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Review problem 1

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- QP dolls, Inc., has developed a new doll it feels could turn into a “collector’s item” through proper advertisement. The PERT/ CPM network shown models the activities (or work packages) of the project.

H

I

B

A

D

F

C

G

E

- Distribution of the total budgeted cost of $5,280,000 and expected completion times (in weeks) for each of the work packages are summarized in the following table

a. Determine an earliest and latest schedule for each work package.

- Consider the situation faced by QP Dolls, Inc. in problem 16. Management is giving some thought to putting extra resources into the project so that it can be completed within one-half year (26 weeks). Accordingly, each work package has been studied, and a set of crash times in weeks and costs has been developed:

Compare

a.Determine a schedule for the work packages which minimizes the total cost of completing the project within 26 weeks. What is the minimum total cost? Go to file ch7-18a.cpm

b. Suppose QP will incur administrative and operational costs totaling $24,000 per week during this project. Modify your model to take into account these costs and solve for the optimal scheduling of the work packages. Did it change from the solution in part (a)? Go to file ch7-18b

Let us first understand how to model the crash

Problem using linear programming. Look at the following

small example.

Normal Crash

Activity Cost Time Cost Time

A1000 10 5000 8

B 2000 12 4500 8

C 1200 7 2200 6

D 1800 8 3000 5

E 500 8 1000 7

8

12

B

D

10

Finish

Due date =20

A

- Define X as the start time of an activity.
- Define Y as the amount of time an activity
- is crashed.

C

E

7

8

Notice that an activity cannot start before all its immediate predecessors are completed.

This observation leads to the following relationship between activity A and B (for example)

XB >= XA + time to complete activity A, or XB >= XA + (10 – YA)

Start time of B

Start time of A

Actual time to complete A

after it was crashed.

This is how you build all the network constraints one constraint for each arc,

covering all the arcs. Also, note that X(FIN) >= XD+(8-YD) and X(FIN) >= XE+(8-YE).

Sometimes the Total budget – Total normal costs has already been

computed, so you need to read the information given carefully.

(5000-1000)/(10-8)

In addition to these constraints, there are constraints on the time each activity can be crashed.

For example, activity C can be crashed by 1 at most, but activity D can be crashed by 3 at most.

Therefore we have to add the following set of constraints

YA <= 2 (10-8)

YB <= 4 (12-8)

…….

The objective function is determined by the case we solve:

Case 1: To minimize crash costs formulate the objective function as follows:

Minimize2000YA+ 625YB+…+500YE and add the due date constraint to the model,

such as X(FIN) = 20. Otherwise, the optimal solution will be “not to crash” and incur “zero” crash costs.

Case 2: To minimize the time to complete the project change the objective to

Minimize X(FIN) [don’t forget that X(FIN) is the duration of the project].

You need to eliminate the constraint X(FIN) =20 since you try to minimize X(FIN), and instead limit the amount of funds used to a given budget by adding the constraint 2000YA+625YB+…+500YE<=Total budget available – Total normal costs.

Case 3:To minimize the total crash costs+ operating costs we assume the operating costs are proportional to the time it takes to complete the project. Suppose the operating costs are $100 per one time unit (a week for example). Then we add 100X(FIN) to the objective function formulated in case 1 and eliminate the constraint X(FIN) = 20.

b. Suppose QP will incur administrative and operational costs totaling $24,000 per week during this project. Modify your model to take into account these costs and solve for the optimal scheduling of the work packages. Did it change from the solution in part (a)?

Note that the objective function

consists of crashing costs +

operational costs.

X(FIN) >= XI+8-YI or

-XI+X(FIN)+YI>=8

The solution did not change

c. Suppose QP only budgeted $6 million for this project. What is the minimum time to complete this project if the weekly fixed costs are included in the model.

6,000,000-5,280,000=720,000

This objective function minimizes

the completion time of the project

for a given budget.

Minimum time = 30 weeks with a crash budget of $720,000

- Golden West Homes is developing a new modular home model. The following table outlines the activities of the project. Times are expressed in days:

a. Determine the expected completion time and the critical path for this project. Go to file ch 7-21c for a solution without a consultant

Expected completion time = 117 days.

Critical path: A B C D E J L N R

b. Because of production considerations, Golden West will lose $ 10,000 if this project is not completed within 114 days. What is the probability that the project will be completed within 114 days?Perform probability analysis with WINQSB (in Results menu): P(Completion in 114 days) = .2957

c. Suppose Golden West were to hire a meeting planning consultant to hold the dealer meetings (activity N). This would cost Golden West an additional $1000, but the activity is guaranteed to take exactly three days. Considering the $10,000 loss if the project is not completed in 114 days, should Golden West hire the consultant? Because there is no variance, all the time estimates of activity N are the same.

With activity N expected to be completed in 3 days,

the project is expected to be completed in 115 days.

The probability to complete in 114 days is .4284.

Expected cost (No consultant)= .2957(0)+(1-.2957)(10000)=$7043

Expected cost (w/consultant)=.4284(0)+(1-.4284)10000)+1000=$6716

Hire the consultant

d. What is the most Golden West should be willing to pay for a meeting planning consultant who could hold dealer meetings in exactly three days?

7043 - 5716 = $1327