2. (a) A coach company has two attractive contracts lined up for a particular date. One contract is to provide coaches for a train replacement service. The train company would need at most 6 coaches. The other contract is to provide additional coaches for a service to France, for which no more than 5 coaches are required. The company has 8 coaches and 12 drivers. Trips to France require 2 drivers per coach, whereas train replacement coaches require only one driver. The company feels that for goodwill, at least one coach should be contracted to each client. The profit is £150 for each coach going to France and £100 for each coach operating as part of the train replacement service.

i. Formulate the problem of profit maximisation subject to constraints as a linear programming problem to help the company to maximise their total profit.

ii. Find an optimal solution to the LP problem graphically and state how many coaches should be provided for each of the two contracts.

(b) A brewer sells three types of beer, denoted by A, B, and C. They can make a profit of £10 on a vat of beer of type A, £5 on each vat of beer of type A, but have a loss of £2 on each vat of type C beer. They have a limited amount of two ingredients, yeast and hops.

They would like to maximise their profit while ensuring they do not use too much yeast or hops. Also, they are committed to a contract to supply a certain amount of type C beer. They have formulated the following linear programming problem to represent their problem:

maximise z = 10xA + 5xB − 2xC

subject to: 3xA + 2xB + xC ≤ 80 (constraint on yeast)

4xA + 2xB + 2xC ≤ 100 (constraint on hops)

xC ≥ 20 (constraint on contract)

xA, xB, xC ≥ 0, integer,

where xA, xB, xC are respectively the number of vats of beer A, B, and C that the brewer should produce.

The final tableau providing an optimal solution to the problem looks as follows:

i. State which constraints are tight and which are slack for this solution. What is the value of P? Provide all the details and explanation.

ii. How would the formulation of the LP/IP problem change if there was an additional requirement that the value of xA should be either 0 or between 10 and 25? Provide all the necessary details and explanation.