Robust Optimization of Multi-product and Multi-class Lot-sizing and Supplier Selection with Uncertain Demand

Document Type : Original Article


1 Assistant Professor, University of Qom.

2 MSc, University of Qom.



In this study, a multi-product and multi-period lot-sizing and supplier selection problem has been considered. The demand of products is multi-class and uncertain. Due to the interchangeability of products, it is possible to satisfy some part of their demand with the alternatives. In the case of an inventory shortage, a specific part of the shortage will be lost sales and the remainder will be backorder. Suppliers can have an all-units quantity discount policy. For confronting uncertain demand, possible modes are defined as scenarios and the robust optimization approach proposed by Mulvey et al. is applied. The objective function of the problem, which is to be minimized, is made up of the total cost of purchasing, transportation, inventory, demand substitution, lost sales, and backorder. The upper and lower bounds for the number of suppliers per product family are defined as a constraint for the implementation of management policies for supplier selection. Dynamic and multi-class demand in a multi-period horizon, together with allowed backlog and lost sales, are features that, to the best of our knowledge, are not yet considered by other researchers. There are many supply chains that have sold the products and should supply required spare parts. The results of this research help attain optimal ordering of the parts. The performance of the model is examined with a numerical example.


  1. Aggarwal, R., & Singh, S. P. (2015). Chance constraint-based multi-objective stochastic model for supplier selection. The International Journal of Advanced Manufacturing Technology79(9-12), 1707-1719.
  2. Alfares, H. K., & Turnadi, R. (2018). Lot sizing and supplier selection with multiple items, multiple periods, quantity discounts, and backordering. Computers & Industrial Engineering116, 59-71.
  3. Alfieri, A., Pastore, E., & Zotteri, G. (2017). Dynamic inventory rationing: How to allocate stock according to managerial priorities. An empirical study. International Journal of Production Economics189, 14-29.
  4. Aouadni, S., Aouadni, I., & Rebaï, A. (2019). A systematic review on supplier selection and order allocation problems. Journal of Industrial Engineering International15, 267-289.
  5. Azadnia, A. H. (2016). A multi-objective mathematical model for sustainable supplier selection and order lot-sizing under inflation. International Journal of Engineering-Transactions B: Applications29(8), 1141.
  6. Bao, L., Liu, Z., Yu, Y., & Zhang, W. (2018). On the decomposition property for a dynamic inventory rationing problem with multiple demand classes and backorder. European Journal of Operational Research, 265(1), 99-106.
  7. Buschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. (2010). Dynamic capacitated lot-sizing problems: a classification and review of solution approaches. Or Spectrum32(2), 231-261.
  8. Chen, S., Xu, J., & Feng, Y. (2010). A partial characterization of the optimal ordering/rationing policy for a periodic review system with two demand classes and backordering. Naval Research Logistics (NRL)57(4), 330-341.
  9. Cheraghalipour, A., & Farsad, S. (2018). A bi-objective sustainable supplier selection and order allocation considering quantity discounts under disruption risks: A case study in plastic industry. Computers & Industrial Engineering118, 237-250.
  10. Choudhary, D., & Shankar, R. (2013). Joint decision of procurement lot-size, supplier selection, and carrier selection. Journal of Purchasing and Supply Management19(1), 16-26.
  11. Choudhary, D., & Shankar, R. (2014). A goal programming model for joint decision making of inventory lot-size, supplier selection and carrier selection. Computers & Industrial Engineering71, 1-9.
  12. Ding, Q., Kouvelis, P., & Milner, J. M. (2007). Dynamic pricing for multiple class deterministic demand fulfillment. IIE Transactions39(11), 997-1013.
  13. Gebennini, E., Zeppetella, L., Grassi, A., & Rimini, B. (2015). Production scheduling to optimize the product assortment in case of constrained capacity and customer-driven substitution. IFAC-PapersOnLine48(3), 1954-1959.
  14. Ghaniabadi, M., & Mazinani, A. (2017). Dynamic lot sizing with multiple suppliers, backlogging and quantity discounts. Computers & Industrial Engineering110, 67-74.
  15. Horri, M.S., & Anjomshoa, A. (2016). Multi-Objective Mathematical Model for Supplier Selection and Order Allocation under Multi-Item Condition. Journal of Industrial Management Perspective, 6(1), 153-179 (In Persian).
  16. Hung, Y. F., & Hsiao, J. Y. (2013). Inventory rationing decision models during replenishment lead time. International Journal of Production Economics144(1), 290-300.
  17. Jans, R., & Degraeve, Z. (2008). Modeling industrial lot sizing problems: a review. International Journal of Production Research46(6), 1619-1643.
  18. Khosroabadi, M., Lotfi, M., & Khademizare, H. (2014). Joint supplier selection and order lot-sizing problem of remanufacturable items with price and transportation cost discounts. Journal of Industrial Engineering Research in Production Systems1(2), 139-153 (In Persian).
  19. Lee, A. H., Kang, H. Y., & Lai, C. M. (2013). Solving lot-sizing problem with quantity discount and transportation cost. International journal of systems science44(4), 760-774.
  20. Li, H., & Thorstenson, A. (2014). A multi-phase algorithm for a joint lot-sizing and pricing problem with stochastic demands. International Journal of Production Research, 52(8), 2345-2362.
  21. Manouchehri, S., Tajdin, A., & Shirazi, B. (2019). Robust Integrated Optimization for Green Closed Loop Supply Chain. Journal of Industrial Management Perspective9(3), 55-85 (In Persian).
  22. Minner, S. (2009). A comparison of simple heuristics for multi-product dynamic demand lot-sizing with limited warehouse capacity. International Journal of Production Economics, 118(1), 305-310.
  23. Mulvey, J. M., Vanderbei, R. J., & Zenios, S. A. (1995). Robust optimization of large-scale systems. Operations research, 43(2), 264-281.
  24. Nourmohamadi Shalke, P., Paydar, M. M., & Hajiaghaei-Keshteli, M. (2018). Sustainable supplier selection and order allocation through quantity discounts. International Journal of Management Science and Engineering Management, 13(1), 20-32.
  25. Pasandideh, S. H. R., Niaki, S. T. A., & Mousavi, S. M. (2013). Two metaheuristics to solve a multi-item multiperiod inventory control problem under storage constraint and discounts. The International Journal of Advanced Manufacturing Technology69(5-8), 1671-1684.
  26. Rabieh, M., Azar, A., Yazdi, M.M., & Haghighi, M.F.F. (2011). Designing a Multi-Objective Resource-Based Mathematical Modeling: An Approach to Supply Chain Risk Reduction (Case Study: Iran Khodro Supply Chain). Journal of Industrial Management Perspective1(1), 57-77 (In Persian).
  27. Rabieh, M., & Esmaelian, M. (2012). Designing a Fuzzy Non-Linear Model of Supplier Selection in Case of Multiple Sourcing. Journal of Industrial Management Perspective1(4), 81-105 (In Persian).
  28. Rahmani, D., Ramezanian, R., Fattahi, P., & Heydari, M. (2013). A robust optimization model for multi-product two-stage capacitated production planning under uncertainty. Applied Mathematical Modelling, 37(20-21), 8957-8971.
  29. Rossi, R., Tarim, S. A., & Bollapragada, R. (2012). Constraint-based local search for inventory control under stochastic demand and lead time. INFORMS journal on computing, 24(1), 66-80.
  30. Sambatt, M., Woarawichai, C., & Naenna, T. (2019). Inventory lot sizing and supplier selection for multiple products, multiple suppliers, multiple periods with storage space using lingo program. In MATEC Web of Conferences (Vol. 259). EDP Sciences.
  31. Shin, H., Park, S., Lee, E., & Benton, W. C. (2015). A classification of the literature on the planning of substitutable products. European Journal of Operational Research246(3), 686-699.
  32. Soto, A. V., Chowdhury, N. T., Allahyari, M. Z., Azab, A., & Baki, M. F. (2017). Mathematical modeling and hybridized evolutionary LP local search method for lot-sizing with supplier selection, inventory shortage, and quantity discounts. Computers & Industrial Engineering, 109, 96-112.
  33. Topkis, D. M. (1968). Optimal ordering and rationing policies in a nonstationary dynamic inventory model with n demand classes. Management Science15(3), 160-176.
  34. Wagner, H. M., & Whitin, T. M. (1958). Dynamic version of the economic lot size model. Management science, 5(1), 89-96.
  35. Wan, G., & Cao, Y. (2018). A continuous cost evaluation approach for periodic review inventory systems with threshold rationing policy. Computers & Industrial Engineering, 126, 75-87.
  36. Wang, D., & Tang, O. (2014). Dynamic inventory rationing with mixed backorders and lost sales. International Journal of Production Economics, 149, 56-67.
  37. Wang, L., & Li, J. (2019). A Robust Weighted Goal Programming Approach for Supplier Selection Problem with Inventory Management and Vehicle Allocation in Uncertain Environment. In International Conference on Management Science and Engineering Management (pp. 295-309). Springer, Cham.
  38. Yu, C. S., & Li, H. L. (2000). A robust optimization model for stochastic logistic problems. International journal of production economics, 64(1-3), 385-397.
  39. Yücel, E., Karaesmen, F., Salman, F. S., & Türkay, M. (2009). Optimizing product assortment under customer-driven demand substitution. European Journal of Operational Research, 199(3), 759-768.
  40. Zanjani, M. K., Ait-Kadi, D., & Nourelfath, M. (2010). Robust production planning in a manufacturing environment with random yield: A case in sawmill production planning. European Journal of Operational Research, 201(3), 882-891.
  41. Zhang, J. L., & Zhang, M. Y. (2011). Supplier selection and purchase problem with fixed cost and constrained order quantities under stochastic demand. International Journal of Production Economics129(1), 1-7.
  42. Petersen, Clifford C. (1971). A Note on Transforming the Product of Variables to Linear Form in Linear Programs. Working Paper, Purdue University.