After Dantzig and Rasmer introduced Vehicle Routing Problem in 1959, this field has been studied with numerous approaches so far. Classical Vehicle Routing Problem can be described as a problem of multiple number of homogeneous vehicles sharing a same starting node and having their own routes to meet the needs of demand nodes. After satisfying all the needs, they go back to the starting node. In order to apply the real world problem, this problem had been developed with additional constraints and pick up & delivery model is one of them. To enhance the effectiveness of pick up & delivery, hub became a popular concept, which often helps reducing the overall cost and improving the quality of service. Lots of studies have suggested heuristic methods to realize this problem because it often becomes a NP-hard problem. However, because of this characteristic, there are not many studies solving this problem optimally. If the problem can be solved in polynomial time, optimal solution is the best option. Therefore, this study proposes a new mathematical model to solve this problem optimally, verified by a real world problem. The main improvements of this study compared to real world case are firstly, make drivers visit every nodes once except hub, secondly, make drivers visit every nodes at the right time, and thirdly, make drivers start and end their journey at their own homes.

1. 서 론
2. 문제 정의 및 최적화 수리 모형
    2.1 문제 정의
    2.2 허브 위치 선정
    2.3 최적화 수리 모형
3. 수치 실험 및 분석
    3.1 경로 분석
    3.2 허브 위치 분석
4. 결 론
References

• Jiyong Kim(공군사관학교 시스템공학과) | 김지용 Corresponding Author