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 A Hybrid IP/GA Approach to the Parallel Production Lines Scheduling Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Huizhi Ren ◽  
Shenshen Sun
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Time Window ◽  
Valid Inequalities ◽  
Mixed Integer ◽  
Scheduling Problem ◽  
Production Lines ◽  
Solution Approach ◽  
Cp Decomposition

A special parallel production lines scheduling problem is studied in this paper. Considering the time window and technical constraints, a mixed integer linear programming (MILP) model is formulated for the problem. A few valid inequalities are deduced and a hybrid mixed integer linear programming/constraint programming (MILP/CP) decomposition strategy is introduced. Based on them, a hybrid integer programming/genetic algorithm (IP/GA) approach is proposed to solve the problem. At last, the numerical experiments demonstrate that the proposed solution approach is effective and efficient.

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 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 Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

2020 ◽  
Author(s):  
Thomas Kleinert ◽  
Martine Labbé ◽  
Fränk Plein ◽  
Martin Schmidt
Keyword(s):  
Branch And Bound ◽  
Optimization Problems ◽  
Valid Inequalities ◽  
Bilevel Optimization ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Valid Inequality ◽  
Single Level ◽  
Primal Dual ◽  
Level Problem

Abstract Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.

scholarly journals An Exact Single-Agent Task Selection Algorithm for the Crowdsourced Logistics

2020 ◽  
Author(s):  
Chung-Kyun Han ◽  
Shih-Fen Cheng
Keyword(s):  
Response Time ◽  
Service Providers ◽  
Time Window ◽  
Valid Inequalities ◽  
Single Agent ◽  
Branch And Cut ◽  
Greedy Heuristic ◽  
Full Time ◽  
Delivery Time ◽  
Logistics Industry

The trend of moving online in the retail industry has created great pressure for the logistics industry to catch up both in terms of volume and response time. On one hand, volume is fluctuating at greater magnitude, making peaks higher; on the other hand, customers are also expecting shorter response time. As a result, logistics service providers are pressured to expand and keep up with the demands. Expanding fleet capacity, however, is not sustainable as capacity built for the peak seasons would be mostly vacant during ordinary days. One promising solution is to engage crowdsourced workers, who are not employed full-time but would be willing to help with the deliveries if their schedules permit. The challenge, however, is to choose appropriate sets of tasks that would not cause too much disruption from their intended routes, while satisfying each delivery task's delivery time window requirement. In this paper, we propose a decision-support algorithm to select delivery tasks for a single crowdsourced worker that best fit his/her upcoming route both in terms of additional travel time and the time window requirements at all stops along his/her route, while at the same time satisfies tasks' delivery time windows. Our major contributions are in the formulation of the problem and the design of an efficient exact algorithm based on the branch-and-cut approach. The major innovation we introduce is the efficient generation of promising valid inequalities via our separation heuristics. In all numerical instances we study, our approach manages to reach optimality yet with much fewer computational resource requirement than the plain integer linear programming formulation. The greedy heuristic, while efficient in time, only achieves around 40-60% of the optimum in all cases. To illustrate how our solver could help in advancing the sustainability objective, we also quantify the reduction in the carbon footprint.

scholarly journals Consistency Cuts for Dantzig-Wolfe Reformulations

2022 ◽  
Author(s):  
Jens Vinther Clausen ◽  
Richard Lusby ◽  
Stefan Ropke
Keyword(s):  
Valid Inequalities ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Linear Programming Relaxation ◽  
Linear Programs ◽  
Integer Programs ◽  
New Family ◽  
Integer Linear Programs ◽  
Integer Solutions ◽  
Online Library

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

scholarly journals Variable and constraint reduction techniques for the temporal bin packing problem with fire-ups

2021 ◽  
Author(s):  
John Martinovic ◽  
Nico Strasdat ◽  
José Valério de Carvalho ◽  
Fabio Furini
Keyword(s):  
Linear Programming ◽  
Bin Packing ◽  
Time Window ◽  
Valid Inequalities ◽  
Packing Problem ◽  
Weighted Sum ◽  
Reduction Techniques ◽  
Bin Packing Problem ◽  
First Time ◽  
Effective Procedures

AbstractThe aim of this letter is to design and computationally test several improvements for the compact integer linear programming (ILP) formulations of the temporal bin packing problem with fire-ups (TBPP-FU). This problem is a challenging generalization of the classical bin packing problem in which the items, interpreted as jobs of given weight, are active only during an associated time window. The TBPP-FU objective function asks for the minimization of the weighted sum of the number of bins, viewed as servers of given capacity, to execute all the jobs and the total number of fire-ups. The fire-ups count the number of times the servers are activated due to the presence of assigned active jobs. Our contributions are effective procedures to reduce the number of variables and constraints of the ILP formulations proposed in the literature as well as the introduction of new valid inequalities. By extensive computational tests we show that substantial improvements can be achieved and several instances from the literature can be solved to proven optimality for the first time.

scholarly journals Mixed-integer programming approaches for the time-constrained maximal covering routing problem

OR Spectrum ◽  
2021 ◽  
Author(s):  
Markus Sinnl
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Valid Inequalities ◽  
Optimal Solution ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Routing Problem ◽  
Maximal Covering ◽  
Solution Algorithms ◽  
Number Of Customers

AbstractIn this paper, we study the recently introduced time-constrained maximal covering routing problem. In this problem, we are given a central depot, a set of facilities, and a set of customers. Each customer is associated with a subset of the facilities which can cover it. A feasible solution consists of k Hamiltonian cycles on subsets of the facilities and the central depot. Each cycle must contain the depot and must respect a given distance limit. The goal is to maximize the number of customers covered by facilities contained in the cycles. We develop two exact solution algorithms for the problem based on new mixed-integer programming models. One algorithm is based on a compact model, while the other model contains an exponential number of constraints, which are separated on-the-fly, i.e., we use branch-and-cut. We also describe preprocessing techniques, valid inequalities and primal heuristics for both models. We evaluate our solution approaches on the instances from literature and our algorithms are able to find the provably optimal solution for 267 out of 270 instances, including 123 instances, for which the optimal solution was not known before. Moreover, for most of the instances, our algorithms only take a few seconds, and thus are up to five magnitudes faster than previous approaches. Finally, we also discuss some issues with the instances from literature and present some new instances.

scholarly journals Electric Vehicle Tour Planning Considering Range Anxiety

2020 ◽  
Vol 12 (9) ◽  
pp. 3685 ◽  
Author(s):  
Rui Chen ◽  
Xinglu Liu ◽  
Lixin Miao ◽  
Peng Yang
Keyword(s):  
Electric Vehicles ◽  
Numerical Experiments ◽  
Time Horizon ◽  
High Reliability ◽  
Route Planning ◽  
Mixed Integer ◽  
Planning Problem ◽  
Mixed Integer Linear Program ◽  
Tour Planning ◽  
Range Anxiety

In this study, the tour planning problem for electric vehicles is investigated. We aim to derive the optimal route and thus, to maximize profitability and minimize range anxiety within the time horizon. To solve this problem, a bi-objective mixed integer model is proposed. Specifically, we first introduced the reliability of route planning and quantified it as a cost with specific functions. The nonlinear model was then converted into a bi-objective mixed integer linear program, and an interactive branch and bound algorithm was adopted. Numerical experiments conducted on different networks have shown that the model that considers range anxiety offers more effective solutions. This means that our model is able to plan the routes with high reliability and low risk of profit loss and accidents.

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