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PERT / CPM:Expected length of a critical path, Expected time and Critical path

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Operations Research (MTH601)
42
Expected length of a critical path:
The expected length of a sequence of independent activities is simply the sum of their separate expected
lengths. This gives us the expected length of the entire project. We have to calculate the expected length te of every
activity with the weights attached to the three time estimates and find the critical path in the manner described
previously. The expected length of the entire project denoted by Te is the length of the critical path (i.e.) the sum of
the, te's of all activities along the critical path.
In the same way, the variance of a sum of independent activity times is equal to the sum of their individual
variances. Since Te is the is the sum of te's along the critical path, then variance of Te equals the sum of all the
variances of the critical activities. The standard deviation of the expected project duration is the square root of the of
the variance Te as calculated above.
At this juncture, consider the following example to illustrate the application of these formulae.
Example
2
1,1,7
1,1,1
1
1,4,7
3
2,5,14
5
3,6,15
2,2,8
4
6
2,5,8
Fig. 20
Activity
Expected time (te)
Std. deviation
Variance
1-2
(1+4+7)/6  = 2
(7-1)/6 = 1
1
1-3
(1+16+7)/6 = 4
(7-1)/6 = 1
1
1-4
(2+8+8)/6  = 3
(8-2)/6 = 1
1
2-5
(1+4+1)/6  = 1
(1-1)/6 = 0
0
3-5
(2+20+14)/6 = 6
(14-2)/6 = 2
4
4-6
(2+20+8)/6 = 5
(8-2)/6 = 1
1
5-6
(3+24+15)/6 = 7
(15-3)/6 = 2
4
For each activity, the optimistic mostly likely and pessimistic time estimates are labeled in the same order.
Using PERT formulae for te and St tabulate the results as above.
To calculate the critical path, list all the three paths with their expected time of completion from figure 21.
1-2-5-6
= 10
42
Operations Research (MTH601)
43
1-3-5-6  = 17
1-4-6
=8
43
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