Solving multiobjective mixed integer convex optimization problems

Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. We present a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. We are able to guarantee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision. As far as we know, the procedure we present is the first deterministic algorithm devised to handle this class of problems. Our numerical experiments show results on biobjective and triobjective mixed integer convex instances

A Supply Chain Problem with Facility Location and Bi-objective Transportation Choices

Abstract A supply chain design problem based on a two-echelon single-product system is addressed. The product is distributed from plants to distribution centers and then to customers. There are several transportation channels available for each pair of facilities between echelons. These transportation channels introduce a cost-time tradeoff in the problem that allows us to formulate it as a bi-objective mixed-integer program. The decisions to be taken are the location of the distribution centers, the selection of the transportation channels and the flow between facilities. Three variations of the classic ε-constraint method for generating optimal Pareto fronts are studied in this paper. The procedures are tested over six different classes of instance sets. The three sets of smallest size were solved completely obtaining their efficient solution set. It was observed that one of three proposed algorithms consistently outperformed the other two in terms of their execution time. Additionally, four schemes for obtaining lower bound sets are studied. These schemes are based on linear programming relaxations of the model. The contribution of this work is the introduction of this model for this new bi-objective optimization problem, and a computational study of implementations of both the ε-constrained methods for obtaining optimal efficient fronts, and the lower bounding schemes

Bi-objective branch-and-cut algorithms based on LP-relaxation and bound sets

Most real-world optimization problems are multi-objective by nature, with conflicting and incomparable objectives. Solving a multi-objective optimization problem requires a method which can generate all rational compromises between the objectives. This paper proposes two distinct bound set based branch-and-cut algorithms for general bi-objective combinatorial optimization problems, based on implicit and explicit lower bound sets, respectively. The algorithm based on explicit lower bound sets computes, for each branching node, a lower bound set and compares it to an upper bound set. The other fathoms branching nodes by generating a single point on the lower bound set for each local nadir point. We outline several approaches for fathoming branching nodes and we propose an updating scheme for the lower bound sets that prevents us from solving the bi-objective LP-relaxation of each branching node. To strengthen the lower bound sets, we propose a bi-objective cutting plane algorithm that adjusts the weights of the objective functions such that different parts of the feasible set are strengthened by cutting planes. In addition, we suggest an extension of the branching strategy "Pareto branching". We prove the effectiveness of the algorithms through extensive computational results

Exact And Representative Algorithms For Multi Objective Optimization

In most real-life problems, the decision alternatives are evaluated with multiple conflicting criteria. The entire set of non-dominated solutions for practical problems is impossible to obtain with reasonable computational effort. Decision maker generally needs only a representative set of solutions from the actual Pareto front. First algorithm we present is for efficiently generating a well dispersed non-dominated solution set representative of the Pareto front which can be used for general multi objective optimization problem. The algorithm first partitions the criteria space into grids to generate reference points and then searches for non-dominated solutions in each grid. This grid-based search utilizes achievement scalarization function and guarantees Pareto optimality. The results of our experimental results demonstrate that the proposed method is very competitive with other algorithms in literature when representativeness quality is considered; and advantageous from the computational efficiency point of view. Although generating the whole Pareto front does not seem very practical for many real life cases, sometimes it is required for verification purposes or where DM wants to run his decision making structures on the full set of Pareto solutions. For this purpose we present another novel algorithm. This algorithm attempts to adapt the standard branch and bound approach to the multi objective context by proposing to branch on solution points on objective space. This algorithm is proposed for multi objective integer optimization type of problems. Various properties of branch and bound concept has been investigated and explained within the multi objective optimization context such as fathoming, node selection, heuristics, as well as some multi objective optimization specific concepts like filtering, non-domination probability, running in parallel. Potential of this approach for being used both as a full Pareto generation or an approximation approach has been shown with experimental studies