Multi-objective Traveling Salesman Problem Research Articles

New techniques to improve neighborhood exploration in pareto local search

Pareto Local Search (PLS) is an extension of local search to multiobjective combinatorial optimization problems (MCOPs). It is widely used as a building block in many multiobjective metaheuristics. However, it suffers from poor anytime behavior and long convergence time. In this paper, we focus on improving the neighborhood exploration mechanism to accelerate PLS. In total, we propose three new search strategies, namely, Promising List, Don’t Look Bits and Continuing Search and implement them on the multiobjective Traveling Salesman Problem (mTSP) and the multiobjective Unconstrained Binary Quadratic Programming (mUBQP) problem. In Promising List, features of problem structure are utilized to guide the local search. In Don’t Look Bits, neighborhood moves that are less promising to produce improving solutions are skipped. In Continuing Search, the agent will start from where previous neighborhood exploration stops to avoid repetitive neighborhood moves. The experimental results verify the effectiveness of the three techniques both individually and collaboratively on most of the test instances. Besides, one proposed PLS variant outperforms two state-of-the-art PLS-based algorithms.

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

As a special type of multi-objective combinatorial optimization problems (MOCOPs), the multi-objective traveling salesman problem (MOTSP) plays an important role in practical fields such as transportation and robot control. However, due to the complexity of its solution space and the conflicts between different objectives, it is difficult to obtain satisfactory solutions in a short time. This paper proposes an end-to-end algorithm framework for solving MOTSP based on deep reinforcement learning (DRL). By decomposing strategies, solving MOTSP is transformed into solving multiple single-objective optimization subproblems. Through linear transformation, the features of the MOTSP are combined with the weights of the objective function. Subsequently, a modified graph pointer network (GPN) model is used to solve the decomposed subproblems. Compared with the previous DRL model, the proposed algorithm can solve all the subproblems using only one model without adding weight information as input features. Furthermore, our algorithm can output a corresponding solution for each weight, which increases the diversity of solutions. In order to verify the performance of our proposed algorithm, it is compared with four classical evolutionary algorithms and two DRL algorithms on several MOTSP instances. The comparison shows that our proposed algorithm outperforms the compared algorithms both in terms of training time and the quality of the resulting solutions.

Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this paper, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto optimal solutions. Then we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.

Multiobjective Combinatorial Optimization Using a Single Deep Reinforcement Learning Model.

This article proposes utilizing a single deep reinforcement learning model to solve combinatorial multiobjective optimization problems. We use the well-known multiobjective traveling salesman problem (MOTSP) as an example. Our proposed method employs an encoder-decoder framework to learn the mapping from the MOTSP instance to its Pareto-optimal set. Specifically, it leverages a novel routing encoder to extract information for both the entire multiobjective aspect and every individual objective from the MOTSP instance. The global embeddings and each objective's embeddings are adaptively aggregated via a routing network to form the subproblems' embedding that can well represent the MOTSP features. Using a modified context embedding, the subproblems' embeddings are fed into a decoder to produce a set of approximate Pareto-optimal solutions in parallel. Additionally, we develop a Top-k baseline to enable more efficient data utilization and lightweight training for our proposed method. We compare our method with heuristic-based and learning-based ones on various types of MOTSP instances, and the experimental results show that our method can solve MOTSP instances in real-time and outperform the other algorithms, especially on large-scale problem instances.

Solution of Uncertain Constrained Multi-Objective Travelling Salesman Problem with Aspiration Level Based Multi Objective Quasi Oppositional Jaya Algorithm

Solution of Uncertain Constrained Multi-Objective Travelling Salesman Problem with Aspiration Level Based Multi Objective Quasi Oppositional Jaya Algorithm

Enhancing the Dhouib-Matrix-4 metaheuristic to generate the Pareto non-dominated set solutions for multi-objective travelling salesman problem: The DM4-PMO method

This paper focuses on the multi-objective Travelling Salesman Problem for which the aim is to find the set of efficient solutions. To obtain this Pareto set's solutions, a novel metaheuristic named DM4-PMO is proposed. The DM4-PMO is based on the first enhancement of the new Dhouib-Matrix-4 (DM4) method and composed of two steps. First, a weighted sum function with a multi variation of weights is used to find the first non-dominated pareto frontier solutions. Second, a lexicographical resolution is applied in some non-dominated solutions found in the first Pareto frontier to generate the final Pareto frontier solutions. The performance of the proposed approach is demonstrated by the experiment on two-objective problems that are taken from TSP-LIB and DIMACS datasets. The test results show that the proposed DM4-PMO is robust, fast, and simply structured, and obtain a Pareto non-dominated set solutions in short computational times using very few user-defined parameters.

Meta-Learning-Based Deep Reinforcement Learning for Multiobjective Optimization Problems.

Deep reinforcement learning (DRL) has recently shown its success in tackling complex combinatorial optimization problems. When these problems are extended to multiobjective ones, it becomes difficult for the existing DRL approaches to flexibly and efficiently deal with multiple subproblems determined by the weight decomposition of objectives. This article proposes a concise meta-learning-based DRL approach. It first trains a meta-model by meta-learning. The meta-model is fine-tuned with a few update steps to derive submodels for the corresponding subproblems. The Pareto front is then built accordingly. Compared with other learning-based methods, our method can greatly shorten the training time of multiple submodels. Due to the rapid and excellent adaptability of the meta-model, more submodels can be derived so as to increase the quality and diversity of the found solutions. The computational experiments on multiobjective traveling salesman problems and multiobjective vehicle routing problems with time windows demonstrate the superiority of our method over most of the learning-based and iteration-based approaches.

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.

On Enhanced Intelligent Water Drops Algorithm for Travelling Salesman Problem under Uncertain Paradigm

Abstract Travelling salesman problem (TSP) is a well known combinatorial optimization problem which has drawn colossal attention due to its eclectic range of applications. In this article, we have proposed two modified versions of intelligent water drops (IWD) algorithm. The first one is the enhanced IWD (e-IWD) algorithm to solve single objective TSP. In the second modification, e-IWD algorithm has been extended to enhanced multi-objective IWD(e-MIWD) algorithm for solving multi-objective TSP. In order to achieve a better exploration capability in both of the proposed algorithms, the soil and velocity parameters of a randomly selected water drop are updated after every iteration of the algorithm when it traverses all the intermediate vertices for a tour. The proposed algorithms have been compared with some other existing similar algorithms on different benchmark instances of TSPs. Furthermore, we have addressed the TSP for both single and multiple objectives under uncertain environment.

A Graph Pointer Network-Based Multi-Objective Deep Reinforcement Learning Algorithm for Solving the Traveling Salesman Problem

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.

Solving a Multi-Objective Travelling Salesman Problem by Genetic Algorithm Based Approach

Solving a Multi-Objective Travelling Salesman Problem by Genetic Algorithm Based Approach

Multi-objective quasi oppositional Jaya algorithm to solve multi-objective solid travelling salesman problem with different aspiration level

Multi-Objective Travelling Salesman Problem (MOTSP) is one of the most crucial problems in realistic scenarios and is difficult to solve by classical methods. However, it can be solved by evolutionary methods. This paper presents an Aspiration Level based Multi-Objective Quasi Oppositional Jaya (AL-based MOQO-Jaya) Algorithm for solving the Multi-Objective Solid Travelling Salesman Problem (MOSTSP). Furthermore, fuzzy judgment was characterised by utilizing the possibility and necessity measures to allow the decision-maker (DM) for optimising different scenarios of the Fuzzy Multi-Objective Solid Travelling Salesman Problem (FMOSTSP). A numerical illustration is provided for 10, 80, 100 and 120 cities, and sensitivity analysis is performed with different shape parameters and aspiration levels. Further the results are compared by CPLEX optimizer and Hybrid GA. The results obtained by AL-based MOQO Jaya are more efficient, the run time of AL-based MOQO is 5.8667, 40.9115, 58.4789 and 60.6882 seconds for 10, 80,100 and 120 cities for FMOSTSP which is quite less as compared to CPLEX and Hybrid GA. To access the performance of the proposed method coverage and hypervolume are calculated. This paper concludes that developed approach has solved MOSTSP and FMOSTSP efficiently with an effective output and provides alternative solutions for decision-making to DM.

Solving the multi-objective Hamiltonian cycle problem using a Branch-and-Fix based algorithm

The Hamiltonian cycle problem consists of finding a cycle in a given graph that passes through every single vertex exactly once, or determining that this cannot be achieved. In this investigation, a graph is considered with an associated set of matrices. The entries of each of the matrix correspond to a different weight of an arc. A multi-objective Hamiltonian cycle problem is addressed here by computing a Pareto set of solutions that minimize the sum of the weights of the arcs for each objective. Our heuristic approach extends the Branch-and-Fix algorithm, an exact method that embeds the problem in a stochastic process. To measure the efficiency of the proposed algorithm, we compare it with a multi-objective genetic algorithm in graphs of a different number of vertices and density. The results show that the density of the graphs is critical when solving the problem. The multi-objective genetic algorithm performs better (quality of the Pareto sets) than the proposed approach in random graphs with high density; however, in these graphs it is easier to find Hamiltonian cycles, and they are closer to the multi-objective traveling salesman problem. The results reveal that, in a challenging benchmark of Hamiltonian graphs with low density, the proposed approach significantly outperforms the multi-objective genetic algorithm.

A plan for transportation and distribution the products based on multi-objective travelling salesman problem in fuzzy environmental

Transportation and distribution are the most important elements in the work system for any company, which are of great importance in the success of the chain work. Al-Rabee factory is one of the largest ice cream factories in Iraq and it is considered one of the most productive and diversified factories with products where its products cover most areas of the capital Baghdad, however, it lacks a distribution system based on scientific and mathematical methods to work in the transportation and distribution processes, moreover, these processes need a set of important data that cannot in any way be separated from the reality of fuzziness industrial environment in Iraq, which led to use the fuzzy sets theory to reduce the levels of uncertainty. The decision-maker has several goals that he aspires to accomplish for two stages, so, the decision-maker adopted in his work system on a multi-objective travelling salesman problem. A network of paths for transportation and distribution of the products has been designed based on a multi-objective travelling salesman problem, by building a mathematical model that finds the best paths for each stage, taking into account the goals required by the decision-maker. The results obtained from the use of (Lingo) software showed the importance of these methods in determining the optimal path for the processes of collecting and transporting milk from their collection centers to the Al-Rabee factory as a first stage, as well as transporting the final products and distributing them from the Al-Rabee factory to the shopping centers as a second stage.

Evolutionary algorithm hybridized with local search and intelligent seeding for solving multi-objective Euclidian TSP

Multi-objective Euclidian TSP (ETSP) has several practical applications such as mobile computing and maritime surveillance. This problem has complexity not only in terms of combinatorial constraints but also in terms of multiple objectives. The local heuristic-based algorithms are extremely efficient in solving the single-objective ETSPs. However, their efficacy is limited in solving the multi-objective ETSPs due to the presence of multiple objectives. To bridge this gap, a two-stage evolutionary algorithm (TSEA) is developed to solve multi-objective ETSPs. In this algorithm, the first stage involves the use of a hybrid local search evolutionary algorithm (HLS-EA) which incorporates the local heuristic of nearest neighbor and 2-opt in the framework of real-coded NSGA-II to solve the individual objectives of multi-objective ETSP. These individual single-objective solutions which represent the corner solutions of the Pareto optimal front are used in the second stage as seed solutions in seeded HLS-EA (SHLS-EA) for solving the corresponding multi-objective ETSP. The developed algorithm is tested on 17 two-objective, 6 three-objective, and 2 four-objective ETSPs to show the superior performance over multi-objective variants of the Lin-Kernighan algorithm. Also, the developed algorithm is compared with several variants of DE and GA to illustrate the superior performance over the variants of evolutionary algorithms. Further, the algorithm is extended to the multi-objective ETSPs up to 10,000 cities.

Safe distance-based vehicle routing problem: Medical waste collection case study in COVID-19 pandemic

In addition to the increasing population and rapid urbanization, the amount and variety of medical waste are rapidly increasing due to the coronavirus disease (COVID-19) pandemic affecting the whole world. COVID-19 does not only increase the amount of medical waste produced, medical wastes generated in the care of COVID-19 carries a high risk of transmission as well. In this regard, the safe and effective management of medical wastes has become a serious health and safety issue. This research aims to determine the safest and shortest transportation routes for medical waste vehicles. The safety scores used in this study were obtained in our previous study. The resulting safety scores were used in a multi-objective traveling salesman problem for deriving two objective functions, which are based on safety scores and total transportation distance. A conciliating solution was obtained by solving this linear programming model. The proposed model faced by health institutions in Istanbul has been applied for a specific district. According to the obtained results, suggestions for the direction of medical waste vehicles have been proposed.

Evolutionary Algorithm Using Random Immigrants for the Multiobjective Travelling Salesman Problem

This paper addresses the Multiobjective Travelling Salesman Problem (MoTSP) with the aim to study the effects of including random immigrants in the population of solutions processed by the evolutionary algorithm. Random immigrants are typically used in evolutionary optimization in order to increase the diversity of the population and to allow the algorithm to explore a larger area of the search space. Introducing random immigrants incurs a certain overhead which is especially significant in combinatorial optimization, because local search procedures are usually employed, which, while effective in improving the solutions, are computationally expensive. In this paper several strategies of introducing new specimens are tested with the aim of improving the effectiveness of the optimization process given a limited computation time. In the experiments the proposed approach was tested on kroABnnn instances of the MoTSP. It was found to improve the results of multiobjective optimization in terms of both the hy-pervolume and the IGD indicators. The most effective immigration strategy turned out to be to decrease the number of immigrants with time.

Multi-objective traveling salesman problem: an ABC approach

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.

An Interactive Regret-Based Genetic Algorithm for Solving Multi-Objective Combinatorial Optimization Problems

We propose a new approach consisting in combining genetic algorithms and regret-based incremental preference elicitation for solving multi-objective combinatorial optimization problems with unknown preferences. For the purpose of elicitation, we assume that the decision maker's preferences can be represented by a parameterized scalarizing function but the parameters are initially not known. Instead, the parameter imprecision is progressively reduced by asking preference queries to the decision maker during the search to help identify the best solutions within a population. Our algorithm, called RIGA, can be applied to any multi-objective combinatorial optimization problem provided that the scalarizing function is linear in its parameters and that a (near-)optimal solution can be efficiently determined when preferences are known. Moreover, RIGA runs in polynomial time while asking no more than a polynomial number of queries. For the multi-objective traveling salesman problem, we provide numerical results showing its practical efficiency in terms of number of queries, computation time and gap to optimality.

Deep Reinforcement Learning for Multiobjective Optimization.

This article proposes an end-to-end framework for solving multiobjective optimization problems (MOPs) using deep reinforcement learning (DRL), that we call DRL-based multiobjective optimization algorithm (DRL-MOA). The idea of decomposition is adopted to decompose the MOP into a set of scalar optimization subproblems. Then, each subproblem is modeled as a neural network. Model parameters of all the subproblems are optimized collaboratively according to a neighborhood-based parameter-transfer strategy and the DRL training algorithm. Pareto-optimal solutions can be directly obtained through the trained neural-network models. Specifically, the multiobjective traveling salesman problem (MOTSP) is solved in this article using the DRL-MOA method by modeling the subproblem as a Pointer Network. Extensive experiments have been conducted to study the DRL-MOA and various benchmark methods are compared with it. It is found that once the trained model is available, it can scale to newly encountered problems with no need for retraining the model. The solutions can be directly obtained by a simple forward calculation of the neural network; thereby, no iteration is required and the MOP can be always solved in a reasonable time. The proposed method provides a new way of solving the MOP by means of DRL. It has shown a set of new characteristics, for example, strong generalization ability and fast solving speed in comparison with the existing methods for multiobjective optimizations. The experimental results show the effectiveness and competitiveness of the proposed method in terms of model performance and running time.