Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 90C29, 90C10; Secondary: 90C57.

 Citation:

• Figure 1.  Search tree of the example

Table 1.  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 $\bar{d}^1$ -$7/6$ -$1/2$ -$10/3$ $80/3$

Table 2.  Optimal simplex table for node 1

 $\mathcal{B}_2$ $x_3$ $x_5$ $x_7$ $RHS$ $x_4$ -$1$ $-1$ $\frac{5}{2}$ $8$ $x_1$ $0$ $2$ -$3$ $1$ $x_6$ $\frac{5}{2}$ $0$ -$1$ $1$ $x_2$ $0$ $0$ $1$ $5$ $\bar{d}^1$ -$\frac{1}{2}$ -$1$ -$\frac{7}{2}$ $\frac{51}{2}$ $\bar{d}^2$ $1$ $0$ $0$ $0$ $\bar{c}^1$ $-1$ $-2$ $5$ $-9$ $\bar{c}^2$ -$\frac{1}{2}$ $0$ -$2$ $10$ $\bar{c}^3$ -$1$ -$2$ $3$ $1$

Table 3.  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 -$\frac{7}{2}$ -$\frac{37}{4}$ $\frac{79}{4}$

Table 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 $\frac{1}{2}$ $\bar{d}^1$ -$5$ -$13$ -$\frac{1}{2}$ $\frac{39}{2}$

Table 5.  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 $-\frac{1}{3}$ $\frac{2}{3}$ $x_6$ $-\frac{2}{3}$ $\frac{5}{2}$ $-\frac{1}{3}$ $\frac{1}{6}$ $\bar{d}^1$ $-\frac{10}{3}$ $-\frac{1}{2}$ $\frac{7}{6}$ $\frac{115}{6}$

Table 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

Table 7.  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

Table 8.  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

Table 9.  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

Table 10.  Random instances execution times

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