Bi-objective Traveling Salesman Problem Research Articles

Deep reinforcement learning for multi-objective combinatorial optimization: A case study on multi-objective traveling salesman problem

Multi-objective combinatorial optimization problems (MOCOPs) widely exist in real applications, and most of them are computationally difficult or NP-hard. How to solve MOCOPs efficiently has been a challenging issue. The heuristic algorithms have achieved good results on MOCOPs, while they require careful hand-crafted heuristics and iterative computing for the solutions. Recently, deep reinforcement learning (DRL) has been employed to solve combinatorial optimization problems, and many DRL-based algorithms have been proposed with promising results. However, it is difficult for these existing algorithms to obtain diverse solutions efficiently for MOCOPs. In this paper, we propose an algorithm named MOMDAM to solve MOCOPs. In MOMDAM, the attention model (AM) is used and can simply modify the encoder to facilitate the construction of solutions with any weight vector, as well as the multiple decoders (MD) are employed to obtain diverse policies to further improve the diversity and convergence of the solutions. Experimental results on the bi-objective traveling salesman problem show that, MOMDAM significantly outperforms some state-of-the-art algorithms in terms of solution quality and running time.

Generalized nash fairness solutions for bi‐objective minimization problems

AbstractIn this article, we consider a particular case of bi‐objective optimization (BOO), called bi‐objective minimization (BOM), where the two objective functions to be minimized take only positive values. As well as for BOO, most of the methods proposed in the literature for solving BOM focus on computing the Pareto‐optimal solutions representing different trade‐offs between two objectives. However, it may be difficult for a central decision‐maker to determine the preferred solutions due to the huge number of solutions in the Pareto set. We propose a novel criterion for selecting the preferred Pareto‐optimal solutions by introducing the concept of ‐Nash Fairness (‐) solutions inspired by the definition of proportional fairness. The ‐ solutions are the feasible solutions achieving some proportional nash equilibrium between the two objectives. The positive parameter is introduced to reflect the relative importance of the first objective to the second one. For this work, we will discuss existential and algorithmic questions about the ‐ solutions by first showing their existence for BOM. Furthermore, the ‐ solution set can be a strict subset of the Pareto set. As there are possibly many ‐ solutions, we focus on extreme ‐ solutions achieving the smallest values for one of the objectives. Then, we propose two Newton‐based iterative algorithms for finding extreme ‐ solutions. Finally, we present computational results on some instances of the bi‐objective travelling salesman problem (BOTSP) and the bi‐objective shortest path problem.

Novel hybrid evolutionary algorithm for bi-objective optimization problems

This work considers the Bi-objective Traveling Salesman Problem (BTSP), where two conflicting objectives, the travel time and monetary cost between cities, are minimized. Our purpose is to compute the trade-off solutions that fulfill the problem requirements. We introduce a novel three-Phase Hybrid Evolutionary Algorithm (3PHEA) based on the Lin–Kernighan Heuristic, an improved version of the Non-Dominated Sorting Genetic Algorithm, and Pareto Variable Neighborhood Search, a multi-objective version of VNS. We conduct a comparative study with three existing approaches dedicated to solving BTSP. To assess the performance of algorithms, we consider 20 BTSP instances from the literature of varying degrees of difficulty (e.g., euclidean, random, mixed, etc.) and different sizes ranging from 100 to 1000 cities. We also compute several multi-objective performance indicators, including running time, coverage, hypervolume, epsilon, generational distance, inverted generational distance, spread, and generalized spread. Experimental results and comparative analysis indicate that the proposed three-phase method 3PHEA is significantly superior to existing approaches covering up to 80% of the true Pareto fronts.

A Grid-Based Two-Stage Parallel Matching Framework for Bi-Objective Euclidean Traveling Salesman Problem

Traveling salesman problem (TSP) is one of the most studied combinatorial optimization problems; several exact, heuristic or even learning-based strategies have been proposed to solve this challenging issue. Targeting on the research problem of bi-objective non-monotonic Euclidean TSP and based on the concept of the multi-agent-based approach, we propose a two-stage parallel matching approaching for solving TSP. Acting as a divide-and-conquer strategy, the merit lies in the simultaneously clustering and routing in the dividing process. Precisely, we first propose the Two-Stage Parallel Matching algorithm (TSPM) to deal with the bi-objective TSP. We then formulate the Grid-Based Two-Stage Parallel Matching (GRAPE) framework, which can synergize with TSPM, exact method, or other state-of-the-art TSP solvers, for solving large-scale Euclidean TSP. According to this framework, the original problem space is divided into smaller regions and then computed in parallel, which helps to tackle and derive solutions for larger-scale Euclidean TSP within reasonable computational resources. Preliminary evaluation based on TSPLIB testbed shows that our proposed GRAPE framework holds a decent quality of solutions in especially runtime for large-scale Euclidean TSP. Meanwhile, experiments conducted on two real-world datasets demonstrate the efficacy and adaptability of our proposed TSPM in solving the bi-objective non-monotonic TSP.

A new artificial bee colony algorithm using a gradual weight method for the bi-objective traveling salesman problems

A new artificial bee colony algorithm using a gradual weight method for the bi-objective traveling salesman problems

Analysis of multiobjective evolutionary algorithms on the biobjective traveling salesman problem (1,2)

Multiobjective evolutionary algorithms have been successfully used to deal with multiobjective combinatorial optimization problems for more than two decades. However, we know little about the performance of multiobjective evolutionary algorithms on multiobjective combinatorial optimization problems in theory so far, especially on NP-hard ones from real-world, since Pareto curves are often of exponential size, meanwhile, evolutionary algorithms rely heavily on the use of randomness and are hard to understand from a theoretical point of view. In this paper, we theoretically investigate the performance of two simple multiobjective evolutionary algorithms with different population diversity mechanisms on the biobjective traveling salesman problem (1,2). It is found that one of them can efficiently find a $\frac {3}{2}$ -approximation Pareto curve for the problem, the best result so far. At the same time, these two multiobjective evolutionary algorithms are proved to be superior to a multiobjective local search algorithm, and the multiobjective local search algorithm is also proven to outperform these two multiobjective evolutionary algorithms as well. Finally, the population diversity is proved to be helpful in reducing the expected runtime of multiobjective evolutionary algorithm.

A novel multi-objective evolutionary algorithm based on subpopulations for the bi-objective traveling salesman problem

This paper presents a new approach called MOEA/NSM (multi-objective evolutionary algorithm integrating NSGA-II, SPEA2 and MOEA/D features). This paper combines the main characteristics of the NSGA-II, SPEA2 and MOEA/D algorithms, and also including 2-opt local search technique to improve the objective space search. The MOEA/NSM algorithm was compared to the other classical approaches using 9 datasets for the bi-objective traveling salesman problem. In addition, experiments were carried out applying the local search in the classical approaches, resulting in a considerable improvement in the results for these algorithms. From the Pareto frontiers resulting from the experiments, we applied the evaluation metrics by hypervolume, Epsilon ( $$\epsilon $$ ), EAF and Shapiro–Wilk statistical hypothesis test. The results showed a better performance of the MOEA/NSM when compared with NSGA-II, SPEA2 and MOEA/D.

A hybrid genetic algorithm for solving bi-objective traveling salesman problems

The traveling salesman problem (TSP) is a typical combinatorial optimization problem, in a traditional TSP only tour distance is taken as a unique objective to be minimized. When more than one optimization objective arises, the problem is known as a multi-objective TSP. In the present paper, a bi-objective traveling salesman problem (BOTSP) is taken into account, where both the distance and the cost are taken as optimization objectives. In order to efficiently solve the problem, a hybrid genetic algorithm is proposed. Firstly, two satisfaction degree indices are provided for each edge by considering the influences of the distance and the cost weight. The first satisfaction degree is used to select edges in a “rough” way, while the second satisfaction degree is executed for a more “refined” choice. Secondly, two satisfaction degrees are also applied to generate new individuals in the iteration process. Finally, based on genetic algorithm framework as well as 2-opt selection strategy, a hybrid genetic algorithm is proposed. The simulation illustrates the efficiency of the proposed algorithm.

Proper balance between search towards and along Pareto front: biobjective TSP case study

In this paper we propose simple yet efficient version of the two-phase Pareto local search (2PPLS) for solving the biobjective traveling salesman problem (bTSP). In the first phase the powerful Lin–Kernighan heuristic is used to generate some high quality solutions being very close to the Pareto front. Then Pareto local search is used to generate more potentially Pareto efficient solutions along the Pareto front. Instead of previously used method of Aneja and Nair we use uniformly distributed weight vectors in the first phase. We show experimentally that properly balancing the computational effort in the first and second phase we can obtain results better than previous versions of 2PPLS for bTSP and at least comparable to the state-of-the art results of more complex MOMAD method. Furthermore, we propose a simple extension of 2PPLS where some additional solutions are generated by Lin–Kernighan heuristic during the run of PLS. In this way we obtain a method that is more robust with respect to the number of initial solutions generated in the first phase.

Multi-Objective Ant Colony Optimization Based on the Physarum-Inspired Mathematical Model for Bi-Objective Traveling Salesman Problems.

Bi-objective Traveling Salesman Problem (bTSP) is an important field in the operations research, its solutions can be widely applied in the real world. Many researches of Multi-objective Ant Colony Optimization (MOACOs) have been proposed to solve bTSPs. However, most of MOACOs suffer premature convergence. This paper proposes an optimization strategy for MOACOs by optimizing the initialization of pheromone matrix with the prior knowledge of Physarum-inspired Mathematical Model (PMM). PMM can find the shortest route between two nodes based on the positive feedback mechanism. The optimized algorithms, named as iPM-MOACOs, can enhance the pheromone in the short paths and promote the search ability of ants. A series of experiments are conducted and experimental results show that the proposed strategy can achieve a better compromise solution than the original MOACOs for solving bTSPs.

Anytime Pareto local search

Pareto Local Search (PLS) is a simple and effective local search method for tackling multi-objective combinatorial optimization problems. It is also a crucial component of many state-of-the-art algorithms for such problems. However, PLS may be not very effective when terminated before completion. In other words, PLS has poor anytime behavior. In this paper, we study the effect that various PLS algorithmic components have on its anytime behavior. We show that the anytime behavior of PLS can be greatly improved by using alternative algorithmic components. We also propose Dynagrid, a dynamic discretization of the objective space that helps PLS to converge faster to a good approximation of the Pareto front and continue to improve it if more time is available. We perform a detailed empirical evaluation of the new proposals on the bi-objective traveling salesman problem and the bi-objective quadratic assignment problem. Our results demonstrate that the new PLS variants not only have significantly better anytime behavior than the original PLS, but also may obtain better results for longer computation time or upon completion.

MOEA/D-ACO: A Multiobjective Evolutionary Algorithm Using Decomposition and AntColony

Combining ant colony optimization (ACO) and the multiobjective evolutionary algorithm (EA) based on decomposition (MOEA/D), this paper proposes a multiobjective EA, i.e., MOEA/D-ACO. Following other MOEA/D-like algorithms, MOEA/D-ACO decomposes a multiobjective optimization problem into a number of single-objective optimization problems. Each ant (i.e., agent) is responsible for solving one subproblem. All the ants are divided into a few groups, and each ant has several neighboring ants. An ant group maintains a pheromone matrix, and an individual ant has a heuristic information matrix. During the search, each ant also records the best solution found so far for its subproblem. To construct a new solution, an ant combines information from its group's pheromone matrix, its own heuristic information matrix, and its current solution. An ant checks the new solutions constructed by itself and its neighbors, and updates its current solution if it has found a better one in terms of its own objective. Extensive experiments have been conducted in this paper to study and compare MOEA/D-ACO with other algorithms on two sets of test problems. On the multiobjective 0-1 knapsack problem,MOEA/D-ACO outperforms the MOEA/D with conventional genetic operators and local search on all the nine test instances. We also demonstrate that the heuristic information matrices in MOEA/D-ACO are crucial to the good performance of MOEA/D-ACO for the knapsack problem. On the biobjective traveling salesman problem, MOEA/D-ACO performs much better than the BicriterionAnt on all the 12 test instances. We also evaluate the effects of grouping, neighborhood, and the location information of current solutions on the performance of MOEA/D-ACO. The work in this paper shows that reactive search optimization scheme, i.e., the "learning while optimizing" principle, is effective in improving multiobjective optimization algorithms.

A Novel, Evolutionary, Simulated Annealing inspired Algorithm for the Multi-Objective Optimization of Combinatorial Problems

This paper states a novel hybrid-metaheuristic based on deterministic swapping, evolutionary algorithms and simulated annealing inspired algorithms for the multi-objective optimization of combinatorial problems. The proposed algorithm is named EMSA. It is an improvement of MODS algorithm. Unlike MODS, EMSA works using a search direction given by the assignation of weights to each objective function of the combinatorial problem to optimize. Lastly, EMSA is tested using well know instances of the Bi-Objective Traveling Salesman Problem (BTSP) from TSPLIB. Its results were compared with MODS metaheuristic (its predecessor). The comparison was made using metrics from the specialized literature such as Spacing, Generational Distance, Inverse Generational Distance and Non-Dominated Generation Vectors. In every case, the EMSA results on the metrics were always better and in some of those cases, the superiority was 100%.

Multi-objective ant colony optimization based on decomposition for bi-objective traveling salesman problems

This paper proposes a framework named multi-objective ant colony optimization based on decomposition (MoACO/D) to solve bi-objective traveling salesman problems (bTSPs). In the framework, a bTSP is first decomposed into a number of scalar optimization subproblems using Tchebycheff approach. To suit for decomposition, an ant colony is divided into many subcolonies in an overlapped manner, each of which is for one subproblem. Then each subcolony independently optimizes its corresponding subproblem using single-objective ant colony optimization algorithm and all subcolonies simultaneously work. During the iteration, each subproblem maintains an aggregated pheromone trail and an aggregated heuristic matrix. Each subcolony uses the information to solve its corresponding subproblem. After an iteration, a pheromone trail share procedure is evoked to realize the information share of those subproblems solved by common ants. Three MoACO algorithms designed by, respectively, combining MoACO/D with AS, MMAS and ACS are presented. Extensive experiments conducted on ten bTSPs with various complexities manifest that MoACO/D is both efficient and effective for solving bTSPs and the ACS version of MoACO/D outperforms three well-known MoACO algorithms on large bTSPs according to several performance measures and median attainment surfaces.

Evolutionary algorithms for solving multi-objective travelling salesman problem

This paper studies the application of evolutionary algorithms for bi-objective travelling salesman problem. Two evolutionary algorithms, including estimation of distribution algorithm (EDA) and genetic algorithm (GA), are considered. The solution to this problem is a set of trade-off alternatives. The problem is solved by optimizing the order of the cities so as to simultaneously minimize the two objectives of travelling distance and travelling cost incurred by the travelling salesman. In this paper, binary-representation-based evolutionary algorithms are replaced with an integer-representation. Three existing EDAs are altered to use this integer-representation, namely restricted Boltzmann machine (RBM), univariate marginal distribution algorithm (UMDA), and population-based incremental learning (PBIL). Each city is associated with a representative integer, and the probability of any of this representative integer to be located in any position of the chromosome is constructed through the modeling approach of the EDAs. New sequences of cities are obtained by sampling from the probabilistic model. A refinement operator and a local search operator are proposed in this piece of work. The EDAs are subsequently hybridized with GA in order to complement the limitations of both algorithms. The effect that each of these operators has on the quality of the solutions are investigated. Empirical results show that the hybrid algorithms are capable of finding a set of good trade-off solutions.

504 分解法に基づく2目的巡回セールスマン問題の解法とその応用(OS9-1 時代を進める「もの・こと」つくりのシステム情報化術(1),OS9 時代を進める「もの・こと」つくりのシステム情報化術)

504 分解法に基づく2目的巡回セールスマン問題の解法とその応用(OS9-1 時代を進める「もの・こと」つくりのシステム情報化術(1),OS9 時代を進める「もの・こと」つくりのシステム情報化術)

Two-phase Pareto local search for the biobjective traveling salesman problem

In this work, we present a method, called Two-Phase Pareto Local Search, to find a good approximation of the efficient set of the biobjective traveling salesman problem. In the first phase of the m...

Speed-up techniques for solving large-scale biobjective TSP

In this paper, we present the Two-Phase Pareto Local Search ( 2PPLS) method with speed-up techniques for the heuristic resolution of the biobjective traveling salesman problem. The 2PPLS method is a state-of-the-art method for this problem. However, because of its running time that strongly grows with the instances size, the method can be hardly applied to instances with more than 200 cities. We thus adapt some speed-up techniques used in single-objective optimization to the biobjective case. The proposed method is able to solve instances with up to 1000 cities in a reasonable time with no, or very small, reduction of the quality of the generated approximations.

A study with neighborhood searches to deal with multiobjective unconstrained permutation problems

This paper introduces a study with neighborhood search algorithms to deal with unconstrained multiobjective permutation problems. Filter-and-fan/path relinking approach designed by us, and the stochastic local search (SLS) developed by Paquete and Stutzle [22], implemented by us, are compared using as study cases the bi-objective quadratic assignment problem, and the bi-objective travelling salesman problem. Our approach is also compared with results published for bi-objective quadratic assignment problem, bi-objective flow shop problem, bi-objective and tri-objective travelling salesman problems. The results obtained show that the filter-and-fan/path relinking approach seems to be promising to tackle multiobjective permutation problems, achieving good and wide distributed approximations to the Pareto-optimal front.

Particle Swarm Optimization applied to the bi-objective Travelling Salesman Problem.

Particle Swarm Optimization applied to the bi-objective Travelling Salesman Problem.