Traveling Sales Person - Family of functions

General Information

Problem Definition

The travelling salesman problem (TSP) or travelling salesperson problem asks the following question:

Given a list of cities and the distances between each pair of cities, which is the shortest possible route that visits each city exactly once and returns to the origin city?

Origin

The traveling sales person problem was studied in the 18th century by mathematicians
Sir William Rowam Hamilton and Thomas Penyngton Kirkman.

A discussion about the work of Hamilton & Kirkman can be found in the book Graph Theory (Biggs et al. 1976).

  • ISBN-13: 978-0198539162
  • ISBN-10: 0198539169

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)

Characteristics

  • The travel costs are symmetric:
    • traveling costs from city A to city B are just as much as traveling from B to A.
  • This problem is an NP-hard optimization problem.
  • To calculate the number of different tours through \(n\) cities:
    • Given a starting city,
    • There are \(n-1\) choices for the second city,
    • And \(n-2\) choices for the third city, etc.
    • Multiplying these together we get \((n-1)! = (n-1) (n-2) . . 1\).
    • Now since our travel costs do not depend on the direction we take around the tour:
      • this number by 2
      • \((n-1)!/2\).

Simulated Annealing Algorithm

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.

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 newSolution 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 {
        newSolution = neighbour(solution);
        If P(E(solution), E(newSolution), T) >= random(0, 1)
            solution = newSolution;
    }

    temperature = temperature * cooling factor;
}

Output: the best solution

pgRouting Implementation

pgRouting’s implementation adds some extra parameters to allow some exit controls within the simulated annealing process.

  • max_changes_per_temperature:
    • Limits the number of changes in the solution per temperature
    • Count is reset to \(0\) when temperature changes
    • Count is increased by :math: 1 when solution changes
  • max_consecutive_non_changes:
    • Limits the number of consecutive non changes per temperature
    • Count is reset to \(0\) when solution changes
    • Count is increased by :math: 1 when solution changes
  • max_processing_time:
    • Limits the time the simulated annealing is performed.
Solution = initial_solution;

temperature = initial_temperature;
WHILE (temperature > final_temperature) {

    DO tries_per_temperature times {
        newSolution = neighbour(solution);
        If Probability(E(solution), E(newSolution), T) >= random(0, 1)
            solution = newSolution;

        BREAK DO WHEN:
            max_changes_per_temperature is reached
            OR max_consecutive_non_changes is reached
    }

    temperature = temperature * cooling factor;
    BREAK WHILE WHEN:
        no changes were done in the current temperature
        OR max_processing_time has being reached
}

Output: the best solution found

Choosing parameters

There is no exact rule on how the parameters have to be chose, it will depend on the special characteristics of the problem.

  • If the computational time is crucial, then limit execution time with max_processing_time.
  • Make the tries_per_temperture depending on the number of cities (\(n\)), for example:
    • Useful to estimate the time it takes to do one cycle: use 1
      • this will help to set a reasonable max_processing_time
    • \(n * (n-1)\)
    • \(500 * n\)
  • For a faster decreasing the temperature set cooling_factor to a smaller number, and set to a higher number for a slower decrease.
  • When for the same given data the same results are needed, set randomize to false.
    • When estimating how long it takes to do one cycle: use false

A recommendation is to play with the values and see what fits to the particular data.

Description of the Control Parameters

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

  • true: Use current time as seed
  • false: Use 1 as seed. Using this value will get the same results with the same data in each execution.

Description of the return columns

Returns SET OF (seq, node, cost, agg_cost)

Column Type Description
seq INTEGER Row sequence.
node BIGINT Identifier of the node/coordinate/point.
cost FLOAT
Cost to traverse from the current node to the next node in the path sequence.
  • 0 for the last row in the path sequence.
agg_cost FLOAT
Aggregate cost from the node at seq = 1 to the current node.
  • 0 for the first row in the path sequence.