7.06e Sensitivity analysis: effect of changing coefficients

A company produces two types of self-assembly wooden bedroom suites, the 'Oxford' and the 'York'. After the pieces of wood have been cut and finished, all the materials have to be packaged. The table below shows the time, in hours, needed to complete each stage of the process and the profit made, in pounds, on each type of suite.

	Oxford	York
Cutting	4	6
Finishing	3.5	4
Packaging	2	4
Profit (£)	300	500

The times available each week for cutting, finishing and packaging are 66, 56 and 40 hours respectively. The company wishes to maximise its profit. Let \(x\) be the number of Oxford, and \(y\) be the number of York suites made each week.

1. Write down the objective function. [1]
2. In addition to $$2x + 3y \leq 33,$$ $$x \geq 0,$$ $$y \geq 0,$$ find two further inequalities to model the company's situation. [2]
3. On the grid in the answer booklet, illustrate all the inequalities, indicating clearly the feasible region. [4]
4. Explain how you would locate the optimal point. [2]
5. Determine the number of Oxford and York suites that should be made each week and the maximum profit gained. [3]

It is noticed that when the optimal solution is adopted, the time needed for one of the three stages of the process is less than that available.

1. Identify this stage and state by how many hours the time may be reduced. [3]

Phil is to buy some squash balls for his club. There are three different types of ball that he can buy: slow, medium and fast. He must buy at least 190 slow balls, at least 50 medium balls and at least 50 fast balls. He must buy at least 300 balls in total. Each slow ball costs £2.50, each medium ball costs £2.00 and each fast ball costs £2.00. He must spend no more than £1000 in total. At least 60% of the balls that he buys must be slow balls. Phil buys \(x\) slow balls, \(y\) medium balls and \(z\) fast balls.

1. Find six inequalities that model Phil's situation. [4 marks]
2. Phil decides to buy the same number of medium balls as fast balls.
   1. Show that the inequalities found in part (a) simplify to give $$x \geq 190, \quad y \geq 50, \quad x + 2y \geq 300, \quad 5x + 8y \leq 2000, \quad y \leq \frac{1}{3}x$$ [2 marks]
   2. Phil sells all the balls that he buys to members of the club. He sells each slow ball for £3.00, each medium ball for £2.25 and each fast ball for £2.25. He wishes to maximise his profit. On Figure 1 on page 14, draw a diagram to enable this problem to be solved graphically, indicating the feasible region and the direction of an objective line. [7 marks]
   3. Find Phil's maximum possible profit and state the number of each type of ball that he must buy to obtain this maximum profit. [4 marks]

The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics{figure_3}

1. Write down the inequalities that define the feasible region. [4]
2. Write down the coordinates of the three vertices of the feasible region. [2]

The objective is to maximise \(2x + 3y\).

1. Find the values of \(x\) and \(y\) at the optimal point, and the corresponding maximum value of \(2x + 3y\). [3]

The objective is changed to maximise \(2x + ky\), where \(k\) is positive.

1. Find the range of values of \(k\) for which the optimal point is the same as in part (iii). [2]