Chapter 17 – Project Management
17–41
A summary of the crashing steps is shown below:
Path
Initial
time
Length after
crashing 1
week
Length after
crashing 2
weeks
Length after
crashing 3
weeks
Length after
crashing 4
weeks
Length after
crashing 5
weeks
A-B-K
C-D-G-M
C-E-H-N
C-E-H-P
C-F-I-J-P
35 wk.
45 wk.
47 wk.
44 wk.
49 wk.
35 wk.
44 wk.
46 wk.
43 wk.
48 wk.
35 wk.
43 wk.
45 wk.
42 wk.
47 wk.
35 wk.
43 wk.
45 wk.
42 wk.
46 wk.
35 wk.
43 wk.
45 wk.
42 wk.
45 wk.
35 wk.
43 wk.
44 wk.
40 wk.
44 wk.
Step 1 Crashing
Step 2 Crashing
Step 3 Crashing
Step 4 Crashing
Step 5 Crashing
Activity Cost
C $5
Activity Cost
C $5
Activity Cost
F $12
Activity Cost
F $15
Activity Cost
E $16
P 20
$36
Chapter 17 – Project Management
17–42
Education.
15. Given:
We have the following activities and crashing costs ($000):
Crashing Costs ($000)
Activity
First
Week
Second
Week
Third
Week
1-2
$18
$22
—
2-5
$24
$25
$25
5-7
$30
$30
$35
7-11
$15
$20
—
11-13
$30
$33
$36
1-3
$12
$24
$26
3-8
—
—
—
8-11
$40
$40
$40
3-9
$3
$10
$12
9-12
$2
$7
$10
12-13
$26
—
—
1-4
$10
$15
$25
4-6
$8
$13
—
6-10
$5
$12
—
10-12
$14
$15
—
Chapter 17 – Project Management
Education.
a. Determine the optimum time-cost crashing plan.
We will continue crashing as long as the crashing cost at a given step ≤ $40.
Step 1:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
35
1-3-8-11-13
32
1-3-9-12-13
20
1-4-6-10-12-13
33
Rank critical activities according to crash costs:
Activity
Cost per Week
to Crash
7-11
$15
1-2
$18
2-5
$24
5-7
$30
11-13
$30
Activity 7-11 should be shortened 1 week because it has the lowest crashing cost ($15) and
this cost is ≤ $40.
Path 1-2-5-7-11-13 will decrease 1 week as shown below.
Activities crashed to this point: Step 1: 7-11 (first week).
Step 2:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
34
1-3-8-11-13
32
1-3-9-12-13
20
1-4-6-10-12-13
33
Activity
Cost per Week
to Crash
1-2
$18
7-11
$20
2-5
$24
5-7
$30
11-13
$30
Chapter 17 – Project Management
17–44
Education.
Step 3:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
33
1-3-8-11-13
32
1-3-9-12-13
20
1-4-6-10-12-13
33
Rank critical activities according to crash costs:
Path
Activity
Cost per Week
to Crash
1-2-5-7-11-13
7-11
$20
1-2
$22
2-5
$24
5-7
$30
11-13
$30
1-4-6-10-12-13
6-10
$5
4-6
$8
1-4
$10
10-12
$14
12-13
$26
Activity 6-10 should be shortened 1 week because it has the lowest crashing cost ($5) on
Path 1-4-6-10-12-13, which will decrease 1 week as shown below.
The combined crashing cost = $25 and is ≤ $40.
Activities crashed to this point: Step 1: 7-11 (first week). Step 2: 1-2 (first week). Step 3: 7-
11 (second week) & 6-10 (first week).
Chapter 17 – Project Management
Step 4:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
32
1-3-8-11-13
32
1-3-9-12-13
20
1-4-6-10-12-13
32
Chapter 17 – Project Management
17–46
Step 5:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
31
1-3-8-11-13
31
1-3-9-12-13
20
1-4-6-10-12-13
31
Critical paths are 1-2-5-7-11–13, 1-3-8-11-13, & 1-4-6-10-12-13.
Rank critical activities according to crash costs:
Path
Activity
Cost per Week
to Crash
1-2-5-7-11-13
1-2
$22
2-5
$24
5-7
$30
11-13
$33
1-3-8-11-13
1-3
$12
11-13
$33
8-11
$40
1-4-6-10-12-13
1-4
$10
6-10
$12
4-6
$13
10-12
$14
12-13
$26
Activity 1-4 could be shortened 1 week because it has the lowest crashing cost ($10) on Path
1-4-6-10-12-13, which would decrease 1 week.
The combined crashing cost = $43 and is > $40. Stop.
The marginal cost of crashing > marginal benefit of crashing. We stop crashing after Step 4.
Chapter 17 – Project Management
17–47
Education.
A summary of the crashing steps is shown below:
Path
Initial
time
Length after
crashing 1
week
Length after
crashing 2
weeks
Length after
crashing 3
weeks
Length after
crashing 4
weeks
1-2-5-7-11-13
1-3-8-11-13
1-3-9-12-13
1-4-6-10-12-13
35
32
20
33
34
32
20
33
33
32
20
33
32
32
20
32
31
31
20
31
Step 1 Crashing
Step 2 Crashing
Step 3 Crashing
Step 4 Crashing
Activity Cost
7-11 $15
Activity Cost
1-2 $18
Activity Cost
7-11 $20
6-10 $ 5
$25
Activity Cost
11-13 $30
4-6 $ 8
$38
Conclusion: Expected project duration = 31 weeks (shortened by 4 weeks).
Total crashing cost = $15 + $18 + $25 + $38 = $96.
Total indirect costs = 31 x $40 = $1,240.
Total cost over 31 weeks = $96 + $1,240 = $1,336(000) = $1,336,000
b. Plot the total-cost curve that describes the least expensive crashing schedule that will reduce
the project length by 6 weeks:
In Part a of this problem, we reduced the project length by 4 weeks. We must decrease the
Chapter 17 – Project Management
Step 5:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
31
1-3-8-11-13
31
1-3-9-12-13
20
1-4-6-10-12-13
31
Rank critical activities according to crash costs:
Path
Activity
Cost per Week
to Crash
1-2-5-7-11-13
1-2
$22
2-5
$24
5-7
$30
11-13
$33
1-3-8-11-13
1-3
$12
11-13
$33
8-11
$40
1-4-6-10-12-13
1-4
$10
6-10
$12
4-6
$13
10-12
$14
12-13
$26
Activity 11-13 should be shortened 1 week on Paths 1-2-5-7-11-13 & 1-3-8-11-13 because it
has the lowest crashing cost ($33) to shorten both paths. Both paths will decrease 1 week as
11 (second week) & 6-10 (first week). Step 4: Activity 11-13 (first week) & Activity 4–6
(first week). Step 5: Activity 11-13 (second week) & Activity 1-4 (first week).
Chapter 17 – Project Management
Step 6:
The paths and their expected duration are shown below:
Path
Expected Duration
1-2-5-7-11-13
30
1-3-8-11-13
30
1-3-9-12-13
20
1-4-6-10-12-13
30
Chapter 17 – Project Management
A summary of the crashing steps is shown below:
Path
Initial
time
Length
after
crashing 1
week
Length
after
crashing 2
weeks
Length
after
crashing 3
weeks
Length
after
crashing 4
weeks
Length
after
crashing 5
weeks
Length
after
crashing 6
weeks
1-2-5-7-11-13
1-3-8-11-13
1-3-9-12-13
1-4-6-10-12-13
35
32
20
33
34
32
20
33
33
32
20
33
32
32
20
32
31
31
20
31
30
30
20
30
29
29
20
30
Step 1
Crashing
Step 2
Crashing
Step 3
Crashing
Step 4
Crashing
Step 5
Crashing
Step 6
Crashing
Activity Cost
7-11 $15
Activity Cost
1-2 $18
Activity Cost
7-11 $20
6-10 $ 5
$25
Activity Cost
11-13 $30
4-6 $ 8
$38
Activity Cost
11-13 $33
1-4 $10
$43
Activity Cost
1-2 $22
1-3 $12
6-10 $12
$46
Conclusion: Expected project duration = 29 weeks (shortened by 6 weeks).
Total crashing cost = $15 + $18 + $25 + $38 + $43 + $46 = $185.
Total indirect costs = 29 x $40 = $1,160.
Total cost over 29 weeks = $185 + $1,160 = $1,345(000) = $1,345,000
The table below summarizes the cumulative crashing cost, indirect cost, and total cost for
project lengths of 35, 34, 33, 32, 31, 30, and 29 weeks:
Project
Length
Cumulative
Weeks
Shortened
Cumulative
Crash Cost
($000)
Indirect Cost
($000)
Total Cost
($000)
35
0
0
35(40) = 1,400
1,400
34
1
15
34(40) = 1,360
1,375
33
2
15 + 18 = 33
33(40) = 1,320
1,353
32
3
33 + 25 = 58
32(40) = 1,280
1,338
31
4
58 + 38 = 96
31(40) = 1,240
1,336
30
5
96 + 43 = 139
30(40) = 1,200
1,339
29
6
139 + 46 = 185
29(40) = 1,160
1,345