A discussion about the work of Hamilton & Kirkman can be found in the book Graph Theory (Biggs et al. 1976).
It is believed that the general form of the TSP have been first studied by Kalr Menger in Vienna and Harvard. The problem was later promoted by Hassler, Whitney & Merrill at Princeton. A detailed description about the connection between Menger & Whitney, and the development of the TSP can be found in On the history of combinatorial optimization (till 1960)
Given a collection of cities and travel cost between each pair, find the cheapest way for visiting all of the cities and returning to the starting point.
The simulated annealing algorithm was originally inspired from the process of annealing in metal work.
Annealing involves heating and cooling a material to alter its physical properties due to the changes in its internal structure. As the metal cools its new structure becomes fixed, consequently causing the metal to retain its newly obtained properties. [C001]
Pseudocode
Given an initial solution, the simulated annealing process, will start with a high temperature and gradually cool down until the desired temperature is reached.
For each temperature, a neighbouring new solution snew is calculated. The higher the temperature the higher the probability of accepting the new solution as a possible bester solution.
Once the desired temperature is reached, the best solution found is returned
Solution ← initial_solution;
temperature ← initial_temperature;
while (temperature > final_temperature) {
do tries_per_temperature times {
snew ← neighbour(solution);
If P(E(solution), E(snew), T) ≥ random(0, 1)
solution ← snew;
}
temperature ← temperature * cooling factor;
}
Output: the best solution
pgRouting’s implementation adds some extra parameters to allow some exit controls within the simulated annealing process.
To cool down faster to the next temperature:
This is done by doing some book keeping on the times solution ← snew; is executed.
Additionally to stop the algorithm at a higher temperature than the desired one:
Note that, if no change was found in the first max_consecutive_non_changes tries, then the simulated annealing will stop.
Solution ← initial_solution;
temperature ← initial_temperature;
while (temperature > final_temperature) {
do tries_per_temperature times {
snew ← neighbour(solution);
If P(E(solution), E(snew), T) ≥ random(0, 1)
solution ← snew;
when max_changes_per_temperature is reached
or max_consecutive_non_changes is reached
BREAK;
}
temperature ← temperature * cooling factor;
when no changes were done in the current temperature
or max_processing_time has being reached
BREAK;
}
Output: the best solution
There is no exact rule on how the parameters have to be chose, it will depend on the special characteristics of the problem.
A recommendation is to play with the values and see what fits to the particular data.
The control parameters are optional, and have a default value.
Parameter  Type  Default  Description 

start_vid  BIGINT  0  The greedy part of the implementation will use this identifier. 
end_vid  BIGINT  0  Last visiting vertex before returning to start_vid. 
max_processing_time  FLOAT  +infinity  Stop the annealing processing when the value is reached. 
tries_per_temperature  INTEGER  500  Maximum number of times a neighbor(s) is searched in each temperature. 
max_changes_per_temperature  INTEGER  60  Maximum number of times the solution is changed in each temperature. 
max_consecutive_non_changes  INTEGER  100  Maximum number of consecutive times the solution is not changed in each temperature. 
initial_temperature  FLOAT  100  Starting temperature. 
final_temperature  FLOAT  0.1  Ending temperature. 
cooling_factor  FLOAT  0.9  Value between between 0 and 1 (not including) used to calculate the next temperature. 
randomize  BOOLEAN  true  Choose the random seed

References
[C001]  Simulated annaeling algorithm for beginners 
Indices and tables