a b s t r a c t
This paper proposed an improved version of the Particle Swarm Optimization (PSO) approach to solve **Traveling** **Salesman** **Problems** (TSP). This **evolutionary** **algorithm** includes two phases. The first phase includes Fuzzy C -Means clustering, a rule-based route permutation, a random swap strategy and a cluster merge procedure. This approach firstly generates an initial non-crossing route, such that the TSP can be solved more efficiently by the proposed PSO **algorithm**. The use of sub-cluster also reduces the complexity and achieves better performance for **problems** with a large number of cities. The proposed Genetic-based PSO procedure is then applied to solve the TSP with better efficiency in the second phase. The proposed Genetic-based PSO procedure is applied to TSPs with better efficiency. Fixed runtime performance was used to demonstrate the efficiency of the proposed **algorithm** for the cases with a large number of cities.

Show more
10 Read more

To speed up the solution evaluation, it is crucial to implement efficient feasi- bility check algorithms with respect to temporal and physical incompatibilities. Savelsbergh (1992) proposed fundamental algorithms for handling time windows in routing **problems**, later improved versions by Cornillier et al (2009). In this pa- per, we implemented two algorithms: T W F easibility Check and W aiting T ime Optimization based on the algorithms proposed by Cornillier et al (2009). The T W F easibility Check is an exact **algorithm** checking the feasibility of the route with regards to time windows. It is called after each removal or insertion move. The purpose of the W aiting T ime Optimization exact **algorithm** is to minimize the waiting time in a given solution. This **algorithm** is called at the end of the search, it postpones the departure time from the depot as much as possible. The capacity and incompatibility constraints are checked through a fast assignment heuristic that we propose to solve the CAP. It is an adapted version of the best-fit heuristic designed for the bin-packing problem. Invoked after each insertion move, it provides feasible assignment of orders while satisfying the loading constraints. Specifically, it first checks the feasibility of a potential order insertion with re- spect to capacity and incompatibility constraints. Then, it assigns the order to a compartment considering compartments according to their increasing residual capacity.

Show more
31 Read more

Basically, a DTSP can be viewed as a sequence of differ- ent static TSP instances that change over time. If the time interval between the changes is long, then a straightforward way is to apply exact methods, e.g., Concorde [ 8 ], or effec- tive heuristics, e.g., Lin–Kernighan [ 35 ], 2-Opt [ 9 ], 3-Opt [ 36 ], to reoptimize whenever a dynamic change occurs, assuming that the changes are detectable. Such methods are capable of finding the global optimum (or close to the global optimum) solution for symmetric TSP cases in seconds. In contrast, the field of **evolutionary** dynamic optimization offers sev- eral metaheuristics that tackle DTSPs [ 10 ], [ 59 ] by using knowledge transferred from previously optimized instances. Such methods are suitable to speed up the reoptimization process when the changes are small to medium [ 6 ], [ 27 ]. Furthermore, they are more appropriate in cases where the time interval between the changes is relatively short, which makes exact or any other computationally expensive methods inappropriate [ 56 ].

Show more
14 Read more

This paper proposes a new discrete firefly **algorithm** for solving TSP problem. It extends continuous artificial swarm (especially FA) optimization **algorithm** to discrete domain. The new **algorithm** makes some definitions based on swap operator and swap sequence. It also adopts disturbance strategy based on variable neighborhood search method to increase search areas and improve the local refinement ability of the **algorithm** and the optimal speed. The experimental results show that the **algorithm** can find relatively satisfactory solution in a short time and improve the efficiency of solving the TSP. In the future, we would find more effective firefly algorithms to improve optimum searching method and solve other combinatorial optimization **problems**.

Show more
DOI: 10.4236/ajor.2018.83010 139 American Journal of Operations Research starting point, via the shortest possible route. Such a path is known as a Hamil- tonian cycle [30]. For centuries, the TSP has attracted researchers’ attention ow- ing to the simplicity of its formulation and constraints. However, despite being easy to describe and understand, the TSP is difficult to solve [31]. Because a vast amount of information has been amassed on the TSP and the behaviors of TSP algorithms are easily observed, the TSP is now recognized as a standard ben- chmarking problem for evaluating new algorithms and comparing their perfor- mances with those of established algorithms. Many real-life **problems** and appli- cations can also be formulated as TSPs, and some optimization **problems** with different structures can be reduced or transformed to variations of TSPs, such as the job scheduling problem, the knapsack problem, DNA sequencing, integrated circuit ( i.e. , VLSI circuits) design, drilling problem, and the satisfiability prob- lem. Finally, a TSP can be classified as a combinatorial optimization problem, as it requires finding the best solution from a finite set of feasible solutions.

Show more
34 Read more

2009), ACO with Sweep **algorithm** (Yousefikhoshbakht and Sedighpour, 2013), threshold accepting and edge recombination (Liu, 2007), particle swarm optimization and local search (Shi, et al., 2007), variable neighborhood descent search and GRASP (Hernandez-Perez, et al., 2009) have greater ability for finding an optimal solution, they have been considered to solve complex **problems**. For example, (Wang, 2010) presented a hybrid **algorithm** in which GA, ACO and a new strategy called GSA was proposed aiming at the key link in the **algorithm**. This **algorithm** converts the genetic solution from GA into information pheromone to distribute in ACO. Furthermore, GSA takes a new matrix which is formed by the combination of the former 90% of individual from genetic solution and 10% of individual by random generation as the basis of the transformation of pheromone value. The best combination of genetic operators in GA was also discussed. Besides, (Weber, 2006) proposed a distributed **algorithm** in which ant colonies and genetic algorithms work independently of each other and only communicate when better solutions are discovered. Ant colonies are used to explore the solution space, while genetic algorithms are used to improve the convergence rate of the search.

Show more
13 Read more

This paper proposes an immune system based genetic **algorithm** to address dy- namic **traveling** **salesman** **problems**. A permutation-based dualism scheme is in- troduced and integrated into the immune operation in order to enhance the exploration capacity of the population to watch over new optima. Furthermore, a memory-based vaccination strategy is presented to drive individuals to exploit promising regions based on the valuable information extracted from the mem- ory. Experimental results show the efficiency of the proposed techniques for the immune-based GA for DTSPs.

10 Read more

The TSP was first defined and studied by mathematician Karl Menger in Vienna in 1930 [2], earlier mentions of the problem had been made in the 1800s but no formal mathematical definition had been formulated until Menger. Flood of Princeton Uni- versity was the first mathematician to popularise the topic of TSP to his colleagues at the RAND Corporation in the 1940s [8]. Solutions to the TSP appeared from the 1950s onwards and in 1972 Kamp proved that the TSP was an NP-Hard problem since it could not be solved in polynomial time [8]. An NP-Hard or NP-Complete problem is classified as a problem which cannot be solved in polynomial time, they usually consist of combinatoric optimisation **problems** such as the TSP [9]. The TSP has become a classic combinatorial optimisation problem [10] with modern researchers continuously attempting to improve on existing results by creating new algorithms or modifying pre- vious algorithms. The appeal of solving the TSP comes from the fact that the problem is easily formulated and understood but a general solution has not been found and is extremely difficult to obtain [2].

Show more
73 Read more

The TSP is a representative of a large class of **problems** known as the combinatorial optimization **problems**. Among them, TSP is one of the most important, since it is very easy to describe, but very difficult to solve. Actually, TSP belongs to the NP-hard class. Hence, an efficient **algorithm** for TSP (that is, an **algorithm** computing, for any TSP instance with m nodes, the tour of least possible cost in polynomial time with respect to m) probably does not exist. More precisely, such an **algorithm** exists if and only if the two computational classes P and NP coincide, a very improbable hypothesis, according to the last years of research developments.

Show more
22 Read more

Travelling **Salesman** Problem (TSP) is a widespread combinatorial optimization problem that falls into a non- deterministic polynomial-time category (NP-hard) [1], [2]. A company in the United States to resolve the path problem using linear programming, a well-known problem in the computer science field presently [3], introduces TSP. TSP is widely used in real-world applications such as vehicle routing, scheduling **problems**, integrated circuits designs, graph theory, and gene ordering [4]. It is a classic problem in combinatorial optimization in the field of computer science that has been solved for decades. It is typically used as a standard problem for testing the efficiency of a newly proposed optimization **algorithm** [5]. TSP also is recognized as one of the combinational optimization **problems** of discovering the best solution out of a finite set of promising

Show more
Optimization of picking route is mainly about how to take out all items from the cargos in each area, making the whole picking path to reach the best. This kind of optimization problem is very similar to the common trav- eling **salesman** problem. This paper links the two **problems**, applying the method of solving the **traveling** sales- man problem to the picking path optimization problem, and a mathematical model is proposed to solve the routing problem.

In this regards to improve results and convergence time for solving TSP, we developed a practical combination strategy for two **evolutionary** algorithms (FA-ACO) based on the ant colony **algorithm** (ACO) and firefly **algorithm** (FA). FA is used for local search because of its fast convergence time to find the local solution. While ACO is used for global search to avoid the local optimum and find the optimal solution based on the local solutions by FA. In this way, we not only avoid local optimum situation but also get the better result with faster convergence time.

Show more
10 Read more

Self Organizing Maps for the TSP and the DTSP
SOM is a type of artificial neural network using unsupervised learning to adapt its neurons. It can be used for mapping high-dimensional data into a low dimensional grid [17]. Therefore it is a useful tool for data visualization that could not be displayed otherwise. It is also used for clustering data and other classification **problems**. SOM for such type of a problem is typically a 2D map. However, SOM for the TSP maps the input space and the targets into the neural network with a one-dimensional array of the output units [18]. The neuron weights and the input signals share the same space. Thus, the connected neuron ring represents the path between the target locations. [19].

Show more
41 Read more

The mDPSO **algorithm** proposed employs the destruction and construction procedure of the iterated greedy **algorithm** (IG) in its mutation phase. Its performance is enhanced by employing a population initialization scheme based on an NEH constructive heuristic for which some speed-up methods previously developed by authors are used for greedy node insertions. Furthermore, the mDPSO **algorithm** is hybridized with local search heuristics to achieve further improvements in the solution quality. To evaluate its performance, the mDPSO **algorithm** is tested on a set of benchmark instances with symmetric Euclidean distances ranging from 51 (11) to 1084 (217) nodes (clusters) from the literature. Furthermore, the mDPSO **algorithm** was able to find optimal solutions for a large percentage of problem instances from a set of test **problems** in the literature. It was also able to further improve 4 out of 9 larger instances from the literature. Both solution quality and computation times are competitive to or even better than the best performing algorithms from the literature.

Show more
23 Read more

In this regards to improve results and convergence time for solving TSP, we developed a practical combination strategy for two **evolutionary** algorithms (FA-ACO) based on the ant colony **algorithm** (ACO) and firefly **algorithm** (FA). FA is used for local search because of its fast convergence time to find the local solution. While ACO is used for global search to avoid the local optimum and find the optimal solution based on the local solutions by FA. In this way, we not only avoid local optimum situation but also get the better result with faster convergence time.

Show more
10 Read more

A computational experiment has been conducted in Table 3 to compare the performance of the proposed **algorithm** with some of the best techniques designed including GA, ACS, PSO (Zhong, Zhang and Chen, 2007) and Bee Colony Optimization (BCO) (Wong, Low and Chong, 2008) for TSP. We executed the **algorithm** on some of the well-known problem instances from one dataset. The sets of data used for the experiment are TSP instances available on the TSP library (http://comopt.ifi.uni- heidelberg.de/software/TSPLIB95/). The selected test **problems** are Euclidean TSP instances where the sizes of the **problems** range from 24 to 200. The first, second and third columns in Table 3 specify the number, name of instance and its size referenced. Moreover, the fourth, fifth and sixth, seventh and eighth columns show the 5 meta-heuristic algorithms. Additionally, in order to recognize the performance of the method, the best known solutions (BKS) in the literature and also on the web, are presented in the last column.

Show more
13 Read more

The **Traveling** **Salesman** Problem (TSP) is one of the most well-known and studied **problems** in the area of combinatorial optimization [1]. It aims to find the shortest tour or the tour with minimum cost for the given number of points, i.e., cities, pieces, vertices, etc., where all the distances between the points are known. Despite the simplicity of the definition of the TSP, its solution is quite difficult. As the number of points increases, the solution time and the difficulty of the **problems** increase correspondingly. Hence, instead of sweeping the whole solution space in order to guarantee the absolute solution, researchers mostly prefer to find quality solutions in short time periods by using heuristic methods.

Show more
with discrete variables. The discrete nature of the problem arises from the fact that each city may be numbered as an integer selection and a non-integer selec- tion has no significance. The number of possible routes is factorially large, so that the solution may not be generated practically via an exhaustive search. The discrete nature of the problem eliminates the use of gradient nonlinear pro- gramming techniques as well as introducing a large number of local minima. The selection of cities without replacement (each city is visited only once) adds another difficulty in generating the solution through the use of a traditional ge- netic based **algorithm**. The problem’s origin dates back to the early days of linear programming and continues to serve as a benchmark for new solution algo- rithms. The problem is representative of a large number of practical optimiza- tion formulations, including electronic circuit design, scheduling, pick-up and delivery and providing home health care or other services. The size of practical applications can range from tens of cities to tens of thousands of cities. While many algorithms can solve **problems** involving tens of cities, most have extreme difficulty with **problems** involving over one hundred cities.

Show more
18 Read more

The applications of the TSP encompass a comprehensive list with the most common ones being vehicle routing problem and its variations (see Laporte, 1992), tool head path planning for drilling VSLI circuit boards, computer wiring (Lenstra and Rinnooy Kan, 1975), hole punching (Reinelt, 1989), several manufacturing contexts, job sequencing, dartboard design (Eiselt and La- porte, 1991), crystallography (Bland and Shallcross, 1989), crew scheduling, etc. Variants to the TSP are used to represent particular situations and to model them accurately. Broadly, routing **problems** can be divided into static and dynamic **problems**. When all required input data of the problem are known before routes are calculated, it is considered a static routing problem. If, on the other hand, some of the input data are obtained during the execution of a route, it is a dynamic problem. In this section, we describe a few variants (see also, Fig. 1).

Show more
39 Read more

The TSP is known to be NP-hard. This means that no known **algorithm** is guaranteed to solve all TSP instances to optimality within reasonable execution time. So in addition to exact solution approaches, a number of heuristics and metaheuristics have been developed to solve **problems** approximately. Heuristics and metaheuristics trade optimality of the solutions that they output with execution times. They are used to find “good” quality solutions within reasonable execution times. The term heuristic is normally used to describe an approximate solution method that is intended for one particular optimization problem, while the term metaheuristic is used to describe a more

Show more
16 Read more