Sensitivity Analysis for Intuitionistic Fuzzy Travelling Salesman Problem

Transcription

1 Volume 119 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Sensitivity Analysis for Intuitionistic Fuzzy Travelling Salesman Problem D. Anuradha 1, K. Kavitha 2 1,2 Department of Mathematics School of Advanced Sciences VIT University, Vellore Tamil Nadu, India. 1 anuradhadhanapal1981@gmail.com 2 kavinphd@gmail.com October 24, 2017 Abstract This paper presents intuitionistic fuzzy travelling salesman problem (IFTSP). A heuristic procedure is proposed for finding the maximum increment and decrement of each edge weight individually without changing the optimality of minimum Hamiltonian cycle. The procedure of the proposed approach is illustrated by numerical example. AMS Subject Classification:... Keywords: Travelling salesman problem, Triangular intuitionistic fuzzy numbers (TIFNr), Hamiltonian cycle, Sensitivity analysis.. 1 Introduction The Travelling Salesman Problem (TSP) is a representative of a huge category of problems called as combinatorial optimization 1 143

2 problems. In TSP, the salesman must visit the entire towns only one time and revisit the origin point of the town to end the trip. The ultimate aim of the problem is to get a tour of minimal length in terms of cost/distance/time on a fully connected graph. In the classical TSP, Hamiltonian cycles are generally called tours. TSP has various applications in diverse engineering and optimization problems. A path of approaching the TSP and especially its case of finding a shortest route is drafted by Dantzig et al. [1]. In literature, many researchers (Bhide et al. [2], Andreae [3], Bockenhauer et al. [4], Blaser et al. [5] have developed numerous algorithms to solve the TSP. The set of efficient solutions for bi-criteria travelling salesman problem by analysing the metrics in various perspectives discussed by Anuradha and Bhavani [6].An intuitionistic fuzzy set (IFS) is one of the generalization of fuzzy set theory (FST). IFS ( [7], [8]) have been found to be helpful to deal with imprecise information. There are cases where due to lack of available data, the assessment of membership values is not continuously applicable and consequently there ruins an indeterministic part on which indecision remains. Certainly FST is not applicable to deal with such problems; rather IFS is more appropriate. Jat et al. [9] have presented the triangular intuitionistic fuzzy numbers and ranking procedure for finding an optimal solution for TSP. Sahaya sudha et al. [10] has been implemented the generalized trapezoidal intuitionistic fuzzy numbers and ranking procedure for finding an optimal solution for TSP. Sensitivity analysis (SA) is one of the most exciting area in optimization. SA is to study the development of the variations of the parameters in the maximization/minimization problems on the maximal/minimal value of the objective function. Post optimality analysis and parametric optimization techniques for integer programming discussed by Geoffrion and Nauss [11]. Tarjan [12] discussed SA of minimum spanning trees (MST) and shortest path trees. An idea of SA for an approximate relaxation of the minimum Hamiltonian path has developed by Libura [13]. A procedure for finding inferior limits of the edge tolerances without changing the optimality of the minimum Hamiltonian path and TSP is discussed by Libura [14]. The cost sensitivity analysis in a transportation problem has discussed by Lucia Cabulea [15]. An inverse-ackermann type lower bound for online 2 144

3 MST verification is studied by Pettie [16]. A procedure for finding the sensitivity of a MST has presented by Pettie [17]. The primary focus of this paper is to find the edge tolerances with respect to the optimal solution of Hamiltonian cycle. This paper is structured as follows: Section 2 projects the basics of IFS. Section 3 overviews the general TSP with its terms and conditions. In Section 4 a simple numerical illustration is chosen to demonstrate the application of the suggested procedure. Finally the paper is concluded in section 5. 2 Preliminaries We need the following definitions of IFS, TIFNr, membership and non-membership function of an intuitionistic fuzzy set/number which can be found in Atanassov [7]. 2.1 Definition Let X is a nonempty set. An IFS A in X is given by a set of ordered triples, à I = { < x, µ A (x), ν A (x) >; x X} where µ A (x), ν A (x) : X [0, 1] define respectively, the degree of membership and degree of non-membership of the element x X to the set A, which is a subset of X, and for every element 0 µ A (x) + ν A (x) Definition A triangular intuitionistic fuzzy number à I is denoted by à I = (a 1, a 2, a 3 )(a 1, a 2, a 3) where a 1 a 1 a 2 a 3 a 3with the following membership function µ I (x) and non-membership à function ν I (x) is given as: à x a 1 a 2 a 1 if a 1 x a 2 a µã(x) = 3 x a 3 a 2 if a 2 x a 3 0 otherwise a 2 x if a a 2 a 1 x a 2 1 x a νã(x) = 2 if a a 2 x a 3 a2 3 1 otherwise 3 145

4 2.3 Definition Let à I = (a 1, a 2, a 3 )(a 1, a 2, a 3) and B I = (b 1, b 2, b 3 )(b 1, b 2, b 3) be any two triangular intuitionistic fuzzy numbers then the arithmetic operations as follows: (i)ãi + B I = (a 1 + b 1, a 2 + b 2, a 3 + b 3 )(a 1 + b 1, a 2 + b 2, a 3 + b 3) (ii) ÃI B I = (a 1 b 3, a 2 b 2, a 3 b 1 )(a 1 b 3, a 2 + b 2, a 3 + b 1) (iii) ÃI B I = (a 1 b 1, a 2 b 2, a 3 b 3 )(a 1b 1, a 2 b 2, a 3b 3) (iv) kãi = (ka 1, ka 2, ka 3 )(ka 1, ka 2, ka 3)fork 0 (v) kãi = (ka 3, ka 2, ka 1 )(ka 3, ka 2, ka 1)fork < 0 (vi) ÃI B I = ( a 1 b 3, a2 b 2, a3 ) ( a 1 b Definition b 3, a2 b 2, a 3 b 1 ) The ranking of a triangular intuitionistic fuzzy number à I = (a 1, a 2, a 3 )(a 1, a 2, a [ 3)is defined as R(ÃI ) = 1 ] (a 3 a 1 )(a2 2a 3 2a 1 )+(a3 a1)(a1+a2+a3)+3(a 2 3 a 2 1 ) 3 a 3 a 1 +a3 a1 2.5 Definition Let ÃI and B I be two triangular intuitionistic fuzzy numbers. The ranking of à I and B I by the R(.) on E, the set of triangular intuitionistic fuzzy numbers is defined as follows: (i)r(ãi ) > R( B I )if and only if ÃI > B I (ii) R(ÃI ) < R( B I )if and only if ÃI < B I (iii) R(ÃI ) = R( B I )if and only if ÃI B I (iv)r(ãi + B I ) = R(ÃI ) + R( B I ) (v) R(ÃI B I ) = R(ÃI ) R( B I ) 3 Travelling salesman problem Suppose a salesperson has to trip n towns. Beginning from a specific town, he has to trip each town only one time and revisit the origin point of the town to end the trip. Our aim is to minimize the total cost of travelling. Mathematical formulation of the TSP is given below. (P) Minimize Z = n i=1 n j=1 c ijx ij 4 146

5 Subject to n x ij = 1, j = 1, 2,..., n and j i (1) i=1 n x ij = 1, i = 1, 2,..., n and i j (2) j=1 x ij + x ji 1, 1 i j n (3) x ip1 +x p1p x p(n 2) i (n 2), 1 i p 1... p (n 2) n (4) { 1; salesman travels from city i to j x ij = (5) 0; otherwise where c ij is the cost of traveling from town i to j; x ij is the link from town i to j ; (1) and (2) make certain that each town is visited only once; (3) is sub tour elimination constraint and eliminates all 2-town sub tours; (4) eliminates all (n 1)-town sub tours. A set X = {x ij, i = 1, 2,..., n; j = 1, 2,..., n} is said to be feasible to problem (P) if X satisfies the conditions (1) to (5). Travelling salesman problem modeled as graph by representing the towns as nodes and the roads connecting the towns as edges. The costs are considered as weights (w ij ) assigned to the edges. Our aim is to find a tour of minimal weight. We need the following definitions which can be found in Narsingh Deo [18]. 3.1 Definition Any graph G = (V, E)consists of two sets of objects namely vertex set Vand edge set E where an edge e k = (v i, v j ) is identified by an unordered pair of vertices v i and v j. 3.2 Definition A connected graph without any circuit is termed as a tree. 3.3 Definition A sub graph T of a connected graph Gis said to be a spanning tree of G if T is a tree containing all the vertices ofg

6 4 Illustration Consider intuitionistic fuzzy travelling salesman problem with five towns. Any IFTSP has to visit all his business towns starting from his residential place and return back to the same place. The IF cost ( 00) between the towns are given in the following table. Table 1: IF Cost Matrix City (23,25,27) (38,40,42) (8,10,12) (10,12,14) (21,25,29) (36,40,44) (6,10,14) (8,12,16) 2 (23,25,27) - (18,20,22) (21,23,25) (9,11,13) (21,25,29) (16,20,24) (19,23,27) (7,11,15) 3 (38,40,42) (18,20,22) - (19,23,27) (31,33,35) (36,40,44) (16,20,24) (17,23,29) (29,33,37) 4 (6,10,14) (19,23,27) (17,23,29) - (16,20,24) (8,10,12) (21,23,25) (19,23,27) (18,20,22) 5 (10,12,14) (9,11,13) (31,33,35) (18,20,22) - (8,12,16) (7,11,15) (29,33,37) (16,20,24) Using definition 2.4, the ranking indices for the cost c ij corresponding to the given IFTSP is given below. Table 2: Cost Matrix City Travelling salesman problem of the above table can be modeled as a graph G. The towns are represented as nodes and the roads connecting the towns as edges. The costs are considered as weights allocated to the edges. The above graph G is a complete graph as every pair of nodes are connected by an edge. Each edge of G has been assigned with weights which are non-negative real numbers. A weighted graph is considered with only one weight at each edge. For weighted graph, one or more minimal spanning trees can be generated. In general, a ST of a weighted connected graph G is said to be MST if its 6 148

7 Figure 1: Graphical TSP with weight total weight is less than or equal to any other spanning tree of G. Though there are many procedures to find a MST of a weighted connected graph, procedure given by Kruskal seem to be simpler. According to Kruskal s procedure, the edges are arranged in the ascending order of its weights for graph G is given below. Table 3: Arrangement of Edges and its Weights G Edge Weight Edge Weight (1,4) 10 (2,4) 23 (2,5) 11 (3,4) 23 (1,5) 12 (1,2) 25 (2,3) 20 (3,5) 33 (4,5) 20 (1,3) 40 The construction of a minimal spanning tree is done by selecting the edges from lower to higher weights such that no edge forms a loop. Continue the selection process until all nodes are included in it. As a result minimal spanning tree T for G is obtained as follows: Now T is the minimal spanning tree with weight 53 ( 00). The TSP resembles like tracing a Hamiltonian cycle, which includes all the vertices of a graph. Here H is the minimal Hamiltonian cycle for graph G with weight 76 ( 00). Sensitivity analysis in Hamiltonian cycle is used to find the maximum increment and decrement of the each edge weight individually without changing the optimality of the solution. To 7 149

8 Figure 2: Minimal spanning tree with weight Figure 3: Minimal Hamiltonian cycle find the sensitivity range for graph H we compute the Modi indices. Modi indices M for graph H is minimum if and only if for every edge T ij =max { } U i, V j. Here Ui and V j are the minimum values of the non-tree edges in M for H. To verify the above said condition, we use the optimality conditions in [Kavitha and Anuradha [19]]. To compute the maximum increment and decrement of the edge weight of (1,4), we choose the non-tree edges (2,4), (5,4), (1,2) and (1,3). Using the optimality condition (i) for the non-tree edges (2,4) and (5,4) we get23 ( w 14 ) 0and 20 ( w 14 ) 0. This provides the minimum value as 6. By using the optimality condition (ii), for the non-tree edges (1,2) and (1,3) we get 25 ( w ) 0 and 40 ( w ) 0. This provides the minimum value as 18. Now, the maximum of minimum value of (1,4) is 18. Proceeding in this same manner, we can find the maximum of minimum value of (1,5) is 18; (2,3) is 14; 8 150

9 Figure 4: Modi indices for graph H (2,5) is 14 and (3,4) is 8. Therefore, the re-optimization of (1,4) in M for H is 28 (by replacing w ij by w ij + w ij ). Similarly, for (1,5) is 30, (2,3) is 34, (2,5) is 25 and (3,4) is 31. Now, the sensitivity analysis for graph H is given below: Figure 5: Sensitivity range for graph H 5 Conclusion This paper ranks triangular intuitionistic fuzzy numbers to solve TSP occurring in practical circumstances. By the heuristic procedure, we obtained the edge tolerances with respect to the optimal solution of Hamiltonian cycle. The proposed approach allows the decision maker to decide what level of exactness is required for edge weight to make the problem beneficial and effective

10 References [1] Dantzig, G., Fulkerson, R. and Johnson, S., Solution of a large-scale Traveling-salesman problem, Journal of the Operations Research Society of America,2(1954), [2] Bhide, S. John, N. Kabuka, M.R., A Boolean Neural Network Approach for the Traveling Salesman Problem, IEEE Transactions on Computers,42(1993), [3] Andreae, T., On the travelling salesman problem restricted to inputs satisfying a relaxed triangle inequality, Networks, 38(2001), [4] Bockenhauer, H.J., Hromkovi, J., Klasing, R., Seibert, S. and Unger, W., Towards the notion of stability of approximation for hard optimization tasks and the travelling salesman problem, Theoretical Computer Science, 285(2002),3-24. [5] Blaser, M., Manthey, B. and Sgall, J. An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality, Journal of Discrete Algorithms, 4(2006), [6] Anuradha, D. and Bhavani,.S, Multi Perspective Metrics for Finding All Efficient Solutions to Bi-Criteria Travelling Salesman Problem, Int. J. of Engineering and Technology, 5(2013), [7] Atanassov, K.T, Intuitionistic fuzzy sets, Fuzzy Sets and Systems,20(1986), [8] Atanassov, K.T., More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33(1989), [9] Jat, R. N., Sharma, S.C., Sanjay Jain, Anchal Choudhary, A fuzzy approach for solving for mixed intuitionistic fuzzy traveling salesman problem, Int. Journal of Mathematical Archive, 6(2015), [10] Sahaya sudha, A., Alice Ange, G., Elizabeth Priyanka, M., Emily Jennifer, S., An intuitionistic fuzzy approach

11 for solving generalized trapezoidal traveling salesman problem, International Journal of Mathematical Trends and Technology,29(2016),9-12. [11] Geoffrion, A.M. and Nauss, R., Parametric and post optimality analysis in integer programming, Management Sci.,23(1977), [12] Tarjan, R.E., Sensitivity analysis of minimum spanning trees and shortest path trees, Inform. Process. Lett.,14(1982), [13] Marek Libura, Sensitivity analysis of optimal solution for minimum Hamiltonian path, Zeszyty Nauk. Politech. Slaskiej: Automatyka, 84(1986), [14] Marek Libura, Sensitivity analysis for Hamiltonian path and travelling salesman problems, Discrete Applied Mathematics, 30(1991), [15] Lucia Cabulea, Sensitivity analysis of costs in a transportation problem, ICTAMI, Alba Iulia, Romania,11(2006), [16] Pettie, S., An inverse-ackermann type lower bound for online minimum spanning tree verification, Combinatorica, 26(2006), [17] Pettie, S., Sensitivity analysis of minimum trees in sub inverse Ackermann time, J. of Graph algorithms and application, 19(2015), [18] Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Eastern Economy Edition, [19] Kavitha, K. Anuradha, D., Heuristic Algorithm for finding Sensitivity Analysis of a More for Less Solution to Transportation Problems, Global Journal of Pure and Applied Mathematics, 11(2015),

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