Inventory Planning in a G/G/1/∞ Queuing System with Multiple Suppliers using Simulation and Response Surface Methodology

Document Type : Original Article


1 Ph.D Student, Tehran University.

2 Professor, Tehran University.


     In this study, a G/G/1/∞ queuing system is considered in which to servicing each costumer, an on-hand inventory is required. Continuous review (r, Q) policy is considered and procurement is done through multiple suppliers and lead time is stochastic with certain probability density functions. During stock out, arriving demands are rejected and their requests are lost also a waiting cost is considered for customers within the system. Our goal in this study is determining economic order quantity (Q) and reorder point (r) whit minimizing summation of various costs includes holding costs, ordering costs, shortage and waiting costs. Due to complexity of exact approaches such as Markov process, we considered a simulation-optimization approach that combines simulation with design of experiments. To evaluate the efficiency of proposed approach, in a special case (two suppliers with exponential lead times) several numerical examples are solved by both exact and simulation-optimization approaches. Comparing the results shows that while our proposed method is much simpler in terms of computational complexity than the Markovian model, offers very close answers to the optimal answer.


1. Adeli, M. & Zandie, M. (2013). Proposing a multi-objectives simulation-optimization approach for an integrated sourcing and inventory model. Journal of Industrial Management Perspective, 11: 89-110 (In Persian). 
2. Amaran, S. & Sahinidis, N. (2016). Simulation optimization: a review of algorithms and applications, Ann Oper Res, 240: 351–380.
3. Badakhshan, E., Pishvaei, S. & Sahebi, H. (2016). An optimization model based on simulitanion to integrated pllaning for cash and physical follows in supply chain. Journal of Industrial Management Perspective, 21: 31-51 (In Persian).
4. Berman, O. & Kim, E. (1999). Stochastic models for inventory management at service facilities. Stat Stoch Model. 15(4): 695–718.
5. Berman, O. & Kim, E. (2001). Dynamic order replenishment policy in internet-based supply chains. Math Meth Oper Res, 53: 371–390.
6. Berman, O. & Sapna, KP. (2000). Inventory management at service facilities for systems with arbitrarily distributed service times. Comm Stat Stoch Model, 16(3, 4): 343–360.
7. Berman, O. & Sapna, KP. (2002). Optimal service rates of a service facility with perishable inventory items. Naval Res Logist, 49: 464– 482.
8. Davoodi, M., Jolai, F. (2015). Design a simulation model for a multi-echelon and multi-products inventory system and comparison with elit models. Journal of Industrial Management Perspective, 19: 9-38 (In Persian).
9. Deepak, T. G., Krishnamoorthy, A., Narayanan, V. C. & Vineetha, K. (2008). Inventory with service time and transfer of customers and inventory. Ann Oper Res. 160: 191–213
10. Hlioui, R., Gharibi, A. & Hajji, A. (2015). Integrated quality strategy in production and raw material replenishment in a manufacturing-oriented supply chain. Int J Adv Manuf Technol. 7: 1-14.
11. Kochel, P. & Nielander, U. (2005). Simulation-based optimization of multi-echelon inventory systems. International Journal of Production Economics, 93-94(1): 505-513.
12. Peidro, D., Mula, J., Poler, R. & F. C. Lario. (2009). Quantitative Models for Supply Chain planning under uncertainty: A review. International Journal of Advanced Manufacturing Technology, 43: 400-420.
13. Razavi, H., Amiri, M. & Seifbarghi, M. (2013). Application of response surface methodology in optimization of a multi-echelon inventory system. Journal of production and operation management. 4(7): 41-54.
14. Rivera Gomez, H., Gharbi, A. & Kenné, JP. (2013). Joint control of production, overhaul, and preventive maintenance for a production system subject to quality and reliability deteriorations. Int J Adv Manuf Technol, 21: 1–20.
15. Saffari, M. & Haji, R. (2009). Queuing system with inventory for two echelon supply chain. CIE Int Conference: 835–838
16. Saffari, M. & Haji, R. (2011). A queuing system with inventory and mixed exponentially distributed lead times. Int J Adv Manuf Technol, 53: 1231–1237.
17. Sajadi, S. M., Seyed Esfahani, M. M. & Sörensen, K. (2011). Production control in a failure-prone manufacturing network using discrete event simulation and automated response surface methodology. Int J Adv Manuf Technol, 53(1–4): 35–46.
18. Schwarz, M., Sauer, C., Daduna, H., Kulik, R. & Szekli, R. (2006). M/M/1 queuing systems with inventory. Queueing Syst 54: 55–78.
19. Schwarz, M. & Daduna, H. (2006). Queuing systems with inventory management with random lead times and with backordering. Math Meth Oper Res, 64: 383–414.
20. Seifbarghi, M., Amiri, M. & Heidari, M. (2008). Estimation of cost function in a two echelon inventory system with lostsale shortage using regrestion. Journal of industrial engineering. 1: 1-10.
21. Tsai, S. & Chen, S. (2016). a Simulation-Based Multi-Objective Optimization Framework: A Case Study on Inventory Management, Omega, 240: 351–380.
22. Ye, W. & You, F. (2016). A computationally efficient simulation-based optimization method with region-wise surrogate modeling for stochastic inventory management of supply chains with general network structures. Computers and Chemical Engineering, 87: 164–179.
23. Zhao, N. & Lian, Z. (2011). A queuing-inventory system with two classes of customers. Int. J. Production Economics 129: 225–231