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Procedía Computer Science 70 (2015) 123 - 129

4th International Conference on Eco-friendly Computing and Communication Systems,

ICECCS 2015

Algorithm for Shortest Path Problem in a Network with Interval-valued Intuitionistic Trapezoidal Fuzzy Number

Gaurav Kumara, Rakesh Kumar Bajajb,*5 Neeraj Gandotrac

aGNA University, Sri Hargobindgarh, Phagwara, Punjab, PIN-174 301, INDIA. bJaypee University of Information Technology, Waknaghat, Solan, H.P., PIN-173 234, INDIA. cShoolini University, Bajhol, Solan, H.P., PIN-173 229, INDIA.

Abstract

The shortest path problem is a classical network optimization problem which has a wide range of applications in various fields of science and engineering. In the present communication, an algorithm to find the shortest path and shortest distance in an interval-valued intuitionistic fuzzy graph with nodes and links being crisp but the edge weights will be Interval-valued Intuitionistic Trapezoidal Fuzzy Numbers (IITFNs). Finally, a numerical example has been provided for illustrating the proposed approach. © 2015 The Authors. Publishedby ElsevierB.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ICECCS 2015

Keywords: Interval-valued Intuitionistic fuzzy sets, Interval-valued Intuitionistic Trapezoidal fuzzy number, ranking function, Interval-valued Intuitionistic fuzzy shortest path problem.

1. Introduction

Shortest path problems are very helpful and widely applied in various fields of science and engineering, e.g., road network applications, transportation, routing in communication channels and scheduling problems. In a classical network problem, weights of the edges are supposed to be real numbers. However, in most practical applications, the parameters are not naturally precise in general. Therefore, in real world situation, they may be considered to be a fuzzy.

Atanassov2 generalized the concept of fuzzy sets1 to interval-valued intuitionistic fuzzy sets. The shortest path problem under an uncertain and imprecise environment was first analyzed by Dubois and Prade3. According to their approach, length of the shortest path can be obtained but the corresponding path in the network may not exist. In literature, various researchers have studied the shortest path problem in different capacity. Klein4 (1991) proposed a dynamical programming recursion-based fuzzy algorithm. The Fuzzy Shortest Path Length (FSPL) in a network by means of a fuzzy linear programming approach was proposed by Lin and Chen5 (1994). Further, two different fuzzy shortest path network problems were developed by Yao and Lin6 (2003), where the first type of fuzzy shortest

* Corresponding author. Tel.: +91-1792-239229;fax: +91-1792-245365. E-mail address: rakesh.bajaj@gmail.com

1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ICECCS 2015 doi:10.1016/j.procs.2015.10.056

path problem uses triangular fuzzy numbers and the second type uses level (1-3, 1 -a) interval valued fuzzy numbers. Okada7 (2004) introduced a new comparison index between the sums of fuzzy numbers by considering interactivity among fuzzy numbers and presented an algorithm to determine the degree of possibility for each arc on a network. Chuang and Kung8 proposed a new algorithm for the discrete fuzzy shortest path problem in a network. Elizabeth and Sujatha10,9 studied the fuzzy shortest path problem based on index ranking and based on level LR type representation of fuzzy interval. One such algorithm for finding the shortest path in network flow with fuzzy arc lengths was proposed by Kumar and Kaur11. Karunambigai et al.12 presented a model based on the concept of dynamic programming to find the shortest paths in intuitionistic fuzzy graphs. Further, Gani and Jabarulla13 also developed a method on searching intuitionistic fuzzy shortest path in a network.

Here, we have proposed a new algorithm for shortest path with interval-valued intuitionistic fuzzy sets. By using the proposed algorithm a decision maker can obtain the shortest path and shortest distance of each node from source node. The paper is organized as follows. In section 2, some basic concepts on interval-valued intuitionistic trapezoidal fuzzy numbers, arithmetic operations and ranking function are reviewed. In section 3, an algorithm is proposed for finding the shortest path and shortest distance in interval-valued intuitionistic fuzzy graph. An illustrative example is provided in section 4 to find the shortest path and shortest distance between the source node and destination node. The last section draws some concluding remarks.

2. Preliminaries

In this section some basic definitions, arithmetic operations and ranking function for interval-valued intuitionistic trapezoidal fuzzy numbers have been discussed.

2.1. Interval-valued Intuitionistic Fuzzy sets and Interval-valued Intuitionistic Fuzzy number

Let X be a finite non-empty set, e.g., X = {x1, x2, x3,.. .,xn}. Let R [0, 1] be the collection of all sub-intervals of [0, 1]. An Interval-valued Intuitionistic Fuzzy Set (IVIFS) a in X is defined with the form

a = {(x, ^a(x), Ya(x)/x € X)};

where : X ^ R [0, 1], ya : X ^ R [0, 1] with the condition 0 < sup(pa(x)) + sup(ya(x)) < 1 for all x € X. The intervals ^a(x) and ya(x) denotes the membership and non-membership degree of x to a, respectively.

Let a = {(x, [£(x), ^(x)], [yL(x), yu(x)])/x € X} be the Interval-valued Intuitionistic Fuzzy Set, where ^(x) = inf (¡la(x)), (x) = sup(jla(x)), y*(x) = inf(Ya(x)), YU(x) = sup(Ya(x)). If ^a(x) = (x) and 7a(x) = YU(x) then IVIFS a reduces to an Intuitionistic Fuzzy Set. For an IVIFS a, the pair {(pa(x), ya(x))} is called an interval-valued Intuitionistic fuzzy number as ([a, ¡3], [n, 5]) where [a, ¡3] c [0, 1], [n, 5] c [0, 1], p + 5 < 1 and ^ is denoted as the set of all IVIFS in Q..

2.2. Interval-valued Trapezoidal Intuitionistic Fuzzy Numbers

Let & be an Interval-valued Intuitionistic Trapezoidal Fuzzy Number, its membership is defined as:

¡la(x) =

x-a b-a ra

d-c Ha 0

for a < x < b, for b < x < c, for c < x < d, otherwise

and its non membership function is

Ya(x) =

b-x+ya(x-a\)

x-c+va(d1 -x)

for a1 < x < b, for b < x < c, for c < x < d1 , otherwise

where 0 < pa < 1 ; 0 < Ya < 1 ; a, b, c, d, a1, d1 e R. Then pa = (([a, b, c, d];pa), ([a1, b, c, d1]; Ya)) is called an intuitionistic trapezoidal fuzzy number. If b = c, then Intuitionistic Trapezoidal Fuzzy Number reduces to Triangular Intuitionistic Fuzzy Number. If pa, ya e int(0, 1), where int(0, 1) denotes all closed subintervals of the interval [0, 1], then a is called an Interval-valued Intuitionistic Trapezoidal Fuzzy Number (IITFN). Let pa = [pL, PU], Ya = [yL, Yu], then IITFN a can be denoted by a = ([a, b, c, d]; [pL, pU][yL, yU]).

2.3. Arithmetic operations between IITFN's

The following are some operational laws of interval-valued intuitionistic fuzzy trapezoidal numbers14. Let «1 = ([«1, b1, c1, d1]; [pn, PuJ[Yli , YuJ) and «2 = ([«2, b2, c2, d2]; [pl2, Pu2][Yl2, Yu2]) be two IITFN, and A > 0, then

• « e «2 = ([([a + a.2, b + b2, c1 + c1, d1 + d2];[pil + Pl2 - Pli Pl2 ,Put + Pu2 - Put Pu2][Yl1 Yl2 ,Yu1 Yu2])]).

• « • «2 = ([([a • a2, b • b2, c1 • c2, d1 • d2]; [pl • Pl2, Put • Pu2hYl1 + Yl2 - YlxYl2, Yux + Yu2 - YuxYu2])]).

• Aa1 = ([Aal, Ab1, Ac1, AdJ; [1 - (1 - pLl)A, 1 - (1 -put)a], [(y^)1, (y^)1]).

• (a1)a = ([(«1)a, (b1)a, (d)a, (d1)a]; [1 - (1 - j^)1, 1 - (1 - j^)1]).

2.4. Ranking function for IITFN's

A ranking function R : IITFN(R) ^ R, where IITFN(R) is a set of all interval-valued intuitionistic fuzzy trapezoidal numbers defined on set of real numbers, maps each interval-valued intuitionistic fuzzy trapezoidal numbers into a real number. The ranking of IITFN's is based on score function of IITFN's. If a = ([a, b, c, d]; [pL, pU][jL, jU]) then score function for a is defined as

S(a) = 2E(Pl - Yl + Pu - Yu),

where E is the expected value of interval-valued intuitionistic fuzzy trapezoidal number. Let a 1 and a 2 be two interval-valued intuitionistic fuzzy trapezoidal numbers, then

• « > a if s«1 > Sa-2;

• « < a if s«

• «1 = «2 if S« = §a~2.

2.5. Interval-valued Intuitionistic Trapezoidal Fuzzy distance

Let T be any connected Interval-valued Intuitionistic fuzzy graph. For any path {P : u1, u2, u3, ... un}, length15 of P is defined as the sum of the weights (Wi) of the arcs in P, where the weights between a pair of vertices is an

interval-valued intuitionistic trapezoidal fuzzy number, i.e., L(P) =2 {Wi-1, Wi}.

For any two nodes u, v in T, let P ={ Pi is a u-v path, i = 1, 2, 3,...,}. The Interval-valued Intuitionistic Trapezoidal Fuzzy distance (IIFT distance) is defined as

Ip{u, v} = minimum{L(Pi); Pi e P, i = 1, 2, 3,..., n}.

3. Algorithm for the Shortest Path

In this section an algorithm is proposed to find the shortest path and shortest distance of each node from source node. The notations used throughout paper are as follows:

G = {1, 2, ..., n} The set of all nodes in the network

Gp( j) The set of all predecessor nodes of node j

Ipi IIFT distance between node i and source node

Ipij IIFT distance between node i and j

Steps of the algorithm are summarized as follows:

• Step 1. Assume Ipi= ([0, 0, 0, 0]; [0, 0][1, 1]) and label the source node as follows:

([0, 0, 0, 0];[0, 0][1, 1])(L(G) = -->•

• Step 2. Enumerate Ipj= minimum {Ipi © Ipij/i e Gp(j)}; j = 2, 3, 4,..., n.

• Step 3. From step 2, if minimum value occurs corresponding to unique value of i, then label the node j as Ipj(x). If minimum value occurs corresponding to more than one values of i, then it represents that there exists more than one path between the source node and the node j but distance along all paths is Ipj, so choose any value of i.

Step 4. Let the destination node be labeled as Ip„(P), then the shortest distance between source node and destination node is Ip„.

Step 5. Now destination node is labeled as Ip„(P). Check label of node p to find the shortest path between source node and destination node. Let it be Ip„(S), now check the label of node q and so on. Repeat same procedure up-to node 1. And the shortest path can be obtained by combining all the nodes.

4. Illustrative Example

Let us consider an interval-valued intuitionistic fuzzy weighted graph given in Figure 1, where the distance between a pair of vertices is an interval-valued intuitionistic trapezoidal fuzzy number. The problem is to find the shortest distance and shortest path between source node and destination node on the network.

Fig. 1: Interval-valued Intuitionistic Trapezoidal Fuzzy Directed Graph

Table 1: Weights of the Graph

Edges Interval-valued Intuitionistic Trapezoidal Fuzzy Distance

1 - 4 [0.1,0.3,0.5,0.6];[0.2,0.4][0.4,0.5]

1 - 2 [0.2,0.4,0.5,0.8];[0.5,0.7][0.1,0.2]

2 - 3 [0.4,0.5,0.6,0.7];[0.3,0.5][0.2,0.4]

3 - 4 [0.3,0.5,0.6,0.8];[0.0,0.3][0.5,0.7]

2 - 5 [0.3,0.4,0.5,0.6];[0.3,0.6][0.3,0.4]

3 - 5 [0.2,0.3,0.4,0.5];[0.1,0.4][0.5,0.6]

4 - 6 [0.1,0.2,0.4,0.6];[0.2,0.4][0.3,0.6]

5 - 6 [0.2,0.3,0.4,0.5];[0.4,0.7][0.2,0.3]

5 - 7 [0.1,0.2,0.4,0.5];[0.6,0.8][0.1,0.2]

6 - 7 [0.2,0.4,0.5,0.7];[0.2,0.5][0.3,0.4]

Solution: From the above proposed algorithm, let us assume Ipi= ([0, 0, 0, 0]; [0, 0][1, 1]) and label source node (1) as ([0, 0, 0, 0]; [0, 0][1, 1])<L(G) = --> and the values of Ip; j = 2, 3, 4, 5, 6, 7 can be obtained as follows:

• Iteration 1. From the above network diagram predecessor of node 2 is node 1, so put values of i and j respectively in step 2.

The value of Ip2 is given by

Ip2 = minimum{Ip1 © Ip12}

= minimum{([0, 0, 0, 0];[0, 0][1, 1]) © ([0.2,0.4,0.5,0.8]; [0.5,0.7][0.1,0.2])} = minimum([0.2,0.4,0.5,0.8]; [0.5,0.7][0.1,0.2]) = ([0.2,0.4,0.5,0.8]; [0.5,0.7][0.1,0.2]).

From the above calculation, minimum occurs corresponding to the node 1 and then label node 2 as

([0.2,0.4,0.5,0.8]; [0.5,0.7][0.1,0.2])<L(G) = 1>.

• Iteration 2. The predecessor of node 3 is node 2, then by using above procedure, the distance between source node and node 3 is

Ips = ([0.6,0.9,1.1,1.5]; [0.65,0.85][0.02,0.08])

and the label for node 3 is

([0.6,0.9,1.1,1.5]; [0.65,0.85][0.02,0.08])<L(G) = 2>.

• Iteration 3. The predecessor nodes for node 4 are the node 1 and 3, then value of Ip4 is given by

Ip4 = minimum{Ip1 © Ip14, Ip3 © Ip34}

= minimum([0, 0, 0, 0];[0, 0][1, 1]) © ([0.1,0.3,0.5,0.6]; [0.2,0.4][0.4,0.5]),

([0.6,0.9,1.1,1.5]; [0.65,0.85][0.02,0.08]) © ([0.3,0.5,0.6,0.8]; [0.0,0.3][0.5,0.7]) = minimum{([0.1, 0.3, 0.5, 0.6]; [0.2, 0.4][0.4, 0.5]), ([0.9,1.4,1.7,2.3]; [0.65,0.895][0.01,0.056])}.

By using ranking method (based on score function), the obtained value of Ip4 is

Ip4 = ([0.1, 0.3, 0.5, 0.6]; [0.2, 0.4][0.4, 0.5])

and the label for Ip4 is

([0.1, 0.3, 0.5, 0.6]; [0.2, 0.4][0.4, 0.5])<L(G) = 1>.

• Iteration 4. The predecessor nodes for node 5 are the node 2 and 3, then value of Ip5 is

Ip5 = minimum{Ip2 ® Ip25, Ip3 ® Ip35}. By using ranking method, the value of Ip4 is

Ip5 = ([0.5, 0.8, 1.0, 1.4]; [0.65, 0.88][0.03, 0.08])

and the label for Ip5 is

([0.5, 0.8, 1.0, 1.4]; [0.65, 0.88][0.03, 0.08])<L(G) = 2>.

• Iteration 5. The predecessor nodes for node 6 are the node 4 and 5, then value of Ip6 is

Ip6 = minimum{Ip4 ® Ip46, Ip5 ® Ip56}.

By using ranking method, the value of Ip6 is Ip6 = ([0.2, 0.5, 0.9, 1.2]; [0.36, 0.64][0.12, 0.30]) and the label for Ip6 is

([0.2, 0.5, 0.9, 1.2]; [0.36, 0.64][0.12, 0.30])<L(G) = 4>.

• Iteration 6. The predecessor nodes for node 7 are the node 5 and 6, then the value of Ip7 is

minimum{Ip5 ® Ip57, Ip6 ® Ip67}. By using ranking method, the value of Ip7 is

Ip7 = ([0.4, 0.9, 1.4, 1.9]; [0.488, 0.82][0.036, 0.12])

and the label for Ip7 is

([0.4, 0.9, 1.4, 1.9]; [0.488, 0.82][0.036, 0.12])<L(G) = 6>.

From above calculations, path 1 ^ 4 ^ 6 ^ 7 is identified as the Shortest Path and the Shortest distance between source node and destination node is

([0.4, 0.9, 1.4, 1.9]; [0.488, 0.82][0.036, 0.12]).

By using the proposed algorithm, shortest path of all nodes from source node and labeling is tabulated below:

Table 2: Shortest Path and Shortest Distance

Node Ipj Shortest path between jth node and source node

2 ([0.2,0.4,0.5,0.8]; [0.5,0.7][0.1,0.2]) 1 ^ 2

3 ([0.6,0.9,1.1,1.5]; [0.65,0.85][0.02,0.08]) 1 ^ 2 ^ 3

4 ([0.1, 0.3, 0.5, 0.6]; [0.2, 0.4][0.4, 0.5]) 1 ^ 4

5 ([0.5, 0.8, 1.0, 1.4]; [0.65, 0.88][0.03, 0.08]) 1 ^ 2 ^ 5

6 ([0.2, 0.5, 0.9, 1.2]; [0.36, 0.64][0.12, 0.30]) 1 ^ 4 ^ 6

7 ([0.4, 0.9, 1.4, 1.9]; [0.488, 0.82][0.036, 0.12]) 1 ^ 4 ^ 6 ^ 7

5. Conclusions

Interval-valued Intuitionistic Fuzzy Shortest Path and Shortest distance are the useful information for Decision Makers in a Shortest Path Problem. In this paper, the distance function has been defined for Interval-valued Intuitionistic Trapezoidal Fuzzy Numbers which helps to identify the shortest path. A new algorithm for solving the interval-valued intuitionistic fuzzy shortest path problem on a network with Interval-valued Intuitionistic Trapezoidal fuzzy length has been proposed. The process of ranking the paths is very useful to make decisions in choosing the best of all possible path alternatives. The procedure of finding shortest path has been well explained and suitably discussed. Further, the implementation of the proposed algorithm is successfully illustrated with the help of an example.

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