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A Critical Path Problem Using Intuitionistic Triangular Fuzzy Number
P.JAYAGOWRI*, G.GEETHARAMANI**
*(Department of Mathematics, Sudharsan Engineering College, and Anna University Chennai, India.) ** (Department of Mathematics, BIT Campus, Anna University Chennai, India.)
ABSTRACT
Critical path method is a network based method planned for development and organization of complex project in real world application. In this paper, a new methodology has been made to find the critical path in a directed acyclic graph, whose activity time uncertain. The vague parameters in the network are represented by intuitionistic triangular fuzzy numbers, instead of crisp numbers. A new procedure is proposed to find the optimal path, and finally illustrative examples is provided to validate the proposed approach.
Keywords

Critical path, Graded mean integration representation of intuitionistic triangular fuzzy number, Intuitionistic triangular fuzzy number, Order relation, Ranking of intuitionistic fuzzy number.
I.
INTRODUCTION
A constructed network is an imperative tool in the development and organizes of definite project execution. Network diagram play a vital role in formative projectcompletion time. In general a project will consist of a number of performance and some performance can be started, only after ultimate some other behavior. There may be some behavior which is selfdetermining of others. Network analysis is a practice which determines the various sequences of behavior in relation to a project and the corresponding project completion time. The method is widely used are the critical path method program assessment and evaluation techniques. The successful finishing of critical path method requires the clear unwavering time duration in each movement. However in real life situation vagueness may arise from a number of possible sources like: due date may be distorted, capital may unavailable weather situation may root several impediments. Therefore the fuzzy set theory can play a significant role in this kind of problems to handle the ambiguity about the time duration of deeds in a project network. To effectively deal with the lack of clarity involved in the process of linguistic predictable times the intuitionistic trapezoidal fuzzy numbers are used to distinguish the fuzzy measures of linguistic values. In the current past, fuzzy critical path problems are addressed by many researchers, like S.Chanas , T.C.Han, J.Kamburowski, Zielinski, L.Sujatha and S.Elizabeth. S.Chanas, and J.Kamburowski,[11] explained fuzzy variables PERT. S.Chanas P.and Zielinski, [10] have also discussed critical path analysis in networks. G.Liang and T.C.Han [4] proposed a fuzzy critical path for project networks. Elizabeth and L.Sujatha [13] discussed a critical path problem under fuzzy Environment. They have deliberated a critical path problem for project networks. C.T Chen and S.F Huang [2] proposed a new model that combines fuzzy set theory with the PERT technique to determine the critical degrees of activities and paths, latest and earliest starting time and floats. The Bellman algorithm seeks to specify the critical path and the fuzzy earliest and latest starting time and floats of activities in a continuous fuzzy network. S.H. Nasution [12] proposed a fuzzy critical path method by considering interactive fuzzy subtraction and by observing that only the nonnegative part of the fuzzy numbers can have physical elucidation. This paper is organized as follows: In section 2, basic definitions of intuitionistic fuzzy set theory have been reviewed. Section 3, give procedures to find out the intuitionistic fuzzy critical path using an illustrative example. In section 4 the obtained results are discussed. Finally, in section 5 some conclusions are drawn.
II.
P
RELIMINARIES
Some basic definitions related to our research work are review. 2.1
Intuitionistic Fuzzy Set Let X be an Universe of discourse, then an Intuitionistic fuzzy set(IFS) A in X is given by A=
X/x)(
A
γ
),(A
μ
x,
x x
, where the function
RESEARCH ARTICLE OPEN ACCESS
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0,1X:(x)
A
μ
and
0,1X:(x)
A
γ
determine the degree of membership and nonmembership of the element
X x
, respectively and for every
X x
,
1.(x)A
γ
(x)A
μ
0
2.2
Triangular Intuitionistic Fuzzy Number An intuitionistic fuzzy number A={<a,b,c> <e,b,f>} is said to be a triangular intuitionistic fuzzy number if its membership function and non membership function are given by
&R c b,a,where
cx b b)(cx)(c bx1 bxaa)(ba)(x(x)A
R f b,e,where
x b b)f ( b)(x bx0 bx)e(b)x(b(x)A
f e
2.3 Graded Mean Integration Representation for Triangular Intuitionistic Fuzzy numbers The membership and nonmembership function of triangular intuitionistic fuzzy numbers are defined as follows.
f x;w
bf bx(x)R bxe;w
e bx b(x)Lcx;w
bcxc(x)R bxa;w
a bax(x)
μ
L
&&
bb
Then
1R 1L
and
are inverse functions of functions L and R respectively,
h/w b)(f b(h)
1
γ
R &h/we)(b b(h)
1
γ
Lh/w b)(cc(h)
1
μ
R &h/wa)(ba(h)
1
μ
L
Then the graded mean integration representation of membership function and nonmembership function are,
&)1(
64(A)
μ
P
cba
)2(3(A)P
f eb
2.4 Arithmetic Operations of Triangular Intuitionistic Fuzzy Number
)3' b,2 b,1' b;3 b,2 b,1 b(IB~and)3'a,2a,1'a;3a,2a,1a(IA~If
are two intuitionistic fuzzy numbers we define, Addition :
)3' b3'a,2 b2a,1' b1'a;3 b3a,2 b2a,1 b1a(IB~IA~
Subtraction:
)1'3',22,3'1';13,22,31(~~
bababa
bababa
I B I A
2.5 Order Relation Consider an order relation among intuitionistic fuzzy numbers.Rule:Let A and B be two triangular intuitionistic fuzzy numbers such that
and)A
γ
;3
γ
c,2
γ
c,1
γ
cA
μ
;3
μ
a,2
μ
a,1
μ
a(A
B
γ
A
γ
andB
μ
A
μ
with)A
γ
;3
γ
d,2
γ
d,1
γ
dA
μ
;3
μ
b,2
μ
b,1
μ
b(B
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(3)3
γ
d3
γ
c;3
μ
b3
μ
a3.2
γ
d2
γ
c;
μ2
b2
μ
a2.1
γ
d1
γ
c;1
μ
b1
μ
a1.
2.6 Ranking of Intuitionstic Triangular Fuzzy umber
f b,e,;c b,a,
'A~number fuzzysticintuitioniTriangular A
Then the graded mean integration representation of membership function and nonmembership function of
A
~
are
.B~A~)B~(
βγ
P)A~(
βγ
Pand)B
~(
αμ
P)A~(
αμ
(iii).B~A~)B~(
βγ
P)A~(
βγ
Pand)B
~(
αμ
P)A~(
αμ
(ii).B~A~)B~(
βγ
P)A~(
βγ
Pand)B
~(
αμ
P)A~(
αμ
P(i)thennumbersfuzzytriangular sticIntuitioni
twoany be)f b,,ec b,,a(B
~)andf b,e,c b,a,(A
~Let32f bf)e(2
β
)A~(
γ
P&32bcc)
α(a
)A~(
μ
P
III.
I
NTUITIONISTIC
F
UZZY
C
RITICAL
P
ATH
M
ETHOD
The following is the Procedure for finding intuitionistic fuzzy critical path. 3.1 Notations N= The set of all nodes in a project network. EST = Earliest Starting time EFT
µij
= Earliest finishing time for membership function EFT
γij
= Earliest finishing time for nonmembership function LFT
µij
= Latest finishing time for membership function LFT
γij
= Latest finishing time for non membership function LST
µij
= Latest starting time for membership function LST
γij
= Latest starting time for nonmembership function TF = Total float T
ij
= The Intuitionistic fuzzy activity time 3.2 Forward Pass Calculation Forward pass calculations are employed to calculate the Earliest Starting Time(EST) in the Project network
(4)nodes precedingof number i
, ji
γ
t)(i
γ
EiMin j
γ
Enodes precedingof number i
, ji
μ
t)(i
μ
EiMax j
μ
E
)5(timeactivityFuzzysticIntuitioni)(
ji
γ
EST ji
γ
EFTfunctionmembershipnonfor timeFinishingEarliest
timeactivityFuzzysticIntuitioni)(
ji
μ
EST ji
μ
EFTfunctionmembershipfor timeFinishingEarliest
3.3 Backward Pass Calculation Backward pass calculations are employed to calculate the Latest Finishing Time (LFT) in the Project network
)6(nodessucceedingof number
,)(nodessucceedingof number
,)(
j jit i Li Max j L j jit j L j Mini L
)7(timeactivityFuzzysticIntuitioni
)( ji
γ
LFT ji
γ
LSTfunctionmembershipnonfor timeFinishingLatest
timeactivityFuzzysticIntuitioni
)( ji
μ
LFT ji
μ
LSTfunctionmembershipfor timeStartingLatest
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Page 3.4 Total Float (TF)
(8) ji
γ
EST ji
γ
LST
γ
TFor ji
γ
EFT ji
γ
LFT
γ
TF ji
μ
EST ji
μ
LST
μ
TFor ji
μ
EFT ji
μ
LFT
μ
TF
3.5 Procedure to Find Intuitionistic Fuzzy Critical Path
Step1:
Construct a network G (V,E) where v is the set of vertices and E is the set of edges. Here G is an acyclic digraph and arc length or edge weight are taken as Intuitionistic Triangular fuzzy numbers.
Step 2:
Expected time in terms of Intuitionistic Triangular fuzzy numbers are defuzzified using equation 1 and 2 in the network diagram.
Step3:
Calculate Earliest Starting time for membership and nonmembership functions ( EST
µij
and EST
γij
respectively) according to Forward pass calculation given in equation (4)
Step4:
Calculate Earliest finishing time for membership and nonmembership functions (EFT
µij
and EFT
γij
respectively )using equation (5).
Step 5 :
Calculate latest finishing time LFT
µij
and LFT
γij
 according to backward pass calculation given in equation(6)
Step 6:
Calculate latest starting time LST
µij
and LST
γij

using equation (7)
Step 7 :
Calculate Total float TF
µ
and TF
γ
using equation (8)
Step 8:
In each activity using (8) whenever one get 0 , such activities are called as Intuitionistic Fuzzy critical activities and the corresponding paths Intuitionistic critical paths.
3.
5 Illustrative example
Consider a small network with 5 vertices and 6 edges shown in figure1, where each arc length is represented as a trapezoidal Intuitionistic fuzzy number
Table1: Results of the network
Activity Intuitionistic Fuzzy Activity time Defuzzified Activity time using equation 1 and 2 for membership and nonmembership functions TF
µ
TF
γ
1→2
<6,8,10> <12,14,15> <8,13.7> 0 2.3
1→3
<7,9,10> <15,17,16> <8.8,16> 12.6 0
1→4
<5,6,7> <16,19,12> <6,15.7> 0 0
2→4
<7,9,11> <10,12,13> <9,11.7> 11 2.3
3→4
<8,10,11> <9,12,14> <9.8, 11.7> 12.6 0
4→5
<9,11,13> <7,9,11> <11,9> 0 0
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Table2: Rank value of total slack intuitionstic fuzzy time of all possible paths for membership function Paths
IFCPM(p
µk
) k=1 to m
Rank value definition 2.6 Rank
1→2→4→5
<22,28,34> 28 II
1→4→5
<14,17,20>
17 I
1→3→4→5
<24,30,34>
29.7 III Table3: Rank value of total slack intuitionstic fuzzy time of all possible paths for non membership function
Paths
IFCPM(p
γk
) k=1 to m
Rank value definition 2.6 Rank
1→2→4→5
<29,35,39> 34.3 II
1→4→5
<23,28,23>
24.7 I
1→3→4→5
<31,38,41>
36.7 III
IV.
R
ESULTS AND
D
ISCUSSION
This paper proposes an algorithm to tackle the problem in intuitionistic fuzzy environment. In this paper the trapezoidal intuitionistic fuzzy number is defuzzified using graded mean integration representation. Now the intuitionistic fuzzy number is converted to crisp number. Then applying the proposed algorithm we find the critical path. The path in intuitionistic fuzzy project network are
1→4→5,1→2→4→5and 1→3→4→5.The critical
path for intuitionistic fuzzy network for both membership and nonmembership function are
1→4→5. Hence the procedure developed in this
paper form new methods to get critical path, in intuitionistic fuzzy environment.
V.
C
ONCLUSION
A new analytical method for finding critical path in an intuitionistic fuzzy project network has been proposed. We have used new defuzzification formula for trapezoidal fuzzy number and applied to the float time for each activity in the intuitionistic fuzzy project network to find the critical path. In general intuitionistic fuzzy models are more effective in determining critical paths in real project networks. This paper, use the Graded mean integration representation the Procedure to find the optimal path in an intuitionistic fuzzy weighted graph having help decision makers to decide on the best possible critical path in intuitionistic fuzzy environments.
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