Discrete Optimization
A simple and effective metaheuristic for the Minimum Latency Problem

https://doi.org/10.1016/j.ejor.2012.03.044Get rights and content

Abstract

The Minimum Latency Problem (MLP) is a variant of the Traveling Salesman Problem which aims to minimize the sum of arrival times at vertices. The problem arises in a number of practical applications such as logistics for relief supply, scheduling and data retrieval in computer networks. This paper introduces a simple metaheuristic for the MLP, based on a greedy randomized approach for solution construction and iterated variable neighborhood descent with random neighborhood ordering for solution improvement. Extensive computational experiments on nine sets of benchmark instances involving up to 1000 customers demonstrate the good performance of the method, which yields solutions of higher quality in less computational time when compared to the current best approaches from the literature. Optimal solutions, known for problems with up to 50 customers, are also systematically obtained in a fraction of seconds.

Highlights

► A new metaheuristic is proposed for the Minimum Latency Problem (MLP). ► Move evaluations can be handled in O(1) time for classical local-search neighborhoods. ► High-quality solutions are found with a simple approach and few parameters. ► Known optimal solutions are rapidly obtained for instances with up to 107 customers.

Introduction

The Minimum Latency Problem (MLP) can be defined as follows. Let G = (V, A) be a complete directed graph, where V = {0,  , n} is the set of vertices and A = {(i, j):i, j  V, i  j} is the set of arcs with associated travel time tij. Vertex 0 stand for the depot while other vertices represent customers. The MLP aims at finding a Hamiltonian circuit that minimizes i=0nl(i), l(i) representing the latency of a vertex i  V, that is, the total travel time to reach i. We also consider the additional constraint that the circuit must start and end at vertex 0 and also that l(0) = 0. The MLP is a variant of the well-known Traveling Salesman Problem (TSP) and it is known in the literature under various other names: Traveling Repairman Problem (Tsitsiklis, 1992), Delivery Man Problem (Fischetti et al., 1993), Cumulative Traveling Salesman Problem (Bianco et al., 1993) and School Bus Driver Problem (Chaudhuri et al., 2003). The constraint on the tour origin at 0 is sometimes not considered in early articles, but this variant can be tackled in our context by adding a dummy depot such that t0j = tj0 = 0 for j  V.

The MLP was proven NP-hard for general metric spaces (Sahni and Gonzalez, 1976) and also when the subjacent structure is an edge-weighted tree (Sitters, 2002). For structures such as paths, unweighted trees and trees with diameter at most 3, polynomial-time algorithms based on dynamic programming have been proposed (Blum et al., 1994, García et al., 2002, Wu et al., 2004). Although the MLP seems to be a simple TSP variant, one can verify that the former has properties that are not present in the latter. One of them is that small local modifications in the configuration of the input points can lead to highly nonlocal changes in the structure of an optimal solution (Blum et al., 1994, Goemans and Kleinberg, 1998). Another feature of the MLP is the nonlocal character of the objective function, as an additional arc inserted in the beginning of the circuit affects the latency of all remaining vertices (Arora and Karakostas, 2003).

Real-life applications of the MLP often arise from distribution systems, where some quality criterion regarding the customer satisfaction must be focused. The MLP considers waiting times (latency) of a service system from the customer’s point of view, i.e., while in the MLP the objective is to minimize the average waiting time of each customer, in the TSP the objective is to minimize the total time required to visit all customers. In view of this, one can say that the MLP is customer oriented, while the TSP is server oriented (Archer and Williamson, 2003). Therefore, the MLP can be employed in the modeling of different types of service systems. Important practical applications can be found in home delivery services (Méndez-Dı´az et al., 2008), logistics for emergency relief (Campbell et al., 2008) and data retrieval in computer networks (Ezzine et al., 2010). However, although the MLP appears in several important settings, this problem has not received sufficient attention in the literature so far. In particular, few efficient heuristics have been designed to tackle problems of realistic size. Moreover, current exact methods are not capable of consistently solving instances with more than 100 customers.

The contributions of this work are twofold. The first one is to present a simple and effective metaheuristic for the MLP, which combines components of Greedy Randomized Adaptive Search Procedure (GRASP) (Feo and Resende, 1995), Iterated Local Search (ILS) (Lourenço et al., 2003) and Variable Neighborhood Descent with Random neighborhood ordering (RVND) (Mladenović and Hansen, 1997, Subramanian et al., 2010). The second contribution is a simple move evaluation procedure that requires O(1) amortized elementary operations. Such procedure can be applied to any neighborhood structure based on a bounded number of arc exchanges and thus to all classical neighborhoods used in the MLP literature. The proposed solution approach is easy to implement and relies on very few parameters. Extensive computational experiments on benchmark instances involving up to 1000 customers underline the remarkable performance of this method, both in terms of solution quality and computational efficiency. Known optimal solutions for problems with up to 107 customers are also systematically obtained in a few seconds.

The remainder of the paper is organized as follows. Section 2 presents some related works. Section 3 describes the proposed algorithm. Section 4 contains the computational results. Finally, Section 5 presents the concluding remarks of this work.

Section snippets

Related works

Several exact and non-exact approaches were proposed to solve the MLP in the literature. However, as shown in the following, exact algorithms are still limited to small problem sizes, while few efficient heuristic procedures have been designed.

Lucena (1990) proposed an early exact enumerative algorithm, relying on a non-linear integer formulation in which lower bounds are derived using a Lagrangian relaxation. Bianco et al. (1993) put forward two exact algorithms that incorporate lower bounds

Proposed algorithm

The simple and efficient metaheuristic proposed here and called GILS-RVND brings together the components of GRASP, ILS and RVND. The pseudocode of the developed approach is presented in Algorithm 1. The method performs IMax iterations (lines 3–21), where in each of which an initial solution is generated using a greedy randomized procedure. The level of greediness is controlled by a parameter α, which is randomly chosen among the values of a given set R. Each initial solution is then improved by

Computational results

The algorithm was coded in C++ (g++ 4.4.3) and executed on an Intel® Core™ i7 2.93 GHz, with 8.0 GB of RAM memory running under GNU/Linux Ubuntu 10.04 (kernel 2.6.32–25). Only a single thread was used in the experiments.

Through preliminary tests, we observed that the values IMax = 10, IILS = min{100, n} and R = {0.00, 0.01, 0.02,  , 0.25} resulted in a good trade-off between solution quality and run time. This parameter setting has thus been used in the following experiments.

GILS-RVND was tested on 9 sets of

Concluding remarks

This paper introduced a new hybrid metaheuristic for the Minimum Latency Problem, which gathers several successful concepts from GRASP, ILS, RVND, along with simple move evaluation procedures in O(1) time. The latter methodology can be applied to any neighborhood structure based on a bounded number of arc exchanges or visit relocations. The overall approach is simple to describe and to implement. Its effectiveness, in terms of both solution quality and computational time, was assessed by

Acknowledgments

We would like to thank Prof. Eduardo Uchoa and Prof. Artur Pessoa for providing access to their BCP code. This research was partially supported by the following Brazilian research agencies: CNPq, CAPES and FAPERJ.

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