A decomposition method for approximating Pareto frontier in Multiobjective Integer Linear Problems

where x is an n-dimensional vector of variables, A is an m × n matrix, b is the RHS vector and the vectors ci (i = 1, ...,m) represent the coefficients of the objective functions (criteria). Let’s denote yi = fi(x), i = 1, ...,m, and let y = (y1, ..., ym) be a vector in the criteria space. The set Y ⊂ Rm composed by all possible criterion vectors y = f(x) when x ∈ X, is known as Feasible Criteria Set (FCS). A point y′ ∈ Y is non-dominated or Pareto optimal (efficient) if and only if there is no y′′ ∈ Y, y′′ ̸= y′, such that y′′ j ≥ y′ j , j = 1, ...,m. The set P(Y ) of all non-dominated points y ∈ Y is called Pareto frontier. A dominance cone for a point v ∈ Rm is defined as D(v) = {z ∈ Rm : z = v −w, w ∈ R+}, where R+ is the non-negative cone in Rm. The information about the Pareto frontier can be very useful for Decision Makers (DM) in Decision Support Systems (DSS) based on multiobjective models. Some examples of such DSSs are given in [4]. When we are only interested in Pareto frontier it is worth considering the Edgeworth-Pareto Hull (EPH) of Y defined as EPH(Y ) = {z ∈ Rm : z = y − w, y ∈ Y, w ∈ R+}. The set EPH(Y ) has the same Pareto frontier that Y but its structure is simpler and therefore EPH(Y ) is easier for constructing and visualizing. Some effective algorithms for constructing and approximating EPH of convex sets are described in chapter 6 in [4]. These algorithms can be applied also for constructing and approximating EPHs of non-convex sets. It is important to remark that it is possible to know the vertices and hyperplanes corresponding to each facet of EPH(Y ) since these algorithms describe the EPH of a set, simultaneously, as a list of vertices and as a system of linear inequalities. When multiobjective programming problems have integer variables, even if the restrictions are linear, the set Y can be non-convex. Unsupported efficient solutions of MOILP, i.e. solutions that do not belong to the frontier of the convex hull of the feasible region, cannot be obtained by optimizing scalar surrogate functions consisting in weightedsums of the objective functions (Weighted Objective Function Method). The other traditional techniques, such as methods based on the Tchebycheff Metric, epsilon-Constraining Method, and some others, have difficulties in the case of MOILP with more than three objective functions [3]. The Convex Edgeworth-Pareto Hull (CEPH) of