Chapter 10, Problem 26AP

A.

Summary Introduction

Interpretation: A network diagram representing the activities of the project is to be constructed.

Concept Introduction: Network diagram is a graphical representation of tasks and events happening in a project. It depicts the work flow and is used to compute the total duration of the project.

Explanation of Solution

Given information: Theactivities to be taken up for the construction of a gas station and the time duration for each are given in the table.

Activity	Time
A	4
B	2
C	3
D	1
E	1
F	2
G	3
H	1
I	5
J	3
K	4
L	2
M	1

The activity times and the precedence relationship are shown in the table:

Activity	Time	Predecessors
A	4
B	2	A
C	3	B
D	1	C
E	1	D
F	2	B
G	3	F
H	1	G
I	5	B
J	3	I
K	4	C, F, I
L	2	K
M	1	L

The network diagram showing the work flows is as follows:

  

B.

Summary Introduction

Interpretation: The critical path is to be identified by computing the earliest and the latest start and finish times.

Concept Introduction: Network diagram is a graphical representation of tasks and events happening in a project. It depicts the work flow and is used to compute the total duration of the project.

Answer to Problem 26AP

The critical path is A − B − I − K − L − M with completion time as 18 weeks

Explanation of Solution

Given information:

Activity	Time
A	4
B	2
C	3
D	1
E	1
F	2
G	3
H	1
I	5
J	3
K	4
L	2
M	1

  

The earliest and latest start and finish times are calculated using the formulae:

Earliest Start (EST) = Maximum EFT of the predecessor activity

Latest Start (LST) = LFT - Duration

Earliest Finish (EFT) = EST + Duration

Latest Finish (LFT) = Minimum LST of the immediate successor

Slack = LFT − EFT (or) LST − EST

Activity	Time	EST	EFT	LST	LFT	Slack
A	4	0	4	0	4	0
B	2	4	6	4	6	0
C	3	6	9	8	11	2
D	1	9	10	16	17	7
E	1	10	11	17	18	7
F	2	6	8	9	11	3
G	3	8	11	14	17	6
H	1	11	12	17	18	6
I	5	6	11	6	11	0
J	3	11	14	15	18	4
K	4	11	15	11	15	0
L	2	15	17	15	17	0
M	1	17	18	17	18	0

The critical path includes the activities with zero slack. So, here the critical path is A − B− I − K − L − M. The total time taken to complete the project is 18 weeks.

C.

Summary Introduction

Interpretation: A Gantt chart of the schedule for the project is to be drawn.

Concept Introduction: Gantt chart is a project management tool that is a type of bar chart. It depicts the project schedule and is used for resource allocation.

Answer to Problem 26AP

A Gantt chart for the project has been drawn.

Explanation of Solution

Given information:

Activity	Time
A	4
B	2
C	3
D	1
E	1
F	2
G	3
H	1
I	5
J	3
K	4
L	2
M	1

Activity times and early start times:

Activity	EST	Time
A	0	4
B	4	2
C	6	3
D	9	1
E	10	1
F	6	2
G	8	3
H	11	1
I	6	5
J	11	3
K	11	4
L	15	2
M	17	1

The Gantt chart depicting the project based on the EST:

  

D.

Summary Introduction

Interpretation: The impact of replacement of the air compressor which takes two weeks on the completion of the project is to be determined.

Concept Introduction: Network diagram is a graphical representation of tasks and events happening in a project. It depicts the work flow and is used to compute the total duration of the project.

Explanation of Solution

Given information:

Activity	Time
A	4
B	2
C	3
D	1
E	1
F	2
G	3
H	1
I	5
J	3
K	4
L	2
M	1

Installing the air compressor is represented by activity G.

Activity	Time	EST	EFT	LST	LFT	Slack
A	4	0	4	0	4	0
B	2	4	6	4	6	0
C	3	6	9	8	11	2
D	1	9	10	16	17	7
E	1	10	11	17	18	7
F	2	6	8	9	11	3
G	3	8	11	14	17	6
H	1	11	12	17	18	6
I	5	6	11	6	11	0
J	3	11	14	15	18	4
K	4	11	15	11	15	0
L	2	15	17	15	17	0
M	1	17	18	17	18	0

The latest start time of the activity G is 14^th week. If the failure is identified before installation, the project will not be delayed as the replacement time (2 weeks) is less than the slack (6 weeks).

If the failure is found during testing of the air compressor, represented by activity H, still the project would not be delayed, as again there is a slack of 6 weeks.

E.

Summary Introduction

Interpretation: The activities that can be completed by the end of the 15^th week without delaying the project completion are to be determined.

Concept Introduction: Network diagram is a graphical representation of tasks and events happening in a project. It depicts the work flow and is used to compute the total duration of the project.

Answer to Problem 26AP

The activities A, B, C, F and I have to be completed by the end of the 15^th week to guarantee project completion without delay.

Explanation of Solution

Given information:

Activity	Time
A	4
B	2
C	3
D	1
E	1
F	2
G	3
H	1
I	5
J	3
K	4
L	2
M	1

The table shows the Earliest and Latest times of the activities

Activity	Time	EST	EFT	LST	LFT	Slack
A	4	0	4	0	4	0
B	2	4	6	4	6	0
C	3	6	9	8	11	2
D	1	9	10	16	17	7
E	1	10	11	17	18	7
F	2	6	8	9	11	3
G	3	8	11	14	17	6
H	1	11	12	17	18	6
I	5	6	11	6	11	0
J	3	11	14	15	18	4
K	4	11	15	11	15	0
L	2	15	17	15	17	0
M	1	17	18	17	18	0

The latest finish times of the activities A, B, C, F and Iam less than 15 weeks. Therefore, these activities have to be completed by the end of the 15^th weeks to guarantee project completion without any delay.

F.

Summary Introduction

Interpretation: The optimal solution for the problem is to be determined using linear programming.

Concept Introduction: Linear programming is a mathematical technique used to achieve the best outcome, given a list of parameters usually restrictions on the resources.

Answer to Problem 26AP

The optimal solution for the problem is obtained using linear programming.

Explanation of Solution

Given information:

Activity	Time
A	4
B	2
C	3
D	1
E	1
F	2
G	3
H	1
I	5
J	3
K	4
L	2
M	1

The various paths of the project and the durations are:

  

The critical path is A − B − I − K − L − M with the completion time of 18 weeks.

Using linear programming, the EST is calculated.

Objective function = =SUM (B2:M2)

Nodes	1	2	3	4	5	6	7	8	9	10	11	12
EST

The constraints

Activity			Time
A	0	>	4
B	0	>	2
C	0	>	3
D	0	>	1
E	0	>	1
F	0	>	2
G	0	>	3
H	0	>	1
I	0	>	5
J	0	>	3
K	0	>	4
L	0	>	2
M	0	>	1
P1	0	>	0
P2	0	>	0
P3	0	>	0

Solver Solution:

Objective Function

120

EST for the nodes:

Nodes	1	2	3	4	5	6	7	8	9	10	11	12
EST	0	4	6	9	8	11	11	11	10	15	17	18