Advanced: Relevant material costs, optimal output and shadow prices using the graphical approach The Milton Carpet Company has been manufactur¬ ing two ranges of carpet for many years, one range for commercial use, the other for private use. The main difference between the two ranges is in the mix of wool and nylon; with the commercial range having 80% wool and 20% nylon and the private range 20% wool and 80% nylon. The designs of each range are the same and each range can be made in 5 different colours. There are variations in the cost of the dyes used for the different colours in the range, but they are all within 5% of each other so the accountant takes an average cost of dyeing in her costing.

The Board has just decided to use up its remaining stocks of wool and nylon and transfer its production over to making acrylic carpets in three months’ time. The company’s objective is to maximize the contribution from the running down of the stocks of wool and nylon over that period subject to any operating constraints. Data concern¬ ing its present carpet range is given below. It is assumed that sufficient demand exists to ensure all production can be sold at the stated price.

There are 24 000 kg of wool and 25 000 kg of nylon in the stores to be used up. At the end of the quarter, when it changes production to the new carpet, any wool or nylon left can be sold for £1 per kg. The production manager forecasts that the machines can operate for a total of 6600 hours during the next quarter.
You are required to:

(a) Formulate the above problem in a linear programming format. Solve the problem and provide the production manager with the required output mix of rolls of private and commercial carpets. State whether any of the raw material has to be sold off as scrap at the end and thus what the total contribution for the quarter’s production should be.
(10 marks)

(b) Show whether or not it will be necessary to recompute the optimum solution if due to economic difficulties, the costs of the dyes for the carpets are to be increased by £110 and £40 per roll of private and commercial carpet respectively. (4 marks)

(c) Describe what would happen in physical and financial terms if the availability of one of the fully utilized resources were to be increased by a small amount. (6 marks)

(d) Comment on the advisability of introducing the concept of opportunity costs into the budgetary control framework, by using the output from the linear programming solution to the optimum production mix.

Per roll Private Commercial use use () () Selling price 2400 3200 Manufacturing costs: Material - Wool 140 700 - Nylon 320 100 Direct labour 90 108 Variable production costs 250 312 Fixed production overheads based on 200% direct labour 180 216 Standard full cost 980 1436 Production requirements: Wool (kg) 440 40 200 Nylon (kg) 160 50 Direct machine time (hours) 30 36