Models for Ordering Multiple Products Subject to Multiple Constraints, Quantity and Freight Discounts
ABSTRACT
One of the most important responsibilities of a supply chain manager is to decide “how much” (or “many”) of inventory items to order and how to transport them. This paper presents four mixed-integer linear programming models to help supply chain managers make these decisions for multiple products subject to multiple constraints when suppliers offer quantity discounts and shippers offer freight discounts. Each model deals with one of the possible combinations of all-units, incremental quantity discounts, all-weight and incremental freight discounts. The models are based on a piecewise linear approximation of the number of orders function. They allow any number of linear constraints and determine if independent or common (fixed) cycle ordering has a lower total cost. Results of computational experiments on an example problem are also presented.
Cite this paper
J. Moussourakis and C. Haksever, "Models for Ordering Multiple Products Subject to Multiple Constraints, Quantity and Freight Discounts," American Journal of Operations Research, Vol. 3 No. 6, 2013, pp. 521-535. doi: 10.4236/ajor.2013.36051.
J. Moussourakis and C. Haksever, "Models for Ordering Multiple Products Subject to Multiple Constraints, Quantity and Freight Discounts," American Journal of Operations Research, Vol. 3 No. 6, 2013, pp. 521-535. doi: 10.4236/ajor.2013.36051.
References
[1] R. Wilson, “24th Annual State of Logistics Report,” Council of Supply Chain Management Professionals, 2013. http://cscmp.org/publication-store/periodicals [2] C. Haksever and J. Moussourakis, “Determining Order Quantities in Multi-Product Inventory Systems Subject to Multiple Constraints and Incremental Discounts,” European Journal of Operational Research, Vol. 184, No. 3, 2008, pp. 930-945.
http://dx.doi.org/10.1016/j.ejor.2006.12.019 [3] W. C. Benton and S. Park, “A classification of Literature on Determining the Lot Size Under Quantity Discounts,” European Journal of Operational Research, Vol. 92, No. 2, 1996, pp. 219-238.
http://dx.doi.org/10.1016/0377-2217(95)00315-0 [4] C. Munson and M. J. Rosenblatt, “Theories and Realities of Quantity Discounts: An Exploratory Study,” Production and Operations Management, Vol. 7, No. 4, 1998, pp. 352-369.
http://dx.doi.org/10.1111/j.1937-5956.1998.tb00129.x [5] R. J. Tersine and S. Barman, “Lot Size Optimization with Quantity and Freight Rate Discounts,” Logistics and Transportation Review, Vol. 27, No. 4, 1991, pp. 319332. [6] R. J. Tersine and S. Barman, “Economic Inventory/ Transport Lot Sizing with Quantity and Freight Rate Discounts,” Decision Sciences, Vol. 22, No. 5, 1991, pp. 1171-1191.
http://dx.doi.org/10.1111/j.1540-5915.1991.tb01914.x [7] F. J. Arcelus and J. E. Rowcroft, “Inventory Policies with Freight Discounts and Disposals,” International Journal of Operations & Production Management, Vol. 11, No. 4, 1991, pp. 89-91.
http://dx.doi.org/10.1108/01443579110143197 [8] M. Diaby and A. Martel, “Dynamic Lot Sizing for Multiechelon Distribution Systems with Purchasing and Transportation Price Discounts,” Operations Research, Vol. 41, No. 1, 1993, pp. 48-59.
http://dx.doi.org/10.1287/opre.41.1.48 [9] R. J. Tersine, S. Barman and R. A. Toelle, “Composite Lot Sizing with Quantity and Freight Discounts,” Computers and Industrial Engineering, Vol. 28, No. 1, 1995, pp. 107-122.
http://dx.doi.org/10.1016/0360-8352(94)00031-H [10] T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, “Economic Lot Size Model for Price-Dependent Demand under Quantity and Freight Discounts,” International Journal of Production Economics, Vol. 48, No. 2, 1997, pp. 141-155.
http://dx.doi.org/10.1016/S0925-5273(96)00085-0 [11] M. A. Darwish, “Joint Determination of Order Quantity and Reorder Point of Continuous Review Model under Quantity and Freight Rate Discounts,” Computers & Operations Research, Vol. 35, No. 12, 2008, pp. 3902-3917.
http://dx.doi.org/10.1016/j.cor.2007.05.001 [12] A. Mendoza and J. A. Ventura, “Incorporating Quantity Discounts to the EOQ Model with Transportation Costs,” International Journal of Production Economics, Vol. 113, No. 2, 2008, pp. 754-765. [13] A. Toptal, “Replenishment Decisions under an All-Units Discount Schedule and Stepwise Freight Costs,” European Journal of Operational Research, Vol. 198, No. 2, 2009, pp. 504-510.
http://dx.doi.org/10.1016/j.ejor.2008.09.037 [14] W. C. Benton, “Quantity Discount Decisions under Conditions of Multiple Items, Multiple Suppliers and Resource Limitations,” International Journal of Production Research, Vol. 29, No. 10, 1991, pp. 1953-1961.
http://dx.doi.org/10.1080/00207549108948060 [15] P. A. Rubin and W. C. Benton, “Jointly Constrained Order Quantities with All-Units Discounts,” Naval Research Logistics, Vol. 40, No. 2, 1993, pp. 255-278.
http://dx.doi.org/10.1002/1520-6750(199303)40:2<255::AID-NAV3220400209>3.0.CO;2-G [16] P. A. Rubin and W. C. Benton, “Evaluating Jointly Constrained Order Quantity Complexities for Incremental Discounts,” European Journal of Operational Research, Vol. 149, No. 3, 2003, pp. 557-570.
http://dx.doi.org/10.1016/S0377-2217(02)00457-5 [17] F. Guder, J. Zydiak and S. Chaudhry, “Capacitated Multiple Item Ordering with Incremental Quantity Discounts,” Journal of the Operational Research Society, Vol. 45, No. 10, 1994, pp. 1197-1205. [18] M. J. Rosenblatt, “Multi-Item Inventory System with Budgetary Constraint: A Comparison between the Lagrangian and the Fixed Cycle Approach,” International Journal of Production Research, Vol. 19, No. 4, 1981, pp. 331-339.
http://dx.doi.org/10.1080/00207548108956661 [19] E. Page and R. J. Paul, “Multi-Product Inventory Situations with One Restriction,” Operational Research Quarterly, Vol. 27, No. 4, 1976, pp. 815-834. [20] F. Güder and J. Zydiak, “Fixed Cycle Ordering Policies for Capacitated Multiple Item Inventory Systems with Quantity Discounts,” Computers and Industrial Engineering, Vol. 38, No. 3, 2000, pp. 67-77.
http://dx.doi.org/10.1016/S0360-8352(00)00029-2 [21] J. Moussourakis and C. Haksever, “A Linear Approximation Approach to Multi-Product Multi-Constraint Inventory Management Problems,” Tamsui Oxford Journal of Management Sciences, Vol. 20, No. 1, 2004, pp. 35-55.
[1] R. Wilson, “24th Annual State of Logistics Report,” Council of Supply Chain Management Professionals, 2013. http://cscmp.org/publication-store/periodicals [2] C. Haksever and J. Moussourakis, “Determining Order Quantities in Multi-Product Inventory Systems Subject to Multiple Constraints and Incremental Discounts,” European Journal of Operational Research, Vol. 184, No. 3, 2008, pp. 930-945.
http://dx.doi.org/10.1016/j.ejor.2006.12.019 [3] W. C. Benton and S. Park, “A classification of Literature on Determining the Lot Size Under Quantity Discounts,” European Journal of Operational Research, Vol. 92, No. 2, 1996, pp. 219-238.
http://dx.doi.org/10.1016/0377-2217(95)00315-0 [4] C. Munson and M. J. Rosenblatt, “Theories and Realities of Quantity Discounts: An Exploratory Study,” Production and Operations Management, Vol. 7, No. 4, 1998, pp. 352-369.
http://dx.doi.org/10.1111/j.1937-5956.1998.tb00129.x [5] R. J. Tersine and S. Barman, “Lot Size Optimization with Quantity and Freight Rate Discounts,” Logistics and Transportation Review, Vol. 27, No. 4, 1991, pp. 319332. [6] R. J. Tersine and S. Barman, “Economic Inventory/ Transport Lot Sizing with Quantity and Freight Rate Discounts,” Decision Sciences, Vol. 22, No. 5, 1991, pp. 1171-1191.
http://dx.doi.org/10.1111/j.1540-5915.1991.tb01914.x [7] F. J. Arcelus and J. E. Rowcroft, “Inventory Policies with Freight Discounts and Disposals,” International Journal of Operations & Production Management, Vol. 11, No. 4, 1991, pp. 89-91.
http://dx.doi.org/10.1108/01443579110143197 [8] M. Diaby and A. Martel, “Dynamic Lot Sizing for Multiechelon Distribution Systems with Purchasing and Transportation Price Discounts,” Operations Research, Vol. 41, No. 1, 1993, pp. 48-59.
http://dx.doi.org/10.1287/opre.41.1.48 [9] R. J. Tersine, S. Barman and R. A. Toelle, “Composite Lot Sizing with Quantity and Freight Discounts,” Computers and Industrial Engineering, Vol. 28, No. 1, 1995, pp. 107-122.
http://dx.doi.org/10.1016/0360-8352(94)00031-H [10] T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, “Economic Lot Size Model for Price-Dependent Demand under Quantity and Freight Discounts,” International Journal of Production Economics, Vol. 48, No. 2, 1997, pp. 141-155.
http://dx.doi.org/10.1016/S0925-5273(96)00085-0 [11] M. A. Darwish, “Joint Determination of Order Quantity and Reorder Point of Continuous Review Model under Quantity and Freight Rate Discounts,” Computers & Operations Research, Vol. 35, No. 12, 2008, pp. 3902-3917.
http://dx.doi.org/10.1016/j.cor.2007.05.001 [12] A. Mendoza and J. A. Ventura, “Incorporating Quantity Discounts to the EOQ Model with Transportation Costs,” International Journal of Production Economics, Vol. 113, No. 2, 2008, pp. 754-765. [13] A. Toptal, “Replenishment Decisions under an All-Units Discount Schedule and Stepwise Freight Costs,” European Journal of Operational Research, Vol. 198, No. 2, 2009, pp. 504-510.
http://dx.doi.org/10.1016/j.ejor.2008.09.037 [14] W. C. Benton, “Quantity Discount Decisions under Conditions of Multiple Items, Multiple Suppliers and Resource Limitations,” International Journal of Production Research, Vol. 29, No. 10, 1991, pp. 1953-1961.
http://dx.doi.org/10.1080/00207549108948060 [15] P. A. Rubin and W. C. Benton, “Jointly Constrained Order Quantities with All-Units Discounts,” Naval Research Logistics, Vol. 40, No. 2, 1993, pp. 255-278.
http://dx.doi.org/10.1002/1520-6750(199303)40:2<255::AID-NAV3220400209>3.0.CO;2-G [16] P. A. Rubin and W. C. Benton, “Evaluating Jointly Constrained Order Quantity Complexities for Incremental Discounts,” European Journal of Operational Research, Vol. 149, No. 3, 2003, pp. 557-570.
http://dx.doi.org/10.1016/S0377-2217(02)00457-5 [17] F. Guder, J. Zydiak and S. Chaudhry, “Capacitated Multiple Item Ordering with Incremental Quantity Discounts,” Journal of the Operational Research Society, Vol. 45, No. 10, 1994, pp. 1197-1205. [18] M. J. Rosenblatt, “Multi-Item Inventory System with Budgetary Constraint: A Comparison between the Lagrangian and the Fixed Cycle Approach,” International Journal of Production Research, Vol. 19, No. 4, 1981, pp. 331-339.
http://dx.doi.org/10.1080/00207548108956661 [19] E. Page and R. J. Paul, “Multi-Product Inventory Situations with One Restriction,” Operational Research Quarterly, Vol. 27, No. 4, 1976, pp. 815-834. [20] F. Güder and J. Zydiak, “Fixed Cycle Ordering Policies for Capacitated Multiple Item Inventory Systems with Quantity Discounts,” Computers and Industrial Engineering, Vol. 38, No. 3, 2000, pp. 67-77.
http://dx.doi.org/10.1016/S0360-8352(00)00029-2 [21] J. Moussourakis and C. Haksever, “A Linear Approximation Approach to Multi-Product Multi-Constraint Inventory Management Problems,” Tamsui Oxford Journal of Management Sciences, Vol. 20, No. 1, 2004, pp. 35-55.