A Simulation-Based Algorithm for the Capacitated Vehicle Routing Problem with Stochastic Travel Times

Algorithm 1

The suggested algorithm of the SBA.

1 Set Solutions = Ø, TotalCosts = Ø, TravelCosts = Ø, PenaltyCosts = Ø, Reliabilities = Ø;

2 Calculate the expected travel time 𝑒𝑡_𝑖𝑗 for each edge (𝑖, 𝑗) (𝑣_𝑖, 𝑣_𝑗 ∈𝑉, 𝑖≠𝑗) in CVRPSTT

  according to its distribution function;

3 For 𝑟=𝑟₀ to 𝑟_max step 𝑟_step, do:

4   Transform CVRPSTT to the well-known CVRP in which the maximum vehicle working

    time is set to 𝑊×𝑟 that cannot be exceeded by any vehicle and the travel time

    between any pair of nodes is replaced with its expected value;

5    Solve the CVRP (𝑟) by using any existing heuristic and insert the obtained solution OS into

     Solutions;

6   Set TotalCost = 0, where TotalCost denotes the expected total cost for solution OS,

     TravelCost = 0, where TravelCost denotes the expected travel cost for solution OS,

     PenaltyCost = 0, where PenaltyCost denotes the expected penalty cost for solution OS, and

     Reliability = 0, where Reliability denotes the reliability for solution OS;

7   For 𝑖=1 to 𝑖_max step 1, do:

8    Construct the simulant travel time matrix, each element of which is generated

     according to its travel time distribution function and indicates the simulant travel time

     for an edge, for example 𝑑𝑡_𝑖𝑗 (0≤𝑖≤𝑛, 0≤𝑗≤𝑛) for the travel time of edge (𝑖, 𝑗);

9    Calculate the total cost, denoted as C1, the travel cost, denoted as C2, the penalty

     cost, denoted as C3, and the reliability, denoted as C4 for solution OS according to

     their definitions in Section 2 by using 𝑑𝑡_𝑖𝑗 to substitute the travel time of each edge

      (𝑖, 𝑗) and using W to represent the maximum vehicle working time that cannot be

     exceeded by any vehicle;

10     Set TotalCost = TotalCost + C1/𝑖_max, TravelCost = TravelCost + C2/𝑖_max,

       PenaltyCost = PenaltyCost + C3/𝑖_max, and Reliability = Reliability + C4/𝑖_max;

11   Next 𝑖;

12   Insert TotalCost into TotalCosts, TravelCost into TravelCosts, PenaltyCost into

   PenaltyCosts, and Reliability into Reliabilities;

13 Next 𝑟;

14 Get the minimum expected total cost from TotalCosts and its corresponding solution CS from

   Solutions.

15 Improve the solution CS by a method that is further described in Algorithm 2;

16 Return the improved solution CS and its corresponding total cost, travel cost, penalty

   cost and reliability, which have been updated in step 15.