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

A comparative study of two key algorithms in multiple objective linear programming

Multiple objective linear programming problems are solved with a variety of algorithms. While these algorithms vary in philosophy and outlook, most of them fall into two broad categories: those that are decision space-based and those that are objective space-based. This paper reports the outcome of a computational investigation of two key representative algorithms, one of each category, namely the parametric simplex algorithm which is a prominent representative of the former and the primal variant of Bensons Outer-approximation algorithm which is a prominent representative of the latter. The paper includes a procedure to compute the most preferred nondominated point which is an important feature in the implementation of these algorithms and their comparison. Computational and comparative results on problem instances ranging from small to medium and large are provided

04461 Abstracts Collection -- Practical Approaches to Multi-Objective Optimization

From 07.11.04 to 12.11.04, the Dagstuhl Seminar 04461 ``Practical Approaches to Multi-Objective Optimization\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Advances and applications in high-dimensional heuristic optimization

“Applicable to most real-world decision scenarios, multiobjective optimization is an area of multicriteria decision-making that seeks to simultaneously optimize two or more conflicting objectives. In contrast to single-objective scenarios, nontrivial multiobjective optimization problems are characterized by a set of Pareto optimal solutions wherein no solution unanimously optimizes all objectives. Evolutionary algorithms have emerged as a standard approach to determine a set of these Pareto optimal solutions, from which a decision-maker can select a vetted alternative. While easy to implement and having demonstrated great efficacy, these evolutionary approaches have been criticized for their runtime complexity when dealing with many alternatives or a high number of objectives, effectively limiting the range of scenarios to which they may be applied. This research introduces mechanisms to improve the runtime complexity of many multiobjective evolutionary algorithms, achieving state-of-the-art performance, as compared to many prominent methods from the literature. Further, the investigations here presented demonstrate the capability of multiobjective evolutionary algorithms in a complex, large-scale optimization scenario. Showcasing the approach’s ability to intelligently generate well-performing solutions to a meaningful optimization problem. These investigations advance the concept of multiobjective evolutionary algorithms by addressing a key limitation and demonstrating their efficacy in a challenging real-world scenario. Through enhanced computational efficiency and exhibited specialized application, the utility of this powerful heuristic strategy is made more robust and evident”--Abstract, page iv

Quality Representation in Multiobjective Programming

In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective programming problems. This manuscript presents two methods for generating discrete representations with equidistant points for multiobjective programs with solution sets determined by convex cones. The Bilevel Controlled Spacing (BCS) method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing. The Constraint Controlled Spacing (CCS) method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. Both methods (under certain assumptions) are proven to produce (weakly) nondominated points. Along the way, several interesting results about obtuse, simplicial cones are also proved. Both the BCS and CCS methods are tested and show promise on a variety of problems: linear, convex, nonconvex (CCS only), two-dimensional, and three-dimensional. Sample Matlab code for two of these examples can be found in the appendices as well as tables containing the generated solution points. The manuscript closes with conclusions and ideas for further research in this field

Heuristic algorithms for solving a class of multiobjective zero-one programming problems

Master'sMASTER OF ENGINEERIN

Bi-objective optimization of the tactical allocation of job types to machines: mathematical modeling, theoretical analysis, and numerical tests

We introduce a tactical resource allocation model for a large aerospace engine system manufacturer aimed at long-term production planning. Our model identifies the routings a product takes through the factory, and which machines should be qualified for a balanced resource loading, to reduce product lead times. We prove some important mathematical properties of the model that are used to develop a heuristic providing a good initial feasible solution. We propose a tailored approach for our class of problems combining two well-known criterion space search algorithms, the bi-directional ε-constraint method and the augmented weighted Tchebycheff method. A computational investigation comparing solution times for several solution methods is presented for 60 numerical\ua0instances

Multi-objective Analysis of Approaches to Dynamic Routing of a Vehicle

We consider a routing problem for a single vehicle serving customer Locations in the course of time. A subset of these customers must necessarily be served, while the complement of this subset contains dynamic customers which request for service over time, and which do not necessarily need to be served. The decision maker’s conflicting goals are serving as many customers as possible as well as minimizing total travel distance. We solve this bi-objective Problem with an evolutionary multi-objective algorithm in order to provide an a-posteriori evaluation tool for enabling decision makers to assess the single objective solution strategies that they actually use in real-time. We present the modifications to be applied to the evolutionary multi-objective algorithm NSGA2 in order to solve the routing problem, we describe a number of real-time single-objective solution strategies, and we finally use the gained efficient trade-off solutions of NSGA2 to exemplarily evaluate the real-time strategies. Our results show that the evolutionary multi-objective approach is well-suited to generate benchmarks for assessing dynamic heuristic strategies. Our findings point into future directions for designing dynamic multi-objective approaches for the vehicle routing problem with time windows