1. Your company has just received an order for a specially designed electric motor. The contract states that if the project is not completed 12 days from now, then beginning on the 13th day, your firm will experience a penalty of $300 per day. The table below shows the activities that need to be completed for this project, along with data on direct costs and precedence relations. (Crash Time represents the minimum activity time if it was completely crashed; the Crash Cost represents the cost of each activity if it is crashed to its minimum.)
| Activity |
Normal Time (days) |
Normal Cost ($) |
Crash Time(days) |
Crash Cost($) |
Immediate Predecessors |
| A |
4 |
1000 |
3 |
1300 |
None |
| B |
7 |
1400 |
4 |
2000 |
None |
| C |
5 |
2000 |
4 |
2700 |
None |
| D |
6 |
1200 |
5 |
1400 |
A |
| E |
3 |
900 |
2 |
1100 |
B |
| F |
11 |
2500 |
6 |
3750 |
C |
| G |
4 |
800 |
3 |
1450 |
D, E |
| H |
3 |
300 |
1 |
500 |
F, G |
a. Draw the network diagram (only draw Activity on Node).
b. Find the critical path(s). What is the normal duration (i.e. without crashing) of the entire project?
c. What is the slack time of activity D?
d. Right now, would you rather incur the late cost of $300 per day, or would you want to crash the project at some cost? What is the project completion time that you recommend for this project? Remember: it will be better to incur the penalty cost if the cost of reducing the project duration is greater than $300 per day. What are the activities that you crash, and by how much? What is the total project cost for your recommended project length?
2. Suppose a project has activities A, B, C and D. Given the table below, assess the current progress of this project. Is the project ahead or behind the schedule? By how much? Is the project over or under budget? By how much?
3. The activities of a project are shown in the following table, with their precedence relations and the three time estimates.
|
|
Three Time Estimates (Weeks) |
| Activity |
Immediate Predecessor(s) |
Optimistic |
Most likely |
Pessimistic |
| A |
- |
2 |
5 |
8 |
| B |
A |
1 |
2 |
3 |
| C |
A |
1 |
2 |
15 |
| D |
B |
1 |
4 |
13 |
| E |
B |
1 |
5 |
9 |
| F |
C |
2 |
5 |
8 |
| G |
E, F |
1 |
2 |
3 |
| H |
D |
1 |
2 |
9 |
| I |
G, H |
1 |
5 |
9 |
a. Find the expected time for activity F.
b. What is the variance for activity C?
c. What is the variance in completion time for the project? (Hint: find critical path).
4. A bank is considering opening a drive-through window for customer service.
Customers are estimated to arrive at the average rate of 15 per hour. The teller who works at the window can service customers at the average rate of one every three minutes. Assume Poisson arrivals and exponential service times.
a. Find the average number of customers in the waiting line (waiting but not being serviced).
b. Find the average waiting time in the system (including waiting in line and in service).
c. Suppose that now the service times are deterministic at the same rate; that is, the teller can service exactly one customer every three minutes. Do you think your answer in (b) will become larger, or smaller, or remain the same? No calculation needed.
5. A bank is considering opening two drive-through windows for customer service.
Customers are estimated to arrive at the average rate of 15 per hour. The tellers who work at the windows are assumed to be identical, and can service customers at the average rate of one every six minutes. Assume Poisson arrivals and exponential service times.
a. Find the average number of customers in the waiting line (waiting but not being serviced).
b. Find the average waiting time in the system (including waiting in line and in service).
c. Suppose that now the inter-arrival times are deterministic at the same rate; that is, every hour exactly 15 customers arrive at the bank. Do you think your answer in (b) will become larger, or smaller, or remain the same, and why? No calculation needed.