Equilibrium pricing, routing and order quantity decisions in a three-level supply chain

Document Type : Research Paper


1 Department of Industrial Engineering, KHATAM University, Tehran, Iran

2 Department of Industrial Engineering, K. N. Toosi University of Technology, Tehran, Iran


In this paper, a supply chain, including a manufacturer, a distributor and some retailers, is considered. The manufacturer produces a single product and outsources distribution operations to a distributor in order to deliver products demanded by retailers. Each retailer has a time window to receive the products and faces the Newsvendor problem with stochastic demand. The manufacturer aims to serve retailers providing that the maximum lateness doesn’t exceed a predetermined value. All players in the supply chain are willing to maximize their own profit. The model simultaneously includes pricing, order quantity and routing decisions. First, the manufacturer announces the whole sale price, then the distributor declares the unit transportation cost to the retailers, and finally each retailer decides on the amount of his order quantity. The profit functions of the players are formulated and linearized; then the solution is determined in three stages using game theory. Finally, a numerical example is presented and the equilibrium decisions of the players are determined using GAMS software.


Main Subjects

Adelman, D. (2004). A price-directed approach to stochastic inventory/routing. Operations Research52(4), 499-514.‏
Al-Khayyal, F., Hwang, S.J., (2007). Inventory constrained maritime routing and scheduling for multi-commodity liquid bulk, Part I: Applications and model. European Journal of Operational Research, 176 (1), 106-130.
Alaei, S., & Setak, M. (2015). Multi objective coordination of a supply chain with routing and service level consideration. International Journal of Production Economics167, 271-281.‏
Andersson, H., (2006). Coordinated Routing – application in location and inventory management. Dissertation, Linköpings Universitet, Sweden.
Andersson, H., Hoff, A., Christiansen, M., Hasle, G., & Løkketangen, A. (2010). Industrial aspects and literature survey: Combined inventory management and routing. Computers & Operations Research37(9), 1515-1536.‏
Azuma, R.M., Coelho, G.P., Von Zuben, F.J., (2011, June). Evolutionary multi-objective optimization for the vendor-managed inventory routing problem. In Evolutionary Computation (CEC), 2011 IEEE Congress on (pp. 1457-1464). IEEE.
Coelho, L.C., Laporte, G., (2013). The exact solution of several classes of inventory-routing problems. Computers and Operations Research, 40 (2), 558-565.
Custódio, A.L., Oliveira, R.C., (2006). Redesigning distribution operations: a case study on integrating inventory management and vehicle routes design. International Journal of Logistics: Research and Applications, 9 (2), 169-187.
Dehghani, E., Behfar, N., & Jabalameli, M. S. (2017). Optimizing location, routing and inventory decisions in an integrated supply chain network under uncertainty. Journal of Industrial and Systems Engineering9(4), 93-111.‏
Etebari, F., & Dabiri, N. (2016). A hybrid heuristic for the inventory routing problem under dynamic regional pricing. Computers & Chemical Engineering95, 231-239.‏
Fattahi, P., Tanhatalab M., & Bashiri, M., (2017). Enhanced chromosome repairing mechanism based genetic algorithm approach for the multi-period perishable production inventory-routing problem. Journal of Industrial and Systems Engineering10, 30-56.‏
Gaur, V., Fisher, M.L., (2004). A periodic inventory routing problem at a supermarket chain. Operations Research, 52 (6), 813-822.
Li, K., Chen, B., Sivakumar, A.I., Wu, Y., )2014(. An inventory–routing problem with the objective of travel time minimization. European Journal of Operational Research, 236(3), 936-945.
Li, Y., Chu, F., & Chen, K. (2017). Coordinated Production Inventory Routing Planning for Perishable Food. Ifac-Papersonline, 50(1), 4246-4251.‏
Liu, S. C., & Chen, J. R. (2011). A heuristic method for the inventory routing and pricing problem in a supply chain. Expert Systems with Applications38(3), 1447-1456.‏
Mirzapour Al-e-hashem, S.M.J., Rekik, Y., )2013(. Multi-product multi-period Inventory Routing Problem with a transshipment option: A green approach. International Journal of Production Economics. DOI: 10.1016/j.ijpe.2013.09.005.
Sayarshad, H. R., & Gao, H. O. (2018). A non-myopic dynamic inventory routing and pricing problem. Transportation Research Part E: Logistics and Transportation Review, 109, 83-98.‏
Soysal M., Çimen, M., Belbağ S., Toğrul E., (2019). A review on sustainable inventory routing. Computers & Industrial Engineering. 132: 395-411.
Vidović, M., Popović, D., Ratković, B., )2014(. Mixed integer and heuristics model for the inventory routing problem in fuel delivery. International Journal of Production Economics, 147, 593-604.