# 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
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##### References:
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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