October  2010, 6(4): 811-823. doi: 10.3934/jimo.2010.6.811

A method for optimizing over the integer efficient set

1. 

USTHB - Faculty of Mathematics - Operations Research Department, Bp 32 El Alia, BEZ, Algiers, 16121, Algeria

2. 

UMONS, Faculté Polytechnique, 9 Rue Houdain-Mons, Mons 7000, Belgium

Received  September 2009 Revised  May 2010 Published  September 2010

In this paper, we are interested in optimizing a linear function on the set of efficient solutions of a Multiple Objective Integer Linear Programming problem ($MOILP $). We propose an exact algorithm for maximizing a linear function denoted $ \phi $ on the set of efficient solutions of a $MOILP$ problem without having to enumerate explicitly all the elements of this set. Two techniques are used: the first is to reduce progressively the admissible domain by adding more constraints eliminating all the dominated points by the current solution; the second, when the new solution obtained by maximizing the function $\phi $ in the reduced area is not efficient, an exploration procedure is applied over the edges incident to this solution in order to find new alternative efficient solutions if they exist. The algorithm produces not only an optimal value of the linear function but also a subset of non-dominated solutions in the direction of $\phi$ that can be helpful in the practice.
Citation: Chaabane Djamal, Pirlot Marc. A method for optimizing over the integer efficient set. Journal of Industrial and Management Optimization, 2010, 6 (4) : 811-823. doi: 10.3934/jimo.2010.6.811
References:
[1]

M. Abbas and D. Chaabane, An algorithm for solving multiple objective integer linear programming problem, RAIRO Operations Research, 36 (2002), 351-364. doi: 10.1051/ro:2003006.

[2]

M. Abbas and D. Chaabane, Optimizing a linear function over an integer efficient set, European Journal of Operational Research, 174 (2006), 1140-1161. doi: 10.1016/j.ejor.2005.02.072.

[3]

H. P. Benson, Existence of efficient solutions for vector maximization problems, Journal of Optimization Theory and Appl, 26 (1978), 569-580. doi: 10.1007/BF00933152.

[4]

H. P. Benson and S. Sayin, Optimization over the efficient set: Four special cases, Journal of Optimization Theory and Appl., 80 (1994), 3-18. doi: 10.1007/BF02196590.

[5]

V. J. Bowman, On the relationship of the Tchebytcheff norm and the efficient frontier of multiple-criteria objectives, Lecture Notes in Economics and Mathematical Systems, 130 (1976), 76-85.

[6]

Chaabane Djamal, "Contribution à l'Optimisation Multicritère en Variables Discrètes," Ph.D thesis, UMONS, Polytechnic Faculty of Mons, Belgium, 2007.

[7]

A. Crema and J. Sylva, A method for finding the set of non-dominated vectors for multiple objective integer linear programs, European Journal of Operational Research, 158 (2004), 46-55. doi: 10.1016/S0377-2217(03)00255-8.

[8]

G. B. Dantzig, On a linear programming combinatorial approach to the traveling salesman problem, Operations Research, 7 (1959), 58-66. doi: 10.1287/opre.7.1.58.

[9]

J. G. Ecker and J. H. Song, Optimizing a linear function over an efficient set, Journal of Optimization Theory and Applications, 83 (1994), 541-563. doi: 10.1007/BF02207641.

[10]

A. M. Geoffrion, Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications, 22 (1968), 618-630. doi: 10.1016/0022-247X(68)90201-1.

[11]

R. Gupta and R. Malhotra, Multi-criteria integer linear programming problem, Cahiers du CERO, 34 (1992), 51-68.

[12]

M. J. Jorge, An algorithm for optimizing a linear function over an integer efficient set, European Journal of Operational Research, 195 (2009), 98-103. doi: 10.1016/j.ejor.2008.02.005.

[13]

J. N. Karaivanova and S. C. Narula, An interactive procedure for multiple objective integer linear programming problems, European Journal of Operational Research, 68 (1993), 344-351. doi: 10.1016/0377-2217(93)90190-X.

[14]

D. Klein and E. Hannan, An algorithm for multiple objective integer linear programming problem, European Journal of Operational Research, 9 (1982), 378-385. doi: 10.1016/0377-2217(82)90182-5.

[15]

N. C. Nguyen, "An Algorithm for Optimizing a Linear Function Over the Integer Efficient Set," Konrad-Zuse-Zentrum fur Informationstechnik Berlin, 1992.

[16]

J. Philip, Algorithms for the vector maximization problem, Mathematical Programming, 2 (1972), 207-229. doi: 10.1007/BF01584543.

[17]

J. Sylva and A. Crema, A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs, European Journal of Operational Reserach, 180 (2007), 1011-1027. doi: 10.1016/j.ejor.2006.02.049.

[18]

Ta Van Tu, Optimization over the efficient set of a parametric multiple objective linear programming problem, European Journal of Operational Reserach, 122 (2000), 570-583. doi: 10.1016/S0377-2217(99)00095-8.

[19]

J. Teghem and P. Kunsch, A survey of techniques for finding efficient solutions to multiobjective integer linear programming, Asia Pacific Journal of Operations Research, 3 (1986), 95-108.

[20]

D. J. White, The maximization of a function over the efficient set via a penalty function approach, European Journal of Operational Research, 94 (1996), 143-153. doi: 10.1016/0377-2217(95)00184-0.

[21]

Y. Yamamoto, Optimization over the efficient set, overview, Journal of Global Optimization, 22 (2002), 285-317.

show all references

References:
[1]

M. Abbas and D. Chaabane, An algorithm for solving multiple objective integer linear programming problem, RAIRO Operations Research, 36 (2002), 351-364. doi: 10.1051/ro:2003006.

[2]

M. Abbas and D. Chaabane, Optimizing a linear function over an integer efficient set, European Journal of Operational Research, 174 (2006), 1140-1161. doi: 10.1016/j.ejor.2005.02.072.

[3]

H. P. Benson, Existence of efficient solutions for vector maximization problems, Journal of Optimization Theory and Appl, 26 (1978), 569-580. doi: 10.1007/BF00933152.

[4]

H. P. Benson and S. Sayin, Optimization over the efficient set: Four special cases, Journal of Optimization Theory and Appl., 80 (1994), 3-18. doi: 10.1007/BF02196590.

[5]

V. J. Bowman, On the relationship of the Tchebytcheff norm and the efficient frontier of multiple-criteria objectives, Lecture Notes in Economics and Mathematical Systems, 130 (1976), 76-85.

[6]

Chaabane Djamal, "Contribution à l'Optimisation Multicritère en Variables Discrètes," Ph.D thesis, UMONS, Polytechnic Faculty of Mons, Belgium, 2007.

[7]

A. Crema and J. Sylva, A method for finding the set of non-dominated vectors for multiple objective integer linear programs, European Journal of Operational Research, 158 (2004), 46-55. doi: 10.1016/S0377-2217(03)00255-8.

[8]

G. B. Dantzig, On a linear programming combinatorial approach to the traveling salesman problem, Operations Research, 7 (1959), 58-66. doi: 10.1287/opre.7.1.58.

[9]

J. G. Ecker and J. H. Song, Optimizing a linear function over an efficient set, Journal of Optimization Theory and Applications, 83 (1994), 541-563. doi: 10.1007/BF02207641.

[10]

A. M. Geoffrion, Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications, 22 (1968), 618-630. doi: 10.1016/0022-247X(68)90201-1.

[11]

R. Gupta and R. Malhotra, Multi-criteria integer linear programming problem, Cahiers du CERO, 34 (1992), 51-68.

[12]

M. J. Jorge, An algorithm for optimizing a linear function over an integer efficient set, European Journal of Operational Research, 195 (2009), 98-103. doi: 10.1016/j.ejor.2008.02.005.

[13]

J. N. Karaivanova and S. C. Narula, An interactive procedure for multiple objective integer linear programming problems, European Journal of Operational Research, 68 (1993), 344-351. doi: 10.1016/0377-2217(93)90190-X.

[14]

D. Klein and E. Hannan, An algorithm for multiple objective integer linear programming problem, European Journal of Operational Research, 9 (1982), 378-385. doi: 10.1016/0377-2217(82)90182-5.

[15]

N. C. Nguyen, "An Algorithm for Optimizing a Linear Function Over the Integer Efficient Set," Konrad-Zuse-Zentrum fur Informationstechnik Berlin, 1992.

[16]

J. Philip, Algorithms for the vector maximization problem, Mathematical Programming, 2 (1972), 207-229. doi: 10.1007/BF01584543.

[17]

J. Sylva and A. Crema, A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs, European Journal of Operational Reserach, 180 (2007), 1011-1027. doi: 10.1016/j.ejor.2006.02.049.

[18]

Ta Van Tu, Optimization over the efficient set of a parametric multiple objective linear programming problem, European Journal of Operational Reserach, 122 (2000), 570-583. doi: 10.1016/S0377-2217(99)00095-8.

[19]

J. Teghem and P. Kunsch, A survey of techniques for finding efficient solutions to multiobjective integer linear programming, Asia Pacific Journal of Operations Research, 3 (1986), 95-108.

[20]

D. J. White, The maximization of a function over the efficient set via a penalty function approach, European Journal of Operational Research, 94 (1996), 143-153. doi: 10.1016/0377-2217(95)00184-0.

[21]

Y. Yamamoto, Optimization over the efficient set, overview, Journal of Global Optimization, 22 (2002), 285-317.

[1]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial and Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[2]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial and Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789

[3]

Tran Ngoc Thang, Nguyen Thi Bach Kim. Outcome space algorithm for generalized multiplicative problems and optimization over the efficient set. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1417-1433. doi: 10.3934/jimo.2016.12.1417

[4]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial and Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[5]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial and Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1

[6]

Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial and Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611

[7]

Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089

[8]

He Huang, Zhen He. A global optimization method for multiple response optimization problems. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022016

[9]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751

[10]

Yanqin Bai, Chuanhao Guo. Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. Journal of Industrial and Management Optimization, 2014, 10 (2) : 543-556. doi: 10.3934/jimo.2014.10.543

[11]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134

[12]

Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial and Management Optimization, 2020, 16 (1) : 25-36. doi: 10.3934/jimo.2018138

[13]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[14]

Yuan-mei Xia, Xin-min Yang, Ke-quan Zhao. A combined scalarization method for multi-objective optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2669-2683. doi: 10.3934/jimo.2020088

[15]

Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022001

[16]

Shungen Luo, Xiuping Guo. Multi-objective optimization of multi-microgrid power dispatch under uncertainties using interval optimization. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021208

[17]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[18]

Yue Qi, Zhihao Wang, Su Zhang. On analyzing and detecting multiple optima of portfolio optimization. Journal of Industrial and Management Optimization, 2018, 14 (1) : 309-323. doi: 10.3934/jimo.2017048

[19]

Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial and Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647

[20]

Lipu Zhang, Yinghong Xu, Zhengjing Jin. An efficient algorithm for convex quadratic semi-definite optimization. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 129-144. doi: 10.3934/naco.2012.2.129

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (130)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]