Problems/Solutions

Problems/Solutions

Problems/Solutions
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Problems

1. A project consists of 10 activities, lettered A through J, as shown in the accompanying precedence diagram, with the activities on the arrows. For each activity, the deterministic time estimate in weeks is shown in the following table.

Activity

Time

Activity

Time

A

B

C

D

E

2

4

3

5

6

F

G

H

I

J

3

4

4

3

1

List all of the paths through the network.
What is the duration of each of the paths?
What is the critical path?
What is the second-most critical path?
What is the slack time for each activity?
Calculate the ES, Ef, LS and LF times for each activity.

2. A project consists of 11 activities, lettered A through K, below. For each activity, the preceding activity is given, and a deterministic estimate of the length of time required to complete it in weeks.

	Activity	Time	Preceding Activity	Activity	Time	Preceding Activity
	A	1 week	-	G	4 weeks	E
	B	3	A	H	6	F
	C	2	A	I	2	G
	D	4	C	J	1	H,I
	E	2	-	K	1	B,D,J
	F	3	E

Draw the precedence network for this project, with the activities on the arrows.
List all of the paths through the network.
What is the duration of each of the paths?
What is the critical path?
What is the slack time for each activity?
The materials required to accomplish activity F have been delayed for two weeks by a strike at the supplier's plant. What effect will this have on the length of time required to complete the project?
An equipment breakdown has delayed activity B for one week. What effect will this have on the length of time required to complete the project?

3. A project consists of 8 activities, lettered A through H. below. For each activity, the preceding activity is given, and a probabilistic estimate of the time required to complete it. Times are in days.

		Preceding	Optimistic	Most Likely	Pessimistic
	Activity	Activity	Time	Time	Time
	A	--	2 days	4 days	6 days
	B	A	3	6	9
	C	A	2	5	11
	D	--	2	10	12
	E	C,	4	8	15
	F	B,E	2	4	12
	G	D	3	4	11
	H	F,G	1	1	1

Determine the expected time for each activity.
Determine the variance for each activity
Draw the PERT network for this project, with the activities on the arrows.
List all of the paths through the network.
What is the duration of each path?
What is the variance of each path?
What is the critical path?
What is the second-most critical path?
Activity D was delayed three days by an earthquake in the area. What effect does this have on the length of time required to complete the project?

4. Refer to Problem 3 and the original critical path before the earthquake.

What is the mean of the probability distribution for the completion time?
What is the standard deviation of the probability distribution for the completion time?
What is the probability of finishing the project within 24 days?
What is the probability of finishing the project within 22 days?
What is the probability that the project will take longer than 28 days?

5. After the earthquake, in Problem 3-i., management faced the problem of how to make up the lost time, and how much it would cost. Estimates of crash times and costs for each activity are given below.

	Activity	Crash Time	Cost per Day
	A	Not possible	-
	B	3 days	$200 per day
	C	not possible	-
	D	not possible*	-
	E	5 days	$300 per day
	F	2 days	$500 per day
	G	not possible	-
	H	not possible	-

*(because of the earthquake)

Which activity should be crashed?
How many days should it be crashed?
How much will it cost?

6. Construct the PERT network for Problem 3 with the activities on the nodes.

Solutions

1. a, b.

Path

Duration (weeks)

A-B-C-I-J

A-D-F-I-J

A-D-G-H-J

E-F-I-J

E-G-H-J

2 + 4 + 3 + 3 + 1 = 13

2 + 5 + 3 + 3 + 1 = 14

2 + 5 + 4 + 4 + 1 = 16

6 + 3 + 3 + 1 = 13

6 + 4 + 4 + 1 = 15

c. The critical path is #3.
d. The second-most critical path is #5.
e. The slack is the difference between the duration of the critical path and the duration of each alternative path. When an activity appears on more than one path, the slack is the smallest difference.

Activity

Path

Slack (wk.)

Activity

Path

Slack (Wk.)

A

B

C

D

E

3

1

1

3

5

0

3

3

0

1

F

G

H

I

J

2

3

3

2

3

2

0

0

2

0

f.

Activity

ES

EF

LS

LF

Slack (wk.)

A

B

C

D

E

F

G

H

I

J

0

2

6

2

0

7

7

11

10

15

2

6

9

7

6

10

11

15

13

16

0

5

9

2

1

9

7

1

2

5

2

9

12

7

7

12

11

15

15

16

0

3

3

0

1

2

0

0

2

0

2. a.

b, c.

Path

Duration (weeks)

A-B-DI-DII-K

A-C-D-DII-K

E-F-H-J-K

E-G-I-J-K

1 + 3 + 0 + 0 + 1 = 5

1 + 2 + 4 + 0 + 1 = 8

2 + 3 + 6 + 1 + 1 = 13

2 + 4 + 2 + 1 + 1 = 10

d. The critical path is #3, with a duration of 13 weeks.

e.

Activity

Path

Slack (wk.)

Activity

Path

Slack (wk.)

A

B

C

D

DI

DII

E

2

1

2

2

1

2

3

5

8

5

5

8

5

0

F

G

G

I

J

K

3

4

3

4

3

3

0

3

0

3

0

0

f. It will delay the completion of the project by 2 weeks.
g. It will have no effect on the completion of the project.

a, b. For activity A: Image141 days.

Repeat for each activity.

For activity A: Image142 .

Repeat for each activity.

Activity

Expected time (days)

Variance

Activity

Expected

Time (days)

Variance

A

B

C

D

4

6

5.5

9

0.4444

1.0000

2.2500

2.7778

E

F

G

H

8.5

5

5

1

3.3611

2.7778

1.7778

0

c.

d, e. There are four paths through the network.

Path

Duration (days)

A-B-DI-F-H

A-C-E-F-H

D-DII-E-F-H

D-G-DIII-H

4 + 6 + 0 + 5 + 1 = 16

4 + 5.5 + 8.5 + 5 + 1 = 24

9 + 0 + 8.5 + 5 + 1 = 23.5

9 + 5 + 0 + 1 = 15

f. The variance for each path is obtained by adding the variances of the activities on the path. For Path #1) Variance = .4444 + 1.0000 + 0 + 2.7778 + 0 = 4.2222. Repeat for each path.

Path

Variance

1)

2)

3)

4)

4.2222

8.8333

8.9167

4.5556

g. The critical path is #2, with a duration , Image143 days.
h. The second-most critical path is #3, with a duration of 23.5 days.
i. The duration along the path, D - DII - E - F - H, is increased to 26.5 days. Because this total is larger than 24 days, this constitutes a new critical path.

4. a. The mean is the longest duration, or 24 days.
b. Image144 .
c. Find the value of z in order to read the table of the areas of the standardized normal curve:

Image145

Image146 .
d. Image147

Image148 .

e.Image149

Image150 .

5. a. Activity E should be crashed because it is on the critical path and crashing it would be less expensive than crashing activity F.
b. Activity E should be crashed for 3 days to make up for the earthquake.
c. The expense will be: 3 days x $200 = $600.

6.

7. From Monday, May 1 to Friday, August 11 is 15 work weeks; set the last LS equal to 15.

Activity

ES

EF

LS

LF

Slack (wk.)

A

B

C

D

DI

DII

E

F

G

H

I

J

K

0

1

1

3

4

7

0

2

2

5

6

11

12

1

4

3

7

4

7

2

5

6

11

18

12

13

7

1

8

0

4

4

2

4

7

7

11

13

14

8

14

10

14

14

14

4

7

11

13

13

14

15

7

10

7

7

10

7

2

2

5

2

5

2

2

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