MATH3081W1
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MATH3081W1
1. (a) Anticipating a busy period due to pandemic, the University IT service allocates three technicians to deal with email queries from staff and students. Assume that the number of email arrivals in any given period follows a Poisson probability distribution with mean rate of 35 emails per hour, and that the handling time of each email follows an exponential distribution with a mean time of 3 minutes for each technician.
(i) (4 marks) Calculate the average number of emails waiting to be handled. (ii) (4 marks) Calculate the average time for an email query getting a reply (from
the time received to the time a response is sent).
(iii) (4 marks) Calculate the probability that an email can be immediately handled after being received.
(b) Due to difficulties to recruit enough students, the University Executive Board
decides that the University needs to make savings and offers a voluntary severance (VS) scheme to all its staff. Each VS application is dealt with individually. One of the IT technicians tasked to deal with emails, applies for the VS. The IT senior management team needs to make the decision whether to approve the technician’s application and let them go or not. They ask you, as an OR expert, for an advice.
(i) (8 marks) How would you advise the managers in this case? How would you justify your recommendation? Could the IT service continue to offer their support to staff and students at a satisfactory level with only two technicians in charge?
You can choose appropriate formulas from the provided formula sheet for the calcula- tions in this question.
2. (a) A supermarket sells 6000 kg frozen fish each month. Each delivery of frozen fish costs f30 independently of the order size, and the holding cost for each kilogram (kg) of frozen fish is f1.5 per month (mostly due to the cost of running freezers). One kilogram of frozen fish costs f3.
(i) (5 marks) Derive the total annual cost in terms of order size.
(ii) (5 marks) Calculate the optimal order quantity and total annual cost. (Assume 1 year comprises 12 equal months, with no variation in sales between months.)
(b) The stock holding cost is calculated with the tax rate of 20% and the cost of placing an order is f15. Demand is steady and the annual demand is 1000 items. Shortages must not occur. The purchase cost depends on the number of items Q in the order as follows: the order of less than 500 items, purchase cost is f0.5 per item, for orders of 500 items or more, purchase cost is f0.45 per item for those beyond 500. That is, the purchase cost function CT (Q) can be written as
CT (Q) = 一(5)Q(00),
Determine the optimal order quantity (your conclusion must be supported by detailed analysis and calculations). (10 marks)
3. A certain project comprises activities A,. . . ,H. It is possible to crash the duration of certain activities. The activities predecessor(s), normal and crash durations are given in the table below. Crashing adds project costs. The table below also includes cost of reduction for each activity.
Activities |
A |
B |
C |
D |
E |
F |
G |
H |
Predecessors |
- |
- |
A |
B,C |
D |
E |
B,C |
F,G |
Normal Duration (weeks) |
3 |
6 |
2 |
5 |
4 |
3 |
9 |
3 |
Crash Duration (weeks) |
1 |
4 |
1 |
3 |
3 |
1 |
5 |
2 |
Cost of Reduction |
800 |
1400 |
500 |
600 |
350 |
760 |
1200 |
1000 |
It is assumed that the cost of reduction = crash cost ← normal cost.
(a) Draw an activity on node network diagram for this project. For each activity
determine early start (ES), late start (LS), early finish (EF), late finish (LF), total
float (TF) and free float (FF).
(b) Deduce the critical path and project completion time.
(c) Draw a time scaled project diagram.
(6 marks) (2 marks)
(4 marks)
(d) Assume that the company wishes to complete the project in 15 weeks. What crashing decisions should be recommended to meet the desired completion time at least possible cost? Your answers must include detailed analysis. (8 marks)
4. (a) Seven jobs are to be scheduled on a single machine. Each job j becomes available for processing at time zero, has a processing time pj and has a due date dj, which are shown in the table below.
Job j |
1 2 3 4 5 6 7 |
pj dj |
6 8 4 3 9 6 5 25 22 15 8 14 14 18 |
(i) Use Moore’s algorithm to find a schedule for which the number of late jobs is minimised. (6 marks)
(ii) If job 5 is constrained to be early, how does this alter your solution? (4 marks)
(b) Seven jobs are to be scheduled on three identical parallel machines. Each job j becomes available for processing at time zero, and has a processing time pj as shown in the table below.
Job j |
1 2 3 4 5 6 7 |
pj |
7 8 4 3 9 6 5 |
(i) If the scheduling objective is to minimise the sum of completion times of the jobs, i.e., problem P ≤ ≤ Cj is to be solved, find an optimal schedule and evaluate the sum of the completion times of the jobs. (5 marks)
(ii) If the scheduling objective is to minimise the maximum completion time and preemption of jobs is allowed, i.e., problem P ≤pmtn≤Cma{ is to be solved, apply McNaughton’s algorithm to find an optimal schedule. (5 marks)
5. (a) Consider a decision problem with three states S1 , S2 , S3 and three decision alternatives d1 , d2 , d3 . The profit associated with each decision is given in the following profit matrix:
|
S1 |
S2 |
S3 |
1 |
70 |
0 |
80 |
2 |
50 |
20 |
70 |
3 |
80 |
10 |
50 |
(i) (4 marks) Find an optimal decision by using the Wald decision criterion and the Savage regret criterion.
(ii) (3 marks) Suppose that the probabilities of the states are P (S1 ) = 0.2, P (S2 ) = 0.3 and P (S3 ) = 0.5.
In addition, it is also possible to obtain perfect information. Determine the value of the perfect information by the criterion of minimal expected cost.
(b) A hospital is deciding on a strategy for diagnosing and treating patients who maybe suffering from colorectal cancer. From the patients visiting the clinic, 20% have colorectal cancer. The hospital predicts that the long-term costs to the health service are as follows:
|
Have cancer (S1 ) |
Don’t have cancer (S2 ) |
Treat (d1 ) Don’t treat (d2 ) |
f10, 000 f55, 000 |
f10, 000 f0 |
The hospital performs a diagnostic test before deciding whether to treat or not to treat. The test costs f5, 000. If they have colorectal cancer, there is a 90% chance that the test will come back positive, and if they do not have colorectal cancer, there is a 5% chance it will come back positive.
(i) (3 marks) Draw a decision tree representing the possible decisions and outcomes of this problem.
(ii) (10 marks) Find optimal decisions which minimise the total expected cost.
2022-08-18