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On the transformation of lexicographic nonlinear multiobjective programs to single objective programs

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Abstract

This paper deals with multiobjective optimization programs in which the objective functions are ordered by their degree of priority. A number of approaches have been proposed (and several implemented) for the solution of lexicographic (preemptive priority) multiobjective optimization programs. These approaches may be divided into two classes. The first encompasses the development of algorithms specifically designed to deal directly with the initial model. Considered only for linear multiobjective programs and multiobjective programs with a finite discrete feasible region, the second one attempts to transform, efficiently, the lexicographic multiobjective model into an equvivalent model, i.e. a single objective programming problem. In this paper, we deal with the second approach for lexicographic nonlinear multiobjective programs.

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References

  • Behringer FA (1972) Lexicographic quasieoncave multiobjective programming. ZOR 21: 103–116

    Article  MathSciNet  Google Scholar 

  • Bhushan M, Rengaswamy R (2004) Lexicographic optimization based sensor network design for Robust fault diagnosis. In: 7th international symposium on dynamics and control of process systems (DYCOPS). Cambridge, Massachusetts

  • Borwein JM (1977) Proper efficient points for maximization with respect to cones. SIAM J Control Optim 15: 57–63

    Article  MATH  Google Scholar 

  • Borwein JM, Zhuang D (1993) Super efficiency in vector optimization. Trans Am Math Soc 338: 105–122

    Article  MathSciNet  MATH  Google Scholar 

  • Calvete HI, Mateo PM (1998) Lexicographic optimisation in generalised network flow problems. J Oper Res Soc 49: 519–529

    MATH  Google Scholar 

  • Desaulniers G (2007) Managing large fixed costs in vehicle routing and crew scheduling problems solved by column generation. Comput Oper Res 34: 1221–1239

    Article  MATH  Google Scholar 

  • Ehrgott M (2005) Multicriteria optimization. Springer, Berlin

    MATH  Google Scholar 

  • Erdoǧan G, Cordeau J, Laporte G (2010) A branch-and-cut algorithm for solving the non-preemptive capacitated swapping problem. Discret Appl Math 158(15): 1599–1614

    Article  MathSciNet  Google Scholar 

  • Gascon V, Villeneuve S, Michelon P, Ferland JA (2000) Scheduling the flying squad nurses of a hospital using a multi-objective programming model. Ann Oper Res 96: 149–166

    Article  MATH  Google Scholar 

  • Geoffrion AM (1968) Proper efficiency and the theory of vector maximization. J Math Anal Appl 22: 616–630

    Article  MathSciNet  Google Scholar 

  • Hartley R (1978) On cone efficiency, cone convexity and cone compactness. SIAM J Appl Math 34: 211–222

    Article  MathSciNet  MATH  Google Scholar 

  • Hernández-Lerma O, Hoyos-Reyes LF (2001) A multiobjective control approach to priority queues. Math Methods Oper Res 53: 265–277

    Article  MathSciNet  MATH  Google Scholar 

  • Ignizio JP (1976) Goal programming and extensions. Heath, Lexington

    Google Scholar 

  • Ignizio JP, Thomas LC (1984) An enhanced conversion scheme for lexicographic, multiobjective integer programs. Eur J Oper Res 18: 57–61

    Article  MathSciNet  MATH  Google Scholar 

  • Isermann H (1974) Proper efficiency and the linear vector maximum problem. Oper Res 22: 189–191

    Article  MathSciNet  MATH  Google Scholar 

  • Isermann H (1982) Linear lexicographic optimization. OR Spektrum 4: 223–228

    Article  MATH  Google Scholar 

  • Jeyakumar V, Lee GM, Dinh N (2006) Characterizations of solution sets of convex vector minimization problems. Eur J Oper Res 174: 1380–1395

    Article  MathSciNet  MATH  Google Scholar 

  • Khorram E, Zarepisheh M, Ghaznavi-ghosoni BA (2010) Sensitivity analysis on the priority of the objective functions in lexicographic multiple objective linear programs. Eur J Oper Res 207(3): 1162–1168

    Article  MathSciNet  MATH  Google Scholar 

  • Mäkelä MM, Nikulin Y (2009) On cone characterizations of strong and lexicographic optimality in convex multiobjective optimization. J Optim Theory Appl 143: 519–538

    Article  MathSciNet  MATH  Google Scholar 

  • Miettinen K (1999) Nonlinear multiobjective optimization. vol 12 of international series in operations research and management science. Kluwer Academic Publishers, Dordrecht

  • Miettinen K, Mäkelä MM (2001) On cone characterizations of weak, proper and Pareto optimality in multiobjective optimization. Math Methods Oper Res 53: 233–245

    Article  MathSciNet  MATH  Google Scholar 

  • Nijkamp P (1980) Environmental policy analysis. Operational methods and models. Wiley, New York

    Google Scholar 

  • Padhiyar N, Bhartiya S (2009) Profile control in distributed parameter systems using lexicographic optimization based MPC. J Process Control 19: 100–109

    Article  MATH  Google Scholar 

  • Pourkarimi L, Zarepisheh M (2007) A dual-based algorithm for solving lexicographic multiple objective programs. Eur J Oper Res 176: 1348–1356

    Article  MathSciNet  MATH  Google Scholar 

  • Rentmeesters M (1998) A theory of multi-objective optimization: comprehensive KuhnTucker conditions for lexicographic and pareto optima. Ph.D. Dissertation, University of California, Irvine

  • Sawaragi Y, Nakayama H, Tanino T (1985) Theory of multiobjective optimization. Academic Press, Orlando

    MATH  Google Scholar 

  • Sherali HD (1982) Equivalent weights for lexicographic multiobjective programs: characterization and computations. Eur J Oper Res 11: 367–379

    Article  MathSciNet  MATH  Google Scholar 

  • Sherali HD, Soyster AL (1983) Preemptive and nonpreemptive multi-objective programming: relationships and counterexamples. J Optim Theory Appl 39: 173–186

    Article  MathSciNet  MATH  Google Scholar 

  • Sun C, Ritchie SG, Tsai K, Jayakrishnan R (1999) Use of vehicle signature analysis and lexicographic optimization for vehicle reidentification on freeways. Transp Res Part C 7: 167–185

    Article  Google Scholar 

  • Turnovec F (1985) Lexicographic optimization problems in production scheduling optimization, theory, methods, applications, vol I. Dum Techniky CSVTS Praha, Prague, pp 295–306

  • Vada J, Slupphaug O, Johansen TA (2001) Optimal prioritized infeasibility handling in model predictive control: parametric preemptive multiobjective linear programming approach. J Optim Theory Appl 109: 385–413

    Article  MathSciNet  MATH  Google Scholar 

  • Weber E, Rizzoli A, Soncini-Sessa R, Castelletti A (2002) Lexicographic optimisation for water resources planning: the case of Lake Verbano, Italy. In: Rizzoli A, Jakeman A (eds) Integrated assessment and decision support—proceedings of the first biennial meeting of the international environmental modelling and software society. Lugano, pp 235–240

  • Zheng XY (1997) Proper efficiency in locally convex topological vector spaces. J Optim Theory Appl 94: 469–486

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Avenue, 15914, Tehran, Iran

    M. Zarepisheh & E. Khorram

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  1. M. Zarepisheh
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  2. E. Khorram
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Correspondence to E. Khorram.

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Zarepisheh, M., Khorram, E. On the transformation of lexicographic nonlinear multiobjective programs to single objective programs. Math Meth Oper Res 74, 217–231 (2011). https://doi.org/10.1007/s00186-011-0360-7

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  • DOI: https://doi.org/10.1007/s00186-011-0360-7

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