# American Institute of Mathematical Sciences

April  2019, 15(2): 667-688. doi: 10.3934/jimo.2018064

## A joint dynamic pricing and production model with asymmetric reference price effect

 College of Management and Economics, Tianjin University, Tianjin 300072, China

* Corresponding author: Jianxiong Zhang

Received  March 2016 Revised  March 2018 Published  June 2018

Reference price plays a significant role in influencing purchase decisions of customers. Due to loss aversion, the asymmetric reference price effect on market demand should be taken into account. This paper develops a joint dynamic pricing and production model with asymmetric reference price effect. In a finite planning horizon, the demand rate is time-varying and depends on price as well as reference price. The decision-making problem with the asymmetric reference price effect turns to be a nonsmooth optimal control problem, which cannot be solved by standard optimal control method. As a special case, we first obtain the joint optimal dynamic pricing and production strategy with symmetric reference price effect by solving the corresponding standard optimal control problem based on Maximum principle. For the case of asymmetric reference price effect, we propose a systematical method on basis of optimality principle to solve the nonsmooth optimal control problem, and obtain the joint strategy. Numerical examples are employed to illustrate the effectiveness of the proposed method. In addition, we assess the sensitivity analysis of system parameters to examine the impacts of asymmetric reference price on optimal pricing and production strategies and total profits.

Citation: Shichen Zhang, Jianxiong Zhang, Jiang Shen, Wansheng Tang. A joint dynamic pricing and production model with asymmetric reference price effect. Journal of Industrial & Management Optimization, 2019, 15 (2) : 667-688. doi: 10.3934/jimo.2018064
##### References:
 [1] E. Adida and G. Perakis, A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders, Naval Research Logistics, 54 (2007), 767-795.  doi: 10.1002/nav.20250.  Google Scholar [2] F. J. Arcelus, S. Kumar and G. Srinivasan, Pricing, rebate, advertising and ordering policies of a retailer facing price-dependent stochastic demand in newsvendor framework under different risk preferences, International Transactions in Operational Research, 13 (2006), 209-227.  doi: 10.1111/j.1475-3995.2006.00545.x.  Google Scholar [3] H. Arslan and S. Kachani, Dynamic pricing under consumer reference-price effects, Wiley Encyclopedia of Operations Research and Management Science, 2011. doi: 10.1002/9780470400531.eorms0273.  Google Scholar [4] I. S. Bakal, J. Geunes and H. E. Romeijn, Market selection decisions for inventory models with price-sensitive demand, Journal of Global Optimization, 41 (2008), 633-657.  doi: 10.1007/s10898-007-9269-3.  Google Scholar [5] W. Bi, G. Li and M. Liu, Dynamic pricing with stochastic reference effects based on a finite memory window, International Journal of Production Research, 55 (2017), 3331-3348.  doi: 10.1080/00207543.2016.1221160.  Google Scholar [6] G. Bitran and R. Caldentey, Commissioned Paper: An overview of pricing models for revenue management, Manufacturing & Service Operations Management, 5 (2003), 203-229.   Google Scholar [7] R. A. Briesch, L. Krishnamurthi and T. Mazumdar, A comparative analysis of reference price models, Journal of Consumer Research, 24 (1997), 202-214.  doi: 10.1086/209505.  Google Scholar [8] L. Caccetta and E. Mardaneh, Joint pricing and production planning of multi-period multi-product systems with uncertainty in demand, Pacific Journal of Optimization, 8 (2012), 121-134.   Google Scholar [9] K. Chen and T. Xiao, Pricing and replenishment policies in a supply chain with competing retailers under different retail behaviors, Computers & Industrial Engineering, 103 (2017), 145-157.  doi: 10.1016/j.cie.2016.11.018.  Google Scholar [10] T. H. Chen, Optimizing pricing, replenishment and rework decision for imperfect and deteriorating items in a manufacturer-retailer channel, International Journal of Production Economics, 183 (2017), 539-550.  doi: 10.1016/j.ijpe.2016.08.015.  Google Scholar [11] C. Y. Dye and C. T. Yang, Optimal dynamic pricing and preservation technology investment for deteriorating products with reference price effects, Omega, 62 (2016), 52-67.  doi: 10.1016/j.omega.2015.08.009.  Google Scholar [12] W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 47pp. Google Scholar [13] G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential dames with nonsmooth (asymmetric) reference-price effects, Operations Research, 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758.  Google Scholar [14] G. Fibich, A. Gavious and O. Lowengart, Optimal price promotion in the presence of asymmetric reference-price effects, Managerial and Decision Economics, 28 (2007), 569-577.   Google Scholar [15] M. Ghoreishi, A. Mirzazadeh and G. W. Weber, et al. 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.  Google Scholar [16] M. Guajardo, M. Kylinger and M. Ronnqvist, Joint optimization of pricing and planning decisions in divergent supply chain, International Transactions in Operational Research, 20 (2013), 889-916.   Google Scholar [17] T. P. Hsieh and C. Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European Journal of Operational Research, 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038.  Google Scholar [18] A. Kabirian, The economic production and pricing model with lot-size-dependent production cost, Journal of Global Optimization, 54 (2012), 1-15.  doi: 10.1007/s10898-011-9737-7.  Google Scholar [19] D. Kahneman and A. Tversky, Prospect Theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-291.   Google Scholar [20] M. U. Kalwani, C. K. Yim and H. J. Rinne, A price expectations model of customer brand choice, Journal of Marketing Research, 27 (1990), 251-262.  doi: 10.2307/3172584.  Google Scholar [21] G. Kalyanaram and J. D. C. Little, An empirical analysis of latitude of price acceptance in consumer package goods, Journal of Consumer Research, 21 (1994), 408-418.  doi: 10.1086/209407.  Google Scholar [22] G. Kalyanaram and R. S. Winer, Empirical generalizations from reference price research, Marketing Science, 14 (1995), 161-169.  doi: 10.1287/mksc.14.3.G161.  Google Scholar [23] L. Krishnamurthi, T. Mazumdar and S. P. Raj, Asymmetric response to price in consumer brand choice and purchase quantity decisions, Journal of Consumer Research, 19 (1992), 387-400.  doi: 10.1086/209309.  Google Scholar [24] J. M. Lattin and R. E. Bucklin, Reference effects of price and promotion on brand choice behavior, Journal of Marketing Research, 26 (1989), 299-310.  doi: 10.2307/3172902.  Google Scholar [25] Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E: Logistics and Transportation Review, 85 (2016), 52-68.  doi: 10.1016/j.tre.2015.11.002.  Google Scholar [26] H. Liu, X. Luo and W. Bi et al., Dynamic pricing of network goods in duopoly markets with boundedly rational consumers, Journal of Industrial and Management Optimization, 13 (2017), 427–445. doi: 10.3934/jimo.2016025.  Google Scholar [27] J. Liu, C. Wu and T. Su, The reference effect newsvendor model with strategic customers, Management Decision, 55 (2017), 1006-1021.  doi: 10.1108/MD-09-2015-0419.  Google Scholar [28] L. Lu, J. Zhang and W. Tang, Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand, International Journal of Systems Science, 47 (2016), 1480-1494.  doi: 10.1080/00207721.2014.938784.  Google Scholar [29] T. Mazumdar and P. Papatla, An investigation of reference price segments, Journal of Marketing Research, 37 (2000), 246-258.   Google Scholar [30] T. Mazumdar, S. P. Raj and I. Sinha, Reference price research: Review and propositions, Journal of Marketing, 69 (2005), 84-102.  doi: 10.1509/jmkg.2005.69.4.84.  Google Scholar [31] J. Nasiry and I. Popescu, Dynamic pricing with loss averse consumers and peak-end anchoring, Operations Research, 59 (2011), 1361-1368.  doi: 10.1287/opre.1110.0952.  Google Scholar [32] K. Pauwels and S. Siddarth, The long-term effects of price promotions on category incidence, brand choice, and purchase quantity, Journal of Marketing Research, 39 (2002), 421-439.   Google Scholar [33] D. Pekelman, Simultaneous price-production decisions, Operations Research, 22 (1974), 788-794.  doi: 10.1287/opre.22.4.788.  Google Scholar [34] I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393.  Google Scholar [35] M. Rabbani, N. P. Zia and H. Rafiei, Joint optimal dynamic pricing and replenishment policies for items with simultaneous quality and physical quantity deterioration, Applied Mathematics and Computation, 287 (2016), 149-160.  doi: 10.1016/j.amc.2016.04.016.  Google Scholar [36] K. N. Rajendran and G. J. Tellis, Contextual and temporal components of reference price, Journal of Marketing, 58 (1994), 22-34.  doi: 10.2307/1252248.  Google Scholar [37] A. Raman and F. M. Bass, A gereral test of reference price theory in the presence of threshold effects, Review of Business and Economics, 47 (2002), 205-226.   Google Scholar [38] R. V. Ramasesh, Lot-sizing decisions under limited-time price incentives: A review, Omega, 38 (2010), 118-135.  doi: 10.1016/j.omega.2009.07.002.  Google Scholar [39] R. T. Rust and A. J. Zahorik, Customer satisfaction, customer retention, and market share, Journal of Retailing, 69 (1993), 193-215.  doi: 10.1016/0022-4359(93)90003-2.  Google Scholar [40] S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, The Netherlands: Kluwer, 2000.  Google Scholar [41] G. Sorger, Reference price formation and optimal marketing strategies, Optimal Control Theory and Economic Analysis, 3 (1988), 97-120.   Google Scholar [42] A. Taudes and C. Rudloff, Integrating inventory control and a price change in the presence of reference price effects: A two-period model, Mathematical Methods of Operations Research, 75 (2012), 29-65.  doi: 10.1007/s00186-011-0374-1.  Google Scholar [43] S. Transchel and S. Minner, Dynamic pricing and replenishment in the warehouse scheduling problem: A common cycle approach, International Journal of Production Economics, 118 (2009), 331-338.  doi: 10.1016/j.ijpe.2008.08.046.  Google Scholar [44] Y. C. Tsao and G. J. Sheen, Joint pricing and replenishment decisions for deteriorating items with lot-size and time-dependent purchasing cost under credit period, International Journal of Systems Science, 38 (2007), 549-561.  doi: 10.1080/00207720701431144.  Google Scholar [45] T. L. Urban, Coordinating pricing and inventory decisions under reference price effects, International Journal of Manufacturing Technology and Management, 13 (2007), 78-94.  doi: 10.1504/IJMTM.2008.015975.  Google Scholar [46] B. L. Wang, X. U. Lei and X. P. Hong, Joint decision on priced produrement and dynamic inventory considering price reference effect, Systems Engineering, 29 (2011), 56-62.   Google Scholar [47] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.   Google Scholar [48] R. S. Winer, A reference price model of brand choice for frequently purchased products, Journal of Consumer Research, 13 (1986), 250-256.  doi: 10.1086/209064.  Google Scholar [49] M. Xue, W. Tang and J. Zhang, Optimal dynamic pricing for deteriorating items with reference-price effects, International Journal of Systems Science, 47 (2016), 2022-2031.  doi: 10.1080/00207721.2014.970598.  Google Scholar [50] P. C. Yang, H. M. Wee and S. L. Chung, et al. 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.  Google Scholar [51] H. Yang, D. Zhang and C. Zhang, The influence of reference effect on pricing strategies in revenue management settings, International Transactions in Operational Research, 24 (2017), 907-924.  doi: 10.1111/itor.12371.  Google Scholar [52] J. Zhang, J. Chen and C. Lee, Coordinated pricing and inventory control problems with capacity constraints and fixed ordering cost, Naval Research Logistics, 59 (2012), 376-383.  doi: 10.1002/nav.21495.  Google Scholar [53] J. Zhang, Q. Gou and L. Liang, Supply chain coordination through cooperative advertising with reference price effect, Omega, 41 (2013), 345-353.  doi: 10.1016/j.omega.2012.03.009.  Google Scholar [54] 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.  Google Scholar [55] J. Zhang, W. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135.  doi: 10.1016/j.omega.2013.07.002.  Google Scholar

show all references

##### References:
 [1] E. Adida and G. Perakis, A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders, Naval Research Logistics, 54 (2007), 767-795.  doi: 10.1002/nav.20250.  Google Scholar [2] F. J. Arcelus, S. Kumar and G. Srinivasan, Pricing, rebate, advertising and ordering policies of a retailer facing price-dependent stochastic demand in newsvendor framework under different risk preferences, International Transactions in Operational Research, 13 (2006), 209-227.  doi: 10.1111/j.1475-3995.2006.00545.x.  Google Scholar [3] H. Arslan and S. Kachani, Dynamic pricing under consumer reference-price effects, Wiley Encyclopedia of Operations Research and Management Science, 2011. doi: 10.1002/9780470400531.eorms0273.  Google Scholar [4] I. S. Bakal, J. Geunes and H. E. Romeijn, Market selection decisions for inventory models with price-sensitive demand, Journal of Global Optimization, 41 (2008), 633-657.  doi: 10.1007/s10898-007-9269-3.  Google Scholar [5] W. Bi, G. Li and M. Liu, Dynamic pricing with stochastic reference effects based on a finite memory window, International Journal of Production Research, 55 (2017), 3331-3348.  doi: 10.1080/00207543.2016.1221160.  Google Scholar [6] G. Bitran and R. Caldentey, Commissioned Paper: An overview of pricing models for revenue management, Manufacturing & Service Operations Management, 5 (2003), 203-229.   Google Scholar [7] R. A. Briesch, L. Krishnamurthi and T. Mazumdar, A comparative analysis of reference price models, Journal of Consumer Research, 24 (1997), 202-214.  doi: 10.1086/209505.  Google Scholar [8] L. Caccetta and E. Mardaneh, Joint pricing and production planning of multi-period multi-product systems with uncertainty in demand, Pacific Journal of Optimization, 8 (2012), 121-134.   Google Scholar [9] K. Chen and T. Xiao, Pricing and replenishment policies in a supply chain with competing retailers under different retail behaviors, Computers & Industrial Engineering, 103 (2017), 145-157.  doi: 10.1016/j.cie.2016.11.018.  Google Scholar [10] T. H. Chen, Optimizing pricing, replenishment and rework decision for imperfect and deteriorating items in a manufacturer-retailer channel, International Journal of Production Economics, 183 (2017), 539-550.  doi: 10.1016/j.ijpe.2016.08.015.  Google Scholar [11] C. Y. Dye and C. T. Yang, Optimal dynamic pricing and preservation technology investment for deteriorating products with reference price effects, Omega, 62 (2016), 52-67.  doi: 10.1016/j.omega.2015.08.009.  Google Scholar [12] W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 47pp. Google Scholar [13] G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential dames with nonsmooth (asymmetric) reference-price effects, Operations Research, 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758.  Google Scholar [14] G. Fibich, A. Gavious and O. Lowengart, Optimal price promotion in the presence of asymmetric reference-price effects, Managerial and Decision Economics, 28 (2007), 569-577.   Google Scholar [15] M. Ghoreishi, A. Mirzazadeh and G. W. Weber, et al. 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.  Google Scholar [16] M. Guajardo, M. Kylinger and M. Ronnqvist, Joint optimization of pricing and planning decisions in divergent supply chain, International Transactions in Operational Research, 20 (2013), 889-916.   Google Scholar [17] T. P. Hsieh and C. Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European Journal of Operational Research, 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038.  Google Scholar [18] A. Kabirian, The economic production and pricing model with lot-size-dependent production cost, Journal of Global Optimization, 54 (2012), 1-15.  doi: 10.1007/s10898-011-9737-7.  Google Scholar [19] D. Kahneman and A. Tversky, Prospect Theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-291.   Google Scholar [20] M. U. Kalwani, C. K. Yim and H. J. Rinne, A price expectations model of customer brand choice, Journal of Marketing Research, 27 (1990), 251-262.  doi: 10.2307/3172584.  Google Scholar [21] G. Kalyanaram and J. D. C. Little, An empirical analysis of latitude of price acceptance in consumer package goods, Journal of Consumer Research, 21 (1994), 408-418.  doi: 10.1086/209407.  Google Scholar [22] G. Kalyanaram and R. S. Winer, Empirical generalizations from reference price research, Marketing Science, 14 (1995), 161-169.  doi: 10.1287/mksc.14.3.G161.  Google Scholar [23] L. Krishnamurthi, T. Mazumdar and S. P. Raj, Asymmetric response to price in consumer brand choice and purchase quantity decisions, Journal of Consumer Research, 19 (1992), 387-400.  doi: 10.1086/209309.  Google Scholar [24] J. M. Lattin and R. E. Bucklin, Reference effects of price and promotion on brand choice behavior, Journal of Marketing Research, 26 (1989), 299-310.  doi: 10.2307/3172902.  Google Scholar [25] Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E: Logistics and Transportation Review, 85 (2016), 52-68.  doi: 10.1016/j.tre.2015.11.002.  Google Scholar [26] H. Liu, X. Luo and W. Bi et al., Dynamic pricing of network goods in duopoly markets with boundedly rational consumers, Journal of Industrial and Management Optimization, 13 (2017), 427–445. doi: 10.3934/jimo.2016025.  Google Scholar [27] J. Liu, C. Wu and T. Su, The reference effect newsvendor model with strategic customers, Management Decision, 55 (2017), 1006-1021.  doi: 10.1108/MD-09-2015-0419.  Google Scholar [28] L. Lu, J. Zhang and W. Tang, Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand, International Journal of Systems Science, 47 (2016), 1480-1494.  doi: 10.1080/00207721.2014.938784.  Google Scholar [29] T. Mazumdar and P. Papatla, An investigation of reference price segments, Journal of Marketing Research, 37 (2000), 246-258.   Google Scholar [30] T. Mazumdar, S. P. Raj and I. Sinha, Reference price research: Review and propositions, Journal of Marketing, 69 (2005), 84-102.  doi: 10.1509/jmkg.2005.69.4.84.  Google Scholar [31] J. Nasiry and I. Popescu, Dynamic pricing with loss averse consumers and peak-end anchoring, Operations Research, 59 (2011), 1361-1368.  doi: 10.1287/opre.1110.0952.  Google Scholar [32] K. Pauwels and S. Siddarth, The long-term effects of price promotions on category incidence, brand choice, and purchase quantity, Journal of Marketing Research, 39 (2002), 421-439.   Google Scholar [33] D. Pekelman, Simultaneous price-production decisions, Operations Research, 22 (1974), 788-794.  doi: 10.1287/opre.22.4.788.  Google Scholar [34] I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393.  Google Scholar [35] M. Rabbani, N. P. Zia and H. Rafiei, Joint optimal dynamic pricing and replenishment policies for items with simultaneous quality and physical quantity deterioration, Applied Mathematics and Computation, 287 (2016), 149-160.  doi: 10.1016/j.amc.2016.04.016.  Google Scholar [36] K. N. Rajendran and G. J. Tellis, Contextual and temporal components of reference price, Journal of Marketing, 58 (1994), 22-34.  doi: 10.2307/1252248.  Google Scholar [37] A. Raman and F. M. Bass, A gereral test of reference price theory in the presence of threshold effects, Review of Business and Economics, 47 (2002), 205-226.   Google Scholar [38] R. V. Ramasesh, Lot-sizing decisions under limited-time price incentives: A review, Omega, 38 (2010), 118-135.  doi: 10.1016/j.omega.2009.07.002.  Google Scholar [39] R. T. Rust and A. J. Zahorik, Customer satisfaction, customer retention, and market share, Journal of Retailing, 69 (1993), 193-215.  doi: 10.1016/0022-4359(93)90003-2.  Google Scholar [40] S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, The Netherlands: Kluwer, 2000.  Google Scholar [41] G. Sorger, Reference price formation and optimal marketing strategies, Optimal Control Theory and Economic Analysis, 3 (1988), 97-120.   Google Scholar [42] A. Taudes and C. Rudloff, Integrating inventory control and a price change in the presence of reference price effects: A two-period model, Mathematical Methods of Operations Research, 75 (2012), 29-65.  doi: 10.1007/s00186-011-0374-1.  Google Scholar [43] S. Transchel and S. Minner, Dynamic pricing and replenishment in the warehouse scheduling problem: A common cycle approach, International Journal of Production Economics, 118 (2009), 331-338.  doi: 10.1016/j.ijpe.2008.08.046.  Google Scholar [44] Y. C. Tsao and G. J. Sheen, Joint pricing and replenishment decisions for deteriorating items with lot-size and time-dependent purchasing cost under credit period, International Journal of Systems Science, 38 (2007), 549-561.  doi: 10.1080/00207720701431144.  Google Scholar [45] T. L. Urban, Coordinating pricing and inventory decisions under reference price effects, International Journal of Manufacturing Technology and Management, 13 (2007), 78-94.  doi: 10.1504/IJMTM.2008.015975.  Google Scholar [46] B. L. Wang, X. U. Lei and X. P. Hong, Joint decision on priced produrement and dynamic inventory considering price reference effect, Systems Engineering, 29 (2011), 56-62.   Google Scholar [47] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.   Google Scholar [48] R. S. Winer, A reference price model of brand choice for frequently purchased products, Journal of Consumer Research, 13 (1986), 250-256.  doi: 10.1086/209064.  Google Scholar [49] M. Xue, W. Tang and J. Zhang, Optimal dynamic pricing for deteriorating items with reference-price effects, International Journal of Systems Science, 47 (2016), 2022-2031.  doi: 10.1080/00207721.2014.970598.  Google Scholar [50] P. C. Yang, H. M. Wee and S. L. Chung, et al. 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.  Google Scholar [51] H. Yang, D. Zhang and C. Zhang, The influence of reference effect on pricing strategies in revenue management settings, International Transactions in Operational Research, 24 (2017), 907-924.  doi: 10.1111/itor.12371.  Google Scholar [52] J. Zhang, J. Chen and C. Lee, Coordinated pricing and inventory control problems with capacity constraints and fixed ordering cost, Naval Research Logistics, 59 (2012), 376-383.  doi: 10.1002/nav.21495.  Google Scholar [53] J. Zhang, Q. Gou and L. Liang, Supply chain coordination through cooperative advertising with reference price effect, Omega, 41 (2013), 345-353.  doi: 10.1016/j.omega.2012.03.009.  Google Scholar [54] 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.  Google Scholar [55] J. Zhang, W. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135.  doi: 10.1016/j.omega.2013.07.002.  Google Scholar
Optimal price $p_s^*$ and reference price $r_s^*$.
Total profit $J_a$ via the intersection time $\tau$.
Optimal price $p_a^*$ and reference price $r_a^*$.
Impact of $\theta$ on the optimal price $p_a^*$ and production $u_a^*$.
Impact of $\theta$ on the total profit $J_a^*$.
Impact of $\delta$ on the optimal price $p_a^*$ and production $u_a^*$.
Impact of $\delta$ on the total profit $J_a^*$.
Impact of $\beta$ on the optimal price $p_a^*$ and production $u_a^*$.
Impact of $\beta$ on the total profit $J_a^*$.
Impact of $\eta$ on the optimal price $p_a^*$ and production $u_a^*$.
Impact of $\eta$ on the total profit $J_a^*$.
Variations in optimal outcomes in the symmetric case.
 $p_s^*$ $u_s^*$ $I_s^*$ $r_s^*$ $J_s^*$ $\delta(0.8;1.0;1.2;1.4)$ $+$ $-$ $+$ $+$ $+$ $\beta(0.25;0.5;0.75;1.0)$ $-$ $-$ $-$ $-$ $-$ $\eta(0.35;0.55;0.75;0.95)$ $-$ $+$ $+$ $-$ $-,+$
 $p_s^*$ $u_s^*$ $I_s^*$ $r_s^*$ $J_s^*$ $\delta(0.8;1.0;1.2;1.4)$ $+$ $-$ $+$ $+$ $+$ $\beta(0.25;0.5;0.75;1.0)$ $-$ $-$ $-$ $-$ $-$ $\eta(0.35;0.55;0.75;0.95)$ $-$ $+$ $+$ $-$ $-,+$
The optimal intersection time $\tau^*$ with different $\theta$.
 $\theta$ 0.05 0.1 0.15 0.2 0.25 0.3 $\tau^*$ 1.14 1.21 1.29 1.36 1.45 1.53
 $\theta$ 0.05 0.1 0.15 0.2 0.25 0.3 $\tau^*$ 1.14 1.21 1.29 1.36 1.45 1.53
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