Searching in Symmetric Solution Space for Permutation-Related Optimization Problems.

The single-row facility layout problem (SRFLP) is concerned with arranging facilities along a straight line so as to minimize the sum of the products of the flow costs and distances among all facility pairs. SRFLP has rich practical applications and is however NP-hard. In this article, we first investigate a dedicated symmetry-breaking approach based on the permutation group theory for reducing the solution space of SRFLP. Relevant symmetry properties are identified through the alternating group of the original solution space or the corresponding coordinate rotation space. Then, a memetic algorithm is proposed to explore promising search regions regarding the reduced solution space. The memetic algorithm employs a problem-specific crossover operator guided by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> -medoids clustering technique to produce meaningful offspring solutions. The algorithm additionally uses a simulated annealing procedure to intensively exploit a given search region and a distance-and-quality-based population management strategy to ensure a reasonable diversity of the population. Experimental results on commonly used benchmark instances and newly introduced large-scale instances with sizes up to 2000 facilities show that the proposed algorithm competes favorably with state-of-the-art SRFLP algorithms. It attains all but one previous best known upper bounds (BKS) and discovers new upper bounds for 33 instances out of the 93 popular benchmark instances.

A decomposition based estimation of distribution algorithm for multiobjective traveling salesman problems

Traveling Salesman Problem (TSP) is one of the most challenging combinatorial optimization problems. As the city number of TSP grows, the feasible solution space size increases factorially. For the small to mid-size TSP, the Lin-Kernighan (D. S. Johnson, 1990) (LK) and Lin-Kernighan Heuristic (C. Walshaw, 2001) (LKH) algorithms are very effective. However, these two algorithms are local search methods which find the best TSP tour in the k-change neighborhoods of the given initial TSP tour. Thus, they can only find a local optimal tour for TSP with complex solution space. Accordingly, the LK and LKH algorithms become very sensitive to the initial solution and often fail to find the global optimal tour within a reasonable time for solving large scale TSP. To remedy this problem, we make use of the global search ability of the immune clonal algorithm. Especially, we combine the two types of approaches (i.e. LK and immune clonal algorithm) to achieve high performance of the immune clonal algorithm, which can be run on loose-coupled computing environment for solving the large scale TSP. The immune clonal algorithm inspired by biological immune system is a type of evolutionary random search algorithms. More and more research achievements indicate that immune clonal algorithm can maintain good population diversity and strong global search capability. Under the searching framework of the immune clonal algorithm, heuristic search strategies can be conveniently employed to enhance its local search capability. Such combinations take into account both global and local search strategies, and thus can realize a good tradeoff between effectiveness and efficiency. Moreover, the parallelizability of the biological immune system ensures the immune clonal algorithm can be run on loosecoupled computing environment which is advantageous to solve massive optimization problems such as the large scale TSP. Simulation and analysis results show that the edges in the intersection set of several local optimal tours obtained by LK approach appear in the global optimal tour with high probability and the probability increases rapidly as the amount of local optimal tours increases. Using this phenomenon, an intersection set based vaccination strategy is designed in this chapter to accelerate the convergence speed of the immune clonal algorithm for TSP. In the immune clonal algorithm, vaccine is a set of genes which are estimations of the genes expected to appear in the global optimal antibody. The proposed approach in this chapter takes the intersection gene set of several memory antibodies as vaccine and injects the set

Traveling Salesman Problems (TSPs) have been a long-lasting interesting challenge to researchers in different areas. The difficulty of such problems scales up further when multiple objectives are considered concurrently. Plenty of work in evolutionary algorithms has been introduced to solve multi-objective TSPs with promising results, and the work in deep learning and reinforcement learning has been surging. This paper introduces a multi-objective deep graph pointer network-based reinforcement learning (MODGRL) algorithm for multi-objective TSPs. The MODGRL improves an earlier multi-objective deep reinforcement learning algorithm, called DRL-MOA, by utilizing a graph pointer network to learn the graphical structures of TSPs. Such improvements allow MODGRL to be trained on a small-scale TSP, but can find optimal solutions for large scale TSPs. NSGA-II, MOEA/D and SPEA2 are selected to compare with MODGRL and DRL-MOA. Hypervolume, spread and coverage over Pareto front (CPF) quality indicators were selected to assess the algorithms’ performance. In terms of the hypervolume indicator that represents the convergence and diversity of Pareto-frontiers, MODGRL outperformed all the competitors on the three well-known benchmark problems. Such findings proved that MODGRL, with the improved graph pointer network, indeed performed better, measured by the hypervolume indicator, than DRL-MOA and the three other evolutionary algorithms. MODGRL and DRL-MOA were comparable in the leading group, measured by the spread indicator. Although MODGRL performed better than DRL-MOA, both of them were just average regarding the evenness and diversity measured by the CPF indicator. Such findings remind that different performance indicators measure Pareto-frontiers from different perspectives. Choosing a well-accepted and suitable performance indicator to one’s experimental design is very critical, and may affect the conclusions. Three evolutionary algorithms were also experimented on with extra iterations, to validate whether extra iterations affected the performance. The results show that NSGA-II and SPEA2 were greatly improved measured by the Spread and CPF indicators. Such findings raise fairness concerns on algorithm comparisons using different fixed stopping criteria for different algorithms, which appeared in the DRL-MOA work and many others. Through these lessons, we concluded that MODGRL indeed performed better than DRL-MOA in terms of hypervolumne, and we also urge researchers on fair experimental designs and comparisons, in order to derive scientifically sound conclusions.

Combining Traveling Salesman and Traveling Repairman Problems: A multi-objective approach based on multiple scenarios

The single-row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange rectangular departments so as to minimize the overall interaction cost. This paper compares the different modelling approaches for (SRFLP) and applies a recent SDP approach for general quadratic ordering problems from Hungerlander and Rendl to (SRFLP). In particular, we report optimal solutions for several (SRFLP) instances from the literature with up to 42 departments that remained unsolved so far. Secondly we significantly reduce the best known gaps and running times for large instances with up to 110 departments.

Using the concept of swap operation and swap sequence on the sequence of paths of a Traveling Salesman Problem(TSP) Artificial Bee Colony (ABC) algorithm is modified to solve multi-objective TSP. The fitness of a solution is determined using a rule following the dominance property of a multi-objective optimization problem. This fitness is used for the selection process of the onlooker bee phase of the algorithm. A set of rules is used to improve the solutions in each phase of the algorithm. Rules are selected according to their performance using the roulette wheel selection process. At the end of each iteration, the parent solution set and the solution sets after each phase of the ABC algorithm are combined to select a new solution set for the next iteration. The combined solution set is divided into different non-dominated fronts and then a new solution set, having cardinality of parent solution set, is selected from the upper-level non-dominated fronts. When some solutions are required to select from a particular front then crowding distances between the solutions of the front are measured and the isolated solutions are selected for the preservation of diversity. Different standard performance metrics are used to test the performance of the proposed approach. Different sizes standard benchmark test problems from TSPLIB are used for the purpose. Test results show that the proposed approach is efficient enough to solve multi-objective TSP.

Single row facility layout problem is an important problem encountered in facility design, factory construction, production optimization, and other areas. At the same time, it is a challenging NP-hard combinatorial optimization problem that has been addressed by many advanced algorithms. In practical scenarios, real-world problems can be cast as single row facility location problem instances with different high-level properties and efficient algorithms that can solve them are sought. This work uses a variant of the self-organizing migration algorithm developed recently for permutation problems to tackle the single row facility layout problem and evaluates its accuracy and performance.

The single row facility layout problem (SRFLP) is an important combinatorial optimization problem where a given set of facilities have to be arranged in a single row to minimize the weighted sum of the distances between all pairs of facil-ities. In this paper, ahybrid method for single row facility layout problem is proposed in which, the simulated annealing (SA) is embedded in the clonal selection algorithm (CSA). The performance of the proposed algorithm is tested on benchmark problems. Computational results show the efficiency of the proposed algorithm compared to other heuristics.

Single Row Facility Layout Problem (SRFLP) consists of arranging a number of rectangular facilities with varying length on one side of a straight line to minimize the weighted sum of the distance between all facility pairs. In this paper we use a Particle Swarm Optimization (PSO) algorithm to solve the SRFLP. We first employ a new coding and decoding technique to efficiently map discrete feasible space of the SRFLP to a continuous space. The proposed PSO will further use this coding technique to explore the continuous solution space. Afterwards, the algorithm decodes the solutions to its respective feasible solution in the discrete feasible space and returns the solutions. Computational results on benchmark problems show the efficiency of the proposed algorithm compared to other heuristics.

The traveling salesman problem (TSP) is one of the most classical NP-hard problems in the combinatorial optimization, as many practical problems, such as scheduling problems and vehicle-routing cost allocation problems can be abstracted. The introduction of multiobjective in the TSP is a very important research topic, which brings serious challenges to the TSP. Currently, genetic algorithms (GAs) are one of the most effective methods to solve the multiobjective traveling salesman problem (MOTSP). However, GA-based algorithms suffer the premature convergence, the insufficient diversity, and nonuniform distribution of solutions when solving the MOTSP, which further restrict the wide application of GA-based algorithms. In order to overcome these problems, this paper proposes an improved method for GAs based on a novel evolutionary computational model, named the Physarum-inspired computational model (PCM). Based on the prior knowledge of the PCM, the initialization of the population in the proposed method is first optimized to enhance the distribution of solutions. Then, the hill climbing (HC) method is used to improve the diversity of individuals and avert running into the local optimum. Compared to the other MOTSP solving algorithms, a series of experimental results demonstrate that our proposed method achieves a better performance.

A deep reinforcement learning algorithm framework for solving multi-objective traveling salesman problem based on feature transformation

Nowadays, due to the inherent complexity of the real optimization problems, it is a challenging issue to develop a solution algorithm to these problems. Single row facility layout problem (SRFLP) is an NP-hard problem of arranging a number of rectangular facilities with varying lengths on one side of a straight line with the aim of minimizing the weighted sum of the distances between all the facility pairs. In this work, the two new algorithms cuckoo optimization (CO) and forest optimization (FO) are applied and compared to solve SRFLP for the first time. The operators of these two algorithms are adapted according to the characteristics of SRFLP, and the results obtained are compared for two groups of benchmark instances of the literature. These groups consist of instances with the number of facilities less and more than 30. The results obtained from the two groups of instances show that the proposed cuckoo optimization algorithm (COA) has a better performance than the proposed forest optimization algorithm (FOA) in both aspects of finding the best solution and the computational time.

The traveling salesman problem (TSP) is a challenging problem in combinatorial optimization. No general method of solution is known, and the problem is NP-hard. In this paper, we consider the multi-objective TSP which encompasses the optimization of two conflicting and competing objectives: here the dual minimization of the total travel distance and total travel time at various traffic flow conditions. It is well known that travellers can experience extra travel time during peak hours (i.e., congestion conditions) compared to free flow conditions (i.e., un-congested conditions), therefore and under some conditions, minimizing traveled time could conflict and compete with travel distance and vice versa. This problem has been studied in the form of a single objective problem, where either the two objectives have been combined in a single objective function or one of the objectives has been treated as a constraint. The purpose of this paper is to find a set of non-dominated solutions (i.e., the sequence of cities) using the notion of Pareto optimality where none of the objective functions can be improved in value without degrading one or more of the other objective values. The traveller then has the chance to choose a solution that fits his/her needs at each congestion level. In this paper, a multi-objective genetic algorithm (MOGA) for searching for efficient solutions is investigated. Here, an initial population composed of an approximation to the extreme supported efficient solutions is generated. A Pareto local search is then applied to all solutions of the initial population. The method is applied to a simulated problem and to a real-world problem where distances and real estimates of the travel duration for multiple origins and destinations for specific transport modes are obtained from Google Maps Platform using a Google Distance Matrix API. Results show that solving a TSP as a multi-objective optimization problem can provide more realistic solutions. The proposed approach can be used for recommending routes based on variable duration matrix and cost.

Facility layout is the arrangement of machines, equipments, or other resources in a manufacturing environment to designate an ideal configuration for minimizing the total cost by affecting the production flow. Layout design has a significant impact on the performance of manufacturing systems, and the layout problems are generally regarded as NP-Hard problems. In the literature, a considerable amount of attention is granted to biology-inspired metaheuristic algorithms in order to find efficient solutions to deal with many optimization problems. In this study, the general features and the mechanism of the Firefly Algorithm are presented initially. In order to illustrate how to adapt the proposed algorithm to a real manufacturing problem, a numerical application is shown for the solution of single row facility layout problem. A candidate solution array for 15 departments is obtained through the presumptions of the proposed algorithm. For a sample size of 500 iterations, 95 % confidence interval is constructed between the values of 8,306.53 and 8,378.22 with a standard error value of 18.288.KeywordsParticle Swarm OptimizationLayout ProblemFirefly AlgorithmMetaheuristic AlgorithmFacility Layout ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.