Transportation problemsEdexcel A-Level Further Maths: Mind map

Exam questions on Transportation problems

1. A company has three warehouses W1, W2 and W3 holding 20, 30 and 25 units of stock, and three shops S1, S2 and S3 needing 15, 35 and 25 units. The cost (£ per unit) of sending stock is: from W1, S1 £4, S2 £6, S3 £8; from W2, S1 £5, S2 £3, S3 £7; from W3, S1 £9, S2 £6, S3 £2. The north-west corner method is used to find an initial solution.
   Calculate the total cost of the north-west corner solution.2 marks
2. In a transportation problem the sources are S1, S2 and S3 and the destinations are D1, D2 and D3. The occupied cells have unit costs S1D1 £4, S1D2 £3, S2D2 £5, S2D3 £6 and S3D3 £4. The unoccupied cells have unit costs S1D3 £8, S2D1 £5, S3D1 £9 and S3D2 £8. Shadow costs $R_i$ (sources) and $K_j$ (destinations) are found from $R_i+K_j=C_{ij}$ for occupied cells with $R_1=0$, and the improvement index of an unoccupied cell is $I_{ij}=C_{ij}-R_i-K_j$.
   Find the improvement indices of the other three unoccupied cells. State whether the current solution is optimal and, if not, which cell should enter.2 marks
3. Three quarries A, B and C supply 25, 45 and 30 tonnes of gravel to three building sites X, Y and Z, which need 20, 40 and 40 tonnes. The cost (£ per tonne) is: from A, X £3, Y £4, Z £9; from B, X £6, Y £6, Z £7; from C, X £2, Y £6, Z £6. The current solution sends 20 tonnes from A to X, 5 from A to Y, 35 from B to Y, 10 from B to Z and 30 from C to Z, at a total cost of £540.
   Using $R_A=0$, find the shadow costs $R_B$, $R_C$, $K_X$, $K_Y$ and $K_Z$, and hence find the improvement index of the unoccupied cell CX.3 marks