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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, 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.

[2]

F. J. ArcelusS. 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.

[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.

[4]

I. S. BakalJ. 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.

[5]

W. BiG. 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.

[6]

G. Bitran and R. Caldentey, Commissioned Paper: An overview of pricing models for revenue management, Manufacturing & Service Operations Management, 5 (2003), 203-229.

[7]

R. A. BrieschL. Krishnamurthi and T. Mazumdar, A comparative analysis of reference price models, Journal of Consumer Research, 24 (1997), 202-214. doi: 10.1086/209505.

[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.

[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.

[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.

[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.

[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.

[13]

G. FibichA. 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.

[14]

G. FibichA. Gavious and O. Lowengart, Optimal price promotion in the presence of asymmetric reference-price effects, Managerial and Decision Economics, 28 (2007), 569-577.

[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.

[16]

M. GuajardoM. Kylinger and M. Ronnqvist, Joint optimization of pricing and planning decisions in divergent supply chain, International Transactions in Operational Research, 20 (2013), 889-916.

[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.

[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.

[19]

D. Kahneman and A. Tversky, Prospect Theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-291.

[20]

M. U. KalwaniC. 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.

[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.

[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.

[23]

L. KrishnamurthiT. 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.

[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.

[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.

[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.

[27]

J. LiuC. 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.

[28]

L. LuJ. 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.

[29]

T. Mazumdar and P. Papatla, An investigation of reference price segments, Journal of Marketing Research, 37 (2000), 246-258.

[30]

T. MazumdarS. 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.

[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.

[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.

[33]

D. Pekelman, Simultaneous price-production decisions, Operations Research, 22 (1974), 788-794. doi: 10.1287/opre.22.4.788.

[34]

I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429. doi: 10.1287/opre.1070.0393.

[35]

M. RabbaniN. 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.

[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.

[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.

[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.

[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.

[40]

S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, The Netherlands: Kluwer, 2000.

[41]

G. Sorger, Reference price formation and optimal marketing strategies, Optimal Control Theory and Economic Analysis, 3 (1988), 97-120.

[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.

[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.

[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.

[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.

[46]

B. L. WangX. U. Lei and X. P. Hong, Joint decision on priced produrement and dynamic inventory considering price reference effect, Systems Engineering, 29 (2011), 56-62.

[47]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.

[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.

[49]

M. XueW. 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.

[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.

[51]

H. YangD. 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.

[52]

J. ZhangJ. 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.

[53]

J. ZhangQ. 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.

[54]

J. ZhangZ. 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.

[55]

J. ZhangW. 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.

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.

[2]

F. J. ArcelusS. 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.

[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.

[4]

I. S. BakalJ. 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.

[5]

W. BiG. 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.

[6]

G. Bitran and R. Caldentey, Commissioned Paper: An overview of pricing models for revenue management, Manufacturing & Service Operations Management, 5 (2003), 203-229.

[7]

R. A. BrieschL. Krishnamurthi and T. Mazumdar, A comparative analysis of reference price models, Journal of Consumer Research, 24 (1997), 202-214. doi: 10.1086/209505.

[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.

[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.

[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.

[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.

[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.

[13]

G. FibichA. 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.

[14]

G. FibichA. Gavious and O. Lowengart, Optimal price promotion in the presence of asymmetric reference-price effects, Managerial and Decision Economics, 28 (2007), 569-577.

[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.

[16]

M. GuajardoM. Kylinger and M. Ronnqvist, Joint optimization of pricing and planning decisions in divergent supply chain, International Transactions in Operational Research, 20 (2013), 889-916.

[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.

[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.

[19]

D. Kahneman and A. Tversky, Prospect Theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-291.

[20]

M. U. KalwaniC. 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.

[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.

[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.

[23]

L. KrishnamurthiT. 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.

[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.

[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.

[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.

[27]

J. LiuC. 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.

[28]

L. LuJ. 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.

[29]

T. Mazumdar and P. Papatla, An investigation of reference price segments, Journal of Marketing Research, 37 (2000), 246-258.

[30]

T. MazumdarS. 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.

[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.

[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.

[33]

D. Pekelman, Simultaneous price-production decisions, Operations Research, 22 (1974), 788-794. doi: 10.1287/opre.22.4.788.

[34]

I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429. doi: 10.1287/opre.1070.0393.

[35]

M. RabbaniN. 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.

[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.

[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.

[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.

[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.

[40]

S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, The Netherlands: Kluwer, 2000.

[41]

G. Sorger, Reference price formation and optimal marketing strategies, Optimal Control Theory and Economic Analysis, 3 (1988), 97-120.

[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.

[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.

[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.

[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.

[46]

B. L. WangX. U. Lei and X. P. Hong, Joint decision on priced produrement and dynamic inventory considering price reference effect, Systems Engineering, 29 (2011), 56-62.

[47]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.

[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.

[49]

M. XueW. 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.

[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.

[51]

H. YangD. 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.

[52]

J. ZhangJ. 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.

[53]

J. ZhangQ. 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.

[54]

J. ZhangZ. 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.

[55]

J. ZhangW. 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.

Figure 1.  Optimal price $p_s^*$ and reference price $r_s^*$.
Figure 2.  Total profit $J_a$ via the intersection time $\tau$.
Figure 3.  Optimal price $p_a^*$ and reference price $r_a^*$.
Figure 4.  Impact of $\theta$ on the optimal price $p_a^*$ and production $u_a^*$.
Figure 5.  Impact of $\theta$ on the total profit $J_a^*$.
Figure 6.  Impact of $\delta$ on the optimal price $p_a^*$ and production $u_a^*$.
Figure 7.  Impact of $\delta$ on the total profit $J_a^*$.
Figure 8.  Impact of $\beta$ on the optimal price $p_a^*$ and production $u_a^*$.
Figure 9.  Impact of $\beta$ on the total profit $J_a^*$.
Figure 10.  Impact of $\eta$ on the optimal price $p_a^*$ and production $u_a^*$.
Figure 11.  Impact of $\eta$ on the total profit $J_a^*$.
Table 1.  Variations in optimal outcomes in the symmetric case.
$p_s^*$ $u_s^*$ $I_s^*$ $r_s^*$ $J_s^*$
$\delta(0.8;1.0;1.2;1.4)$ $+$ $-$ $+$ $+$ $+$
$\beta(0.25;0.5;0.75;1.0)$ $-$ $-$ $-$ $-$ $-$
$\eta(0.35;0.55;0.75;0.95)$ $-$ $+$ $+$ $-$ $-,+$
$p_s^*$ $u_s^*$ $I_s^*$ $r_s^*$ $J_s^*$
$\delta(0.8;1.0;1.2;1.4)$ $+$ $-$ $+$ $+$ $+$
$\beta(0.25;0.5;0.75;1.0)$ $-$ $-$ $-$ $-$ $-$
$\eta(0.35;0.55;0.75;0.95)$ $-$ $+$ $+$ $-$ $-,+$
Table 2.  The optimal intersection time $\tau^*$ with different $\theta$.
$\theta$ 0.05 0.10 0.15 0.20 0.25 0.30
$\tau^*$ 1.14 1.21 1.29 1.36 1.45 1.53
$\theta$ 0.05 0.10 0.15 0.20 0.25 0.30
$\tau^*$ 1.14 1.21 1.29 1.36 1.45 1.53
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