Multiobjective Integer and Mixed-Integer Linear Programming

The introduction of discrete variables into multiobjective programming problems leads to all-integer or mixed-integer problems that are more difficult to tackle, even if they have linear objective functions and constraints. The feasible set is no longer convex, and the additional difficulties go beyond those of changing from single objective linear programming to integer programming. Thus, in many cases the problems cannot be handled by adaptations of MOLP methods to deal with integer variables. In addition, there are approaches specifically designed for multiobjective pure integer problems that do not apply to the multiobjective mixed-integer case. Therefore, even for the linear case, techniques for dealing with multiobjective integer/mixed-integer linear programming (MOILP/MOMILP) involve more than the combination of MOLP approaches with integer programming techniques. In this chapter we focus on MOILP/MOMILP problems formulated as (6.1): $$ \begin{array}{l}\left.\begin{array}{c}\hfill \max \kern0.5em {z}_1={f}_1\left(\mathbf{x}\right)={\mathbf{c}}_1\mathbf{x}\hfill \\ {}\hfill \dots \hfill \\ {}\hfill \kern0.5em \max \kern0.5em {z}_p={f}_p\left(\mathbf{x}\right)={\mathbf{c}}_p\mathbf{x}\hfill \end{array}\right\}\kern1em "\mathrm{Max}"\kern0.5em z=\mathbf{f}\left(\mathbf{x}\right)=\mathbf{C}\mathbf{x}\\ {}\mathrm{s}.\mathrm{t}\kern0.75em \mathbf{x}\in X = \left\{\mathbf{x}\in {\mathbb{R}}^n:\mathbf{Ax} = \mathbf{b},\mathbf{x}\ge \mathbf{0},{x}_j\in\ {\mathbb{N}}_0,j\in I\right\}\end{array} $$ where I is the set of indices of the integer variables, I ⊆ {1,…,n}, I ≠ ∅. It is assumed that X is bounded and non-empty. Let Z denote the feasible region in the objective space, that is, Z = f(X). If all decision variables are integer then the multiobjective problem is all-integer (MOILP), which is a special case of the multiobjective mixed-integer case. In what follows we will refer to MOMILP as the general case, in which integrality constraints are imposed on all or a subset of the decision variables. For basic concepts concerning this type of problems, namely the characterization of efficient/nondominated solutions and the distinction between supported and unsupported nondominated solutions, please refer to Chap. 2 .