A company makes two products, X and Y. Product X has a contribution of 124 per unit

Question:

A company makes two products, X and Y. Product X has a contribution of £124 per unit and product Y £80 per unit. Both products pass through two departments for processing and the times in minutes per unit are:

Currently there is a maximum of 225 hours per week available in department 1 and 200 hours in department 2. The company can sell all it can produce of X but EEC quotas restrict the sale of Y to a maximum of 75 units per week.
The company, which wishes to maximize contribution, currently makes and sells 30 units of X and 75 units of Y per week.
The company is considering several possibilities including:
(i) Altering the production plan if it could be proved that there is a better plan than the current one;
(ii) Increasing the availability of either department 1 or department 2 hours. The extra costs involved in increasing capacity are £0.5 per hour for each department;
(iii) Transferring some of their allowed sales quota for Product Y to another company. Because of commitments the company would always retain a minimum sales level of 30 units.
(a) Calculate the optimum production plan using the existing capacities and state the extra contribution that would be achieved compared with the existing plan;
(b) Advise management whether they should increase the capacity of either department 1 or department 2 and, if so, by how many hours and what the resulting increase in contribution would be over that calculated in the improved production plan;
c) Calculate the minimum price per unit for which they could sell the rights to their quota, down to the minimum level, given the plan in (a) as a starting point.