需求双向替代与生产转换下动态批量决策及预测时阈研究

doi:10.16381/j.cnki.issn1003-207x.2021.0044

摘要/Abstract

摘要：

本文研究了存在需求双向替代和生产转换情形下的两产品动态批量决策和预测时阈问题。构建了包含时变的生产转换成本、替代成本以及两产品的生产成本和库存成本在内的成本最小化模型。在最优解性质的基础上，设计出前向动态规划算法求解问题。利用边际成本分析方法得出两产品生产点及再生点的单调性特征，并建立了相应的两产品再生集，最后给出求解预测时阈和相应的决策时阈的充分条件。运用数值实验分析了需求单向替代和双向替代情形下预测时阈的相对大小、同一周期最多发生一次生产转换与可以发生多次转换情形下预测时阈的相对大小，需求增长性与波动性对预测时阈的影响以及生产转换成本与替代成本同时变动时对预测时阈的影响。发现需求双向替代将增加预测时阈的长度，而同一周期可以发生多次生产转换将降低预测时阈的长度。预测时阈随着生产转换成本的增加而显著递增，随着需求增长参数和需求波动参数以及替代成本的增加而递减。

关键词: 预测时阈, 需求双向替代, 生产转换, 动态批量

Abstract:

Demand （or product） substitution allows a customer who comes to purchase a specific product to accept or choose another available product when her original choice is unavailable or not offered. The notion of substitution has recently been recognized as an effective tool to improve the efficiency of multi-product inventory systems by taking advantage of the inherent flexibility. There are a large number of settings in which costs are incurred only when the product being produced is changed. Such a cost is referred to as the changeover cost. The flexibility associated with demand substitution can be used to address inefficiencies resulting from significant changeover costs.The dynamic lot sizing problem and forecast horizon under production changeovers and two-way demand substitution are analyzed. The optimal decisions of the production for a T-period problem are computed to minimize the production changeovers cost，variable production cost， substitution cost and holding cost. Based on the structural properties of the optimal solution， a forward dynamic programming algorithm is developed to solve the problem. By the marginal cost analysis method， the sufficient condition is given to establish the monotonicity of production points for two products， then the regeneration set is established for two products and the sufficient condition is given to obtain the forecast horizon. Finally， a computational experiment is used to compare the length of forecast horizon under one-way and two-way demand substitution. The length of forecast horizon under at most one changeover and multiple changeovers in a period are also compared. The length of forecast horizon will be lengthened under two-way demand substitution， and will be shortened under multiple changeovers in a period. In addition， the length of forecast horizon increases with changeover costs， and decreases with the parameters of growth and variability and substitution costs.

Key words: forecast horizon, two-way demand substitution, production changeovers, dynamic lot sizing

中图分类号:

F253.4

靖富营,慕银平,晁祥瑞. 需求双向替代与生产转换下动态批量决策及预测时阈研究[J]. 中国管理科学, 2023, 31(8): 226-238.

Fu-ying JING,Yin-ping Mu,Xiang-rui CHAO. Forecast Horizon of Dynamic Lot Sizing Model Under Production Changeovers and Two-Way Demand Substitution[J]. Chinese Journal of Management Science, 2023, 31(8): 226-238.

图/表 14

表1

符号定义"

符号	定义
$T$	时间周期；
$P (t)$	$t$ -周期问题
$d i t$	第 $t$ 期产品 $i$ 的需求， $d i t ≥ 0$ ， $i = 1,2$ ， $1 ≤ t ≤ T$
$x i t$	第 $t$ 期产品 $i$ 的生产数量， $x i t ≥ 0$ ， $i = 1,2$ ， $1 ≤ t ≤ T$
$I i t$	第 $t$ 期期末（第 $t + 1$ 期期初）产品 $i$ 的库存数量， $I i t ≥ 0$ ， $i = 1,2$ ， $1 ≤ t ≤ T$
$y i j t$	第 $t$ 期产品 $i$ 替代产品 $j$ 的数量， $y i j t ≥ 0$ ， $i, j = 1,2$ ， $i ≠ j$ ， $1 ≤ t ≤ T$
$c i t$	第 $t$ 期产品 $i$ 的单位生产成本， $c i t ≥ 0$ ， $i = 1,2$ ， $1 ≤ t ≤ T$
$K i j t$	第 $t$ 期生产由产品 $i$ 转换到产品 $j$ 的转换成本， $K i j t ≥ 0$ ， $i = 1,2$ ， $i ≠ j$ ， $1 ≤ t ≤ T$
$ω i j t$	第 $t$ 期产品 $i$ 替代产品 $j$ 的替代成本， $ω i j t ≥ 0$ ， $i = 1,2$ ， $i ≠ j$ ， $1 ≤ t ≤ T$
$h i t$	第 $t$ 期产品 $i$ 的单位库存持有成本， $h i t ≥ 0$ ， $i = 1,2$ ， $1 ≤ t ≤ T$

表1

图1

图2

图3

图4

图5

图6

图7

图8

图9

图10

图11

图12

图13

参考文献 41

1	王滔，颜波，李鸿媛. 网络间接渠道下替代产品竞争的决策分析［J］. 中国管理科学， 2017， 25（4）： 42-51.
	Wang Tao， Yan Bo， Li Hongyuan. Decision analysis of substitute product’s competition under the indirect internet channel［J］. Chinese Journal of Management Science， 2017， 25（4）： 42-51.
2	李友东，夏良杰，王锋正. 基于产品替代的低碳供应链博弈与协调模型［J］. 中国管理科学， 2019， 27（10）： 66-76.
	Li Youdong， Xia Liangjie， Wang Fengzheng. Game and coordination model for low-carbon supply chain with product substitution［J］. Chinese Journal of Management Science， 2019， 27（10）： 66-76.
3	张柯慧，杨华龙，辛禹辰，等. 考虑顾客缺失反应的多产品两阶段JIT联合订货［J］. 中国管理科学， 2020， 28（10）： 118-130.
	Zhang Kehui， Yang Hualong， Xin Yuchen， et al. Multi-item/two-stage JIT joint replenishment considering customer response to stock-out［J］. Chinese Journal of Management Science， 2020， 28（10）： 118-130.
4	Balakrishnan A， Geunes J. Requirements planning with substitutions： exploiting Bill-of-Materials flexibility in production planning［J］. Manufacturing and Service Operations Management， 2000， 2（2）： 166-185.
5	Hsu V N， Li C L， Xiao Wenqiang. Dynamic lot size problem with one-way product substitution［J］. IIE Transactions， 2005， 37（3）： 201-215.
6	Yaman H. Polyhedral analysis for the two-item uncapacitated lot-sizing problem with one-way substitution［J］. Discrete Applied Mathematics， 2009， 157（14）： 3133-3151.
7	Yaman H. Erratum to： polyhedral analysis for the two-item uncapacitated lot-sizing problem with one-way substitution［J］. Discrete Applied Mathematics， 2011， 159（10）： 1058-1058.
8	Lang J C， Domschke W. Efficient reformulations for dynamic lot-sizing problems with product substitution［J］. OR Spectrum， 2010， 32（2）： 263-291.
9	Li Yongjian， Chen Jian， Cai Xiaoqiang. Uncapacitated production planning with multiple product types， returned product remanufacturing， and demand substitution［J］. OR Spectrum， 2006， 28（1）： 101-125.
10	Pineyro P， Viera O. The economic lot-sizing problem with remanufacturing and one-way substitution［J］. International Journal of Production Economics， 2010， 124（2）： 482-488.
11	Pineyro P， Viera O. Note on “The economic lot-sizing problem with remanufacturing and one-way substitution”［J］. International Journal of Production Economics， 2014， 156： 167-168.
12	Geunes J. Solving large-scale requirements planning problems with component substitution options［J］. Computers & Industrial Engineering， 2003，44（3）： 475-491.
13	Li Yongjian， Chen Jian， Cai Xiaoqiang. Heuristic genetic algorithm for capacitated production planning problems with batch processing and remanufacturing［J］. International Journal of Production Economics， 2007，105（2）： 301-307.
14	胡海菊，李勇建. 考虑再制造和产品需求可替代的短生命周期产品动态批量生产计划问题［J］. 系统工程理论与实践， 2007， 27（12）： 76-84.
	Hu Haiju， Li Yongjian. Dynamic production planning problem for short-life-cycle products with product remanufacturing and demand substitution［J］. Systems Engineering-Theory & Practice， 2007，12（12）： 76-84.
15	Blocher J D， Chand S， Sengupta K. The changeover scheduling problem with time and cost considerations： analytical results and a forward algorithm［J］. Operations Research， 1999， 47（4）： 559-569.
16	Karmarkar U S， Schrage L. The deterministic dynamic product cycling problem［J］. Operations Research， 1985， 33（2）： 326-345.
17	Gascon A， Leachman R C. A dynamic programming solution to the dynamic， multi-item， single-machine scheduling problem［J］. Operations Research， 1988， 36（1）： 50-56.
18	Magnanti T L， Vachani R. A strong cutting plane algorithm for production scheduling with changeover costs［J］. Operations Research， 1990， 38（3）： 456-473.
19	Fleischmann B. The discrete lot-sizing and scheduling problem［J］. European Journal of Operational Research， 1990， 38（3）： 337-348.
20	Salomou M， Kroon L G， Kuik R， et al. Some extensions of the discrete lot sizing and scheduling problem［J］.Management Science， 1991， 37（7）：801-812-486.
21	Cattrysse D， Salomon M， Kuik R， et al. A dual ascent and column generation heuristic for the discrete lot sizing and scheduling problem with setup times［J］. Management Science， 1993， 39（4）： 477-486.
22	Fleischmann B. The discrete lot-sizing and scheduling problem with sequence-dependent setup costs［J］. European Journal of Operational Research， 1994， 75（2）： 395-404.
23	Gicquel C， Lisser A， Minoux M. An evaluation semidefinite programming based approaches for discrete lot-sizing problems［J］. European Journal of Operational Research， 2014， 237： 498-507.
24	Gicquel C， Minoux M. Multi-product valid inequalities for the discrete lot-sizing and scheduling problem［J］. Computers & Operations Research， 2015， 54： 12-20.
25	Ceschia S， Gaspero L D， Schaerf A. Solving discrete lot-sizing and scheduling by simulated annealing and mixed integer programming［J］. Computers & Industrial Engineering， 2017， 114： 235-243.
26	Lang J C， Shen Zuojun. Fix-and-optimize heuristics for capacitated lot-sizing with sequence-dependent setups and substitutions［J］. European Journal of Operational Research， 2011， 214（3）： 595-605.
27	Dawande M， Gavirneni S， Mu Yinping， et al. On the interaction between demand substitution and production changeovers［J］. Manufacturing and Service Operations Management， 2010， 12（4）： 682-691.
28	Zeppetella L， Gebennini E， Grassi A， et al. Optimal production scheduling with customer-driven demand substitution［J］. International Journal of Production Research， 2017， 55（6）： 1692-1706.
29	Glassey C R. Minimum change-over scheduling of several products on one machine［J］. Operations Research， 1968， 16（2）： 342-352.
30	Mitsumor S. Optimum production scheduling of multicommodity in flow line［J］. IEEE transactions on systems， man， and cybernetics， 1972， 2（4）： 486-493.
31	Driscoll W C， Emmons H. Scheduling production on one machine with changeover costs［J］. AIIE Transactions， 1977， 9（4）： 388-395.
32	Bruno J， Downey P. Complexity of task sequencing with deadlines， setup times， and changeover costs［J］. SIAM Journal on computing， 1978， 7（4）， 393-404.
33	Mitsumor S. Optimum scheduling for load balance of two-machine production lines［J］. IEEE transactions on systems， man， and cybernetics， 1981， 11（6）： 400-409.
34	Hu T C， Kuo Y S， Ruskey F. Some optimum algorithms for scheduling problems with changeover costs［J］. Operations Research， 1987， 35（1）： 94-99.
35	Blocher J D， Chand S. An improved lower bound for the changeover scheduling problem［J］. IIE Transactions， 1996， 28（11）： 901-909.
36	Blocher J D， Chand S. A forward branch-and search algorithm and forecast horizon results for the changeover scheduling problem［J］. European Journal of Operational Research， 1996， 91（3）： 456-470.
37	Zhou Feng， Blocher J D， Hu Xinxin， et al. Optimal single machine scheduling of products with components and changeover cost ［J］. European Journal of Operational Research， 2014， 233： 75-83.
38	慕银平. 需求替代的两产品动态批量最优预测时阈研究［J］. 管理科学学报， 2011， 14（3）： 10-23.
	Mu Yinping. Forecast horizon of dynamic lot-size for two products under demand and substitution［J］. Journal of Management Science in China， 2011，14（3）： 10-23.
39	Lundin R A， Morton T E. Planning horizons for the dynamic lot size model： Zabel vs. protective procedures and computational results［J］. Operations Research， 1975， 23（4）： 711-734.
40	靖富营. 需求替代和生产转换下动态批量决策与预测时阈研究［D］. 成都：电子科技大学， 2019.
	Jing Fuying. Studies on the decision of dynamic lot sizing and forecast horizon under demand substitution and production changeovers［D］. Chengdu： University of Electronic Science and Technology of China， 2019.
41	Mu Yinping， Jing Fuying. Forecast Horizons under demand substitution and production changeovers： a generalized model［R］. Working Paper， University of Electronic Science and Technology of China， 2021.