AOTP Firouzabad Institute of Higher Education Annals of Optimization Theory and Practice 2588-3666 Firouzabad Institute of Higher Education 58381 10.22121/aotp.2018.119731.1010 Original Article Linear programming problem with generalized interval-valued fuzzy numbers Linear programming problem with generalized interval-valued fuzzy numbers Taghaodi Rohollah Department of Mathematics, Kashan Branch, Islamic Azad University, Kashan, Iran Kardani Fatemeh Department of Industrial Engineering, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran 01 05 2018 1 2 1 9 17 01 2018 02 03 2018 Copyright © 2018, Firouzabad Institute of Higher Education. 2018 http://aotp.fabad-ihe.ac.ir/article_58381.html

In this paper, we concentrate on linear programming problems in which the cost vector, the technological coefficients and the right-hand side values are interval-valued generalized trapezoidal fuzzy numbers. To the best of our knowledge, till now there is no method described in the literature to find the optimal solution of the linear programming problems with interval-valued generalized trapezoidal fuzzy numbers. We apply the signed distance for defuzzification of this problem. The crisp problem obtained after the defuzzification is solved by the linear programming methods. Finally, we give an illustrative example and its numerical solutions.

Linear programming problem Generalized trapezoidal fuzzy number Interval-valued generalized trapezoidal fuzzy number
Allahviranloo, T., Lotfi, F. H., Kiasary, M. K., Kiani, N. A., & Alizadeh, L. (2008). Solving fully fuzzy linear programming problem by the ranking function. Applied mathematical sciences, 2(1), 19-32. Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management science, 17(4), B-141. Baranifar, S. (2018). A credibility-constrained programming for closed-loop supply chain network design problem under uncertainty. Annals of Optimization Theory and Practice, 1(1), 69-83. Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (1990). Linear programming and network flows. John Wiley, New York, Second Edition.  Chen, J. H. C. S. M., & Chen, S. M. (2006). A new method for ranking generalized fuzzy numbers for handling fuzzy risk analysis problems. In Proceedings of the Ninth Conference on Information Sciences, 1196–1199. Chen, S.H. (1985). Operations on fuzzy numbers with function principal. Tamkang Journal of Management Science, 6 (1), 13–25. Chen, T. Y. (2012). Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights. Applied Mathematical Modelling, 36, 3029–3052. Chen,S.H. (1985). Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems, 17, 13–129. Ebrahimnejad, A. (2016). Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sadhana, 41 (3), 299–316.  Farhadinia, B. (2014). Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems. Applied Mathematical Modelling, 38, 50–62. Gorzalczany, M. B. (1987). A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems, 21(1), 1–17. Hong, D. H., Lee, S. (2002). Some algebraic properties and a distance measure for interval-valued fuzzy numbers. Information Sciences, 148 (1), 1–10. Turksen, I. B. (1996). Interval-valued strict preference with Zadeh triples. Fuzzy Sets and Systems, 78 (2), 183–195. Wang, G., Li, Xi. (1998). The applications of interval-valued fuzzy numbers and interval-distribution numbers.   Fuzzy Sets and Systems, 98, 331–335. Wei, S. H., & Chen, S. M. (2009). Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Systems with Applications, 36(2), 2285-2299. Wang, G., & Li, X. (2001). Correlation and information energy of interval-valued fuzzy numbers. Fuzzy sets and systems, 103, 69-175. Zadeh, A. (1965). Fuzzy Sets. Information and Control, 8, 338-353.
AOTP Firouzabad Institute of Higher Education Annals of Optimization Theory and Practice 2588-3666 Firouzabad Institute of Higher Education 63490 10.22121/aotp.2018.130632.1013 Original Article A Fuzzy multi-product two-stage supply chain network design with possibility of direct shipment A Fuzzy multi-product two-stage supply chain network design with possibility of direct shipment Molla-Alizadeh-Zavardehi Saber Department of Industrial Engineering, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran Shoja Abbas Department of Industrial Engineering, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran 01 05 2018 1 2 11 22 13 03 2018 28 05 2018 Copyright © 2018, Firouzabad Institute of Higher Education. 2018 http://aotp.fabad-ihe.ac.ir/article_63490.html

The configuration of the supply chain network (SCN) is one of the strategic issues that have a major impact on the overall performance of the supply chain. A well designed SCN leads to an ability to reduce the supply chain total cost. These purposes are influenced by the supply chain strategy, which is based either on direct or indirect supply or shipment. In the case of direct shipment, the products are directly transported from the point of origin to the customers. In the classic transportation problems, it is usually assumed that the transportation time and costs are certain. Most existing mathematical models neglect the presence of uncertainty within a programming environment. This uncertainty might come about because of traffic jam, machine malfunctioning, defect in raw material, interpretation of various events and etc. These emprise parameters can be considering as fuzzy numbers. In this study, for the first time a mathematical model for a responsive, multi-product two-stage, SCN with possibility of direct shipment is proposed. Because of the unpredictable factors that mentioned above, cost coefficients are considered as trapezoidal fuzzy numbers. Therefore, for validation, the proposed model is coded by GAMS software. The results showed that relevant model is valid.

Supply chain network Direct shipment mathematical model Fuzzy theory
Basirzadeh, H. & Abbasi, R. (2008). A new approach for ranking fuzzy numbers based on cuts. Journal of Applied Mathematics and Informatics, 11, 767–78. Chanas, S. & Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems, 82, 299–305. Ebrahimnejad, A. (2016). A New Method for Solving Fuzzy Transportation Problem with LR Flat Fuzzy Numbers. Information Sciences, 357, 108-124. Farahani, R.Z., Rezapour, S., Drezner, T. & Fallah, S. (2014). Competitive supply chain network design: An overview of classifications, models, solution techniques and applications. Omega, 45, 92-118. Gao, S.P. & Liu, S.Y. (2004). Two-phase fuzzy algorithms for multi-objective transportation problem. The Journal of Fuzzy Mathematics, 12, 147-155. Giri, P.K., Maiti, M.K. & Maiti, M. (2015). Fully fuzzy fixed charge multi-item solid transportation problem. Applied Soft Computing, 27, 77-91. Hirsch, W.M. & Dantzig, G.B. (1968). The fixed charge problem. Naval Research Logistics, 15, 413–424. Jimenez, F. & Verdegay, J.L. (1998). Uncertain solid transportation problems. Fuzzy Sets and Systems, 100, 45–57. Jimenez, F. & Verdegay, J.L. (1999). Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. European Journal of Operational Research, 117, 485–510. Kocken, H. G. & Sivri M. (2016). a simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem. Applied Mathematical Modelling, 40, 4612-4624. Li, Y., Ida, K. & Gen, M. (1997). Improved genetic algorithm for solving multiobjective solid transportation problem with fuzzy numbers. Computers & Industrial Engineering, 33, 589–592. Liu, P., Yang, L., Wang, L. & Shukai. L. (2014). A solid transportation problem with type-2 fuzzy variables. Applied Soft Computing, 24, 543–558. Lin, L. Gen, M. & Wang, X. (2009). Integrated multistage logistics network design by using hybrid evolutionary algorithm. Computers & Industrial Engineering, 56, 854–873. Molla-Alizadeh-Zavardehi, S., Sadi Nezhad, S., Tavakkoli-Moghaddam, R. & Yazdani, M. (2013). Solving a fuzzy fixed charge solid transportation problem by metaheuristics. Mathematical and Computer Modelling, 57, 1543–1558. Mahmoodirad, A. & Sanei, M. (2016). Solving a multi-stage multi-product solid supply chain network design problem by meta-heuristics. Scientia Iranica E, 23, 1429-1440. Omar, M.S. & Samir, A.A. (2003). A parametric study on transportation problem under fuzzy environment. Engineering Journal of the University of Qatar, 15, 165-176. Pishvaee, M.S., Rabbani, M. (2011). A graph theoretic-based heuristic algorithm for responsive supply chain network design with direct and indirect shipment. Advances in Engineering Software, 42, 57–63. Pramanik, S., Jana, D.K. & Maiti, M. (2013). Multi-objective solid transportation problem in imprecise environments. Journal of Transportation Security, 6, 131-150. Pramanik, S., Jana, D. K., Mondal ,S.K. & Maiti, M. (2015). A fixed-charge transportation problem in two-stage supply chain network in Gaussian type-2 fuzzy environments. Information Sciences, 325, 190-214. Rani D. & Gulati, T.R. (2014). A new approach to solve unbalanced transportation problems in imprecise environment. Journal of Transportation Security, 7, 277-287. Sakawa, M. & Yano, H. (1986). Interactive fuzzy decision making for multiobjective nonlinear programming using augmented minimax problems. Fuzzy Sets and Systems, 20, 31-43. Sakawa, M., Yano, H. & Yumine, T. (1987). An Interactive fuzzy satisfying method for multi-objective linear programming problems and its application. IEEE Transactions on Man, Systems, and Cybernetics, 17, 654–661. Samanta, B. & Roy, T.K. (2005). Multi-objective entropy transportation model with trapezoidal fuzzy number penalties, sources and destination. Journal of Transportation Engineering, 131, 419–428. Singh, S. & Gupta, G. (2014). A new approach for solving cost minimization balanced transportation problem under uncertainty. Journal of Transportation Security, 7, 339-345. Sanei, M. Mahmoodirad, A. & Niroomand, S. (2016). Two-Stage Supply Chain Network Design Problem with Interval Data. International Journal of e-Navigation and Maritime Economy, 5, 074 – 084. Yang, L. & Liu, L. (2007). Fuzzy fixed charge solid transportation problem and algorithm. Applied Soft Computing, 7, 879–889.