Annals of Pure and Applied Mathematics
Vol. 5, No.1, 2013, 26-36
ISSN: 2279-087X (P), 2279-0888(online)
Published on 13 November 2013
www.researchmathsci.org
Shortest Path Problem on Intuitionistic Fuzzy Network
Saibal Majumder1 and Anita Pal2
2
Department of Mathematics, National Institute of Technology, Durgapur
Durgapur-713209, West Bengal, India
1
Email:
[email protected], 2Email:
[email protected]
Received 17September 2013; accepted 5 October2013
L Baba
Abstract. Finding shortest paths in graphs has been the area of interest for many
researchers. Shortest paths are one of the fundamental and most widely used concepts in
networks. In this paper the authors present an algorithm to find an Intuitionistic Fuzzy
Shortest Path (IFSP) in a directed graph in which the cost of every edges are represented
by a trapezoidal intuitionistic fuzzy numbers(TrIFNs) which is the most generalized form
of Trapezoidal Fuzzy Numbers(TrFNs) consisting of degree of acceptance and degree of
rejection. The proposed algorithm uses Induced Intutionistic Trapezoidal Fuzzy Order
Weighted Geometric (I-ITFOWG) operator for finding Intuitionistic Fuzzy Shortest Path
(IFSP). A numerical example is given to explain the proposed algorithm.
Keywords: Intuitionistic Fuzzy Set, Trapezoidal Intuitionistic Fuzzy Number, Centroid
point.
AMS Mathematics Subject Classification (2010): 05C72
1. Introduction
Shortest Path problems are among the fundamental problems studied in computational
geometry and other areas including graph algorithms, Geographical Information Systems
(GIS), network optimization etc. The classical shortest path problems having certain
edge length have been studied intensively by many researchers but in real world
application we have to deal with many uncertain information. The shortest path problem
under the uncertain environment was first analyzed by Dubois and Prade [3]. In this
paper we will be dealing with intuitionistic trapezoidal fuzzy numbers which are
expressed as the edge costs for the underlined directed graph. We proposed an algorithm
which will first find out all possible paths between the source and sink vertices of the
directed graph, then I-ITFOWG operator is applied to the set of TrIFNs comprising in
each of those paths to find an aggregated ITrFN. We then calculate the centroid of each
of these TrIFN. The TrIFN with largest centroid value is considered to be the aggregated
ITrFN with maximum acceptance and minimum rejection degree. And the path
26
Shortest Path Problem on an Intuitionistic Fuzzy Network
corresponding to that aggregated TrIFN is recognized to be the shortest path of the
network.
The paper is organized as follows:
Section 2 discussses briefly the preliminary concepts of TrIFN, Ordered Weighted
Averaging (OWA) operator, Geometric Mean (GM) operator, Ordered Weighted
Geometric (OWG) operator Induced Intuitionistic Fuzzy Ordered Weighted Geometric
(I-ITFOWG) operator. Section 3 explain about ranking of trapezoidal intuitionistic fuzzy
numbers based on its centroid point. Section 4 detailed our proposed algorithm to find the
shortest path of a network whose edge costs are trapezoidal intuitionistic fuzzy numbers.
Section 5 gives a numerical explanation of our proposed algorithm and finally we
conclude our paper in section 6.
2. Preliminaries
Atanassov [1] first introduce the concept of intuitionistic fuzzy set as follows:
Let X be the universe of discourse then an intuitionistic fuzzy set I on U is defined as
I = x, μ x , υ x |x ∈ X where functions μ x : X → [0,1] and υ x : X → [0,1] are
the degree of membership and non membership of the element x ∈ X and ∀x ∈ X, 0 ≤
μ x + υ x ≤ 1.
2.1. Definition Intuitionistic Fuzzy Set
Atanassov [1] first introduce the concept of intuitionistic fuzzy set (IFS) as follows:
Let be the universe of discourse then an intuitionistic fuzzy set on is defined as
,!
| ∈
: → [0,1] and !
: → [0,1]
=
,
where functions
are the degree of membership and non membership of the element ∈ and ∀ ∈
,0 ≤
+!
≤ 1.
2.2. Definition Trapezoidal Intuitionistic Fuzzy Number
A Trapezoidal Intuitionistic Fuzzy Number (TrIFN) s# = 〈 s% , s& , s' , s( ; w,# , u,# 〉 is a
special case of intuitionistic fuzzy number defined on the set of real number ℜ, whose
membership and non-membership functions are defined as:
3456
57 456
85#
5< 43
5< 45;
85# ;06 9:907
57 43 GH5# 3456
;06 9:907
57 456
I5# ;07 9:90;
345; GH5# 5< 43
;0; 9:90<
5< 45;
C
E
= ;:>06 ?@A :B;0<
D % ;:>06 ?@A :B;0<
The graphical representation of which is shown in Fig. 1. The values w,# and u,#
represents the maximum membership degree and minimum non-membership degree
respectively such that0 ≤ w,# ≤ 1, 0 ≤ u,# ≤ 1 and 0 ≤ w,# + u,# ≤ 1 holds. The
parameters w,# and u,# defines the confidence level and the non-confidence level of the
TrIFN s# = 〈 s% , s& , s' , s( ; w,# , u,# 〉 respectively.
0̃
=2
;07 9:90;
85# ;0; 9:90<
C and !0̃
F
=
Let ρ,# x = 1 − w,# x − u,# x , which is called hesitation degree or
intuitionistic fuzzy index of whether x belongs to s#. If the hesitation degree is small then
knowledge whether x belongs to s# is more certain, while if the hesitation degree is large
then knowledge on that is more uncertain.
27
Saibal Majumder and Anita Pal
1.0
wu#
xu#
0.5
uv
uy
uz
u{
Figure 1: Membership degree and non-membership degree of TrIFN
2.3. Arithmetic operations of two TrIFN
Let s# = 〈 s% , s& , s' , s( ; w,# , u,# 〉 and t̃ = 〈 t% , t & , t ' , t ( ; wPQ , uPQ 〉 be the two TrIFN and η
be a real number. Then,
s# ⊕ t̃ = 〈 s% + t% , s& + t & , s' + t ' , s( + t ( ; w,# + wPQ − w,# wPQ , u,# uPQ 〉
s# ⊗ t̃ = 〈 s%t %, s& t & , s' t ' , s( t ( ; w,# wPQ, u,# + uPQ − u,# uPQ 〉
ηs# = 〈 ηs% , ηs& , ηs' , ηs( ; 1 − 1 − w,# U , u,# U 〉
2.4. Definition Support of TrIFN
The support of a TrIFN s# = 〈 s% , s& , s' , s( ; w,# , u,# 〉 for the membership function and
non-membership function are defined as supW s# = {x|w,# x ≥ 0} and sup[ s# =
{x|u,# x ≤ 1} respectively.
2.5. Definition Ordered Weighted Averaging (OWA) Operator [7]
An OWA operator of dimension m is a function τ, τ: ℜ^_ → ℜ^ which has associated a
set of weights or weighting vector ω = [ω% , ω& , … , ω_ ]b ∀ωc ∈ [0,1] and ∑c ωc =
1. Furthermore, τ y% , y& , … , y_ = ωb . B = ∑_
hj% ωh bh where B is the associated ordered
lm
value vector and each element bh ∈ B, is the k largest value among yh j = 1,2,3, … , m .
2.6. Definition Geometric Mean Operator [7]
A geometric mean operator of dimension m is a function ρ, ρ: ℜ^_ → ℜ^ such that
ρ y% , y& , … , y_ = ∏_
rj% yr
6
s
.
2.7. Definition Ordered Weighted Geometric (OWG) operator [8]
The OWG operator is developed based on the concept of OWA operator and the
geometric mean operator so this operator inherits the advantages of both the OWA
operator and the geometric mean operator. Mathematically we defined an OWA operator
of dimension m is a function δ, δ: ℜ^_ → ℜ^ that has associated with it a weight vector,
ω = [ω% , ω& , … , ω_ ]b ∀ωc ∈ [0,1] and ∑c ωc = 1
such
that
δ y% , y& , … , y_ =
28
Shortest Path Problem on an Intuitionistic Fuzzy Network
}~
∏_
where cr is the k lm largest value among yr k = 1,2,3, … , m . One of the
rj% cr
most important factor in the OWG operator is to determine its associated weights. Xu [4]
develop a normal distribution based method which is defined as ωc =
€
4
∑s
~„6 €
•4‚s
7ƒ7
s
4
~4‚s
7ƒ7
s
,
k = 1,2, … , m
where μ_ is the mean of the collection of 1,2, … , m and σ_ > 0 is the standard
%^‡
and
deviation of the collection of 1,2, … , m. So mathematically we define μ_ =
σ_ =
%
ˆ ∑_
_ Šj%
&
l − μ_ & .
2.8. Definition Induced Intuitionistic Trapezoidal Fuzzy Ordered Weighted
Geometric Operator [4]
•Œ i = 1,2, … , n be a collection intuitionistic trapezoidal fuzzy numbers. An induced
If φ
intuitionistic trapezoidal fuzzy ordered weighted geometric (I-ITFOWG) operator of
dimension n is a mapping I − ITFOWG: ε‡ → ε which has an associated vector ω =
[ω% , ω& , … , ω‡ ]b such that ∀ωc ∈ [0,1] and ∑‡cj% ωc = 1.
}6
}7
• –(%)
Moreover, I − ITFOWG} 〈u% , φ
• % 〉, 〈u& , φ
• & 〉, … , 〈u‡ , φ
• ‡ 〉) = φ
⊗φ
• –(&)
…⊗φ
• ‡–(‡)
where ω = ω% , ω& , … , ω‡
0,1 and ∑‡hj% ωh = 1.Let
b
is the weighted vector of φ
• h (j = 1,2, … , n) ∀ωh ∈
σ(1), σ(2), … , σ(j − 1), σ(j), σ(j + 1), … , σ(n)
is
a
• –(h) is the φ
• c value of the ITFOW
permutation of 1,2,3, … , n such that u–(h—%) ≥ u–(h) . φ
lm
• c 〉 having the j largest uc (uc ∈ 0,1 ), where uc in 〈uc , φ
• c 〉 is referred to as the
pair 〈uc , φ
• c as the intuitionistic trapezoidal fuzzy numbers. Now if we
order inducing variable and φ
consider s#c (i = 1,2, … , n) as a collection intuitionistic trapezoidal fuzzy numbers then
their aggregated value by using the I − ITFOWG operator is also an intuitionistic fuzzy
number and is expressed as,
I − ITFOWG} (s#% , s#& , … , s#‡ ) =
}~
}~
}~
}~
‡
‡
‡
‡
‡
~
}~
〈˜∏‡rj% a}
–(h) , ∏rj% b–(h) , ∏rj% c–(h) , ∏rj% d–(h) ™ ; ∏rj% w,#ƒ(~) , 1 − ∏rj%(1 − u,#ƒ(~) ) 〉
where ω = ω% , ω& , … , ω‡
0,1 and ∑‡cj% ωc = 1.
b
is the weight vector of I − ITFOWG operator ∀ωc ∈
3. Ranking intuitionistic fuzzy numbers
Let s# = 〈(s% , s& , s' , s( ); w,# , u,# 〉 be a trapezoidal intutionistic fuzzy number and the area
covered by it’s membership and non-membership grade is depicted in Fig. 2. The total
area is divided into five parts namely, x% x ' pš z% , z% pš y( y% , y% y( x(x& , x & x( qš z&
and z& qš y' y& . The coordinates of the corner point of the plotted graph are listed below:
29
Saibal Majumder and Anita Pal
¯z
1.0
wu#
²š
xu#
0.5
·µ
°z
·¶
°{
¯{
³š
´µ
´¶
¯v
± v °v
¯y ±y
°y
Fig. 2: Representation of area covered under membership and non-membership function
of TrIFN
x% = s%, 0 , y% = s& , 0 , x & = s' , 0 , y& = s( , 0 , x' = s% , 1 , y' = s( , 1 , x( =
s' , w,# , y( = s& , w,#
, W —, [ ^,
, W —, [•# ^,;
W•#
W•#
pš = ˜ 6 •# 6 •# 7 ,
,
™ , qš = ˜ < W•# —[< ^%
™ , z% =
W —[ ^%
W —[ ^%
W —[ ^%
˜
•#
•#
,6 W•# —,6 [•# ^,7
W•#—[•# ^%
•#
•#
, 0™ , z& = ˜
point(X ,# , Y,# ) of TrIFN such
where,
x% = Ÿ,
•6 ¦•# 4•6 §•# G•7
¦•# 4§•# G6
6
¦•# 4§•# G6
•6 ¦•# 4•6 §•# G•7
¦•# 4§•# G6
x & = Ÿ,
6
<
Ÿ •< ¦•# 4•< §•#G•; g © dx
,
¦•# 4§•# G6
y% =
W•#
Ÿ= y
¦•#
¦•# 4§•# G6
-Ÿ=
y& = Ÿ=
Ÿ
%
W•#
•#
•#
•#
•#
, 0™. Das and Guha [9] proposed the centroid
¢
¢
, Y,# =
Ÿ £¡ £ ¢£
.
Ÿ ¡ £ ¢£
Now, X,# =
xg ¥ dx + Ÿ •6¦•# 4•6§•#G•7 xf¥ dx + Ÿ« xw,# dx + Ÿª
«
<
Ÿ •< ¦•# 4•< §•#G•; xg © dx
,
,< W•# —,< [•# ^,;
W•# —[•# ^%
Ÿ ¡
that X ,# =
Ÿ¡
ª
¦•# 4§•# G6
g ¥ dx + Ÿ •6 ¦•#4•6§•#G•7 f¥ dx + Ÿ« w,# dx + Ÿª
«
h© − h¥ dy +
yh¥ dy + Ÿ
%
¦•# 4§•# G6
7
and Y,# =
•< ¦•# 4•< §•# G•;
¦•# 4§•# G6
•< ¦•# 4•< §•# G•;
¦•# 4§•# G6
£6
£7
xf© dx +
f© dx +
¦•#
%
%
¦ 4§ G6
-Ÿ= ys( dy − Ÿ= •# •# yh© dy − Ÿ ¦•# yI© dy® +
¦•# 4§•# G6
¦•#
¦•# 4§•# G6
yI¥ dy − Ÿ= ys% dy®
%
¦•#
¦•# 4§•# G6
h© − h¥ dy + -Ÿ= s( dy − Ÿ=
%
ª
6
h© dy − Ÿ
%
¦•#
¦•# 4§•# G6
¦•#
¦•# 4§•#G6
yt © dy® + -Ÿ=
h¥ dy +
t dy − Ÿ= s% dy® and f¥ : [s% , s& ] → [0, w,# ], f¥ : [s' , s( ] → [0, w,# ] are leftmost and
¦•#
¥
¦•# 4§•# G6
%
rightmost part of membership grade and [s% , s& ] → [0, u,# ] , g © : [s' , s( ] → [0, u,# ] are
leftmost and rightmost part of non-membership grade of TrIFN respectively depicted in
30
Shortest Path Problem on an Intuitionistic Fuzzy Network
Fig. 2. h¥ : [0, w,# ] → [s% , s& ], h© : [0, w,# ] → [s' , s( ] are the inverse function of f¥and f©
respectively. t ¥ : [0, u,# ] → [s% , s& ], t © : [0, u,# ] → [s' , s( ] are the inverse function of
g ¥ and g © respectively. The analytical expression of each of these above functions are
given below:
f¥ x =
g¥ x =
h¥ y =
t¥ y =
—,6
W —,<
; s% ≤ x ≤ s&
f© x = •#
; s' ≤ x ≤ s(
,7 —,6
,; —,<
—,7 ^[•# ,6 —
—,; ^[•# ,< —
; s% ≤ x ≤ s& g ¥ x =
; s' ≤ x ≤ s(
,6 —,7
,< —,;
,7 —,6 £
,< —,; £
s% + W
; 0 ≤ y ≤ w,# h© y = s( − W
; 0 ≤ y ≤ w,#
•#
•#
,6 —,7 £^ ,7 —,6 [•#
,< —,; £^ ,; —,< [•#
; u,# ≤ y ≤ 1 t © y =
; u,# ≤ y ≤
%—[
%—[
W•#
•#
•#
1
Now the ranking of any two TrIFN r# and s# [9] is done based on following criteria :
¹´ º̃ > 0̃ , »¼½¾ ¿̃ > À̃
¹´ º̃ < 0̃ , »¼½¾ ¿̃ < À̃
¹´ º̃ = 0̃ , »¼½¾
¹´ º̃ > Â0̃ , »¼½¾ ¿̃ > À̃
½Ãu½ ¹´ º̃ < Â0̃ , »¼½¾ ¿̃ < À̃
½Ãu½ º̃ = Â0̃ , »¼½¾ ¿̃ = À̃
4. Proposed Shortest Path Algorithm
We consider a connected acyclic network having a source vertex u and and a sink
vertex z. Each edge i − j of the network represents the cost (or distance) parameter
between vertices i and j. We consider these parameters to determine the shortest path in a
network. The edges of our network are associated with a pair of ordered inducing
variable and TrIFN, 〈uh , φ
• h 〉. The significance of such a pair in the context of the
connected network is as follows:
• c〉
The order inducing variable, uc (⊆ 0,1 ) determines the k lm largest edge 〈uc , φ
which is included in a path from source to sink of the network such that for any two
• h , its membership
edges included in a path, u–(h—%) ≥ u–(h) ; whereas for each TrIFN φ
grade represents the acceptance degree to which an edge i − j will be included in the
shortest path between source to sink and it’s non-membership grade represents the
rejection degree to which an edge i − j will be accepted from the shortest path between
source to sink. The new algorithm is detailed below:
Input: Let G = (V, E) be a directed acyclic network having intuitionistic fuzzy costs
associated with its edges.
Output: A shortest path between the source and the sink vertices of the
Network is traced out based on comparison of centroid point of the
aggregated trapezoidal intuitionistic fuzzy numbers for all the paths that exists between
the source and sink of the network under consideration.
31
Saibal Majumder and Anita Pal
Step-1: Trace all existing paths between source and sink of an acyclic connected
network having intuitionistic fuzzy costs associated with its edges.
Step-2: Rearrange every edges of a path between two terminal vertices in non-increasing
order with respect to their order inducing variable.
Step-3: Derive the associated weight vector [8] Ë = [Ë% , Ë& , … , ËÌ ]Í for − ÎÏÐÑÒ
operator, where each ËÓ > 0 , ∑Ì
Ój% ËÓ = 1 and Ô is the number of edges of the network
path under consideration. Calculate the aggregated TrIFN,
I − ITFOWG} ˜φ
• –Õ & , … , φ
• –Õ _ ™ for path pc . Repeat Step-2 and Step-3 for all
• –Õ % , φ
•
•
•
the traced paths.
Step-4: Determine the centroid pont of all the aggregated TrIFN. The network path
corresponding to the aggregated TrIFN having largest X or Y value of its centroid point
gives the shortest path of the network.
Step-5: Add φ
• r ′s associated with the edges of the selected shortest path to get the
shortest path length.
5. Numerical Explanation
A connected acyclc network has been considered in Fig. 3 with vertices u, v, x, y, w and z,
with u the source vertex and the z the sink vertex. Every edge weights of the network are
• h 〉 that are listed in
expressed as a pair of ordered inducing variable and TrIFN 〈uh , φ
Table 1. Our main objective is to search the shortest path between u and z by applying
our algorithm illustrated in section 4.
x
y
u
z
v
w
Fig. 3: Connected acyclic network
Edges ¹ − ×
〈x× , Ø
• ×〉
Ù−
〈0.90, X(0.3,0.4,0.6,0.8)Z; (0.8,0.1)〉
Ù−Þ
〈0.60, X(0.2,0.3,0.5,0.7)Z; (0.5,0.1)〉
−á
〈0.35, X(0.2,0.3,0.5,0.6)Z; (0.6,0.4)〉
Þ−â
〈0.55, X(0.2,0.3,0.5,0.6)Z; (0.6,0.4)〉
32
Shortest Path Problem on an Intuitionistic Fuzzy Network
Þ−
〈0.14, X(0.1,0.2,0.7,0.8)Z; (0.4,0.5)〉
Þ−á
〈0.38, X(0.15,0.35,0.45,0.72)Z; (0.3,0.42)〉
−â
〈0.42, X(0.6,0.7,0.8,0.9)Z; (0.5,0.27)〉
á−â
〈0.30, X(0.1,0.3,0.5,0.8)Z; (0.4,0.3)〉
Ù−á
〈0.82, X(0.4,0.6,0.7,0.8)Z; (0.7,0.2)〉
á−å
〈0.75, X(0.4,0.7,0.8,0.9)Z; (0.6,0.2)〉
â−å
〈0.40, X(0.2,0.5,0.6,0.7)Z; (0.5,0.3)〉
Table 1: List of edges and their weights
weights
Now applying our algorithm we trace out all the paths between the source and sink of the
considered network which are listed below:
ã% ≔ Ù − − á − å; ã& ≔ Ù − − â − å; ã' ≔ Ù − Þ − â − å; ã( : Ù − á − â − å;
ãæ ≔ Ù − Þ − á − â − å;
ãç ≔ Ù − Þ − − á − å;
ãè ≔ Ù − Þ − − â − å;
ãé ≔ Ù − Þ − á − å; ãê ≔ Ù − − á − â − å; ã%= ≔ Ù − á − å; ã%% ≔ Ù − Þ −
− á − â − å.
Considering ã% which consist of edges Ù − , − á and á − å. We rearrange these edges
in non-increasing order based on their respective order inducing variable. Therefore,
Ù − , − á and á − å are arranged to Ù − , á − å and − á .
So the rearranged edges are,
ë#ìí % = [Ù − ] = 〈0.90, X(0.3,0.4,0.6,0.8)Z; (0.8,0.1)〉;ë#ìí (&) = á − å =
6
6
〈0.75, X(0.4,0.7,0.8,0.9)Z; (0.6,0.2)〉 and
ë#ìí
6 (')
=
− á = 〈0.35, X(0.2,0.3,0.5,0.6)Z; (0.6,0.4)〉
The normal distribution based weight vector for path ã% is calculated as Ë =
0.2429,0.5142,0.2429 Í .
Now calculating aggregated trapezoidal intuitionistic fuzzy number using
operator we get,
33
− ÎÏÐÑÒ
I − ITFOWG ˜φ
• –Õ
6
%
,φ
• –Õ
Saibal Majumder and Anita Pal
6
&
,φ
• –Õ
6
'
™=
}~
}~
}~
}~
~
∏'
∏'
∏'
〈˜∏'rj% a}
∏'
• ƒÕ (~) , 1
–Õ6 (r) , rj% b–Õ6 (r) , rj% c–Õ6 (r) , rj% d–Õ6 (r) ™ ; rj% wî
6
∏'rj%(1 − u (~) )}~ 〉
Õ
−
6
=〈X(0.3=.&(&ê × 0.4=.æ%(& × 0.2=.&(&ê ), (0.4=.&(&ê × 0.7=.æ%(& × 0.3=.&(&ê ), (0.6=.&(&ê ×
0.8=.æ%(& × 0.5=.&(&ê ), (0.8=.&(&ê × 0.9=.æ%(& × 0.6=.&(&ê )Z; (0.8=.&(&ê × 0.6=.æ%(& ×
0.6=.&(&ê ), (1 − X(1 − 0.1)=.&(&ê × (1 − 0.2)=.æ%(& × (1 − 0.2)=.&(&ê Z)〉
= 〈(0.315204,0.497373,0.665520,0.792583); 0.6434,0.2323〉.
The aggregated trapezoidal intuitionistic fuzzy numbers for the all the paths are listed in
Table 2.
Paths
from Aggregated ðñòóô using ò − òðóõö÷ operator
source(x) to
sink(±)
〈(0.315204,0.497373,0.665520,0.792583); 0.6434,0.2323〉
Ù− −á−å
Ù−
−â−å
〈(0.388274,0.563079,0.695656,0.822822); 0.5604,0.2397〉
Ù−Þ−â−å
〈(0.246363,0.393778,0.621358,0.749751); 0.5,0.2030〉
Ù−á−â−å
〈(0.2,0.461653,0.595908,0.7469138); 0.5139,0.2769〉
Ù−Þ−á−â
−å
Ù−Þ− −á
−å
Ù−Þ− −â
−å
Ù−Þ−á−å
〈(0.162654,0.377359,0.513453,0.721618); 0.404959,0.206274〉
Ù− −á−â
−å
Ù−á−å
〈(0.191277,0.374131,0.547722,0.691797); 0.553229,0.309889〉
Ù−Þ− −á
−â−å
〈(0.157174,0.323594,542089,0.699827); 0.489029,0.338282〉
〈(0.2,0.321265,0.566576,0.704543); 0.5291,0.2985〉
〈(0.262408,0.450108,0.659718,0.779366); 0.483002,0.299104〉
〈(0.220700,0.382616,0.546306,0.749171); 0.461653,0.213916〉
〈(0.4,0.648074,0.748331,0.848528); 0.648074,0.2〉
Table 2: List of aggregated TrIFN for different paths
Now, we find out the centroid point [9] of each of the aggregated ο Ïølisted in Table 2.
34
Shortest Path Problem on an Intuitionistic Fuzzy Network
Aggregated ðñòóô Øù··
sink
²¹
of all paths from source to Centroid
path
Point ¹th
〈(0.315204,0.497373,0.665520,0.792583); 0.6434,0.2323〉
(0.551259,0 .345403)
〈(0.388274,0.563079,0.695656,0.822822); 0.5604,0.2397〉
(0.601941,0 .332687)
〈(0.246363,0.393778,0.621358,0.749751); 0.5,0.2030〉
(0.496098,0 308039)
〈(0.2,0.461653,0.595908,0.7469138); 0.5139,0.2769〉
(0.463530,0 335227)
〈(0.162654,0.377359,0.513453,0.721618); 0.404959,0.206274〉
(0.441321,0.319675)
〈(0.2,0.321265,0.566576,0.704543); 0.5291,0.2985〉
(0.453922,0.318196)
〈(0.262408,0.450108,0.659718,0.779366); 0.483002,0.299104〉
(0.513091,0.318800)
〈(0.220700,0.382616,0.546306,0.749171); 0.461653,0.213916〉
(0.489474,0.317353)
〈(0.191277,0.374131,0.547722,0.691797); 0.553229,0.309889〉
(0.438326,0.335173)
〈(0.4,0.648074,0.748331,0.848528); 0.648074,0.2〉
(0.616970,0.347077)
〈(0.157174,0.323594,542089,0.699827); 0.489029,0.338282〉
(0.427513,0.324039)
Table 3: Centroid points of aggregated TrIFNs
Hence, applying ranking algorithm of TrIFN we rank the TrIFN in the following order:
φúûû(ü66 ) ≼ φúûû(üþ ) ≼ φúûû(ü ) ≼ φúûû(ü ) ≼ φúûû(ü< ) ≼ φúûû(ü ) ≼ φúûû(ü; ) ≼
φúûû(ü ) ≼ φúûû(ü6 ) ≼ φúûû(ü7 ) ≼ φúûû(ü6 ) . Since X î (Õ ) value of the centroid point
6
of φúûû(ü6 ) is largest so the path corresponding to φúûû(ü6 ) is, u − y − z which is
recognized to be the shortest path from source to sink of the network and its length is
〈(0.8,1.3,1.5,1.7); 0.88,0.04〉.
6. Conclusion
In this paper we have proposed an algorithm to find the shortest paths of a network. The
edge weights of the network is represented as a pair of ordered inducing variable and
TrIFN, 〈Ù , ë# 〉. The TrIFNs used in this paper can be consider as the more generalized
version of TrFNs since TrIFN considers belongingness and non-belongingness of an
event as a matter of which the uncertain information can be expressed in a much better
way with respect to TrFN. In our algorithm of shortest path problem we have used
I − ITFOWG operator to find the aggregated TrIFNs and have ranked them based on the
centroid values of those TrIFNs and finally find the shortest path having largest centroid
point value and have also calculated its path length.
35
Saibal Majumder and Anita Pal
From the view point of computational complexity this problem can be well
considered as NP −hard. Execution of Step-1, Step-2 and Step-4 of the proposed
algorithm may be a hard task in a large scale network, since these steps of the algorithm
may take most of the computational time. In the future, improvements can be done in
terms of efficient comparisons of TrIFN and data structure for this algorithm.
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