Optimality For Instances Research Articles

The Electric Vehicle Routing and Overnight Charging Scheduling Problem on a Multigraph

In the electric vehicle (EV) routing and overnight charging scheduling problem, a fleet of EVs must serve the demand of a set of customers with time windows. The problem consists in finding a set of minimum cost routes and determining an overnight EV charging schedule that ensures the routes’ feasibility. Because (i) travel time and energy consumption are conflicting resources, (ii) the overnight charging operations take considerable time, and (iii) the charging infrastructure at the depot is limited, we model the problem on a multigraph where each arc between two vertices represents a path with a different resource consumption trade-off. To solve the problem, we design a branch-price-and-cut algorithm that implements state-of-the-art techniques, including the ng-path relaxation, subset-row inequalities, and a specialized labeling algorithm. We report computational results showing that the method solves to optimality instances with up to 50 customers. We also present experiments evaluating the benefits of modeling the problem on a multigraph rather than on the more classical 1-graph representation. History: Accepted by Andra Lodi, Area Editor for Design and Analysis of Algorithms—Discrete. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada through the Discovery grants [Grant RGPIN-2023-03791]. It was also partially funded by HEC Montréal through the research professorship on Clean Transportation Analytics. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0404 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0404 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Node based compact formulations for the Hamiltonian p‐median problem

AbstractIn this paper, we introduce, study and analyze several classes of compact formulations for the symmetric Hamiltonian ‐median problem (HMP). Given a positive integer and a weighted complete undirected graph with weights on the edges, the HMP on is to find a minimum weight set of elementary cycles partitioning the vertices of . The advantage of developing compact formulations is that they can be readily used in combination with off‐the‐shelf optimization software, unlike other types of formulations possibly involving the use of exponentially sized sets of variables or constraints. The main part of the paper focuses on compact formulations for eliminating solutions with less than cycles. Such formulations are less well known and studied than formulations which prevent solutions with more than cycles. The proposed formulations are based on a common motivation, that is, the formulations contain variables that assign labels to nodes, and prevent less than cycles by stating that different depots must have different labels and that nodes in the same cycle must have the same label. We introduce and study aggregated formulations (which consider integer variables that represent the label of the node) and disaggregated formulations (which consider binary variables that assign each node to a given label). The aggregated models are new. The disaggregated formulations are not, although in all of them new enhancements have been included to make them more competitive with the aggregated models. The two main conclusions of this study are: (i) in the context of compact formulations, it is worth looking at the models with integer node variables, which have a smaller size. Despite their weaker LP relaxation bounds, the fewer variables and constraints lead to faster integer resolution, especially when solving instances with more than 50 nodes; (ii) the best of our compact models exhibit a performance that, overall, is comparable to that of the best methods known for the HMP (including branch‐and‐cut algorithms), solving to optimality instances with up to 226 nodes within 1 h. This corroborates our message that the knowledge of the inequalities for preventing less than cycles is much less well understood.

An Exact Approach for a Variant of the FS-TSP

In this work we focus on the flying sidekick traveling salesman problem (FS-TSP). The FS-TSP arises in the last-mile distribution context and it is a variant of the TSP aimed at determining the distribution plan of a driver-operated truck assisted by a drone (unmanned aerial vehicle, UAV), where the synchronization between the two vehicles allows to parallelize the delivery operations, so providing a reduction of the overall completion time. Several variants of the FS-TSP have been considered in literature, most of them sharing the assumption that launching and rendezvous position of a drone sortie must be different. In this work, we relax this assumption allowing the truck to wait for the drone at the launching position (FS-TSP*). This introduces an important flexibility element to deal with deliveries in rural or in urban areas with no-fly zones. We propose an original and compact integer linear programming formulation which allows to exactly solve the FS-TSP* and consequently also the FS-TSP. The results on several test instances show that our method can effectively solve to optimality instances with up to 20 customers and quantify the advantages of the FS-TSP* with respect to FS-TSP in terms of overall completion time.

A multiperiod drayage problem with customer-dependent service periods

We investigate a multiperiod drayage problem in which customers request transportation services over several days, possibly leaving the carrier some flexibility to change service periods. We compare three approaches for the problem: a path-based model with all feasible routes, a “Price-and-Branch” algorithm in which the pricing is formulated as a collection of shortest path problems in a cunningly constructed acyclic network, and a compact arc-flow formulation based on this network. The experiments shows that the latter formulation is the most efficient, and can solve to optimality instances of real-world size (and beyond) in time compatible with typical operational constraints. Also, the models allow us to assess that limited amounts of flexibility from customers can significantly improve routing costs for the carrier while decreasing customers’ cost as well.

Shift Planning Under Delay Uncertainty at Air France: A Vehicle-Scheduling Problem with Outsourcing

Airlines must operate many jobs in airports, such as passenger check-in or runway tasks. In airlines’ hubs, airlines generally choose to perform these jobs with their own agents. Shift planning aims at building the sequences of jobs operated by the airline agents and has been widely studied given its impact on operating costs. The impact of delayed flights is generally not taken into account despite the propagation of flight delays along these sequences: If a flight is late, then the agents doing the corresponding jobs are delayed, and may arrive late to their next jobs and delay the corresponding flights. Since delay costs are much higher than the costs of outsourcing jobs, if the agent who is supposed to operate a job is still working elsewhere when the job begins, then airlines tend to outsource the job to their own dedicated team or to a third party. We introduce a stochastic version of the shift-planning problem that takes into account outsourcing costs due to delay. It can be seen as a natural stochastic generalization of the vehicle-scheduling problem in which delayed jobs are outsourced. We propose a column-generation approach to solve it, whose key element is the pricing subproblem algorithm, modeled as a stochastic resource-constrained shortest-path problem. Numerical results on Air France industrial instances show the benefits of using our stochastic version of the shift-planning problem and the efficiency of the solution method. Moving to the stochastic version enables Air France to reduce total operating costs by 3.5%–4.8% on instances with more than 200 jobs, and our algorithm can solve to near optimality instances with up to 400 jobs.

Scheduling jobs with release dates on identical parallel machines by minimizing the total weighted completion time

This paper addresses the problem of scheduling a set of jobs that are released over the time on a set of identical parallel machines, aiming at the minimization of the total weighted completion time. This problem, referred to as P|rj|∑wjCj, is of great importance in practice, because it models a variety of real-life applications. Despite its importance, the P|rj|∑wjCj has not received much attention in the recent literature. In this work, we fill this gap by proposing mixed integer linear programs and a tailored branch-and-price algorithm. Our branch-and-price relies on the decomposition of an arc-flow formulation and on the use of efficient exact and heuristic methods for solving the pricing subproblem. Computational experiments carried out on a set of randomly generated instances prove that the proposed methods can solve to the proven optimality instances with up to 200 jobs and 10 machines, and provide very low gaps for larger instances.

An exact algorithm for robust influence maximization

We propose a Branch-and-Cut algorithm for the robust influence maximization problem. The influence maximization problem aims to identify, in a social network, a set of given cardinality comprising actors that are able to influence the maximum number of other actors. We assume that the social network is given in the form of a graph with node thresholds to indicate the resistance of an actor to influence, and arc weights to represent the strength of the influence between two actors. In the robust version of the problem that we study, the node thresholds are affected by uncertainty and we optimize over a worst-case scenario within a given robustness budget. Numerical experiments show that we are able to solve to optimality instances of size comparable to other exact approaches in the literature for the non-robust problem, but in addition to this we can also tackle the robust version with similar performance.

A Decomposition Method for Finding Optimal Container Stowage Plans

In transportation of goods in large container ships, shipping industries need to minimize the time spent at ports to load/unload containers. An optimal stowage of containers on board minimizes unnecessary unloading/reloading movements, while satisfying many operational constraints. We address the basic container stowage planning problem (CSPP). Different heuristics and formulations have been proposed for the CSPP, but finding an optimal stowage plan remains an open problem even for small-sized instances. We introduce a novel formulation that decomposes CSPPs into two sets of decision variables: the first defining how single container stacks evolve over time and the second modeling port-dependent constraints. Its linear relaxation is solved through stabilized column generation and with different heuristic and exact pricing algorithms. The lower bound achieved is then used to find an optimal stowage plan by solving a mixed-integer programming model. The proposed solution method outperforms the methods from the literature and can solve to optimality instances with up to 10 ports and 5,000 containers in a few minutes of computing time. The online appendix is available at https://doi.org/10.1287/trsc.2017.0795 .

Exact approaches for the orderly colored longest path problem: Performance comparison

In this paper we consider a particular graph-optimization problem. Given an edge-colored graph and a set of constraints on the sequence of the colors, one is to find the longest path whose colored edges obey the constraints on the sequence of the colors. In the actual formulation, the problem generalizes already known NP-Complete problems, and, evidently, the alternating path problem in edge colored graphs. Recent literature has shown several contexts where such problem may be useful to model interesting applications, and has proposed exact IP models and related algorithms. We extend on these existing models and extensively test new formulations for the problem, showing how one of the newly developed model clearly exhibits better performance, allowing to solve at optimality instances of significant sizes.

Modelling and symmetry breaking in scheduling problems on batch processing machines

Problems of scheduling batch-processing machines to minimise the makespan are widely exploited in the literature, mainly motivated by real-world applications, such as burn-in tests in the semiconductor industry. These problems consist of grouping jobs in batches and scheduling them on machines. We consider problems where jobs have non-identical sizes and processing times, and the total size of each batch cannot exceed the machine capacity. The processing time of a batch is defined as the longest processing time among all jobs assigned to it. Jobs can also have non-identical release times, and in this case, a batch can only be processed when all jobs assigned to it are available. This paper discusses four different versions of batch scheduling problems, considering a single processing machine or parallel processing machines and considering jobs with or without release times. New mixed integer linear programming formulations are proposed as enhancements of formulations proposed in the literature, and symmetry breaking constraints are investigated to reduce the size of the feasible sets. Computational results show that the proposed formulations have a better performance than other models in the literature, being able to solve to optimality instances only considered before to be solved by heuristic procedures.

Optimal planning of buffer sizes and inspection station positions

The problem of buffer sizing and inspection stations positioning in unreliable production lines is a complex mixed integer nonlinear optimization problem. In this problem, we have a production line with n machines and n fixed-size (storage) buffers in series. The machines produce parts that are either conforming or nonconforming, and the line includes inspection stations that reject the nonconforming pats. The goal is to find the optimal buffer sizes, the number and positions of the inspection stations, and satisfy the customer demand on conforming parts while minimizing the total cost. We present in this paper an exact method to solve this complex manufacturing problem. We also present new theoretical results on buffer-size bounds, stationarity, and cost function convexity permitting to significantly reduce the problem complexity. These theoretical and algorithmic developments allow solving to optimality instances with up to 30 machines tools developed previously cannot solve.

A Decomposition Algorithm for the Consistent Traveling Salesman Problem with Vehicle Idling

The consistent traveling salesman problem aims to identify minimum-cost routes to be followed by a single vehicle so as to provide a set of customers with service that adheres to arrival-time consistency across the multiple time periods of a planning horizon. In this paper, we address this problem via a new, exact algorithm that decomposes the problem into a sequence of single-period traveling salesman problems with time windows within a branch-and-bound search procedure. The new algorithm is highly competitive, as it is able to solve to guaranteed optimality instances with up to 100 customers requiring service over a five-period horizon, effectively doubling the size of instances that were solvable by the previous state of the art. Furthermore, and in contrast to all previous approaches, the new algorithm accounts for route duration limits, whenever applicable, as well as incorporates the flexibility for the vehicle to idle before providing service to a customer. We in fact show that, for the benchmark instances we considered, the cost of implementing consistent routes can be reduced significantly if the vehicle is allowed to idle en route, further motivating the need for algorithmic schemes to incorporate this realistic option.

An exact algorithm for a Vehicle-and-Driver Scheduling Problem

This article introduces a combinatorial optimization problem that consists of assigning tasks to machines and operators, and sequencing the tasks assigned to each one. Two configurations exist. Machines alternate configurations, while the operators must start and finish the process in the same configuration. Moreover, machines and operator have limited capacities. The sequencing of the tasks must guarantee that each one is performed by a machine and an operator at the same time, and it is determined in order to minimize an overall cost function. Two critical aspects of the problem are the need of synchronizing the machine and the operator performing each task, and the need of minimizing the changeovers, which are pairs of tasks done consecutively by the same machine but by different operators. The problem is modeled as a vehicle routing problem with two types of vehicles and with two depots. We propose a mixed integer programming formulation, and introduce valid inequalities to strengthen its linear programming relaxation. We describe separation routines for these inequalities and design a branch-and-cut algorithm for the problem. The algorithm is tested on a set of benchmark instances showing that it is able to solve to optimality instances with up to 50 customers.

The traveling salesman problem with pickup, delivery, and ride-time constraints

In the traveling salesman problem with pickup, delivery, and ride-time constraints (TSPPD-RT), a vehicle located at a depot is required to service a number of requests where the requests are known before the route is formed. Each request consists of (i) a pickup location (origin), (ii) a delivery location (destination), and (iii) a maximum allowable travel time from the origin to the destination (maximum ride-time). The problem is to design a tour for the vehicle that (i) starts and ends at the depot, (ii) services all requests, (iii) ensures that each request's ride-time does not exceed its maximum ride-time, and (iv) minimizes the total travel time required by the vehicle to service all requests (objective function). A capacity constraint that may be present is that the weight or volume of the undelivered requests on the vehicle must always be no greater than the vehicle's capacity. In this article, we concurrently analyze the TSPPD-RT with capacity constraints and without capacity constraints. We describe two mathematical formulations of the problem. These formulations are used to derive new lower bounds on the solution to the problem. Then, we provide two exact methods for finding the optimal route that minimizes the total travel cost. Our extensive computational analysis on both versions of the TSPPD-RT shows that the proposed algorithms are capable of solving to optimality instances involving up to 50 requests. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(2), 95–110 2016

Minimizing customers’ waiting time in a vehicle routing problem with unit demands

In this paper we study a new variant of the unit demand vehicle routing problem with the aim of minimizing the sum of customers waiting times to receive service. These kinds of problems are relevant in applications where the customers' waiting time is essential or is more important than the vehicles' travel time. We modify a known formulation for the multiple traveling salesman problem adapting it to the addressed problem. The derived mixed integer formulation is able to solve to optimality instances up to 40 nodes. We also develop a metaheuristic algorithm based on Iterated Greedy approach. We implement two variants of the metaheuristic algorithm using two different strategies in the constructive phase. For instances up to 40 nodes the proposed algorithms found almost all the optimal solutions and outperformed the results obtained by the formulation for instances with unknown optimal solution. In general, both versions of the metaheuristic algorithm are very fast and have a good performance too for instances ranging from 50 to 100 nodes.

A customer-centric routing problem with multiple trips of a single vehicle

In this study, we introduce a routing problem with multiple uses of a single vehicle and service time in demand points, minimizing the sum of clients’ waiting time to receive service. This problem is relevant in the distribution of aid in disaster-stricken communities, in the recollection and/or delivery of perishable goods and personnel transportation, among other situations, where reaching clients to perform service, fast and fair, is a priority. We consider vehicle capacity and travel distance constraints, forcing multiple use of the vehicle during the planning horizon. This paper presents two mixed integer formulations for this problem, based on a multi-level network, as well as a metaheuristic algorithm. The proposed models can solve to optimality instances with up to 30 clients. The proposed metaheuristic algorithm obtains high-quality solutions in short computational times.

The Rural Postman Problem with time windows

The Rural Postman Problem with Time Windows for the undirected case is introduced. The problem occurs in the monitoring of roads for black-ice detection. Different formulations are proposed and tested on sets of instances adapted from the literature. A cutting plane algorithm based on valid inequalities for the Traveling Salesman Problem (TSP) with Time Windows and the Precedence Constrained TSP is presented as solution method and tested on a set of real-life networks. Computational results show that this approach is able to solve to optimality instances with up to 104 required edges. At the end of the article the formulations for the undirected case are extended to the directed case. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(3), 169–180 2014

Spanning trees with variable degree bounds

In this paper, we introduce and study a generalization of the degree constrained minimum spanning tree problem where we may install one of several available transmission systems (each with a different cost value) in each edge. The degree of the endnodes of each edge depends on the system installed on the edge. We also discuss a particular case that arises in the design of wireless mesh networks (in this variant the degree of the endnodes of each edge depend on the transmission system installed on it as well as on the length of the edge). We propose three classes of models using different sets of variables and compare from a theoretical perspective as well as from a computational point of view, the models and the corresponding linear programming relaxations. The computational results show that some of the proposed models are able to solve to optimality instances with 100 nodes and different scenarios.

A Novel Multiinstance Learning Approach for Liver Cancer Recognition on Abdominal CT Images Based on CPSO-SVM and IO

A novel multi-instance learning (MIL) method is proposed to recognize liver cancer with abdominal CT images based on instance optimization (IO) and support vector machine with parameters optimized by a combination algorithm of particle swarm optimization and local optimization (CPSO-SVM). Introducing MIL into liver cancer recognition can solve the problem of multiple regions of interest classification. The images we use in the experiments are liver CT images extracted from abdominal CT images. The proposed method consists of two main steps: (1) obtaining the key instances through IO by texture features and a classification threshold in classification of instances with CPSO-SVM and (2) predicting unknown samples with the key instances and the classification threshold. By extracting the instances equally based on the entire image, the proposed method can ignore the procedure of tumor region segmentation and lower the demand of segmentation accuracy of liver region. The normal SVM method and two MIL algorithms, Citation-kNN algorithm and WEMISVM algorithm, have been chosen as comparing algorithms. The experimental results show that the proposed method can effectively recognize liver cancer images from two kinds of cancer CT images and greatly improve the recognition accuracy.

A column generation approach for the split delivery vehicle routing problem

AbstractIn this article we present a branch‐and‐price‐and‐cut method for the solution of the split delivery vehicle routing problem (SDVRP). The SDVRP is the problem to serve customers with a fleet of capacitated vehicles at minimum traveling cost. With respect to the classical vehicle routing problem, where each customer is visited exactly once, in the SDVRP a customer may be visited any number of times. The exact method we propose is based on a decomposition of the problem where the possible routes, with the delivery quantities, are generated in the subproblem. The generated routes are also used to find a heuristic solution to the problem. We consider both the case where the fleet of vehicles is unlimited and the case where the fleet is limited to the minimum possible number of vehicles. We solve to optimality instances with larger size with respect to previous approaches, find new best solutions to several benchmark instances and reduce the optimality gap on most of the benchmark instances. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 58(4), 241–254 2011