A column generation and Combinatorial Benders Decomposition algorithm for the Selective Dial-A-Ride-Problem

Highlights

• Column generation and Benders decomposition to solve the Selective Dial-A-Ride Problem.

• Exploiting symmetry and integrality of objective function with variable fixing.

• Algorithm outperforms current best methods on existing benchmark instances.

Abstract

In Pickup-and-Delivery Problems (PDP), one must design a set of vehicle routes that visit matching pickup and delivery locations. The Selective Dial-A-Ride Problem (SDARP) is a PDP which aims to serve as many pickup–delivery pairs as possible, where each pair has a time window and a maximum ride time. This paper introduces Extended Fragments to solve the SDARP. Extended Fragments stem from a subclass of vehicle routes which can be directly enumerated. An Extended Fragment formulation for the SDARP is proposed and solved using Combinatorial Benders Decomposition, augmented by a time discretisation and a novel variable-fixing technique. This algorithm, along with other recent ‘fragment’-based methods for PDP, is an a priori column generation method, combined with Combinatorial Benders Decomposition. The new method solves the existing benchmark for the SDARP, including 11 previously-unsolved instances.

Introduction

In the Vehicle Routing Problem (VRP), the objective is to assign routes to one or more homogeneous vehicles of limited capacity which achieve minimal travel costs. The Pickup and Delivery Problem with Time Windows (PDPTW) is a VRP in which each pickup location must be paired with a delivery location on the same route; this constitutes one request. The pickup must also precede the delivery. Additionally, the travel time between locations is taken into account and the availability of locations is restricted by a time window. The Dial-a-Ride Problem (DARP) (Savelsbergh and Sol, 1995) is a PDPTW variant with a limit on the amount of time that requests can spend on the vehicle. This feature is sometimes referred to as dynamic time windows and is relevant in ride sharing applications as a measure to reduce user inconvenience. In most VRP and PDPTW variants, all requests must be served and the objective is to minimise the total travel cost between locations. This paper addresses the Selective Dial-a-Ride Problem (SDARP) (Riedler and Raidl, 2018) which disregards travel costs, allows requests to go unserved and instead maximises the total number of requests served. Compared to the DARP, the SDARP has a highly symmetric objective function.

Branch-and-Price (BP) is a prolific technique in VRP and PDP literature. In BP a Mixed Integer Program (MIP) is solved using Branch-and-Bound where Dantzig–Wolfe Decomposition is used to solve the Linear Programming relaxations at each node. Care must be taken to create strong branching rules that are easily enforceable in the column generation subproblem (Barnhart et al., 1998). In PDP applications, the approach typically involves variables which describe a single feasible route, a set partitioning MIP master problem which chooses one or more routes to use and a pricing subproblem which generates new feasible routes. The subproblem is usually solved using a labelling algorithm over partial routes, with resources such as total travel time, current vehicle capacity and total travel cost. In Branch-and-Price-and-Cut (BPC), the BP algorithm is augmented by adding valid inequalities to the master problem as cutting planes throughout the Branch-and-Bound tree, however, this also adds dual variables which complicate the pricing subproblem. Desrochers et al. (1992) presented the first BP algorithm to solve the VRP with Time Windows. Ropke and Cordeau (2009) later proposed a BPC algorithm to solve the PDPTW before Gschwind and Irnich (2015) introduced ride-time labelling resources for an efficient BPC implementation for the DARP.

More recently, Logic-Based Benders Decomposition (LBBD) (Hooker and Ottosson, 2003) and Branch-and-Check (BaC) which combines MIP and Constraint Programming (CP) (Thorsteinsson, 2001), have gained traction as methods for solving PDP. In LBBD for MIP, a master problem MIP is formulated along with additional side constraints (not necessarily inequalities) which must be satisfied. At integer solutions to the master problem, these side constraints are checked using a feasibility subproblem. If they are satisfied then the current solution is optimal, otherwise an inequality eliminating this infeasibility must be added to the master problem. Combinatorial Benders Decomposition (CBD) (Codato and Fischetti, 2006) is a specialisation of LBBD to MIP master problems where the side constraints are linear inequalities conditioned on the value of individual binary master problem variables. In PDP and related problems, LBBD, BaC and CBD approaches are usually loosely based on a flow MIP model, which assigns a binary decision variable to each pair of locations, indicating the successor of each location within a route. In a complete flow MIP formulation, various resources such as vehicle capacity or time are modelled with Miller–Tucker–Zemlin (MTZ) (Miller et al., 1960) constraints which are notorious for their poor linear programming relaxation.

The early work on the DARP by Cordeau and Laporte (2003) uses Benders cuts as valid inequalities for a complete flow MIP in a Branch-and-Cut (BC) algorithm. Benders Decomposition is not mentioned in the work, but the inequalities developed are similar to those presented in the CBD algorithm of Ropke et al. (2007). Ropke et al. (2007) also do not relate their work to Benders Decomposition, but it can be easily formulated as such. Riedler and Raidl (2018) proposed a BaC algorithm to solve the SDARP, using a MIP master problem which assigns requests to vehicles and a CP subproblem to construct vehicle routes to serve those requests. A similar MIP/CP hybrid approach was taken by Corréa et al. (2007) to solve the Autonomous Guided Vehicle Problem (AVG), a VRPTW variant for routing autonomous guided vehicles on a factory floor. In their work, the master problem was a CP which assigned an ordered set of requests to vehicles and the timing subproblem was solved instead with a MIP. Fachini and Armentano (2020) solve the VRP with Time Windows and Heterogeneous vehicles using LBBD, assigning requests to vehicles in a MIP master problem like Riedler and Raidl (2018) but solving the subproblem with a MIP as well.

In problems with a complete formulation available such as the PDPTW, Benders Decomposition approaches can vary in how difficult, or complex, the master problem is. Leaving most of constraints (such as pairing, precedence, capacity, timing) to the subproblems has the advantage of making the master problem quick to solve, but can lead to large numbers of cuts. Many of these cuts may only be needed at a few Branch-and-Bound nodes and are otherwise useless. In particular in some LBBD algorithms, the Benders cut may be the so-called no-good cut, which cuts off one and only one subproblem-infeasible solution to the master problem. This can lead to a poor overall LP relaxation and slow progress in the Branch-and-Bound tree. On the other hand, leaving too many of the constraints in the master problem will decrease the effectiveness of the decomposition, by creating a large and slow-to-solve master problem.

In contrast, the decomposition in Branch-and-Price implementations for PDPTW variants is invariably the same. The master problem variables are complete routes, satisfying all intra-route constraints, and the satisfiability of these constraints is left to the subproblem. Intra-route constraints are those whose satisfiability depends only on the ordering of locations within a route, such as pairing, precedence, timing and capacity. Inter-route constraints are the remaining constraints such as path elementarity or maximum number of vehicles. For example, Veenstra et al. (2017) develop a BPC method to solve the PDPTW with Handling, a PDPTW variant which differs significantly from the DARP, but the master problem is nearly identical to that of a BPC algorithm for the DARP (Gschwind and Irnich, 2015).

Lam and Hentenryck (2016) combine BP with BaC (Branch-and-Price-and-Check — BPaC) to solve the PDPTW with Local Congestion. Local congestion limits the number of vehicles that can visit a group of locations simultaneously. The BPaC master- and subproblems are the same as in a PDPTW BP algorithm, but Benders cuts are added to eliminate combinations of routes which are infeasible due to congestion. However, all intra-route constraints are still handled by the pricing subproblem – the Benders cuts are used only to handle inter-route constraints.

The method described in this paper is a hybridisation of column generation and CBD. We focus on the main idea of Dantzig–Wolfe in BP, where the structure of a variable in the master problem implies some of the constraints in the original problem. As mentioned above, in set partitioning models for PDP all intra-route constraints are modelled by the variables. However, instead of moving all intra-route constraints inside the variables as in Lam and Hentenryck (2016), the idea we describe here is to exclude those intra-route constraints which will drastically increase the number of variables. The aim is to define variables that contain some of the constraints while still remaining directly enumerable.

The first example of this is the work of Alyasiry et al. (2019) which solves the PDPTW with Last-In-First-Out (LIFO) Loading, where each vehicle maintains a LIFO stack of onboard requests. Delivery locations must be served in the order they appear on the stack. The authors design variables called “fragments” which are paths that (amongst other things) satisfy the intra-route constraints of pairing, precedence, vehicle capacity and LIFO stacking. Moving the timing constraints (time windows and travel time) into the variables as well would mean generating complete routes, which are too numerous to enumerate directly. Fragments on the other hand can easily be enumerated, largely due to the LIFO constraint.

Rist and Forbes (2021) propose a similar approach for the DARP. The lack of a LIFO constraint in the DARP makes enumerating fragments intractable, so instead they define “Restricted Fragments”. These Restricted Fragments contain the same constraints (except the LIFO constraint) as fragments but are less numerous. The tradeoff is that the master problem contains additional constraints. In both of these works, the intra-route constraints which remain in the master problem are the timing constraints: time windows, travel times and ride times (for the DARP). Instead of using continuous time variables and MTZ constraints in the master problem, CBD is used. The timing constraints are left to a feasibility subproblem and Benders cuts are added to eliminate infeasible combinations of (restricted) fragments. In this paper we propose a similar method for the SDARP, but move the travel time and ride time constraints into the variables and leave only the time windows to the Benders subproblem.

A summary of different approaches to solving the DARP and related problems is shown in Table 1. The last three columns describe how each work handles the different constraints of the relevant problem. The approaches can be broadly grouped into three categories: Dantzig–Wolfe Decomposition (including BP and BPC), Benders Decomposition and hybrid approaches. This table is by no means complete; it is intended to show a variety of strategies for handling constraints for vehicle routing problems with a mix of different intra- and inter-route constraints. The first three algorithms listed in the table (including the one presented in this paper) avoid moving all intra-route constraints inside the master problem variables. It is for this reason that they do not require a pricing subproblem as in Lam and Hentenryck (2016), as the variables may be directly enumerated. This allows for valid inequalities without additional complexity from dual variables as well as some new variable fixing strategies.

The remainder of the paper is structured as follows. Section 2 formally defines the SDARP and establishes the notation for the following sections. Section 3 details the MIP formulation on which our algorithm is based, before applying Combinatorial Benders Decomposition in Section 4. In Section 5 we develop a variable-fixing scheme aimed at reducing the size of the MIP. We test our method in Section 6, showing it to be superior in performance to Logic Based Benders Decomposition and investigate adding a secondary objective function. Finally, Section 7 presents our concluding remarks on the method.