Optimizing a bi-objective vendor-managed inventory of multi-product EPQ model for a green supply chain with stochastic constraints

Document Type: Research Paper


Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran


In this paper, a bi-objective multi-product single-vendor single-buyer supply chain problem is studied under green vendor-managed inventory (VMI) policy based on the economic production quantity (EPQ) model. To bring the model closer to real-world supply chain, four constraints of model including backordering cost, number of orders, production budget and warehouse space are considered stochastic. In addition to holding, ordering and backordering costs of the VMI chain, the unused storage space cost is also added to the total cost of the chain. To observe environmental requirements and decrease the adverse effects of greenhouse gases emissions (GHGs) on the earth and human’s life, green supply chain is utilized to reduce the GHGs emissions through storage and transportation activities in the second objective function. Three multi-objective decision making methods namely, LP-metric, Goal attainment and multi-choice goal programming with utility function (MCGP-U) are implemented in different sizes to solve the presented model as well. Two multi-criteria decision making (MCDM) approach and statistical analysis are applied to compare the outcomes of three proposed solving methods. GAMS/BARON software is utilized to minimize the values of the objective functions. At the end, numerical examples are presented to represent the application of the mentioned methodology. To come up with more insights, sensitivity analysis is executed on the main parameters of proposed model.  


Main Subjects

Achabal DD, McIntyre SH, Smith SA, & Kalyanam K. (2000) A decision
support system for vendor managed inventory. J Retail 76: 430–454.

Alfares, H. K., & Attia, A. M. (2017). A supply chain model with vendor-managed inventory, consignment, and quality inspection errors. International Journal of Production Research, 55(19), 5706-5727.

Azaron, A., Katagiri, H., Kato, K., & Sakawa, M. (2006). Modelling complex assemblies as a queueing network for lead time control. European Journal of Operational Research, 174(1), 150-168.

Azaron, A., Katagiri, H., Kato, K., & Sakawa, M. (2007). A multi-objective discrete reliability optimization problem for dissimilar-unit standby systems. OR Spectrum, 29(2), 235.

Azaron, A., Katagiri, H., & Sakawa, M. (2007). Time-cost trade-off via optimal control theory in Markov PERT networks. Annals of Operations Research, 150(1), 47-64.

Bakeshlu, E. A., Sadeghi, J., Poorbagheri, T., & Taghizadeh, M. (2014). Optimizing a bi-objective inventory model for a two-echelon supply chain management using a tuned meta-heuristic algorithm. Production & Manufacturing Research, 2(1), 156-166.

Bazan, E., Jaber, M. Y., & Zanoni, S. (2015). Supply chain models with greenhouse gases emissions, energy usage and different coordination decisions. Applied Mathematical Modelling, 39(17), 5131-5151.

Bonney, M., & Jaber, M.Y. (2011).  Environmentally responsible inventory models: nonclassical models for a non-classical era, Int. J. Prod.Econ.133(1)43–53.

Branke, J., Branke, J., Deb, K., Miettinen, K., & Slowiński, R. (Eds.). (2008). Multiobjective optimization: Interactive and evolutionary approaches (Vol. 5252). Springer Science & Business Media.

Büyüközkan, G., & Cifci, G. (2013). An integrated QFD framework with multiple formatted and incomplete preferences: a sustainable supply chain application, Appl. Soft Comput. 13 3931–3941.

Çetinkaya, S., & Lee, C. Y. (2000). Stock replenishment and shipment scheduling for vendor-managed inventory systems. Management Science, 46(2), 217-232.

Chang, C. T. (2011). Multi-choice goal programming with utility functions. European Journal of Operational Research, 215(2), 439-445.

Darwish, M.A., Odah, O.M. )2010(. Vendor managed inventory model for single-vendor multi-retailer supply chains. European Journal of Operational Research.204 (3),473–484.

Disney, S.M., & Towill, D.R. (2002). A procedure for the optimization of the dynamic response of a vendor managed inventory system, Computers & Industrial Engineering 43 27–58.

Disney, S.M., & Towill, D.R. (2003).  The effect of vendor managed inventory (VMI)
dynamics on the Bullwhip Effect in supply chains, International Journal of
Production Economics 85, 199–215.

Dong, Y., & Xu, K. (2002). A supply chain model of vendor managed inventory. Transportation research part E: logistics and transportation review, 38(2), 75-95.

Duckstein, L., & Opricovic, S. (1980). Multiobjective optimization in river basin development. Water resources research, 16(1), 14-20.

Fugate, B., Sahin, F., & Mentzer, J. T. (2006). Supply chain management
coordination mechanisms. Journal of Business Logistics, 27, 129–161.

Gembicki, F. W. (1974). Vector optimization for control with performance and parameter sensitivity indices. Ph. D. thesis, Case Western Reserve Univ.

Gharaei, A., Karimi, M., & Shekarabi, S. A. H. (2018). An integrated multi-product multi-buyer supply chain under the penalty, green, and VMI-CS policies: OA/ER/AP algorithm. Applied Mathematical Modelling. doi:10.1016/j.apm.2018.11.035. 

Golpîra, H., Najafi, E., Zandieh, M., & Sadi-Nezhad, S. (2017). Robust bi-level optimization for green opportunistic supply chain network design problem against uncertainty and environmental risk. Computers & Industrial Engineering, 107, 301-312.

Golpîra, H. (2020). Optimal integration of the facility location problem into the multi-project multi-supplier multi-resource Construction Supply Chain network design under the vendor managed inventory strategy. Expert Systems with Applications, 139, 112841. doi:10.1016/j.eswa.2019.112841.

Han, J., Lu, J., & Zhang, G. (2017). Tri-level decision-making for decentralized vendor-managed inventory. Information Sciences, 421, 85-103.

Hemmati, M., Fatemi Ghomi, S. M. T., & Sajadieh, M. S. (2017). Vendor managed inventory with consignment stock for supply chain with stock-and price-dependent demand. International Journal of Production Research, 55(18), 5225-5242.

Hwang, C. L., & Yoon, K. (1981). Methods for multiple attribute decision making. In Multiple attribute decision making (pp. 58-191). Springer, Berlin, Heidelberg.

Hwang, C.L., Lai, Y.J., Liu, T.Y.)1993(. A new approach for multiple objective decision making. Computers and Operational Research. 20, 889–899. doi:10.1016/0305-0548(93)90109-v.

Jiang, Y., Li, B., Qu, X., & Cheng, Y. (2015). A green vendor-managed inventory analysis in supply chains under carbon emissions trading mechanism. Clean technologies and environmental policy, 18(5), 1369-1380.

Kaasgari, M. A., Imani, D. M., & Mahmoodjanloo, M. (2017). Optimizing a vendor managed inventory (VMI) supply chain for perishable products by considering discount: Two calibrated meta-heuristic algorithms. Computers & Industrial Engineering, 103, 227-241.

Kaipia, R., & Tanskanen, K. (2003). Vendor managed category management—an outsourcing solution in retailing. Journal of Purchasing and Supply Management, 9(4), 165-175.

Karimi, M., Niknamfar, A. H., & Pasandideh, S. H. R. (2016). Two-stage single period inventory management for a manufacturing vendor under green-supplier supply chain. International Journal of System Assurance Engineering and Management, 8(4), 704-718.

Khan, M., Jaber, M. Y., Zanoni, S., & Zavanella, L. (2016). Vendor managed inventory with consignment stock agreement for a supply chain with defective items. Applied Mathematical Modelling, 40(15-16), 7102-7114.

Khalilpourazari, S., Pasandideh, S. H. R., & Niaki, S. T. A. (2016). Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP, SFS, SA, and WCA. Applied Soft Computing, 49, 770-791.

Kleywegt, A. J., Nori, V. S., & Savelsbergh, M. W. (2002). The stochastic inventory routing problem with direct deliveries. Transportation Science, 36(1), 94-118.

Lee, S., & Kim, D. (2014). An optimal policy for a single-vendor single-buyer integrated production–distribution model with both deteriorating and defective items. International Journal of Production Economics, 147, 161-170.

Liao, S. H., Hsieh, C. L., & Lai, P. J. (2011). An evolutionary approach for multi-objective optimization of the integrated location–inventory distribution network problem in vendor-managed inventory. Expert Systems with Applications, 38(6), 6768-6776.

Li, Y., Wan, Z., & Liu, J. (2017). Bi-level programming approach to optimal strategy for vendor-managed inventory problems under random demand. The ANZIAM Journal, 59(2), 247-270.

Lu, X., Shang, J., Wu, S. Y., Hegde, G. G., Vargas, L., & Zhao, D. (2015). Impacts of supplier hubris on inventory decisions and green manufacturing endeavors. European Journal of Operational Research, 245(1), 121-132.

Mirzapour Al-E-Hashem, S. M. J., Malekly, H., & Aryanezhad, M. B. (2011). A multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain under uncertainty. International Journal of Production Economics, 134(1), 28-42.

Mokhtari, H., & Rezvan, M. T. (2017). A single-supplier, multi-buyer, multi-product VMI production-inventory system under partial backordering. Operational Research, 1-21.

Montgomery, D. C., Bazaraa, M. S., & Keswani, A. K. (1973). Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly, 20(2), 255-263.

Nia, A. R., Far, M. H., & Niaki, S. T. A. (2014). A fuzzy vendor managed inventory of multi-item economic order quantity model under shortage: An ant colony optimization algorithm. International Journal of Production Economics, 155, 259-271.

Nia, A. R., Far, M. H., & Niaki, S. T. A. (2015). A hybrid genetic and imperialist competitive algorithm for green vendor managed inventory of multi-item multi-constraint EOQ model under shortage. Applied Soft Computing, 30, 353-364.

Park, Y. B., Yoo, J. S., & Park, H. S. (2016). A genetic algorithm for the vendor-managed inventory routing problem with lost sales. Expert Systems with Applications, 53, 149-159.

Pasandideh, S. H. R., Niaki, S. T. A., & Far, M. H. (2014). Optimization of vendor managed inventory of multiproduct EPQ model with multiple constraints using genetic algorithm. The International Journal of Advanced Manufacturing Technology, 71(1-4), 365-376.

Pasandideh, S. H. R., Niaki, S. T. A., & Nia, A. R. (2010). An investigation of vendor-managed inventory application in supply chain: the EOQ model with shortage. The International Journal of Advanced Manufacturing Technology, 49(1-4), 329-339.

Pasandideh, S. H. R., Niaki, S. T. A., & Nia, A. R. (2011). A genetic algorithm for vendor managed inventory control system of multi-product multi-constraint economic order quantity model. Expert Systems with Applications, 38(3), 2708-2716.

Pasandideh, S. H. R., Niaki, S. T. A., & Niknamfar, A. H. (2014). Lexicographic max–min approach for an integrated vendor-managed inventory problem. Knowledge-Based Systems, 59, 58-65.

Radha, V., Praveen Prakash, A. )2016(. Fuzzy optimization of green vendor managed inventory of multi item multi constraint EOQ model under shortage. Global Journal of Pure and Applied Mathematics.12 (1), 315-319.

Rahim, M.K.I.A., Radzuan, K., Yaakob, M. )2016(. Integrated analysis of inventory management and transportation system for the single period problem. Advanced Science Letters.22 (9), 2113-2116.

Rabbani, M., Rezaei, H., Lashgari, M., & Farrokhi-Asl, H. (2018). Vendor managed inventory control system for deteriorating items using metaheuristic algorithms. Decision Science Letters, 7(1), 25-38.

Razmi, J., Rad, R. H., & Sangari, M. S. (2010). Developing a two-echelon mathematical model for a vendor-managed inventory (VMI) system. The International Journal of Advanced Manufacturing Technology, 48(5-8), 773-783.

Sadeghi, J., Mousavi, S. M., & Niaki, S. T. A. (2016). Optimizing an inventory model with fuzzy demand, backordering, and discount using a hybrid imperialist competitive algorithm. Applied Mathematical Modelling, 40(15-16), 7318-7335.

Sadeghi, J., Mousavi, S. M., Niaki, S. T. A., & Sadeghi, S. (2013). Optimizing a multi-vendor multi-retailer vendor managed inventory problem: Two tuned meta-heuristic algorithms. Knowledge-Based Systems, 50, 159-170.

Sadeghi, J., Mousavi, S. M., Niaki, S. T. A., & Sadeghi, S. (2014). Optimizing a bi-objective inventory model of a three-echelon supply chain using a tuned hybrid bat algorithm. Transportation Research Part E: Logistics and Transportation Review, 70, 274-292.

Sadeghi, J., Sadeghi, S., & Niaki, S. T. A. (2014). A hybrid vendor managed inventory and redundancy allocation optimization problem in supply chain management: An NSGA-II with tuned parameters. Computers & Operations Research, 41, 53-64.

Sadeghi, J., & Niaki, S. T. A. (2015). Two parameter tuned multi-objective evolutionary algorithms for a bi-objective vendor managed inventory model with trapezoidal fuzzy demand. Applied Soft Computing, 30, 567-576.

Schaefer B., & Konur, D. (2015) Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions. Transp Res E-Log 80: 142–165.

Setak, M., & Daneshfar, L. (2014). An inventory model for deteriorating items using vendor-managed inventory policy. International Journal of Engineering-Transactions A: Basics, 27(7), 1081-1090.

Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., & Shankar, R. (2007). Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies, third ed., McGraw-Hill, New York.

Soni, H., Gor, A., & Patel, H. (2018). Vendor managed inventory model for non-instantaneous deteriorating product with quadratic demand allowing partial backlogging. Uncertain Supply Chain Management, 6(3), 321-334.

Stellingwerf, H. M., Kanellopoulos, A., Cruijssen, F. C. A. M., & Bloemhof, J. M. (2019). Fair gain allocation in eco-efficient vendor-managed inventory cooperation. Journal of Cleaner Production. doi:10.1016/j.jclepro.2019.05.232.

Sumrit, D. (2019). Supplier selection for vendor-managed inventory in healthcare using fuzzy multi-criteria decision-making approach, Decision Science Letters. 233–256.

Szidarovszky, F., Gershon, M. E., & Duckstein, L. (1986). Techniques for multiobjective decision making in systems management (Vol. 2). Elsevier Science Ltd.

Taleizadeh, A. A., Noori-daryan, M., & Cárdenas-Barrón, L. E. (2015). Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items. International Journal of Production Economics, 159, 285-295.

Tat, R., Taleizadeh, A. A., & Esmaeili, M. (2015). Developing economic order quantity model for non-instantaneous deteriorating items in vendor-managed inventory (VMI) system. International Journal of Systems Science, 46(7), 1257-1268.

Taleizadeh, A. A., Shokr, I., & Joali, F. (2020). Optimizing vendor-managed inventory systems with limited storage capacity and partial backordering under stochastic demand. RAIRO - Operations Research, 54(1), 179–209. doi:10.1051/ro/2018090.

Tersine, R.J. (1993). Principles of Inventory and Materials Management, fourth ed., Prentice Hall PTR.

Tsou C.S., Hsu C.H., Chen J.H., & Yeh C.C. (2010) Approximating tradeoff
surfaces for inventory control through evolutionary multi objective optimization. In: IEEE international conference on advanced management science (ICAMS) vol 3, pp 652–655.

Xiao, N., & Rao, Y. L. (2016). Multi-product multi-period inventory routing optimization with time window constrains. International Journal of Simulation Modelling, 15(2), 352-364.

Xiao, T., & Xu, T. (2013). Coordinating price and service level decisions for a supply chain with deteriorating item under vendor managed inventory. International Journal of Production Economics, 145(2), 743-752.

Yao, Y., Evers, P. T., & Dresner, M. E. (2007). Supply chain integration in vendor-managed inventory. Decision support systems, 43(2), 663-674.

Yoon, K. (1987). A reconciliation among discrete compromise solutions. Journal of the Operational Research Society, 38(3), 277-286.

Yu, Y., Wang, Z., & Liang, L. (2012). A vendor managed inventory supply chain with deteriorating raw materials and products. International Journal of Production Economics, 136(2), 266-274.

Yu, Y., Hong, Z., Zhang, L. L., Liang, L., & Chu, C. (2013). Optimal selection of retailers for a manufacturing vendor in a vendor managed inventory system. European Journal of Operational Research, 225(2), 273-284.

Zanoni, S., Mazzoldi, L., Jaber, M.Y. (2014). Vendor-managed inventory with consignment stock agreement for single vendor–single buyer under the
emission-trading scheme, Int. J. Prod. Res. 52 (1), 20–31.

Zeleny, M., & Cochrane, J. L. (1973). Multiple criteria decision making. University of South Carolina Press.

Zhu, C. B., Ye, Y. H., & Dai, X. (2007). The three-echelon stochastic inventory routing problem with direct deliveries [J]. Systems Engineering-Theory & Practice, 12, 003.