D2 June 2005 Q3
3. Three warehouses \(W\), \(X\) and \(Y\) supply televisions to three supermarkets \(J\), \(K\) and \(L\). The table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. The warehouses have stocks of 34, 57 and 25 televisions respectively, and the supermarkets require 20, 56 and 40 televisions respectively. The total cost of transporting the televisions is to be minimised.
| \(J\) | \(K\) | \(L\) | |
|---|---|---|---|
| \(W\) | 3 | 6 | 3 |
| \(X\) | 5 | 8 | 4 |
| \(Y\) | 2 | 5 | 7 |
Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints. (7)
| Scheme | Marks |
|---|---|
| Let \(x_{ij}\) be number of units transported from \(i\) to \(j\) where \(i \in \{W, X, Y\}\) (warehouse) and \(j \in \{J, K, L\}\) (supermarket) | B1 |
| objective minimise “\(C\)” \(= 3x_{WJ} + 6x_{WK} + 3x_{WL} +\) | B1 |
| \(5x_{XJ} + 8x_{XK} + 4x_{XL} +\) | B1 |
| \(2x_{YJ} + 5x_{YK} + 7x_{YL}\) | |
| subject to \(x_{WJ} + x_{WK} + x_{WL} = 34\) \(x_{XJ} + x_{XK} + x_{XL} = 57\) \(x_{YJ} + x_{YK} + x_{YL} = 25\) | M1 A1 |
| \(x_{WJ} + x_{XJ} + x_{YJ} = 20\) \(x_{WK} + x_{XK} + x_{YK} = 56\) \(x_{WL} + x_{XL} + x_{YL} = 40\) | A1 |
| \(x_{ij} \geqslant 0\quad \forall\ i \in \{W, X, Y\}\) and \(j \in \{J, K, L\}\) | B1 |
| (7 marks) |
Notes
(Corrected from the printed mark scheme: \(x_{WL}\) is printed as \(x_{UL}\) in the first constraint, and \(x_{XJ}\) is printed as \(x_{XY}\) in the fourth.)