Document Type : Original Article


School of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran.


This paper addresses an integrated multi-echelon location-allocation-inventory problem in a ‎stochastic supply chain. In a bid to be more realistic, the demand and lead time are considered ‎to be hemmed in by uncertainty. To tackle the proposed supply chain network design problem, ‎a two-phase approach based on queuing and optimization models is devised. The queuing ‎approach is first deployed, which is able to cope with inherent uncertainty of parameters. ‎Afterwards, the proposed supply chain network design problem is formulated using a mixed‏-‏integer nonlinear model. Likewise, the convexity of the model is proved and the optimal ‎inventory policy as closed-form is acquired. Inasmuch as the concerned problem belongs to ‎NP-hard problems, two meta-heuristic algorithms are employed, which are capable of ‎circumventing the complexity burden of the model. The numerical examples evince the efficient ‎and effective performance of the solving algorithms. Lastly, sensitivity analyses are conducted ‎through which interesting insights are gained.


Ahmadi-Javid, A., and Seddighi, A. H., (2012)."A location-routing-inventory model for designing multisource distribution networks", Engineering Optimization,Vol.44, No. 6, pp. 637-656.
Ahmadi Javid, A., and Azad, N., (2010)."Incorporating location, routing and inventory decisions in supply chain network design", Transportation Research Part E: Logistics and Transportation Review,Vol. 46, No. 5, pp. 582-597. doi: 10.1016/j.tre.2009.06.005 
Alshamsi, A., and Diabat, A., (2018). "Large-scale reverse supply chain network design: An accelerated Benders decomposition algorithm", Computers and Industrial Engineering,Vol. 124, pp. 545-559.
AmalNick, M., and Qorbanian, R., (2017). "Dynamic pricing using wavelet neural network under uncertain demands", Decision Science Letters, Vol. 6, No. 3, pp. 251-260. 
Baek, J. W., and Moon, S. K., (2014)."The M/M/1 queue with a production-inventory system and lost sales", Applied Mathematics and Computation, Vol.233, pp. 534-544. 
Baumol, W. J., and Wolfe, P., (1958)."A warehouse-location problem", Operations Research,Vol. 6, No. 2, pp. 252-263.
Berman, O., and Kim, E., (2001). "Dynamic order replenishment policy in internet-based supply chains", Mathematical Methods of Operations Research, Vol.53, No. 3, pp. 371-390.
Berman, O., Krass, D., and Tajbakhsh, M. M., (2012)."A coordinated location-inventory model", European Journal of Operational Research,Vol. 217, No. 3, pp. 500-508. doi: 10.1016/j.ejor.2011.09.039
Chao, X., Xu, Y., and Yang, B., (2012). "Optimal policy for a production–inventory system with setup cost and average cost criterion", Probability in the Engineering and Informational Sciences,Vol. 26, No. 4, pp. 457-481.
Chen, F. Y., and Feng, Y., (2006)."Optimization and optimality of (s, S) stochastic inventory systems with non-quasiconvex costs", Probability in the Engineering and Informational Sciences,Vol. 20, No. 2, pp. 287-306. 
Chew, E. P., Lee, L. H., and Rajaratnam, K., (2007). Evolutionary algorithm for an inventory location problem, Evolutionary Scheduling (pp. 613-628): Springer. 
Daskin, M. S., Coullard, C. R., and Shen, Z.-J. M., (2002)."An inventory-location model: Formulation, solution algorithm and computational results", Annals of operations research,Vol.110, No. 4, pp. 83-106.
Diabat, A., Abdallah, T., and Henschel, A., (2015)."A closed-loop location-inventory problem with spare parts consideration", Computers and Operations Research,Vol. 54, pp. 245-256.
Diabat, A., Battaïa, O., and Nazzal, D., (2015)."An improved lagrangian relaxation-based heuristic for a joint location-inventory problem", Computers and Operations Research, Vol. 61, pp. 170-178. 
Diabat, A., and Theodorou, E., (2015)."A location–inventory supply chain problem: Reformulation and piecewise linearization", Computers and Industrial Engineering,Vol. 90, pp. 381-389. 
Díaz-Mateus, Y., Forero, B., López-Ospina, H., and Zambrano-Rey, G., (2018). "Pricing and lot sizing optimization in a two-echelon supply chain with a constrained Logit demand function", International Journal of Industrial Engineering Computations, Vol. 9, No. 2, pp. 205-220. 
Eiben, A. E., and Smith, J. E., (2003). Introduction to evolutionary computing, Berlin: Springer.
Federgruen, A., and Zheng, Y.-S., (1993)."Optimal control policies for stochastic inventory systems with endogenous supply", Probability in the Engineering and Informational Sciences,Vol. 7, No. 2,  pp. 257-272
Frizelle, G., and Jaber, M., (2009)."Seeing inventory as a queue", Inventory Management: Non-Classical Views, pp. 151-172. 
Gunasekaran, A., Patel, C., and Tirtiroglu, E., (2001)."Performance measures and metrics in a supply chain environment", International journal of operations and production Management,Vol. 21, pp. 71-87. 
Hill, R. M., (1999)."On the suboptimality of (S−1, S) lost sales inventory policies", International Journal of Production Economics,Vol. 59, No. 1–3, pp. 387-393. doi:
Holland, J. H., (1975). "Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence", Ann Arbor, Michigan: The University of Michigan Press. 
Jindal, P., and Solanki, A., (2016). "Integrated vendor-buyer inventory models with inflation and time value of money in controllable lead time", Decision Science Letters, Vol. 5, No. 1, pp. 81-94.
Kalpakam, S., and Shanthi, S., (2001)."A perishable inventory system with modified (S−1,S) policy and arbitrary processing times", Computers and Operations Research,Vol. 28, No. 5, pp. 453-471. doi:
Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P., (1983)."Optimization by simulated annealing", science, Vol.220, No. 4598, pp. 671-680.
Krishnamoorthy, A., Manikandan, R., and Lakshmy, B., (2013). "A revisit to queueing-inventory system with positive service time", Annals of Operations Research, pp. 1-16.
Miranda, P. A., and Garrido, R. A., (2009)."Inventory service-level optimization within distribution network design problem", International Journal of Production Economics,Vol. 122, No. 1, pp. 276-285. doi: 10.1016/j.ijpe.2009.06.010
Mirchandani, P. B., and Francis, R. L., (1990). Discrete location theory: Wiley.
Nagurney, A., (2006). Supply chain network economics: dynamics of prices, flows and profits: Edward Elgar Publishing.
Nekooghadirli, N., Tavakkoli-Moghaddam, R., Ghezavati, V., and Javanmard, S., (2014)."Solving a new bi-objective location-routing-inventory problem in a distribution network by meta-heuristics", Computers and Industrial Engineering,Vol. 76, pp. 204-221.
Ozsen, L., Daskin, M. S., and Coullard, C. R., (2009)."Facility location modeling and inventory management with multisourcing", Transportation Science, Vol.43, No. 4, pp. 455-472.
Park, S., Lee, T.-E., and Sung, C. S., (2010)."A three-level supply chain network design model with risk-pooling and lead times", Transportation Research Part E: Logistics and Transportation Review,Vol. 46, No. 5, pp. 563-581.
Rabbani, M., Keyhanian, S., Hasannia, M., Eskandari, M., and Jalali, M., (2016). "Impact of end of lease contracts' option on joint pricing and inventory decisions of remanufacturable leased products", International Journal of Industrial Engineering Computations, Vol. 7, No. 2, pp. 191-204. 
Rabbani, M., Aliabadi, L., Heidari, R., and Farrokhi-Asl, H., (2017)."Applying queuing theory for a reliable integrated location inventory problem under facility disruption risks", Journal of Industrial Engineering and Management Studies, Vol.4, No. 2, pp. 1-18.
Ramezani, S., and Naderi, K., (2018)."A multi-objective optimization model for locating the distribution warehouses in the logistic network supply chain: A case study of Bonny Chow Co", Journal of Industrial Engineering and Management Studies, Vol.5, No. 1, pp. 26-42.
Sadjadi, S. J., Makui, A., Dehghani, E., and Pourmohammad, M., (2016). "Applying queuing approach for a stochastic location-inventory problem with two different mean inventory considerations", Applied Mathematical Modelling, Vol.40, No. 1, pp. 578-596.
Saffari, M., Asmussen, S., and Haji, R., (2013)."The M/M/1 queue with inventory, lost sale, and general lead times", Queueing Systems, No.75, No. 1, pp. 65-77.
Schultz, C. R., (1990)."On the optimality of the (S — 1,S) policy", Naval Research Logistics (NRL),Vol. 37, No. 5, pp. 715-723. doi: 10.1002/1520-6750(199010)37:5<715::AID-NAV3220370510>3.0.CO;2-U
Schwarz, M., and Daduna, H., (2006)."Queueing systems with inventory management with random lead times and with backordering", Mathematical Methods of Operations Research, Vol.64, No. 3, pp. 383-414.
Shahabi, M., Unnikrishnan, A., Jafari-Shirazi, E., and Boyles, S. D., (2014). "A three level location-inventory problem with correlated demand", Transportation Research Part B: Methodological,Vol. 69, pp. 1-18.
Shen, Z.-J. M., Coullard, C., and Daskin, M. S., (2003)."A joint location-inventory model", Transportation science, Vol.37, No. 1, pp. 40-55.
Shu, J., Teo, C.-P., and Shen, Z.-J. M., (2005)."Stochastic transportation-inventory network design problem", Operations Research, Vol.53, No. 1, pp. 48-60.
Sigman, K., and Simchi-Levi, D., (1992). "Light traffic heuristic for anM/G/1 queue with limited inventory", Annals of Operations Research,Vol. 40, No. 1, pp. 371-380.
Stevens, G. C., (1989)."Integrating the supply chain", International Journal of Physical Distribution and Materials Management,Vol. 19, No. 8, pp. 3-8.
Taguchi, G., Chowdhury, S., and Taguchi, S., (2000). Robust engineering: McGraw-Hill Professional.
Tragantalerngsak, S., Holt, J., and Rönnqvist, M., (2000).An exact method for the two-echelon, single-source, capacitated facility location problem, European Journal of Operational Research, Vol.123, Vol. 3, pp. 473-489.