需求损失下两产品联合生产动态批量决策及预测时阈
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摘要
研究需求损失下两产品联合生产(采购)动态批量决策问题.在各周期成本变动情形下分析多周期动态批量决策的预测时阈和决策时阈,构建包含联合启动成本、两产品的单独启动成本、库存持有成本、变动生产成本和需求损失成本在内的成本最小化模型.在最优解结构特性的基础上,设计出前向动态规划算法求解问题,通过建立两产品生产点的单调性和建立生产集,给出求解预测时阈和决策时阈的充分条件.通过数值算例分析预测时阈求解的具体过程,表明所构建模型的有效性.
This paper investigates a two-product dynamic lot-size problem under joint production and lost sales, and analyzes forecast and decision horizons for time-varying costs. We formulate a cost minimization model including the joint setup costs, individual setup costs, variable costs, holding costs and lost sale costs over a finite horizon of T periods. Based on some properties of the optimal solution, we develop an efficient forward dynamic programming algorithm to solve the problem. Based on the monotonicity of the production point and production set for both products, a sufficient condition for obtaining forecast horizon and decision horizon is given through a numerical example, the details of computing the forecast horizon is analyzed, which shows the effectiveness of the proposed model.
This paper investigates a two-product dynamic lot-size problem under joint production and lost sales, and analyzes forecast and decision horizons for time-varying costs. We formulate a cost minimization model including the joint setup costs, individual setup costs, variable costs, holding costs and lost sale costs over a finite horizon of T periods. Based on some properties of the optimal solution, we develop an efficient forward dynamic programming algorithm to solve the problem. Based on the monotonicity of the production point and production set for both products, a sufficient condition for obtaining forecast horizon and decision horizon is given through a numerical example, the details of computing the forecast horizon is analyzed, which shows the effectiveness of the proposed model.
引文
[1]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 Y P. Forecast horizon of dynamic lot-size for two products under demand and substitution[J]. J of Management Science in China, 2011, 14(3):10-23.)
[2] Wagner H M, Whitin T M. Dynamic version of the economic lot size model[J]. Management Science, 1958,5(1):89-96.
[3] Chand S, Hsu V N, Sethi S. Forecast, solution, and rolling horizons in operations management problems:A classified bibliography[J]. Manufacturing and Service Operations Management, 2002, 4(1):25-43.
[4] Dawande M, Gavirneni S, Naranpanawe S, et al. Forecast horizons for a class of dynamic lot-size problems under discrete future demand[J]. Operations Research, 2007,55(4):688-702.
[5] Dawande M, Gavirneni S, Naranpanawe S, et al. Discrete forecast horizons for two-product variants of the dynamic lot-size problem[J]. Int J of Production Economics, 2009,120(2):430-436.
[6] Dawande M, Gavirneni S, Mu Y P, et al. On the interaction between demand substitution and production changeovers[J]. Manufacturing and Service Operations Management, 2010, 12(4):682-691.
[7] Ghate A, Smith R L. Optimal backlogging over an infinite horizon under time varying convex production and inventory costs[J]. Manufacturing and Service Operations Management, 2009, 11(2):362-368.
[8] Zhao T G, Yang D W, Cai X M, et al. Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast[J]. Water Resources Research,2012, 48(1):1-15.
[9] Bardhan A, Dawande M, Gavirneni S, et al. Forecast and rolling horizons under demand substitution and production changeovers:Analysis and insights[J]. IIE Transactions, 2013, 45(3):323-340.
[10] You J Y, Yu C W. Theoretical error convergence of limited forecast horzion in optimal reservoir operating decisions[J]. Water Resources Research, 2013, 49(3):277-297.
[11] Teyarachakul S, Chand S, Tzur M. Lot sizing with learning and forgetting in setups:Analytical results and insights[J]. Naval Research Logistics, 2017, 63(2):93-108.
[12] Arena C, Cannarozzo M, Mazzola M R. Exploring the potential and the boundaries of the rolling horizon technique for the management of reservoir systems with one-year behaviour[J]. Water Resources Management,2017, 31(3):1-18.
[13] Akbalik A, Penz B, Rapine C. Multi-item uncapacitated lot sizing problem with inventory bounds[J].Optimization Letters, 2015, 9(1):143-154.
[14] Akbalik A, Penz B, Rapine C. Capacitated lot sizing problems with inventory bounds[J]. Annals of Operations Research, 2015, 229(1):1-18.
[15] Melo R A, Ribeiro C C. Formulations and heuristics for the multi-item uncapacitated lot-sizing problem with inventory bounds[J]. Int J of Production Research, DOI:10.1080/00207543.2016.1215567.
[16] Cheung M, Elmachtoub A N, Levi R, et al.The submodular joint replenishment problem[J].Mathematical Programming, 2016, 158(1/2):207-233.
[17] Li Y J, Cai X Q, Xu L, et al. Heuristic approach on dynamic lot-sizing model for durable products with end-of-use constraints[J]. Annals of Operations Research,2015, 242(2):265-283.
[18] Cuha J O, Konstantaras I, Melo R A, et al. On multi-item economic lot-sizing with remanufacturing and uncapacitated production[J]. Applied Mathematical Modelling, 2017, 50(6):678-686.
[19] Sandbothe R A, Thompson G L. A forward algorithm for the capacitated lot size model with stockouts[J].Operations Research, 1990, 38(3):474-486.
[20] Sandbothe R A, Thompson G L. Decision horizons for the capacitated lot size model with inventory bounds and stockouts[J]. Computers&Operations Research, 1993,20(5):455-465.
[21] Aksen D, Altinkemer K, Chand S. The single item lot-sizing problem with immediate lost sales[J]. European J of Operational Research, 2003, 147(3):558-566.
[22] Aksen D. Loss of customer goodwill in the uncapacitated lot-sizing problem[J]. Computer and Operations Research, 2007, 34(9):2805-2823.
[23] Absi N, Sidhoum S K. The multi-item capacitated lot-sizing problem with setup times and shortage costs[J].European J of Operational Research, 2008, 185(3):1351-1374.
[24] Absi N, Sidhoum S K. The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs[J]. Computers&Operations Research,2009, 36(11):2926-2936.
[25] Absi N, Sidhoum S K, Peres S D. Uncapacitated lot-sizing problem with production time window, early production,backlog and lost sale[J]. Int J of Production Research,2011, 49(9):2551-2566.
[26] Narenji M, Ghomi S M T F, Nooraie S V R. Grouping in decomposition method for multi-item capacitated lot-szing problem with immediate lost sales and joint and item-dependent setup cost[J]. Int J of System Science,2011, 42(3):489-498.
[27] Gayon J P, Massonnet G, Rapine C, et al. Constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost-sales[J]. European J of Operational Research, 2016, 250(1):155-163.
[28] Chu F, Chu C C, Liu X. Lot sizing models with backlog and out-sourcing[C]. IEEE Int Conf on Systems, Man,Cybernetics. Hague, 2004:4342-4347.
[29] Chu F, Chu C C. Polynomial algorithms for single item lot-sizing models with bounded inventory and backlogging, or outsourcing[J]. IEEE Trans on Automation Science and Engineering, 2007, 4(2):233-251.
[30] Chu F, Chu C. Single-Item dynamic lot-sizing models with bounded inventory and outsourcing[J]. IEEE Trans on Systems, Man, and Cybernetics, 2008, 38(1):70-77.
[31] Chu C C, Chu F, Zhong J H, et al. A polynomial algorithm for lot-sizing problem with backlogging,outsourcing and limited inventory[J]. Computer and Industrial Engineering, 2013, 64(1):200-210.
[32] Zhong J H, Chu F, Chu H B, et al. Polynomial dynamic programming algorithms for lot sizing models with bounded inventory and stockout and or backlogging[J].J of Systems Science and Systems Engineering, 2016,25(3):370-397.
[33] Liu X, Chu F, Chu C, et al. Lot sizing with bounded and inventory and lost sales[J]. Int J of Production Research,2007, 45(24):5881-5894.
[34] Liu X. A polynomial time algorithm for production planning with bounded inventory[J]. Int J of Advanced Manufacturing Technology, 2008, 39(7):774-782.
[35] Liu X, Tu Y L. Production planning with limited inventory capacity and allowed stockout[J]. Int J of Production Economics, 2008, 111(1):180-191.
[36] Berk E, Toy A O, Hazir O. Single item lot-sizing problem for a worm/cold process with immediate lost sales[J].European J of Operational Research, 2008, 187(3):1251-1267.
[37] Hwang H C, Heuvel W V D, Wagelmans A P M. The economic lot-sizing problem with lost sales and bounded inventory[J]. IIE Transactions, 2013, 45(8):912-924.
[38]黄玲,钟金宏,杨善林.考虑延期交货、转包和非减库存能力约束的单产品批量模型[J].系统工程理论与实践, 2007, 27(9):87-95.(Huang L, Zhong J H, Yang S L. Single item lot sizing models with backlogging and outsourcing and non-decreasing inventory capacity[J]. System Engineering—Theory&Practice, 2007, 27(9):87-95.)
[39]黄玲,钟金宏,倪志伟,等.允许缺货和转包的受限批量模型的多项式算法[J].系统仿真学报, 2008, 20(11):2800-2805.(Huang L, Zhong J H, Ni Z W, et al. Polynomial algorithm for capacitated lot sizing models with shortage and outsouring[J]. J of Systems Simulation, 2008, 20(11):2800-2805.)
[40]钟金宏,黄玲,李兴国.带延期交货和固定成本的库存受限转包模型[J].系统管理学报, 2010, 19(5):578-583.(Zhong J H, Huang L, Li X G. Outsourcing models with backlogging and fixed costs and inventory capacity[J]. J of Systems and Management, 2010, 19(5):578-583.)
[41]王能民,孙青林,孙林岩.考虑外包的单产品再制造批量决策[J].运筹与管理, 2011, 20(5):162-168.(Wang N M, Sun Q L, Sun L Y. Single-item dynamic lot sizing problem with remanufacturing and outsourcing[J].Operations Research and Management Science, 2011,20(5):162-168.)
[42]徐娟,汪小京,刘志学.不同顾客环境下生产及外包的动态批量问题[J].系统工程, 2014, 32(10):63-70.(Xu J, Wang X J, Liu Z X. Dynamic lot sizing problem of production and outsourcing under the environment of different customer classes[J]. Systems Engineering, 2014,32(10):63-70.)
[43]伍景琼,蒲云,伍锦群.价格波动下钢铁企业原材料采购多期优化模型[J].控制与决策, 2013, 28(9):1383-1387.(Wu J Q, Pu Y, Wu J Q. Raw material purchasing multi-period optimization model with fluctuating prices in steel enterprise[J]. Control and Decision, 2013, 28(9):1383-1387.)
[44]周炳海,彭涛.混流装配生产线准时化物料补给调度方法[J].控制与决策, 2017, 32(6):976-982.(Zhou B H, Peng T. Scheduling methods of just in time material replenishment in mixed model assembly lines[J].Control and Decision, 2017, 32(6):976-982.)
[45]计国君,韩尚清.基于第三方的易变质产品库存决策[J].控制与决策, 2015, 30(4):663-669.(Ji G J, Han S Q. Deteriorating item inventory decision-making research based on third-party[J].Control and Decision, 2015, 30(4):663-669.)
[46] 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.
[2] Wagner H M, Whitin T M. Dynamic version of the economic lot size model[J]. Management Science, 1958,5(1):89-96.
[3] Chand S, Hsu V N, Sethi S. Forecast, solution, and rolling horizons in operations management problems:A classified bibliography[J]. Manufacturing and Service Operations Management, 2002, 4(1):25-43.
[4] Dawande M, Gavirneni S, Naranpanawe S, et al. Forecast horizons for a class of dynamic lot-size problems under discrete future demand[J]. Operations Research, 2007,55(4):688-702.
[5] Dawande M, Gavirneni S, Naranpanawe S, et al. Discrete forecast horizons for two-product variants of the dynamic lot-size problem[J]. Int J of Production Economics, 2009,120(2):430-436.
[6] Dawande M, Gavirneni S, Mu Y P, et al. On the interaction between demand substitution and production changeovers[J]. Manufacturing and Service Operations Management, 2010, 12(4):682-691.
[7] Ghate A, Smith R L. Optimal backlogging over an infinite horizon under time varying convex production and inventory costs[J]. Manufacturing and Service Operations Management, 2009, 11(2):362-368.
[8] Zhao T G, Yang D W, Cai X M, et al. Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast[J]. Water Resources Research,2012, 48(1):1-15.
[9] Bardhan A, Dawande M, Gavirneni S, et al. Forecast and rolling horizons under demand substitution and production changeovers:Analysis and insights[J]. IIE Transactions, 2013, 45(3):323-340.
[10] You J Y, Yu C W. Theoretical error convergence of limited forecast horzion in optimal reservoir operating decisions[J]. Water Resources Research, 2013, 49(3):277-297.
[11] Teyarachakul S, Chand S, Tzur M. Lot sizing with learning and forgetting in setups:Analytical results and insights[J]. Naval Research Logistics, 2017, 63(2):93-108.
[12] Arena C, Cannarozzo M, Mazzola M R. Exploring the potential and the boundaries of the rolling horizon technique for the management of reservoir systems with one-year behaviour[J]. Water Resources Management,2017, 31(3):1-18.
[13] Akbalik A, Penz B, Rapine C. Multi-item uncapacitated lot sizing problem with inventory bounds[J].Optimization Letters, 2015, 9(1):143-154.
[14] Akbalik A, Penz B, Rapine C. Capacitated lot sizing problems with inventory bounds[J]. Annals of Operations Research, 2015, 229(1):1-18.
[15] Melo R A, Ribeiro C C. Formulations and heuristics for the multi-item uncapacitated lot-sizing problem with inventory bounds[J]. Int J of Production Research, DOI:10.1080/00207543.2016.1215567.
[16] Cheung M, Elmachtoub A N, Levi R, et al.The submodular joint replenishment problem[J].Mathematical Programming, 2016, 158(1/2):207-233.
[17] Li Y J, Cai X Q, Xu L, et al. Heuristic approach on dynamic lot-sizing model for durable products with end-of-use constraints[J]. Annals of Operations Research,2015, 242(2):265-283.
[18] Cuha J O, Konstantaras I, Melo R A, et al. On multi-item economic lot-sizing with remanufacturing and uncapacitated production[J]. Applied Mathematical Modelling, 2017, 50(6):678-686.
[19] Sandbothe R A, Thompson G L. A forward algorithm for the capacitated lot size model with stockouts[J].Operations Research, 1990, 38(3):474-486.
[20] Sandbothe R A, Thompson G L. Decision horizons for the capacitated lot size model with inventory bounds and stockouts[J]. Computers&Operations Research, 1993,20(5):455-465.
[21] Aksen D, Altinkemer K, Chand S. The single item lot-sizing problem with immediate lost sales[J]. European J of Operational Research, 2003, 147(3):558-566.
[22] Aksen D. Loss of customer goodwill in the uncapacitated lot-sizing problem[J]. Computer and Operations Research, 2007, 34(9):2805-2823.
[23] Absi N, Sidhoum S K. The multi-item capacitated lot-sizing problem with setup times and shortage costs[J].European J of Operational Research, 2008, 185(3):1351-1374.
[24] Absi N, Sidhoum S K. The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs[J]. Computers&Operations Research,2009, 36(11):2926-2936.
[25] Absi N, Sidhoum S K, Peres S D. Uncapacitated lot-sizing problem with production time window, early production,backlog and lost sale[J]. Int J of Production Research,2011, 49(9):2551-2566.
[26] Narenji M, Ghomi S M T F, Nooraie S V R. Grouping in decomposition method for multi-item capacitated lot-szing problem with immediate lost sales and joint and item-dependent setup cost[J]. Int J of System Science,2011, 42(3):489-498.
[27] Gayon J P, Massonnet G, Rapine C, et al. Constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost-sales[J]. European J of Operational Research, 2016, 250(1):155-163.
[28] Chu F, Chu C C, Liu X. Lot sizing models with backlog and out-sourcing[C]. IEEE Int Conf on Systems, Man,Cybernetics. Hague, 2004:4342-4347.
[29] Chu F, Chu C C. Polynomial algorithms for single item lot-sizing models with bounded inventory and backlogging, or outsourcing[J]. IEEE Trans on Automation Science and Engineering, 2007, 4(2):233-251.
[30] Chu F, Chu C. Single-Item dynamic lot-sizing models with bounded inventory and outsourcing[J]. IEEE Trans on Systems, Man, and Cybernetics, 2008, 38(1):70-77.
[31] Chu C C, Chu F, Zhong J H, et al. A polynomial algorithm for lot-sizing problem with backlogging,outsourcing and limited inventory[J]. Computer and Industrial Engineering, 2013, 64(1):200-210.
[32] Zhong J H, Chu F, Chu H B, et al. Polynomial dynamic programming algorithms for lot sizing models with bounded inventory and stockout and or backlogging[J].J of Systems Science and Systems Engineering, 2016,25(3):370-397.
[33] Liu X, Chu F, Chu C, et al. Lot sizing with bounded and inventory and lost sales[J]. Int J of Production Research,2007, 45(24):5881-5894.
[34] Liu X. A polynomial time algorithm for production planning with bounded inventory[J]. Int J of Advanced Manufacturing Technology, 2008, 39(7):774-782.
[35] Liu X, Tu Y L. Production planning with limited inventory capacity and allowed stockout[J]. Int J of Production Economics, 2008, 111(1):180-191.
[36] Berk E, Toy A O, Hazir O. Single item lot-sizing problem for a worm/cold process with immediate lost sales[J].European J of Operational Research, 2008, 187(3):1251-1267.
[37] Hwang H C, Heuvel W V D, Wagelmans A P M. The economic lot-sizing problem with lost sales and bounded inventory[J]. IIE Transactions, 2013, 45(8):912-924.
[38]黄玲,钟金宏,杨善林.考虑延期交货、转包和非减库存能力约束的单产品批量模型[J].系统工程理论与实践, 2007, 27(9):87-95.(Huang L, Zhong J H, Yang S L. Single item lot sizing models with backlogging and outsourcing and non-decreasing inventory capacity[J]. System Engineering—Theory&Practice, 2007, 27(9):87-95.)
[39]黄玲,钟金宏,倪志伟,等.允许缺货和转包的受限批量模型的多项式算法[J].系统仿真学报, 2008, 20(11):2800-2805.(Huang L, Zhong J H, Ni Z W, et al. Polynomial algorithm for capacitated lot sizing models with shortage and outsouring[J]. J of Systems Simulation, 2008, 20(11):2800-2805.)
[40]钟金宏,黄玲,李兴国.带延期交货和固定成本的库存受限转包模型[J].系统管理学报, 2010, 19(5):578-583.(Zhong J H, Huang L, Li X G. Outsourcing models with backlogging and fixed costs and inventory capacity[J]. J of Systems and Management, 2010, 19(5):578-583.)
[41]王能民,孙青林,孙林岩.考虑外包的单产品再制造批量决策[J].运筹与管理, 2011, 20(5):162-168.(Wang N M, Sun Q L, Sun L Y. Single-item dynamic lot sizing problem with remanufacturing and outsourcing[J].Operations Research and Management Science, 2011,20(5):162-168.)
[42]徐娟,汪小京,刘志学.不同顾客环境下生产及外包的动态批量问题[J].系统工程, 2014, 32(10):63-70.(Xu J, Wang X J, Liu Z X. Dynamic lot sizing problem of production and outsourcing under the environment of different customer classes[J]. Systems Engineering, 2014,32(10):63-70.)
[43]伍景琼,蒲云,伍锦群.价格波动下钢铁企业原材料采购多期优化模型[J].控制与决策, 2013, 28(9):1383-1387.(Wu J Q, Pu Y, Wu J Q. Raw material purchasing multi-period optimization model with fluctuating prices in steel enterprise[J]. Control and Decision, 2013, 28(9):1383-1387.)
[44]周炳海,彭涛.混流装配生产线准时化物料补给调度方法[J].控制与决策, 2017, 32(6):976-982.(Zhou B H, Peng T. Scheduling methods of just in time material replenishment in mixed model assembly lines[J].Control and Decision, 2017, 32(6):976-982.)
[45]计国君,韩尚清.基于第三方的易变质产品库存决策[J].控制与决策, 2015, 30(4):663-669.(Ji G J, Han S Q. Deteriorating item inventory decision-making research based on third-party[J].Control and Decision, 2015, 30(4):663-669.)
[46] 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.