Multi-objective Knapsack Research Articles

Supported nondominated points as a representation of the nondominated set: An empirical analysis

AbstractThe nondominated set of a multiple objective discrete optimization problem is known to contain unsupported nondominated points, which outnumber the supported ones and are more difficult to obtain. We treat supported nondominated points as a representation and analyse their quality using different metrics beyond their sheer numbers. Under different data generation schemes on multiobjective knapsack and assignment problems, we observe that supported nondominated points almost always provide a good representation of the entire nondominated set.

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems. The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of items

Identifying preferred solutions in multiobjective combinatorial optimizationproblems

We develop an evolutionary algorithm for multiobjective combinatorial optimization problems. The algorithm aims at converging the preferred solutions of a decision-maker. We test the performance of the algorithm on the multiobjective knapsack and multiobjective spanning tree problems. We generate the true nondominated solutions using an exact algorithm and compare the results with those of the evolutionary algorithm. We observe that the evolutionary algorithm works well in approximating the solutions in the preferred regions.

Integer cat swarm optimization algorithm for multiobjective integer problems

In the literature, several variants of cat swarm optimization (CSO) algorithm are reported. However, CSO for integer multiobjective optimization problems (MOPs) has not yet been investigated. Owing to the frequent occurrence of integer MOPs and their importance in practical design problems, in this work, we investigate a new CSO approach for solving purely integer MOPs. This new approach named as multiobjective integer cat swarm optimization (MO-ICSO) algorithm incorporates the modified version of the CSO algorithm for MOPs. This approach is comprised of the concepts of rounding the floating points to the nearest integer numbers and the probabilistic updating (PU) technique. It uses the idea of Pareto dominance for finding the non-dominated solutions and an external archive for storing these solutions. We demonstrate the power of this new approach via its quantitative analysis and sensitivity test of its several parameters using different performance metrics performed over multiobjective multidimensional knapsack problem and several standard test functions. The simulation results argue that the proposed MO-ICSO approach can be a better candidate for solving the integer MOPs.

A pruned Pareto set for multi-objective optimisation problems via particle swarm and simulated annealing

A Pareto optimal set, which is obtained from solving multi-objective optimisation problems, usually contain a large number of optimal solutions. This situation poses a challenge for decision makers in choosing a suitable solution from a large number of overlapping and complex Pareto solutions. This paper proposes a new procedure for solving multi-objective optimisation problems by reducing the size of the Pareto set. The procedure is divided into two major stages. In the first stage, the multi-objective simulated annealing algorithm is used to solve a multi-objective optimisation problem by constructing the Pareto optimal set. In the second stage, the automatic clustering algorithm is used to prune the Pareto set. This procedure is implemented to solve two multi-objective optimisation problems, namely, the 0/1 multi-objective multi-dimensional knapsack problem and the multi-objective inventory system. The procedure enables the decision maker to select an appropriate solution efficiently.

A MULTI-OBJECTIVE APPROACH TO THE ASSIGNMENT OF STOCK KEEPING UNITS TO UNIDIRECTIONAL PICKING LINES

An order picking system in a distribution centre consisting of parallel unidirectional picking lines is considered. The objectives are to minimise the walking distance of the pickers, the largest volume of stock on a picking line over all picking lines, the number of small packages, and the total penalty incurred for late distributions. The problem is formulated as a multi-objective multiple knapsack problem that is not solvable in a realistic time. Population-based algorithms, including the artificial bee colony algorithm and the genetic algorithm, are also implemented. The results obtained from all algorithms indicate a substantial improvement on all objectives relative to historical assignments. The genetic algorithm delivers the best performance.

Computing the nadir point for multiobjective discrete optimization problems

We investigate the problem of finding the nadir point for multiobjective discrete optimization problems (MODO). The nadir point is constructed from the worst objective values over the efficient set of a multiobjective optimization problem. We present a new algorithm to compute nadir values for MODO with $$p$$ p objective functions. The proposed algorithm is based on an exhaustive search of the $$(p-2)$$ ( p - 2 ) -dimensional space for each component of the nadir point. We compare our algorithm with two earlier studies from the literature. We give numerical results for all algorithms on multiobjective knapsack, assignment and integer linear programming problems. Our algorithm is able to obtain the nadir point for relatively large problem instances with up to five-objectives.

Finding highly preferred points for multi-objective integer programs

This article develops exact algorithms to generate all non-dominated points in a specified region of the criteria space in Multi-Objective Integer Programs (MOIPs). Typically, there are too many non-dominated points in large MOIPs and it is not practical to generate them all. Therefore, the problem of generating non-dominated points in the preferred region of the decision-maker is addressed. To define the preferred region, the non-dominated set is approximated using a hyper-surface. A procedure is developed that then finds a preferred hypothetical point on this surface and defines a preferred region around the hypothetical point. Once the preferred region is defined, all non-dominated points in that region are generated. The performance of the proposed approach is tested on multi-objective assignment, multi-objective knapsack, and multi-objective shortest path problems with three and four objectives. Computational results show that a small set of non-dominated points is generated that contains highly preferred points in a reasonable time.

A combined approach for fuzzy multi-objective multiple knapsack problems for defence project selection

In this study, a model representing military requirements as scenarios and capabilities is offered. Pair-wise comparisons of scenarios are made according to occurrence probabilities by using the Analytical Hierarchy Process (AHP). The weights calculated from AHP are used as the starting weights in a Quality Function Deployment (QFD) matrix. QFD is used to transfer war fighter requirements into the benefit values of projects. Two levels of QFD matrices are used to evaluate new capability areas versus capabilities and capabilities versus projects. The benefit values of the projects are used in a multi-objective problem (multi-objective multiple knapsack problem) that considers the project benefit, implementation risks and environmental impact as multiple objectives. Implementation risk and environmental impact values are also calculated using the same combined AHP and QFD methodology. Finally, the results of the fuzzy multi-objective goal programming suggest a list of projects that offers optimal benefit when carried out within multiple budgets.

A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems

Most real-life decision-making activities require more than one objective to be considered. Therefore, several studies have been presented in the literature that use multiple objectives in decision models. In a mathematical programming context, the majority of these studies deal with two objective functions known as bicriteria optimization, while few of them consider more than two objective functions. In this study, a new algorithm is proposed to generate all nondominated solutions for multiobjective discrete optimization problems with any number of objective functions. In this algorithm, the search is managed over (p−1)-dimensional rectangles where p represents the number of objectives in the problem and for each rectangle two-stage optimization problems are solved. The algorithm is motivated by the well-known ε-constraint scalarization and its contribution lies in the way rectangles are defined and tracked. The algorithm is compared with former studies on multiobjective knapsack and multiobjective assignment problem instances. The method is highly competitive in terms of solution time and the number of optimization models solved.