Logic-based Benders decomposition for the heterogeneous fixed fleet vehicle routing problem with time windows
Introduction
The vehicle routing problem (VRP) plays a key role in the optimization of logistic distribution management and is one of the main themes addressed in the field of operational research. In its classical version, introduced by Dantzig and Ramser (1959), we are given a set of geographically dispersed customers, each one with a demand, a single depot and a homogeneous vehicle fleet with fixed capacity. The objective is to determine minimum-cost routes, starting from and ending at the depot, such that the demand of each customer is satisfied, each customer is visited only once, and the total demand of each route does not exceed the capacity of a vehicle.
Over the past decades, research has been conducted on models and algorithms for a large number of practical VRP variants and, as a consequence, a vast literature has been developed as evidenced by the books edited by Golden et al., 2008, Toth and Vigo, 2014, see also the state of the art taxonomy and review provided by Eksioglu et al., 2009, Braekers et al., 2016.
Real-world VRP applications, together with complex objective functions and constraints, have been termed rich vehicle routing problems (RVRP) in the literature, see the surveys proposed by Drexl, 2012, Caceres-Cruz et al., 2015, Lahyani et al., 2015. The last provides a taxonomy for the RVRP literature with respect to real-life attributes such as load splitting constraints, time restrictions and vehicle characteristics.
This paper deals with two of the most common VRP attributes occurring in practice, namely time windows and the use of heterogeneous fleets. The resulting problem is called the heterogeneous fixed fleet vehicle routing problem with time windows (HFFVRPTW). Time windows restrict the time interval in which the service at each customer starts, and can be found in many applications of ground, maritime and air transportation problems. A recent survey on the VRP with time windows (VRPTW) is provided in Desaulniers, Madsen, and Ropke (2014).
A heterogeneous fleet consists of multiple types of vehicles with capacity constraints expressed in terms of weight, volume, number of pallets, stacks and compartments. This is a relevant VRP attribute since the assumption of homogeneous fleets is not realistic in most industrial applications (Hoff, Andersson, Christiansen, Hasle, & Løkketangen, 2010). To illustrate, the magazine Transport Topics specialized in logistics and trucking news presents the top 100 largest private carriers in North America for 2018 (Transport Topics, 2020). PepsiCo Inc. is placed in the first position with 11,100 tractors, 17,500 trailers, 3500 straight trucks, 20,600 pickup trucks and cargo vans. Another example is Amazon that is increasing its online grocery pickup and delivery in a 2-hour time window. For this purpose, they require a driver with a 4-door, mid-sized sedan or a larger vehicle, such as a truck with a covered bed, SUV or a van. Typically, the driver is hired for 3–6 h and the routes are determined by the Amazon software (Amazon, 2020).
Given the significance of such attributes, we propose exact algorithms based on logic-based Benders decomposition (LBBD) (Hooker, 2000, Hooker and Ottosson, 2003) and a variant, called branch-and-check (BAC) (Thorsteinsson, 2001), for the HFFVRPTW. Our algorithms are based on a mathematical formulation that suggests a two-phase method called cluster-first and route-second (Fisher & Jaikumar, 1981). Such approach is general enough to encompass both exact and heuristic methods, for further details see the comprehensive survey of Archetti and Speranza (2014) on decomposition matheuristics for VRP variants. In our application, the Benders master problem is a generalized assignment problem (GAP) with binary variables that indicate the assignment of each customer to a type of vehicle. When such variables are fixed, the Benders independent subproblems have the special structure of the traveling salesman problem with time windows (TSPTW). Optimality cuts and/or feasibility cuts are generated from the solutions of subproblems and are added to the master problem which is solved in the subsequent iteration.
The main contributions of this paper are threefold:
- 1.
To the best of our knowledge, we present the first exact algorithms for the standard HFFVRPTW.
- 2.
We suggest several enhancements for the proposed algorithms and provide a detailed analysis of their effectiveness.
- 3.
We conduct extensive computational experiments on benchmark instances and assess the performance of the best proposed algorithm against state-of-the-art methods.
The remainder of the article is organized as follows. Section 2 summarizes related work, while Section 3 introduces the problem description. Section 4 describes the exact algorithms based on LBBD and BAC. Section 5 details the suggested algorithmic enhancements. Computational results are reported in Section 6, and conclusions are outlined in Section 7.
Section snippets
Related work
There are two major VRP variants involving heterogeneous fleets, namely the fleet size and mix VRP (FSMVRP) and the heterogeneous fixed fleet VRP (HFFVRP). The first one was introduced in the seminal paper of Golden, Assad, Levy, and Gheysens (1984) and comprises both fleet composition and routing tasks, whereas the second considers a fleet with limited number of vehicles, as suggested by Taillard (1999). The objective of these problems includes vehicle fixed and/or variable costs, and their
Problem description
The HFFVRPTW is defined on a complete graph where the node set consists of the depot 0 and the set of n customers , while the set represents the edges between the nodes. A heterogeneous fleet of vehicles is positioned at the depot in order to supply the customers. The set represents the distinct types of vehicles. For each type , the set represents the vehicles available at the depot, each having capacity and an associated
Logic-based Benders decomposition for the HFFVRPTW
Benders decomposition (Benders, 1962) was originally proposed for MIP problems and applies techniques of projection, outer linearization, and relaxation to reformulate a problem in a more amenable form. This approach partitions a MIP problem into two simpler ones: the master problem, a relaxed version of the original problem including its integer variables; and the subproblem, a linear program parametrized by the values of integer variables temporarily fixed in the master problem. At each
Algorithmic enhancements
Extensive research has been carried out to explore ways to accelerate the straightforward application of Benders decomposition, which may require too much computing time and memory. For a thorough review on this topic, see Rahmaniani, Crainic, Gendreau, and Rei (2017). In this section, we describe algorithmic enhancements that improve the convergence of LBBD and BAC.
Computational experiments
This section describes the computational results of the proposed algorithms, which were programmed in C# and run on an CPU Intel® Core™ i7-6500U 2.5 GHz with 16 GB RAM. As MIP solver, we used the Gurobi v.6.05 software. We start by presenting the benchmark instances and the parameters used within our algorithms in 6.1 Instance generation, 6.2 Parameter setting, respectively. Section 6.3 assesses the performance of LBBD versus BAC, and Section 6.4 analyses the impact of the algorithmic
Conclusions
We have proposed two exact algorithms under the cluster-first and route-second approach for the HFFVRPTW, namely LBBD and BAC. Both apply a Benders decomposition scheme that leads to a GAP master problem and independent TSPTW subproblems. Valid optimality and feasibility cuts are devised to guarantee the convergence of such methods to optimality in a finite number of iterations, and various algorithmic enhancements are presented. Furthermore, extensive computational tests were carried out on
CRediT authorship contribution statement
Ramon Faganello Fachini: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization, Project administration. Vinícius Amaral Armentano: Conceptualization, Methodology, Resources, Writing - review & editing, Supervision, Funding acquisition.
Acknowledgments
This research was funded by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq - Brazil, grants 141064/2015-3 and 305577/2014-0) and the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP - Brazil, grants 2016/01860-1 and 2016/06566-4). We are grateful for the valuable suggestions made by the editor and referees.
References (68)
- et al.
A survey on matheuristics for routing problems
EURO Journal on Computational Optimization
(2014) - et al.
Conflict graphs in solving integer programming problems
European Journal of Operational Research
(2000) - et al.
A branch-and-cut-and-price algorithm for the multi-depot heterogeneous vehicle routing problem with time windows
Transportation Research Part C: Emerging Technologies
(2011) - et al.
The vehicle routing problem: State of the art classification and review
Computers & Industrial Engineering
(2016) - et al.
A well-scalable metaheuristic for the fleet size and mix vehicle routing problem with time windows
Expert Systems with Applications
(2009) - et al.
A computational study of conflict graphs and aggressive cut separation in integer programming
Electronic Notes in Discrete Mathematics
(2015) - et al.
Time constrained routing and scheduling
- et al.
A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows
European Journal of Operational Research
(2007) - et al.
The vehicle routing problem: A taxonomic review
Computers & Industrial Engineering
(2009) - et al.
The fleet size and mix vehicle routing problem
Computers & Operations Research
(1984)
Industrial aspects and literature survey: Fleet composition and routing
Computers & Operations Research
Vehicle routing problem with a heterogeneous fleet and time windows
Expert Systems with Applications
A hybrid evolutionary algorithm for heterogeneous fleet vehicle routing problems with time windows
Computers & Operations Research
Thirty years of heterogeneous vehicle routing
European Journal of Operational Research
Rich vehicle routing problems: From a taxonomy to a definition
European Journal of Operational Research
The Benders decomposition algorithm: A literature review
European Journal of Operational Research
Boosting an exact logic-based Benders decomposition approach by variable neighborhood search
Electronic Notes in Discrete Mathematics
Solving the fleet size and mix vehicle routing problem with time windows via adaptive memory programming
Transportation Research Part C: Emerging Technologies
Solving a selective dial-a-ride problem with logic-based Benders decomposition
Computers & Operations Research
Propagating logic-based Benders’ decomposition approaches for distributed operating room scheduling
European Journal of Operational Research
An exact algorithm for integrated planning of operations in dry bulk terminals
Transportation Research Part E: Logistics and Transportation Review
A unified solution framework for multi-attribute vehicle routing problems
European Journal of Operational Research
Local truckload pickup and delivery with hard time window constraints
Transportation Research Part B: Methodological
Logic-based Benders decomposition for an inventory-location problem with service constraints
Omega
Linear time dynamic-programming algorithms for new classes of restricted TSPs: A computational study
INFORMS journal on Computing
Routing a heterogeneous fleet of vehicles
Scatter search for the fleet size and mix vehicle routing problem with time windows
Central European Journal of Operations Research
Particle swarm optimization algorithm for a vehicle routing problem with heterogeneous fleet, mixed backhauls, and time windows
Journal of intelligent manufacturing
Partitioning procedures for solving mixed-variables programming problems
Numerische Mathematik
An effective multirestart deterministic annealing metaheuristic for the fleet size and mix vehicle-routing problem with time windows
Transportation Science
A simple metaheuristic for the fleet size and mix problem with time windows
Algorithm 457: Finding all cliques of an undirected graph
Communications of the ACM
Rich vehicle routing problem: Survey
ACM Computing Surveys (CSUR)
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2022, Computers and Operations ResearchCitation Excerpt :Hooker and Ottosson (2003) further generalized the classic Benders decomposition by allowing the SP to take any form instead of just LP, with cuts generated based on logic information. This method, known as LBBD, has shown its good performance in solving operating room planning (Roshanaei et al., 2017), vehicle routing (Fachini and Armentano, 2020), and machine scheduling problems (Emde et al., 2019; Sun et al., 2019; Li et al., 2022a). We first decompose the studied DPSL-DD problem, where the MP determines the machine location and job assignment variables.