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An algorithm for multiparametric 0-1-Integer Programming problemsrelative to a generalized min max objective function

Published online by Cambridge University Press:  28 January 2009

José Luis Quintero  and
Alejandro Crema
José Luis Quintero
Affiliation:
Departamento de Matemáticas Aplicadas, Facultad de Ingeniería, Universidad Central de Venezuela, Apartado 47002, Caracas 1041-A, Venezuela; [email protected]
Alejandro Crema
Affiliation:
Escuela de Computación, Facultad de Ciencias, Universidad Central de Venezuela, Apartado 47002, Caracas 1041-A, Venezuela; [email protected]
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Abstract

The multiparametric 0-1-Integer Programming (0-1-IP)problem relative to the objective function is a family of 0-1-IP problemswhich are relatedby havingidentical constraint matrix and right-hand-sidevector. In this paper we present an algorithm to perform a completemultiparametric analysis relative to a generalized min max objectivefunction such that the min sum and min max are particular cases.

Keywords

0-1-Integer Programmingmultiparametric programmingBottleneck problem.
Type
Research Article
Information
RAIRO - Operations Research , Volume 43 , Issue 1 , January 2009 , pp. 1 - 12
Copyright
© EDP Sciences, ROADEF, SMAI, 2009

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References

Adams, W.P. and Forrester, R.J., A simple recipe for concise mixed 0-1 linearizations. Oper. Res. Lett. 33 (2005) 55–61. CrossRef
Adams, W.P., Forrester, R.J. and Glover, F.W., Comparisons and enhancement strategies for linearizing mixed 0-1-quadratic programs. Discrete Optim. 1 (2004) 99–120. CrossRef
Alves, M.J. and Climaco, J., A review of interactive methods for multiobjective integer and mixed-integer programming. Eur. J. Operat. Res. 180 (2007) 99–115. CrossRef
H. Arsham, http://home.ubalt.edu/ntsbarsh/Business-stat/Refop.htm.
V.J. Bowman Jr, On the relationship of the Tchebycheff norm and the efficient frontier of multiple-criteria objectives. in Multiple Criteria Decision Making, edited by H. Thiriez, S. Zionts, Lect. Notes Economics Math. Systems, edited by H. Thiriez, S. Zionts, Springer-Verlag, Berlin (1976) 76–86.
Crema, A., A contraction algorithm for the multiparametric integer linear programming problem. Eur. J. Oper. Res. 101 (1997) 130–139. CrossRef
Crema, A., An algorithm for the multiparametric 0-1-integer linear programming problem relative to the objective function. Eur. J. Oper. Res. 125 (2000) 18–24. CrossRef
Crema, A., An algorithm for the multiparametric 0-1-integer linear programming problem relative to the constraint matrix. Oper. Res. Lett. 27 (2000) 1–46. CrossRef
Crema, A., The multiparametric 0-1-Integer Linear Programming problem: A unified approach. Eur. J. Oper. Res. 139 (2002) 511–520. CrossRef
Geoffrion, A.M. and Nauss, R., Parametric and postoptimality analysis in integer linear programming. Manag. Sci. 23 (1977) 453–466. CrossRef
H.J. Greenberg, An annoted bibliography for post-solution analysis in mixed integer propgramming and combinatorial optimization, Advances in Computational and Stochastic Optimization, Logic Programming and Heuristic Search, edited by David L. Woodruff, Kluwer Academic Publishers, Boston, MA (1998) 97–148.
H.J. Greenberg, An annoted bibliography for post-solution analysis in mixed integer programming and combinatorial optimization, http://www-math.cudenver.edu/ hgreenbe/papers/mipbib.pdf.
IBM Optimization Library C, Application Programming Interface. Available at http://www-306.ibm.com/software/data/bi/osl/pubs/library/featCAPI.htm.
Jenkins, L., Parametric mixed integer programming: an application to solid waste management. Manag. Sci. 28 (1982) 1270–1284. CrossRef
Jenkins, L., Using parametric integer programming to plan the mix of an air transport fleet. INFOR 25 (1987) 117–135.
Jenkins, L., Parametric-objective integer programming using knapsack facets and Gomory cutting planes. Eur. J. Oper. Res. 31 (1987) 102–109. CrossRef
Jenkins, L., Parametric methods in integer linear programming. Ann. Oper. Res. 27 (1990) 77–96. CrossRef
Laumanns, M., Thiele, L. and Zitzler, E., An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon constraint method. Eur. J. Oper. Res. 169 (2006) 932–942. CrossRef
Li, Z. and Ierapetritou, M.G., A new methodology for the general multiparametric mixed integer linear programming (MILP) problems. Ind. Eng. Che. Res. 46 (2007) 5141–5151. CrossRef
Martello, S. and Toth, P., The bottleneck generalized assignment problem. Eur. J. Oper. Res. 83 (1995) 621–638. CrossRef
Optimization Soubroutine Library, release 2, Guide and Reference, IBM (1992).
Oral, M. and Kettani, O., A linearization procedure for quadratic and cubic mixed integer problems. Oper. Res. 40 (1992) S109–S116. CrossRef
Quintero, J.L. and Crema, A., An algorithm for multiparametric min max 0-1-integer programming problem relative to the objective function. RAIRO Oper. Res. 39 (2005) 243–252. CrossRef
Steuer, R.E. and Choo, E.U., An interactive weighted Tchebycheff procedure for multiple objective programming. Math. Program. 26 (1983) 326–344. CrossRef
Sylva, J. and Crema, A., A method for finding the set of non-dominated vectors for multiple objective integer linear programs. Eur. J. Oper. Res. 158 (2004) 46–55. CrossRef
Sylva, J. and Crema, A., A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs. Eur. J. Oper. Res. 180 (2007) 1011–1027. CrossRef
Sylva, J. and Crema, A., Enumerating the set of non-dominated vectors in multiple objective integer linear programming. RAIRO Oper. Res. 42 (2008) 371–388. CrossRef
B. Thiongane, A. Nagih and G. Plateau, Theoretical and algorithmic study for parametric 0-1 linear programs relative to the objective function. Research Report LIPN 2003-01.