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Instructions to Candidates:

• Write your candidate number on the answer booklet.
• Answer all questions.
• All questions carry equal marks.
• Students may bring and use their own calculator, which must be from the Casio FX- 83, Casio FX-85, HP10s or HP10s+ series. No other type of calculator is allowed.
• Please note that questions are printed on both sides of the examination paper.

Materials Provided:

• Answer booklets.
• Graph paper

This examination paper cannot be removed from the Examination Room

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Question 1

The XY-company manufactures two different products, named X and Y. Those products are made from two different materials, which are called A and B. To produce one unit of product X, 3 kg of material A and 6 kg of material B are needed. To make one unit of product Y, the company needs 5 kg of material A and 2 kg of material B. Altogether, the amount of material they can get hold of each week is limited to 120 kg of material A and 120 kg of material B. Each unit X that is manufactured can be sold at a price of £60 and each unit Y at a price of £200. The company wants to maximize their weekly revenue.

1. Formulate a linear programming model for this problem. Define the variables you use and explain the objective function and the restrictions. [25%]

2. Plot the feasible region to the model and identify on your graph an optimal solution. On the graph clearly indicate the constraints, extreme points of the feasible region and the objective function contour line passing through the optimal point. [25%]

3. Explain the meaning of “binding constraints” in general, then with respect to the XY-company’s problem, and then to the graphical solution found in part b). [25%]

4. LINDO arrives at the following solution:

OBJECTIVE FUNCTION VALUE

1) 4800.000

VARIABLE VALUE REDUCED COST

X 0.000000 60.000000

Y 24.000000 0.000000

ROW SLACK OR SURPLUS DUAL PRICES

Mtrl_A) 0.000000 40.000000

Mtrl_B) 72.000000 0.000000

Explain all the information that you get from this part of the output. Link the information from the output to the graph from part b). [25%]

Question 2

JFE manufactures three types of steel at different plants. The time required for manufacturing one ton of steel (regardless of type) and the costs at each plant are shown in the table below. Each week, 100 tons of each type of steel (1, 2, and 3) must be produced. Each plant is open 48 hours per week.

	Cost of producing 1 ton of:	Time required (minutes)
		Steel 1	Steel 2	Steel 3
Plant 1	£50	£40	£25	18
Plant 2	£45	£35	£30	16
Plant 3	£52	£32	£24	15

1. Formulate a transportation problem that can be used to minimise the cost of meeting JFE’s weekly demand. [Hint: To formulate supply constraints use the time required to manufacture one ton of steel in combination with the total time available to find out the weekly production capacity (in tons of steel) at each plant.] [40%]

2. If there was a requirement that each plant should produce in total at least 75 tons of steel per week, what additional constraints would you have to add to your model to reflect this requirement? [20%]

3. If there was a requirement for Steel 2 produced at Plant 1 to be at least 50% of the total amount of steel produced at that plant, how would you formulate this requirement into your model? [15%]

4. Suppose now that the time required to produce one ton of steel depend on the type of steel as well as on the plant at which it is produced. (See the table below). How would you reformulate supply constraint to reflect these time requirements? [25%]

	Time (minutes)
	Steel 1	Steel 2	Steel 3
Plant 1	20	18	15
Plant 2	18	16	15
Plant 3	18	15	12

Question 3

PART A [40%]

The distance (in miles) between the 5 cities A, B, C, D and E are shown in the table below. It is necessary to build a state road system that connects all these cities. Assume that for various reasons no road can be built connecting A and B, and no road can be built connecting E and C. Use the minimum spanning tree algorithm to find the minimum length of road required.

Distances	A	B	C	D	E
A	–	136	225	167	61
B		–	282	212	77
C			–	109	311
D				–	184

PART B [60%]

The east-west motorway system can accommodate the capacities shown on the network below (in thousands of vehicles per hour). What is the maximal flow in vehicles per hour through the system? How many vehicles per hour must travel over each road (arc) to obtain maximal flow?

Question 4

As a fund manager of an investment company, you are helping a client with her investment portfolio. Diversification strategy requires any portfolio to be split across three different mutual funds: GF, BF and MF while respecting the following limits:

• Between 10% and 40% to be allocated within GF.
• Between 10% and 70% to be allocated within BF.
• No less than 20% to be allocated within MF.

At the moment, the client is not certain about the total amount of money to invest but she is happy to split her portfolio across the three funds using the specified limits.

1. Let G, B and M represent the amount of money invested into GF, BF and MF, respectively. Use only these three variables to formulate constraints that would ensure that the above limits are respected. [25%]

Risk factors are estimated to be: 9% for GF, 6% for BF and 1% for MF.

2. If the limits on the three funds must be respected, what is the minimum and what is the maximum possible average risk level for any portfolio? [25%]

Expected returns are 16% for GF, 12% for BF and 5% for MF. Your client has decided to invest £400,000 but she has two objectives: to maximise expected return and at the same time to minimise average risk level of the portfolio. Your client’s ideal targets are at least 12% for the expected return and at most 5% for the average risk level. The client has also informed you that she is ready to accept 1% of additional risk if that is going to result in at least 2% extra in expected return.

3. Use variables G, B and M and any necessary deviation variables to formulate a complete goal programming model, which you could use to find the best investment strategy for your client. [50%]

END OF PAPER