Application of project scheduling and Cost analysis Using PERT and CPM

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SHIV SHAKTI

International Journal of in Multidisciplinary and

Academic Research (SSIJMAR)

Vol. 4, No. 2, April 2015 (ISSN 2278 – 5973)

Application of project scheduling and Cost analysis

Using PERT and CPM

S.Shanmugasundaram

Assistant Professor, Department of Mathematics, Government Arts College (A), Salem.(TN)

V.Mathan Kumar

Assistant Professor, Department of Mathematics, Sri Ganesh College of Arts & Science, Salem.

Impact Factor = 3.133 (Scientific Journal Impact Factor Value for 2012 by Inno Space Scientific Journal Impact Factor)

Global Impact Factor (2013)= 0.326 (By GIF)

Indexing:

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Abstract:

Crashing is reducing project time by expending additional resources. Crashing the network is the

contracting or compressing the network that means to reduce project duration at minimum cost with

minimum project duration. It is found that there is a project schedule to determine the optimum cost

and time by CPM/PERT network analysis. In this paper we use optimum crashing method to

minimize the required cost while attaining a specified completion time.

Keywords: Crashing, Critical Path, Cost slope value, Time-cost Trade off

I-INTRODUCTION

Project management is a tool that is used by many companies to help improve performance and

competitiveness. Projects and their execution, in general, require resources. An important aspect of

project management is risk management. Different types of risk are present in any given project, but

the emphasis of this research will be focused on schedule/time risk and associated costs. The

schedule/time risk essentially implies not completing project activities on time, resulting in a late

completion of the project. Late project completion generally has negative effects for the company such

as penalty costs and customer dissatisfaction. Two of the most popular approaches to Project

management are the critical path method (CPM) and Project Evaluation and Review Technique

(PERT), which were developed short after the World War II. The critical path method (CPM) and

Project Evaluation and Review Technique (PERT) methods have been used since the 1950s to estimate

the completion time of a project.

II-Crashing

Crashing is a compression technique applied in project scheduling to shorten the project or activity

duration by exploring alternatives to apply maximum schedule compression at least additional cost.

An example of crashing is adding resources to the critical path tasks. The project manager is

frequently confronted with having to reduce the scheduled completion time of a project to meet a

deadline. In other words, the manager must finish the project sooner than indicated by the

CPM/PERT network analysis. Project crashing is a method for shortening the project duration by

reducing the time of one (or more) of the critical project activities to less than its normal activity time.

This reduction in the normal activity time is referred to as crashing.

III-Activity time-cost relationship

A simple representation of the possible relationship between the duration of an activity and its direct

costs appears in Figure 1. Considering only this activity in isolation and without reference to the

project completion deadline, a manager would choose a duration which implies minimum direct cost,

called the normal duration. At the other extreme, a manager might choose to complete the activity in

the minimum possible time, called crashed duration, but at a maximum cost. The linear relationship

shown in the Figure 1 between these two points implies that any intermediate duration could also be

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Fig 1: Linear time/cost trade-off for an activity

may represent the ideal or optimal trade-off between time and cost for this activity. The slope of the

line connecting the normal point (lower point) and the crash point (upper point) is called the cost

slope of the activity. The slope of this line can be calculated mathematically by knowing the

coordinates of the normal and crash points.

Cost slope = crash cost – normal cost / normal duration – crash duration

IV-Steps to approach the Crashing Network

1. Calculate cost slope of each activity of the project.

2. Identify critical path and find the expected duration of the project.

3. Select among the activities on critical path an activity which has least cost slope. For more than

one critical path, select one activity on each critical path such that total cost of crashing all these

activities is minimum among all such combinations of activities.

4. Reduce the activity time of the selected activity progressively until either crashed time reached or

the previous non-critical parallel path becomes critical.

5. Proceed above steps until there is at least one critical path on which none of the activities can be

further crashed.

Example:

The following table gives data on normal time and cost, Crash time and cost for a project.

TABLE: 1

Activity

Normal Crash

Time

(weeks)(NT)

Cost(Rs.)

(NC)

Time

(weeks)(CT)

Cost(Rs.)

(CC)

1-2 8 7000 3 10000

1-3 4 6000 2 8000

2-3 0 0 0 0

2-5 6 9000 1 11500

3-4 7 2500 5 3000

4-6 12 10000 8 16000

5-6 15 12000 10 16000

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6-8 5 10000 5 10000

7-8 14 6000 7 7400

7-9 8 6000 5 12000

8-9 6 6000 4 7800

Total Cost=Rs.86500

The indirect cost per week is Rs.1000

1. Draw the network for the project & critical path

2. Find the optimum time and optimum cost

3. Determine minimum total time & corresponding cost.

TABLE: 2

Activity

Normal Crash

∆T ∆C ∆C/∆T Time

(weeks)

(NT)

Cost(Rs.)

(NC)

Time

(weeks)

(CT)

Cost(Rs.)

(CC)

1-2 8 7000 3 10000 5 3000 600

1-3 4 6000 2 8000 2 2000 1000

2-3 0 0 0 0 0 0 0

2-5 6 9000 1 11500 5 2500 500

3-4 7 2500 5 3000 2 500 250

4-6 12 10000 8 16000 4 6000 1500

5-6 15 12000 10 16000 5 4000 800

5-7 7 12000 6 14000 1 2000 2000

6-8 5 10000 5 10000 0 0 0

7-8 14 6000 7 7400 7 1400 200

7-9 8 6000 5 12000 3 6000 2000

8-9 6 6000 4 7800 2 1800 900

1. Draw the network of the proposed project.

6 7

8 8

0 15 14

6

4

7 12 5

Critical path starting towards end

1

8

3 4 6

2 5 7

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Activity Duration Total

1-2 2-5 5-7 7-9 8 6 7 8 29

1-2 2-5 5-7 7-8 8-9 8 6 7 14 6 41

1-2 2-5 5-6 6-8 8-9 8 6 15 5 6 40

1-2 2-3 3-4 4-6 6-8 8-9 8 0 7 12 5 6 38

1-3 3-4 4-6 6-8 8-9 4 7 12 5 6 34

1→2→5→7→9 =29 weeks,

1→2→5→6→8→9 =40 weeks,

1→2→3→4→6→8→9 =38 weeks,

1→4→4→6→8→9 =34 weeks,

1→2→5→7→8→9 =41 weeks it is critical path (CPM) of the project duration.

2. Optimum time and Optimum cost.

Step:1

Crashing in activity 7-8 in 1 week

Now we are identified their activity in critical path which can be crashed with the lowest cost

slope that are to be choose as list activity 7-8 ,∆C/∆T =200.

1-2 2-5 5-7 7-9 8 6 7 8 29

1-2 2-5 5-7 7-8 8-9 8 6 7 13 6 40 1-2 2-5 5-6 6-8 8-9 8 6 15 5 6 40

1-2 2-3 3-4 4-6 6-8 8-9 8 0 7 12 5 6 38

1-3 3-4 4-6 6-8 8-9 4 7 12 5 6 34

Step:2

Crashing in activity 2-5 in 2 weeks

1-2 2-5 5-7 7-9 8 4 7 8 27

1-2 2-5 5-7 7-8 8-9 8 4 7 13 6 38 1-2 2-5 5-6 6-8 8-9 8 4 15 5 6 38 1-2 2-3 3-4 4-6 6-8 8-9 8 0 7 12 5 6 38

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Step:3

1-2 2-5 5-7 7-9 4 4 7 8 23

1-2 2-5 5-7 7-8 8-9 4 4 7 13 6 34 1-2 2-5 5-6 6-8 8-9 4 4 15 5 6 34 1-2 2-3 3-4 4-6 6-8 8-9 4 0 7 12 5 6 34 1-3 3-4 4-6 6-8 8-9 4 7 12 5 6 34

Step:4

1-2 2-5 5-7 7-9 4 4 7 8 23

1-2 2-5 5-7 7-8 8-9 4 4 7 13 4 32 1-2 2-5 5-6 6-8 8-9 4 4 15 5 4 32 1-2 2-3 3-4 4-6 6-8 8-9 4 0 7 12 5 4 32 1-3 3-4 4-6 6-8 8-9 4 7 12 5 4 32

Step:5

Crashing in activity 2-5 in 2 weeks and activity 3-4 in 2 weeks

1-2 2-5 5-7 7-9 4 2 7 8 21

1-2 2-5 5-7 7-8 8-9 4 2 7 13 4 30 1-2 2-5 5-6 6-8 8-9 4 2 15 5 4 30 1-2 2-3 3-4 4-6 6-8 8-9 4 0 5 12 5 4 30 1-3 3-4 4-6 6-8 8-9 4 5 12 5 4 30

Step:6

Crashing in activity 1-2 in 1 week and activity 1-3 in 1 week

1-2 2-5 5-7 7-9 3 2 7 8 20

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Step:7

Crashing in activity 2-5 in 1 week and activity 4-6 in 1 week

1-2 2-5 5-7 7-9 3 1 7 8 19 1-2 2-5 5-7 7-8 8-9 3 1 7 13 4 28 1-2 2-5 5-6 6-8 8-9 3 1 15 5 4 28 1-2 2-3 3-4 4-6 6-8 8-9 3 0 5 11 5 4 28 1-3 3-4 4-6 6-8 8-9 3 5 11 5 4 28

Step:8

Crashing in activity 7-8 in 3 weeks, activity 5-6 in 3 weeks and activity 4-6 in 3 weeks

1-2 2-5 5-7 7-9 3 1 7 8 19

1-2 2-5 5-7 7-8 8-9 3 1 7 10 4 25 1-2 2-5 5-6 6-8 8-9 3 1 12 5 4 25 1-2 2-3 3-4 4-6 6-8 8-9 3 0 5 8 5 4 25 1-3 3-4 4-6 6-8 8-9 3 5 8 5 4 25

The above results are summarized in the table below:

Activity

Crashed

No.of

Week

Crash _Repr

es ent step Weeks saved in project P rojec t Dur ati on Normal Direct cost (DC) Rs. Indirect cost (IC) Rs. Crash Cost (CC) Rs. Total cost= CC+IC+CC Rs.

CPM - Fig:4 - 41 86500 41X1000

=41000 0 127500

7-8 1 step:1 1 40 86500 40X1000

=40000

1X200=20

0 126700

2-5 2 step:2 2 38 86500 38X1000

=38000

200+

(2X500)

=1200

125700

1-2 4 step:3 4 34 86500 34X1000

=34000

1200+

(4X600)

=3600

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8-9 2 step:4 2 32 86500 32X1000 =32000

3600+

(2X900)

=5400

123900

2-5 2

step:5 2 30 86500 30X1000 =30000

5400+ (2X500) +(2X250) =6900

123400 3-4 2

1-2 1

step:6 2 29 86500 29X1000

=29000

6900+

(1X600)

+(1X1000)

=8500

124000

1-3 1

2-5 1

step:7 1 28 86500 28X1000

=28000

8500+

(1X500)

+(1X1500)

=10500

125000

4-6 1

7-8 3

step:8 3 25 86500 25X1000 =25000

10500+ (3X200)+ (3X800)+ (3X1500) =18000

132500 5-6 3

4-6 3

V-Results

The above table indicates,

the normal duration of the project is 41weaks and the corresponding cost is Rs.1, 27,500/-,

the optimum duration of the project is 30weaks and the corresponding cost is Rs.1,23,400/-,

the minimum duration of the project is 25weaks and the corresponding cost is Rs.1,32,500/.

VI- Conclusion

The time network is shown in above steps all the activities are fully crashed. Thus the minimum

project duration is 25 weeks. The maximum (complete) crashing of the network from 41 weeks to 25

weeks. We see that all the activities on the critical path which could be crashed as per the given

condition have been possible to further crash the project network.

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