Edexcel A-Level Mathematics AS D1.5 Linear Programming Questions

Practise linear programming by formulating constraints, graphing feasible regions and choosing optimal continuous or integer solutions.

Exam points

Translate resource, ratio and total conditions into simplified linear inequalities.
Graph constraints, label the feasible region and test vertices with the objective function.
Adjust the optimum when variables must be integers and check feasibility in context.

Covered syllabus points

D1.5.1 - Formulation of problems as linear programs D1.5.2 - Graphical solution of two variable problems D1.5.3 - Consideration of problems where solutions must have integer

[Maximum number: 12]

The head of a Mathematics department needs to order three types of paper. The three types of paper are plain, lined and graph.

All three types of paper are sold in reams. (A ream is 500 sheets of paper.)
Based on the last academic year the head of department formed the following constraints.
- At least half the paper must be lined
- No more than 15% of the paper must be graph paper
- The ratio of plain paper to graph paper must be 5: 2

The cost of each ream of plain, lined and graph paper is £ 5, £ 12 and £ 15 respectively. The head of department has at most £ 834 to spend on paper.

The head of department wants to maximise the total number of reams of paper ordered.
Let x, y and z represent the number of reams of plain paper, lined paper and graph paper ordered respectively.

(a)

Formulate this information as a linear programming problem in x and y only, stating the objective and listing the constraints as simplified inequalities with integer coefficients.

The head of department decides to order exactly 42 reams of lined paper and still wishes to maximise the total number of reams of paper ordered.

[ 7 ]

$\frac12(x+y+z)\le y \quad(\Rightarrow x-y+z\le0)$ $\frac3{20}(x+y+z)\ge z \quad(\Rightarrow 3x+3y-17z\ge0)$ 2x=5z $5x+12y+15z\le834$
Eliminating z from the objective x+y+z and at least one correct constraint, or stating the objective and eliminating z from at least two correct constraints.
Maximise P=1.4x+y, subject to
$\begin{aligned} 11x+12y&\le834\\ 7x-5y&\le0\\ 19x-15y&\le0\\ x&\ge0,\quad y\ge0 \end{aligned}$
(7)

(b)

Determine

[ 5 ]

(i)

the total number of reams of paper to be ordered,

[ 2 ]

Substitute y=42 into the LP:
$x\le30,\quad x\le30,\quad x\le\frac{630}{19}$
so x=30.
Total number of reams ordered:
1.4(30)+42=84

(ii)

the number of reams of graph paper to be ordered.

[ 3 ]

[Maximum number: 17]

Figure for Question 2 — Edexcel A-Level Mathematics AS

Martin is making three types of cake for a picnic. The three types of cake are carrot cake, apple cake and chocolate cake. Along with other ingredients,
- each carrot cake contains 275 grams of flour, 300 grams of sugar and 5 eggs
- each apple cake contains 200 grams of flour, 400 grams of sugar and 2 eggs
- each chocolate cake contains 100 grams of flour, 400 grams of sugar and 3 eggs

If Martin makes only one type of cake then he has enough time to prepare 15 carrot cakes or 20 apple cakes or 30 chocolate cakes.

Martin has 5.5 kilograms of flour and 70 eggs available and he has promised the picnic organisers that he will make at least 18 cakes in total.

Martin plans to make a selection of these cakes and wants to minimise the total amount of sugar that he uses.

Let x be the number of carrot cakes made, y the number of apple cakes made and z the number of chocolate cakes made.

(a)

Formulate this information as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients.

[ 6 ]

Minimise ( P= ) 300 x+400 y+400 z
Subject to:

$275 x+200 y+100 z, 5500(\Rightarrow 11 x+8 y+4 z,, 220)$
5 x+2 y+3 z, 70
$\frac{x}{15}+\frac{y}{20}+\frac{z}{30}, 1 \Rightarrow 4 x+3 y+2 z,, 60$
x+y+z..18

( $x, y, z \ldots 0$ )

B1 (6)

(b)

A further constraint is that y=2z.

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

[ 1 ]

(c)

The constraint y=2z reduces the problem to the following

\begin{aligned} & \text { Minimise } P=300 x+600 y \\ & \text { subject to } \quad \begin{aligned} 11 x+10 y & \leqslant 220 \\ 10 x+7 y & \leqslant 140 \\ x+y & \leqslant 15 \\ 2 x+3 y & \geqslant 36 \\ x & \geqslant 0, y \geqslant 0 \end{aligned} \end{aligned}

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

[ 4 ]

(d)

Use the objective line method to find the optimal number of each type of cake that Martin should make, and the amount of sugar used.

[ 4 ]

(e)

Determine how much flour and how many eggs Martin will have left over after making the optimal number of cakes.

[ 2 ]

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)