scholarly journals Cover Inequalities for a Vehicle Routing Problem with Time Windows and Shifts

2019 ◽  
Vol 53 (5) ◽  
pp. 1354-1371 ◽  
Author(s):  
Said Dabia ◽  
Stefan Ropke ◽  
Tom van Woensel
Keyword(s):  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Valid Inequalities ◽  
Time Windows ◽  
Branch And Cut ◽  
Master Problem ◽  
Loading Capacity ◽  
Routing Problem ◽  
Cover Inequalities ◽  
And Shifts

This paper introduces the vehicle routing problem with time windows and shifts (VRPTWS). At the depot, several shifts with nonoverlapping operating periods are available to load the planned trucks. Each shift has a limited loading capacity. We solve the VRPTWS exactly by a branch-and-cut-and-price algorithm. The master problem is a set partitioning with an additional constraint for every shift. Each constraint requires the total quantity loaded in a shift to be less than its loading capacity. For every shift, a pricing subproblem is solved by a label-setting algorithm. Shift capacity constraints define knapsack inequalities; hence we use valid inequalities inspired from knapsack inequalities to strengthen the linear programming relaxation of the master problem when solved by column generation. In particular, we use a family of tailored robust cover inequalities and a family of new nonrobust cover inequalities. Numerical results show that nonrobust cover inequalities significantly improve the algorithm.

scholarly journals The Robust Vehicle Routing Problem with Time Windows: Compact Formulation and Branch-Price-and-Cut Method

2019 ◽  
Vol 53 (4) ◽  
pp. 1043-1066 ◽  
Author(s):  
Pedro Munari ◽  
Alfredo Moreno ◽  
Jonathan De La Vega ◽  
Douglas Alem ◽  
Jacek Gondzio ◽  
...  
Keyword(s):  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Time Windows ◽  
Real Life ◽  
Set Partitioning ◽  
Branch And Cut ◽  
Small Scale ◽  
Routing Problem ◽  
Compact Formulation ◽  
Cut Method

We address the robust vehicle routing problem with time windows (RVRPTW) under customer demand and travel time uncertainties. As presented thus far in the literature, robust counterparts of standard formulations have challenged general-purpose optimization solvers and specialized branch-and-cut methods. Hence, optimal solutions have been reported for small-scale instances only. Additionally, although the most successful methods for solving many variants of vehicle routing problems are based on the column generation technique, the RVRPTW has never been addressed by this type of method. In this paper, we introduce a novel robust counterpart model based on the well-known budgeted uncertainty set, which has advantageous features in comparison with other formulations and presents better overall performance when solved by commercial solvers. This model results from incorporating dynamic programming recursive equations into a standard deterministic formulation and does not require the classical dualization scheme typically used in robust optimization. In addition, we propose a branch-price-and-cut method based on a set partitioning formulation of the problem, which relies on a robust resource-constrained elementary shortest path problem to generate routes that are robust regarding both vehicle capacity and customer time windows. Computational experiments using Solomon’s instances show that the proposed approach is effective and able to obtain robust solutions within a reasonable running time. The results of an extensive Monte Carlo simulation indicate the relevance of obtaining robust routes for a more reliable decision-making process in real-life settings.

The Robust Vehicle Routing Problem with Time Window Assignments

2021 ◽  
Vol 55 (2) ◽  
pp. 395-413
Author(s):  
Maaike Hoogeboom ◽  
Yossiri Adulyasak ◽  
Wout Dullaert ◽  
Patrick Jaillet
Keyword(s):  
Travel Time ◽  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Service Providers ◽  
Time Window ◽  
Time Windows ◽  
Branch And Cut ◽  
Optimal Time ◽  
Routing Problem ◽  
Solution Approach

In practice, there are several applications in which logistics service providers determine the service time windows at the customers, for example, in parcel delivery, retail, and repair services. These companies face uncertain travel times and service times that have to be taken into account when determining the time windows and routes prior to departure. The objective of the proposed robust vehicle routing problem with time window assignments (RVRP-TWA) is to simultaneously determine routes and time window assignments such that the expected travel time and the risk of violating the time windows are minimized. We assume that the travel time probability distributions are not completely known but that some statistics, such as the mean, minimum, and maximum, can be estimated. We extend the robust framework based on the requirements’ violation index, which was originally developed for the case where the specific requirements (time windows) are given as inputs, to the case where they are also part of the decisions. The subproblem of finding the optimal time window assignment for the customers in a given route is shown to be convex, and the subgradients can be derived. The RVRP-TWA is solved by iteratively generating subgradient cuts from the subproblem that are added in a branch-and-cut fashion. Experiments address the performance of the proposed solution approach and examine the trade-off between expected travel time and risk of violating the time windows.

scholarly journals Mathematical models for the periodic vehicle routing problem with time windows and time spread constraints

2020 ◽  
Vol 11 (1) ◽  
pp. 10-23
Author(s):  
Hande Öztop ◽  
Damla Kizilay ◽  
Zeynel Abidin Çil
Keyword(s):  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Optimization Problems ◽  
Valid Inequalities ◽  
Time Windows ◽  
Mixed Integer ◽  
Routing Problem ◽  
Periodic Vehicle Routing Problem ◽  
Time Spread ◽  
Periodic Vehicle Routing

The periodic vehicle routing problem (PVRP) is an extension of the well-known vehicle routing problem. In this paper, the PVRP with time windows and time spread constraints (PVRP-TWTS) is addressed, which arises in the high-value shipment transportation area. In the PVRP-TWTS, period-specific demands of the customers must be delivered by a fleet of heterogeneous capacitated vehicles over the several planning periods. Additionally, the arrival times to a customer should be irregular within its time window over the planning periods, and the waiting time is not allowed for the vehicles due to the security concerns. This study, proposes novel mixed-integer linear programming (MILP) and constraint programming (CP) models for the PVRP-TWTS. Furthermore, we develop several valid inequalities to strengthen the proposed MILP and CP models as well as a lower bound. Even though CP has successful applications for various optimization problems, it is still not as well-known as MILP in the operations research field. This study aims to utilize the effectiveness of CP in solving the PVRP-TWTS. This study presents a CP model for PVRP-TWTS for the first time in the literature to the best of our knowledge. Having a comparison of the CP and MILP models can help in providing a baseline for the problem. We evaluate the performance of the proposed MILP and CP models by modifying the well-known benchmark set from the literature. The extensive computational results show that the CP model performs much better than the MILP model in terms of the solution quality.

A Branch-and-Cut Procedure for the Vehicle Routing Problem with Time Windows

2002 ◽  
Vol 36 (2) ◽  
pp. 250-269 ◽  
Author(s):  
Jonathan F. Bard ◽  
George Kontoravdis ◽  
Gang Yu
Keyword(s):  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Time Windows ◽  
Branch And Cut ◽  
Routing Problem

A Branch-Price-and-Cut Algorithm for the Two-Echelon Vehicle Routing Problem with Time Windows

2021 ◽  
Author(s):  
Tayeb Mhamedi ◽  
Henrik Andersson ◽  
Marilène Cherkesly ◽  
Guy Desaulniers
Keyword(s):  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Time Window ◽  
Valid Inequalities ◽  
Time Windows ◽  
A Priori ◽  
Solution Process ◽  
High Capacity ◽  
Sensitivity Analyses ◽  
Routing Problem

In this paper, we propose an exact branch-price-and-cut (BPC) algorithm for the two-echelon vehicle routing problem with time windows. This problem arises in city logistics when high-capacity and low-capacity vehicles are used to transport items from depots to satellites (first echelon) and from satellites to customers (second echelon), respectively. The aim is to determine a set of least-cost first- and second-echelon routes such that the load on the routes respect the capacity of the vehicles, each second-echelon route is supplied by exactly one first-echelon route, and each customer is visited by exactly one second-echelon route within its time window. We model the problem with a route-based formulation where first-echelon routes are enumerated a priori, and second-echelon routes are generated using column generation. The problem is solved using BPC. To generate second-echelon routes, one pricing problem per satellite is solved using a labeling algorithm which keeps track of the first-echelon route associated with each (partial) second-echelon route considered. Furthermore, to speed up the solution process, we introduce effective deep dual-optimal inequalities and apply known valid inequalities. We perform extensive computational experiments on benchmark instances and show that our method outperforms a state-of-the-art algorithm. We also conduct sensitivity analyses on the different components of our algorithm and derive managerial insights related to the structure of the first-echelon routes.

scholarly journals A Branch-and-Cut-and-Price Algorithm for the Electric Vehicle Routing Problem with Multiple Technologies

2021 ◽  
Vol 2 (1) ◽  
Author(s):  
Alberto Ceselli ◽  
Ángel Felipe ◽  
M. Teresa Ortuño ◽  
Giovanni Righini ◽  
Gregorio Tirado
Keyword(s):  
Vehicle Routing ◽  
Electric Vehicle ◽  
Vehicle Routing Problem ◽  
Parallel Implementation ◽  
General Purpose ◽  
Branch And Cut ◽  
Master Problem ◽  
Routing Problem ◽  
Exact Optimization ◽  
Price Algorithm

AbstractWe provide an exact optimization algorithm for the electric vehicle routing problem with multiple recharge technologies. Our branch-and-cut-and-price algorithm relies upon a path-based formulation, where each column in the master problem represents a sequence of customer visits between two recharge stations instead of a whole route. This allows for massive decomposition, and parallel implementation of the pricing phase, exploiting the large number of independent pricing sub-problems. The algorithm could solve instances with up to thirty customers, nine recharge stations, five vehicles and three technologies to proven optimality. Near-optimal heuristic solutions were obtained with a general-purpose MIP solver from the columns generated at the root node.

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