Omega 39 (2011) Contents lists available at ScienceDirect. Omega. journal homepage:

Transcription

1 Omega 39 (2011) Contents lsts avalable at ScenceDrect Omega journal homepage: Supply chan confguraton for dffuson of new products: An ntegrated optmzaton approach Mehd Amn a,n, Hatao L b a Department of Marketng and Supply Chan Management, Fogelman College of Busness & Economcs, Unversty of Memphs, Memphs, TN , USA b Department of Logstcs and Operatons Management, College of Busness Admnstraton, Unversty of Mssour St. Lous, St. Lous, MO , USA artcle nfo Artcle hstory: Receved 13 July 2009 Accepted 30 July 2010 Avalable onlne 7 August 2010 Keywords: Operatons management Marketng Product lfe cycle Producton plannng and control Mathematcal programmng abstract We develop an ntegrated/hybrd optmzaton model for confgurng new products supply chans whle explctly consderng the mpact of demand dynamcs durng new products dffuson. The hybrd model smultaneously determnes optmal producton/sales plan and supply chan confguraton. The producton and sales plan provdes decsons on the optmal tmng to launch a new product, as well as the producton and sales quantty n each plannng perod. The supply chan confguraton provdes optmal selecton of optons and safety stock level kept at each supply chan functon. Extensve computatonal experments on randomly generated testbed problems ndcate that the hybrd modelng and soluton approach sgnfcantly outperforms non-hybrd alternatve modelng and soluton approaches under varous dffuson and supply chan topologes. We provde nsghts on optmal producton/sales plan and supply chan confguraton for new products durng ther dffuson process. Also, manageral mplcatons relevant to effectveness of the hybrd approach are dscussed. Publshed by Elsever Ltd. 1. Introducton In ths paper, we consder the scenaro where a frm needs to confgure ts supply chan before launchng a new product. To respond to customer demand effcently, the frm s supply chan confguraton encompasses decsons ncludng selecton of supplers; manufacturng and transportaton modes; as well as locatons n supply chan network to place approprate levels of safety stocks. Under a fxed producton and supply capacty n the ntermedate term, the frm mght be overwhelmed by potentally rapd growth of demand for the new product due to marketng actvtes and postve word-of-mouth. Examples nclude Apple s Phone [1] and Nntendo s W [2], where both manufacturers were ht by the rapd growth of demand for these nnovatve products. Often, such an mpact affects not only the manufacturer tself, but also ts vendors and supplers through the supply chan. Example ncludes Apple s PowerMac G4 [3], where Motorola, as the suppler of G4 chps, was not able to catch up wth the rapd growth of demand for the popular computer. Another potental scenaro for the frm s to experence a slow growth n demand for the new product and hence resultng n major fnancal rsks. For example, ntal sales of Sony s Playstaton 2 (PS2) were more than ten tmes that of the orgnal PS s ntroducton fve years earler n Correspondng author. Tel.: E-mal address: mamn@memphs.edu (M. Amn). [4]. However the launch of Playstaton 3 (PS3) was not successful for Sony and hence resulted n $1.8B annual loss n ts game dvson and layoff of 3% of ts workforce [5]. Manufacturers are often able to save nventory cost by not keepng any ntal stock before launchng the new product, but they (and related players n the supply chan) may suffer later when supples of the new product are outpaced by the fast growth of demand. Often, the savng on nventory cost may not compensate the cost due to lost demand. On the other hand, when a frm experences demand below expectaton, the nventory cost of safety stocks located at dfferent ters of supply chan network has a negatve effect on effcency. Thus when launchng a new product, effcency n terms of cost and speed s not the only qualty a successful supply chan can own. As noted by Lee [6], supply chans that fal to adapt to changes n market structure wll not gan sustanable compettve advantage. These have motvated us to model marketng-supply chan nteractons, and n partcular, the nteracton between new product s dffuson and the correspondng supply chan s confguraton. The dynamcs of customer demand durng dffuson of new products are well-captured by the classcal Bass model [7]. Kumar and Swamnathan [8] and Ho et al. [9] have shown that the customer demand pattern durng new product dffuson wll affect the manufacturer s producton plannng decsons durng the new product s lfetme. They extend the classcal Bass model by consderng producton capacty of the frm, so that the demand of a new product may not be completely met due to the producton capacty lmt. Ther model s used to fnd optmal /$ - see front matter Publshed by Elsever Ltd. do: /j.omega

2 314 M. Amn, H. L / Omega 39 (2011) producton and sales plans that maxmze proft durng the new product s lfetme, spannng from one to two years. They fnd that when supply constrant s present, the rapd growth of customer demand durng dffuson may motvate manufacturer to buldup ntal nventory and delay launchng of the new product. We call ther model the new product dffuson (NPD) model n the sequel. The NPD model focuses on the nteractons between manufacturng and marketng/sales decsons wthn a frm by assumng a fxed per-unt product cost, but gnores other functons of the frm s supply chan lke procurement, sourcng, assembly and dstrbuton. Graves and Wllems [10] proposed a model optmzng the supply chan confguraton for a new product, whch we call the supply chan confguraton (SCC) model. In ths model, a frm selects optons for each functon (components, parts, or processes requred) n the supply chan to mnmze the system-wde total supply chan cost. Avalable optons often dffer n lead tme and drect cost added. For nstance, parts and raw materals can be purchased from dfferent supplers. Goods can be shpped va regular ground shppng or next day delvery. The SCC model also allows coordnaton among supply chan players by optmally determnng ther nbound and outbound servce tmes, thus the nventory postonng through the supply chan. New product demand s assumed to be known n the form of mean and standard devaton for the entre plannng horzon (usually 9 months 1 year). Because demand s exogenously gven, the queston of how the demand trajectory durng new product dffuson wll mpact supply chan confguraton s not addressed by the SCC model. The problems addressed by the NPD and SCC models are closely related. Both problems are tactcal n nature. Durng the new product s lfe cycle, the frm s producton and sales plan s only part of the bg pcture. Gven the expanded complexty and scope of modern supply chans, t s rare to have a sngle frm beng nvolved through all stages of sourcng, manufacturng, assembly, transportaton, warehousng and delvery. Thus the frm s facng more mportant and wder scope of decsons on how to confgure ts entre supply chan to allow products as well as the requred parts and components (descrbed by the new product s bll-of-materals or BOM) to be sourced, manufactured and delvered n an effcent and responsve manner. Therefore, there s mert n developng an ntegrated optmzaton model to study the optmal supply chan confguraton decson n concert wth dynamc process of new product dffuson. On one hand, the demand pattern (n terms of mean and varaton) durng new product dffuson has an explct mpact on supply chan confguraton. Specfcally, the mean customer demand serves as the external demand to be satsfed by the supply chan network, and the varaton of demand drectly mpacts the amount of safety stock to carry (or nventory postonng) through the supply chan. On the other hand, the confguraton of supply chan may n turn affect the optmal dffuson pattern of the new product. Ths s due to the fact that n the general supply chan settngs, the per-unt cost of product should be calculated as an accumulatve cost due to selecton of supplers/vendors and manufacturng/transportaton modes through dfferent supply chan stages such as sourcng, assembly, transportaton, etc. Ths mples that the per-unt cost assumed to be constant n manufacturng plannng models durng dffuson, as n [8], can be extended and generalzed to the so-called unt manufacturng cost (UMC) determned through confgurng the correspondng supply chan [10]. In ths study, we present a hybrd model to confgure a new product s supply chan by consderng the dynamcs of dffuson process through the product lfe cycle. Both the demand/supply pattern and unt-product cost are endogenously determned n one model, as opposed to beng exogenous as assumed n the separate models. Our model offers a decson support tool for smultaneously optmzng an nnovator s producton plannng n a mult-perod settng and supply chan confguraton. It also provdes a modelng framework to desgn a supply chan whch s not only cost effcent, but also adaptve to the changng market demand durng new products lfetme. As we wll show, solutons that optmze supply chan performance from ether the NPD aspect or the SCC pont of vew alone would not obtan optmal solutons. Lower supply chan confguraton cost s often acheved by a myopc polcy,.e. sellng as much as possble n each tme perod. Ths leads to less varaton of the realzed demand, thus less safety stock. However, such myopc polcy may perform poorly from the dffuson perspectve due to loss of demand. On the other hand, a buldup polcy,.e. delayng the launch of the new product and buldng some ntal nventory, may generate hgher sales revenue from the dffuson perspectve; but too many buldup perods may lead to ncreased amount of safety stocks and hence ncrease supply chan confguraton cost. Determnng the optmal number of buldup perods wll not be an ntutve task wthout the ad of an ntegratve optmzaton model n whch both supply chan confguraton decsons and new product dffuson outcomes are smultaneously consdered. The remander of the paper s organzed as follows. Secton 2 revews the relevant lterature. Secton 3 presents the ntegrated model. The computatonal expermental study and results are presented n Secton 4. Secton 5 draws conclusons and suggests future research drectons. 2. Related lterature Our work s related to the research lterature of supply chan confguraton desgn n operatons management and new product dffuson n marketng. In ths secton, we provde a revew of the supply chan confguraton and new product dffuson process lterature. Also, we dscuss the mert and objectves of the current study. Supply chan confguraton desgn s tradtonally understood as determnng the optmal manufacturng and dstrbuton network of a frm at the strategc level (see, e.g., [11,12]). Such strategc decson problems focus on desgnng physcal supply chan networks and often have a long-term mpact for the frm from 5 to 10 years. In today s fast changng and compettve busness envronment, however, a new product s lfe cycle (often around or less than 1 year) s much shorter than the scale of strategc plannng horzon. When desgnng supply chan for new products, t s crtcal that the supply chan adapts to the changng envronment n terms of demand, lead tme and cost n the ntermedate run. Thus the supply chan confguraton problem, addressed n ths paper, focuses on tactcal level decsons wth a plannng horzon that matches a new product s lfe cycle. Such tactcal supply chan confguraton problems are able to model all echelons n the supply chan and optmze the system-wde supply chan performance [10], as opposed to optmzng two or three echelons n strategc network desgn problems. The tactcal supply chan confguraton problem on multechelon supply chans has ts root n the so-called nventory postonng problem, whch studes where n the supply chan to keep safety stock. Early models on nventory postonng, e.g., [13 16], optmze safety stock levels for an exstng supply chan by assumng that the opton chosen at each supply chan functon s fxed. Graves and Wllems [10] developed a model that smultaneously determnes the safety stock placement and opton

3 M. Amn, H. L / Omega 39 (2011) selecton decsons. Instead of takng the lead tme and cost at each supply chan functon as gven, they treat the lead tme and drect cost added of a functon as decson varables dependent upon the opton chosen for that functon. Ther model mnmzes the system-wde total supply chan costs ncludng not only the safety stock cost, but also the cost of goods sold (COGS) and the holdng cost of ppelne nventory. A byproduct of ther model s the optmal unt manufacturng cost (UMC) calculated as the cumulatve cost through all stages of the supply chan. One assumpton made by the model n [10] s that the demand pattern of new products durng the plannng horzon s exogenously gven. That s, the mean and standard devaton of demand s assumed to be known. Thus ther model does not explctly account for the mpact of demand dynamcs durng new product dffuson, gven that the length of plannng horzon matches wth the length of a new product s lfe cycle. Adopted from dffuson models for the spread of technologcal nnovatons n the economcs lterature [17], the Bass model [7] has been wdely used n marketng to forecast the demand trajectory of new products. We refer to [18 20] for a comprehensve survey on applcatons of Bass model and ts varants. There have been growng nterests n applyng Bass model to study the underlyng demand n producton plannng n operatons management. Kurawarwala and Mastuo [21] studed an nventory plannng problem wth demand dynamcs characterzed by the Bass model over the new product s entre lfe cycle. Ther model does not ncorporate capacty constrant from the supply sde. Ho et al. [9] proposed an optmzaton model for jontly analyzng the dynamcs of supply and demand. Ther model addresses capacty szng, tme to market and demand fulfllment polcy under an explct supply constrant. They fnd that n a make-to-stock producton envronment, a myopc polcy s always optmal. Kumar and Swamnathan [8] consdered a more general problem settng, n whch a varety of scenaros from complete backloggng to lost sales are captured. They showed that the buldup polcy s robust and very close to optma on average, and under certan scenaros the myopc polcy may devate far from optma. Other work ncorporatng supply constrants n new product dffuson model ncludes that of Swam and Kharnar [22], where new products are avalable n lmted quantty untl a known expraton date. The aforemetoned studes focused on optmzng jont new product dffuson and producton plannng over the new product s lfetme. They mplctly assume that the per-unt cost has ncorporated the costs of materals, sourcng of parts/components, and transportaton nvolved through dfferent stages of the correspondng supply chan. Thus the exstng studes have focused on operatons plannng at one echelon of the supply chan,.e. the producton stage. To our best knowledge, the mpact of demand dynamcs durng new product dffuson on the confguraton of the entre multechelon supply chan has not been addressed. Hence, the key objectve of ths study s to develop an ntegrated optmzaton model by whch mult-echelon supply chan confguraton decsons are made whle explctly consderng demand dynamcs resultng from the new product dffuson process throughout ts lfe cycle. The ntegrated optmzaton model s capable of jontly determnng the optmal producton/sales plan and supply chan confguraton n terms of nventory postonng and opton selected at each supply chan functon. On one hand, the model allows endogenously determnng optmal demand/supply trajectory from the dffuson aspect for the supply chan confguraton sde. On the other hand, the model permts optmal confguraton of supply chan and hence provdng the optmal per-unt product cost. Ths n turn lnks the supply chan opton selecton decsons to producton plannng durng dffuson process. Our work s a frst attempt to model the nteracton between new product dffuson and supply chan confguraton. Related research on desgnng flexble supply chans (cf. [23]) adapted to the dynamcs and varablty of demand can be found elsewhere, e.g. manufacturng network desgn [24] and sourcng [25]. 3. Optmzaton model Consder a frm whch plans to launch a new product. Wth a short product lfe cycle and a long lead tme for capacty expanson, the frm has a fxed producton capacty throughout the lfe cycle of the product. The frm has to: (a) plan ts producton and sales over the product lfe cycle; and (b) confgure the correspondng supply chan n terms of opton selected and nventory level kept at each supply chan functon. We formally descrbe the problem n Secton 3.1. Secton 3.2 presents the mathematcal formulaton. Model complexty, computatonal ssues and soluton approach are dscussed n Secton Problem descrpton We dscretze the dffuson process nto T tme perods and assume that the frm s ready to start producton at t¼0. Once the dffuson process starts, the frm wll decde how much to produce (r t ) and sell (y t ) at tme perod t. For the supply process, we consder the cumulatve producton R t and the nventory of avalable products I t at tme perod t. The total producton up to tme t equals the sum of nventory I t and cumulatve sales Y t. The nstantaneous and cumulatve demand at tme t s denoted by d t and D t, respectvely. The demand process follows that descrbed by a modfed Bass model due to Kumar and Swamnathan [8] d t ¼ pðm D t Þþ q m Y tðm D t Þ where p stands for the coeffcent of nnovaton and q represents the coeffcent of mtaton. The frm faces a fxed market potental m. The demand d t at tme t s expressed as a fracton of the remanng potental adopters m D t consstng of two components: one due to the mpact of mass-meda or coeffcent of nnovaton p, and the other proportonal to the cumulatve sales Y t due to postve word-of-mouth or coeffcent of mtaton q. Notce the dfference of (1) from the classcal Bass model s that, the mpact of word of mouth s nfluenced by the number of people who have successfully bought the product up to tme t (Y t ), not necessarly by all the people who demanded the product up to tme t (D t ), although ths assumpton may not always hold n practce. Eq. (1) s especally useful to capture the unmet demand under explct supply capacty constrant. The unmet demand up to t can be represented as D t Y t, whch can be completely lost or backlogged at a percentage of x. The supply of the new product nvolves a set V¼{1, 2, y, N} of N functons. A functon AV refers to a component, part or a process requred by the new product as descrbed n bll-ofmaterals (BOM). Functon N refers to the end product. We assume that the desgn of the product has completed,.e. the BOM structure s gven as a graph G(V,E) wth E beng a set of arcs descrbng the demand dependences n BOM. An arc (,j)ae has a weght of r j specfyng the unts of requred by 1 unt of j. Wthout loss of generalty, we assume that r j equal 1. We also assume that the external demand occurs only at the end product N. The mean and standard devaton of such external demand s determned endogenously through the new product dffuson process descrbed earler. Any other functon has only nternal demand m from ts mmedate successors, whch can be ð1þ

4 316 M. Amn, H. L / Omega 39 (2011) calculated as follows: m ¼ X r j m j ð,jþ A E Followng Graves and Wllems [16], each supply chan functon quotes a guaranteed outbound servce tme s out by whch wll satsfy ts demand. The tme for functon to receve all the requred nputs from ts predecessors s called the nbound servce tme s n, whch equals the maxmum of outbound servce tmes quoted to functon by ts predecessors. The net replenshment tme t of functon,.e. the tme requred to provde, s gven by the followng: t ¼ s n þp s out ð3þ The amount of safety stock SS held at functon can be calculated as pffffffff SS ¼ ks t ð4þ where s refer to functon s mean and standard devaton of demand, respectvely. The constant k s the z-value assocated wth a pre-specfed servce level. As n Graves and Wllems [10], a set of O optons are avalable for each functon AV. Each opton dffers n lead tme and drect cost added, reflectng the tme-cost tradeoff. Exactly one opton needs to be selected for each functon. The frm s supply chan confguraton problem nvolves determnng whch suppler/ vendor and manufacturng/dstrbuton opton to choose for ts supply chan functons. The new producton dffuson process and supply chan confguraton problem are related n the followng two ways: Lnk 1: Unlke n [8] where the per-unt product cost s assumed to be 1, the per-unt product cost n our model refers to the unt manufacturng cost (UMC, [10]), whch wll be determned by the confguraton (optons chosen) for the new product s supply chan. Lnk 2: The product dffuson process wll shape the demand/ sales pattern for the entre plannng horzon, by determnng the mean, m, and standard devaton, d, of demand for the new product. The ntegrated/hybrd supply chan confguraton problem for dffuson of new products s defned as the problem of choosng a feasble producton plan and confguraton of the correspondng supply chan over the new product s entre lfe cycle, so that the total net proft s maxmzed. A mxed-nteger nonlnear programmng (MINLP) formulaton of the problem s presented next Model formulaton In ths secton, we present three sets of decson varables relevant to supply chan confguraton, new product dffuson process and decson varables lnkng supply chan confguraton and dffuson process. Next, we dscuss an ntegrated optmzaton model depctng supply chan confguraton and product dffuson process. Model parameters ð2þ p w h O P k C k L k sellng prce rato watng cost rate per unt backlogged per unt tme nventory holdng cost rate per tme perod number of canddate optons for functon lead tme of the kth opton for functon drect cost added of the kth opton for functon due date of the outbound servce tme of the end product z-value determned by user-specfed servce level The coeffcent of nnovaton p and mtaton q can be determned usng regresson as descrbed n Bass [7]. The ssue of fndng the best ft parameters for a new product s a non-trval one and requres analyzng the sales hstory data of a smlar product n the same product famly or ndustry, whch goes beyond the scope of ths paper. Our current study consders a generc product havng an average p and q beng 0.03 and 0.4, respectvely, based on the study of Sultan et al. [26] on 213 applcatons of dffuson models n 15 artcles. The length of plannng horzon s chosen to be 30 as n [8]. The sellng prce rato p s a markup over the product s cost. The nventory holdng cost rate h s the percentage of nventory cost over the value of the goods, whch can be provded the accountng department of the frm. Lead tme and drect cost added of each opton are readly avalable from supplers/vendors of the supply chan. The due date of outbound servce tme of the end product s specfed by customers. Decson varables related to the supply chan confguraton x k ¼1, f functon s kth opton s selected, 0 otherwse s n, nbound servce tme of functon,.e. the tme to receve all nputs from ts supplers s out, outbound servce tme of functon,.e. the tme by whch wll satsfy ts demand c Z0, drect cost added for functon c, cumulatve cost for functon p AZ +, lead tme of functon Decson varables related to product dffuson y t Z0, sales of product at tme t Y t Z0, cumulatve sales of product at tme t d t Z0, demand at tme t D t Z0, cumulatve demand of product at tme t L t Z0, cumulatve number of backlogged orders at tme t I t Z0, nventory at tme t r t Z0, producton at tme t R t Z0, cumulatve producton at tme t Decson varables connectng supply chan confguraton and product dffuson m N, mean supply of the end product (functon N) over the dffuson horzon s N, standard devaton of supply of the end product over the dffuson horzon m ntal sze of potental adopter populaton of product T number of tme perods of the plannng horzon V set of functons n the supply chan N number of functons n the supply chan (N¼9V9) E set of arcs n the supply chan x fracton of unmet demand that s backlogged p, q coeffcents of nnovaton and mtaton K producton capacty per tme perod Objectve functon The objectve functon may be formulated as follows: Maxmze: Total Net Proft¼Total Lfe-Cycle Revenue Supply Chan Confguraton Costs, where, Total Lfe Cycle Revenue ¼ XT c N ½py t r t wl t hi t Š t ¼ 0 ð5þ

5 M. Amn, H. L / Omega 39 (2011) Supply Chan Confguraton Costs ¼ Ppelne Stock Cost þsafety Stock Cost Ppelne Stock Cost ¼ XN Safety Stock Cost ¼ XN ¼ 1 ¼ 1 hðc c =2Þp m qffffffffffffffffffffffffffffffffffffffffffffffffff hc ks s n þp s out ð6þ ð7þ ð8þ X O k ¼ 1 m ¼ x k ¼ 1 8 ¼f1,..., Ng ð24þ X j:ð,jþ A E m j 8 ¼f1,..., Ng ð25þ s n, s out Z0 and nteger 8 ¼f1,..., Ng ð26þ x k Af0,1g 8 ¼f1,..., Ng, kaf1,..., O g ð27þ The objectve functon s to maxmze the total net proft over the new product s lfetme as the dfference between the total lfecycle revenue and supply chan confguraton costs. In (5), the total lfe-cycle revenue s calculated as the total sales revenue mnus three tems: the cost of goods produced, watng cost of backlogged demand and nventory holdng cost of fnshed goods. Notce that the per-unt product cost s represented by the cumulatve cost of the end product c N, whch s determned through confgurng the correspondng supply chan. Ths reflects Lnk 1 n the hybrd model. The supply chan confguraton cost s defned as the sum of ppelne stock cost n (7) and safety stock cost n (8). The ppelne stock cost (7) follows the Lttle s Law as n [10], where the term c c =2 denotes the average value of goods over ts lead tme p. The safety stock cost (8) s obtaned combnng (3) and (4). Constrants The set of constrants ncludes constrants for product dffuson, supply chan confguraton and the lnkng constrants. We state and descrbe these constrants below. Constrants on product dffuson R t þ 1 ¼ R t þr t 8t ¼f0,1,..., T 1g ð9þ D t þ 1 ¼ D t þd t 8t ¼f0,1,..., T 1g ð10þ Y t þ 1 ¼ Y t þy t 8t ¼f0,1,..., T 1g ð11þ d t ¼ pðm D t Þþ q m Y tðm D t Þ 8t ¼ f0,1,..., Tg ð12þ I t ¼ R t Y t 8t ¼f0,1,..., Tg ð13þ L t þ 1 ¼ xðl t þd t y t Þ 8t ¼f0,1,..., Tg ð14þ L t Z0 and L 0 ¼ 0 I t Z0 and I 0 ¼ 0 r t rk and R 0 ¼ 0 Constrants on supply chan confguraton p ¼ XO k ¼ 1 c ¼ XO k ¼ 1 c X j:ðj,þ A A ð15þ ð16þ ð17þ P k x k 8 ¼f1,..., Ng ð18þ C k x k 8 ¼f1,..., Ng ð19þ c j c ¼ 0 8 ¼f1,..., Ng ð20þ s n Zs out j 8 ¼f1,..., Ng, j : ðj,þae ð21þ s n þp s out Z0 8 ¼f1,..., Ng ð22þ s out N rl ð23þ Lnkng constrants m N ¼ Y T =ðt þ1þ vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ux s T N ¼ t ðy t m N Þ 2 =T t ¼ 0 ð28þ ð29þ Constrants (9) through (17) are smlar to those n Kumar and Swamnathan [8]. Constrants (9) through (11) calculate the cumulatve producton, demand and sales of the product n each tme perod, respectvely. Constrant (12) follows (1) as explaned earler. Constrant (13) calculates the nventory of fnshed product n each tme perod as the dfference between cumulatve producton and cumulatve sales. Constrant (14) states that a fracton x of the unmet demand s nstantaneously backlogged. By adjustng x, the model s able to capture varous scenaros rangng from complete loss of sales (x¼0) to complete backlog (x¼1). Constrants (15) through (17) specfy the bounds of the decson varables relevant to the product dffuson sde. Constrants (18) through (27) are smlar to those n the supply chan confguraton optmzaton model of Graves and Wllems [10]. Constrants (18) and (19) calculate the lead tme and drect cost added of each functon, respectvely. Constrant (20) calculates the cumulatve cost added for each functon. Constrant (21) states that the nbound servce tme of a functon must be no less than the outbound servce tme of ts mmedate predecessors. Constrant (22) ensures that the net replenshment tme of a functon s non-negatve. Constrant (23) guarantees than the deadlne of the new product (end functon N) be satsfed for delvery to customers. Constrant (24) assgns exactly one opton to a functon. Constrant (25) follows (2) and calculates the nternal demand of each functon assumng r j equal 1. Constrants (28) and (29) calculate the mean and standard devaton of sales of the new product over ts entre lfe cycle, respectvely. Notce that the demand pattern now becomes endogenously determned by the producton plan durng the new product dffuson process. They establsh Lnk 2 between the product dffuson and supply chan confguraton Complexty and soluton methods The above MINLP s NP-hard as the objectve functon nvolves mnmzng a concave functon n (7), and t s well-known that the general problem of mnmzng a concave functon s NP-hard [27]. Further computatonal complexty arses due to the exstence of nteger varables and nonlnear equaltes as n (12) and (29). Varous computatonal algorthms have been developed to tackle MINLPs. We refer to Floudas [28] for a comprehensve survey of MINLP algorthms. An MINLP wth a large number of nonlnear equalty constrants can be handled by the so-called outer approxmaton wth equalty relaxaton (OA/ER, [29]) algorthm. The OA/ER algorthm reles on the assumpton that the expressons n an MINLP are convex. Vswanathan and Grossmann [30] generalzed the OA/ER algorthm to nclude an

6 318 M. Amn, H. L / Omega 39 (2011) augmented penalty (AP) functon, called OA/ER/AP, to avod the lmtatons mposed by the convexty assumpton made n OA/ER. Gven the presence of concave functons n our MILP model, OA/ER/AP fts better for our needs. The OA/ER/AP algorthm has been mplemented n the DICOPT solver by GAMS [31]. It starts by relaxng the nteger requrement of decson varables and solvng the relaxed NLP problem. If the soluton to ths relaxed problem s nteger, the search stops. Otherwse, t contnues wth a sequence of NLP subproblems by fxng the nteger varables, and mxed-nteger lnear program (MIP) master problems generated by augmented penalty functon. Integer cuts are added to the MIP to exclude prevous vsted nteger solutons. When objectve functon s convex, solvng the MIP master problem provdes an upper bound on the objectve functon (for a maxmzaton problem); when objectve functon s concave as n our case, however, solvng the master problem may not provde a vald upper bound. Thus the OA/ER/AP algorthm may not guarantee optmal solutons for our MINLP model and wll serve as a heurstc approach for solvng our hybrd model. For mplementaton and techncal detals about the DICOPT solver, we refer the readers to [32]. In order to evaluate soluton qualty, we adapt a lnear approxmaton approach descrbed n [33] and [34] to approxmate the concave term (8) n the objectve functon. In our mplementaton, we use one lnear pece to approxmate (8). Wth such approxmaton, the objectve functon becomes convex (as two other terms (5) and (7) are convex), for whch a local optma s also global optma when all nteger varables are relaxed. The optmal soluton (or upper bound) to ths approxmated/ relaxed problem provdes an upper bound to the orgnal problem. 4. Computatonal expermental study We conduct a computatonal expermental study to analyze the performance of our ntegrated optmzaton model under varous topologes characterzed by problem parameters from both the product dffuson and supply chan confguraton perspectves. We compare solutons obtaned from the ntegrated model wth those found by dfferent heurstc polces to draw observatons and nsghts concernng the addressed problem Desgn of experments The computatonal expermental desgn for ths study ncludes two sets of factors (parameters): set one characterzng the supply chan network, and set two defnng the parameters mpactng new product dffuson process. We consder seral supply chans wth number of echelons beng 2, 4 or 8. We also consder dfferent cost and lead tme accrual functons, whch captures the speed at whch lead tme and cost cumulate along the supply chan. Followng Graves and Wllems [10], three profles: f(x)¼x 0.25, x and x 2 are ncluded, where x denotes the cumulatve poston x n a seral supply chan, and f(x) s a functon whch calculates the lead tme or cumulatve cost of all low-cost optons for poston x. We refer to [10] for detals about the functonal form of f(u), and dscussons on how dfferent profles capture varous supply chan structures rangng from the tradtonal consumer-electroncs-manufacturng to orgnal equpment manufacturng (OEM). To elmnate the undesred effect of dfferent magntudes of lead tme and value of products, we scale the cost and lead tme of low cost opton confguraton to $100 and 100 days, respectvely. We follow [8] for the choce of dffuson sde parameters. We consder hgh, medum and low prce ratos p¼1.1, 1.2 and 1.3. In a smlar way, we choose nventory holdng cost rate h¼0.001, and 0.01, and watng cost rate w¼0.005, and We consder full backloggng (x¼1), lost sales (x¼0) and partal-backloggng (x¼0.5). Ths generates a total of 2187 random testbed problem nstances. Other parameters are chosen as follows. As n [8], we set the market potental m¼3000 to be exhausted n T¼30 perods, wth each perod representng two to four weeks. Ths results n an entre product lfe cycle of about 1 to 2 years. The producton capacty per perod s chosen to be K¼100. The coeffcent of nnovaton p, and coeffcent of mtaton q are set to be 0.03 and 0.4, respectvely, whch represents a generc product based on the study of [26]. Each problem nstance s randomly generated and solved by dfferent soluton methods. In addton to our hybrd model, a set of two-layer supply chan strateges are consdered. Layer one strategy nvolves the product dffuson sde and ncludes myopc and buldup strateges as consdered n Kumar and Swamnathan [8]. In the myopc strategy, producton and sales of the product starts n the frst perod, whle the buldup strategy ncludes nventory buldup perods followed by launchng the new product. In our experment, we pck three possble number of buldup perods: 2, 6 and 12. Thus there are four layer one strateges. Then for each of these four strateges, we consder two sub-strateges concernng the supply chan confguraton sde as n Graves and Wllems [10],.e. the lowest cost (MnCost) and lowest lead tme (MnLT) confguraton. Ths leads to a total of eght alternatve heurstcs. For brevty, we name these soluton methods as follows. Hybrd Model: It smultaneously determnes the optmal sales/ demand plan and supply chan confguraton. The hybrd model s solved usng the DICOPT solver n GAMS. Myopc+MnCost: Ths polcy keeps no buldup perod for the dffuson process and chooses the least cost opton for each supply chan functon. Myopc+MnLT: Ths polcy keeps no buldup perod for the dffuson process and chooses the least lead tme opton for each supply chan functon. Buldup(n)+MnCost: Ths polcy keeps n buldup perods for the dffuson process and chooses the least cost opton for each supply chan functon, where n refers to 2, 6 or 12. Buldup(n)+MnLT: Ths polcy keeps n buldup perods for the dffuson process and chooses the least lead tme opton for each supply chan functon, where n refers to 2, 6 or 12. Table 1 summarzes the parameters and ther values used n the expermental study Computatonal expermental results We frst present results on the overall performance of dfferent soluton approaches, followed by results on the effect of problem parameters. Also, our dscussons present key fndngs. Table 1 Summary of expermental parameters and ther levels. Parameters Levels Number of echelons N {2, 4, 8} Lead tme accrual functon {x 0.25,x, x 2 } Cumulatve cost accrual functon {x 0.25,x, x 2 } Sellng prce rato p {1.1, 1.2, 1.3} Inventory holdng cost rate h {0.001, 0.005, 0.01} Watng cost rate w {0.005, 0.008, 0.01} Percentage of backlogged x {0, 0.5, 1} Number of buldup perod n 2, 6, 12 Intal sze of adopter m 3000 Plannng horzon T 30 Producton capacty per perod K 100 Coeffcent of nnovaton p 0.03 Coeffcent of mtaton q 0.4

7 M. Amn, H. L / Omega 39 (2011) Overall performance Table 2 summarzes the cost and revenue tems found by each soluton method. The Buldup6+MnLT heurstc generates the hghest total revenue on average, but also ncurs the hghest average COGS; whle the Myopc+MnCost heurstc ncurs the lowest COGS on average, but only acheves the lowest average total sales revenue. Increasng the number of buldup perods may reduce the cost of lost sales (e.g., Buldup12+MnCost s the best n ths category), but ncurs more nventory cost for holdng fnshed products n stock; the myopc polcy, on the other hand, does not ncur any cost for fnshed goods nventory, but suffers hgher cost due to unsatsfed demand. For the ppelne stock, a mnmum lead tme confguraton heurstc mght be advantageous as t results n shorter supply chan cycle tme (the tme an entty takes to traverse the entre supply chan [35]), but hgher cumulatve costs of work-n-process nventory at the same tme; a mnmum cost confguraton heurstc mght be advantageous as t results n lower cumulatve costs of work-n-process nventory, but longer cycle tme. Such a tradeoff depends on the cost and lead tme structure of specfc problem settngs. For the safety stock, a smoothed demand pattern s desrable as t leads to less varaton of demand, thus less safety stock cost. In our experment, Myopc+MnLT appears to have best performance on supply chan confguraton n terms of ppelne stock (column 5) and safety stock costs (column 6). Strkngly, our hybrd model does not excel n any ndvdual category of these revenue and cost measures. Table 3 reports soluton qualty of dfferent approaches at a more aggregate level. The Myopc+MnLT performs best n terms of supply chan confguraton cost due to lower SPC and lower SSC (columns 5 and 6 n Table 1). However, t has a medum performance n total lfe-cycle revenue (column (7) n Table 2). Our hybrd model generates the hghest total lfe-cycle revenue on average (wth modest varaton compared wth others). And despte ts hgher spendng on the supply chan confguraton cost, the hybrd model acheves hghest average total net proft (wth an average gap of 2.57% from upper bound). An nterestng fndng s that best solutons for the ntegrated new product dffuson process and ts supply chan confguraton problem do not necessarly lead to lowest total supply chan costs. (The Myopc+MnLT polcy results n least supply chan confguraton cost, but has an average of 6.67% devaton from upper bound.) That s, purely mnmzng the supply chan confguraton cost wthout consderng the dffuson process may lead to local optma. We may draw the followng nsght from Tables 2 and 3. Observaton 1: Dfferent performance measures on new product dffuson and ts supply chan confguraton nteract and have tradeoffs wth each other. They must be consdered n a unfed framework to acheve optmal solutons; optmzng each ndvdual measure may only lead to local optma. Observaton 1 provdes useful nsghts for managng producton/sales and supply chan n practce. An optmal producton/ sales plan or supply chan confguraton for a new product does not necessarly lead to maxmum amount of sales revenue from the marketng perspectve, or mnmum amount of costs from the supply chan confguraton perspectve. Producton/sales plannng and supply chan confguraton for new products need to be consdered smultaneously to obtan optmal solutons. Fg. 1 through Fg. 3 depct the topologes of relatve soluton qualty for dfferent heurstc approaches. For the product Table 2 Summary of soluton cost and revenue tems found by each approach. Supply chan strategy TSR (1) COGS (2) TWC (3) TIC (4) SPC (5) SSC (6) Hybrd model (49.52) (32.56) 4.06 (5.69) 1.31 (2.28) 3.68 (3.26) 1.70 (1.78) Myopc+MnCost (67.12) (48.10) 4.57 (6.29) 0 (0) 4.89 (4.41) 1.81 (1.99) Myopc+MnLT (69.03) (49.59) 4.76 (6.54) 0 (0) 3.55 (3.21) 1.57 (1.72) Buldup2+MnCost (48.21) (31.62) 3.85 (5.39) 1.48 (2.03) 5.07 (4.48) 2.14 (2.22) Buldup2+MnLT (48.78) (31.97) 3.98 (5.58) 1.60 (2.21) 3.71 (3.29) 1.88 (1.96) Buldup6+MnCost (47.68) (31.28) 3.65 (5.11) 2.43 (2.05) 5.08 (4.49) 2.48 (2.61) Buldup6+MnLT (48.02) (31.44) 3.77 (5.29) 2.58 (2.26) 3.72 (3.30) 2.21 (2.37) Buldup12+MnCost (64.35) (45.46) 3.51 (4.78) 4.22 (3.37) 4.87 (4.36) 3.04 (3.29) Buldup12+MnLT (65.61) (46.62) 3.63 (4.95) 4.35 (3.44) 3.57 (3.21) 2.80 (3.20) 1. TSR total sales revenue, COGS cost of goods sold, TWC total cost due to watng, TIC total nventory cost for holdng fnshed products, SPC supply chan ppelne stock cost, SSC supply chan safety stock cost. 2. Numbers are n thousands, numbers n parenthess represent standard devaton. 3. Bold entry ndcates the best performance for the correspondng tem (column). Table 3 Descrptve statstcs of soluton qualty of each soluton approach. Total lfe-cycle revenue (7)¼(1) (2) (3) (4) Supply chan confguraton costs (8)¼(5)+(6) Total lfe-cycle net proft (9)¼(7) (8) Mn Max Avg. Std. Mn Max Avg. Std. Mn Max Avg. Std. Hybrd model (0) (19) (2.57) Myopc+MnCost (3) (31) (12.04) Myopc+MnLT (0) (19) (6.67) Buldup2+MnCost (1) (32) (8.63) Buldup2+MnLT (0) (34) (3.08) Buldup6+MnCost (0) (55) (10.01) Buldup6+MnLT (0) (53) (4.54) Buldup12+MnCost (1) (57) (17.5) Buldup12+MnLT (0) (56) (12.2) Numbers (except those n parenthess) are n thousands. 2. Bold entry ndcates the best performance for the correspondng tem (column). 3. Numbers n parenthess are the percentage of gap between upper bound calculated as gap¼100n(upper bound obj. value)/upper bound.

8 320 M. Amn, H. L / Omega 39 (2011) dffuson sde (Fg. 1), the MnLT heurstc on average outperforms the MnCost heurstc (gven the cost and lead tme structures assumed n our experment). For the same supply chan confguraton heurstc, as the producton dffuson polcy evolves from myopc (number of buldup perods equals 0) to the buldup polcy up to 12 perods, the soluton qualty frst mproves, then deterorates. Ths s probably due to the polcy s tradeoff between backlog watng (or lost sale) cost and nventory holdng cost. If there s no buldup perod, the frm s too aggressve n stmulatng the demand, ncurrng large cost due to backlog or lost sales although nventory holdng s low; f there are too many buldup perods, the frm becomes too conservatve whle spendng too much on nventory holdng cost. In our experment, the best number of buldup perods occurs at 2. We then have Observaton 2. Pared t-test was conducted to test ths hypothess. The p-value s less than 0.01, whch supports the hypothess at a confdence level of 99%. Fg. 1. Average percentage mprovement of hybrd model over heurstc approaches on total revenue. Observaton 2: For the product dffuson process alone, nether myopc nor buldup polcy wth many buldup perods may result n maxmum net revenue. For the supply chan confguraton sde alone (Fg. 2), as the product dffuson polcy evolves from myopc to the buldup polcy wth up to 12 perods, ts performance on supply chan confguraton cost becomes worse. Ths s probably caused by the hgher varaton of demand and sales due to the buldup perods; whle the myopc polcy n general leads to smoother demand and sales pattern wth smaller varaton, whch may not be desrable for the new producton dffuson process alone, but s preferable for plannng and confgurng the assocated supply chan. In partcular, smaller varaton of demand and sales lead to less safety stock cost through the supply chan (see column 6 n Table 1). Ths leads to Observaton 3. Pared t-test was conducted to test ths hypothess, wth the p-value beng less than Observaton 3: For the supply chan confguraton sde alone, myopc polcy outperforms buldup polces n supply chan confguraton cost: the more buldup perods, the hgher the supply chan confguraton cost. Observaton 3 reveals the relatonshp between the number of buldup perods from the producton plannng sde and the supply chan confguraton sde. A smooth and even producton/sales plan results n less nventory holdng costs for the supply chan, but may suffer sgnfcant loss of sales by neglectng the dynamcs of market demand. On the other hand, a producton plan wth certan number of buldup perods may ncrease sales revenue, but also ncurs more supply chan confguraton costs due to ncreased varaton of sales. Fg. 3 shows the relatve performance on the overall soluton qualty of total lfe-cycle net proft. The cost savngs from the myopc polcy s not suffcent to compensate the shortage on net revenue of the myopc polcy, thus the myopc polcy fals to acheve the qualty of the hybrd soluton. On the other hand, the surplus of net revenue from the buldup polcy (wth many buldup perods) s not suffcent to cover the extra spendng on supply chan confguraton, thus the buldup polcy (wth many perods) may also fal to obtan optmal solutons. Observaton 4 follows. Pared t-test was conducted to test ths hypothess, wth the p-value beng less than Fg. 2. Average percentage mprovement of hybrd model over heurstc approaches on supply chan confguraton cost. Fg. 3. Average percentage mprovement of hybrd model over heurstc approaches on total net proft.

9 M. Amn, H. L / Omega 39 (2011) Fg. 4. Four best soluton methods when nventory holdng cost rate vares. Fg. 5. Four best soluton methods when percentage of backlogged demand vares. Observaton 4: For the ntegrated product dffuson process and supply chan confguraton problem, nether myopc nor buldup polcy wth many buldup perods may perform well Effect of parameters The effect of nventory holdng cost rate s llustrated by Fg. 4. None of the heurstcs performs better than the hybrd model n all ranges of nventory cost rate. The second best soluton s Buldup2+MnLT, whch consstently outperforms Myopc+MnLT, although Myopc+MnLT leads to lower supply chan confguraton cost. Ths agan verfes the mportance of ntegratng product dffuson and supply chan confguraton. We also observe that as nventory holdng cost rate ncreases, the dfference of soluton qualty between myopc heurstcs and hybrd model decreases; whereas the dfference of soluton qualty between buldup heurstcs and hybrd model ncreases. Observaton 5 follows. Observaton 5: As nventory holdng cost rate ncreases, the beneft of hybrd model over buldup polcy ncreases; as nventory holdng cost rate decreases, the beneft of hybrd model over myopc polcy ncreases. One-way ANOVA was conducted to test ths hypothess, wth the p-value beng less than Inventory holdng cost rate has mpacts on both product dffuson and supply chan confguraton sdes. As nventory holdng cost rate ncreases, buldup polcy leads to not only ncreased amount of fnshed goods nventory on the product dffuson sde, but also the ppelne and safety stock on the supply chan confguraton sde (due to ncreased demand/ producton varaton). Thus a hgh nventory holdng cost rate wll make buldup polcy less attractve. For the myopc polcy, ts man advantage s savngs on nventory costs of fnshed goods, ppelne and safety stock: as myopc polcy attempts to sell as much as possble, t ncurs no fnshed goods nventory, and results n lower safety stock cost due to smaller demand/ producton varaton. Such advantage of myopc polcy wll be amplfed when the nventory holdng cost rate s hgh. The effect of percentage of backlogged demand s llustrated by Fg. 5. When percentage of backlogged demand (x) s ether hgh or low,.e. no loss of demand or complete loss of demand, there s less flexblty to shape the demand/producton pattern, thus the advantage of hybrd model over heurstc polces s small; when x s around the mddle (0.5), the hybrd model s advantage ncreases. Observaton 6 follows. Observaton 6: The benefts of hybrd model are more pronounced when the percentage of backlogged demand s nether too hgh nor too low. Two-sample t-test was conducted to test Observaton 6, wth the p-value beng less than The choce of x depends on the specfc product and ndustry the model s addressng. A value of 0 represents an extreme where the unmet demand s completely lost. Examples of such case nclude hghly seasonable goods. A value of 1 presents the other extreme that the unmet demand s completely backlogged. Examples of such case nclude varous consumer goods for whch the demand s not seasonable. The practcal mplcaton of Observaton 8 s that for average value of x, the hybrd model s expected to sgnfcantly outperform heurstc decson rules. We also examne effects of the parameters from the supply chan confguraton sde on total net proft. It s found that for varous sze supply chan networks and products rangng from the tradtonal consumer-electroncs-manufacturng to an OEM, the hybrd model consstently outperforms heurstc decson rules. 5. Conclusons and future research In ths paper, we develop an ntegrated optmzaton model for confgurng new products supply chans whle explctly consderng the mpact of demand dynamcs durng new products dffuson. It smultaneously determnes optmal producton/sales plan and supply chan confguraton. The producton and sales plan provdes decsons on the optmal tmng to launch a new product, as well as the producton and sales quantty n each plannng perod. The supply chan confguraton provdes optmal selecton of optons and safety stock level kept at each supply chan functon. The ntegrated model mnmzes the total lfecycle proft durng a new product s entre lfe cycle. An n-depth computatonal experment, ncludng 2187 randomly generated testbed problem nstances, was conducted to examne the performance characterstcs of our ntegrated optmzaton model versus seven alternatve heurstc polces under varous dffuson and supply chan topologes. We obtan a number of manageral nsghts regardng producton/sales plannng and supply chan confguraton for new products. (1) An optmal producton/sales plan or supply chan confguraton for a new product does not necessarly lead to maxmum amount of sales revenue from the marketng perspectve, or mnmum amount of costs from the supply chan confguraton perspectve.

10 322 M. Amn, H. L / Omega 39 (2011) An optmal producton/sales plan and supply chan confguraton durng new products lfe cycle balances the tradeoffs among varous cost and revenue components, and can only be acheved through an ntegrated optmzaton approach. (2) A smooth and even producton/sales plan results n less nventory holdng costs for the supply chan, but may suffer sgnfcant loss of sales by neglectng the dynamcs of market demand. On the other hand, a producton plan wth certan number of buldup perods may ncrease sales revenue, but also ncurs more supply chan confguraton costs due to ncreased varaton of sales/demand. Due to (1) and (2), our hybrd approach s more advantageous than models that consder producton dffuson or supply chan confguraton ndependently. The hybrd optmzaton approach s also robust for supply chan networks wth dfferent topologes characterzed by the number of functons n the network and the lead tme/drect cost profle of the new product. We fnd that as the nventory holdng cost rate ncreases, t becomes more costly for the buldup polcy to avod sales loss, thus the beneft of hybrd model over buldup polcy ncreases. When the percentage of backlogged demand (x) devates from two extream cases,.e. when x s nether hgh nor low as often the case n the real world, the beneft of hybrd model also ncreases. Ths research opens a number of extenson opportuntes for analyzng the ntegrated new product dffuson and supply chan confguraton problem. Here, we menton a few of these potental research extensons. Frst, our current study assumes a generc product wth an average coeffcent of nnovaton (p) and coeffcent of mtaton (q). It wll be nterestng to study the mpact of dfferent product dffuson characterstcs captured by pared p and q for specfc products/ndustres. Some ndustryspecfc decsons and/or constrants may also be modeled. For nstance, desgnng supply chans n the fashon ndustry may emphasze responsveness n terms of the supply chan cycle tme, thus a deadlne on the cycle tme may be necessary to be ncluded as a constrant. Products n the hgh-tech ndustry may have a dfferent cost and lead tme accrual profle than that of consumer durable goods. Second, some assumptons n the supply chan confguraton can be relaxed. For example, one could relax the sngle-sourcng assumpton and allow multple optons to be assgned to a functon. Thrd, the current model consders only sngle product, t wll be nterestng to extend the model to consder a famly of new products sharng the same BOM. Ths would requre that we consder the dffuson process of multple correlated products smultaneously. Fourth, n real-world settngs compettors mght offer smultaneously smlar products n the marketplace wth the ntenton of capturng as much market share as possble. Obvously, the compettve envronment mpacts the supply chan confguraton decsons and the dynamc demand pattern emerges from the new product dffuson process. Then from the computatonal perspectve, our current work reles on a commercal optmzaton solver to solve the ntegrated model. When problem sze becomes large, soluton tmes by solvers are not tractable. Thus a promsng future research wll be developng more advanced soluton methods, e.g. varous metaheurstcs [36], to solve large sze problems both effectvely and effcently. Acknowledgements The authors thank the assocate edtor and three anonymous referees for ther helpful comments and suggestons that mprove the contents and presentaton of the paper. Mehd Amn s supported n part by the FedEx Insttute of Technology and the Enterprse Smulaton and Optmzaton Lab (esol) at the Unversty of Memphs. Hatao L s supported n part by the Center for Transportaton Studes (CTS) at the Unversty of Mssour St. Lous. References [1] Busness Week, About that phone shortage, 2008, Aprl 4. [2] Fnancal Tmes, Untmely W shortage hts Nntendo, 2007, December 24. [3] New York Post, Shortage chps apple net supplers delays wll hold results down, 1999, September 21. [4] New York Tmes, Playstaton sales zoom, 2000, March 7. [5] Los Angeles Tmes, Sony to cut game workers n US, 2007, June 7. [6] Lee HL. The trple-a supply chan. Harvard Busness Revew 2004:1 12. October. [7] Bass FM. A new product growth for model consumer durables. Management Scence 1969;15(5): [8] Kumar S, Swamnathan JM. Dffuson of nnovatons under supply constrants. Operatons Research 2003;51(6): [9] Ho T-H, Savn S, Terwesch C. Managng demand and sales dynamcs n new product dffuson under supply constrant. Management Scence 2002;48(2): [10] Graves SC, Wllems SP. Optmzng the supply chan confguraton for new products. Management Scence 2005;51(8): [11] Geoffron AM, Graves GW. Multcommodty dstrbuton system desgn by benders decomposton. Management Scence 1974;20(5): [12] Daskn MS. Network and dscrete locaton: models, algorthms, and applcatons. New York: John Wley & Sons, Inc.; [13] Smpson KF. In-process nventores. Operatons Research 1958;6(6): [14] Mnner S. Dynamc programmng algorthms for mult-stage safety stock optmzaton. OR Spectrum 1997;19: [15] Inderfurth K, Mnner S. Safety stocks n mult-stage nventory systems under dfferent servce measures. European Journal of Operatonal Research 1998;106: [16] Graves SC, Wllems SP. Optmzng strategc safety stock placement n supply chans. Manufacturng & Servce Operatons Management 2000;2(1): [17] Mansfeld E. Techncal change and the rate of mtaton. Econometrca 1961;29(4): [18] Mahajan V, Muller E, Bass FM. New product dffuson models n marketng: A revew and drectons for research. Journal of Marketng 1990;54:1 26. [19] Mahajan V, Muller E, Bass FM, Wnd J. New product dffuson models. Thousand Oaks, CA: Sage; [20] Bass FM. Comments on a new product growth for model consumer durables the bass model. Management Scence 2004;50: [21] Kurawarwala AA, Mastuo H. Forecastng and nventory management of short lfe-cycle products. Operatons Research 1996;44(1): [22] Swam S, Kharnar PJ. Dffuson of products wth lmted supply and known expraton date. Marketng Letters 2003;14(1): [23] Chandra C, Grabs J. Role of flexblty n supply chan desgn and modelng ntroducton to the specal ssue. Omega 2009;37: [24] Francas D, Mnner S. Manufacturng network confguraton n supply chans wth product recovery. Omega 2009;37: [25] Yu H, Zeng AZ, Zhao L. Sngle or dual sourcng: decson-makng n the presence of supply chan dsrupton rsks. Omega 2009;37: [26] Sultan F, Farley JU, Lehmann DR. A meta-analyss of applcatons of dffuson models. Journal of Marketng Research 1990;27(1):70 7. [27] Sahn S. Computatonally related problems. SIAM Journal on Computng 1974;3: [28] Floudas CA. Nonlnear and mxed-nteger optmzaton: fundamentals and applcatons. New York: Oxford Unversty Press; [29] Kocs GR, Grossmann IE. Relaxaton strategy for the structural optmzaton of process flowsheets. Industral and Engneerng Chemstry Research 1987;26: [30] Vswanathan J, Grossmann IE. A combned penalty functon and outer approxmaton method for mnlp optmzaton. Computers & Chemcal Engneerng 1990;14: [31] GAMS, Gams user s gude, GAMS Development Corporaton, Washngton DC, [32] Grossmann IE, Vswanathan J, Vecchette A. DICOPT User s Manual. GAMS Development Corporaton; [33] Magnant TL, Shen Z-J, Shu J, Smch-Lev D, Teo C-P. Inventory placement n acyclc supply chan networks. Operatons Research Letters 2006;34: [34] Shu J, Karm IA. Effcent heurstcs for nventory placement n acyclc networks. Computers & Operatons Research 2009;36: [35] Hopp WJ. Supply chan scence. New York: McGraw Hll; [36] Glover F, Kochenberger G. Handbook of metaheurstcs. Sprnger; 2005.