OPTIMAL DESIGN OF A MULTITIER REWARD SCHEME. Amir Gandomi *, Saeed Zolfaghari **


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1 OPTIMAL DESIGN OF A MULTITIER REWARD SCHEME Amir Gandomi *, Saeed Zolfaghari ** Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Ontario * Tel.: x7702, ** Tel.: x7735, Various forms of customer loyalty programs have been designed and adopted in real world markets. A principal element of the loyalty programs design is the reward structure. In this study, an optimization model is developed for a commonlyused yet underexplored type of reward structure known as the multitier reward scheme. Multitier reward schemes offer disproportionately higher rewards to more loyal customers. We study a threelevel tiered scheme. The design elements are breakpoints and the reward value at each tier. We model an asymmetric duopoly market where only one firm adopts the multitier loyalty scheme. Customer choice behavior is modeled using the binarylogit model. The utility components are the price, reward and the distance to a reward. The customers accumulated purchase through different periods turns out to form a Markov chain. The transition probabilities are obtained to formulate the firm s revenue function. The revenue function is maximized in terms of the loyalty program s design elements. The structural properties of the optimal solution yield useful insights into the profitability of multitier loyalty programs. Keywords: Marketing, Optimization, Loyalty Programs, Multitier reward, Stochastic Programming.. Introduction Because of the wellestablished link between customer loyalty and profitability (Reichheld & Teal, 200; Smith & Sparks, 2009), acquiring loyal customers is of growing importance. Loyalty programs are one of the marketing strategies to build and enhance customers' loyalty and thereby increase a firm's longterm profitability (Gandomi & Zolfaghari, 20). Since early 980 s, the contemporary forms of loyalty programs have increasingly grown in popularity and have expanded in various industries including airlines, credit card companies, retail and hotel chains (Kim, Shi, & Srinivasan, 200; Kumar, 2008). On the other end, consumers participation in such programs has considerably evolved during the past few years. The 20 COLLOQUY Loyalty Census reveals that the number of loyalty memberships in the US exceeds 2 billion, netting out to more than 8 memberships per household (COLLOQUY, 20). This indicates a 4.7% increase in memberships in the US since Given the ubiquity of loyalty programs in practice, academics have recently shown interest in the area. The approaches to research on loyalty programs can be classified into two broad categories. While some researchers use empirical methods to assess the effect of loyalty programs on customers buying behavior (Kivetz & Simonson, 2002; B. Sharp & A. Sharp, 997), some other adopt mathematical models to analyze the effects of loyalty programs on a firm s profitability and market conditions (e.g., Kim et al., 200; Singh, Jain, & Krishnan, 2008; Gandomi & Zolfaghari, 20). However, virtually all existing empirical and analytical studies on loyalty program have considered a particular reward structure known as the linear reward. Specifically, multitier reward scheme has remained understudied in the OR/marketing literature. Based on a multitier reward scheme, more loyal customers earn disproportionately higher rewards. That is, multitier schemes offer different rewards to different levels of loyalty. In other words, the reward per dollar spent is not fixed and depends on some measure of customers purchase history. Some examples of such reward systems are Microsoft s Xbox Live Reward in the US, Shopper s Optimum Card in Canada and British Airways Executive Club in the UK. 992
2 The reward structure, as one of the main components of a loyalty program s design (Kumar, 2006, p. 72), is a key driver of the loyalty program s effectiveness. Thus, the lack of research on multitier reward schemes is a significant gap in the loyalty programs literature. In this study, a mathematical model is developed to address part of this gap. Specifically, we develop a model to optimize the profitability of the commonlyseen 3level reward schemes where customers loyalty status is determined based on their accumulated purchase at the end of a selling cycle. The decision variables are the breakpoints and reward values at each tier. We model a duopoly market where only one firm offers loyalty discounts. Customers choose one of the firms and buy one unit of the product in each period. Their choice behavior is modeled using binarylogit model where the utility is a function of the offered price, the reward value and the distance to the reward. By including the distance component in the utility, we address the fact that customers accelerate their purchasing process as they progress toward earning a particular reward (Kivetz, Urminsky, & Zheng, 2006). It will be shown that the customers accumulated purchase evolves as a Markov chain. The transition probabilities of the Markov chain are obtained in terms of the decision variables and are used to formulate the firm s expected revenue function. The revenue function is optimized in terms of the reward scheme s design elements. The optimal solution yields some useful insights into the profitability of multitier reward systems. The rest of the paper is organized as follows: section 2 describes the underlying assumptions and the model formulation. Section 3 presents some sample results derived from the model. We conclude in section 4 with a summary and the future directions. 2. The model formulation The market is served by two firms where only one firm offer loyalty rewards (Firm A). Firms sell the same good/service through the selling cycle which is divided into 4 discrete time periods. The market size is normalized to one and remains constant across periods. Suppose that both Firm A and Firm B offer the same price,, throughout the selling horizon. is incorporated as a parameter in the model. Firm adopts a threetier reward scheme. Customers progress through reward tiers based on their accumulated purchase at Firm. The reward is determined based on a customer s loyalty status at the end of the selling horizon. Customers who make enough purchases to achieve Tier 2 status earn a reward of and those who make it to Tier 3 win a reward of. and are the breakpoints that specify the limits of each loyalty level. Customers whose total purchase exceeds earn the Tier 3 reward. Tier 2 status is granted to accumulated purchases falling in the range,. If the total purchase is less than, no loyalty reward is offered. Note that and are the monetary values of the breakpoints.,, and are incorporated as Firm s decision variables in the model. In each period, customers buy one unit of the product, either at Firm or at Firm. Here, the binarylogit model is employed to model customers choice behavior. First, let,,,,...,4, () be a customer s utility from buying at Firm in period. is the marginal sensitivity of the utility to the price level and, is the random component of the utility which captures the heterogeneity in customers preferences. Similarly, we can formulate customers utility from buying at Firm. As mentioned earlier, we assume that each individual s utility depends on two additional factors: the reward level in the next tier and the 993
3 distance to the next tier. The values of both factors in a period depend on a customer s accumulated purchase up until the previous period. Let represent the reward value in the next loyalty tier. Based on Firm s reward scheme described earlier, it follows that:,,,4, (2), where denotes the total money spent by the customer at Firm up to period. We assume that 0, that is, customers initiate accumulating points at the beginning of the selling horizon. Moreover, let denote the distance to the next loyalty level. can be written as a function of as follows:,,,...,4. (3) 0, Having defined and, we can now formulate the net utility derived from making a purchase at Firm in period,,, as follows:,,,,...,4. (4) In the above equation, and denote the utility sensitivity to the price and to the distance from the next reward, respectively and, is the random disturbance term. In every period, customers choose between Firm and Firm. Intuitively, a customer chooses Firm over Firm in period if,,. Hence, the probability that the customer buys at Firm in period,,, is given by,,,,,,,...,4. (5) Now, we adopt the commonly used assumption that, and,,...,4) are i.i.d random variables and,,,...,4) has a logistic distribution with mean zero and scale parameter (BenAkiva & Lerman, 985, p. 07). Let. denote the CDF of the distribution. From Equation (5), it follows that,,,...,4. (6) Note that and are piecewise functions and depend upon, the customer s accumulated purchase in period. To formulate Firm s expected revenue function, we need to derive the probability distribution of the accumulated purchase at the end of the selling horizon. That is, we must find Pr for any 0,,4. It can be shown that,..,4 satisfies the conditional independence property and hence evolves according to a Markov chain. For notational simplicity, let,, 4 represent a customer s total number of purchases at Firm up to period. Clearly,,..,4 is also a Markov chain with the state space Ω0,,2,3, 4. Let be the transition probability matrix whose, element is, Pr,,,4, and 0,,4. (7) In other words,, is the probability that a customer s total number of purchases at Firm reaches given that he/she has bought units of the product at Firm up to the previous period. Considering the fact that each customer buys exactly one unit of the product in each period either at Firm or at Firm,, in the above equation can be expressed as 994
4 ,,,,, 0, 0,,4, (8) where, can be derived based on equations (2), (3) and (6) as,,,, 0,,4. (9) Now, Pr can be found using the 4step transition probability matrix of,. To simplify the notation, let,,,,5, (0) be a vector containing the elements of the first row of. Thus, the entry of represents the probability that a customer buys products over the entire selling cycle. Having found the total purchase probabilities, now we can formulate Firm s expected revenue function,, as follows:, () where denotes the expected cost of reward that Firm incurs during the selling horizon. Based on Firm A s reward structure, it can be shown that where. (2) :, 0,,4 and :, 0,,4. (3) Thus, the revenue function in Equation () can be rewritten as. (4) The purpose is to study the structural properties of Firm s optimal reward scheme. Thus, one can optimize Firm s revenue function,, in terms of,, and. Note that depends also on the model parameters (i.e.,, and ). So, the optimal solution will depend on parameter values. For analytical convenience, we normalize the price to and change and correspondingly, so that the choice probabilities remain the same. Subsequently, after obtaining the optimal solution, the values of decision variables and revenue functions must be scaled back with the same factor. The optimization model can be written as follows: 995
5 , (5),,, subject to:, (5a) 2, (5b), (5c) 0, (5d) 0, (5e), (5f), (5g),,, 0. (5h) The first two constraints state that at least two purchases are needed to achieve the second tier and minimum three purchases are required to earn the Tier3 loyalty status. Constraint (5c) guarantees that the reward will not exceed the offered price in any period. Constraints (5d) and (5e) ensure that the net utility that loyalty program creates is nonnegative. Constraints (5f) and (5g) refer to the fact that under optimal conditions, the value of the next reward must be less than or equal to the amount of money a customer must pay to gain it. In other words, the distance to the next reward must be less than the reward itself. Otherwise, a customer may basically earn money by making an additional purchase. Thus, without these two constraints, the assumption that customers make only one purchase in each period becomes implausible. 3. Sample results It can be shown that the above model is a nonconvex NLP. One can employ the interiorpoint algorithm proposed by Byrd, Gilbert, & Nocedal (2000) to find the optimal solution. Various analyses can be performed using the developed model. For instance, Figure () presents the effects of and on the optimal breakpoint values. As can be seen, and both increase with and decrease with. That is, as the sensitivity to distance from a reward increases, the firm is better off to set higher requirements for both Tier 2 and Tier 3 loyalty levels. On the other hand, as the customer s sensitivity to price increases, the optimal breakpoints decrease Figure. Optimal values of and under different levels of and 996
6 4. Conclusion In this paper, an analytical model was developed to optimize the structure of a 3tier loyalty reward scheme. The decision variables were the breakpoints and reward values of each tier. The purpose was to optimize the total revenue function. The formulation resulted in a nonlinear programming. The optimal solution of the model was found at some arbitrary values of and, the utility sensitivity coefficients. The results can be used to perform various analyses. For example, it is useful to study the effects of and on the optimal values of reward at each tier and each firm s overall profitability. Moreover, we can assume that Firm offers a lower price and then compare the effectiveness of loyalty programs with that of the lower price strategy. In our model, the selling cycle was broken into 4 discrete time periods. One can extend the model to a more general case where the cycle is divided into periods. The model can also be expanded by removing the assumption that the offered price is constant across periods. In fact, this assumption was made to ensure that customers accumulated purchases follow the Markov chain. The stochastic dynamic programming approach can be used to formulate and optimize the revenue function under the variable price assumption. Finally, one can incorporate the offered prices of each firm as decision variables and study the equilibrium conditions using game theoretic models. References BenAkiva, M., & Lerman, S. R. (985). Discrete choice analysis: Theory and application to travel demand (st ed.). The MIT Press. Byrd, R. H., Gilbert, J. C., & Nocedal, J. (2000). A trust region method based on interior point techniques for nonlinear programming. Mathematical Programming, 89(), COLLOQUY. (20, April). The billion member march: The 20 COLLOQUY loyalty census. Retrieved July 5, 20, from Paper.pdf Gandomi, A., & Zolfaghari, S. (20). A stochastic model on the profitability of loyalty programs. Computers & Industrial Engineering, In Press, Corrected Proof. Kim, B., Shi, M., & Srinivasan, K. (200). Reward programs and tacit collusion. Marketing Science, 20(2), Kivetz, R., & Simonson, I. (2002). Earning the right to indulge: Effort as a determinant of customer preferences toward frequency program rewards. Journal of Marketing Research, 39(2), Kivetz, R., Urminsky, O., & Zheng, Y. (2006). The goalgradient hypothesis resurrected: Purchase acceleration, illusionary goal progress, and customer retention. Journal of Marketing Research, 43(), Kumar, V. (2006). Customer relationship management: A database approach. Hoboken, N.J: John Wiley & Sons. Kumar, V. (2008). Managing customers for profit: Strategies to increase profits and build loyalty. Indianapolis, IN: Wharton School Publishing. Reichheld, F. F., & Teal, T. (200). The loyalty effect: the hidden force behind growth, profits, and lasting value. Boston, MA: Harvard Business Press. Sharp, B., & Sharp, A. (997). Loyalty programs and their impact on repeatpurchase loyalty patterns. International Journal of Research in Marketing, 4(5), Singh, S. S., Jain, D. C., & Krishnan, T. V. (2008). Customer loyalty programs: are they profitable? Management Science, 54(6), Smith, A., & Sparks, L. (2009). It s nice to get a wee treat if you ve had a bad week : Consumer motivations in retail loyalty scheme points redemption. Journal of Business Research, 62(5),
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