a) Using your own LP model with two variables and minimization objective function, illustrate graphically how you can find the shadow price of a binding constraint.

b) Explain how ranked goal programming works and briefly explain why this approach may be preferred over the weighted goal programming approach.

c) Why are integer linear programming models (as compared with standard linear programming models) in general more difficult to solve?

d) Explain what is meant by the reduced cost of a decision variable and briefly comment why this information may be of interest to a decision maker.

e) What is the difference between decision making under certainty, decision making under risk and decision making under uncertainty?

f) Describe two applications of the shortest path problem.

g) A box contains 8 red balls, 10 green balls and 2 white balls. A ball is removed from the box and replaced; the balls are mixed and then another ball is removed. What is the probability of:

1. two green balls being drawn?
2. a red ball on the second draw?
3. A red or white ball on the first draw, and a green or white ball on the second draw?
4. a red ball on the second draw, given that a white ball was drawn on the first?
5. Re-examine part (1) in case the first ball is not replaced in the box.

h) Nottingham University must purchase 1100 computers from three potential suppliers. Supplier 1 charges £500 per computer plus a (fixed) delivery charge of
£5000. Supplier 2 charges £350 per computer plus a (fixed) delivery charge of
£4000. Supplier 3 charges £250 per computer plus a (fixed) delivery charge of
£6000. Supplier 1 will sell at most 500 computers; supplier 2, at most 900; and supplier 3, at most 400. Formulate an integer linear programming model to minimise the cost of purchasing the needed computers. Clearly define the decision variables and explain the objective function and constraints.