Research Article A Multiple-Starting-Path Approach to the Resource-Constrained th Elementary Shortest Path Problem Hyunchul Tae and Byung-In Kim Department of Industrial and Management Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Republic of Korea Correspondence should be addressed to Byung-In Kim; bkim@postech.ac.kr Received 12 September 2014; Accepted 9 March 2015 Academic Editor: Dong Ngoduy Copyright © 2015 H. Tae and B.-I. Kim. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te resource-constrained elementary shortest path problem (RCESPP) aims to determine the shortest elementary path from the origin to the sink that satisfes the resource constraints. Te resource-constrained kth elementary shortest path problem (RCKESPP) is a generalization of the RCESPP that aims to determine the kth shortest path when a set of −1 shortest paths is given. To the best of our knowledge, the RCKESPP has been solved most efciently by using Lawler’s algorithm. Tis paper proposes a new approach named multiple-starting-path (MSP) to the RCKESPP. Te computational results indicate that the MSP approach outperforms Lawler’s algorithm. 1. Introduction Te vehicle routing problem (VRP) is a well-known combi- natorial problem of determining the optimal routes used by a feet of vehicles to visit all vertices with the minimum cost. One of the most efective exact approaches for the VRP is branch-and-price (B&P). A B&P solves a linear relaxation of the set covering formulation of the VRP by means of column generation method at each node. Te method solves the set covering relaxation by decomposing it to master and auxiliary problems. Whenever a master problem is solved, the dual values of its constraints are allocated to vertices as prizes. Ten, an auxiliary problem is solved to fnd a column with a negative reduced cost. In the VRP, the auxiliary problem exhibits a form of the resource-constrained elementary shortest path problem (RCESPP). Te RCESPP aims to determine the shortest elementary path from the origin to the sink that satisfes the resource constraints. In the RCESPP, the cost of a path is calculated as the sum of the travel costs of the traversed arcs minus the sum of the prizes of the visited customers. Because of the prizes, the graph of the RCESPP may contain negative arcs and cycles. Te RCESPP is strongly NP-Hard [1] and has been solved most efciently by the dynamic programming (DP) algorithms [2–5]. Some VRP researches [6, 7] have opted to relax the elementary constraint of the RCESPP in their B&P because its relaxed version can be solved much faster. However, others [8, 9] opted not to because the nonrelaxed version promises the tighter bounds of nodes in a B&P. In addition, in some VRP variants such as the team orienteering problem [10], the relaxation should be avoided because it brings a malfunction [2, 11] to a B&P. In this paper, we also do not relax the elementary path constraint. Among the various types of the resource constraints, this research considers the most representative ones, namely, the vehicle capacity constraint and the vertex time window constraint. Te RCESPP can be defned as follows. Let a weighted digraph =(,) be given, where  and  denote sets of vertices and arcs, respectively. Each vertex V  ∈ has a demand   and a vehicle has capacity . A vehicle should depart from the source V 0 ∈ and end at the sink V +1 ∈. A vehicle can visit a subset of vertices only if the sum of demands of the visited vertices does not exceed . A vehicle takes travelling time  , to traverse an arc (,) ∈  and service time   to serve V  ∈. A vehicle can visit V  ∈ only between its time windows [  ,  ] and must wait until   if the vehicle arrives before   . A vehicle pays the travel cost  , when it traverses (,)∈ and collects the prize   when it visits V  ∈. From now on, we denote the cost of an arc Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 803135, 7 pages http://dx.doi.org/10.1155/2015/803135