Multi-objective bi-level supply chain network order allocation problem under fuzziness


In this paper we addressed supply chain network (SCN) as bi-level programming problem in which the primary objective is to determine optimal order allocation of products where the customer’s demands and supply for the products are fuzzy. In the proposed SCN model, we suppose that the first level (leader) and second level (follower) operate two separate groups of SCN. The leader, who moves first, determines quantities shipped to retailers, and then, the follower decides his quantities rationally. The leader’s objective is to minimize the total transportation costs, and similarly, the follower’s objective is to minimize the total delivery time of the SCN and at the same time balancing the optimal order allocation from each source, plant, retailer and warehouse respectively. The fuzzy goal programming approach has been used to achieve the highest degree of the membership goals by minimizing the deviational variables so that most satisfactory or the preferred solution for both the levels to be obtained. A numerical example is given to demonstrate the proposed methodology.

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  1. 1.

    Feili, H., Khoshdoon, M.: A fuzzy optimization model for supply chain production planning with an atotal aspect of decision making. J. Math. Comput. Sci. 2(1), 65–80 (2011)

    Article  Google Scholar 

  2. 2.

    Liu, S.T., Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153(3), 661–674 (2004)

    Article  Google Scholar 

  3. 3.

    Liang, T.F.: Distribution planning decisions using interactive fuzzy multi-objective linear programming. Fuzzy Sets Syst. 157(10), 1303–1316 (2006)

    Article  Google Scholar 

  4. 4.

    Liang, T.F.: Applying fuzzy goal programming to production/transportation planning decisions in a supply chain. Int. J. Syst. Sci. 38(4), 293–304 (2007)

    Article  Google Scholar 

  5. 5.

    Sakawa, M., Nishizaki, I., Uemura, Y.: Fuzzy programming and profit and cost allocation for a production and transportation problem. Eur. J. Oper. Res. 131(1), 1–15 (2001)

    Article  Google Scholar 

  6. 6.

    Selim, H., Araz, C., Ozkarahan, I.: Collaborative production–distribution planning in supply chain: a fuzzy goal programming approach. Transp. Res. Part E Logist. Transp. Rev. 44(3), 396–419 (2008)

    Article  Google Scholar 

  7. 7.

    Aliev, R.A., Fazlollahi, B., Guirimov, B.G., Aliev, R.R.: Fuzzy-genetic approach to aggregate production-distribution planning in supply-chain management. Inf. Sci. 177(20), 4241–4255 (2007)

    Article  Google Scholar 

  8. 8.

    Chen, S.P., Chang, P.C.: A mathematical programming approach to supply chain models with fuzzy parameters. Eng. Optim. 38(6), 647–669 (2006)

    Article  Google Scholar 

  9. 9.

    Torabi, S.A., Hassini, E.: An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy Sets Syst. 159(2), 193–214 (2008)

    Article  Google Scholar 

  10. 10.

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

  11. 11.

    Gen, M., Tsujimura, Y., Ida, K.: Method for solving multiobjective aggregate production planning problem with fuzzy parameters. Comput. Ind. Eng. 23(1–4), 117–120 (1992)

    Article  Google Scholar 

  12. 12.

    Gumus, A.T., Guneri, A.F., Keles, S.: Supply chain network designusing an integrated neuro-fuzzy and MILP approach: a comparative design study. Expert Syst. Appl. 36(10), 12570–12577 (2009)

    Article  Google Scholar 

  13. 13.

    Bilgen, B.: Application of fuzzy mathematical programming approach to the production allocation and distribution supply chain network problem. Expert Syst. Appl. 37(6), 4488–4495 (2010)

    Article  Google Scholar 

  14. 14.

    Fahimnia, B., Farahani, R.Z., Marian, R., Luong, L.: A review and critique on integrated production-distribution planning models and techniques. J. Manuf. Syst. 32(1), 1–19 (2013)

    Article  Google Scholar 

  15. 15.

    Jolai, F., Razmi, J., Rostami, N.K.M.: A fuzzy goal programming and metaheuristic algorithms for solving integrated production: distribution planning problem. CEJOR 19(4), 547–569 (2011)

    Article  Google Scholar 

  16. 16.

    Paksoy, T., Pehlivan, N.Y.: A fuzzy linear programming model for the optimization of multi-stage supply chain networks with triangular and trapezoidal membership functions. J. Franklin Inst. 349(1), 93–109 (2012)

    Article  Google Scholar 

  17. 17.

    Garai, A., Mandal, P., Roy, T.K.: Intuitionistic fuzzy T-sets based optimization technique for production-distribution planning in supply chain management. OPSEARCH 53(4), 950–975 (2016)

    Article  Google Scholar 

  18. 18.

    Abo-Sinna, M.A., Baky, I.A.: Fuzzy goal programming procedure to bilevel multiobjective linear fractional programming problems. Int. J. Math. Math. Sci. 2010, 148975 (2010).

    Article  Google Scholar 

  19. 19.

    Baky, I.A.: Fuzzy goal programming algorithm for solving decentralised-level multi-objective programming problems. Fuzzy Sets Syst. 160(18), 2701–2713 (2009)

    Article  Google Scholar 

  20. 20.

    Bialas, W.F., Karwan, M.H.: Two-level linear programming. Manag. Sci. 30(8), 1004–1020 (1984)

    Article  Google Scholar 

  21. 21.

    Baky, I.A., Eid, M.H., El Sayed, M.A.: Bi-level multi-objective programming problem with fuzzy demands: a fuzzy goal programming algorithm. Opsearch 51(2), 280–296 (2014)

    Article  Google Scholar 

  22. 22.

    Birla, R., Agarwal, V.K., Khan, I.A., Mishra, V.N.: An alternative approach for solving bi-level programming problems. Am. J. Oper. Res. 7(03), 239 (2017)

    Google Scholar 

  23. 23.

    Bagloee, S.A., Asadi, M., Sarvi, M., Patriksson, M.: A hybrid machine-learning and optimization method to solve bi-level problems. Expert Syst. Appl. 95, 142–152 (2018)

    Article  Google Scholar 

  24. 24.

    Osman, M.S., Emam, O.E., Elsayed, M.A.: Interactive approach for multi-level multi-objective fractional programming problems with fuzzy parameters. Beni-Suef Univ. J. Basic Appl. Sci. 7(1), 139–149 (2018)

    Article  Google Scholar 

  25. 25.

    Osman, M.S., Emam, O.E., El Sayed, M.A.: Solving multi-level multi-objective fractional programming problems with fuzzy demands via FGP approach. Int. J. Appl. Comput. Math. 4(1), 41 (2018)

    Article  Google Scholar 

  26. 26.

    Golpîra, H., Najafi, E., Zandieh, M., Sadi-Nezhad, S.: Robust bi-level optimization for green opportunistic supply chain network design problem against uncertainty and environmental risk. Comput. Ind. Eng. 107, 301–312 (2017)

    Article  Google Scholar 

  27. 27.

    Jalil, S.A., Javaid, S., Muneeb, S.M.: A decentralized multi-level decision making model for solid transportation problem with uncertainty. Int. J. Syst. Assur. Eng. Manag (2018).

    Article  Google Scholar 

  28. 28.

    Muneeb, S.M., Adhami, A.Y., Asim, Z., Jalil, S.A.: Bi-level decision making models for advertising allocation problem under fuzzy environment. Int. J. Syst. Assur. Eng. Manag (2018).

    Article  Google Scholar 

  29. 29.

    Adhami, A.Y., Muneeb, S.M., Nomani, M.A.: A multi-level decision making model for the supplier selection problem in a fuzzy situation. Oper. Res. Decis. 27(4), 5–26 (2017)

    Google Scholar 

  30. 30.

    Muneeb, S.M., Adhami, A.Y., Jalil, S.A., Asim, Z.: Decentralised bi-level decision planning model for municipal solid waste recycling and management with cost reliability under uncertain environment. Sustain. Prod. Consum (2018).

    Article  Google Scholar 

  31. 31.

    Amirtaheri, O., Zandieh, M., Dorri, B., Motameni, A.R.: A bi-level programming approach for production-distribution supply chain problem. Comput. Ind. Eng. 110, 527–537 (2017)

    Article  Google Scholar 

  32. 32.

    Kolak, Oİ., Feyzioğlu, O., Noyan, N.: Bi-level multi-objective traffic network optimisation with sustainability perspective. Expert Syst. Appl. 104, 294–306 (2018)

    Article  Google Scholar 

  33. 33.

    Jin, S.W., Li, Y.P., Nie, S.: An integrated bi-level optimization model for air quality management of Beijing’s energy system under uncertainty. J. Hazard. Mater. 350, 27–37 (2018)

    Article  Google Scholar 

  34. 34.

    Parvasi, S.P., Mahmoodjanloo, M., Setak, M.: A bi-level school bus routing problem with bus stops selection and possibility of demand outsourcing. Appl. Soft Comput. 61, 222–238 (2017)

    Article  Google Scholar 

  35. 35.

    Zeng, Q., Zhang, B., Fang, J., Chen, Z.: A bi-level programming for multistage co-expansion planning of the integrated gas and electricity system. Appl. Energy 200, 192–203 (2017)

    Article  Google Scholar 

  36. 36.

    Ryu, J., Dua, V., Pistikopoulos, E.N.: A bilevel programming framework for enterprise-wide process networks under uncertainty. Comput. Chem. Eng. 28(6–7), 1121–1129 (2004)

    Article  Google Scholar 

  37. 37.

    Chang, Y., Lee, C.: Machine scheduling with job delivery coordination. Eur. J. Oper. Res. 158(2), 470–487 (2004)

    Article  Google Scholar 

  38. 38.

    Lejeune, M.A.: A variable neighbourhood decomposition search method for supply chain management planning problems. Eur. J. Oper. Res. 175(2), 959–976 (2006)

    Article  Google Scholar 

  39. 39.

    Sadigh, A.N., Mozafari, M., Karimi, B.: Manufacturer–retailer supply chain coordination: A bi-level programming approach. Adv. Eng. Softw. 45(1), 144–152 (2012)

    Article  Google Scholar 

  40. 40.

    Nishi, T., Yoshida, O.: Optimization of multi-period bilevel supply chains under demand uncertainty. Procedia CIRP 41, 508–513 (2016)

    Article  Google Scholar 

  41. 41.

    Calvete, H.I., Galé, C., Oliveros, M.J.: Bilevel model for production–distribution planning solved by using ant colony optimization. Comput. Oper. Res. 38(1), 320–327 (2011)

    Article  Google Scholar 

  42. 42.

    Camacho-Vallejo, J.F., Cordero-Franco, Á.E., González-Ramírez, R.G.: Solving the bilevel facility location problem under preferences by a stackelberg-evolutionary algorithm. Math. Probl. Eng. 2014, 430243 (2014).

    Article  Google Scholar 

  43. 43.

    Huang, B., Liu, N.: Bilevel programming approach to optimizing a logistic distribution network with balancing requirements. Transp. Res. Record J. Transp. Res. Board 1894, 188–197 (2004)

    Article  Google Scholar 

  44. 44.

    Aryanezhad, M.B., Roghanian, E.A.: Bilevel linear multi-objective decision making model with interval coefficients for supply chain coordination. Int. J. Eng. Sci. 19(1–2), 67–74 (2008)

    Google Scholar 

  45. 45.

    Jianhua, Y.: Analysis on bi-level programming model in supply chain distribution problem. In: 2012 Fifth International Conference on Intelligent Computation Technology and Automation (ICICTA), pp. 94–97. IEEE (2012)

  46. 46.

    Yang, J., Hao, Z.: The study on supply chain distribution optimization based on bi-level programming model. In: 2009 International Conference on Information Management, Innovation Management and Industrial Engineering, Vol. 3, pp. 7–10. IEEE (2009)

  47. 47.

    Sun, H.J., Gao, Z.Y.: An optimization model for two-echelon distribution network design in supply chain based on bi-level programming. J. Ind. Eng. Eng. Manag. 1, 017 (2004)

    Google Scholar 

  48. 48.

    Zhigang, Z., Xinyi, G.: Bi-level programming method for distribution network model in supply chain. Univ. Shanghai Sci. Technol. 28, 300–302 (2006)

    Google Scholar 

  49. 49.

    Liu, S.T., Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153(3), 661–674 (2004)

    Article  Google Scholar 

  50. 50.

    Chakraborty, D., Jana, D.K., Roy, T.K.: Arithmetic operations on generalized intuitionistic fuzzy number and its applications to transportation problem. Opsearch 52(3), 431–471 (2015)

    Article  Google Scholar 

  51. 51.

    Nishad, A.K., Singh, S.R.: Goal programming for solving fractional programming problem in fuzzy environment. Appl. Math. 6(14), 2360 (2015)

    Article  Google Scholar 

  52. 52.

    Kuwano, H.: On the fuzzy multi-objective linear programming problem: goal programming approach. Fuzzy Sets Syst. 82(1), 57–64 (1996)

    Article  Google Scholar 

  53. 53.

    Ebrahimnejad, A.: Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sādhanā 41(3), 299–316 (2016)

    Google Scholar 

  54. 54.

    Liu, S.T.: Fractional transportation problem with fuzzy parameters. Soft. Comput. 20(9), 3629–3636 (2016)

    Article  Google Scholar 

  55. 55.

    Abbasbandy, S., Hajjari, T.: A new approach for ranking of trapezoidal fuzzy numbers. Comput. Math Appl. 7(3), 413–419 (2009)

    Article  Google Scholar 

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Correspondence to Srikant Gupta.

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Gupta, S., Ali, I. & Ahmed, A. Multi-objective bi-level supply chain network order allocation problem under fuzziness. OPSEARCH 55, 721–748 (2018).

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  • Supply chain network
  • Multi-objective optimization
  • Bi-level programming problem
  • Trapezoidal fuzzy number
  • \(\alpha\)-cut approach
  • Fuzzy goal programming