Fuzzy Shortest Path with α- Cuts

Transcription

1 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 58 Issue 3 Jue 2018 Fuzzy Shortest Path with α- Cuts P. Sadhya Assistat Professor, Deptt. Of Mathematics, AIMAN College of Arts ad Sciece for Wome, Trichy, Tamiladu, Idia Abstract I this paper, we preset a algorithm for computig a shortest path i a acyclic etwork i which each edge is assiged to a o-trapezoidal fuzzy umber. α-cuts are used to fid the fuzzy path legths. I a proposed algorithm, Euclidea distace is used to fid the shortest path. Cosequetly, a shortest path is obtaied from source ode to destiatio ode. Key words - No-trapezoidal fuzzy umber, α-cut, Euclidea distace, shortest path I. INTRODUCTON The shortest path problem is oe of the most fudametal ad well-kow combiatorial optimizatio problems that appear i may applicatios as a sub problem. The legth of arcs i the etwork represets travellig time, cost, distace or other variables. I real life applicatios, these arc legths could be ucertai ad to determie the exact value of these arc legths is very difficult or sometimes difficult for decisio maker. I such a situatio fuzzy shortest path problem seems to be more realistic, where the arc legths are characterized by fuzzy umbers. Dubois ad Prade [3] first itroduce fuzzy shortest path problem. Okada ad Soper [7] developed a algorithm based o multiple labelig approach which is useful to geerate umber of o-domiated paths. Applyig fuzzy mi cocept they have itroduced a order relatio betwee fuzzy umbers. Applyig extesio priciple Klei[5] has give a algorithm which results domiated path o a acyclic etwork. The remaider of the paper is orgaized as follows. I sectio 2, basic cocepts ad defiitios are give. It also explais ways of computig α-cuts for fuzzy umbers. I sectio 3, we preset a algorithm for fidig fuzzy shortest path i a acyclic etwork i which each edge is assiged to a o-trapezoidal fuzzy umber. A illustratio is give for the proposed algorithm i sectio 4. A. Fuzzy Set II. CONCEPTS A fuzzy set A of a uiversal set X is defied by its membership fuctio μ A : X 0,1 which assigs a real umber μ A (x) i the iterval [0,1] to each elemet x X, where the value of μ A (x) at x shows the grade of membership of x i A. B. α-cut Give a fuzzy set A i X ad ay real umber α [0,1], the the α-cut or α-level or cut worthy set of A, deoted by α A is the crisp set α A = {x X/ μ A (x) α} 1) Example: Let A be a fuzzy set whose membership fuctio is give as μ A, a x b c x c b, b x c To fid the α cut of A, we first set α 0,1 to both left ad right referece fuctio of A. That is, α = ad α = c x c b Expressig x i terms of α we have b a α + a ad c (c b)α which gives the α-cuts of A as α A = b a α + a, c (c b)α C. No-trapezoidal fuzzy umber I SSN: Page 195

2 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 58 Issue 3 Jue 2018 Let X be the uiversal set. The o-trapezoidal fuzzy umber A(a, b, c, d ) is defied by the membership fuctio μ A, a x b 1, b x c, c x d The diagrammatic represetatio of o-trapezoidal fuzzy umber is give by a b c d Fig. 1 D. α cut for o-trapezoidal fuzzy umber To fid the α cut of A(a, b, c, d ), we first set α 0,1 to both left ad right referece fuctio of A. That is, α = x a ad α = Expressig x i terms of α, we have b a α + a ad d (d c)α which gives the α- cuts of A as α A = b a α + a, d (d c)α E. Additio of two fuzzy umbers Let X = a, b, c ad Y = p, q, r be two fuzzy umbers whose membership fuctios are μ X, a x b c x c b, b x c μ Y x p q p, p x q r x r q, q x r The α X = b a α + a, c c b α ad α Y = q p α + p, r (r q)α are the α-cuts of fuzzy umbers X ad Y respectively. To calculate additio of fuzzy umbers X ad Y usig iterval arithmetic α X + α Y = b a α + a, c c b α + q p α + p, r r q α = a + p + b a + q p α, c + r (c b + r q)α [1] F. Additio of two o-trapezoidal fuzzy umbers Let X = a, b, c, d ad Y = p, q, r, s be two o-trapezoidal fuzzy umbers whose membership fuctios are μ X, a x b 1, b x c, c x d I SSN: Page 196

3 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 58 Issue 3 Jue 2018 μ Y x p q p, p x q 1, q x r s x s r, r x s The α X = b a α + a, d (d c)α ad α Y = q p α + p, s (s r)α are the α-cuts of o-trapezoidal fuzzy umbers X ad Y respectively. To calculate additio of o-trapezoidal fuzzy umbers X ad Y usig iterval arithmetic α X + α Y = b a α + a, d (d c)α + q p α + p, s (s r)α G. Euclidea Distace = b a α + a + q p α + p, d d c α + s (s r)α [2] Let A = x 1, y 1 ad B = x 2, y 2 be two itervals. The the Euclidea distace D(A, B) is defied as D A, B = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 H. Miimum value of α-cuts Let α A = a, b ad α B = p, q be two α-cuts. The miimum value of α A ad α B is give by I. Network Termiology MV = mi α A, α B = mi a, p, mi (b, q) Cosider a directed etwork G(V, E), cosistig of a fiite set of odes V = 1,, ad a set of m directed edges E V V. Each edge is deoted by a ordered pair (i, j), where i, j V. Each edge is assiged to a otrapezoidal fuzzy umber. We calculate the α-cuts for each ad every edge i the etwork by the formula α X = b a α + a, d d c α where X a, b, c, d is a o-trapezoidal fuzzy umber ad α 0,1. III. ALGORITHM Step: 1 Fidig possible paths (i) Fid all the possible paths P i from source ode to destiatio ode i the give acyclic etwork. (ii) Assig the umber of possible paths i the give acyclic etwork to N. Step: 2 Computatio of legth of paths (i) Set α value betwee 0 ad 1. (ii) Fid α -cuts for every edge. (iii) Fid the legths L i of all possible paths by addig the α-cuts of the correspodig edges. Step: 3 Compariso of paths (i) Let L mi = L 1 (ii) For i = 2 to N MV = mi L mi, L i D 1 = D MV, L mi D 2 = D MV, L i If D 1 < D 2 the L mi = L mi Otherwise L mi = L i Step:4 Shortest Path Shortest path is the correspodig path of L mi I SSN: Page 197

4 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 58 Issue 3 Jue 2018 IV. NUMERICAL EXAMPLE The followig fuzzy etwork explais the proposed algorithm. A (1, 4, 9, 16) 2 C (4, 36, 49, 64) 3 F (4, 9, 16, 25) 1 6 B (9, 25, 49, 81) Step:1 Fidig possible paths 3 D (4, 16, 36, 64) 5 E (16, 25, 36, 49) Fig. 2 Example Network G (9, 25, 36, 81) (i) Fid all the possible paths P i from the source ode to the destiatio ode i the give acyclic etwork. There are three possible paths P 1 A C F P 2 : B E G P 3 : A D G (ii) Assig the umber of possible paths i the give acyclic etwork to N. Here the umbers of possible paths are 3. N = 3 Step:2 Computatio of legth of paths (i) Set α value betwee 0 ad 1. Let α = 0.73 (ii) Fid α-cuts for every edge Now, A = 1, 4, 9, 16 ie., A = 1 2, 2 2, 3 2, 4 2 α A = b a α + a, d (d c)α α A = 2 1 α + 1 2, 4 (4 3)α 2 α A = (α + 1) 2, (4 α) 2 = [2.99, 10.69] Similarly, α B = 2α + 3 2, 9 2α 2 = 19.89, α C = 4α + 2 2, 8 α 2 = 24.21, α D = 2α + 2 2, 8 2α 2 = 11.97, α E = α + 4 2, 7 α 2 = 22.37, α F = α + 2 2, 5 α 2 = 7.45, α G = 2α + 3 2, 9 3α 2 = 19.89, (iii) Fid the legths L i of all possible paths by addig the α-cuts of the correspodig edges L 1 = 34.65, L 2 = 62.15, L 3 = 34.85, Step:3 Compariso of paths (i) Let L mi = L 1 L mi = 34.65,81.77 (ii) For i = 2 to N MV = mi[l mi, L i ] I SSN: Page 198

5 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 58 Issue 3 Jue 2018 Whe i = 2, MV = mi[l mi, L 2 ] MV = mi 34.65, 81.77, 62.15, = 34.65, D 1 = D(MV,L 2 ) = D 34.65, 81.77, 34.65, = D 1 = 0 D 2 = D(MV, L 2 ) = D 34.65, 81.77, 62.15, = D 2 = If D 1 < D 2 the L mi = L mi. Otherwise L mi = L i Here D 1 < D 2. L mi = L mi = 34.65, Whe i = 3, MV = mi L mi, L 3 = 34.65,81.77 D 1 = 0 D 2 = D 1 < D 2. The L mi = 34.65,81.77 = L 1 Step:4 Shortest Path Shortest path is the correspodig path of L mi P 1 is the shortest path. ie., A C F is the shortest path from source ode to destiatio ode. V. CONCLUSION I this paper, we preseted a algorithm for computig a shortest path i a acyclic etwork usig the α- cuts i which each edge is assiged to a o-trapezoidal fuzzy umber. Cosequetly, the shortest path is obtaied from source ode to destiatio ode. But this algorithm does t work for a cyclic etwork. REFERENCES [1] Bojadziev, G., Bojadziev, M (1995), Fuzzy sets, Fuzzy logic, applicatio, World Scietific. [2] T. N. Chuag, J. Y. Kug, The Fuzzy shortest path legth ad the correspodig shortest path i a etwork, Comput. Oper. Res. 32 (2005), [3] D. Dubois, H. Prade, Fuzzy Sets ad Systems: Theory ad Applicatios, Academic Press, New York, [4] Kaufma A., ad Gupta, M.M(1984) Itroductio to Fuzzy Arithmetic, Theory ad Applicatios, Va Nostrad Reihold Co. Ic., Workigham, Berkshire. [5] C. M. Klei, Fuzzy shortest paths, Fuzzy Sets ad Systems 39 (1991), [6] C. Li, M. S. Cher, The fuzzy shortest path problem ad its most vital arcs, Fuzzy Sets ad Systems 58 (1993) [7] S. Okada ad T. Soper, A shortest path problems as a etwork with fuzzy arc legths, Fuzzy Sets ad Systems 109 (2000), [8] Palash Dutta, Hrishikesh Boruah, Tazid Ali, Fuzzy Arithmetic with ad without usig α-cut method: A comparative study, Iteratioal joural of latest Treds i Computig, Volume 2, Issue 1, March (2011), [9] A. Tajdi, I. Mahdavia, N. Mahdavi-Amiri, B. Sadeghpour-Gildeh, Computig a fuzzy shortest path i a etwork with mixed fuzzy arc legths usig α-cuts, Computers ad Mathematics with Applicatios, i press, [10] L.A. Zadeh, Fuzzy Logic ad the Calculi of Fuzzy Rules, Fuzzy Graphs, Fuzzy Probabilities, Computers & Mathematics with Applicatios, Volume 37, p. 35, I SSN: Page 199