Project SchedulingProbabilistic PERT

PERT Probability Approach to Project SchedulingActivity completion times are seldom known with cetainty.

PERT is a technique that treats activity completion times as random variables.

Completion time estimates can be estimated using the Three Time Estimate approach. In this approach, three time estimates are required for each activity:

Results from statistical studiesSubjective best estimates

a = an optimistic time to perform the activity P(Finish < a) < .01m = the most likely time to perform the activity (mode)b = a pessimistic time to perform the activity P(Finish > b) < .01

3-Time Estimate ApproachProbability DistributionWith three time estimates, the activity completion time can be approximated by a Beta distribution.Beta distributions can come in a variety of shapes:

Mean and Standard Deviation forActivity Completion TimesThe best estimate for the mean is a weighted average of the three time estimates with weights 1/6, 4/6, and 1/6 respectively on a, m, and b.Since most of the area is with the range from a to b (b-a), and since most of the area lies 3 standard deviations on either side of the mean (6 standard deviations total), then the standard deviation is approximated by Range/6.

PERT AssumptionsAssumption 2There are enough activities on the critical path so that the distribution of the overall project completion time can be approximated by the normal distribution.

Assumption 1A critical path can be determined by using the mean completion times for the activities. The project mean completion time is determined solely by the completion time of the activities on the critical path.

Assumption 3The time to complete one activity is independent of the completion time of any other activity.

The Project Completion Time DistributionThe three assumptions imply that the overall project completion time is normally distributed, with: = Sum of the s on the critical path

2 = Sum of the 2 s on the critical path

The Probability Approach

Sheet1

ActivityOptimisticMost LikelyPessimistic

A7686120

B121518

C456

D151833

E182124

F162630

G101322

H241832

I222750

J384360

Sheet1

mss2mss2

907.3353.73907.3353.73

151.001.00151.001.00

50.330.1150.330.11

203.009.00203.009.00

211.001.00211.001.00

252.335.43252.335.43

142.004.00142.004.00

281.331.77281.331.77

304.6721.81304.6721.81

453.6713.47453.6713.47

Sheet2

Sheet3

Distribution For Klone ComputersThe project has a normal distribution.The critical path is A-F-G-D-J.

Standard Probability QuestionsWhat is the probability the project will be finished within 194 days? P(X < 194)Give an interval within which we are 95% sure of completing the project.X values, xL, the lower confidnce limit, and xU, the upper confidnce limit, such that P(XxU) = .025What is the probability the project will be completed within 180 days?P(X < 180)What is the probability the project will take longer than 210 days.P(X > 210)By what time are we 99% sure of completing the project?X value such that P(X < x) = .99

Excel Solutions

Using the PERT-CPM Template for Probabilistic ModelsInstead of calculating and by hand, the Excel template may be used.Instead of entering data in the and columns, input the estimates for a, m , and b into columns C, D, and E.The template does all the required calculationsAfter the problem has been solved, probability analyses may be performed.

Call SolverClick SolveGo to PERT OUTPUT worksheet

Call SolverClick Solve

Cost Analysis Using theExpected Value ApproachSpending extra money, in general should decrease project duration.

But is this operation cost effective?

The expected value criterion can be used as a guide for answering this question.

Cost Analyses Using Probabilities Suppose an analysis of the competition indicated:If the project is completed within 180 days, this would yields an additional profit of $1 million.If the project is completed in 180 days to 200 days, this would yield an additional profit of $400,000.

KLONE COMPUTERS - Cost analysis using probabilitiesCompletion time reduction can be achieved by additional training.

Two possible activities are being considered.Sales personnel training: (Activity H) Cost $200,000; New time estimates are a = 19, m= 21, and b = 23 days.Technical staff training: (Activity F)Cost $250,000; New time estimates are a = 12, m = 14, and b = 16.

Which, if either option, should be pursued?

Analysis of Additional Sales Personnel Training Sales personnel training (Activity H) is not a critical activity.Thus any reduction in Activity H will not affect the critical path and hence the distribution of the project completion time.

This option should not bepursued at any cost.

Analysis of Additional Technical Staff TrainingTechnical Staff Training (Activity F) is on the critical path so this option should be analyzed.One of three things will happen:The project will finish within 180 days:Klonepalm will net an additional $1 millionThe project will finish in the period from 180 to 200 days Klonepalm will net an additional $400,000The project will take longer than 200 daysKlonepalm will not make any additional profit.

The Expected Value ApproachFind the P(X < 180), P(180 < X < 200), and P(X > 200) under the scenarios thatNo additional staff training is doneAdditional staff is doneFor each scenario find the expected profit:

Subtract the two expected values. If the difference is less than the cost of the training, do not perform the additional training.Caution: These are expected values (long run average values). But this approach serves as a good indicator for the decision maker to consider.

Expected Additional Profit

1000000(P(X

The CalculationsThe PERT-CPM template can be used to calculate the probabilities.

= 194 = 9.255 = 189 = 9.0185Total = $335,751Total = $450,976Net increase = $450,976-$335,751 = $115,225This is less than the $250,000 required for training.Do not perform theadditional training!

Review3-Time Estimate Approach for PERTEach activity has a Beta distributionCalculation of Mean of each activityCalculation Variance and Standard Deviation for each activityAssumptions for using PERT approachDistribution of Project CompletionTimeNormalMean = Sum of means on critical pathVariance = Sum of variances on critical pathUsing the PERT-CPM templateUsing PERT in cost analyses