The Travelling Salesman Problem. in Fuzzy Membership Functions 1. Abstract

Transcription

1 Chapter 7 The Travelling Salesman Problem in Fuzzy Membership Functions 1 Abstract In this chapter, the fuzzification of travelling salesman problem in the way of trapezoidal fuzzy membership functions is discussed and finally the optimal solution in terms of same membership functions have been verified. Chapter 7 ends with a conclusion section. The concluding section bears the summary of the results of the thesis. 1 The contents of this chapter form the material of the paper published in International Journal of Fuzzy Mathematics, International Institute of Fuzzy Mathematics, Los Angels, USA, Vol. 21, No. 1, (2013). 144

2 Functions Introduction Travelling salesman problem (TSP) is one of the challenging real - life problems, attracting researchers of many fields including artificial intelligence, operations research and algorithm design and analysis. The problem has been well studied till now under different heading has been solved with different approaches including genetic algorithms and linear programming, Linear programming is designed to deal with crisp parameters, but information about real life systems is often available in the form of vague descriptions. Fuzzy methods are designed to handle vague terms, and are most suited to finding optimal solutions to problems with vague parameters. In TSP the salesman takes decision of selecting an optimal and feasible route between any couple of cities on the basis of expected measures. In most of the real world problems it is not possible to have all constraints and resources in exact from rather they are in expected or vague form. This leads to the concept of fuzzy logic which enables us to emulate the human reasoning process and make decisions based on vague or imprecise data, and fuzzy programming gives the methodology of solving the problems in fuzzy environment. An ideal solution method would solve every travelling salesman problem to optimality, but this is not practical in most large problems. While advances have been made in solving the TSP, those advances have come at the cost of more complicated computer code. The complexity involves not only the length of the code, but the required

3 Functions 146 nesting and data structures. It is required to meet the aspiration level of a decision maker under which the current optimal solution remains still optimal and feasible. In 2004, Makani Das and Hemanta K. Baruah had developed in solving procedure for linear programming problem in the way of triangular fuzzy membership functions [68]. Further we have applied the same method for solving the transportation problem in the way of trapezoidal fuzzy membership functions and also the assignment problem with the help of fuzzy Hungarian method in trapezoidal membership functions. The organization of the chapter is detailed below. In section 7.2, we introduced the travelling salesman problem in terms of fuzzified form. We have considered the whole problem in terms of trapezoidal fuzzy membership functions of travelling salesman problem, after finding the optimal solution for the minimum cost of the sequence of the shortest route between the couple of cities and also this cost is verified with the aid of trapezoidal fuzzy membership functions which are stated in section 7.3. We are solving a TSP with the help of trapezoidal membership functions and its arithmetic operations. Solving procedure has been applied from the way of fuzzy assignment problem. The optimal solution in terms of fuzzy numbers which has been verified in the nature of fuzzy membership functions. The fuzzified version of the problem has been discussed with the aid of a numerical example with results and discussion.

4 Functions Travelling Salesman Problem in Fuzzified Form The most frequently considered objective of the travelling salesman problem is to determine an optimal order for travelling all the cities so that the total fuzzy cost is minimized. Consider the situation when decision maker has to determine the fuzzy optimal solution of travelling salesman problem with minimized fuzzy cost, time and overall distance. The individual functions can be formed for all the objectives of decision maker. Let [x (1) ij,x(2) ij,x(3) ij,x(4) ij [x (1) ij,x(2) ij,x(3) ij,x(4) ij ] (=) ] represents the link from city i to city j and [-δ,0,δ,4δ] if city j is visited from city i 0, otherwise Let [c (1) ij,c(2) ij,c(3) ij,c(4) ij ] be the cost of travelling from city i to city j, the overall fuzzy cost of a particular route is the sum of the costs on the links comprising the route. Since the decision maker has to minimize the overall fuzzy cost of travelling, so he can set goal for the total estimated fuzzy cost of the entire route.

5 Functions Formulation and Solution of Fuzzified Travelling Salesman Problem We shall consider a numerical example which will make clear the techniques for the solution of fuzzy travelling salesman with the help of fuzzy assignment problem and its algorithm for the aid of trapezoidal membership functions. A travelling salesman has to visit 5 cities. He wishes to start from a particular city, visit each city once and then return to his starting point. The travelling cost in terms of fuzziness for each city from a particular city is given below. What is the sequence of visit of the salesman, so that the fuzzy cost is minimized?

6 Functions 149 Table 7.1: Travelling Salesman Problem A B C D E A - [1,3,5,7] [3,5,7,9] [0,2,4,6] [1,3,5,7] B [1,3,5,7] - [2,4,6,8] [0,2,4,6] [1,3,5,7] C [4,6,8,10] [3,5,7,9] - [4,6,8,10] [2,4,6,8] D [0,2,4,6] [0,2,4,6] [4,6,8,10] - [4,6,8,10] E [1,3,5,7] [1,3,5,7] [2,4,6,8] [4,6,8,10] - Locate the row, and subtract the minimum element from every element including the minimum element with the help of fuzzy arithmetic s, Table 7.2: Travelling Salesman Tableau (Initial Step) A B C D E A - [-5,-1, 3,7] [-3,1,5,9] [-6,-2, 2,6] [-5,-1, 3,7] B [-5,-1,3,7] - [-4,0,4,8] [-6,-2, 2,6] [-5,-1, 3,7] C [-4,0,4,8] [-5,-1, 3,7] - [-4,0,4,8] [-6,-2, 2,6] D [-6,-2,2,6] [-6,-2, 2,6] [-2, 2,6,10] - [-2,2, 6,10] E [-6,-2,2,6] [-6,-2, 2,6] [-5,-1,3,7] [-3,1,5,7] - Since the column c is not having at least one fuzzy zero, subtract the minimum element from every element in the column.

7 Functions 150 Table 7.3: Travelling Salesman Tableau (Continued) A B C D E A - [-5,-1,3,7] [-10,-2,6,14] [-6,-2,2,6] [-5,-1,3,7] B [-5,-1,3,7] - [-11,-3,5,13] [-6,-2,2,6] [-5,-1,3,7] C [-4,0,4,8] [-5,-1,3,7] - [-4,0,4,8] [-6,-2,2,6] D [-6,-2,2,6] [-6,-2,2,6] [-9,-1,7,15] - [-2,2,6,10] E [-6,-2,2,6] [-6,-2,2,6] [-12,-4,4,12] [-3,1,5,9] - Since the number of lines covering the fuzzy zero 4 (dark numbers - it represented as a seperate line) is less the order of the fuzzy matrix (5). The fuzzy optimal solution is not obtained. Subtract the minimum element from every element and add in the intersection of lines (not from the element in the line). Table 7.4: Travelling Salesman Tableau (Continued) A B C D E A - [-12,-4,4,12] [-11,-3,5,13] [-6,-2,2,6] [-5,-1,3,7] B [-12,-4,4,12] - [-12,-4,4,12] [-6,-2,2,6] [-5,-1,3,7] C [-11,-3,5,13] [-12,-4,4,12] - [-4,0,4,8] [-6,-2,2,6] D [-6,-2,2,6] [-6,-2,2,6] [-9,-1,7,15] - [-7,1,9,17] E [-6,-2,2,6] [-6,-2,2,6] [-12,-4,4,12] [-7,1,9,17] -

8 Table 7.5: Travelling Salesman Tableau (Continued) A B C D E A - [-12,-4,4,12] [-11,-3,5,13] [-6,-2,2,6] [-5,-1,3,7] B [-12,-4,4,12] - [-12,-4,4,12] [-6,-2,2,6] [-5,-1,3,7] C [-11,-3,5,13] [-12,-4,4,12] - [-4,0,4,8] [-6,-2,2,6] D [-6,-2,2,6] [-6,-2,2,6] [-9,-1,7,15] - [-7,1,9,17] E [-6,-2,2,6] [-6,-2,2,6] [-12,-4,4,12] [-7,1,9,17] - Now draw the minimum lines to cover the fuzzy zeros. Number of lines (dark numbers - it represented as a seperate line) covering fuzzy zeros is equal to the order of the fuzzy matrix. There fore, the fuzzy optimal solution is obtained. Then the fuzzy assignment can be made Ch.7 The Travelling Salesman Problem in Fuzzy Membership Functions 151

9 Table 7.6: Travelling Salesman Tableau (Continued) A B C D E A - [-12,-4,4,12] [-11,-3,5,13] [-6,-2,2,6] [-5,-1,3,7] B [-12,-4,4,12] - [-12,-4,4,12] [-6,-2,2,6] [-5,-1,3,7] C [-11,-3,5,13] [-12,-4,4,12] - [-4,0,4,8] [-6,-2,2,6] D [-6,-2,2,6] [-6,-2,2,6] [-9,-1,7,15] - [-7,1,9,17] E [-6,-2,2,6] [-6,-2,2,6] [-12,-4,4,12] [-7,1,9,17] - Ch.7 The Travelling Salesman Problem in Fuzzy Membership Functions 152

10 Functions 153 Assign unit fuzzy zero in the row or column first, no other fuzzy zero can be marked. The fuzzy assignment is City A to City B - [1,3,5,7] City B to City D - [0,2,4,6] City C to City E - [2,4,6,8] City D to City A - [0,2,4,6] City E to City C - [2,4,6,8] or, City A to City D - [0,2,4,6] City B to City A - [0,2,4,6] City C to City E - [2,4,6,8] City D to City B - [0,2,4,6] City E to City C - [2,4,6,8] Therefore, the total fuzzy minimum cost is = {[c (1) 12,c(2) 12,c(3) 12,c(4) 12 ]+[c(1) 24,c(2) 24,c(3) 24,c(4) 24 ]+[c(1) 35,c(2) 35,c(3) 35,c(4) 35 ]+ [c (1) 41,c(2) 41,c(3) 41,c(4) 41 ]+[c(1) 53,c(2) 53,c(3) 53,c(4) 53 ]} = {[1,3,5,7] + [0,2,4,6] + [2,4,6,8] + [0,2,4,6] + [2,4,6,8]} = [5,15,25,35] or, ={[c (1) 14,c(2) 14,c(3) 14,c(4) 14 ] + [c(1) 21,c(2) 21,c(3) 21,c(4) 21 ] + [c(1) 35,c(2) 35,c(3) 35,c(4) 35 ] + [c (1) 42,c(2) 42,c(3) 42,c(4) 42 ]+ [c (1) 53,c(2) 53,c(3) 53,c(4) 53 ]} = {[0,2,4,6] + [0,2,4,6] +[ 2,4,6,8] + [0,2,4,6] + [2,4,6,8]} = [5,15,25,35] (7.1)

11 Table 7.7: Travelling Salesman Tableau (Continued) A B C D E A - [-12,-4,4,12] [-22,-6,10,26] [-6,-2,2,6] [-5,-1,3,7] B [-12,-4,4,12] - [-11,-3,5,13] [-6,-2,2,6] [-5,-1,3,7] C [-11,-3,5,13] [-12,-4,4,12] - [-4,0,4,8] [-6,-2,2,6] D [-6,-2,2,6] [-6,-2,2,6] [-9,-1,7,15] - [-7,1,9,17] E [-6,-2,2,6] [-6,-2,2,6] [-12,-4,4,12] [-7,1,9,17] - Ch.7 The Travelling Salesman Problem in Fuzzy Membership Functions 154

12 Functions 155 The fuzzy optimum assignment does not provide solution for the case of travelling salesman problem. Therefore, the route is City A to City D, City D to City B, City B to City A. City B is not allowed to follow City A, and he can t come to the origin. (He can start from any city) Now consider the next minimum element [-5,-1, 3,7] and mark this instead of [-6,-2,2,6]. Now the sequence of visit of the salesman route is City A to City E - [1,3,5,7] City E to City C - [2,4,6,8] City C to City B - [3,5,7,9] City B to City D - [0,2,4,6] City D to City A - [0,2,4,6] Therefore, the fuzzy total setup cost is ={[c (1) 15,c(2) 15,c(3) 15,c(4) 15 ]+[c(1) 53,c(2) 53,c(3) 53,c(4) 53 ]+[c(1) 32,c(2) 32,c(3) 32,c(4) 32 ]+ [c (1) 24,c(2) 24,c(3) 24,c(4) 24 ]+[c(1) 41,c(2) 41,c(3) 41,c(4) 41 ]} ={[1,3,5,7]+[2,4,6,8]+[3,5,7,9]+[0,2,4,6]+[0,2,4,6]} = [6,16,26,36] (7.2) We have to find fuzzy membership functions (f.m.fs) of C ij then the fuzzy travelling cost as follows:

13 Functions 156 { x 1 2 } if 1 x 3 μ c15 (x) (=) 1 if 3 x 5 { x 7 2 } if 5 x 7 0, otherwise To compute the interval of confidence for each level α the trapezoidal shapes will be described by functions of α in the following manner. Here α = (x(α) 1 1) 2 and α = (x(α) 2 7) 2 There fore, c 15 = [x (α) 1,x(α) 2 ] = [2α+1, 2α+1] (7.3) Exactly in the similar way, we have to write { x 2 2 } if 2 x 4 μ c53 (x) (=) 1 if 4 x 6 { x 8 2 } if 6 x 8 0, otherwise There fore, c 53 = [x (α) 1,x(α) 2 ] = [2α+2, 2α+8] (7.4)

14 Functions 157 Then { x 3 2 } if 3 x 5 μ c32 (x) (=) 1 if 5 x 7 { x 9 2 } if 7 x 9 0, otherwise There fore, c 53 = [x (α) 1,x(α) 2 ] = [2α+3, 2α+9] (7.5) and here μ c24 (x) = μ c41 (x) which implies that,c 24 = [2α, 2α+6] we can write (using (7.3),(7.4),(7.5) and (7.6)) fuzzy minimum cost =[C 15 +C 53 +C 32 +C 24 +C 41 ] = [8α+6,, 8α+30] The equations to be solved are; 8α+6 X1 = 0 8α+30 X2 = 0 We are retain only two roots α in [0, 1] From the above two equations we get, α = [X 1 6] 8 and α = [30 X 2] 8 (7.6) (7.7)

15 Functions 158 Therefore, { x 6 8 } if 6 x 16 μ Min.Cost. (x) (=) 1 if 16 x 26 { 30 x 8 } if 26 x 36 0, otherwise which is the required fuzzy membership functions of fuzzy travelling minimum cost (using 7.2). 7.4 Results and Discussion Using the proposed method the total fuzzy travelling minimum cost is [6, 16, 26, 36], which can be physically interpreted as follows: (1) The least amount of cost is 6. (2) The most possible amount of cost lies between 16 and 26. (3) The greatest amount of cost is 36. That is, The optimal fuzzy travelling cost will be always greater than 6 and less than 36 and maximum chances are that the cost will be between 16 and 26. The variations in cost with respect to chances are shown in the figure 7.1. Similarly the obtained fuzzy optimal solutions x ij may be physically interpreted.

16 Functions 159 Figure: 7.1. Membership function of fuzzy number representing the total fuzzy travelling minimum cost (i) According to decision maker the total fuzzy travelling minimum cost will be greater than Rs 6 and less than Rs 36. (ii) Decision maker in favour of that the total fuzzy travelling minimum cost will be greater than or equal to Rs 16 and less than or equal to Rs 26. (iii) The percentage of the favourness of the decision maker for the remaining values of total fuzzy travelling minimum cost can be obtained as follows: Let x represent the value of the total fuzzy travelling minimum cost then the percentage of the favourness of the decision maker for x = μ min.cost. (x),

17 Conclusion 160 where { x 6 8 } if 6 x 16 μ Min.Cost. (x) (=) 1 if 16 x 26 { 30 x 8 } if 26 x 36 0, otherwise 7.5 Conclusion Fuzzy set theory is certainly not a philosopher s stone which solves all the problems that confronted us today. But it has a considerable potential for practical as well as for mathematical applications. This thesis extensively examines the concept of fuzzy membership functions and their utility in decision making under fuzzy environment. The results of the thesis dwell upon the concept of trapezoidal fuzzy numbers. The fuzzy number is defined in terms of extension principle, accordingly this notion is applied to solve some optimization problems and to find optimal solution which can be verified with the help of trapezoidal membership functions.

18 Conclusion 161 In the thesis, we have focused primarily on fuzzy methodologies and fuzzy systems, as they bring basic ideas to the area of fuzzy membership functions. The other constituents of fuzzy optimization techniques are also surveyed here but for details we refer to the existing vast literature. In the first part of the study we have presented an overview of developments in the individual parts of fuzzy membership functions. The entire fuzzy system operation is based on the membership functions. The sense of reasoning is very important in forming the membership functions. For each constituent of membership functions we briefly have overviewed its basic background, arithmetic operations, main classification and interpretation on all the three types of fuzzy numbers, methodologies for finding alpha cut and recent developments with the help of fuzzy extension principle. The extension principle is one of the most basic ideas in fuzzy set theory. It provides a general method for extending crisp mathematical concepts to address fuzzy quantities, such as real algebra operations on fuzzy numbers. These operations are computationally effective generalizations of interval analysis. Although the set of real fuzzy numbers equipped with an extended addition or multiplication is no longer a group, many structural properties of the resulting fuzzy numbers are

19 Conclusion 162 preserved in the process. Also the main literature, professional journals and technical newsletters and other relevant information are mentioned in the part of detailed literature. In the second part of the study we have extensively studied the subject of fuzzy optimization; we have dealt with a class of fuzzy linear programming problems and again investigated feasible and optimal solutions - the necessary tools for dealing with such problems applying in terms of trapezoidal fuzzy membership functions. Inthiswaywehaveshownthattheclassofcrisp(classical)linear programming problems can be embedded into the class of fuzzy linear programming problem ones. Moreover, we have defined and proposed the new simplex computational procedure with suitable numerical example in terms of fuzzy membership functions in which all the parameters and variables of all the problems should be considered as a trapezoidal membership functions. Further, we have investigated special classes of linear programming problems such as transportation, least time transportation, assignment problems and its application of the travelling salesman problem. Henceforth we have studied and introduced new fuzzified forms in all the chapters and solved the problems with suitable

20 Conclusion 163 new proposed methods which are investigated with the aid of trapezoidal fuzzy membership functions in a suitable and relevant numerical example with results and discussion. Finally, the optimal solutions of every problems are verified in terms of trapezoidal fuzzy membership functions. It is very new in the environment of fuzzy optimization techniques which gives better accuracy when compared with existing methods.

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