A NEW APPROACH FOR SOLVING TRAVELLING SALESMAN PROBLEM WITH FUZZY NUMBERS USING DYNAMIC PROGRAMMING

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 11, November2018, pp , Article ID: IJMET_09_11_097 Available online at ISSN Print: andISSN Online: IAEME Publication Scopus Indexed A NEW APPROACH FOR SOLVING TRAVELLING SALESMAN PROBLEM WITH FUZZY NUMBERS USING DYNAMIC PROGRAMMING V. MYTHILI Department of Mathematics, Prince Shri Venkateshwara Padmavathy Engineering College, Chennai, Research Scholar, Mathematics Division, School of Advanced Sciences, VIT Chennai Campus, Chennai, India. M. KALIYAPPAN Mathematics Division, School of Advanced Sciences, VIT Chennai Campus, Chennai, India. S. HARIHARAN Department of Mathematics, Amrita School of Engineering, Coimbatore, Amrita Vishwa Vidyapeetham, India. S. DHANASEKAR Mathematics Division, School of Advanced Sciences, VIT Chennai Campus, Chennai, India. ABSTRACT Travelling Salesman Problem (TSP) is to detect the shortest closed tour such the teach city is visited exactly on ceinan n-city. The TSP problem is a special kind of an assignment model that eliminates sub-tours. In Dynamic Programming the optimum solution of a multivariable problem is obtained by decomposing sub-problem of a single variable. In this paper, Dynamic programming technique is applied to solve a Fuzzy Travelling Salesman Problem (FTSP) and MATLAB program is developed. Keywords: Fuzzy number, Triangular fuzzy number, Trapezoidal fuzzy number, LR fuzzy numbers, Fuzzy arithmetic operations, Fuzzy Travelling Salesman Problem, Fuzzy optimal solution. Cite this Article: V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar, A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming International Journal of Mechanical Engineering and Technology, 9(11), 2018, pp editor@iaeme.com

2 V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar 1 INTRODUCTION Zadeh [1] introduced concept of fuzzy sets in the year Fuzzy concepts have been used in various fields such as Engineering, Management, and Science etc. Travelling Salesman Problem is a classical problem in Graph Theory and Combinatorial Optimization and it has been studied since On a typical business trip, a travelling salesman visits various towns and cities. If he wants to avoid having to pass through the same city twice, he needs a Hamiltonian cycle. FTSP is a Travelling salesman problem where the cost or distance will be given in fuzzy numbers instead of crisp numbers. Hansen [2] applied tabu search algorithm to solve fuzzy TSP. Jaszkiewicz [3] applied genetic local search algorithm for solving fuzzy TSP. Yan et al., [4] used the evolutionary algorithm to solve fuzzy TSP. Sepideh Fereidouni [5] used multi objective linear programming technique to get the optimal solution. Angel et al., [6] proposed dynamic search algorithm to solve fuzzy TSP. Paquete et al., [7] developed Pareto local search algorithm which is the exted version of the local search algorithm. Rehmat et al., [8] used fuzzy linear programming to solve the problem. Mukherjee et al., [9] proposed a new approach to solve fuzzy TSP. Chaudhuri et al., [10] used fuzzy linear programming for solving fuzzy TSP. Amit kumar et al., [11] proposed an algorithm for solving fuzzy TSP with LR-fuzzy parameters. Dhanasekar et al.,[14] applied Hungarian algorithm to solve fuzzy TSP with elementwise subtraction of fuzzy numbers. Since fuzzy TSP is a polynomial time problem, more number of algorithms are still proposed to solve the fuzzy TSP Dynamic programming algorithm sare often used for optimization. It will examine the previously solved sub-problems and will combine their solutions to gives the best solution for the given problem. In this paper, Dynamic programming algorithm issued to solve FTSP. The proposed method is easy to apply and understand compared to other existing methods and it is also a systematic, complete enumeration technique. The proposed algorithm is validated through examples with existing methods. Inthispaper, Section2 presents the fuzzy numbers and their ranking techniques. Procedure for dynamic programming for solving Travelling Salesman Problem and numerical examples are being presented in Section3. MATLAB implementation for the above approach is provided in Section4. 2. PRELIMINARIES Definition 2.1 A fuzzy set [12] can be mathematically constructed by assigning to each possible individual in the universe of discourse a value representing its grade of membership. Definition 2.2 A membership function μ of a fuzzy number [12] is piecewise continuous, convex and normal. Definition 2.3 A fuzzy number A=,,,with membership function of the form 1 μ = 0 h!" Is called a trapezoidal fuzzy number [12]. In this if b = c then it is called triangular fuzzy number editor@iaeme.com

3 A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming Figure 1. a) Triangular Fuzzy Number b) Trapezoidal Fuzzy Number Definition2.4 The Fuzzy Operations [12] of fuzzy numbers are defined as Fuzzy Addition Fuzzy Subtraction #, #, #, # ) + $, $, $, $ ) = # % $, # % $, # % $, # % $ ) #, #, # ) + $, $, $ ) = # % $, # % $, # % $ ) #, #, #, # ) - $, $, $, $ ) = # $, # $, # $, # $ ) #, #, # ) - $, $, $ ) = # $, # $, # $ ) Definition2.5. Ranking of fuzzy numbers Ranking of fuzzy numbers plays an important role in decision making, risk analysis and optimization. In modelling real world problems, fuzzy numbers are used to represent the performance of alternatives. In decision making scenario, selection of alternatives according to the need of the problem is the main objective. Selection of alternatives means ordering of fuzzy numbers. Even though many methods are available to order the fuzzy numbers, all methods do not agree with each other. Due to lack of best method many authors are working on it still. Abbasbandy and Hajjari [16] introduced an approach of ranking trapezoidal fuzzy numbers based on left and right spread sat some level. Yager s ranking technique is based upon the idea of associating with a fuzzy number a scalar value, its valuation and using this valuation fuzzy numbers are compared and ordered. Hence this technique is better when compared with other techniques. Here we used Yager sranking technique to compare fuzzy numbers. The Yager s ranking [13] of a fuzzy number A is given by # α &'()*=+ 0.5(. / %( 0 α α where ( 0 α = Lower α- level cut and (. α = Upper α-level If&'()* &'1* then() 1. Definition 2.6 A Fuzzy Travelling Salesman Problem is defined as : : Subject to : : 1. 8;# 89 =1, 9;# 89 1 min5=667) ;# 9;# editor@iaeme.com

4 V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar % 98 1, 1! => % 9? %?8 2, 1! = A> 4. 8BC % BC B D % % BFGD 8 >2, 1! H # H $ H :J$ > The matrix representation is given by City 1 City 2 City n City 1 7) #$ 7) #: City 2 7) $# 7) $: City n 7) :# 7) :$ Constraints (1) assure that each city is visited only once. All 2- city sub tours are excluded by Constraint (2). Constraint (3) ensures that all 3-city sub tours are eliminated. All (n-2) city sub tours are eliminated by Constraints (4). 3. DYNAMIC PROGRAMMING FOR FUZZY TRAVELLING SALESMAN PROBLEM Dynamic Programming is a mathematical tool for the treatment of many complex problems. The basic theory and applications of Dynamic Programming is presented by Richard E. Bellman in [19]. Dynamic Programming algorithms are applied for many optimization problems. The Dynamic Programming algorithm for the 4 4 matrix is given as follows Consider the TSP with fuzzy parameters of 4 4 matrix City 1 City 2 City 3 City 4 City 1 7) #$ 7) #K 7) #L City 2 7) $# 7) $K 7) $L City 3 7) K# 7) K$ 7) KL City 4 7) L# 7) L$ 7) LK Here the cost to move City 1 to City 2 is 7) #$. Similarly we do the calculations for various stages as follows Stage 1 M)2,N1O 7) #$ M)3,N1O 7) #K Stage 2 M)4,N1O 7) #L M)3,N2O 7) $K %7) #$ M)4,N2O 7) $L %7) #$ M)2,N3O 7) K$ %7) #K M)4,N3O 7) KL %7) #K M)2,N4O 7) L$ %7) #L Stage 3 M)3,N4O 7) LK %7) #L editor@iaeme.com

5 A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming M)4,N2,3O min 7) $L %M)2,N3O,7) KL %M)3,N2O M)3,N2,4O min 7) $K %M)2,N4O,7) LK %M)4,N2O Stage 4 M)2,N3,4O min 7) K$ %M)3,N4O,7) L$ %M)4,N3O M)1,N2,3,4O min 7) $# %M)2,N3,4O,7) K# %M)3,N2,4O,7) L# %M)4,N2,3O According to the minimum fuzzy value the route is determined. Numerical examples Example 1 Consider the following fuzzy TSP discussed in [11]: City A City B City C City D City A ( ) (9,10,1,3) (6,8,3,5) (8,9,1,3) City B (9,10,2,4) ( ) (10,11,3,1) (4,5,1,3) City C (7,8,1,3) (10,11,3,4) ( ) (7,8,2,3) City D (9,10,3,5) (9,11,3,4) (6,8,1,5) ( ) Solution: Here the cost to move from City A to City B is7) #$ 9,10,1,3i.e.,M)2,N1OSimilarly we do the calculations for various stages as follows Stage 1 M)2,N1O 7) #$ 9,10,1,3 M)3,N1O 7) #K 6,8,3,5 M)4,N1O Stage 2 7) #L 8,9,1,3 M)3,N2O 7) $K %7) #$ 10,11,3,1%9,10,1,3 19,21,4,4 M)4,N2O 7) $L %7) #$ 4,5,1,3%9,10,1,3 13,15,2,6 M)2,N3O 7) K$ %7) #K 10,11,3,4%6,8,3,5 16,19,6,9 M)4,N3O 7) KL %7) #K 7,8,2,3%6,8,3,5 13,16,5,8 M)2,N4O 7) L$ %7) #L 9,11,3,4%8,9,1,3 17,20,4,7 Stage 3 M)3,N4O 7) LK %7) #L 6,8,1,5%8,9,1,3 14,17,2,8 M)4,N2,3O V!> 7) $L %M)2,N3O,7) KL %M)3,N2O V!> 4,5,1,3 % 16,19,6,9,7,8,2,3 % 19,21,4,4 V!> 20,24,7,12,26,29,6,7 20,24,7, editor@iaeme.com

6 V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar M)3,N2,4O V!> 7) $K %M)2,N4O,7) LK %M)4,N2O V!> 10,11,3,1 % 17,20,4,7,6,8,1,5 % 13,15,2,6 V!> 27,31,7,8,19,23,3,11 19,23,3,11 M)2,N3,4O V!> 7) K$ %M)3,N4O,7) L$ %M)4,N3O V!>'10,11,3,4% 14,17,2,8,9,11,3,4% 13,16,5,8* min'24,28,5,12,22,27,8,12* 22,27,8,12 Stage 4 M)1,N2,3,4O V!> 7) $# %M)2,N3,4O,7) K# %M)3,N2,4O,7) L# %M)4,N2,3O V!>9,10,2,4%22,27,8,12,7,8,1,3%19,23,3,11,9,10,3,5%20,24,7,12 V!>'31,37,10,16,26,31,4,14,29,34,10,17* 26,31,4,14 Therefore the optimal tour is and also !" i.e.,( 1 W 7 ( and the fuzzy optimal cost is26,31,4,14. Example 2 Consider the following fuzzy TSP discussed in [11]: Solution City A City B City C City D City A ( ) (20,5,4) (15,5,5) (11,3,2) City B (20,5,4) ( ) (30,5,3) (10,3,3) City C (15,5,5) (30,5,3) ( ) (20,10,12) City D (11,3,2) (10,3,3) (20,10,2) ( ) Here the cost to move from City A to City B is 7) #$ 20,5,4i.e,M)2,N1O Similarly we do the calculations for various stages as follows Stage 1 M)2,N1O 7) #$ 20,5,4 Stage 2 M)3,N1O 7) #K 15,5,5 M)4,N1O 7) #L 11,3,2 M)3,N2O 7) $K %7) #$ 30,5,3%20,5,4 50,10,7 M)4,N2O 7) $L %7) #$ 10,3,3%20,5,4 30,8,7 M)2,N3O 7) K$ %7) #K 30,5,3%15,5,5 45,10,8 M)4,N3O 7) KL %7) #K 20,10,12%15,5,5 35,15,17 M)2,N4O 7) L$ %7) #L 10,3,3%11,3,2 21,6, editor@iaeme.com

7 A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming M)3,N4O 7) LK %7) #L 20,10,2%11,3,2 31,13,4 Stage 3 M)4,N2,3O V!> 7) $L %M)2,N3O,7) KL %M)3,N2O V!> 10,3,3 % 45,10,8,20,10,12 % 50,10,7 V!> 55,13,11,70,20,19 55,13,11 M)3,N2,4O V!> 7) $K %M)2,N4O,7) LK %M)4,N2O V!> 30,5,3 % 21,6,5,20,10,2 % 30,8,7 V!> 51,11,8,50,18,9 51,11,8 M)2,N3,4O V!> 7) K$ %M)3,N4O,7) L$ %M)4,N3O V!>30,5,3%31,13,4,10,3,3%35,15,17 V!> 61,18,7,45,18,20 Stage 4 45,18,20 M)1,N2,3,4O V!> 7) $# %M)2,N3,4O,7) K# %M)3,N2,4O,7) L# %M)4,N2,3O V!>20,5,4%45,18,20,15,5,5%51,11,8,11,3,2%55,13,11 V!> 65,23,24,66,16,13,66,16,13 66,16,13 Therefore the optimal tour is a n d a l s o i.e.,( W 1 7 ( and also ( 7 W 1 ( the fuzzy optimal cost is 66,16,13. Example 3 Consider the following fuzzy TSP discussed in [17]: Solution: City A City B City C City D City A ( ) (1,2,3) (8,9,10) (9,10,11) City B (0,1,2) ( ) (5,6,7) (3,4,5) City C (14,15,16) (6,7,8) ( ) (7,8,9) City D (5,6,7) (2,3,4) (11,12,13) ( ) Here the cost to move from City A to City B is 7) #$ 1,2,3i.e., M)2,N1O Similarly we do the calculations for various stages as follows Stage editor@iaeme.com

8 V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar M)2,N1O 7) #$ 1,2,3 Stage 2 M)3,N1O 7) #K 8,9,10 M)4,N1O 7) #L 9,10,11 M)3,N2O 7) $K %7) #$ 5,6,7%1,2,3 6,8,10 M)4,N2O 7) $L %7) #$ 3,4,5%1,2,3 4,6,8 M)2,N3O 7) K$ %7) #K 6,7,8%8,9,10 14,16,18 M)4,N3O 7) KL %7) #K 7,8,9%8,9,10 15,17,19 M)2,N4O 7) L$ %7) #L 2,3,4%9,10,11 11,13,15 Stage 3 M)3,N4O 7) LK %7) #L 11,12,13%9,10,11 20,22,24 M)4,N2,3O V!> 7) $L %M)2,N3O,7) KL %M)3,N2O V!> 3,4,5 % 14,16,18,7,8,9 % 6,8,10 V!> 17,20,23,13,16,19 13,16,19 M)3,N2,4O V!> 7) $K %M)2,N4O,7) LK %M)4,N2O V!> 5,6,7 % 11,13,15,11,12,13 % 4,6,8 V!> 16,19,22,15,18,21 15,18,21 Stage 4 M)2,N3,4O V!> 7) K$ %M)3,N4O,7) L$ %M)4,N3O V!>6,7,8%20,22,24,2,3,4%15,17,19 V!> 26,29,32,17,20,23 17,20,23 M)1,N2,3,4O V!> 7) $# %M)2,N3,4O,7) K# %M)3,N2,4O,7) L# %M)4,N2,3O V!>0,1,2 % 17,20,23,14,15,16 % 15,18,21,5,6,7% 13,16,19 min'17,21,25,29,33,37,18,22,26* 17,21,25 Therefore the optimal tour is i.e.( 7 W 1 ( the fuzzy optimal cost is 17,21,25. Example 4 Consider the following fuzzy TSP discussed in [18]: City A City B City C City D City A ( ) (4,6,8,10) (5,7,9,11) (6,8,10,12) editor@iaeme.com

9 A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming City B (4,6,8,10) ( ) (2,4,6,8) (1,3,5,7) City C (5,7,9,11) (2,4,6,8) ( ) (3,5,7,9) City D (6,8,10,12) (1,3,5,7) (3,5,7,9) ( ) Solution: Here the cost to move from City A to City B is7) #$ 4,6,8,10i.e.,M)2,N1O.Similarly we do the calculations for various stages as follows Stage 1 M)2,N1O 7) #$ 4,6,8,10 M)3,N1O 7) #K 5,7,9,11 Stage 2 M)4,N1O 7) #L (6, 8, 10, 12) M)3,N2O 7) $K %7) #$ 2,4,6,8%4,6,8,10 6,10,14,18 M)4,N2O 7) $L %7) #$ 1,3,5,7%4,6,8,10 5,9,13,17 M)2,N3O 7) K$ %7) #K 2,4,6,8%5,7,9,11 7,11,15,19 M)4,N3O 7) KL %7) #K 3,5,7,9%5,7,9,11 8,12,16,20 M)2,N4O 7) L$ %7) #L 1,3,5,7%6,8,10,12 7,11,15,19 Stage 3 M)3,N4O 7) LK %7) #L 3,5,7,9%6,8,10,12 9,13,17,21 M)4,N2,3O V!> 7) $L %M)2,N3O,7) KL %M)3,N2O V!> 1,3,5,7 % 7,11,15,19,3,5,7,9 % 6,10,14,18 V!> 8,14,20,26,9,15,21,27 8,14,20,26 M)3,N2,4O V!> 7) $K %M)2,N4O,7) LK %M)4,N2O V!> 2,4,6,8 % 7,11,15,19,3,5,7,9 % 5,9,13,17 V!> 9,15,21,27,8,14,20,26 8,14,20,26 M)2,N3,4O V!> 7) K$ %M)3,N4O,7) L$ %M)4,N3O V!>2,4,6,8%9,13,17,21,1,3,5,7%8,12,16,20 V!> 11,17,23,29,9,15,21,27 Stage 4 9,15,21,27 M)1,N2,3,4O V!> 7) $# %M)2,N3,4O,7) K# %M)3,N2,4O,7) L# %M)4,N2,3O V!>4,6,8,10%9,15,21,27,5,7,9,11%8,14,20,26,6,8,10,12%8,14,20, editor@iaeme.com

10 V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar V!> 13,21,29,37,13,21,29,37,14,22,30,38 13,21,29,37 Therefore the optimal tour is and also i.e.,( 7 W 1 ( and also ( 1 W 7 ( the fuzzy optimal cost is 13,21,29,37. Table 1 Comparison Table Example Existing Algorithms Optimal Solution Example I (26,31,4,14) Example II Example III Example IV (66,16,13) Dhanasekar et al., [15] (17,21,25) AbhaSinghal and Priyanka Pandey [17] (13,21,29,37) Srinivasan and Geetharamani [18] Dynamic Programming method Optimal Solution (26,31,4,14) Route conditions satisfied. (66,16,13) Route conditions satisfied. (17,21,25) Route conditions satisfied (13,21,29,37) Route conditions satisfied 4 MAT LAB CODE The following MATLAB code will provide optimal solution for the 4x4 (matrix) fuzzy TSP clc clear all % This program is to calculate the minimal cost using Dynamic Programming % Enter the array size of the problem in n % Enter the triangular or trapezoidal fuzzy number as [a,b,c]or [a,b,c,d] n = input('enter the size: '); c = input('enter 1 for symmetric: '); if c==1 for i=1:n cell(i,i).s = inf; for j=i+1:n cell(i,j).s=input('enter the upper triangular entries: within [ cell(j,i).s=cell(i,j).s; else for i=1:n cell(i,i).s = inf; for j=1:n if i~=j cell(i,j).s=input('enter the row wise entries within [ ]: '); ] '); editor@iaeme.com

11 A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming M=[cell(1,1).S cell(1,2).s cell(1,3).s cell(1,4).s;cell(2,1).s cell(2,2).s cell(2,3).s cell(2,4).s; cell(3,1).s cell(3,2).s cell(3,3).s cell(3,4).s;cell(4,1).s cell(4,2).s cell(4,3).s cell(4,4).s]; % Stage 1 for i=2:n stgone(1,i).f=cell(1,i).s; % Stage 2 m1=1; for i=2:n for j=2:n if j~=i stgtwo(j,i).f1 = cell(i,j).s+cell(m1,i).s; % Stage 3 k=n; n1 = 1; for i = 2:n-1 for j=i+1:n t1= mean(cell(i,k).s+stgtwo(i,j).f1); t2= mean(cell(j,k).s+stgtwo(j,i).f1); t3=min(t1,t2); if(t3==t1) stgthree(m1,n1).f2=cell(i,k).s+stgtwo(i,j).f1; else stgthree(m1,n1).f2=cell(j,k).s+stgtwo(j,i).f1; k = k-1; n1 = n1+1; t7 = inf; for i=2:n t(i-1) = mean(cell(i,m1).s+stgthree(m1,n-i+1).f2); t7=min(t(i-1),t7); temp = t(i-1); if t7 == temp stgfour(1,1).f3 = cell(i,m1).s+stgthree(m1,n-i+1).f editor@iaeme.com

12 V. Mythili, M. Kaliyappan, S. Hariharan and S. Dhanasekar 5. CONCLUSION In this paper we have proposed a new algorithm through Dynamic Programming to solve a Fuzzy Travelling Salesman Problem. As the multivariate problem is decomposed into stages the working procedure becomes easier and the task is made simpler. Few numerical examples have been discussed and the above results are match with the current existing techniques. REFERENCES [1] Zadeh, L.A., Fuzzysets, InformationControl, 8(1965) [2] Hansen, M.P., Use of substitutes canalizing functions to guide local search based heuristics: The case of MOTSP, J. Heuristics, 6 (2000) [3] Jaszkiewicz, A., Genetic local search for multi objective combinatorial optimization, European Journal of Operational Research, 137(2002) [4] Yan, Z., Zhang, L., Kang, L., and Lin, G., A new MOEA for multi-objective TSP and its convergence property analysis, Proceedings of Second International Conference, Springer Verlag, Berlin, (2003) [5] Sepideh, F., Travelling salesman problem by using a fuzzy multi-objective linear programming, African Journal of Mathematics and Computer Science Research 4(11) (2003) [6] Angel, E., Bampis, E., and Gourvs, L., Approximating the Pareto curve with local search for the bicriteria TSP(1, 2) problem., Theoretical Computer Science, 310 (2004) [7] Paquete, L., Chiarandini, M., and Stytzle, T., Pareto local optimum sets in the Biobijective travelling salesman problem: an experimental study. Met heuristics for multi objective optimization, Lecture Notes in Economics and Mathematical Systems, 535, Springer, Berlin, (2004) [8] Rehmat, A., Saeed, H., and Cheema, M.S., Fuzzy multi objective linear programming approach for travelling salesman problem, Pakistan Journal of Statistics Operation Research., 3(2) (2007) [9] Mukherjee, S., and Basu, K., Application fuzzy ranking method for solving assignment problems with fuzzy costs, International Journal of Computational and Applied Mathematics, 5 (2010) [10] Chaudhuri, A., and De, K., Fuzzy multi-objective linear programming for traveling sales man problem, African Journal of Mathematics and Computer Science Research, 4(2) (2011) [11] Amitkumar and Anilgupta, Assignment and Travelling salesman problems with co.eff as LR fuzzy parameters, International Journal of Applied Science and Engineering, 10(3) (2012) [12] Klir, G.J., and Yuan, B., Fuzzy sets and fuzzy logic theory and applications, Prentice Hall, [13] Yager, R.R., Ranking fuzzy subsets over the unit interval, Proceedings of 17th IEEE International Conference on Decision and Control, San Diego, California, (1978) [14] Dhanasekar, S., Hariharan, S., and Sekar, P., Fuzzy Hungarian MODI Algorithm to solve Fully Fuzzy Transportation Problems, International journal of fuzzy systems, (2016) SCIE.DOI /s [15] Dhanasekar, S., Hariharan, S., and Sekar, P., Classical Travelling Salesman Problem (TSP) based approach to solve fuzzy TSP using Yager sranking, International journal of Computer Applications (IJCA),74(13)(2013)1-4 [16] Abbasbandy, S., and Hajjari, T., A new approach for ranking of trapezoidal fuzzy numbers, Computers and Mathematics with Applications, 57(3) (2009) editor@iaeme.com

13 A New Approach for Solving Travelling Salesman Problem with Fuzzy Numbers Using Dynamic Programming [17] AbhaSinghal and Priyanka Pandy., Travelling Salesman Problems by Dynamic Programming Algorithm, International Journal of Scientific Engineering and Applied Science, 2 (2016) [18] Srinivasan, A., and Geetharamani, G., Proposed method for solving FTSP, International journal of Application or Innovation in Engineering and Management, 3 (2014) [19] Bellman, R., Dynamic Programming, Princeton University Press, Princeton, New Jersy, editor@iaeme.com