# American Institute of Mathematical Sciences

January  2021, 17(1): 117-131. doi: 10.3934/jimo.2019102

## Biobjective optimization over the efficient set of multiobjective integer programming problem

 1 USTHB, Department of Operational Research, Bp 32 El Alia, 16111 Algiers, Algeria 2 USTHB, LaROMaD Laboratory, Bp 32 El Alia, 16111 Algiers, Algeria

Received  May 2018 Revised  April 2019 Published  January 2021 Early access  September 2019

In this article, an exact method is proposed to optimize two preference functions over the efficient set of a multiobjective integer linear program (MOILP). This kind of problems arises whenever two associated decision-makers have to optimize their respective preference functions over many efficient solutions. For this purpose, we develop a branch-and-cut algorithm based on linear programming, for finding efficient solutions in terms of both preference functions and MOILP problem, without explicitly enumerating all efficient solutions of MOILP problem. The branch and bound process, strengthened by efficient cuts and tests, allows us to prune a large number of nodes in the tree to avoid many solutions. An illustrative example and an experimental study are reported.

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

show all references

##### References:
Search tree of the example
Optimal simplex table for node 0
 $\mathcal{B}_1$ $x_1$ $x_3$ $x_5$ $RHS$ $x_4$ $5/6$ -$1$ $2/3$ $53/6$ $x_2$ $1/3$ $0$ $2/3$ $16/3$ $x_6$ $1/3$ $5/2$ -$2/3$ 2/3 $\bar{d}^1$ -$7/6$ -$1/2$ -$10/3$ $80/3$
 $\mathcal{B}_1$ $x_1$ $x_3$ $x_5$ $RHS$ $x_4$ $5/6$ -$1$ $2/3$ $53/6$ $x_2$ $1/3$ $0$ $2/3$ $16/3$ $x_6$ $1/3$ $5/2$ -$2/3$ 2/3 $\bar{d}^1$ -$7/6$ -$1/2$ -$10/3$ $80/3$
Optimal simplex table for node 1
 $\mathcal{B}_2$ $x_3$ $x_5$ $x_7$ $RHS$ $x_4$ -$1$ $-1$ $\frac{5}{2}$ $8$ $x_1$ $0$ $2$ -$3$ $1$ $x_6$ $\frac{5}{2}$ $0$ -$1$ $1$ $x_2$ $0$ $0$ $1$ $5$ $\bar{d}^1$ -$\frac{1}{2}$ -$1$ -$\frac{7}{2}$ $\frac{51}{2}$ $\bar{d}^2$ $1$ $0$ $0$ $0$ $\bar{c}^1$ $-1$ $-2$ $5$ $-9$ $\bar{c}^2$ -$\frac{1}{2}$ $0$ -$2$ $10$ $\bar{c}^3$ -$1$ -$2$ $3$ $1$
 $\mathcal{B}_2$ $x_3$ $x_5$ $x_7$ $RHS$ $x_4$ -$1$ $-1$ $\frac{5}{2}$ $8$ $x_1$ $0$ $2$ -$3$ $1$ $x_6$ $\frac{5}{2}$ $0$ -$1$ $1$ $x_2$ $0$ $0$ $1$ $5$ $\bar{d}^1$ -$\frac{1}{2}$ -$1$ -$\frac{7}{2}$ $\frac{51}{2}$ $\bar{d}^2$ $1$ $0$ $0$ $0$ $\bar{c}^1$ $-1$ $-2$ $5$ $-9$ $\bar{c}^2$ -$\frac{1}{2}$ $0$ -$2$ $10$ $\bar{c}^3$ -$1$ -$2$ $3$ $1$
Optimal simplex table for node 3
 $\mathcal{B}_3$ $x_5$ $x_6$ $x_9$ $RHS$ $x_4$ 1 $\frac{5}{2}$ $\frac{21}{4}$ $\frac{21}{5}$ $x_1$ 2 -3 $\frac{15}{2}$ $\frac{11}{2}$ $x_3$ 0 0 -1 1 $x_2$ 0 1 $\frac{5}{2}$ $\frac{7}{2}$ $x_7$ 0 -1 -$\frac{5}{2}$ $\frac{3}{2}$ $x_8$ 0 -1 -$\frac{5}{2}$ $\frac{1}{2}$ $\bar{d}^1$ -1 -$\frac{7}{2}$ -$\frac{37}{4}$ $\frac{79}{4}$
 $\mathcal{B}_3$ $x_5$ $x_6$ $x_9$ $RHS$ $x_4$ 1 $\frac{5}{2}$ $\frac{21}{4}$ $\frac{21}{5}$ $x_1$ 2 -3 $\frac{15}{2}$ $\frac{11}{2}$ $x_3$ 0 0 -1 1 $x_2$ 0 1 $\frac{5}{2}$ $\frac{7}{2}$ $x_7$ 0 -1 -$\frac{5}{2}$ $\frac{3}{2}$ $x_8$ 0 -1 -$\frac{5}{2}$ $\frac{1}{2}$ $\bar{d}^1$ -1 -$\frac{7}{2}$ -$\frac{37}{4}$ $\frac{79}{4}$
Optimal simplex table for node 4
 $\mathcal{B}_4$ $x_6$ $x_9$ $x_{10}$ $RHS$ $x_4$ $1$ $\frac{3}{2}$ -$\frac{1}{2}$ $\frac{11}{2}$ $x_2$ $1$ $\frac{5}{2}$ 0 $\frac{7}{2}$ $x_3$ $0$ -$1$ $0$ $1$ $x_1$ $0$ $0$ $1$ $5$ $x_5$ -$\frac{3}{2}$ -$\frac{15}{4}$ -$\frac{1}{2}$ $\frac{1}{4}$ $x_7$ -$1$ -$\frac{5}{2}$ $0$ $\frac{3}{2}$ $x_8$ -$1$ -$\frac{5}{2}$ 0 $\frac{1}{2}$ $\bar{d}^1$ -$5$ -$13$ -$\frac{1}{2}$ $\frac{39}{2}$
 $\mathcal{B}_4$ $x_6$ $x_9$ $x_{10}$ $RHS$ $x_4$ $1$ $\frac{3}{2}$ -$\frac{1}{2}$ $\frac{11}{2}$ $x_2$ $1$ $\frac{5}{2}$ 0 $\frac{7}{2}$ $x_3$ $0$ -$1$ $0$ $1$ $x_1$ $0$ $0$ $1$ $5$ $x_5$ -$\frac{3}{2}$ -$\frac{15}{4}$ -$\frac{1}{2}$ $\frac{1}{4}$ $x_7$ -$1$ -$\frac{5}{2}$ $0$ $\frac{3}{2}$ $x_8$ -$1$ -$\frac{5}{2}$ 0 $\frac{1}{2}$ $\bar{d}^1$ -$5$ -$13$ -$\frac{1}{2}$ $\frac{39}{2}$
Optimal simplex table for node 5
 $\mathcal{B}_5$ $x_5$ $x_9$ $x_{10}$ $RHS$ $x_4$ $\frac{2}{3}$ -1 $\frac{5}{6}$ $\frac{29}{6}$ $x_1$ 0 0 -1 6 $x_3$ 0 -1 0 1 $x_2$ $\frac{2}{3}$ 0 $\frac{1}{3}$ $\frac{10}{3}$ $x_7$ $-\frac{2}{3}$ 0 $-\frac{1}{3}$ $\frac{5}{3}$ $x_8$ $-\frac{2}{3}$ 0 $-\frac{1}{3}$ $\frac{2}{3}$ $x_6$ $-\frac{2}{3}$ $\frac{5}{2}$ $-\frac{1}{3}$ $\frac{1}{6}$ $\bar{d}^1$ $-\frac{10}{3}$ $-\frac{1}{2}$ $\frac{7}{6}$ $\frac{115}{6}$
 $\mathcal{B}_5$ $x_5$ $x_9$ $x_{10}$ $RHS$ $x_4$ $\frac{2}{3}$ -1 $\frac{5}{6}$ $\frac{29}{6}$ $x_1$ 0 0 -1 6 $x_3$ 0 -1 0 1 $x_2$ $\frac{2}{3}$ 0 $\frac{1}{3}$ $\frac{10}{3}$ $x_7$ $-\frac{2}{3}$ 0 $-\frac{1}{3}$ $\frac{5}{3}$ $x_8$ $-\frac{2}{3}$ 0 $-\frac{1}{3}$ $\frac{2}{3}$ $x_6$ $-\frac{2}{3}$ $\frac{5}{2}$ $-\frac{1}{3}$ $\frac{1}{6}$ $\bar{d}^1$ $-\frac{10}{3}$ $-\frac{1}{2}$ $\frac{7}{6}$ $\frac{115}{6}$
Optimal simplex table for node 6
 $\mathcal{B}_6$ $x_9$ $x_{10}$ $x_{11}$ RHS $x_4$ -1 $-\frac{1}{2}$ 1 5 $x_1$ 0 1 0 5 $x_3$ -1 0 0 1 $x_2$ 0 0 0 3 $x_5$ 0 $-\frac{1}{2}$ $-\frac{3}{2}$ 1 $x_8$ 0 0 -1 1 $x_{6}$ $\frac{5}{2}$ 0 -1 $\frac{1}{2}$ $x_7$ 0 0 -1 2 $\bar{d}^1$ $-\frac{1}{2}$ $-\frac{1}{2}$ -5 17 $\bar{d}^2$ 1 0 0 1 $\bar{c}^1$ -1 -1 2 -2 $\bar{c}^2$ $-\frac{1}{2}$ 0 -2 $\frac{11}{2}$ $\bar{c}^3$ -1 -1 0 4
 $\mathcal{B}_6$ $x_9$ $x_{10}$ $x_{11}$ RHS $x_4$ -1 $-\frac{1}{2}$ 1 5 $x_1$ 0 1 0 5 $x_3$ -1 0 0 1 $x_2$ 0 0 0 3 $x_5$ 0 $-\frac{1}{2}$ $-\frac{3}{2}$ 1 $x_8$ 0 0 -1 1 $x_{6}$ $\frac{5}{2}$ 0 -1 $\frac{1}{2}$ $x_7$ 0 0 -1 2 $\bar{d}^1$ $-\frac{1}{2}$ $-\frac{1}{2}$ -5 17 $\bar{d}^2$ 1 0 0 1 $\bar{c}^1$ -1 -1 2 -2 $\bar{c}^2$ $-\frac{1}{2}$ 0 -2 $\frac{11}{2}$ $\bar{c}^3$ -1 -1 0 4
Optimal simplex table for node 8
 $\mathcal{B}_8$ $x_5$ $x_9$ $x_{11}$ RHS $x_4$ -1 -1 $\frac{5}{2}$ 4 $x_2$ 0 0 1 3 $x_3$ 0 -1 0 1 $x_1$ 2 0 -3 7 $x_6$ 0 $\frac{5}{2}$ -1 $\frac{1}{2}$ $x_7$ 0 0 -1 2 $x_8$ 0 0 -1 1 $x_{10}$ 2 0 -3 1 $\bar{d}^1$ -1 $-\frac{1}{2}$ $-\frac{7}{2}$ 18 $\bar{d}^2$ 0 1 0 1 $\bar{c}^1$ -2 -1 5 0 $\bar{c}^2$ 0 $-\frac{1}{2}$ -2 $\frac{11}{2}$ $\bar{c}^3$ -2 -1 3 6
 $\mathcal{B}_8$ $x_5$ $x_9$ $x_{11}$ RHS $x_4$ -1 -1 $\frac{5}{2}$ 4 $x_2$ 0 0 1 3 $x_3$ 0 -1 0 1 $x_1$ 2 0 -3 7 $x_6$ 0 $\frac{5}{2}$ -1 $\frac{1}{2}$ $x_7$ 0 0 -1 2 $x_8$ 0 0 -1 1 $x_{10}$ 2 0 -3 1 $\bar{d}^1$ -1 $-\frac{1}{2}$ $-\frac{7}{2}$ 18 $\bar{d}^2$ 0 1 0 1 $\bar{c}^1$ -2 -1 5 0 $\bar{c}^2$ 0 $-\frac{1}{2}$ -2 $\frac{11}{2}$ $\bar{c}^3$ -2 -1 3 6
Optimal simplex table for the node 10
 $\mathcal{B}_{10}$ $x_6$ $x_{10}$ $x_{13}$ RHS $x_5$ $-\frac{3}{2}$ $-\frac{1}{2}$ $-\frac{15}{4}$ 4 $x_4$ 1 $-\frac{1}{2}$ $\frac{3}{2}$ 4 $x_3$ 0 0 -1 2 $x_7$ -1 0 $-\frac{5}{2}$ 4 $x_8$ -1 1 $-\frac{5}{2}$ 3 $x_9$ 0 0 -1 1 $x_2$ 1 0 $\frac{5}{2}$ 1 $x_1$ 0 1 0 5 $\bar{d}^1$ -5 $-\frac{1}{2}$ -13 $\frac{13}{2}$ $\bar{d}^2$ 0 0 1 2 $\bar{c}^1$ 2 -1 4 1 $\bar{c}^2$ -2 0 $-\frac{11}{2}$ 1 $\bar{c}^3$ 0 -1 -1 3
 $\mathcal{B}_{10}$ $x_6$ $x_{10}$ $x_{13}$ RHS $x_5$ $-\frac{3}{2}$ $-\frac{1}{2}$ $-\frac{15}{4}$ 4 $x_4$ 1 $-\frac{1}{2}$ $\frac{3}{2}$ 4 $x_3$ 0 0 -1 2 $x_7$ -1 0 $-\frac{5}{2}$ 4 $x_8$ -1 1 $-\frac{5}{2}$ 3 $x_9$ 0 0 -1 1 $x_2$ 1 0 $\frac{5}{2}$ 1 $x_1$ 0 1 0 5 $\bar{d}^1$ -5 $-\frac{1}{2}$ -13 $\frac{13}{2}$ $\bar{d}^2$ 0 0 1 2 $\bar{c}^1$ 2 -1 4 1 $\bar{c}^2$ -2 0 $-\frac{11}{2}$ 1 $\bar{c}^3$ 0 -1 -1 3
Optimal simplex table for node 11
 $\mathcal{B}_{11}$ $x_4$ $x_6$ $x_{13}$ RHS $x_5$ -1 $-\frac{5}{2}$ $-\frac{21}{4}$ 0 $x_3$ 0 0 -1 2 $x_7$ 0 -1 $-\frac{5}{2}$ 4 $x_8$ 0 -1 $-\frac{5}{2}$ 3 $x_{10}$ 2 2 3 7 $x_{11}$ 0 -1 $-\frac{5}{2}$ 2 $x_{12}$ 0 -1 $-\frac{5}{2}$ 1 $x_9$ 0 0 -1 1 $x_2$ 1 0 $\frac{5}{2}$ 1 $x_1$ 2 2 3 13 $\bar{d}^1$ -1 -6 $-\frac{29}{2}$ $\frac{21}{2}$ $\bar{d}^2$ 0 0 3 6 $\bar{c}^1$ -2 0 1 9 $\bar{c}^2$ 0 -2 $-\frac{11}{2}$ 1 $\bar{c}^3$ -2 -2 -4 11
 $\mathcal{B}_{11}$ $x_4$ $x_6$ $x_{13}$ RHS $x_5$ -1 $-\frac{5}{2}$ $-\frac{21}{4}$ 0 $x_3$ 0 0 -1 2 $x_7$ 0 -1 $-\frac{5}{2}$ 4 $x_8$ 0 -1 $-\frac{5}{2}$ 3 $x_{10}$ 2 2 3 7 $x_{11}$ 0 -1 $-\frac{5}{2}$ 2 $x_{12}$ 0 -1 $-\frac{5}{2}$ 1 $x_9$ 0 0 -1 1 $x_2$ 1 0 $\frac{5}{2}$ 1 $x_1$ 2 2 3 13 $\bar{d}^1$ -1 -6 $-\frac{29}{2}$ $\frac{21}{2}$ $\bar{d}^2$ 0 0 3 6 $\bar{c}^1$ -2 0 1 9 $\bar{c}^2$ 0 -2 $-\frac{11}{2}$ 1 $\bar{c}^3$ -2 -2 -4 11
Random instances execution times
 [1] Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557 [2] Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170 [3] Najeeb Abdulaleem. $V$-$E$-invexity in $E$-differentiable multiobjective programming. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021014 [4] Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697 [5] Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353 [6] Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385 [7] Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial & Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497 [8] Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1873-1884. doi: 10.3934/jimo.2019033 [9] Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027 [10] Zhenbo Wang, Shu-Cherng Fang, David Y. Gao, Wenxun Xing. Global extremal conditions for multi-integer quadratic programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 213-225. doi: 10.3934/jimo.2008.4.213 [11] Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67 [12] Mohamed A. Tawhid, Ahmed F. Ali. A simplex grey wolf optimizer for solving integer programming and minimax problems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 301-323. doi: 10.3934/naco.2017020 [13] Matthew H. Henry, Yacov Y. Haimes. Robust multiobjective dynamic programming: Minimax envelopes for efficient decisionmaking under scenario uncertainty. Journal of Industrial & Management Optimization, 2009, 5 (4) : 791-824. doi: 10.3934/jimo.2009.5.791 [14] Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102 [15] Zutong Wang, Jiansheng Guo, Mingfa Zheng, Youshe Yang. A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle. Journal of Industrial & Management Optimization, 2015, 11 (1) : 13-26. doi: 10.3934/jimo.2015.11.13 [16] Mahmoud Ameri, Armin Jarrahi. An executive model for network-level pavement maintenance and rehabilitation planning based on linear integer programming. Journal of Industrial & Management Optimization, 2020, 16 (2) : 795-811. doi: 10.3934/jimo.2018179 [17] René Henrion, Christian Küchler, Werner Römisch. Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 363-384. doi: 10.3934/jimo.2008.4.363 [18] Louis Caccetta, Syarifah Z. Nordin. Mixed integer programming model for scheduling in unrelated parallel processor system with priority consideration. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 115-132. doi: 10.3934/naco.2014.4.115 [19] Elham Mardaneh, Ryan Loxton, Qun Lin, Phil Schmidli. A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1601-1623. doi: 10.3934/jimo.2017009 [20] Edward S. Canepa, Alexandre M. Bayen, Christian G. Claudel. Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming. Networks & Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783

2020 Impact Factor: 1.801

## Tools

Article outline

Figures and Tables