两种情况下的Newsboy问题最优订购研究
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摘要
Newsboy问题(报童问题)一直是库存控制管理中研究的热点之一,经典的报童问题是指在单周期内,商品的需求为随机状态下,寻找一种商品订购数量,使系统预期利润最大或成本费用最小.本文主要从以下两方面对报童问题进行了扩展:
在本文第二章,我们首先建立了经典报童问题的费用模型,并得到了使得费用最小的订购量Q所应满足的条件.其次,根据经典报童模型,我们建立了需求变量为一般随机分布情形下的两次订购总费用模型,而且从理论上证明了阶段一和阶段二需求分布相互独立的情况下,零售商的期望费用函数存在最优解.通过数值实例我们验证了模型的求解过程,而且对一次订购和二次订购这两种订购策略进行了比较,得到了相关管理启示.
可替代问题就是指在计划期内,若某一商品发生缺货,则可以被另外的物品以一定的替换率补充销售.目前,可替代产品库存管理已逐渐成为供应链库存管理研究方面的一个热点.本文第三章中研究了三种可替代商品的经济订购数量问题.文中首先建立了三种商品相互替代的模型,根据实际的库存与缺货状况,将其分为八种情况,并分别求出这八种情况下的利润和期望利润函数,然后得到了整个过程的总利润函数,最后,给出了数值实例和灵敏度分析.
Newsboy problem is always the hot topic in inventory management. The classical newsboy problem is to find a product order quantity that maximizing expected total profit or minimizing expected total expense in a single cycle while the demand is random. This article expands the Newsboy problem in two situatiations as follows:
In the second chapter, firstly we establish a classical Newsboy cost model, and obtain the optimal order quantity that minimizing the total cost. Secondly, the two-ordering total expense models are established according to the classical Newsboy model by considering that the demand distributions have general distribution forms. The existence and the uniqueness of optimal ordering policies are proved under the assumption that the first-period and the second-period demands are independent. Numerical examples illustrate the solution procedure. And some managerial insights are obtained as well through the comparison of two ordering policies.
As the inventory control of alternative products has become a hot research in supply chain inventory management, the third chapter of this article considers the economic order quantity problem with three substituting commodities, in which shortage of one commodity may be replenished by another item with the fixed substituting rate in planned horizon. Firstly, we establish a model in which three kinds of commodities can be mutually replaced; Secondly, according to the actual inventory condition, we divide the sample space of random variables into eight cases, then derive the profit of each case, and abtain the total profit; Finally, the numerical examples and sensitivity analyses are provided.
在本文第二章,我们首先建立了经典报童问题的费用模型,并得到了使得费用最小的订购量Q所应满足的条件.其次,根据经典报童模型,我们建立了需求变量为一般随机分布情形下的两次订购总费用模型,而且从理论上证明了阶段一和阶段二需求分布相互独立的情况下,零售商的期望费用函数存在最优解.通过数值实例我们验证了模型的求解过程,而且对一次订购和二次订购这两种订购策略进行了比较,得到了相关管理启示.
可替代问题就是指在计划期内,若某一商品发生缺货,则可以被另外的物品以一定的替换率补充销售.目前,可替代产品库存管理已逐渐成为供应链库存管理研究方面的一个热点.本文第三章中研究了三种可替代商品的经济订购数量问题.文中首先建立了三种商品相互替代的模型,根据实际的库存与缺货状况,将其分为八种情况,并分别求出这八种情况下的利润和期望利润函数,然后得到了整个过程的总利润函数,最后,给出了数值实例和灵敏度分析.
Newsboy problem is always the hot topic in inventory management. The classical newsboy problem is to find a product order quantity that maximizing expected total profit or minimizing expected total expense in a single cycle while the demand is random. This article expands the Newsboy problem in two situatiations as follows:
In the second chapter, firstly we establish a classical Newsboy cost model, and obtain the optimal order quantity that minimizing the total cost. Secondly, the two-ordering total expense models are established according to the classical Newsboy model by considering that the demand distributions have general distribution forms. The existence and the uniqueness of optimal ordering policies are proved under the assumption that the first-period and the second-period demands are independent. Numerical examples illustrate the solution procedure. And some managerial insights are obtained as well through the comparison of two ordering policies.
As the inventory control of alternative products has become a hot research in supply chain inventory management, the third chapter of this article considers the economic order quantity problem with three substituting commodities, in which shortage of one commodity may be replenished by another item with the fixed substituting rate in planned horizon. Firstly, we establish a model in which three kinds of commodities can be mutually replaced; Secondly, according to the actual inventory condition, we divide the sample space of random variables into eight cases, then derive the profit of each case, and abtain the total profit; Finally, the numerical examples and sensitivity analyses are provided.
引文
[1] Harris F W. 1915. Economic order quantity model [J]. Management Science, 35(3):898-900.
[2] Salameh M K, Abdul-Malak M U, Jaber M Y. 1993. Mathematical modeling of the effect of human learning in the finite production inventory model [J]. Applied Mathematical Modelling, 17(11):613-615.
[3] Jaber M Y, Salameh M K. 1995. Optimal lot sizing under learning considerations: Shortages allowed and backordered [J]. Applied Mathematical Modelling, 19(5): 307-310.
[4] Aull-Hyde R L. 1996. A backlog inventory model during restricted sale periods [J]. Journal of the Operational Research Society, 47(9): 1192-1200.
[5]周永务.1996.考虑费用时值的库存系统的EOQ模型[J].系统工程理论与实践,16(8):96-102.
[6] Chung K J, Lin C N. 2001. Optimal inventory replenishment models for deteriorating items taking account of time discounting [J]. Computers and Operations Research, 28(1): 67-83.
[7]杨善林,周永务.2003.两货栈库存模型:考虑时变需求和价格折扣[J].系统工程学报,18(16):498-505.
[8] Zhou Y W. 2003. A multi-warehouse inventory model for items with time-varying demand and shortages [J]. Computers and Operations Research, 30(14): 2115-2134.
[9] Padmanabhan G, Vrat P. 1995. EOQ models for perishable items under stock dependent selling rate [J]. European Journal of Operational Research, 86(2): 281-292.
[10] Ray J, Chaudhuri K S. 1997. An EOQ model with stock dependent demand, shortage, inflation and time discounting [J]. International Journal of Production Economics, 53(2): 171-180.
[11] Chung K J, Chu P, Lan S P. 2000. A note on EOQ models for deteriorating items under stock dependent selling rate [J]. European Journal of Operational Research, 124(3): 550-559.
[12] Liao H C, Tsai C H, Su C T. 2000. An inventory model with deteriorating items under inflation when a delay in payment is permissible [J]. International Journal of Production Economics, 63(2): 207-214.
[13] Chang C T. 2004. Inventory models with stock-dependent demand andnonlinear holding costs for deteriorating items [J]. Asia-Pacific Journal of Operational Research, 21(4): 435-446.
[14] Alfares H K. 2007. Inventory model with stock-level dependent demand rate and variable holding cost [J]. International Journal of Production Economics, 108(1-2): 259-265.
[15] Hwang H, Shinn S. 1997. Retailer’s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments [J]. Computers and Operations Research, 24(6): 539-547.
[16] Abad P L, Aggarwal V. 2005. Incorporating transport cost in the lot size and pricing decisions with downward sloping demand [J]. International Journal of Production Economics, 95(3): 297-305.
[17] Chan G H, Song Y. 2003. A dynamic analysis of the single-item periodic stochastic inventory system with order capacity [J]. European Journal of Operational Research, 146(3): 529-542.
[18] Kogan K, Lou S. 2003. Multi-stage Newsboy problem: A dynamic model [J]. European Journal of Operational Research, 149(2): 448-458.
[19]陈旭.需求信息更新条件下易逝品的批量定货策略[J].管理科学学报,2005,8(5):38-42.
[20] Lau A H, Lau H S. 1998. Decision models for single-period products with two ordering opportunities [J]. International Journal of Production Economics, 55(1): 57-70.
[21] Goyal S K, Morin D, Nebebe F. 1992. The finite horizon trended inventory replenishment problem with shortages [J]. Journal of the Operational Research Society, 43(12): 1173-1178.
[22] Abad P L. 1988b. Joint price and lot-size determination when supplier offers incremental quantity discounts [J]. Journal of the Operational Research Society, 39(6): 603-607.
[23] Billington P J, Blackburn J, Maes J. 1994. Multi-item, lot-sizing in capacitated multi-stage serial systems [J]. IIE Transactions, 26(2): 12-18.
[24] Haksever C, Moussourakis J. 2005. A model for optimizing multi-product inventory systems with multiple constraints [J]. International Journal of Production Economics, 97(1): 18-30.
[25] Dave U, Patel L K. 1981. (T , S i)Policy inventory model for deteriorating items with time proportional demand [J]. Journal of the Operational Research Society, 32(1): 137-142.
[26] Xu H, Wang H. 1992. Optimal inventory policy for perishable items with timeproportional demand [J]. IIE Transactions, 24(5): 105-110.
[27] Gupta R, Vrat P. 1986. Inventory models for stock dependent consumption rate [J]. Opsearch, 23(1): 19-24.
[28] Vrat P, Padmanabhan G. 1990. An EOQ model for items with stock dependent consumption rate and exponential decay [J]. Engineering Costs and Production Economics, 18(3): 241-246.
[29] Su C T, Tong L T, Liao H C. 1996. An inventory model under inflation for stock dependent consumption rate and exponential decay [J]. Opsearch, 33(2): 71-82.
[30] Liao H C, Tsai C H, Su C T. 2000. An inventory model with deteriorating items under inflation when a delay in payment is permissible [J]. International Journal of Production Economics, 63(2): 2097-214.
[31] Rao U. 2003. Properties of the periodic review ( R , T ) inventory control policy for stationary, stochastic demand [J]. Manufacturing and Service Operations Management, 5(1): 37-53.
[32] Janakiraman G, Roundy R O. 2004. Lost-sales problems with stochastic lead times: Convexity results for base-stock policies [J]. Operations Research, 52(5): 795-803.
[33] Sankarasubramanian E, Kumaraswamy S. 1983. Optimal order quantity for pre-determined level of profit [J]. Management Science, 29(4): 512-514.
[34] Lau A H, Lau H S. 1988. Maximizing the probability of achieving a target profit level in a two-product Nwesboy problem [J]. Decision Science, 19(2): 392-408.
[35] Khouja M. 1996. A note on the Newsboy problem with an emergency supply option [J]. Journal of the Operational Research Society, 47(12): 1530-1534.
[36] Weng Z K. 2004. Coordinating order quantities between the manufacturer and the buyer: A generalized newsvendor mode [J]l. European Journal of Operational Research, 156(1): 148-161.
[37]王圣东,周永务.2009.带有两次订购机会且两阶段需求相关的Newsboy模型[J].控制与决策,24(5):706-710.
[38] Yue J, Chen B, Wang M S. 2006. Expected value of distribution information for the newsvendor problem [J]. Operations Research, 54(6): 1128-1136.
[39] Eynan A, Kropp D H. Effective and simple EOQ-like solutions for stochastic demand periodic review systems [J]. European Journal of Operational Research, 2007, 180(3): 1135-1143.
[40] Kogan K. Scheduling parallel machines by the dynamic Newsboy problem [J].Computers and Operations Research, 2004, 31(3): 429-443.
[41] Khouja M, Mehrez A, Rabinowitz G. Two-item Newsboy problem with substitutability [J]. International Journal of Production Economics, 1996, 44(3): 267-275.
[42]蔡连侨,陈剑,严厚民.2003a.可替代产品的库存模型研究(I):最优订购量[J].系统工程理论与实践,23(6):63-68.
[43]蔡连侨,陈剑,严厚民.2003b.可替代产品的库存模型研究(II):基本性质[J].系统工程理论与实践,23(8):59-68.
[44] Haksever C, Moussourakis J. 2005. A model for optimizing multi-product inventory systems with multiple constraints [J]. International Journal of Production Economics, 97(1): 18-30.
[45] Bhattacharya D K. 2005. On multi-item inventory [J]. European Journal of Operational Research, 162(3): 786-791.
[2] Salameh M K, Abdul-Malak M U, Jaber M Y. 1993. Mathematical modeling of the effect of human learning in the finite production inventory model [J]. Applied Mathematical Modelling, 17(11):613-615.
[3] Jaber M Y, Salameh M K. 1995. Optimal lot sizing under learning considerations: Shortages allowed and backordered [J]. Applied Mathematical Modelling, 19(5): 307-310.
[4] Aull-Hyde R L. 1996. A backlog inventory model during restricted sale periods [J]. Journal of the Operational Research Society, 47(9): 1192-1200.
[5]周永务.1996.考虑费用时值的库存系统的EOQ模型[J].系统工程理论与实践,16(8):96-102.
[6] Chung K J, Lin C N. 2001. Optimal inventory replenishment models for deteriorating items taking account of time discounting [J]. Computers and Operations Research, 28(1): 67-83.
[7]杨善林,周永务.2003.两货栈库存模型:考虑时变需求和价格折扣[J].系统工程学报,18(16):498-505.
[8] Zhou Y W. 2003. A multi-warehouse inventory model for items with time-varying demand and shortages [J]. Computers and Operations Research, 30(14): 2115-2134.
[9] Padmanabhan G, Vrat P. 1995. EOQ models for perishable items under stock dependent selling rate [J]. European Journal of Operational Research, 86(2): 281-292.
[10] Ray J, Chaudhuri K S. 1997. An EOQ model with stock dependent demand, shortage, inflation and time discounting [J]. International Journal of Production Economics, 53(2): 171-180.
[11] Chung K J, Chu P, Lan S P. 2000. A note on EOQ models for deteriorating items under stock dependent selling rate [J]. European Journal of Operational Research, 124(3): 550-559.
[12] Liao H C, Tsai C H, Su C T. 2000. An inventory model with deteriorating items under inflation when a delay in payment is permissible [J]. International Journal of Production Economics, 63(2): 207-214.
[13] Chang C T. 2004. Inventory models with stock-dependent demand andnonlinear holding costs for deteriorating items [J]. Asia-Pacific Journal of Operational Research, 21(4): 435-446.
[14] Alfares H K. 2007. Inventory model with stock-level dependent demand rate and variable holding cost [J]. International Journal of Production Economics, 108(1-2): 259-265.
[15] Hwang H, Shinn S. 1997. Retailer’s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments [J]. Computers and Operations Research, 24(6): 539-547.
[16] Abad P L, Aggarwal V. 2005. Incorporating transport cost in the lot size and pricing decisions with downward sloping demand [J]. International Journal of Production Economics, 95(3): 297-305.
[17] Chan G H, Song Y. 2003. A dynamic analysis of the single-item periodic stochastic inventory system with order capacity [J]. European Journal of Operational Research, 146(3): 529-542.
[18] Kogan K, Lou S. 2003. Multi-stage Newsboy problem: A dynamic model [J]. European Journal of Operational Research, 149(2): 448-458.
[19]陈旭.需求信息更新条件下易逝品的批量定货策略[J].管理科学学报,2005,8(5):38-42.
[20] Lau A H, Lau H S. 1998. Decision models for single-period products with two ordering opportunities [J]. International Journal of Production Economics, 55(1): 57-70.
[21] Goyal S K, Morin D, Nebebe F. 1992. The finite horizon trended inventory replenishment problem with shortages [J]. Journal of the Operational Research Society, 43(12): 1173-1178.
[22] Abad P L. 1988b. Joint price and lot-size determination when supplier offers incremental quantity discounts [J]. Journal of the Operational Research Society, 39(6): 603-607.
[23] Billington P J, Blackburn J, Maes J. 1994. Multi-item, lot-sizing in capacitated multi-stage serial systems [J]. IIE Transactions, 26(2): 12-18.
[24] Haksever C, Moussourakis J. 2005. A model for optimizing multi-product inventory systems with multiple constraints [J]. International Journal of Production Economics, 97(1): 18-30.
[25] Dave U, Patel L K. 1981. (T , S i)Policy inventory model for deteriorating items with time proportional demand [J]. Journal of the Operational Research Society, 32(1): 137-142.
[26] Xu H, Wang H. 1992. Optimal inventory policy for perishable items with timeproportional demand [J]. IIE Transactions, 24(5): 105-110.
[27] Gupta R, Vrat P. 1986. Inventory models for stock dependent consumption rate [J]. Opsearch, 23(1): 19-24.
[28] Vrat P, Padmanabhan G. 1990. An EOQ model for items with stock dependent consumption rate and exponential decay [J]. Engineering Costs and Production Economics, 18(3): 241-246.
[29] Su C T, Tong L T, Liao H C. 1996. An inventory model under inflation for stock dependent consumption rate and exponential decay [J]. Opsearch, 33(2): 71-82.
[30] Liao H C, Tsai C H, Su C T. 2000. An inventory model with deteriorating items under inflation when a delay in payment is permissible [J]. International Journal of Production Economics, 63(2): 2097-214.
[31] Rao U. 2003. Properties of the periodic review ( R , T ) inventory control policy for stationary, stochastic demand [J]. Manufacturing and Service Operations Management, 5(1): 37-53.
[32] Janakiraman G, Roundy R O. 2004. Lost-sales problems with stochastic lead times: Convexity results for base-stock policies [J]. Operations Research, 52(5): 795-803.
[33] Sankarasubramanian E, Kumaraswamy S. 1983. Optimal order quantity for pre-determined level of profit [J]. Management Science, 29(4): 512-514.
[34] Lau A H, Lau H S. 1988. Maximizing the probability of achieving a target profit level in a two-product Nwesboy problem [J]. Decision Science, 19(2): 392-408.
[35] Khouja M. 1996. A note on the Newsboy problem with an emergency supply option [J]. Journal of the Operational Research Society, 47(12): 1530-1534.
[36] Weng Z K. 2004. Coordinating order quantities between the manufacturer and the buyer: A generalized newsvendor mode [J]l. European Journal of Operational Research, 156(1): 148-161.
[37]王圣东,周永务.2009.带有两次订购机会且两阶段需求相关的Newsboy模型[J].控制与决策,24(5):706-710.
[38] Yue J, Chen B, Wang M S. 2006. Expected value of distribution information for the newsvendor problem [J]. Operations Research, 54(6): 1128-1136.
[39] Eynan A, Kropp D H. Effective and simple EOQ-like solutions for stochastic demand periodic review systems [J]. European Journal of Operational Research, 2007, 180(3): 1135-1143.
[40] Kogan K. Scheduling parallel machines by the dynamic Newsboy problem [J].Computers and Operations Research, 2004, 31(3): 429-443.
[41] Khouja M, Mehrez A, Rabinowitz G. Two-item Newsboy problem with substitutability [J]. International Journal of Production Economics, 1996, 44(3): 267-275.
[42]蔡连侨,陈剑,严厚民.2003a.可替代产品的库存模型研究(I):最优订购量[J].系统工程理论与实践,23(6):63-68.
[43]蔡连侨,陈剑,严厚民.2003b.可替代产品的库存模型研究(II):基本性质[J].系统工程理论与实践,23(8):59-68.
[44] Haksever C, Moussourakis J. 2005. A model for optimizing multi-product inventory systems with multiple constraints [J]. International Journal of Production Economics, 97(1): 18-30.
[45] Bhattacharya D K. 2005. On multi-item inventory [J]. European Journal of Operational Research, 162(3): 786-791.