GoNDEF: an exact method to generate all non-dominated points of multi-objective mixed-integer linear programs

Abstract

Most real-world problems involve multiple conflicting criteria. These problems are called multi-criteria/multi-objective optimization problems (MOOP). The main task in solving MOOPs is to find the non-dominated (ND) points in the objective space or efficient solutions in the decision space. A ND point is a point in the objective space with objective function values that cannot be improved without worsening another objective function. In this paper, we present a new method that generates the set of ND points for a multi-objective mixed-integer linear program (MOMILP). The Generator of ND and Efficient Frontier (GoNDEF) for MOMILPs finds that the ND points represented as points, line segments, and facets consist of every type of ND point. First, the GoNDEF sets integer variables to the values that result in ND points. Fixing integer variables to specific values results in a multi-objective linear program (MOLP). This MOLP has its own set of ND points. A subset of this set establishes a subset of the ND points set of the MOMILP. In this paper, we present an extensive theoretical analysis of the GoNDEF and illustrate its effectiveness on a set of instance problems.

Appendices

Appendix 1: The formulation of our illustrative instance given in Fig. 1

We provide a mathematical formulation for Fig. 1. Note that there may be other formulations to provide a feasible region corresponding to Fig. 1. Let M be a sufficiently large positive number, $x=(x_1,x_2,x_3) \ge 0$, and $y=(y_1,y_2,y_3) \in \{0, 1\}^3$.

$$\begin{aligned} & \max \quad z(x,y)=(x_1,x_2,x_3) \\ & \text{s.t.} \end{aligned}$$

$$\begin{aligned} & x_1 \le 6+M(1-y_1),\\& x_2 \le 6+M(1-y_1),\\& 7-M(1-y_1) \le x_1+x_2 \le 9+M(1-y_1),\\& 4-M(1-y_1) \le x_3 \le 10+M(1-y_1),\end{aligned}$$

$$\begin{aligned} & -6-M(1-y_2) \le x_1 - x_2 \le 8+M(1-y_2),\\& 8-M(1-y_2) \le x_1 + x_2,\\& 3-M(1-y_2) \le x_3,\\& 3x_1+3x_2+2x_3 \le 42+M(1-y_2), \end{aligned}$$

$$\begin{aligned} & 4-M(1-y_3) \le x_1 \le 8+M(1-y_3),\\& x_2 \le 2+M(1-y_3),\\& x_3 \le 5+M(1-y_3),\\& y_1+y_2+y_3 = 1,\\ & x_i \ge 0, \; \forall i=1,2,3, \end{aligned}$$

$$\begin{aligned} & y_i \in \{0,1\}, \; \forall i=1,2,3.\end{aligned}$$

Appendix 2: Generating instance problems

In the following mathematical formulation we have k objective functions, m constraints, q binary variables, and n continuous positive variables. The size of instance is displayed as $k \times m \times (n+q)$. $U_j$ is an integer value that shows the upper bound of variable $y_j$ for $j=1,\ldots ,q$.

$$\begin{aligned} max \; \; z_t(x,y)= & \sum _{i=1}^{n} c_i^t x_i + \sum _{j=1}^{q} f_j^t y_j, \; \forall t=1, \ldots , k \\ {\text{s.t.}}&\\ &\sum _{i=1}^{n} a_{ij} x_i + a_j^{\prime } y_j \le b_j, \; \forall j=1, \ldots ,q, \\ & \sum _{i=1}^{n} a_{ij} x_i \le b_j, \; \forall j=q + 1, \ldots , m-1,\\ & \sum _{j=1}^{q} y_j \le \frac{q}{3},\\ & x_i \in {\mathcal{R}}^{+}, \; \forall \; i=1, \ldots , n,\\ & y_j \in \{0,1, \ldots , U_j\}, \; \forall \; j=1, \ldots , q,\\ \end{aligned}$$

where, in the described benchmarks, the objective function coefficients of the continuous variables, binary variables, the right hand sides of the constraints, and the matrix of coefficients (for both continuous and binary variables) are drawn from uniformly distributed random numbers in the ranges [$-10, 10$], [$-200, 200$], [50, 100], and [$-1, 20$], respectively. In addition, the sparsity of coefficient matrix is 40 percent.