A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows

The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) is a popular algorithm for solving Multi-Objective Problems (MOPs). The main component of MOEA/D is to decompose a MOP into easier sub-problems using a set of weight vectors. The choice of the number of weight vectors significantly impacts the performance of MOEA/D. However, the right choice for this number varies, given different MOPs and search stages. We adaptively change the number of vectors by removing unnecessary vectors and adding new ones in empty areas of the objective space. Our MOEA/D variant uses the Consolidation Ratio to decide when to change the number of vectors and to decide where to add or remove these weighted vectors. We investigate the effects of this adaptive MOEA/D against MOEA/D with a poorly chosen set of vectors, a MOEA/D with fine-tuned vectors and MOEA/D with Adaptive Weight Adjustment on two commonly used benchmark functions. We analyse the algorithms in terms of hypervolume, IGD and entropy performance. Our results show that the proposed method is equivalent to MOEA/D with fine-tuned vectors and superior to MOEA/D with poorly defined vectors. Thus, our adaptive mechanism mitigates problems related to the choice of the number of weight vectors in MOEA/D, increasing the final performance of MOEA/D by filling empty areas of the objective space and avoiding premature stagnation of the search progress.

Given that the definition of the multi-objective optimization problem is raised when number of objectives is increased in number at the optimization problem, where not only the number of objectives but also the computational resources which are needed to solve the problem, is also more desired. Therefore, novel approaches had required to solve multi-objective optimization problem in a reasonable time. One of this novel approach is utilization of the decomposition method with the evolutionary algorithm/operator. This algorithm was called multi-objective evolutionary algorithm based on decomposition (MOEA/D). Later on, variants have been proposed to improve the performance of the MOEA/D algorithm. However, a general comparison between these variants has needed for demonstrate the performance of these algorithm. For this reason, in this research the variants of MOEA/D algorithms have implemented on benchmark problems (DTLZ and MaF) and the performances has compared with each other. Two metrics had selected to evaluate/compare the performances of the variants. The metrics are IGD and Spread metrics. The results at the end of the implementations suggest that adaptive weighting idea is the most promising idea to increase the performance of the MOEA/D algorithm.

Multiobjective evolutionary algorithms based on decomposition (MOEA/D) decompose a multiobjective optimization problem (MOP) into a group of subproblems and optimizes them at the same time. The reproduction method in MOEA/D, which generates offspring solutions, has crucial effect on the performance of algorithm. As the difficulties of MOPs increases, it requires much higher efficiency for the reproduction methods in MOEA/D. However, for the complex optimization problems whose PS shape is complicated, the original reproduction method used in MOEA/D might not be suitable to generate excellent offspring solutions. In order to improve the property of the reproduction method for MOEA/D, this paper proposes an external archive matching strategy which selects solutions’ most matching archive solutions as parent solutions. The offspring solutions generated by this strategy can maintain a good convergence ability. To balance convergence and diversity, a perturbed learning scheme is used to extend the search space of the solutions. The experimental results on three groups of test problems reveal that the solutions obtained by MOEA/D-EAM have better convergence and diversity than the other four state-of-the-art algorithms.

Learning of fuzzy parameters for system modeling using evolutionary algorithms is an interesting topic. In this paper, two optimal design and tuning of Interval type-2 fuzzy logic system are proposed using hybrid learning algorithms. The consequent parameters of the interval type-2 fuzzy logic system in both the hybrid algorithms are tuned using Kalman filter. Whereas the antecedent parameters of the system in the first hybrid algorithm is optimized using the multi-objective particle swarm optimization (MOPSO) and using the multi-objective evolutionary algorithm Based on Decomposition (MOEA/D) in the second hybrid algorithm. Root mean square error and maximum absolute error as the two accuracy objective are utilized to find the Pareto-optimal solution with the MOPSO and MOEA/D respectively. The proposed hybrid multi-objective designs of the interval type-2 fuzzy logic system are utilized to the prediction of solar photovoltaic output. It is observed that MOEA/D outperforms MOPSO in this case in terms of quality of results and its diversity. Finally, one point is selected from the obtained Pareto front and its performance is illustrated.

Although a number of multiobjective evolutionary algorithms have been proposed over the last two decades, not much effort has been made to deal with variable linkages in multiobjective optimization. Recently, we have suggested a general framework of multiobjective evolutionary algorithms based on decomposition (MOEA/D) [1]. MOEA/D decomposes a MOP into a number of scalar optimization subproblems by a conventional decomposition method. The optimal solution to each of these problems is a Pareto optimal solution to the MOP under consideration. An appropriate decomposition could make these individual Pareto solutions evenly distribute along the Pareto optimal front. MOEA/D aims at solving these scalar optimization subproblems simultaneously. In this paper, we propose, under the framework of MOEA/D, a multiobjective differential evolution based decomposition (MODE/D) for tackling variable linkages. Our experimental results show that MODE/D outperforms several other MOEAs on several test problems with variable linkages.

Multi-Objective Evolutionary Algorithm based on decomposition (MOEA/D) has been proposed for a decade in solving complex multi-objective problems (MOP) by decomposing it into a set of single-objective problems. MOEA/D-AGR is one of the improved algorithm introduced recently to substitute the original replacement scheme in MOEA/D with a new adaptive global replacement (GR) scheme so that the neighborhood replacement size Tr is increased among the generation to achieve the shifting of focus from solution diversity to convergence. However, the new replacement scheme only considers the one-way convergence of the objective solutions among all sub-problems. It is hard to re-achieve the solution diversity once the algorithm reaches another steeper landscape of solution from a flatten one while it is focusing on convergence. This paper proposes a new adaptive GR scheme to prolong the period of Tr increment so that it can fit the re-increment of fitness landscape. To compensate the shorten period for solution convergence, local searching is adapted for those individuals which has stopped improving its solution value by Simulated Annealing (SA) algorithm. In order to suppress the degree of local searching at the early stage of focusing solution diversity, Fuzzy Logic is used here to coordinate the frequency of local searching according to the average change of objective solution values. To demonstrate the performance of the proposed algorithm, several common benchmark MOPs are used in this paper for comparing with the several state-of-the-art MOEA/D algorithms in terms of IGD. The performance investigation found that the performance of the proposed algorithm was generally better than the other compared algorithms.

There are a large number of multi-objective optimization problems in real-world applications, like in games, that need to be solved in real time. In order to meet this pressing need, we suggests a method of parallelizing the multi-objective evolutionary algorithm based on decomposition (MOEA/D). Furthermore, a novel task decomposition strategy and scalarizing method without the ideal point are proposed for meeting the requirements of real-time and precision of the game. By combining the novel scalarizing function and GPU-based CUDA technology with the MOEA/D, a parallel MOEA/D for real-time multi-objective optimization problems is developed, namely P-MOEA/D. Experimental studies on ZDT and DTLZ benchmark problems suggest that the P-MOEA/D algorithm is efficient and fast.

This paper proposes an idea of using heuristic local search procedures specific for single-objective optimisation in multiobjectie evolutionary algorithms (MOEAs). In this paper, a multiobjective evolutionary algorithm based on decomposition (MOEA/D) hybridised with a multi-start single-objective metaheuristic called greedy randomised adaptive search procedure (GRASP). In our method a multiobjetive optimisation problem (MOP) is decomposed into a number of single-objecive subproblems and optimised in parallel by using neighbourhood information. The proposed GRASP alternates between subproblems to help them escape local Pareto optimal solutions. Experimental results have demonstrated that MOEA/D with GRASP outperforms the classical MOEA/D algorithm on the multiobjective 0–1 knapsack problem that is commonly used in the literature. It has also demonstrated that the use of greedy genetic crossover can significantly improve the algorithm performance.

In optimization problems it is often necessary to perform an optimization based on more than one objective. The goal of the multiobjective optimization is usually to find an approximation of the Pareto front which contains solutions that represent the best possible trade-offs between the objectives. In a multiobjective scenario it is important to both improve the solutions in terms of the objectives and to find a wide variety of available options. Evolutionary algorithms are often used for multiobjective optimization because they maintain a population of individuals which directly represent a set of solutions of the optimization problem. multiobjective evolutionary algorithm based on decomposition (MOEA/D) is one of the most effective multiobjective algorithms currently used in the literature. This paper investigates several methods which increase the selective pressure to the outside of the Pareto front in the case of the MOEA/D algorithm. Experiments show that by applying greater selective pressure in the outwards direction the quality of results obtained by the algorithm can be increased. The proposed methods were tested on nine test instances with complicated Pareto sets. In the tests the new methods outperformed the original MOEA/D algorithm.

This paper investigates the multi-objective uncertain multi-stage weapon target assignment (UMWTA) problem in which weapons' kill probabilities are considered to be uncertain from practical aspects. Here uncertainties are introduced by assuming that these probabilities depend on random parameters which are impacted by various factors. The multi-objective formulation of UMWTA problems covers the objectives of minimizing the risk of missing hostile targets measured by the conditional value-at-risk (CVaR) measure and minimizing the ammunition consumption. With the aim of determining referenced Pareto fronts, an approximated linear formulation of the UMWTA problem is put forward based on problem-specific characteristics. Two state-of-the-art decomposition-based multi-objective evolutionary algorithms (DMOEAs), i.e., MOEA/D-AWA and DMOEA -ɛC are used to solve the formulated problem. In view of the inefficiency of the standard comparison mechanism in DMOEAs, a hierarchical comparison strategy which takes into account Pareto dominance relationships and aggregated objective values simultaneously is proposed and embedded in MOEA/D-AWA and DMOEA-ɛC. MOEA/D-AWA and DMOEA-ɛC with the hierarchical comparison strategy are denoted as MOEA/D-AWA-H and DMOEA-ɛC-H, respectively. Numerical experiments have been performed on two sets of UMWTA instances. Experimental results confirm the effectiveness of the proposed hierarchical comparison strategy and demonstrate the superiority of DMOEA-ɛC-H over MOEA/D-AWA-H on the majority of test instances.

A triple‐objective optimal design method for AC filters in high voltage direct current (HVDC) system based on an improved multi‐objective evolutionary algorithm based on decomposition (MOEA/D) is proposed. A triple‐objective optimisation mathematical model including total harmonic distortion (THD) on AC side voltage, investment cost of AC filters and telephone harmonic form factor (THFF) is established. An improved MOEA/D is proposed to realise the triple‐objective optimal design for AC filters more efficiently. The improved MOEA/D adds a particular mutation operator, which can not only to prevent MOEA/D from getting stuck at local optimum but also accelerate the convergence to the optimum solution while the algorithm approaches the neighbourhood of the optimum solution. A three‐dimensional fuzzy evaluation system is designed to select the optimum solution of the triple‐objective optimisation. A case of ±500 kV HVDC system is simulated. Taking the ±500 kV HVDC system as the prototype system, a dynamic simulation experimental platform of the rectification system of the direct current transmission system with AC filters is designed and established based on the similarity principle. Both the simulation and experimental results verify that the proposed method have excellent performance.

Multi-objective energy-efficient dense deployment in Wireless Sensor Networks using a hybrid problem-specific MOEA/D

Wireless sensor networks design requires high quality location assignment and energy efficient power assignment for maximizing the network coverage and lifetime. Classical deployment and power assignment approaches optimize these two objectives individually or by combining them together in a single objective or by constraining one and optimizing the other. In this article a multi-objective deployment and power assignment problem (DPAP) is formulated and a multi-objective evolutionary algorithm based on decomposition (MOEA/D) is specialized. Following the MOEA/D's framework the above multiobjective optimization problem (MOP) is decomposed into many scalar single objective problems. The sub-problems are solved simultaneously by using neighborhood information. Additionally, unique problem-specific, parameter-rising, genetic operators and local search heuristics were designed specifically for the DPAP. In addition, a new encoding scheme is designed to represent a WSN based on the DPAP's design variables. Simulation results show that MOEA/D provides a high quality set of alternative solutions without any prior knowledge on the objectives preference.

Research on multiobjective optimization problems becomes one of the hottest topics of intelligent computation. In order to improve the search efficiency of an evolutionary algorithm and maintain the diversity of solutions, in this paper, the learning automata (LA) is first used for quantization orthogonal crossover (QOX), and a new fitness function based on decomposition is proposed to achieve these two purposes. Based on these, an orthogonal evolutionary algorithm with LA for complex multiobjective optimization problems with continuous variables is proposed. The experimental results show that in continuous states, the proposed algorithm is able to achieve accurate Pareto-optimal sets and wide Pareto-optimal fronts efficiently. Moreover, the comparison with the several existing well-known algorithms: nondominated sorting genetic algorithm II, decomposition-based multiobjective evolutionary algorithm, decomposition-based multiobjective evolutionary algorithm with an ensemble of neighborhood sizes, multiobjective optimization by LA, and multiobjective immune algorithm with nondominated neighbor-based selection, on 15 multiobjective benchmark problems, shows that the proposed algorithm is able to find more accurate and evenly distributed Pareto-optimal fronts than the compared ones.

Multi-objective Estimation of Distribution Algorithms (MOEDAS) have been successfully applied to solve Multi-objective Optimization Problems (MOPs) since they are able to model dependencies between variables of the problem and then sample new solutions to guide the search to promising areas. A state-of-the-art optimizer for single-objective continuous functions that also uses probabilistic modeling is the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Different variants of CMA-ES have been proposed for MOPs however most of them are based on Pareto dominance as the main selection criterion. Recently, a new multi-objective CMA-ES called MOEA/D-CMA was proposed combining the strengths of CMA-ES with those of the multi-objective evolutionary algorithm based on decomposition (MOEA/D). Nowadays, however, researchers on MOEAs agree that combining Pareto and decomposition can be beneficial for the search on MOPs. As a result, a new MOEA has been proposed, called MOEA/DD. This algorithm modifies the MOEA/D by including a new Pareto dominance update mechanism that brings more diversity into the search. In this study, we extend the MOEA/D-CMA by replacing its update mechanism by the one of MOEA/DD. The hypothesis is that this update mechanism will improve the performance of MOEA/D-CMA as it improved MOEA/D. MOEA/D-CMA and MOEA/DD-CMA are implemented and evaluated through an experimental study. The experimental study involves two well-known families of benchmark problems whose objective numbers scale from two to fifteen. Then, an extensive statistical analysis of the results is made to extract sound, statistically supported conclusions about the performance of the algorithms as the number of objectives scales.