Crashing PERT Networks Using Mathematical Programing

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International Journal Of Project Management 19 (2001)

Submitted By:

Ibrahim H. Malubhaiwala (40) Jasmeet Singh Bhatia (41)Manish Pashine (53)Mayank Kumar Sharma (54)

Ghaleb Y. Abbasi, Adnan M. Mukattash

Advance Project Management Assignment



Introduction Aim of the study is to develop an approach for crashingpessimistic time in PERT networks

Crashing is done by investing additional amount of money in order to reduce variance and projectdurationThis increases the probability of completing the

project in the given timeConcept of crashing in CPM is applied to PERTnetworks



Model ConstructionProbability of certain project meeting a time T, for aparticular event is equal to

where,



The amount of money ri is invested in each activity atcritical path, to reduce the pessimistic time from b to ,the expected project duration from and thestandard deviation from to respectively After investing additional amount of money, the newexpected duration and new variance for each activity

would be



The new expected duration of activities lying on theCP would be



The new variance would be

And the amount of money invested on all activities



The probability of realizing the terminal node



Or,



Since the amount of money invested will reduce theexpected time, the new expected time for criticalactivities is

The equation of new expected time becomes

And the new variance becomes



Reducing the certain variance by certain percentmeans reducing the standard deviation by the samepercent, hence above equation can be written as



Also from eq (16) we get

By substituting eq (20) and (21), the following isobtained

This means the amount of money invested in certainactivities will reduce pessimistic time and its variance



The new expected duration of CP activities

After approximating function, the above eq.becomes



Since the decrease in variance is the function of then the new variance of CP activities would be

After approximating function, above eq. can be written as



After investing amount on each CP activity, thenew expected duration and new variance are thefunction of . The probability of realizing theterminal node is where,

Where,



Then Z becomes



Eq. (32) represents the mathematical model, whichrepresents the objective function and the constraints



The whole model can be re-written as follows

such that



Linearizing the Mathematical

ModelEq. (33) can be expanded using Taylors expansion formulti dimensional function. Then the linearizedmodel would be



From above Eq. the first the first term is constant

Hence the Eq. (34) can be re-written as

Subject to



A program was developed to substitute the valuesof and in the linearizedconstructed model. The output of this program isthe linear equation, which is the objectivefunction of the following form:

where is constant and is the coefficient of theadditional amount of money which must beinvested on the first activity on CP



Algorithm development



Contd..



Case Study



Table shows time estimates of all the activities



Values of qi and si are suggested by experts for eachactivity By substituting the values of in thedevelopment model we get the objective function



Next by specifying the value of , we get theconstraint of the objective function



Then we continue the steps of the algorithm, by finding the value of the objective function Z andsubstituting in Eq. (37) we get Z=-0.36476 and

Also we continue by linearizing the model around toget the following objective function



ConclusionThe objective of the paper was to increase theprobability of realizing the last node by minimizingpessimistic time, which led to decrease in projectduration and deviationBy investing additional amount of money theprobability of completing the project increased from20.7% to 35%.Moreover the scheduled time decreased from 421 daysto 413 days

Crashing PERT Networks Using Mathematical Programing

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