Dual objective optimization

5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} y & \geqslant 20 \\ x + y & \geqslant 25 \\ 5 x + 2 y & \leqslant 100 \\ y & \leqslant 4 x \\ y & \geqslant 2 x \end{aligned}$$

1. On Figure 1 opposite, draw a suitable diagram to represent the inequalities and indicate the feasible region.
2. Use your diagram to find the minimum value of \(P\), on the feasible region, in the case where:
   1. \(P = x + 2 y\);
   2. \(P = - x + y\). In each case, state the corresponding values of \(x\) and \(y\).

7. A linear programming problem is modelled by the following constraints $$\begin{aligned} 8 x + 3 y & \leqslant 480 \\ 8 x + 7 y & \geqslant 560 \\ y & \geqslant 4 x \\ x , y & \geqslant 0 \end{aligned}$$

1. Use the grid provided in your answer book to represent these inequalities graphically. Hence determine the feasible region and label it R . The objective function, \(F\), is given by $$F = 3 x + y$$
2. Making your method clear, determine
   1. the minimum value of the function \(F\) and the coordinates of the optimal point,
   2. the maximum value of the function \(F\) and the coordinates of the optimal point.

4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-005_1349_1395_408_294}

1. On the feasible region, find:
   1. the maximum value of \(2 x + 3 y\);
   2. the maximum value of \(3 x + 2 y\);
   3. the minimum value of \(- 2 x + y\).
2. Find the 5 inequalities that define the feasible region.

4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-04_1349_1395_408_294}

1. On the feasible region, find:
   1. the maximum value of \(2 x + 3 y\);
   2. the maximum value of \(3 x + 2 y\);
   3. the minimum value of \(- 2 x + y\).
2. Find the 5 inequalities that define the feasible region.

5 [Figure 2, printed on the insert, is provided for use in this question.]
The Jolly Company sells two types of party pack: excellent and luxury.
Each excellent pack has five balloons and each luxury pack has ten balloons.
Each excellent pack has 32 sweets and each luxury pack has 8 sweets.
The company has 1500 balloons and 4000 sweets available.
The company sells at least 50 of each type of pack and at least 140 packs in total.
The company sells \(x\) excellent packs and \(y\) luxury packs.

1. Show that the above information can be modelled by the following inequalities. $$x + 2 y \leqslant 300 , \quad 4 x + y \leqslant 500 , \quad x \geqslant 50 , \quad y \geqslant 50 , \quad x + y \geqslant 140$$ (4 marks)
2. The company sells each excellent pack for 80 p and each luxury pack for \(\pounds 1.20\). The company needs to find its minimum and maximum total income.
   1. On Figure 2, draw a suitable diagram to enable this linear programming problem to be solved graphically, indicating the feasible region and an objective line.
   2. Find the company's maximum total income and state the corresponding number of each type of pack that needs to be sold.
   3. Find the company's minimum total income and state the corresponding number of each type of pack that needs to be sold.

5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} x & \geqslant 1 \\ y & \geqslant 3 \\ x + y & \geqslant 5 \\ x + y & \leqslant 12 \\ 3 x + 8 y & \leqslant 64 \end{aligned}$$

1. On the grid below, draw a suitable diagram to represent the inequalities and indicate the feasible region.
2. Use your diagram to find, on the feasible region:
   1. the maximum value of \(3 x + y\);
   2. the maximum value of \(2 x + 3 y\);
   3. the minimum value of \(- 2 x + y\). In each case, state the coordinates of the point corresponding to your answer.
      [0pt] [6 marks]