Abstract
The current paper focuses on a multiobjective linear programming problem with interval objective functions coefficients. Taking into account the minimax regret criterion, an attempt is being made to propose a new solution i.e. minimax regret solution. With respect to its properties, a minimax regret solution is necessarily ideal when a necessarily ideal solution exists; otherwise it is still considered a possibly weak efficient solution. In order to obtain a minimax regret solution, an algorithm based on a relaxation procedure is suggested. A numerical example demonstrates the validity and strengths of the proposed algorithm. Finally, two special cases are investigated: the minimax regret solution for fixed objective functions coefficients as well as the minimax regret solution with a reference point. Some of the characteristic features of both cases are highlighted thereafter.
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References
Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms. 3rd Edn. Wiley, New Jersey
Bazaraa MS, Jarvis JJ (2010) Linear programming and network flows. 4th Edn. Wiley, New Jersey
Birge JR, Louveaux F (1993) Introduction to stochastic programming. Physica-Verlag, New York
Bitran GR (1980) Linear multiobjective problems with interval coefficients. Manag Sci 26: 694–706
Das SK, Goswami A, Alam SS (1999) Multiobjective transportation problem with interval cost, source and destination parameters. Eur J Oper Res 117: 100–112
Dong C, Huang GH, Cai YP, Xu Y (2011) An interval-parameter minimax regret programming approach for power management systems planning under uncertainty. Appl Energy 88: 2835–2845
Ehrgott M (2005) Multicriteria optimization. 2nd Edn. Springer, Berlin
Feidler M, Nedoma J, Ramik J, Rohn J, Zimmermann K (2006) Linear optimization problems with inexact data. Springer, Berlin
Giove S, Funari S, Nardelli C (2006) An interval portfolio selection problem based on regret function. Eur J Oper Res 170: 253–264
Hladík M (2008) Tolerances in portfolio selection via interval linear programming. In: Proceedings of the 26th international conference on mathematical methods in economics, MME08, pp 185–191
Hladík M (2010) On necessarily efficient solutions in interval multiobjective linear programming, In: Proceedings of the 25th Mini-EURO conference on uncertainty and robustness in planning and decision making, URPDM 2010, pp 1–10
Ida M., Generation of efficient solutions for multiobjective linear programming with interval coefficients. In: Proceedings of the SICE annual conference, SICE’96, pp 1041–1044
Ida M (1999) Necessary efficient test in interval multiobjective linear programming. In: Proceedings of the eighth international fuzzy systems association world congress, pp 500–504
Ida M (2003) Portfolio selection problem with interval coefficients. Appl Math Lett 16: 709–713
Ida M (2004) Solutions for the portfolio selection problem with interval and fuzzy coefficients. Reliab Comput 10: 389–400
Inuiguchi M, Kume Y (1991) Goal programming problems with interval coefficients and target intervals. Eur J Oper Res 52: 345–360
Inuiguchi M, Sakawa M (1995) Minimax regret solution to linear programming problems with an interval objective function. Eur J Oper Res 86: 526–536
Inuiguchi M, Sakawa M (1996) Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test. Fuzzy Sets Syst 78: 321–341
Inuiguchi M, Higashitani H, Tanino T (1999) On computation methods of the minimax regret solution for linear programming problems with uncertain objective function coefficients. In: Proceedings of IEEE international conference, pp 979–984
Lai KK, Wang SY, Xu JP, Zho SS, Fang Y (2002) A class of linear interval programming problems and its application to portfolio selection. IEEE Trans Fuzzy Syst 10: 698–704
Lai Y-J, Hwang C-L (1996) Fuzzy multiple objective decision making: methods and applications, Lecture Notes in Economics and Mathematical Systems, vol 404. Springer, Berlin
Loulou R, Kanudia A (1999) Minimax regret strategies for greenhouse gas abatement: methodology and application. Oper Res Lett 25: 219–230
Mausser HE, Laguna M (1998) A new mixed integer formulation for the maximum regret problem. Int Trans Oper Res 5: 389–403
Oliveira C, Antunes CH (2007) Multiple objective linear programming models with interval coefficients—an illustrative overview. Eur J Oper Res 181: 1434–1463
Oliveira C, Antunes CH (2009) An interactive method of tackling uncertainty in interval multiple objective linear programming. J Math Sci 161: 854–866
Prókopa A (1995) Stochastic programming. Kluwer, Boston
Sakawa M (1993) Fuzzy sets and interactive multiobjective optimization. Plenum Press, New York
Savage LJ (1951) The theory of statistical decision. J Am Stat Assoc 46: 55–67
Shimizu K, Aiyoshi E (1980) Necessary conditions for min-max problems and algorithms by a relaxation procedure. IEEE Trans Automat Control AC-25: 62–66
Urli B, Nadeau R (1992) An interactive method to multiobjective linear programming problems with interval coefficients. INFOR 30: 127–137
Wang ML, Wang HF (2001) Interval analysis of a fuzzy multiobjective linear programming. Int J Fuzzy Syst 34: 558–568
Wierzbicki AP (1982) A mathematical basis for satisficing decision making. Math Model 3: 391–405
Wu HC (2009) The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions. Eur J Oper Res 196: 49–60
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Rivaz, S., Yaghoobi, M.A. Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients. Cent Eur J Oper Res 21, 625–649 (2013). https://doi.org/10.1007/s10100-012-0252-9
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DOI: https://doi.org/10.1007/s10100-012-0252-9