An efficient genetic algorithm for solving the generalized traveling salesman problem

Optimization problem like Travelling Salesman Problem (TSP) can be solved by applying Genetic Algorithm (GA) to obtain perfect approximation in time. In addition, TSP is considered as a NP-hard problem as well as an optimal minimization problem. Selection, crossover and mutation are the three main operators of GA. The algorithm is usually employed to find the optimal minimum total distance to visit all the nodes in a TSP. Therefore, the research presents a new crossover operator for TSP, allowing the further minimization of the total distance. The proposed crossover operator consists of two crossover point selection and new offspring creation by performing cost comparison. The computational results as well as the comparison with available well-developed crossover operators are also presented. It has been found that the new crossover operator produces better results than that of other cross-over operators.

The Traveling Salesman Problem (TSP) is an NP-hard graph search problem. Despite having numerous modifications of the original abstract problem, Time Dependent Traveling Salesman Problem (TD TSP) was one of the most realistic extensions under real traffic conditions. In TD TSP the edges between nodes are assigned higher costs (weights), if they were traveled during the rush hour periods, or crossed the traffic jam regions, such as the city center(s). In this paper we introduce an even more real-life motivated approach, the Intuitionistic Fuzzy Time Dependent Traveling Salesman Problem (IFTD TSP), which is a further extension of the TSP, and also of the classic TD TSP, with the additional notion of using intuitionistic fuzzy sets for the definition of uncertain costs, time, and space of the rush hour—traffic jam region affecting graph sections. In IFTD TSP we use fuzzy memberships and non-memberships sets for estimating the vague costs between nodes in order to quantify the behavior of traffic jam regions, and the rush hour periods. Since intuitionistic fuzzy sets are generalizations of classic fuzzy sets, our approach may be considered an extension and substitution of the original abstract TD TSP problem, even, of the (classic) Fuzzy TD TSP. Lastly, DBMEA (Discrete Bacterial Memetic Evolutionary Algorithm) was applied on the IFTD TSP model, the results of the simulation runs based on some extensions of the benchmarks generated from the original TD TSP data set showed quite good and promising preliminary results.

Traveling salesman problem (TSP) means that a travelling salesman needs to promote products in n cities (including the city where he lives). After visiting each city (each city can be visited once), he returns to the departure city. Let’s suppose that there is one road to connect each two cities. What is the best route to follow in order to minimize the distance of the journey? TSP has been proven to be a NP-hard problem, i.e. failure of finding a polynomial time algorithm to get a optimal solution. TSP is easy to interpret, yet hard to solve. This problem has aroused many scholars’ interests since it was put forward in 1932. However, until now, no effective solution has been found. Though TSP only represents a problem of the shortest ring road, in actual life, many physical problems are found to be the TSP. Example 1, postal route. Postal route problem is a TSP. Suppose that a mail car needs to collect mails in n places. Under such circumstances, you can show the route through a drawing containing n+1 crunodes. One crunode means a post office which this mail car departures from and returns to. The remaining n crunodes mean the crunodes at which the mails need to be collected. The route that the mail car passes through is a travelling route. We hope to find a travelling route with the shortest length. Example 2, mechanical arm. When a mechanical arm is used to fasten the nuts for the ready-to-assembling parts on the assembly line, this mechanical arm will move from the initial position (position where the first nut needs to be fastened) to each nut in proper order and then return to the initial position. The route which the mechanical arm follows is a travelling route in the drawing which contains crunodes as nuts; the most economical travelling route will enable the mechanical arm to finish its work within the shortest time. Example 3, integrated circuit. In the course of manufacturing the integrated circuit, we often need to insert thousands of electrical elements. It will consume certain energy when moving from one electrical element to the other during manufacturing. How can we do to arrange the manufacturing order to minimum the energy consumption? This is obviously a solution for TSP. Except for the above examples, problems like route distribution of transportation network, choice of tourist route, laying of pipelines needed for city planning and engineering construction are interlinked with the problems of finding the shortest route. So, it is of significance to make a study on the problem of the shortest route. This renders us a use value. As finding a solution for TSP plays an important role in the real life, since the TSP appeared, it has attracted many scholars to make a study on it.

In view of Traveling Salesman Problem (TSP) is a NP hard problem, Lion Swarm Optimization (LSO) Algorithm is a new swarm intelligence optimization algorithm. Therefore, in this paper, it uses improved hybrid LSO algorithm to solve TSP. Because the basic LSO algorithm is mainly used to solve the continuous problem, and the TSP is a discrete combinatorial optimization problem, this paper improves the LSO algorithm, introduces the greedy strategy, and combines with the Population Competitive Search (PCS) algorithm in the later stage of the search. As a result, the convergence speed is accelerated and the search efficiency is improved. The classic TSP example is tested, and the simulation results show that the algorithm proposed in this paper has strong stability and is an effective method for solving TSP.

Generalized traveling salesman problem (GTSP) is an extension of classical traveling salesman problem (TSP), which is a combinatorial optimization problem and an NP-hard problem. In this paper, an efficient discrete state transition algorithm (DSTA) for GTSP is proposed, where a new local search operator named \textit{K-circle}, directed by neighborhood information in space, has been introduced to DSTA to shrink search space and strengthen search ability. A novel robust update mechanism, restore in probability and risk in probability (Double R-Probability), is used in our work to escape from local minima. The proposed algorithm is tested on a set of GTSP instances. Compared with other heuristics, experimental results have demonstrated the effectiveness and strong adaptability of DSTA and also show that DSTA has better search ability than its competitors.

In the first part we are going to discuss about the Travelling Salesman Problem. As we are aware of that the Travelling Salesman Problem is an NP-hard problem thus it is impossible to solve it through the normal algorithm or say deterministic algorithm. If the number of cities in Travelling Salesman Problem is very small say 4-5 then we can think of solving it through brute force approach but the number of cities become 100-200 then it is impossible to comprehend as the number of possible solution or routes increases exponentially relative to the number of cities. Then we discuss about the benefits of Genetic Algorithm and how we can apply the study or use the benefit of Genetic Algorithm to solve Travelling Salesman Problem. Also we are going to go through every step of Genetic Algorithm and see very closely how all the work is done on every step. Then we are going to discuss about the background we have gone through like what kind of work or development has been done ever since in every other research papers. Then we are going to discuss about our methodology in a very detailed manner i.e. the actual method or process we have used or proposed in our paper. As we were going through it we realize as why we take some values of crossover or mutation rating as optimal for some problem not optimal for other, how do we decide it and why we do it, what difference does it make to our implementation. We have tried to answer this kind of question through our paper. And finally we go through the conclusive part of our work and talk about possibilities in future or possible work that can be done in the near future.

Objectives: A well-known NP-complete problem is the travelling salesman problem (TSP). It has numerous engineering and scientific applications. In this article, we have proposed a multi-conveyance TSP where different conveyances are present to travel from one city to another city. This is an extension to classical TSP. In this TSP, the salesman visits all the cities only once during his/her tour, using different conveyances to travel from one city to another. The cost of travelling between cities using various modes of conveyance varies. The objective of this research is to find the minimum cost tour using an ant colony optimization (ACO) based approach by satisfying the constraints of the proposed multi-conveyance TSP. Method: The considered TSP has been solved using a novel ACO technique. The proposed ACO is adopted with the Roulette-wheel selection and “tuning solution” techniques. We have used a few benchmark datasets from TSPLIB to check the effectiveness of the proposed algorithm. The experimental findings for a few benchmarks TSP instances show that almost always, the proposed ACO is able to find a better result. Then, we used some redefined and randomly generated datasets for experiments. The experimental outcomes for various input datasets are also very encouraging. Findings: The goal of the proposed TSP is to find a complete tour with a minimum cost without exceeding the total travel cost and total travel time. Thus, there are two novel constraints in the classical TSP. Novelty: A unique aspect of the proposed research is the use of several conveyance facilities and a fixed total travel time and cost. This is new, as these three factors are integrated into a single TSP model. Keywords: Travelling Salesman Problem; Travel Cost; Travel Time; MultiConveyance TSP; Ant Colony Optimization

Traveling salesman problem (TSP) is a problem of determining the shortest path for a salesman to take to visit all cities. Although a small number of cities is easy to solve, as the number of cities increases, it’s not possible to solve in polynomial time as it was a combinatorial nondeterministic polynomial (NP-hard) problem. Hence, this project is implementing a genetic algorithm (GA) to solve TSP using Python programming. The focus of this paper is to analyze the GA using order crossover (OX) and random crossover (RX) and propose a combination mechanism, direct combination (OX-RX) and Dynamic Linear combination (OXRX-Linear) to optimize TSP. We test GA for OX and RX in a random set of cities, up to 75 total cities. Then compare the result of the proposed combination OX-RX and OX-RXLinear. The result shows that both proposed combined mechanisms OX-RX and OX-RX-Linear improve the performance of GA in solving TSP.

Travelling Salesman Problem(TSP) is an NP hard problem, which has been addressed via several heuristic algorithms like genetic algorithms (GA) etc. The paper presents TSP solution using Teaching Learning Based optimization(TLBO) which includes many similarities like the other nature-inspired algorithms. In addition, the strategy adds an auxiliary operator to regulate the sequence of array elements in mutation process with partial-mapped crossover used over the selected candidates of the population. Experimental study shows that introduced strategy can provide better convergence. The prominent feature of the strategy is in its stability and superior ways to handle computational complexity while dealing with TSP optimization as compared to GA.

The traveling salesman problem (TSP) is one of the most studied problems in combinatorial optimization. Given a set of nodes and the distances between them, it consists in finding the shortest route that visits each node exactly once and returns to the first. Nevertheless, more flexible and applicable formulations of this problem exist and can be considered. The Steiner TSP (STSP) is a variant of the TSP that assumes that only a given subset of nodes must be visited by the shortest route, eventually visiting some nodes and edges more than once. In this paper, we adapt some classical TSP constructive heuristics and neighborhood structures to the STSP variant. In particular, we propose a reduced 2‐opt neighborhood and we show that it leads to better results in smaller computation times. Computational results with an implementation of a GRASP heuristic using path‐relinking and restarts are reported. In addition, ten large test instances are generated. All instances and their best‐known solutions are made available for download and benchmarking purposes.

A new heuristic algorithm is proposed by this paper, on multi-objective optimization using metaheuristics and TSP (travelling salesman problems). Basic thinking behind this algorithm is minimizing the TSP path or tour by dividing the entire tour into blocks that are overlapped to each other and then improve each individual block separately. Although it is unproven that a good solution have small improvement chances if a node moved far way to a position compared to its original solution. By intensively searching each block, further improvement is possible in TSP path or tour that never be supported in various search methods and genetic algorithm. Proposed algorithm and computational experiment performance was tested, and these tests are carried out with support of already present instances of problem. According to the results represented by paper, the computation verifies that proposed algorithm can solve TSPs efficiently. Proposed algorithm is then used for selecting optimal test cases, thousands of those test cases which are selected after confirming that they identify bugs and they itself selected from a repository of test cases; these thousand test cases are those test cases which are selected from several thousand test cases because they detect bugs. Few test cases from repository act as milestones (nodes) and having certain weight associated with each, proposed algorithm based on TSP implemented over selected result and select the optimal result or path or solution. These selected optimal test cases or selected path are further used to perform the regression testing, by applying those test cases selected by proposed algorithm in order to remove most of the faults or bugs effectively, i.e. take less time and identify almost all the bugs with few test cases. Hence this proposed algorithm assures most effective solution for regression testing test case selection.

The probabilistic travelling salesman problem (PTSP) is an extension of the well-known travelling salesman problem (TSP), which has been extensively studied in the field of combinatorial optimization. The goal of the TSP is to find the minimum length of a tour to all customers, given the distances between all pairs of customers whereas the objective of the PTSP is to minimize the expected length of the a priori tour where each customer requires a visit only with a given probability (Bertsimas, 1988; Bertsimas et al., 1990; Jaillet, 1985). The main difference between the PTSP and the TSP is that in the PTSP the probability of each node being visited is between 0.0 and 1.0 while in TSP the probability of each node being visited is 1.0. Due to the fact that the element of uncertainty not only exists, but also significantly affects the system performance in many real-world transportation and logistics applications, the results from the PTSP can provide insights into research in other probabilistic combinatorial optimization problems. Moreover, the PTSP can also be used to model many real-world applications in logistical and transportation planning, such as daily pickup-delivery services with stochastic demand, job sequencing involving changeover cost, design of retrieval sequences in a warehouse or in a cargo terminal operations, meals on wheels in senior citizen services, trip-chaining activities, vehicle routing problem with stochastic demand, and home delivery service under e-commerce (Bartholdi et al., 1983; Bertsimas et al., 1995; Campbell, 2006; Jaillet, 1988; Tang & Miller-Hooks, 2004). Early PTSP computational studies, dating from 1985, adopted heuristic approaches that were modified from the TSP (e.g., nearest neighbor, savings approach, spacefilling curve, radial sorting, 1-shift, and 2-opt exchanges) (Bartholdi & Platzman, 1988; Bertsimas, 1988; Bertsimas & Howell, 1993; Jaillet, 1985, 1987; Rossi & Gavioli, 1987). With its less than satisfactory performance in yielding solution quality, researchers in the recent years switch to metaheuristic methods, such as ant colony optimization (Bianchi, 2006; Branke & Guntsch, 2004), evolutionary algorithm (Liu et al., 2007), simulated annealing (Bowler et al., 2003), threshold accepting (Tang & Miller-Hooks, 2004) and scatter search (Liu, 2006, 2007, 2008). Because the genetic algorithm (GA), a conceptual framework of the population-based metaheuristic method, has been shown to yield promising outcomes for solving various complicated optimization problems in the past three decades (Back et al., 1997; Davis, 1991; O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems, widely studied due to its numerous industrial applications in logistics, supply chain management, and resource planning. Over the years, various metaheuristic algorithms have been developed to provide near-optimal solutions within a reasonable computational time. While traditional methods such as Genetic Algorithms (GA) and Simulated Annealing (SA) have been applied widely to the TSP, recent developments in metaheuristics have introduced new methods that can ensure improved performance. In this study, we compare three new metaheuristics: Arctic Puffin Optimization (APO), Human Evolutionary Optimization (HEO), and Ship Rescue Optimization (SRO). These methods, inspired by puffin's behavior, human evolutions, and naval rescue, respectively, have demonstrated high optimization capability in various areas. However, these have not been extensively tried on solving the TSP yet. This research seeks to establish the performance of these algorithms by comparing their convergence rate, solution quality, and computational efficiency with well-established metaheuristics. The results of the experiment expose the strengths and weaknesses of such innovative approaches, showing their potential as competitive alternatives to solve complex combinatorial optimization problems

Traveling salesman problem (TSP) is a well-known NP-hard combinatorial optimization problem. It has been solved by a number of exact and approximate algorithms and serves as a testbed for new heuristic and metaheuristic optimization algorithms. However, it is often not easy to evaluate the hardness (complexity) of a TSP instance. Simple measures such as the number of cities or the minimum (maximum) route length do not capture the internal structure of a TSP instance sufficiently. In this work, we propose a new method for the assessment of TSP instance complexity based on clustering. The new approach is evaluated on a set of randomized TSP instances with different structure and its relation to the performance of a selected metaheuristic TSP solver is studied.

Genetic algorithms (GAs) are search algorithms based on population genetics and natural selection concepts. Maintaining population variety in GAs is critical for ensuring global exploration and mitigating the risks of premature convergence. Rapid convergence to local optima is one such challenge in the application of genetic algorithms. To address this issue, we provide Cave-Surface GA (CSGA), an alternative method based on the Dual Population Genetic Algorithm and inspired by the genetic variety observed in Mexican cavefish. Through inter-population crossbreeding, CSGA increases diversity via a secondary population (Cave population) and facilitates the exchange of information between populations, effectively counteracting premature convergence. Several experiments are carried out utilizing benchmark instances of the Traveling Salesman Problem (TSP) obtained from TSPLIB, a well-known TSP problem library. Our experimental results over many TSP instances show that CSGA outperforms both classic GAs and other GAs that use diversity preservation techniques, such as Multipopulation GA (MPGA). CSGA has the potential to give promising solutions to challenging optimization issues like TSP.