5 There is a one-way system in Manchester. Mia is parked at her base, \(B\), in Manchester and intends to visit four other places, \(A , C , D\) and \(E\), before returning to her base. The following table shows the distances, in kilometres, for Mia to drive between the five places \(A , B , C , D\) and \(E\). Mia wants to keep the total distance that she drives to a minimum.
| \backslashbox{From}{To} | \(\boldsymbol { A }\) | \(\boldsymbol { B }\) | \(\boldsymbol { C }\) | \(D\) | E |
| A | - | 1.7 | 1.9 | 1.8 | 2.1 |
| B | 3.1 | - | 2.5 | 1.8 | 3.7 |
| \(\boldsymbol { C }\) | 3.1 | 2.9 | - | 2.7 | 4.2 |
| \(\boldsymbol { D }\) | 2.0 | 2.8 | 2.1 | - | 2.3 |
| E | 2.2 | 3.6 | 1.9 | 1.7 | - |
- Find the length of the tour \(B E C D A B\).
- Find the length of the tour obtained by using the nearest neighbour algorithm starting from \(B\).
- Write down which of your answers to parts (a) and (b) would be the better upper bound for the total distance that Mia drives.
- On a particular day, the council decides to reverse the one-way system. For this day, find the length of the tour obtained by using the nearest neighbour algorithm starting from \(B\).