In the name of Allah the Most Gracious the Most Merciful

Construction Management and Planning

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Float

Total Float is Calculated by Subtracting Early Start and Duration from the Activity’s Late Finish Time An Activity that has Zero Total Float is a Critical Activity An Activity’s Start may be Critical Even Though an Activity itself may not be Critical Start Float may be Calculated by Subtracting an Activity’s Early Start Time from Its Late Start Time

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Float

Finish Float is Calculated by Subtracting an Activity Early Finish Time from its Late Finish Time

Project Scheduling

Probabilistic PERT

History of PERT

• Project Evaluation and Review Technique (PERT) – U S Navy (1958) for the POLARIS missile program – Multiple task time estimates (probabilistic nature) – Activity-on-arrow network construction – Non-repetitive jobs (R & D work) 6

PERT Probability Approach to Project Scheduling

• Activity completion times are seldom known with certainty.

• Completion time estimates can be estimated using the

Three Time Estimate approach

. In this approach, three time estimates are required for each activity:

a = an optimistic time to perform the activity m = the most likely time to perform the activity b = a pessimistic time to perform the activity

PERT

– pessimistic time (a) - the time the activity would take if things did not go well – most likely time ( m) - the consensus best estimate of the activity’s duration – optimistic time (b) - the time the activity would take if things did go well 8

3-Time Estimate Approach Probability Distribution

• With three time estimates, the activity completion time can be approximated by a

Beta distribution

.

• Beta distributions can come in a variety of shapes:

a m b a m b a m b

Mean and Standard Deviation for Activity Completion Times

• The 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.

  = the mean completion time = the standard deviation = a + 4m + b 6 = b 6 a

The Project Completion Time Distribution

The 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

PERT analysis

• Draw the network.

• Analyze the paths through the network and find the critical path.

• The length of the critical path is the mean of the project duration probability distribution which is assumed to be normal • The standard deviation of the project duration probability distribution is computed by adding the variances of the critical activities (all of the activities that make up the critical path) and taking the square root of that sum • Probability computations can now be made using the normal distribution table.

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Probability computation

• Determine probability that project is completed within specified time

Z = x -

 

where



= t p = project mean time



= standard deviation x = (proposed ) specified time

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Normal Distribution of Project Time

Probability

Z

 

= t

p x

Time

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Immed. Optimistic Most Likely Pessimistic Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.) A - B - 4 6 8 1 4.5 5 C A 3 3 3 D A 4 5 6 E A 0.5 1 1.5

F B,C 3 4 5 G B,C 1 1.5 5 H E,F 5 6 7 I E,F 2 5 8 J D,H 2.5 2.75 4.5

K G,I 3 5 7 15

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PERT Example

Activity Expected Time ( μ) (a + 4m + b)/6 A 6 B C D E F 4 3 5 1 4 Variance ( σ 2 ) [(b – a)/6] 2 4/9 4/9 0 1/9 1/36 1/9 G H I J K 2 6 5 3 5 4/9 1/9 1 1/9 4/9 17

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σ 2 path = σ 2 A + σ 2 C + σ 2 F + σ 2 I + σ 2 K = 4/9 + 0 + 1/9 + 1 + 4/9 = 2 σ path = 1.414

Mean project completion time = 23

z

Proposed completion time = 24 = (24 - 23)/  (24-23)/1.414 = .71

From the Standard Normal Distribution table: P(z < .71) = .5 + .2612 = .7612

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Using the PERT-CPM Template for Probabilistic Models

• Instead of calculating µ and  Excel template may be used.

by hand, the • Instead of entering data in the µ and  columns, input the estimates for a, m, and b into columns C, D, and E.

– The template does all the required calculations – After the problem has been solved, probability analyses may be performed.

Enter a, m, b instead of



Call Solver Click Solve Go to PERT OUTPUT worksheet

Call Solver Click Solve

To get a cumulative probability, enter a number here

P(Project is completed in less than 180 days)