Multiobjective Multidimensional Knapsack Problem Research Articles

Indicator Weighted Based Multi-Objective Approach using Self-Adaptive Neighborhood Operator

Balancing diversity and convergence seems to be a difficult task when solving multi-objective optimization problems (MOPs). For addressing this issue, researchers’ interest has been drawn to hybrid approaches since this cooperation allows benefiting from the advantages of both approaches and gains better trade-offs overall. Considering such fact, this paper aims at introducing a hybrid approach as a synergy of a multi-directional Ant Colony Optimization algorithm with a Local Search method based on a weighted version of epsilon Indicator using a self-adaptive neighborhood operator coined as Indicator Weighted Based Local Search with Ant Colony Optimization (IWBLS/ACO) to handle the knapsack problem within the multi-objective framework. In IWBLS/ACO, initial solutions are created by the ant colony. Then, the enhancement phase is ensured by the local search procedure. The algorithm is evolving based on different configurations of the epsilon quality indicator through different weight vectors. Moreover, we propose in this work, a novel self-adaptive neighborhood operator which changes automatically and dynamically as the IWBLS algorithm runs. The proposed IWBLS/ACO was tested on widely used Multi-objective Multidimensional Knapsack Problem (MOMKP) instances and compared with powerful state-of-the-art algorithms. Experimental results highlight that the proposed approach can lead to finding a good compromise between exploration and exploitation.

Water wave optimization for combinatorial optimization: Design strategies and applications

Although the community of nature-inspired computing has witnessed a wide variety of metaheuristics, it often requires considerable effort to adapt them to different combinatorial optimization problems (COPs), and few studies have been devoted to reducing this burden. This paper proposes a systematic approach that consists of a set of basic steps and strategies for adapting water wave optimization (WWO), a simple and generic metaheuristic, to concrete heuristic algorithms for different COPs. Taking advantages of the generic algorithmic framework, designers can only focus on adapting the prorogation operator and the wavelength calculation method according to the combinatorial properties of the given problem, and thus easily derive efficient problem-solving algorithms. We illustrate and test our approach on the flow-shop scheduling problem (FSP), the single-objective multidimensional knapsack problem (MKP), and the multi-objective MKP, and then present an application to a machine utilization optimization problem for a large manufacturing enterprise. The results demonstrate that our approach can derive concrete algorithms that are competitive to the state-of-the-arts. Our approach also provides insights into the adaptation of other metaheuristics and the development of new metaheuristics for COPs.

A Pareto evolutionary algorithm based on incremental learning for a kind of multi-objective multidimensional knapsack problem

Multidimensional knapsack problem (MKP) is a classic combinatorial optimization problem arising from many practical applications. To be closer to real-life applications, this paper focuses on a special kind of multi-objective MKP (MOMKP), where the optimization objectives are to maximize the total profit and minimize the maximum consumption of multidimensional resources. To solve it, a Pareto evolutionary algorithm based on Incremental Learning (PEAIL) is proposed, whose components mainly consist of two parts, i.e., the online Incremental Learning and the nondominated sorting. PEAIL is unique in that it can extract the historical information of the search behavior and then feed them back to the main evolutionary framework. Firstly, problem-dependent heuristics, including genetic operators and a repair mechanism for infeasible solutions, are proposed and discussed. Secondly, an online Incremental Learning approach is put forward to learn the probability model of excellent solutions during the iteration process. Thereby, we can predict promising individuals from this probability model, which are used for further strengthening PEAIL’s search ability. Thirdly, a simple yet effective competition-based improvement mechanism is proposed to refine the offspring of PEAIL. Finally, results of experiments on 45 benchmark instances and a real-life case study demonstrate the effectiveness and practical values of the proposed PEAIL.

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 gradual weight-based ant colony approach for solving the multiobjective multidimensional knapsack problem

The multiobjective multidimensional knapsack problem (MOMKP) is an extension of the multiobjective knapsack problem that consists in selecting a subset of items in order to maximize m objective functions. The MOMKP creates an additional difficulty than the monodimensional version caused by the fact of respecting more than one constraint simultaneously. In this paper, we propose to solve the MOMKP with an ant colony optimization approach based on a gradual weight generation method, named Gw-ACO. Here, the weight vectors are gradually distributed in the objective space and change relatively to the optimization process. This enables ants to target, at each cycle, different regions in order to try to achieve almost all solutions covering the Pareto front. To evaluate the suggested Gw-ACO approach, a set of experiments is performed on MOMKP benchmark instances and compared with well-known state-of-the-art metaheuristic approaches. The obtained experimental results show that Gw-ACO is significantly better and able to achieve a well distribution all over the Pareto-optimal front.

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.

Real Estate Property Maintenance Optimization Based on Multiobjective Multidimensional Knapsack Problem

AbstractThe real estate property maintenance problem (REPMP) deals with the problem of the maintenance of real estate property buildings’ components using a limited budget in a limited period to achieve multiple and often conflicting objectives. Maintenance of the components requires a multiannual maintenance action plan optimized simultaneously on different criteria relative to buildings’ maintenance objectives (the quality of service, client satisfaction, regulatory compliance, etc.) while respecting a predetermined budget. This article presents an innovative modeling of REPMP into a multiobjective multidimensional knapsack problem and its optimization using a generalization of the discrete multiobjective particle swarm optimization algorithm and the Pareto optimality. An application of this methodology using data from a real estate property of 22 buildings of a major French company is presented and discussed.

A technique based on trade-off maps to visualise and analyse relationships between objectives in optimisation problems

Understanding the relationships between objectives in a multiobjective optimisation problem is important for developing tailored and efficient solving techniques. In particular, when tackling combinatorial optimisation problems with many objectives that arise in real-world logistic scenarios, better support for the decision maker can be achieved through better understanding of the often complex fitness landscape. This paper makes a contribution in this direction by presenting a technique that allows a visualisation and analysis of the local and global relationships between objectives in optimisation problems with many objectives. The proposed technique uses four steps: first the global pairwise relationships are analysed using the Kendall correlation method; then the ranges of the values found on the given Pareto front are estimated and assessed; next these ranges are used to plot a map using Gray code, similar to Karnaugh maps, that has the ability to highlight the trade-offs between multiple objectives; and finally local relationships are identified using scatter-plots. Experiments are presented for three different combinatorial optimisation problems: multiobjective multidimensional knapsack problem, multiobjective nurse scheduling problem and multiobjective vehicle routing problem with time windows. Results show that the proposed technique helps in the gaining of insights into the problem difficulty arising from the relationships between objectives.

An improved version of a core based algorithm for the multi-objective multi-dimensional knapsack problem: A computational study and comparison with meta-heuristics

In this paper we propose an improved version of a core based algorithm for the multi-objective extension of one of the most well-known combinatorial optimization problems, the multi-dimensional knapsack problem. The original algorithm was designed only for bi-objective problems combining an extension of the core concept and a branch-and-bound algorithm. It is a deterministic algorithm and the core concept exploits the “divide and conquer” solution strategy which proves very effective in such problems. The new version is not limited to bi-objective problems; it can effectively handle problems with more than two objective functions and it has features that greatly accelerate the solution process. Namely, these features are the use of a heuristic based on the Dantzig bound in the fathoming process and the better housekeeping of the incumbent list through filtering of solutions. The key parameters of the new algorithm are (a) the size of the core and (b) the number of provided cores. Varying these parameters the user can easily tune the size of the obtained Pareto set. A comparison with evolutionary algorithms, like NSGA–II, SPEA2 and MOEA/D, has been made according to run time and the most widely used metrics (set coverage, set convergence etc). Our new version performs better than the most popular evolutionary algorithms and it is comparable to very recent state-of-the-art metaheuristics, like Multi-objective Memetic Algorithm based on Decomposition (MOMAD), for multi-objective programming.

Local search-based heuristics for the multiobjective multidimensional knapsack problem

In real optimization problems it is generally desirable to optimize more than one performance criterion (or objective) at the same time. The goal of the multiobjective combinatorial optimization (MOCO) is to optimize simultaneously r > 1 objectives. As in the single-objective case, the use of heuristic/metaheuristic techniques seems to be the most promising approach to MOCO problems because of their efficiency, generality and relative simplicity of implementation. In this work, we develop algorithms based on Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Local Search (ILS) metaheuristics for the multiobjective knapsack problem. Computational experiments on benchmark instances show that the proposed algorithms are very robust and outperform other heuristics in terms of solution quality and running times.

Multi-directional local search

This paper introduces multi-directional local search, a metaheuristic for multi-objective optimization. We first motivate the method and present an algorithmic framework for it. We then apply it to several known multi-objective problems such as the multi-objective multi-dimensional knapsack problem, the bi-objective set packing problem and the bi-objective orienteering problem. Experimental results show that our method systematically provides solution sets of comparable quality with state-of-the-art methods applied to benchmark instances of these problems, within reasonable CPU effort. We conclude that the proposed algorithmic framework is a viable option when solving multi-objective optimization problems.

Solving the bi-objective multi-dimensional knapsack problem exploiting the concept of core

This paper deals with the bi-objective multi-dimensional knapsack problem. We propose the adaptation of the core concept that is effectively used in single-objective multi-dimensional knapsack problems. The main idea of the core concept is based on the “divide and conquer” principle. Namely, instead of solving one problem with n variables we solve several sub-problems with a fraction of n variables (core variables). The quality of the obtained solution can be adjusted according to the size of the core and there is always a trade off between the solution time and the quality of solution. In the specific study we define the core problem for the multi-objective multi-dimensional knapsack problem. After defining the core we solve the bi-objective integer programming that comprises only the core variables using the Multicriteria Branch and Bound algorithm that can generate the complete Pareto set in small and medium size multi-objective integer programming problems. A small example is used to illustrate the method while computational and economy issues are also discussed. Computational experiments are also presented using available or appropriately modified benchmarks in order to examine the quality of Pareto set approximation with respect to the solution time. Extensions to the general multi-objective case as well as to the computation of the exact solution are also mentioned.

A multiobjective programming approach for selecting non-independent transportation investment alternatives

This article presents a new method for SElecting Non-independent TRansportation investment Alternatives (SENTRA). This method utilizes effective distance heuristic algorithm which attempts to maximize the achieved objectives needed to satisfy available resources. Since transportation investment planning cannot avoid dealing with issues of interdependence among alternatives, this paper will consider four types of investment alternatives: independence, complementarity, substitution and common complementary substitution. Transportation investment alternative selection problem can be formulated in terms of the 0–1 multiobjective multidimensional knapsack problem. Possessing the characteristics of NP-completeness, strict computation is not necessary for the optimal solution, but simple computation for near-optimal solution is expected. The method is proposed in this paper so as to attain the near-optimal solution, which, aside from ranking the selected transportation investment alternatives, can easily perform sensitivity analysis. Finally, an example is presented to illustrate the method.