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Chapter: Operations Research: An Introduction - Network Models

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CPM AND PERT: Network Representation

Each activity of the project is represented by an arc pointing in the direction of progress in the project. The nodes of the network establish the precedence relation-ships among the different activities.

Network Representation

 

Each activity of the project is represented by an arc pointing in the direction of progress in the project. The nodes of the network establish the precedence relation-ships among the different activities.

 

Three rules are available for constructing the network.

 

Rule 1. Each activity is represented by one, and only one, arc.

Rule 2. Each activity must be identified by two distinct end nodes.

 

Figure 6.39 shows how a dummy activity can be used to represent two concurrent activities, A and B. By definition, a dummy activity, which normally is depicted by a



dashed are, consumes no time or resources. Inserting a dummy activity in one of the four ways shown in Figure 6.39, we maintain the concurrence of A and B, and provide unique end nodes for the two activities (to satisfy rule 2).

 

Rule 3. To maintain the correct precedence relationships, the following questions must be answered as each activity is added to the network:

 

                                                                     i.What activities must immediately precede the current activity?

 

                                                                  ii.What activities must follow the current activity?

 

                                                                iii.What activities must occur concurrently with the current activity?

 

The answers to these questions may require the use of dummy activities to ensure correct precedences among the activities. For example, consider the following segment of a project:

 

a) Activity C starts immediately after A and B have been completed.

 

b)  Activity E starts only after B has been completed.

 

Part (a) of Figure 6.40 shows the incorrect representation of the precedence relation-ship because it requires both A and B to be completed before E can start. In part (b), the use of a dummy activity rectifies the situation.

 

 

Example 6.5-1

 

A publisher has a contract with an author to publish a textbook. The (simplified) activities associated with the production of the textbook are given below. The author is required to submit to the publisher a hard copy and a computer file of the manuscript. Develop the associated net-work for the project.



Figure 6.41 provides the network describing the precedence relationships among the different activities. Dummy activity (2,3) produces unique end nodes for concurrent activities A and B. It is convenient to number the nodes in ascending order in the direction of progress in the project.

 

 

PROBLEM SET 6.5A

 

1. Construct the project network comprised of activities A to L with the following prece-dence relationships:

 

a)     A, B, and C, the first activities of the project, can be executed concurrently.

 

b)    A and B precede D.

 

c)     B precedes E, F, and H.

 

d)    F and C precede G.

 

e)     E and H precede I and J.

 

f)    C, D, F, and J precede K.

 

g)     K precedes L.

 

h)    I, G, and L are the terminal activities of the project.

 

2. Construct the project network comprised of activities A to P that satisfies the following precedence relationships:

 

a.     A, B, and C, the first activities of the project, can be executed concurrently.

 

b.     D, E, and F follow A.

 

c.      I and G follow both Band D.

 

d.     H follows both C and G.

 

e.      K and L follow 1.

 

f.       J succeeds both E and H.

 

g.     M and N succeed F, but cannot start until both E and H are completed.

 

h.     succeeds M and I.

 

i.       P succeeds 1, L, and o.

 

j.       K, N, and P are the terminal activities of the project.

 

*3. The footings of a building can be completed in four consecutive sections. The activities for each section include (1) digging, (2) placing steel, and (3) pouring cqncrete. The digging of one section cannot start until that of the preceding section has been completed. The same restriction applies to pouring concrete. Develop the project network.

 

4. In Problem 3, suppose that 10% of the plumbing work can be started simultaneously with the digging of the first section but before any concrete is poured. After each section of the footings is completed, an additional 5% of the plumbing can be started provided that the preceding 5% portion is complete. The remaining plumbing can be completed at the end of the project. Construct the project network.

 

5. An opinion survey involves designing and printing questionnaires, hiring and training personnel, selecting participants, mailing questionnaires, and analyzing the data. Construct the project network, stating all assumptions.

 

6. The activities in the following table describe the construction of a new house. Construct the associated project network.


7. A company is in the process of preparing a budget for launching a new product. The fol-lowing table provides the associated activities and their durations. Construct the project network.


8. The activities involved in a candlelight choir service are listed in the following table. Construct the project network.


9. The widening of a road section requires relocating ("reconductoring") 1700 feet of 13.8-kV overhead primary line. The following table summarizes the activities of the project. Construct the associated project network.


10. The following table gives the activities for buying a new car. Construct the project network.



 

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