Integrated inventory model with stochastic lead time and controllable variability for milk runs

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July 2012, 8(3): 657-672. doi: 10.3934/jimo.2012.8.657

Integrated inventory model with stochastic lead time and controllable variability for milk runs

Xin Zhou ^1, , Liangping Shi ^1, and Bingzhi Huang ^1,

1.	Department of Customs Management, Shanghai Customs College, Shanghai, 201204, China, China, China

Received May 2011 Revised January 2012 Published June 2012

Abstract
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The current study deals with the lead-time variability reduction problem in a multi-supplier and single-buyer system with a milk-run delivery network. Under the assumption of finite-range stochastic lead time, we consider that the lead-time variance can be shortened at an extra crashing cost. Aiming to minimize the joint system costs, an integrated inventory model is formulated with a capacity constraint, and a solution procedure is developed to simultaneously optimize lead-time variance, replenishment cycle time, reorder point, and the integer numbers of shipment per production cycle for multiple suppliers. We also show that the buyer does a tradeoff between the crashing cost and the sum of holding and shortage costs for the decision making of the optimal lead-time variability. Numerical examples and sensitivity analysis are presented to validate the proposed model.

Keywords: supply chain., stochastic lead time, integrated model, variability reduction, Inventory.

Mathematics Subject Classification: Primary: 90B05, 90B15; Secondary: 90C9.

Citation: Xin Zhou, Liangping Shi, Bingzhi Huang. Integrated inventory model with stochastic lead time and controllable variability for milk runs. Journal of Industrial & Management Optimization, 2012, 8 (3) : 657-672. doi: 10.3934/jimo.2012.8.657

References:

[1]	M. Ben-Daya and M. Hariga, Integrated single vendor single buyer model with stochastic demand and variable lead time,, International Journal of Production Economics, 92 (2004), 75. doi: 10.1016/j.ijpe.2003.09.012.
[2]	H. Chang, L. Ouyang, K. Wu and C. Ho, Integrated vendor-buyer cooperative inventory models with controllable lead time and ordering cost reduction,, European Journal of Operational Research, 170 (2006), 481. doi: 10.1016/j.ejor.2004.06.029.
[3]	J. M. Chen and T. Chen, The multi-item replenishment problem in a two-echelon supply chain: the effect of centralization versus decentralization,, Computers and Operations Research, 32 (2005), 3191. doi: 10.1016/j.cor.2004.05.007.
[4]	S. Chopra, G. Reinhardt and M. Dada, The effect of lead time uncertainty on safety stocks,, Decision Sciences, 35 (2004), 1. doi: 10.1111/j.1540-5414.2004.02332.x.
[5]	C. F. Daganzo, The distance traveled to visit N points with a maximum of C stops per vehicle: An analytic model and an application,, Transportation Science, 18 (1984), 331. doi: 10.1287/trsc.18.4.331.
[6]	T. Du, F. K. Wang and P. Lu, A real-time vehicle-dispatching system for consolidating milk runs,, Transportation Research, 43 (2007), 565. doi: 10.1016/j.tre.2006.03.001.
[7]	Y. Gerchak and M. Parlar, Investing in reducing lead-time randomness in continuous-review inventory models,, Engineering Costs and Production Economics, 21 (1991), 191. doi: 10.1016/0167-188X(91)90032-W.
[8]	C. H. Glock, A comment: "Integrated single vendor-single buyer model with stochastic demand and variable lead time,", International Journal of Production Research, 122 (2009), 790. doi: 10.1016/j.ijpe.2009.06.032.
[9]	C. H. Glock, The joint economic lot size problem: A review,, International Journal of Production Economics, 135 (2012), 671. doi: 10.1016/j.ijpe.2011.10.026.
[10]	S. K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment,, Decision Sciences, 19 (1988), 236. doi: 10.1111/j.1540-5915.1988.tb00264.x.
[11]	M. G. Guler and T. Bilgic, On coordinating an assembly system under random yield and random demand,, European Journal of Operational Research, 196 (2009), 342. doi: 10.1016/j.ejor.2008.03.002.
[12]	M. Hariga and M. Ben-Daya, Some stochastic inventory models with deterministic variable lead time,, European Journal of Operational Research, 113 (1999), 42. doi: 10.1016/S0377-2217(97)00441-4.
[13]	X. J. He, J. G. Kim and C. H. Jack, The cost of lead-time variability: The case of the exponential distribution,, International Journal of Production Economics, 97 (2005), 113. doi: 10.1016/j.ijpe.2004.05.007.
[14]	R. M. Hill, The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem,, International Journal of Production Research, 37 (1999), 2463. doi: 10.1080/002075499190617.
[15]	M. A. Hoque and S. K. Goyal, An optimal policy for a single-vendor single-buyer integrated production-inventory system with capacity constraint of the transportation equipment,, International Journal of Production Economics, 65 (2000), 305. doi: 10.1016/S0925-5273(99)00082-1.
[16]	P. N. Joglekar, Comments on: A quantity discount pricing model to increase vendor profits,, Management Science, 34 (1988), 1391. doi: 10.1287/mnsc.34.11.1391.
[17]	C. J. Liao and C. H. Shyu, An analytical determination of lead time with normal demand,, International Journal of Operations and Production Management, 11 (1991), 72. doi: 10.1108/EUM0000000001287.
[18]	M. J. Liberatore, Planning Horizons for a stochastic lead-time inventory model,, Operations Research, 26 (1977), 927.
[19]	A. K. Maiti, M. K. Maiti and M. Maiti, Inventory model with stochastic lead-time and price dependent demand incorporating advance payment,, Applied Mathematical Modeling, 33 (2009), 2433. doi: 10.1016/j.apm.2008.07.024.
[20]	B. F. Moghadam and S. M. Seyedhosseini, A particle swarm approach to solve vehicle routing problem with uncertain demand: A drug distribution case study,, International Journal of Industrial Engineering Computations, 1 (2010), 55.
[21]	L.-Y. Ouyang and H.-C. Chang, Lot size reorder point inventory model with controllable lead time and set-up cost,, International Journal of Systems Science, 33 (2002), 635. doi: 10.1080/00207720210136685.
[22]	M. J. Paknejad, F. Nasri and J. F. Affisco, Lead-time variability reduction in stochastic inventory models,, European Journal of Operational Research, 62 (1992), 311. doi: 10.1016/0377-2217(92)90121-O.
[23]	M. J. Paknejad, F. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times,, Journal of Applied Mathematics and Decision Sciences, 3 (2005), 177.
[24]	J. C. Pan and J. S. Yang, A study of an integrated inventory with controllable lead time,, International Journal of Production Research, 40 (2002), 1263. doi: 10.1080/00207540110105680.
[25]	S. Sadjadi, M. Jafari and T. Amini, A new mathematical modeling and a genetic algorithm search for milk run problem (an auto industry supply chain case study),, International Journal of Advanced Manufacturing Technology, 44 (2009), 194. doi: 10.1007/s00170-008-1648-5.
[26]	G. P. Sphicas and F. Nasri, An inventory model with finite-range stochastic lead times,, Naval Research Logistics, 31 (1984), 609. doi: 10.1002/nav.3800310410.
[27]	J. C. Yu, H. M. Wee and K. J. Wang, Supply chain partnership for three-echelon deteriorating inventory model,, Journal of Industrial and Management Optimization, 4 (2008), 827.

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References:

[1]	M. Ben-Daya and M. Hariga, Integrated single vendor single buyer model with stochastic demand and variable lead time,, International Journal of Production Economics, 92 (2004), 75. doi: 10.1016/j.ijpe.2003.09.012.
[2]	H. Chang, L. Ouyang, K. Wu and C. Ho, Integrated vendor-buyer cooperative inventory models with controllable lead time and ordering cost reduction,, European Journal of Operational Research, 170 (2006), 481. doi: 10.1016/j.ejor.2004.06.029.
[3]	J. M. Chen and T. Chen, The multi-item replenishment problem in a two-echelon supply chain: the effect of centralization versus decentralization,, Computers and Operations Research, 32 (2005), 3191. doi: 10.1016/j.cor.2004.05.007.
[4]	S. Chopra, G. Reinhardt and M. Dada, The effect of lead time uncertainty on safety stocks,, Decision Sciences, 35 (2004), 1. doi: 10.1111/j.1540-5414.2004.02332.x.
[5]	C. F. Daganzo, The distance traveled to visit N points with a maximum of C stops per vehicle: An analytic model and an application,, Transportation Science, 18 (1984), 331. doi: 10.1287/trsc.18.4.331.
[6]	T. Du, F. K. Wang and P. Lu, A real-time vehicle-dispatching system for consolidating milk runs,, Transportation Research, 43 (2007), 565. doi: 10.1016/j.tre.2006.03.001.
[7]	Y. Gerchak and M. Parlar, Investing in reducing lead-time randomness in continuous-review inventory models,, Engineering Costs and Production Economics, 21 (1991), 191. doi: 10.1016/0167-188X(91)90032-W.
[8]	C. H. Glock, A comment: "Integrated single vendor-single buyer model with stochastic demand and variable lead time,", International Journal of Production Research, 122 (2009), 790. doi: 10.1016/j.ijpe.2009.06.032.
[9]	C. H. Glock, The joint economic lot size problem: A review,, International Journal of Production Economics, 135 (2012), 671. doi: 10.1016/j.ijpe.2011.10.026.
[10]	S. K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment,, Decision Sciences, 19 (1988), 236. doi: 10.1111/j.1540-5915.1988.tb00264.x.
[11]	M. G. Guler and T. Bilgic, On coordinating an assembly system under random yield and random demand,, European Journal of Operational Research, 196 (2009), 342. doi: 10.1016/j.ejor.2008.03.002.
[12]	M. Hariga and M. Ben-Daya, Some stochastic inventory models with deterministic variable lead time,, European Journal of Operational Research, 113 (1999), 42. doi: 10.1016/S0377-2217(97)00441-4.
[13]	X. J. He, J. G. Kim and C. H. Jack, The cost of lead-time variability: The case of the exponential distribution,, International Journal of Production Economics, 97 (2005), 113. doi: 10.1016/j.ijpe.2004.05.007.
[14]	R. M. Hill, The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem,, International Journal of Production Research, 37 (1999), 2463. doi: 10.1080/002075499190617.
[15]	M. A. Hoque and S. K. Goyal, An optimal policy for a single-vendor single-buyer integrated production-inventory system with capacity constraint of the transportation equipment,, International Journal of Production Economics, 65 (2000), 305. doi: 10.1016/S0925-5273(99)00082-1.
[16]	P. N. Joglekar, Comments on: A quantity discount pricing model to increase vendor profits,, Management Science, 34 (1988), 1391. doi: 10.1287/mnsc.34.11.1391.
[17]	C. J. Liao and C. H. Shyu, An analytical determination of lead time with normal demand,, International Journal of Operations and Production Management, 11 (1991), 72. doi: 10.1108/EUM0000000001287.
[18]	M. J. Liberatore, Planning Horizons for a stochastic lead-time inventory model,, Operations Research, 26 (1977), 927.
[19]	A. K. Maiti, M. K. Maiti and M. Maiti, Inventory model with stochastic lead-time and price dependent demand incorporating advance payment,, Applied Mathematical Modeling, 33 (2009), 2433. doi: 10.1016/j.apm.2008.07.024.
[20]	B. F. Moghadam and S. M. Seyedhosseini, A particle swarm approach to solve vehicle routing problem with uncertain demand: A drug distribution case study,, International Journal of Industrial Engineering Computations, 1 (2010), 55.
[21]	L.-Y. Ouyang and H.-C. Chang, Lot size reorder point inventory model with controllable lead time and set-up cost,, International Journal of Systems Science, 33 (2002), 635. doi: 10.1080/00207720210136685.
[22]	M. J. Paknejad, F. Nasri and J. F. Affisco, Lead-time variability reduction in stochastic inventory models,, European Journal of Operational Research, 62 (1992), 311. doi: 10.1016/0377-2217(92)90121-O.
[23]	M. J. Paknejad, F. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times,, Journal of Applied Mathematics and Decision Sciences, 3 (2005), 177.
[24]	J. C. Pan and J. S. Yang, A study of an integrated inventory with controllable lead time,, International Journal of Production Research, 40 (2002), 1263. doi: 10.1080/00207540110105680.
[25]	S. Sadjadi, M. Jafari and T. Amini, A new mathematical modeling and a genetic algorithm search for milk run problem (an auto industry supply chain case study),, International Journal of Advanced Manufacturing Technology, 44 (2009), 194. doi: 10.1007/s00170-008-1648-5.
[26]	G. P. Sphicas and F. Nasri, An inventory model with finite-range stochastic lead times,, Naval Research Logistics, 31 (1984), 609. doi: 10.1002/nav.3800310410.
[27]	J. C. Yu, H. M. Wee and K. J. Wang, Supply chain partnership for three-echelon deteriorating inventory model,, Journal of Industrial and Management Optimization, 4 (2008), 827.

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