本文研究静态市场下的以质保期和价格为决策变量的最优控制决策问题。考虑了初次购买和重复购买两个过程以及重复购买过程是否受价格影响两种情况。模型中的生产成本考虑了产品生产过程中的学习效应, 质保成本使用了免费维修质保策略并假设产品两次故障之间的时间服从韦伯分布。在完整建模的基础上, 应用最大化原理对基本模型进行了求解, 之后本文给出了一种基本模型的具体应用场景, 并运用Lingo11软件进行了数值试验。最后对主要研究结论进行了应用讨论。
In this paper, a decision model is developed to determine the optimal price and warranty length for a static demand market. The demand function of the model characterizes both the repurchasing process and the initial purchase process. The cost functions consider both the learning effect and free repair warranty policy. And it is assumed that the time between two failures follows Weibull distribution. Two scenarios are provided, which are based on whether the repurchasing process is affected by price. The maximum principle method is used to obtain optimal solutions. The optimal policies for a special case are further investigated and Lingo11 is used to do some numerical tests. In this paper, it is found that price and warranty have to be considered in the same time, the best policy is to set high price and long warranty in the beginning and reduce them along the product life circle. Considering the repurchasing process, manufacture can get more profit and consumer can have better warranty. Finally, the management implication get from the model solutions is discussed.
[1] Blischke, W R. Mathematical models for analysis of warranty policies [J]. Mathematical and Computer Modelling, 1990, 13(7):1-16.
[2] Murthy D N P, Blischke W R. Product warranty management-III:A review of mathematical models [J]. European Journal of Operational Research, 1992, 63(1):1-34.
[3] Murthy D N P, Djamaludin I. New product warranty:A literature review [J]. International Journal of Production Economics, 2002, 79(3):231-260.
[4] Thomas M U, Rao S S. Warranty economic decision models:A summary and some suggested directions for future research [J]. Operations Research, 1999, 47(6):807-820.
[5] Murthy D N P, Solem O, Roren T. Product warranty logistics:Issues and challenges [J]. European Journal of Operational Research, 2004, 156(1):110-126.
[6] Karim M R, Suzuki K. Analysis of warranty claim data:A literature review [J]. International Journal of Quality & Reliability Management, 2005, 22(7):667-686.
[7] Emons W. The theory of warranty contracts [J]. Journal of Economic Surveys, 1989, 3(1):43-57.
[8] Glickman T S, Berger P D. Optimal price and protection period decisions for a product under warranty [J]. Management Science, 1976, 22(12):1381-1390.
[9] Zhou Zhefong, Li Yanjun, Tang K. Dynamic pricing and warranty policies for products with fixed lifetime [J]. European Journal of Operational Research, 2009, 196(3):940-948.
[10] Lin Peichun, Shue L Y. Application of optimal control theory to product pricing and warranty with free replacement under the influence of basic lifetime distributions [J]. Computers & Industrial Engineering, 2005, 48(1):69-82.
[11] Wu C C, Lin Peichun, Chou Chaoyu. Determination of price and warranty length for a normal lifetime distributed product [J]. International Journal of Production Economics, 2006, 102(1):95-107.
[12] Teng J T, Thompson G L. Optimal strategies for general price-quality decision models of new products with learning production costs [J]. European Journal of Operational Research, 1996, 93(3):476-489.
[13] Wu C C, Chou Chaoyu, Huang C. Optimal burn-in time and warranty length under fully renewing combination free replacement and pro-rata warranty [J]. Reliability Engineering & System Safety, 2007, 92(7):914-920.
[14] Wu C C, Chou Chaoyu, Huang C. Optimal price, warranty length and production rate for free replacement policy in the static demand market [J]. Omega, 2009, 37(1):29-39.
[15] Lin Peichun, Wang J, Chin S S. Dynamic optimization of price, warranty length and production rate [J]. International Journal of Systems Science, 2009, 40(4):411-420.
[16] Mesak H I. Modeling monopolist pricing and protection period decisions for new products under warranty [J]. Optimal Control Applications and Methods, 1996, 17(4):231-252.
[17] Bass F M. A new product growth for model consumer durables [J]. Management Science, 1969, 15(5):215-227.
[18] Blischke W R, Karim M R, Murthy D N P. Cost models for one-dimensional warranties in warranty data collection and analysis [M]. London:Springer London, 2011.
[19] Joachim K. Evolutionary model of an anonymous consumer durable market [J]. Physica A:Statistical Mechanics and its Applications, 2011, 390(14):2692-2715.
[20] 王海云, 尚志田. 重复购买的产品生命周期模型研究[J]. 中国管理科学, 2002, 10(2):24-29.
[21] Mesak H I, Berg W D. Incorporating price and replacement purchases in new product diffusion models for consumer durables [J]. Decision Sciences, 1995, 26(4):425-450.
[22] Huang Hongzhong, Liu Zhijie, Murthy D N P. Optimal reliability, warranty and price for new products [J]. IIE Transactions, 2007, 39(8):819-827.
