Two-Stage Ordering Policy under Buyer’s Minimum-Commitment Quantity Contract

ABSTRACT

In this paper we consider a two-stage ordering problem with a buyer’s minimum commitment quantity contract. Under the contract the buyer is required to give a minimum-commitment quantity. Then the manufacturer has the obligations to supply the minimum-commitment quantity and to provide a shortage compensation policy to the buyer. We formulate a dynamic optimization model to determine the manufacturer’s two stage order quantities for maximizing the expected profit. The conditions for the existence of the optimal solution are defined. And we also develop a procedure to solve the problem. Numerical examples are given to illustrate the proposed solution procedure and sensitivity analyses are performed to find managerial insights.

In this paper we consider a two-stage ordering problem with a buyer’s minimum commitment quantity contract. Under the contract the buyer is required to give a minimum-commitment quantity. Then the manufacturer has the obligations to supply the minimum-commitment quantity and to provide a shortage compensation policy to the buyer. We formulate a dynamic optimization model to determine the manufacturer’s two stage order quantities for maximizing the expected profit. The conditions for the existence of the optimal solution are defined. And we also develop a procedure to solve the problem. Numerical examples are given to illustrate the proposed solution procedure and sensitivity analyses are performed to find managerial insights.

Cite this paper

nullH. Hsu and Z. Chen, "Two-Stage Ordering Policy under Buyer’s Minimum-Commitment Quantity Contract,"*American Journal of Operations Research*, Vol. 1 No. 3, 2011, pp. 84-99. doi: 10.4236/ajor.2011.13012.

nullH. Hsu and Z. Chen, "Two-Stage Ordering Policy under Buyer’s Minimum-Commitment Quantity Contract,"

References

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[2] S. Sethi, H. Yan and H. Zhang, “Inventory and Supply Chain Management with Forecast Updates,” Kluwer Aca- demic Publishers, Boston, 2005.

[3] E. J. Durango-Cohen and C. A. Yano, “Supplier Commitment and Production Decisions under a Forecast- Commitment Contract,” Management Science, Vol. 52, No. 1, 2006, pp. 54-67. doi:10.1287/mnsc.1050.0471

[4] Y. Bassok and R. Anupindi, “Analysis of Supply Contracts with Total Minimum Commitment,” IIE Transactions, Vol. 29, No. 5, 1997, pp. 373-381. doi:10.1080/07408179708966342

[5] R. Anupindi and Y. Bassok, “Approximations for Multiproduct Contracts with Stochastic Demands and Business Volume Discounts: Single Supplier Case,” IIE Transactions, Vol. 30, No. 8, 1998, pp. 723-734. doi:10.1080/07408179808966518

[6] S. Sethi and G. Sorger, “A Theory of Rolling Horizon Decision Making,” Annals of Operations Research, Vol. 29, No. 1-4, 1991, pp. 387-416. doi:10.1007/BF02283607

[7] Y. Bassok and R. Anupindi, “Analysis of Supply Contracts with Commitments and Flexibility,” Naval Research Logistics, Vol. 55, No. 5, 2008, pp. 459-477. doi:10.1002/nav.20300

[8] Z. Lian and A. Deshmukh, “Analysis of Supply Contracts with Quantity Flexibility,” European Journal of Operational Research, Vol. 196, No. 2, 2009, pp. 526-533. doi:10.1016/j.ejor.2008.02.043

[9] G. Gallego and O. Ozer, “Integrating Replenishment Decisions with Advance Demand Information,” Management Science, Vol. 47, No. 10, 2001, pp. 1344-1360. doi:10.1287/mnsc.47.10.1344.10261

[10] Y. Aviv, “A Time-Series Framework for Supply-Chain Inventory Management,” Operations Research, Vol. 51, No. 2, 2003, pp. 210-227. doi:10.1287/opre.51.2.210.12780

[11] A. Dvoretzky, J. Kiefer and J. Wolfowitz, “The Inventory Problem: II. Case of Unknown Distribution of Demand,” Econometrica, Vol. 20, No. 3, 1952, pp. 450-466.

[12] H. Scarf, “Bayes Solution of the Statistical Inventory Problem,” Annals of Mathematical Statistics, Vol. 30, No. 2, 1959, pp. 490-508. doi:10.1214/aoms/1177706264

[13] H. Scarf, “Some Remarks on Bayes Solutions to the Inventory Problem,” Naval Research Logistics, Vol. 7, No. 4, 1960, pp. 591-596. doi:10.1002/nav.3800070428

[14] M. A. Lariviere and E. L. Porteus, “Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales,” Management Science, Vol. 45, No. 3, 1999, pp. 346-363. doi:10.1287/mnsc.45.3.346

[15] G. D. Eppen and A. V. Iyer, “Improved Fashion Buying with Bayesian Updates,” Operations Research, Vol. 45, No. 6, 1997, pp. 805-819.

[16] J. Wu, “Quantity Flexibility Contracts under Bayesian Updating,” Computers & Operations Research, Vol. 32, No. 5, 2005, pp. 1267-1288.

[17] A. V. Iyer and M. E. Bergen, “Quick Response in Manufacturer-Retailer Channels,” Management Science, Vol. 43, No. 4, 1997, pp. 559-570.

[18] T. M. Choi, D. Li and H. Yan, “Optimal Two-Stage Ordering Policy with Bayesian Information Updating,” Journal of the Operation Research Society, Vol. 54, No. 8, 2003, pp. 846-859.

[19] O. Johnson and H. Thompson, “Optimality of Myopic Inventory Policies for Certain Dependent Demand Processes,” Management Science, Vol. 21, No. 11, 1975, pp. 1303-1307. doi:10.1287/mnsc.21.11.1303

[20] W. H. Hausmann, “Sequential Decision Problems: A Model to Exploit Existing Forecast,” Management Science, Vol. 16, No. 2, 1969, pp. B93-B111. doi:10.1287/mnsc.16.2.B93

[21] D. Heath and P. Jackson, “Modeling the Evolution of Demand Forecast with Application to Safety Stock Ana- lysis in Production/Distribution Systems,” IIE Transactions, Vol. 26. No. 3, 1994, pp. 17-30. doi:10.1080/07408179408966604

[22] Y. Wang and B. Tomlin, “To Wait or Not to Wait: Optimal Ordering under Lead Time Uncertainty and Forecast Updating,” Naval Research Logistics, Vol. 56, No. 8, 2009, pp. 766-779. doi:10.1002/nav.20381

[23] H. Gurnani and C. S. Tang, “Optimal Ordering Decision with Uncertain Cost and Demand Forecast Updating,” Management Science, Vol. 45, No. 10, 1999, pp. 1456- 1462. doi:10.1287/mnsc.45.10.1456

[24] K. L. Donohue, “Efficient Supply Contract for Fashion Goods with Forecast Updating and Two Production Modes,” Management Science, Vol. 46, No. 11, 2000, pp. 1397- 1411. doi:10.1287/mnsc.46.11.1397.12088

[25] H. Y. Huang, S. Sethi and H. Yan, “Purchase Contract Management with Demand Forecast Updates,” IIE Transactions, Vol. 37, No. 8, 2005, pp. 775-785. doi:10.1080/07408170590961157

[26] H. Chen, J. Chen and Y. Chen, “A Coordination Mechanism for a Supply Chain with Demand Information Updating,” International Journal of Production Economics, Vol. 103, No. 1, 2006, pp. 347-361. doi:10.1016/j.ijpe.2005.09.002

[27] H. Yan, K. Liu and A. Hsu, “Order Quantity in Dual Supply Mode with Updating Forecasts,” Production and Operations Management, Vol. 12, No. 1, 2003, pp. 30-45. doi:10.1111/j.1937-5956.2003.tb00196.x

[28] S. Sethi, H. Yan and H. Zhang, “Quantity Flexibility Contracts: Optimal Decisions with Information Updates,” Decision Sciences, Vol. 35, No. 4, 2004, pp. 691-712. doi:10.1111/j.1540-5915.2004.02873.x

[1] R. Anupindi and Y. Bassok, “Supply Contracts with Quan- tity Commitments and Stochastic Demand,” In: S. Tayur, R. Ganeshan and M. Magazine, Eds., Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Boston, 1999.

[2] S. Sethi, H. Yan and H. Zhang, “Inventory and Supply Chain Management with Forecast Updates,” Kluwer Aca- demic Publishers, Boston, 2005.

[3] E. J. Durango-Cohen and C. A. Yano, “Supplier Commitment and Production Decisions under a Forecast- Commitment Contract,” Management Science, Vol. 52, No. 1, 2006, pp. 54-67. doi:10.1287/mnsc.1050.0471

[4] Y. Bassok and R. Anupindi, “Analysis of Supply Contracts with Total Minimum Commitment,” IIE Transactions, Vol. 29, No. 5, 1997, pp. 373-381. doi:10.1080/07408179708966342

[5] R. Anupindi and Y. Bassok, “Approximations for Multiproduct Contracts with Stochastic Demands and Business Volume Discounts: Single Supplier Case,” IIE Transactions, Vol. 30, No. 8, 1998, pp. 723-734. doi:10.1080/07408179808966518

[6] S. Sethi and G. Sorger, “A Theory of Rolling Horizon Decision Making,” Annals of Operations Research, Vol. 29, No. 1-4, 1991, pp. 387-416. doi:10.1007/BF02283607

[7] Y. Bassok and R. Anupindi, “Analysis of Supply Contracts with Commitments and Flexibility,” Naval Research Logistics, Vol. 55, No. 5, 2008, pp. 459-477. doi:10.1002/nav.20300

[8] Z. Lian and A. Deshmukh, “Analysis of Supply Contracts with Quantity Flexibility,” European Journal of Operational Research, Vol. 196, No. 2, 2009, pp. 526-533. doi:10.1016/j.ejor.2008.02.043

[9] G. Gallego and O. Ozer, “Integrating Replenishment Decisions with Advance Demand Information,” Management Science, Vol. 47, No. 10, 2001, pp. 1344-1360. doi:10.1287/mnsc.47.10.1344.10261

[10] Y. Aviv, “A Time-Series Framework for Supply-Chain Inventory Management,” Operations Research, Vol. 51, No. 2, 2003, pp. 210-227. doi:10.1287/opre.51.2.210.12780

[11] A. Dvoretzky, J. Kiefer and J. Wolfowitz, “The Inventory Problem: II. Case of Unknown Distribution of Demand,” Econometrica, Vol. 20, No. 3, 1952, pp. 450-466.

[12] H. Scarf, “Bayes Solution of the Statistical Inventory Problem,” Annals of Mathematical Statistics, Vol. 30, No. 2, 1959, pp. 490-508. doi:10.1214/aoms/1177706264

[13] H. Scarf, “Some Remarks on Bayes Solutions to the Inventory Problem,” Naval Research Logistics, Vol. 7, No. 4, 1960, pp. 591-596. doi:10.1002/nav.3800070428

[14] M. A. Lariviere and E. L. Porteus, “Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales,” Management Science, Vol. 45, No. 3, 1999, pp. 346-363. doi:10.1287/mnsc.45.3.346

[15] G. D. Eppen and A. V. Iyer, “Improved Fashion Buying with Bayesian Updates,” Operations Research, Vol. 45, No. 6, 1997, pp. 805-819.

[16] J. Wu, “Quantity Flexibility Contracts under Bayesian Updating,” Computers & Operations Research, Vol. 32, No. 5, 2005, pp. 1267-1288.

[17] A. V. Iyer and M. E. Bergen, “Quick Response in Manufacturer-Retailer Channels,” Management Science, Vol. 43, No. 4, 1997, pp. 559-570.

[18] T. M. Choi, D. Li and H. Yan, “Optimal Two-Stage Ordering Policy with Bayesian Information Updating,” Journal of the Operation Research Society, Vol. 54, No. 8, 2003, pp. 846-859.

[19] O. Johnson and H. Thompson, “Optimality of Myopic Inventory Policies for Certain Dependent Demand Processes,” Management Science, Vol. 21, No. 11, 1975, pp. 1303-1307. doi:10.1287/mnsc.21.11.1303

[20] W. H. Hausmann, “Sequential Decision Problems: A Model to Exploit Existing Forecast,” Management Science, Vol. 16, No. 2, 1969, pp. B93-B111. doi:10.1287/mnsc.16.2.B93

[21] D. Heath and P. Jackson, “Modeling the Evolution of Demand Forecast with Application to Safety Stock Ana- lysis in Production/Distribution Systems,” IIE Transactions, Vol. 26. No. 3, 1994, pp. 17-30. doi:10.1080/07408179408966604

[22] Y. Wang and B. Tomlin, “To Wait or Not to Wait: Optimal Ordering under Lead Time Uncertainty and Forecast Updating,” Naval Research Logistics, Vol. 56, No. 8, 2009, pp. 766-779. doi:10.1002/nav.20381

[23] H. Gurnani and C. S. Tang, “Optimal Ordering Decision with Uncertain Cost and Demand Forecast Updating,” Management Science, Vol. 45, No. 10, 1999, pp. 1456- 1462. doi:10.1287/mnsc.45.10.1456

[24] K. L. Donohue, “Efficient Supply Contract for Fashion Goods with Forecast Updating and Two Production Modes,” Management Science, Vol. 46, No. 11, 2000, pp. 1397- 1411. doi:10.1287/mnsc.46.11.1397.12088

[25] H. Y. Huang, S. Sethi and H. Yan, “Purchase Contract Management with Demand Forecast Updates,” IIE Transactions, Vol. 37, No. 8, 2005, pp. 775-785. doi:10.1080/07408170590961157

[26] H. Chen, J. Chen and Y. Chen, “A Coordination Mechanism for a Supply Chain with Demand Information Updating,” International Journal of Production Economics, Vol. 103, No. 1, 2006, pp. 347-361. doi:10.1016/j.ijpe.2005.09.002

[27] H. Yan, K. Liu and A. Hsu, “Order Quantity in Dual Supply Mode with Updating Forecasts,” Production and Operations Management, Vol. 12, No. 1, 2003, pp. 30-45. doi:10.1111/j.1937-5956.2003.tb00196.x

[28] S. Sethi, H. Yan and H. Zhang, “Quantity Flexibility Contracts: Optimal Decisions with Information Updates,” Decision Sciences, Vol. 35, No. 4, 2004, pp. 691-712. doi:10.1111/j.1540-5915.2004.02873.x