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Alas, I won't be claiming the big prize! I think you missed the preamble to the question, which makes it clear that in the solution to the practical problem I am facing I am using a BRUTE FORCE approach.
As I'm sure you're aware it is possible to solve the problem by trying every possible permutation of locations to be visited, ONLY if the number of locations is small.
In my case I am dealing with a maximum of 8 as the people I'm working with run 9 seater mini-buses (driver & 8 passengers).
So the completely mathematically inelegant approach I'm taking is simply to calculate all possible permutations of the drop off/pick-up points (max 8) and then use the Dijkstzra algorithm to calculate the distance between each successive pair of drop off/pick-up locations and sum these to reach a total distance for each permutation.
In this way I am guaranteed to find the shortest path.
The people I'm working with run 20 mini-buses, which can theoretically do 6 trips a day. If each of them were to carry 8 passengers each being picked up/dropped off a a different location, the maximum calculation time on my aging twin core 32bit PC is about 15 minutes.
This is much quicker than the task can be done manually and of course relies on the person doing it to have an intimate knowledge of the local area.
In reality, this extreme case is never likely to occur, but I would like to reduce the time the script spends doing the Dijkstra calculations.