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

  • * Corresponding author: Jianxiong Zhang

    * Corresponding author: Jianxiong Zhang 
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  • 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.

    Mathematics Subject Classification: Primary: 49J52, 49M30.


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  • 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.

    $p_s^*$ $u_s^*$ $I_s^*$ $r_s^*$ $J_s^*$
    $\delta(0.8;1.0;1.2;1.4)$ $+$ $-$ $+$ $+$ $+$
    $\beta(0.25;0.5;0.75;1.0)$ $-$ $-$ $-$ $-$ $-$
    $\eta(0.35;0.55;0.75;0.95)$ $-$ $+$ $+$ $-$ $-,+$
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    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
     | Show Table
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