Cmjv02i06p0820 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (6), 820-826 (2011)

Fuzzy Shortest Path Problem Based On Similarity Degree Using Trapezoidal Fuzzy Number L. SUJATHA1 and R. SATTANATHAN2 1

Department of Mathematics, Auxilium College (Autonomous), Vellore. India 2 Department of Mathematics, D.G. Vaishnav College, Chennai. India ABSTRACT Many researchers have focused on the Fuzzy Shortest Path Problem in a network since it is significant to various applications. It offers a Decision Maker the Shortest Path in a network with Fuzzy arc length. In this paper the arc length of the network is taken as Trapezoidal Fuzzy Number and it has been proposed to give an Algorithm for Fuzzy Shortest Path Problem (FSPP) using Trapezoidal Fuzzy Number where the Shortest Path (SP) is identified with the highest Similarity Degree (SD). Keywords: Shortest Path, Network, Triangular Fuzzy Number, Trapezoidal Fuzzy Number, Similarity Degree, Decision Maker.

1. INTRODUCTION In many transportation, routing, communications, and other applications, graphs (network) emerge naturally as a mathematical model of the observed realworld system. Moreover Shortest Paths are one of the simplest and most widely used concepts in non-fuzzy networks. Conventionally, the arc lengths in a network are assumed to be crisp. In the network considered here, the lengths of the arcs are assumed to represent transportation time or cost and usually it will be difficult for the Decision Maker to represent the arc length precisely. Therefore, the arc length in a

network can be regarded as a fuzzy set. Fuzzy set theory, proposed by Zadeh17 is frequently utilized to deal with uncertainty problem. Many literatures about the Fuzzy Shortest Path can be found in15, 4, 12, 2. Dubois and Prade5 first introduced Fuzzy Shortest Path Problem on a Network. They tried to find the Fuzzy Shortest Path length although they did not develop an approach to determine the Shortest Path. Klein8 proposed a dynamical programming recursion-based fuzzy algorithm that specified each arc length within an integer value from one to a fixed number. Okada and Soper16 addressed their Fuzzy Shortest Path approach by modifying the multiple labeling methods for

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traditional Shortest Path algorithm for Fuzzy graph whose link weight is L-R type Fuzzy number. Lin and Chern11 developed a Fuzzy linear programming approach to calculate the Fuzzy Shortest Path and the most vital arc. Kung9 proposed Fuzzy Shortest Length Heuristic Procedure where the Shortest Path was decided by examining the Intersection Area (IA). Hence Fuzzy Similarity Degree10,3 was utilized to get the Shortest Path in the case of Triangular Fuzzy Number. Various similarity measures were presented to evaluate the Similarity Degree between two fuzzy sets6, 7, 13.

membership function (degree) is maximum (=1). ~

A Trapezoidal Fuzzy Number A can be defined by a quadruplet (a, b, c, d) defined on R with the membership function defined as µ Ã (x) = (x-a) / (b-a) , a ≤ x = 1 , b ≤ x = (d-x) / (d-c) , c ≤ x = 0 , otherwise

≤ b ≤ c ≤ d

2. PRE-REQUISITES Definition 1: A Triangular Fuzzy Number ~

A Triangular Fuzzy Number A can be defined by a triplet (a, b, c) defined on R with the membership function defined as µ Ã (x) = (x-a) / (b-a) , a ≤ x = (c-x) / (c-b) , b ≤ x =0 , otherwise

≤ b ≤ c

Fig. 2: A trapezoidal fuzzy number

Definition 3 (Mahdavi14): Basic operation on a Fuzzy Trapezoidal Number ~

If A = (a1, b1, c1, d1) and B = (a2, b2, c2, d2) are two trapezoidal fuzzy numbers then the addition operation on it are as follows: ~

A+ B = (a1+a2, b1+b2, c1+c2, d1+d2) ~

Fig. 1: A triangular fuzzy number

Definition 4 (Mahdavi14): For two Trapezoidal Fuzzy Numbers ~

Definition 2: A Trapezoidal Fuzzy Number

L 1 = (a1, b1, c1, d1) and L 2 = (a2, b2, c2, d2),

This shape is originated from the fact that there are several points whose

L min = min( L 1 , L 2 )

= min(a1,a2),min(b1,b2),min(c1,c2),min(d1,d2))

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Crisp graph with fuzzy weights (Type V) (Blue1):

if L i ∩ L min ≠ ∅ where bi ≤ x ≤ c ( I )

A fifth type of graph fuzziness occurs when the graph has known vertices and edges, but unknown weights (or capacities) on the edges. Thus only the weights are fuzzy.

Formulation for SD ( L min , L i ) where c < x < d, ai < x < bi Let yd be the membership function

Applications: To plan the quickest automobile route from one city to another. Unfortunately, the map gives distances, not travel times, so it is not known exactly how long it takes to travel any particular road segment. 2.1. Similarity Degree using Trapezoidal Fuzzy Number A new Similarity Degree has been defined for Trapezoidal Fuzzy Number which is utilized to identify the Shortest Path in Fuzzy sense. The procedure is given as follows: ~

(d − ai )2 1 = 2 (d − c) + (bi − ai )

Since c < x < d, yd = (d - x) / (d - c) Since ai < x < bi, From (1)

From (2)

(1)

yd = (x - ai) / (bi - ai) (2)

yd (d - c) = d - x x = d + (c - d) yd

(3)

yd (bi - ai ) = x - ai x = (bi - ai) yd + ai

(4)

Equating (3) and (4) d + (c - d) yd = (bi - ai) yd + ai [(c - d) - (bi - ai)] yd = ai - d

⇒ yd =

ai − d d − ai = (c − d ) − (bi − ai ) ( d − c ) + (bi − ai )

= height = h Thus we obtain the IA as 1/2 x (d - a i ) x

SD ( L min , L i ) = 0 if L i ∩ L min = ∅

Let, L min = A1= (a, b, c, d) and L i = A2 = (ai, bi, ci, di) be two Trapezoidal Fuzzy Numbers. If a ≤ ai , b ≤ bi , c ≤ ci , d ≤ d i then the Similarity Degree (SD) between A1 and A2 can be calculated as follows: ~

d − ai = ½ x ( d − c ) + (bi − ai )

base x height =1/2 x

(d − ai )2 (d − c) + (bi − ai )

if L i ∩ L min ≠ ∅ where c < x < d, ai < x < bi =

1 x1x [(d - ai) + (c - bi)] 2

The Similarity Degree diagrams for two ~

Trapezoidal Fuzzy Numbers L min = A1 and ~

L i = A2 are given in the following Figures:

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Fig.3

Fig.4

Fig.5

3. ALGORITHM FOR FUZZY SHORTEST PATH PROBLEM Step 1: Construct a fuzzy graph (network) G = (V, E) with pure Type V fuzziness, where V is the set of vertices (or nodes) and E is the set of edges (or arcs). Here G is an acyclic digraph.

Step 2: Consider all possible paths from source vertex s to the destination vertex d and find ~

the corresponding path lengths 1,2,â&#x20AC;Ś,n for n possible paths using

Li , i =

Definition 3 and let L i = (ai, bi, ci, di).

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Step 3: Find the Fuzzy Shortest length ~

L min using Definition 4 and let L min = (a, b, c, d) Step 4: Find the Similarity Degree SD ~

( L min , L i ) for i = 1,2,â&#x20AC;Ś,n using ( I ) and Ranking is given to the paths based on the ~

SD( L min , L i ). Step 5: Identify the Shortest Path with the ~

E(2-5) = (51,79,85,91), F(3-5) = (43,55,60,87), G(4-5) = (32,40,46,53), H(4-6) = (88,92,134,146), I(5-6) = (75,110,111,130) Step 2: From Fig. 6 we get the possible paths and the corresponding path lengths as follows: Path 1-2-3-5-6 with A+C+F+I =

highest SD( L min , L i ). Illustrative Example As a real time application of this model the following example is considered: A person wishes to find the shortest possible route from Chennai (s) to Thiruchirapalli (d). Given a road map of Tamil Nadu on which the distance between each pair of successive intersections is marked, to determine the shortest route from source vertex s to the destination vertex d. Step 1: Construct a network G = (V, E) with six vertices and nine edges.

(201,262,282,351) = L 1 Path 1-2-4-5-6 with A+D+G+I = ~

(196,253,279,318) = L 2 Path 1-2-4-6 with A+D+H = ~

(177,195,256,281) = L 3 Path 1-2-5-6 with A+E+I = ~

(159,234,246,281) = L 4 Path 1-3-5-6 with B+F+I = ~

(160,222,232,287) = L 5 ~

Fig. 6 A network with fuzzy arc lengths

Let vertex 1 be Chennai, vertex 2 be Tambaram, vertex 3 be Vellore, vertex 4 be Villupuram, vertex 5 be Thiruva-nnamalai and vertex 6 be Thiruchirapalli. In Fig. 6: A(1-2) = (33,45,50,60), B(1-3) = (42,57,61,70), C(2-3) = (50,52,61,74), D(24) = (56,58,72,75),

Step 3: L min = (a, b, c, d) = (159,195, 232,281) Step 4: Results of the Network are given in the following table Table : Results of network Path 1-2-3-5-6 1-2-4-5-6 1-2-4-6 1-2-5-6 1-3-5-6

SD ( L min , L i ) using ( I ) 29.09 34.08 70.50 60.02 65.50

Ranking

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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L. Sujatha, et al., J. Comp. & Math. Sci. Vol.2 (6), 820-826 (2011)

Step 5: The Shortest Path is 1-2-4-6, since the corresponding path has the highest Similarity Degree (= 70.50) using ( I )

4. CONCLUSION Fuzzy Shortest Path Length and Shortest Path are the useful information for the Decision Makers in a Fuzzy Shortest Path Problem. In this paper Similarity Degree (SD) has been defined for Trapezoidal Fuzzy Numbers which helps to identify the Shortest Path between Fuzzy arc length and the Fuzzy Shortest Path length. In section 3, an Algorithm for Fuzzy Shortest Path Problem has been developed where the Shortest Path has been identified with the highest SD using ( I ). Also ranking given to the paths based on highest SD helps the Decision Maker to identify the preferable path alternatives. An Illustrative example has been included to demonstrate the proposed approach. Hence it has been finally concluded that the Algorithm developed for Fuzzy Shortest Path Problem in the current research is an alternative method to get the Shortest Path.

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Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)