- The table shows the least distances, in km, between six towns, A, B, C, D, E and F.
| A | B | C | D | E | F |
| A | - | 122 | 217 | 137 | 109 | 82 |
| B | 122 | - | 110 | 130 | 128 | 204 |
| C | 217 | 110 | - | 204 | 238 | 135 |
| D | 137 | 130 | 204 | - | 98 | 211 |
| E | 109 | 128 | 238 | 98 | - | 113 |
| F | 82 | 204 | 135 | 211 | 113 | - |
Liz must visit each town at least once. She will start and finish at A and wishes to minimise the total distance she will travel.
- Starting with the minimum spanning tree given in your answer book, use the shortcut method to find an upper bound below 810 km for Liz's route. You must state the shortcut(s) you use and the length of your upper bound.
- Use the nearest neighbour algorithm, starting at A , to find another upper bound for the length of Liz's route.
- Starting by deleting F, and all of its arcs, find a lower bound for the length of Liz's route.
- Use your results to write down the smallest interval which you are confident contains the optimal length of the route.