问答题 The Cosmetic Co is a company producing a variety of cosmetic creams and lotions. The creams and lotions are sold to a variety of retailers at a price of $23·20 for each jar of face cream and $16·80 for each bottle of body lotion. Each of the products has a variety of ingredients, with the key ones being silk powder, silk amino acids and aloe vera. Six months ago, silk worms were attacked by disease causing a huge reduction in the availability of silk powder and silk amino acids. The Cosmetic Co had to dramatically reduce production and make part of its workforce, which it had trained over a number of years, redundant. The company now wants to increase production again by ensuring that it uses the limited ingredients available to maximise profits by selling the optimum mix of creams and lotions. Due to the redundancies made earlier in the year, supply of skilled labour is now limited in the short-term to 160 hours (9,600 minutes) per week, although unskilled labour is unlimited. The purchasing manager is confident that they can obtain 5,000 grams of silk powder and 1,600 grams of silk amino acids per week. All other ingredients are unlimited. The following information is available for the two products: Cream Lotion Materials required: silk powder (at $2·20 per gram) 3 grams 2 grams – silk amino acids (at $0·80 per gram) 1 gram 0·5 grams – aloe vera (at $1·40 per gram) 4 grams 2 grams Labour required: skilled ($12 per hour) 4 minutes 5 minutes – unskilled (at $8 per hour) 3 minutes 1·5 minutes Each jar of cream sold generates a contribution of $9 per unit, whilst each bottle of lotion generates a contribution of $8 per unit. The maximum demand for lotions is 2,000 bottles per week, although demand for creams is unlimited. Fixed costs total $1,800 per week. The company does not keep inventory although if a product is partially complete at the end of one week, its production will be completed in the following week. Required:

问答题 (a)On the graph paper provided, use linear programming to calculate the optimum number of each product that the Cosmetic Co should make per week, assuming that it wishes to maximise contribution. Calculate the total contribution per week for the new production plan. All workings MUST be rounded to 2 decimal places. (14 marks)

【正确答案】Optimum production plan Define the variables Let x = no. of jars of face cream to be produced Let y = no. of bottles of body lotion to be produced Let C = contribution State the objective function The objective is to maximise contribution, C C = 9x + 8y State the constraints Silk powder 3x + 2y ≤ 5,000 Silk amino acids 1x + 0·5y ≤ 1,600 Skilled labour 4x + 5y ≤ 9,600 Non-negativity constraints: x, y ≥ 0 Sales constraint: y ≤ 2,000 Draw the graph Silk powder 3x + 2y = 5,000 If x = 0, then 2y = 5,000, therefore y = 2,500 If y = 0, then 3x = 5,000, therefore x = 1,666·7 Silk amino acids 1x +0·5y = 1,600 If x = 0, then 0·5y = 1,600, therefore y = 3,200 If y = 0, then x = 1,600 Skilled labour 4x + 5y = 9,600 If x = 0, then 5y = 9,600, therefore y = 1,920 If y = 0, then 4x = 9,600, therefore x = 2,400

【答案解析】

问答题 (b)Calculate the shadow price for silk powder and the slack for silk amino acids. All workings MUST be rounded to 2 decimal places. (6 marks)

【正确答案】Shadow prices and slack The shadow price for silk powder can be found by solving the two simultaneous equations intersecting at point c, whilst adding one more hour to the equation for silk powder. 4x +5y = 9,600 x 3 3x + 2y = 5,001 x 4 12x + 15y = 28,800 12x + 8y = 20,004 Subtract the second one from the first one 7y = 8,796, therefore y = 1,256·57 3x + (2 x 1,256·57) = 5,001. Therefore x = 829·29 C = (9 x 829·29) + (8 x 1,256·57) = $17,516·17 Original contribution = $17,514·34 Therefore shadow price for silk powder is $1·83 per gram. The slack for amino acids can be calculated as follows: (828·58 x 1) + (0·5 x 1,257·14) = 1,457·15 grams used. Available = 1,600 grams. Therefore slack = 142·85 grams.

【答案解析】