The aim of this paper is to present mathematical models optimizing all materials flows in supply chain. In this research a fuzzy multi-objective nonlinear mixed- integer programming model with piecewise linear membership function is applied to design a multi echelon supply chain network (SCN) by considering total transportation costs and capacities of all echelons with fuzzy objectives. The model that is proposed in this study has 4 fuzzy functions. The first function is minimizing the total transportation costs between all echelons (suppliers, factories, distribution centers (DCs) and customers). The second one is minimizing holding and ordering cost on DCs. The third objective is minimizing the unnecessary and unused capacity of factories and DCs via decreasing variance of transported amounts between echelons. The forth is minimizing the number of total vehicles that ship the materials and products along with SCN. For solving such a problem, as nodes increases in SCN, the traditional method does not have ability to solve large scale problem. So, we applied a Meta heuristic method called Genetic Algorithm. The numerical example is real world applied and compared the results with each other demonstrate the feasibility of applying the proposed model to given problem, and also its advantages are discussed.

Z, H & Afshari, M. (2012). Optimizing a multi-echelon supply chain network flow using nonlinear fuzzy multi-objective integer programming: Genetic algorithm approach.Management Science Letters, 2(6), 1871-1884.

Alves, M.J. & Climaco, J. (2007). A review of interactive methods for multi objective integer and mixed-integer programming approach. European Journal of Operational Research, 180, 99-115.

Banerjee, A. (1986). A joint economic lot-size model for purchaser and vendor. Decision Sciences, 17, 292 – 311.

Cao, C., Gu, X., & Xin, Z. (2010). Stochastic chance constrained mixed-integer nonlinear programming models & the solution approaches for refinery short-term crude oil scheduling problem. Applied Mathematical Modelling, 34(11), 3231–3243.

Chang, C.T. (2007). Binary behavior of fuzzy programming with piecewise linear membership functions. IEEE Transactions on Fuzzy Systems, 15 (3), 342-349.

Chen, L., & Lee, W. (2004). Multi objective optimization of multi echelon supply chain networks with uncertain product demands & prices. Computers and Chemical Engineering, 28, 1131-1144.

Cheshmberah, M., Zahedi, M.R., Hadizadeh, A., &  Tofighi, S.M. (2011). A mathematical model for optimum single-commodity distribution in the network of chain stores: a case study of food industry. Management Science Letter, 1(4), 575-582.

Cohen, M.A., & Lee, H.L. (1989). Resource deployment analysis of global manufacturing and distribution networks. Journal of Manufacturing and Operations Management, 2, 81-104.

Davis, L. (1991). The Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York.

Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization, & Machine Learning. Addison-Wesley, Reading, MA.

Ermis, M., Sahingoz, O.K. & Ulengin, F. (2004). Mobile agent based supply chain modeling with neural network controlled services. Proceedings of 4th International ICSC Symposium on Engineering of Intelligent Systems, Island of Madeira Portugal, pp.1–8.

Falcone, M.A., Lopes, H.S., & dos Santos Coelho, L. (2008), Supply chain optimisation using Evolutionary Algorithms. International Journal of Computer Applications in Technology, 158-166.

Fortes, J. (2009). Green Supply Chain Management: A Literature Review. Otago Management Graduate Review, 7, 51-62.

Hill, R.M. (1999). The optimal production & shipment policy for the single-vendor single-buyer integrated production-inventory model. International Journal of Production Research, 37, 2463-2475.

Hu, C.F. & Fang, S.C. (1999). Solving fuzzy inequalities with piecewise linear membership functions, IEEE Transactions on Fuzzy Systems, 7 (2), 230-235.

Holland, J.H. (1975). Adaptation in Natural & Artificial Systems. The University of Michigan Press, Ann Arbor, IL.

Lambert, Douglas M. (editor) (2008). Supply Chain Management: Processes, Partnerships, Performance, 3rd Edition, The Hartely Press Inc., USA.

Lau, A., & Lau, H. (2001). Some two-echelon style-goods inventory models with asymmetric market information. European Journal of Operational Research, 134, 29–42.

Liang, T.F. (2008). Fuzzy multi-objective production/distribution planning decisions with multi-product and multi-time period in supply chain. Computers and Industrial Engineering, 55, 676-694.

Liang, T.F., & Cheng, H.W. (2009). Application of fuzzy sets to manufacturing/distribution planning decisions with multi-product and multi-time period in supply chains. Expert Systems with Applications 36, 3367-3377.

Liang, T.F. (2006). Distribution planning decisions using interactive fuzzy multi-objective linear programming. Fuzzy Sets and Systems, 157, 1303-1316.

Meier, R. L., Williams, M. R. & Singley, R B. (2004), Supply Chain Management: Strategic Factors From The Buyers’ Perspective. Journal of Industrial Technology, 20(2), 1-8.

Melachrinoudis, E. & Min, H. (2000). The dynamic relocation and phase-out of a hybrid, two-echelon plant/warehousing facility: a multiple objective approach. European Journal of Operational Research, 123(1), 1-15.

Michalewicz, Z. (1994). Genetic Algorithms + Data Structures = Evolution Programs. AI Series, Springer, New York.

Muckstadt, J.A., Roundy, R. O. (1987). Multi-Item, One- Warehouse, Multi-Retailer Distribution Systems. Management Science, 33, 1613-1621.

Narasimhan, R. & Mahapatra, S. (2004). Decision models in global supply chain management. Industrial Marketing Management, 33, 21-27.

Paksoy, T. (2005). Distribution network design & optimization in supply chain management: under material requirements constraints a strategic production-distribution model, Journal of Selcuk University Social Sciences Institute 14, 435-454, in Turkish.

Paksoy, T., Pehlivan, N.Y., & Özceylan, E. (2010). A Fuzzy Multi Objective Mixed Integer Programming Model for Multi Echelon Supply Chain Network Design and Optimization, 3rd Conference on Nonlinear Science and Complexity, July 28-31, Ankara.

Pasternack, B. (2001). The capacitated newsboy problem with revenue sharing. Journal of Applied Mathematics and Decision Sciences, 5, 21–33.

Peidro, D., J. Mula & Poler, R. (2007). Supply chain planning under uncertainty: a fuzzy linear programming approach. Fuzzy Systems Conference, 2007 FUZZ-IEEE, 2007 IEEE International, pp. 1-6.

Peidro, D., Mula, J., Poler, R., & Verdegay, J.L. (2009). Fuzzy optimization for supply chain planning under supply, demand and process uncertainities. Fuzzy Sets and Systems 160 (18), 2640-2657.

Rezaei, J., & Davoodi, M. (2008). A deterministic, multi-item inventory model with supplier selection and imperfect quality. Applied Mathematical Modelling 32, 2106–2116.

Ross, A. D. (2000). Performance-based strategic resource allocation in supply networks. International Journal of Production Economics, 63(2), 255-266.

Srivastava, S. (2007). Green supply-chain management: A state-of-the-art literature review. International Journal of Management Reviews, 9(1), 53-80.

Schwarz, L.B. (1973). A simple continuous review deterministic one-warehouse N-retailer inventory problem. Management Science, 19, 555-566

Syam, S.S. (2002). A model and methodologies for the location problem with logistical components. Computers & Operations Research, 29, 1173-1193.

Van Opijnen, M., & Oldenziel, J. (2011), Responsible supply chain management: Potential success factors and challenges for addressing prevailing human rights and other CSR issues in supply chains of EU-based companies.

Vasant, P., Nagarajan, R., & Yacoob, S. (2005). Fuzzy linear programming with vague objective coefficients in an uncertain environment. Journal of the Operational Research Society, 56, 597-603.

Verma, A. &, Nitin, S. (2011). A Conceptual Framework for Supply Chain Competitiveness, International Journal of Human and Social Sciences, 6, 1, 5-10.

Yang, P. C. & Wee, H. M. (2000). Economic ordering policy of deteriorated item for vendor and buyer: An integrated approach. Production Planning and Control, 11,474-480.

Zandhessam, H., Afshari, M. A., Siahkali, M.J., & Ayazi, S.A. (2011). A hybrid analytical network process and fuzzy goal programming for supplier selection: A case study of auto part maker. Management Science Letter, 1(4), 583-594

Zimmermann H.J. (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45-55

Appendix( Membership functions for echelons)

Non-increasing linear membership function for DC





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Non-increasing linear membership function for first Manufacturer’s fuzzy capacity

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Non-increasing linear membership function for first supplier





1