coimbatore- sr. elizabeth & mrs. sujatha
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ppt of coimbatoreTRANSCRIPT
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
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
NEW DEFINITIONSNEW DEFINITIONS
19
Maximum operation for two interval numbers in mean- width notation
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
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
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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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