Comparision of Critical Path Problem under Fuzzy Environment with Conventional Method

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Comparision of Critical Path Problem under Fuzzy Environment with Conventional Method

^*

C. Rajendran

¹

and M. Ananthanarayanan

²

1

Assistant Professor, Department of Mathematics,

Govt. Arts College for Men, Nandanam, Chennai-35, INDIA.

email: [email protected]

2

Associate Professor,

P.G. Department of Mathematics, A.M Jain College Chennai-114, INDIA.

(Received on: July 31, 2018) ABSTRACT

In this paper, we propose a new approach to critical path analysis in a project network, whose activity timing are uncertain. The uncertain parameters in the project network are represented by fuzzy numbers. We used ranking method and defuzzification formula on fuzzy numbers. The numerical example is given and we compared the fuzzy crtical path with conventional method.

Keywords: Fuzzy Critical path method, Fuzzy project network, Fuzzy numbers, Fuzzy arithmetic, trapezoidal fuzzy numbers, Magnitude measure, Area measure, critical path.

1. INTRODUCTION

A project network is defined as a set of activities that must be performed according precedence constraint stating which activities must start after the completion specified other activities. The critical path is the one from the start of the project to finish of project where the slack times are all zeros. The operation time for each activity in the fuzzy project network is characterized as a positive triangular fuzzy numbers. The main purpose of critical path method (CPM) is to evaluating project performance and to identifying the critical activities on the critical path so that the available resources could be utilized on these activities in the project network in order to reduce the project completion time.

In Accordance with critical path method the forward pass yields the fuzzy earliest start

and earliest finish times. He backward pass in performed to calculate the fuzzy latest start and

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latest finish time

^1,2

. The longest path is called the critical path in the network. CPM has been used in business management, factory production etc

^3,4

extended the fuzzy arithmetic operations model to compute EST of each activity in a project network

^5,6

used fuzzy arithmetic operations to compute the EST of each activity in aproject network.

Zadeh

⁷

introduced an alternative way to deal with imprecise data to employ the concept of fuzziness Dubois et al.

⁸

presented a project network defined as a set of activities that must be performed according to procedure constraints starting which must start after completion of other specified activities.

In this paper we presents three difference approaches to find the critical path in fuzzy project network. First we to perform critical path analysis in fuzzy environment and others are by conventional method.

2. PRELIMINARIES

The aim of this section is to present some notations and results whch are useful in our further consideration.

2.1. Fuzzy Set.

Let 𝑋 be a set. A fuzzy set 𝐴 on 𝑋 is defined to be a function A: 𝑋 → [0,1] or 𝜇

𝐴

: 𝑋 → [0, 1].

Equivalently, A fuzzy set 𝐴 is defined to be the class of objects having the following representation: = {(𝑥,𝜇

𝐴

(𝑥), 𝑥 ∈ 𝑋)} where 𝜇

𝐴

: 𝑋 → [0,1] is a function called the membership function of 𝐴.

2.2. Fuzzy Number. The fuzzy number 𝐴 is a fuzzy set whose membership function satisfies the following conditions:

(1) 𝜇

𝐴

(𝑥) is piecewise continuous.

(2) A fuzzy set 𝐴 of the universe of discourse 𝑋 is convex.

(3) A fuzzy set of the universe of discourse 𝑋 is called a normal fuzzy set if 𝑥

𝑖

∈ 𝑋 exists.

2.3. Triangular Fuzzy numbers

A fuzzy number with membership function in the form

µ

A

(x)=

{

𝑥−𝑎

𝑏−𝑎

, 𝑎 ≤ 𝑥 ≤ 𝑏

𝑐−𝑥

𝑐−𝑏

, 𝑏 ≤ 𝑥 ≤ 𝑐 0, 𝑜𝑡h𝑤𝑖𝑠𝑒

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is called a triangular fuzzy number A= (a,b,c).

2.4 Trapezoidal Fuzzy Number.

A fuzzy number A in R is said to be a Trapezoidal fuzzy number if its membership function or

𝜇

𝐴

(x):R → [0, 1] has the following characteristics

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µ

A

(x)=

{

𝑥−𝑎₁

𝑎₂−𝑎₁

, 𝑎

₁

≤ 𝑥 ≤ 𝑎

₂

1, 𝑎

₂

≤ 𝑥 ≤ 𝑎

₃

𝑥−𝑎₄

𝑎₃−𝑎₄

, 𝑎

₃

≤ 𝑥 ≤ 𝑎

₄

0, 𝑜𝑡h𝑤𝑖𝑠𝑒

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We denote this trapezoidal fuzzy numbers by A=(a

1

,a

2

,a

3

,a

4

). We use F (R) to denote the set of all trapezoidal fuzzy number

2.5. Arithmetic operation on trapezoidal fuzzy numbers

⁹

Let 𝐴 =(𝑎

1

, 𝑎

2

, 𝑎

3

,𝑎

4)

and B= (𝑏

1

, 𝑏

2

, 𝑏

3

, 𝑏

4)

be any two trapezoidal numbers in F(R). The arithmetic operations on A and B are defined as

A * B ={ a

i

*b

i

/ a

i

∈A , b

i

∈B} (3)

where * ∈[+, -,*,÷]

In particular, for any two trapezoidal fuzzy number 𝐴 =(𝑎

1

, 𝑎

2

, 𝑎

3

,𝑎

4)

and B= (𝑏

1

, 𝑏

2

, 𝑏

3

, 𝑏

4)

we defines

Addition: A + B = (𝑎

1

, 𝑎

2

, 𝑎

3

,

4

,) + (𝑏

1

, 𝑏

2

, 𝑏

3

, 𝑏

4

,)

= (𝑎

1

+b

1

, a

2

+b

2

, a

3

+b

3

, a

4

+b

4

) (4) Subtraction: A- B= ( 𝑎

1

, 𝑎

2

, 𝑎

3

,𝑎

4

) - (𝑏

1

, 𝑏

2

, 𝑏

3

, 𝑏

4

)

= (𝑎

1

-b

1

, a

2

-b

2

, a

3

-b

3

, a

4

-b

4

) (5) 3. FUZZY CRITICAL PATH ANALYSIS

Step 1: Construct network diagram according to Fulkerson rule

Step 2: Calculate Earliest starting time according to forward pass calculation i.e., E

j

= Max

i

{E

i

+ D ~

_ij

}, i = no of preceding nodes Step 3: Calculate Earliest finishing time EFT = EST + NT

Step 4: Calculate Latest Starting time according to backward pass calculation.

i.e., L

i

= Min

j

{L

i

 D ~

_ij

}, j = number of succeeding nodes.

Step 5: Calculate the Latest Finishing time LFT = LST  NT Step 6: Calculate Total Floating time TFT = LFT  EFT Step 7: If TFT = 0 those activities are called critical activities.

3.1 Procedure for Fuzzy critical path Problem Forward Pass Calculation:

Forward pass calculations are employed to calculate the earliest starting time (E𝑆̃T) in the

project network.

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E𝑺̃T

j

= Max

i

(E𝑆̃T + t

ij

) , i= number of proceeding nodes Earliest finishing time E𝐹̃T= E𝑆̃T

j

+ t

ij

Backward Pass Calculation:

Backward pass calculation are employed to calculate the latest finishing time (LFT) in the project network

L𝐹̃T

i

= Min

j

(L𝐹̃T- t

ij

) , j = number of succeeding nodes Latest Starting Time LST = LFT - t

ij

Total float:

T𝐹̃T =L𝐹̃T –E𝐹̃T or L𝑆̃T -E𝑆̃T

If T𝐹̃T =0, those activity are called critical activities and the corresponding path is the fuzzy critical path

4. NUMERICAL EXAMPLE

Suppose that there is a project network with the set of fuzzy events 𝑣 = {1, 2, 3, 4, 5, 6} the fuzzy activity time for each activity is shown in table 1(all duration are in hours)

Table 1: Activity duration of each activity in a fuzzy project network.

Activity Fuzzy activity time

1-2 (5,8,12,15)

1-3 (10,25,35,40)

1-5 (15,20,26,30)

2-4 (14,19,29,34)

2-5 (18,23,27,30)

3-4 (22,26,32,37)

3-6 (20,24,30,35)

4-5 (25,28,32,38)

4-6 (35,38,42,45)

5-6 (32,37,43,48)

Figure 1: Fuzzy Project network

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Table 2: Using the fuzzy activity duration calculation of total float for each activity in fuzzy project network and determine the critical path

Activity Fuzzy activity time

EST EFT LST LFT TF

1-2 (5,8,12,15) (0,0,0,0) (5,8,12,15) (13,24,26,28) (18,32,38,43) (13,24,26,28) 1-3 (10,25,35,40) (0,0,0,0) (10,25,35,40) (0,0,0,0) (10,25,35,40) (0,0,0,0) 1-5 (15,20,26,30) (0,0,0,0) (15,20,26,30) (42,59,73,85 ) (57,79,99,115 ) (42,59,73,85) 2-4 (14,19,29,34) (5,8,12,15) 19,27,41,49 ) (18,32,38,43 ) (32,51,67,77 ) (13,24,26,28) 2-5 (18,23,27,30) (5,8,12,15) (23,31,39,45 ) ( 39,56,72,85) (57,79,99,115 ) (34,48,60,70) 3-4 (22,26,32,37) (10,25,35,40) ( 32,51,67,77) (10,25,32,40 ) (32,51,67,77 ) (0,0,0,0) 3-6 (20,24,30,35) (10,25,35,40) (30,49,65,75 ) (69,92,112,128 ) ( 89,116,142,163) (50,67,77,88) 4-5 (25,28,32,38) (32,51,67,77) ( 57,79,99,115) (32,51,67,77 ) ( 57,79,99,115) (0,0,0,0) 4-6 (35,38,42,45) (32,51,67,77) (67,89,109,122 ) ( 54,78,100,118) (89,116,142,163) (22,27,33,41) 5-6 (32,37,43,48) (57,79,99,115) ( 89,116,142,163) ( 57,79,99,115) (89,116,142,163 ) (0,0,0,0)

Here path 1-3-4-5-6 is identifies as the fuzzy critical path.

5. DESCRIPTION OF THE MODEL

Trapezoidal Fuzzy numbers are converted into expected time(normal time). These expected time treated as the time between the nodes and fuzzy critical path is calculated by using conventional method.

Definition 5.1: [10] A trapezoidal fuzzy number Ã= (a

1

,a

2

,a

3

,a

4

) can be approximated as a fuzzy number S(µ,σ), µ denotes the mean of Ã, σ denotes the standard deviation of Ã as the membership function of Ã interms of mean and the standard deviation is defined as follows.

µ

Ã(x)

= = {

𝑥−(µ−𝜎)

𝜎

, 𝑖𝑓 µ − 𝜎 ≤ 𝑥 ≤ µ

(µ+𝜎)−𝑥

𝜎

, 𝑖𝑓 µ ≤ 𝑥 ≤ µ + 𝜎

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Where µ and σ calculated as follows σ =

^2(𝑎⁴^−𝑎¹^)+𝑎³^−𝑎³

4

, µ=

^𝑎¹^+𝑎²^+𝑎₄ ³^+𝑎⁴

Definition 5.2

Let Ã = (a, b, c, d) be a trapezoidal fuzzy numbers such that a<b<c<d. It is converted to triangular fuzzy numbers as Ã = (a, b

1

=

^𝑏+𝑐

2

,d) such that a<b

1

<d. The magnitude measures of the triangular fuzzy number Ã =(a,b

1

,d) with parametric form

Ã

α

=[A

L

(α ),A

R

( α)]=[(b

1

-a) α+ a, d-(d-b

1

)α]

is given by

Mag(Ã) = ∫

₀¹

^A^L^{(α) + A}^R^{(α) + b1}

2

αdα , αϵ[0,1]

=

^{a + 7b1+d}

12

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Definition 5.3:[11]

Let Ã = (a,b,c,d; λ) be a level λ trapezoidal fuzzy numbers such that a<b<c<d, 0<λ<1.

If (c-b) ǂ λ, the area measure of Ã =

^{𝜆(𝑐−𝑏−𝑎+𝑑)}

2

and if ( c-b) = λ the area measures of Ã=

𝜆(𝑏−𝑎+2𝜆+𝑑−𝑐) 2

5.4 Verification using area measures

We consider the network diagram (figure 1) and assume the are length as trapezoidal fuzzy numbers as given in the numerical example. Now the expected times in terms of trapezoidal fuzzy numbers are defuzzified using area measure (let λ=1) for all activities in the network diagram. Then the usual crtical path method is followed to identify the fuzzy critical path (refer table 4)

Table 3: Results of the fuzzy project network based on magnitude measures Activity Fuzzy activity time

(Trapezoidal fuzzy numbers)

Fuzzy activity time converted in terms of triangular fuzzy numbers

Defuzzified Activity time using magnitude

1-2 (5,8,12,15) (5,10,15) 7.5

1-3 (10,25,35,40) (10,30,40) 21.67

1-5 (15,20,26,30) (15,23,30) 17.17

2-4 (14,19,29,34) (14,24,34) 18

2-5 (18,23,27,30) (18,25,30) 18.58

3-4 (22,26,32,37) (22,29,37) 21.83

3-6 (20,24,30,35) (20,27,35) 20.33

4-5 (25,28,32,38) (25,30,38) 22.75

4-6 (35,38,42,45) (35,40,45) 30

5-6 (32,37,43,48) (32,40,48) 30

Figure 2: Project network 3 17.17 18.58 22.75 30

7.5 18 30

21.67 21.83 20.33

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From the above calculation we can identified the critical path 1-3-4-5-6.

Table 4: Results of the fuzzy project network based on Area measures Activity Fuzzy activity time(trapezoidal

fuzzy numbers)

Defuzzified activity time using area measures

1-2 (5,8,12,15) 7

1-3 (10,25,35,40) 20

1-5 (15,20,26,30) 10.5

2-4 (14,19,29,34) 15

2-5 (18,23,27,30) 8

3-4 (22,26,32,37) 10.5

3-6 (20,24,30,35) 10.5

4-5 (25,28,32,38) 8.5

4-6 (35,38,42,45) 7

5-6 (32,37,43,48) 11

Here path 1-3-4-5-6 is identified as critical path.

RESULT AND DISCUSSION

In this paper various ranking methods are applied to determine the critical path under fuzzy environment. It was found the result (critical path) obtained through three different procedure remains the same. Hence the procedures developed in this paper are alternative methods to get the fuzzy critical path

CONCLUSION

In this paper we have computed total float time for each path in the fuzzy project network to find the critical path for a given fuzzy project network. A numerical example has particularly provided to explain the ranking method defuzzification method for fuzzy numbers comparing these method the result are presented in the ranking measures which help the decision makers to decide the best possible critical path in fuzzy environment.

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