Multi-objective Branch Research Articles

On Improvements of Multi-Objective Branch and Bound

On Improvements of Multi-Objective Branch and Bound

Augmenting bi-objective branch and bound by scalarization-based information

While branch and bound based algorithms are a standard approach to solve single-objective (mixed-)integer optimization problems, multi-objective branch and bound methods are only rarely applied compared to the predominant objective space methods. In this paper we propose modifications to increase the performance of multi-objective branch and bound algorithms by utilizing scalarization-based information. We use the hypervolume indicator as a measure for the gap between lower and upper bound set to implement a multi-objective best-first strategy. By adaptively solving scalarizations in the root node to integer optimality we improve both, upper and lower bound set. The obtained lower bound can then be integrated into the lower bounds of all active nodes, while the determined solution is added to the upper bound set. Numerical experiments show that the number of investigated nodes can be significantly reduced by up to 83% and the total computation time can be reduced by up to 80%.

Multi-objective Branch and Bound based on Decomposition

Multi-objective Branch and Bound based on Decomposition

Pareto-based branch and bound algorithm for multiobjective optimization of a safety transformer

PurposeThis paper aims to propose a multiobjective branch and bound (MOBB) algorithm with a new criteria for the branching and discarding of nodes based on Pareto dominance and contribution metric.Design/methodology/approachA multiobjective branch and bound (MOBB) method is presented and applied to the bi-objective combinatorial optimization of a safety transformer. A comparison with exhaustive enumeration and non-dominated sorting genetic algorithm (NSGA2) confirms the solutions.FindingsIt appears that MOBB and NSGA2 are both sensitive to their control parameters. The parameters for the MOBB algorithm are the number of starting points and the number of solutions on the relaxed Pareto front. The parameters of NSGA2 are the population size and the number of generations.Originality/valueThe comparison with exhaustive enumeration confirms that the proposed algorithm is able to find the complete set of non-dominated solutions in about 235 times fewer evaluations. As this last method is exact, its confidence level is higher.

Multi-objective branch and bound

Branch and bound is a well-known generic method for computing an optimal solution of a single-objective optimization problem. Based on the idea “divide to conquer”, it consists in an implicit enumeration principle viewed as a tree search. Although the branch and bound was first suggested by Land and Doig (1960), the first complete algorithm introduced as a multi-objective branch and bound that we identified was proposed by Kiziltan and Yucaoglu (1983). Rather few multi-objective branch and bound algorithms have been proposed. This situation is not surprising as the contributions on the extensions of the components of branch and bound for multi-objective optimization are recent. For example, the concept of bound sets, which extends the classic notion of bounds, has been mentioned by Villarreal and Karwan (1981). But it was only developed for the first time in 2001 by Ehrgott and Gandibleux, and fully defined in 2007.This paper describes a state-of-the-art of multi-objective branch and bound, which reviews concepts, components and published algorithms. It mainly focuses on the contributions belonging to the class of optimization problems who has received the most of attention in this context from 1983 until 2015: the linear optimization problems with zero-one variables and mixed 0–1/continuous variables. Only papers aiming to compute a complete set of efficient solutions are discussed.

Multi-objective parallel machine scheduling problem with job deterioration and learning effect under fuzzy environment

The current paper investigates a non-identical parallel machine multi-objective scheduling problem in which both the deterioration and learning effects have been considered. Due to uncertainty of the parameters in real-world systems, processing times and due dates of jobs are represented here with triangular fuzzy numbers. Using the credibility measure, a nonlinear mathematical model is provided based on fuzzy chance-constrained programming (FCCP) with the aim to minimize two objective functions, namely total earliness/tardiness (ET) and maximum completion time of jobs (makespan). Since it is a mixed integer nonlinear mathematical model, there is no guarantee that the solution will obtain a global optimum. Therefore, a multi-objective branch and bound algorithm is provided by introducing an effective lower bound in order to obtain a Pareto optimal front. Computational results show that the algorithm proposed is especially useful to solve large-scale problems.

Self-Sensing Composites and Optimization of Composite Structures in Japan

I review research on self-sensing and structural optimizations of laminated carbon/epoxy composites in Japan. Self-sensing is one of the multiple functions of composites; i.e., carbon fiber is used as a sensor as well as reinforcement. I present a controversial issue in self-sensing and detail research results. Structural optimization of laminated CFRP composites is indispensable in reducing the weights of modern aerospace structural components. I present a modified efficient global search method using the multi-objective genetic algorithm and fractal branch and bound method. My group has focused its research on these subjects and our research results are presented here.

A Hybrid Method for Solving the Multi-objective Assignment Problem

Most of the well known methods for solving multi-objective combinatorial optimization problems deal with only two objectives. In this paper, we develop a metaheuristic method for solving multi-objective assignment problems with three or more objectives. This method is based on the dominance cost variant of the multi-objective simulated annealing (DCMOSA) and hybridizes neighborhood search techniques which consist of either a local search or a multi-objective branch and bound search (here the multi-objective branch and bound search is used as a local move to a fragment of a solution).

Bucket elimination for multiobjective optimization problems

Multiobjective optimization deals with problems involving multiple measures of performance that should be optimized simultaneously. In this paper we extend bucket elimination (BE), a well known dynamic programming generic algorithm, from mono-objective to multiobjective optimization. We show that the resulting algorithm, MO-BE, can be applied to true multi-objective problems as well as mono-objective problems with knapsack (or related) global constraints. We also extend mini-bucket elimination (MBE), the approximation form of BE, to multiobjective optimization. The new algorithm MO-MBE can be used to obtain good quality multi-objective lower bounds or it can be integrated into multi-objective branch and bound in order to increase its pruning efficiency. Its accuracy is empirically evaluated in real scheduling problems, as well as in Max-SAT-ONE and biobjective weighted minimum vertex cover problems.

The multiscenario lot size problem with concave costs

The dynamic single-facility single-item lot size problem is addressed. The finite planning horizon is divided into several time periods. Although the total demand is assumed to be a fixed value, the distribution of this demand among the different periods is unknown. Therefore, for each period the demand can be chosen from a discrete set of values. For this reason, all the combinations of the demand vector yield a set of different scenarios. Moreover, we assume that the production/reorder and holding cost vectors can vary from one scenario to another. For each scenario, we consider as the objective function the sum of the production/reorder and the holding costs. The problem consists of determining all the Pareto-optimal or non-dominated production plans with respect to all scenarios. We propose a solution method based on a multiobjective branch and bound approach. Depending on whether shortages are considered or not, different upper bound sets are provided. Computational results on several randomly generated problems are reported.

A multiobjective optimization approach to thermal generating units maintenance scheduling

This paper presents a new approach to preventive maintenance scheduling of thermal generating units in electric power systems. The problem has been formulated as a combinatorial optimization task, with explicit and simultaneous treatment of multiple objectives: minimization of fuel costs, maximization of reliability, and minimization of constraints violations. The optimization is performed by the multiobjective branch and bound algorithm with successive approximations, embedded in an outer loop where in each iteration a new initial solution is generated in a pseudorandom manner. This procedure was the basis for development of the multiobjective maintenance scheduling software package. A realistic example of annual maintenance scheduling of 21 thermal generating units demonstrates the effectiveness of the proposed methodology.

Multiobjective programming in power system optimization: new approach to generator maintenance scheduling

This paper discusses the need for and the applicability of multiobjective programming methodology in solving various power system problems. The advantages of a multiobjective optimization approach over the conventional single-objective approach are demonstrated in the case of thermal generating units maintenance scheduling. The proposed optimization model is based on the original multiobjective branch and bound algorithm. The following objectives: power system reliability maximization, fuel costs minimization, and minimization of constraints violations are simultaneously considered. A realistic example of annual maintenance scheduling of 21 thermal generating units illustrates the proposed methodology.

A multiobjective branch and bound method for network‐structured water resources planning problems

A multiobjective branch and bound algorithm is proposed for use in analyzing multiobjective fixed charge network flow problems which are found commonly in water resources planning situations. Also proposed is a multiobjective imputed value analysis which makes use of the branch and bound tree structure and allows the comparison of the importance of facilities in the network as represented by individual arcs or sets of arcs. The mathematical formulation and the analysis procedure of the method are described, and the potential usefulness of the method is demonstrated using a hypothetical example problem dealing with a regional wastewater treatment and residual management system. A Fortran program for implementing the algorithm is available from the first author.