Mathematical Problems in Engineering

Volume 2015, Article ID 803135, 7 pages

http://dx.doi.org/10.1155/2015/803135

## A Multiple-Starting-Path Approach to the Resource-Constrained th Elementary Shortest Path Problem

Department of Industrial and Management Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Republic of Korea

Received 12 September 2014; Accepted 9 March 2015

Academic Editor: Dong Ngoduy

Copyright © 2015 Hyunchul Tae and Byung-In Kim. This 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.

#### Abstract

The resource-constrained elementary shortest path problem (RCESPP) aims to determine the shortest elementary path from the origin to the sink that satisfies the resource constraints. The resource-constrained *k*th elementary shortest path problem (RCKESPP) is a generalization of the RCESPP that aims to determine the *k*th shortest path when a set of shortest paths is given. To the best of our knowledge, the RCKESPP has been solved most efficiently by using Lawler’s algorithm. This paper proposes a new approach named multiple-starting-path (MSP) to the RCKESPP. The computational results indicate that the MSP approach outperforms Lawler’s algorithm.

#### 1. Introduction

The vehicle routing problem (VRP) is a well-known combinatorial problem of determining the optimal routes used by a fleet of vehicles to visit all vertices with the minimum cost. One of the most effective 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. The 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. Then, an auxiliary problem is solved to find 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). The RCESPP aims to determine the shortest elementary path from the origin to the sink that satisfies 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. The RCESPP is strongly NP-Hard [1] and has been solved most efficiently 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. The RCESPP can be defined as follows. Let a weighted digraph be given, where and denote sets of vertices and arcs, respectively. Each vertex has a demand and a vehicle has capacity . A vehicle should depart from the source and end at the sink . 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 . A vehicle can visit 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 . From now on, we denote the cost of an arc as for simplicity. Let be a set of all possible paths from to that satisfies the vehicle capacity and vertex time window constraints. Let represent the cost of a path . The optimal path of the RCESPP or can be found by solving the following problem:

The resource-constrained th elementary shortest path problem (RCKESPP) is a generalization of the RCESPP that aims to determine the th shortest path when a set of shortest paths or is given, where represents the th shortest path. The optimal path of the RCKESPP or can be found by solving the following problem:

The RCKESPP has many practical applications. Göthe-Lundgren et al. [12] used a constraint generation method to solve the vehicle routing game, in which the vehicle routing cost is allocated to the customers as fairly as possible. The separation problem of the method can be viewed as the RCKESPP. Liu and Ramakrishnan [13] viewed the RCKESPP as a quality of service (QOS) routing problem in the telecommunication industry. van der Zijpp and Catalano [14] viewed the RCKESPP as a path enumeration problem in the transportation industry. Shi [15] viewed the RCKESPP as a robust and stable routing problem in an automated storage and retrieval system. Boussier et al. [11] implicitly showed that the RCKESPP could emerge as a pricing problem in the column generation method for the team orienteering problem. However, Boussier et al. [11] chose the RCESPP as a pricing problem instead of the RCKESPP because they believed that the latter was more difficult to solve than the former.

In an instance with a reasonable number of vertices, enumerating every member of is nearly impossible. Thus, Lawler's [16] algorithm has been used to solve the RCKESPP [14, 15], which does not require to fill in . To the best of our knowledge, the RCKESPP has been solved most efficiently by using Lawler’s algorithm. This paper proposes a new approach to the RCKESPP named multiple-starting-path (MSP) approach in Section 2. Section 3 reports the computational results which indicate that MSP approach outperforms Lawler’s algorithm. Section 4 concludes the paper.

#### 2. MSP Approach

This section begins with introducing a new generalization of the RCESPP, the RCESPP with multiple-starting-paths (RCESPP-MSP). The RCESPP-MSP aims to determine the shortest path from the given starting paths to that satisfies the resource constraints. Let be a set of starting paths, where each represents a starting path from to (this paper uses an alphabet “” to denote a path from to and “” to denote a path from to ). Let be the shortest path from to that satisfies the resource constraints. The optimal path of the RCESPP-MSP or can be found by solving the following problem:

In the following, Section 2.1 describes the DP algorithm for the RCESPP-MSP and identifies the properties of the RCESPP-MSP. Section 2.2 shows how the RCKESPP is reduced to the RCESPP-MSP.

##### 2.1. RCESPP-MSP

From now on, we refer to the RCESPP-MSP with a set of starting paths as . can be solved by the DP algorithms [2–5] which were originally developed for the RCESPP. For example, the most basic DP algorithm of Feillet et al. [2] can solve as Procedure 1.