차량경로문제의 경로분할모형에 관한 연구

A Route-Splitting Approach to the Vehicle Routing Problem

차량경로문제의 Set-Partitioning 모형에 적용된 열생성 프로세스에서 차량경로를 생성하는 하위문제는 순회외판원 문제와 같은 combinatorial 구조를 가지므로 연산상의 어려움이 크다. 본 논문은 각 차량경로를 분할하여 하위문제에서 분할된 부분경로를 생성하는 경로분할모형을 소개한다. 열생성이 용이해지는 반면 주문제가 복잡해지는 단점이 있으나 이 모형은 set-partitioning 모형으로 다루기 힘든 크기의 VRP에 접근하도록 한다. 경로분할모형은 최대 199곳의 수요지를 갖는 Symmetric VRP의 실험문제에서 평균93.5%, 수요지수 최대 70곳의 Asymmetric VRP의 실험문제에서 평균 97.6%의 하한값을 도출해 특히 Asymmetric VRP의 경우에서 잘 알려진 다른 하한값 기법들보다 우수함을 보였다. 개발된 Branch-and-Price 프로세스로는 도출된 하한값을 사용하여 수요지 최대 48곳의Asymmetric VRP의 최적해를 구할 수 있었다. 경로분할모형은 성능이 비교되는 다른 모형과 달리 다른 크기의 차량을 다룰 수 있는 장점이 있고, Asymmetric VRP 문제에서는 현재 가장 우수한 하한값을 제시한다. 이러한 점에서 본 모형은 향후 연구가치가 있다고 판단된다.

The vehicle routing problem (VRP) is to determine a set of feasible vehicle routes, one for each vehicle, such that each customer is visited exactly once and the total distance travelled by the vehicles is minimized. A feasible route is defined as a simple circuit including the depot such that the total demand of the customers in the route does not exceed the vehicle capacity. While there have been significant advances recently in exact solution methodology, the VRP is not a well solved problem. We find most approaches still relying on the branch and bound method. These approaches employ various methodologies to compute a lower bound on the optimal value. We introduce a new modelling approach, termed route-splitting, for the VRP that allows us to address problems whose size is beyond the current computational range of set-partitioning models. The route-splitting model splits each vehicle route into segments, and results in more tractable subproblems. Lifting much of the burden of solving combinatorially hard subproblems, the route-splitting approach puts more weight on the LP master problem, Recent breakthroughs in solving LP problems (Nemhauser, 1994) bode well for our approach. Lower bounds are computed on five symmetric VRPs with up to 199 customers, and eight asymmetric VRPs with up to 70 customers. while it is said that the exact methods developed for asymmetric instances have in general a poor performance when applied to symmetric ones (Toth and Vigo, 2002), the route splitting approach shows a competent performance of 93.5% on average in the symmetric VRPs. For the asymmetric ones, the approach comes up with lower bounds of 97.6% on average. The route-splitting model can deal with asymmetric cost matrices and non-identical vehicles. Given the ability of the route-splitting model to address a wider range of applications and its good performance on asymmetric instances, we find the model promising and valuable for further research.