CSC Co is a health food company producing and selling three types of high－energy products:

CSC Co is a health food company producing and selling three types of high-energy products: cakes, shakes and cookies, to gyms and health food shops. Shakes are the newest of the three products and were first launched three months ago. Each of the three products has two special ingredients, sourced from a remote part the world. The first of these, Singa, is a super-energising rare type of caffeine. The second, Betta, is derived from an unusual plant believed to have miraculous health benefits.

CSC Co’s projected manufacture costs and selling prices for the three products are as follows:

For each of the three products, the expected demand for the next month is 11,200 cakes, 9,800 cookies and 2,500 shakes.

The total fixed costs for the next month are $3,000.

CSC Co has just found out that the supply of Betta is going to be limited to 12,000 grams next month. Prior to this, CSC Co had signed a contract with a leading chain of gyms, Encompass Health, to supply it with 5,000 shakes each month, at a discounted price of $5·80 per shake, starting immediately. The order for the 5,000 shakes is not included in the expected demand levels above.

Required:

(a) Assuming that CSC Co keeps to its agreement with Encompass Health, calculate the shortage of Betta, the resulting optimum production plan and the total profit for next month. (6 marks)

One month later, the supply of Betta is still limited and CSC Co is considering whether it should breach its contract with Encompass Health so that it can optimise its profits.

Required:

(b) Discuss whether CSC Co should breach the agreement with Encompass Health.

Note: No further calculations are required. (4 marks)

Several months later, the demand for both cakes and cookies has increased significantly to 20,000 and 15,000 units per month respectively. However, CSC Co has lost the contract with Encompass Health and, after suffering from further shortages of supply of Betta, Singa and of its labour force, CSC Co has decided to stop making shakes at all. CSC Co now needs to use linear programming to work out the optimum production plan for cakes and cookies for the coming month. The variable ‘x’ is being used to represent cakes and the variable ‘y’ to represent cookies.

The following constraints have been formulated and a graph representing the new production problem has been drawn:

Singa: 0·25x + 0·5y ≤ 12,000

Betta: 0·5x + 0·2y ≤ 12,500

Labour: 0·1x + 0·12y ≤ 3,000

x ≤ 20,000

y ≤ 15,000

x, y ≥ 0

Required:

(c) (i) Explain what the line labelled ‘C = 2·6x + 1·75y’ on the graph is and what the area represented by the points 0ABCD means. (4 marks)

(ii) Explain how the optimum production plan will be found using the line labelled ‘C = 2·6x + 1·75y’ and identify the optimum point from the graph. (2 marks)

(iii) Explain what a slack value is and identify, from the graph, where slack will occur as a result of the optimum production plan. (4 marks)

Note: No calculations are needed for part (c).