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

Shichen Zhang,Wansheng Tang,Jianxiong Zhang,Jiang Shen

doi:10.3934/jimo.2018064

Abstract

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.

Highlights

Reference price, as a cognitive price benchmark which depends on past product prices and affects purchase decisions of consumers ( Kalwani [20]), has received a great deal of attention recently
The asymmetric reference effects in a continuous and dynamic environment leads to a nonsmmoth optimization problem, which cannot be solved by the standard optimal control method, and we propose a systematical method on basis of optimality principle to solve it
Asymmetric reference price effect, as an important factor to be considered in pricing models, will lead to a nonsmooth optimization problem which

Summary

Introduction

As a cognitive price benchmark which depends on past product prices and affects purchase decisions of consumers ( Kalwani [20]), has received a great deal of attention recently. In this paper, we address the joint optimal pricing and production decision-making problem to maximize the profit for a firm in a finite planning horizon where both reference price and inventory level are dynamic, while simultaneously considering the asymmetric reference price effect. The asymmetry of the reference price effect on demand stemming from the loss aversion leads to the nonsmooth inventory dynamics and the non-continuity of objective functional with respect to the reference price, and brings about a nonsmooth optimal control problem (8) which cannot be solved by using the standard optimal control method. The optimal pricing policy p∗a, production policy u∗a, inventory level Ia∗ and reference price ra∗ during the interval [0, τ ], respectively, are given by p∗a(t) k1 (c0 + c1 −c2)er1t +k2 (c0 + c1 +c2)e−r1t +k3 (c0 +c1 +c3)er3t.

Conclusions

I r λ 1