Write a linear program for determining the minimum cost of completing this project in half a year (=26 weeks). Show your work and explain in details.

Caterpillar Inc. is an American corporation which designs, manufactures, markets and sells machinery, engines, financial products and insurance to customers via a worldwide dealer network Caterpillar equipment sets the standard for our industry. They will remain the leader by continuing to help their customers meet their needs with durable and reliable equipment. Caterpillar is envisioning introduction of a new grader which brings the latest emissions reduction technology to the most durable, productive and comfortable motor graders on the market. A feasibility study will be conducted, if the study supports introduction of the new the M Series 3, a manufacturing building is to be secured and a project leader will be hired. Once the building is secured, Caterpillar will be committed to the project. Therefore an advertising company will be selected and the raw materials for the manufacturing process will be ordered. When, in addition to securing the building, a project leader has been named, manufacturing personnel will be hired. After the manufacturing personnel have been hired and the raw materials purchased, an iconic model an iconic model of the M Series 3 class will be produced. Following the completion of the prototype, work will begin on a production run of 100 the M Series 3 class. When both the iconic model has been built and the advertising personnel have been selected, a rigorous advertising campaign will be launched.

The next phase of the project is the development phase, which will be complete with the production of the first 100 the M Series 3 class and the initiation of the advertising campaign. In order to help their final decision, Caterpillar has conducted two separate costs analyses. One is effective under current (normal) conditions, while the other is effective if the development phase of the activities is “crashed”. Below are the summarized information (Times are in weeks.)

Activity

Normal Time

Normal Cost

Crash Time

Crash Cost

Feasibility Study (A)

6

$80,000

5

$100,000

Building Purchased (B)

4

$100,000

4

$100,000

Project Leader Hired (C)

3

$50,000

2

$100,000

Advertising Personnel Selected (D)

6

$150,000

3

$300,000

Materials Ordered (E)

3

$180,000

2

$250,000

Manufacturing Personnel Hired (F)

10

$300,000

7

$480,000

Iconic Model Manufactured (G)

2

$100,000

2

$100,000

Production Run of 100 (H)

6

$450,000

5

$800,000

Advertising Campaign (I)

8

$350,000

4

$650,000

a- Set up the PERT network for the Caterpillar problem.

b- Identify the Critical Path. Calculate the ES, EF, and the LS, LF for every activity and determine the slacks for all activities.

c- Which activity has the largest slack? Interpret.

d- Let’s assume that project has a variance of 16 weeks:

i. What is the probability of finishing the project in less than 30 weeks?

ii. What is the probability of finishing the project in more than 50 weeks?

iii. How many weeks will it take for the middle 50%?

a.  Write a linear program for determining the minimum cost of completing this project in half a year (=26 weeks).  Show your work and explain in details.

b. Interpret the meaning of the shadow price that would be associated with the constraint that set the maximum completion time to 26 weeks

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