Intuitionistic fuzzy technique to find the critical path

In this paper, intuitionistic fuzzy number has been applied to find the critical path for the project scheduling problems with the aid of interval valid intuitionistic fuzzy numbers (IVIFN). Decision maker’s risk attitude index and decision maker’s risk index ranking value has also been utilized to find the critical path. In order to explain intuitionistic fuzzy technique to find critical path, numerical example has been illustrated in this paper.

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Rank value Using equation 1 β=0.602 x1=-39,... and Rank value Using equation 1 β=0.35 x1=-45,...: 1 →2 →4 vs 1 →2 →3 →4 vs 1 →3 →4Close ×

Replotted from Table 3 (page 4) of the original paper. Every bar is the paper's own table value.

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Introduction

A constructed network is an imperative tool in the development, and it organizes a definite project execution. Network diagram plays a vital role in formative project-completion time. 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. Fuzzy set theory as developed by Zadeh [11] and the concept of fuzzy numbers presented by Dubois and Prade [4] are applied to present a fuzzily defined system. Chanas and Zielinski [12] proposed a method to make critical path analysis in the network with fuzzy activity times (interval activity times, fuzzy numbers of L-R type) by directly applying the extension principle [11] to the classical criticality notion treated as a function of activity duration time in the network. And two methods of calculation of the path degree of criticality are presented. Slyeptsov and Tyshchuk [13] presented an efficient method of computation of fuzzy time windows for late start and finish times of operations in the problem of fuzzy network. G. Liang and T.C. Han [6] proposed a fuzzy critical path for project networks. Elizabeth and L. Sujatha [5] discussed a critical path problem under fuzzy Environment. 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. S.H. Nasution [8] proposed a fuzzy critical path method by considering interactive fuzzy subtraction and by observing that only the non-negative part of the fuzzy numbers can have physical work. Basic definitions of intuitionistic fuzzy set theory have been reviewed. K.Atanassov [1] proposed a procedure to find out the intuitionistic fuzzy critical path using an illustrative example. The results are discussed. Finally, some conclusions are drawn.

Preliminaries 2.1 Definition: Fuzzy Set

Let X be the universal set A fuzzy set A in X represented by A={(𝑥, 𝜇 𝐴 (𝑥))/𝑥𝜀𝑋},where the function𝜇 𝐴 (𝑥):X→ [0,1] is the membership degree of element x in the fuzzy set A.

Definition: Intuitionistic Fuzzy Set

Let X be an Universe of discourse, then an Intuitionistic Fuzzy Set (IFS) A in X is given by A={(𝑥, 𝜇 𝐴 (𝑥), 𝜐 𝐴 (𝑥))/𝑥𝜀𝑋},where the function 𝜇 𝐴 (𝑥):X→ [0,1] and 𝜐 𝐴 (𝑥):X→ [0,1] determine the degree of membership and non-membership of the element𝑥𝜀𝑋, respectively and for every 𝑥𝜀𝑋,0 ≤ 𝜇 𝐴 (𝑥) + 𝜐 𝐴 (𝑥) ≤ 1.

Definition: Intuitionistic Fuzzy Number

An IFN A is  An IF subset of the real line.  Normal ie, there is an 𝑥 0 𝜀𝑅such that 𝜇 𝐴 (𝑥 0 ) =1 (so 𝜐 𝐴 (𝑥 0 ) =0).  Convex for the Membership function 𝜇 𝐴 (𝑥). ie, 𝜇 𝐴 (𝜆𝑥 1 + (1 -𝜆)𝑥 2 ) ≥ 𝑚𝑖𝑛( 𝜇 𝐴 (𝑥 1 ), 𝜇 𝐴 (𝑥 2 )) for every 𝑥 1 , 𝑥 2 𝜀𝑅, 𝜆𝜀[0,1].  Concave for the non-membership function ie 𝜐 𝐴 (𝜆𝑥 1 + (1 -𝜆)𝑥 2 ) ≤ 𝑚𝑎𝑥( 𝜐 𝐴 (𝑥 1 ), 𝜐 𝐴 (𝑥 2 )) for every 𝑥 1 , 𝑥 2 𝜀𝑅, 𝜆𝜀[0,1].

Definition: Trapezoidal Intuitionistic Fuzzy Number

An Intuitionistic Fuzzy Number 𝐴 = {< 𝑎, 𝑏, 𝑐, 𝑑 >< 𝑎 ′ , 𝑏 ′ , 𝑐 ′ , 𝑑 ′ >}is said to be a trapezoidal intuitionistic fuzzy number if its membership function and non-membership function are given by

Definition: Algebraic Operations of any Two Trapezoidal Intuitionistic Fuzzy Number Addition

Definition: Algebraic Operations of any Two Trapezoidal Intuitionistic Fuzzy Number Addition

Definition: Decision Maker's Risk Attitude Index

For an Intuitionistic fuzzy critical path analysis problem [14] , using the Trapezoidal Intuitionistic fuzzy numbers such as 𝑇 𝑖𝑗 =< 𝑎 𝑖𝑗 , 𝑏 𝑖𝑗 𝑐 𝑖𝑗 , 𝑑 𝑖𝑗 > to denote the intuitionistic fuzzy activity time of activity 𝐴 𝑖𝑗 the decision maker's risk attitude index 𝛽can be obtained by

where ACT and t denote the set of all actives and the number of actives in a project network, respectively.

Definition: Decision Maker's Risk Index Ranking Value

The ranking value 𝑅(𝐴 𝑖 ) of the trapezoidal intuitionistic fuzzy number 𝐴 𝑖 can be obtained as follows:

𝑅(𝐴 𝑖 ) = 𝛽[(𝑑 𝑖 -𝑥 1 )/(𝑥 2 -𝑥 1 -𝑐 𝑖 + 𝑑 𝑖 )] + (1 -𝛽){1 -[(𝑥 2 -𝑎 𝑖 )/(𝑥 2 -𝑥 1 + 𝑏 𝑖 -𝑎 𝑖 )]}. . . . . . . . . . . . (2) Where 𝛽is the decision maker's risk attitude index, 𝑥 1 = 𝑚𝑖𝑛{ 𝑎 1 , 𝑎 2, . . . . . . , 𝑎 𝑛 } and 𝑥 2 = 𝑚𝑎𝑥{ 𝑑 1 , 𝑑 2, . . . . . . , 𝑑 𝑛 }

Definition: Rules for Ranking Trapezoidal Intuitionistic Fuzzy Number

Now, we rank the intuitionistic fuzzy numbers 𝐴 𝑖 and 𝐴 𝑗 according to the following rules [9] :

Definition: An Intuitionistic Fuzzy Completion Time

Assume that there exists a path P c in a project network such that IFCPM(P c )=min{IFCPM(P i )/𝑃 𝑖 𝜀𝑃} then the path P c is an intuitionistic fuzzy critical path.

Intuitionistic fuzzy critical path analysis 3.1 Notation

Intuitionistic fuzzy critical path analysis 3.1 Notation

N :

The set of all nodes in a project network. A ij : The activity between nodes i and j. 𝑇 𝜇𝑖𝑗 , 𝑇 𝜐𝑖𝑗 : The intuitionistic fuzzy activity time for membership function and non-membership function.

Figures and tables from the paper

Activity Aij	Intutionistic Fuzzy Activity time Tij
1-2	<3, 5, 5, 7> <4, 5, 5, 8>
1-3	<5, 10, 10, 15> <6, 10, 10, 16>
2-3	<1, 3, 4, 5> <2, 3, 4, 5>
2-4	<2, 4, 5, 6> <3, 4, 5, 7>
3-4	<6, 8, 10, 11> <7, 8, 10, 12>

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The fuzzy activity time for each activity in the project network shown as above figure

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Original paper tableSource PDF · page 4 ↗

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Total slack intuitionistic fuzzy time for each activity

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Paths	IFCP(Pµk) K=1–m	Rank value Using equation 1 β=0.602 x1=-39, x2=47	Rank	IFCP(Pνk) K=1-m	Rank value Using equation 1 β=0.35 x1=-45, x2=11	Rank
1 →2 →4	<-14,7,15,37>	0.5606	III	<-26,-10,-6,9>	0.5827	III
1 →2 →3 →4	<-39,-4,10,47>	0.5360	II	<-45,-20,-12,11>	0.4487	II
1 →3 →4	<-30,-4,4,30>	0.4953	I	<-43,-21,-14,4>	0.4318	I

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Rank value of total slack intuitionistic fuzzy time of all possible paths

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Original paper tableSource PDF · page 4 ↗

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