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A project manager has compiled a list of major activities that will be required to install a computer information system in her firm. The list includes estimated completion times for activities and precedence relationships.
a. Construct a network diagram. You can use either AOA or AON (see Example 5).
b. If the project is finished within 26 weeks of its start, the project manager will receive a bonus of $1,000; and if the project is finished within 27 weeks of its start, the bonus will be $500. Find the probability of each bonus.
a)
To construct: A network diagram.
Answer to Problem 12P
Answer
Network diagram:
Explanation of Solution
Given information:
| Activity | Immediate predecessor | Optimistic time | Most likely time | Pessimistic time |
| A | 2 | 4 | 6 | |
| D | A | 6 | 8 | 10 |
| E | D | 7 | 9 | 12 |
| H | E | 2 | 3 | 5 |
| F | A | 3 | 4 | 8 |
| G | F | 5 | 7 | 9 |
| B | 2 | 2 | 3 | |
| I | B | 2 | 3 | 6 |
| J | I | 3 | 4 | 5 |
| K | J | 4 | 5 | 8 |
| C | 5 | 8 | 12 | |
| M | C | 1 | 1 | 1 |
| N | M | 6 | 7 | 11 |
| O | N | 8 | 9 | 13 |
| End | H, G, K, O |
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 | |
| A | 2 | 4 | 6 | 4 | 0.667 | 0.444 |
| D | 6 | 8 | 10 | 8 | 0.667 | 0.444 |
| E | 7 | 9 | 12 | 9.17 | 0.833 | 0.694 |
| H | 2 | 3 | 5 | 3.17 | 0.500 | 0.250 |
| F | 3 | 4 | 8 | 4.5 | 0.833 | 0.694 |
| G | 5 | 7 | 9 | 7 | 0.667 | 0.444 |
| B | 2 | 2 | 3 | 2.17 | 0.167 | 0.028 |
| I | 2 | 3 | 6 | 3.33 | 0.667 | 0.444 |
| J | 3 | 4 | 5 | 4 | 0.333 | 0.111 |
| K | 4 | 5 | 8 | 5.33 | 0.667 | 0.444 |
| C | 5 | 8 | 12 | 8.17 | 1.167 | 1.361 |
| M | 1 | 1 | 1 | 1 | 0.000 | 0.000 |
| N | 6 | 7 | 11 | 7.5 | 0.833 | 0.694 |
| O | 8 | 9 | 13 | 9.5 | 0.833 | 0.694 |
| End |
Network diagram:
The activity on arrow diagram is constructed. The activities are mentioned on the arrows. The arrows in the diagram are mentioned in the precedence order. The arrows indicates the successor for each node.
b)
To determine: The probability of each bonus.
Answer to Problem 12P
Explanation of Solution
Given information:
- If the project is finished in 26 weeks the bonus is $1,000.
- If the project is finished in 27 weeks the bonus is $500.
Calculation of expected duration, variance and standard deviation for each path:
Path A-D-E-H:
Path A-F-G:
Path B-I-J-K:
Path C-M-N-O:
Bonus of $1,000 (less than or equal to 26 weeks):
Calculation of probability of completion in less than or equal to 26 weeks:
Calculation of z value for all paths:
Formula:
Path A-D-E-H:
From the standard normal distribution table,
The probability value for (z = 1.23) is 0.8907.
Path A-F-G:
Since z value is greater than +3.00, probability of completion is 1.0000.
Path B-I-J-K:
Since z value is greater than +3.00, probability of completion is 1.0000.
Path C-M-N-O:
From the standard normal distribution table,
The probability value for (z = -0.10) is 0.4602.
Probability of completion in less than or equal to 26 weeks:
The probability of achieving a bonus of $1,000 is 0.4099.
Bonus of $500 (less than or equal to 27 weeks):
Calculation of probability of completion in less than or equal to 27 weeks:
Calculation of z value for all paths:
Formula:
Path A-D-E-H:
From the standard normal distribution table,
The probability value for (z = 1.96) is 0.9750.
Path A-F-G:
Since z value is greater than +3.00, probability of completion is 1.0000.
Path B-I-J-K:
Since z value is greater than +3.00, probability of completion is 1.0000.
Path C-M-N-O:
From the standard normal distribution table,
The probability value for (z = 0.50) is 0.6915.
Probability of completion in less than or equal to 27 weeks:
The probability of achieving a bonus of $500 is 0.6742.
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Chapter 17 Solutions
Loose-leaf for Operations Management (The Mcgraw-hill Series in Operations and Decision Sciences)
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