Using Metric Distance Ranking Method to Find Intuitionistic Fuzzy

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2015, Article ID 952150, 12 pages http://dx.doi.org/10.1155/2015/952150

Research Article Using Metric Distance Ranking Method to Find Intuitionistic Fuzzy Critical Path P. Jayagowri1 and G. Geetharamani2 1

Department of Mathematics, Sudharsan Engineering College, Tamil Nadu 622501, India Department of Mathematics, Anna University Chennai, BIT Campus, Tamil Nadu 620024, India

2

Correspondence should be addressed to P. Jayagowri; [email protected] Received 1 April 2015; Revised 4 June 2015; Accepted 28 June 2015 Academic Editor: Humberto Bustince Copyright © 2015 P. Jayagowri and G. Geetharamani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Network analysis is a technique which determines the various sequences of activities concerning a project and the project completion time. The popular methods of this technique which is widely used are the critical path method and program evaluation and review techniques. The aim of this paper is to present an analytical method for measuring the criticality in an (Atanassov) intuitionistic fuzzy project network. Vague parameters in the project network are represented by (Atanassov) intuitionistic trapezoidal fuzzy numbers. A metric distance ranking method for (Atanassov) intuitionistic fuzzy numbers to a critical path method is proposed. (Atanassov) Intuitionistic fuzzy critical length of the project network is found without converting the (Atanassov) intuitionistic fuzzy activity times to classical numbers. The fuzzified conversion of the problem has been discussed with the numerical example. We also apply four different ranking procedures and we compare it with metric distance ranking method. Comparison reveals that the proposed ranking method is better than other raking procedures.

1. Introduction Critical path method is a network based method designed for planning and managing of complicated project in real world applications. According to the critical path, the decision maker can control the time and the cost of the project and improve the efficiency of resource allocation to ensure the project quality. In many situations, project can be complicated and challenging to manage. There are many cases where the activity times may not be presented in a precise manner. An alternative way to deal with imprecise data is to employ the concept of fuzziness by vague activity times that can be represented by fuzzy sets. Fuzzy set theory proposed by Zadeh [1] can play a significant role in solving such a management problem. Chanas and Zielinski [2] 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 to the usual criticality notion treated as a function of activity duration time in

the network. Chanas and Kamburowski [3] explained fuzzy variables in PERT. Chen and Huang [4] 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. Based on signed distance ranking of fuzzy numbers, Yao and Lin [5] found out fuzzy critical path method based on signed distance ranking method. Styeptsov and Tyshchuk [6] created a proficient method of calculation of fuzzy time windows for late start and finish times of operations in the problem of fuzzy network. Nasution [7] 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. Ravi Shankar et al. [8] proposed new defuzzified formula to find critical path in a fuzzy project network. Elizabeth and Sujatha [9, 10] introduced new ranking methods to find fuzzy critical path problem for project network. In this paper a new algorithm is proposed to find (Atanassov) intuitionistic fuzzy critical path without converting

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the (Atanassov) intuitionistic fuzzy activity times to crisp number. Here we propose a metric distance ranking method to an (Atanassov) intuitionistic fuzzy critical path method for a project network problem. Finally we compare the proposed ranking method with existing method and conclude the paper. This paper is organized as follows. In Section 2, basic definitions of (Atanassov) intuitionistic fuzzy set theory have been reviewed and some new definitions are framed for (Atanassov) intuitionistic fuzzy critical path method. Section 3 gives new procedures to find out the (Atanassov) intuitionistic fuzzy critical path using an illustrative example. Different ranking approach has been given and results are discussed. Finally Section 4 concludes the paper.

2. Preliminaries In this section some basic definitions and ranking function are reviewed. A new ranking approach is introduced for (Atanassov) intuitionistic trapezoidal fuzzy number.

(𝜇a (x)v a (x)) 1

0.5

0

0

a1 a

c

b

d d1

X

Figure 1: Trapezoidal intuitionistic fuzzy number.

Definition 4 (arithmetic operations of trapezoidal (Atañ𝐼 = {(𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 ); assov) intuitionistic fuzzy number). If 𝐴 󸀠 󸀠 󸀠 󸀠 𝐼 (𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 )} and 𝐵̃ = {(𝑏1 , 𝑏2 , 𝑏3 , 𝑏4 ); (𝑏1󸀠 , 𝑏2󸀠 , 𝑏3󸀠 , 𝑏4󸀠 )} are two (Atanassov) intuitionistic fuzzy numbers, we define: Addition:

Definition 1 (fuzzy set (Zadeh [1])). Let a nonempty fuzzy set 𝐴 in 𝑥 be characterized by a membership function 𝜇𝐴 (𝑥) for all 𝑥 ∈ 𝐸 which associates with each point in 𝐸, a real number in the interval [0, 1], with the value of 𝜇𝐴 (𝑥) at 𝑥 representing the “grade of membership” of in 𝑥 in 𝐴.

̃𝐼 + 𝐵̃𝐼 = {(𝑎1 + 𝑏1 , 𝑎2 + 𝑏2 , 𝑎3 + 𝑏3 , 𝑎4 + 𝑏4 ) ; 𝐴 (2) (𝑎1󸀠

Definition 2 (Atanassov intuitionistic fuzzy set [11]). Let 𝑋 be universe of discourse; then an intuitionistic fuzzy set (IFS) 𝐴 in 𝑋 is given by 𝐴 = {(𝑥, 𝜇𝐴 (𝑥), 𝛾𝐴(𝑥))/𝑥 ∈ 𝑋}, where the functions 𝜇𝐴 (𝑥) : 𝑋 → [0, 1] and 𝛾𝐴(𝑥) : 𝑋 → [0, 1] determine the degree of membership and nonmembership of the element 𝑥 ∈ 𝑋, respectively, and for every 𝑥 ∈ 𝑋, 0 ≤ 𝜇𝐴 (𝑥) + 𝛾𝐴(𝑥) ≤ 1. Definition 3 (trapezoidal intuitionistic fuzzy number). An (Atanassov) intuitionistic fuzzy number 𝐴 = {(𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 )(𝑏1 , 𝑏2 , 𝑏3 , 𝑏4 )} is said to be a trapezoidal (Atanassov) intuitionistic fuzzy number (see Figure 1) if its membership function and nonmembership function are given by (𝑥 − 𝑎1 ) { , 𝑎1 ≤ 𝑥 ≤ 𝑎2 { { { (𝑎2 − 𝑎1 ) { { { { { 𝑎2 ≤ 𝑥 ≤ 𝑎3 𝜇𝐴 (𝑥) = {1, { { { { { { { { (𝑥 − 𝑎4 ) , 𝑎3 ≤ 𝑥 ≤ 𝑎4 , { (𝑎3 − 𝑎4 )

+ 𝑏2󸀠 , 𝑎3󸀠

+ 𝑏3󸀠 , 𝑎4󸀠

+ 𝑏4󸀠 )}

Subtraction: ̃𝐼 − 𝐵̃𝐼 = {(𝑎1 − 𝑎󸀠 , 𝑎2 − 𝑎󸀠 , 𝑎3 − 𝑎󸀠 , 𝑎4 − 𝑎󸀠 ) ; 𝐴 4 3 2 1 (3) (𝑏1 − 𝑏4󸀠 , 𝑏2

− 𝑏3󸀠 , 𝑏3

− 𝑏2󸀠 , 𝑏4

− 𝑏1󸀠 )} .

Definition 5 (arithmetic operations on two trapezoidal (Atañ1 = {(𝑎𝜇 , assov) intuitionistic fuzzy activity times). Let IFAT 1 ̃2 = {(𝑎𝜇 , 𝑏𝜇 , 𝑐𝜇 , 𝑏𝜇 , 𝑐𝜇 , 𝑑𝜇 )(𝑎𝛾 , 𝑏𝛾 , 𝑐𝛾 , 𝑑𝛾 )} and IFAT 1

1

1

1

1

1

1

2

2

⋅ (𝑎𝛾 1 , 𝑏𝛾 1 , 𝑐𝛾 1 , 𝑑𝛾 1 ) + (𝑎𝜇 2 , 𝑏𝜇 2 , 𝑐𝜇 2 , 𝑑𝜇 2 )

𝑏1 ≤ 𝑥 ≤ 𝑏2

⋅ (𝑎𝛾 2 , 𝑏𝛾 2 , 𝑐𝛾 2 , 𝑑𝛾 2 )}

𝑏2 ≤ 𝑥 ≤ 𝑏3

= {(𝑎𝜇 1 + 𝑎𝜇 2 , 𝑏𝜇 1 + 𝑏𝜇 2 , 𝑐𝜇 1 + 𝑐𝜇 2 , 𝑑𝜇 1 + 𝑑𝜇 2 ) ;

𝑏3 ≤ 𝑥 ≤ 𝑏4 .

2

𝑑𝜇 2 )(𝑎𝛾 2 , 𝑏𝛾 2 , 𝑐𝛾 2 , 𝑑𝛾 2 )} be any two (Atanassov) intuitionistic fuzzy activity times. (i) For addition operation (+) we have the following. Let ̃ 1 + IFAT ̃2 = {(𝑎𝜇 , 𝑏𝜇 , 𝑐𝜇 , 𝑑𝜇 ) IFAT 1 1 1 1

(1) (𝑥 − 𝑏1 ) { , { { { (𝑏 { { 2 − 𝑏1 ) { { { 𝛾𝐴 (𝑥) = {1, { { { { { { { (𝑥 − 𝑏4 ) { , { (𝑏3 − 𝑏4 )

+ 𝑏1󸀠 , 𝑎2󸀠

(𝑎𝛾 1 + 𝑎𝛾 2 , 𝑏𝛾 1 + 𝑏𝛾 2 , 𝑐𝛾 1 + 𝑐𝛾 2 , 𝑑𝛾 1 + 𝑑𝛾 2 )} .

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(ii) For subtraction operation (−), we have the following. Let ̃ 2 = {(𝑎𝜇 , 𝑏𝜇 , 𝑐𝜇 , 𝑑𝜇 ) ̃1 − IFAT IFAT 1 1 1 1 ⋅ (𝑎𝛾 1 , 𝑏𝛾 1 , 𝑐𝛾 1 , 𝑑𝛾 1 ) + (𝑎𝜇 2 , 𝑏𝜇 2 , 𝑐𝜇 2 , 𝑑𝜇 2 ) ⋅ (𝑎𝛾 2 , 𝑏𝛾 2 , 𝑐𝛾 2 , 𝑑𝛾 2 )}

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Then the values of the membership and nonmembership function for the trapezoidal (Atanassov) intuitionistic fuzzy number are as follows: 𝑉𝜇 (𝐴) =

𝑎1 + 2𝑏1 + 2𝑐1 + 𝑑1 , 6

𝑉𝛾 (𝐴) =

𝑏1󸀠 + 𝑐1󸀠 𝑑1󸀠 − 𝑐1󸀠 − 𝑏1󸀠 + 𝑎1󸀠 + 2 6

= {(𝑎𝜇 1 − 𝑑𝜇 2 , 𝑏𝜇 1 − 𝑐𝜇 2 , 𝑐𝜇 1 − 𝑏𝜇 2 , 𝑑𝜇 1 − 𝑎𝜇 2 ) ;

̃ = {(𝑎1 , 𝑏1 , 𝑐1 , 𝑑1 ) (𝑎󸀠 , 𝑏1 , 𝑐1 , 𝑑󸀠 )} , 𝐴 1 1

(𝑎𝛾 1 − 𝑑𝛾 2 , 𝑏𝛾 1 − 𝑐𝛾 2 , 𝑐𝛾 1 − 𝑏𝛾 2 , 𝑑𝛾 1 − 𝑎𝛾 2 )} .

𝐵̃ = {(𝑎2 , 𝑏2 , 𝑐2 , 𝑑2 ) (𝑎2󸀠 , 𝑏2 , 𝑐2 , 𝑑2󸀠 )}

Definition 6 (ranking of Atanassov intuitionistic trapezoidal fuzzy number [12]). Ranking procedure was introduced by Li [13], which was based on ratio of value and ambiguity index. Li procedure was more generalized and applicability was wider, but the ratio ranking method lacks in linearity property. For ratio ranking method, 𝑅 (̃ 𝑎 + ̃𝑏, 𝜆) ≠ 𝑅 (̃ 𝑎, 𝜆) + 𝑅 (̃𝑏, 𝜆) .

(11)

(6)

De and Das [14] tried to rectify this by taking linear sum of value and ambiguity indices using the following definition. Definition 7 (see [14]). Let 𝑎̃ = {(𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 )𝑤𝑎̃, 𝑢𝑎̃} be a trapezoidal (Atanassov) intuitionistic fuzzy number. A value index and ambiguity index for the trapezoidal (Atanassov) intuitionistic fuzzy number 𝑎̃ are defined as follows:

be two trapezoidal (Atanassov) intuitionistic fuzzy number. If 𝑎1 > 𝑑2 then 𝐴 > 𝐵. (i) If 𝑉𝜇(𝐴) > 𝑉𝜇(𝐵) then 𝐴 > 𝐵; (ii) if 𝑉𝜇(𝐴) < 𝑉𝜇(𝐵) then 𝐴 < 𝐵; (iii) if 𝑉𝜇(𝐴) = 𝑉𝜇(𝐵) then 𝐴 is equivalent to 𝐵. Definition 9 (Euclidean Ranking (ER) technique for trapezoidal (Atanassov) intuitionistic fuzzy numbers [10]). Let 𝐿 𝑖 = {(𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖 )(𝑎𝑖󸀠 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖󸀠 )} be the 𝑖th fuzzy path length and let 𝐿 max = {(𝑎, 𝑏, 𝑐, 𝑑)(𝑎󸀠 , 𝑏, 𝑐, 𝑑󸀠 )} = (𝐿 max , 𝑙max ) be the fuzzy longest length then the Euclidean Ranking of 𝐿 𝑖 is denoted by

𝑎) + 𝜆 (𝑉𝛾 (̃ 𝑎) − 𝑉𝜇 (̃ 𝑎)) , 𝑎, 𝜆) = 𝑉𝜇 (̃ 𝑉 (̃

(7)

ER (𝐿 𝑖 ) = (√(𝑎 − 𝑎𝑖 ) + (𝑏 − 𝑏𝑖 ) + (𝑐 − 𝑐𝑖 ) + (𝑑 − 𝑑𝑖 ) ) ,

𝑎) + 𝜆 (𝐴 𝛾 (̃ 𝑎) − 𝐴 𝜇 (̃ 𝑎)) , 𝐴 (̃ 𝑎, 𝜆) = 𝐴 𝛾 (̃

(8)

(√(𝑎󸀠 − 𝑎𝑖󸀠 ) + (𝑏 − 𝑏𝑖 ) + (𝑐 − 𝑐𝑖 ) + (𝑑󸀠 − 𝑑𝑖󸀠 ) )

respectively, where 𝜆 ∈ [0, 1] is a weight which represents the decision maker’s preference information. Limited to the above formulation, the choice 𝜆 = 1/2 appears to be a reasonable one. One can choose 𝜆 according to the suitability of the subject. 𝜆 ∈ [0, 1/2) indicates decision maker’s pessimistic attitude towards uncertainty while 𝜆 ∈ (1/2, 1] indicates decision maker’s optimistic attitude towards uncertainty. With our choice 𝜆 = 1/2, the value and ambiguity indices for trapezoidal (Atanassov) intuitionistic fuzzy number reduce to the following:

1 𝐴 (̃ 𝑎, ) = 2

2

2

2

2

2

2

2

2

(9) .

If 𝐴 and 𝐵 are trapezoidal (Atanassov) intuitionistic fuzzy numbers, then 𝐴 ≥ 𝐵 if and only if ER(𝐴) ≤ ER(𝐵). Definition 10 (similarity ranking for trapezoidal (Atanassov) intuitionist fuzzy numbers [15]). For two (Atanassov) intuitionist fuzzy path lengths, 𝐿 1 = {(𝑎1 , 𝑏1 , 𝑐1 , 𝑑1 ) (𝑙1 , 𝑚1 , 𝑛1 , 𝑜1 )} ,

1 1 𝑎, ) . 𝑅 (̃ 𝑎) = 𝑉 (̃ 𝑎, ) + 𝐴 (̃ 2 2

min (𝑏1 , 𝑏2 ) , min (𝑐1 , 𝑐2 ) , min (𝑑1 , 𝑑2 )) . (10)

Definition 8 (see [12]). Now let 𝐴 𝛼 and 𝐴 𝛽 be any 𝛼-cut and 𝛽-cut set of an trapezoidal (Atanassov) intuitionistic fuzzy ̃ = {(𝑎1 , 𝑏1 , 𝑐1 , 𝑑1 )(𝑎󸀠 , 𝑏󸀠 , 𝑐󸀠 , 𝑑󸀠 )}, respectively. number 𝐴 1 1 1 1

(13)

For membership function, ̃ min (𝑎, 𝑏, 𝑐, 𝑑) = min (𝐿 1 , 𝐿 2 ) = (min (𝑎1 , 𝑎2 ) , 𝐿 𝜇

The proposed ranking method is as follows:

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= (ER of membership function; ER of nonmembership function) .

𝐿 2 = {(𝑎2 , 𝑏2 , 𝑐2 , 𝑑2 ) (𝑙2 , 𝑚2 , 𝑛2 , 𝑜2 )} .

𝑉𝜇 (̃ 𝑎) + 𝑉𝛾 (̃ 𝑎) 1 , 𝑉 (̃ 𝑎, ) = 2 2 𝐴 𝜇 (̃ 𝑎) + 𝐴 𝛾 (̃ 𝑎)

2

(14)

For nonmembership function, ̃ max (𝑙, 𝑚, 𝑛, 𝑜) = max (𝐿 1 , 𝐿 2 ) = (max (𝑙1 , 𝑙2 ) , 𝐿 𝛾 max (𝑚1 , 𝑚2 ) , max (𝑛1 , 𝑛2 ) , max (𝑜1 , 𝑜2 )) .

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Let 𝐿 𝑖 = {(𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖 )(𝑎𝑖󸀠 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖󸀠 )} be the 𝑖th fuzzy path length. Then the similarity ranking of trapezoidal (Atanassov) intuitionist fuzzy numbers for both membership function and nonmembership function is

𝐿−1 𝛾 (ℎ) = 𝑏1 −

(𝑏1 − 𝑏2 ) ℎ , 𝑤

𝑅𝛾−1 (ℎ) = 𝑏4 +

(𝑏3 − 𝑏4 ) ℎ . 𝑤 (18)

̃ min ) SD (𝐿 𝑖 , 𝐿 𝜇 0, { { { { 2 { {1 (𝑑 − 𝑎𝑖 ) { { , { { { { 2 (𝑑 − 𝑐) + (𝑏𝑖 − 𝑎𝑖 ) ={ { { { { { 1 { { [(𝑑 − 𝑎𝑖 ) + (𝑐 − 𝑏𝑖 )] , { { {2 { {

̃ min ≠ 𝜑, 𝐿𝑖 ∩ 𝐿 𝜇

𝑃𝜇 (𝐴) =

where 𝑐 < 𝑥 < 𝑑, 𝑎𝑖 < 𝑥 < 𝑏𝑖 𝐿𝑖 ∩

̃ min 𝐿 𝜇

where 𝑏𝑖 ≤ 𝑥 ≤ 𝑐,

(16)

̃ max = 𝜑 𝐿𝑖 ∩ 𝐿 𝛾 𝐿𝑖 ∩

̃ max 𝐿 𝛾

̃󸀠 ) = 𝑃𝛼 (𝐵 ̃󸀠 ) and 𝑃𝛽 (𝐴 ̃󸀠 ) = 𝑃𝛽 (𝐵 ̃󸀠 ) ∴ 𝐴 ̃󸀠 ≈ 𝐵 ̃󸀠 . (iii) 𝑃𝜇𝛼 (𝐴 𝜇 𝛾 𝛾

̃ max ≠ 𝜑, 𝐿𝑖 ∩ 𝐿 𝛾

Then the following ranking method is introduced in this paper.

where 𝑏𝑖 ≤ 𝑥 ≤ 𝑐.

Let 𝐿 1 and 𝐿 2 be two (Atanassov) intuitionistic fuzzy numbers then 𝐿 1 > 𝐿 2 if and only if the similarity ranking of membership function of 𝐿 1 is greater than the similarity ranking of membership function of 𝐿 2 and the similarity ranking of nonmembership function of 𝐿 1 is less than the similarity ranking of nonmembership function of 𝐿 2 . Definition 11 (graded mean integration representation for trapezoidal (Atanassov) intuitionistic fuzzy number [16]). The membership and nonmembership functions of trapezoidal (Atanassov) intuitionistic fuzzy numbers are defined by Definition 3 as follows:

𝑥 − 𝑎4 𝑤; 𝑎3 − 𝑎4

𝑎2 ≤ 𝑥 ≤ 𝑎4 ,

𝑅𝜇 (𝑥) =

𝑏1 ≤ 𝑥 ≤ 𝑏2 ,

𝑥 − 𝑏4 𝑤; 𝑏3 − 𝑏4

𝑏2 ≤ 𝑥 ≤ 𝑏4 .

𝑅𝛾 (𝑥) =

(𝑎2 − 𝑎1 ) ℎ , (ℎ) = 𝑎1 + 𝑤

𝑅𝜇−1

(𝑎4 − 𝑎3 ) ℎ , (ℎ) = 𝑎4 − 𝑤

1

̃𝜇 , 𝐵̃𝜇 ) = [∫ [𝐴 𝜇𝐿 (𝛼) − 𝐵𝜇𝐿 (𝛼)]2 𝑑𝛼 𝐷 (𝐴 0

1

2

1/2

+ ∫ [𝐴 𝜇𝑅 (𝛼) − 𝐵𝜇𝑅 (𝛼)] 𝑑𝛼] 0

, (20)

1

2

̃𝛾 , 𝐵̃𝛾 ) = [∫ [𝐴 𝛾𝐿 (𝛼) − 𝐵𝛾𝐿 (𝛼)] 𝑑𝛼 𝐷 (𝐴 1

2

+ ∫ [𝐴 𝛾𝑅 (𝛼) − 𝐵𝛾𝑅 (𝛼)] 𝑑𝛼] 0

Then 𝐿−1 and 𝑅−1 are inverse functions of functions 𝐿 and 𝑅, respectively. One has 𝐿−1 𝜇

̃ = {(𝑎1 , 𝑏1 , 𝑐1 , 𝑑1 ); Definition 12 (metric distance). Let 𝐴 (𝑎1󸀠 , 𝑏1 , 𝑐1 , 𝑑1󸀠 )} and 𝐵̃ = {(𝑎2 , 𝑏2 , 𝑐2 , 𝑑2 ); (𝑎2󸀠 , 𝑏2 , 𝑐2 , 𝑑2󸀠 )} be two (Atanassov) intuitionistic trapezoidal fuzzy numbers. Then the metric distance between 𝐴 and 𝐵 can be calculated as follows:

0

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𝑥 − 𝑏1 𝑤; 𝐿 𝛾 (𝑥) = 𝑏2 − 𝑏1

̃󸀠 = {(𝑎, 𝑏, 𝑐, 𝑑)(𝑒, 𝑏, 𝑓, 𝑔)} and 𝐵̃󸀠 = {(𝑎󸀠 , 𝑏, 𝑐, 𝑑󸀠 )(𝑒󸀠 , Let 𝐴 󸀠 𝑏, 𝑓, 𝑔 )} be any two (Atanassov) intuitionistic triangular fuzzy numbers; then ̃󸀠 ) > 𝑃𝛼 (𝐵 ̃󸀠 ) and 𝑃𝛽 (𝐴 ̃󸀠 ) > 𝑃𝛽 (𝐵 ̃󸀠 ) ∴ 𝐴 ̃󸀠 > 𝐵 ̃󸀠 ; (ii) 𝑃𝜇𝛼 (𝐴 𝜇 𝛾 𝛾

where 𝑐 < 𝑥 < 𝑑, 𝑎𝑖 < 𝑥 < 𝑏𝑖

𝑎1 ≤ 𝑥 ≤ 𝑎2 ,

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̃󸀠 ) < 𝑃𝛼 (𝐵 ̃󸀠 ) and 𝑃𝛽 (𝐴 ̃󸀠 ) < 𝑃𝛽 (𝐵 ̃󸀠 ) ∴ 𝐴 ̃󸀠 < 𝐵 ̃󸀠 ; (i) 𝑃𝜇𝛼 (𝐴 𝜇 𝛾 𝛾

≠ 𝜑,

𝑥 − 𝑎1 𝑤; 𝐿 𝜇 (𝑥) = 𝑎2 − 𝑎1

𝑎1 + 2𝑎2 + 2𝑎3 + 𝑎4 , 6

𝑏 + 2𝑏2 + 2𝑏3 + 𝑏4 𝑃𝛾 (𝐴) = 1 . 6

≠ 𝜑,

̃ max ) SD (𝐿 𝑖 , 𝐿 𝛾 0, { { { { 2 { {1 (𝑑 − 𝑎𝑖 ) { { , { { { { 2 (𝑑 − 𝑐) + (𝑏𝑖 − 𝑎𝑖 ) ={ { { { { { 1 { { [(𝑑 − 𝑎𝑖 ) + (𝑐 − 𝑏𝑖 )] , { { {2 { {

Then the graded mean integration representation of membership function and nonmembership function is

̃ min = 𝜑 if 𝐿 𝑖 ∩ 𝐿 𝜇

1/2

,

where 𝐴 𝜇𝐿 (𝛼), 𝐵𝜇𝐿 (𝛼), 𝐴 𝜇𝑅 (𝛼), and 𝐵𝜇𝑅 (𝛼) are 𝛼-cut intervals ̃ and 𝐵, ̃ respectively, and 𝐴 𝛾𝐿 (𝛼), of membership function of 𝐴 𝐵𝛾𝐿 (𝛼), 𝐴 𝛾𝑅 (𝛼), and 𝐵𝛾𝑅 (𝛼) are 𝛼-cut intervals of nonmem̃ and 𝐵, ̃ respectively. bership function of 𝐴 In order to rank (Atanassov) intuitionistic fuzzy numbers, let us take the (Atanassov) intuitionistic trapezoidal fuzzy ̃ and 0 number 𝐵̃ = 0; then the metric distance between 𝐴 is calculated as follows: 1

̃𝜇 , 0) = [∫ [𝐴 𝜇𝐿 (𝛼)]2 𝑑𝛼 𝐷 (𝐴 0

1

2

+ ∫ [𝐴 𝜇𝑅 (𝛼)] 𝑑𝛼] 0

1/2

Journal of Applied Mathematics

5

1

2

3.1. Notations. The notations that will be used throughout the paper are

= [∫ [𝑎1 + (𝑏1 − 𝑎1 ) 𝛼] 𝑑𝛼 0

1

1/2

2

𝑁: the set of all nodes in a project network,

+ ∫ [𝑑1 − (𝑑1 − 𝑐1 ) 𝛼] 𝑑𝛼] 0

=[

𝑎12

+

𝑏12

+

𝑐12

+

𝑑12

− 𝑎1 𝑏1 − 𝑐1 𝑑1

3 1

EST: earliest starting time,

1/2

]

̃𝜇𝑖𝑗 : the activity between nodes 𝑖 and 𝑗 for member𝐴 ship function,

,

̃𝛾𝑖𝑗 : the activity between nodes 𝑖 and 𝑗 for nonmem𝐴 bership function,

2

̃𝛾 , 0) = [∫ [𝐴 𝛾𝐿 (𝛼)] 𝑑𝛼 𝐷 (𝐴 0

1

2

̃ 𝜇𝑖𝑗 : (Atanassov) intuitionistic fuzzy activity time IFET for membership function,

1/2

+ ∫ [𝐴 𝛾𝑅 (𝛼)] 𝑑𝛼] 0

1

̃ 𝛾𝑖𝑗 : (Atanassov) intuitionistic fuzzy activity time IFET for nonmembership function,

2

= [∫ [𝑎1󸀠 + (𝑏1 − 𝑎1󸀠 ) 𝛼] 𝑑𝛼 0

+∫

1

0

[𝑑1󸀠

− (𝑑1󸀠

2

̃ 𝜇𝑗 : the earliest starting (Atanassov) intuitionistic IFES fuzzy time for membership function of node 𝑗,

1/2

− 𝑐1 ) 𝛼] 𝑑𝛼]

𝑎󸀠2 + 𝑏12 + 𝑐12 + 𝑑1󸀠2 + 𝑎1󸀠 𝑏1 + 𝑑1󸀠 𝑐1 =[ 1 ] 3

̃ 𝛾𝑗 : the earliest starting (Atanassov) intuitionistic IFES fuzzy time for nonmembership function of node 𝑗,

1/2

. (21)

Here, ̃ > 𝐵̃ iff 𝐷 (𝐴, ̃ 0) > 𝐷 (𝐵, ̃ 0) . 𝐴

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3. Calculating Intuitionistic Fuzzy Time Values and Critical Path Analysis in (Atanassov) Intuitionistic Fuzzy Project Network [17] An (Atanassov) intuitionistic fuzzy project network is an acyclic directed graph, where the vertices represent events, and the direct edges represent the activities to be performed in a project network. An (Atanassov) intuitionistic fuzzy ̃𝜇 , ̃𝜇 , 𝐴 ̃𝛾 )(𝑇 ̃ = {(̃V𝜇 , ̃V𝛾 )(𝐴 project network is represented by 𝑁 ̃𝛾 )}. 𝑇 Let ̃V = (̃V𝜇 , ̃V𝛾 ) = {(̃V𝜇1 , ̃V𝜇2 , . . . , ̃V𝜇𝑛 )(̃V𝛾1 , ̃V𝛾2 , . . . , ̃V𝛾𝑛 )} be the set of (Atanassov) intuitionistic fuzzy vertices, where (̃V𝜇1 , ̃V𝛾1 ) and (̃V𝜇𝑛 , ̃V𝛾𝑛 ) are the tail and head events of the project, and each (̃V𝜇𝑖 , ̃V𝛾𝑖 ) belongs to some path from ̃⊂ 𝑉 ̃×𝑉 ̃ be the set of a directed (̃V𝜇1 , ̃V𝛾1 ) to (̃V𝜇𝑛 , ̃V𝛾𝑛 ). Let 𝐴 ̃ edge 𝐴 = {(𝑎𝜇𝑖𝑗 , 𝑎𝛾𝑖𝑗 ) = (V𝜇𝑖 , V𝜇𝑗 )(V𝛾𝑖 , V𝛾𝑗 )/for V𝜇𝑖 , V𝜇𝑗 , V𝛾𝑖 , ̃ that represents the activities to be performed in V𝛾𝑗 ∈ 𝑉} the project. Activity (̃ 𝑎𝜇𝑖𝑗 , 𝑎̃𝛾𝑖𝑗 ) is then represented by one and only one arrow with tail event (̃V𝜇𝑖 , ̃V𝛾𝑖 ) and a head event 𝑎𝜇𝑖𝑗 , 𝑎̃𝛾𝑖𝑗 ), an (Atanassov) intu(̃V𝜇𝑗 , ̃V𝛾𝑗 ). For each activity (̃ ̃ is defined as the (Atanassov) itionistic fuzzy number ̃𝑡𝑖𝑗 ∈ 𝑇 intuitionistic fuzzy time required for the completion of (̃ 𝑎𝜇𝑖𝑗 , 𝑎̃𝛾𝑖𝑗 ). A critical path is the longest path from the initial event (̃V𝜇1 , ̃V𝛾1 ) to the terminal event (̃V𝜇𝑛 , ̃V𝛾𝑛 ) of the project network, and an activity (̃ 𝑎𝜇𝑖𝑗 , 𝑎̃𝛾𝑖𝑗 ) on a critical path is called a critical activity.

̃ 𝜇𝑖 : the latest finishing (Atanassov) intuitionistic IFLF fuzzy time for membership function of node 𝑖, ̃ 𝛾𝑖 : the latest finishing (Atanassov) intuitionistic IFLF fuzzy time for nonmembership function of node 𝑖, ̃ 𝜇𝑖𝑗 : the total slack (Atanassov) intuitionistic fuzzy IFTS ̃𝜇𝑖𝑗 , time of 𝐴 ̃ 𝛾𝑖𝑗 : the total slack (Atanassov) intuitionistic fuzzy IFTS ̃𝛾𝑖𝑗 , time of 𝐴 𝑃𝑘 : the 𝑘th (Atanassov) intuitionistic fuzzy path, TF: total float, IFCPM (𝑃𝑘 ): the total slack (Atanassov) intuitionistic fuzzy time of path 𝑃𝑘 in a project network, 𝑇𝑖𝑗 : the (Atanassov) intuitionistic fuzzy activity time. Property 1 (forward pass calculation). Forward pass calculations are employed to calculate the earliest starting time (EST) in the project network. Set the initial node to zero for starting, ̃ 𝛾1 = (0, 0, 0, 0), ̃ 𝜇1 = (0, 0, 0, 0) and IFES that is, IFES ̃ 𝜇𝑖 (+) FET ̃ 𝜇𝑖𝑗 ] , ̃ 𝜇𝑗 = Max [IFES IFES 𝑖

𝑖 = number of preceding nodes, 𝑗 ≠ 1, 𝑗 ∈ 𝑁, ̃ 𝛾𝑖 (+) FET ̃ 𝛾𝑗 = Min [IFES ̃ 𝛾𝑖𝑗 ] , IFES

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𝑖

𝑖 = number of preceding nodes, 𝑗 ≠ 1, 𝑗 ∈ 𝑁. Ranking value is utilized to identify the maximum and minimum value.

6

Journal of Applied Mathematics Earliest finishing time for membership function is ̃ 𝜇𝑖𝑗 = [earliest starting intuitionistic fuzzy time for membership function IFET + (Atanassov) intuitionistic fuzzy activity time] .

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Earliest finishing time for nonmembership function is ̃ 𝛾𝑖𝑗 = [earliest starting intuitionistic fuzzy time for nonmembership function IFET + (Atanassov) intuitionistic fuzzy activity time] .

Property 2 (backward pass calculation). Backward pass calculations are employed to calculate the latest finishing time ̃ 𝜇𝑛 and ̃ 𝜇𝑛 = IFES (LFT) in the project network. Set IFLF ̃ 𝛾𝑛 . One has ̃ 𝛾𝑛 = IFES IFLF ̃ 𝜇𝑖𝑗 ] , ̃ 𝜇𝑖 = Min [IFLF ̃ 𝜇𝑗 (−) IFET IFLF 𝑗

𝑖 ≠ 𝑛, 𝑖 ∈ 𝑁, 𝑗 = number of succeeding nodes,

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̃ 𝛾𝑖𝑗 ] , ̃ 𝛾𝑗 (−) IFET ̃ 𝛾𝑖 = Max [IFLF IFLF 𝑗

𝑖 ≠ 𝑛, 𝑖 ∈ 𝑁, 𝑗 = number of succeeding nodes. (26) Ranking value is utilized to identify the maximum and minimum value. Latest starting time for membership function is

̃ 𝜇𝑖𝑗 = [latest finishing intuitionistic fuzzy time for membership function (−) IFLS (Atanassov) intuitionistic fuzzy activity time] .

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Latest finishing time for nonmembership function is ̃ 𝛾𝑖𝑗 = [latest finishing intuitionistic fuzzy time for nonmembership function (−) (Atanassov) IFLS intuitionistic fuzzy activity time] .

̃𝜇𝑖𝑗 , 𝐴 ̃𝛾𝑖𝑗 ), 𝑖 < Property 3 (total float (TF)). For the activity (𝐴 𝑗, total (Atanassov) intuitionistic fuzzy slack is

IFCPM (𝑝𝛾𝑘 ) =

∑ 1≤𝑖