Table 2: Travel and visit durations Time in minutes from i to j Time in minutes 1 2 3 4 5 6 Town Visit duration 123456 0 19 23 18 20 25 1 64 19 0 22 13 22 11 3 23 22 0 60 17 20 18 13 60 0 55 17 20 22 17 2345 2 162 88 120 55 0 12 73 25 11 20 17 12 0 6 106 (a) Travel time matrix (b) Visit duration per town Table 3: Available emergency vehicles and emergencies per town Emergency Number of Town Vehicles Emergencies 1 2 1 2 3 4 5 6 62530 3 1 4 1 2
The North West province includes six towns that need clinic visit services. Because some towns are nearby, a single
station may serve at most two communities. The stipulation is that the station must be within 18 minutes of driving
time from the towns it serves. Each clinic has an estimated time that the clinic visits vehicle attends to patients. An
optimal route is required to ensure that the team complete their route before 3:00 p.m. and that 2 hours are left to
complete their administration. The team will only leave their station at 09:00 as daily stock-taking is required before
any clinical visits can be completed. As per law, the personnel have a 1-hour lunch break where no patients are
assisted, or driving has been completed. For simplicity, it is assumed that the team will travel directly to the allocated
location and back to the station. The table below gives the six towns' driving and visiting times in minutes.
Each of these towns will require emergency services during medical emergencies or accidents; the protocol for
allocating the emergency services’ vehicles is to assign the closest available vehicle. If an emergency vehicle has
been allocated, it is seen as in use for the day and cannot be assigned to another emergency. Emergencies are grouped
per town, and each town has a specified number of emergency vehicles available if the town is seen as a service
station. The number of vehicles available and emergencies per station is given in the table below. Note: A station
must be assigned to utilise the emergency vehicles.
The cost associated with the travel duration is R2.00 per minute, and the station's setup cost is R2000.00 per station.
(a) Formulate an integer linear programming (ILP) whose solution will produce the smallest setup and
travelling cost. Find the optimum solution.
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