A Hybrid Simulated Annealing Algorithm for Travelling Salesman Problem with Three Neighbor Generation Structures Misagh Rahbari, Ali Jahed

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Misagh Rahbari, Ali Jahed. A Hybrid Simulated Annealing Algorithm for Travelling Salesman Prob- lem with Three Neighbor Generation Structures. 10th International Conference of Iranian Operations Research Society (ICIORS 2017), University of Mazandaran, May 2017, Babolsar, Iran. hal-01962049

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A Hybrid Simulated Annealing Algorithm for Travelling Salesman Problem with Three Neighbor Generation Structures

Misagh Rahbari 1,* , Ali Jahed 2 1 Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran Email: [email protected] 2 Department of Industrial Engineering, Islamic Azad University South Tehran Branch , Tehran, Iran Email: [email protected]

Abstract Travelling salesman problem (TSP) has been considered as one of the most complicated problems. The problem is NP-Hard and practical large-scale instances cannot be solved by exact algorithms within acceptable computational times. The aim of this study is to presents a hybrid method using tabu search and simulated annealing technique to solve TSP called hybrid simulation annealing (HSA). The proposed HSA algorithm incorporates three neighborhood structures, called swap, insertion and reversion, to explore different possibilities of neighbor solution. This proposed HSA not only prevents revisiting the solution but also maintains the stochastic nature. Finally, the performance of the proposed HSA is examined against tabu search and simulation annealing technique, and the preliminary results indicate that the HSA is capable of solving real-world problems, efficiently.

Keywords: Travelling salesman problem, Simulated annealing, Tabu search, Meta-heuristics methods

1. Introduction Most problems in discrete optimization concern the problems in NP-hard classes which are difficult to find the best solution in reasonable time. We can use two approaches for finding the solution. First exact algorithm to find the best solution and the other approach is to use meta-heuristics algorithms. Some examples of first approach is Multi-agent systems in production planning and control scheduling of mixed-model assembly lines [1]. Agent based on flow shop scheduling [2],[3],[4]. The second method is based on meta-heuristic optimization. These techniques generate different sets of solutions and filter out some low scored solutions in order to maximize or minimize the objective function. The geometrical structure of the objective function is not involved as a part of solution finding process. Without considering the actual geometrical structure of the objective function, it is almost impossible to achieve the best solution. However, the objective functions of some combinatorial optimization problems are rather difficult to define due to their nature. For example, the travelling salesman problem (TSP) concerns the sequence of cities and the total travelling distance. The best sequence of cities is the desired solution to be achieved. Hence, a set of travelling sequences must be used as a set of generated points scattered throughout the manifold of objective function for optimizing the function. But in general, the objective function of this problem concentrates only the total distance and lacks the sequence of traversed cities. Furthermore, this problem can be transformed to various NP- complete problems such as DNA sequencing, planning and logistics. Previously, the rever several proposed algorithms to solve the problems related to travelling salesman problems such as vehicle routing [5], scheduling problems [6]. A number of meta- heuristic algorithms have been implemented to find the optimal path by various researchers. Some of the techniques are Ant colony Optimization (ACO) [7],[8],[9], Genetic Algorithm (GA)[10],[11]. Other newly emerging techniques have been also applied to TSPs such as Intelligent Water Drop (IWD)

algorithm [12] and River Formation Dynamics (RFD) [13]. Some improvements of ant colony optimization to overcome the problems of pheromone update and being stuck at local optimal solution are also proposed. For instance, PS–ACO [14] modified the pheromone updating rules of ACO by using the mechanism of PSO.ACS–TSPTW based on the ACO technique to solve the travelling salesman problem with time windows (TSPTW) [15]. GA–PSO–ACO [16] used the randomness, rapidity, and wholeness of the PSO and GA in the first step to obtain sub – optimal solutions for adjusting the initial allocation of pheromone in ACO. The problem of travelling salesman combines two interesting instantaneous aspects. The first aspect is the sequence of objects which are traversed cities. The second aspect is the minimum total travelling distance.

2. Describe model

The TSP model is shown here:

Parameters

cij :travel costs from city i to city j n: number of city

Decision variables 1 the path goes from city i to city j X ij :  0 otherwise

ui : Positive variable for each city to avoid subtours

nn min cXij ij i0 j  i , j  0 Subject to:

01X ij i, j 0,1,...... , n n  X ij 1 jn 0,1,...... , i j,0 i n  X ij 1 in 0,1,...... , i j,0 j

ui u j  nX ij  n 1 2 i  j  n

The objective shows that we want to minimize the cost. The first constraint shows that all of the variables are binary. Second constraint shows that there is one city that goes to other city. The Third constraint shows that there is on exit from each city. The last constraints enforce that there is only a single tour covering all cities, and not two or more disjointed tours that only collectively cover all cities.

3. Definition of algorithms

3.1 SIMULATED ANNEALING ALGORITHM Simulated annealing (SA) is a generic probabilistic meta-heuristics for combinatorial optimization problem of locating a good approximation to the global optimum of a given function in a relatively large search space. During the search, SA not only accepts better solutions but also the worse solutions but with a decreasing probability. At higher temperatures, the probability of accepting worse solutions is much higher. But, as the temperature decreases, the probability of accepting worse solution decreases. The probability of acceptance is assigned the value:

Probability of acceptance = exp(/)ZT

A randomly generated number is used to test whether the move is accepted. Finally to decreases and update temperature, we multiply the current temperature by a constant ALPHA:

Update temperature = T 

The SA consists of two loops. The inner loop runs till maximum number of acceptance or study neighbors for current temperature reached. The outer loop check for the stopping condition to be met. Each time the inner loop is finished, the temperature is updated using an update temperature formula till primary temperature to become equal freezing temperature.

3.2 TABU SEARCH ALGORITHM Tabu search (TS) is an optimization technique for solving permutation problem. TS moves from a current solution to the best solution in its neighborhood by use long-term memory (LTM). Long-term memory (LTM) has one main purposes: to drive the search towards the regions of the solution space not yet explored and with high potential of containing good solutions the formula to determine the best replacement for generation neighbor:

Random (i, j) = rand/Long-term memory (LTM) (i, j) (n1, n2) = the indices of maximum value in random

Short-term memory (STM) has one main purposes: to prevent the return to the most recently visited solutions in order to avoid cycling. The exit strategy of STM as a first in first out (FIFO). Finally after the generation neighbor we update STM and LTM till maximum of iteration is reached.

3.3 HYBRID SIMULATION ANNEALING ALGORITHM

The proposed hybrid simulation annealing (HAS) meta-heuristic is introduced for travelling salesman problem based on the hybridization of two algorithms, namely SA and TS. In the algorithm by the combination of simulated annealing (SA) and tabu search (TS), the number of solution revisits has been remarkably decreased. By using the proposed HAS, a number of solution revisits can be decreased by providing a short- term memory while keeping the stochastic nature of the SA algorithm also the proposed HAS, not only accepts better solutions but also the worse solutions but with a decreasing probability.

The proposed HAS has a number of advantages, including stochastic feature avoiding cycling and Short-term memory to escape from local optima. These characteristics limit the search from a previously visited solution and improve the performance of the conventional SA remarkably.

3.4 THREE NEIGHBOUR GENERATION STRUCTURE

In using of meta-heuristics for TSP, the most widely used and the simplest representation solution is permutation encoding. In permutation encoding the order of the numbers in the array represents the visiting order of the cities. In this section, we present three neighbor generation structure that using in proposed HAS.

3.4.1 SWAP

Selecting two random positions in permutation encoding representation solution and swapping elements of these positions is the easiest and most widely used way of generating of neighbor solutions. In order to help to understand, in the Figure 1 we introduce the swap method. For example, if the third and fifth element be selected then their position will be replace by each other.

Figure 1.neighbor generation with swap operator

3.4.2 REVERSION

Selecting two random positions in permutation encoding representation solution and reversing the direction between two randomly chosen elements. In order to help to understand, in the Figure 2 we introduce the reversion method. For example, if the second and fifth element be selected then reversing the direction between their position.

Figure 2.Neighbor Generation with Reversion Operator

3.4.3 INSERTION

Selecting two random positions in permutation encoding representation solution and with due attention to number of chosen elements, reversing the direction between two randomly chosen elements. In order to help to understand, in the Figure 3. we introduce the reversion method. For example, if the second and fifth element be selected then reversing the direction between their position.

Figure 3.Neighbor Generation with Insertion Operator in 2 State

4.Computational results In this section, we present details of the implementation of the proposed HSA and compares the results with TS and SA. For the testing of these techniques, 3 example problems from the literature were used. All these techniques were written in MATLAB programming language and every technique was run 50 times for every problem. For the comparison of the techniques, GAP of the average results with due attention to the optimal solution was used.

average()() Z optimal Z GAP% 100 optimal() Z

Table 1 shows details of our results.

Name of Problem Name of Number Average Optimal GAP techniques number instances of city Z Z (%)

1 Gr17[17] 17 2119.28 2085 1.64

2 Gr48[17] 48 5312.9 5046 5.28 SA

3 Gr96[17] 96 62779.78 55209 13.71

1 Gr17[17] 17 2110.3 2085 1.21

2 Gr48[17] 48 5332.86 5046 5.68 TS

3 Gr96[17] 96 65622.28 55209 18.86

1 Gr17[17] 17 2096.8 2085 0.56

2 Gr48[17] 48 5281.2 5046 4.6 HSA

3 Gr96[17] 96 61510.32 55209 11.41

Table 1.The summary of comparison of HSA with TS and SA

5. Conclusion

In this study, first we have presented travelling salesman problem then presented two well known technique, tabu search and simulated annealing. We have presented a hybrid simulation annealing (HAS) meta-heuristic based on combination of two well-known techniques, tabu search and simulated annealing to solve travelling salesman problem. Also, we introduce three neighbor generation structure that used in proposed HAS called swap, insertion and reversion. To validate and verify the proposed HSA, the technique were tested on 3well known benchmark problem sets from literature. The proposed HAS has been illustrated to be efficient in finding optimal solutions in most cases.

Acknowledgments

The first author greatly appreciates to the Research Deputy of University of Mazandaran for their support.

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