D2 June 2005 Q1
1. Freezy Co. has three factories \(A\), \(B\) and \(C\). It supplies freezers to three shops \(D\), \(E\) and \(F\). The table shows the transportation cost in pounds of moving one freezer from each factory to each outlet. It also shows the number of freezers available for delivery at each factory and the number of freezers required at each shop. The total number of freezers required is equal to the total number of freezers available.
| \(D\) | \(E\) | \(F\) | Available | |
|---|---|---|---|---|
| \(A\) | 21 | 24 | 16 | 24 |
| \(B\) | 18 | 23 | 17 | 32 |
| \(C\) | 15 | 19 | 25 | 14 |
| Required | 20 | 30 | 20 |
(a) Use the north-west corner rule to find an initial solution. (2)
(b) Obtain improvement indices for each unused route. (5)
(c) Use the stepping-stone method once to obtain a better solution and state its cost. (4)
| Scheme | Marks | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 A1 | ||||||||||||||||
| (2) |
| Scheme | Marks |
|---|---|
| \(S_A = 0\quad S_B = -1\quad S_C = 7\) | M1 |
| \(D_D = 21\quad D_E = 24\quad D_F = 18\) | A1 |
| \(I_{13} = I_{AF} = 16 - 0 - 18 = -2\) | |
| \(I_{21} = I_{BD} = 18 + 1 - 21 = -2\) | M1 |
| \(I_{31} = I_{CD} = 15 - 7 - 21 = -13\ (*)\) | A1ft |
| \(I_{32} = I_{CE} = 19 - 7 - 24 = -12\) | A1ft |
| (5) |
Notes
(Corrected from the printed mark scheme: the shadow cost \(D_D = 21\) is printed as \(D_P = 21\).)
| Scheme | Marks | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| eg \(CD(+) \to AD(-) \to AE(+) \to BE(-) \to BF(+) \to CF(-)\qquad \theta = 14\) | M1 A1ft | ||||||||||||||||
cost £1384 | A1ft A1 | ||||||||||||||||
| (4) | |||||||||||||||||
| (11 marks) |