Decision Mathematics 2: Transportation problemsEdexcel A-Level Further Maths: Topic test

20 questions, 54 marks

Edexcel A-Level Further Maths

Decision Mathematics 2: Transportation problems topic test

Total 54 marks

Name

Class

Date

1
A grain merchant moves grain from three farms, F1, F2 and F3, which have 30, 40 and 20 tonnes available, to three mills, X, Y and Z, which need 25, 35 and 30 tonnes. The cost, in £ per tonne, is: F1 to X 6, F1 to Y 8, F1 to Z 11; F2 to X 9, F2 to Y 5, F2 to Z 7; F3 to X 10, F3 to Y 12, F3 to Z 4.
(a)
Using the north-west corner method, how many tonnes are sent from F1 to mill Y?
[1 mark]
A $25$
B $5$
C $10$
D $0$
(b)
How many occupied cells are there in the initial solution from the north-west corner method?
[1 mark]
A $3$
B $4$
C $6$
D $5$
(c)
Find the total cost of the initial solution from the north-west corner method.
[2 marks]
Total for question 1: 4 marks
2
A brewery has three depots, D1, D2 and D3, holding 45, 35 and 30 crates of lemonade. Three shops, X, Y and Z, order 30, 40 and 25 crates. The cost, in £ per crate, is: D1 to X 4, D1 to Y 7, D1 to Z 9; D2 to X 6, D2 to Y 3, D2 to Z 8; D3 to X 5, D3 to Y 6, D3 to Z 2. Total stock is greater than total orders.
(a)
A dummy destination is added to balance the problem. What is its demand, in crates?
[1 mark]
A $15$
B $95$
C $110$
D $205$
(b)
What cost is entered in each cell of the dummy column?
[1 mark]
AThe largest cost in the table, £9
BThe mean of the real costs
C£0
DThe cost of the nearest depot
(c)
Use the north-west corner method on the balanced problem, with the dummy destination as the last column, to find an initial allocation. Find its total cost.
[2 marks]
Total for question 2: 4 marks
3
A fruit grower has three greenhouses, G1, G2 and G3, with 20, 25 and 15 boxes of tomatoes available for three stores, X, Y and Z, which need 12, 23 and 25 boxes. The cost, in £ per box, is: G1 to X 5, G1 to Y 3, G1 to Z 8; G2 to X 3, G2 to Y 4, G2 to Z 6; G3 to X 9, G3 to Y 8, G3 to Z 2. The current solution has G1X 12, G1Y 8, G2Y 15, G2Z 10 and G3Z 15 occupied. The shadow costs are $R_i$ for the greenhouses and $K_j$ for the stores.
(a)
Find the shadow costs, taking $R_1=0$ .
[3 marks]
(b)
Find the improvement index of each empty cell and state which cell should enter the solution.
[4 marks]
Total for question 3: 7 marks
4
A water company moves bottled water from three plants, A, B and C, holding 30, 40 and 30 thousand litres, to three towns, X, Y and Z, which need 30, 50 and 20 thousand litres. The cost, in £ per thousand litres, is: A to X 4, A to Y 6, A to Z 9; B to X 5, B to Y 3, B to Z 7; C to X 8, C to Y 5, C to Z 4.
(a)
Use the north-west corner method to find an initial solution and show that it is degenerate. Explain how to deal with the degeneracy before shadow costs are found, and find the cost of the initial solution.
[6 marks]
(b)
A zero is placed in cell BX. Taking $R_A=0$ , find the shadow costs and the improvement indices of the empty cells, and hence show that the solution is optimal. State the minimum cost.
[6 marks]
Total for question 4: 12 marks
5
A company sends goods from two factories, F1 and F2, which can supply 40 and 35 tonnes, to three customers, X, Y and Z, who need 20, 30 and 25 tonnes. Let $x_{ij}$ be the number of tonnes sent from factory $i$ to customer $j$ , where $j=1,2,3$ for X, Y, Z. The cost, in £ per tonne, is: F1 to X 3, F1 to Y 5, F1 to Z 4; F2 to X 6, F2 to Y 2, F2 to Z 7. The problem is formulated as a linear programme.
(a)
How many decision variables does the linear programme have?
[1 mark]
A $5$
B $3$
C $6$
D $2$
(b)
Which equation is the constraint for customer Y?
[1 mark]
A $x_{12}+x_{22}=30$
B $x_{21}+x_{22}+x_{23}=30$
C $x_{12}+x_{22}=35$
D $x_{11}+x_{12}+x_{13}=30$
(c)
Write down the objective function, and the constraint for the amount leaving F2.
[2 marks]
Total for question 5: 4 marks
6
A transportation problem has three sources, S1, S2 and S3, and three destinations, X, Y and Z. The cost, in £ per unit, is: S1 to X 5, S1 to Y 4, S1 to Z 6; S2 to X 9, S2 to Y 7, S2 to Z 10; S3 to X 8, S3 to Y 7, S3 to Z 9. The current solution has S1X 18, S1Y 7, S2Y 15, S2Z 10 and S3Z 12 occupied. The shadow costs are $R_i$ for the sources and $K_j$ for the destinations, with $R_1=0$ .
(a)
What is the value of $K_Z$ ?
[1 mark]
A $10$
B $3$
C $4$
D $7$
(b)
What is the improvement index of the empty cell S1Z?
[1 mark]
A $1$
B $-1$
C $6$
D $13$
(c)
The cell S1Z enters the solution. Use the stepping-stone method to find the exiting cell and the number of units that move round the loop.
[2 marks]
Total for question 6: 4 marks
7
A firm sends flour from three mills, S1, S2 and S3, with 20, 30 and 25 tonnes available, to three bakeries, X, Y and Z, which need 20, 35 and 15 tonnes. A dummy destination W is added because supply is greater than demand. The cost, in £ per tonne, is: S1 to X 3, S1 to Y 5, S1 to Z 6; S2 to X 7, S2 to Y 4, S2 to Z 8; S3 to X 2, S3 to Y 6, S3 to Z 2. Every cell in the dummy column has cost £0.
(a)
State the demand of W. Use the north-west corner method to find an initial solution, and show that it is degenerate.
[3 marks]
(b)
A zero is placed in cell S1Y. Taking $R_1=0$ , find the shadow costs. Find the improvement index of S3X and state what it shows about the solution.
[4 marks]
Total for question 7: 7 marks
8
A council moves waste from three collection sites, H1, H2 and H3, which have 40, 30 and 35 tonnes, to three plants, X, Y and Z, which can take 25, 40 and 40 tonnes. The cost, in £ per tonne, is: H1 to X 2, H1 to Y 4, H1 to Z 3; H2 to X 7, H2 to Y 4, H2 to Z 9; H3 to X 5, H3 to Y 10, H3 to Z 2.
(a)
Use the north-west corner method to find an initial solution and its cost. Use shadow costs and improvement indices to show that this solution is not optimal, and state which cell should enter.
[6 marks]
(b)
The cell H1Z enters the solution from part (a). Use the stepping-stone method to find the improved solution and its cost. Show that the improved solution is optimal.
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).