Graphical solution of linear programs

Figure 5 shows the constraints of a linear programming problem in x and y, where R is the feasible

Write down the inequalities that form region R.

Find the exact coordinates of the vertices of the feasible region.

The objective is to maximise P, where P = 2 x + 3 y

Use point testing to find the optimal vertex, V, of the feasible region.

The objective is changed to maximise Q, where Q = 2x + Given that is a constant and V is still the only optimal vertex of the feasible region,

find the range of possible values of .

Charlie needs to buy storage containers. There are two different types of storage container available, standard and deluxe. Standard containers cost £20 and deluxe containers cost £65. Let x be the number of standard containers and y be the number of deluxe containers. The maximum budget available is £520

Write down an inequality, in terms of x and y, to model this constraint.

Three further constraints are:

Add lines and shading to Diagram 1 in the answer book to represent all four constraints. Hence determine the feasible region and label it R.

The capacity of a deluxe container is 50% greater than the capacity of a standard container. Charlie wishes to maximise the total capacity.

State an objective function, in terms of x and y.

Use the objective line method to find the optimal vertex, V, of the feasible region. You must make your objective line clear and label the optimal vertex V.

Calculate the exact coordinates of vertex V.

Determine the number of each type of container that Charlie should buy. You must make your method clear and calculate the cost of purchasing the storage containers.

A linear programming problem in x and y is described as follows. Minimise C = 2 x + 3 y subject to

x < 8

Add lines and shading to Diagram 1 in the answer book to represent these constraints.

Hence determine the feasible region and label it R.

Use the objective line (ruler) method to find the exact coordinates of the optimal vertex, V, of the feasible region. You must draw and label your objective line clearly.

Calculate the corresponding value of C at V.

The objective is now to maximise 2 x + 3 y, where x and y are integers.

Write down the optimal values of x and y and the corresponding maximum value of 2 x + 3

A further constraint, y kx, where k is a positive constant, is added to the linear programming problem.

Determine the least value of k for which this additional constraint does not affect the feasible region.

Harry wants to rent out boats at his local park. He can use linear programming to determine the number of each type of boat he should buy. Let x be the number of 2-seater boats and y be the number of 4-seater boats. One of the constraints is

Explain what this constraint means in the context of the question.

Another constraint is

Explain what this constraint means in the context of the question.

A third constraint is

Represent these three constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R.

Each 2-seater boat costs £100 and each 4-seater boat costs £300 to buy. Harry wishes to minimise the total cost of buying the boats.

Write down the objective function, C, in terms of x and y.

Determine the number of each type of boat that Harry should buy. You must make your method clear and state the minimum cost.

A company makes two types of garden bench, the ‘Rustic’ and the ‘Contemporary’. The company wishes to maximise its profit and decides to use linear programming. Let x be the number of ‘Rustic’ benches made each week and y be the number of ‘Contemporary’ benches made each week. The graph in Figure 6 is being used to solve this linear programming problem. Two of the constraints have been drawn on the graph and the rejected region shaded out.

Write down the constraints shown on the graph giving your answers as inequalities in terms of x and y.

It takes 4 working hours to make one ‘Rustic’ bench and 3 working hours to make one ‘Contemporary’ bench. There are 120 working hours available in each week.

Write down an inequality to represent this information. Market research shows that ‘Rustic’ benches should be at most of the total benches made each week.

Write down, and simplify, an inequality to represent this information. Your inequality must have integer coefficients.

Add two lines and shading to Diagram 1 in your answer book to represent the inequalities of (b) and (c). Hence determine and label the feasible region, R.

The profit on each ‘Rustic’ bench and each ‘Contemporary’ bench is £45 and £30 respectively.

Write down the objective function, P, in terms of x and y.

Determine the coordinates of each of the vertices of the feasible region and hence use the vertex method to determine the optimal point.

State the maximum weekly profit the company could make.

Lethna is producing floral arrangements for an awards ceremony. She will produce two types of arrangement, Celebration and Party. Let x be the number of Celebration arrangements made. Let y be the number of Party arrangements made. Figure 6 shows three constraints, other than x, y 0 The rejected region has been shaded. Given that two of the three constraints are y 30 and x 60,

write down, as an inequality, the third constraint shown in Figure 6.

Each Celebration arrangement includes 2 white roses and 4 red roses. Each Party arrangement includes 1 white rose and 5 red roses. Lethna wishes to use at least 70 white roses and at least 200 red roses.

Write down two further inequalities to represent this information.

Add two lines and shading to Diagram 1 in the answer book to represent these two inequalities.

Hence determine the feasible region and label it R.

The times taken to produce each Celebration arrangement and each Party arrangement are 10 minutes and 4 minutes respectively. Lethna wishes to minimise the total time taken to produce the arrangements.

Write down the objective function, T, in terms of x and y.

Use point testing to find the optimal number of each type of arrangement Lethna should produce, and find the total time she will take.

60

Figure 6

A company is going to hire out two types of car, standard and luxury. Let x be the number of standard cars it should buy. Let y be the number of luxury cars it should buy. Figure 6 shows three constraints, other than x, y 0 Two of these are x 20 and y 8

Write, as an inequality, the third constraint shown in Figure 6. The company decides that at least of the cars must be luxury cars.

Express this information as an inequality and show that it simplifies to

You must make the steps in your working clear.

Each time the cars are hired they need to be prepared. It takes 5 hours to prepare a standard car and it takes 6 hours to prepare a luxury car. There are 300 hours available each week to prepare the cars.

Express this information as an inequality.

Add two lines and shading to Diagram 1 in the answer book to illustrate the constraints found in parts (b) and (c).

Hence determine the feasible region and label it R.

The company expects to make £80 profit per week on each car. It therefore wishes to maximise P = 80 x + 80 y, where P is the profit per week.

Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must clearly draw and label your objective line and the vertex V.

Given that P is the expected profit, in pounds, per week, find the number of each type of car that the company should buy and the maximum expected profit.

Figure 2

Figure 2 shows the constraints of a linear programming problem in x and y, where R is the feasible region.

Write down the inequalities that form region R. The objective is to maximise 3 x + y.

Find the optimal values of x and y. You must make your method clear.

Obtain the optimal value of the objective function. Given that integer values of x and y are now required,

write down the optimal values of x and y.

The graph in Figure 6 is being used to solve a linear programming problem. Two of the constraints have been drawn on the graph and the rejected regions shaded out.

Write down the constraints shown on the graph.

Two further constraints are

Add two lines and shading to Graph 1 in your answer book to represent these constraints. Hence determine the feasible region and label it R.

The objective is to

Draw a profit line on Graph 1 and use it to find the optimal solution. You must label your profit line clearly.