Parametric objective analysis

5 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-04_1118_816_404_662}

1. Write down four inequalities that define the feasible region. The objective is to maximise \(P = 5 x + 3 y\).
2. Using the graph or otherwise, obtain the coordinates of the vertices of the feasible region and hence find the values of \(x\) and \(y\) that maximise \(P\), and the corresponding maximum value of \(P\). The objective is changed to maximise \(Q = a x + 3 y\).
3. For what set of values of \(a\) is the maximum value of \(Q\) equal to 3?

3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{7ca6d572-d776-4ad7-a0ed-9ec43c975585-03_908_915_392_614}

1. Write down the inequalities that define the feasible region. The objective is to maximise \(P _ { 1 } = x + 6 y\).
2. Find the values of \(x\) and \(y\) at the optimal point, and the corresponding value of \(P _ { 1 }\). The objective is changed to maximise \(P _ { k } = k x + 6 y\), where \(k\) is positive.
3. Calculate the coordinates of the optimal point, and the corresponding value of \(P _ { k }\) when the optimal point is not the same as in part (ii).
4. Find the range of values of \(k\) for which the point identified in part (ii) is still optimal.

1 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{cec8d4db-4a72-43a3-88f3-ff9df2a11d2c-2_885_873_388_635}

1. Write down the inequalities that define the feasible region. The objective is to maximise \(P _ { m } = x + m y\), where \(m\) is a positive, real-valued constant.
2. In the case when \(m = 2\), calculate the values of \(x\) and \(y\) at the optimal point, and the corresponding value of \(P _ { 2 }\).
3. (a) Write down the values of \(m\) for which point \(A\) is optimal.
   (b) Write down the values of \(m\) for which point \(B\) is optimal.

3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{372c062a-793f-4fb8-a769-957479f5fce7-05_846_833_365_614} The vertices of the feasible region are \(A ( 3.5,2 ) , B ( 1.5,3 ) , C ( 0.5,1.5 ) , D ( 1,0.5 )\).
The objective is to maximise \(P = x + 3 y\).

1. Find the coordinates of the optimum vertex and the corresponding value of \(P\).
2. Find the optimum point if \(x\) and \(y\) must both have integer values. The objective is changed to maximise \(P = x + k y\).
3. If \(k\) is positive, explain why the optimum point cannot be at \(C\) or \(D\).
4. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the optimum point.

7 A linear programming problem is
Maximise \(P = 4 x + y\) subject to $$\begin{aligned} 3 x - y & \leqslant 30 \\ x + y & \leqslant 15 \\ x - 3 y & \leqslant 6 \end{aligned}$$ and \(x \geqslant 0 , y \geqslant 0\)

1. Use a graphical method to find the optimal value of \(P\), and the corresponding values of \(x\) and \(y\). An additional constraint is introduced.
   This constraint means that the value of \(y\) must be at least \(k\) times the value of \(x\), where \(k\) is a positive constant.
2. 1. Determine the set of values of \(k\) for which the optimal value of \(P\) found in part (a) is unchanged.
   2. Determine, in terms of \(k\), the values of \(x , y\) and \(P\) in the cases when the optimal solution is different from that found in part (a).

6 Sasha is making three identical bead bracelets using amber, brown and red coloured beads. Sasha has 20 amber beads, 12 brown beads and 10 red beads. Each bracelet must use exactly 12 beads.
The profit from selling a bracelet is 6 pence for each amber bead used plus 2 pence for each brown bead used plus 3 pence for each red bead used. Sasha wants to maximise the total profit from selling the three bracelets.

1. Express Sasha's problem as a linear programming formulation in two variables \(a\) and \(b\), where \(a\) represents the number of amber beads in each bracelet and \(b\) represents the number of brown beads in each bracelet.
2. Determine how many beads of each colour will be used in each bracelet.
3. By listing all the feasible solutions, identify an aspect of the optimal solution, other than the profit, that is different from all the other feasible solutions. The beads that are not used in making the bracelets can be sold. The profit from selling each amber bead is \(k\) pence, where \(k\) is an integer, but nothing for each brown or red bead sold. All the previous constraints still apply. Instead of maximising the profit from the bracelets, Sasha wants to maximise the total profit from selling the bracelets and any left over beads. You are given that the optimal solution to the earlier problem does not maximise the total profit from selling the bracelets and any left over beads.
4. Determine the least possible value of Sasha's maximum total profit.
5. Why might Sasha not achieve this maximum profit?

7. \begin{figure}[h] \end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines have been shown in Figure 3. Given that \(k\) is a positive constant,

1. determine, in terms of \(k\) where necessary, the inequalities that define \(R\). The objective is to maximise \(P = 5 x + k y\) Given that the value of \(P\) is 38 at the optimal vertex of \(R\),
2. determine the possible value(s) of \(k\). You must show algebraic working and make your method clear.
   (Total 11 marks)

6. \begin{figure}[h] \end{figure} The graph in Figure 2 is being used to solve a linear programming problem in \(x\) and \(y\). The three constraints have been drawn on the graph and the rejected regions have been shaded out. The three vertices of the feasible region \(R\) are labelled \(\mathrm { A } , \mathrm { B }\) and C .

1. Determine the inequalities that define \(R\).
   (2) The objective function, \(P\), is given by $$P = a x + b y$$ where \(a\) and \(b\) are positive constants.
   The minimum value of \(P\) is 8 and the maximum value of \(P\) occurs at C .
2. Find the range of possible values of \(a\). You must make your method clear.
   (5)

4. \begin{figure}[h] \end{figure} Figure 2 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines are shown on the graph.

1. Determine the inequalities that define the feasible region.
2. Find the exact coordinates of the vertices of the feasible region. The objective is to maximise \(P\), where \(P = 2 x + k y\)
3. For the case \(k = 3\), use the point testing method to find the optimal vertex of the feasible region and state the corresponding value of \(P\).
4. Determine the range of values for \(k\) for which the optimal vertex found in (c) is still optimal.

6. A linear programming problem in \(x\) and \(y\) is described as follows. Maximise \(P = k x + y\), where \(k\) is a constant
subject to: \(\quad 3 y \geqslant x\) $$\begin{aligned} x + 2 y & \leqslant 130 \\ 4 x + y & \geqslant 100 \\ 4 x + 3 y & \leqslant 300 \end{aligned}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it \(R\).
2. For the case when \(k = 0.8\)
   1. use the objective line method to find the optimal vertex, \(V\), of the feasible region. You must draw and label your objective line and label vertex \(V\) clearly.
   2. calculate the coordinates of \(V\) and hence calculate the corresponding value of \(P\) at \(V\). Given that for a different value of \(k , V\) is not the optimal vertex of \(R\),
3. determine the range of possible values for \(k\). You must make your method and working clear.

8. \begin{figure}[h] \end{figure} The graph in Figure 4 is being used to solve a linear programming problem. The four constraints have been drawn on the graph and the rejected regions have been shaded out. The four vertices of the feasible region \(R\) are labelled \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .

1. Write down the constraints represented on the graph.
   (2) The objective function, P , is given by $$\mathrm { P } = x + k y$$ where \(k\) is a positive constant. The minimum value of the function P is given by the coordinates of vertex A and the maximum value of the function P is given by the coordinates of vertex D .
2. Find the range of possible values for \(k\). You must make your method clear.
   (Total 8 marks)

5. \begin{figure}[h] \end{figure} Figure 5 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.

1. Write down the inequalities that form region \(R\).
2. Find the exact coordinates of the vertices of the feasible region. The objective is to maximise \(P\), where \(P = 2 x + 3 y\)
3. Use point testing to find the optimal vertex, V, of the feasible region. The objective is changed to maximise \(Q\), where \(Q = 2 x + \lambda y\) Given that \(\lambda\) is a constant and V is still the only optimal vertex of the feasible region,
4. find the range of possible values of \(\lambda\).

6. \begin{figure}[h] \end{figure} Figure 4 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The vertices of the feasible region are \(A ( 4,7 ) , B ( 5,3 ) , C ( - 1,5 )\) and \(D ( - 2,1 )\).

1. Determine the inequality that defines the boundary of \(R\) that passes through vertices \(A\) and \(C\), leaving your answer with integer coefficients only. The objective is to maximise \(P = 5 x + y\)
2. Find the coordinates of the optimal vertex and the corresponding value of \(P\). The objective is changed to maximise \(Q = k x + y\)
3. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the only optimal vertex.

4. \begin{figure}[h] \end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.

1. Write down the inequalities that define \(R\). The objective is to maximise \(P\), where \(P = 3 x + y\)
2. Obtain the exact value of \(P\) at each of the three vertices of \(R\) and hence find the optimal vertex, \(V\). The objective is changed to maximise \(Q\), where \(Q = 3 x + a y\). Given that \(a\) is a constant and the optimal vertex is still \(V\),
3. find the range of possible values of \(a\).

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]

A linear programming problem is set up to maximise \(P = ax + y\) where \(a\) is a constant. \(P\) is maximised subject to three linear constraints which form the triangular feasible region shown in the diagram below. \includegraphics{figure_8} The vertices of the triangle are \((1, 6)\), \((5, 11)\) and \((13, 9)\) \(P\) is maximised at \((5, 11)\) Find the range of possible values for \(P\) [5 marks]