Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 Neighborig Optimal Solutio for Fuzzy Travellig Salesma Problem D. Stephe Digar 1, K. Thiripura Sudari 2 1 Research Scholar (PG), Research Departmet of Mathematics, TBML College, Porayar, Idia 2 Research Scholar (PG), Departmet of Mathematics, Poompuhar College, Melaiyur, Idia E-mail- Ksudari_1982@yahoo.com Abstract - A ew method is itroduced to fid fuzzy optimal solutio for fuzzy Travellig salesma problems. I this method, ituitioistic trapezoidal fuzzy umbers are utilized to fid the fuzzy optimal solutio. This proposed method provides some of other fuzzy salesma problem very eighbour optimal solutio called fuzzy eighbourig optimal salesma. A relevat umerical example is also icluded Key words - Ituitioistic fuzzy umber, ituitioistic trapezoidal fuzzy umber, fuzzy salesma algorithm, fuzzy optimal solutio 1. INTRODUCTION Travellig salesma problem is a well-kow NP-hard problem i combiatorial optimizatio. I the ordiary form of travellig salesma problem, a map of cities is give to the salesma ad he has to visit all the cities oly oce ad retur to the startig poit to complete the tour i such a way that the legth of the tour is the shortest amog all possible tours for this map. The data cosists of weights assiged to the edges of a fiite complete graph ad the objective is to fid a cycle passig through all the vertices of the graph while havig the miimum total weight. There are differet approaches for solvig travellig salesma problem. Almostevery ew approach for solvig egieerig ad optimizatio problems has bee tried o travellig salesma problem. May methods have bee developed for solvig travellig salesma problem. These methods cosist of heuristic methods ad populatio based optimizatio algorithms etc. Heuristic methods like cuttig plaes ad brach ad boud ca optimally solve oly small problems whereas the heuristic methods such as 2-opt, 3-opt, Markov chai, simulated aealig ad tabu search are good for large problems. Populatio based optimizatio algorithms are a kid of ature based optimizatio algorithms. The atural systems ad creatures which are workig ad developig i ature are oe of the iterestig ad valuable sources of ispiratio for desigig ad ivetig ew systems ad algorithms i differet fields of sciece ad techology. Particle Swarm Optimizatio, Neural Networks, Evolutioary Computatio, At Systems etc. are a few of the problem solvig techiques ispired from observig ature. Travellig salesma problems i crisp ad fuzzy eviromet have received great attetio i recet years [1-4, 5,6,7,8,9,10,11]. With the use of LR fuzzy umbers, the computatioal efforts required to solve fuzzy assigmet problems ad fuzzy travellig salesma problem are cosiderably reduced [12]. I this paper, we itroduce ew method for fidig a fuzzy optimal solutio as well as of alterative solutios which is very ear to fuzzy optimal solutio for the give fuzzy travellig salesma problem. I sectio 2,recall the defiitio of ituitioistic trapezoidal fuzzy umber ad some operatios. I sectio 3, we preseted fuzzy travellig salesma problem ad algorithm. I sectio 4 umerical example. I sectio 5, coclusio is also icluded. 2. PRELIMINARIES I this sectio, some basic defiitios ad arithmetic operatios are reviewed. 307 www.ijergs.org
Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 2.1. INTUITIONISTIC FUZZY NUMBER Let a set X be fixed a Ifs A i X is a object havig the form A = {(X, μ A x, θ A x )/x X} where μ A x : X [0,1] ad θ A x : X [0,1],defie the degree of membership ad degree of o-membership respectively of the elemet x X to the set A,which is a subset of X, for every elemet of x X, 0 μ A x + θ A (x) 1. 2.2. DEFINITION A IFS A, defied o the uiversal set of real umbers R, is said to be geeralized IFN if its membership ad omembership fuctio has the followig characteristics: (i) (ii) μ A (x) : R [0, 1] is cotiuous. μ A (x) = 0 for all x (, a 1 ] [a 4, ). (iii) μ A (x) is strictly icreasig o [a 1,a 2 ] ad strictly decreasig o [a 3,a 4 ]. (iv) μ A (x) = w 1 for all x [a 2,a 3 ]. (v) ν A (x) : R [0, 1] is cotiuous. (vi) ν A (x) = w 2 for all x [b 2,b 3 ]. (vii) ν A (x) is strictly decreasig o [b 1,b 2 ] ad strictly icreasig o [b 3,b 4 ]. (viii) ν A (x) = w 1, for all x (, b 1 ] [b 4, ) ad w = w 1 +w 2, 0 < w 1. 2.3. DEFINITION A geeralized ituitioistic fuzzy umber A is said to be a geeralized trapezoidal ituitioistic fuzzy umber with parameters, b 1 a 1 b 2 a 2 a 3 b 3 a 4 b 4 ad deoted by A = (b 1,a 1,b 2,a 2,a 3,b 3,a 4,b 4 ;w 1,w 2 ) if its membership ad omembership fuctio is give by μ A (x) = w 1 (x a 1 ) (a 2 a 1 ), a 1 x a 2 w 1, a 2 x a 3 w 1 x a 4 a 3 a 4, a 3 x a 4 0, otherwise ad ν A (x) = w 2 (b 2 x), b (b 2 b 1 ) 1 x b 2 w 2, b 2 x b 3 w 2 x b 3, b b 4 b 3 x b 4 3 w 1, otherwise Geeralized trapezoidal ituitioistic fuzzy umber is deoted by A GITrFN = (b 1, a 1, b 2, a 2, a 3, b 3, a 4, b 4 ; w 1, w 2 ). Fig.1 membership ad o-membership fuctio of GITrFN. 308 www.ijergs.org
Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 2.4 DEFINITION We defie a rakig fuctior : F(R) R which maps each fuzzy umber i to the real lie; F(R) represets the set of all ituitioistic trapezoidal fuzzy umbers. If R be ay liear rakig fuctio, the R A = b 1+a 1 +b 2 +a 2 +a 3 +b 3 +a 4 +b 4.. 8 2.5 ARITHMETIC OPERATIONS umbers R. Let are as follows: I this sectio, arithmetic operatios betwee two ituitioistic trapezoidal fuzzy umbers, defied o uiversal set of real A = b 1, a 1, b 2, a 2, a 3, b 3, a 4, b 4 ad B = (d 1, c 1, d 2, c 2, c 3, d 3, c 4, d 4 ) ituitioistic trapezoidal fuzzy umbers, ImageA = ( b 4, a 4, b 3, a 3, a 2, b 2, a 1, b 1 ). A + B = (b 1 + d 1, a 1 + c 1, b 2 + d 2, a 2 + c 2, a 3 + c 3, b 3 + d 3, a 4 + c 4, b 4 + d 4. A B = b 1 d 4, a 1 c 4, b 2 d 3, a 2 c 3, a 3 c 2, b 3 d 2, a 4 c 1, b 4 d 1. if λ is ay scalar, the λa = λb 1, λa 1, λb 2, λa 2, λa 3, λb 3, λa 4, λb 4. if λ > 0. = (λb 4, λa 4, λb 3, λa 3, λa 2, λb 2, λa 1, λb 1 ). if λ < 0. A B = b 1 σ, a 1 σ, b 2 σ, a 2 σ, a 3 σ, b 3 σ, a 4 σ, b 4 σ if R B > 0 = ( b 4 σ, a 4 σ, b 3 σ, a 3 σ, a 2 σ, b 2 σ, a 1 σ, b 1 σ), if R B < 0 A B = b 1 a 1, b 2, a 2, a 3, b 3, a 4, b 4 σ σ σ σ σ σ σ σ ifr B 0 R B > 0, = ( b 4 σ, a 4 σ, b 3 σ, a 3 σ, a 2 σ, b 2 σ, a 1 σ, b 1 σ ) ifr B 0, R B < 0. Where σ = (d 1 + c 1 + d 2 + c 2 + c 3 + d 3 + c 4 + d 4 )/8. 3. FUZZY TRAVELLING SALESMAN PROBLEMS The fuzzy travellig sales ma problem is very similar to the fuzzy assigmet problem expect that i the former, there is a additioal restrictios. Suppose a fuzzy salesma has to visit cities. He wishes to start from a particular city, visit each city oce, ad the retur to his startig poit.the objective is to select the sequece i which the cities are visited i such a way that his total fuzzy travellig time is miimized. Sice the salesma has to visit all cities, the fuzzy optimal solutio remais idepedet of selectio of startig poit. The mathematical form of the fuzzy travellig salesma is give below Miimize z = i=1 j =1 k=1 d ij x ijk i j Subject to i=1 = 1, i j, k=1,2, m j=1 x ijk j=1 = 1, i=1,2, m k=1 x ijk 309 www.ijergs.org
Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 i=1 = 1, j=1,2, k=1 x ijk x ijk i j = i j x ij (k+1) for all j ad k x ijk = 1, if kth directed arc is from city i to j 0, other wise Where i,j ad k are itegers that vary betwee 1 ad. A fuzzy assigmet i a row is said to be a miimum fuzzy assigmet if the fuzzy cost of the fuzzy assigmet is miimum i the row. A tour of a fuzzy travellig salesma problem is said to be miimum tour if it cotais oe or more miimum fuzzy assigmets. 3.1 ALGORITHM Step 1 Step 2 Fid the miimum assigmets for each row i the fuzzy cost matrix below ad above of leadig diagoal elemets. Fid all possible miimum tour ad their fuzzy costs. Step 3: Fid the miimum of the all fuzzy costs of the miimum possible tours say z. Step 4: The tour correspodig to z is the fuzzy optimal tour ad z is the fuzzy optimal value of the tour. 4. EXAMPLE Cosider the followig fuzzy travellig salesma problem so as to miimize the fuzzy cost cycle. A B C D A - (-3,-1,0,2,3,4,5,6) (1,2,3,4,6,7,8,9) (-10,-6,5,6,10,15,17,19) B (-3,-1,0,2,3,4,5,6) - (-3,0,2,3,4,5,6,7) (-6,4,6,8,10,12,14,16) C (1,2,3,4,6,7,8,9) (-3,0,2,3,4,5,6,7) - (0,1,2,3,5,6,7,8) D (-10,-6,5,6,10,15,17,19) (-6,4,6,8,10,12,14,16) (0,1,2,3,5,6,7,8) - The miimum fuzzy costs i each row ad their elemets are give below R(c 12 ) = 2 R(c 13 ) = 5 R(c 14 ) = 7 R(c 21 ) = 2 R(c 23 ) = 3 R(c 24 ) = 8 R(c 31 ) = 5 R(c 32 ) = 3 R(c 34 ) = 4. R(c 41 ) = 7 R(c 42 ) = 8 R(c 43 ) = 4 1 st row c 12 : AB 2 d row c 23 : BC 3 rd row c 34 : CD All possible cycles which cotais oe or more miimum elemets are give below 310 www.ijergs.org
Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 Cycle 1 2 3 4 1 AB BC CD DA 2 AB BD DC CA 3 AC CB BD DA 4 AC CD DB BA 5 AD DC CB BA 6 AD DB BC CA The fuzzy cost of the each of the miimum tours with their miimum elemets is give below Cycle Tour z R(z ) 1 A B c D A (-16,-6,9,14,22,30,35,40) 16 2 A B D C A (-8,6,11,17,24,29,34,39) 19 3 A C B D A (-18,0,16,21,30,39,45,51) 23 4 A C D B A (-8,6,11,17,24,29,34,39) 19 5 A D c B A (-16,-6,9,14,22,30,35,40) 16 6 A D B C A (-18,0,16,21,30,39,45,51) 23 Best tours are A B c D A ad A D c B A.The miimum total distace travelled is 16. Satisfactio tours are A B D C A ad A C D B A.The total distace travelled is 19. The worst tours are A C B D A ad A D B C A The total distace travelled is 23. 5. CONCLUSION Usig the proposed method, we ca solve a fuzzy travellig salesma problem. The proposed method is very easy to uderstad ad apply ad also provides ot oly to a fuzzy optimal solutio for the problem ad also, to list some other alterative solutios to the problem which are very ear to fuzzy optimal solutio of the problem. REFERENCES: [1] Adreae, T. 2001. O the travellig salesma problem restricted to iputs satisfyig a relaxed triagle iequality.networks, 38: 59-67. [2] Blaser, M., Mathey, B., ad Sgall, J.2006. A improved approximatio algorithm for the asymmetric TSP with stregtheed triagle iequality. Joural of Discrete Algorithms, 4: 623-632. [3] Bockehauer, H. J., Hromkovi, J., Klasig, R., Seibert, S., ad Uger, W. 2002. Towards the otio of stability of approximatio for hard optimizatio tasks ad the travellig salesma problem. Theoretical Computer 311 www.ijergs.org
Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 Sciece, 285:3-24. [4] Chadra, L. S. ad Ram, L. S. 2007. O the relatioship betwee ATSP ad the cycle cover problem. Theoretical Computer Sciece, 370: 218-228. [5] Crisa, G. C. ad Nechita, E. 2008. Solvig Fuzzy TSP with At Algorithms. Iteratioal Joural of Computers,Commuicatios ad Cotrol, III (Suppl.Amit Kumar ad Aila Gupta It. J. Appl. Sci. 170 Eg., 2012. 10, 3 issue: Proceedigs of ICCCC 2008), 228-231. [6] Fischer, R. ad Richter, K. 1982. Solvig a multiobjective travellig salesma problem by Dyamic programmig. Optimizatio, 13:247-252. [7] Melamed, I. I. ad Sigal, I. K. 1997. The liear covolutio of criteria i the bicriteria travellig salesma problem. Computatioal Mathematics ad Mathematical Physics, 37: 902-905. [8] Padberg, M. ad Rialdi, G. 1987.Optimizatio of a 532-city symmetric travellig salesma problem by brach ad cut. Operatios Research Letters, 6:1-7. [9] Rehmat, A., Saeed H., ad Cheema, M.S. 2007. Fuzzy multi-objective liear programmig approach for travellig salesma problem. Pakista Joural of Statistics ad Operatio Research, 3: 87-98. [10] Segupta, A. ad Pal, T. K. 2009. Fuzzy Preferece Orderig of ItervalNumbers i Decisio Problems. Berli. [11] Sigal, I. K. 1994. A algorithm for solvig large-scale travellig salesma problem ad its umerical implemetatio. USSR Computatioal Mathematics ad Mathematical Physics, 27: 121-127. [12] Zimmerma, H. J. 1996. Fuzzy Set Theory ad its Applicatio. Bosto 312 www.ijergs.org