Each year, a shoe manufacturing company faces demands (which must be met on time) for pairs of shoes as shown in the table below.

Quarter 1 Quarter 2 Quarter 3 Quarter 4

Demand 6000 3000 8000 1000

Employees work three consecutive quarters and then receive one quarter off. For example, a worker might work during quarters 3 and 4 of one year and quarter 1 of the next year. During a quarter in which an employee works, he or she can produce up to 500 pairs of shoes. Each worker is paid £5000 per quarter. At the end of each quarter, a holding cost of £10 per pair of shoes is incurred.

a) Formulate algebraically a linear programme to minimise the cost per year (labour plus holding) of meeting the demands for shoes. To simplify the model, assume that at the end of each year, the ending inventory is 0. (You can assume that a given worker gets the same quarter off during each year.)

b) Formulate a spreadsheet model for the problem and solve. Describe your solution.

c) Suppose the company can pay a flat fee for a training program that increases the productivity of all of its workers. Determine how much the company would be willing to pay for a training program that increases worker productivity from 500 pairs of shoes per quarter to P pairs of shoes per quarter, where P varies from 525 to 700 in increments of 25.