Linear Programming (LP) problems | Edexcel A Level Further Maths: Decision 1 Exam Questions & Answers 2017 [PDF]

7 marks

Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies.

Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order.

At least three-fifths of all the flowers must be roses.
For every 2 hydrangeas there must be at most 3 peonies.
The total number of flowers must be exactly 1000

The cost of each rose is £1, the cost of each hydrangea is £5 and the cost of each peony is £4

Ben wants to minimise the cost of the flowers.

Let $x$ represent the number of roses, let $y$ represent the number of hydrangeas and let $z$ represent the number of peonies that he will order.
Formulate this as a linear programming problem in $x$ and $y$ only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients.

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3 marks

Ben decides to order the minimum number of roses that satisfy his constraints.

(i) Calculate the number of each type of flower that he will order to minimise the cost of the flowers.

(ii) Calculate the corresponding total cost of this order.

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5 marks

A publisher plans to produce three versions of the same book: a paperback, a hardcover, and a deluxe edition.

Each paperback takes 4 minutes to print and 1 minute to bind
Each hardcover takes 8 minutes to print and 5 minutes to bind
Each deluxe edition takes 15 minutes to print and 12 minutes to bind

The printing machine is available for at most 150 hours and the binding machine must be used for at least 60 hours.

The publisher decides to produce

at least 1600 books in total
at least three times as many paperbacks as hardcovers

The profit on each paperback sold is £8, the profit on each hardcover sold is £20 and the profit on each deluxe edition sold is £40

Let $x, y$ and $z$ represent the number of paperbacks, hardcovers and deluxe editions produced.

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

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6 marks

The publisher decides to solve this linear programming problem by using the two-stage simplex method.

Set up an initial tableau for solving this problem using the two-stage simplex method.

As part of your solution, you must show how

the constraints have been made into equations by using slack variables, exactly two surplus variables and exactly two artificial variables
the rows for the two objective functions are formed

Empty table with six columns labeled b.v., x, y, z, and Value. Rows include variables P and I, with spaces for additional data entries.

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5 marks

The following tableau is obtained after two iterations of the first stage of the two-stage simplex method.

b.v.	$x$	$y$	$z$	$s_{1}$	$s_{2}$	$s_{3}$	$s_{4}$	$a_{1}$	$a_{2}$	Value
$s_{1}$	$0$	$0$	$0$	$1$	$1$	3	$0$	$- 1$	$- 3$	$600$
$z$	$0$	$\frac{4}{11}$	$1$	$0$	$- \frac{1}{11}$	$\frac{1}{11}$	$0$	$\frac{1}{11}$	$- \frac{1}{11}$	$\frac{2000}{11}$
$x$	$1$	$\frac{7}{11}$	$0$	$0$	$\frac{1}{11}$	$- \frac{12}{11}$	$0$	$- \frac{1}{11}$	$\frac{12}{11}$	$\frac{15600}{11}$
$s_{4}$	$0$	$\frac{40}{11}$	$0$	$0$	$\frac{1}{11}$	$- \frac{12}{11}$	$1$	$- \frac{1}{11}$	$\frac{12}{11}$	$\frac{15600}{11}$
$P$	$0$	$- \frac{4}{11}$	$0$	$0$	$- \frac{32}{11}$	$- \frac{56}{11}$	$0$	$\frac{32}{11}$	$\frac{56}{11}$	$\frac{204800}{11}$
$I$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$0$

Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the second stage of the two-stage simplex method to obtain a new tableau. Make your method clear by stating the row operations you use.

Table for a linear programming tableau with columns for variables, slack variables, value, and row operations. Six rows, starting with b.v. and P.

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2 marks

After three iterations of the second stage of the two-stage simplex method, the following tableau is obtained.

b.v.	$x$	$y$	$z$	$s_{1}$	$s_{2}$	$s_{3}$	$s_{4}$	Value
$s_{2}$	$0$	$0$	$0$	$1$	$1$	$3$	$0$	$600$
$z$	$0$	$0$	$1$	$\frac{1}{10}$	$0$	$\frac{1}{2}$	$- \frac{1}{10}$	$100$
$x$	$1$	$0$	$0$	$- \frac{3}{40}$	$0$	$- \frac{9}{8}$	$- \frac{7}{40}$	$1125$
$y$	$0$	$1$	$0$	$- \frac{1}{40}$	$0$	$- \frac{3}{8}$	$\frac{11}{40}$	$375$
$P$	$0$	$0$	$0$	$\frac{29}{10}$	$0$	$\frac{7}{2}$	$\frac{1}{10}$	$20500$

Given that the publisher produces the optimal number of each version of the book,

(i) state the maximum profit the publisher can earn,

(ii) find the number of hours the binding machine will be used.

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1 mark

Give a reason why the publisher may not earn the profit stated in (d)(i).

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Linear Programming (LP) problems (Edexcel A Level Further Maths: Decision 1): Exam Questions