2. The table shows the least times, in seconds, that it takes a robot to travel between six points in an automated warehouse. These six points are an entrance, A , and five storage bins, \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F . The robot will start at A , visit each bin, and return to A . The total time taken for the robot's route is to be minimised.
| A | B | C | D | E | F |
| A | - | 90 | 130 | 85 | 35 | 125 |
| B | 90 | - | 80 | 100 | 83 | 88 |
| C | 130 | 80 | - | 108 | 106 | 105 |
| D | 85 | 100 | 108 | - | 110 | 88 |
| E | 35 | 83 | 106 | 110 | - | 75 |
| F | 125 | 88 | 105 | 88 | 75 | - |
- Show that there are two nearest neighbour routes that start from A . You must make the routes and their lengths clear.
- Starting by deleting F , and all of its arcs, find a lower bound for the time taken for the robot's route.
- Use your results to write down the smallest interval which you are confident contains the optimal time for the robot's route.