Redalyc.Combined Mutation Operators of Genetic ... - Semantic Scholar

Redalyc Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal

Deep, Kusum; Mebrahtu, Hadush Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem International Journal of Combinatorial Optimization Problems and Informatics, vol. 2, núm. 3, septiembre-diciembre, 2011, pp. 1-23 International Journal of Combinatorial Optimization Problems and Informatics Morelos, México Available in: http://www.redalyc.org/src/inicio/ArtPdfRed.jsp?iCve=265219635002

International Journal of Combinatorial Optimization Problems and Informatics ISSN (Printed Version): 2007-1558 [email protected] International Journal of Combinatorial Optimization Problems and Informatics México

How to cite

Complete issue

More information about this article

Journal's homepage

www.redalyc.org Non-Profit Academic Project, developed under the Open Acces Initiative

© International Journal of Combinatorial Optimization Problems and Informatics, Vol. 2, No.3, Sep-Dec 2011, pp. 1-23, ISSN: 2007-1558.

Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem Kusum Deep, Hadush Mebrahtu Department of Mathematics, Indian Institute of Technology, Roorkee, India [email protected], [email protected]

Abstract. In this study two combined mutation operators have been developed to increase the performance of Genetic Algorithm that helps to find the minimum cost in the known Travelling Salesman problem (TSP).In order to do this the combined mutation operators which are named as Inverted Exchange and Inverted Displacement are compared with four existing mutation operators. These are programmed in C++ and algorithms, ten bench mark test problems and five variations of order crossover are used. Based implemented on a set of benchmark test problems taken from the TSPLIB. The crossover operator used here is the order crossover using two of its existing and three of its new variations. In general six mutations on the numerical and graphical analysis it indicates that the Inverted Displacement has a definite supremacy over the existing four mutations except for few instances considered here. The Inversion mutation is second best performer. All optimal values found in the six mutations are obtained when the new variations of order crossover are used. Keywords: The Travelling Salesman problem; Genetic Algorithms; Inverted Exchange and Inverted Displacement Mutations

1. Introduction. The Travelling Salesman problem (TSP) represents a classic NP-hard combinatorial optimization problem and good, traditional approximation algorithms have been developed to tackle it down [1]. However, in evolutionary computation, the TSP serves as the simplest case of a variety of combinatorial problems which are of enormous relevance to industrial scheduling problems [2]. Since any problem which belongs to the NP-class can be formulated as TSP several evolution inspired algorithms used it as a battleground to develop combinatorial-specific search operators. Even though the Travelling Salesman problem is a very hard problem to solve, solving this problem even using approximation algorithms is an important part of applications in many areas including vehicle routing, computer wiring, over hauling gas turbine engines in aircraft, machine sequencing and scheduling, and frequency assignment in communication networks. Applications in statistical data analysis include clustering and ordering objects. Specially Clustering and ordering using TSP solvers in nowadays becoming much known in biostatistics. For example, [3] describe an application for ordering genes and [4] use a TSP solver for clustering proteins. Following [5] the Travelling Salesman problem can be stated mathematically as follows: If a travelling salesman were to visit a given finite set of cities with cost (cij) of travelling from city i to j exactly once and return to its original city, then which tour would obtain the minimum cost? Formally let {1, 2, 3, …, n} be labels of the n cities , then the total cost TC of a TSP tour is: n −1

TC (n) = ∑ C i ,i +1 + C n,1 i =1

Here we consider cij = cji which is a symmetric TSP.

Received April 16, 2011 / Accepted June 14, 2011 Editorial Académica Dragón Azteca (EDITADA.ORG)

(1)

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Combinatorial optimization problems have attracted many researchers in recent decades due to their practical relevance and their considerable difficulties. Usually two different categories of approaches are used to solve these problems, namely, exact and heuristic methods. Exact methods apply analytical and mathematical approaches to solve the problems. Due to the considerable complexity of the problems, the efficiency of exact methods is limited to small scale problems. However exact methods yield analytically optimal solutions. Heuristic methods are employed to solve problems that are difficult to solve by using exact methods. They use simple intuitive techniques to make the search limited to potentially better solutions. Although heuristic methods can find a solution in a short time compared to exact methods, the analytically optimal solution is not guaranteed. Some of the new heuristic approaches that are being adapted to solve the NP-hard TSP are Genetic algorithm [6], Particle swarm [7], Ant colony [8] and simulated annealing [9]. Genetic algorithms (GAs) are population based search techniques which mimics the principles of natural selection and natural genetics laid by Charles Darwin. Since Holland introduced the genetic algorithm (GA) in the early 1970’s, many researchers have become interested in it as a non traditional method of solving real life problems. As it is a promising heuristic approach to locate near optimal solution in large spaces, it is not surprising that it is a target of many researchers. In Genetic Algorithm, a population of potential solutions termed as chromosomes/individuals are evolved over successive generations using a set of genetic operators called selection, crossover and mutation. First of all, based on some criteria, every chromosome is assigned a fitness value and then the selection operator is applied to choose relatively fit chromosomes to be part of the reproduction process. In reproduction process new individuals are created through application of operators. Large number of operators have been developed for improving the performance of GA, because the performance of algorithm depends on the ability of these operators. One of the operators, mutation operator is used to maintain adequate diversity in the population of chromosomes and avoid premature convergence. Whereas crossover operator, blends the genetic information between chromosomes to explore the search space. TSP as opposed to most problems tackled by GAs is a pure ordering problem; accordingly specialized permutation operators like mutation must be developed for this problem. TSP tours can have different representations, such as ordinal, path, adjacency and binary matrix representation. From these, the path representation is the most natural representation of a tour and this paper focuses on it. In this paper the performance of six mutation operators that is four existing and two combined mutation operators are compared. From the existing four mutations inversion mutation performed best and insertion mutation has shown weak performance relatively. Here exchange and displacement mutations are combined with inversion mutation with their properties. The combined mutation operators are inverted exchange mutation and inverted displacement mutation. In order to accomplish the performance comparison eleven algorithms are applied. Six algorithms of mutation, five algorithms of variations of order crossover operator (two existing and three new) and ten TSPLIB [10] problems are used. This paper is organized as follows: Section 2 is a literature review on mutation operators for TSP. The proposed two mutation operators and the existing four mutation operators are explained in section 3. Encoding, Initial population Generation and Selection operator are discussed in Section 4. Section 5 describes the variations of order crossover used here. In section 6 experimental setup is discussed. Results and discussion are described in section 7. Finally conclusion and future work is presented in section 8.

2. Literature review on mutation operators for TSP According to [11] Evolutionary algorithms are randomized search techniques aimed at simulating the natural evolution of asexual species. In this model individuals were created via random mutation to the existing individuals. Holland and his students extended this model by allowing “sexual reproduction” i.e. the combination or crossover of genetic materials from two parents to create a new offspring. These algorithms were Genetic algorithms but they were not designed to solve combinatorial optimization problems like TSP. The TSP as opposed to most problems tackled by GAs is a pure ordering problem.

2

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Mutation involves the modification of the value of each ‘gene’ of a solution with some probability pm, (the mutation probability). The role of mutation in GAS has been that of restoring lost or unexplored genetic material into the population to prevent the premature convergence of the GA to suboptimal solutions. The mutation operator randomly selects a position in the chromosome and changes the corresponding allele, thereby modifying information. The need for mutation comes from the fact that as the less fit members of successive generations are discarded; some aspects of genetic material could be lost forever. By performing occasional random changes in the chromosomes, GAs ensure that new parts of the search space are reached, which reproduction and crossover alone couldn’t fully guarantee. In doing so, mutation ensures that no important features are prematurely lost, thus maintaining the mating pool diversity. For the TSP, the classical mutation operator does not work. In comparison to crossover the probability of mutation is less than the probability of crossover in solving TSP by genetic algorithm, which otherwise would be a random walk. In the last 2-3 decades several mutation operators for TSP are proposed by researchers. Here are some of them. Exchange mutation by [12] selects two positions at random and swaps the cities on these positions. This mutation is essentially the 2-opt heuristic for TSP. Inversion Mutation [13] selects two positions within a chromosome/tour and then inverts the substring between these two positions. Insertion Mutation [14] selects a city at random and inserts it in a random position. Displacement Mutation [6] selects a sub tour at random and inserts it in a random .Insertion can be viewed as a special case of displacement in which the substring contains only one city. Heuristic mutation by [15] was designed with neighborhood technique in order to produce an improved offspring. Greedy Swap Mutation (GSM) [16] also selects better result and therefore comes up closer to the solution quickly. However, when greedy methods reach a local solution it is very difficult to jump up any other unprecedented solutions. Therefore greedy methods produce local solutions that are so far from optimal. As to the selection of mutation and mutation rate it is not uncommon for a large amount of time to be spent tuning these parameters in order to optimize the GA performance [17] and [18]. Often the parameters selected will only be optimal for a particular problem type or problem instance [19].In order to minimize this problem Self-Adaptation of Mutation Operators is proposed by [20]. Even though [21], pointed out the importance of hill climbing mutation operator, other mutation operators have also been seen to perform well with different crossover operators.

3. The most known existing and proposed mutations In the last 2-3 decades several mutation operators for TSP are proposed by researchers. Here four most known existing mutation operators are explained below. In addition to these the two proposed mutation operators will be described. The four most known existing mutation operators are Inversion mutation, Displacement mutation, Exchange mutation and Insertion mutation. The proposed mutation operators are Inverted Exchange mutation (IEM) and Inverted Displacement mutation (IDM).

3.1 The most known existing mutation operators ►Inversion mutation Inversion Mutation selects two positions within a chromosome/tour at random and then inverts the cities in the substring between these two positions. Example: Consider the following tour/chromosome (P) P = (2 3 4 5 6 1 7 9 8) If a sub tour 4 5 6 is selected at random using two positions P = (2 3 │4 5 6 │ 1 7 9 8) and inverting the sub tour 4 5 6 the mutated tour (O) will be O = (2 3 6 5 4 1 7 9 8) ►Displacement mutation Displacement Mutation selects a sub tour at random and inserts it at a random position outside the sub tour .Insertion can be viewed as a special case of displacement in which the substring contains only one city. Example: Consider the following tour/chromosome (P) P = (9 4 2 6 3 8 7 1 5) If the randomly selected sub tour is 4 2 6 and the randomly selected insertion position is between 1 and 5,that is 3

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. P = (9 4 2 6 3 8 7 1 │ 5) then the mutated tour (O) will be O= (9 3 8 7 1 4 2 6 5)

►Exchange mutation Exchange mutation selects two positions at random and swaps the cities on these positions. This mutation is essentially the 2-opt heuristic for TSP. Example: Consider the following tour/chromosome (P) P = (5 6 8 2 3 7 9 1 4) If positions 2 and 8 are selected at random city 6 and city 1 interchange their positions then the mutated tour (O) will be O = (5 1 8 2 3 7 9 6 4) ►Insertion mutation Insertion Mutation selects a city at random and inserts it at random position. Example: Consider the following tour/chromosome (P) P = (4 5 8 2 3 6 7 9 1) If the city selected at random is 5 and the random position is between 7 and 9 that is P = (4 5 8 2 3 6 7 │ 9 1) then the mutated tour (O) will be O = (4 8 2 3 6 7 5 9 1)

3.2 The proposed mutation operators ►Inverted Exchange mutation • First select two positions within a chromosome/tour at random and then invert the cities in the substring between these two positions. • Second after selecting a city/node from the cities in the inverted substring randomly then exchange this city with a randomly chosen city outside the inverted substring. Example: Let us consider the following tour/chromosome (P) P = (6 1 9 4 5 3 2 8 7) First if a substring 1 9 4 is selected at random at positions between 6 and 1 and 4 and 5 P = (6 │ 1 9 4 │ 5 3 2 8 7) and inverting the substring 1 9 4 we have a chromosome (R) R = (6 │ 4 9 1 │ 5 3 2 8 7) Second if for instance 9 from the inverted substring and 8 from the remaining are selected randomly for exchange then the mutated tour (O) would be O = (6 4 8 1 5 3 2 9 7) ►Inverted Displacement mutation • The first step is to select two positions within a chromosome/tour at random and then invert the cities in the substring between these two positions. • In the second step the substring obtained by the first step is inserted at a random position outside the substring. Example: Consider the following tour/chromosome (P) P = (5 6 9 3 4 7 8 1 2) In the first step if the randomly selected two positions are between 9 and 3 and 8 and 1 P = (5 6 9 │ 3 4 7 8 │ 1 2) and inverting the substring 3 4 7 8 then we have a chromosome (Q) Q = (5 6 9 │ 8 7 4 3 │ 1 2) In the second step if the randomly selected position for displacement is say between 5 and 6 then the mutated tour (O) would be O = (5 8 7 4 3 6 9 1 2)

4

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23.

4. Encoding, Initial population generation and Selection operator used herein 4.1 Encoding The problem in applying the encoding process to a specific problem is actually finding an appropriate representation of the solution that will be easy to use in the crossover process. The encoding process is often the most difficult aspect of solving combinatorial optimization problem using Genetic Algorithms. There are different possible encoding methods in Genetic Algorithm. The common encodings are binary encoding, real encoding, matrix encoding and permutation encoding. In this paper permutation encoding is used. In permutation encoding a chromosome represents a tour of the salesman. A chromosome Pt (t = 1, 2, …, m; m is the population size) is represented as: Pt = (C1, C2, C3, …, Cn). Where: Ci is the i-th city to be visited, i = 1, 2, …, n. Such type of representation is called permutation representation (encoding).All the cities are sequentially numbered starting from 1. The route between the cities is described with an array. Each element of the array represents the number of the city. The array represents the sequence in which the cities are traversed to make up a tour. Each chromosome must contain each and every city exactly once. The permutation representation is the most natural representation of a tour in solving the Travelling Salesman problem. Example: Considering a TSP with 9 cities, a tour: 2→4→8→6→5→3→7→1→9 can be represented using the path representation as (2 4 8 6 5 3 7 1 9) This chromosome represents the tour stating from city 2 to city 4, city 4 to city 8, city 8 to city 6, city 6 to city 5, city 5 to city 3, city 3 to city 7, city 7 to city 1, city 1 to 9 and city 9 to city 2. Chromosome describes the order of cities, in which the salesman will visit them.

4.2 Initial population generation The choice of initialization strategy is an important aspect in genetic algorithm one must not forget. Generating an appropriate initial population has arguably just as large an impact on the Genetic Algorithms as the operators used. Instead of simply defaulted to random initialization if one starts with a carefully chosen but diverse population of individuals the results would be much better. The nearest-neighbor heuristic for creating initial population, have the advantage that they only contain a few severe mistakes, while there are long distances connecting cities with short distances. Therefore since such tours can serve as good starting tours, our initial population is generated using nearest–neighbor initialization strategy [22]. The nearest-neighbor algorithm is as follows: Given: • • • • • • •

A set of m points. A query point, p. An initial minimum computed distance, min-distance For each point, mi., i = 1,2, …, m-1: Compute the distance between the points p and mi. If the computed distance, d, is less than min-distance, then set min-distance = d Solution: The value of min-distance.

►Fitness The fitness function plays an important role in genetic algorithms because it is used to decide how good a chromosome is. Generally for different types of problems different fitness functions are used. Our fitness function is simple one in which distance, cost or time of complete tour is used as fitness value. Minimum cost stands for good fitness value and it has more chances to be selected for crossover and mutation. C = CA1, A2 + CA2, A3+ CA3, A4+…+ CAn-1, An + CAn, A1 Where C is the total distance, cost or time of complete tour, CA1, A2 is the distance, cost or time of path travelled from city A1 to A2. 5

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. The steps for the fitness measure are as follows: • Traverse the cities according to the sequence in a tour. • Calculate the distance, cost or time of CAi, Ai+1 if not given. Hence it is a matter of finding the total distance, cost or time C, n −1

C = ∑ C Ai , Ai +1 + C An , A1 i =1

•

We can calculate the fitness, Fit (Pt) (t =1, 2, …, n) of the chromosome / tour in the population as

1 TotalDistance 1 = C

Fit ( Pt ) =

(2)

4.3 Selection Operator There are different Selection operators in Genetic Algorithms. The common Selection operators are Roulette wheel Selection, Elitist Selection, Rank Selection and Tournament Selection. In this paper one of the most known selection operator known as the roulette wheel selection is applied to solve the Travelling Salesman problem for the first generation. The parent chromosomes are selected for mating via "Roulette wheel" as follows • Sum up the fitness values of all chromosomes in the population. • Generate a random number between 0 and the sum of the fitness values. • Select the chromosome whose fitness value added to the sum of the fitness values of the previous chromosomes is greater than or equal to the random number. In the next generations selection by elitism is used. After two parents produce two children by the variations of order crossover operator they will be four individuals. From these four tours/ individuals the first and the second fit individuals are selected for the next generation after mutation.

5. The crossover operator used here From the fact that the order of cities (not the position of cities) is important for tours the order crossover which was first proposed in [23] is used for its simplicity and convenience to application. The order crossover had two basic variants.

5.1 The two basic existing variants ►The first basic variant of order crossover (Ox1) The first basic variant is the standard crossover (Ox1) that build off springs by choosing a subsequence of a tour from one parent and preserving the relative order of cities from the other parent. For example, consider two parents P1 and P2 (with two points marked by “│”). P1 = (2 3 │4 5 6 │ 7 8 9 1) and P2 = (9 6 │ 8 2 3 │ 5 1 4 7) Produce off springs in the following way. First the segments between the cut points are copied in to off springs. O1 = (- - │4 5 6 │ - - - -) and O2 = (- - │8 2 3 │ - - - -) Then, starting from the second cut point of one parent, the cities from the other parent are copied in the same order, omitting those which already exist. Reaching the end of the string /chromosome/ we continue from the first place of the string.

6

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. The sequence of cities in the second parent (from the second cut point) is 5 1 4 7 9 6 8 2 3 From this, after removal of cities 4, 5 and 6 which lie between the two cut points of the first parent we get 1 7 9 8 2 3 Placing this sequence in the first offspring (starting from the second cut point), we have O1 = (2 3 │ 4 5 6│ 1 7 9 8) Similarly O2 = (5 6 │8 2 3 │ 7 9 1 4). ►The second basic variant of order crossover (Ox2) The second basic variant of order crossover (Ox2) proposed by Davis [23] is similar as to cut point selection of the first (Ox1), except the repairing procedure. Two cut points are selected and the elements between them are copied. The rest elements are copied from the beginning of the second parent respecting their relative order omitting those which already exist in the first parent. Consider the following example. P1 = (2 3 │4 5 6 │ 1 7 9 8) and P2 = (5 6 │ 8 2 3 │ 7 9 1 4) Copy the segments between the cut points to get O1 = (- - │ 4 5 6 │ - - - -) O2 = (- - │ 8 2 3 │ - - - -)

and

The sequence of cities in the second parent is 5 6 8 2 3 7 9 1 4 Removing 4, 5 and 6 we get 8 2 3 7 9 1 Placing this sequence in the first offspring from left to right according to the order we have O1 = (8 2 4 5 6 3 7 9 1) Similarly O2 = (4 5 8 2 3 6 7 9 1)

5.2. The three additional new variations of order crossover The idea of variations of order crossover (VOX) is inspired from the fact Davis used cut point positions. Why does he use the same cut point positions in both parents? As a result of this [24] proposed three new additional variations of order crossover and their performance were far better than the existing variations. The variations are designed using cut point analysis on the standard ordered crossover operator Ox1. ►The first new variation (Ox3 ) In this variation [24] the cut points in both parents are at different positions, but size of substrings between the cut points in both parents is the same. Obviously Ox1 is a special case of Ox3. The repairing procedure is the same as Ox1.For example consider P1 = (9 4 2 6│ 3 8 7 │ 1 5) and P2 = (2│1 9 4 │ 7 6 3 5 8) Replacing 6 5 2 1 9 4 starting from the second cut point in P1 we have the first offspring as O1 = (2 1 9 4 3 8 7 6 5 ) Replacing 5 2 6 3 8 7 starting from the second cut point in P2 we have the second offspring as O2 = (7 1 9 4 5 2 6 3 8) ►The second new variation (Ox4) Here not only the positions of the cut points are different but the sizes of the substrings between the cut points in both parents are also different [24]. The remaining steps are similar to Ox1.Consider the following example P1 = (8 2 7 3 │ 6 9 1 │ 4 5) and 7

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. P2 = (4 │ 9 5 2 1 │ 6 7 8 3) Replacing 7 8 3 4 5 2 after the second cut point in P1 and 4 8 7 3 6 off springs as O1 = (3 4 5 2 6 9 1 7 8 ) O2 = (6 9 5 2 1 4 8 7 3 )

after the second cut point in P2 we have the two

►The third new variation (Ox5) The only difference from Ox1 is, here [24] used two pairs of cut points. The rest is the same as Ox1.For example let us Consider P1 = (2 │ 8 4 │ 1 6 │ 5 3 7│ 9) and P2 = (3 │ 5 7 │ 9 2 │ 4 6 8│ 1) Replacing 1 9 2 6 after the fourth cut point in P1 and 9 2 1 3 after the fourth cut point in P2 we have the two off springs as O1 = (9 8 4 2 6 5 3 7 1 ) and O2 = (2 5 7 1 3 4 6 8 9 ) These Variations have been studied by the Authors in [24].

6. Experimental setup Getting the most appropriate combination of parameters occurring in a GA is termed as parameter tuning [25] is considered to be the most important and perhaps the difficult task. In case of permutation coded GA parameter tuning is generally more difficult as compared to binary coded GA due to the simple reason that the number of tunable parameters occurring in a permutation coded GA are usually more than that occurring in binary GA. This difficulty also increases in both the permutation and binary GA as we take larger and larger test suit into consideration. In order to overcome this difficulty an extensive experiment has been carried out for all the six GAs. These six GAs are Inversion, Displacement, Exchange, Insertion, Inverted Exchange and Inverted Displacement mutations. These Genetic Algorithms are tested using ten TSPLIB [10] bench mark test problems. The problems are eil51, eil76, kroA100, lin105, rat195, gr24, swiss42, kroB150, gil262 and eil101. The final parameter values for all the GAs are 0.9 as probability of crossover and 0.01 as probability of mutation. We do not claim that these parameter values are the best for any problem in general. But these values are selected and recommended since they are found to repeatedly good results for most of the problems and hence they are appropriate values to choose if we talk about the overall performance of the algorithms in general. Initial populations in all algorithms are obtained by nearest neighbor initialization strategy. The selection is done using roulette wheel selection for the first generation. For the remaining generations selection is done with elitism of size one i.e. if the best individual of the parent population is better than the best individual in offspring population then it replaces the best individual of offspring population. For each GA 20 independent runs with different seed numbers are taken and the objective function values, average number of function evaluations, execution time of successful runs, mean and standard deviations are recorded. For a particular problem and a particular algorithm, a run is said to be successful run here if the best objective function value found in that run lies within 5% accuracy of the best known objective function value of that problem. The maximum numbers generations are also fixed to be 20000 for all the GAs. All the algorithms are implemented in C++ and the experiments for all computations are performed on a 1.66 GHz processor PC with 2 GB of RAM. The solution quality is measured by calculating the percentage of excess above the optimal value reported in TSPLIB [10].We used the formula given by:

Error (%) =

Obtained value - optimal value x100 optimal value

8


7. Results and discussion In this section the results of GA are compared and discussed. The performance analysis of each of the six mutation algorithms and ten problems from TSPLIB[10] are summarized using tables and graphs below. Table 1. Result of eil51 obtained for six Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion

Inverted Exchange Inverted Displacement

min max time min max time min max time min max time min max time min max time

438.1 748 258 439.5 758.9 228 432.4 829.5 257 456.1 721.6 232 438.1 863 225 431.6 758 256

447 679 195 434.4 749.3 190 463.9 787.4 193 472.7 710.2 197 435.1 834.4 198 443.1 761.4 194

431.1 753.9 250 439.2 727.3 228 447.1 805.4 249 434.4 746.3 242 430.7 914.5 257 431.1 733.9 250

431.1 747 249 441.2 759 238 440.1 834.6 245 439.5 748.7 237 433.5 965.4 254 430.2 753.7 250

431.1 801.6 294 434 787 252 444.9 824.7 287 449.9 784.4 279 441 914.1 298 431.1 754.3 294

431.1 (1.1) 434 (1.8) 432.4 (1.5) 434.4 (1.9) 430.7 (1.1) 430.2 (0.9)

As can be observed from Table 1 the Inverted Displacement mutation has shown a better performance than the remaining five mutation operators for eil51 with 0.9% error. The Inverted Exchange mutation is also good for showing better performance than the remaining four mutations for eil51. In order to clearly see the comparative convergence process of the six mutations for eil51 see the following fig1 below. In

Fig.1. Convergence comparison of the mutations in eil51

9

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 2. Result of eil76 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



558.6 862.2 540 566.2 929.5 496 564.3 968.9 536 564.7 898.6 524 562.3 1114.1 351 560.9 910.9 537

563.8 829.3 427 578.7 867.8 431 595.5 967.9 423 570.2 822.9 420 566.3 1028.1 432 569.1 910.9 423

558.5 889.5 528 564.7 915.1 512 564.3 962 550 559.8 893 492 554.5 1067.2 522 554.5 936.9 515

556.1 885.1 519 562.2 962.1 481 566 1044 528 566.6 943.8 493 554.5 1078.5 521 552.3 941.2 513

557 908.9 624 568.6 943.2 552 568.8 1014.3 624 568.3 910.5 571 561 1062.2 600 561.7 984.2 613

556.1 (3.3) 562.2 (4.4) 564.3 (4.8) 559.8 (4) 554.5 (3) 552.3 (2.6)

Still in Table2 the Inverted Displacement and Inverted Exchange mutations are best and next best performers respectively. The minimum values are obtained using Ox4 in both mutations. Interms of time Displacement mutation takes less time relatively after 20000 generations. The following figure shows the comparative convergence of the six mutations clearly for eil76.

Fig.2. Convergence comparison of the mutations for eil76.

10

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 3. Result of kroA100 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



21380.4 37499.9 928 22494.2 39866.7 882 22128.2 49465.1 927 22128.2 37371.5 883 21539.6 53556.6 949 21427.1 43589 928

21481.3 37229.5 786 22291.3 39962.9 758 22570.7 47780.3 766 22292.8 36505.8 764 21381.8 54658.5 762 21818.2 40280.2 763

21307.4 37551.1 909 21682 39396.5 847 21598.7 45056.7 928 22209.8 38843.5 853 21390.8 51723 888 21294.4 40132.6 908

21484.2 40592.9 905 21682 40789.4 850 21682 47581.2 894 22128.2 42799 856 21846.7 55750.7 884 21316.4 41640.8 897

21307.4 44611.2 1079 22655.8 45707.5 997 22482.3 49577.3 1074 21906.5 45870.5 991 21294.4 52599.8 1050 21497.7 44881.9 1076

21307.4 (0.1) 21682 (1.8) 21598.7 (1.4) 21906.5 (2.9) 21294.4 (0.05) 21294.4 (0.05)

In Table 3 it is observed that both the Inverted Exchange by (Ox5) and Inverted Displacement by (Ox3) obtained the same value with a 0.05% error but the time Inverted Displacement mutation took to obtain this value is less than that of Inverted Exchange mutation. Accordingly Inverted Displacement mutation still performs better than the remaining mutations for kroA100. Graphical representation of Table 3 is shown below as Fig3.

Fig.3. Convergence comparison of the mutations in kroA100

11

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 4. Result of lin105 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



14568.5 28468.5 897 15193.9 28790.2 863 14756.8 32931.6 906 14547.5 26893.4 843 14764.6 38345.7 899 14660 31794.1 880

14699.3 26019.4 739 15530 27134.3 757 15330.2 32539.7 735 15223.5 25942.9 738 15068.1 37168.3 738 15149.5 27279.1 751

14439.8 26806.6 886 14612.9 28047.3 845 14619.9 33130.1 901 14559 28067.4 842 14497.7 37564 872 14484.7 29260.9 876

14429.1 27819.4 1093 14554.5 30965.5 826 14612.1 33335.3 895 14651.4 28867.6 840 14497.7 35994.5 870 14405.7 30500 871

14524.7 30978.1 1239 14715.5 32283.5 977 14708.7 35142.8 1075 14632.4 30099.7 998 14550.6 37280.5 1038 14555.7 32054.8 1033

14429.1 (0.3) 14554.5 (1.2) 14612.1 (1.6) 14547.5 (1.1) 14497.7 (0.8) 14405.7 (0.1)

As is seen in Table 4 the Inverted Displacement mutation has performed best than the rest five mutation operators relatively. The value obtained is within a 0.1% error using Ox4.Inversion mutation is found to be next best performer for lin105. Fig.4 shows the comparative convergence of the six mutation operators for the problem lin105.

Fig.4. Convergence comparison of the mutations in lin105

12

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 5. Result of rat195 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



2454.3 3630.9 1804 2440.6 3718.8 1818 2442.2 4413.8 1785 2440.9 3629.3 1735 2418.7 4475.8 1772 2413.6 3816.8 1827

2389.1 3621.2 1566 2449.1 3745.8 1540 2527.6 4446.5 1536 2483.3 3499.7 1535 2414.5 4851.7 1543 2427 3783.1 1552

2386.2 3604.8 1774 2432.2 3780.3 1718 2471 4247.4 1724 2419.5 3662 1626 2441.3 4588.1 1698 2413.6 3739.5 1707

2364.3 3743.6 1748 2440 3887.3 1743 2408.1 4324 1728 2394.7 3954.1 1694 2418.4 4884.6 1709 2429.2 3959.3 1723

2443.4 4190.4 2198 2479.1 4324.6 2157 2492.8 4464.6 2108 2508.3 4028.9 1972 2404.7 4896.9 2066 2364.2 3986.2 2125

2364.3 (1.7) 2432.2 (4.7) 2408.1 (3.6) 2394.7 (3) 2404.7 (3.5) 2364.2 (1.7)

What are observed in Table 5 are the values obtained by Inverted Displacement mutation and Inversion mutation very much nearer. The values obtained by Inverted Displacement is 0.1 units less than the value obtained by Inversion. In addition to this the time Inverted Displacement took to get value is less than the time Inversion took. This implies the Inverted Displacement mutation still performs well using (Ox5). In Fig 5 the comparative convergence shown by the graphs Inverted Displacement mutation and Inversion mutation almost coincide starting at about 18000 generations.

Fig.5. Convergence comparison of the mutations for rat195

13

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 6. Result of gr24 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion


min max time min max time min max time min

1272 2407 17 1340 2539 32 1323 2680 31 1323

1286 2438 13 1340 2349 13 1366 2549 12 1430

1272 2511 9 1272 2404 1 1272 2529 9 1323

1272 2525 8 1323 2591 30 1323 2860 28 1272

1272 2597 11 1323 2790 39 1323 3000 39 1314

max time min max time min max time

2497 31 1272 3154 9 1272 2948 17

2373 13 1272 2686 9 1314 2306 13

2560 30 1272 3054 8 1272 2551 1

2581 8 1272 2914 9 1272 2447 2

2951 38 1272 3029 11 1272 2773 10

1272 (0) 1272 (0) 1272 (0) 1272 (0)

1272 (0) 1272 (0)

In Table 6 all the six mutations has achieved the required known optimal value of gr24 with no error as in TSPLIB [10].They are compared simply by their time of convergence. Accordingly since the computation using Inverted Displacement mutation took one second which is the smallest time still it performs better.

Fig.6. Convergence comparison of the mutations for gr24 As seen in Fig.6 with the exception of insertion mutation the remaining mutations converge to the same value after 5000 generations. While after 10000 generations all mutations converged to the known optimal value.

14

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 7. Result of swiss42 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



1273 2179 3 1273 2312 19 1273 2653 18 1273 2369 4 1273 2816 4 1273 2238 4

1273 2178 14 1384 2641 34 1301 2495 27 1294 2049 28 1273 2748 8 1273 2097 18

1273 2119 3 1273 2332 17 1273 2531 16 1273 2273 17 1273 2897 17 1273 2295 2

1273 2385 17 1273 2312 19 1273 2603 17 1273 2273 3 1273 2927 16 1273 2577 2

1273 2666 5 1273 2586 5 1273 2681 5 1273 2443 2 1273 3081 23 1273 2526 7

1273 (0) 1273 (0) 1273 (0) 1273 (0) 1273 (0) 1273 (0)

In Table 7 all the six mutations has achieved the required known optimal value of swiss42with no error as in TSPLIB [10].They are compared simply by their time of convergence. Accordingly since the computation using Inverted Displacement mutation took two seconds which is the smallest time still it performs better. Here the performance of Insertion mutation is found to be the same as that of the Inverted Displacement mutation.

Fig.7. Convergence comparison of the mutations for swiss42 In Fig.7 it can be seen that all the six mutations had different results up to the first one thousand but after that all converged to the same known optimal value.

15

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 8. Result of kroB150 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



27263.4 48149.3 782 28747.5 46527.5 787 28782 52204.3 834 27604.7 45607.4 833 26990.7 59300.5 804 27320.4 51752.4 792

27360.1 46081.3 660 28769.9 46222.9 728 29259.2 53779.5 709 28652.7 44138.2 713 27286.9 63830.9 663 27708.3 47142.5 662

26900.9 46491.7 748 27087 44719.7 831 27845.3 54256.4 795 27594.5 45946.1 814 26783.3 59946.8 757 26910.3 49458.2 756

26675.9 47391.4 751 27397.2 49528.2 825 27193.9 54698.1 789 27403 49042.5 779 27363.6 58747.3 759 26967.1 51015.9 762

26878.4 49305.4 913 27845.6 52525.1 1007 27797.8 52495.6 974 27834.6 49185.2 948 26775.8 61646.3 910 27169.8 50269.4 919

26675.9 (2) 27087 (3.6) 27193.9 (4) 27403 (4.8) 26775.8 (2.4) 26910.3 (2.9)

Up to now it has been seen that the Inverted Displacement mutation was the best performer in all the seven known TSPLIB instances. But know as can be observed from Table8 the Inversion mutation performs best and the Inverted exchange mutation is next best. The Inverted Displacement mutation is third with a 2.9% error.

Fig.8. Convergence comparison of the mutations for kroB150 Fig 8 shows that, especially after 10000 generations the computation using Inversion mutation shows faster convergence to the minimum value than the Inverted exchange and Inverted Displacement mutations.

16

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 9. Result of gil262 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



2537.2 3735.5 1412 2640.2 3782.4 1525 2618.6 4198.6 1399 2626.3 3827.1 1391 2554.3 4483.4 1699 2619.6 3655.3 1426

2536.7 3717.2 1242 2765.4 3713 1330 2692.5 4111.3 1222 2740.7 3599.3 1215 2638 4585 1467 2580.4 3670 1243

2529 3621.6 1314 2606.3 3685.5 1401 2555.9 3912.9 1289 2583.5 3691.6 1280 2554.1 4636 1549 2546.1 3633.8 1287

2506.7 3799.5 1326 2577.4 3999.9 1552 2620.7 4227.5 1313 2600.6 3856.8 1309 2662.6 4666 1550 2559.3 3802.5 1313

2509.5 4204.8 1648 2760.1 4202.3 1856 2610.8 4439.8 1633 2549.2 4240.6 1623 2575.4 4577.9 1991 2565.2 4106.2 1639

2506.7 (5.4) 2577.4 (8.3) 2555.9 (7.4) 2549.2 (7.1) 2554.1 (7.4) 2546.1 (7)

Results in Fig 9 for the problem gil262 indicates that the computation using Inversion mutation in combination with fourth variation of order crossover operator has shown a better performance than the remaining five mutations. Computation using Inverted Displacement mutation in combination with the third variation of order crossover operator is second best performer.

Fig.9. Convergence comparison of the mutations for gil262 In Fig. 9 what can be observed is the convergence of Inverted mutation. After 6000 generations it is clearly seen to converge faster in comparison to the remaining mutations.

17

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 10. Result of eil101 obtained for different Mutation operators before or at 20000 generations Mutation Cost and Ox1 Ox2 Ox3 Ox4 Ox5 Best and Operators time /sec %

Inversion

Displacement

Exchange

Insertion



668.5 1024.5 916 677.6 1053.8 871 714.6 1176 924 691 1013.4 866 676.7 1200.7 901 678.1 1005.5 923

660.3 924.1 749 715 1064.8 752 683.5 1080 764 664.4 971.4 772 670.4 1221.3 760 661 986.7 758

660.3 972.4 872 675.5 1024.9 848 670.2 1124 899 669.1 986.9 872 667.7 1234.8 888 655.8 1019 867

644.5 998.3 881 697.8 1066.5 851 657.4 1138.8 882 660.8 1056.1 851 658.2 1213.9 892 654.2 1074.2 867

656.1 1162.8 1512 689 1163.3 993 682.9 1222.1 1043 680.6 1060.6 990 668.8 1262.8 1049 663.7 1214.4 1044

644.5 (2.4) 675.5 (7.3) 657.4 (4.5) 660.8 (5) 658.2 (4.6) 654.2 (4)

In Table 10 computational results indicate that the performance of Inversion mutation is best in relative to the remaining five mutation operators. The crossover used to obtain its result is the fourth variation of order crossover operator. The second best performer is the Inverted Displacement mutation and the fourth variation of order crossover operator is the crossover used here. With the exception of Displacement mutation results obtained by Exchange, Insertion and Inverted Exchange are not that much exaggerated.

Fig.10. Convergence comparison of the mutations for eil101 Fig 10 shows the fast convergence and slow convergence of Inversion and Displacement mutations respectively. The fast convergence of Inversion mutation is observed after 4000 generations. Notations of mutation operators used here below are as follows: Inver (Inversion), Dis (Displacement), Exch (Exchange), Inser (Insertion), InverExch (Inverted Exchange) and InverDis (Inverted Displacement) in Tables 11-15.In Fig11, Fig12 and Fig13 1,2,3,4,5 and 6 represents Inversion, Displacement, Exchange, Insertion, Inverted Exchange and Inverted Displacement mutations respectively. 18

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. The following Tables show the number of successful runs, Average No. of function evaluation of successful runs, Average time of execution of successful runs in seconds, Mean of objective function values of successful runs and Standard Deviation of successful runs out of 20 runs. In Table11 The Inversion, Inverted Exchange and Inverted Displacement mutations have shown 20 successful runs out of 20 runs in seven instances in the remaining three instances Inversion mutation is better and can be observed in Fig.11. In addition the box plots of the number of successful runs, Average No. of function evaluation of successful runs, Average time of execution of successful runs in seconds are also presented. Table 11. Number of successful runs. Number of successful runs (out of 20 runs)

Problem Inver 20 20 20 20 18 19 20 20 20 19

Eil51 Eil76 kroA100 Lin105 Eil101 gil262 kroB150 Gr24 Swiss42 Rat195

Dis 11 11 20 20 4 11 6 20 20 7

Exch 15 13 19 20 7 12 3 20 20 10

Inser 19 16 19 19 7 9 3 20 18 11

InverExch 20 20 20 20 12 13 20 20 20 15

InverDis 20 20 20 20 17 13 20 20 20 13

100 90 80

Success Rate

70 60 50 40 30 20

1

2

3

4

5

6

Algorithm

Fig.11. Performance analysis of the mutations for successful runs in relation to success rate.

19

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. As can be seen from Table 12 the average number of function evaluation of successful runs when Inverted Displacement mutation show less number of function evaluations in five instances. Displacement and Exchange mutations show better function evaluation in two instances each. This can be seen in Fig.12. Table 12. Average number of function evaluation of successful runs Average number of function evaluation of successful runs)

Problem

Inver 1930055 3175164 4619966 4702266 4620480 10462857 5850342 864174.7 1439323 7402356


Dis 1930012 3175392 4600355 4702019 4620059 10464520 5850105 863919.4 1438906 7402361

Exch 1930055 3174746 4600152 4702323 4620892 10464964 5850875 863890.5 1439989 7402642

Inser 1930131 3174899 4600331 4702779 4620479 10464117 5851399 864281 1439841 7402704

InverExch 1930139 3175349 4600163 4702543 4620571 10464571 5850104 863861.3 1440681 7402709

InverDis 1930003 3175277 4600716 4701929 4620304 10464297 5849971 863724.4 1439878 7402326

6

x 10

10

Average Function Evaluation

9 8 7 6 5 4 3 2 1 1

2

3

4

5

6

Algorithm

Fig.12. Performance analysis of the mutations for Average number of function evaluation of successful runs. In Table 13 the genetic algorithm that used Inverted Displacement mutation performed well by giving the smallest Average time of execution of successful runs in four instances. It is also described by Fig.13.

20

Kusum Deep and Hadush Mebrahtu / Combined Mutation Operators of Genetic Algorithm for the Travelling Salesman problem, IJCOPI Vol. 2, No. 3, Sep-Dec 2011, pp. 1-23. Table 13. Average time of execution of successful runs in seconds Average time of execution of successful runs in seconds

Problem

Inver 191.5 375.1 598.4 605.5 555.9 1325.7 902.3 30.5 63.2 968.3


Dis 235.8 347.9 550.4 578.7 641.5 1341.1 887 27.3 90.1 977.7

Exch 194.4 332 558.1 565.4 609.4 1304 873 28.5 79.8 968.4

Inser 186.6 343.9 782.3 621.8 606.7 1656.2 861.6 30.4 80.3 983.9

InverExch 184.7 340.7 781.9 578.8 597 1302 1222.2 28 65.8 1168.5

InverDis 179.8 336 582 575 570.8 1319.2 869.1 26.7 60.7 1188.5

1600 1400

time (sec)

1200 1000 800 600 400 200 0 1

2

3

4

5

6

Algorithm

Fig.13. Performance analysis of the mutations for Average time of execution of successful runs in seconds

21


Problem Eil51 Eil76 kroA100 Lin105 Eil101 gil262 kroB150 Gr24 Swiss42 Rat195

Problem Eil51 Eil76 kroA100 Lin105 Eil101 gil262 kroB150 Gr24 Swiss42 Rat195

Table 14. Mean of objective function values of successful runs Mean of objective function values of successful runs

Inver 432.6 557.7 21436.3 14435.2 649.8 2525.4 26805.4 1272 1284.7 2400.9

Dis 447.8 563.2 21708.4 14649.3 659.7 2574.9 27251.5 1289.8 1283.9 2429.2

Exch 440.9 563 21647.5 14528 655.1 2568.9 27256.4 1287.3 1273 2426.2

Inser 436.8 564 21964.8 14789.9 655.7 2569.6 27305.8 1297.5 1279.4 2422.5

InverExch 432.1 557.2 21560.9 14532.9 655.4 2567.8 26904.1 1272 1273 2403.3

InverDis 432.2 557.4 21417.7 14431.2 653.2 2547.3 26909.3 1272 1273 2403.7

Table 15. Standard deviation of objective function values of successful runs Standard Deviation of successful runs

Inver 3.2 3.4 182.7 42.2 13.4 23.3 169.3 0 24 19

Dis 4.9 1.8 130.3 141.2 0.6 11.4 188.4 24.9 22.4 9.4

Exch 4.9 2.8 106.3 59.5 4.5 12.9 135.2 23.9 0 9.7

Inser 5.6 1.8 189.7 155 3.6 17.7 124 26.1 18.7 14.1

InverExch 3.2 4.6 221 54.1 4.5 19.5 224 0 0 24.8

InverDis 2.4 4.5 149.5 55.9 4.4 24.9 188.4 0 0 22.3

7. Conclusion and future work In this paper two mutation operators known as Inverted Exchange and Inverted Displacement are introduced. The Inverted Exchange mutation is a combination of Inversion and Exchange mutations and the Inverted Displacement is a combination of Inversion and Displacement mutations. The performance of these mutation operators are compared with four existing mutations. For this comparison ten TSPLIB [10] are used. The results in section 6 indicate the superiority of Inverted Displacement mutation by winning in seven problems. In three problems the minimal costs are obtained using Inversion mutation. In general after the application of six mutation operators, five different variations of order crossover on ten TSPLIB [10] test problems the following are concluded. First the performance of the Inverted Displacement mutation is found to be superior in finding a better minimal values than the remaining mutations used here. Second it is observed that the Inverted mutation operator outperforms the existing Displacement, Exchange and Insertion mutation operators. Third it has been confirmed that the three new variations of order crossover proposed by [24] are superior to Davis’s two existing basic variants of order crossover. This is because all the optimal values found by all mutations used in this paper for the ten problems are also obtained by the new variations. Based on this study, it is expected that in future the Inverted Displacement mutation operator shows a great potential for future research using other crossover operators.

22


References 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25.

Papadimitriou, C. H. and Steiglitz, K. Combinatorial Optimization: algorithms and complexity, Prentice Hall (1982). Bonachea, D., E. Ingerman, J. Levy, and S. McPeak, An Improved Adaptive Multi-Start Approach to Finding NearOptimal Solutions to the Euclidean TSP. In D. Whitley, D.Goldberg, E. Cantu-Paz, L. Spector, I. Parmee, and H.G.Beyer, eds., Proceedings of the Genetic and Evolutionary Computation Conference, 143-150, Las Vegas, Nevada,Morgan Kaufmann (2000). Ray, S.S., Bandyopadhyay S., Pal S.K. Gene Ordering in Partitive Clustering using Microarray Expressions. Journal of Biosciences, Vol. 32, No. 5 (2007) 1019-1025. Johnson, O., Liu, J. A Travelling Salesman Approach for Predicting Protein Functions. Source Code for Biology and Medicine, Vol. 1, No. 3 (2006) 1-7. Sur-Kolay, S., Banerjee, S. and Murthy, C.A.: Flavours of Travelling Salesman problem In VLSI Design. 1st Indian international conference on Artificial Intelligence, Hyderabad (2003). Michalewicz, Z. Genetic Algorithms + Data Structures = Evolution programs, Berlin: Springer-Verlag (1992). Kennedy, J., Eberhart, R.C.: Swarm Intelligence. Morgan Kauffman publishers, San Francisco (2001). Dorigo, M. and Gambardella, L.M. Ant Colony System: A Cooperative Learning Approach to the Travelling Salesman Problem, IEEE Transactions on Evolutionary Computation, Vol. 1 (1997) 53-66. Laarhoven, P.Van. and Aarts,E.H.L. Simulated Annealing: Theory and Applications, Kluwer Academic (1987). TSPLIB, http://www.iwr.uniheidelberg.de/groups/comopt/software/TSPLIB95/ Fogel, L.J. Owens, A.J. and Walsh, M.J.: Artificial Intelligence through Simulated Evolution. John Wiley (1966). Banzhaf, W. The molecular traveling salesman, Biological Cybernetics, Vol. 64 (1990) 7–14. Fogel, D. A parallel processing approach to a multiple travelling salesman problem using evolutionary programming, Proceedings of the Fourth annual Symposium on Parallel Processing, (Fullerton, California), pp. 318–326. (1990). Fogel, D. B. An evolutionary approach to the travelling salesman problem,Biological Cybernetics, Vol. 60, No. 2 (1988) 139-144. Gen, M. and Cheng, R. Genetic Algorithm and Engineering design, John wiley and sons, New york (1997) 118-127. Louis, S. J., Tang, R.. Interactive genetic algorithms for the traveling salesman problem. In Proceedings of the 1999 genetic and evolutionary computing conference (GECCO), pp. 385-392 (1999). Eiben, A., Michalewicz, Z., Schoenauer, M., and Smith, J. Parameter control in evolutionary algorithms. In F. G. Lobo, C. F. Lima, and Z. Michalewicz (Eds.), Parameter setting in Evolutionary Algorithms, Vol. 54 of Studies in Computational Intelligence, pp. 19-46, Berlin:Springer Verlag. (2007). Meyer-Nieberg, S. and Beyer, H.-G. Self adaptation in evolutionary algorithms. In F. G.Lobo, C. F. Lima, and Z. Michalewicz (Eds.), Parameter Setting in Evolutionary Algorithms, pp. 47–76. Berlin: Springer (2007). Eiben, A. and Smith, J. Introduction to evolutionary computation. Berlin: Springer. (2003). Serpell, M. and Smith, J.E. Self-Adaptation of Mutation Operator and Probability for Permutation Representations in Genetic Algorithms. Evolutionary Computation, Vol. 18, No. 3 (2010) 491–514. Suh, J.Y., Gucht, D.V. Incorporating Heuristic Information into Genetic Search, in Proceedings of the Second Int. Conf. On Genetic Algorithms, Massachusetts Institute of Technology, Cambridge, MA, pp. 100-107 (1987) Athitos, V. and Sclaroff, S. Approximate Nearest Neighbor Retrieval Using Euclidean Embeddings. Approximate Nearest Neighbors Methods for Learning and Vision, (2003). Davis, L. Handbook of Genetic Algorithms, New York: Van Nostrand Reinhold (1991). Kusum D. and Mebrahtu, H. New Variations of Order Crossover for Travelling Salesman Problem. International Journal of Combinatorial Optimization problems and Informatics. Vol. 2, No. 1, Jan-April (2011) 2-13. Kusum, D. and Thakur, M. A Real Coded Multi Parent Genetic Algorithm for Function Optimization, Journal of Hybrid Computing Research, Vol. 1, No. 2, July-Dec. (2008) 67-83.

23