Integer Programming Formulation Research Articles

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

Physical 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.

Towards a circular economy with waste‐to‐resource system optimization

AbstractMotivated by mounting pressures to achieve environmental sustainability, and the emergence of online waste exchange platforms, in this work we propose an optimization‐based framework for studying waste‐to‐resource formation in the industry. We first develop a linear programming market clearing model comprising self‐interested agents participating in waste‐to‐resource trading. We then embed this in a bi‐level capacity optimization problem under uncertainties in the agents' reserve prices, the goal of which is to maximize the economic savings achieved by waste‐to‐resource. We show that the resulting robust optimization models admit linear mixed integer programming formulations that can be readily computed using available solvers. Furthermore, we provide the formulas for applying an equitable levy and incentive scheme, to enable capacity cost‐sharing among the agents. Numerical studies based on a case of organic waste streams demonstrate the usefulness of the proposed models in generating valuable insights for decision support in the design of waste‐to‐resource markets.

The cumulative vehicle routing problem with arc time windows

In this paper, we introduce a new variant of the cumulative Vehicle Routing Problem (cum-VRP), where certain arcs of the undirected complete graph can only be utilized within specified time constraints. The problem is referred to as the Cumulative Vehicle Routing Problem with Arc Time Windows (Cum-VRPATW) and emerges in routing situations with flow disruptions across road segments, which generally emerges from military and civilian transportation. We devise a mixed integer programming (MIP) formulation to model the problem and solve it using CPLEX. To effectively address this problem, an effective algorithm is designed. It is a hybrid algorithm that combines principles of the Greedy Randomized Adaptive Search Procedure (GRASP) and decomposition strategies reducing computational complexity. Also, we propose two indexes, the RatioIn and the RatioTWL index that can identify difficult problems. Furthermore, we compare the CPLEX results vis-a-vis the new hybrid heuristic we have developed. Computational tests on converted Solomon (1987) data sets show that the new hybrid heuristic provides extremely good performance results for the cum-VRPATW large size test problems, justifying its effectiveness and robustness.

Valid inequalities for the non-unit demand capacitated minimum spanning tree problem with arc time windows and flow costs

In this paper, we introduce the non-unit demand capacitated minimum spanning tree problem with arc time windows and flow costs. The problem is a variant of the capacitated minimum spanning tree problem with arc time windows (CMSTP_ATW). We devise a mixed integer programming (MIP) formulation to model the problem and solve it using CPLEX. Furthermore, we propose three sets of inequalities, and we prove that they are valid. These valid inequalities tighten the model and lead to better lower bounds. To examine the quality of the solutions obtained, we convert the original data sets of Solomon (1987, “Algorithms for the Vehicle Routing and Scheduling Problem with Time Window Constraints.” Operations Research 35 (2): 254–265. https://doi.org/10.1287/opre.35.2.254) to approximate the non-unit demand CMSTP_ATW instances and provide results for the problems with 100 nodes. We execute extensive computational experiments, and the results show the positive effect of the inclusion of valid inequalities in the MIP.

An exact algorithm for Two-Echelon Location-Routing problem with simultaneous pickup and delivery

This study considers an extension of the two-Echelon Location-Routing Problem (2E-LRP) namely two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD). A Mixed Integer Programming (MIP) formulation is developed and strengthened by the use of valid inequalities to solve the 2E-LRPSPD, and then a Branch and Cut-based exact algorithm (B&C) is presented. To improve the upper bounds, Iterated Local Search and Variable Neighborhood Search-based metaheuristic algorithms, namely, ILS-VNS and VNS, are developed and embedded into the B&C. Computational results, conducted on instances produced from the literature, demonstrate that the B&C yields tight lower and upper bounds, and also is capable of solving some instances optimally with up to 50 customers and 10 satellites in a reasonable time. Furthermore, sensitivity analysis reveals that the heuristic approaches and an increase in the runtime limit assist the algorithm to reach tight the upper bounds, whereas the number of potential depots does not have any effect on the performance of the algorithm. Besides, further computational analysis on the 2E-LRPSPD with a single depot, which is studied in the literature, shows that the B&C improves some best-known solutions while proving the optimality of some instances. Finally, the B&C is applied to a two-echelon supermarket distribution system in Türkiye. The computational results show that simultaneous pickup and delivery operations, as well as depot location decisions are critical elements to consider in the design of cost-effective distribution networks.

BlendMR: A Computational Method to Create Ambient Mixed Reality Interfaces

Mixed Reality (MR) systems display content freely in space, and present nearly arbitrary amounts of information, enabling ubiquitous access to digital information. This approach, however, introduces clutter and distraction if too much virtual content is shown. We present BlendMR, an optimization-based MR system that blends virtual content onto the physical objects in users’ environments to serve as ambient information displays. Our approach takes existing 2D applications and meshes of physical objects as input. It analyses the geometry of the physical objects and identifies regions that are suitable hosts for virtual elements. Using a novel integer programming formulation, our approach then optimally maps selected contents of the 2D applications onto the object, optimizing for factors such as importance and hierarchy of information, viewing angle, and geometric distortion. We evaluate BlendMR by comparing it to a 2D window baseline. Study results show that BlendMR decreases clutter and distraction, and is preferred by users. We demonstrate the applicability of BlendMR in a series of results and usage scenarios.

Shapley–Scarf Housing Markets: Respecting Improvement, Integer Programming, and Kidney Exchange

In a housing market of Shapley and Scarf, each agent is endowed with one indivisible object and has preferences over all objects. An allocation of the objects is in the (strong) core if there exists no (weakly) blocking coalition. We show that, for strict preferences, the unique strong core allocation “respects improvement”—if an agent’s object becomes more desirable for some other agents, then the agent’s allotment in the unique strong core allocation weakly improves. We extend this result to weak preferences for both the strong core (conditional on nonemptiness) and the set of competitive allocations (using probabilistic allocations and stochastic dominance). There are no counterparts of the latter two results in the two-sided matching literature. We provide examples to show how our results break down when there is a bound on the length of exchange cycles. Respecting improvements is an important property for applications of the housing markets model, such as kidney exchange: it incentivizes each patient to bring the best possible set of donors to the market. We conduct computer simulations using markets that resemble the pools of kidney exchange programs. We compare the game-theoretical solutions with current techniques (maximum size and maximum weight allocations) in terms of violations of the respecting improvement property. We find that game-theoretical solutions fare much better at respecting improvements even when exchange cycles are bounded, and they do so at a low efficiency cost. As a stepping stone for our simulations, we provide novel integer programming formulations for computing core, competitive, and strong core allocations. Funding: P. Biró gratefully acknowledges financial support from the Hungarian Scientific Research Fund, OTKA [Grant K143858] and the Hungarian Academy of Sciences [Grant LP2021-2]. F. Klijn gratefully acknowledges financial support from AGAUR–Generalitat de Catalunya [Grants 2017-SGR-1359 and 2021-SGR-00416] and the Spanish Agencia Estatal de Investigación [Grants ECO2017-88130-P and PID2020-114251GB-I00] (funded by MCIN/AEI/10.13039/501100011033) and the Severo Ochoa Programme for Centres of Excellence in R&D (Barcelona School of Economics) [Grant CEX2019-000915-S]. Research visits related to this work were financed by COST Action [Grant CA15210 ENCKEP], supported by COST (European Cooperation in Science and Technology), http://www.cost.eu/ .

Min–max optimization of node‐targeted attacks in service networks

AbstractThis article considers resilience of service networks that are composed of service and control nodes to node‐targeted attacks. Two complementary problems of selecting attacked nodes and placing control nodes reflect the interaction between the network operator and the network attacker. This interaction can be analyzed within the framework of game theory. Considering the limited performance of the previously introduced iterative solution algorithms based on non‐compact problem models, new compact integer programming formulations of the node attack optimization problem are proposed, which are based on the notion of pseudo‐components and on a bilevel model. The efficiency of the new formulations is illustrated by the numerical study that uses two reference networks (medium‐size and large‐size), and a wide range of the sizes of attacks and controllers placements.

Optimum Scheduling of a Multi-Machine Flexible Manufacturing System Considering Job and Tool Transfer Times without Tool Delay

In order to minimize makespan (Cmax) without causing tool delay with the fewest copies of each tool type, this study investigates the concurrent scheduling of automated guided vehicles (AGVs), machines (MCs), tool transporter (TT), and tools in a multi-machine flexible manufacturing system (FMS). The tools are housed in a central tool magazine (CTM), accessible to and utilized by several machines. AGVs and the tool transporter (TT) move jobs and tools between machines. Since it involves allocating tool copies and AGVs to job operations, sequencing job operations on machines, and related trip operations, such as AGVs’ and TT’s empty trip and loaded trip times, this simultaneous scheduling problem is highly complicated. This issue is resolved using the symbiotic organisms search algorithm (SOSA), based on the symbiotic interaction strategies that organisms adapt to survive in the ecosystem. This study proposes a mixed nonlinear integer programming formulation to address this problem. Verification is performed using an industrial problem from a manufacturing organization. The results show that employing two copies for two tool types out of 22 tool kinds and one copy for the remaining tool types results in no tool delay, which causes a reduction in the Cmax as well as cost. The industries that can benefit directly from this study are consumer electronics manufacturers, original equipment manufacturers, automobile manufacturers, and textile machine producers. The results demonstrate that the SOSA provides promising results compared to the flower pollination algorithm (FPA).

Handling the significance of regression coefficients via optimization

Subset selection in multiple linear regression is to identify a subset of explanatory variables that are most relevant to the response variable while discarding the remaining variables. Traditionally, a regression model is obtained by solving a given optimization problem, and then, some statistical tests and evaluations are performed to confirm its validity and to verify the satisfaction of qualifying requirements. Practically, human intervention is necessary at this stage. To prevent the need for manual inspection, we propose a new mixed integer programming formulation for the multiple linear regression subset selection problem. Our model enforces the statistical significance and the multicollinearity through explicit constraints along with several other well-known criteria. In contrast to existing approaches that incorporate these criteria gradually through a cutting-plane structure, this effort considers them explicitly within the formulation. The effectiveness of the approach is evaluated over different real datasets and compared with the existing methods in the literature. The results demonstrate that the significance of coefficients can be provided without any considerable change in the value of mean squared error.