Mathematical modelling of a decentralized multi-echelon supply chain network considering service level under uncertainty

Document Type : Article


1 Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, P.O. Box 4716685635, Iran

2 Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran.

3 Department of Industrial Engineering, Babol Noshirvani University of Technology, Babol, Iran.


We study a multi-time, multi-product and multi-echelon supply chain aggregate procurement, production and distribution planning problem and discuss the implications of formulating a tri-level model to integrate procurement, production and distribution, maintaining the existing hierarchy in the decision process. In our model, there are three different decision makers controlling the procurement, production and the distribution processes in the absence of cooperation because of different optimization strategies. First, we present a hierarchical tri-level programming model to deal with decentralized supply chain problems. Then, an algorithm is presented to solve the proposed model. A numerical illustration is provided to show the applicability of the optimization model and the proposed algorithm. In order to evaluate the application of the model and the proposed algorithm, ten sets of small and large problems are randomly generated and tested. The experimental results show that our proposed fuzzy-stochastic simulation based hierarchical interactive particle swarm optimization (Sim-HIPSO) performs well in finding good approximate solutions within reasonable computation times.


Main Subjects

1. Farahani, R.Z., Rezapour, S., Drezner, T., and Fallah, S. "Competitive supply chain network design: An overview of  classifications, models, solution techniques and applications", Omega, 45, pp. 92-118 (2014).
2. Calvete, H.I., Gale, C., and Iranzo, J.A. "Planning of a decentralized distribution network using bilevel optimization", Omega, 49, pp. 30-41 (2014).
3. Ma, Y., Yan, F., Kang, K., and Wei, X. "A novel integrated production-distribution planning model with con ict and coordination in a supply chain network", Knowledge-Based Systems, 105, pp. 119-133 (2016).
4. Park, B.J., Choi, H.R., and Kang, M.H. "Integration of production and distribution planning using a genetic algorithm in supply chain management", Analysis and Design of Intelligent Systems Using Soft Computing Techniques, 41, pp. 416-426 (2007).
5. Kazemi, A., Zarandi, M.F., and Husseini, S.M. "A multi-agent system to solve the productiondistribution planning problem for a supply chain: a genetic algorithm approach", The International Journal of Advanced Manufacturing Technology, 44, pp. 180- 193 (2009).
6. Jolai, F., Razmi, J., and Rostami, N. "A fuzzy goal programming and meta heuristic algorithms for solving integrated production: distribution planning problem", Central European Journal of Operations Research, 19, pp. 547-569 (2011).
7. Liu S., Sawik, T., and Papageorgiou, L.G. "Corrigendum to multiobjective optimisation of production, distribution and capacity planning of global supply chains in the process industry", The International Journal of Management Science, 44, pp. 149-170 (2014).
8. Sarrafha, K., Rahmati, S.H.A., Niaki, S.T.A., and Zaretalab, A. "A bi-objective integrated procurement, production, and distribution problem of a multiechelon supply chain network design: A new tuned MOEA", Computers & Operations Research, 54, pp. 35-51 (2015).
9. Chang, Y.H. "Adopting co-evolution and constraintsatisfaction concept on genetic algorithms to solve supply chain network design problems", Expert Systems with Applications, 37, pp. 6919-6930 (2010).
10. Guo, Z., Zhang, D., Leung, S., and Shi, L. "A bilevel evolutionary optimization approach for integrated production and transportation scheduling", Applied Soft Computing, 42, pp. 215-228 (2016).
11. Marinakis, Y. "An improved particle swarm optimization algorithm for the capacitated location routing problem and for the location routing problem with stochastic demands", Applied Soft Computing, 37, pp. 680-701 (2015).
12. Wang, K.J. and Lee, C.-H. "A revised ant algorithm for solving location-allocation problem with risky demand in a multi-echelon supply chain network", Applied Soft Computing, 32, pp. 311-321 (2015).
13. Chu, Y., You, F., Wassick, J.M., and Agarwal, A. "Integrated planning and scheduling under production uncertainties: Bi-level model formulation and hybrid solution method", Computers & Chemical Engineering, 72, pp. 255-272 (2015).
14. Wang, K.J., Makond, B., and Liu, S.Y. "Location and allocation decisions in a two-echelon supply chain with stochastic demand - A genetic-algorithm based solution", Expert Systems with Applications, 38, pp. 6125-6131 (2011).
15. Amirtaheri, O., Zandieh, M., and Dorri, B. "A bi-level programming model for decentralized manufacturerdistributer supply chain considering cooperative advertising",Scientia Iranica, 25(2), pp. 891-910 (2018).
16. Calvete, H.I., Gale, C., and Oliveros, M.-J. "Bilevel model for production-distribution planning solved by using ant colony optimization", Computers and Operations Research, 38, pp. 320-327 (2011).
17. Camacho-Vallejo, J.-F., Munoz-Sanchez, R., and Gonzalez-Velarde, J.L. "A heuristic algorithm for a supply chain's production-distribution planning", Computers and Operations Research, 61, pp. 110-121 (2015).
18. Xu, X., Meng, Z., and Shen, R. "A tri-level programming model based on conditional value-at-risk for three-stage supply chain management", Computers and Industrial Engineering, 66, pp. 470-475 (2013).
19. Mirzapour Al-E-Hashem, S., Malekly, H., and Aryanezhad, M. "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, pp. 28-42 (2011).
20. Azaron, A., Brown, K., Tarim, S., and Modarres, M. "A multi-objective stochastic programming approach for supply chain design considering risk", International Journal of Production Economics, 116, pp. 129-138 (2008).
21. Yongheng, J., Rodriguez, M.A., Harjunkoski, I., and Grossmann, I.E. "Optimal supply chain design and management over a multi-period horizon under demand uncertainty, Part II: A Lagrangean decomposition algorithm", Computers & Chemical Engineering, 62, pp. 211-224 (2014).
22. Song, D.P., Dong, J.X., and Xu, J. "Integrated inventory management and supplier base reduction in a supply chain with multiple uncertainties", European Journal of Operational Research, 232, pp. 522-536 (2014).
23. Pasandideh, S.H.R., Niaki, S.T.A., and Asadi, K.  Biobjective optimization of a multi-product multi-period three-echelon supply chain problem under uncertain environments: NSGA-II and NRGA", Information Sciences, 292, pp. 57-74 (2015).
24. da Silva, A.F. and Marins, F.A.S. "A fuzzy goal programming model for solving aggregate productionplanning problems under uncertainty: A case study in a Brazilian sugar mill", Energy Economics, 45, pp. 196-204 (2014).
25. Zhang, G., Shang, J., and Li, W. "Collaborative production planning of supply chain under price and demand uncertainty", European Journal of Operational Research, 215, pp. 590-603 (2011).
26. Saranwong, S. and Likasiri, C. "Bi-level programming model for solving distribution center problem: a case study in Northern Thailand's sugarcane management", Computers & Industrial Engineering, 103, pp. 26-39 (2017).
27. Saranwong, S. and Likasiri, C. "Product distribution via a bi-level programming approach: algorithms and a case study in municipal waste system", Expert Systems with Application, 44, pp. 78-91 (2016).
28. Yue, D. and You, F. "Stackelberg-game-based modeling and optimization for supply chain design and operations: A mixed integer bilevel programming framework", Computers and Chemical Engineering, 102, pp. 81-95 (2017).
29. Fard, A.M.F. and Hajaghaei-Keshteli, M. "A tri-level location-allocation model for forward/reverse supply chain", Applied Soft Computing, 62, pp. 328-346 (2018).
30. Dempe, S., Foundations of Bilevel Programming, Springer Science & Business Media (2002).
31. Bard, J., Practical Bilevel Optimization: Applications and Algorithms, Kluwer Academic Press, Dordrecht, Netherlands (1998).
32. Bard, J.F. and Falk, J.E. "An explicit solution to the multi-level programming problem", Computers & Operations Research, 9, pp. 77-100 (1982).
33. Bard, J.F. and Moore, J.T. "A branch and bound algorithm for the bilevel programming problem", SIAM Journal on Scientific and Statistical Computing, 11, pp. 281-292 (1990).
34. Talbi, E.G., Metaheuristics for Bi-level Optimization, Springer (2013). 
35. Zadeh, L.A. "Fuzzy sets", Information and Control, 8, pp. 338-353 (1965).
36. Dubois, D. and Prade, H. "Systems of linear fuzzy constraints", Fuzzy Sets and Systems, 3, pp. 37-48 (1980).
37. Camacho-Vallejo, J.-F., Cordero-Franco, A.E., and Gonzalez-Ramirez, R.G. "Solving the bilevel facility location problem under preferences by a Stackelbergevolutionary algorithm", Mathematical Problems in Engineering, 201, pp. 30-44 (2014).
38. Kennedy, J. and Eberhart, R.C., Swarm Intelligence, San Francisco, USA: Morgan Kaufmann Publishers (2001).
39. Liu, B., Uncertain Programming, John Wiley & Sons, Inc. (1999).
40. Jia, L., Wang, Y., and Fan, L. "Multi-objective bilevel optimization for production-distribution planning problems using hybrid genetic algorithm [J]". Integrated Computer-Aided Engineering, 21(1), pp. 77-90 (2014).