Lower bounds for large traveling umpire instances: New valid inequalities and a branch-and-cut algorithm

2016 ◽  
Vol 72 ◽  
pp. 147-159 ◽  
Author(s):  
Lucas de Oliveira ◽  
Cid C. de Souza ◽  
Tallys Yunes
Keyword(s):  
Lower Bounds ◽  
Valid Inequalities ◽  
Branch And Cut

A New Formulation for the Dial-a-Ride Problem

2021 ◽  
Author(s):  
Yannik Rist ◽  
Michael A. Forbes
Keyword(s):  
Time Window ◽  
Valid Inequalities ◽  
Route Planning ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Planning Problem ◽  
Current State ◽  
New Formulation ◽  
First Time ◽  
Delivery Location

This paper proposes a new mixed integer programming formulation and branch and cut (BC) algorithm to solve the dial-a-ride problem (DARP). The DARP is a route-planning problem where several vehicles must serve a set of customers, each of which has a pickup and delivery location, and includes time window and ride time constraints. We develop “restricted fragments,” which are select segments of routes that can represent any DARP route. We show how to enumerate these restricted fragments and prove results on domination between them. The formulation we propose is solved with a BC algorithm, which includes new valid inequalities specific to our restricted fragment formulation. The algorithm is benchmarked on existing and new instances, solving nine existing instances to optimality for the first time. In comparison with current state-of-the-art methods, run times are reduced between one and two orders of magnitude on large instances.

scholarly journals Production, Replenishment and Inventory Policies for Perishable Products in a Two-Echelon Distribution Network

2020 ◽  
Vol 12 (11) ◽  
pp. 4735
Author(s):  
Mingyuan Wei ◽  
Hao Guan ◽  
Yunhan Liu ◽  
Benhe Gao ◽  
Canrong Zhang
Keyword(s):  
Inventory Management ◽  
Valid Inequalities ◽  
Distribution Network ◽  
Planning Horizon ◽  
Branch And Cut ◽  
Computational Time ◽  
Perishable Products ◽  
Inventory Cost ◽  
Routing Problem ◽  
Selection Of

The research on production, delivery and inventory strategies for perishable products in a two-echelon distribution network integrates the production routing problem (PRP) and two-echelon vehicle routing problem (2E-VRP), which mainly considers the inventory and delivery sustainability of perishable products. The problem investigated in this study is an extension of the basic problems, and it simultaneously optimizes production, replenishment, inventory, and routing decisions for perishable products that will deteriorate over the planning horizon. Additionally, the lead time has been considered in the replenishment echelon, and the unit inventory cost varying with the inventory time is considered in the inventory management. Based on a newly designed model, different inventory strategies are discussed in this study: old first (OF) and fresh first (FF) strategies both for the first echelon and second echelon, for which four propositions to model them are proposed. Then, four valid inequalities, including logical inequalities, a ( ℓ , S , W W ) inequality, and a replenishment-related inequality, are proposed to construct a branch-and-cut algorithm. The computational experiments are conducted to test the efficiency of valid inequalities, branch-and-cut, and policies. Experimental results show that the valid inequalities can effectively increase the relaxed lower bound by 4.80% on average and the branch-and-cut algorithm can significantly reduce the computational time by 58.18% on average when compared to CPLEX in small and medium-sized cases. For the selection of strategy combinations, OF–FF is suggested to be used in priority.

The two-edge connected hop-constrained network design problem: Valid inequalities and branch-and-cut

Networks ◽  
2006 ◽  
Vol 49 (1) ◽  
pp. 116-133 ◽  
Author(s):  
David Huygens ◽  
Martine Labbé ◽  
A. Ridha Mahjoub ◽  
Pierre Pesneau
Keyword(s):  
Network Design ◽  
Design Problem ◽  
Valid Inequalities ◽  
Branch And Cut ◽  
Network Design Problem

scholarly journals Integer Programming, Constraint Programming, and Hybrid Decomposition Approaches to Discretizable Distance Geometry Problems

2021 ◽  
Author(s):  
Moira MacNeil ◽  
Merve Bodur
Keyword(s):  
Integer Programming ◽  
Constraint Programming ◽  
State Of The Art ◽  
Valid Inequalities ◽  
Distance Geometry ◽  
The State ◽  
Bp Algorithm ◽  
Branch And Cut ◽  
Vertex Order ◽  
Hybrid Decomposition

Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in [Formula: see text] such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one, which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is on finding optimal BP trees for a class of discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting-plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low-density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem. Summary of Contribution: We present a new model to solve a combinatorial optimization problem on graphs, MIN DOUBLE, which comes from the highly active area of distance geometry and has applications in a wide variety of fields. We use integer programming (IP) and present the first constraint programming (CP) models and hybrid decomposition methods, implemented as a branch-and-cut procedure, for MIN DOUBLE. Through an extensive computational study, we show that our approaches advance the state of the art for MIN DOUBLE. We accomplish this by not only combining generic techniques from IP and CP but also exploring the structure of the problem in developing valid inequalities and variable fixing rules. Our methods significantly improve the solvability of MIN DOUBLE, which we believe can also provide insights for tackling other problem classes and applications.

scholarly journals Last mile deliveries with lockers: formulations and algorithms

2021 ◽  
Author(s):  
Giovanni Buzzega ◽  
Stefano Novellani
Keyword(s):  
Valid Inequalities ◽  
Time Windows ◽  
Home Delivery ◽  
Branch And Cut ◽  
Routing Problems ◽  
Last Mile ◽  
Multiple Vehicles ◽  
Recent Method ◽  
The Difference

Abstract In this paper we consider the use of lockers in parcel delivery, a recent method used in the last mile logistics. Lockers are pick up points made of several cells that are located in several points of a city where customers can collect their parcels as an alternative to home delivery. We study routing problems in which one or multiple vehicles are used to deliver parcels directly to customers or to lockers. We also study the influence of the introduction of lockers when these problems include time windows. We propose a set of novel formulations for these problems, some valid inequalities, and a branch-and-cut algorithm. Moreover, we investigate the difference between the routing problems with lockers and the classical routing problems.

Team Orienteering with Time-Varying Profit

2021 ◽  
Author(s):  
Qinxiao Yu ◽  
Yossiri Adulyasak ◽  
Louis-Martin Rousseau ◽  
Ning Zhu ◽  
Shoufeng Ma
Keyword(s):  
Valid Inequalities ◽  
Iterated Local Search ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Orienteering Problem ◽  
Time Varying ◽  
High Quality ◽  
Hybrid Heuristic ◽  
Team Orienteering Problem ◽  
Team Orienteering

This paper studies the team orienteering problem, where the arrival time and service time affect the collection of profits. Such interactions result in a nonconcave profit function. This problem integrates the aspect of time scheduling into the routing decision, which can be applied in humanitarian search and rescue operations where the survival rate declines rapidly. Rescue teams are needed to help trapped people in multiple affected sites, whereas the number of people who could be saved depends as well on how long a rescue team spends at each site. Efficient allocation and scheduling of rescue teams is critical to ensure a high survival rate. To solve the problem, we formulate a mixed-integer nonconcave programming model and propose a Benders branch-and-cut algorithm, along with valid inequalities for tightening the upper bound. To solve it more effectively, we introduce a hybrid heuristic that integrates a modified coordinate search (MCS) into an iterated local search. Computational results show that valid inequalities significantly reduce the optimality gap, and the proposed exact method is capable of solving instances where the mixed-integer nonlinear programming solver SCIP fails in finding an optimal solution. In addition, the proposed MCS algorithm is highly efficient compared with other benchmark approaches, whereas the hybrid heuristic is proven to be effective in finding high-quality solutions within short computing times. We also demonstrate the performance of the heuristic with the MCS using instances with up to 100 customers. Summary of Contribution: Motivated by search and rescue (SAR) operations, we consider a generalization of the well-known team orienteering problem (TOP) to incorporate a nonlinear time-varying profit function in conjunction with routing and scheduling decisions. This paper expands the envelope of operations research and computing in several ways. To address the scalability issue of this highly complex combinatorial problem in an exact manner, we propose a Benders branch-and-cut (BBC) algorithm, which allows us to efficiently deal with the nonconcave component. This BBC algorithm is computationally enhanced through valid inequalities used to strengthen the bounds of the BBC. In addition, we propose a highly efficient hybrid heuristic that integrates a modified coordinate search into an iterated local search. It can quickly produce high-quality solutions to this complex problem. The performance of our solution algorithms is demonstrated through a series of computational experiments.

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 Strengthened Formulations and Valid Inequalities for Single Delay Management in Public Transportation

2019 ◽  
Vol 53 (5) ◽  
pp. 1271-1286
Author(s):  
Veronica Dal Sasso ◽  
Luigi De Giovanni ◽  
Martine Labbé
Keyword(s):  
Linear Programming ◽  
Public Transportation ◽  
Mixed Integer Linear Programming ◽  
Transportation Networks ◽  
Valid Inequalities ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Management Problem ◽  
Delay Management ◽  
Initial Delay

The delay management problem arises in public transportation networks, often characterized by the necessity of connections between different vehicles. The attractiveness of public transportation networks is strongly related to the reliability of connections, which can be missed when delays or other unpredictable events occur. Given a single initial delay at one node of the network, the delay management problem is to determine which vehicles have to wait for the delayed ones, with the aim of minimizing the dissatisfaction of the passengers. In this paper, we present strengthened mixed integer linear programming formulations and new families of valid inequalities. The implementation of branch-and-cut methods and tests on a benchmark of instances taken from real networks show the potential of the proposed formulations and cuts.

scholarly journals Asymmetric Multidepot Vehicle Routing Problems: Valid Inequalities and a Branch-and-Cut Algorithm

2021 ◽  
Vol 69 (2) ◽  
pp. 380-409
Author(s):  
Michiel A. J. uit het Broek ◽  
Albert H. Schrotenboer ◽  
Bolor Jargalsaikhan ◽  
Kees Jan Roodbergen ◽  
Leandro C. Coelho
Keyword(s):  
Vehicle Routing ◽  
Valid Inequalities ◽  
Directed Graphs ◽  
Branch And Cut ◽  
Vehicle Routing Problems ◽  
Problem Size ◽  
Routing Problems ◽  
Multiple Depots ◽  
Number Of Customers ◽  
New Inequalities

In “Asymmetric Multidepot Vehicle Routing Problems: Valid Inequalities and a Branch-and-Cut Algorithm,” Uit het Broek, Schrotenboer, Jargalsaikhan, Roodbergen, and Coelho present a generic branch-and-cut framework to solve routing problems with multiple depots on directed graphs. They present new valid inequalities that eliminate subtours, enforce tours to be linked to the same depot, and enforce bounds on the number of customers in a vehicle tour. This is embedded in a branch-and-cut scheme that also contains generalized and adapted versions of valid inequalities that are well known for related routing problems. The authors show that the new inequalities tighten root node relaxations considerably. In combination with a simple but effective upper-bound procedure, only requiring a MIP solver and a smart reduction of the problem size, the authors show that the overall framework solves instances of considerably larger size to optimality than have been reported in the literature.

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