I am implementing the algorithms found in the pdf document downloadable at this link An Effective Simulated Annealing Algorithm for Solving the Traveling Salesman Problem.
I'm trying to understand the meaning of the summation found at the very end of the first column on page 4 of the pdf (step 3 of the second stage of the algorithm).
I have no real mathematical background but, as far as I can understand, the upper bound can be assumed to be the number of cities (excluded the one randomly chosen). I'm also assuming that the lack of elements to sum in the formula defaults to 1 (one), meaning that the result of the summation is nothing more than a mere count of the cities (minus one; the x1 randomly extracted).
Am I correct? If yes I wonder why not to write a simple n-1?
I hope is clear where my bafflement come from.
Thanks in advance.
I figured it out, I think.
I don't know if it is common practice in a mathematical grammatic sense, but it seems to me that the denominator in formula(8) is not a "real" summation.
Anyway the meaning of that sigma, as far as I can understand, should be: sum all the roads distances excluding the ones leading to or departing from city x1.
In other words "eliminate" the city x1 from the map, then sum all the remaining roads lengths.
It makes sense, algorithmically speaking, I think.
Let me know if you see some flaw in my thought.