
Concept explainers
Three recent college graduates have formed a
a. Draw the precedence diagram.
b. What is the probability that the project can be completed in 24 days or less? In 21 days or less?
c. Suppose it is now the end of the seventh day and that activities A and B have been completed while activity D is 50 percent completed. Time estimates for the completion of activity D are 5, 6, and 7. Activities C and H are ready to begin. Determine the probability of finishing the project by day 24 and the probability of finishing by day 21.
d. The partners have decided that shortening the project by two days would be beneficial, as long as it doesn’t cost more than about $20,000. They have estimated the daily crashing costs for each activity in thousands, as shown in the following table. Which activities should be crashed, and what further analysis would they probably want to do?
a)

To draw: A precedence diagram.
Answer to Problem 7P
Precedence diagram:
Explanation of Solution
Given information:
Activity | Immediate Predecessor | Optimistic time | Most likely time | Pessimistic time |
A | 5 | 6 | 7 | |
B | 8 | 8 | 11 | |
C | A | 6 | 8 | 11 |
D | 9 | 12 | 15 | |
E | C | 5 | 6 | 9 |
F | D | 5 | 6 | 7 |
G | F | 2 | 3 | 7 |
H | B | 4 | 4 | 5 |
I | H | 5 | 7 | 8 |
End | E, G, I |
Activity | First crash | Second crash |
C | $ 8.00 | $ 10.00 |
D | $ 10.00 | $ 11.00 |
E | $ 9.00 | $ 10.00 |
F | $ 7.00 | $ 9.00 |
G | $ 8.00 | $ 9.00 |
H | $ 7.00 | $ 8.00 |
I | $ 6.00 | $ 8.00 |
Precedence diagram:
The precedence diagram is drawn from the first task till the last task. The activities are placed from left to right. The directions are represented with arrows to indicate the relationship between activities. The arrows are represented with the activity name.
b)

To determine: The probability at which the projected can be completed in 24 days or less and 21 days or less.
Answer to Problem 7P
24 days or less = 0.9686
21 days or less = 0.2350
Explanation of Solution
Given information:
Activity | Immediate Predecessor | Optimistic time | Most likely time | Pessimistic time |
A | 5 | 6 | 7 | |
B | 8 | 8 | 11 | |
C | A | 6 | 8 | 11 |
D | 9 | 12 | 15 | |
E | C | 5 | 6 | 9 |
F | D | 5 | 6 | 7 |
G | F | 2 | 3 | 7 |
H | B | 4 | 4 | 5 |
I | H | 5 | 7 | 8 |
End | E, G, I |
Activity | First crash | Second crash |
C | $ 8.00 | $ 10.00 |
D | $ 10.00 | $ 11.00 |
E | $ 9.00 | $ 10.00 |
F | $ 7.00 | $ 9.00 |
G | $ 8.00 | $ 9.00 |
H | $ 7.00 | $ 8.00 |
I | $ 6.00 | $ 8.00 |
Formula to calculate expected time and variance:
Calculation of expected time 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 | 5 | 6 | 7 | 6 | 0.333 | 0.111 |
B | 8 | 8 | 11 | 8.5 | 0.500 | 0.250 |
C | 6 | 8 | 11 | 8.17 | 0.833 | 0.694 |
D | 9 | 12 | 15 | 12 | 1.000 | 1.000 |
E | 5 | 6 | 9 | 6.33 | 0.667 | 0.444 |
F | 5 | 6 | 7 | 6 | 0.333 | 0.111 |
G | 2 | 3 | 7 | 3.5 | 0.833 | 0.694 |
H | 4 | 4 | 5 | 4.17 | 0.167 | 0.028 |
I | 5 | 7 | 8 | 6.83 | 0.500 | 0.250 |
Calculation of expected duration, variance and standard deviation for each path:
A-C-E:
D-F-G:
B-H-I:
Calculation of z value for all paths:
Formula:
24 days or less:
A-C-E:
Since z value is greater than +3.00, probability of completion is 1.00.
D-F-G:
From the standard normal distribution table,
The probability value for (z = 1.86) is 0.9686.
B-H-I:
Since z value is greater than +3.00, probability of completion is 1.00.
Probability of completion in 24 days or less:
The probability at which the project can be completed in 24 days or less is 0.9686.
21 days or less:
A-C-E:
From the standard normal distribution table,
The probability value for (z = 0.45) is 0.6736.
D-F-G:
From the standard normal distribution table,
The probability value for (z = -0.37) is 0.3557.
B-H-I:
From the standard normal distribution table,
The probability value for (z = 2.07) is 0.9808.
Probability of completion in 21 days or less:
The probability at which the project can be completed in 21 days or less is 0.2350.
c)

To determine: The probability of completing the project by day 24 and day 21.
Answer to Problem 7P
Day 24 = 0.9328
Day 21 = 0.0186
Explanation of Solution
Given information:
- At the end of 7th day activities A and B are completed and D is 50% completed.
- Time estimates of activity D completion are 5, 6 and 7.
- Activities C and H are ready to begin.
Activity | Immediate Predecessor | Optimistic time | Most likely time | Pessimistic time |
A | 5 | 6 | 7 | |
B | 8 | 8 | 11 | |
C | A | 6 | 8 | 11 |
D | 5 | 6 | 7 | |
E | C | 5 | 6 | 9 |
F | D | 5 | 6 | 7 |
G | F | 2 | 3 | 7 |
H | B | 4 | 4 | 5 |
I | H | 5 | 7 | 8 |
End | E, G, I |
Activity | First crash | Second crash |
C | $ 8.00 | $ 10.00 |
D | $ 10.00 | $ 11.00 |
E | $ 9.00 | $ 10.00 |
F | $ 7.00 | $ 9.00 |
G | $ 8.00 | $ 9.00 |
H | $ 7.00 | $ 8.00 |
I | $ 6.00 | $ 8.00 |
Formula to calculate expected time and variance:
Calculation of expected time 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 | 5 | 6 | 7 | 6 | 0.333 | 0.111 |
B | 8 | 8 | 11 | 8.5 | 0.500 | 0.250 |
C | 6 | 8 | 11 | 8.17 | 0.833 | 0.694 |
D | 5 | 6 | 7 | 6 | 0.333 | 0.111 |
E | 5 | 6 | 9 | 6.33 | 0.667 | 0.444 |
F | 5 | 6 | 7 | 6 | 0.333 | 0.111 |
G | 2 | 3 | 7 | 3.5 | 0.833 | 0.694 |
H | 4 | 4 | 5 | 4.17 | 0.167 | 0.028 |
I | 5 | 7 | 8 | 6.83 | 0.500 | 0.250 |
Revised project diagram:
Calculation of expected duration, variance and standard deviation for each path:
C-E:
D-F-G:
H-I:
Calculation of z value for all paths:
Formula:
24 days or less:
C-E:
From the standard normal distribution table,
The probability value for (z = 2.34) is 0.9904.
D-F-G:
From the standard normal distribution table,
The probability value for (z = 1.57) is 0.9418.
H-I:
Since z value is greater than +3.00, probability of completion is 1.00.
Probability of completion in 24 days or less:
The probability at which the project can be completed in 24 days is 0.9328.
21 days or less:
C-E:
From the standard normal distribution table,
The probability value for (z = -0.47) is 0.3192.
D-F-G:
From the standard normal distribution table,
The probability value for (z = -1.57) is 0.0582.
H-I:
Since z value is greater than +3.00, probability of completion is 1.00.
Probability of completion in 21 days or less:
The probability at which the project can be completed in 21 days is 0.0186.
d)

To determine: The activities that should be crashed and further analysis.
Explanation of Solution
Given information:
- The partners want to shorten the project by 2 days as long as the cost is not more than $20,000.
Activity | Immediate Predecessor | Optimistic time | Most likely time | Pessimistic time |
A | 5 | 6 | 7 | |
B | 8 | 8 | 11 | |
C | A | 6 | 8 | 11 |
D | 5 | 6 | 7 | |
E | C | 5 | 6 | 9 |
F | D | 5 | 6 | 7 |
G | F | 2 | 3 | 7 |
H | B | 4 | 4 | 5 |
I | H | 5 | 7 | 8 |
End | E, G, I |
Activity | First crash | Second crash |
C | $ 8.00 | $ 10.00 |
D | $ 10.00 | $ 11.00 |
E | $ 9.00 | $ 10.00 |
F | $ 7.00 | $ 9.00 |
G | $ 8.00 | $ 9.00 |
H | $ 7.00 | $ 8.00 |
I | $ 6.00 | $ 8.00 |
Paths and expected duration:
Paths | Expected Duration |
C-E | 21.50 |
D-F-G | 22.50 |
H-I | 18.00 |
The critical path is D – F – G.
The activities are crashed based on the cost of crash given.
Activity | Cost |
F | $7 |
G | $8 |
D | $10 |
Step 1:
Activity F has the lowest crashing cost ($7,000) and will be crashed first for 1 day. The expected duration of D-F-G will be 21.50 days.
Step 2:
Path | Expected Duration |
C-E | 21.50 |
D-F-G | 21.50 |
H-I | 18.00 |
Now there are two critical paths C-E and D-F-G.
The critical activities are arranged in the order of low crash costs.
Path | Activity | Cost |
C-E | C | $8 |
F | $9 |
Path | Activity | Cost |
D-F-G | D | $8 |
F | $9 | |
G | $10 |
One activity in each path is chosen to crash.
Activity C is crashed for 1 day since it has the lowest crashing cost ($8,000) on path C-E. The expected duration of path C-E is now 20.50 days.
Activity G is crashed for 1 day since it has the lowest crashing cost ($8,000) on path D-F-G. The expected duration of path D-F-G is now 20.50 days.
Calculation of total crashing cost:
The total cost of crashing is over the budget of $20,000 ($23,000 > $20,000). Hence, the partners will have to determine if crashing the project by 1 day or 2 days is really beneficial or not.
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Chapter 17 Solutions
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