Concept explainers
The following precedence diagram reflects three time estimates in weeks for each activity. Determine:
a. The expected completion time for each path and its variance.
b. The probability that the project will require more than 49 weeks.
c. The probability that the project can be completed in 46 weeks or less.
a)
To determine: The expected completion time for each path and its variance.
Explanation of Solution
Given information:
Activity | Optimistic time | Most likely time | Pessimistic time |
1 to 2 | 8 | 8 | 8 |
2 to 3 | 11 | 12 | 13 |
2 to 4 | 5 | 6 | 7 |
2 to 5 | 10 | 12 | 14 |
3 to 8 | 9 | 10 | 12 |
4 to 6 | 14 | 18 | 26 |
4 to 7 | 13 | 13 | 13 |
5 to 9 | 7 | 10 | 12 |
6 to 11 | 8 | 10 | 14 |
7 to 11 | 10.5 | 13 | 15.5 |
8 to 11 | 5 | 7 | 10 |
9 to 10 | 10 | 11 | 12 |
10 to 11 | 6 | 6 | 6 |
Formula to calculate expected time and variance:
Calculation of mean and variance:
Activity | Optimistic time | Most likely time | Pessimistic time | Expected time | Standard deviation | Variance |
A | B | C | (A+(4*B)+C)/6 | (C-A)/6 | (C-A)^2/6^2 | |
1 to 2 | 8 | 8 | 8 | 8 | 0.000 | 0.000 |
2 to 3 | 11 | 12 | 13 | 12 | 0.333 | 0.111 |
2 to 4 | 5 | 6 | 7 | 6 | 0.333 | 0.111 |
2 to 5 | 10 | 12 | 14 | 12 | 0.667 | 0.444 |
3 to 8 | 9 | 10 | 12 | 10.17 | 0.500 | 0.250 |
4 to 6 | 14 | 18 | 26 | 18.67 | 2.000 | 4.000 |
4 to 7 | 13 | 13 | 13 | 13 | 0.000 | 0.000 |
5 to 9 | 7 | 10 | 12 | 9.83 | 0.833 | 0.694 |
6 to 11 | 8 | 10 | 14 | 10.33 | 1.000 | 1.000 |
7 to 11 | 10.5 | 13 | 15.5 | 13 | 0.833 | 0.694 |
8 to 11 | 5 | 7 | 10 | 7.17 | 0.833 | 0.694 |
9 to 10 | 10 | 11 | 12 | 11 | 0.333 | 0.111 |
10 to 11 | 6 | 6 | 6 | 6 | 0.000 | 0.000 |
Calculation of expected duration, variance and standard deviation for each path:
Path 1-2-3-8-11:
Path 1-2-4-6-11:
Path 1-2-4-7-11:
Path 1-2-5-9-11:
b)
To determine: The probability that the project will require more than 49 weeks.
Answer to Problem 11P
Explanation of Solution
Calculation of probability of completion in more than 49 weeks:
Calculation of z value for all paths:
Formula:
Greater than 49 weeks:
Path 1-2-3-8-11:
Since z value is greater than +3.00, probability of completion is 1.00.
Path 1-2-4-6-11:
From the standard normal distribution table,
The probability value for (z = 2.261) is 0.9960.
Path 1-2-4-7-11:
Since z value is greater than +3.00, probability of completion is 1.00.
Path 1-2-5-9-10-11:
From the standard normal distribution table,
The probability value for (z = 1.94) is 0.9738.
Probability of completion in less than or equal to 49 weeks:
Calculation of probability of completion in greater than 49 weeks:
The probability that the project will require more than 49 weeks is 0.0301.
c)
To determine: The probability that the project will be completed in 46 weeks or less.
Answer to Problem 11P
Explanation of Solution
Calculation of probability of completion in 46 weeks or less:
Calculation of z value for all paths:
Formula:
Path 1-2-3-8-11:
Since z value is greater than +3.00, probability of completion is 1.00.
Path 1-2-4-6-11:
From the standard normal distribution table,
The probability value for (z = 1.33) is 0.9082.
Path 1-2-4-7-11:
Since z value is greater than +3.00, probability of completion is 1.00.
Path 1-2-5-9-10-11:
From the standard normal distribution table,
The probability value for (z = -0.74) is 0.2296.
Probability of completion in less than or equal to 46 weeks:
The probability that the project will be completed in 46 weeks or less is 0.2085.
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Chapter 17 Solutions
EBK OPERATIONS MANAGEMENT
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