ttp:// Reports/ MDA.pdf (page 15, accessed in July, 2011) 3

16 Miles, a primary competitor of Aeroplan founded in 1992, as more tan 9.5 million active mile collector accounts, representing approximately two-tirds of all Canadian ouseolds. Te program is owned and operated by anoter loyalty-service-oriented company named LoyaltyOne, Inc. Air Miles offers its members more tan 1,200 different leisure, entertainment, mercandise, travel, and oter lifestyle rewards wen members sop at one of more tan one undred brand-name sponsors of te program 1. Despite te wide adoption of LRPs in te business world, te increasing economic impacts of tese programs, and te increase of te complexities in managing tem, tere are few academic models tat specifically deal wit LRPs to support planning and operational decision making. Te majority of te existing papers are limited in teir coverage of marketing-oriented LRP management problems. Many LRP operational issues ave not yet been fully explored. Tese issues relate to, for instance, prediction and control of liability; sort-, medium-, or long-term planning for rewards and points supply; accumulation and redemption demand forecasting; contract design for coordination between LRP ost and partners; and revenue assessment. Tis motivated us to devote our researc efforts to addressing operational issues in LRP management. 1.2 Researc Objectives A crucial operational issue faced by LRPs is tat of planning and managing te supply of rewards (and points) efficiently and effectively in order to acieve management goals - suc as meeting customer demand, improving customer satisfaction, lowering operational costs, or generating iger profits - wile taking into account bot internal (e.g., resource limitations, 1 ttp://lovalty.com/busmcss/air-rniles-rcward-program (accessed in July, 2011) 4

17 management requirements or targets) and external dynamics (e.g., demand uncertainties or competition treats). In fact, LRP managers rely on good planning for rewards-supply to maintain a balance between te customer service levels and te overall costs of rewarding customers, and to assess te growt of te program and te risk level associated wit tis growt. Te lack of availability of rewards at te time of redemption results in a poor service level and/or an increase of te reward supply costs to meet customer demands, since te LRP ost will ave to acquire te additional rewards at a iger cost. On te oter and, too muc availability will result in a iger cost as well (altoug te level of customer service would be ig in tat case). Te unused rewards availability will result in a penalty wenever te LRP ost decides to reduce or cancel teir reservation of rewards or return unused rewards to reward suppliers (i.e., LRP partners). In effect, in setting up long-term contracts wit partners, te LRP ost must decide te volume of rewards to purcase in advance. Moreover, good planning for rewards-supply provides LRP managers wit te ability to develop promotion plans tat seek for better management of redemption demand among multiple partners, or for better management policies to mitigate te risks associated wit te increases of liability. It is for tese reasons tat we explore te planning of te supply of rewards in tis study. More specifically, we focus on examining aggregate rewards-supply decisions and te associated LRP performance for a single-period planning orizon. We study te planning issue from only te LRP ost point of view. Our researc objectives are defined as follows: 5

18 (1) Develop an analytical model to cope wit rewards-supply planning decisions in te presence of (a) multiple commercial partners wo are involved in te redemption and accumulation business to offer various points collection and redeeming options to LRP members; (b) multiple resource constraints suc as budget and capacity constraints; (c) multiple management concerns including LRP profitability, liability control, and demand uncertainties. (2) Examine te impacts of internal dynamics and external uncertainties on LRP profitability and an LRP ost's rewards-supply planning decisions. Te internal dynamics tat we focus on in tis study are te canges in rewards budget and te canges in target liability. Te external uncertainties tat we focus on are LRP members' accumulation and redemption demand uncertainties. (3) Investigate te role of cooperative advertising in dealing wit internal dynamics and external uncertainties in LRP rewards-supply planning. It is a common practice in LRPs tat LRP members can receive "bonus points" wen tey purcase specified products or services from some LRP partners during a certain time period. Tis kind of advertising/promotion activities is offered by an LRP ost as a type of cooperative advertising between te LRP ost and its partners. As pointed out by practitioners 1 : "te timing and amount of bonus points increase (point) liability, and sould be used strategically to drive beaviors tat continuously increase spending or create a deeper sense of loyalty"), bonus points not only ave an impact on LRP members' accumulation demand but also on liability. 1 Sneed, G.L "Points Liability - Enoug, or Too Muc?" ttp://newsoct05.maritzloyalty.us/..index.ptml (accessed in July, 2011) 6

19 (4) Investigate te role of option contracts in dealing wit te internal dynamics and external uncertainties in rewards-supply planning. In te supply cain and operations management (SC/OM) literature, option contracts are widely regarded as a type of SC coordination mecanism between commercial suppliers (e.g., manufacturers) and buyers (e.g., retailers). Researcers ave pointed out tat option contracts can provide more flexibility in coping wit demand uncertainties. In an LRP, redemption partners can be viewed as te suppliers of rewards and te LRP ost can be viewed as a buyer of rewards. Terefore, instead of resorting to a traditional supply contract, suc as wolesale-priceonly, we assess weter option contracts can provide more flexibility to cope wit internal dynamics and external uncertainties in rewards-supply planning problem. Tis study contributes to LRP literature and practices by addressing te following researc questions: 1) How can te rewards-supply planning problem (RSPP) be formulated matematically and solved effectively so as to ensure tat good quality solutions can be obtained for largescale problems in a reasonable computational time? 2) How does demand variability affect LRP performance in terms of LRP profitability, liability, and cost of rewards? Do te impacts of demand variability vary among different model settings? 3) How does budget tigtness affect LRP ost decisions as well as LRP profitability and liability? Do te impacts of budget tigtness vary among different model settings and under different levels of demand variability? 7

20 4) How does te liability control tigtness affect LRP ost decisions as well as LRP profitability and reward costs? Do te impacts of liability control tigtness vary among different model settings and under different levels of demand variability? 1.3 Outline of te Tesis Te tesis is divided into seven capters. Te next capter reviews te literature tat is most relevant to tis study. Te literature review is based on tree areas: loyally reward programs, option contracts, and cooperative advertising. In addition, a typology framework is presented to provide a guideline for understanding te features of LRPs. Capter 3 first describes te teoretical background used to develop our analytical models, and ten provides a detailed explanation of te matematical models, including model assumptions, problem description, and model formulation. Capter 4 reports on te solution metodology. It starts wit a brief introduction to stocastic linear programming and ten presents a euristic-based solution procedure for solving our models. Te callenges involved in te implementation of te solution procedure are also discussed. Capter 5 describes te design of our numerical studies carried out troug computer simulation. Te computational results under eac model setting, as well as across te model settings, are presented and analyzed in Capter 6. Finally, Capter 7 igligts te main findings and managerial implications, summarizes te contributions and limitations of tis study, and discusses future researc directions. 8

21 Capter 2 Literature Review Tis capter provides an overview of previous studies on tree subjects: LRPs, option contracts, and cooperative advertising. It begins by reviewing literature on LRPs and examining te status quo of LRPs in today's marketplace. Tis leads to proposing a typology framework to capture te variety of LRPs. Te type of LRPs tat tis researc focuses on is ten presented in detail. Next, te literature on supply cain contracts is discussed, especially option contracts and teir modeling in te literature. After tat, te concept of cooperative advertising is introduced and a brief review of te literature on advertising budget allocation and advertising-sales response function follows. Eac section contains a summary of te relevance of previous studies and of tis researc work. Because LRPs, cooperative advertising, and option contracts are tree broad researc areas, we limit our literature review to te concepts and models tat are relevant to our study. 2.1 Loyalty Reward Programs In tis section, we first introduce te concept of LRP and provide a typology framework of LRPs. Ten a survey of tirty-nine well-known LRPs in today's marketplace is provided. After tat, we review and discuss te main researc streams in te LRP literature. Our view of LRPs in tis study is also presented Overview LRPs are also known as loyalty programs, rewards programs, "frequent-sopper" programs, "frequent-guest" programs, and "frequent-flier" programs. "Loyalty" and "reward" are te core 9

22 concepts in tese programs. More specifically, loyalty is te primary purpose of LRPs and reward is te key instrument for attaining it. According to Oliver (1999), customer loyally is a deeply eld commitment to buy or patronize a preferred product or service consistently in te future. Altoug academia still debates te meaning of customer loyalty (e.g., Dick and Basu, 1994; Palmer, 1996; O'Malley, 1998), tere is no doubt tat customer loyalty as become an important asset to a firm (Liu, 2007; O'Brien and Jones, 1995). It as long been proven in psycology studies (e.g., Ebert, 2003) tat reward as a strong impact on a person's decision making as well as on beavior modification. In LRPs, rewards refer to all kinds of incentives suc as discounts, rebates, free goods, or special services. Tese rewards are designed to encourage customers to keep doing business wit one firm or a group of firms sponsoring te same LRP rater tan wit competitor firms. In te literature, tere is no universal definition of LRPs because researcers view tem in different ways. Some of te researcers (e.g., Taylor and Neslin, 2005; Berman, 2006; Nunes and Dreze, 2006) take a broad view and consider any business/marketing program tat uses certain formats of rewards to enance repeated purcases by customers as LRPs. Using tis definition, irrespective of weter a program uses a simple format like "buy a cup of coffee ten times and get one free at te elevent time" or applies a complex structure as in te case of Aeroplan, it will be designated as LRP. However, oter researcers old te view tat some of te simple format programs suc as "20% off towards your next purcase" are not true LRP programs because tese programs do not reward loyal beavior on te basis of a customer's purcase istory of products/services. In oter words, tese programs focus on sort-term profits 10

23 rater tan long-term customer assets (Sugan, 2005). Terefore, we propose a different definition of LPRs, wic is based on literature and our studies. A business/marketing program may be considered as a loyalty reward program (LRP) wen it displays all of te following features: Te program targets customers' long-term profitability or customers' life-time value. Te program requires customer enrollment. Te program collects customer information and records customers' purcase istory of products/services troug membersip cards, co-branded credit cards, or identification numbers (e.g., login ID) tat customers use. Te program rewards repeated customer purcase beavior on te basis of customer's purcasing istory. Te program as clear reward scemes explicitly stating ow customers will be rewarded and te benefits tat customers can obtain after tey join te program. Despite te increasing use of LRPs worldwide, and proliferation of a large variety of LRPs in recent years, few studies ave addressed classification of LRPs, except Kadar and Kotanko (2001), Bagdoniene and Jakstaite (2006), and Berman (2006). Kadar and Kotanko (2001) studied LRPs in terms of organizational structure and classified LRPs into tree categories: exclusive one-company programs, inclusive company-specific programs, and cross-company programs. Exclusive one-company programs refer to programs operated and controlled solely by one firm; inclusive company-specific programs refer to programs wic ave been extended to include a 11

24 number of partners; and cross-company programs refer to te programs created by a group of companies togeter. Focusing on te customer perspective, Bagdoniene and Jakstaite (2006) classified LRPs into open or closed programs, programs for end customers or for intermediate customers, and direct or indirect programs. In an open program all customers are eligible to join te program, as long as tey purcase goods or services from te LRP firm. In a closed LRP program only desired or invited customers can join. Programs for end customers are te business to consumers (B2C) LRPs, wereas tose for intermediate customers are business to business (B2B) LRPs. Direct LRPs provide customers some financial benefits as rewards, suc as permanent discounts, gift cards, or free products. In contrast, indirect LRPs provide customer rewards tat are non-financial privileges (e.g., pre-board services or free access to te business lounge at airports) (Bagdoniene and Jakstaite, 2006). Berman (2006) in is study discussed four types of LRPs in terms of reward sceme: members receive additional discount at register, members receive one unit free wen tey purcase n units, members receive rebates or points based on cumulative purcases, and members receive targeted offers and mailings. We found tat altoug tese classifications touced different aspects of LRPs, none of tem capture te full expanse of te variety of LRPs in today's marketplace. Terefore, we address tis issue and propose a compreensive framework for LRP classification in te next section. Our typology framework was developed based on business to customers (B2C) LRPs only Typology Framework for LRPs Te typology tat we propose ere is based on te relationsip view of an LRP system. In general, tere are five fundamental entities involved in an LRP system (see Figure 2.1). Te ost 12

25 is te business entity tat launces or owns te program. LRP is te program itself Partners (also known as sponsors) refer to te business entities tat join te program to provide redemption and/or accumulation options. Members are te consumers wo participate in te program to redeem rewards troug purcasing products or services from osts or partners/sponsors. Service providers are te business entities tat provide service or tecnical support for te program but do not communicate wit te members directly (Nsakanda et a, 2006). v4=l Dimensions: \ Redemption ; (and accumulation) Sceme Dimensioni: Organizational Structure Figure 2.1: General LRP system (source: Nsakanda et a, 2006) Te variety of LRPs can be viewed as arising from te different relationsips among tese entities. As bot organizational structure and redemption (and accumulation) sceme are te key determinants of te relationsips among tese entities, we treat tem as two dimensions of te framework. Organizational structure defines te relationsips among entities in te lower triangle; wile te redemption (and accumulation) sceme defines te relationsips among te entities in te upper triangle, especially among members and te oter tree entities (i.e., ost, LRP, and partners). 13

26 Typology Dimension One Organizational Structure Organizational structure defines te relationsips among LRP ost, LRP service providers, and LRP partners. Along tis dimension, LRPs are grouped into tree categories: Type A, single sponsor programs; Type B, multi-sponsor programs; and Type C, joint programs. Under types A and B, tere is a subdivision, wic we ave adopted from Gudmundsson et al. (2002) (see Figure 2.2). Gudmudsson et al. (2002) identified tree internal structure models of airline frequent flier programs (FFPs). We find tat tese structure models are also common in LRPs oter tan FFPs. Te graps in Figure 2.2 sow te differences among tese structure models. Type A Single-sponsor LRPs Host Host Host Type A-I Type A-II Type A-III Figure 2.2: Organizational structure models: Type A (I, II, III) (source: Gudmundsson et al., 2002) Types A-I, A-II, and A-III illustrate structures were an LRP is fully and solely owned by an LRP ost. In Type A-I, te LRP is an internal unit of te LRP ost and fully managed by te ost. Type A-II represents te structure were an LRP is partially managed by an LRP ost and some of te management functions are outsourced to oter firms or a tird party; wereas in te Type A-III model, all of te management functions of te LRP are outsourced. Te "exclusive one-company programs" mentioned in Kadar and Kotanko (2001) are quite similar to te Type A-I LRPs tat we define ere. 14

27 In te above models, te LRP ost is te sole sponsor offering accumulation and redemption to LRP members. In general, suc LRPs are limited in flexibility and are narrow in scope. We noticed tat in recent years many existing LRPs ave been restructured to contribute to value growt. Some of tose tird-party service agents in Type A-III model ave replaced te traditional LRP ost enterprises (e.g., airline companies or retail companies) to become LRP osts temselves. Meanwile, in order to compete effectively, LRP osts ave started to offer products and services in different categories troug partnersip wit oter non-lrp firms (e.g., Hofer, 2008). Terefore, Type B and Type C structure models (see Figure 2.3) ave appeared in recent years and ave become more and more popular in large-scale LRPs. Type B Multi-sponsor LRPs Type C Joint LRPs Type B-I Type B-II Type B-III Type B-IV Figure 2.3: Organizational structure models, Types B (I, II, III, IV) and Type C Type B models (I, II, and III) are extensions of Type A models (I, II, and III). In tese Type B models, LRPs follow te same ownersip and management structures as tose in Type A models, but ave multi-redemption and/or multi-accumulation partners/sponsors. Type B-IV model represents te structure were LRP and related services are te focal business of te ost firm (e.g., Aeroplan, Air Miles). Type B LRPs are known as multi-sponsor programs or coalition programs. Te "more inclusive company-specific program" in Kadar and Kotanko (2001) is similar to a Type B-I LRP. 15

28 In contrast to tese structure models, Type C model represents te structure in wic an LRP does not belong to any individual firm (i.e., no sole ost) and is formed wen a number of firms band togeter to develop a joint program, known in te literature as joint LRP program 1 or crosscompany program (e.g., Kadar and Kotanko, 2001). In Types B and C, customer loyalty is no longer built around a product or a company but around te LRP program and te associated reward system. Suc programs not only possess significant advantages in operational scale and offer a wide range of benefits to members, but more importantly tey can leverage teir customer bases for cross-selling (Kadar and Kotanko, 2001). In tese LRPs, eac company brings different capabilities to te table and eac may take away a different form of value. In tis way, tey are structured as win-win solutions for bot LRP ost and LRP partners. Typology Dimension Two Redemption (and Accumulation) Sceme Te second dimension of te typology framework is based on redemption (and accumulation) sceme. Redemption (and accumulation) sceme primarily defines te relationsip between LRP members and te LRP. It is also identified in te literature as an element tat is essential to te administration and positioning of LRPs. Fundamentally, tere are two strategies for designing redemption (and accumulation) scemes: static and dynamic. Te term 'static' refers to tose scemes tat do not cange over time. Tey usually take te form of "one sceme applicable for all LRP members". Tis type of sceme is 1 "A tang of bitter-sweet loyalty". Brand Strategy, Oct

29 used quite often in large retailing industries suc as Te Bay, Soppers' Drug Mart, and Loblaws. Common rewards offered to LRP members are cas back and gift cards. Tese firms ave a large customer base and tey essentially deal wit products. Teir marginal cost and revenue per customer are low. In tese firms, LRPs are viewed purely as marketing tools. Dynamic reward scemes are quite often used by enterprises tat specialize in LRPs or large service-industry companies tat ave a large number of partners. Te pricing, type, and timing of redemption (and accumulation) in tese scemes cange over time, and are structured differently for different member segments. Dynamic reward scemes offer LRP members more accumulation and redemption coices. On te oter and, members need to put in muc more effort in order to take advantage of te 'ever-canging' scemes. Altoug te design and administration costs are iger, dynamic scemes ave muc more flexibility and capability to improve te profitability of LRPs. Compared to te dynamic scemes, te static scemes are muc simpler, requiring lower design and administration costs, and lesser learning effort on te part of members. Overall, LRPs can be classified as eiter using a static sceme strategy or using a dynamic sceme strategy. In addition, no matter wic design strategy is used, an LRP can be furter classified based on four oter sceme-related criteria: reward medium, redemption (and accumulation) timing, reward type, and redemption (and accumulation) grid. Reward medium. In many LRPs, te relationsip between a member's purcase effort and a final outcome is mediated by te presence of an intermediate currency known as 'reward medium' or 'loyalty unit' (i.e., points or oter excange units). As Duffy (1998) pointed out, te 17

30 communication between an LRP ost and LRP members may get confusing wen no proper unified medium is used. Points, miles, or voucers are te most popular reward mediums tat are used to link members' spending to rewards. Most existing LRPs use a single-medium (e.g., expenses -> points -> rewards), te rest are eiter no-medium LRPs (e.g., expenses -^ rewards) or multi-medium LRPs (e.g., expenses -> points -> voucers -^ rewards). Hence, reward medium refers to weter a program currency is used or not and wic program currency is used. Te paper by Si and Soman (2004) was te first study to examine te effectiveness of an LRP from te reward-medium perspective. Tey proposed an analytical model to formulate customer valuation on single-medium (i.e., points only) and multi-medium (i.e., points and voucers) LRPs and conducted laboratory experiments to compare te impact of single-medium vs. multi-medium on member valuation of LRPs. Tey found tat te multi-medium LRP is more attractive to LRP members, wic in turn results in a positive effect on member purcasing beavior. In real-life LRPs, altoug most of te programs use at least one reward-medium, tere are a few programs in wic LRP members are rewarded witout using any reward-medium (e.g., M&M Max, Reservation Rewards, ETR Rewards, see in Appendix A.l). In tose programs, LRP members are rewarded directly based on teir spending. Terefore, LRPs can be classified based on reward medium as: no-medium LRPs, single-medium LRPs, or multi-medium LRPs. Reward type. Reward type refers to te type of rewards offered by an LRP. Tis criterion as been studied in previous LRP literature. Dowling and Uncles (1997) classified LRPs into eiter direct-reward LRPs or indirect-reward LRPs. A direct-reward LRP refers to an LRP offering 18

31 rewards tat directly support te value proposition of te products and/or services tat te LRP ost or sponsors provide (e.g., gift card, free tickets). Te main purpose of offering a directreward is to keep customer loyalty on a single product, one company, or one LRP brand (e.g., Aeroplan). An indirect-reward LRP is defined as an LRP presenting rewards tat indirectly cause te LRP members to buy products or services. Tis type of reward as no linkage wit products/services. Usually, tis type of reward is money-oriented. Discounts, rebates, and casback are te typical formats of indirect rewards. As te money tat customers get back can be used elsewere, indirect rewards in most cases cannot lock a customer to furter purcases or to use te products or services provided by an LRP. Following Dowling and Uncles' classification, Kim et al. (2001) examined te decisions on selecting direct- or indirect- rewards (i.e. firm's own products/services vs. cas) wen te firm faces different customers (eavy vs. ligt users or price sensitive vs. insensitive users). In today's marketplace, fewer LRPs offer indirect-reward solely because it is believed tat direct-reward is more appropriate for creating loyal customers. Some LRPs offer bundles of direct- and indirect- rewards to teir members (Nunes and Park, 2003). Terefore, under tis criterion, LRPs can be grouped into LRPs offering indirect rewards, LRPs offering direct rewards, or LRPs offering mixed rewards (i.e. bundles of direct- and indirect- rewards). Redemption (and accumulation) timing. Redemption (and accumulation) timing is used by some researcers to refer to weter an LRP offers immediate or delayed rewards (e.g., Dowling and Uncles, 1997, Zang et al, 2000; Yi and Jeon, 2003; Ke and Lee, 2006). Delayed rewards are benefits and incentives tat are obtained or are redeemable at a later date from te point of 19

32 sale. Conversely, immediate rewards refer to benefits tat are experienced at te point of transaction. Examples of immediate rewards include direct-mail coupons, discounts, or price cuts offered to customers at te point of transaction (Dowling and Uncles, 1997). However, as immediate rewards do not relate to a customer's cumulative purcasing beaviour over time, tey are less effective in retaining consumers tan delayed rewards (Zang et al, 2000). From our point of view, redemption (and accumulation) timing involves te variable of time in te redemption (and accumulation) sceme. For example, in some LRPs, points accumulated or rewards available for redemption expire after some days. Obviously, using redemption (and accumulation) timing can increase flexibility in LRP operations; owever, it creates time pressures on LRP members. Members must remain active in order to keep teir status in te program. For instance, members are required to acquire or redeem points, sometimes up to a certain minimum level witin a certain time period. Tis type of requirements may ave negative impact on members' valuation of te program. Atak (2005) found tat te potential of an LRP to attract members is determined by te value of te rewards it offers, as well as by te timing of te rewards available. In reality, weter to use redemption (and accumulation) timing or not depends on an LRP's overall strategy. Some LRPs use redemption (and accumulation) timing eavily, wile oters coose not to use it at all 1. Terefore, based on te redemption (and accumulation) timing criterion, LRP can be classified into two categories: LRPs using reward timing or LPRs not using reward timing. Wells, Jennifer, "Exporting te Loyalty Business: Hoarding, Frustrating, Winning," ttp:// [Accessed Sept. 30, 2008] 20

33 Redemption (and accumulation) grid. Redemption (and accumulation) grid refers to te detailed reward prices (and accumulation options) offered for members. Redemption (and accumulation) grid can be stated in terms of number of points or miles, amount of LRP members' spending, members' purcase frequency, or members' portfolio. Here portfolio refers to, for instance, te combination of members' spending, product category, and membersip 'status' or 'tier'. Terefore, based on tis criterion, LRPs can be classified as LRPs using an amount-based grid, LRPs using a frequency-based grid, or LRPs using a portfolio-based grid. To our knowledge, earlier researc work as not discussed tis criterion formally. Organizational _ structure Single-sponsor: Type A (I, II, m) Multi-sponsor: Type B (I, II, in, IV) Joint: Type C LRP typology framework Redemption (and accumulation) - sceme Design strategy static dynamic no-medium Reward medium j. singfe-medium multi-medium Reward type Redemption (and accumulation) timing direct indirect mixed use not use Redemption (and " based on amount accumulation) grid - based on frequency. based on portfolio Figure 2.4: Typology framework for LRPs Based on te above discussion, we present a two-dimensional typology framework in Figure 2.4 below. Tis framework elps researcers and practitioners to distinguis different facets of LRPs 21

34 systematically and to identify te key caracteristics tat are important for LRP design and implementation. Due to te large number of LRPs in today's marketplace, a sample of tirty-nine well-known LRPs in Canada (see Appendix A.l) is surveyed according to tis typology framework. Among te tirty-nine LRPs, seventeen programs are of Type A, twenty programs are of Type B and te two remaining programs are of Type C. In te table of Appendix A. 1, we also pointed out weter eac LRP runs as a profit center or a cost center. Generally speaking, te LRP tat runs as a profit center is an independent business entity and its management focus is on creating revenue directly troug te program and te associated business. Taken in tis sense, only Type B-IV and Type C LRPs can be profit centers, and te oter types of LRPs are eiter cost centers or semi-cost centers. In a cost center LRP, management focus is on te LRP's contributions to te ost firm's focal business. In a semi-cost center LRP, te management focus remains te same as tat in a cost center LRP; but te ost firm also attempts to create extra revenue troug LRP operations directly. For example, te LRP ost may sell points or offer LRP-related services to commercial partners (Nunes and Dreze, 2006) Literature Review of LRPs In tis section, we provide a brief review of LRP literature from tree perspectives: researc topic, researc metodology, and types of LRPs tat ave been examined. 22

35 Researcers ave addressed various issues regarding LRPs. However, te majority of tese studies are from marketing and economics fields. Economics researc ave focused on understanding te function of LRPs and te underlining mecanism, suc as switcing cost, point economics, etc. (e.g., Klemperer 1987, 1995; Caminal and Matutes, 1990; Carlsson and Lofgren, 2006). In marketing literature, tere are tree main streams of LRP researc. One stream of researc aims to provide management insigts on designing and implementing LRPs (e.g., Kadar and Kotanko, 2001; Sugan, 2005; Berman, 2006). Te second stream of researc focuses on te sort- and/or long-term impact of LRPs on consumer purcase beaviour, attitude, and decision (e.g., Sarp and Sarp 1997; Lewis, 2005; Meyer-Waarden, 2008). Te tird stream of researc explores te influence of LRPs on a firm's market competition (e.g., Caminal and Matutes, 1990; Kim et al, 2001; Sing et al, 2008). Altoug all tese studies are relevant to te general management of LRPs, te models proposed in tese papers do not support planning and operational decision-making. Tis as left LRP managers trying to address te callenges raised in LRP operations suc as revenue or cost optimization, rewards supply planning, accumulation and redemption demands forecasting, etc. by using teir experience and intuition rater tan analytical-based approaces. However, wit te growt in bot size and complexity of LRPs, it is impossible for LRP managers to make proper operational decisions witout using any analytical approaces. In recent years, a few studies ave appeared in te literature tat attempt to address some specific issues in LRP operations. Kim et al. (2004) focused on te adoption and designing issues of LRPs in te context of capacity management. Tey examined te interdependency between 23

36 excess capacity and price competition in te market. Tey sowed tat for tose firms wo face demand uncertainty and strict sort-term capacity constraint, an LRP can be used as a flexible tool to adjust available capacity to market demand. Teir analysis includes a discussion on reward amount, but te main purpose is to examine te firms' dynamic price decisions. Gandomi and Zolfagari (2011) developed a stocastic consumer beavior model to investigate te effectiveness of LRPs. Tey examined ow customers' valuations and satisfaction levels affect LRP profitability in a two-period time orizon. Tey proposed a customer-oriented model in wic customer valuations of te programs and customer satisfaction levels are modeled as stocastic variables. Teir computational results sow tat te effectiveness of LRPs depends on te mean and variance of customer satisfaction levels. Firms may not be always better off wen tey offer a loyalty program. Te above OM-oriented LRP researc sows te increasing attention in te OM literature to explore te dynamics in customer buying beaviour and te triggers beind tese dynamics. Te autors focused on te value of LRPs on a firm's focal business rater tan te management of te LRP itself and LRPs are treated as cost centers. All tese studies are deeply rooted in previous marketing or economic teories and analytical models. In two recent papers, Diaby and Nsakanda (2008) and Nsakanda et a (2010), a closer look at LRP operations was given from a different angle. Te autors considered te LRP as a dynamic system were LRP members are grouped into membersip tiers at te beginning of eac qualifying period and member accumulation and redemption activities trigger member migration from one tier to anoter. Diaby and Nsakanda (2008) focused on determining te "breakage 24

37 rate" 1 in Type A-I or Type B-I LRPs. Tey developed a quantitative model and a simulation process in wic te accumulation and redemption of points follow a stocastic process. Tey also discussed key management issues wen an LRP is run as a profit center. Following te same system view of LRPs, Nsakanda et al. (2010) proposed a predictive model of redemption demand and liability level in LRPs tat ad not been explored in any previous LRP literature. Tey also discussed practical implementation of te model in te context of a frequent flyer program. As te autors pointed out, altoug teir metodology was developed based on a simple multi-tier LRP (i.e., Type A-I wit one accumulation and redemption coice in eac tier) it can be expanded to accommodate a more complex LRP structure (e.g., Type B or Type C) and reward sceme (e.g., using timing or dynamic rewards strategy). Wit respect to researc metodologies, descriptive studies (e.g., Berman, 2006; Sugan, 2005), case studies (e.g., Duffy, 1998; Ho et al, 2009, Ivanuskiene and Auruskeviciene, 2009), empirical studies (e.g., Sarp and Sarp, 1997; Taylor and Neslin, 2005; Park, 2010), and modeling-based studies (e.g., Kim et al, 2001; Lewis, 2005; Labbi and Berrospi, 2007; Sing et al., 2008; Diaby and Nsakanda, 2008; Nsakanda et al, 2010) ave all been applied in LRP researc. Among modeling-based LRP researc, some papers developed game-teoretical models to explore te impact of te LRP on market competition (e.g., Kim et al., 2001, 2004; Sing et a, 2008). Some papers proposed consumer decision and beaviour models to examine te profitability, effectiveness, or attractiveness of LRPs (e.g., Lewis, 2005; Sing et al., 2008; Gandomi and Zolfagari, 2011). Anoter set of papers applied te stocastic process in teir models to address diverse issues suc as allocating promotion budget to targeting LRP members '"Breakage rate" refers to accumulated points tat end up not being redeemed by LRP members In te LRP industry, breakage rate is claimed as part of te ost's revenue in te financial statements (e g, Aeroplan annual report 2007, page 24, ttp /Jwffn groupeaeroplan com/pages/invreports pp) 25

38 (e.g., Labbi and Berrospi, 2007), calculating breakage rate (e.g., Diaby and Nsakanda, 2008) and estimating redemption and liability (e.g., Nsakanda et a, 2010). Regarding te types of LRPs, altoug many researcers realized tat LRPs come in great varieties, few of tem incorporated tis consideration into teir studies. Te types of LRPs were eiter set to te most simplified form or ignored. In Table 2.1, we provide a summary of te main streams of LRP literature in terms of researc topics, types of LRPs, researc metodologies and key researc findings. 26

39 Table 2.1: Summary of te main streams of LRP literature Main Streams Literature & Researc Topic Type of LRP (*see Note 1) Researc Metodologies Key Researc Findings Customer switcing cost Klemperer (1987), (1995) Switcing cost Type A-I, -/-/direct/-/- Game teoretical model LRPs (e.g. repeat-purcase coupons, frequent flyer programs) increase switcing cost. Caminal & Matutes (1990) Switcing cost and marketing competition Type A-I, -/-/indirect/-/- Game teoretical model LRPs tend to decrease te competitiveness of markets. Carlsson & Lofgren (2006) N/A Empirical study LRPs (i.e. frequent flyer programs) increase switcing cost. Switcing cost LRP and customer beaviour Sarp & Sarp (1997) Customer loyalty Type B-I, dynamic/onemedium/ mixed/no timing/ amount-based Empirical study Altoug to cange te fundamental repeat-purcase patterns of markets is very difficult, it is possible to alter tose patterns at least to a small degree troug LRPs. Zang et al. (2000) Optimal coice of promotion tools Type A-I, -/-/indirect/-/- Consumer beaviour model and empirical study In a market dominated by "switcer" it is more profitable for a firm to use rear-load incentive (e.g, LRPs), wereas in a market wit ig inertia, it is more profitable for a firm to use front-load incentive. Kadar & Kotanko (2001) LRP design and implementation Types A, B and C -/-/-/-/- Descriptive study & Scientific literature analysis Properly designed and executed, LRP can bring tremendous benefits to te firms offering tem. Note 1: In tis column of te table, we listed all te dimensions and criteria tat ave been discussed or mentioned in te literature in te format of "organization structure, reward design strategy/ medium/ type/use timing or not/ reward grid". We found tat in many studies, not all te classification criteria were mentioned. If none of te classification dimensions were mentioned in te literature, "N/A" is used to denote tis case; if some of te classification criteria were not mentioned in te literature, ten "-/" is used to indicate tis case. 27

40 Table 2.1: Summary of te main streams of LRP literature (Cont.) Main Streams Literature & Researc Topic Type of LRP (*see Note 1) Researc Metodologies Key Researc Findings LRP and customer beaviour Kivetz & Simonson (2002) Effect of customer effort level on teir preferences toward LRP rewards -/ static / indirect/-/- Empirical study Higer program requirements sift preferences in favour of luxury rewards rater tan necessary rewards. Tis effect is also observed wen te program requirements are eld constant but te individual consumer's effort is iger. Yi & Jeon (2003) Customer loyalty -/-/direct vs. indirect/ -/- Empirical study Effects of LRP on customer loyalty depending on customer involvements. Lewis (2004) LRP and sort-term promotions on customer retention N/A Consumer beaviour model, simulation and experiment study LRP effectively increases repeat-purcase rates. Atak (2005) Impact of LRP elements on airline customers N/A Empirical study LRP elements are of great importance for LRP members, and also te level of members' perceived satisfaction as a uge effect on te level of customer loyalty. Lewis (2005) strategic consumer beaviour Type A-I, -/-/indirect/-/- Consumer beaviour model and empirical study A structural dynamic model is developed to enance accurate forecasts of customer value. Sugan (2005) LRP design and implementation N/A Descriptive study & Scientific literature analysis Pointed out tat many current so-called loyalty programs appear unrelated to te cultivation of customer brand loyalty and te creation of customer assets. A true program sould invest in customer wit te expectation of greater future revenue. 28

41 Table 2.1: Summary of te main streams of LRP literature (Cont.) Main Streams Literature & Researc Topic Type of LRP (*see Note 1) Researc Metodologies Key Researc Findings LRP and customer beaviour Liu (2007) Long-term impact on consumer purcase beaviour and loyalty Type A-I, static/one-medium/-/- /amount-based Empirical study LRP as positive effects on bot ligt and moderate buyers' purcase frequency and transaction size. Meyer-Waarden (2008) Customer purcase beaviour Type A-I, -/one-medium/ mixed /-/ amount-based Empirical study Tere is a positive association between LRP and customer purcase beaviour. Gandomi and Zolfagari (2011) Impact of consumer valuations and satisfaction levels on LRP profitability Type A-I, -/-/ indirect/-/ amount-based Consumer beaviour model and Experiment study Firm may not always be better off to offer an LRP. Effectiveness of LRPs depends on te stocastic parameters: mean and variance of te customer satisfaction levels. Design and implementation of LRP Duffy (1998) Customer loyalty N/A Case study, descriptive study, and scientific literature analysis Provided a consistent framework for developing a loyalty strategy and program. Discussed te selection of reward types and te design of te reward grid. Si & Soman (2004) Effectiveness of LRP in terms of medium usage Type A-I, static /one or twomedium/ mixed /-/ amount-based Customer beaviour model and Experiment study Multi-medium LRPs outperform functionally equivalent single-medium programs. Banasiewicz (2005) LRP planning and analytics N/A Descriptive study and Scientific literature analysis A general LRP planning approac is proposed and te importance of robust customer insigts to program planning and its ongoing management is demonstrated. 29

42 Table 2.1: Summary of te main streams of LRP literature (Cont.) Main Streams Literature & Researc Topic Type of LRP (*see Note 1) Researc Metodologies Key Researc Findings Design and implementation of LRP Berman (2006) LRP design and implement ~5 -/-/mixed/-/- Descriptive study and Scientific literature analysis Classified LRPs based on reward type and reward grid. A firm must assess te suitability of eac program type and ten carefully plan, evaluate and constantly revise te program. Bagdoniene & Jakstaite (2006) LRP purpose, means and development -/one-medium vs. nomedium /direct vs. indirect /-/- Descriptive study and Scientific literature analysis Tere is no absolute constant dependence between customer loyalty and customer satisfaction. LRPs are distinguised by great variety. Ke & Lee (2006) Customer satisfaction and reward preference -/-/direct vs. indirect/- /- Empirical study Wen customers are satisfied, tey prefer delayed, direct rewards to immediate, direct rewards. Wen customers are not satisfied, te conclusion is opposite. Ivanuskiene & Auruskeviciene (2009) LRP callenges in retail banking industry Type A-I, static/one- medium/mixed/-/- Descriptive study and Case study Te majority of te analyzed LRPs reward teir customers by using only a discount on te transaction costs, wic leads to a constant battle for te price. Ho et al. (2009) How to develop an effective LRP in retail industry (e.g., grocery store) Type A-I, static /one or twomedium/ mixed /-/ amount-based Case study Customers prefer different types of rewards. A "ybrid" reward structure was recommended to incorporating competition and eterogeneity in customers' preferences. Park (2010) Effect of LRP in airline industry N/A Case study and Empirical study Developed a conceptual model to explore te relationsips between LRP and airline service quality, pricing, passenger satisfaction, airline image, and airline selection troug a case study of te Korean airline industry. 30

43 Table 2.1: Summary of te main streams of LRP literature (Cont.) Main Streams Literature & Researc Topic Type of LRP Note 1) (*see Researc Metodologies Key Researc Findings Impact of LRP on a firm's focal business and/or market competition Kim etal. (2001) Pricing strategies Type A-I, -/-/ indirect /-/ amountbased Game teoretical model LRPs ave ability to weaken price competition and increase switcing cost. Kimefo/. (2004) Capacity management Type A-I, -/-/direct/-/- Game teoretical model By offering capacity rewards, firms can effectively reduce available capacities and set iger price in te market. Taylor & Neslin (2005) Sort and long-term sales impact Type A-I, -/-/direct/-/- Empirical study LRPs create impacts of points-pressure and rewardedbeaviour on customers' purcases. Sing et al. (2008) Profitability & marketing competition -/-/indirect/-/- Consumer beaviour model and Game teoretical model A market equilibrium exist wen one firm offers LRP and te oter doesn't. LRP operations Labbi & Berrospi (2007) Advertising budget allocation Type A-I, -/-/-/-/- Markov process model and experimental study A quantitative metodology is developed to allocation advertising /promotion budget in LRPs. Diaby & Nsakanda (2008) Breakage rate Type A-I or Type B-I, -/one medium / mixed / use timing /- Markov process model and Simulation model A quantitative approac is proposed to determine breakage rate in LRPs. Nsakanda et al. (2010) Redemption and liability estimation Type A-I, -/one medium /indirect or direct/ use timing / - Markov process model and Simulation model An analytical model is developed to estimate redemption and liability in LRPs. 31

44 Summary Our study is complementary to existing LRP literature. Te type of LRPs tat we focus on is te Type B-IV LRP wit static / one-medium / mixed / no timing / amount-based redemption (and accumulation) sceme. Altoug Type B-IV is one of te most popular LRP structures in te marketplace, it as not been investigated in te previous literature. In Type B-IV, te LRP is an independent business entity operated by a ost firm as a profit center. Te ost's focal business is to provide a brand-name LRP service to LRP members troug partnersip in bot redemption and accumulation operations. Tere is commonly more tan one partner involved in eiter redemption or accumulation operations in te system to provide goods and/or service in different categories. Points are used as te single universal reward media in te program to record members' purcase efforts. Tere is no limitation on wat types of rewards can be offered in te system. In oter words, redemption partners may offer direct, indirect, or mixed rewards. No time variable is used in te redemption (and accumulation) sceme and te redemption (and accumulation) grid is amount-based. As seen in Appendix A.l, practical applications of Type B-IV LRPs are increasing in recent years. It as become te most popular form of large-scale LRPs in today's marketplace. In tis type of LRPs, te ost firms run te programs as focal business and attempt to maximize te value created troug "rewards and points" business. All LRP partners are linked by te program to provide accumulation and/or redemption services to fulfill customer requirements for points and rewards. Furtermore, troug tis cain system, te LRP partners also attempt to compete wit oter non-lrp firms in te market and gain extra profit. Terefore, Type B-IV LRPs can be viewed as rewards-points supply cains (RSCs). 32

45 A rewards-points supply cain (RSC) and a traditional supply cain (TSC) sare many common caracteristics. Generally speaking, a TSC consists of multiple independent business entities suc as vendors, manufacturers, distributors, retailers, etc. Similarly, an RSC typically also consists of independent business entities including an LRP ost (i.e., te firm tat owns te program), LRP redemption partners (i.e., firms tat join te program to provide members wit redemption options), LRP accumulation partners (i.e., firms tat join te program to provide members wit accumulation options), etc. Te revenue flows in bot te TSC and te RSC are created by end consumers and sared among all business entities in te cain systems. However, important differences exist between TSCs and RSCs. In a TSC, products provided by te business entities are in te same goods/service categories. In an RSC, rewards in different goods/service categories are provided by redemption partners and points are a special type of products provided by an LRP ost and distributed by accumulation partners. End consumer demands relate to products in te TSC, wile in te RSC, end consumer demands relate to bot points and rewards. Wit regard to te system structure, TSCs are mainly sequentially based. Following te production flow, te business entities involved are operated sequentially. Downstream entities play a key role in interactions wit end consumers. In contrast, te structures of RSCs are parallel-based. In an RSC, te LRP ost is at te center of te cain system. Redemption and accumulation partners are te LRP ost's multiple cannel partners and tey operate independently. All of tese entities in te RSC interact directly wit end consumers (i.e., members wo join te LRP). Anoter key difference between a TSC and an RSC is tat costs and revenues are generated in a different time order. In te TSC, costs associated wit production occur first, and ten revenues are generated troug selling products. In te RSC, revenues are generated troug selling points first (tis applies only to an LRP operating as a 33

46 profit center), and ten costs associated wit rewards occur later wen LRP members redeem teir points for rewards. Table 2.2 summarizes te similarities and differences between a traditional supply cain (TSC) and a rewards-points supply cain (RSC). Te common caracteristics between TSCs and RSCs motivate us to apply traditional SC models to study LRPs. Meanwile, te unique features of LRPs allow us to explore te special problems associated wit LRP operations. Table 2.2: Traditional supply cain versus "rewards-points" supply cain Independent business entities in te system Information flow Product flow Revenue flow Cain Structure Cost and revenue occur in a different time order Demands in te system TSC Suppliers, manufacturers, distributors, retailers, etc. Product-oriented (e.g., number of products produced, number of products sold, eac product's inventory level, demand for eac product) Products provided by te entities in te multiple cannels are te same (i.e., eiter same branded or in te same goods/service categories) Comes from end-consumers and sared by SC entities Sequential-based. Business entities are operated sequentially following production flow. Downstream entities play a key role in te interaction wit end consumers. Cost associated wit production occurs first, and ten revenue is generated troug selling products. End consumer demands are on products only. RSC LRP ost, redemption partners, accumulation partners, etc. Reward- and point-oriented (e.g., number of points issued, amount of rewards, LRP members' accumulation and redemption demand, liability) Rewards in different goods/service categories are provided by redemption partners; points are te semi-tangible product provided by LRP ost Comes from end-consumers and sared by SC entities Parallel-based, ost firm is te center of te system. Redemption and accumulation partners are ost's multiple cannel members, operated separately. All entities in te system plays important role in te interaction wit end consumers. Revenue is generated first troug selling points, and ten cost associated wit rewards occurs later wen customers redeem teir points. End consumer demands are on bot points and rewards. 34

47 2.2 Supply Cain Contracts Because we view an LRP system as a rewards-points supply cain, exploring te impact of SC contracts on LRP operations and te corresponding decision making becomes anoter aspect of our researc focus Overview SC contracts are widely recognized as a necessary means for governing buyer-supplier relationsips in te supply cain, coordinating te decisions of te supply cain partners, saring te risks arising from various sources of uncertainty, or facilitating long-term partnersips (Tsay et al. 1999). Generally speaking, an SC contract is a binding agreement settled partially or fully prior to a B2B excange tat specifies contractual relations among te contract parties. It may also offer suitable incentives, troug various terms and conditions, on material, information, and fund flows in order to facilitate te B2B excange and influence te beaviour and decisions of contract parties along te supply cain (Tirole, 1988; Liu et al., 2005; Park et al., 2006). A variety of supply cain contracts witin and across industries ave been investigated in te academic literature and contributions ave been reported on a number of issues related to teir role. Tey are critical in coping wit internal dynamics and external uncertainties in supply cains, in influencing te beaviour and decisions of contract parties along te supply cain, and in facilitating supply cain coordination. Tey are also very important in contract selection, design, evaluation, and implementation. 35

48 Wolesale-price contracts, two-part tariff linear price contracts, two-part tariff non-linear price contracts are examples of supply cain contracts tat are defined on te basis of pricing policies (e.g., Corbett and Tang, 1999). Buyback or return contracts, quantity-flexibility contracts, backup agreement contracts, revenue-saring contracts, option contracts, sared-saving contracts and teir many variations are oter examples of supply cain contracts defined on te basis of incentives or purposes (e.g., Eppen and Iyer, 1997; Li and Kouvelis, 1999; Barnes-Scuster et al, 2002; Tsay, 2002; Cacon and Lariviere, 2005; Corbett et al, 2005). Most existing studies on supply cain contracts ave focused on supply cains tat deal wit tangible products. Te analysis in most studies primarily assumes a supply cain involving a single retailer and a single supplier. For compreensive survey papers on supply cain contracts, we refer readers to Tsay et al (1999), Lariviere (1999), and Cacon (2003). Altoug various SC contracts ave been discussed in te literature, in tis study, we focus our attention on option contracts. Option contracts are widely viewed as a edging instrument against supply and demand uncertainties and ence improve overall supply cain performance (e.g., Ritcen and Tapiero, 1986; Wu et al, 2005; Xu and Nozick, 2009; Ceng et al, 2011). Some researcers (e.g., Barnes-Scuster et al, 2002) ave sowed tat quantity flexibility contracts (Anupini and Bassok, 1998), backup agreements (Eppen and Iyer, 1997), and pay-to-delay capacity reservation (Brown and Lee, 1997) are all special cases of option contracts Option Contracts Option contracts tat were mainly used as a risk management mecanism in te financial area ave attracted enormous attention in te recent OM/SC literature. In te context of OM/SC, an 36

49 option contract refers to a contract in wic a supplier (e.g., manufacturer) allows a buyer (e.g., retailer) to purcase up to a given quantity of a product during a specified time interval at specified pricing (Xu and Nozick, 2009). In tis type of contract, capacity is commonly regarded as an option to be exercised in te future to produce needed goods. A capacity reservation fee (i.e., option price) and execution fee (i.e., exercise price) are te two key parameters in te contracts. A typical option setting follows a two-pase structure. In te first pase, an option contract specifying a reservation fee, r, and an execution fee, e, is offered by a supplier. Ten, based on tese fees, te buyer cooses optional ordering quantity, Q, wic is matced by te supplier's capacity. At tis stage, te actual demand is unknown. Wile te reservation fee is immediately payable, te execution fee is due wen te option is exercised (after demand uncertainty is resolved). In te second pase, te buyer decides on te exercise amount and pays te exercise fee after observing te realized demand. Option contracts ave been widely applied in industries suc as semiconductor, aerospace, toys, apparel, electronic, telecommunication, etc. (e.g. Cang, 1996; Cole, 1998; Barnes-Scuster et al., 2002; Jin and Wu, 2007). Suc option contracts are also known as capacity reservation contracts in te literature. Generally speaking, suppliers are motivated to use tis type of contracts by "reducing potential cost troug early commitments," wile buyers are motivated by "ensuring availability during demand upsides" (Wu et al., 2005). Ritcen and Tapiero (1986) are considered to be te first to propose a two-period model to study an option contract to cooperate wit uncertain future demands and prices. In teir model, te buyer as te coice of 37

50 carrying inventory from te first period to te second and can utilize first-period demand to update te forecast for second-period demand. Te buyer ten exercises te options based on te updated information at te beginning of te second period. Te autors focus teir analysis on te buyer's perspective and derive optimal contract policies. Tey also explored te buyer's ordering decision under different scenarios (i.e., risk neutrality vs. risk aversion and independent vs. dependent price and demand). Considering a similar option contract, as in Ritcen and Tapiero (1986), oter researcers studied various SC management issues. For example, Barnes-Scuster et al. (2002) examined optimal ordering policies and teir implementations on SC coordination. Te autors discussed te necessary conditions to acieve cannel coordination wen a return incentive is considered in te contract and te supplier is limited to using a linear pricing policy in contract design. Tey found tat coordination conditions depended not only on te cost and salvage parameters but also on te correlation coefficient parameter of te demand. Tey sowed tat options are useful instruments for increasing flexibility and cannel profitability. Option contracts in a model wit a single capacitated seller and a single buyer were also studied by Serel et al. (2001), Wu et al. (2002), Spinier and Hunczermeier (2006), Wang and Liu (2007), Ceng et al. (2009) and Zao et al, (2010). Wit multiple suppliers and/or multiple buyers, researcers are interested in te strategic interactions between te supplier(s) and te buyer(s) (e.g., Erkoc and Wu, 2005; Jin and Wu, 2007) or te competitive beavior of independent players wit option contracts in an SC system (e.g., Wu and Kleindorfer, 2005; Perakis and Zaretsky, 2008; Xu and Nozick 2009; Martinez-de-Albeniz and Simci-Levi, 2009). 38

51 Serel et al. (2001) extended te buyer's decision wit option contract to a multi-period setting troug a periodic review inventory control model. Te autors examined te value of te option contract to bot buyers and suppliers. In tis model, option price and exercise price are not decision variables but predetermined parameters. Jin and Wu (2007) investigated te capacity coordination issue in a ig tec manufacturing supply cain. Tey proposed an option contract in wic te buyer's reservation fee (option price) is deductible from te purcasing price (exercise price). Tey developed a single-period newsvendor-based analytical model for a scenario of one supplier - one buyer. Tey also examined te capacity allocation issue tat arises wen one supplier faces two buyers wo ave te same marginal profit but a different demand distribution in teir respective markets. In te same context, Erkoc and Wu (2005) proposed an option contract integrating partial deduction and cost saring tat can acieve coordination. Ceng et al. (2009) presented a capacity option contract tat includes a firm purcase commitment wit exogenous wolesale price. Te retail price is a constant and demand is random. Similar to Erkoc and Wu (2005), Ceng et al. observed tat options will be appealing for te buyer only if te reservation (option) fee is below a certain tresold. Wu et al. (2002) considered a case in wic bot supplier and buyer ave access to spot markets to sell or to buy non-storable goods. However, bot parties embark on a capacity option contract due to uncertainty in te future spot market prices. Te paper derived te seller's and buyer's optimal contracting strategies troug a game teory framework. Golovackina and Bradley (2003) also studied a capacity option contract in te presence of a sport market. But unlike Wu et al. wo assumed tat demand is deterministic, Golovackina and Bradley considered a case in wic te buyer (i.e., a manufacturer) must fulfill periodic stocastic demand. Spinier and 39

52 Hunczermeier (2006) extended Wu et al.'s framework to value options by taking into account te uncertainty in te buyer's future demand and te seller's future marginal costs. Tey sowed tat bot parties are better off under option contracts compared to oter market scemes. Te buyer's demand for options depends on te correlation between buyer demand and spot price. Buzacott et al. (2011) proposed a mean-variance optimization model to examine a supplier's optimal stocking level under option contracts in te presence of demand uncertainty due to spot market price. Burnetas and Ritcken (2005) studied te option contract in a supply cain were te demand distribution is influenced by retailers' pricing decisions (a downward-sloping demand curve wit a stocastic parameter). Tey found tat te introduction of option contracts causes te wolesale price to increase and te volatility of te ratail price to decrease. Tey also outlined te conditions under wic manufacturer and retailer will benefit respectively from option contracts. In a setting of multiple suppliers, Wu and Kleindorfer (2005) extended te model in Wu et al. (2002) to examine te situation were several suppliers compete to provide capacity to a single buyer. Tey investigated te optimal portfolios of contracting and spot market transactions for buyers and te suppliers, and tey determined te market equilibrium pricing strategies. Xu and Nozick (2009) provided a solution procedure, a two-stage mixed integer stocastic programming, to optimize supplier selection in a multi-period transportation model using option contracts. In teir model, te buyer is required to purcase a fixed amount per period from some of te suppliers (equivalent to minimum purcase requirement). Similar to Xu and Nozick 40

53 (2009), Perakis and Zaretsky (2008) also considered a competitive SC in a multi-period setting. In teir model, tey treated te spot market as a dummy supplier wit stocastic costs and unlimited capacity. Tey sowed tat te equilibrium policies coordinate te SC. Martinez-de- Albeniz and Simci-Levi (2009) analyzed pricing strategy wen suppliers are competing troug option price and flexibility. Tey found tat in market equilibrium a variety of suppliers coexist, and tese suppliers offer different prices. However, unlike te finding of Perakis and Zaretsky, Martinez-de-Albeniz and Simci-Levi found tat teir equilibrium solution cannot acieve SC coordination. Zao et al. (2010) included in teir work te supply cain members' risk references and negotiating powers. Tey analyzed ow tese factors impact te profit allocation between te retailer and te manufacturer in a retailer-led supply cain in wic te retailer olds te decision rigt on pricing and takes te initiative to coordinate te manufacturer's production quantity. Table 2.3 provides a summary of te main contributions of option contracts in te SC/OM literature in terms of system settings, analytical models, contract structures, and sources of uncertainties. Te system settings refer to supply cain structure (i.e., ow many suppliers, buyers, and products are involved in te contractual relationsips as well as te time period tat eac contract covers). We found tat in te existing literature, te basic system setting in many option contract studies is tat of one buyer, one supplier, one product, and one time period ("1/1/1/1"). 41

54 Table 2.3: Summary of te main contributions of option contracts in OM/SC literature Literature System setting (supplier / buyer / product / time period) and from wic perspective Contract structure (*See Note 2) Source of uncertainty Analytical model Ritcen & Tapiero (1986) 1/1/1/1 buyer [r,e] demand & spot market price Mean-variance approximations of expected utility from retailer's perspective Serel, etal. (2001) 1+ spot/1/l/n buyer, Supplier and cannel [r,e] demand periodical review inventory model Barnes-Scuster et al. (2002) 1/1/1/1 buyer, supplier and cannel [r,e] demand Stocastic inventory model Wuetal. (2002) l/l(n)/l/l buyer, supplier M] spot market price Game teoretical model Golovackina & Bradley (2003) 1/1/1/1 supplier [r,e] demand, spot market price Game teoretical model Erkoc & Wu (2005) 1/1(2)/1/1 buyer, supplier and cannel [r,e] demand Newsvendor-based model Burnetas & Ritcken (2005) 1/1/1/1 buyer, supplier [r,e] demand Game teoretical model Wu & Kleindorfer (2005) n/1/1/1 buyer, supplier [r, e, L] demand, spot market price Game teoretical model 42

55 Table 2.3: Summary of te main contributions of option contracts in OM/SC literature (Cont.) Literature System setting (supplier / buyer / product / time period) and from wic perspective Contract structure (*See Note 2) Source of uncertainty Analytical model Spinier & Hunczermeier (2006) 1/1/1/1 buyer, supplier [r,e] demand, seller's marginal cost, spot market price Game teoretical model Jin&Wu (2007) 1/1(2)/1/1 buyer, supplier and cannel [r, e, C] demand Newsvendor-based model and Game teoretical model Perakis & Zaretsky (2008) n/l/l/n buyer, supplier and cannel [r,e] demand Game teoretical model Martinez-de-Albeniz & Simci-Levi (2009) n/1/1/1 supplier [r,e] demand Game teoretical model Xu & Nozick, (2009) Ceng et al. (2009) Zao et al. (2010) Buzacott et al. (2011) n/l/l/n buyer 1/1/1/1 buyer, supplier and cannel 1/1/1/1 buyer, supplier and cannel 1/1/1/1 supplier [R,e] [r, e] wit minimum order/purcase commitment [r,e] [r, e] wit minimum order/purcase commitment demand demand demand demand, spot market price Multi-period transportation model wit option contract Newsvendor-based model Newsvendor-based model and Game teoretical model Mean-variance model *Note 2: "r" is reservation fee in te form of unit price, "e" is exercise fee in te form of unit price, "L" is seller's capacity reserved for contract market (vs. spot market), "C" is excess capacity in addition to buyer's reservation amount, and "R" is reservation fee in te form of lump sum payment. 43

56 In Table 2.3, we also listed perspective(s) from wic te researcers addressed issues related to option contracts. We found tat Ritcen and Tapiero (1986) and Xu and Nozick (2009) studied option contracts only from te buyer's perspective, wile Golovackina and Bradley (2003), Martinezde-Albeniz and Simci-Levi (2009) and Buzacott et al. (2011) from only te supplier's perspective. In te papers focusing on cannel coordination issues, te supplier, te buyer, and te cannel perspectives are all included in te analysis (e.g., Erkoc and Wu, 2005; Perakis and Zaretsky, 2008). In most studies, te structures of option contracts are caracterized by two parameters: r and e. Te parameter r refers to te option price, also known as reservation fee. It is paid by a buyer wen te buyer reserves te options before knowing actual demand. Te parameter e refers to te exercise price, wic is paid by te buyer wen te buyer exercises te options after knowing te demand Summary Since LRP can be viewed as a "rewards and points" supply cain tat carries common features of TSC, option contracts are also applicable for LRP management to target demand and/or supply uncertainties. In te existing literature, no one as introduced option contracts in LRP management, wic opens an opportunity for us to add contributions to te literature by considering tis dimension. Under te option contract setting, an LRP ost (i.e. te reward buyer) and LRP redemption partners (i.e. te reward suppliers) interacts wit eac oter in two pases. In te first pase, an option contract specifying a reservation fee (option price) r, an execution fee e, and a reservation quantity (options to purcase) Q, is agreed upon by bot parties. At tis stage, te demand is 44

57 unknown. Based on te reservation fee, te ost cooses Q, wic is matced by te redemption partner's capacity. In te second pase, te ost decides on te exercise amount (options to exercise) and pays te exercise amount after observing te realized demand. Rater tan examining strategic beavior between te LRP redemption partners and te LRP ost in rewards-supply contracting, we focus our attention on te LRP ost's rewards-ordering decisions based on te ost's own financial and managerial constraints and te option contracts offered by te different rewards suppliers. In oter words, option price and exercise price are not decision variables, but predetermined parameters; and te option price is deductable from te exercise price. 2.3 Cooperative Advertising Similar to oter marketing programs, advertising and promotion are te common tools used by LRP osts and LRP partners to increase sales and to target potential customers. In tis section, we first present te concept of cooperative advertising (including promotion) (CA), and ten provide a brief review of CA literature. We also discuss budget allocation issues and different formulations of sales response functions tat are relevant to CA Overview CA as been adopted as a business practice for many years. It is essentially a financial arrangement under wic two parties agree on ow te sponsorsip and costs of mutual promotion and advertising will be defrayed (Crimmins, 1984). Te tree principal categories of 45

58 cooperative advertising are orizontal, ingredient-producer, and vertical (Young and Greyser, 1982). In te SC/OM literature, vertical cooperative advertising (including promotion) is te most prevalent of tese types. It refers to an interactive arrangement were a manufacturer pays for some or all of te costs of local advertising implemented by a retailer of te manufacturer's product (Huang et al., 2002). Terefore, CA in te SC/OM literature mainly refers to cost saring of vertical advertising (e.g., manufacturer sares retailer's advertising and promotion expenses). CA is usually used by a manufacturer to motivate immediate sales at te retail level and to strengten te image of te brand (Huang et al, 2002). Tere are two strategies to realize tis cost saring among SC members (e.g., manufacturers and retailers). Te use of side payment is one strategy. Manufacturers decide on te size of a lump sum payment for eac retailer, and retailers decide ow muc of tis side payment to use in promoting te manufacturer's product. A portion of tis side payment is normally pocketed by te retailer. However, due to a sift in power from manufacturers to retailers and/or te competition in te marketplace, even toug te manufacturers know te retailers may pocket muc of te payment tey are still willing to pay (Kim and Staelin, 1999). It was reported in te researc tat te percentage pocketed ranges on average from 30 to 50 percent (Armstrong, 1991; MacClaren, 1992; Agrawal, 1996). Te use of participation rate is te oter strategy. It as been discussed more frequently in te recent CA literature (e.g., Bergen and Jon, 1997; Huang et al, 2002; Karray and Zaccour, 2007; Xie and Wei, 2009; He et al, 2009, He et al, 2011). Participation rate specifies te proportion of te retailer's advertising and promotion costs tat te manufacturer sares. Because participation rate ties to a retailer's actual advertising expenditures, generally speaking, it overcomes te weakness of side payment, and it as become 46

59 te most prominent strategy used by manufacturers. However, since it requires te retailer to self-report on advertising expenditure, it causes more administration costs. In analytical models based on tis payment strategy, te manufacturer decides on is te amount of te participation rate wile te retailer decides on te amount of advertising spending Cooperative Advertising and Budget Allocation In te SC/OM domain, te majority of te researc attempts to investigate te role of CA in SC management and SC coordination of suppliers and retailers (e.g., Huang et al, 2002; Yue et al, 2006, Karray and Zaccour, 2007; Xie and Wei, 2009; He et al, 2009). For example, bot Huang et al. (2002) and Yue et al, (2006) derived optimal levels of CA expenditures for manufacturers, retailers, and cannels under a simple one-to-one SC setting (i.e., one manufacturer and one supplier). But Yue et al. (2006) went one step furter. Tey focused on te situation were a manufacturer offers price deductions directly to customers instead of troug a retailer and explored te optimal levels of CA expenditures in tis situation. Xie and Wei (2009) examined te impact of te retail price rater tan manufacturer's price discount on customer demand in a CA model wit te same SC setting as tat used in te previous two studies. Karray and Zaccour (2007) considered a SC formed by two suppliers and two retailers. Tey assessed an efficient CA plan by comparing te scenarios tat eiter no manufacturer or one or bot manufacturers offer(s) CA to retailers. Unlike previous studies, He et al. (2009) investigated manufacturers' and retailers' CA decisions troug an analytical model tat integrates a stocastic sales advertisingresponse function rater tan a deterministic advertising-response function. He et al. (2011) extended te model proposed in He et al. (2009) by considering retail competition. We refer 47

60 readers interested in knowing more about CA in a broader context to Neslin (2006), Stewart and Kamins (2002), and Ailawadi et al. (2009) for recent surveys. Anoter important decision issue relating to CA is ow to allocate advertising budget among different submarkets or different retailers. Tese submarkets or retailers are caracterized by geograpic areas, market segments, product categories, media instruments, or seasonal canges. Extensive studies ave examined budget allocation problems. For example, Holtausen and Assmus (1982) presented an advertising budget allocation model for different market segments from a manufacturer's point of view. Tey studied te budget allocation issues under sales response uncertainty. Advertising correlations among marketing segments are also considered in te model. Amed (1984) investigated a budget allocation and advertising media selection problem in te single-product context. Te autor developed an efficient integer programming model to solve te problem. Basu and Batra (1988) considered CA for multiple brands. Tey discussed te issue of ow to allocate an advertising budget among individual brands in order to maximize te overall corporate profit wen corporate promotional resources are limited. In teir analytical model, sales response for eac brand is determined by te advertising budget allocated as well as te price elasticity. Doyle and Saunders (1990) focused on allocating retailer's decision on allocating advertising resources to different products wit te consideration of influence factors suc as manufacturer's allowance, cross-product effects, and store-wide traffic effects. Bockstael et al. (1992), Vande Kamp and Kaiser (2000), and Kinnucan and Myrland (2002) examined te optimal timing of advertising allocations. Seasonal variations in price, demand, and product substitution are te prime concerns in tese 48

61 studies. In more recent researc, Labbi and Berrospi (2007) proposed an optimization model for budget allocation in te context of LRP. Tey introduced a customer segmentation approac tat is based on customer value and loyalty metrics as well as customers' response beavior. Tey also used a Markov process to simulate te dynamics of customer status over time, namely te movement of customers among tree states: first-time/not frequent, repeated purcase, and loyal customers. A teoretical underpinning of CA and budget allocation models is te relationsip between advertising inputs (e.g., advertising budget or expenditures) and advertising outputs (e.g., sales response). Simon and Arndt (1980) identified te caracteristics of tis relationsip troug a survey of more tan one undred empirical studies of advertising response functions. Teir survey sowed tat te response functions proposed in te majority of researc on advertising follow one of te two proposed sapes: (1) a non-negative concave-downward curve and (2) an S-saped curve. Tey conclude tat te former curve is te one found most often. Based on Simon and Arndt's study, Tull et al. (1986) proposed tree different aggregate advertising-sales response functions (namely, a concave saturation sales response function, a diminising returns function and a quadratic response function) wit respect to different operating ranges used by companies for advertising. Te autors integrated tese functions respectively wit a singleperiod profit maximization model to study te decisions on te optimal advertising budget level. Basu and Batra (1988) proposed a general sales response function wic allows bot concave and S-saped relationsips between te advertising budget and sales response. Mantrala et al. (1992) analyzed budget allocation problem among multiple submarkets. Tese submarkets are caracterized by advertising-sales response functions tat are similar to tose used in Tull et al. 49

62 (1986). Altoug te complexity and difficulty in modeling sales-advertising relationsip as led to a multitude of researc papers on tis topic (e.g. Agrali and Geunes, 2009; Vakratsas et al, 2004), we found tat concave saturate-based response functions are te ones most frequently used in te literature (e.g., Naert and Leeflang, 1978; Holtausen and Assmus, 1982; Mantrala et al, 1992; Huang, et al, 2002) Cooperative Advertising in LRP Operations It is a common practice in LRP management for an LRP ost to offer CA to accumulation (and redemption partners) to boost consumers' accumulation (and redemption) demands. In tis paper, we focus our attention on a type of CA tat is uniquely associated wit LRPs. It is known as "bonus points". Bonus points refer to te CA agreement in wic te LRP ost offers LRP members extra points wen te members purcase certain products from te LRP partners or participate in certain partners' promotional activities. Unlike te common CA strategies tat we discussed in te previous section, an LRP ost doesn't give tis benefit (i.e., bonus points) to LRP partners. Instead, te ost gives bonus points to LRP members directly. For example, Aeroplan supports is financial partner CIBC by offering 15,000 points to customers on teir first application for an Aerogold Visa Card. Air Miles allows members to earn triple reward miles wen tey sop online troug te Air Miles website. In a recent study 1, several different formats of bonus points were discussed: enrolment-related vs. selected members-related, and flat bonus (i.e. fixed) vs. percentage bonus (i.e. based on accrual points te member earned). 1 "Ensuring Customer Loyalty: Designing Next-Generation Loyalty Programs", Oracle Wite Paper, Feb.,

63 To LRP osts, 'bonus points' is a two-edged sword. In te sort run, bonus points can directly contribute to LRP value growt and encourage LRP partners to promote te LRP besides teir own products. But in te long run, bonus points increase redemption costs and points-related liability, wic may cause potential risks in te LRP management. Terefore, LRP osts need to manage bonus-point type of CA properly in order to get te most out of it. Unlike using CA in traditional SCs, in an LRP system, since te LRP ost can communicate wit LRP members (i.e. consumers) directly troug its media network (e.g., s, websites, or call centers), te ost doesn't ave to rely on te partners' advertising effort and media network to motivate te LRP. In oter words, te ost can estimate and observe te sales response to bonus-point type CA troug tracing LRP members' purcase istory. In tis sense, te ost may control te impact of a bonus-point type of CA troug te proper selection of te amount of bonus points assigned to eac partner. Tis is te essential difference between common CA and LRP CA. To te best of our knowledge, te impact of bonus points on te operational decisions of LRPs as not been explored in te LRP literature Summary In tis section, we reviewed te literature tat is relevant to te second focus of tis study - bonus points, a unique type of CA associated wit LRP management. We first introduced te concept of CA and te two common CA strategies (i.e. lump sum payment and percentage rate) tat ave been examined in te literature. We ten focused on te literature related to budget allocation since CA budget allocation is one of te central concerns to a manufacturer/supplier wen e plans to offer CA to multiple retailers/buyers. Meanwile, we overviewed sales advertising-response functions, a key determinant of model validity in te development of CA 51

64 related analytical models. Finally, we pinpointed te differences between bonus-point type of CA and te oter common types of CA. In our study, we examine LRP ost's decisions on te amount of bonus points to offer to different LRP accumulation partners. Unlike previous CA and budget allocation researc, we integrate tis problem into te ost's decisions on rewards-ordering by considering te "delayed" cost of offering bonus points. Te sales response function proposed in our model is on te same line of te concave saturate-based functions. 52

65 Capter 3 Researc Framework and Matematical Models In tis capter, we first explain te teoretical foundation of tis researc, and ten we discuss te matematical formulation of te LRP rewards-supply aggregate planning problem under various settings. 3.1 Loyalty Reward Programs - "Rewards-Points" Supply Cains Let us consider an LRP system were a ost firm runs it as a focal business tat seeks to maximize te value created troug te "rewards & points" business. Multiple LRP partners participate in tis system to fulfill te requirements of LRP members for points and rewards. Based on tis point of view, te LRP system can be described by te following conceptual model (Figure 3.1): Figure 3.1: Conceptual model of a rewards-points supply cain 53

66 In tis conceptual model, one LRP ost firm as multiple LRP redemption partners (Rj, j l,...,j), multiple LRP accumulation partners (A i = l,...,i), and LRP members, te end consumers wo collect points and redeem tem later for rewards. Hence, LRP members ave two types of demand: point-accumulation demand (Df,i-l,...,I) and point-redemption demand (Dj,j \,...,J).Df refers to LRP members' overall (aggregate) demand for collecting points from te LRP accumulation partner A,. Df refers to LRP members' overall (aggregate) demand for redeeming points for rewards offered by te LRP redemption partner Rj. LRP members' accumulation demand towards eac LRP accumulation partner drives te points business between te LRP ost and te LRP accumulation partners (e.g., LRP ost sells points to an LRP accumulation partner A,). Meanwile, LRP members' redemption demand towards eac LRP redemption partner drives te rewards business between te LRP ost and te LRP redemption partners (e.g., LRP ost purcases rewards from an LRP redemption partner Rj). In tis pointsrewards supply cain, points are circulated as LRP currency among all te entities (i.e., LRP ost, LRP partners, and LRP members). Meanwile, points and rewards are te sort- and longterm incentives to drive business forward and create value in te wole cain. How does tis rewards-points supply cain create values for all te entities involved in te system? To answer tis question, we examine te value creation processes at bot accumulation and redemption sides. At te accumulation side, wen an LRP member purcases a product or a service from an LRP accumulation partner, te value from points business is created. To te LRP member, te value obtained is te points tat were earned (in addition to te product or te service purcased) troug te purcase of te product or te service. To te LRP accumulation partner, te value obtained is te revenue generated troug te sale of te product or te service. 54

67 However, as te LRP accumulation partner does not directly issue te points earned by te LRP member, tose points ave to be purcased from te LRP ost, te issuer of points. By selling points to te accumulation partner, te LRP ost sares te revenue obtained from te purcase of te product or te service by te LRP member. Te sum of te revenue sares from all LRP accumulation partners is te value tat te LRP ost obtains troug te points business. At te redemption side, te business starts wen an LRP member redeems points for a reward. Since te LRP ost does not produce te reward, it as to be purcased from an LRP redemption partner to meet te LRP members' redemption demand. Te LPR ost carges te LRP member a certain amount of points for te rewards. Terefore, to te LPR ost, te value obtained troug te rewards business is te difference between te value 1 of te reward (measured by te value of points tat te LRP member gives out for te reward) and te actual amount tat te LRP ost spends to purcase te reward. To te LRP redemption partner, te value obtained from tis redemption business is te revenue generated by selling te reward to te LRP ost. Te reward tat te LRP member receives is te value obtained from redemption. Figure 3.2 illustrates te value creation process in a rewards-points supply cain. We assume tat te accumulation grid is defined as follows: "For every $1 spending at an LRP accumulation partner (A,), an LRP member can collect one point." Te redemption grid is defined as follows: "A LRP member obtains a S0.01 value of reward per point." (e.g., a reward wort $150 requires 15,000 points). 1 Te value of rewards quite often depends on te bencmarks, for example, te retail prices of te rewards in te market. 55

68 Terefore, te value created wit te points business is te revenue generated per point at te accumulation side. Let p A denotes te revenue generated per point at te accumulation side, in tis example,p A =$\. Partner ^*S^ V^ ^*S^ ^-^^ Partner pernor 1 point LRP Members 1 point V=$i Figure 3.2: Illustration example of value creation At te accumulation side, te revenue per point will be sared among te LRP accumulation partner (A,) and te LRP ost. Te LRP ost sells points to te LRP accumulation partner (A t ) at price w 4 per point, w 4 = $0.05. Te accumulation partner A/s sare of te value created wit te points business is: p A - w 4, wic is $0.95 per point in tis example. Te LRP ost's sare of te value is W 4, wic is $0.05 per point in te example. At te redemption side, let p R denotes te value of reward per point, p R = $0.01. Te LRP ost buys a reward (transformed to te equivalent value of points) at price w R per point from te LRP redemption partner (Rj), w R = $0,008 in te example. As suc, te redemption partner (Rj) receives its sare of te value created by te rewards business, wic is w R. Wen an LRP member redeems points for a reward, te value created troug te rewards business is p R - w R, from te LRP ost point of view, wic is $0.01-$0.008= $0,002 per point in te example. Redemption business will be profitable only wen te redemption cost per point, w R is less tan te value of reward per point, 56

69 PR. Overall, te LRP value created troug transacting 1 point is $0.05+$0.002=$0.052 from te LRP ost point of view. As we can see, in tis rewards-points supply cain, managing rewards and points plays an important role in te LRP value creation and sustainable development. In te remainder of tis study, te overall value tat an LRP ost obtains troug tis rewards-points supply cain is called "LRP profitability". It includes te LRP ost's sares of values created from bot te accumulation and redemption sides of business. 3.2 LRP Rewards-Supply Aggregate Planning Models Te LRP ost's aggregate planning problem for rewards-supply consists of determining te optimal procurement of rewards from multiple LRP redemption partners to maximize te LRP profitability (as measured by te value creation) subject to system constraints and managerial requirements. Te system constraints refer to te capacity constraint tat an LRP redemption partner as on offering rewards, and te LRP ost's overall budget for purcasing rewards. Te managerial requirements mainly refer to te LRP ost's control on liability, te outstanding balance of points representing te total value of future financial arrangements (i.e., future redemption obligation). Liability is widely recognized in te LRP industry as a risk indicator for LRP sustainability. For many LRPs, especially large-size LRPs wit a uge customer base, business practices sow tat a "ig liability" results in iger redemption uncertainties, iger management costs, and iger 57

70 risks (e.g., yperinflation and devaluation of points 1 ). Terefore, in tis study, we explore ow to maintain liability witin a reasonable range tat is most beneficial to te LRP ost. In te following sections, we introduce first a matematical model for te LRP rewards-supply planning problem witout considering cooperative advertising (troug bonus points) or option contracts. Ten we extend tis model to integrate cooperative advertising and option contract decisions. In eac section we first explain our modeling assumptions, and ten we introduce te model formulation LRP Rewards-Supply Planning Problem witout Bonus Points In tis section, we report te matematical formulation of te LRP rewards-supply planning problem witout bonus points offering Modeling Assumptions In our modeling te following assumptions are considered: Assumption (1): Assumption (2): Assumption (3): Assumption (4): Assumption (5): Relationsips between LRP partners and an LRP ost are governed by contracts. LRP redemption partners ave capacity limitations on offering rewards. LRP ost as no capacity limitation on issuing points. LRP members' accumulation demands will always be met. LRP members' accumulation and redemption demands are not known wit certainty but ave known probability distributions and bot demands are price-independent. Hofer, D Reinforcing te value of frequent flyer miles. Wite paper, Loylogic Inc wwft.loylogic com 58

71 Assumption (6): Assumption (7): Assumption (8): Assumption (9): LRP members' demands towards LRP accumulation partners are independent from eac oter. LRP members' demands towards LRP redemption partners are independent from eac oter. LRP members' demands towards LRP redemption partners are independent from LRP members' demands towards LRP accumulation partners. Eac LRP accumulation partner's ordering quantity of points is equal to te mean value of LRP members' accumulation demands towards tat accumulation partner. Assumption (10): Static redemption scemes are adopted by LRP redemption partners during te planning time period. Assumption (11): Static accumulation scemes are adopted by LRP accumulation partners during te planning time period. Assumption (12): Wolesale-price contracts are used as te governing mecanism between an LRP ost and LRP accumulation partners. Assumption (13): Wolesale-price contracts are used as te governing mecanism between an LRP ost and LRP redemption partners. Assumption (14): Cooperative advertising troug bonus points is not considered Assumption (1) indicates tat, similar to te traditional supply cain operations, te business relationsips between LRP partners (bot for redemption and/or accumulation) and an LRP ost are primarily governed by contracts. Altoug many different types of contracts ave been discussed in te literature (e.g., Tsay et a, 1999; Cacon, 2003), we limit our study in tis section to te wolesale-price contracts because of its fit for LRPs (see assumptions (12) and (13)). Assumption (2) sows tat te amount of rewards (e.g., free airline tickets or vacation packages) provided by redemption partners are limited by te available capacities. 59

72 Assumptions (3) and (4) relate to te unique features of points. As points are a kind of information symbol for recording and counting LRP members' purcase efforts in te LRP system, te LRP ost as almost no "production" related costs or "resource" related capacity limitations on offering points. As suc, te LRP ost as no capacity limitation on issuing points. Furtermore, unlike tangible products, production and movement of points are not limited by time and pysical space. Points are never "stock-out" in te sense tat tere is almost no time lag between producing points and meeting te customer accumulation demand on points. Terefore, LRP members' demands for accumulation will always be met. Assumptions (5) to (7) indicate tat te rewards offered by redemption partners are not substitutable, as well as te accumulation options offered by accumulation partners. Besides tat, bot accumulation and redemption demands are random parameters following known probability distributions. Assumption (8) indicates tat during te planning orizon, LRP members' redemption demands at te aggregate level are not influenced by LRP members' accumulation demands. Assumption (9) describes tat altoug accumulation demand is a random parameter, an LRP accumulation partner knows te mean value of te accumulation demand and uses it to determine ordering quantity of points. Terefore, tis assumption indicates tat LRP accumulation partners are risk-neutral. Assumptions (10) to (11) suggest tat te redemption and accumulation scemes are predetermined and do not cange over te planning orizon. Te scemes are structured troug 60

73 fonnats suc as "earn 3 points per dollar spending", "every 1000 points are wort $1 value of reward", or "redeeming 5000 points can get $10 discount", etc Problem Description and Model Formulation Te notations used in tis section are as follows: (1) Indices: Rj A H LRP redemption partners, wo offer rewards to LRP members, j=l,2,...,j LRP accumulation partners, wo allow LRP members to collect points troug teir purcasing of products or services, i=l,2,...,i LRP ost, wo runs an LRP as a profit center. (2) Decision variables: <ij LRP ost/ts ordering quantity of rewards from LRP partner Rj. (3) Parameters in objective function: Df LPR members' accumulation demand towards LRP accumulation partner A l. qf LRP partner^, 's ordering quantity of points. W A Wolesale price per unit of points tat LRP ost H carges to LRP accumulation partner A t W' A Price per unit of points tat LPR ost H carges to LRP accumulation partner 4 wen accumulation demand is over 4 's ordering quantity. W R 1 p R Wolesale unit price of rewards tat LRP redemption partner Rj carges to LRP ost//. Per unit point value of rewards offered by LRP redemption partner Rj. 61

74 D R LRP members' redemption demand towards LRP redemption partner Rj. V R Per unit sortage penalty cost of LRP redemption partner R/s rewards. S R Per unit salvage value of LRP redemption partner R/s rewards. (4) Parameters in constraints: / 0 Liability in points at te beginning of te planning orizon. / Target liability in points at te end of te planning orizon. /' L UB Actual liability in points at te end of te planning orizon. Upper bound of liability control limits for te planning orizon. Q R LRP redemption partner R/s capacity limitation on offering rewards. W R LRP ost fts budget limitation on purcasing rewards. Te problem tat we are concerned wit ere is to determine, given te ordering quantities of points from LRP accumulation partners A (i.e., qf), te LRP ost's optimal ordering quantity of rewards from LRP redemption partner Rj (i.e., q ) in order to maximize te LRP profitability (as measured by its value creation); all of tis is subject to te LRP redemption partner R/s capacity limitations on offering rewards, te LRP ost's overall budget for purcasing rewards, and te LRP ost's control on liability. One ting wort mentioning ere is tat, based on Assumption (4), te LRP ost allows te accumulation partners to meet all te accumulation demands (in points) tat are above te 62

75 accumulation partners ordering quantities of points. However, te LRP ost also as an intention to limit te overall amount of points tat te accumulation partners pass to LRP members. Pricing is an effective strategy applied by te LRP ost to encourage or discourage te accumulation partners' usage of points and ten subsequently control te overall accumulation level. By carging a iger price on te extra ordering quantities of points, te LRP ost forces te accumulation partners to sare a greater part of te potential cost of points, wic will occur later as te redemption cost to te LRP ost. Terefore, a reasonable assumption ere is tat w\ A > wf. Hence, under te wolesale-price contract setting, te LRP ost guarantees eac LRP accumulation partner A, a wolesale unit price of points wf. Te LRP accumulation partner A, decides on te quantity qf of points to order, wic is te mean value of te accumulation demand Df tat accumulation partner A, predicts. At te end of te planning orizon, if te accumulation demand Df is iger tan te LRP accumulation partner A,'s ordering quantity ( qf ), te per unit excess demand is purcased at wf, te unit price of points given out to meet te excess demand. Te LRP ost's profitability (i.e., value creation) function at te accumulation side is formulated as follows (BP-A): Te first term in (BP-A), denotes te revenue obtained troug te selling of points to an accumulation partner A,. Te second term denotes te extra revenue tat te LRP ost H gains 63

76 troug te accumulation partner A,'s extra ordering of points wen Df is iger tan te ordering quantity qf. At te redemption side, eac LRP redemption partner R, guarantees te LPR ost a wolesale unit cost of points redeemed, w. Te LRP ost decides on te quantity of rewards (in points) q*, to order during te planning orizon from LRP redemption partner Rj at te given wolesale unit cost, wf. At te end of te planning orizon, if te LRP members' redemption demand (Dj) towards LRP redemption partner R, is iger tan te LRP ost ordering quantity q R }, te excess demand is assumed to be lost and te under-stocking cost v per unit of points is incurred. On te contrary, wen D* is less tan q, te excess ordering quantity is sold at a over-stocking unit sale price s. Te LRP ost ITs profitability (i.e., value creation) function at te redemption side is defined as follows (BP-R): *»<*> =t(p* «W H *tf -< *W -fayt(t*tf -VI) (3-2) 7=1 7=1 In (BP-R), [Dj-q*] + denotes under-stocking quantity wen te redemption demand is iger tan te LRP ost /fs ordering quantity of rewards from a redemption partner Ry, wile [q* -D*] + denotes te over-stocking quantity wen te redemption demand is lower tan te ordering quantity of rewards. Overall, (BP-R) is a linear function consisting of four terms. Te first term indicates te value of rewards offered by eac partner Rj. Te second term indicates te 64

77 LRP ost ffs purcasing cost of rewards. Te tird term refers to te under-stocking cost of rewards. Finally, te fourt term refers to te salvage value of over-stocking rewards. Combining (BP-A) and (BP-R), te problem of planning te supply of rewards witout bonus points can be formulated as follows (ereafter Problem BP): n (q*;d%df) = max E[X {A) +X HW ] max 7=1 L 1=1 ±( P * xmin{,;,d;}-v; XLD; -,; ]++ < x [9 ; -z>; ]+ ) (3.3) subject to: q^<q^,fovj = l,...,j (3.4) 7=1 1 -<L UB, were/ = l 0 + (q?+[d? -q?] + )-tq* l 0 1=1 7=1 q] >0, fory = l,...,j (3.5) (3.6) (3.7) Constraints (3.4) indicate tat eac LRP redemption partner as a capacity limitation on te quantity of rewards offered to an LRP ost. Constraint (3.5) indicates tat te LRP ost as an overall budget limit for purcasing rewards tat cannot be exceeded. Constraint (3.6) is te liability control constraint. It is formulated as te ratio of te target liability (/) at te end of te planning orizon and te initial liability (/ ) at te beginning of te planning orizon. As discussed in te previous section, points earned by LRP members are stored in members' accounts. To te LRP ost, tese points are counted as a liability until tey are redeemed by LRP 65

78 members for rewards. Terefore, te overall target liability at te end of te planning orizon (/) is equal to te initial liability (/ 0 ) at te beginning of te planning orizon plus te overall amount of points earned by members during te planning orizon, ^iqf i.. 1=1 +\D? -^1 ) > mmus te overall "target amount" of points redeemed by members for rewards during te same time j period, ^<Z*. L UB is introduced as te upper bound of te liability ratio in Constraint (3.6). If 7=1 te LRP ost wises to reduce te liability (/), te value ofl UB sould be set to no more tan 1. If te LRP ost plans to increase te liability (/) for te planning orizon, te value of L UB sould be set to larger tan 1. Oterwise, a value equal to 1 is given to maintain te liability at te same level as before. Constraints (3.7) refer to te non-negativity constraints. J Note tat ere ^ q R is called "target redemption" because te actual amount of redemption 7=1 depends not only on LRP ost IT ordering quantities of rewards qf, but also on te redemption J demands Df. Te actual redemption is equal to/^min(gf,z)f J, wic can be reformulated as: J. ^[Qj ilj ~Dj] ) Terefore, te actual liability ( /') at te end of te planning orizon is 7=1 computed as follows: 7=1 / ^ + t(^+[^-^l)-t(?f-k-^l) (3-8)!=1 7=1 66

79 3.2.2 LRP Rewards-Supply Planning Problem wit Bonus Points We introduce in tis section, te modeling of te LRP rewards-supply planning problem for a case were te LRP ost offers cooperative advertising (CA) troug bonus points to LRP accumulation partners. Like oter promotion and advertising strategies, te purpose of using bonus points is to boost end consumers' demand (i.e., LRP members' accumulation demand) for certain products or services. In te rewards-supply planning problem wit bonus points, we transform te CA budget allocation problem into an equivalent bonus points offering decision problem. Te amount of bonus points assigned to LRP accumulation partners represents te CA effort Modeling Assumptions In te modeling of tis problem, assumptions (1) to (13) are still valid, but assumption (14) is relaxed and te following additional assumptions are considered: Assumption (15): Te unit cost of offering bonus points to LRP accumulation partners is a constant. Assumption (16): Te LRP members' accumulation demand is not known wit certainty and consists of two parts: te initial random demand and te demand induced by offering bonus points. Assumption (17): Tere are no carry-over effects of te bonus points offered (as advertising inputs) in past periods. Assumption (18): Te impact of te CA effort (i.e., bonus points offering) on LRP members' accumulation demand towards eac LRP accumulation partner is non-negative and deterministic. Assumption (19): Tere are no cross-effects of bonus points offering among LRP accumulation partners. 67

80 Assumption (15) indicates tat tere are no differences in te costs of offering bonus points to LRP accumulation partners, as te costs associated wit issuing points and teir transfer are very low (negligible). Assumption (16) indicates tat eac accumulation demand consists of two parts. Te first part is te initial random demand represented by a probability distribution. Tis initial unknown demand indicates te accumulation demand obtained at zero level of CA effort (i.e., bonus points are not offered at all). Te second part is represented by a saturation-based response function of CA efforts allocated to eac accumulation partner. Te second part measures te effect of CA (i.e., bonus points) on te initial random demand. Assumptions (17) to (19) relate to te advertising-response function. Tey indicate tat te analysis of bonus points, similar to Holtansen (1982) and Mantrala et al. (1992), is confined to a single-period decision model as "te effects of advertising input usually do not extend beyond a few monts" (Doyle and Saunders 1990). Terefore, we assume tat te carry-over effects of advertising inputs (i.e., bonus points) in te past period do not exist. Furtermore, te effect of bonus points on te initial random demand is always nonnegative and tere is no randomness in tat effect. In oter words, demand uncertainties come only from te initial demands. Assumption (19) guarantees tat te sales-response function we ave proposed is applicable Problem Description and Model Formulation Te following additional notations are used in our modeling: (1) Decision variables: 68

81 b A Te amount of bonus points offered to LRP accumulation partner A,. (2) Parameters: k A Elasticity of LRP ost/ts CA effort (i.e., bonus points offering) on accumulation demand towards LRP accumulation partners A,'s products. Overead cost per unit of points for offering bonus points to eac accumulation partner A,. Te maximum proportion of accumulation demand/)' 4 tat can be reaced by CA effort, 0 < #* < 1. Y A df LRP members' overall accumulation demand towards LRP accumulation partner A,. Additional accumulation demand due to te offering of bonus points. In te setting of an LRP system wit bonus points, Yf is taken to be a function of te initial demand Df, as well as an additional demand df, due to te offering of bonus points. Yf 1 is formulated as follows (EP1-Y): A^-uA r. nsua ^0. kfxbf, weno<bf< Yf = Df + df, were df = rmn{kf x bf, cj>f *Df} = 0f*Df, wenb, XD, K (3.9) In (EP1-Y), LRP ost IPs CA effort is represented by te number of bonus points (bf) offered to LRP accumulation partners. Terefore, te effect of te CA effort can be captured by a piecewise linear function wit a tresold representing te market potential or saturation level of demand 69

82 influenced by bonus points (see Figure 3.3). Tis tresold is defined as a percentage of initial accumulation demand: <j>f x Df. i i i i i i? 4t*»i bf Figure 3.3: Effect of bonus points on accumulation demands As discussed in te literature (e.g., Mantrala et a, 1992), te type of sales-response functions sown in Figure 3.3 is applicable only wen tere are no cross-effects of CA budget allocations to individual submarkets and competitive efforts among submarkets can be ignored (ence, Assumption (19) is stated as above). Meanwile, te competition between LRP accumulation partners is small enoug to ignore it in an LRP system because, generally speaking, accumulation options are rarely overlapping or substitutable. As suc, we can adopt tis type of sales response functions as a part of te LRP member demand function. Hence, te rewards-supply planning problem we are concerned wit ere is determining, given te ordering quantities of points from LRP accumulation partners A, (i.e., qf, i=l,2,...,i), te 70

83 LRP ost's optimal ordering quantities of rewards from LRP redemption partners Rj (i.e., q R, j=l,2,...,j) as well as te bonus points offering among LRP accumulation partners A, (i.e., bf, i=l,2,...,i ) in order to maximize te LRP profitability (as measured by its value creation), subject to LRP redemption partners R/s capacity constraints on offering rewards, te LRP ost's overall redemption budget for rewards, and te LRP ost's liability control requirement. Te LRP ost Ifs value creation function considering bonus points offering at te accumulation side can be formulated as follows (EP1-A): ^)=Z(^x^+^x[^-^] + - c " x^) (3-10) 1=1 Te meanings of te first and te second terms in te function are te same as in (BP-A), except tat te accumulation demand is Y A and not Df. Tese terms state tat te LPR ost revenue from points-business at te accumulation side is obtained troug te selling of points, and te LRP accumulation partners' extra ordering of points wen Y A t is iger tan LRP accumulation partner A,'s ordering quantity of points qf. Te tird term, cf x bf, refers to te costs (e.g. overead or administration cost) incurred wen LRP ost H offers bonus points to LRP accumulation partner A,. At te redemption side, te LRP ost ITs value creation function remains te same as in te previous model, but it is now restated as (EP1-R): 71

84 * =Z(^;xmin{,;,z);}- w ;x 9 ;_v? X[ D;-<a)+I«4?; -^] + ) o.n) Combining (EPl-A) and (EPl-R), te LRP rewards-supply planning problem wit bonus points offerings can be formulated as follow s (ereafter Problem EP1): n H (q%b?-d«,d*) = m a *E[7t H(A)+ 7T HW ] = max < ±(w?xq?-c?><b?) + E\±(w*><[Y*- q?l)\-±w«>< q «+ L '=1 J 7=1 ±( P ; xmin{,;,d;}-v; x[z>; - q *\ +< x # -z>; ]+ ) (3.12) subject to: q*<q% forj = l,...,j J=I 1 -<L UB,werel = l 0 + t(qf +[Y, A -^] +^)- (< ) l o i=\ J=l Y, A = Df + min {kf x bf, tf x D?} q] >0, forj = \,...,J bf>0, forz = l,...,/ (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) In comparison to Problem BP tat deals solely wit te ordering quantity decisions, tis model also deals wit te bonus points offering decisions. Constraint (3.15) in tis model is a modification of te constraint (3.6) in Problem BP. In constraint (3.15), te target liability (/) is still equal to te initial liability (/ 0 ) at te beginning of te planning orizon plus te overall amount of points earned by members during te planning orizon (i.e., overall accumulation) minus te overall "target amount" of points redeemed by members for rewards during te same 72

85 time period. Te overall accumulation in Problem EPl includes tree components: te LRP accumulation partners' ordering quantities of points (qf), te extra ordering amount of points ( ["^-^1 ), and te bonus points offered by LRP ost (bf). Te "target" redemption in J Constraint (3.15) is te same as tat in Constraint (3.6), wic is2^,^ Constraint (3.16) defines te accumulation demand tat includes te initial random demand and te extra demand driven by bonus points. Constraints (3.4), (3.5), and (3.7) in Problem BP are respectively relabeled as constraints (3.13), (3.14), and (3.17) in tis model. Constraint (3.18) is te non-negativity constraint for te decision variable &/* LRP Rewards-Supply Planning Problem wit Option Contracts In Problem BP and Problem EPl, we assume tat te relationsips between LRP redemption partners and te LRP ost are governed solely by wolesale-price-only contracts. However, as pointed out in te SC/OM literature (see Capter 2), tis type of contracts cannot offer enoug flexibility to cope wit demand uncertainties, wereas option contracts ave been viewed in te literature as an effective tool tat can bring more flexibility into coordination among SC entities. Terefore, as an extension, we introduce in tis section te modeling of te LRP rewards-supply planning problem to te case were option contract is used in rewards-supply contracting Modeling Assumptions In our modeling, Assumptions (1) to (12) and (14) are still valid, but Assumption (13) is relaxed and te following additional assumption is considered: 73

86 Assumption (20): Option contracts are used as te governing mecanism in rewardssupply contracting. Assumption (20) indicates tat instead of wolesale-price contracts, option contracts are used as te governing mecanism in te business relationsips between an LRP ost and LRP redemption partners Problem Description and Model Formulation Te following are te additional notations used in tis extension: (1) Decision variables: qx<> LRP ost /Ts initial ordering quantity of rewards from redemption partner Rj 1 (transformed into te ordering quantity of points). m«i 1 Ri 1 Number of units of options tat LRP ost H purcases from redemption partner RJ- Number of units of options tat LRP ost H exercises from redemption partner Rj. (2) Parameters: via w*> g^ Per unit price tat LRP redemption partner Rj carges LRP ost H to purcase rewards (in points). Per unit price tat LRP redemption partner Rj carges LRP ost H to purcase options (in points). Per unit price tat LRP redemption partner Rj carges LRP ost H to exercise options (in points). Te rewards-supply planning problem wit option contracts consists of determining, given te ordering quantities of points from LRP accumulation partners A, (i.e., qf, i=l,2,...,i), te LRP 74

87 ost FTs optimal ordering quantities of rewards from LRP redemption partners Rj (i.e., q*, j=l,2,...,j), and determining te optimal numbers of options to purcase (i.e., m" 0,j=l,2,...,J) and to exercise (i.e., ey 1,j=l,2,...,J) in order to maximize te LRP profitability (as measured by its value creation), subject to LRP redemption partners' capacity constraints on offering rewards, te LRP ost's overall redemption budget for rewards, and te LRP ost's liability control requirement. In a typical option contract, te buyer (i.e., te LRP ost in te context of tis study), is allowed to adjust te ordering quantity after observing te actual demand, wereas te wolesale-price contract doesn't provide tis kind of flexibility. More specifically, te LRP ost H as to make tree types of decisions wit regards to rewards ordering quantities. At te initial ordering time (e.g., before sales begin), te LRP ost H orders qf units of rewards from eac partner R, at a unit wolesale-price of w^. Tis ordering quantity usually cannot be canged at a later time. It is known as a "firm order" or "minimum purcase commitment". However, te LRP ost H is allowed to purcase m^ units of options up to a certain amount, M y ' (i.e., 0<m y ' <M y '), at a unit option price of vt^1. At a later ordering time (e.g., a time closer to te beginning of sales or in te middle of te sales period), te LRP ost //may coose to exercise all te units of options or part of tem, qf', at a per unit exercise price of ej, were qf* < mf 1. Terefore, under option contracts, te LRP ost H does not get te ordering flexibility for free; instead, te LRP ost H as to pay extra money to obtain tat flexibility. 75

88 Hence, under te option contracts setting, LRP ost ITs value creation function and rewardssupply decisions at te redemption side are modified as (EP2-R): p* x min \qf + q* 1, D* } - w* 0 x q* - wf x m* 1 - e* 1 x q* K H(R) = 1 (3.19) -V>[Z>;-(^+^)] 7=1 ++ <X[^-D;] + Te first term in (EP2-R) indicates te value of rewards offered by eac redemption partner Rj. Te second term indicates te LRP ost's purcasing cost of rewards. Te tird and fourt terms indicate te LRP ost's costs for purcasing and exercising options. Te fift term refers to te under-stocking cost of rewards. Finally, te sixt term refers to te salvage value of overstocking rewards. At te accumulation side, te LRP ost ITs value creation function remains te same as before, but restated as (EP2-A): ^)=ZK x?>^x[^-^l) (3.20) Combining (EP2-A) and (EP2-R), te rewards-supply planning problem wit option contracts is modeled as follows (ereafter Problem EP2): n (^,mf,q*;d",d?) = max E\n B{A) + n H{R) ] = max < j=\ ±p^rrnn{q^+qf,d^-±v^[df-(q^+q^} + ±s^[ q^-d^l L «=i + (3.21) subject to: 76

89 q*<q?,forj = l,...,j (3.22) J E( w^ x^ + W J' xm j' 7=1 +e? x^') - w * (3-23) m* 1 <M*, for j = l,...,j (3.24) f <L UB,werel = l 0 + (q?+[d? -qt\)-fj^ 'O 1=1 7=1 +^) ( 3 " 25 > ^'<m y \ fory = l,...,j (3.26) qf<max{0,d«-q^}, forj = l,...,j (3.27) qt>,m*z0, for7=1,...,j (3.28) ^'>0, fory=l,...,j (3.29) In Problem EP2, Constraint (3.22) indicates tat eac redemption partner R } as a capacity limitation on te quantity of rewards offered to LRP ost H. Constraint (3.23) indicates tat te LRP ost as an overall budget limit for purcasing rewards and options tat cannot be exceeded. Constraint (3.24) indicates tat eac redemption partner as a limitation on te overall quantity of options offered to te LRP ost. Constraint (3.25) is te liability constraint. Te target redemption in tis constraint is different from tat in Constraint (3.4) in Problem BP and in Constraint (3.15) in Problem EP1. In Constraint (3.25), te target redemption is te sum of te initial ordering quantity of rewards and te number of options to exercise. Constraint (3.26) represents a logical assumption associated wit te option contracts: te number of options {q^ ) tat te LRP ost H can exercise sould be no more tan te number of options tat te LRP ost H purcased (mf ). Constraint (3.27) represents anoter logical assumption associated wit option contracts, wic states tat, after observing te "true" redemption demand, te options exercised will not exceed te difference between te redemption demand and te initial ordering 77

90 quantity of rewards. In oter words, if te "true" redemption demand is less tan te initial ordering quantity, ten no options will be exercised. Constraints (3.28) and (3.29) refer to te non-negativity constraints. 3.3 Summary In tis capter, we ave proposed tree matematical programming models to examine te LRP rewards-supply planning problem under different settings. More specifically, te first model {Problem BP) was developed to cope wit rewards-supply planning decisions in te presence of multiple commercial partners offering various redemption and accumulation opportunities to LRP members, and multiple management concerns (e.g., LRP ost profitability, liability control, demand uncertainties, budget, and capacity limitations). Te second model {Problem EP1) is a generalization of te first model in te sense tat it incorporates bonus-point decisions. Te tird model {Problem EP2) is also a generalization of te first model to consider option contracts in rewards-supply contracting. Considering te demand uncertainties involved in tese models as well as te underlying linearity assumptions, our models lend temselves to stocastic linear programming models. We discuss our solution metodology in te following capter. 78

91 Capter 4 Solution Metodology In tis capter, we will sow ow our matematical models can be solved troug te stocastic programming (SP) approac. We begin wit a brief introduction of key caracteristics of SP and its implementation, and ten propose a sampling-based euristic solution procedure in te context of tis study. We also present te reformulations of te rewards-supply planning models (i.e., Problem BP, Problem EP1, and Problem EP2) according to te solution procedure and discuss some computational implementation issues related to our solution procedure. 4.1 Stocastic Programming and Its Implementation SP is a well-known optimization approac used to incorporate uncertainty in te decision process. Te study of SP dates back to te 1950s (e.g., Beale, 1955; Dantzig, 1955). SPs ave been applied to a wide range of researc problems involving uncertainties. Examples of tese researc problems include production planning wit a random yield (Zanjani et al, 2007), supply cain network design under demand, capacity, transportation cost and supply uncertainties (Santoso et al, 2005), assignment problems wit resource-constrained uncertainties (Toktas et al., 2006), resource portfolio planning under demand uncertainty (Wang et al., 2008), and quantitative analysis of multi-periodic supply cain contracts wit options (Delft and Vial, 2004). Most of te models proposed in tese studies focus on minimizing (or maximizing) te expectation of te resultant distributions of random components. In tis researc, we follow te same logic to deal wit te randomness involved in our models. As pointed out by many researcers (e.g., Birge and Louveaux, 1997; Ruszczynski and Sapiro, 2003), te advantage of 79

92 using SP to deal wit randomness is tat te optimal solution obtained is more robust wit respect to variations of random elements tan te optimal solution of te corresponding deterministic problem in wic random elements are replaced by teir mean values. Te simplest form of SP results wen some of te parameters in a linear program are represented by random variables (eiter discrete or continuous). Our models are quite similar to tat simple form, wic is known as stocastic linear programming (SLP) in te literature. A general format of SLP is defined as follows (ereafter Problem-SLP): max n H =c T x z (4.1) s.t.: Axz = b (4.2) T(co)xz<(o)) (4.3) z>0 (4.4) were c T is te transpose of te coefficient vector in te objective function (4.1). z is te vector of te decision variables. Constraints (4.2) and (4.3) represent deterministic and stocastic constraints in Problem-SLP, respectively. A and b are te coefficient matrix and te rigt and side vector of te deterministic constraints (4.2). co is te vector of random parameters and r(&>) and (co) are te coefficient matrix and rigt-and side vector of te stocastic constraints (4.3) involving random parameters. Due to te randomness in te problem data, te value of te objective function in Problem-SLP is no longer deterministic, wic brings more callenges in solving it. Various studies ave been conducted on SLP formulations, solution tecniques, and teoretical solution properties. For an extensive discussion of tese topics, 80

93 readers are referred to Kail et al. (1994, 2005), Birge and Louveaux (1997), Sen and Higle (1999), and Ruszczynski and Sapiro (2003). In our study, due to te special structure, our models lend temselves to a specific type of SLP models, known as two-stage SLP wit recourse (2SLPR). Te key caracteristic of tis type of SLP models is tat decision variables are classified into two stages according to weter tey are implemented before or after an outcome of a (vector valued) random variable is observed. In oter words, a set of decisions (z) is taken in te first stage witout full information on te random vector. Later, full information is received on te possible realization of te random vector; ten a second stage action (also called recourse action) is taken. Te second stage decisions (y) allow us to model a response to eac of te observed outcomes of te random vector, wic constitute te recourse. A 2SLPR format of Problem-SLP is structured as follows (ereafter Problem-2SLPR): max n H =c r xz + $(z) (4.5) s.t.: Axz = b (4.6) z>0 (4.7) were 0(z) = E? [g(z,«;)]is te expected value of te linear recourse function, and g(z,o))is defined by te second stage problem or, in oter terminology, by te recourse problem as follows: max g(z,o))=q(co) T xy(co) (4.8) s.t: 81

94 W(a))xy(co) = (co)-t(a>)xz (4.9) y{co)>0 (4.10) were ^(co) = [W(o)),T(a>),(a>),q(ca)j is te vector of stocastic components in te second stage. Problem-2SLPR is suitable for decision models wit a single-period randomness and reaction (e.g., newsvendor-based models). Our models (i.e., Problem BP, Problem EPl, and Problem EP2) satisfy tis caracteristic. Terefore, tey can be rewritten into te 2SLPR formats as sown below in sections 4.1.1,4.1.2 and 4.1.3, respectively Model Reformulation for Problem BP Te model considering te rewards-supply planning problem witout bonus points (i.e., Problem BP) can be reformulated into a 2SLPR format as follows (ereafter Problem BP-2SLPR): 1=1 ]=\ subject to: q«<q«,forj = l,...,j (4.12) t«x^)<^ (4.13) q«>0, forj = l,...,j (4.14) were g(q R,a>) corresponds to solving te following model: g(l%co) = g(<l%d?,d«) = (4.15) y=i 82

95 subject to: / i J -<L UB, were I = l 0 + Jf j (qf+1f')-^qf (4.16) 'O i=l 7=1 lf + -lf-=qf-df, for i=l,...,/ (4.17) I? + -I?-=q?-B?, fovj = l,...,j (4.18) lf\lf',lf\ If' >0, fori = \,...,IJ = l,...,j (4.19) In tis reformulation, co = {. >/*, )*} is te vector of random parameters. Te vector of first-stage decision variables is x - V) and te vector of second-stage decision variables is y(co) = llf +,lf',lf +,lf'\, were lf +,lf',lf +,lf' are new decision variables defined in te second stage. lf + and 7* + denote te quantities: [^-Z); 4 ] andt^-d*] respectively, wereas, if' and /*" denote te quantities: \Df -qf~\ and [D* -q*~\ respectively. Te values of tese second-stage decision variables depend on te variation of demand. Te term minl^,dj\ in Problem BP is replaced by q R } -Ij + in Problem BP-2SLPR, for simplicity. Constraint (4.16) is te liability constraint, wic as te same formulation as Constraint (3.6) in Problem BP except tat [pf -q?~\ is replaced by if' in Problem BP-2SLPR. Constraint (4.17) is constructed to define te values of if', 1'f +, so tat wen Df>qf, if' = [pf - qf ] and If + =0, and wen Df < qf, lf + = [qf -Df~\ and lf~ = 0. Constraint (4.18) is constructed to define te values of if- and I* +, so tat wen Df >q R, if' =[pf - qf ~\md lf + =0, and 83

96 wen Dj<q*, I* + = [^ -D*] and I* =0. Constraints (4.19) are te non-negativity constraints for te second stage decision variables Model Reformulation for Problem EP1 Te model corresponding to te LRP rewards-supply planning problem wit bonus points (i.e., Problem EPl) can be reformulated into 2SLPR format as follows (ereafter Problem EP1-2SLPR): M ^ n ^ ^ x ^ - c f ^ j + ^ x ^ - ^ x ^ l + E^gj^fl,)] (4.20) subject to: constraints (4.12) - (4.14), and bf>0, for i = l,...,/ (4.21) were g(q*,bf,a>\ corresponds to solving te following model: g(q«,b?,co) = g( q %b?,d?,d*) = max (^ x//) + X K x/f +< x/- - p * x/f) ( 4 " 22 > i-l y=l subject to: constraints (4.18) - (4.19), and l -<L UB, w e rel = l 0 + l (qf+i?-+b?)-f t q* (4.23) <o <=i j=i kf x bf < </>? x Df, for i = 1,...,/ (4.24) I? + -I?- = q?-(d?+k?xbf), for» = 1,...,/ (4.25) 84

97 Te first stage model in Problem EP1-2SLPR includes two sets of decision variables: (1) te ordering quantity of rewards (qf) and (2) te amount of bonus points (bf ) offered to eac accumulation partner. Te impact of bonus points on demand (kfxbf) is integrated into te second stage model. Constraint (4.23) is te same as Constraint (3.15) in Problem EP1 except tat Yf qf is replaced by if'. Constraint (4.24) indicates tat te maximum impact of bonus points on te random accumulation demand cannot exceed <j>f x Df. Constraint (4.25) is used to define te values for if- and lf + so tat wen Df>qf, If' = [Y, A - qf ] and lf + =0, and wen Df < qf, lf + = [qf -Y, A ]analf- = Model Reformulation for Problem EP2 Te model corresponding to te LRP rewards-supply planning problem wit option contracts (i.e., Problem EP2) can be reformulated into 2SLPR format as follows (ereafter Problem EP2-2SLPR): Maxn =XK x^) + i(pj ~ w^) x 1^ -i, * 1 * * 1 +E;[g(qf>,m%<o)] (4.26) subject to: ^<0,\ for 7 = 1,..., J (4.27) m R ; < M* 1, for j = I,..., J (4.28) q«,m«>>0, for j =!,...,/ (4.29) 85

98 were giqf 0, mf 1, an corresponds to solving te following model: g(tf,m?,a>) = g(q*,m?,d*,d*) (4 30) =Max ±(wf */,'-) + ±(p? x tf -$ xtf ) + (-vf x/;- +,; x/- -^ x/-) i=l y=l y=l subject to: constraints (4.17) and (4.19), and f<l [/s,were/ = / 0+ X(^+ / ")-Z(^+^1) (4-31) '0 '=1 7=1 ( w* x ^ + w*> x m* 1 + e^ x #*') < W R (4.32) /; + -/;-=(^+^)- jd ;, fory=l,...,j (4.33) /^_/; =^_Z);, fory=l,...,/ (4.34) ^m,*', for./=l,...,j ^</f", fory = l,...j /^Vf >0, for y =!,...,/ (4.35) (4.36) (4.37) In Problem EP2-2SLPR te first-stage decision variables arex = ^,wm and te second stage decision variables are y(a>) = \qf,i* +,I?~,I* +,lf,l"*,l"~}. Te first-stage model involves optimal decisions on initial ordering quantities of rewards as well as te purcase of options. Te second stage model involves optimal decisions on balancing overstocking and under-stocking costs as well as on te number of units of options to be exercised based on initial ordering quantities, purcased options, and observed scenarios of demand realizations. Constraints (3.22) - (3.24), and (3.28) in Problem EP2 are relabeled as constraints (4.27) - (4.29) and (4.32) in tis model. Constraint (4.31) is te same as constraint (3.25) except tat \Df -<jf \ is replaced by 86

99 if. Similar to Constraint (4.18) in Problem BP-2SLPR, Constraint (4.33) is constructed to define te values of if and if, so tat wen/)* > [qf + qf ), if = [Df - qf - qf ], and wen Df <[qf + qf), If = [qf + qf - Df ]. Constraint (4.34) is constructed to define te values of If and/ y u+, so tat wen Df > qf,lf = [Df -qf ] +, and wen Df < qf, lf + = [qf -Df~\ +. Constraint (4.36) is te reformulation of Constraint (3.27) in Problem EP2. Constraints (4.37) are non-negativity constraints for te second stage decision variables, if* and if. 4.2 Solution Procedure In tis section, we propose and explain te solution procedure used to solve te rewards-supply planning models. Generally speaking, solving an SP model is not an easy task due to te randomness involved in te model. Solution metodologies for solving SLP are divided into two main categories: (1) exact metods tat include te L-saped approac (e.g., Kail et al, 1994 and 2005; Birge and Louveaux, 1997) and te regularized decomposition approac (e.g., Ruszczynski and Swietanowski, 1997), and (2) approximate metods tat are based on sampling tecniques, suc as te sampling average approximation (SAA) approac (e.g., Sapiro et al, 1998; Mak et al, 1999; Kleywegt et al, 2001) or te stocastic decomposition approac (e.g., Higle and Sen, 1996). Te exact metods attempt to find exact solutions for an SLP optimization problem. Tese metods are only feasible for situations were tere exists only a finite number of realizations of random parameters (i.e., modeled as discrete random variables). In contrast, approximate approaces attempt to find approximate solutions tat are close to te optimal solution witin a reasonable error range. One standard approximate approac is to apply te 87

100 Monte Carlo sampling procedure to generate n realizations of random parameters and to solve te approximated problem based on samples of tose random parameters. In our models, redemption and accumulation demands are assumed to be continuous random parameters, and consequently yield an infinite number of demand realizations. In addition to tat, wit te increase of te number of redemption and accumulation partners, te number of demand parameters increases as well. Eac of tose demand parameters follows different demand distributions (in terms of te values of te distribution parameters). As suc, approximate approac is more suitable for solving our models. We coose te approximate approac tat is based on te SAA sceme. Tis approac is also known as an "external" approximation approac in te literature (e.g., Ruszczyfiski and Sapiro, 2003; Mak et a, 1999). Compared to oter approximation tecniques, te external approac as te following advantages, as pointed out by Ruszczynski and Sapiro (2003): It is easier to implement because it separates sampling procedures and optimization tecniques. It is more universal in te sense tat any optimization algoritm tat is developed for a considered class of SLPs can be applied to te constructed SAA problem in a straigtforward way. A quite well-developed statistical inference of te SAA metod is available tat, in turn, gives a possibility of error estimation, validation analysis, and ence stopping rules. 88

101 Te structures of our models are complex, and random demands are assumed to be independent and follow different probability distributions. Terefore, it is more appropriate to apply an external approximation approac, suc as SAA Sample Average Approximation (SAA) Reformulation In te external approac wit te SAA sceme, a random sample a> l,a> 2,...,(o N of Af realizations (scenarios) of te random vector is generated outside of an optimization procedure and te expectation of te second-stage objective function is approximated by te sample average 1 N function: <t N (z) = ^glz, '). Based on tis approac, Problem-2SLPR is approximated by N 1=1 te following SAA problem (ereafter Problem-2SLPR-SAA): 1 N max^ =c T xz+ ^q((o")xy(co n ) (4.38) s.t: constraints (4.6), (4.7) and W(co")xy(co n ) = (co n )-T(co n )xz, farn = l,...,n (4.39) y(o)")>0, forn = l,...,n (4.40) For a particular realization of te random parameters, Problem-2SLPR-SAA is a deterministic problem tat can be solved by an appropriate deterministic algoritm. However, wat we obtained troug Problem-2SLPR-SAA is an approximate solution rater tan te actual solution. Note tatis[ ff ] = /r ff. Previous studies ave sown tat under mild regularity conditions, a fairly good approximate solution to a true SLP problem can be obtained troug SAA (Kleywegt et al, 2001; Ruszczynski and Sapiro, 2003). We now sow ow te SAA sceme applies to te models discussed in tis study. 89

102 SAA Model for Problem BP-2SLPR Te SAA model of Problem BP-2SLPR is a deterministic linear programming model tat can be stated as follows (ereafter Problem BP-2SLPR-SAA): Max ft H = 1=1 ly S=l,=1 J J=\ (4.41) S=l J=\ subject to: q«<q% fory = l,...,/ (4.42) Z(w;x?;)<^ 7=1 q] >0,forj = l,...,j tw)^z(^)+tw')-( z t»- 1 ) x/ o' forj=1 '-^ y=l (=1 i=l», r«- TR+ T^R <+C-C=^' fory=l,...,j,5 = l,...,iv 7^-=max{Z)j-^,0}, fori = l,...,i,s = \,...,N I«+,I«>0,forj = l,...,j,s = l,...,n (4.43) (4.44) (4-45) (4.46) (4.47) (4.48) In Problem BP-2SLPR-SAA, te second stage decision variables lf +,I, ~ J* +, and/*" are redefined as I ls + >I IS ~>I JS + > and/ ys "for eac sample of random demand realizations. Te optimal values of tese variables are te sample average of te optimal values of /,/,I 1S,IJ',I p obtained under eac sample of random demand realizations. Constraints (4.16) - (4.19) in 90

103 Problem BP-2SLPR, wic involve te second stage decision variables, are redefined in terms of eac sample of random demand realizations as constraints (4.45) - (4.48). Note tat, given a sample of random accumulation demand, e.g., Df s, te values of if* and I*~ are determined before solving te model as qf are now known parameters. In oter words, if Df is greater tan qf, ten /,f =0 and lf~=df s -qf ; oterwise, lf + =qf-df and if-=0. Terefore, constraints (4.17) in Problem BP-2SLPR are modified as constraints (4.47) and lf + and if'are not decision variables any more in Problem BP-2SLPR-SAA SAA Model for Problem EP1-2SLPR Te SAA model for Problem EP1-2SLPR is also a deterministic linear programming problem formulated as follows (ereafter Problem EP1-2SLPR-SAA): MaxJr H =±(^><qf-cfxbf) + ±( P^^- W^q^ + ^±±( W^lf s -) + 1=1 j=l " S=\ 1=1 -LYY(-v*x/*-+j*x/* + -p*x/* + ) \J LuLu\ J J* J P rj js ) (4.49) -<V s=l i=\ subject to: qf<q% forj = l,...,j ±(w«xq«)<w* 7=1 qf >0, for/ = l,...,j bf>0, for i =!,...,! (4.50) (4.51) (4.52) (4.53) 91

104 (q«)- (l?; + b?)>j:(q?)-(l UB -l)><l 0,fors = l,...,n (4.54) y=l 1=1 1=1 kfxbf < tfxdi, for i = l,...,i,s = l,...,n (4.55) I«~-I?s + -kfxb?=d? s -qf, fori = l,...,i,s = l,...,n (4.56) I R ;-C + VJ= D JS> forj = l,...,j,s = l,...,n (4.57) CJ?s~Jjs + >C'> 0 > f0ti = l,...,i, j = \,...,J,s = \,...,N (4.58) In te same way as in Problem BP-2SLPR-SAA, te second stage decision variables lf +,lf',ij +, and/"" in Problem EP1-2SLPR-SAA are redefined as lf,i*~,i* +, and/j for eac sample of random demand realizations. Te optimal values of tese variables are te sample average of te optimal values of I ls +,1^' JJ J p obtained under eac sample of random demand realizations. Constraints (4.18) - (4.19) and (4.23) - (4.25) in Problem EP1-2SLPR are redefined as constraints (4.54) - (4.58) for eac sample of random demand realizations because tese constraints involve second stage decision variables. Te values of I* + and I*~ depend on te random demands Df s as well as te bonus points (bf). Since bf is a decision variable in tis model, I* + and I*~ are decision variables as well SAA Model for Problem EP2-2SLPR Te SAA problem of (EP2-2SLPR) is a deterministic linear programming problem formulated as follows (ereafter Problem EP2-2SLPR-SAA): 92

105 ,=1 j=\ j=l JV s=l I=1 (4.59) + izzk -< K +^ZSK <+*; *C -pj «C) M s=l j=l -< V S=\ J=\ subject to: q^<q% for y = l,..., J (4.60) m* 1 < M* 1, for j = I,..., J (4.61) q*,n$>0, for/ = l,...,j (4.62) ZW+^)^Z(^) + t(^')-(a«-l)x'o.for5 = l )...,^ (4.63) j=i ;=i 1=1 J Z( w^ x^ +W J 1 xm * 1 +e? *q*)^wr > fors = l,...,n (4.64) ^ < iwf, for/ = 1,.../, s = 1,..., JV (4.65) q* < Il~, for j = 1,...J, s = l,...,n (4.66) 7f =max{^-^,0}, for i = l,...,/,.s = l,...,tf (4.67) / " + - / ""=(^+^,)- jd "' fou=l,...,/,5 = l,...,jv (4.68) C- J JS=^- D *> forj = l,...,j,s = l,...,n (4.69) /;/> ^>C' C * 0, for ; = 1,..., J,s = \,...,N (4.70) g >0,fory = l,...,j,s = l,...,jv (4.71) D In Problem EP2-2SLPR-SAA, q } \ is defined as te quantities of exercised options for eac sample of random demand realizations. Te optimal value of q^ is estimated as te sample average of te optimal values of q J. obtained under eac sample of random demand realizations. 93

106 Meanwile, constraints (4.17), (4.19), and (4.31) - (4.37) in Problem EP2-2SLPR are restructured for eac sample of random demand realizations as constraints (4.63) - (4.71). Te solving procedure for obtaining approximate solutions of our models as well as te evaluation procedure for te approximate solutions is presented next SAA-based Heuristic Solution Procedure Te basic idea of te solving procedure consists of generating an approximate solution, wic is te solution of a number of instances, say M of SAA problems, eac wit N sampled scenarios. Te quality of a candidate solution is ten tested by bounding te optimality gap between te true objective value and te expected objective value troug standard statistical procedures. A sampling evaluation procedure based on common random numbers (CRN) is used to construct te confidence interval for te optimality gap. CRN means tat te same N sampled scenarios are used to generate te optimal gap as well as to evaluate te objective function value. According to te work of Mak et al. (1999) and Ruszczynski and Sapiro (2003), CRN can provide a significant variance reduction over naive sampling. Moreover, CRN can also eliminate te "negative gap" penomenon (Mak et al., 1999; Feng et al., 2010). Te complete solving procedure is described as follows (ereafter Algoritm 1): Step 1: Generate i.i.d. batces of samples. Generate M independent identically distributed (i.i.d.) sample replications of random demand realizations, eac of size N, i.e., of... co", m=l,..,m; 94

107 <=[AT - D N m j - D u -D i\» n=l,...,n were D% denotes te nt sample of redemption demand of partner R/s in te mt sample replication, and D^ denotes te «t sample of accumulation demand of partner A,'s in te mt sample replication. Step 2: Solve te corresponding SAA problems. Compute te statistical upper bound estimation of te optimal value of te objective function. Terefore, for eac sample replication m, m=l,..., M, solve te corresponding SAA problem (e.g., Problem BP-2SLPR-SAA). Let n and x" be te corresponding optimal objective function value and te vector of optimal solution, respectively. It is 1 M 1 M well known tat T^w is an unbiased estimator of^ (i.e.,e(x N )- T^ ) and meanwile, N is an upwards biased estimator of;r* (x* denotes te optimal value of te true problem) in te case of maximization. As te corresponding SAA problems can be seen as relaxation problems (i.e., wit fewer feasible constraints/cuts) of te true 1 M 1 M TT problem, we ave ft N >n*. Hence, ~^n^>7u*, wic indicates tat ^^ N provides a valid statistical upper bound estimation for te optimal valuer* of te true problem (see Mak et al, 1999, for more details). Step 3: Generate candidate solutions to compute te statistical lower bound estimation by completing te following two sub-steps: 95

108 Step 3.1: Solve te same SAA problems (e.g., Problem BP-2SLPR-SAA) but use a sample wit sample size N' larger tan N. Let x^.be te corresponding optimal solution vector. Here, te sample of size N' is generated independently of te samples used to obtainx. Step 3.2: Estimate te true objective function value 7r(x) wit a candidate solution x = x N, for all replications of samples wit sample size N as follows: 1 K 1 N ^(x) = c T x+ Y J g(x,(o^ = c T x N,+ Y J g(x N 0)^, form = l,...,m (4.72) Tis step involves solving M independent second-stage sub-problems g(x N,,co ) given x=x N,. Let ^(x)be te corresponding objective function value. Ifxis a feasible solution of te problem sown in te formulation (4.72), ten n (x) < X. In addition, it can be sown tat given x,ft (x) is an unbiased estimator of te true objective function value ^"(x)and E\K (X)1 = n(x) < n, were n is te true optimal objective function 1 M value (see Mak et al, 1999). Terefore, ^>f (x)provides a valid statistical lower M m=\ bound estimation for te optimal valuer* of te true problem. Step 4: Compute te optimality gap for te candidate solution x as follows: G;(x) = 7t m N-^{x) = ^-7^(x N ), form = l,...,m (4.73) were A are generated in Step 2 and ft^ (x w,) are generated in Step 3. 96

109 Step 5: Generate te point estimator and te confidence interval (CI) of te optimality gap. Te point estimator of te optimality gap of te candidate solution x - x N, is obtained by calculating te sample mean as follows: 1 M 1 M 1 M 1 M 1 M 1 M _ Since ^ ^ > ^ ^ (*), we ave G^ (x) > 0. Te sample variance of te mean lvl m=l iyi m=\ optimality gap can be computed as follows: 4(-)~Z(^(-)-^(x)) Z&W-GXfa)) (4.75) Terefore, te approximate 100(1-a) % CI for te mean of te optimality gap can be obtained as follows: 0, G%(3c) + s e ], were s G = ^ *i G (*) (4. 76) Te algoritm described in steps 1-5 assumes tat te candidate solution x = x N, will always be a feasible solution of problem (4.72). However, in Step 3.2, wen solving M independent secondstage sub-problems g(x N,, a ) wit givenx N,, we may face an issue of infeasibility. Due to te randomness involved in te models, te optimal solution x N, of, for example, Problem BP- 2SLPR-SAA wit sample size N\ may not be a feasible solution of Problem BP-2SLPR-SAA wit sample size N. In order to deal wit tis infeasibility issue, te following euristic procedure is proposed (ereafter Algoritm 2): 97

110 (a) Given a candidate solution x = x" N,, solve te second-stage sub-problems g{x N,,co") for m = 1,...,Mas sown in relation (4.72). (b) If a feasible solution is found, continue to steps (4) and (5). (c) If a feasible solution is not found, ten identify te constraint(s) tat caused te infeasibility, and add tem as feasible cuts into te SAA problem used to generate x N, in Step 3.1. Solve te SAA problem again as in Step 3.1, but wit te extra feasible cuts to generate a new candidate solution. Obviously, if a new candidate solution, for example xl,,, can be found, it must be a feasible solution of all M second-stage sub-problems g(x.,fl>;) Implementation Issues of te Solution Procedure We found tat tere are tree implementation callenges involved in te above SAA-based euristic solution procedure. Callenge 1: In Step 1 of Algoritm 1, an important decision is made about te sample size. According to Kleywegt et al. (2001), teoretically speaking, te sample size can be estimated as: N> Xgmax, xlog (s-sf \ a J (4.77) were cr^ is te maximum variance of a certain function value (see Kleywegt et al, 2001, for more details), X is te feasible set of te true problem, and \X\ denotes te number of elements in te setx. Tis estimation guarantees tat te SAA solution x wit an absolute optimal gap of 5 ( 98

111 S e [0, s]) to te SAA problem is a solution wit an absolute optimal gap of s to te true problem wit a probability of at least I-a. In oter words, \7t N (x)-7r* < 8 => p (^r(x)-^j < s \=l-a. However, as te autors pointed out, tere are two drawbacks to tis estimate: (1) It may be too conservative for practical applications and (2) for many problems eiter cr^ax or \X\ or bot may be difficult to obtain. Furtermore, te trade-off between te improvement of te solution quality of te SAA problem and te increase of computational complexity for solving te SAA problem sould be taken into account in te coice of sample size N. Terefore, euristic approaces are quite often used in te literature to determine te sample size in te numerical tests for various SAA problems (e.g., Kleywegt et a, 2001). In Algoritm 1, iger is te sample size N, better is te estimated solution quality of te SAA problem; wereas, iger is te number of replications M, smaller is te confidence interval of te optimality gap. Te sample size N and te number of replications M in tis study will be empirically determined. Callenge 2: In Step 2 of Algoritm 1, if te SAA problems cannot be solved directly in a reasonable amount of computational time due to te size or te complexity of te problems ten some advanced solving tecniques, suc as decomposition or relaxation metods, will need to be used to reduce te computational time. In our case, in order to determine weter suc advanced tecniques are necessary or not, a computational simulation is conducted considering different problem sizes (see Capter 5). Callenge 3: To implement Algoritm 2, we explore te special structures of our SLP models as tere are only a few sets of constraints tat may lead to infeasible solutions. We call tose 99

112 constraints candidate infeasible constraints. For example, in Problem BP-2SLPR-SAA only te liability control constraints (4.45) are candidate infeasible constraints. Secondly, given a candidate solution jc^,, after generating samples of random demand realizations for te secondstage sub-problems, candidate infeasible constraints can immediately be detected witout solving te second-stage sub-problems. Terefore, witout adding too muc extra computational effort, we can easily identify te "infeasible" constraints and add tem as feasible cuts into te SAA problem (e.g., Problem BP-2SLPR-SAA) being solved in step 3.1. Te number of feasible cuts tat we need to add into our SAA problem depends on te number of sample replications M and te sample size N. Hence, in te worse case, te maximum number of feasible cuts is Mx N. 4.3 Summary In tis capter, we give a brief review of SLP and ten discuss its implementation in te context of rewards-supply planning problems. We present te 2SLPR reformulations of te analytical models (i.e., Problem BP, Problem EP1, and Problem EP2). After tat, we propose a samplingbased euristic solution procedure to solve SP models wit continuous random parameters. Our solution procedure is a modification of te standard SAA sceme discussed in te literature. In order to implement te solution procedure, we rewrite te 2SLPR-format models into SAA models. Te SAA models can provide approximate solutions for te original stocastic models. A summary list of te models developed in Capter 3 and tis capter is sown in Table 4.1. Finally, some tecnical callenges involved in te solution procedure are discussed. Te computational studies are reported in te next two capters. 100

113 Table 4.1: Summary of te models System Setting Analytical Model 2SLPR Format SAA Model LRP partnersips are governed by wolesale price contracts solely witout bonus points offering Problem BP Formulation (3.3)- (3.7) Problem BP-2SLPR Formulation (4.11)-(4.19) Problem BP-2SLPR-SAA Formulation (4.41)-(4.48) LRP partnersips are governed by wolesale price contracts solely wit bonus points offering Problem EP1 Formulation (3.12)- (3.18) Problem EP1-2SLPR Formulation (4.20)-(4.25) Problem EP1-2SLPR-SAA Formulation (4.49)-(4.58) LRP partnersips are governed by wolesale price contracts at te accumulation side and by option contracts at te redemption side witout bonus points offering Problem EP2 Formulation (3.21)- (3.29) Problem EP2-2SLPR Formulation (4.26)-(4.37) Problem EP2-2SLPR-SAA Formulation (4.59)-(4.71) Model Foundation single-period constrained newsvendor models two stage stocastic linear programming wit recourse sample average approximate models 101

114 Capter 5 Design of Numerical Studies In order to answer our researc questions, four sets of numerical studies were carried out. In tis capter, we describe te design of eac of tem. We first introduce a problem generation procedure tat is applicable to all of our numerical studies. Ten, we present te design of te studies for examining te effectiveness of our solution metodology, and te design of te studies for investigating te impacts of demand variability, budget tigtness, and liability control on an LRP ost's decisions and LRP performance. All te results obtained from our numerical studies are reported and analyzed in te next capter. From tis capter on, our models previously named as Problem BP-2SLR-SAA, Problem EP1-2SLR-SAA, and Problem EP2-2SLR-SAA are abbreviated to BP, EP1, and EP2, respectively. 5.1 Procedure for Generating Testing Problems In te absence of readily available industrial data and academic testing problems, we designed a set of randomly generated problems tat are empirically driven. Table 5.1 reports te main input parameters and teir values used in generating te testing problems. Based on tese parameters, ten testing cases of different sizes were generated randomly. Furtermore, in eac testing case, we also considered te randomness involved in te accumulation and redemption demands. Tus, for eac testing case, ten problem instances were carried out and independent demand samples were generated for eac problem instance. 102

115 Table 5.1: Problem generation parameters Parameters Range of number of LRP redemption partners (NR) Range of number of LRP accumulation partners (NA) Average redemption cost of rewards (AvgMQ Range of market/retailing price factor of rewards (M?) Range of back order price factor of points (BF) Range for capacity factor of rewards (CA) Salvage value factor of rewards (SR) Sortage cost factor of rewards (VR) Average wolesale price of points (AvgMP) Upper bound of liability ratio (L UB ) Price factor for purcasing options (OP) Price factor for exercising options (EP) Parameter Value(s) In generating eac testing case and problem instances, LRP accumulation partners and redemption partners were grouped into tree categories (Type A, Type B, and Type C) to reflect te different levels of business activities. Type A refers to te most important partners of te LRP ost, Type B refers to te less important partners, and Type C refers to te least important partners. Table 5.2 summarizes te input parameters used to generate additional input data per partner type. We limit our analysis of accumulation and redemption demands following eiter normal distributions or uniform distributions. For eac redemption or accumulation partner, samples of redemption demand realizations (Z/orZ)^) are generated randomly based on eiter normal or uniform distributions wit parameter values sown in Table

116 Table 5.2: Problem generation parameters per partner type Parameters Probability distribution of partner types Redemption cost factor (FR) Range of wolesale price factors per unit of points ordered (FA) Range for redemption/accumulation demand distribution (uniform distribution) Demand variability parameter (DV) for uniform distribution Mean and standard deviation for redemption/accumulation demand (normal distribution) DV for normal distribution Maximum percentage of accumulation demand increase caused by CA effort troug bonus points offering (<j>^) A 20% Partner Type B 40% ,0.10,0.15,0.20,0.25 C 40% (45, 6) (25, 3) (7.5, 1) ,0.5,1.0,1.5, Elasticity of CA effort troug bonus points offering on accumulation demand (kf) Te oter input parameters for our problems are obtained according to te scemes described below: a) Wolesale price (Wf) of rewards in terms of dollar value per unit points is te product of redemption cost factor (FR) and te average redemption cost of rewards in terms of dollar value per unit points (AvgMC). b) Market/Retail price (p ) of rewards in terms of dollar value per unit points is te product of wolesale price of rewards (w*) and te market price factor (MR) tat is generated randomly witin te given range. c) Salvage value (s* ) of rewards is te product of wolesale price of rewards ( M? ) and salvage value factor (SR). 104

117 d) Sortage cost (v* ) of rewards is te product of wolesale price of rewards (w/ ) and sortage cost factor (VR). e) Wolesale price (wf) per unit points ordered by accumulation partners is te product of wolesale price factor of points (FA) tat is generated randomly witin te given range and te average wolesale price of points in terms of dollar value per unit points (AvgMP). f) Back order price (w] A ) per unit points requested by accumulation partners is te product of te wolesale price (wf) and te back order price factor of points (BF) tat is generated randomly witin te given range. g) Te rigt and side of te capacity constraint (Q ) for eac redemption partner is generated based on te ranges of te corresponding redemption demand parameters (DR) from te redemption partner and te capacity factor of te rewards. Te capacity factor (CA) is generated randomly witin te given range, ) Te available budget (W R ) for purcasing rewards is required to be determined carefully to ensure problem feasibility and to guarantee a certain level of computational complexity in solving our problem instances. As te maximum possible ordering quantity of rewards from eac redemption partner is limited by te capacity of rewards offered by te corresponding redemption partner (Q*), we use Of and te wolesale price (w* ) to determine te value of W R. i) Te initial liability (/ 0 ) is determined based on te information of accumulation demands. We assume tat before te planning orizon of te testing problems, only accumulation occurs and no redemption, and te accumulation demand follows te same demand distribution as te current planning orizon. Terefore, te initial liability is determined by 105

118 te sum of accumulation demands of points from all accumulation partners. Since te accumulation demand is a random parameter in te model, we use te sum of te upper bounds of te accumulation demands from all accumulation partners as te value of te initial liability. j) In EP2, te option purcase price (w^1) is te product of te price factor for purcasing options (OP) and te wolesale price of rewards (w } ). Te option exercise price, e 1 is te product of te price factor for exercising options (EP) and wolesale price of rewards (v^ ). 5.2 Testing te Effectiveness of te Solution Metodology In tis section, a set of numerical studies is developed to evaluate te effectiveness of our solution metodology for solving realistically sized rewards-supply planning problems in te range of LRPs tat exist in today's marketplace, considering bot te number of LRP redemption partners and te number of LRP accumulation partners. Te effectiveness of te proposed solution procedure is examined from two perspectives: model solvability and quality of stocastic solutions. Model solvability refers to te extent to wic our solution procedure can solve problems of various sizes witin a reasonable computation times (i.e., computational efficiency) wile providing approximated solutions of good quality (ie., a smaller solution gap). In order to evaluate te model solvability, tree model outputs are generated: (1) te estimated optimality gap (in percentage) between a candidate solution and an estimated true optimal solution, (2) te 95% confidence internal of te estimated optimal gap (in percentage), and (3) te computational time required to solve eac testing problem (CPU time). 106

119 Tese outputs are te key indicators of model solvability (Mak et al, 1999; Kleywegt et al, 2000; Ruszczynski & Sapiro, 2003). Wit smaller solution gaps and sorter computational times, te model solvability is better. Meanwile, tese outputs are also used to determine te sample size and sample replications for te rest of te numerical studies. Te sample size and sample replications are determined by te tradeoff between te solution gap and te computational efficiency. Quality of stocastic solutions seeks to examine weter using a stocastic programming approac is appropriate and wortwile. In order to evaluate te quality of stocastic solutions, solutions of te stocastic optimization models are quite often compared to tose resulting from deterministic optimization models involving mean values of te uncertain problem parameters (e.g., Santoso et al, 2005; Zanjani et al, 2007; Cen-Ritzo et al, 2010). Te difference between te expected values of stocastic models and te corresponding mean-value models is known as te value of te stocastic solution (VSS) (Birge and Louveaux, 1997). As long as te VSS is greater tan zero, te stocastic models are no worse tan te deterministic models. Tis indicates tat stocastic models are wortwile to resort to. Higer is te VSS, more appropriate are te stocastic models wen compared to te deterministic models Model Solvability Te caracteristics of te testing problems generated for examining te model solvability are sown in Table 5.3, were TC refers to te problem case designed to represent real life LRP systems of different sizes. NR refers to te number of LRP redemption partners, NA refers to te number of accumulation partners. Te size of an LRP system is determined by NR and NA. N 107

120 refers to te number of sample demand realizations (i.e., sample size), M refers to te number of sample replications, ND refers to te number of decision variables, and NC refers to te number of constraints. In order to determine te proper sample size and te proper number of replications, we consider tree candidate sample sizes: N= 30, 60, and 90; and two candidate sample replications: M=30 and 45 for eac testing case. As listed in Table 5.3, ten problem cases were developed. Te number of LRP accumulation partners and te number of LRP redemption partners in eac problem case were arbitrarily determined based on our study on LRPs in te Canadian marketplace (see Capter 2). For a problem case, four testing problems were generated to represent te different combinations of te candidate sample sizes and te candidate sample replications. For a testing problem, te number of decision variables and te number of constraints, as sown in Table 5.3, are different for eac model (i.e., BP, EP1, and EP2). For instance, Problem No.1.1 as 61 decision variables and 62 constraints in BP model, 122 decision variables and 122 constraints in te EP1 model, and 152 decision variables and 182 constraints in te EP2 model. Te largest testing problem for our models as 45,200 decision variables and 36,380 constraints (i.e., EP2 model, Problem No. 10.4). 108

121 Table 5.3: Summary caracteristics of te set of random generated problems TC No NR NA Problem No N M ND BP EP EP NC BP EP EP

122 5.2.2 Quality of Stocastic Solutions In order to calculate te value of stocastic solution (VSS) for eac of our models (i.e., BP, EP1, and EP2), te following steps are followed: Step 1: Replace random parameters D/, D, A wit te mean values of >/, D, A respectively. Solve te corresponding models (i.e., BP, EP1, and EP2) and find te optimal solutions x MVP. Step 2: Compute te estimated true objective function values, n N {x mp ) for x MVP. Step 3: Let x N (x)be te objective function value of te SAA reformulation problem for te solutionx. Te VSS for eac candidate solution (x) is calculated as: VSS = fi lf (x)-fi tl (x uyf ) (5.1) VSS in terms of percentage is calculated as: vss ( % ) = \(2s xl0 (5-2) Te quality of stocastic solutions was evaluated on te set of randomly generated problems sown in Table 5.3 wit sample size of N=60 and sample replications of M=30 (i.e., Problem No. 1.2, 2.2, 3.2,4.2, 5.2, 6.2, 7.2, 8.2, 9.2, and 10.2). 5.3 Testing te Impacts of Demand Variability Our second set of numerical studies aims to analyze te situation were internal constraints remain te same, but external uncertainty (i.e., demand uncertainly) varies. In tis section, we 110

123 describe te design of te studies for testing te impacts of demand variability on LRP performance. In LRP management, demand uncertainly is one of te primary callenges tat LRP managers ave to face. Unlike daily goods (e.g., grocery) on wic te overall customer demand is generally quite stable, LRP members' demands on accumulation and redemption vary due to various reasons. Terefore, some of te questions of interest to managers are: Does demand variability ave an impact on LRP performance? If yes, is tat impact negative or positive? Does tat impact remain te same or not under different model settings (i.e., BP, EP1 and EP2)? Given a demand distribution, demand variability (i.e., te degree of demand randomness) is quite often measured by te variance of te demand distribution. In order to cange te degree of demand randomness, we define a demand variability parameter, DV, and use it to cange te variances of demand distributions in our testing problems. Wit te increase in te values of DV, demand variability increases. Te values of te demand variability parameter (DV) and te resulting canges in te demand distributions wit te values of DV are listed in Tables Table 5.4: Demand variability parameter and its values used for normal distributions Normal <7 2 =cr 0 2 xdv Mo CT o DV=0.1 DV=0.5 DV=1.0 DV=1.5 DV=1.8 DV=2 Type A TypeB TypeC

124 Table 5.5: Demand variability parameter and its values used for uniform distributions Type A TypeB TypeC Uniform KA) (40, 50) (20, 30) (5, 10) M fl== //+ V3 x <j, b = //-V3x a, were a = pixdv DV=5% (41.1,48.9) (22.8, 27.2) (6.9,8.1) DV=10% (37.2, 52.8) (20.7, 29.3) (6.2, 8.8) DV=15% (33.3,56.7) (18.5,31.5) (5.6, 9.4) DV=20% (29.4, 60.6) (16.3,33.7) (4.9, 10.1) DV=25% (25.5, 64.5) (14.2,35.8) (4.3, 10.7) In Table 5.4, fi Q denotes te initial values of te mean of te normal distribution for eac type of LRP accumulation and redemption partners, and al denotes te initial values of te variance of te normal distribution for eac type of accumulation and redemption partners. Wit a value of DV, te variances of te normal distributions ( a 2 ) are canged by using te relation a 2 =O-QXDV, and te means of te normal distributions remain te same. In Table 5.5, a 0 and b 0 denotes te initial lower bound and upper bound values of te uniform distributions for different types of LRP partners. Wit a value of DV, te variance of a uniform distribution (a 2 ) is obtained as follows: ff 2 =(//xdv) 2 (5.3) and ten te lower bound (a) and te upper bound ( b) of te uniform distribution is obtained as follows: fl = // + -v/3x<j, and b = //-v3xcr (5.4) 112

125 5.4 Testing te Impacts of Budget Tigtness To evaluate te impact of budget tigtness, te upper bound of te budget constraint (i.e., W R in constraint (4.43), constraint (4.51), and constraint (4.64)) in our models is redefined as follows: W R x(l-a) (5.5) were a denotes te budget tigtness parameter, 0 < a < 1. a - 0 indicates tat tere is no cange in te budget availability, a < 1 indicates tat tere is a cange in te budget availability (i.e., reduction). As suc, te level of budget tigtness is determined by te budget tigtness parameter a. Te iger is te value of or, te tigter is te budget availability. We develop our analysis by comparing te outputs of our models wit a low level of budget tigtness to te outputs of te models wit a ig level of budget tigtness. Te outputs include LRP profitability, actual liability ratio, and ordering quantities of rewards. In order to compare wit te previous analysis in tis capter, we limit our study to tree testing cases (i.e., TC No.4: NR=20 and NA=40, TC No. 6: A«=40 and NA=40, and TC No.8: JVR=65 and NA=70), and tree levels of demand variability (low, medium, and ig). Tese tree test cases represent te most common sizes of LRPs in real world applications. Wit a normally distributed demand, te demand variability parameter DV=0.1 represents a low level of demand variability, DV=1.0 represents a medium level of demand variability, and DV=1.8 represents a ig level of demand variability. Wit a uniformly distributed demand, DV=0.05 represents a low level of demand variability, DV=0.15 represents a medium level of demand variability, and DV=0.25 represents a ig level of demand variability. 113

126 For eac testing problem, at eac level of te demand variability (i.e., low, medium, and ig), we perform ypotesis tests on te differences in te mean values of LRP profitability, liability ratio, and ordering quantities of rewards in te tigt budget and loose budget cases. We also perform ypotesis tests on te differences in te mean values of profitability, liability ratio, and ordering quantities of rewards across different levels of demand variability (i.e., low, medium, and ig) for cases of bot tigt and loose budgets. In addition, we perform ypotesis tests on te differences in te mean values of profitability, liability ratio, and ordering quantities of rewards across te model settings (i.e., BP, EP1, and EP2). Te purpose of conducting tese ypotesis tests is to examine weter tere are significant impacts of budget tigtness, demand variability, and model settings on LRP performance and te LRP ost's decisions. Te null ypotesis for eac test is stated as follows: HI: Wit a given level of demand variability (i.e., low, medium, and ig) tere is no significant difference in te mean LRP profitability between a setting wit low budget tigtness and a setting wit ig budget tigtness. H 2: Wit a given level of demand variability (i.e., low, medium, and ig), tere is no significant difference in te mean liability ratio between a setting wit low budget tigtness and a setting wit ig budget tigtness. H 3: Wit a given level of demand variability (i.e., low, medium, and ig), tere is no significant difference in te mean rewards ordering quantity between a setting wit low budget tigtness and a setting wit ig budget tigtness. H 4: Wit a given level of budget tigtness (i.e., low and ig), tere is no significant difference in te mean values of LRP profitability between settings operating under different demand variability levels. 114

127 H 5: Wit a given level of budget tigtness (i.e., low and ig), tere is no significant difference in te mean liability ratio between settings operating under different demand variability levels. H 6: Wit a given level of budget tigtness (i.e., low and ig), tere is no significant difference in te mean rewards ordering quantities between settings operating under different demand variability levels. H 7: Wit a given level of demand variability (i.e., low, medium, and ig) as well as a level of budget tigtness (i.e., low and ig) tere is no significant difference in te mean LRP profitability between different model settings (i.e., BP, EP1, and EP2). H 8: Wit a given level of demand variability (i.e., low, medium, and ig) as well as a level of budget tigtness (i.e., low and ig), tere is no significant difference in te mean liability ratio between different model settings (i.e., BP, EP1, and EP2). H 9: Wit a given level of demand variability (i.e., low, medium, and ig) as well as a level of budget tigtness (i.e., low and ig), tere is no significant difference in te mean rewards ordering quantities between different model settings (i.e., BP, EP1, and EP2). Hypotesis tests HI - H6 are conducted under eac model setting given bot uniformly distributed demand and normally distributed demand. Hypotesis tests H7 - H9 are conducted across te model settings given bot uniformly distributed demand and normally distributed demand. Overall, 378 ypotesis tests are conducted by employing all pair-wise comparisons tecnique using SPSS (PASW Statistics 18). Te confidence level (CL) is set to 95% for all ypotesis tests. Intuitively, we would like to see tat te null ypotesis would be rejected, wic indicates tat te budget tigtness does ave an impact on LRP performance and LRP ost rewards-supply decisions. 115

128 5.5 Testing te Impacts of Liability Control Tigtness As discussed in Capter 3, te liability associated wit unredeemed points is a key concern in te management of LRPs. In reality, many companies ave experienced a dramatic increase of te liability wit te growt of LRPs. For example, Aeroplan stated tat te estimated future redemption cost for te liability amounts to $1,277 million. According to its annual report for te fiscal year 2010, tis amount is even iger tan te revenue from selling points in te year 2010 ($1,033 million). Marriott International, Inc. also reported tat teir liability for te Marriott Rewards program was $1,536 million at year-end Hence, one of te callenges in LRP operations is to plan for te supply of rewards wile keeping te liability witin a certain range. To evaluate te impacts of liability control, te upper bound of te liability ratio (LUB) in liability constraints (i.e., constraints (4.45), (4.54), and (4.63) in BP, EP1, and EP2) is redefined as follow: L m =yxr Q (5.6) were y denotes te liability tigtness parameter, and r Q denotes te initial liability ratio. A four-step procedure described below is developed to conduct our numerical studies on liability control: 1 Marriott International annual report ttp:' files sareolder.com/downloads/mar/ x0x283218'c30cblc0-3ad a4e3-5a6f12d41ae7/mamott 2008 AR.pdf (accessed in July, 2011) 116

129 Step 1: Wit te model outputs from BP, EP1, and EP2, generate te initial liability ratio (r 0 ) for eac test case and eac level of demand variability by calculating, i.e., set I r 0 =y Step 2: Based on te values of /and r 0, reset te values of te upper bound of te liability ratio (LUB) using te above formulation (5.6). Step 3: Relax te budget and capacity constraints in BP, EP1, and EP2 to solve eac model wit te new upper bound of te liability ratio as obtained in Step 2. Step 4: Generate te model outputs based on te solutions obtained in Step 3. Te model outputs include: (1) LRP profitability, (2) cost of rewards, (3) ordering quantity of rewards, and (4) budget usage ratio of te current budget usage (obtained from te model witout te budget and capacity constraints) to te initial budget usage (obtained from te model wit te budget and capacity constraints), and to te maximum available budget ( W R ), respectively. In Step 2, in order to understand te sole impact of liability control, we relax te capacity as well as te budget constraints (i.e., constraints 4.42, 4.43, 4.50, 4.51, 4.61, 4.64) in our models and ten generated te model outputs given various values of LUB- Te values of LUB used in te numerical study for eac model are listed in Tables (5.6) - (5.8). As sown in tose tables, in eac of tese models y = 0.7 represents te situation were te liability control level is tigt (i.e., te upper bound of te liability ratio in te liability constraint is less tan te initial liability ratio), y =1.3 represents te situation were te liability control level is loose (i.e., te upper 117

130 bound of te liability ratio in te liability constraint is larger tan te initial liability ratio). As in Section 5.4, we limit our analysis to tree problem testing cases: TC No. 4, 6, and 8. Table 5.6: Values of LUB used for BP setting (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case 4 Case 4 V Y DV r DV r Case 6 Case 6 DV r DV r Case 8 Case 8 V V DV r DV r

131 Table 5.7: Values of LUB used for EPl setting Table 5.8: Values of LUB used for EP2 setting (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand DV Case 4 V ro DV Case 4 V ro DV r Case 4 V DV r Case 4 V DV r Case 6 V DV Case 6 V ro DV r Case 6 V DV Case 6 V ro DV r Case 8 V DV Case 8 V ro DV r Case 8 Y DV r Case 8 V

132 Capter 6 Results and Analysis In tis capter, we present and discuss te results obtained from our numerical studies, wic are about: (1) te effectiveness of our models and solution metodology, and (2) te impacts of demand uncertainty, budget tigtness, and liability control on LRP performance and LRP ost's decisions, respectively. We discuss our findings for eac model setting (i.e., BP, EP1, and EP2) as well as across model settings. All of our numerical studies were carried out troug computer simulations on a Tosiba Tecra M10 notebook equipped wit 2.26 GHZ processor. A computer simulation (coded in C++) was developed to generate testing problems and implement te solution procedure for our SLP models (te C++ code is available upon request). Te IBM Optimization Subroutine Library (OSL, version 3.0) was used to solve te corresponding SAA problems. 6.1 Testing te Effectiveness of te Solution Metodology Our first set of numerical studies sougt to evaluate te effectiveness of our solution metodology for solving realistically sized (large-scale) rewards-supply planning problems Model Solvability Figures illustrate te model outputs (i.e., optimality gaps, confidence intervals of te optimality gaps, and CPU times) for eac testing problem defined in Table 5.3 (see Section 5.2 in Capter 5). "MeanG(%)" in Figure 6.1 refers to te sample average of te optimality gap of a 120

133 candidate solution in terms of percentage (i.e., MeanG(%) = G M (x) N M _ xloo, were G^(x) is 1 M te mean optimality gap for a candidate solution x, and J\n% is te upper bound estimation for te optimal objective value of te true problem). "CIU(%)" in Figure 6.2 refers to te average upper bound of te confidence interval of te optimality gap in terms of percentage (i.e., CIU(%) _G%(x) + e c 1 M -xloo, were s G is te alf-widt of te confidence interval. "CPU (mins)" in Figure 6.3 refers to te average computational time (in minutes) spent to solve eac testing problem. Te value of demand variability parameter (DV) is set to 1.8 for normally distributed demand and 0.25 for uniformly distributed demand. 121

134 Figure 6.1: Estimated mean gaps wit different sample sizes and sample replications (a) Under BP Setting M e a n G Uniform K!Kv! Xv_ i «v^w. " ' ^ ^gi te3^; ^i uz**^as»^ ^l"*»i^^,sbs **Bfa, ^ '^^'4mm^mX N30M30 N60M30 N60M45 N90M45 Sample Size (N) and Sample Replications (M) TCNo.l TCNo.2 TC No.3 TCNo.4 TCNo.5 TCNo.6 TCNo.7 TCNo.8 TCNO.9 TCNO.10 M e n G {») Normal -j^^^-shbffs^^ N33M30 N60M30 N60M45 N9QM45 Sample Size (N) and Sample Replications (M) TCNo.1 TCNo.2 TC No J TCNo.4 TCNo.5 TCNo.6 TCNo.7 TC No.8 TCNo.9 TCNO.10 (b) Under EP1 Setting Uniform N30VI30N60M30N60M45 N90M45 Sample Size (N) and Sample Replications (M) TCNo.1 TCNo.2 TCNo.3 TCNo.4 TCNo.5 TCNo.6 TCNo.7 TCNo.8 TCNo.9 TCNo.10 M e a n Normal H30M30 N60M30N60M45 N90M45 Sample Si*e (N) and Sample Replications (M) (c) Under EP2 Setting Uniform N30M30 N63M30 N60M45 N90M45 Sample Size (N) and Sample Replications (M) TCNo.1 TCNo.2 TCNoJ TCNO.4 TCNo.5 TCNo.6 TCNo.7 TCNo.8 TCNo.9 TCNo.10 M e a n G Normal N30M30 N60M30 N60M45 N90M45 Sample Size (N) and Sample Replications (M) TCNo.1 TCNo.2 TCNo.3 TC NO.4 TCNo.5 TC No.6 TCNo.7 TCNo.8 TCNo.9 TCNO

135 Figure 6.2: Confidence interval upper bounds of te estimated mean gaps wit different sample sizes and sample replications (a) Under BP Setting Uniform N30M3ON6OVI3ON6OM4S N90M45 Sample Site (N) and Sample Replications (M) TCNo.1 TCNo.2 TCNo.3 TCNO.4 TCNO.S TCNo.6 TCNo.7 TCNo.8 ICNo/J TCNo.10 C u {*) S 0.00 Normal N3QM30 N60M30NGOM45 N0OV145 Sample Sae (N) and Sample Replicatiom (M) (b) Under EPl Setting Uniform N30M30 N60M33 NE0M45 N9GM4S Sample Size (N) and Sample Replications (M) Normal N3GIV30 N63M30N60VI45 N90M45 Sample Se (N) and Sample Replications (M) -TCNo.1 -TCNo.2 -TCNo.3 -TCNo.4 -TC NoS -TC No.6 -TC No.7 -TCNo.8 TCNo.9»TC NO.10 (c) Under EP2 Setting Uniform N30M30 NfiOA/UONEOfrMJi N<]0M4S Sample Site N) and < sample Replications (nn) TcslCasol TcstCdsc2 TcstCasc3 TcslCasc4 TcslCascS TctCusra ToslCase7 TcslCiboS Test Case*) TeM.GlM.-lO C u m O.'.O Normal N30M30 N50VI50 N60M4S n9gm4s Sample Si<e(N) and sample Replications (M) 123

136 Figure 6.3: CPU times wit different sample sizes and sample replications (a) Under BP Setting G (mins) Uniform XL ZZ- N30WI3ON6OM30N6OM45N9OM45 Sample Size [N)and Sample Replications (M) TCNo.l TCN0.2 TCNo.3 TCNo.4 TCNo.5 TCN0.6 TCNo.7 TCN0.8 TCNo.9 TCNo.10 Normal N30M30NeOM30N60M45N90M45 Sample Se (N)and Sample Replications (M) TCNo.l TCNo.2 TCNo.3 TC No.4 TCNo.5 TCNo.6 TC No.7 TCNo.8 TCNo.9 TCNo.10 (b) Under EP1 Setting Uniform TC No.l Normal TCNo.l TC No.2 TC No.2 TC No.3 TC No.3 TC No.4 TC No.4 TC No.5 TC No.6 {minsl TC No.5 TC No.6 0.3C N 30M30 N60M30N60M45 N90M^5 TC No.7 TC No.8 TC No.9 N30M30N60IV30I\60M45N90M45 TC No.7 TC No.8 TC No.9 Sample Size (N) and Sample Replications {M TC No.10 Sample Size (N) and Sample Replications (M) TCNO.10 (c) Under EP2 Setting 7.D0 Uniform N30IV30N60I/3DN60M45 N9CM45 Sample Size {N) and Sample Replications (M) It NO.l ICN0.2 TC No.3 TC No.4 TC No.5 TC No.e TC No.7 TC No.8 TC No.G TCNo G 12.0G c p S.OG U 6.0G {mlns 4 o G?.no O.OG Normal ^ zr= N30M30 NEOM30N60M45 N90IV45 Sample Size {H and Sample Replications (M) It No.l IC NO.2 TC No.3 TC No.4 TC No.5 TCNO.6 TCNo.7 TC No.8 TCNo.9 TC No

137 According to te model outputs illustrated in Figures (for more detailed model outputs please refer to Appendices B.l - B.3), wit only a modest number of samples (e.g. 7V=30), te proposed solution procedure provides ig quality estimated SAA solutions to te true stocastic programming problems (i.e., Problem BP-2SLPR, Problem-EPl-2SLPR, and Problem-EP2-2SLPR). For instance, wit sample size N=30, and number of replications M=30, te estimated mean optimal gaps (MeanG(%)) and te estimated upper bound of te confidence interval (CIU(%)) of te mean optimal gaps are lower tan 0.25% in BP and EP2, and tese values are lower tan 2.5% in EP1. Regarding te determination of te sample size and sample replications for solving BP, EP1, and EP2 models, we found tat iger improvement of te optimality gaps in terms of MeanG(%) and CIU(%) occurs wen sample size N increases from 30 to 60, compared to te improvement of te optimal gap wen N canges from 60 to 90 and M canges from 30 to 45. On te oter and, comparing te increase in CPU times wen N canges from 60 to 90 and M canges from 30 to 45, lower increase in CPU times occurs wen sample size N increases from 30 to 60 and M remains at 30. Tese trends remain te same regardless of te demand distribution (uniform versus normal). Hence, in te remainder of tis work, all numerical studies are based on a sample size of ^=60 and a sample replication of M=30. In comparison to oter tested sample sizes and sample replications, tese offer a better tradeoff between te quality of solutions and te computational efficiency Quality of Stocastic Solutions Te comparison between candidate solutions of stocastic models and te solutions of te meanvalue models are reported in Tables considering normal distribution (wit demand 125

138 variability parameter DV= 0.1, 1.0, and 1.8) and uniform distribution (wit demand variability parameter DV= 0.05, 0.15, and 0.25), respectively. Results from Tables sow tat VSS(%) in all te testing problems are greater tan zero, wic indicates tat te estimated optimal values of te objective function (i.e. LRP profitability) for all candidate solutions of te stocastic programming models are greater tan tose of te mean-value-based models (see te calculation of VSS(%) in te formulation (5.2)). Tis reveals tat stocastic programming solutions are superior to te corresponding mean-value solutions in terms of te objective function values. Tese solutions also lead to significantly smaller mean gaps (i.e., MeanG(%)) and a smaller variability of te gaps (i.e., StdG). Moreover, wit te increases in demand variability, te quality of te stocastic solutions is more superior to te quality of te mean-value model solutions. Te value of te stocastic solution in percentage (i.e., VSS(%)) increases from less tan 0.5% in te case of low demand variability to more tan 10% in te case of ig demand variability. All of tese results sow tat te stocastic model is an effective tool to solve rewards-supply planning problems wit te presence of random demand. Overall, te results from all testing problems sow tat our solution metodology can solve realistically sized rewards-supply planning problems in reasonable time and it can generate estimated optimal solutions wit an optimality gap tat is small enoug. Moreover, te expected values of our stocastic models are better tan te expected values of te corresponding meanvalue models, especially wen te variability of te stocastic parameter in our models (i.e., redemption and accumulation demands) increases. 126

139 Table 6.1: Quality (a) wit Uniformly Distributed Demand TC # DV Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) vss (%) solutions of BP model (b) wit Normally Distributed Demand TC # DV Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%)

140 Table 6.1: Quality of stocastic solutions of BP model (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC # DV 0.05 Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%) 2.09 TC # DV 0.1 Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%)

141 Table 6.2: Quality of stocastic solutions of EPl model (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC # DV Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%) TC # DV Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%)

142 Table 6.2: Quality of stocastic solutions of EPl model (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC # DV 0.05 Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%) 2.00 TC # DV 0.1 Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%)

143 Table 6.3: Quality of (a) wit Uniformly Distributed Demand TC # DV Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%) solutions of EP2 model (b) wit Normally Distributed Demand TC # DV Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%)

144 Table 6.3: Quality of stocastic solutions of EP2 model (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC # DV 0.05 Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%) 2.44 TC # DV 0.1 Stocastic Model StdG MeanG (%) Mean Value Model StdG MeanG (%) VSS (%)

145 6.2 Testing te Impacts of Demand Variability In tis section, we report te results of te numerical studies tat we carried out to investigate te impacts of demand variability on LRP performance. Ten problem testing cases (TC No. 1-10) are used in te studies to represent realistically sized planning problems Under BP Setting In Table 6.4, we report LRP profitability (i.e. ost firm's value creation), actual liability ratio (V), and budget usage (in percentage) under BP setting wit demand variability canging from small to large. As sown in Table 6.4, in one alf of te testing cases (i.e., TC No. 2, 3, 5, 6, and 8), LRP profitability increases wit te increases of demand variability, wile in te oter alf of te testing cases (i.e., TC No.l, 4, 7, 9, and 10), LRP profitability decreases wit te increase of demand variability. Te reason is tat profitability from te accumulation side always increases wit te increase of variability in accumulation demand due to te assumption tat accumulation demand will always met. Meanwile, profitability from te redemption side always decreases wit te increase of variability in redemption demand due to te increased cost of purcasing rewards (i.e., te need to order more rewards wen demand variability is iger). Wen te increase in te accumulation-side profitability is iger tan te decrease in te redemption-side profitability, te overall profitability will increase. Tis is more likely to appen wen te number of LRP redemption partners (NR) is less tan te number of LRP accumulation partners (NA). Wen te increase in te accumulation-side profitability is lower tan te decrease in te 133

146 redemption-side profitability, te overall profitability will decrease. Tis is more likely to appen wen NR is greater tan NA. However, te average increasing or decreasing rates of LRP profitability among different testing cases are quite similar (e.g., te average increasing rate is 1.623% and te average decreasing rate is 1.191% wit uniformly distributed demand; 1.763% and 1.212% wit normally distributed demand). Table 6.4: Summary table of computational results wit different levels of demand variability under BP setting (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. DV 0.05 LRP Profitability Liability Ratio 1.02 Budget Used(%) TC No. DV 0.1 LRP Profitability Liability Ratio 1.01 Budget Used(%)

147 Table 6.4: Summary table of computational results wit different levels of demand variability under BP setting (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No DV LRP Profitability Liability Ratio Budget Used(%) TC No DV LRP Profitability Liability Ratio Budget Used(%)

148 Wit respect to te actual liability ratio (/'), in some testing cases (i.e., TC No. 2, 3, 5, and 8) te liability ratio is strictly larger tan 1.0 wit different DVs, wile in oter testing cases te liability ratio is around 1.0 (i.e., cases 1, 6, 9, and 10) or strictly less tan 1.0 (i.e., cases 4 and 7), wit different DVs. Overall, wit te increase of demand variability, te liability ratio always increases in eac of te testing cases. Tis is because wen demand variability increases, te overall accumulation will increase, wile te overall redemption quantity will decrease. For example, in te testing case 10, te accumulation (i.e., te number of accumulated points) increases from at DV=0.05 to at DV=0.25 given uniformly distributed demand and from at DV=0.1 to at DV=1.8 given normally distributed demand. Te redemption (i.e., te number of redeemed points) decreases from to given uniformly distributed demand, and from to given normally distributed demand. Wen demand variability canges from small to large, te increasing rates of te liability ratio in terms of percentage are between 6% and 11% in te case of uniformly distributed demand, and between 7% and 12% in te case of normally distributed demand. In regards to te budget usage, it increases wit te demand variability in all problem testing cases given bot normally distributed and uniformly distributed demand. Wit increasing demand variability, te increasing rate of te budget usage (in percentage) is between 12% and 29% in te case of uniformly distributed demand, and between 16% and 36% in te case of normally distributed demand. Compared to te canges in te profitability and in te liability ratio, canges in te budget usage are remarkably larger due to te increases in overall rewards ordering quantity wen demand variability increases. In effect, te overall rewards ordering quantity increases wit te increase of demand variability, wic results in an increase of budget 136

149 usage for eac testing case. For example, in TC No. 1, te total ordering quantity of rewards increases from at DV=0.05 to at DV=0.25 in uniformly distributed cases and from at DV=0.1 to at DV=1.8 in normally distributed cases. Te above computational results and analysis indicate tat under BP setting, demand variability does ave a negative impact on LPR profitability (i.e., iger DVs, lower profitability), especially wen te number of redemption partners is greater tan te number of accumulation partners (i.e., NR>NA). Te negative impact of demand variability on an LRP ost firm's liability ratios (i.e., iger DVs, iger liability) seems more noticeable compared to te LRP profitability, as te ost firm's liability ratio increases in all testing cases, wit te increases in demand variability given normally distributed demand or uniformly distributed demand. Also, demand variability as a negative impact on budget usage (i.e., iger DVs, iger budget usage). Te increases in budget usage in terms of percentage ave reaced up to 35% in te testing cases. We also find tat te impacts of demand variability on LRP performance in terms of profitability, liability ratio, and budget usage are consistent between normally distributed demand and uniformly distributed demand Under EP1 Setting In Table 6.5, we report LRP profitability, actual liability ratio (/'), and budget usage (in percentage) under EP1 setting wit demand variability canging from small to large. 137

150 Table 6.5: Summary table of computational results wit different levels of demand variability under EPl setting (b) wit Normally Distributed Demand TC No DV LRP Profitability Liability Ratio Budget Used(%) (a) wit Uniformly Distributed Demand TC No DV LRP Profitability Liability Ratio Budget Used(%)

151 Table 6.5: Summary table of computational results wit different levels of demand variability under EPl setting (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. DV LRP Profitability Liability Ratio Budget Used(%) TC No. DV LRP Profitability Liability Ratio Budget Used(%) Based on te computational outputs, we observe tat under EPl setting, LRP profitability decreases wit te increases in demand variability in all testing cases wit a few exceptions (i.e., TC No. 2 and 3). Te average rate of decrease (in percentage) of LRP profitability is 2.02% wit uniformly distributed demand and 3.4% wit normally distributed demand. In testing cases 2 and 3, wit uniformly distribution demands, te LRP profitability decreases wit DV canges from 0.05 to 0.10 and ten increases wit DV canges from 0.10 to As explained in Section 6.2.1, LRP profitability from te accumulation side always increases wit te increase of 139

152 variability in accumulation demand and LRP profitability from te redemption side always decreases wit te increase of variability in redemption demand. Wen te increase in te accumulation-side profitability is iger tan te decrease in te redemption-side profitability, te overall profitability will increase. On te oter and, wen te increase in te accumulationside profitability is lower tan te decrease in te redemption-side profitability, te overall profitability will increase. In regards to te actual liability ratio, in some of te testing cases (i.e., TC No.l, 2, 3, 5, 6, and 8) te liability ratio is strictly larger tan 1.0, wile in te rest of te testing cases te liability ratio is eiter strictly less tan 1.0 (i.e., TC No. 4) or around 1.0 (i.e., TC No. 7, 9, and 10). Under EP1 setting, te impact of demand variability on liability ratio seems muc smaller tan tat under BP setting. Wit te canges in demand variability, te maximum cange in liability ratio is around 0.05 in EP1, wereas in BP, te maximum cange in te liability ratio is around 0.1. Wit respect to te impact of demand variability on budget usage, te same penomena are observed as tose under BP setting. Te increases in budget usage in terms of percentage are between 12% and 29% wit uniformly distributed demand, and between 16% and 36% wit normally distributed demand. Tis indicates tat increases in demand variability force te LRP ost to increase te ordering quantities of rewards to maximize LRP profitability, wic in turn increases te budget usage. In tis setting, in addition to te ordering quantities of rewards, te LRP ost will also need to decide on te number of bonus points to offer to eac LRP accumulation partner. Te 140

153 computational results for bonus points are reported in Appendix C. 1. In te table, "TQA" refers to te total volume of points ordered by LRP accumulation partners in eac testing case. As we mentioned in Section 5.1, Capter 5, TQA is one of te input parameters in our models. "TBA" refers to te total volume of bonus points to offer to te accumulation partners. In eac testing case, te overall volume of bonus points to offer decreases wit te increases in demand variability. Wit uniformly distributed demand, wen te value of demand variability parameter (DV) canges from 0.05 to 0.25, TBA reduces from around 8.5% of TQA to around 5.5% of TQA. Wit normally distributed demand, wen te value of demand variability parameter (DV) canges from 0.1 to 1.8, TBA reduces from around 9% of TQA to 1.7% of TQA. Tose results indicate tat wen demand variability increases, offering bonus points is not beneficial for improving LRP profitability in rewards-supply planning. Based on te above computational results and analysis, we observe tat demand variability as a negative impact on LRP profitability, liability, and budget usage under EP1 setting. Te increase in demand variability forces an LRP ost to consider purcasing more rewards from redemption partners and offering fewer bonus points to accumulation partners, wic indicates tat demand variability does influence te LRP ost's decisions on bot rewards ordering quantities and offering bonus points Under EP2 Setting In Table 6.6, we report LRP profitability, actual liability ratio (/'), and budget usages under EP2 setting wit demand variability canging from small to large. 141

154 Table 6.6: Summary table of computational results wit different levels of demand variability under EP2 Setting (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No DV Profitability Liability Ratio Budget Used(%) TC No DV Profitability Liability Ratio Budget Used(%)

155 Table 6.6 Summary table of computational results wit different levels of demand variability under EP2 Setting (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. DV Profitability Liability Ratio Budget Used(%) TC No. DV Profitability Liability Ratio Budget Used(%) Our computational results sow tat LRP profitability increases wit te increases in demand variability in all testing cases except testing TC No.4. In tat testing case, te profitability increases sligtly and ten decreases sligtly wit te increasing demand variability. Te reason beind tese results is te same as tat under BP setting: profitability from te accumulation business always increases wit te increase of demand variability due to te assumption tat te accumulation demand will always been met. Meanwile, profitability from te redemption business always decreases wit te increase of demand variability due to te increased cost of 143

156 purcasing rewards (i.e., te need to order more rewards wen demand variability is iger). Wen te increase in te accumulation-side profitability is iger (lower) tan te decrease in te redemption-side profitability, te overall profitability will increase (decrease). However, unlike in BP model, in EP2 model te increases in demand variability most of te time enance LRP profitability rater tan lower it. Te average and maximum increases in LRP profitability among all te testing cases are 2.14% and 5.05% wit uniformly distributed demand. Wit normally distributed demand tey are 1.98% and 5.44%. Wit regard to liability ratio, overall it increases wit te increases in demand variability. Tis indicates tat wen demand variability is larger, LRP liability is expected to increase due to increased accumulation and decreased redemption. Tese results are very similar to tose tat we obtained under BP setting. Wen demand variability increases, te increases in te liability ratio in terms of percentage are between 5% and 8% wit uniformly distributed demand, and between 6% and 9% wit normally distributed demand. Similar results are obtained wit budget usage: wit te increases in demand variability, te budget usage increases in all testing cases wit normally distributed or uniformly distributed demands. Te canges in budget usage are muc larger wen demand variability is larger. Wit te igest level of demand variability, te increases in budget usage in terms of percentage are between 12% and 18% wen demand is uniformly distributed and between 16% and 24% wen demand is normally distributed. 144

157 Under EP2 setting, in addition to te initial ordering quantity of rewards, an LRP ost as to decide on te total number of options to purcase and to exercise. We find tat an LRP ost firm's best decision, wen facing an increase in demand variability, is to increase te overall rewards ordering quantity (including bot te initial ordering quantity of rewards and te total number of options to purcase). Tis also results in increases in budget usage. Appendix C.2 reports te computational results related to options to purcase and to exercise. In tis table, "TQR" refers to te total number of rewards (in points) to order, "TOR" refers to te total number of options (in points) to purcase, and "TOE" refers to te total number of options (in points) to exercise. Wit uniformly distributed demand, wen te value of demand variability parameter (DV) canges from 0.05 to 0.25, TOR increases from around 5% of TQR to around 21% of TQR. Wit normally distributed demand, wen te value of demand variability parameter (DV) canges from 0.1 to 1.8, TOR increases from around 1.7% of TQR to around 22% of TQR. Meanwile, te number of options to exercise is around 18% - 26% of te number of options to purcase wen demand is uniformly distributed, and 13% - 20% wen demand is normally distributed. Tese results suggest tat option contract does play a role in dealing wit demand variability. Based on te above computational results and analysis, we find tat demand variability does ave negative impacts on LRP liability and budget usage (i.e., iger demand variability results in iger LRP liability and budget usage). Meanwile, te increases in demand variability also force te LRP ost to consider purcasing more rewards from te redemption partners (i.e., bot te initial ordering quantity and te number of options to purcase ave increased). Tis suggests tat demand variability does influence an LRP ost's decisions on bot rewards ordering 145

158 quantities and te number of options to purcase. Anoter interesting finding is tat demand variability also causes canges in LRP profitability, but iger demand variability results in iger LRP profitability under EP2 setting. Tis indicates tat using option contracts can mitigate te negative impacts of demand variability on LRP profitability Comparisons across BP, EP1, and EP2 Model Settings In Sections we examine te impacts of demand variability under BP, EPl, and EP2 model settings, respectively. Tis section investigates te impacts of demand variability on LRP performance (in terms of LRP profitability, liability ratio, and budget usage) across BP, EPl, and EP2 model settings. Appendix C.3 sows tat, altoug te LRP profitability curves of BP, EPl, and EP2 models are sligtly different in different testing cases, te overall pattern seems quite similar. Wen demand variability is lower (e.g., DV=0.1, or DV=0.05), BP and EP2 models generate almost te same profitability, wile EPl model generates muc iger profitability due to bonus points offering. However, wit te increases in demand variability, te advantage of EPl model in terms of profitability is weakening. In contrast to tat, in almost all te testing cases EP2 model results in iger profitability tan BP and EPl models under ig demand variability (e.g., DV=1.8, or DV=0.25). Tese results suggest tat adopting option contracts is more attractive wen demand variability is iger. 146

159 Appendix C.4 sows tat te liability ratio is quite different among different testing cases (e.g., te liability ratio is less tan 0.8 in testing case 4, wile it is more tan 1.2 in testing case 1). Overall, in eac of te testing cases, te liability ratios under BP and EP2 settings are almost te same. However, due to te bonus points offering (i.e., creating a iger accumulation), EP1 model always generates a iger liability ratio tan BP and EP2 models do. However, wen demand variability increases, in comparison to EP1 model, te advantage of BP and EP2 models in terms of liability ratio is not very obvious. Tese results suggest tat an option contract does play a role in keeping liability relatively low and generating relatively ig profitability, especially wen te LRP ost faces ig demand variability. Wit regards to budget usage, as reported in Appendix C.5, we observe tat it increases wit te increases in demand variability. In BP and EP1 models, budget usage is exactly te same in eac of our testing cases. However, EP2 model results in lower budget usage tan BP and EP1 models, and te differences in budget usage are more notable wit iger demand variability (e.g., see TC No. 1, 2, 4, 7, and 8). In addition to tat, EP2's advantage in budget usage seems to be weakened in larger-sized problems (e.g., see TC No.9 and 10). Tese results suggest tat option contracts do play a role in keeping budget usage relatively lower wen te LRP ost faces iger demand variability. However, te LRP ost does need to pay more attention to te design of option contracts in te presence of quite a large number of LRP partners Summary Te computational results from te numerical experiments reveal tat ig demand variability does ave some negative impacts, especially under BP setting, on LRP performance (in terms of 147

160 profitability, liability ratio, and budget usage). Altoug EPl setting (i.e., offering bonus points) can acieve iger profitability wen demand variability is relatively low, it also results in iger liability and iger budget usage tan EP2 setting (i.e., using option contracts). Terefore, overall, using option contracts one can acieve better LRP performance tan not using option contracts and/or offering bonus points, especially in a situation were demand variability is ig. LRP managers sould pay some attention to te negative impacts of demand variability and take some actions to reduce te negative impacts troug eiter matcing demand variability (e.g., using option contract as in EP2 setting), especially wen facing relatively ig demand variability or leveraging demand variability (e.g., using promotions suc as bonus points, as in EPl setting) wen facing relatively low demand variability. 148

161 6.3 Testing te Impacts of Budget Tigtness Tis set of numerical studies as been conducted to examine te impacts of redemption budget tigtness on LRP performance in terms of LRP profitability and liability, and te LRP ost's decisions. We also investigate weter te impacts vary among different model settings (i.e., BP, EP1, and EP2) or under different demand variability levels. Te redemption budget is one of te internal dynamics in LRPs. Given tat te budget usage in all our original testing problems (i.e., tose considered in Section 6.2) is lower tan 100%, we used tem as basis for comparison wit budget tigtness parameter a = 0 (i.e., low budget tigtness). We set te tigtness parameter a = 0.3 for ig budget tigtness, since at tat level, te budget usage in most of our testing problems reaces to or above 100% Under BP Setting Table 6.7 sows te computational outputs from bot tigt and loose budget limits under BP setting. We find tat no matter wat te demand distribution (uniform or normal) is, te average LRP profitability and te average ordering quantity of rewards wit a tigt budget (i.e., a = 0.3) are always lower tan tose wit loose budget. Te average actual liability ratio (/') in eac testing case is iger in te tigt budget case tan in te loose budget case. Moreover, given different level of demand variability, te decrease in LRP profitability wit te increase in budget tigtness (i.e., a canges from 0 to 0.3) is witin te range of 4.4% to 11.6%. Hence, wen demand variability is iger, LRP profitability is lower. On te oter and, te increases 149

162 in liability ratio due to te increase in budget tigtness are witin te range of 7.4% %, and te increases in ordering quantity of rewards are witin te range of 12.9% %. Overall, wit a tigt budget (i.e., a = 0.3) te LRP profitability decreases wit te increases in demand variability, wile te liability ratio and te ordering quantity of rewards increase. In addition to tat, wit te increases in demand variability, te decreasing rate of te profitability and te increasing rate of liability ratio and ordering quantity of rewards are iger in te case of a tigt budget compared to te case of a loose budget. 150

163 Table 6.7: Impacts of budget tigtness under BP setting (a) wit Uniformly Distributed Demand Case4(NR20NA15) Case 6 (NR40NA40) Case 8 (NR65NA70) BG-L(O.O) (0.3) BG-L(O.O) (0.3) BG-L(O.O) (0.3) DV Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) LRP Profitability Liability Ratio L(0.1) M(1.0) H(1.8) L(0.1) M(1.0) H(1.8) Rewards Ordering Q L(0.1) M(1.0) H(1.8)

164 Table 6.7: Impacts of budget tigtness under BP setting (Cont.) (b) wit Normally Distributed Demand Case4(NR20NA15) Case 6 (NR40NA40) Case 8 (NR65NA70) BG-L(O.O) (0.3) BG-L(O.O) (0.3) BG-L(O.O) (0.3) DV Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) LRP Profitability L(0.1) M(1.0) H(1.8) Liability Ratio L(0.1) M(1.0) H(1.8) Rewards Ordering Q L(0.1) M(1.0) H(1.8)

165 Te results of te ypotesis tests related to BP model are summarized in Appendices D.l and D.2, in wic we provide te confidence intervals generated for all pair-wise comparisons. In Appendix D.l, i2 refers to ig budget tigtness (i.e., ), and il refers to low budget tigtness (i.e., BG-L). Te confidence intervals are reported as: (Ua-Uu) ± alf-widt, were U, refers to te mean values of our computational results (e.g., LRP profitability, liability ratio, and ordering quantity). For instance, in Appendix D.l (a), testing case 4, te confidence interval for LPR profitability is ± " " refers to te difference of te mean LRP profitability wit a tigt budget limit from tat of a loose budget limit. "1.079" refers to te alf-widt of te confidence interval associated wit te difference of te mean LRP profitability. Similarly, in Appendix D.2 refers to te demand variability level, eiter medium (i.e., ) or ig (i.e., DV-H), and jl also refers to te demand variability level, eiter low (i.e., DV-L) or medium (i.e., ). On one and, te testing results (as reported in Appendix D.l) sow tat, given a level of demand variability (i.e., low, medium, or ig), te mean values of LRP profitability, liability ratio, and ordering quantity of rewards are significantly different (at 95% CL) in bot te tigt and loose budget cases. Tis confirms tat budget tigtness does ave impacts on LRP profitability, liability ratios, and te LRP ost's rewards-supply decisions. On te oter and, wit eiter low or ig budget tigtness, te mean values of LRP profitability, liability ratio, and LRP ost firm's rewards-ordering decisions (as reported in Appendix D.2) are significantly different (at 95% CL) across different levels of demand variability, except in te testing case 8 (i.e., TC No.8). In tat testing case, wit te tigt budget (i.e., or = 0.3), te LRP ost's overall 153

166 ordering quantity of rewards seems not significantly different among various levels of demand variability (i.e., te confidence interval includes zero, as igligted by "*" in Appendix D.2). Tese results indicate tat, statistically speaking, demand variability does ave a significant impact on LRP profitability, liability ratio, and te LRP ost firm's rewards-ordering decisions. However, in some cases, wen te budget is very tigt, te optimal ordering quantity decisions will be limited by te available amount of budget (one of te internal constraints), and te LRP system cannot fully respond to external uncertainties (e.g., demand variability). Based on te above computational results and statistical analyses, we find tat budget tigtness does ave a negative impact on LRP profitability and liability ratio (i.e., iger budget tigtness, lower LRP profitability, and iger liability ratio) regardless any canges in demand variability. However, it seems tat BP model is more sensitive to budget tigtness wen te demand variability is iger. Furtermore, troug ypotesis tests and all pair-wise comparisons, we statistically sow tat te impacts of budget tigtness on LRP profitability, te liability ratio, and te LRP ost's ordering decisions are significant at a 95% confidence level considering different levels of demand variability. We also statistically sow tat te canges in demand variability ave a significant impact on LRP profitability, te liability ratio, and te LRP ost's ordering decisions at a 95% confidence level in most of te testing cases Under EP1 Setting As sown in Table 6.8, in eac testing case, te mean LRP profitability and te mean ordering quantity of rewards are lower, and te mean liability ratio is iger in te case of tigt budget (i.e., a ), compared to te case of loose budget (i.e., a = 0). Te decrease in te mean LRP 154

167 profitability due to te increase in budget tigtness is witin te range of 4.2% %. On te oter and, te increase in te mean liability ratio due to te increase in budget tigtness is witin te range of 6.5% %, and te increase in te mean ordering quantity of rewards is witin te range of 12.6% %. Overall, wit a tigt budget (e.g., a = 0.3), LRP profitability decreases wit te increase in demand variability, wile te liability ratio and te ordering quantity of rewards increase. In addition, wit te increase in demand variability, te decreasing rate of te mean LRP profitability and te increasing rates of te mean liability ratio and te mean ordering quantity of rewards are iger in te case of a tigt budget compared to te case of a loose budget. 155

168 Table 6.8: Impacts of budget tigtness under EPl setting (a) wit Uniformly Distributed Demand Case4(NR20NA15) Case 6 (NR40NA40) Case 8 (NR65NA70) BG-L(O.O) (0.3) BG-L(O.O) (0.3) BG-L(O.O) (0.3) DV Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) LRP Profitability L(0.05) M(0.15) H(0.25) Liability Ratio L(0.05) M(0.15) H(0.25) Rewards Ordering Q L(0.05) M(0.15) H(0.25)

169 Table 6.8: Impacts of budget tigtness under EPl setting (Cont.) (b) wit Normally Distributed Demand Case4(NR20NA15) Case 6 (NR40NA40) Case 8 (NR65NA70) BG-L(O.O) (0.3) BG-L(O.O) (0.3) BG-L(O.O) (0.3) DV Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) LRP Profitability Liability Ratio L(0.1) M(1.0) H(1.8) L(0.1) M(1.0) H(1.8) Rewards Ordering Q L(0.1) M(1.0) H(1.8)

170 Te results of te ypotesis tests related to EPl model are summarized in Appendix D.3 and Appendix D.4. We find tat te same conclusions can be drawn under EPl setting as were drawn under BP setting. All testing problems sow tat tere are significant differences among te mean values of LRP profitability, liability ratio, and ordering quantities of rewards in tigt and loose budget cases and among different levels of demand variability, wit a few exceptions (as igligted by "*" in Appendix D.4). Overall, wit regard to te impacts of budget tigtness under EPl setting, we find tat budget tigtness does ave negative impacts on LRP profitability and liability ratio, no matter wat te canges in demand variability. In addition, troug ypoteses tests and all pair-wise comparisons, we statistically sow tat te impacts of budget tigtness on LRP profitability, liability ratio, and LRP ost's decisions are significant at a 95% confidence level considering different levels of demand variability. We also statistically sow tat te canges in demand variability ave a significant impact on LRP profitability, liability ratio, and te LRP ost's decisions at a 95% confidence level in most of te testing cases Under EP2 Setting As sown in Table 6.9, te mean LRP profitability and te mean ordering quantity of rewards are lower, and te mean liability ratio is iger in te case of tigt budget (i.e., a = 0.3) compared to te case of loose budget (i.e., a = 0). Te decrease in te mean LRP profitability due to te increase in te budget tigtness is witin te range of 4.2% %. On te oter and, te increase in te mean liability ratio due to te increase in te budget tigtness is witin te range of 6.8% %, and te increases in te mean ordering quantity of rewards are witin te range 158

171 of 10.7% %. Overall, wen te budget is tigt (i.e., a = 0.3), te mean LRP profitability decreases wit te increases in demand variability, wile te mean liability ratio and te mean ordering quantity of rewards increase. In addition to tat, wit te increases in demand variability, te decreasing rate of te LRP profitability and te increasing rates of liability ratios and ordering quantity of rewards are iger in te case of tigt budget compared to te case of loose budget. r 159

172 Table 6.9: Impacts of budget tigtness under EP2 setting (a) wit Uniformly Distributed Demand Case4(NR20NA15) Case 6 (NR40NA40) Case 8 (NR65NA70) BG-L(O.O) (0.3) BG-L(O.O) (0.3) BG-L(O.O) (0.3) DV Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) LRP Profitability Liability Ratio Rewards Ordering Q L(0.05) M(0.15) H(0.25) L(0.05) M(0.15) H(0.25) L(0.05) M(0.15) H(0.25)

173 Table 6.9: Impacts of budget tigtness under EP2 setting (Cont.) (b) wit Normally Distributed Demand Case4(NR20NA15) Case 6 (NR40NA40) Case 8 (NR65NA70) BG-L(O.O) (0.3) BG-L(O.O) (0.3) BG-L(O.O) (0.3) DV Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) Mean Std(%) LRP Profitability Liability Ratio Rewards Ordering Q L(0.1) M(1.0) H(1.8) L(0.1) M(1.0) H(1.8) L(0.1) M(1.0) H(1.8)

174 Te results of te ypotesis tests related to EP2 model are summarized in Appendix D.5 and Appendix D.6. We find tat same conclusions can be drawn under EP2 setting as tose drawn under BP and EPl settings. All testing results sow tat tere are significant differences among te mean values of LRP profitability, liability ratio, and ordering quantities of rewards in cases of bot tigt and loose budget as well as among different levels of demand variability, wit a few exceptions (as igligted by "*" in Appendix D.6). Overall, we can draw te same conclusions as tose under BP and EPl settings: budget tigtness does ave significant negative impacts on LRP profitability and liability ratios at a 95% confidence level wit te canges in demand variability. However, it seems tat EP2 model is more sensitive to budget tigtness wen demand variability is iger. Furtermore, troug ypotesis tests and all pair-wise comparisons, we statistically sow tat te impacts of budget tigtness on LRP profitability, liability ratios and te LRP ost's decisions are significant at a 95% confidence level wit different levels of demand variability. We also statistically sow tat te canges in demand variability ave a significant impact on LRP profitability, liability ratio, and te LRP ost's decisions at a 95% confidence level in most of te testing problems Comparison across BP, EP1, and EP2 Model Settings As sown in Figure 6.4, LRP profitability decreases under eac model setting wen budget tigtness increases. It seems tat LRP profitability is sligtly more sensitive to demand variability in te case of tigt budget. Wit a lower level of demand variability, BP and EP2 models generate similar LRP profitability, but EPl model generates iger LRP profitability tan BP and EP2 models. On te oter and, wen demand variability level is ig, EP2 model 162

175 generates iger LRP profitability, but BP and EPl models generate similar LRP profitability in bot tigt and loose budget cases. Figure 6.4: LRP profitability under BP, EPl, and EP2 settings LRP Profitability ~BP_BG-L BP «-*- EP1BG-L EPl ~*~ EP2 BG-L «EP (a)case4:nr20na15 (wit uniformly distributed demand) fl.fm-..^,n^..^..!^'..^^.^^f-j,s?,,,,^f>,^^, e ^ M H DV LRP Profitability BP_BG-L IB BP_ -*- EP1BG-L -* EPl_-*» EP2_BG-L EP2_ (b)case4:nr20na15 (wit normally distributed demand) 163

176 Figure 6.4: LRP profitability under BP, EPl, and EP2 settings (Cont.) LPR Profitability BP_BG-L B BP «&- EP1_BG-L EPl -*- EP2 BG-L S EP2 (c) Case 6: NR40NA40 (wit uniformly distributed demand) LPR Profitability -BP BG-L BP -A- EPl BG-L EPl -*- EP2 BG-L-»EP2 (d) Case 6: NR40NA40 (wit normally distributed demand)

177 Figure 6.4: LRP profitability under BP, EPl, and EP2 settings (Cont.) LRP Profitability BP_BG-L ta BP -»<A~ EP1_BG-L EPl ~*C«- EP2 BG-L -EP2 (e) Case 8: NR65NA70 (wit uniformly distributed demand) LRP Profitability BP_BG-L P BP -*- EP1BG-L EPl *- EP2 BG-L EP2 (f) Case 8: NR65NA70 (wit normally distributed demand) Wit respect to actual liability ratio, it increases wit te increase in budget tigtness and te demand variability under eac model setting. EPl model always incurs a iger liability ratio tan te oter two models. Furtermore, wen demand variability increases, te liability ratio in 165

178 BP and EP2 models seems to get closer to te liability ratio in EPl model, no matter wat te canges in budget tigtness. Tis indicates tat in comparison to EPl model, te advantages of BP and EP2 models (in terms of liability ratio) are mitigated wen te demand variability level is ig. Tese penomena are observed in all testing cases (see Figure 6.5). 166

179 Figure 6.5: Liability ratios under BP, EPl, and EP2 settings Liability Ratio BP_BG-L m BP EPl *- EP2 BG-L' EP1_BG-L»EP2 (a)case4:nr20na15 (wit uniformly distributed demand) A- "»?,«.«.'= ST "HI** tmste* M DV H (b)case4:nr20na15 (wit normally distributed demand) Liability Ratio BP_BG-L BP -A- EPl BG-L EPl -* EP2 BG-L *-EP2 x J --sss ~z>rm ^^m lx ^ *» ass. *&» m» '> gzs> 3BK> MS8S» «*&*J k ig^^^"** M DV 88SS> *** " ^ ^ K H 167

180 Figure 6.5: Liability ratios under BP, EPl, and EP2 settings (Cont.) Liability Ratio -BPBG-L -HEH-BP - *- EP1JBG-L EPl -*- EP2 BG-L EP2 (c) Case 6: NR40NA40 (wit uniformly distributed demand) Liability Ratio (d) Case 6: NR40NA40 (wit normally distributed demand) BP BG-L EP1BG-L - *- EP 1BG-L EPl -* EP2 BG-L * EP2 a^mjs^ &M$0$&M iffjssw M DV H 168

181 Figure 6.5: Liability ratios under BP, EPl, and EP2 settings (Cont.) Liability Ratio _BP_BG-L E3 BP_ -A- EP1BG-L EPl -*- EP2 BG-L -EP2 (e) Case 8: NR65NA70 (wit uniformly distributed demand) Liability Ratio -BP_BG-L «BP EPl -*- EP2 BG-L EP1BG-L EP2 (f) Case 8: NR65NA70 (wit normally distributed demand)

182 Wit respect to te ordering quantity of rewards, it is te same under BP and EPl settings. As sown in Figure 6.6, wen te level of budget tigtness is low, te ordering quantities of rewards may increase wit te increase in demand variability under BP, EPl, and EP2 settings. However, wen te level of budget tigtness is ig, te ordering quantity of rewards eiter remains te same (i.e., in BP and EPl models) or decrease sligtly (i.e., in EP2 model). Tis is because te ordering quantity of rewards is limited by te available amount of budget. Ordering quantities of rewards determined under EP2 setting are never more tan tose under BP and EPl settings, no matter wat te canges in budget tigtness and demand variability. 170

183 Figure 6.6: Ordering quantity of rewards under BP, EPl, and EP2 settings Ordering Quantity of Rewards BP_BG-L IB BP_ ~il,<~ EP1BG-L EPl -«~ EP2 BG-L EP2 700 (a)case4:nr20na15 (wit uniformly distributed demand) "fs^*,«.«sfl M H DV Ordering Quantity of Rewards -BP_BG-L BP_»*- EP1BG-L EPl -*- EP2 BG-L EP2 (b)case4:nr20na15 (wit normally distributed demand) *~i&. r»-«- - *> M DV H 171

184 Figure 6.6: Ordering quantity of rewards under BP, EPl, and EP2 settings (Cont.) Ordering Quantity of Rewards - -BPBG-L n BP_ -<A~ EP1BG-L EPl -*- EP2 BG-L i» EP (c) Case 6: NR40NA40 (wit uniformly distributed demand) \*iS^tSS'>Sz" m "" J~ M DV H Ordering Quantity of Rewards BPBG-L BP_ -A- EP1BG-L EPl -*- EP2 BG-L EP (d) Case 6: NR40NA40 (wit normally distributed demand) M DV H 172

185 Figure 6.6: Ordering quantity of rewards under BP, EPl, and EP2 settings (Cont.) Ordering Quantity of Rewards (e) Case 8: NR65NA70 (wit uniformly distributed demand) ~.BP_BG-L M BP_ -<A» EP1BG-L * EPl * EP2 BG-L~«EP2 ij&asssrsr&b M DV H (f) Case 8: NR65NA70 (wit normally distributed demand) Ordering Quantity of Rewards BPBG-L B BP_ -A- EP1_BG-L EPl -*- EP2 BG-L EP2 fiajg' M DV H 173

186 Te results for all pair-wise comparisons and ypotesis tests (see Appendix D.7) sow tat given te canges in budget tigtness and demand variability, te impacts of te model settings (i.e., BP, EP1, and EP2) on LRP profitability, liability ratio, and ordering quantities of rewards are significantly different in most of te testing cases wit an exception in te testing case 4 (i.e., TC No.4) wit normally distributed demand. In tat testing case, wit a low level of demand variability (e.g., DV=0.1) and a ig level of budget tigtness (e.g., a=0.3), te ordering quantities of rewards across te tree models ave no significant differences. In addition, te ordering quantities of rewards are te same under BP and EP1 settings and te corresponding confidence intervals include zero (as igligted by "*" in Appendices D.7). Tese results indicate tat offering bonus points doesn't affect te LRP ost's rewards-ordering decisions. According to tese computational outputs and statistical analyses, we conclude tat wit a ig level of budget tigtness, EP2 model still acieves iger LRP profitability tan BP and EP1 models wen demand variability increases, wereas EP1 model seems better tan BP and EP2 models wen te demand variability level is low. Wit respect to liability, EP1 model generates a iger liability tan BP and EP2 models given a level of budget tigtness. BP and EP2 models seem to be better tan EP1 model in terms of liability, but tis advantage is mitigated wen demand variability increases. Wit regard to an LRP ost's decisions on ordering quantities of rewards, under BP and EP1 settings te ordering quantities of rewards are te same; wereas under EP2 setting, te ordering quantities are sligtly lower tan tose under BP and EP1 settings. 174

187 6.3.5 Summary In tis section, we investigate te impacts of budget tigtness on LRP profitability, liability, and LRP ost firm's rewards-ordering decisions. We conduct ypotesis tests troug all pair-wise comparisons under eac model setting and across model settings. In addition to tat, we also examine te impacts of budget tigtness under different demand variability levels. We observe tat: (1) Budget tigtness does ave a negative impact on LRP profitability and LRP liability (i.e., iger budget tigtness, lower LRP profitability and iger liability ratio) under BP, EP1, and EP2 settings, respectively. (2) Budget tigtness does ave negative impacts on LRP profitability and LRP liability (i.e., iger budget tigtness, lower LRP profitability and iger liability ratios) at all levels of demand variability (i.e., low, medium, and ig). (3) Wit a level of budget tigtness, EP2 model always acieves a iger LRP profitability wen demand variability level is ig; owever, EP1 model seems to be better tan BP and EP2 models in terms of LPR profitability wen demand variability level is low. (4) Wit a level of budget tigtness, EP1 model always generates a iger liability tan BP and EP2 models. BP and EP2 models seem to be better tan EP1 model in terms of LRP liability, but tis advantage is mitigated wen demand variability increases. (5) Wit a level of budget tigtness, under BP and EP1 settings, ordering quantities of rewards are always te same; owever, under EP2 model setting te ordering quantities are sligtly lower tan tose under BP and EP1 model settings. 175

188 Tese observations are consistent wit wat we expect to see (i.e., null ypoteses are rejected). Overall, internal dynamics suc as budget tigtness do ave an impact on LRP profitability, liability, and LRP ost firm's rewards-ordering decisions. Given te model inputs, troug our analytical models we can examine ow likely and at wic level te impact will occur. In terms of LRP profitability and liability, using option contracts still seems better tan using wolesaleprice-only contracts in rewards-supply contracting wen te level of budget tigtness is ig. Meanwile, te advantages of adopting option contracts are more obvious wen demand variability increases. Offering bonus points doesn't affect LRP ost's rewards-ordering decisions and it acieves iger LRP profitability only wen te level of demand variability is low, no matter wat te canges in budget tigtness. 176

189 6.4 Testing te Impacts of Liability Control Troug te set of numerical experiments in tis section, we seek to examine te impacts of liability control, anoter important internal dynamic, on te LRP ost's ordering decisions, LRP profitability, and budget usage. We also investigate weter te impacts varies between different model settings and under different demand variability levels Under BP Setting Table 6.10 summarizes te results of BP wen budget constraint (4.43) and capacity constraints (4.42) are relaxed and te liability control becomes tigt (i.e., y =0.7). We observe tat LRP profitability decreases wit te increase of liability control tigtness (i.e., y canges from 1.3 to 0.7). Wen te liability control is tigt (i.e., y =0.7), iger target redemption is required. In oter words, te estimated optimal ordering quantities of rewards and te cost of purcasing rewards (i.e., budget usage) are expected to increase, wic results in decrease in te LRP profitability at te redemption side. As suc, te overall LRP profitability decreases. On te oter and, just as in te case of te loose liability control (i.e., y = 1.3), wit te increase in demand variability, te LRP profitability decreases in te testing case 4 but increases in te testing cases 6 and 8 wen liability control is tigt. 177

190 Table 6.10: Computational results under BP Setting (a) LRP Profitability Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (b) Costs of Rewards Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H(0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3)

191 Table 6.10: Computational results under BP setting (Cont.) (c) Ordering quantity of rewards Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) Sown in Table 6.11 are te budget usage ratios generated in Step 4 of te procedure designed for conducting te numerical studies in tis section (see Section 5.5 in Capter 5 for more details of te procedure). In Table 6.11, te budget ratio of Type I refers to te ratio of te current budget usage (obtained in te model witout budget and capacity constraints) to te initial budget usage (obtained in te model wit budget and capacity constraints). Te budget ratio of Type II refers to te ratio of te current budget usage to te maximum budget available (W R ). Wit respect to te budget usage, we observe tat increased liability control tigtness results in a iger budget usage. As suc, te budget ratio of Type I in eac testing case is greater tan 1.0. We also observe tat te budget ratio of Type I increases wit te increase in liability control tigtness (i.e., y canges from 1.3 to 0.7). Furtermore, in some cases, te current budget usage (obtained in te model witout budget and capacity constraints) is even larger tan te maximum amount of budget (i.e., te rigt-and side of te budget constraint: W 1 *) (see budget ratio of Type 179

192 II in Table 6.11), especially wen te liability control is tigt (i.e., y =0.7) and te demand variability level is ig. Table 6.11 Comparison of te budget usages under BP setting (a) Budget Ratio (Type I) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (b) Budget Ratio (Type II) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3)

193 Tis examination of te budget usage, considering te target liability control requirement (e.g., defined by liability control parameter y), provides a means of determining weter or not te budget constraint would be violated. In oter words, if te model results sow tat te budget usage is lower tan te maximum budget available (i.e., ff 8 ), ten we know tat we can meet te target liability control requirement by purcasing more rewards. If te model results sow tat te budget usage is larger tan te maximum budget available, ten we can tell wat te extra amount of budget required to meet te target liability control requirement will be Under EP1 Setting Table 6.12 summarizes te results of EP1 wen budget constraint (4.51) and capacity constraints (4.50) are relaxed and te liability control becomes tigt (i.e., y =0.7). We observe tat LRP profitability decreases wit te increase of liability control tigtness (i.e., y canges from 1.3 to 0.7). Wen liability control is tigt (e.g., y =0.7), iger target redemption is expected. In oter words, te estimated optimal ordering quantities of rewards and te cost for purcasing rewards (i.e., budget usage) increase, wic results in decrease in LRP profitability at te redemption side. As suc, te overall LRP profitability decreases. On te oter and, just as in te case tat te liability control is loose (i.e., y = 1.3), wit te increase in demand variability, te LRP profitability decreases in te testing case 4, and decreases first (e.g., DV=0.15) and ten increases (e.g., DV=0.25) in testing cases 6 and 8 in te case tat liability control is tigt. Tis result suggests tat as long as te budget and te capacities for rewards are not limited, te LRP ost may reduce te negative impact of demand uncertainty on LRP profitability wit a larger number of accumulation and redemption partners (e.g., in Case 6, tere are 40 accumulation partners, and 40 redemption partners; and in Case 8, tere are 70 accumulation partners, and

194 redemption partners), even wen te ost attempts to implement a tigter liability control requirement. Table 6.12 Computational results under EPl setting (a) LRP Profitability Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (b) Costs of Rewards Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0 15) H(0.25) H(0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3)

195 Table 6.12 Computational results under EPl setting (Cont.) (c) Ordering quantity of rewards Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H(0.7) L(1.3) (d) Bonus points offering Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) As sown in Table 6.12, we also observe tat canges in liability control parameter ave no impact on te LRP ost's decisions on offering bonus points. Te estimated optimal decisions on 183

196 offering bonus points remain te same, regardless any canges in te liability control requirement. Table 6.13 Comparison of te budget usage under EPl setting (a) Budget Ratio (Type I) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H(0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (b) Budget Ratio (Type II) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H(0.7) L(1.3) LC tigtness (y) H(0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3)

197 Te budget usage ratio for eac testing cases is reported in Table We observe tat increased liability control tigtness results in iger budget usage. As suc, te budget ratio of Type-I in most testing problems is greater tan 1.0. We also observe tat te budget ratio of Type-II increases wit te increase in liability control tigtness (i.e., y canges from 1.3 to 0.7). Moreover, in some cases te current budget usage (obtained in te model witout budget and capacity constraints) is even larger tan te maximum amount of budget (i.e., te rigt and side of te budget constraint: W^) (see te budget ratio of Type-II in Table 6.13), especially wen te liability control is tigt (i.e., y =0.7) and te demand variability level is ig. Te same type of budget-related analysis as tat discussed in Section can be done under EP1 setting Under EP2 Setting Table 6.14 provides a summary of EP2 model results obtained wen te budget constraint (4.64) and te capacity constraint (4.61) are relaxed and te liability control becomes tigt (i.e., y =0.7). Similar to wat as been observed and discussed under BP and EP1 settings, iger redemption is required wen te liability control gets tigter. We also observe tat under EP2 setting, a iger redemption results in an increase in te optimal ordering quantity of rewards and in te optimal number of options to purcase and to exercise. As suc, te corresponding costs for purcasing rewards and options and for exercising options increase and te overall LRP profitability is reduced due to te decreases in te redemption side LRP profitability. Meanwile, wen te liability control is tigt (i.e., y = 0.7), te number of options to purcase as well as te number of options to exercise increase considerably. In Table 6.14(d), for example, in te testing case 4, given DV=0.05, te optimal solution sows tat te number of options to purcase increases from 5.19% of te initial ordering quantity of rewards (= ) wen y =1.3 to 185

198 17.25% of te initial ordering quantity of rewards (= ) wen y =0.7 (see Table 6.14(d) and (e)). Meanwile, te number of options to exercise increases from 18.97%) of te options to purcase (= 26.51) wen y =1.3 to 90.58%> of te options to purcase (= 91.63) wen y =0.7 (see Table 6.14(f) and (g)). Table 6.14 Computational results under EP2 setting (a) LRP Profitability Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (b) Costs of Rewards (including purcasing and exercising costs of options) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H(0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) M(1.0) H(1.8)

199 Table 6.14 Computational results under EP2 setting (Cont.) (c) Initial ordering quantity of rewards Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (d) Number of options to purcase Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3)

200 Table 6.14 Computational results under EP2 setting (Cont.) (e) Ratio of number of options to purcase and initial ordering quantity of rewards (%) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) D L(0.05) M(0.15) H(0.25) H (0.7) 17.25% 14.07% 15.08% L(1.3) 5.19% 15.13% 24.64% H (0.7) 31.80% 28.61% 32.65% L(1.3) 4.83% 14.12% 23.12% H(0.7) 43.23% 39.07% 40.53% L(1.3) 4.91% 14.52% 23.90% Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) 19.62% 1.95% 16.03% 19.08% 19.62% 33.33% LC tigtness (y) H (0.7) L(1.3) 34.36% 1.70% 29.24% 16.28% 34.52% 28.48% LC tigtness (y) H(0.7) L(1.3) 46.04% 1.78% 39.57% 17.22% 43.58% 30.45% (f) Number of options to exercise Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3)

201 Table 6.14 Computational results under EP2 setting (Cont.) (g) Ratio of number of options to exercise and Number of options to purcase (%) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) 90.58% 63.72% 36.55% L(1.3) 18.97% 19.42% 19.79% H (0.7) 94.42% 83.08% 68.22% L(1.3) 18.95% 19.41% 19.88% H (0.7) 96.35% 88.94% 80.91% L(1.3) 19.42% 20.05% 20.57% Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) 96.54% 14.29% 55.52% 14.86% 31.35% 15.06% LC tigtness (y) H (0.7) L(1.3) 98.35% 15.64% 82.67% 15.78% 64.45% 16.02% LC tigtness (y) H(0.7) L(1.3) 98.91% 15.79% 88.77% 16.16% 76.96% 16.52% Under EP2 setting, te budget usage ratio for eac testing case is reported in Table We find tat te results under EP2 setting are similar to tose obtained under BP and EP1 settings. Te increase in liability control tigtness results in a iger budget usage. As suc, te budget ratio of Type I in all testing problems is greater tan or equal to 1.0. We also observe tat te budget ratio of Type II increases wit te increase in liability control tigtness (i.e., y canges from 1.3 to 0.7). Moreover, in some cases, te current budget usage (obtained in te model witout budget and capacity constraints) is even larger tan te maximum budget available (i.e., te rigt and side of te budget constraint: JF*) (see budget ratio of Type II in Table 6.15), especially wen te liability control is tigt (i.e., y =0.7) and demand variability level is ig. Te same type of budget related analysis tat was discussed in section can be done under EP2 setting. 189

202 Table 6.15 Comparison of te budget usage under EP2 setting (a) Budget Ratio (Type I) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) M(0.15) H(0.25) H (0.7) L(1.3) H(0.7) L(1.3) H (0.7) L(1.3) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) M(1.0) H(1.8) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) (b) Budget Ratio (Type II) Case 4 Case 6 Case 8 LC tigtness (y) LC tigtness (y) LC tigtness (y) (wit uniformly distributed demand) DV L(0.05) H (0.7) L(1.3) H (0.7) L(1.3) H (0.7) L(1.3) M(0.15) H(0.25) Case 4 Case 6 Case 8 (wit normally distributed demand) DV L(0.1) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) LC tigtness (y) H (0.7) L(1.3) M(1.0) H(1.8) Overall, te impacts of liability control are quite consistent under BP, EPl, and EP2 settings. A summary is provided as follows:

203 (1) Wen te liability control gets tigter, LRP profitability seems to be reduced due to te increasing costs for purcasing rewards. (2) Wen te liability control gets tigter, iger target redemption is required. Tis causes te estimated optimal ordering quantities of rewards to increase. Under EP2 setting, te initial ordering quantities and te number of options to purcase and to exercise increase wit te increase in liability control tigtness. Under EP1 setting, te LRP ost's decisions on bonus points are not influenced by te canges in liability control requirement. (3) Wen te liability control gets tigter, te budget usage increases. Compared to te budget usage under te original model settings (wit budget and capacity constraints, and loose liability control), and under te current model settings (i.e., tigt liability control), te budget usage is always larger tan tat under te original model setting. Especially wen te demand variability level is ig, to meet a liability control requirement, te budget usage may exceed te maximum budget available (i.e., W 11 ) Comparison across BP, EP1, and EP2 Model Settings In order to compare te impacts of liability control across our model settings (BP, EP1 and EP2), we examine te canges in LRP profitability, cost of rewards, and ordering quantity of rewards in terms of percentage wen te liability control parameter canges from loose (i.e., y =1.3) to tigt (i.e., y =0.7), and budget and capacity constraints are relaxed. Let P(0.7), C(0.7), 2(0.7) denote te LRP profitability, cost of rewards, and te ordering quantity of rewards respectively, given y =0.7. Let P(1.3), C(1.3), Q(13) denote te LRP profitability, cost of rewards, and te 191

204 ordering quantity of rewards, respectively, given y =1.3. Te following tree measures of te impact of te liability control are calculated: P(07)-P(13) Canges in LRP profitability (%) = v " ; ^-^-x 100, C(0 7)-C(l 3) Canges in cost of rewards (%) =, ^-^-xioo, and 6 v ' C(1.3) Canges in ordering quantity of rewards (%) = ^ ' ' ^-ixloo >c\ ) Tables report on tese canges under BP, EPl, and EP2 settings. Given a level of demand variability, iger decrease (in percentage) in te LRP profitability and iger increase (in percentage) in te cost of rewards are observed under EP2 setting tan under BP and EPl settings. Regarding te ordering quantities of rewards (including te options to exercise under EP2 setting), wen te level of demand variability is ig, a iger increase (in percentage) in te ordering quantity of rewards is observed under EP2 setting tan under BP and EPl settings. However, wen te level of demand variability is low, EPl model is observed to ave a iger increase (in percentage) tan BP and EP2 models. Tese results suggest tat EP2 model seems more sensitive to canges in te liability control parameter. In oter words, EP2 model does not sow any advantage in controlling te liability. 192

205 Table 6.16 Comparison of te impacts of liability control on LRP profitability across different model settings (a) wit Uniformly Distributed Demand Case 4 Case 6 Case 8 DV BP EP1 EP2 BP EP1 EP2 BP EP1 EP2 L(0.05) M(0.15) H(0.25) -0.61% -0.53% -0.38% -0.71% -0.62% -0.45% -1.18% -1.00% -0.86% -1.43% -1.33% -1.12% -1.61% -1.50% -1.25% -3.24% -2.61% -2.43% -1.71% -1.60% -1.40% -1.88% -1.76% -1.53% -3.95% -3.40% -3.14% (b) wit Normally Distributed Demand Case 4 Case 6 Case 8 DV BP EP1 EP2 BP EP1 EP2 BP EP1 EP2 L(0.1) M(1.0) H(1.8) -0.64% -0.54% -0.37% -0.72% -0.61% -0.40% -1.18% -1.00% -0.86% -1.44% -1.36% -1.15% -1.60% -1.49% -1.22% -3.24% -2.61% -2.43% -1.73% -1.60% -1.37% -1.88% -1.72% -1.44% -3.95% -3.40% -3.14% Table 6.17 Comparison of te impacts of liability control on cost of rewards across different model settings (a) wit Uniformly Distributed Demand Case 4 Case 6 Case 8 DV BP EP1 EP2 BP EP1 EP2 BP EP1 EP2 L(0.05) M(0.15) H(0.25) 7.86% 6.59% 5.00% 9.16% 7.61% 5.71% 8.97% 9.51% 9.39% 15.55% 14.02% 11.84% 18.03% 15.93% 13.13% 18.84% 16.38% 14.92% 19.24% 17.20% 14.71% 21.90% 19.15% 16.04% 23.57% 21.17% 19.25% (b) wit Normally Distributed Demand Case 4 Case 6 Case 8 DV BP EP1 EP2 BP EP1 EP2 BP EP1 EP2 L(0.1) M(1.0) H(1.8) 8.45% 8.05% 7.05% 9.69% 8.83% 7.36% 8.68% 10.16% 11.06% 16.08% 15.77% 14.41% 18.63% 17.24% 15.03% 19.99% 16.93% 16.56% 20.05% 18.68% 16.71% 22.85% 20.24% 17.46% 24.57% 21.70% 20.84% 193

206 Table 6.18 Comparison of te impacts of liability control on ordering quantity of rewards across different model settings (a) wit Uniformly Distributed Demand Case 4 Case 6 Case 8 DV BP EPl EP2 BP EPl EP2 BP EPl EP2 L(0.05) M(0.15) H(0.25) 18.56% 14.49% 9.85% 21.72% 16.99% 11.57% 18.98% 16.41% 14.18% 33.34% 28.57% 22.67% 38.78% 32.75% 25.49% 33.46% 30.45% 27.90% 44.36% 38.13% 31.13% 50.62% 42.72% 34.26% 45.21% 41.72% 38.89% (b) wit Normally Distributed Demand Case 4 Case 6 Case 8 DV BP EPl EP2 BP EPl EP2 BP EPl EP2 L(0.1) M(1.0) H(1.8) 20.17% 16.08% 11.47% 23.19% 17.97% 12.24% 20.18% 17.47% 16.92% 34.91% 30.83% 25.49% 40.49% 34.06% 26.86% 34.92% 31.82% 30.41% 46.66% 39.93% 32.84% 53.24% 43.57% 34.58% 46.80% 43.08% 41.39% Figure 6.7 illustrates te ratio of budget usage under EPl and EP2 settings and te budget usage under BP model setting in eac testing case. Let B(BP), B(EP1), and B(EP2) denote te budget usages under BP, EPl, and EP2 settings, respectively. Let R(BP), R(EP1), and R(EP2) denote te ratio of budget usage under different model settings. Te ratio illustrated in Figure 6.7 is calculated by using te follow relations: R(BP) = B(BP) / B(BP) R(EP1) = B(EP1) / B(BP), and R(EP2) = B(EP2)/B(BP) Under different levels of liability control (i.e., y =0.7 or 1.3), we observed tat in most cases, EPl model generates te igest budget usage among te tree models, wereas EP2 model generates te lowest budget usage. However, wen te demand variability level is low, EP2

207 model requires more budget tan EPl and BP models (e.g., in testing cases 6 and 8, DV=0.05 or DV=0.1). Wen te demand variability level is ig, EP2 models seems better tan bot BP and EPl models because it acieves te same liability control requirement wile requiring a lower budget. Figure 6.7 Comparison of budget usage ratios across different model settings (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case 4 (NR20NA15) i.uj Case 4 (NR30NA15) HBP Budget Usage EP2_DV0.05 EP2_DV0.15 EP2_DV0.25 EP1_DV0.05 EP1_DV0.15 EP1_DV0.25 Jl 7 = Ji y= a BP Budget Usage EP2DV0.1 HEP2_DV1.0 EP2JDV1.8 BEP1_DV0.1 EP1_DV1.0 EP1_DV1.8 7 = U - ~ 7=

208 Figure 6.7 Comparison of budget usage ratios across different model settings (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case 6 (NR40NA40) l.uj Case 6 (NR40NA40) l.uj HBP Budget Usage EP2DV0.05 EP2_DV0.15 EP2_DV0.25 EP1_DV0.05 EP1_DV0.15 EP1DV0.25 Jl 7 = : i Jl n 7= BP Budget Usage EP2DV0.1 EP2_DV1.0 EP2_DV1.8 EP1_DV0.1 EP1_DV1.0 EP1DV1.8 7 = Jl 7=

209 Figure 6.7 Comparison of budget usage ratios across different model settings (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case 8 (NR65NA70) Case 8 (NR65NA70) l.uj BP Budget Usage 11 7 = i 7= HBP Budget Usage Ji 7 = = EP2DV EP2DV EP2_DV0.15 EP2DV EP2JDV1.0 EP2DV EP1DV EP1_DV EP1_DV EP1_DV EP1_DV EP1_DV More on Management Insigts from Liability Control Analysis Te budget usage reported in Sections may, in some cases, exceed te maximum amount of budget available wit a targeted liability ratio (see te budget ratio of Type II in Tables 6.15, 6.17, and 6.19). In tose cases, te simplest way for te LRP ost to meet teir targeted liability is to seek an additional budget in te ranges suggested by our models. However, in practice, tis may not always be possible for te following reasons: Te extra budget is not available 197

210 Te LRP ost does not want to spend more to buy additional rewards Te extra amount of rewards are not available due to LRP capacity constraints In tat regard, te LRP ost may resort to some managerial policies (e.g., cange te redemption sceme to carge more points for redeeming a reward) to increase te overall redemption and acieve te targeted liability level witout increasing te budget usage. Our models provide an analytical tool for LRP ost firms to detect weter tere is any conflict between meeting te target liability level and satisfying various resource and management constraints at te planning stage and to decide wic managerial policy would be te most appropriate to apply under different scenarios. We illustrate tis under BP setting (similar analysis can be performed under EP1 and te EP2 settings). Let q represent te estimated optimal ordering quantities of rewards from partner Rj given te targeted liability ratio LUB and relaxed budget and capacity constraints (i.e., constraints (4.43), (4.51), (4.64), (4.42), (4.50) and (4.61)), and let q R represents te estimated optimal ordering quantity of rewards from partner R } given budget and capacity constraints and a relaxed liability constraint. Since #*are te ordering quantities of rewards under budget and capacity constraints, we ave q R <q Rb for j=\,...j; qf can be understood as te targeted redemption level tat satisfies te targeted liability control requirement and <7 ; can be understood as te maximum redemption level wit te budget and capacity constraints. 198

211 However, as mentioned in Capter 3, wat really matters is te actual redemption level rater tan te maximum redemption level. Terefore, let q Ra represent te actual redemption level for rewards from partner R ;, ten q Ra = minyq R,Dj), wic indicates tat te value of ^"depends on q* as well asd* and q Ra < q R. Terefore, in order to acieve te targeted redemption level q Rb and to satisfy te liability control requirement as well as te budget and capacity constraints at te same time, several managerial policies can be applied to make q R " &s close to q Rb as possible. Case 1: Wen q R <D R <q Rb or q R <q Rb <D R, in oter words, q R " = q R <q Rb, te LRP manager may apply managerial policies suc as carging more points for redeeming a reward to increase te actual redemption ( qf 1 ) of rewards to te target redemption level q Rb witout violating te budget and capacity constraints. Case 2: Wen D R <q R < qf, in oter words, q Ra < q* < q Rb, te LRP manager may need to boost redemption demand (e.g., troug a promotion) to first make Z)^as close to g^as possible. Ten managerial policies sould be applied (e.g., carging more points for redeeming a reward) to increase te redemption (q R ) of rewards from te redemption partner j so tat te target redemption level q Rb can be reaced witout violating te budget and capacity constraints. However, te LRP manager sould pay more attention to te managerial policies to apply in tis case because some of tem (e.g., carging more points for redeeming a reward) may discourage 199

212 customer redemption beavior and, terefore, ave a negative impact on enancing redemption demand to te maximum redemption level q* Summary In sort, based on te above computational results and analyses, we conclude tat: (1) Increasing te liability control tigtness will reduce LRP profitability no matter wic model setting is used. (2) Increasing te liability control tigtness results in iger target redemption and an increase in te supply of rewards. Terefore, in order to meet a liability control target, te costs for rewards (i.e., te budget usage) may also increase and sometimes even exceed te maximum redemption budget available. (3) Under EP1 setting, since te LRP ost firm offers bonus points to accumulation partners, a iger accumulation may be expected. Terefore, to meet te same target level of liability, a iger redemption is required wit a bonus points offering (i.e., EP1) tan witout a bonus points offering (i.e., BP and EP2). (4) In comparison wit BP and EP1 models, it seems tat EP2 model does not ave advantages in dealing wit te canges in liability control, altoug wit te decreases in demand variability, EP2 model performs better. Tis indicates tat te notable advantages of using option contracts ave not been observed in meeting te liability control requirements. 200

213 In addition, if meeting te target liability requires te spending for rewards to exceed te maximum redemption budget available, various managerial policies can be used to deal wit tat conflict. Using our analytical models, te LRP ost firm can detect weter tere is any conflict between meeting te targeted liability level and satisfying various resource and management constraints at te planning stage. 201

214 Capter 7 Conclusions and Future Researc Directions In tis study, we ave intended to develop quantitative analytical models for rewards-supply aggregate planning in loyalty rewards programs (LRPs) tat operate as profit centers. We ave investigated te LRP ost's rewards-supply planning decisions in te presence of multiple LRP accumulation partners, multiple redemption partners, multiple resource constraints (e.g., budget, capacity), and multiple management concerns including LRP profitability, liability control, and demand uncertainties. We ave proposed tree analytical models to examine te impacts of internal dynamics (budget tigtness and liability control tigtness), and external uncertainties (demand uncertainties) on LRP ost's rewards-ordering decisions and LRP performance under different system settings. Te first model {Problem BP) is developed under te setting were only wolesale price contracts are used in bot points-sale contracting and rewards-supply contracting. Te second model {Problem EPl) extends te first model by adding te consideration of a common cooperative advertising practice (i.e., troug bonus points) in te rewards-supply planning decisions. Te tird model {Problem EP2) extends te first model by considering option contracts in te rewards-supply contracting. We ave captured, wit eac of tese models, some of te relationsips among independent variables and dependent variables. Te independent variables refer to model inputs suc as maximum available budget, target liability ratio, and demand variability. Te dependent variables refer to model outputs suc as LRP profitability, budget usage, liability ratio, ordering quantities of rewards and bonus points offering, number of units of options to purcase, and number of units of options to exercise. 202

215 A euristic-based and SAA sceme-based stocastic programming solution procedure is developed to obtain solutions for te tree analytical models. Using a computer simulation, numerical experiments, and statistical analysis, we ave analyzed te impacts of te internal dynamics and external uncertainties under eac model setting as well as across model settings considering different realistically sized problems. Tis process as yielded a number of managerial insigts about wic system setting is more effective in different situations. 7.1 Findings and implications Our teoretical framework, computational results, and analysis lead to tree important findings and managerial implications. Te first finding is tat demand uncertainties ave negative impacts on LRP performance (i.e., LRP profitability, liability ratio, and budget usage). Wen demand variability is low, EP1 model generates iger LRP profitability, but also iger liability. Wen demand variability is ig, EP2 model is better tan te oter two models. Furtermore, wit te increases in demand variability, te LRP ost's estimated optimal ordering quantities of rewards increase, but te estimated optimal amount of bonus points offered to accumulation partners decreases. In addition, EP2 model results in lower budget usage tan BP and EP1 models; tat difference is more obvious wen demand variability is ig. However, EP2 model's advantage in terms of budget usage seems to weaken in large-size problems. Tese results indicate tat option contract does play a role in coping wit demand uncertainties. Tis finding is consistent wit te findings in te previous studies on option contracts (e.g., Barnes-Scuster et a, 2002; Erkoc and Wu, 2005; Jin and Wu, 2007). To an LRP ost, adopting an option contract in rewards-supply 203

216 contracting results in relatively lower liability and relatively iger profitability. Te advantage of using te option contract is more obvious wen demand variability is ig. In contrast, offering bonus points seems not to be preferred wen demand variability is ig. Indeed, bonus points are a double-edged sword to te LRP ost. On one and, offering bonus points may bring in iger LRP profitability, especially wen demand variability is low. On te oter and, offering bonus points will result in iger liability no matter wat te canges in demand variability. One implication of tis finding is tat wen an LRP ost faces ig demand uncertainties, an effective strategy to reduce te negative impacts of tose uncertainties is to adopt option contracts in rewards-supply contracting. In addition to tat, in order to take te best advantage of option contracts, te LRP ost sould pay attention to te design of te option contracts, especially wen tere is a large number of LRP redemption partners involved in te business. Offering bonus points may not be a good strategy to deal wit demand uncertainties. A LRP ost sould be conservative in using bonus points wen demand uncertainty is ig. Te second finding is tat a tigt budget, an internal dynamic, as an impact on LRP profitability and liability, and on te LRP ost's rewards-supply decisions. Altoug tis is an intuitive finding, te corresponding computational experiments demonstrate tat wit given model inputs, we can examine ow likely and at wat level te impact of a tigt budget will be. Moreover, wit tigt budget constraints, te responses of eac model to te canges in demand variability are consistent wit te responses obtained wit loose budget constraints. Tis indicates tat altoug a tigt budget (i.e., a smaller available budget) weakens te effectiveness of using 204

217 option contracts to reduce te negative impact of demand variability; in comparison wit wolesale-price-only contracts, option contracts are still better. Besides tat, no matter wat te canges in te budget tigtness, an LRP ost's decisions on offering bonus points are made independently from rewards-ordering decisions. One implication of te second finding is tat troug our analytical models, an LRP ost as te capability to evaluate te impact of canges in te available budget for purcasing rewards on te profitability and liability as long as all te model input parameters can be determined or estimated. In a case were an LRP ost as a tigt budget for purcasing rewards, option contracts are still a better coice to cope wit demand uncertainties and reduce te negative effect of te uncertainties on LRP performance in rewards-supply contracting. Te tird finding is tat increasing te liability control level results in increases in te ordering quantity of rewards and budget usages, and decrease in LRP profitability. In order to meet te same target level of liability control, a iger redemption is required under EP1 setting (wit bonus points offering) tan under BP and EP2 settings (i.e., witout bonus points offering). Furtermore, in comparison to wolesale-price-only contracts, our numerical study does not indicate tat option contracts are better for supporting te liability control target, altoug LRP performance improves in EP2 model wit te increases in demand variability. Terefore, supply contracts (i.e., wolesale price contracts and option contracts) and bonus points may not be elpful in acieving a liability control target. 205

218 One implementation of te tird finding is tat adding a liability control target in te rewardssupply planning may elp te LRP ost to detect potential conflicts between controlling liability and satisfying resource constraints, suc as budget and capacity constraints. Moreover, as long as all model input parameters can be determined or estimated, an LRP ost can use te model outputs to evaluate te impact of canges in liability control on oter LRP constraints, and ten make furter managerial decisions on ow to deal wit te issues eiter troug increasing te redemption budget (e.g., our models elp in determining te range of te required budget) or adopting oter managerial practices (e.g., cange redemption sceme). 7.2 Limitations and Contributions A few limitations of tis study sould be recognized. Tese limitations also point to some future researc opportunities on tis topic. a. In tis study, we assume tat redemption and accumulation demands follow eiter normal or uniform distributions. Oter types of teoretical demand distributions (e.g., exponential, bi-normal, lognormal, etc.) sould also be examined in order to generalize te findings of te study. b. Te analysis and findings are based on solutions and outputs from our analytical models wit given inputs. Terefore, if our models are used wit individual data, te values of all input parameters sould be determined and estimated accordingly. c. Te focus in tis study is solely on te LRP ost's profitability in rewards-supply planning decisions. Te profitability of LRP partners is not considered. Te model can be furter extended to consider te coordination between te LRP ost and te LRP 206

219 partners by improving te overall rewards-points supply cain profitability (e.g., te sum of te profitability of an LRP ost and its LRP partners), d. We ave examined bonus points and option contracts separately. We did not explore settings tat consider bot bonus points and option contracts. However, by combining problem EP1 and problem EP2, a model tat includes simultaneously bonus points and option contracts can be developed. Despite tese limitations, tis study contributes to te development of te literature on LRP from te following four perspectives: 1. Our researc syntesizes and extends te concept of supply cain management in te context of loyalty reward programs. LRPs are analyzed as a rewards-points supply cain wit some unique features. A compreensive typology framework for LRP classification is proposed tat provides a guideline for academics and practitioners to understand te variety of LRPs in today's marketplace. Furtermore, based on te framework, 39 well known LRPs in te Canadian marketplace were examined to provide illustrative examples for different types of LRPs. Tis researc focuses on a specific type of LRP tat as not been studied in previous LRP literature. 2. Tis is te first study of its kind to investigate rewards-supply planning decisions in LRP management troug a quantitative modeling approac. Constrained newsvendor-based stocastic linear programming (SLP) models were developed to model te rewardssupply planning decisions wit multiple managerial concerns and different system 207

220 settings. Moreover, a euristic-based solution procedure based on Sample Average Approximation sceme was proposed to find te best solutions. Using SLP and its corresponding solution metod to solve te operational issues in LRPs is also a novel application to te existing LRP literature. 3. Tis researc provides numerical evidence of te relationsips between internal dynamics, external uncertainties, and an LRP ost firm's rewards-supply planning decisions and LRP profitability, respectively. To expand current knowledge, tis study evaluates te impacts of internal dynamics and external uncertainties on te LRP ost's decisions and LRP performance under tree different system settings. Tese system settings relate to two LRP practices: rewards-supply contracting and cooperative advertising. We demonstrate tat a supply contract (e.g., an option contract) plays a significant role in coping wit demand uncertainties in rewards-supply aggregate planning. Tis finding is consistent wit previous analytical studies on supply contracts. In regards to cooperative advertising troug bonus points, it is a common practice in LRP operations to improve cooperation between te LRP ost firm and its commerical partners. Tis study provides numerical evidence and management insigts on conditions under wic offering bonus points is a good coice. 4. Tis researc makes a significant contribution to enancing a better understanding of LRPs. We investigate te type of LRP system tat is muc closer to te actual examples deployed in te current marketplace tan te LRP systems discussed in oter literature. Consequently, we bring forward valuable managerial insigts for LRP ost firms to support teir decisions on rewards-supply wit multiple managerial concerns suc as 208

221 LRP profitability, liability control, demand uncertainties, and budget usage. Tis study also provides tested analytical models for judging ow te internal dynamics and external uncertainties impact te LRP ost's rewards-supply decisions and LRP profitability for real applications. In addition, te presented analysis may guide LRP ost firms in te negotiation process wit teir commercial partners regarding rewards-supply contracts and elp tem to seek te possibility of improving cooperation in te partnersips to acieve win-win solutions. 7.3 Future Researc Directions A number of lines of researc could be undertaken to provide a better understanding of LPR operations and te rewards/points-supply planning problems. For instance, some possible future researc directions are: Single-period model vs. multiple-period model In tis study, te analytical models tat we propose are single-period models, in wic all te input parameters except of te liability ratio are assumed to be independent from te parameters in eiter te previous or te following time periods. Our single-period models can be applied to solve multiple-period rewards-supply planning problems directly as long as te assumption of independence (all te input parameters in eac time period are independent from te parameters in te oter time periods) is still valid. In te case were tis assumption is not valid any more, we need to extend our models to consider te dependency among parameters in different time periods. Te common dependencies among parameters in te multi-period LRP rewards-supply planning problems are: 209

222 (1) Te accumulation and redemption demands in eac time period may influence te demands in te subsequent time periods. (2) Offering bonus points in a certain time period may ave a delayed effect on accumulation and redemption demands. Tese delayed effects could occur in te following time periods rater tan te current time period. (3) Te inventory of some imperisable rewards in te current time period can be used as part of te rewards-supply in following time periods. Supply Contracts Te present study assumes tat te relationsips between te LRP ost and accumulation partners are governed by te wolesale-price contracts. Signing a wolesaleprice contract is more suitable in te case were te LRP ost wants to keep a simple relationsip wit accumulation partners. However, in te case were te LRP ost wants to ave a closer relationsip wit LRP accumulation partners, oter types of contracts may be more suitable to govern te relationsips between te LRP ost and te accumulation partners. For instance, te LRP ost and an accumulation partner can sign a revenue-saring type of contract. Te LRP ost will decide te ordering quantity of points for te accumulation partner and commit on te partner's minimum revenue increase. But at te same time, te accumulation partner as to sare te increased revenue wit te LRP ost. Signing revenue-saring type of contracts may allow te LRP ost to ave more flexibility in managing te accumulation business. In comparison wit te wolesale-price contracts, applying revenue-saring type of contracts requires more managerial efforts and expenses. An extension of Problem BP tat includes te consideration of revenue-saring contracts in accumulation business can be formulated as Problem BP1 (See Appendix E.l for te model notations and formulation). Te 210

223 same structure of revenue saring types of contracts can be embedded in Problem EP1 and Problem EP2 models. In addition, in te redemption business we consider option contracts wit a simple structure tat as been discussed in te existing literature. We tink tat te option structure wit multi-layers can be applied to improve te flexibility in te rewards-supply contracting. For example, a treelayer option structure can be designed as follows: Option level 1: 0 < m* 1 < M* 1, unit option price is w* 1, and unit exercise price is e* 1 Option level 2: 0 < m* 2 < Mj 2, unit option price is w* 2, and unit exercise price is e* 2 Option level 3: 0 < m^ < M* 3, unit option price is wf*, and unit exercise price is e* 3 were M* 3 > M* 2 > M* 1 and w^ > w* 2 > w 1 }'. Different upper bounds for te number of options are offered in te contract so tat te buyer (i.e., LRP ost H) can ave different coices. To incorporate multi-layer option contracts, our analytical models sould be modified as stocastic mixed integer programming models. A matematical model for Problem EP2 wit multi-layer option contracts is formulated as Problem EP3 (see Appendix E.2 for te model notations and formulation). Decomposition approaces may need to be developed in order to solve large-sized problems witin a reasonable amount of computational time. Aggregate model vs. disaggregate model We ave developed rewards-supply planning models at te aggregate level in tis study. In oter words, we consider te LRP ost's decisions on aggregate supply of rewards based on aggregate demands for accumulation and redemption in te planning orizon. Te aggregate planning models are suitable for long-term and intermediate-term planning. However, for sort-term rewards-supply planning problems, 211

224 disaggregate planning models are necessary. Disaggregate planning models can include more customer-based demand varieties and partner-based varieties of te supply of rewards: for example, to incorporate LRP member segmentation or LRP partner segmentation in te decisions on dealing wit redemption timing (i.e., wen to redeem) and redemption structure (i.e., were and ow muc to redeem). Modeling random parameters In tis study, redemption and accumulation demands are assumed to be random parameters following certain teoretical probability distributions (i.e., uniform, normal). To te best of our knowledge, no one as discussed in te literature on wat would be te most representative distributions of accumulation and redemption demands. An empirical study can be undertaken to find out weter tere are representative distributions of accumulation and redemption demands for various types of loyalty reward programs. Explore oter solution metods Altoug in te study, we did not discuss te case tat te random parameters follow discrete probability distributions, our SAA-based euristic solution procedure is still applicable for tat case. Alternatively, exact approac (e.g., Birge and Louveaux, 1997) can be explored to solve some special cases of te rewards-supply planning problem described in te study. Discrete-event simulation is anoter suitable approac tat can be used to deal wit te rewards-supply planning problem. Hence, it offers anoter avenue tat can be pursued and te solutions obtained can ten be compared to tose from te approximate approac implemented in tis study. 212

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234 Appendix A 222

235 Appendix A.l: Sample of LRPs in today's marketplace in Canada Name oflrp Aeroplan Air Miles PC points Soppers Optimum Program Canadian Tire Money American Airline AAdvantage United Airline Mileage Plus Save-On-More Casino Niagara Players Prestige Club Casino Rama Player's Passport program Casino Windsor Players Prestige Club Blockbuster Reward Delta Privilege Organizational Structure Type B-IV Type B-IV Type A-I Type A-I Type A-I Type B-(I, II, or III) Type B-(I, II, or III) TypeC Type A-I Type A-I Type A-I Type A-I Reward Sceme Dynamic/ onemedium/ direct / use timing/ amountbased, frequencybased and portfoliobased Dynamic/ onemedium/ direct / no timing/ amountbased Static/one-medium / direct/ no timing/ amount-based Static/one-medium / indirect/ no timing/ amount-based Static/one-medium / indirect/ no timing/ amount-based Dynamic/onemedium/direct/use timing/portfoliobased Dynamic/onemedium/direct/ use timing / amountbased, frequencybased, and portfoliobased Dynamic/onemedium/direct/use timing/amountbased Static/one-medium /mixed /no timing/ portfolio-based Static/one-medium/ indirect/ no timing/ portfolio-based Industry base Travel Travel Grocery Grocery Dept. store Travel Travel Te same as (9) Static/ one-medium /mixed / use timing/ frequency-based Static/ no-medium /direct / no timing / frequency-based Grocery Entertainment (Casino) Entertainment (Casino) Entertainment (Video rent) Hotel Profit or Cost center Profit center Profit center Cost center Cost center Cost center Semi-cost center Semi-cost center Semi-cost center Cost center Cost center Cost center Cost center 223

236 Appendix A.l: Sample of LRPs in today's marketplace in Canada (Cont.) Esso Extra Best Western Rewards Carlson Hotels Worldwide Goldpointsplus HBC Rewards Hilton Honours CAA Dollar icoke Futura Rewards Program Marriott Rewards M & M Max OLGC Winner's Circle Pennington PS Club Advantage Petro-Points Te Soe Company web points Type A-(I, II, or III)* Type B-(I, II, or in) Type B-(II or III)* Type B-(I, II, or III)* Type B-(I, II, or III)* Type B-(I, II, or III)* Type B-(I, II or III)* Type B-IV Type B-(I, II, or III)* TypeA-(I,IL or III)* Type A-I Type A-I Type B-(I, II or III)* Type A-(I, II, or III)* Static/one-medium /direct / no timing /amount-based Static/ one-medium (wit option) /mixed / no timing /amountbased Static/one-medium /mixed / no timing /amount-based Dynamic (for rewards oter tan gift card)/ onemedium / mixed /no timing / amountbased Dynamic/ twomedium (wit option)/ mixed / no timing/ portfolio Static/ one-medium/ mixed / no timing / portfolio-based Dynamic/ onemedium/ mixed / no timing / portfoliobased Dynamic/nomedium/ indirect / no timing /amountbased Dynamic/ onemedium (wit option) / mixed/ no timing / portfoliobased Dynamic/ no medium / mixed / no timing / amountbased Static/ one-medium/ mixed/ no timing / amount-based Static/ no-medium/ direct/ no timing/ amount-based Static/ one-medium/ mixed/ no timing / amount-based Static/one-medium/ indirect/ no timing/ amount-based Gas Hotel Hotel Dept. store Hotel Travel Food/beverage LRP service Hotel Grocery Entertainment (Casino) Dept. store Gas Specialty store Cost center Semi-cost center Semi-cost center Semi-cost center Semi-cost center Cost center Cost center Profit center Semi-cost center Cost center Cost center Cost center Cost center Cost center 224

237 Appendix A.l: Sample of LRPs in today's marketplace in Canada (Cont.) RBC Rewards Reservation- Rewards Sears Club Scotia Bank SCENE Starwood Preferred Guest ZipRewards American Express Membersip Rewards Asia Miles Harra's Total Rewards Best Buy's Reward Zone ETR Rewards Club S obeys Type B-(I, II, or III) Type B-IV Type A-I TypeC Type B-(I, II, or III)* Type A-I Type B-( I or II)* Type B-IV Type B-(I, II, or III)* Type A-(I, II or III)* Type A-I Type A-(I, II or III)* Static/ two-medium/ mixed/ no timing / amount-based Static/ no-medium/ indirect/ no timing/ amount-based Static/ one-medium/ direct/ no timing/ amount-based Static/ one-medium/ mixed/ no timing/ amount-based Dynamic/onemedium/ mixed/ no timing/ portfoliobased Static/one-medium/ direct/ no timing/ portfolio-based Dynamic/ onemedium/ mixed / no timing/ portfoliobased Dynamic/ one medium/ mixed / no timing / portfoliobased Dynamic/ onemedium/ direct/ use timing/ portfoliobased Static/ one-medium/ direct/ no timing/ amount-based Static/no-medium/ direct/ use timing/ amount-based Static/one-medium/ direct/ use timing/ amount-based Finance LRP service Dept. store Finance & Entertainment Hotel Entertainment (video rent) Finance LRP service Entertainment (Casino) Speciality store Transportation Grocery Semi-cost center Profit center Cost center Semi-cost center Semi-cost center Cost center Semi-cost center Profit center Cost center Cost center Cost center Cost center Note: * represents te case tat available information about te LRP is not enoug to tell wic structure model (I, II, III) tat te program belongs to. 225

238 Among tese 39 LRPs: Industry Travel Hotel Finance Grocery Dept. store Entertainment Specialty store Gas Food/beverage LRP service Transportation LRPs 1,2,6,7,19 13,15,16,18,22,32 28,31,34 3,4, 8,23 5,17,25,30,39 9, 10, 11,12,24,31,33,36 27,37 14, ,29,35 38 Total Type of LRPs Type A TypeB TypeC LRPs A-I: 3,4, 5, 9, 10,11, 12, 13, 24, 25, 30, 33, 38 A-I, II or III: 14, 23, 37, 39 B-I, II or III: 6,7, 15,16, 17, 18, 19, 20,22, 26, 27, 28 32, 34, 36 B-IV: 1,2,21,29,35 8,31 Total

239 Appendix B 227

240 Appendix B.l: Computational outputs for examining BP model solvability and determining sample size and sample replications (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No Problem No MeanG (%) CIU (%) CPU (mins) TC No Problem No MeanG (%) CIU (%) CPU (mins)

241 Appendix B.l: Computational outputs for examining BP model solvability and determining sample size and sample replications (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. 9 Problem No MeanG (%) CIU (%) CPU (mins) TC No. 9 Problem No MeanG (%) CIU (%) CPU (mins) Appendix B.2: Computational outputs for examining EPl model solvability and determining sample size and sample replications (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. Problem No. MeanG (%) CIU (%) CPU (mins) TC No. Problem No. MeanG (%) CIU (%) CPU (mins)

242 Appendix B.2: Computational outputs for examining EPl model solvability and determining sample size and sample replications (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No Problem No MeanG (%) cru (%) CPU (mins) TC No Problem No MeanG (%) CIU (%) CPU (mins)

243 Appendix B.3: Computational outputs for examining EP2 model solvability and determining sample size and sample replications (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No Problem No MeanG (%) CIU (%) CPU (mins) TC No Problem No MeanG (%) CIU (%) CPU (mins)

244 Appendix B.3: Computational outputs for examining EP2 model solvability and determining sample size and sample replications (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. 8 9 Problem No MeanG (%) CIU (%) CPU (mins) TC No. 8 9 Problem No MeanG (%) CIU CPU (mins)

245 Appendix C 233

246 Appendix C.l: Summary table of bonus points to offer wit different levels of demand variability under EP1 Setting (a) wit Uniformly Distributed demand (b) wit Normally Distributed Demand TC No. DV TQA TBA TBA (%) TQA V ; TC No. DV TQA TBA TBA (%) TQA V ; % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

247 Appendix C. 1: Summary Table of Bonus Points to Offer wit Different Levels of Demand Variability under EP1 Setting (Cont.) (a) wit Uniformly Distributed demand (b) wit Normally Distributed Demand TC No. DV TQA TBA TBA (%) TQA V ; TC No. DV TQA TBA TBA (%) TQA V ; % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

248 Appendix C.2: Summary table of options to purcase and to exercise wit different demand variability under EP2 Setting (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. 1 DV TQR TOR TOE TOR (%) TQR V ' 5.28% 10.29% 15.04% 19.57% 23.88% 5.02% TOE (%) TOR v ; 18.93% 18.91% 18.92% 18.91% 18.92% 18.85% TC No. 1 DV TQR TOR TOE TOR (%) TQR V ; 2.19% 10.74% 20.92% 26.85% 25.60% 1.92% TOE (%) TOR v ; 13.29% 13.27% 13.27% 12.42% 10.85% 14.56% % 18.85% % 14.58% % 18.92% 22.61% 4.86% 9.47% 13.70% 17.77% 21.10% 5.19% 10.25% 14.87% 18.86% 21.67% 18.98% 19.27% 19.65% 19.44% 19.49% 19.67% 20.07% 21.88% 18.97% 19.26% 19.68% 20.16% 20.45% % 24.45% 25.01% 1.78% 8.75% 15.62% 19.95% 20.83% 1.95% 9.70% 18.24% 22.63% 22.58% 14.87% 15.32% 14.96% 16.09% 16.29% 17.11% 17.18% 17.47% 14.29% 14.63% 15.20% 15.20% 15.30%

249 Appendix C.2: Summary table of options to purcase and to exercise wit different demand variability under EP2 setting (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. DV 0.05 TQR TOR TOE 5.38 TOR (%) TQR V ; 5.04% TOE (%) TOR v ; 19.51% TC No. DV 0.1 TQR TOR 9.90 TOE 1.51 TOR (%) TQR V ' 1.85% TOE (%) TOR v ' 15.27% % 19.74% % 15.65% % 20.24% % 16.25% % 22.05% % 17.72% % 25.84% % 20.46% % 18.95% % 15.64% % 19.16% % 15.69% % 19.43% % 15.81% % 19.97% % 16.03% % 20.55% % 16.19% % 19.27% % 15.75% % 19.71% % 15.86% % 20.10% % 16.28% % 20.83% % 16.83% % 21.08% % 16.97% % 19.42% % 15.79% % 19.73% % 15.90% % 20.15% % 16.25% % 20.83% % 16.68% % 21.17% % 17.11%

250 Appendix C.2: Summary table of options to purcase and to exercise wit different demand variability under EP2 setting (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand TC No. DV 0.05 TQR TOR TOE TOR (%) TQR V ; 4.98% TOE (%) TOR v ; 19.45% TC No. DV 0.1 TQR TOR TOE 5.47 TOR (%) TQR V ; 1.72% TOE (%) TOR v ' 15.62% % 19.85% % 15.89% % 20.47% % 16.33% % 21.67% % 17.54% % 23.33% % 18.30% % 19.36% % 15.42% % 19.79% % 15.68% % 20.41% % 16.10% % 21.60% % 17.12% % 22.85% % 17.98%

251 Appendix C.3: LRP profitability comparisons across BP, EPl, and EP2 (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case 1:N KIN Al Case 1: NK1NA GO #»BP «HB EP A CPU ««#»» BP HB-»EPI *~CPII Case2:NR3NA5 Case2:NR3NA W&&&&** 01 "** BP _{ _[p ep - 1 -EPl 1.5 EPII 1.8 Case 3: NR10NA10 Case3:NR10NA **$P ,,^# g*»* a *** G E OOS S #-BP -B-LP -i-lhi BP - -LHI LPI Case4:NR20NA15 Case4:NR2DNA SOO #-0P -H-EPI -A-CPII -BP - -EPl EPII

252 Appendix C.3: LRP profitability comparisons across BP, EPl, and EP2 (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case5:NR25NA30 Case5:NR25NA SOO BP EPl 0.2 i-epii »BP 1 EPl EPII Case6:NR40IMA40 Case6:NR40NA S BP - -EPl «*»EPII Case7:NR60NA55 Case7:NR60NA "^^T fr"si8wrabftoimm»rit3^^^ 0.0S #~BP -«-EPI -* EPII BP HH-EPI 1.5 -EPII 1.8 Case8:NR65NA70 Case8:NR65NA E *#~BP -B-EP1 -*-EPII BP -H-EPI 1.5 -EPII

253 Appendix C.3: LRP profitability comparisons across BP, EPl, and EP2 (Cont.) (a) wit Uniformly Distributed Demand (b) wit Normally Distributed Demand Case 9: NR8QNA80 Case9:NR80NA80 900OO BP D»l 0.2 _ CPii 0.25 o.i 0.5 -BP - 1 EPl 1.S f»epll 1.8 CaselO:NR100NA Case 10: NR100NA O*"- " ^ BP EPl EPII BP -f}-epi»epii

254 Appendix C.4: Liability ratio comparisons across BP, EPl, and EP2 (a) wit Uniformly Distributed Demand 1.GO (IBP [PI MEPII Casel: NR1NA lilt ft (b) wit Normally Distributed Demand S > HBP EPl S-f CPII Casel: NR1NA S BBP acpi l.nq g 1.20 S BBP HCPII ME 3! MEPII Case2: NR3NA5 IS Ml 311 lit! Ca$e3: NR10NA10 I I A-ali-A-ttHil- Ml II II a HBP U EPl 61 CPII i.eo l.go o.eo 0.10 HP CPI it CPII Case2: NR3NA rj. JHlltl ISM ihm ] ill ifliiilat n ill 0.S IS Case3; NR10NA >>M. RAj :] I IIIIIII , Case 4: NR20NA15 Case4: NR20NA g s 1.00 I HBP II EPl ig* CPII G S si a BP CPI HI CPII S

255 Appendix C.4: Liability ratio comparisons across BP, EPl, and EP2 (Cont.) o (a) wit Uniformly Distributed Demand HBP BEPI HEPII Case5: NR25NA30 iiinlil 81 1 IP! ill fill Ii WW fir n (b) wit Normally Distributed Demand _ 1.60,2 m > HBP HCPI mcpii Case5: NR25NA Case6: NR40NA40 Case6: NR40NA U.DU 0.40 BP EPl HLPII / S U 1.01S jmyl i.a Case7: NR60NA55 Case7: NR60NA a te 1.20 ** > 1.00 Z 0.80 a IBP ICPI HEPII lililiiiii * * S HUP IEPT HCPII E

256 Appendix C.4: Liability ratio comparisons across BP, EPl, and EP2 (Cont.) (a) wit Uniformly Distributed Demand E (IBP HCPI tacpii Case8: NR65NA70 0.0S (b) wit Normally Distributed Demand BP nr.pi in CPII Case8: NRG5NA70 i _ H _ _Jj-_ l IE it rtfk CaseS: NR80NA80 Case9: NR80NA U4U - => ffl BP H EPl HEPII C.l E HBP C 3 I HE'll , ^ S81 1.G13 0.G S HBP EPl a CPII CaselO: NR100NA95 ipfipiplpi i GO HBP BCPI HCPII CaselO: NR100NA95 irppinfiil *

257 Appendix C.5: Budget usage comparisons across BP, EPl, and EP2 (a) wit Uniformly Distributed Demand Casel: NR1NA1 (b) wit Normally Distributed Demand Casel: NR1NA Zwzl BP D EPl A~EP1I Case2: NR3NA5 Case2: NR3NA S «i, BP -S"»EPI *B»EP1I 0.1 OS IS l.a»bp '' e-'cpi ««*»EPII Case3: NR10NA10 Case 3: NR10NA10 s 9 n «3 a «>» 8b bb GO «#«BP -S-EPI -*-EPII s aa 10 V ff at <)(> 8b bb GO BP -B-EPI Case4: NR20NA15 Case4: NR20NA15 s M ID => o m 'HI S 70 bb SXSSX^SBSB g P IIIIQMIII [P 0.25

258 Appendix C.5: Budget usage comparisons across BP, EPl, and EP2 (Cont.) (a) wit Uniformly Distributed Demand Case5: IMR25NA30 (b) wit Normally Distributed Demand CaseS: NR25NA30 ST «AD 3 * a ><> as so « Z^f^ pjs^^il S 1 1.S 1.8 BP -S-CPI ~*~EPII CaseG: NR40NA40 Case6: NR4DNA40 3T V ** am ax bb BP -8-EPI -*-EPII CPU Case7: NR60NA55 Case?: NR60NA «-# BP B-EPI «*»«EPII CPI ~*~CPII Case8: NR65NA70 Case8: NR65NA70 w M 3 «o a fab 60 "gp^ asrafsasgp fng«*[p *BB^& EPII o.i 0.5 BP

259 Appendix C.5: Budget usage comparisons across BP, EPl, and EP2 (Cont.) (a) wit Uniformly Distributed Demand Case9:NR80NA80 (b) wit Normally Distributed Demand Case9: NR80NA80 a* 3 V m w bb 60 * 3 3 m <W 8b j^uj^gjg_ draa^kb&bp NNiigmmtPI ^ # ^ CPU # BP -a EPI «*»EPII CaselO: NR100NA95 CaselO: NR100NA O.OS S sj BP -g- EPl *»*r»epii BP -a EPl ««ir»epii

260 Appendix D 248

261 Appendix D.l: Comparison of te impacts of budget tigtness wit a given level of demand variability under BP setting (Confidence intervals for HI, H2, and H3, (U i2 -Uii) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of Demand Variability (Normal, DV-L) Profitability U BG-L Case 4 Case 6 12 Case 8 ii ±1.079 Profitability BG-L ±0.662 Profitability BG-L ± Liability Ratio U BG-L ± Liability Ratio U BG-L ± Liability Ratio ii BG-L ± Ordering Quantity BG-L ± Ordering Quantity BG-L ±0.197 Ordering Quantity BG-L ±0.105 (b) wit Uniformly Distributed Demand and Low Level of Demand Variability (Uniform, DV-L) Profitability Case 4 Case 6 Case 8 U ii BG-L ±5.437 Profitability BG-L ±4.061 Profitability BG-L ±3.060 Liability Ratio BG-L ± Liability Ratio U BG-L ± Liability Ratio U BG-L ± Ordering Quantity _ ii BG-L ± Ordering Quantity U BG-L ±0.206 Ordering Quantity BG-L ±

262 Appendix D.l: Comparison of te impacts of budget tigtness wit a given level of demand variability under BP setting (Confidence intervals for HI, H2, and H3, (U i2 -Uii) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Medium Level of Demand Variability (Normal, ) Profitability Case 4 Case 6 Case 8 k k k k k BG-L ±6.601 Profitability BG-L ± Profitability BG-L ± Liability Ratio k BG-L k ± Liability Ratio k BG-L k ± Liability Ratio k BG-L k ± Ordering _ Quantity BG-L k ±2.208 Ordering Quantity k BG-L k ±1.558 Ordering Quantity k BG-L k ±2.131 (d) wit Uniformly Distributed Demand and Medium Level of Demand Variability (Uniform, ) Profitability k BG-L Case 4 Case 6 Case 8 k k k k ± Profitability BG-L ± Profitability BG-L k ± Liability Ratio k BG-L k ± Liability Ratio k BG-L k ± Liability Ratio *'/ BG-L k ± Ordering Quantity k BG-L k ±0.534 Ordering Quantity k BG-L k ±0.770 Ordering _ Quantity k BG-L k ±

263 Appendix D.l: Comparison of te impacts of budget tigtness wit a given level of demand variability under BP setting (Confidence intervals for HI, H2, and H3, (UirUu) ± alf-widt) (Cont.) (e) wit Normally Distributed Demand and Hig Level of Demand Variability (Normal, DV-H) Profitability - Case 4 Case 6 Case 8 U ii U Profitability - Profitability - BG-L ± BG-L ± BG-L ± Liability Ratio ii BG-L ± Liability Ratio ii BG-L ± Liability Ratio BG-L ± Ordering Quantity U BG-L ± Ordering Quantity 'i BG-L ± Ordering Quantity ii BG-L ±1.503 (f) wit Uniformly Distributed Demand and Hig Level of Demand Variability (Uniform, DV-H) Profitability - Case 4 Case 6 Case 8 12 ii ii Profitability - Profitability - BG-L ± BG-L ± BG-L ± Liability Ratio ii BG-L ± Liability Ratio U BG-L ± Liability Ratio BG-L ± Ordering Quantity BG-L ± Ordering Quantity U BG-L ±1.060 Ordering Quantity U BG-L ±

264 Appendix D.2: Comparison of te impacts of demand variability wit a given level of budget tigtness under BP setting (Confidence intervals for H4, H5, and H6, (Ui2-Uu) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of Budget Tigtness (Normal, BG-L) Profitability DV-L Case 4 Case 6 Case ± DV-H ± ± Profitability DV-L ± DV-H ± ± Profitability Ji DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H 0.1 ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±3.137 Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ±1.512 DV-H ± ±

265 Appendix D.2: Comparison of te impacts of demand variability wit a given level of budget tigtness under BP setting (Confidence intervals for H4, H5, and H6, (Ui2-Uu) ± alf-widt) (Cont.) (b) wit Uniformly Distributed Demand and Low Level of Budget Tigtness (Uniform, BG-L) Case 4 Case 6 Case 8 Profitability DV-L ± DV-H ± ± Profitability DV-L DV-H ± ± ,914 ±21,008* Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio Jl DV-L ± DV-H ± ± Liability Ratio Ji DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±

266 Appendix D.2: Comparison of te impacts of demand variability wit a given level of budget tigtness under BP setting (Confidence intervals for H4, H5, and H6, (Ui2-U u ) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Hig Level of Budget Tigtness (Normal, ) Case 4 Case 6 Case 8 Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L 2.13 ± DV-H ± ± Ordering Quantity DV-L DV-H 1M± ± 1.915* " ±

267 Appendix D.2: Comparison of te impacts of demand variability wit a given level of budget tigtness under BP setting (Confidence intervals for H4, H5, and H6, {Ua-Ua) ± alf-widt) (Cont.) (d) wit Uniformly Distributed Demand and Hig Level of Budget Tigtness (Uniform, ) Profitability DV-L Case ±9.715 DV-H ± ±9.715 Profitability DV-L Case ± DV-H ± ± Profitability DV-L Case ± DV-H ± ± Liability Ratio Jl DV-H DV-L ± ± ± Liability Ratio Ji DV-L 0.07 ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L 3.81 ± DV-H ± ±0.455* Ordering Quantity Ji DV-L ± DV-H ± ± Ordering Quantity Ji DV-L ± 0.898* DV-H ± 0.898* ±«"; 0.898* 255

268 Appendix D.3: Comparison of te impacts of budget tigtness wit a given level of demand variability under EPl setting (Confidence intervals for HI, H2, and H3, (Ua-U a ) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of Demand Variability (Normal, DV-L) Profitability Case 4 Case 6 Case 8»'/ BG-L ±1.080 Profitability BG-L ±0.667 Profitability BG-L ±1.237 Liability Ratio U BG-L ± Liability Ratio U BG-L ± Liability Ratio U BG-L ± Ordering _ Quantity U BG-L ±0.667 Ordering Quantity U BG-L ±0.197 Ordering Quantity ii BG-L ±0.105 (b) wit Uniformly Distributed Demand and Low Level of Demand Variability (Uniform, DV-L) Profitability U BG-L Case ±5.437 Profitability BG-L Case ±4.061 Profitability U BG-L Case ± Liability Ratio BG-L ± Liability Ratio ii BG-L ± Liability Ratio U BG-L ± Ordering Quantity BG-L ± Ordering Quantity ii BG-L ±0.206 Ordering Quantity BG-L ±

269 Appendix D.3: Comparison of te impacts of budget tigtness wit a given level of demand variability under EPl setting (Confidence intervals for HI, H2, and H3, (Ui2-U u ) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Medium Level of Demand Variability (Normal, ) Profitability ii BG-L Case 4 Case 6 Case 8 ii '/ ±6.601 Profitability BG-L ±6.105 Profitability BG-L ± Liability Ratio ii BG-L ± Liability Ratio ii BG-L ± Liability Ratio ii BG-L ± Ordering Quantity ii BG-L ±2.208 Ordering Quantity ii BG-L ±1.547 Ordering Quantity ii BG-L ±2.131 (d) wit Uniformly Distributed Demand and Medium Level of Demand Variability (Uniform, ) Profitability BG-L Case 4 Case 6 Case 8 ii ± Profitability BG-L ± Profitability BG-L ± Liability Ratio BG-L ± Liability Ratio U BG-L ± Liability Ratio ii BG-L ± Ordering _ Quantity ii BG-L ±0.534 Ordering Quantity ii BG-L ±0.770 Ordering Quantity ii BG-L ±

270 Appendix D.3: Comparison of te impacts of budget tigtness wit a given level of demand variability under EPl setting (Confidence intervals for HI, H2, and H3, (Ui 2 -U a ) ± alf-widt) (Cont.) (e) wit Normally Distributed Demand and Hig Level of Demand Variability (Normal, DV-H) Profitability - Case 4 Case 6 Case 8 ii ij U Profitability - Profitability - BG-L ± BG-L ± BG-L ± Liability Ratio ii BG-L ± Liability Ratio U BG-L ± Liability Ratio U BG-L ± Ordering Quantity U BG-L ±3.072 Ordering Quantity ii BG-L ±2.041 Ordering Quantity ii BG-L ±1.503 (f) wit Uniformly Distributed Demand and Hig Level of Demand Variability (Uniform, DV-H) Profitability - Case 4 Case 6 Case 8»'/ ii Profitability - Profitability - BG-L ± BG-L ± BG-L ± Liability Ratio U BG-L ± Liability Ratio BG-L ± Liability Ratio ii BG-L ± Ordering Quantity BG-L ± Ordering Quantity BG-L ±1.060 Ordering Quantity *'/ BG-L ±

271 Appendix D.4: Comparison of te impacts of demand variability wit a given level of budget tigtness under EPl setting (Confidence intervals for H4, H5, and H6, (Ui 2 -Uu) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of Budget Tigtness (Normal, BG-L) Profitability DV-L Case ± DV-H ± ± Profitability Ji DV-L Case 6 Case ± DV-H ± ± Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±3.137 Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ±1.512 DV-H ± ±

272 Appendix D.4: Comparison of te impacts of demand variability wit a given level of budget tigtness under EPl setting (Confidence intervals for H4, H5, and H6, (Ui2-Uu) ± alf-widt) (Cont.) (b) wit Uniformly Distributed Demand and Low Level of Budget Tigtness (Uniform, BG-L) Profitability DV-L Case 4 Case 6 Case ± DV-H ± ± Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±0.894 Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±

273 Appendix D.4: Comparison of te impacts of demand variability wit a given level of budget tigtness under EPl setting (Confidence intervals for H4, H5, and H6, (Uir-Uu) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Hig Level of Budget Tigtness (Normal, ) Profitability DV-L Case 4 Case 6 Case ± DV-H ± ± Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H 0.01 ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L 1.06 ± DV-H ± ± Ordering Quantity DV-L 1M± DV-H ± " ±

274 Appendix D.4: Comparison of te impacts of demand variability wit a given level of budget tigtness under EPl setting (Confidence intervals for H4, H5, and H6, (Ui2-Uu) ± alf-widt) (Cont.) (d) wit Uniformly Distributed Demand and Hig Level of Budget Tigtness (Uniform, ) Case 4 Case 6 Case 8 Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H 0.08 ± ± Liability Ratio DV-L 0.03 ± DV-H ± ± Ordering Quantity Ji DV-L DV-H ± ± ''' ±0.455* Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L - DV-H 0.037,,4*508* 0.898* 0,898* xx ± 0.898* ' 262

275 Appendix D.5: Comparison of te impacts of budget tigtness wit a given level of demand variability under EP2 setting (Confidence intervals for HI, H2, and H3, (U i2 -Uu) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of Demand Variability (Normal, DV-L) Profitability U BG-L Case 4 Case 6 Case 8 ii ± Profitability BG-L ±0.918 Profitability BG-L ±1.206 Liability Ratio U BG-L ± Liability Ratio U BG-L ± Liability Ratio ii BG-L ± Ordering _ Quantity U BG-L ' ±0.123 Ordering _ Quantity ii BG-L ±1.388 Ordering Quantity ii BG-L ±1.451 (b) wit Uniformly Distributed Demand and Low Level of Demand Variability (Uniform, DV-L) Profitability U BG-L Case 4 Case 6 Case 8 '; ±5.174 Profitability BG-L ±4.061 Profitability BG-L ±4.197 Liability Ratio ii BG-L ± Liability Ratio '/ BG-L ± Liability Ratio ii BG-L ± Ordering _ Quantity ii BG-L ±0.183 Ordering Quantity ii BG-L ±0.787 Ordering Quantity ii BG-L ±

276 Appendix D.5: Comparison of te impacts of budget tigtness wit a given level of demand variability under EP2 setting (Confidence intervals for HI, H2, and H3, (Ua-U a ) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Medium Level of Demand Variability (Normal, ) Profitability Case 4 Case 6 Case 8 U U BG-L ±9.615 Profitability BG-L ±9.286 Profitability BG-L ± Liability Ratio U BG-L ± Liability Ratio BG-L ii ± Liability Ratio BG-L ± Ordering Quantity 'i BG-L ±0.788 Ordering Quantity BG-L ±1.189 Ordering Quantity ii BG-L ±1.097 (d) wit Uniformly Distributed Demand and Medium Level of Demand Variability (Uniform, ) Profitability Case 4 Case 6 Case 8 U ii U BG-L ± Profitability BG-L ± Profitability BG-L ± Liability Ratio ii BG-L ± Liability Ratio ii BG-L ± Liability Ratio '/ BG-L ± Ordering Quantity ii BG-L ±0.816 Ordering Quantity ii BG-L ±0.985 Ordering Quantity ii BG-L ±

277 Appendix D.5: Comparison of te impacts of budget tigtness wit a given level of demand variability under EP2 setting (Confidence intervals for HI, H2, and H3, (U i2 -Uii) ± alf-widt) (Cont.) (e) wit Normally Distributed Demand and Hig Level of Demand Variability (Normal, DV-H) Profitability - Case 4 Case 6 12 Case 8 U Profitability - Profitability - BG-L ± BG-L ± BG-L ± Liability Ratio BG-L ± Liability Ratio U BG-L ± Liability Ratio ii BG-L ± Ordering Quantity ii BG-L ± Ordering Quantity U BG-L ii ±2.195 Ordering Quantity '; BG-L ±1.388 (f) wit Uniformly Distributed Demand and Hig Level of Demand Variability (Uniform, DV-H) Profitability - Case 4 Case 6 Case 8 '/ U U Profitability - Profitability - BG-L ± BG-L ± BG-L ± Liability Ratio BG-L ± Liability Ratio BG-L ± Liability Ratio BG-L ± Ordering Quantity U BG-L ±1.187 Ordering Quantity BG-L ± Ordering Quantity BG-L ±

278 Appendix D.6: Comparison of te impacts of demand variability wit a given level of budget tigtness under EP2 setting (Confidence intervals for H4, H5, and H6, (U i2 -U u ) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of Budget Tigtness (Normal, BG-L) Profitability DV-L Cas >e ± ?* DV-H ± ± Profitability DV-L Case 6 Case ± DV-H ± ± Profitability Ji DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity Ji DV-L DV-H ± ± ±

279 Appendix D.6: Comparison of te impacts of demand variability wit a given level of budget tigtness under EP2 setting (Confidence intervals for H4, H5, and H6, (Ui2-Uu) ± alf-widt) (Cont.) (b) wit Uniformly Distributed Demand and Low Level of Budget Tigtness (Uniform, BG-L) Profitability Case 4 Case 6 Case 8 ii DV-H DV-H \, ±_ ± ± ± ± DV-L x * Profitability DV-L Profitability DV-L ± ± DV-H ± ± Liability Ratio DV-L 0.02 ± DV-H ± ± Liability Ratio Jl DV-H DV-L ± ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ±1.156 DV-H ± ±1.156 Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±

280 Appendix D.6: Comparison of te impacts of demand variability wit a given level of budget tigtness under EP2 setting (Confidence intervals for H4, H5, and H6, (U i2 -Uu) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Hig Level of Budget Tigtness (Normal, ) Profitability DV-L Case 4 Case 6 Case ± DV-H ± ± Profitability DV-L ± DV-H ± ± Profitability DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H 0.08 ± ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±

281 Appendix D.6: Comparison of te impacts of demand variability wit a given level of budget tigtness under EP2 setting (Confidence intervals for H4, H5, and H6, (Ui2-Uu) ± alf-widt) (Cont.) (d) wit Uniformly Distributed Demand and Hig Level of Budget Tigtness (Uniform, ) Profitability DV-L Case 4 Case 6 Case ± ± Profitability DV-L Profitability DV-L DV-H ± ± DV-H ± ± DV-H ± i '' U2U* ± Liability Ratio DV-L ± DV-H ± ± Liability Ratio DV-L 0.06 ± DV-H ± ± Liability Ratio DV-L DV-H ± ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ± Ordering Quantity DV-L ± DV-H ± ±

282 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (U i2 -Uu) ± alf-widt) (a) wit Normally Distributed Demand and Low Level of budget tigtness and low level of Demand Variability (Normal, BG-L, DV-L) ( Case 4 Case 6 Case 8 Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ±1.512 EP ± ±1.512 Profitability BP EPl EPl ±2.25 EP ± ±2.25 Liability Ratio Ji BP EPl EPl ± EP ± * ± Liability Ratio Ji BP EPl EPl ± EP ± * ± Liability Ratio Jl BP EPl EPl ± EP at, * ± Ordering Quantity BP EPl EPl ±0.242* EP ± ± Ordering Quantity BP EPl EPl EP ±0.283* ± ± Ordenng Quantity BP EPl EPl EP ±0.159* ± ±

283 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (b) wit Uniformly Distributed Demand and Low Level of budget tigtness and low level of Demand Variability (Uniform, BG-L, DV-L) Case 4 Case 6 Case 8 Profitability BP EPl EPl ±9.577 EP ± ±9.577 Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ±4.288 EP ± ±4.288 Liability Ratio Jl BP EPl EPl ± EP ± ± Liability Ratio Jl BP EPl EPl ± EP ± ± Liability Ratio Jl BP EPl EPl ± EP ± ± Ordering Quantity BP EPl EPl EP ' ' ±0.261* ± ±0.261 Ordering Quantity BP EPl EPl ± 0.4?9* EP ± ± Ordering Quantity BP EPl EPl, OJ0O ±1.015* EP ± ±

284 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (c) wit Normally Distributed Demand and Low Level of budget tigtness and Medium level of Demand Variability (Normal, BG-L, ) Profitability BP EP1 Case 4 Case 6 Case 8 EP ± EP ± ± Profitability BP EP1 EP ± EP ± ± Profitability BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Ordering Quantity BP EP1 EP1 0.00ft ± 2.624* EP ± ± Ordering Quantity BP EP1 EP ± 2.604* EP ± ± Ordering Quantity BP EP1 EP ^" A 1.474* EP ± ±

285 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Ua) ± alf-widt) (Cont.) (d) wit Uniformly Distributed Demand and Low Level of budget tigtness and Medium level of Demand Variability (Uniform, BG-L, ) Profitability BP EP1 Case 4 Case 6 Case 8 EP1 EP1 ii Profitability BP ± ± Profitability BP EP ± ± EP1 EP ± ± EP1 EP ± ]2 EP ± ± Liability Ratio ii BP EP1 EP ± EP ± ± Liability Ratio ii BP EP1 EP ± EP ± ± Liability Ratio ii BP EP1 EP ± EP ± ± Ordering Quantity Ji BP EP1 EP * 0.809* EP ± ± Ordering Quantity ii BP EP1 EP ± 1.475* EP ± ± Ordering Quantity ii BP EP1 EP * 2.735*.12 EP ± ±

286 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (e) wit Normally Distributed Demand and Low Level of budget tigtness and Hig level of Demand Variability (Normal, BG-L, DV-H) Profitability BP EPl ( 3ase4 EPl ± EP ± ± Profitability BP EPl Case 6 EPl ± EP ± ± Profitability BP EPl Case 8 EPl ± EP ± ± Liability Ratio Ji BP EPl EPl ± EP ± ± Liability Ratio Ji BP EPl EPl ± EP ± ± Liability Ratio BP EPl EPl ± EP ± ± Ordering Quantity BP EPl EPl EP ±3.98*. ± ±3.98 Ordering Quantity BP EPl EPl ±2.712* EP ± ±2.712 Ordering Quantity BP EPl EPl EP < ± ± 1,993* ±

287 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (f) wit Uniformly Distributed Demand and Low Level of Budget Tigtness and Hig Level of Demand Variability (Uniform, BG-L, DV-H) Case 4 Case 6 ( Case 8 Profitability BP EPl EPl ± EP ± ± Profitability ii BP EPl EPl ± EP ± ± Profitability ii BP EPl EPl ± EP ± ± Liability Ratio ii BP EPl EPl ± EP ± ± Liability Ratio ii BP EPl EPl ± EP ± ± Ordering Quantity Ji BP EPl EPl EP « ± ±1.488% ± Ordering Quantity ii BP EPl EPl ± 1.744* EP ± ±1.744 Ordering Quantity ii BP EPl EPl ±3.286* EP ± ±

288 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (g) wit Normally Distributed Demand and Hig Level of Budget Tigtness and Low Level of Demand Variability (Normal,, DV-L) Profitability BP EP1 Case 4 Case 6 Case 8 EP1 EP ± Profitability BP ± Profitability BP EP ± ± EP1 EP ± ± EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Ordering Quantity - BP EP1 $ 32 EP1 EP v %256 ±0.863* ±0.863* MU56 ±0.863* Ordering Quantity BP EP1 EP ± 1308* EP ± ± Ordering Quantity BP EP1 EP V '' ± 1.314* EP ± ±

289 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ua-Uu) ± alf-widt) (Cont.) () wit Uniformly Distributed Demand and Hig Level of Budget Tigtness and Low Level of Demand Variability (Uniform,, DV-L) Case 4 Case 6 Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ±6.211 EP ± ±6.211 Profitability BP EPl EPl ± EP ± ± Liability Ratio Ji BP EPl EPl ± EP ± ± Liability Ratio Ji BP EPl EPl ± EP ± ± Liability Ratio BP EPl EPl ± EP ± ± Ordering Quantity BP EPl EPl EP ± 0.207* ± ± Ordering Quantity BP EPl EPl ± 0.170* EP ± ±0.170 Ordering Quantity BP EPl EPl ±1.475* EP ± ±

290 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Uir-Uu) ± alf-widt) (Cont.) (i) wit Normally Distributed Demand and Hig Level of Budget Tigtness and Medium Level of Demand Variability (Normal,, ) Case 4 ( Dase 6 < 3ase 8 Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ± EP ± ± Liability Ratio BP EPl EPl ± EP ± ± Liability Ratio Ji BP EPl EPl ± EP ± ± Ordering Quantity BP EPl! EPl EP , s ±0,533* " ± ±0.533 Ordering Quantity BP EPl EPl ±1412* EP ± ±1.112 Ordering Quantity BP EPl EPl EP ±2,170*. ± ±

291 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (U i2 -Uu) ± alf-widt) (Cont.) (j) wit Uniformly Distributed Demand and Hig Level of Budget Tigtness and Medium Level of Demand Variability (Uniform,, ) Profitability BP EP1 Case 4 Case 6 Case 8 EP ± EP ± ± Profitability BP EP1 EP ± EP ± ± Profitability BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Ordering Quantity - Ji BP EP1 EP1 " MOO ±0.449* EP ± ± Ordering Quantity - BP EP1 EP ±0.541* EP ± ± Ordering Quantity - BP EP1.12 EP j ± 1.098* EP ± ±

292 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (k) wit Normally Distributed Demand and Hig Level of Budget Tigtness and Hig Level of Demand Variability (Normal,, DV-H) Profitability BP EP1 Case 4 Case 6 Case 8 EP1 EP ± Profitability BP ± Profitability BP EP ± ± EP1 EP ± ± EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Liability Ratio BP EP1 EP ± EP ± ± Ordering Quantity - BP EP1 EP1 15 o.ooo ±0.653* EP ± ± Ordering Quantity - BP EP1 EP ± 1.709* EP ± ± Ordering Quantity BP EP1 EP l ±1.874* " EP ± ±

293 Appendix D.7: Comparison of te impacts of te model settings wit a given level of budget tigtness and a given level of demand variability (Confidence intervals for H7, H8, and H9, (Ui2-Uu) ± alf-widt) (Cont.) (1) wit Uniformly Distributed Demand and Hig Level of Budget Tigtness and Hig Level of Demand Variability (Uniform,, DV-H) Ca se4 Case 6 Case 8 Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ± EP ± ± Profitability BP EPl EPl ± EP ± ± Liability Ratio BP EPl EPl ± EP ± ± Liability Ratio BP EPl EPl ± EP ± ± Ordering Quantity BP EPl EPl ± 0.60!* EP ± ± Ordering Quantity BP EPl EPl EP ±0.803*< x ± ± Ordering Quantity BP EPl EPl EP ^ * ±1.350*'" ± ±

294 Appendix E 282

295 Appendix E.l: Extended matematical model for Problem BP wit revenue-saring type of contracts Te additional notations used in te extended model are defined as follows: (1) Indices: A, LRP accumulation partners wo coose a wolesale price contract A 2i LRP accumulation partners wo coose a revenue-saring type contract (2) Parameters in objective function: Z),f LPR members' accumulation demand towards LRP accumulation partner A u. D 2i LRP members' accumulation demand towards LRP accumulation partner A 2l. qf t LRP accumulation partner A u 's ordering quantity of points. q 2i LRP accumulation partner A 2t 's ordering quantity of points. w* Wolesale price per unit of points tat LRP ost H carges LRP accumulation partner A u. wf, Price per unit of points tat LPR ost H carges LRP accumulation partner A u wen accumulation demand is more tan LRP partner A u 's ordering quantity. w 2i Revenue saring in terms of te price per unit of points tat LRP ost H carges partner A 2l. w 2i Extra revenue saring in terms of te price per unit of points tat LRP ost H carges LRP partner A wen te actual revenue increase is iger tan tat to wic H as committed. s* Penalty per unit of points tat LRP ost H pays to LRP partner A 2l wen te actual revenue increase is lower tan tat to wic H as committed.

296 Te model is formulated as follows ((ere after Problem BPl): U H (q«;z);,af,al) = max E[x HW tf jz^ + ^ ^ A l max E + E t(< *q +< *\p -qt\) +E (w xmin{^ Al} + < x[al "<] + "** * >* ~K\) subject to: (1) Liability control constraints: / I <L were/ =l 0 +±(qf 1 +[^-^] + ) + ^(min{^,^} + [^-^] + )-S(min {^,^s}) li=l 2i=l j=l (2) LRP redemption partners' capacity limitations on rewards supply: q*zqf,forj = l,...,j (3) LRP ost's budget constraint on purcasing rewards: IK"?;)^1 J=I (4) Non-negative constraints: q* >Q,forj = l,...,j (7.1) (7.2) (7.3) (7.4) (7.5) Te objective function in Problem BPl consists of two parts: te LRP ost's profitability from te redemption business, n H, R) \q^ ',Df ), and te LRP ost's profitability from te accumulation business, ^(/4) (;Z)^,Z)^). Te profitability function 7t HW \a^ \D*\ in Problem BPl is te same as te profitability function in Problem BP. At te accumulation side, te LRP ost's profitability comes from points-related business wit bot LRP accumulation partners A and partners A 2l : 284

297 + t(<xmin{^,al} + w-x[al-^] + )-t(^x[? 2 1-Al] + ) LRP ost H offers a wolesale-price contract to partners A],. Terefore, te meaning of te first two terms in tat profitability function is te same as te meaning of (BP-A). LRP ost//offers a "revenue-saring" type of contract to LRP accumulation partner A 2l, in wic H commits to increase A 2,, s revenue to a minimum level during a specified time period. In return, A 2l will commit to pay back to H a sare of revenue increase. Te tird term, wf t xminjg^, A*,}, refers to te sare of revenue increase tat A 2l pays back to ost H. Te amount of te sare is converted in terms of te points tat ave been accumulated. Te fourt term, w'f t x A*, -q 2 A, refers to te amount of extra sare of te revenue increase wen D*, > qf, It indicates te situation were an actual increase in revenue is iger tan te minimum committed amount. If te actual revenue increase is lower tan te amount tat H committed (i.e.jq^ _All )' men LRP ost H will pay a unit penalty cost ofsf r Te last term in te function, s f, x lit ~ A>1 ] ' refers to te overall penalty cost tat LRP ost H pays to partner A 2l wen Al < I2, 285

298 Appendix E.2: Extended matematical model for Problem EP2 wit multi-layer of contracts Te following are additional notations used in te extended model: (1) Additional indices: k Option layer offered by partner Rj.k= l,2,...,k (2) Additional decision variables: m R k Number of layer k options tat LRP ost //purcases from redemption partner Rj. q R t y R t Number of layer k options tat LRP ost H exercises. Weter LRP ost H cooses layer k option from redemption partner Rj. (3) Additional parameters in objective function and constraints: w R " Option price per unit of points tat redemption partner Rj carges LRP ost H for option k. er k Exercise price per unit of points tat redemption partner R } carges LRP ost H wen H exercises te option k. Te model is formulated as follows (ere after Problem EP3): = max < 7=1 L /t=l IXxmin tf + 5>,\Df -2>;x ;=i 4=1 7=1 1=1 K ahtf+stf k=\ 7=1 + (7.7) subject to: (1) Liability constraints: 286

299 (7.8) were/ =/ 0 + tk+[a'-^l)- k+t^" 1=1 y=l V *=1 J (2) Redemption partners' capacity limitations on rewards supply: (7.9) (3) Host //'s overall budget constraint on purcasing rewards: I J f 7=1 V *=1 / (7.10) (4) Upper bound of option quantity tat ost //is allowed to purcase for eac option layer k: m R ; <M**y% for* = l,...kj = \,...,J (7.11) (5) Logic constraints: 2>*=1, fory=w *=i ^ <m R ", for = l,..x (7.12) (7.13) (6) Non-negativity constraints: (7.14) qf > 0 v y s * = 0 or 1 (7.15) (7.16) In tis model, te meaning of eac term in te objective function is te same as te meaning of eac term in te objective function of Problem EP2. Te meanings of constraints (7.7) - (7.10) and (7.12) are te same as tose inproblem EP2. In constraints (7.10) and (7.12), we define a binary variable y Rt, for eac redemption partner R,, and eac option level k, to 287

300 define a condition in wic altoug eac partner if, offers multiple options, LRP ost H is allowed to coose only one option layer. Terefore, Constraint (7.12) represents te condition and Constraint (7.10) defines te upper bound of option quantity tat LRP ost H is allowed to purcase for eac option k from partner if,-. 288