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First PhaseANALYSIS AND DEVELOPMENT

Project Network analysis by using FPERTmethod adopting activity operation times and

probability distribution.

Presented by:SUSHANTA KUMER ROY

Supervisor: PROF. DR. MD. SHARIF UDDINJahangirnagar University

email: [email protected]

July 8, 2015

Presented by:SUSHANTA KUMER ROY PERT IN FUZZY ANALYSIS


Overview

PERT technique has so many drawbacks; the main one, is toobtaining the three times estimates. In real situation, operationstimes of activities in a project network may be difficult to defineand estimate exactly. However, it is very important to be operationtimes more accurate. In this presentation,I will present a new andeffective approach to make operation times more accurate than theconventional method,by using Fuzzy activity times and probability.Triangular fuzzy numbers are used to express the operation timesfor all activities in the project network.Also a new effectivetechnique has to establish for checking the criticality of the activityand new FPERT (Fuzzy PERT) method .Finally, an example is be presented to verify the FPERT method

using the proposed effective technique.



Background

Fuzzy PERT was originated by Chanas and Kamburowski [1].Other authors who used the fuzzy pert are: Chanas[2], Gazdik[3],Lootsma[4], Buckley[5], MacCahon[6] and many like them .Chanas[5] proposed approach can describes the total duration timeof project network via a membership function which is completelyconserves all the fuzziness of activity times, and the critical pathsunder different possibility levels can also be obtained. Clearly,when activity times in a project network are fuzzy, the totalduration time needed to complete the project will be fuzzy also. Inour proposed method, able to calculate fuzzy activity time for eachactivity more accurately than conventional method, also the fuzzytotal duration of project Network. The basic idea is to apply herethe combination of concepts of the α cuts (or α level sets), Zadehsextension principle [7] and our proposed method.



Historical Development of Fuzzy in Project Management

1 Prade (1979) : (FPERT ) 17 node network model for schedulingacademic programs.

2 Chanas and Kamburowski (1981) : (FPERT) 11 activity, 9 nodenetwork.

3 Dubois and Prade (1985) : ( FPERT ) tutorial on fuzzy PERT.

4 Kaufmann and Gupta (1988) : ( Fuzzy CPM ) tutorial on fuzzyCPM.

5 McCahon and Lee (1988) : ( Fuzzy PERT ) triangular activitytimes.

6 Lootsma (1989) : ( Fuzzy PERT ) compares stochastic PERTand fuzzy PERT.

7 Buckley (1989) : ( Fuzzy PERT ) discrete and continuous pos-sibility distributions.

8 DePorter and Ellis (1990) : (Fuzzy CPM) project crashing for-mulation.



Historical Development of Fuzzy in Project Management

9 McCahon (1993) : (Fuzzy PERT ) compares fuzzy network andPERT over four basic network configurations of 4 to 8 activities.

10 Nasuation (1994): (Fuzzy CPM) For study slack.

11 Hapke et al. (1994): ( Project scheduling Support System ) 53activity Network for resource allocation in software developmentstudies on fuzzy slack.

12 Lorterapong (1994): (Fuzzy CPM ) fuzzy resource constrainedproject scheduling.

13 Chang et al. (1995): ( Solution Procedure for fuzzy project )uses fuzzy Delphi method, also uses combine comparison andcomposite method.

14 Shipley et al. (1996): (Fuzzy PERT): 8 activity network forselling and producing a television commercial.



OUT LINE AND OBJECTIVE OF RESEARCH

My proposed method can serve the pursuit of calculation ofminimum time for completion of a large scale project. For meetingthis purpose, we have to follow the following steps sequentially:

Define three time estimates for any project by experienced ofproject manager.

Convert these times steps to fuzzy activity times.

α cuts should be followed in this case such that the three timeestimates will re-estimate in this case.

Now we apply Monte carlo simulation for creating pseudo pointsin between most optimistic and pessimistic time.

In this situation we calculate new time estimate for most opti-mistic time by using random generations so the time calculationfollow new estimates for three time estimates.

New adaptive FPERT method can followed in this step.



History of PERTPERT was developed by the US Navy for the planning and controlof the Polaris missile program and the emphasis was on completingthe program in the shortest possible time. In addition PERT hadthe ability to cope with uncertain activity completion times.

What is PERT ?

In PERT activities are shown as a network of precedence relation-ships using activity-on-arrow network construction

Multiple time estimates

Probabilistic activity times

USED IN

Project management - for non-repetitive jobs (research and devel-opment work), where the time and cost estimates tend to be quiteuncertain. This technique uses probabilistic time estimates.



Some Definitions:

1. Fuzzy Set:

If A is a function from X into the interval [0,1], then A is called afuzzy set.

2. α cuts of Fuzzy set :

If A is a fuzzy set, by α -cuts αε [0,1] the authors mean the sets A[ α ]=[x ε X: A(X) ≥ α].The α -cuts of A are close intervals. If A[α]=B[ α], ∀αε [0,1] for arbitrary fuzzy sets A and B, Then it holdsthat A=B .

3. Normalize Fuzzy Set:

The fuzzy set A is normalized if there exits for all t ε [0,1] andx1, x2εX ,A(tx1 + (1− t)x2) ≥ minA(x1),A(x2)



Some Definitions:

4. Support of Fuzzy Set:

The support of a fuzzy set A within a universal set X is the crisp setthat contains all the elements of X that have nonzero membershipgrades in A. Clearly the support of A is exactly the same as the α-cut for A=0. The support (A)=

⋃aε(0,1] A[α] = {x : A(x) > 0}.

5.Fuzzy Number:

If A be any fuzz set then A is a fuzzy member if the followingconditions hold:

i The fuzzy set A is normalized

ii The fuzzy set A is a convex

iii The fuzzy set A is upper semi continuous.

iv The support of A is compact.



Some Definitions:6. EXTENSION PRINCILPAL OF FUZZY SETS:

For any given function f: X → Y induces two functions f:F(X)→ F(Y) and f −1 : F (X ) → F(Y) which are defined by [f(A)](y):supx/y=f (X ),∀AεF (Y )and[f −1B(x) : B(f (x))]∀BεF (Y )Based on the Extension Principle any crisp function can be fuzzyfied.

7. FUZZY MEMBERSHIP FUNCTION

A membership function (MF) is a curve that defines how each pointin the input space is mapped to a membership value (or degreeof membership) between 0 and 1. The input space is sometimesreferred to as the universe of discourse, a fancy name for a simpleconcept.For any set X, a membership function on X is any functionfrom X to the real unit interval [0,1].Membership functions on X represent fuzzy subsets of X. The mem-bership function which represents a fuzzy setA is usually denoted byµA.



Some Membership functions



Frame Work of the Presentation

The method proposed in this presentation is described in theflowchart presented in below which shows the development of thepresentation.



Distribution of Operation Times

In this case we will use Monte Carlo Simulation which generatessome pseudo points in between optimistic and pessimistic time.Then we will able to plot a curve between the time of completionand the number of jobs completed in that time. These curves willbe any one of the following:

Fig: DistributionPresented by:SUSHANTA KUMER ROY PERT IN FUZZY ANALYSIS


Time Calculation Procedure

Step: 1

Define three time estimates for any project by Project manager.

The value of three time steps given by [17] are as follows:

Activity No. O Mo P Mean Time µ S.D σ

1 2.90 4.49 6.08 4.49 1.19

2 4.01 5.70 7.39 5.70 0.85

3 3.49 4.51 5.54 4.51 0.17

4 3.09 3.68 4.27 3.68 0.06

5 3.97 4.43 4.89 4.43 0.03

6 3.91 4.84 5.77 4.84 0.14

7 4.67 5.22 5.76 5.21 0.05

8 5.5 6.27 7.04 6.27 0.33

9 4.8 5.68 6.55 5.67 0.13



Phase:1 Reliable Time Calculation Phase

Step 2:

Value of the three time estimates after α cuts.(Not More the0.5ε(0, 1) )

Activity No. O Mo P Mean Time µ S. D. σ

1 2.40 4.49 5.58 4.49 0.43

2 3.51 5.70 6.89 5.70 0.49

3 2.99 4.51 5.04 4.51 0.27

4 2.59 3.68 3.77 3.68 0.07

5 3.47 4.43 4.39 4.43 0.04

6 3.41 4.84 5.27 4.84 0.16

7 4.17 5.22 5.26 5.21 0.06

8 5.00 6.27 6.54 6.27 0.11

9 4.30 5.68 6.05 5.67 0.14



Step 3:

Monte Carlo Simulation For Creating Pseudo Points in Distributions:

To generate normally distributed realizations of activity durations,we can use a two steps procedure [20]. First, we generateuniformly distributed random variables, in the interval from zero toone. For example, a general formula for random numbergeneration can be of the form:ui = fractional part of

[(π + ui−1)5

]where π = 3.14159265 and ui−1 was the previously generatedrandom number or seed number.µ= µx + s.sint with s = σx

√−2lnui−1

where, xk is the normal realization, µx is the mean of x , σx is thestandard deviation of x , and ui and ui−1 are the two uniformlydistributed random variable realizations.Here we use triangulardistribution to calculate mean and variance.



Step-4

Value after performing Monte Carlo Simulation

AC µ σ s= σx√−2lnu1 t= 2 πu2 s.sint µ

1 4.49 0.43 0.57 5.46 0.05 4.54

2 5.70 0.49 0.65 5.46 0.04 5.74

3 4.51 0.27 0.37 5.46 0.03 4.54

4 3.68 0.07 0.09 5.46 0.009 3.69

5 4.43 0.04 0.06 5.46 0.006 4.44

6 4.48 0.16 0.21 5.46 0.02 4.86

7 5.21 0.06 0.08 5.46 0.001 5.22

8 6.27 0.11 0.15 5.46 0.0002 6.27

9 5.67 0.14 0.19 5.46 0.00004 5.67



step-5

Update table for three activity times:

AC No. O Mo P σ

1 2.40 4.54 5.58 0.43

2 3.52 5.74 6.89 0.49

3 2.99 4.54 5.04 0.27

4 2.59 3.68 3.77 0.07

5 3.47 4.43 4.39 0.04

6 3.41 4.86 4.86 0.16

7 4.17 5.22 5.27 0.06

8 5.00 6.27 6.54 0.11

9 4.30 5.67 6.04 0.14



Phase: 2 Fuzzy PERT Calculation Phase

Step-6

Network Diagram for FPERT Method :

Fig: Network Diagram



Calculation by our proposed method

Step-7

Some of the notation here we introduced for our total calculation.These are as follows:ESi = Earliest start time of the activityEFi = Earliest finish time of the activityLSi = Latest start time of the activityLSi = Latest finish time of the activityduration of each activity : dj andFloat time of the activity : FTi

Calculation : ESi= maxjεP(i)

{ESj + dj

}; where P(i) is the set

of predecessors for activity iand EFi= ESi+ dj

again, LFi= minjεS(i)

{LFj − dj

}also LSi= LFj - di



Table for Calculating Earliest start, Earliest finish,Latest start and latest finish by our method

Step-8: Project time Calculation table



Float Time and Criticality Calculation

Step-9

The Activity Float time and Criticality of each activity on the Projectare as follows:

Activity No. Float time Criticality of activity

1 (-5.66, 2.50, 10.77) 0.69

2 (-7.68, 0.00 , 7.59) 0.84

3 (-6.97, 1.30, 4.23 ) 1.00

4 (-6.97, 0.74, 3.91) 0.90

5 (-7.59, 0.00 , 3.29) 1.00

6 (-6.97, 1.30, 2.30) 0.84

7 (-5.69, 1.79, 3.65) 0.76

8 (-7.59, 0.00, 1.75) 1.00

9 (-7.59, 0.00, 7.59) 1.00



Net work progress

The progress of Network done by Our proposed method and Themethod from where we select the data has two different views.The comparison of this two method has shown below:



Critical Path and Path Criticality

Step-10

Project completion Path and Alternative probabilities

Activity no. Critical Path Path Criticality

1 ST → 1 → 2→ 6→ 9 → ED 0.692 ST → 2 → 4 → 6→ 9→ ED 0.843 ST→ 2→ 4 → 7 → 9 → ED 0.904 ST → 2 → 4 → 8 → 9→ ED 0.845 ST → 2 → 5 → 8 → 9→ ED 1

Table: Critical Path of the Project activity



Observations

Step-11

Critical Path and Comparison with Our Proposed Method andmethod discussed in Ref. [17]

Method Our Proposed Method Method[17] Per.

Ac.No Critical Path π(PK ) π(PK ) Incr.

1 ST →1 → 3 → 6 → 9 → ED 0.69 0.02 0.67

2 ST → 2 → 4 → 6→ 9→ ED 0.84 0.54 0.30

3 ST→ 2→ 4 → 7 → 9 → ED 0.90 0.22 0.68

4 ST → 2 → 4 → 8 → 9→ ED 0.84 0.54 0.30

5 ST → 2 → 5 → 8 → 9→ ED 1 1 Opt.



Possible Critical Path for the project



ANALYSIS

The above table shows that, both methods can clarify the criticalpath of the project. In the analysis of conventional FPERT methodpresents, the time required for completion of time is 25.87 whereasour proposed technique takes only 23.87 which is 8.34 percent lessthan the method discussed in [17]. Moreover, Path 2 and Path 4have the same degree of criticality whereas variance of Path 2 isgreater than that of Path 4. So, in this case Project managerwould take the decision to select Path 4 if there arises anyuncertainty.Here, it is state that if the differences of optimistic andMost optimistic, and those of pessimistic and Most optimistic arelarge, our proposed method will give more effective result thanother conventional technique in that case.



Conclusion

This presentation shows, how any project manager can estimatemore effective and accurate schedule for a large scale project byusing Fuzzy activity analysis. It includes the process of choosingfuzzified beta distribution and level of α cuts for any project withmean and variance. By increasing the accuracy of estimation,proposed method expressed the way of calculating three operationtimes more accurately. Moreover, the method discussed aboveshows an alternative technique of finding Critical Path by testingactivity Criticality in the project network. It expresses the way toshift any Critical Path to another by adopting necessaryrequirement and budget, also which one is best for this shifting. Inaddition to, our proposed method can reveals all possible CriticalPath and their Criticality for the project completion. So it is easyto take any quick decision for project manager at any stage ofproject completion by measuring the criticality.



Advantages of Research

1 It shows the most optimistic time for project completion withoutany kind of further analysis.

2 It facilitate the calculation of Most Optimistic time for eachactivity accurately.

3 It also presents the easy way of calculation criticality of everycritical path.

4 It express the frequent analysis for quick decision of any projectManager when there is any uncertainty.

5 It does not require crashing for optimum time for the project.

6 Overall, it shows the more reliable and effect approach for Timeanalysis of any Large scale Projects.



REFERENCES

1 S. Chanas and J. Kamburowski , The use of fuzzy variables inPERT, Fuzzy Sets and Systems 5(1981), 11-19.

2 S. Chanas, Fuzzy Sets in a classical operation Research problemin: M.M.Gupta, E. Sanchez ,Approximate reasoning in Dece-sion analysis,North-Holland Amsterdam,(1982),pp.351-363

3 I. Gazdik, Fuzzy network planning, IEEE Transactions Reliabil-ity R-32(3) (1983),304-313.

4 F.A Lootsma, Stochastic and Fuzzy PERT, European Journalof Operation Research 43(1989), 174-183

5 J.J Buckley, Fuzzy Probility and Statistics, Springer-Verlag,Berlin Heidelberg, printed in the Netherlands, 2006.

6 C.S MacCahon, Using PERT as an approximation of fuzzyproject Network analysis, IEE:E Trans Engineering Management40 (1993), 646-669

7 L.A Zadeh, Fuzzy Sets as a basis for a theory of possibility,Fuzzy Sets and Systems 1 (1978) 3-28.



REFERENCES

8 H.J . Zimmermann, Fuzzy Set Theory and Its Applications,fourth ed. Kluwer-Nijhoff, Boston-2001

9 G.J Klir and B.Yuan, Fuzzy Sets and Fuzzy logic: Theory andApplications, Prentice Hall, Englewood Cliffs, New Jersey, 1995

10 L.A Zadeh, Fuzzy sets , Information and Control 8 (1965) 338-353

11 L.A Zadeh, Toward a generalized theory of uncertainty (GTU)an online, Information Sciences 17(2005) 1-40.

12 A. Kaufimann, M.M Gupta, Introduction to Fuzzy arithmetic:Theory and Applications, International Thomson Computer Press,London,1991

13 D. Dubois, H. Prade, Fuzzy sets and Systems: Theory andapplications, Academy press, 1980

14 H.J. Zimmerman, Fuzzy Set Theory and Its Applications, sec-ond ed., Kluwer Academic Publishers, Boston,1991.



REFERENCES

15 E.S. Lee, R.J. Li, Comparison of fuzzy numbers based on theprobability measures of fuzzy events, Computers and Mathe-matics Applications 15 (1988) 887-896

16 Chen-Tung Chen, Sue-Fen Huang, Applying fuzzy method formeasuring criticality in project network, Information Sciences177 (2007) 2448-2458

17 Kanstantinos A. Chrysafis, Basil K. Papadopoulos, Approachingactivity duration in PERT by means of fuzzy sets theory andstatistics, Journal of Intelligent and Fuzzy Systems 26 (2014)577-587.

18 T.T. Soong, Fundamentals of Probability and Statistics for En-gineers, John Wiley and Sons Ltd, The authorsst Sussex, Eng-land,2004.

19 Wilks, S., 1942, Statistical Prediction with Special Referenceto the Problem of Tolerance Limits, Ann. Math. Stat. 13 400

20 Bratley, Paul, Bennett L. Fox and Linus E. Schrage, A Guideto Simulation Springer-Verlag, 1973



REFERENCES

21 Jackson, M.J., Computers in Construction Planning and Con-trol, Allen and Unwin, London, 1986.

22 C.S MacCahon, E.S. Lee, Project Network Analysis with FuzzyActivity Times.Comp. Math. Applic. Vol. 15, No. 10, pp.829-838, 1988 .

23 M. Sharif Uddin, M. Nazrul Islam, Aminur R. Khan, SushantaK. Roy and Muhammad A. Malek Estimation of Shortest Pos-sible Time and Scheduling Critical Path of a Project BasedUpon Node Labeling Jahangirnagar University Journal of Sci-ence, Volume 31, Issue 2, December 2008.

24 C. P. Pappis, N. I. Karacapilidis, A comparative assessment ofmeasures of similarity of fuzzy values, Fuzzy sets and systems,1993, 56, pp. 171-174.

25 W.J. Stevenson, Operation Management, seventh ed. McGraw-Hill,2002.



ANY QUESTIONS ?



THANK YOU


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