﻿ 零售商多种价格策略下的最优订购模型
 控制与决策  2020, Vol. 35 Issue (5): 1231-1239 0

### 引用本文 [复制中英文]

[复制中文]
MENG Zhi-qing, MU Yun-zhi, XU Lei-yan, ZHENG Min-chao. Optimal ordering strategy for retailers under multiple-price strategy[J]. Control and Decision, 2020, 35(5): 1231-1239. DOI: 10.13195/j.kzyjc.2018.1257.
[复制英文]

### 文章历史

1. 浙江工业大学 管理学院，杭州 310023;
2. 浙江理工大学 科技与艺术学院，浙江 绍兴 312369;
3. 浙江财经大学 教务处，杭州 310018

Optimal ordering strategy for retailers under multiple-price strategy
MENG Zhi-qing 1, MU Yun-zhi 1,2, XU Lei-yan 1,3, ZHENG Min-chao 1
1. School of Management, Zhejiang University of Technology, Hangzhou 310023, China;
2. Keyi College of Zhejiang Sci-Tech University, Shaoxing 312369, China;
3. Academic Affairs Office, Zhejiang University of Finance and Economics, Hangzhou 310018, China
Abstract: It is a popular discount strategy for retailers to sell the same product at multiple prices simultaneously (multiple-price strategy). In order to determine the order quantity of the retailer such that the expected profit is maximized, the classical newsboy problem is extended, and a new newsboy model is established wherein multiple prices correspond to uncertain consumer demand. Two aspects of with and without order quantity constraint are discussed respectively. The Lagrangian multiplier method is applied to solve the order quantity constraint problem, and an algorithm for solving the approximate optimal total order quantity is designed. Numerical results show that the multiple-price strategy is better than a single price strategy, and order quantity constraint has an impact on the choice of the retailers' multiple-price strategy. The retailer's multiple-price strategy is also affected by the price discount coefficient and the difference in demand in the case of order quantity constraint.
Keywords: newsboy problem    optimal ordering strategy    Lagrangian multiplier method    multiple-price    order quantity constraint
0 引言

Lu等[16]对多种价格策略问题的研究做出了重要贡献, 他们基于需求不确定的假定, 分析了采用双重价格策略的零售商如何确定最优定价和库存策略以实现期望利润最大化.诚然, 文献[16]对于多种价格策略下零售商最优订购问题的研究具有重要的启发意义, 但是该研究仅考虑了两种价格情况, 并没有考虑订购量约束条件.为此, 在文献[16]的基础上, 本文考虑订购量约束, 研究更一般化的数量折扣方式(零售商同时采用多种销售价格), 得到为实现利润最大化目标的零售商最优订购策略, 并分析多种价格策略相较于单一价格策略的优势.本文所提的多种价格同时销售的零售商最优订购策略能够在一定程度上填补多种价格策略研究的空白, 是对现有收益管理研究的有益补充; 本文从无约束和有约束两种情况下分析同一阶段多种销售价格的问题, 对相应的结果进行对比分析, 能更加突出所得结论和贴近零售商运行现实.论文的主要贡献是:建立了在多种销售价格下一个新的报童模型, 获得了采用多价格策略销售同一种产品的近似最优订购算法.

1 模型描述与求解 1.1 无约束期望利润订购模型

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

n=1时, 式(6)即为零售商采用单一价格策略时的最优总订购量, 此时与经典报童模型的结论相同.

1.2 考虑有订购量约束的期望利润订购模型

 (7)

 (8)

 (9)

 (10)

pi=(1-(i-1)d)p1代入式(7)的目标函数, 易知期望利润关于d的一阶导数小于0, 于是有以下结论.

step 1:确定拉格朗日乘子λ的取值区间, 设, 则λminλλmax.

step 2:在λ的取值区间内均匀地选取M+1个值, 设Δλ=(λmax-λmin)/M, 将λ的取值区间划分成长度为ΔλM个子区间, λ(k)取值为

step 3:将λ(k)代入式(9)计算qik, i=1, 2, …, n, k=0, 1, …, M, 计算总订购量=0, 1, …, M.

step 4:确定拉格朗日乘子的近似最优值, 寻找最接近a=.

step 5:计算订购量约束情况下零售商的近似最优总订购量和相应的期望利润, 比较a.

a, 则近似最优总订购量为=, 将qik*代入式(7)的目标函数, 计算零售商的期望利润

, 根据前文对定理2的说明, 取总订购量为a, 若, 则零售商的期望利润为

2 数值分析

2.1 参数设置

2.2 最优订购策略

 图 1 零售商的总订购量
 图 2 零售商的期望利润

2.3 订购成本对零售商多种价格策略的影响

 图 3 单位订购成本对零售商期望利润的影响

2.4 折扣系数对零售商多种价格策略的影响

 图 4 价格折扣系数对零售商期望利润的影响

2.5 需求差异性对零售商多种价格策略的影响

3 最优策略的启示

4 结论

1) 考虑转移概率, 探究消费者面对同种产品有不同数量价格时的购买倾向;

2) 采用多种价格策略时, 考虑需求预测的零售商最优订购决策;

3) 本文研究了风险中性零售商的最优订购策略, 后续的研究方向是从决策者行为的角度研究风险厌恶零售商采用多种价格策略时的最优订购决策.

 [1] Lee H L, Rosenblatt M J. A generalized quantity discount pricing model to increase supplier's profits[J]. Management Science, 1986, 32(9): 1177-1185. DOI:10.1287/mnsc.32.9.1177 [2] Lin C S, Kroll D E. The single-item newsboy problem with dual performance measures and quantity discounts[J]. European Journal of Operational Research, 1997, 100(3): 562-565. DOI:10.1016/S0377-2217(96)00158-0 [3] Altintas N, Erhun F, Tayur S. Quantity discounts under demand uncertainty[J]. Management Science, 2008, 54(4): 777-792. DOI:10.1287/mnsc.1070.0829 [4] Chen X. Inventory centralization games with price-dependent demand and quantity discount[J]. Operations Research, 2009, 57(6): 1394-1406. DOI:10.1287/opre.1080.0615 [5] Chen S P, Ho Y H. Analysis of the newsboy problem with fuzzy demands and incremental discounts[J]. International Journal of Production Economics, 2011, 129(1): 169-177. DOI:10.1016/j.ijpe.2010.09.014 [6] Chen S P, Ho Y H. Optimal inventory policy for the fuzzy newsboy problem with quantity discounts[J]. Information Sciences, 2013, 228(7): 75-89. [7] Chung W, Talluri S, Narasimhan R. Optimal pricing and inventory strategies with multiple price markdowns over time[J]. European Journal of Operational Research, 2015, 243(1): 130-141. DOI:10.1016/j.ejor.2014.11.020 [8] Taleizadeh A A, Stojkovska I, Pentico D W. An economic order quantity model with partial backordering and incremental discount[J]. Computers & Industrial Engineering, 2015, 82: 21-32. [9] Tamjidzad S, Mirmohammadi S H. Optimal (r, Q) policy in a stochastic inventory system with limited resource under incremental quantity discount[J]. Computers & Industrial Engineering, 2017, 103: 59-69. [10] Khouja M. The newsboy problem under progressive multiple discounts[J]. European Journal of Operational Research, 1995, 84(2): 458-466. DOI:10.1016/0377-2217(94)00053-F [11] Khouja M. The newsboy problem with multiple discounts offered by suppliers and retailers[J]. Decision Sciences, 1996, 27(3): 589-599. DOI:10.1111/j.1540-5915.1996.tb01827.x [12] Feng Y, Xiao B. Optimal policies of yield management with multiple predetermined prices[J]. Operations Research, 2000, 48(2): 332-343. DOI:10.1287/opre.48.2.332.13373 [13] 王丽颖, 巩天啸, 陈丽华, 等. 二级市场季节性商品的订购和销售决策[J]. 管理科学学报, 2014, 17(5): 35-42. (Wang L Y, Gong T X, Chen L H, et al. Decisions for ordering seasonal goods and switching to secondary markets[J]. Journal of Management Sciences in China, 2014, 17(5): 35-42.) [14] Khouja M, Mehrez A. A multi-product constrained newsboy problem with progressive multiple discounts[J]. Computers & Industrial Engineering, 1996, 30(1): 95-101. [15] Moon I, Yoo D K, Saha S. The distribution-free newsboy problem with multiple discounts and upgrades[J]. Mathematical Problems in Engineering, 2016, 2016: 1-11. [16] Lu Y, Chen Y, Song M, et al. Optimal pricing and inventory control policy with quantity-based price differentiation[J]. Operations Research, 2014, 62(3): 512-523. DOI:10.1287/opre.2013.1240 [17] Zhang G. The multi-product newsboy problem with supplier quantity discounts and a budget constraint[J]. European Journal of Operational Research, 2010, 206(2): 350-360. DOI:10.1016/j.ejor.2010.02.038 [18] 禹海波. 需求不确定性对最小化成本和最大化利润报童问题的影响[J]. 系统工程理论与实践, 2014, 34(7): 1756-1768. (Yu H B. Effect of demand uncertainty on newsvendor problems with minimization cost and maximization profit[J]. Systems Engineering — Theory & Practice, 2014, 34(7): 1756-1768.)