This paper presents a parallel memetic algorithm for solving the vehicle routing problem with time windows (VRPTW). The VRPTW is a well-known NP-hard discrete optimization problem with two objectives. The main objective is to minimize the number of vehicles serving customers scattered on the map, and the second one is to minimize the total distance traveled by the vehicles. Here, the fleet size is minimized in the first phase of the proposed method using the parallel heuristic algorithm (PHA), and the traveled distance is minimized in the second phase by the parallel memetic algorithm (PMA). In both parallel algorithms, the parallel components co-operate periodically in order to exchange the best solutions found so far. An extensive experimental study performed on the Gehring and Homberger's benchmark proves the high convergence capabilities and robustness of both PHA and PMA. Also, we present the speedup analysis of the PMA.
Evolutionary algorithms have attracted the scientific attention to solve the VRPTW due to their high search capabilities. Genetic algorithms were applied by Cheng and Wang @cite_9 , @cite_0 and @cite_8 . The evolution strategies were proposed by Gehring and Homberger @cite_30 , @cite_31 and @cite_17 . The sequential and parallel memetic algorithms (MAs), combining the evolutionary algorithms for more distant search with the local optimization and refinements algorithms for local exploitation, were described by Berger and Barkaoui @cite_3 , @cite_24 and Nagata and Br " a ysy @cite_2 . The MA based on the edge-assembly crossover operator (EAX) has been proposed by Nagata and Br " a ysy @cite_2 .
@cite_30 @cite_8 @cite_9 @cite_3 @cite_0 @cite_24 @cite_2 @cite_31 @cite_17
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This article deals with the vehicle routing problem with time windows (VRPTW). This problem consists in determining a least-cost set of trips to serve customers during specific time windows. The proposed solution method is a memetic algorithm (MA), a genetic algorithm hybridised with a local search. Contrary to most papers on the VRPTW, which minimize first the number of vehicles, our method is also able to minimize the total distance travelled. The results on 56 classical instances are compared to those of the best metaheuristics. The efficiency of the MA is similar for the classical criterion, but it becomes the best algorithm available for the total distance, being much faster and improving 20 best-known solutions. In this paper, we present an effective memetic algorithm for the vehicle routing problem with time windows (VRPTW). The paper builds upon an existing edge assembly crossover (EAX) developed for the capacitated VRP. The adjustments of the EAX operator and the introduction of a novel penalty function to eliminate violations of the time window constraint as well as the capacity constraint from offspring solutions generated by the EAX operator have proven essential to the heuristic's performance. Experimental results on Solomon's and Gehring and Homberger benchmarks demonstrate that our algorithm outperforms previous approaches and is able to improve 184 best-known solutions out of 356 instances. Many different algorithms have been developed in the last few decades for solving complex real-world search and optimization problems. The main focus in this research has been on the development of a single universal genetic operator for population evolution that is always efficient for a diverse set of optimization problems. In this paper, we argue that significant advances to the field of evolutionary computation can be made if we embrace a concept of self-adaptive multimethod optimization in which multiple different search algorithms are run concurrently, and learn from each other through information exchange using a common population of points. We present an evolutionary algorithm, entitled A Multialgorithm Genetically Adaptive Method for Single Objective Optimization (AMALGAM-SO), that implements this concept of self adaptive multimethod search. This method simultaneously merges the strengths of the covariance matrix adaptation (CMA) evolution strategy, genetic algorithm (GA), and particle swarm optimizer (PSO) for population evolution and implements a self-adaptive learning strategy to automatically tune the number of offspring these three individual algorithms are allowed to contribute during each generation. Benchmark results in 10, 30, and 50 dimensions using synthetic functions from the special session on real-parameter optimization of CEC 2005 show that AMALGAM-SO obtains similar efficiencies as existing algorithms on relatively simple unimodal problems, but is superior for more complex higher dimensional multimodal optimization problems. The new search method scales well with increasing number of dimensions, converges in the close proximity of the global minimum for functions with noise induced multimodality, and is designed to take full advantage of the power of distributed computer networks. A key search mechanism in Evolutionary Algorithms is the mixing or juxtaposing of partial solutions present in the parent solutions. In this paper we look at the efficiency of mixing in genetic algorithms (GAs) and estimation-of-distribution algorithms (EDAs). We compute the mixing probabilities of two partial solutions and discuss the effect of the covariance build-up in GAs and EDas. Moreover, we propose two new Evolutionary Algorithms that maximize the juxtaposing of the partial solutions present in the parents: the Recombinative Optimal Mixing Evolutionary Algorithm (ROMEA) and the Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA). A challenge in evolutionary computation is to create representations as evolvable as those in natural evolution. This paper hypothesizes that extinction events, i.e. mass extinctions, can significantly increase evolvability, but only when combined with a divergent search algorithm, i.e. a search driven towards diversity (instead of optimality). Extinctions amplify diversity-generation by creating unpredictable evolutionary bottlenecks. Persisting through multiple such bottlenecks is more likely for lineages that diversify across many niches, resulting in indirect selection pressure for the capacity to evolve. This hypothesis is tested through experiments in two evolutionary robotics domains. The results show that combining extinction events with divergent search increases evolvability, while combining them with convergent search offers no similar benefit. The conclusion is that extinction events may provide a simple and effective mechanism to enhance performance of divergent search algorithms. While evolutionary computation and evolutionary robotics take inspiration from nature, they have long focused mainly on problems of performance optimization. Yet evolution in nature can be interpreted as more nuanced than a process of simple optimization. In particular, natural evolution is a divergent search that optimizes locally within each niche as it simultaneously diversifies. This tendency to discover both quality and diversity at the same time differs from many of the conventional algorithms of machine learning, and also thereby suggests a different foundation for inferring the approach of greatest potential for evolutionary algorithms. In fact, several recent evolutionary algorithms called quality diversity (QD) algorithms(e.g. novelty search with local competition and MAP-Elites) have drawn inspiration from this more nuanced view, aiming to fill a space of possibilities with the best possible example of each type of achievable behavior. The result is a new class of algorithms that return an archive of diverse, high-quality behaviors in a single run. The aim in this paper is to study the application of QD algorithms in challenging environments (in particular complex mazes) to establish their best practices for ambitious domains in the future. In addition to providing insight into cases when QD succeeds and fails, a new approach is investigated that hybridizes multiple views of behaviors (called behavior characterizations) in the same run, which succeeds in overcoming some of the challenges associated with searching for QD with respect to a behavior characterization that is not necessarily sufficient for generating both quality and diversity at the same time. In this paper we review the exact algorithms proposed in the last three decades for the solution of the vehicle routing problem with time windows (VRPTW). The exact algorithms for the VRPTW are in many aspects inherited from work on the traveling salesman problem (TSP). In recognition of this fact this paper is structured relative to four seminal papers concerning the formulation and exact solution of the TSP, i.e. the arc formulation, the arc-node formulation, the spanning tree formulation, and the path formulation. We give a detailed analysis of the formulations of the VRPTW and a review of the literature related to the different formulations. There are two main lines of development in relation to the exact algorithms for the VRPTW. One is concerned with the general decomposition approach and the solution to certain dual problems associated with the VRPTW. Another more recent direction is concerned with the analysis of the polyhedral structure of the VRPTW. We conclude by examining possible future lines of research in the area of the VRPTW. Recently, the hybridization between evolutionary algorithms and other metaheuristics has shown very good performances in many kinds of multiobjective optimization problems (MOPs), and thus has attracted considerable attentions from both academic and industrial communities. In this paper, we propose a novel hybrid multiobjective evolutionary algorithm (HMOEA) for real-valued MOPs by incorporating the concepts of personal best and global best in particle swarm optimization and multiple crossover operators to update the population. One major feature of the HMOEA is that each solution in the population maintains a nondominated archive of personal best and the update of each solution is in fact the exploration of the region between a selected personal best and a selected global best from the external archive. Before the exploration, a selfadaptive selection mechanism is developed to determine an appropriate crossover operator from several candidates so as to improve the robustness of the HMOEA for different instances of MOPs. Besides the selection of global best from the external archive, the quality of the external archive is also considered in the HMOEA through a propagating mechanism. Computational study on the biobjective and three-objective benchmark problems shows that the HMOEA is competitive or superior to previous multiobjective algorithms in the literature. Many real-world optimization problems are difficult to solve as they do not possess the nice mathematical properties required by the exact algorithms. Evolutionary algorithms are proven to be appropriate for such problems. In this paper, we propose an improved differential evolution algorithm that uses a mix of different mutation operators. In addition, the algorithm is empowered by a covariance adaptation matrix evolution strategy algorithm as a local search. To judge the performance of the algorithm, we have solved well-known benchmark as well as a variety of real-world optimization problems. The real-life problems were taken from different sources and disciplines. According to the results obtained, the algorithm shows a superior performance in comparison with other algorithms that also solved these problems. In the field of evolutionary computation, there has been a growing interest in applying evolutionary algorithms to solve multimodal optimization problems (MMOPs). Due to the fact that an MMOP involves multiple optimal solutions, many niching methods have been suggested and incorporated into evolutionary algorithms for locating such optimal solutions in a single run. In this paper, we propose a novel transformation technique based on multiobjective optimization for MMOPs, called MOMMOP. MOMMOP transforms an MMOP into a multiobjective optimization problem with two conflicting objectives. After the above transformation, all the optimal solutions of an MMOP become the Pareto optimal solutions of the transformed problem. Thus, multiobjective evolutionary algorithms can be readily applied to find a set of representative Pareto optimal solutions of the transformed problem, and as a result, multiple optimal solutions of the original MMOP could also be simultaneously located in a single run. In principle, MOMMOP is an implicit niching method. In this paper, we also discuss two issues in MOMMOP and introduce two new comparison criteria. MOMMOP has been used to solve 20 multimodal benchmark test functions, after combining with nondominated sorting and differential evolution. Systematic experiments have indicated that MOMMOP outperforms a number of methods for multimodal optimization, including four recent methods at the 2013 IEEE Congress on Evolutionary Computation, four state-of-the-art single-objective optimization based methods, and two well-known multiobjective optimization based approaches. Abstract We consider the usage of evolutionary algorithms for multiobjective programming (MOP), i.e. for decision problems with alternatives taken from a real-valued vector space and evaluated according to a vector-valued objective function. Selection mechanisms, possibilities of temporary fitness deterioration, and problems of unreachable alternatives for such multiobjective evolutionary algorithms (MOEAs) are studied. Theoretical properties of MOEAs such as stochastic convergence with probability 1 are analyzed. Evolutionary computation techniques have had limited capabilities in solving large-scale problems due to the large search space demanding large memory and much longer training times. In the work presented here, a genetic programming like rich encoding scheme has been constructed to identify building blocks of knowledge in a learning classifier system. The fitter building blocks from the learning system trained against smaller problems have been utilized in a higher complexity problem in the domain to achieve scalable learning. The proposed system has been examined and evaluated on four different Boolean problem domains: 1) multiplexer, 2) majority-on, 3) carry, and 4) even-parity problems. The major contribution of this paper is to successfully extract useful building blocks from smaller problems and reuse them to learn more complex large-scale problems in the domain, e.g., 135-bit multiplexer problem, where the number of possible instances is 2 135 ≈ 4 × 10 40 , is solved by reusing the extracted knowledge from the learned lower level solutions in the domain. Autonomous scaling is, for the first time, shown to be possible in learning classifier systems. It improves effectiveness and reduces the number of training instances required in large problems, but requires more time due to its sequential build-up of knowledge. In this paper, a hybrid Taguchi-genetic algorithm (HTGA) is applied to solve the problem of tuning both network structure and parameters of a feedforward neural network. The HTGA approach is a method of combining the traditional genetic algorithm (TGA), which has a powerful global exploration capability, with the Taguchi method, which can exploit the optimum offspring. The Taguchi method is inserted between crossover and mutation operations of a TGA. Then, the systematic reasoning ability of the Taguchi method is incorporated in the crossover operations to select the better genes to achieve crossover, and consequently enhance the genetic algorithms. Therefore, the HTGA approach can be more robust, statistically sound, and quickly convergent. First, the authors evaluate the performance of the presented HTGA approach by studying some global numerical optimization problems. Then, the presented HTGA approach is effectively applied to solve three examples on forecasting the sunspot numbers, tuning the associative memory, and solving the XOR problem. The numbers of hidden nodes and the links of the feedforward neural network are chosen by increasing them from small numbers until the learning performance is good enough. As a result, a partially connected feedforward neural network can be obtained after tuning. This implies that the cost of implementation of the neural network can be reduced. In these studied problems of tuning both network structure and parameters of a feedforward neural network, there are many parameters and numerous local optima so that these studied problems are challenging enough for evaluating the performances of any proposed GA-based approaches. The computational experiments show that the presented HTGA approach can obtain better results than the existing method reported recently in the literature. In genetic programming, a search algorithm is expected to produce a program that achieves the desired final computation state (desired output). To reach that state, an executing program needs to traverse certain intermediate computation states. An evolutionary search process is expected to autonomously discover such states. This can be difficult for nontrivial tasks that require long programs to be solved. The semantic backpropagation algorithm proposed in this paper heuristically inverts the execution of evolving programs to determine the desired intermediate computation states. Two search operators, random desired operator and approximately geometric semantic crossover, use the intermediate states determined by semantic backpropagation to define subtasks of the original programming task, which are then solved using an exhaustive search. The operators outperform the standard genetic search operators and other semantic-aware operators when compared on a suite of symbolic regression and Boolean benchmarks. This result and additional analysis conducted in this paper indicate that semantic backpropagation helps evolution to identify the desired intermediate computation states and makes the search process more efficient. Genetic Algorithms (GAs) can efficiently produce high quality results for hard combinatorial real world problems such as the Vehicle Routing Problem (VRP). Genetic Vehicle Representation (GVR), a recent approach to solving instances of the VRP with a GA, produces competitive or superior results to the standard benchmark problems. This work extends GVR research by presenting a more precise mathematical model of GVR than in previous works and a thorough comparison of GVR to Path Based Representation approaches. A suite of metrics that measures GVR's efficiency and effectiveness provides an adequate characterization of the jagged search landscape. A new variation of a crossover operator is introduced. A previously unmentioned insight about the convergence rate of the search is also noted that is especially important to the application of a priori and dynamic routing for swarms of Unmanned Aerial Vehicles (UAVs). Results indicate that the search is robust, and it exponentially drives toward high quality solutions in relatively short time. Consequently, a GA with GVR encoding is capable of providing a state-of-the-art engine for a UAV routing system or related application. Over the last few decades, many different evolutionary algorithms have been introduced for solving constrained optimization problems. However, due to the variability of problem characteristics, no single algorithm performs consistently over a range of problems. In this paper, instead of introducing another such algorithm, we propose an evolutionary framework that utilizes existing knowledge to make logical changes for better performance. The algorithmic aspects considered here are: the way of using search operators, dealing with feasibility, setting parameters, and refining solutions. The combined impact of such modifications is significant as has been shown by solving two sets of test problems: (i) a set of 24 test problems that were used for the CEC2006 constrained optimization competition and (ii) a second set of 36 test instances introduced for the CEC2010 constrained optimization competition. The results demonstrate that the proposed algorithm shows better performance in comparison to the state-of-the-art algorithms. This paper introduces the Localized Optimization Framework (LOF). This framework is an iterative procedure between two phases, Optimization and De-optimization. Optimization is done on the problem parts rather than the problem as a whole, while de-optimization is done on the whole problem. To test our hypothesis, we have chosen a genetic algorithm as an optimization methodology and Vehicle Routing Problem with Time Windows (VRPTW) as a domain space. We call this new scheme the Localized Genetic Algorithm (LGA). We demonstrate that the LGA is, on average, able to produce better solutions than most of the other heuristics on small scale problems of VRPTW. Furthermore the LGA has attained several new best solutions on popular datasets. Evolutionary algorithms are among the metaheuristic search methods that have been applied to the structural test data generation problem. Fitness evaluation methods play an important role in the performance of evolutionary algorithms and various methods have been devised for this problem. In this paper, we propose a new fitness evaluation method based on pairwise sequence comparison also used in bioinformatics. Our preliminary study shows that this method is easy to implement and produces promising results. Randomized search heuristics like evolutionary algorithms are mostly applied to problems whose structure is not completely known but also to combinatorial optimization problems. Practitioners report surprising successes but almost no results with theoretically well-founded analyses exist. Such an analysis is started in this paper for a fundamental evolutionary algorithm and the well-known maximum matching problem. It is proven that the evolutionary algorithm is a polynomial-time randomized approximation scheme (PRAS) for this optimization problem, although the algorithm does not employ the idea of augmenting paths. Moreover, for very simple graphs it is proved that the expected optimization time of the algorithm is polynomially bounded and bipartite graphs are constructed where this time grows exponentially. Evolutionary and genetic algorithms (EAs and GAs) are quite successful randomized function optimizers. This success is mainly based on the interaction of different operators like selection, mutation, and crossover. Since this interaction is still not well understood, one is interested in the analysis of the single operators. Jansen and Wegener [Proceedings of GECCO'2001, 2001, pp. 375-382] have described so-called real royal road functions where simple steady-state GAs have a polynomial expected optimization time while the success probability of mutation-based EAs is exponentially small even after an exponential number of steps. This success of the GA is based on the crossover operator and a population whose size is moderately increasing with the dimension of the search space. Here new real royal road functions are presented where crossover leads to a small optimization time, although the GA works with the smallest possible population size--namely 2.