1. Department X
Department X contains an area of study that has grown in popularity over recent years. This area of study is delivered through two pathways, one academic and one vocational. Both pathways are over-subscribed and a decision has to be made on how many groups of each pathway should be offered in the next academic year. Both the academic pathway and the vocational pathway require the same number of contact hours (21 hours/week); however they differ in specialist staffing requirements and the amount of Laboratory time required. The vocational pathway requires a minimum of 9 hours of Laboratory time whereas the academic pathway only requires a minimum of 3 hours. The remaining time for both pathways is delivered in classrooms. Access to both laboratories and classrooms is limited with a maximum of 90 hours and 234 hours respectively. The need for specialist staff input restricts the number of vocational and academic groups to a maximum of 10 and 6 respectively. The head of department has been informed that a given recruitment meets targets that each vocational group contributes £5,000 to central costs and each academic group contributes £4,500. All heads of department have been instructed to maximise contribution to central costs.
Task
Linear programming is a technique that can be applied to situations where the optimum use of scarce resources is required. The technique requires constraints to be stated as inequalities and represented graphically. The first step is to identify which of the three graphical representations and stated inequalities accurately represents the information provide for department X. When you have selected a graph you will notice a shaded area, this represents the area of feasibility where all inequalities are satisfied. Advise the head of department on the possible combinations of academic and vocational groups and how that might affect contribution. Comments would also be welcome on any underutilisation of resources and suggestions for growing the area of study.

2. Graph A

3. Graph B

4. Graph C

5. Department X
Graph C and the stated inequalities accurately represents the information provide for Department X
All possible combinations of vocational and academic groups lie within the area of feasibility. The combination that provides the best contribution is 6 vocational groups and 9 academic groups.
Contribution with this combination of academic and vocational groups
C = 4500A V = (4500 x 9) + (5000 x 6) = £70,500
Examination of graph C will reveal that the laboratory constraint is not influencing the area of feasibility therefore the laboratory is underutilised. Suggestions of how to grow the area could include additional classroom capacity which would reduce the underutilisation of the laboratory. Suggestion may also include increases in capacity of specialist staffing which would take advantage of the increased classroom capacity.