# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018112

## Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers

 1 School of Business, Central South University, Changsha 410083, China 2 Institute of Big Data and Internet Innovation, Hunan University of Commerce, Changsha 410205, China 3 School of Business, Central South University, Changsha 410083, China

* Corresponding author: Xiaohong Chen

Received  September 2016 Revised  May 2018 Published  August 2018

Recognizing that strategic consumers have become increasingly common in the perishable products market, we develop a two-stage pricing model for a monopolistic firm with two classes of inventory strategies: non-replenishment and replenishment. First, the retailer mapping out his pricing policy, and then consumers determining their buying behavior given the retailers policy. Our results indicate that the game equilibrium exists between retailers and consumers in both cases. For a given realized price and inventory policy, the consumer's space is split into several areas by the optimal threshold functions. Inventory replenishment decisions are affected by market demand and a decline factor of consumers reservation value. The retailers profit loss is not necessarily related to the consumers waiting behavior but results from the ignorance of this behavior when pricing.

Citation: Guodong Yi, Xiaohong Chen, Chunqiao Tan. Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018112
##### References:
 [1] F. J. Arcelus, N. H. Shah and G. Srinivasan, Retailer's pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives, International Journal of Production Economics, 81 (2003), 153-162. doi: 10.1016/S0925-5273(02)00269-4. [2] Y. Aviv and A. Pazgal, Optimal pricing of seasonal products in the presence of forward-looking consumers, Manufacturing and Service Operations Management, 10 (2008), 339-359. doi: 10.1287/msom.1070.0183. [3] D. E. Bell and D. Starr, Filene's basement, HBS Case Collection, 594-018(1993). [4] D. Besanko and W. L. Winston, Optimal price skimming by a monopolist facing rational consumers, Management Science, 36 (1990), 555-567. doi: 10.1287/mnsc.36.5.555. [5] G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science, 43 (1997), 64-79. doi: 10.1287/mnsc.43.1.64. [6] J. I. Bulow, Durable-goods monopolists, Journal of Political Economy, 90 (1982), 314-332. doi: 10.1086/261058. [7] G. P. Cachon and R. Swinney, Purchasing, pricing, and quick response in the presence of strategic consumers, Management Science, 55 (2008), 497-511. doi: 10.1287/mnsc.1080.0948. [8] G. P. Cachon and R. Swinney, The value of fast fashion: Quick response, enhanced design, and strategic consumer behavior, Management Science, 57 (2011), 778-795. doi: 10.1287/mnsc.1100.1303. [9] C. T. Chang, M. C. Cheng and L. Y. Ouyang, Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments, Applied Mathematical Modelling, 39 (2015), 747-763. doi: 10.1016/j.apm.2014.07.002. [10] R. H. Coase, Durability and monopoly, Journal of Law and Economics, 15 (1972), 143-149. doi: 10.1086/466731. [11] S. Dasuab, Dynamic pricing when consumers are strategic: Analysis of posted and contingent pricing schemes, European Journal of Operational Research, 204 (2010), 662-671. [12] J. Du, J. Zhang and G. Hua, Pricing and inventory management in the presence of strategic customers with risk preference and decreasing value, International Journal of Production Economics, 164 (2015), 160-166. doi: 10.1016/j.ijpe.2015.02.013. [13] C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging, Omega, 35 (2007), 184-189. doi: 10.1016/j.omega.2005.05.002. [14] W. Elmaghraby, A. Gülcü and P. Keskinocak, Designing optimal preannounced markdowns in the presence of rational customers with multiunit demands, Manufacturing and Service Operations Management, 10 (2007), 126-148. doi: 10.1287/msom.1070.0157. [15] G. Gallego, R. Phillips and O. ahin, Strategic management of distressed inventory, Production and Operations Management, 17 (2008), 402-415. [16] M. Ghoreishi, A. Mirzazadeh, G. W. Weber and I. Nakhai-Kamalabadi, Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns, Journal of Industrial and Management Optimization, 11 (2015), 933-949. doi: 10.3934/jimo.2015.11.933. [17] P. Hu, S. Shum and M. Yu, Joint inventory and markdown management for perishable goods with strategic consumer behavior, Operations Research, 64 (2016), 118-134. doi: 10.1287/opre.2015.1439. [18] H. Hwang and S. W. Shinn, Retailer's pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments, Computers and Operations Research, 24 (1997), 539-547. doi: 10.1016/S0305-0548(96)00069-X. [19] M. Landsberger and I. Meilijson, Intertemporal price discrimination and sales strategy under incomplete information, Rand Journal of Economics, 16 (1985), 424-430. [20] Y. Levin, J. Mcgill and M. Nediak, Dynamic pricing in the presence of strategic consumers and oligopolistic competition, Management Science, 55 (2008), 32-46. doi: 10.1287/mnsc.1080.0936. [21] Y. Levin, J. Mcgill and M. Nediak, Optimal dynamic pricing of perishable items by a monopolist facing strategic consumers, Production and Operations Management, 19 (2010), 40-60. doi: 10.1111/j.1937-5956.2009.01046.x. [22] C. Liang, M. Cakanyildirim and S. P. Sethi, Analysis of product rollover strategies in the presence of strategic customers, Management Science, 60 (2014), 1033-1056. [23] Q. Liu and D. Zhang, Dynamic pricing competition with strategic customers under vertical product differentiation, Management Science, 59 (2013), 84-101. doi: 10.1287/mnsc.1120.1564. [24] R. Maihami and B. Karimi, Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts, Computers and Operations Research, 51 (2014), 302-312. doi: 10.1016/j.cor.2014.05.022. [25] R. Maihami and I. N. Kamalabadi, Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand, International Journal of Production Economics, 136 (2012), 116-122. doi: 10.1016/j.ijpe.2011.09.020. [26] G. Mcwilliams, Analyzing customers, best buy decides not all are welcome, Wall Street Journal Online, 2004. [27] A. J. Mersereau and D. Zhang, Markdown pricing with unknown fraction of strategic customers, Manufacturing and Service Operations Management, 14 (2012), 355-370. doi: 10.1287/msom.1120.0376. [28] N. L. Stokey, Rational expectations and durable goods pricing, Bell Journal of Economics, 12 (2000), 112-128. doi: 10.2307/3003511. [29] N. L. Stokey, Intertemporal price discrimination, Quarterly Journal of Economics, 93 (1979), 355-371. doi: 10.2307/1883163. [30] X. Su, Intertemporal pricing with strategic customer behavior, Management Science, 53 (2007), 726-741. doi: 10.1287/mnsc.1060.0667. [31] X. Su and F. Zhang, On the value of commitment and availability guarantees when selling to strategic consumers, Management Science, 55 (2009), 713-726. [32] A. A. Taleizadeh and M. Noori-Daryan, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009. [33] H. M. Wee and S. T. Law, Replenishment and pricing policy for deteriorating items taking into account the time-value of money, International Journal of Production Economics, 71 (2001), 213-220. doi: 10.1016/S0925-5273(00)00121-3. [34] P. C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, 157 (2004), 389-397. doi: 10.1016/S0377-2217(03)00241-8. [35] P. C. Yang, H. M. Wee, S. L. Chung and Y. Y. Huang, Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand, Journal of Industrial and Management Optimization, 9 (2013), 769-787. doi: 10.3934/jimo.2013.9.769. [36] D. Yang, E. Qi and Y. Li, Quick response and supply chain structure with strategic consumers, Omega, 52 (2015), 1-14. doi: 10.1016/j.omega.2014.10.006. [37] R. Yin and C. S. Tang, The implications of customer purchasing behavior and in-store display formats, Decisions Operations and Technology Management. [38] R. Yin, Y. Aviv, A. Pazgal and C. S. Tang, Optimal markdown pricing: Implications of inventory display formats in the presence of strategic customers, Management Science, 55 (2009), 1391-1408. doi: 10.1287/mnsc.1090.1029. [39] J. Zhang, Z. Bai and W. Tang, Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277. doi: 10.3934/jimo.2014.10.1261.

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##### References:
 [1] F. J. Arcelus, N. H. Shah and G. Srinivasan, Retailer's pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives, International Journal of Production Economics, 81 (2003), 153-162. doi: 10.1016/S0925-5273(02)00269-4. [2] Y. Aviv and A. Pazgal, Optimal pricing of seasonal products in the presence of forward-looking consumers, Manufacturing and Service Operations Management, 10 (2008), 339-359. doi: 10.1287/msom.1070.0183. [3] D. E. Bell and D. Starr, Filene's basement, HBS Case Collection, 594-018(1993). [4] D. Besanko and W. L. Winston, Optimal price skimming by a monopolist facing rational consumers, Management Science, 36 (1990), 555-567. doi: 10.1287/mnsc.36.5.555. [5] G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science, 43 (1997), 64-79. doi: 10.1287/mnsc.43.1.64. [6] J. I. Bulow, Durable-goods monopolists, Journal of Political Economy, 90 (1982), 314-332. doi: 10.1086/261058. [7] G. P. Cachon and R. Swinney, Purchasing, pricing, and quick response in the presence of strategic consumers, Management Science, 55 (2008), 497-511. doi: 10.1287/mnsc.1080.0948. [8] G. P. Cachon and R. Swinney, The value of fast fashion: Quick response, enhanced design, and strategic consumer behavior, Management Science, 57 (2011), 778-795. doi: 10.1287/mnsc.1100.1303. [9] C. T. Chang, M. C. Cheng and L. Y. Ouyang, Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments, Applied Mathematical Modelling, 39 (2015), 747-763. doi: 10.1016/j.apm.2014.07.002. [10] R. H. Coase, Durability and monopoly, Journal of Law and Economics, 15 (1972), 143-149. doi: 10.1086/466731. [11] S. Dasuab, Dynamic pricing when consumers are strategic: Analysis of posted and contingent pricing schemes, European Journal of Operational Research, 204 (2010), 662-671. [12] J. Du, J. Zhang and G. Hua, Pricing and inventory management in the presence of strategic customers with risk preference and decreasing value, International Journal of Production Economics, 164 (2015), 160-166. doi: 10.1016/j.ijpe.2015.02.013. [13] C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging, Omega, 35 (2007), 184-189. doi: 10.1016/j.omega.2005.05.002. [14] W. Elmaghraby, A. Gülcü and P. Keskinocak, Designing optimal preannounced markdowns in the presence of rational customers with multiunit demands, Manufacturing and Service Operations Management, 10 (2007), 126-148. doi: 10.1287/msom.1070.0157. [15] G. Gallego, R. Phillips and O. ahin, Strategic management of distressed inventory, Production and Operations Management, 17 (2008), 402-415. [16] M. Ghoreishi, A. Mirzazadeh, G. W. Weber and I. Nakhai-Kamalabadi, Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns, Journal of Industrial and Management Optimization, 11 (2015), 933-949. doi: 10.3934/jimo.2015.11.933. [17] P. Hu, S. Shum and M. Yu, Joint inventory and markdown management for perishable goods with strategic consumer behavior, Operations Research, 64 (2016), 118-134. doi: 10.1287/opre.2015.1439. [18] H. Hwang and S. W. Shinn, Retailer's pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments, Computers and Operations Research, 24 (1997), 539-547. doi: 10.1016/S0305-0548(96)00069-X. [19] M. Landsberger and I. Meilijson, Intertemporal price discrimination and sales strategy under incomplete information, Rand Journal of Economics, 16 (1985), 424-430. [20] Y. Levin, J. Mcgill and M. Nediak, Dynamic pricing in the presence of strategic consumers and oligopolistic competition, Management Science, 55 (2008), 32-46. doi: 10.1287/mnsc.1080.0936. [21] Y. Levin, J. Mcgill and M. Nediak, Optimal dynamic pricing of perishable items by a monopolist facing strategic consumers, Production and Operations Management, 19 (2010), 40-60. doi: 10.1111/j.1937-5956.2009.01046.x. [22] C. Liang, M. Cakanyildirim and S. P. Sethi, Analysis of product rollover strategies in the presence of strategic customers, Management Science, 60 (2014), 1033-1056. [23] Q. Liu and D. Zhang, Dynamic pricing competition with strategic customers under vertical product differentiation, Management Science, 59 (2013), 84-101. doi: 10.1287/mnsc.1120.1564. [24] R. Maihami and B. Karimi, Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts, Computers and Operations Research, 51 (2014), 302-312. doi: 10.1016/j.cor.2014.05.022. [25] R. Maihami and I. N. Kamalabadi, Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand, International Journal of Production Economics, 136 (2012), 116-122. doi: 10.1016/j.ijpe.2011.09.020. [26] G. Mcwilliams, Analyzing customers, best buy decides not all are welcome, Wall Street Journal Online, 2004. [27] A. J. Mersereau and D. Zhang, Markdown pricing with unknown fraction of strategic customers, Manufacturing and Service Operations Management, 14 (2012), 355-370. doi: 10.1287/msom.1120.0376. [28] N. L. Stokey, Rational expectations and durable goods pricing, Bell Journal of Economics, 12 (2000), 112-128. doi: 10.2307/3003511. [29] N. L. Stokey, Intertemporal price discrimination, Quarterly Journal of Economics, 93 (1979), 355-371. doi: 10.2307/1883163. [30] X. Su, Intertemporal pricing with strategic customer behavior, Management Science, 53 (2007), 726-741. doi: 10.1287/mnsc.1060.0667. [31] X. Su and F. Zhang, On the value of commitment and availability guarantees when selling to strategic consumers, Management Science, 55 (2009), 713-726. [32] A. A. Taleizadeh and M. Noori-Daryan, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009. [33] H. M. Wee and S. T. Law, Replenishment and pricing policy for deteriorating items taking into account the time-value of money, International Journal of Production Economics, 71 (2001), 213-220. doi: 10.1016/S0925-5273(00)00121-3. [34] P. C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, 157 (2004), 389-397. doi: 10.1016/S0377-2217(03)00241-8. [35] P. C. Yang, H. M. Wee, S. L. Chung and Y. Y. Huang, Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand, Journal of Industrial and Management Optimization, 9 (2013), 769-787. doi: 10.3934/jimo.2013.9.769. [36] D. Yang, E. Qi and Y. Li, Quick response and supply chain structure with strategic consumers, Omega, 52 (2015), 1-14. doi: 10.1016/j.omega.2014.10.006. [37] R. Yin and C. S. Tang, The implications of customer purchasing behavior and in-store display formats, Decisions Operations and Technology Management. [38] R. Yin, Y. Aviv, A. Pazgal and C. S. Tang, Optimal markdown pricing: Implications of inventory display formats in the presence of strategic customers, Management Science, 55 (2009), 1391-1408. doi: 10.1287/mnsc.1090.1029. [39] J. Zhang, Z. Bai and W. Tang, Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277. doi: 10.3934/jimo.2014.10.1261.
Customer types for the case without inventory replenishment1
Customer types for the case with inventory replenishment2
Impact of Pricing Strategies on Consumer Waiting Behavior
Variation of the Profits of a Seller with an Initial Stock
Impact of Q and $\lambda$ on the Profits of Sellers
Profits of a Seller with $\alpha$ Change
Pricing Strategy and Retailer Expected Profits Under Both Scenarios3
 Q $[p_1, p_2]$ Expected profits Distribution of expected profits 0 [0.419, 0.392] 0.909 (45.4%, 9.8%, 44.8%) 1 [0.548, 0.362] 0.790 (1.1%, 42.8%, 37.5%, 13.4%, 5.1%)
 Q $[p_1, p_2]$ Expected profits Distribution of expected profits 0 [0.419, 0.392] 0.909 (45.4%, 9.8%, 44.8%) 1 [0.548, 0.362] 0.790 (1.1%, 42.8%, 37.5%, 13.4%, 5.1%)
Optimal Pricing and Expected Profit When Customers Are Non-strategic4
 Q $[p_1, p_2]$ Expected profits Distribution of expected profits 0 [0.454 0.357] 0.933 (56.2%, 43.8%) 1 [0.487 0.312] 0.862 (49.8%, 38.7%, 11.5%)
 Q $[p_1, p_2]$ Expected profits Distribution of expected profits 0 [0.454 0.357] 0.933 (56.2%, 43.8%) 1 [0.487 0.312] 0.862 (49.8%, 38.7%, 11.5%)
Influence of Consumer Waiting Behavior on Seller Profit6
 Case Ⅰ (%) Case Ⅱ(%) Q $\alpha=0.5$ $\alpha=1$ $\alpha=1.5$ $\alpha=0.5$ $\alpha=1$ $\alpha=1.5$ 0 -2.61 -10.04 -38.74 -4.95 -11.7 -33.31 1 -9.21 -16.73 -40.92 -4.64 -20.52 -71.89 5 7.08 22.32 52.68 -12.30 -17.79 -43.36
 Case Ⅰ (%) Case Ⅱ(%) Q $\alpha=0.5$ $\alpha=1$ $\alpha=1.5$ $\alpha=0.5$ $\alpha=1$ $\alpha=1.5$ 0 -2.61 -10.04 -38.74 -4.95 -11.7 -33.31 1 -9.21 -16.73 -40.92 -4.64 -20.52 -71.89 5 7.08 22.32 52.68 -12.30 -17.79 -43.36
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