Decision Mathematics 1: Linear programmingEdexcel A-Level Further Maths: Topic test

1. 1
   A print shop makes $x$ posters and $y$ flyers each day. Each poster needs 3 ml of ink and 2 minutes of cutting. Each flyer needs 1 ml of ink and 3 minutes of cutting. There are 60 ml of ink and 70 minutes of cutting time available each day.

   (a)

   Which inequality models the ink available each day?

   [1 mark]

   • A$3x+y\le60$
   • B$x+3y\le60$
   • C$3x+y\ge60$
   • D$2x+3y\le60$

   (b)

   On one day the shop makes 10 posters and 15 flyers. How much ink is left unused?

   [1 mark]

   • A$45$ ml
   • B$5$ ml
   • C$15$ ml
   • D$75$ ml

   (c)

   Write the cutting-time constraint as an equation by introducing a slack variable $t$, and state what $t$ represents.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A potter makes $x$ vases and $y$ bowls each week. The profit is £50 per vase and £40 per bowl, so the total profit is $P=50x+40y$ pounds. The clay available limits production to $x+y\le8$ and the kiln time to $2x+y\le12$, with $x\ge0$ and $y\ge0$.

   (a)

   What is the largest value of $x$ that satisfies both constraints?

   [1 mark]

   • A$8$
   • B$6$
   • C$12$
   • D$4$

   (b)

   What is the gradient of the objective line $P=50x+40y$?

   [1 mark]

   • A$\frac{5}{4}$
   • B$-\frac{4}{5}$
   • C$\frac{4}{5}$
   • D$-\frac{5}{4}$

   (c)

   Use the vertex method to find the number of vases and bowls that give the greatest weekly profit, and state that profit.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A caterer prepares $x$ trays of samosas and $y$ trays of spring rolls each day. The profit is £5 per tray of samosas and £6 per tray of spring rolls, so the profit is $P=5x+6y$ pounds. Preparation time limits production to $x+2y\le24$ and oven time to $x+y\le14$. Slack variables $r$ and $s$ are added to the two constraints and the Simplex algorithm is used to maximise $P$.

   (a)

   Write down the initial Simplex tableau.

   [3 marks]

   (b)

   Perform one complete iteration of the Simplex algorithm, stating your pivot, and show the new tableau. State, with a reason, whether the solution is now optimal.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school canteen mixes $x$ kg of lentils and $y$ kg of rice for each batch of meals. Each kilogram of lentils supplies 2 units of protein and 1 unit of fibre. Each kilogram of rice supplies 1 unit of protein and 2 units of fibre. A batch must supply at least 8 units of protein and at least 7 units of fibre, and its total mass must be at most 10 kg. Lentils cost £3 per kg and rice costs £2 per kg. The canteen wishes to minimise the cost $C$ pounds of a batch.

   (a)

   Formulate this as a linear programming problem. Then write each constraint as an equation, introducing slack, surplus and artificial variables as necessary, ready for the two-stage Simplex method.

   [6 marks]

   (b)

   Solve the problem graphically to find the cheapest mix of lentils and rice, giving the cost of that batch. Show the coordinates of every vertex of the feasible region and the cost at each.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A greengrocer buys $x$ boxes of mangoes and $y$ boxes of papayas each week and must buy a whole number of boxes of each. The profit is £8 per box of mangoes and £9 per box of papayas, so the profit is $P=8x+9y$ pounds. Storage space limits the purchases to $2x+y\le19$ and the weekly budget, in hundreds of pounds, limits them to $x+2y\le21$, with $x\ge0$ and $y\ge0$.

   (a)

   Which of these points is not in the feasible region?

   [1 mark]

   • A$(5,8)$
   • B$(6,7)$
   • C$(6,8)$
   • D$(7,5)$

   (b)

   What is the profit at the integer point $(6,7)$?

   [1 mark]

   • A£$111$
   • B£$101$
   • C£$114\frac{1}{3}$
   • D£$120$

   (c)

   The two constraint lines meet at $\left(\frac{17}{3},\frac{23}{3}\right)$. Find the number of boxes of each fruit that maximises the profit, and the maximum profit.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A pottery makes $x$ tea sets and $y$ dinner sets each day. The profit is $P=9x+5y$ in hundreds of pounds. The kiln, glazing and packing constraints are $x+y\le8$, $x+3y\le12$ and $2x+y\le15$, with slack variables $r$, $s$ and $t$ respectively. After two iterations of the Simplex algorithm the tableau is $\begin{array}{c|ccccc|c} \text{b.v.} & x & y & r & s & t & \text{Value} \\ \hline y & 0 & 1 & 2 & 0 & -1 & 1 \\ s & 0 & 0 & -5 & 1 & 2 & 2 \\ x & 1 & 0 & -1 & 0 & 1 & 7 \\ \hline P & 0 & 0 & 1 & 0 & 4 & 68 \end{array}$

   (a)

   What is the value of $P$ shown in the tableau?

   [1 mark]

   • A$63$
   • B$7$
   • C$\frac{135}{2}$
   • D$68$

   (b)

   Which constraint is not fully used at this solution?

   [1 mark]

   • AThe kiln constraint
   • BThe glazing constraint
   • CThe packing constraint
   • DNone of them

   (c)

   Explain how the tableau shows that the solution is optimal, and state the number of tea sets and dinner sets made.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A nursery grows $x$ trays of tomato plants and $y$ trays of pepper plants each week. The profit is $P=4x+3y$ pounds. The constraints are $x+y\ge4$, $2x+y\le14$ and $x+2y\le12$, with $x\ge0$ and $y\ge0$. The nursery wishes to maximise $P$.

   (a)

   Write the constraints as equations using surplus variable $s_1$, slack variables $s_2$ and $s_3$ and an artificial variable $t_1$. State the objective of the first stage of the two-stage Simplex method.

   [3 marks]

   (b)

   Using the big-M method, write down the modified objective function. Eliminate $t_1$ to find the objective row of the initial tableau for the columns $x$, $y$, $s_1$, $s_2$, $s_3$, $t_1$ and the value.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A soap maker produces $x$ boxes of bath bombs and $y$ boxes of soaps each day. Each box of bath bombs needs 3 litres of oil, 2 hours of moulding and 2 hours of packing. Each box of soaps needs 2 litres of oil, 1 hour of moulding and 4 hours of packing. Each day there are 16 litres of oil, 10 hours of moulding time and 24 hours of packing time available. The profit is £8 per box of bath bombs and £5 per box of soaps, and the soap maker wishes to maximise the daily profit $P$ pounds.

   (a)

   Formulate this as a linear programming problem, find the coordinates of every vertex of the feasible region and use the vertex method to find the best production plan and the maximum profit.

   [6 marks]

   (b)

   Use the Simplex algorithm, with slack variables $r$, $s$ and $t$ for oil, moulding and packing, to confirm your answer to part (a). Show the initial tableau and every iteration, and state the unused packing time.

   [6 marks]

   Total for question 8: 12 marks

End of questions