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PERT / CPM:Expected time and Critical path

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Operations Research (MTH601)
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2
2
1
1
4
3
6
5
3
7
4
5
6
Expected time and Critical path.
Fig. 21
The longest path is the critical path and it consists of activities 1-3, 3-5 and 5-6 with a length Te = 4 + 6 + 7
= 17. The variance of the critical path is then VT = Variance of 1-3 + variance of 3-5 + variance of 5-6.
VT = 1 + 4 + 4 = 9
Standard deviation of duration of the critical path
σ =  VT = 9 = 3
So far what we have done with PERT model is to recognize uncertainty by using three time estimates and
these are reduced to a single time estimate for finding the critical path. It can be used to find early start-early finish
programme, late start-late finish programme and slack. The variability of the time estimates for each activity is also
reduced to a standard deviation and variance and this is used to find the standard deviation of expected completion
time for the project.
Probability of completing a project with a given date
One may wonder how the calculations made to find Te, ST and VT are useful to a project manager. These
parameters serve as a very useful tool for a project manager to estimate the probability of completing a project with
a given date.
We have seen that the time required for an activity is uncertain and hence it is a random variable. Its
average or expected value (te) can be estimated on the basis of an assumption regarding the type of probability
distribution, and three points on this distribution namely optimistic, most likely and pessimistic time estimates.
We estimate the average, or expected, projected length, Te by adding the expected activity durations along
the critical path. As the te's are all random variables, then so is Te. It is assumed that Te is to follow approximately a
normal distribution. This gives the probability of completing a project with a given date.
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Operations Research (MTH601)
45
Referring to the example illustrated above, the expected project length is 17 days and a standard
deviation of 3 days. These are the two parameters useful to calculate the probability if the due date is to be met. For
example, if the due date is T and this is deviated from the mean by T-Te and the same can be expressed as the ratio
of standard deviation as (T-Te)/S.D. This is defined as the standard normal variates denoted by Z.
It is known that in a normal distribution the area under the normal curve gives the probability. For Z = 0, T
= Te and hence if the due date is exactly the expected project length, the probability of completing the project is 50%
as represented by the area to the left of the central line.
⎡T - Te ⎤
⎡T - Te ⎤
z=⎢
z=0
z=⎢
⎥
⎥
⎣ σ  ⎦
⎣ σ  ⎦
Fig. 22
Fig 23
If in this example the due date is 20 days, Z = (20 - 17)/3 = 1. For the value of Z = 1 the area between Z = 0
and Z = 1 is estimated (or as found from the statistical tables) as 0.3413. This is indicated in figure 222 by the
shaded portion. Hence the probability of completing the project in 20 days will be 0.50 + 0.3413 = 0.8413 or
84.13%. Similarly if the due date is 14 days, the corresponding value of Z = - 1. For this value of Z = -1 also, area is
0.3413 but to the left of the mean as indicated in figure 23.
The probability of completing the project in 14 days will be =0.5000 - 0.3413 = 0.1587 or 15.87%.
Example The following table lists the jobs of a network with their time estimates.
Job
Duration (days)
i-j
Optimistic
Most likely
Pessimistic
1 2
3
6
15
1 6
2
5
14
2 3
6
12
30
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Operations Research (MTH601)
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2 4
2
5
8
3 5
5
11
17
4 5
3
6
15
6 7
3
9
27
5 8
1
4
7
7 8
4
19
28
(a)
Draw the project network.
(b)
Calculate the length and variance of the critical path.
(c)
What the is approximate probability that the jobs on the critical path will be completed by the due date of 42
days?
(d)
What due date has about 90% chance of being met?
Solution:
Before proceeding to draw the project network, let us calculate the expected time of activity te, standard
deviation and variance of the expected time of activity using
(to + 4tm + t  p )
te =
6
S.D = (t  p - to ) / 6;
Variance = (S.D)2
Activity
te (Days)
S.D (Days)
Variance
1-2
7
2
4
1-6
6
2
4
2-3
14
4
16
2-4
5
1
1
3-5
11
2
4
4-5
7
2
4
6-7
11
4
16
5-8
4
1
1
7-8
18
4
16
(a)
Project Network:
3
14
11
2
5
7
5
7
4
1
4
8
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Operations Research (MTH601)
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6
11
18
6
7
Expected time and Critical path
Fig. 24
(b)
There are three paths:
1-2-3-5-8 = 36 days
1-2-4-5-8 = 23 days
1-6-7-8
= 35 days
1-2-3-5-8 is the longest path and hence the critical path.
Expected length of the critical path is 36 days. The variance for 1-2, 2-3, 3-5 and 5-8 are 4, 16, 4 and 1
respectively and variance of the projection duration is 25 and hence
Standard deviation of the project duration = 25 = 5 days.
(c)
Due date = 42 days (T)
Expected duration = 36 days (Te) and S.D = 5 days (o)
Z = (T - Te ) / S.D. = (42 - 36) / 5 = 1.2
The area under the normal curve for Z = 1.2 is 0.3849.
Therefore, the probability of completing the project in 42 days
= 0.5000 + 0.3849
= 0.8849
= 88.49%
Exercise
(1)
A project has the following characteristics:
Activity
Optimistic time
Pessimistic time
Most likely time
1-2
1
5
1.5
2-3
1
3
2
2-4
1
5
3
3-5
3
5
4
4-5
2
4
3
4-6
3
7
5
5-7
4
6
5
6-7
6
8
7
7-8
2
6
4
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Operations Research (MTH601)
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7-9
5
8
6
8-10
1
3
2
9-10
3
7
5
Construct a network. Find the critical path and variance for each event. Find the project duration at 95%
probability.
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Table of Contents:
  1. Introduction:OR APPROACH TO PROBLEM SOLVING, Observation
  2. Introduction:Model Solution, Implementation of Results
  3. Introduction:USES OF OPERATIONS RESEARCH, Marketing, Personnel
  4. PERT / CPM:CONCEPT OF NETWORK, RULES FOR CONSTRUCTION OF NETWORK
  5. PERT / CPM:DUMMY ACTIVITIES, TO FIND THE CRITICAL PATH
  6. PERT / CPM:ALGORITHM FOR CRITICAL PATH, Free Slack
  7. PERT / CPM:Expected length of a critical path, Expected time and Critical path
  8. PERT / CPM:Expected time and Critical path
  9. PERT / CPM:RESOURCE SCHEDULING IN NETWORK
  10. PERT / CPM:Exercises
  11. Inventory Control:INVENTORY COSTS, INVENTORY MODELS (E.O.Q. MODELS)
  12. Inventory Control:Purchasing model with shortages
  13. Inventory Control:Manufacturing model with no shortages
  14. Inventory Control:Manufacturing model with shortages
  15. Inventory Control:ORDER QUANTITY WITH PRICE-BREAK
  16. Inventory Control:SOME DEFINITIONS, Computation of Safety Stock
  17. Linear Programming:Formulation of the Linear Programming Problem
  18. Linear Programming:Formulation of the Linear Programming Problem, Decision Variables
  19. Linear Programming:Model Constraints, Ingredients Mixing
  20. Linear Programming:VITAMIN CONTRIBUTION, Decision Variables
  21. Linear Programming:LINEAR PROGRAMMING PROBLEM
  22. Linear Programming:LIMITATIONS OF LINEAR PROGRAMMING
  23. Linear Programming:SOLUTION TO LINEAR PROGRAMMING PROBLEMS
  24. Linear Programming:SIMPLEX METHOD, Simplex Procedure
  25. Linear Programming:PRESENTATION IN TABULAR FORM - (SIMPLEX TABLE)
  26. Linear Programming:ARTIFICIAL VARIABLE TECHNIQUE
  27. Linear Programming:The Two Phase Method, First Iteration
  28. Linear Programming:VARIANTS OF THE SIMPLEX METHOD
  29. Linear Programming:Tie for the Leaving Basic Variable (Degeneracy)
  30. Linear Programming:Multiple or Alternative optimal Solutions
  31. Transportation Problems:TRANSPORTATION MODEL, Distribution centers
  32. Transportation Problems:FINDING AN INITIAL BASIC FEASIBLE SOLUTION
  33. Transportation Problems:MOVING TOWARDS OPTIMALITY
  34. Transportation Problems:DEGENERACY, Destination
  35. Transportation Problems:REVIEW QUESTIONS
  36. Assignment Problems:MATHEMATICAL FORMULATION OF THE PROBLEM
  37. Assignment Problems:SOLUTION OF AN ASSIGNMENT PROBLEM
  38. Queuing Theory:DEFINITION OF TERMS IN QUEUEING MODEL
  39. Queuing Theory:SINGLE-CHANNEL INFINITE-POPULATION MODEL
  40. Replacement Models:REPLACEMENT OF ITEMS WITH GRADUAL DETERIORATION
  41. Replacement Models:ITEMS DETERIORATING WITH TIME VALUE OF MONEY
  42. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS
  43. Dynamic Programming:Analysis of the Result, One Stage Problem
  44. Miscellaneous:SEQUENCING, PROCESSING n JOBS THROUGH TWO MACHINES
  45. Miscellaneous:METHODS OF INTEGER PROGRAMMING SOLUTION