Profitability of Loyalty Programs in the Presence of Uncertainty in Customers Valuations

Transcription

1 Proceeding of the 0 Indutrial Engineering Reearch Conference T. Doolen and E. Van Aken, ed. Profitability of Loyalty Program in the Preence of Uncertainty in Cutomer Valuation Amir Gandomi and Saeed Zolfaghari Department of Mechanical and Indutrial Engineering Ryeron Univerity, Toronto, Ontario M5B K3, Canada Abtract Effectivene of cutomer loyalty program ha been the focal point of ome recent analytical and empirical tudie in the marketing literature. Analytical tudie have mainly conidered a two-period model through which cutomer gain a loyalty reward if they purchae in both period. In thi paper, we build upon previou analytical tudie by incorporating a tochatic valuation. Specifically, we aume that cutomer valuation for the product in the firt period follow the normal ditribution. Valuation in the econd period, however, depend on the level of atifaction with product or ervice purchaed in the firt period. The additive form i choen to build the valuation in the econd period baed on valuation in the firt period and the atifaction level. The atifaction level i alo modeled a a normally ditributed random variable. The purpoe i to maximize the firm revenue function. The formulation reult in a tochatic programming problem with a nonlinear non-convex objective function. The olution i found in term of the model parameter. The reult reveal that if the coefficient of variation of atifaction level turn out to be poitive and le than a certain threhold, it i optimal not to offer a loyalty reward. Keyword Loyalty program, cutomer atifaction, valuation, marketing, nonlinear programming, tochatic programming. Introduction Loyalty program are one of the marketing trategie to build and enhance cutomer loyalty and thereby increae a firm long-term profitability. Since the time American Airline launched AAdvantage, the firt contemporary loyalty program (ee []), loyalty program have proliferated in variou indutrie including airline, credit card companie, retail and hotel chain (ee, e.g., [-4]). Cutomer participation in loyalty program ha alo grown teadily during the pat decade (ee, e.g., [5-7]). Depite their wide-pread ue, academic have not reached conenu on whether loyalty program are effective in all cae (ee [8-0]). In other word, it i not apparent that loyalty program are able to influence the etablihed buying pattern of cutomer and increae a firm long-term profitability. Specifically, empirical tudie have reported contradictory finding on the effectivene of loyalty program (ee, [], [9] and []). While ome reearcher have found a ignificant impact (e.g., e.g., [-5]), other claim that loyalty program are unlikely to change etablihed buying behavior of cutomer (e.g., [8], [6], [7]). Analytical tudie, on the other hand, are carce in thi field. Thi fact i evident from Kim et al. [], the firt publihed analytical reearch on loyalty program, who view their work a an initial tep, and clearly far removed from the ideal model in which the implication directly tranlate into managerial practice (p. 3). Kim et al. [] have conidered a duopoly market tructure and, uing a game-theoretic approach, have found that competitive market price generally increae by adopting loyalty program. In a more recent tudy, Singh et al. [0] have examined an aymmetric duopoly market in which only one of the firm i offering a loyalty reward. By finding equilibrium in thi market under certain condition, Singh et al. [0] have proved that it i not alway optimal to adopt a loyalty program in repone to the competitor imilar program. In thi paper, we develop an analytical model to invetigate the profitability of loyalty program when cutomer atifaction level i taken into account. Previou analytical tudie have not conidered the cutomer purchae experience a a factor in their deciion to buy. Thi indicate the implicit aumption that cutomer alway return to

2 the firm with the ame valuation for the product being offered, no matter whether they are atified or diatified with their pat purchae. In contrat to thi underlying aumption, empirical tudie have found that atified and diatified cutomer perceive loyalty program in different way (ee [8]). Yi [9] alo tate that many tudie found that cutomer atifaction influence purchae intention a well a pot-purchae attitude (p. 04). Thu, cutomer atifaction level i an important factor that may affect the performance of a loyalty program in driving repeat purchae behavior and increaing the firm profitability. We alo contribute to the exiting literature on loyalty program by incorporating a tochatic valuation into the model. Previou tudie in the field have either not conidered the valuation a a factor in cutomer deciion making (e.g., Singh et al. [0]) or have aumed that cutomer valuation for the product i ufficiently high that it exceed the product price (e.g., [0], []). However, we aume that cutomer valuation for the product i normally ditributed. Thu, our model capture the heterogeneity in cutomer preference. The objective of the model i to maximize the firm expected revenue function in term of it deciion variable. The model i formulated a a tochatic programming problem with a nonlinear non-convex objective function, which mut be olved in term of three parameter. Here, we employ the interior-point algorithm propoed by Byrd et al. [] to find the local optimum. More pecifically, the model i olved for ome pre-determined et of parameter value. Subequently, the functional form of the optimal olution will be derived by analyzing the relationhip between optimal olution and the correponding parameter value. To enure the obtained optimum are global maxima, we apply the multi-tart procedure of Ugray et al. [], which i claimed to be one of the mot effective global optimization algorithm (ee [3]). The reult reveal remarkable inight into the effectivene of loyalty program. It will be hown that depending on the mean and variance of cutomer atifaction/diatifaction level, the firm may be better off not to offer a loyalty reward. More pecifically, the firm will maximize it profit if it maintain a poitive atifaction level among cutomer with a variance le than a certain threhold. The ret of thi paper i organized a follow: The model i formulated in ection. Section 3 decribe the procedure to olve the model and the obtained reult. We conclude in ection 4 with a ummary of the finding and ome direction for future reearch.. The model The model conit of a firm elling a good or ervice through two period. The firm precommit to the price of the product in each period (i.e., p and p ) and the loyalty reward (i.e., r). Cutomer earn the loyalty reward if they make purchae in both period. The reward i offered in the form of an abolute value in the econd period. The objective i to maximize the firm total profit in term of it deciion variable, that i, p, p and r. Cutomer deciion-making i modeled uing the urplu they obtain from their purchae. The urplu i the difference between a cutomer valuation for the product and the amount he/he mut pay to buy it. Similar to Biyalogorky et al. [4], it i aumed that a cutomer buy one unit of the product, if the offer yield a nonnegative urplu. Cutomer are aumed to be forward-looking; that i, they conider the future gain/lo while making deciion in period. Conequently, the urplu from the purchae in period (S ), can be expreed a the um of two term: the urplu reulting from buying in the firt period and the expected urplu from the purchae in the econd period, that i, S v p v p r, () where v denote the cutomer valuation in the firt period and γ denote the probability of a firt-period buyer to repatronize the firm in the econd period. Thu, γ repreent the buyer intention to return to the firm in period. Thi i to capture the effect of cutomer who ign up for the loyalty program, but fail to how up in the next period, mainly becaue of a low conumption rate. γ will be treated a a parameter in the model. Cutomer may differ in their valuation, but it i aumed that they have enough information to determine the product utility. The firm, on the other hand, i unaware of each individual valuation; however, the aggregate-level valuation ditribution i known to the firm. Here, we conider the pecific, but commonly tudied, cae where v i normally ditributed. A mentioned above, a cutomer buy if he/he earn a nonnegative urplu from the purchae. Auming that v i normalized to follow a tandard normal ditribution, from Equation (), one can find the probability of a cutomer making a purchae a follow:

3 p p r Pr S 0 () where (.) denote the tandard normal CDF. fraction of cutomer who have not accepted the offer in the firt period exit the firm market. γ fraction of buyer, on the other hand, proceed to the econd period. Since thee cutomer have made a purchae in period, they are eligible for the loyalty reward. Thu, the firt-period buyer hould pay p -r for the product in period. Subequently, a cutomer urplu from buying a unit of the product in period i: S v p r (3) where v i the cutomer valuation for the product in period. Unlike the previou tudie, here v i neither equal to nor independent from v. To model thi dependency, it i aumed that v = v +, where repreent the hift in each individual cutomer valuation. Eentially, erve a the cutomer atifaction/diatifaction level with the purchae in period.thi aumption i conitent with the finding of Homburg et al. [5] who have explored the effect of cutomer atifaction on their valuation uing empirical reearch. A negative ignifie diatifaction with the product itelf and/or with the firm quality of ervice. A poitive, on the contrary, indicate that the cutomer ha been atified with the pat purchae. Here, we allow for variability in cutomer atifaction level. In other word, we aume that cutomer atifaction may vary independently from their initial valuation. Specifically, we aume, at the aggregate level, i normally ditributed with parameter (μ, σ ). Since v and are added up to form v, they mut have been normalized with the ame factor. Thu, without lo of generality, we aume μ and σ are obtained by normalizing the atifaction level with the ame factor ued to normalize v. There are many endogenou and exogenou factor influencing cutomer perceived atifaction. A a reult, it i not plauible to aume μ and σ are primarily known to the firm. So, we model μ and σ a parameter, and through a enitivity analyi, we will evaluate the effect of thee parameter on the performance of loyalty program. A cutomer in the econd period make a purchae when S 0. Since v = v +, v and v are clearly dependent, and thu, o are S and S. Hence, a cutomer probability of earning a nonnegative urplu in the econd period i conditioned on the fact that hi/her urplu ha been nonnegative in the firt period. That i, PrS 0 S 0, (4) where i the probability of making a purchae in period. Subtituting S and S from Equation () and (3) into the above expreion and rearranging them, it follow that: p p r Prv p r v. (5) Applying conditional probability theory, the above equation can be rewritten a: p p r Prv, v p r. (6) p p r Prv In order to find the explicit expreion of a a function of the model variable and parameter, one mut find the joint ditribution of v and v. It can be proved that v and v follow a bivariate normal ditribution with the following mean vector and covariance matrix: 0 M and. (7) The reulting pdf can be found a: V V V f v,, exp v V V V. (8) Now, baed on the above pdf, in Equation (6) can be retated a:

4 p r Gandomi, Zolfaghari p p r v, v. (9) p p r A mentioned earlier, denote the probability of a firt-period buyer to repurchae in the econd period. f V, V dv dv The market ize in the firt period i normalized to one. Similar to Singh et al. (008), we aume that a new group of cutomer join the firm market in the econd period. The ize of thi light-uer egment i aumed to be γ. Thi aumption i mainly made to avoid market expanion or contraction affecting the reult. Light uer, in fact, make up for the miing firt-period buyer who fail to proceed to period (with the ize γ). Thee cutomer are one-time buyer, o they do not take the loyalty reward into account. Thu, their urplu from making a purchae i: l l S v. (0) p l v i the light-uer valuation for the product. It i reaonable to aume v l follow the ame ditribution a v, ince light uer are imilar to cutomer in period in the ene that they have not experienced the product yet. So, a light-uer make a purchae with the probability of: l Pr S l 0 p. () Now, we can formulate the firm expected revenue function baed on the cutomer probabilitie of buying the product in each period, a follow: l R p p r p. () R i a function of the firm deciion variable (i.e., p, p and r) and the model parameter (i.e., γ, μ and σ ).The purpoe i to maximize the revenue function with repect to the deciion variable. The optimal olution i expected to depend on the parameter. Such a olution yield valuable inight into how cutomer atifaction influence the optimal tructure of a loyalty program. The optimization i, of coure, ubject to the non-negativity of the price and reward. Moreover, the offered reward cannot exceed the product price in period. The reulting maximization problem i: p p r Maximize R p p, p and r V V V p r exp V dv dv p p r p r p p (3) Subject to: r p (3a) p p and r 0 (3b), 3. Reult The revenue function in Equation (3) i a highly nonlinear function. Evaluating the eigenvalue of it Heian matrix, it can be een that R i not a concave function. In more detail, there are ome point at which ome eigenvalue of the Heian of R are poitive. Therefore, ince the Heian i a ymmetric matrix, a poitive eigenvalue violate the concavity condition. Since R i not generally a concave (nor a convex) function, tandard optimization algorithm are not guaranteed to converge to the global optimum. To overcome thi challenge, firt we employ the interior-point algorithm propoed by Byrd et al. (000) to find the local maxima. Subequently, a global earch method will be invoked to invetigate if any better olution exit. A mentioned earlier, the optimal olution will be a function of the parameter, γ, μ and σ. In order to derive the optimum in term of the parameter, the model i olved for pre-determined et of parameter value. Then, we extract the function by evaluating the relationhip between the obtained optimum and the correponding value of parameter. More pecifically, ten value in the range [0,], ten point in the range [-,] and five value in the interval [0.,] were choen for γ, μ and σ, repectively. The value are all equally paced in their repective range.

5 Next, the model wa olved for all 500 combination of parameter value. Evaluating the relationhip between the optimal value of the deciion variable (i.e., p, p and r ) and the parameter value, it can be een that: p r 0.758, (4) p (5) From the obtained optimal olution, it can be inferred that r, the optimal reward value, depend on γ, μ and σ. In order to derive r in term of parameter, we can ubtitute p and p from Equation (4) and (5) into the original model (Equation (3)), and find the maximum with repect to r. The reulting optimization model i: Maximize R r 0.6 r V V V exp r V dv dv r 0.6 (6) Subject to: 0 r (6a) Figure illutrate R in the above model veru r for ome pecific value of γ, μ and σ. Baed on the Contraint 6a, the domain of R i retricted to r 0, From thi figure, it can be een that r p i the revenue-maximizing reward. However, depending on the parameter value, lower reward may alo lead to the maximum revenue. That i, there might be an optimal range of reward value. For example, Figure (a) how that when μ =0., σ =0. and γ=0.9, roughly any r in the range [0,0.758] i optimal. Note that baed on Equation (4), p increae with r. That i, the price of the product in period mut be adjuted baed on the offered reward in the econd period. In other word, depending on the parameter value, the model may yield alternate optimal olution. In ummary, the optimal olution to the model in Equation (3) can be formulated a follow: p r p r r,0.758, (7) l where ρ l i the lower bound of the optimal loyalty reward. ρ l depend on γ, μ and σ. Thi relationhip will be addreed later and it will be hown that ρ l range from 0 to The olution in Equation (7) yield R = which can be ued a a bai to verify whether it i a global optimum. More pecifically, one can employ the multitart procedure propoed by Ugray et al. (007) to check if any better olution exit. The reult of thi analyi indicate that the olution in Equation (7) can be the global maximum. One can analyze the effect of each parameter on the optimal reward uing Figure. From thi figure, it can be inferred that ρ l decreae a μ increae. That i, a the average of the individual atifaction level increae, a broader range of reward value become optimal. Moreover, by comparing Figure (a) and (b), it can be een that a higher variability in cutomer atifaction level reult in a higher ρ l. That i, ρ l increae with σ. The effect of γ on the optimal range of r can be evaluated uing Figure (e) and (f). A can be een, revenue function on thee figure reach the maximum level nearly at the ame r value. Thi indicate that γ doe not have a ignificant effect on ρ l. In ummary, ρ l decreae with μ and increae with σ while it remain nearly contant at different level of γ. Thi implie that, regardle of the cutomer repurchae intention, the firm will benefit from a broader range of optimal loyalty reward value if it manage to monotonically increae atifaction among cutomer. Moreover, the top curve in Figure (a) ugget that ρ l can be equal to zero. ρ l =0 ignifie the optimal reward range of [0,0.758] which include the particular point r =0. Thu, if there exit ρ l =0, the firm will achieve the maximum revenue even if it doe not offer any reward to loyal cutomer. In fact, r=0 generate a higher profit compared to other r value in the optimal range. Thi i becaue by not offering loyalty reward the firm will not incur the fixed expene aociated with adopting a loyalty program. By finding the maximum σ value that yield ρ l =0 in different μ level, it can be oberved that when the average of cutomer atifaction/diatifaction level turn out to be

6 poitive and coefficient of variation of atifaction level ( ) i le than 0.3, r=0 i optimal. A a reult, under the mentioned condition, the firm i better off to tick to the lower price trategy intead of adopting a loyalty program (baed on the optimal olution in Equation (7), the price in period decreae a the loyalty reward decreae). μ = 0. μ = 0 μ = 0. μ = 0. μ = 0 μ = 0. (a) γ=0.9, σ =0. (b) γ=0.9, σ =0. σ = 0. σ = 0.3 σ = 0.5 σ = 0. σ = 0.3 σ = 0.5 (c) γ=0.9, μ = 0. (d) γ=0.9, μ =0. γ = 0. γ = 0.3 γ = 0.5 γ = 0. γ = 0.3 γ = 0.5 (e) μ = 0. σ = 0. (f) μ = 0. σ = 0. Figure : R veru r for different value of parameter

7 4. Summary and concluion In thi paper, an analytical model wa developed to evaluate the profitability of loyalty program. The model conit of a revenue-maximizing firm elling a good or ervice through two period. Cutomer earn a loyalty reward in the form of an abolute dicount on the product in the econd period if they purchae in both period. Cutomer who reject the offer in the firt period will leave the firm market. Thoe who buy in period may alo leave the market with a certain probability denoted by γ. γ, in fact, repreent cutomer intention to rebuy the product. The value of γ depend on different factor like the product category and the overall conumption level. γ i incorporated a a parameter in the model. One of the ditinctive feature of our model i that cutomer atifaction level i incorporated a a factor in their deciion making in the econd period. The atifaction level i modelled a a normally ditributed random variable which i ummed up with the cutomer valuation in period to form their valuation in period. The mean and tandard deviation of the atifaction level are modelled a parameter. Cutomer valuation in the firt period i aumed to follow the tandard normal ditribution. Thu, our model capture the heterogeneity in cutomer preference a well a in buyer perceived quality. The objective of the model i to maximize the firm revenue in term of it deciion variable, that i, the price of the product in the firt and econd period and the loyalty reward amount. A mentioned above, the model conit of three parameter on which the optimal olution depend. To derive the optimal olution in term of parameter, the model wa olved for ome pre-pecified value of parameter. Subequently, by evaluating the obtained reult, optimal olution wa formulated a a function of parameter. The obtained reult yield ueful inight into the profitability of loyalty program. Specifically, it wa oberved that under certain condition the firm may obtain the maximum profit without adopting a loyalty program. Thee condition refer to the mean and variance of cutomer atifaction level. Particularly, if the mean of atifaction level turn out to be poitive with a tandard deviation le than a certain threhold, not offering reward yield the optimal profit. Thu, if the firm homogenouly maintain atifaction among all cutomer, it i optimal not to offer a loyalty reward. The variable cot of the product are not included in our model. However, it can be een variable cot will not alter the framework of our finding. Thi model can be further extended by incorporating competition. By allowing a cutomer not to buy from the firm, it i implicitly aumed that there are other eller in the market that will meet the cutomer demand. However, the effect of other eller action i not conidered in the firm pricing deciion. Reference. Liu, Y., 007, The long-term impact of loyalty program on conumer purchae behavior and loyalty, Journal of Marketing, 7 (4), Kim, B., Shi, M., and Srinivaan, K., 00, Reward program and tacit colluion, Marketing Science, 0 (), Kumar, V., 008, Managing cutomer for profit: Strategie to increae profit and build loyalty. Indianapoli, IN: Wharton School Publihing. 4. Pauler, G., and Dick, A., 006, Maximizing profit of a food retailing chain by targeting and promoting valuable cutomer uing Loyalty Card and Scanner Data, European Journal of Operational Reearch, 74 (), Dekay, F., Toh, R. S., and Raven, P., 009, Loyalty program: Airline outdo hotel, Cornell Hotel and Retaurant Adminitration Quarterly, 50 (3), McCall, M., and Voorhee, C., 00, The driver of loyalty program ucce: An organizing framework and reearch agenda, Cornell Hopitality Quarterly, 5 (), Smith, A., and Spark, L., 009, "It' nice to get a wee treat if you've had a bad week": Conumer motivation in retail loyalty cheme point redemption, Journal of Buine Reearch, 6 (5), Dowling, G. R., and Uncle, M., 997, Do cutomer loyalty program really work?, Sloan Management Review, 38 (4), Meyer-Waarden, L., 007, The effect of loyalty program on cutomer lifetime duration and hare of wallet, Journal of Retailing, 83 (), 3-36.

8 0. Singh, S. S., Jain, D. C., and Krihnan, T. V., 008, Cutomer loyalty program: are they profitable?, Management Science, 54 (6),05-.. Meyer-Waarden, L., 008, The influence of loyalty programme memberhip on cutomer purchae behaviour., European Journal of Marketing, 4 (), Nako, S. M., 99, Frequent flyer program and buine traveller: An empirical invetigation, Logitic and Tranportation Review, 8(4), Paingham, J., 998, Grocery retailing and the loyalty card, Journal of the Market Reearch Society, 40(), Smith, A., Spark, L., Hart, S., and Tzoka, N., 003, Retail loyalty cheme: reult from a conumer diary tudy, Journal of Retailing and Conumer Service, 0(), Taylor, G. A., and Nelin, S. A., 005, The current and future ale impact of a retail frequency reward program, Journal of Retailing, 8(4), Mägi, A. W., 003, Share of wallet in retailing: the effect of cutomer atifaction, loyalty card and hopper characteritic, Journal of Retailing, 79 (), Sharp, B., and Sharp, A., 997, Loyalty program and their impact on repeat-purchae loyalty pattern, International Journal of Reearch in Marketing, 4(5), Keh, H. T., and Lee, Y. H., 006, Do reward program build loyalty for ervice?: The moderating effect of atifaction on type and timing of reward, Journal of Retailing, 8(), Yi, Y., 989, A critical review of conumer atifaction, Review of Marketing, 4, Caminal, R., and Matute, C., 990, Endogenou witching cot in a duopoly model, International Journal of Indutrial Organization, 8(3), Byrd, R. H., Gilbert, J. C., and Nocedal, J., 000, A trut region method baed on interior point technique for nonlinear programming, Mathematical Programming, 89(), Ugray, Z., Ladon, L., Plummer, J., Glover, F., Kelly, J., and Martí, R., 007, Scatter earch and local NLP olver: A multitart framework for global optimization., INFORMS Journal on Computing, 9(3), Ladon, L., Duarte, A., Glover, F., Laguna, M., and Martí, R., 00, Adaptive memory programming for contrained global optimization, Computer and Operation Reearch, 37(8), Biyalogorky, E., Gertner, E., and Libai, B., 00 Cutomer referral management: Optimal reward program, Marketing Science, 0(), Homburg, C., Kochate, N., and Hoyer, W. D., 005, Do atified cutomer really pay more? A tudy of the relationhip between cutomer atifaction and willingne to pay, Journal of Marketing, 69(),