Chapter 17, Problem 15P

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A construction project has indirect costs totaling $40,000 per week. Major activities in the project and their expected times in weeks are shown in this precedence diagram.

Crashing costs for each activity are:

a. Determine the optimum time–cost crashing plan.

b. Plot the total-cost curve that describes the least expensive crashing schedule that will reduce the project length by six weeks.

Summary Introduction

To determine: The optimum cost-saving plan.

Introduction:

Project crashing:

It is method to shorten the total time taken for a project by reducing the time taken for one or more activities on the critical path. The reduction in the normal time taken is known as crashing.

Answer to Problem 15P

The activities to be crashed are 7-11, 1-2, 6-10, 11-13, and 4-6. The total crashing cost is $1,336,000.

Explanation of Solution

Given information:

• Indirect cost is $40,000 per week.

Activity	Crash cost first week ($000)	Crash cost second week ($000)	Crash cost third week ($000)
1 to 2	18	22
2 to 5	24	25	25
5 to 7	30	30	35
7 to 11	15	20
11 to 13	30	33	36
1 to 3	12	24	26
3 to 8
8 to 11	40	40	40
3 to 9	3	10	12
9 to 12	2	7	10
12 to 13	26
1 to 4	10	15	25
4 to 6	8	13
6 to 10	5	12
10 to 12	14	15

Project crashing:

Calculation of expected duration of each path:

Path 1-2-5-7-11-13:

$\begin{array}{c}\text{Expected duration}=5+8+7+4+11\\ =35\end{array}$

Path 1-3-8-11-13:

$\begin{array}{c}\text{Expected duration}=4+12+5+11\\ =32\end{array}$

Path 1-3-9-12-13:

$\begin{array}{c}\text{Expected duration}=4+6+9+1\\ =20\end{array}$

Path 1-4-6-10-12-13:

$\begin{array}{c}\text{Expected duration}=3+12+9+8+1\\ =33\end{array}$

Step 1:

Critical path is 1-2-5-7-11-13.

The activities are ranked according to the cost per week to crash.

Activity	Cost ($)
7-11	15
1-2	18
2-5	24
5-7	30
11-13	30

Activity 7-11 will be crashed first by 1 week since it has the lowest crashing cost ($15) and this cost is ≤ 40. Path 1-2-5-7-11-13 will decrease by 1 week.

Step 2:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13.

The activities are ranked according to the cost per week to crash.

Activity	Cost ($)
1-2	$18
7-11	$20
2-5	$24
5-7	$30
11-13	$30

Activity 1-2 will be crashed first by 1 week since it has the lowest crashing cost ($18) and this cost is ≤ 40. Path 1-2-5-7-11-13 will decrease by 1 week.

Step 3:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13 and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
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 7-11 will be crashed first by 1 week since it has the lowest crashing cost ($20). Path 1-2-5-7-11-13 will decrease by 1 week.

Activity 6-10 will be crashed first by 1 week since it has the lowest crashing cost ($5). Path 1-4-6-10-12-13 will decrease by 1 week.

The combined crash cost ($25) is ≤ $40.

Step 4:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13, 1-3-8-11-13, and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
1-2-5-7-11-13	1-2	$22
	2-5	$24
	5-7	$30
	11-13	$30
1-3-8-11-13	1-3	$12
	11-13	$30
	8-11	$40
1-4-6-10-12-13	4-6	$8
	1-4	$10
	6-10	$12
	10-12	$14
	12-13	$26

Activity 11-13 will be crashed first by 1 week since it has the lowest crashing cost ($30). Paths 1-2-5-7-11-13 and 1-3-8-11-13 will decrease by 1 week.

Activity 4-6 will be crashed first by 1 week since it has the lowest crashing cost ($8). Path 1-4-6-10-12-13 will decrease by 1 week.

The combined crash cost ($38) is ≤ $40.

Step 5:

The paths and new expected duration are:

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 path is 1-2-5-7-11-13, 1-3-8-11-13, and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
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 could be crashed first by 1 week since it has the lowest crashing cost ($33). Paths 1-2-5-7-11-13 and 1-3-8-11-13 will decrease by 1 week.

Activity 1-4 could be crashed first by 1 week since it has the lowest crashing cost ($10). Path 1-4-6-10-12-13 will decrease by 1 week.

The combined crash cost ($43) is ≥ $40.

Since the marginal cost of crashing is greater than the marginal benefit of crashing, crashing will be stopped at step 4.

The final project duration time is 31 weeks. The activities that are crashed are:

Activity 7-11 (First week)

Activity 7-11 (Second week)

Activity 1-2

Activity 6-10

Activity 11-13

Activity 4-6

Calculation of total crashing cost:

The total crashing cost is calculated by summing the crashing cost involved all the steps and the indirect costs every week.

$\begin{array}{c}\text{Total crashing cost}=15+18+25+38+\left(31\text{ weeks}\times \$40\right)\\ =96+1,240\\ =\$1,336\end{array}$

The activities to be crashed are: 7-11, 1-2, 6-10, 11-13, and 4-6. The total crashing cost is $1,336,000.

Summary Introduction

To Plot: The cost curve with the least expensive crashing which will reduce the project by 6 weeks.

Introduction:

Project crashing:

It is method to shorten the total time taken for a project by reducing the time taken for one or more activities on the critical path. The reduction in the normal time taken is known as crashing.

Answer to Problem 15P

Cost curve:

Explanation of Solution

Given information:

• Indirect cost is $40,000 per week.

Activity	Crash cost first week ($000)	Crash cost second week ($000)	Crash cost third week ($000)
1 to 2	18	22
2 to 5	24	25	25
5 to 7	30	30	35
7 to 11	15	20
11 to 13	30	33	36
1 to 3	12	24	26
3 to 8
8 to 11	40	40	40
3 to 9	3	10	12
9 to 12	2	7	10
12 to 13	26
1 to 4	10	15	25
4 to 6	8	13
6 to 10	5	12
10 to 12	14	15

Project crashing:

Calculation of expected duration of each path:

Path 1-2-5-7-11-13:

$\begin{array}{c}\text{Expected duration}=5+8+7+4+11\\ =35\end{array}$

Path 1-3-8-11-13:

$\begin{array}{c}\text{Expected duration}=4+12+5+11\\ =32\end{array}$

Path 1-3-9-12-13:

$\begin{array}{c}\text{Expected duration}=4+6+9+1\\ =20\end{array}$

Path 1-4-6-10-12-13:

$\begin{array}{c}\text{Expected duration}=3+12+9+8+1\\ =33\end{array}$

Step 1:

Critical path is 1-2-5-7-11-13.

The activities are ranked according to the cost per week to crash.

Activity	Cost ($)
7-11	15
1-2	18
2-5	24
5-7	30
11-13	30

Activity 7-11 will be crashed first by 1 week since it has the lowest crashing cost ($15) and this cost is ≤ 40. Path 1-2-5-7-11-13 will decrease by 1 week.

Step 2:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13.

The activities are ranked according to the cost per week to crash.

Activity	Cost ($)
1-2	$18
7-11	$20
2-5	$24
5-7	$30
11-13	$30

Activity 1-2 will be crashed first by 1 week since it has the lowest crashing cost ($18) and this cost is ≤ 40. Path 1-2-5-7-11-13 will decrease by 1 week.

Step 3:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13 and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
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 7-11 will be crashed first by 1 week since it has the lowest crashing cost ($20). Path 1-2-5-7-11-13 will decrease by 1 week.

Activity 6-10 will be crashed first by 1 week since it has the lowest crashing cost ($5). Path 1-4-6-10-12-13 will decrease by 1 week.

The combined crash cost ($25) is ≤ $40.

Step 4:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13, 1-3-8-11-13, and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
1-2-5-7-11-13	1-2	$22
	2-5	$24
	5-7	$30
	11-13	$30
1-3-8-11-13	1-3	$12
	11-13	$30
	8-11	$40
1-4-6-10-12-13	4-6	$8
	1-4	$10
	6-10	$12
	10-12	$14
	12-13	$26

Activity 11-13 will be crashed first by 1 week since it has the lowest crashing cost ($30). Paths 1-2-5-7-11-13 and 1-3-8-11-13 will decrease by 1 week.

Activity 4-6 will be crashed first by 1 week since it has the lowest crashing cost ($8). Path 1-4-6-10-12-13 will decrease by 1 week.

The combined crash cost ($38) is ≤ $40.

Step 5:

The paths and new expected duration are:

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 path is 1-2-5-7-11-13, 1-3-8-11-13, and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
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 will be crashed first by 1 week since it has the lowest crashing cost ($33). Paths 1-2-5-7-11-13 and 1-3-8-11-13 will decrease by 1 week.

Activity 1-4 will be crashed first by 1 week since it has the lowest crashing cost ($10). Path 1-4-6-10-12-13 will decrease by 1 week. The combined crash cost is ($43).

Step 6:

The paths and new expected duration are:

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

Critical path is 1-2-5-7-11-13, 1-3-8-11-13, and 1-4-6-10-12-13.

The activities are ranked according to the cost per week to crash.

Path	Activity	Cost ($)
1-2-5-7-11-13	1-2	$22
	2-5	$24
	5-7	$30
	11-13	$36
1-3-8-11-13	1-3	$12
	11-13	$36
	8-11	$40
1-4-6-10-12-13	6-10	$12
	4-6	$13
	10-12	$14
	1-4	$15
	12-13	$26

Activity 1-2 will be crashed first by 1 week since it has the lowest crashing cost ($22). Path 1-2-5-7-11-13 will decrease by 1 week.

Activity 1-3 will be crashed first by 1 week since it has the lowest crashing cost ($12). Path 1-3-8-11-13 will decrease by 1 week.

Activity 6-10 will be crashed first by 1 week since it has the lowest crashing cost ($12). Path 1-4-6-10-12-13 will decrease by 1 week. The combined crash cost is ($436.

The final project duration time is 29 weeks. The activities that are crashed are:

Activity 7-11 (First week)

Activity 7-11 (Second week)

Activity 1-2 (First week)

Activity 1-2 (Second week)

Activity 11-13 (First week)

Activity 11-13 (Second week)

Activity 4-6

Activity 6-10

Activity 1-4

Activity 1-3

Calculation of total crashing cost:

The total crashing cost is calculated by summing the crashing cost involved all the steps and the indirect costs every week.

$\begin{array}{c}\text{Total crashing cost}=15+18+25+38+\left(29\text{ weeks}\times \$40\right)\\ =185+1,160\\ =\$1,345\end{array}$

Summarization of total costs for different project lengths:

Project Length	Cumulative Weeks shortened	Cumulative crash cost ($000)	Indirect cost ($000)	Total cost ($000)
A	B	C	$\left(\text{D}=\text{A}\times \text{40}\right)$	E = C+D
35	0	$                              -	$   1,400.00	$ 1,400.00
34	1	$                         15.00	$   1,360.00	$ 1,375.00
33	2	$                         33.00	$   1,320.00	$ 1,353.00
32	3	$                         58.00	$   1,280.00	$ 1,338.00
31	4	$                         96.00	$   1,240.00	$ 1,336.00
30	5	$                       139.00	$   1,200.00	$ 1,339.00
29	6	$                       185.00	$   1,160.00	$ 1,345.00

Cost curve:

The cost curve is plotted by taking the project length on the X-axis and the total cost on the Y-axis.