Coimbatore- Sr. Elizabeth & Mrs. Sujatha

Fuzzy Critical Path in

a Network

L. SujathaAsst. Prof. Dept. of Mathematics Auxilium College, Vellore – 6

Dr. (Sr.) Elizabeth SebastianVice Principal ,Auxilium College, Vellore – 6

ABSTRACTABSTRACTFour different procedures are presented to obtain the fuzzy critical path in an acyclic network.

The optimal solution obtained through the procedures proposed in this paper coincides with the existing earlier results.

Keywords: Network (Graph), Fuzzy trapezoidal numbers, α–cut interval numbers, Signed distance measure, Centroid measure, Metric distance, Ranking degree, Mean-Width notation of -cut interval numbers, Critical path, Decision Maker.

INTRODUCTIONINTRODUCTION

Critical Path - one of the most important

problem

Wide range of applications in planning and

scheduling projects.

Organization of the paper

BASIC DEFINITIONS1

3

Chen and Cheng’s membership function

4

A = (a1, a2, a3, a4; λ ), 0 < λ ≤ 1, a1< a2 < a3 < a4

2

λ = 1

BASIC DEFINITIONS

-Cut interval (Chen and Cheng (2005))

5

6

BASIC DEFINITIONS

8

9

10

11

Operations on -Cut interval (Kwang (2005))

7

BASIC DEFINITIONS

Signed Distance of ‘b’ Closed interval [a,b]

measured from ‘0’ - F. T. Lin(2001)

13

12

BASIC DEFINITIONS Interval numbers in terms of mean - width

notation (Nayeem and Pal (2009)) 14

15

16

17

BASIC DEFINITIONS

Addition operation in mean-width notation (Nayeem and Pal (2009))

18

NEW DEFINITIONSNEW DEFINITIONS

19

Maximum operation for two interval numbers in mean- width notation

NEW DEFINITIONSNEW DEFINITIONS

21

22

20

The signed distance of α-cut interval number

NEW DEFINITIONSNEW DEFINITIONS

23

24

25

Mean and Standard deviation of α-cut interval number

NEW DEFINITIONSNEW DEFINITIONS

26

27

Centroid measure for α-cut interval number

NEW DEFINITIONSNEW DEFINITIONS

29

30

28

Metric distance

33

31

32

34

Ranking degree

NEW DEFINITIONSNEW DEFINITIONS

35

36

37

α-cut interval numbers in terms of mean-width notation

NEW DEFINITIONSNEW DEFINITIONS

38

39

40

Acceptability Index

NEW DEFINITIONSNEW DEFINITIONS

43

42

41

NEW DEFINITIONSNEW DEFINITIONS

Area Measure (Elizabeth and Sujatha (2011))

Procedure for Fuzzy Critical Path Problem

Forward Pass Calculation

Backward Pass Calculation

44

47

46

45

48Total Float

Step 1: Construct a network G(V,E) , arc lengths or edge weights are taken as trapezoidal fuzzy numbers which in turn converted in terms of α-cut interval numbers

Step 2: Calculate Earliest starting time according to forward pass calculation

Step 3: Calculate Earliest finishing time using

Step 4: Calculate Latest finishing time according to backward pass calculation

Step 5: Calculate Latest starting time using

45

47

PROCEDURE 1

PROCEDURE 1Step 6: Calculate Total Float using

Step 7: Calculate Centroid measure or Signed distance measure for each activity using

Step 8: If Centroid measure = Signed distance measure = 0, those activities are called critical activities and the corresponding path is the critical path.

48

27

Consider, a civil building construction project:

1 - Excavation and Foundation, 2 - Columns and Beams, 3 – Brick Work, 4 – Flooring,5 – Roof concrete, 6 – Plastering, 7 – Painting.

EXAMPLE : 1EXAMPLE : 1

Figure 1 Network of Civil

Project

RESULTS OF THE NETWORK BASED ON CENTROID MEASURE

RESULTS OF THE NETWORK BASED ON CENTROID MEASURE

Path p3 : 1-3-4-7 is identified as the critical path

RESULTS OF THE NETWORK BASED ON AREA MEASURE (Verification)

Path p3 : 1-3-4-7 is identified as the critical path

Step 1: is same as in procedure 1.

Step 2: Calculate all possible paths pi , i=1 to n from source vertex ‘s’ to the destination vertex ‘d’ and the corresponding path lengths Li , i=1 to n using addition operation and set

Step 3: Calculate metric distance for each possible path lengths D (Li , 0) for i = 1 to n using

Step 4: The path having the maximum metric distance is identified as the critical path

PROCEDURE 2

29

8

Step 1 and Step 2 are same as in procedure 2.

Step 3: Calculate Lmax using and set

Step 4: Calculate Ranking degree using for each possible path lengths Li that is, i = 1to n

Step 5: The path having the minimum Ranking degree is identified as the critical path

10

33

PROCEDURE 3

Step 1 and Step 2 : are same as in procedure 2.

Step 3 :The path lengths Li, i=1 to n given in terms of α-cut interval numbers are converted into mean-width notation using and set

Step 4:Calculate Lmax in terms of mean-width notation using and set

Step 5: Calculate Acceptability index between Li and Lmax using

Step 6:The path having the minimum Acceptability Index is identified as the critical path

36

39

PROCEDURE 4

40

EXAMPLEEXAMPLEPaths D(Li, 0) Ranking

pp11 : 1-2-4-7 : 1-2-4-7 201.3 3

p2 : 1-2-5-7 211.5 2

p3 : 1-3-4-7 227.5 1

p4 : 1-3-6-7 183.7 4

Paths R(Lmax≥Li) Ranking

pp11 : 1-2-4-7 : 1-2-4-7 17.517.5 3

p2 : 1-2-5-7 10 2

p3 : 1-3-4-7 0 1

p4 : 1-3-6-7 30.5 4

Metric Distance Ranking Degree

EXAMPLEEXAMPLEPaths A(Li<Lmax) Ranking

pp1 1 : 1-2-4-7 : 1-2-4-7 0.61 3

p2 : 1-2-5-7 0.37 2

p3 : 1-3-4-7 0 1

p4 : 1-3-6-7 0.98 4

Acceptability Index

RESULTS AND DISCUSSIONS

• In this paper some procedures are developed to find the optimal paths in a fuzzy weighted graph

• Coincides with the existing earlier result (Soltani and Haji (2007))

• It is an alternative way to identify the critical path in fuzzy sense.

• Planning and controlling the complex projects.

• Linguistic variables for activity durations, whereas this

specification do not exist in crisp models.

• Fuzzy models are more effective in determining the critical

path in a real project network.

CONCLUSION

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REFERENCESREFERENCES10. Lin, F.T.(2001). A shortest path network problem in a fuzzy environment,

IEEE international fuzzy system conférence, 1096-1100.

11. Mon, D.L., Cheng, C.H. & Lu, H.C. (1995). Application of fuzzy distributions on project management. Fuzzy Sets and Systems, 73, 227-234.

12. Nayeem, S.M.A., & Pal, M. (2009). Near-shortest simple paths on a network with imprecise edge weights, Journal of Physical Sciences, 13, 223-228.

13. Soltani, A., & Haji, R. (2007). A project scheduling method based on fuzzy

theory, Journal of Industrial and Systems Engineering, 1(1), 70-80.

THANK YOUTHANK YOU