Integer Programming Formulation Research Articles

The home health care routing with heterogeneous electric vehicles and synchronization

AbstractThis paper studies the problem of heterogeneous electric vehicles, fast chargers, and synchronized jobs that have time windows in home healthcare routing and scheduling. We consider a problem that aims to establish daily routes and schedules for healthcare nurses to provide a variety of services to patients located in a scattered area. Each nurse should be assigned to an electric vehicle (EV) from a heterogeneous fleet of EVs to perform the assigned jobs within working hours. We consider three different types of EVs in terms of battery capacity and energy consumption. We aim to minimize the total cost of energy consumption, fixed nurse cost, and costs arising from the patients that cannot be served within the working day. We model the problem as a mixed integer programming formulation. We develop a hybrid metaheuristic based on a greedy random adaptive search procedure heuristic, to generate good quality initial solutions quickly, and an adaptive variable neighborhood search algorithm to generate high quality solutions in reasonable time. The hybrid metaheuristic employs a set of new advanced efficient procedures designed to handle the complex structure of the problem. Through extensive computational experiments, the performance of the mathematical model and the hybrid metaheuristic are evaluated. We conduct analyses on the robustness of the metaheuristic and the performance contribution of employing adaptive probabilities. We analyze the impact of problem parameters such as competency requirements, job duration, and synchronized jobs.

The Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station: Compact formulation and a General Variable Neighborhood Search metaheuristic

In this article, we introduce the Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station (VRPCDSD), which integrates pickup and delivery routing decisions with the cross-docking scheduling of the unloading and reloading of requests. We present a compact Mixed Integer Programming (MIP) formulation and a General Variable Neighborhood Search (GVNS) metaheuristic for the problem. The proposed GVNS employs several routing and scheduling neighborhoods that require problem-specific adjustments due to the integrated characteristic of the problem. Computational experiments indicate that the MIP formulation can solve instances with up to 31 nodes (15 requests) within the specified time limit when using a standard commercial solver. Furthermore, the GVNS metaheuristic can obtain high-quality solutions that at least match those encountered by the solver for the tested instances, achieving considerably large improvements over the latter especially as the number of requests increases. In fact, GVNS is able to find the optimal solution for all the instances for which the optimal is known. In addition, GVNS is a viable alternative for tackling larger instances of VRPCDSD and is very robust in that it obtains low deviations from the best-encountered solutions when multiple executions are performed. We also present some insights on how the different cross-docking operation costs impact the optimal solutions.

On approximating a new generalization of traveling salesman problem

The current best-known performance guarantee for the extensively studied Traveling Salesman Problem (TSP) of determinate approximation algorithms is 32, achieved by Christofides' algorithm 47 years ago. This paper investigates a new generalization problem of the TSP, termed the Minimum-Cost Bounded Degree Connected Subgraph (MBDCS) problem. In the MBDCS problem, the goal is to identify a minimum-cost connected subgraph containing n=|V| edges from an input graph G=(V,E) with degree upper bounds for particular vertices. We show that for certain special cases of MBDCS, the aim is equivalent to finding a minimum-cost Hamiltonian cycle for the input graph, same as the TSP. To appropriately solve MBDCS, we initially present an integer programming formulation for the problem. Subsequently, we propose an algorithm to approximate the optimal solution by applying the iterative rounding technique to solution of the integer programming relaxation. We demonstrate that the returned subgraph of our proposed algorithm is one of the best guarantees for the MBDCS problem in polynomial time, assuming P≠NP. This study views the optimization of TSP as finding a minimum-cost connected subgraph containing n edges with degree upper bounds for certain vertices, and it may provide new insights into optimizing the TSP in future research.

Mathematical programming-based heuristic for highway patrol drone scheduling problem

Recently, highway drone patrols have been introduced in several countries to improve the efficiency of continuous patrols to enforce traffic laws and respond to accidents on highways. Since highways stretch over a vast area, a large number of drones are needed to patrol simultaneously, so it is necessary to develop an optimized flight plan to make them operate efficiently. This paper addresses the problem of determining an operational schedule for multiple highway patrol drones in order to satisfy varying patrol demands on different highway sections and at different times. Two integer programming formulations, namely the basic formulation and the path-based formulation, are proposed for the problem. However, these formulations have limitations in generating high quality solutions within a reasonable time frame. To overcome these limitations and find high quality solutions in a short time, we propose a column generation-based heuristic algorithm that combines the advantages of both formulations. Computational experiments were conducted on two real-world Korean highway instances to verify the effectiveness of the proposed heuristic algorithm. The results show that the proposed heuristic is highly effective and can be successfully applied to repetitive scheduling environments in real-world highway drone patrol systems.

Integrated stochastic optimisation of stope design and long-term production scheduling at an operating underground copper mine

ABSTRACT Conventional underground long-term mine planning is based on a deterministic stepwise framework, which is unable to effectively manage the synergies between the mine planning components or to manage the orebody risk in production schedules and forecasts. The present study enhances prior integrated optimisation approaches to a variation of the sublevel longhole open stoping mining method using backfilling, through a new two-stage stochastic integer programming formulation. A comparison to a sequential stochastic approach shows different stope layouts and extraction sequences for a copper mine with secondary elements being gold and uranium. The integrated approach shows lower horizontal development costs and 6% higher net present value.

Joint stochastic optimisation of stope layout, production scheduling and access network

The three main optimisation components of sublevel stoping methods are stope layout, production schedule (or stope sequencing) and access networks. The joint optimisation of these components could further add value to an underground mining project. This potential has not been considered in the literature due to computational difficulties, and the problem was solved sequentially. This paper proposes a new joint optimisation model to integrate these components. In addition, the proposed optimisation model incorporates stochastic simulations to capture uncertainty and variability associated with the grades of the related mineral deposits mined. The optimisation model is based on a two-stage stochastic integer programming (SIP) formulation that maximises the project's net present value (NPV) and minimises the planned dilution. Applying the proposed method at a small copper deposit shows that the SIP outperforms the results obtained from mixed integer programming. For a seven-year mine life, the SIP model generated ∼20% more NPV, demonstrating the importance of developing a joint optimisation formulation and accounting for grade uncertainty and variability.

Integer optimization models and algorithms for the multi-period non-shareable resource allocation problem

The resource allocation problem (RAP) determines a solution to optimally allocate limited resources to several activities or tasks. In this study, we propose a novel resource allocation problem referred to as multi-period non-shareable resource allocation problem (MNRAP), which is motivated by the characteristics of resources considered in the stem cell culture process for producing stem cell therapeutics. A resource considered in the MNRAP has the following three characteristics: (i) resource consumption required to perform an activity and available resource capacity may change over time; (ii) multiple activities cannot share one resource; and (iii) resource requirements can be satisfied through the combination of different types of resources. The MNRAP selects some of the given activities to maximize the overall profit under limited resources with these characteristics. To address this problem, pattern-based integer programming formulations based on the concept of resource patterns are proposed. These formulations attempt to overcome the limitations of a compact integer programming formulation, the utilization of which is challenging for large-scale problems owing to their complexity. Further, based on a branch-and-price approach to solving pattern-based formulations, effective heuristic algorithms are proposed to provide high-quality solutions for large instances. Moreover, through computational experiments on a wide range of instances, including real-world instances, the superiority of the proposed formulations and heuristic algorithms is demonstrated.

A branch-and-cut approach to solve the Fault Diagnosis Problem with Lazy Spread and imperfect system information

This paper presents a new approach to solving the Fault Diagnosis Problem with Lazy Spread (FDPL) that arises in many fault-tolerant real-world systems with few opportunities for maintenance during their operations and significant failure interactions between the subsystems/components. As opposed to cascading faults that spread to most of the system almost instantaneously, FDPL considers fault-resistant systems where the spread of failures is relatively slow (lazy), i.e., only a small fraction of the components are faulty at the time of inspection, and accurate diagnosis of the faulty components is of critical importance to restore system performance and stop further damage. Introducing an extra level of difficulty, FDPL needs to be solved under imperfect (missing and wrong) system information in most real-world settings. To address this challenging prediction problem, we use graph theory concepts to develop an integer programming formulation and devise an efficient branch-and-cut algorithm for its solution. Extensive numerical experiments on realistic problem instances attest to the superior performance of our approach in terms of both computational efficiency and prediction accuracy compared to the state-of-the-art in the literature.

The wildfire suppression problem with multiple types of resources

The frequency and impact of wildfires have considerably increased in the past decade, due to the extreme weather conditions as well as the increased population density. The aim of this study is to introduce, model, and solve a wildfire suppression problem that involves multiple types of fire suppression resources and their operational characteristics. Two integer programming (IP) formulations, a basic IP and its reformulation with combinatorial Benders’ cuts, are presented. The performances of the proposed formulations are evaluated on a set of randomly generated instances. The results indicate that the proposed formulations are able to obtain high quality upper and lower bounds. Extensive numerical experiments are performed to analyse the effects of several operational constraints on the computational performance of the models. A case study arising in Yatağan district of Muğla province of Türkiye is presented.

Determining cost-efficient controls of electrical energy storages using dynamic programming

Volatile electrical energy prices are a challenge and an opportunity for small and medium-sized companies in energy-intensive industries. By using electrical energy storage and/or an adaptation of production processes, companies can significantly profit from time-depending energy prices and reduce their energy costs.We consider a time-discrete optimal control problem to reach a desired final state of the energy storage at a certain time step. Thereby, the energy input is discrete since only multiples of 100 kWh can be purchased at the EPEX SPOT market. We use available price estimates to minimize the total energy cost by a rounding based dynamic programming approach. With our model non-linear energy loss functions of the storage can be considered and we obtain a significant speed-up compared to the integer (linear) programming formulation.

Solving the patient admission scheduling problem using constraint aggregation

Patient admission scheduling (PAS) consists of assigning patients to beds over a planning horizon to maximize treatment efficiency, patient satisfaction, and hospital utilization while meeting all necessary medical constraints and considering patient preferences as much as possible. There are several different variants of the PAS problem in the literature, which differ mainly in the constraints that must be satisfied (hard) or can be violated (soft). Due to the intrinsic difficulty of the PAS problem, solving large integer programming (IP) models to optimality is challenging. In this paper, we consider the widely studied variant of the PAS problem that has the maximum number of soft constraints, and focus on how to reduce the size of IP formulations of the PAS problem to improve the solving efficiency. We employ a two-stage optimization method where the first stage builds reduced models by constraint aggregation to improve the typical formulation of the PAS problem. Experimental results on the 13 benchmark instances in the literature indicate that our method can obtain new improved solutions (new upper bounds) for 6 instances, including one proven optimal solution. For the 5 other instances whose optimal solutions are known, our approach can reach these known optimal solutions in a shorter computation time compared to the existing methods. In addition, we apply our method to the original PAS problem, which has the maximum number of hard constraints, and perform computational experiments on the same 13 benchmark instances. Our method yields 5 new best solutions and proves optimality for 6 instances.

Scheduling conferences using data on attendees’ preferences

Conference organisers often face the challenge of scheduling a scientific program. This problem usually involves many talks that must be scheduled in parallel, subject to time and space limitations. This paper adopts an Attendee-Based-Perspective to the conference scheduling problem, in which we collect (anonymised) data on attendees’ preferences and use these as a main driver to schedule the talks. We test three optimization approaches for this problem, based on integer programming formulations. The main approach divides the problem into two stages: the first stage schedules predefined thematic sessions and the second stage schedules talks within these sessions. We report results using real data instances of three conferences. The results show that our main approach can produce solutions swiftly, accommodating the requirements of the organisers while allowing attendees to attend most of their preferred talks. Our work has been used in practice to generate the actual schedule of these three conferences.

Integrated Stochastic Underground Mine Planning with Long-Term Stockpiling: Method and Impacts of Using High-Order Sequential Simulations

The integrated optimization of stope design and underground mine production scheduling is an approach that has been shown to effectively leverage the synergies among these two underground mine planning components to generate truly optimal stope layouts and extraction sequences. The existing stochastic integrated methods, however, do not include several elements of a mining complex, such as stockpiles, due to the computational complexity and non-linearity that they might add to the optimization of the long-term mine plan. Additionally, sequential simulation methods that rely on two-point statistics and Gaussian distribution assumptions are commonly used to generate the input realizations of the mineral deposit. These methods, however, are not able to properly characterize complex spatial geometries or the high-grade connectivity of non-Gaussian and non-linear natural phenomena. The present work proposes an extension of previous developments on the integrated stope design and underground mine scheduling optimization through an expanded stochastic integer programming formulation that incorporates long-term stockpiling decisions. An application of the proposed method at an operating underground copper mine compares the cases in which the geological simulated orebody models are based on high-order and Gaussian sequential simulation methods. The extraction sequence and related final stope design are shown to be physically different. It is seen that the optimization process takes advantage of the better representation of high-grade connectivity when high-order sequential simulations are used, by targeting the areas with grades that follow the mill’s blending requirements and by making less use of the stockpiles. Overall, a 4% higher copper metal production and a resultant 6% higher net present value are observed when high-order sequential simulations are used.

Tighter bounds on the minimum broadcast time

Given a connected graph and a subset of its vertices referred to as the sources, the minimum broadcast time problem asks for the shortest time necessary for communicating a message from the sources to all other vertices in the graph. Information exchange is possible only between neighbors, and each vertex can transmit the message to at most one neighbor at a time. Since early works on complexity theory, the problem has been known to be NP-hard. Contributions from the current text to the understanding of the minimum broadcast time problem are threefold. Through considerations of the shortest distances between sources and other vertices, a new lower bound on the broadcast time is derived. Analytical expressions of this bound are given in the single source instances of several graph classes. Fast procedures for computing upper bounds are studied next, including both the construction of feasible solutions, and the improvement of existing ones. Finally, with a focus on a new stable-set interpretation of the problem, integer programming formulations are studied, and for their theoretical interest, associated facet-defining valid inequalities are given. The computational performance of the novel methodology is evaluated in numerical experiments applied to standard benchmark instances and to instances larger than those studied in preceding recent works.

Customized GRASP for rehabilitation therapy scheduling with appointment priorities and accounting for therapist satisfaction

AbstractPhysical therapy in acute care hospitals plays an important role in the rehabilitation of patients. Nevertheless, the profession must deal with staff shortages caused by a lack of potential employees and absenteeism which are results of high physical and mental workloads. The therapist shortage negatively affects the total number of daily appointments the department can fulfill. For appointments that can be successfully scheduled, continuity of care with the same therapist cannot be guaranteed for individual patients. Lack of continuity of care negatively influences the therapist's satisfaction. Therapist preferences for individual appointments in general cannot always be guaranteed when designing schedules, which also hurts satisfaction. This paper develops a multi-criteria model for the daily therapy appointment-scheduling problem. The primary objective is to minimize the total sum of priority violations for unscheduled appointments. To improve therapist satisfaction, we consider therapist preferences including continuity of care as a secondary objective. Here, our integer programming formulation aims to minimize the total sum of preference violations for scheduled appointments. We are dealing with an operational planning problem with a daily planning horizon. The operational objective is to achieve therapist schedules in at most two hours. The therapists’ schedules together need to include several hundred appointments for a planning day. Due to intractability, the developed integer program cannot provide schedules for such problem sizes. Therefore, we develop a customized Greedy Randomized Adaptive Search Procedure (GRASP) with six innovative local search operations to improve an initially constructed solution. We test the heuristic algorithm on realistic data instances. The metaheuristic provides high-quality schedules for various problem sizes in short runtimes, i.e., within minutes. Comparisons with the optimal solutions for small problem instances show very good results of the GRASP with a similar number of scheduled appointments and good adherence to continuity of care and therapist preference requirements.

Mathematical programming formulations for the reclaimer scheduling problem with sequence-dependent setup times and availability constraints

In the context of Industry 4.0, which encompasses advanced technologies and interconnected systems, the integration of optimization techniques assumes a crucial role in addressing resource management challenges across various sectors, including manufacturing systems. In bulk ports, which are an integral part of the manufacturing and logistics chain, one of the most critical resource management challenges is the scheduling of automated or semi-automated reclaimers. These machines are responsible for reclaiming dry bulk materials for loading onto vessels using ship-loaders. The efficient operation of reclaimers directly impacts the overall efficiency of the port. However, the current works for the reclaimer scheduling problem (RSP) overlook the necessity of maintenance activities and assume continuous machine availability. Nonetheless, the inclusion of preventive maintenance activities is critical to ensuring optimal scheduling decisions that minimize production disruptions and downtime. To address these challenges, this paper proposes a novel approach that considers periodic preventive maintenance activities of the reclaimers. We consider two cases to tackle this problem. The first case involves the optimization of material handling within a system of two stockpads and one reclaimer machine. For this case, two novel Mixed Integer Programming (MIP) formulations are developed to drive efficient scheduling decisions and maximize productivity. Furthermore, for the second case representing a more complex system comprising three stockpads and two reclaimers, a novel MIP formulation is devised. The effectiveness of the proposed mathematical formulations is rigorously tested through the use of randomly generated instances based on a real-world case.

Combinatorial settlement planning

In this article, we consider a combinatorial settlement model on a rectangular grid where at least one side (east, south, or west) of each house must be exposed to sunlight without obstructions. We are interested in maximal configurations, where no additional houses can be added. For a fixed $m\times n$ grid, we explicitly calculate the lowest number of houses, and give close to optimal bounds on the highest number of houses that a maximal configuration can have. Additionally, we provide an integer programming formulation of the problem and solve it explicitly for small values of $m$ and $n$.

Modeling and solving the fuel distribution problem with unloading precedence and loading sequence considerations

AbstractThis paper presents a real-world liquid fuel distribution problem involving a heterogeneous fleet of multi-compartment vehicles servicing a set of orders of different fuels. Two new features are introduced that affect significantly the performance of the fuel distribution process in terms of safety and efficiency: (i) loading trucks so that the payload remains balanced throughout each phase of a delivery route, and (ii) sequencing the requests for loading trucks at the depot loading facilities. A Mixed Integer Programming formulation is presented and an Adaptive Large Neighbourhood Search algorithm with various novel features is developed and benchmarked. A new loading model is formed and solved for allocating the ordered items (fuel) to vehicle compartments, as part of constructing/repairing delivery routes. The computational performance of the proposed solution approach has been tested on a series of benchmark problems. Moreover, a series of experiments were performed in order to assess the effect of the balanced loading constraints on the traveled distance. The results indicate that the effect of this type of constraint on the total traveled distance is kept at a reasonable level, reaching a maximum increase of 4.37%. The computational tools presented in this work may accommodate the dispatchers’ work in producing efficient and safe delivery routes while managing efficiently any potential bottleneck in the truck loading facilities.

Correlation Clustering Problem Under Mediation

In the context of community detection, correlation clustering (CC) provides a measure of balance for social networks as well as a tool to explore their structures. However, CC does not encompass features such as the mediation between the clusters, which could be all the more relevant with the recent rise of ideological polarization. In this work, we study correlation clustering under mediation (CCM), a new variant of CC in which a set of mediators is determined. This new signed graph clustering problem is proved to be NP-hard and formulated as an integer programming formulation. An extensive investigation of the mediation set structure leads to the development of two efficient exact enumeration algorithms for CCM. The first one exhaustively enumerates the maximal sets of mediators in order to provide several relevant solutions. The second algorithm implements a pruning mechanism, which drastically reduces the size of the exploration tree in order to return a single optimal solution. Computational experiments are presented on two sets of instances: signed networks representing voting activity in the European Parliament and random signed graphs. History: Accepted by Van Hentenryck, Area Editor for Pascal. Funding: This work was supported by Fondation Mathématique Jacques Hadamard [Grant P-2019-0031]. 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.2022.0129 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0129 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Formulations for the maximum common edge subgraph problem

The Maximum Common Edge Subgraph Problem (MCES), given two graphs G and H, asks about finding a maximum common subgraph between those two graphs with a maximal number of edges. We present new integer programming formulations for MCES and compare the performances of those formulations with earlier formulations from the literature.