American Institute of Mathematical Sciences

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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. J. Liberatore, Planning Horizons for a stochastic lead-time inventory model,, Operations Research, 26 (1977), 927.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. J. Liberatore, Planning Horizons for a stochastic lead-time inventory model,, Operations Research, 26 (1977), 927.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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