A Hybrid Symbiotic Organisms Search Algorithm with Variable Neighbourhood Search for Solving Symmetric and Asymmetric Traveling Salesman Problem

Combinatorial optimization has been frequently used to solve both problems in science, engineering, and commercial applications. One combinatorial problems in the field of transportation is to find a shortest travel route that can be taken from the initial point of departure to point of destination, as well as minimizing travel costs and travel time. When the distance from one (initial) node to another (destination) node is the same with the distance to travel back from destination to initial, this problems known to the Traveling Salesman Problem (TSP), otherwise it call as an Asymmetric Traveling Salesman Problem (ATSP). The most recent optimization techniques is Symbiotic Organisms Search (SOS). This paper discuss how to hybrid the SOS algorithm with variable neighborhoods search (SOS-VNS) that can be applied to solve the ATSP problem. The proposed mechanism to add the variable neighborhoods search as a local search is to generate the better initial solution and then we modify the phase of parasites with adapting mechanism of mutation. After modification, the performance of the algorithm SOS-VNS is evaluated with several data sets and then the results is compared with the best known solution and some algorithm such PSO algorithm and SOS original algorithm. The SOS-VNS algorithm shows better results based on convergence, divergence and computing time.


Introduction
Combinatorial optimization from the past has often been used to solve problems in science, engineering, and commercial applications. Since the Mathematical model began to be used to help solve problems in life real, algorithms have become important in recent years due to their flexibility to find a solution. One of the combinatorial problems in the field of transportation is looking for a shortest route that can be traveled from the starting point to the destination point, as well as minimize travel costs and travel time. Such problems such as this is known as Traveling Salesman Problem (TSP). Author Error! Reference source not found. suggests that TSP issues have become a concern over the past two decades and various approaches have been proposed for solve the problem. Then research to solve these TSP issues continue to be developed with modifications and with new approaches such as particle swarm optimization [2], Neural network for Euclidean TSP [3], 2-opt based approach on differential evolution [4]. Simulated annealing with greedy search [5], Ant colony with multidepot Multiple TSP [6], Evolutionary Algorithm [7], Lower tolerance branch and bound [8], and Genetic algorithm [9]. The asymmetric traveling salesman problem (ATSP) is a well-known combinatorial optimization problem, in which bi-directional distances between a pair of cities are not necessarily identical. An optimal solution to the ATSP minimizes the total distance traveled to visit all cities in a list [10] [11]. The asymmetric traveling salesman problem is defined on a directed graph G = (V , A), where V ={1, . . . , n} is the vertex set, A={(i, j ) : i, j ∈ V } is the arc set, and a non-symmetric cost matrix (cij ) is defined on A [12]. For the simple equation [13]describe Let cij be the distance from city i to city j . If cij = cji for all i and j , then this problem is known as the symmetric traveling salesman problem; otherwise, it is designated as an asymmetric traveling salesman problem (ATSP). According [14] and [15], the asymmetric TSP, which is more general in principle and has numerous real-world applications. In contrast to the symmetric TSP, the distance from one city to another in the ATSP may not necessarily be the same as the distance on the reverse direction. Asymmetric traveling salesman problem is one of a class of difficult problems in combinatorial optimization that is representative of a large number of scientific and engineering problems. ATSP and its variants are commonly used models for formulating many practical applications in manufacturing scheduling problem. For example, the scheduling problem in a discrete manufacturing is mainly concerned with how to determine the sequence of jobs so as to minimize the total set-up cost [16].

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One of new metaheuristics is the Symbiotic Organism Search (SOS). This method was developed by Cheng and Prayogo [14] that is inspired by the biological relationship between organisms on ecological systems. Obstacles which is encountered in the application of SOS algorithm to solve the combinatorial problem is on the parasitic phase of SOS algorithm requires to create a parasitic organism using the dimensions of the objective function to be sought (Continuous function), whereas TSP and ATSP does not have dimension of purpose function for it is necessary there is another suitable mechanism to replace the mechanism in the parasitic phase. Authors [17] also suggested modifications should be made to the phase parasites by adding more specific search algorithms to overcome combinatorial problems. The proposed mechanism for replacing this parasitic phase is to use mutation mechanism. The mutation mechanism allows for bringing in new individuals who are not from the previous population with refers to the change or replacement of elements of the solution vector [18].
This paper focusses on solving TSP and ATSP using SOS algorithm by modifying the SOS algorithm to be applicable to TSP and ATSP. The modification is done by adding mutation mechanism to replace the mechanism at that parasitic phase to find the best route. SOS is hybridized with Variable Neighbourhood Search algorithm as a local search to generate the better initial leader.

Symbiotic Organisms Search (SOS)
SOS is one of new metaheuristics that is inspired by the interaction behavior seen among organisms in universe. The organism has the basic nature of being unable to live alone so have a dependence on other species to maintain sustainability. This dependence-based relationship is known as a symbiosis. There is some forms of symbiosis i.e symbiotic of mutualism, symbiotic of commensalism and symbiotic of parasitism. Here is the flow of SOS: This SOS phase mimics mutualistic relationships between organisms in nature. SOS describes Xi is an organism compatible with members of the ecosystem. Other organisms Xj Then randomly selected from the ecosystem to interact with Xi. Both organisms Engage in mutualistic relationships with the goal of improving survival together with benefits in the ecosystem. New candidate solutions for Xi and Xj are calculated based on the mutualistic symbiosis between the organisms Xi and Xj, which are modeled on equations (2) and (3) − Commensalism Phase Similar to the phase of mutualism, an organism Xj is randomly selected from the ecosystem to interact with Xi. In this case, the Xi organism tries to get advantage of interaction. However, organism Xj alone does not profit or suffer relationship. New candidate solutions from Xi are calculated according to the symbiosis between organisms commensal Xi and Xj, which are modeled in Eq. (2). Following the rules, Xi organism updated only if the new fittnes are better than the previous interaction fittnes.
− Parasitism Phase In SOS, the Xi organism is given a role similar to anopheles mosquito through the creation of an artificial parasite called '' Parasite Vector ''. Parasite Vector is made in space search by duplicating the Xi organism, then modify selected at random dimensions using random numbers. Organism Xj is randomly selected from the ecosystem and serves as the host for vector parasites. Parasite Vector tries to replacing Xj in the ecosystem. Both organisms are then evaluated to measure their fitness value. If Parasite Vector has a better fitness value, it will kill the organism Xj and assume its position in the ecosystem. If the fitness value of Xj is better, Xj will have immunity from parasites and Parasite Vector will no longer can live in the ecosystem. − Mutation on TSP Mutations are possible to elicit new non-originating individuals of cross breeding results. Mutation refers to the change of order or replacement of elements from solution vector (on the TSP problem) by generating a new value (function optimization). This process is done by choosing which individual to mutate with using random numbers. Individuals who are selected then mutate by placing a sequence of routes from the individual. Mutation can be swap, slide and flip .

Variable Neighbourhood Search (VNS)
According to [19], VNS's basic idea is to change the environment in searching for better solutions. VNS starts from a descent method to reach the local minimum, then investigate systematically or randomly, causing the environment to be further away from this solution. Each time, one or more points in the current environment is used as the initial solution for local descent. One jump point from the current solution as a new reference if and only if a better solution is found. VNS is not like the trajectory method on Simulated Annealing or Taboo Search and does not include prohibited steps in the process. Although simplicity is more specific.

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International The SOS-VNS algorithm is a hybrid of SOS and VNS algorithm. The VNS is a local search algorithm widely used for solving combinatorial problem. The steps of the hybrid SOS-VNS algorithm are then described in algorithm shown in Figure 1.   Table 2 compares the running results of Particle Swarm Optimization (PSO), original Symbiotic organisms Search (SOS) and Symbiotic organisms Search with Variable Neighbourhood Search (SOS-VNS) on a set of bechmark TSP and ATSP in TSPLIB. Convergence is use to measure the ability of the algorithm to find result near the optimal solution. The smaller of convergence value shown the algorithm is able to find the near optimal solution. While divergence is used to see how much deviation generated from the algorithm. Smaller level of divergence shown the more robust algorithm. Computation time (comp time) is used to evaluated the efficientcy of optimization algorithm. All of algorithm are written in MATLAB code and limit of iteration of each is 50000 on Intel(R) Core(TM) i3 processor 2.27GHz.
Based on the results of 10 runs are carried out for each algorithm, overall SOS-VNS algorithm can be said to be superior than PSO and SOS original. Convergence value of the SOS-VNS algorithm is superior to all data sets. This proves that the SOS-VNS algorithm is capable of generating a set of solutions closer to the optimal solution than PSO and SOS original. In terms of SOS-VNS divergence it also shows that better than PSO and SOS original, with smaller divergence values than for all data sets. For computing time SOS-VNS also shown that it more eficient than PSO and SOS original.

Conclusion
In this paper, a novel and hybrid SOS with VNS is proposed as a new approach for solving symetric and asymetric TSP. This new algorithm is compared with PSO and the original SOS. The result show excellent results. Using convergence, divergence, and computation time, SOS-VNS is better than PSO and original SOS. From 7 data sets tested, SOS-VNS is able to solve all data sets.