Options
The vehicle routing problem
Author
Tan, Lynda Ai Pin
Supervisor
Shutler, Paul
Abstract
The Vehicle Routing Problem (VRP) involves the design of a set of vehicle routes that operates with minimum cost. In many practical examples, the capital costs involved are usually dominant. In my particular variant, the assumption is that most of the capital cost is from purchasing vehicles, so the objective will be to minimize the number of vehicles required.
To solve this problem, we adopted heuristic methods, due to the computational complexity of this problem type. Since optimal solutions are almost impossible to obtain with a reasonable time for large number of cities, near-optimal solutions were sought instead, by using upper and lower bounds.
To compute the upper bound, we found a tour of the pick-up points using a greedy algorithm and then constructed a schedule using a sector partitioning approach. To compute the lower bound, certain constraints were relaxed and a simpler but unrealizable problem was solved. Then the 3-opt algorithm was used to improve both the upper and the lower bounds. These algorithms were tested by a program written in FORTRAN 77 and we analyzed the results by plotting a histogram of the ratios of the upper to lower bounds, for a number of randomly generated instances of the problem. These ratios reflect the relative difference between the two bounds, which in turn indicates the quality of the heuristic solution.
The results for simulations run on similar sets of constraints improved when the 3-opt algorithm was used. However, the histograms did not remain the same for simulations run on different sets of constraints, using the 3-opt algorithm. The lower bound was found to be the likely culprit for the poorer results.
Though the 3-opt algorithms used have shown improvements in the results, we shall show how our approach may be extended to improve the results still further.
To solve this problem, we adopted heuristic methods, due to the computational complexity of this problem type. Since optimal solutions are almost impossible to obtain with a reasonable time for large number of cities, near-optimal solutions were sought instead, by using upper and lower bounds.
To compute the upper bound, we found a tour of the pick-up points using a greedy algorithm and then constructed a schedule using a sector partitioning approach. To compute the lower bound, certain constraints were relaxed and a simpler but unrealizable problem was solved. Then the 3-opt algorithm was used to improve both the upper and the lower bounds. These algorithms were tested by a program written in FORTRAN 77 and we analyzed the results by plotting a histogram of the ratios of the upper to lower bounds, for a number of randomly generated instances of the problem. These ratios reflect the relative difference between the two bounds, which in turn indicates the quality of the heuristic solution.
The results for simulations run on similar sets of constraints improved when the 3-opt algorithm was used. However, the histograms did not remain the same for simulations run on different sets of constraints, using the 3-opt algorithm. The lower bound was found to be the likely culprit for the poorer results.
Though the 3-opt algorithms used have shown improvements in the results, we shall show how our approach may be extended to improve the results still further.
Date Issued
1997
Call Number
QA9.58 Tan
Date Submitted
1997