Mixed-integer Programming Techniques Research Articles

Benders decomposition for the discrete ordered median problem

Ordered median optimization has been proven to be a powerful tool to generalize many well-known problems from the literature. In Location Theory, the Discrete Ordered Median Problem (DOMP) is a facility location problem where clients are first ranked according to their allocation cost to the nearest open facility, and then these costs are multiplied by a suitable weight vector λ. That way, DOMP generalizes many well-known discrete location problems including p-median, p-center or centdian. In this article, we also allow negative entries of λ, allowing us to derive models for better addressing equity and fairness in facility location, for modeling obnoxious facility location problems or for including other client preference models. We present new mixed integer programming models for DOMP along with algorithmic enhancements for solving the DOMP to optimality using mixed integer programming techniques. Specifically, starting from state-of-the-art formulations from the literature, we present several Benders decomposition reformulations applied to them. Using these approaches, new state-of-the-art results have been obtained for different ordered weighting vectors.

Large neighborhood local search method with MIP techniques for large-scale machining scheduling with many constraints

AbstractThis study addresses the problem of scheduling machining operations in a highly automated manufacturing environment while considering the work styles of workers. In actual manufacturing, many aspects of the operation must be considered, such as constraints related to the works to be machined in the machining schedule and the states of the workers. To derive good solutions for such a large-scale problem with many constraints within a realistic amount of computation time, we develop an optimization technique based on a mixed-integer programming (MIP)-based large neighborhood local search method for the machining scheduling problem. Then, computer experiments on a problem based on actual machining requirements are performed to verify the validity of the proposed method.

MIP formulations for induced graph optimization problems: a tutorial

AbstractGiven a graph and a subset of its vertices , the subgraph induced by in G is that with vertex set and edge set formed by all the edges in E linking two vertices in . Mixed integer programming (MIP) approaches are among the most successful techniques for solving induced graph optimization problems, that is, those related to obtaining maximum or minimum (weighted or not) induced subgraphs with certain properties. In this tutorial, we provide a literature review of some of these problems. Furthermore, we illustrate the use of MIP formulations and techniques for solving combinatorial optimization problems involving induced graphs. We focus on compact formulations and those with an exponential number of constraints that can be effectively solved using branch‐and‐cut procedures. More specifically, we revisit applications of their use for problems of finding induced forests (which correspond to the complement of feedback vertex sets), trees, paths, as well as quasi‐clique partitionings.

Efficient MIP techniques for computing the relaxation complexity

The relaxation complexity {{,textrm{rc},}}(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning. We employ efficient mixed-integer programming techniques to compute a robust and numerically more practical variant of the relaxation complexity. Our proposed models require row or column generation techniques and can be enhanced by symmetry handling and suitable propagation algorithms. Theoretically, we compare the quality of our models in terms of their LP relaxation values. The performance of those models is investigated on a broad test set and is underlined by their ability to solve challenging instances that could not be solved previously.

Mixed-integer programming techniques for the minimum sum-of-squares clustering problem

The minimum sum-of-squares clustering problem is a very important problem in data mining and machine learning with very many applications in, e.g., medicine or social sciences. However, it is known to be NP-hard in all relevant cases and to be notoriously hard to be solved to global optimality in practice. In this paper, we develop and test different tailored mixed-integer programming techniques to improve the performance of state-of-the-art MINLP solvers when applied to the problem—among them are cutting planes, propagation techniques, branching rules, or primal heuristics. Our extensive numerical study shows that our techniques significantly improve the performance of the open-source MINLP solver SCIP. Consequently, using our novel techniques, we can solve many instances that are not solvable with SCIP without our techniques and we obtain much smaller gaps for those instances that can still not be solved to global optimality.

A Refined Siphon-Based Deadlock Prevention Policy for a Class of Petri Nets

Resource allocation systems (RASs) exist in various fields of modern society. The deadlock control problem is a crucial issue in control theory of RAS. This work is concentrated on a special class of shared resource and process-oriented Petri nets whose initial marking can have only a token in every resource place. Using mixed-integer programming (MIP) and iterative siphon control, we present a two-stage deadlock prevention policy. In particular, a modified MIP technique is developed for the first stage to compute a specific type of emptiable siphons and a siphon control method introducing monitors with related arcs whose weights all equal to one is established in the second stage. This policy leads to a maximally permissive liveness-enforcing supervisor and such an obtained controlled net is ordinary. Moreover, it avoids the exhaustive enumeration of siphons and the reachability analysis. Examples are provided to explain the policy.

Assessing flexibility options in electricity market clearing

This work presents a model to co-optimize the energy and reserves markets, taking into account the penetration and participation of various flexibility providers in both markets. In particular, a detailed unit commitment model has been developed based on mixed-integer programming techniques incorporating energy storage systems with both charging and discharging options, electric vehicles with both grid-to-vehicle and vehicle-to-grid modes, and demand response programs for cost-optimal energy and ancillary services scheduling. The balancing services considered include Frequency Containment Reserves (FCR), automatic Frequency Restoration Reserves (aFRR), and manual Frequency Restoration Reserves (mFRR), in both upward and downward directions. The impact of all these flexibility providers on operational and economic aspects has been assessed through an illustrative case study of a power system with high penetration of renewable energy sources, including thermal and hydroelectric power units. The results highlight the superiority of results when considering the participation of all flexibility providers, especially in the ancillary services market, in terms of economic competitiveness, renewable energy curtailment, associated CO2 emissions, and utilization of costly energy resources. The growing share of flexibility providers in both energy management and reserve provision mix highlights the importance of these sources for power mixes with low carbon content. The methodological framework developed can be employed by system operators, market participants, and policymakers to provide price signals and optimize their resources and portfolios.

Robust Optimization-Based Optimal Operation of Islanded Microgrid Considering Demand Response

This paper presents a new robust scheduling model for an islanded microgrid (MG) considering demand response. The model is expressed as a min–max bilevel optimization problem that tries to minimize the total costs of MG including operation cost of conventional distributed generators, energy storages, renewable energy sources (RES), cost of load shifting, and interruptible/non-interruptible load shedding in the worst situation of uncertainties. The uncertainties associated with renewable power generations and MG demand are modeled via robust optimization method. A hybrid method based on the genetic algorithm (GA) and mixed-integer programming technique is utilized to solve the bilevel optimization problem. The proposed model is utilized on a typical MG, and the outcomes are analyzed to show the effectiveness of the proposed method.

Systematic Design of Solvent Recovery Pathways: Integrating Economics and Environmental Metrics

The increasing trend of solvent usage by chemical industries has caused an upsurge in hazardous waste solvents and, consequently, increased their toxicity potential within the environment. The increase in overall carbon and ecological footprints due to conventional waste handling techniques such as incineration and onsite and offsite disposal can hardly be overstated. Even though solvent recovery methods present a better alternative to these techniques, analysis is generally centered on economics. This paper presents a framework that evaluates the economic and sustainability metrics of solvent recovery processes. In the first approach, standalone software (SimaPro) was used to evaluate the impact of the recovery process. The framework was extended by incorporating the economics, Sustainable Process Index (SPI), and Emergy metrics. Thus, solvent recovery challenges have been transformed into a multiobjective optimization problem that is solved through superstructure and mixed-integer nonlinear programming techniques. The applicability of the framework is illustrated by two comprehensive case studies with varying complexities, and the results are compared with conventional methods. The results indicate that about 76–85% of the ecological and energy burden can be prevented if solvent recovery is the preferred option to incineration.

Integrated Optimization for Biofuel Management Associated with a Biomass-Penetrated Heating System under Multiple and Compound Uncertainties

The biofuel management of a biofuel-penetrated district heating system is complicated due to its association with multiple and polymorphic uncertainties. To handle uncertainties and system dynamic complexities, an inexact two-stage compound-stochastic mixed-integer programming technique is proposed, innovatively based on the integration of different uncertain optimization approaches. The proposed technique can not only address the inexact recourse problems sourced from multiple and compound uncertainties existing in the pre-regulated biofuel supply–demand match mode, but can also quantitatively analyze the conflicts between the economic target that minimizes the system cost and the risk preference that maximizes the heating service satisfaction. The developed model is applied to a real-world biofuel management case study of a district heating system to obtain the optimal biofuel management schemes subject to supply–demand, policy requirement constraints, and the financial minimization objective. The results indicate that biofuel allocation and expansion schemes are sensitive to the multiple and compound uncertainty inputs, and the corresponding biofuel-deficit change trends of three heat sources are obviously distinct with the system’s condition, varying due to the complicated interactions of the system’s components. Beyond that, a potential trade-off relationship between the heating cost and the constraint-violation risk can be obtained by observing system responses with thermalization coefficient varying.

Search for a unique Nash equilibrium in two public goods games: mixed integer programming technique

ABSTRACT We provide the methods for searching for the unique Nash equilibrium using mixed integer programming techniques. We also simulate the model parameters using this technique and derive the numerical solution to show the characteristics of a key player providing both public goods as suggested by Bergstrom et al. (1986). We present two key results. First, when the number of players decreases, the appearance rate of the unique equilibrium accompanying a key player increases. Second, when the variance of the preference or price parameters decreases, the appearance rate of the unique equilibrium accompanying a key player increases.

Increased renewable energy penetration in national electrical grids constraints and solutions

This manuscript presents the impact of high penetration of renewables into the energy mix of the national electrical grids, Jordan's as a case study. The generation-consumption balance is main interest of the network operator and hence, taken as key aspect of the study. The shortcomings of installing renewables include the intermittency and unpredictability. To improve the load sharing strategy between these renewable energy resources (RESs) and conventional energy generation plants, energy storage is suggested. Sizing and design of storage elements is considered. Different scenarios are realistically assumed to integrate energy storage into the network operation and economic aspects of the entire dispatch problem are analyzed. The analysis of and simulation of the unit commitment problem is formulated and carried out using PLEXOS software. Mixed integer programming technique is adapted to solve for the optimum solution of each scenario. Results show that high penetration of RESs into the national electrical grid would considerably change the optimum operation of the network and different dispatch scenarios are needed to cope with that. It will be shown too that energy storage is highly recommended for increased penetration levels to meet the load demand economically. Detailed problem statement and scenarios simulation is presented and validated.

Mixed-integer quadratic programming reformulations of multi-task learning models

<abstract><p>In this manuscript, we consider well-known multi-task learning (MTL) models from the literature for linear regression problems, such as clustered MTL or weakly constrained MTL. We propose novel reformulations of the training problem for these models, based on mixed-integer quadratic programming (MIQP) techniques. We show that our approach allows to drive the optimization process up to certified global optimality, exploiting popular off-the-shelf software solvers. By computational experiments on both synthetic and real-world datasets, we show that this strategy generally leads to improvements in terms of the predictive performance of the models, if compared to the classical local optimization techniques, based on alternating minimization strategies, that are usually employed. We also suggest a number of possible extensions of our model that should further improve the quality of the obtained regressors, introducing, for example, sparsity and features selection elements.</p></abstract>

A sustainable smart multi-type biofuel manufacturing with the optimum energy utilization under flexible production

Biofuel manufacturing from renewable biomass through a smart manufacturing system is an important alternative to fossil fuels, which helps to decrease dependability on conventional fuel and decrease carbon emanations. By utilizing efficient labors, smart machines, and minimized energy utilization, the conventional biofuel manufacturing framework can be converted into a smart sustainable manufacturing framework. The existing conventional fuel demand can be replaced by biofuel. This study efforts to make pure biofuel with less amount of carbon emanations and energy utilization through a smart multi-type biofuel manufacturing framework, where the demand of the biofuel is selling price dependent. The different energy costs including air handling cost, lighting cost are calculated in this work-in-process inventory. The carbon emissions cost is included in every stage of this model. A variable demand has introduced for the maximization of profit. To reduce energy consumption and carbon emissions, a two-stage inspection cost with a variable manufacturing rate is taken to make the manufacturing process flexible such that the amount of impure biofuel is minimized. Although a random manufacturing rate is applied through a smart manufacturing system, still impure biofuel is manufactured. The impure biofuel is remanufactured again through refining just after the well-planned manufacturing ends. In this model, the multi-delivery technique is used such that a fixed amount of biofuel of n segments of the pure biofuel is transported to market places at predetermined intervals at the time of delivery frame. The classical optimization technique is used for continuous and differentiable variables and the mixed integer programming technique is utilized for discrete variables to maximize the total profit globally. To validate the model’s usefulness, four numerical observations are studied. It is shown that by utilizing the said procedure, the percentage of impure biofuel can be decreased truly through the minimized energy consumption. The graphical representation and the sensitivity experiment show the impact of every parameter with the whole cost of the research paper. Numerical results assist to get the maximum profit and the optimum selling price of biofuel globally in a sustainable smart multi-type biofuel manufacturing system.

Development of transportation cost functions and optimization of transportation networks for solar-aided utilization of CO2

The present work aims to establish a methodology for the optimization of CO2 transportation networks by developing transportation levelized-cost functions for the mixture CO2:N2 (96:4%) within the ranges {25–2000 kg/s} and {50–2000 km}, locating the optimal solar cities (locations-sinks) to receive CO2 for further transformation into fine chemicals, and mapping the optimal routes which present the minimum transportation and utilization cost. Potential economies of scale of merging same-route sinks into a joint pipeline are scrutinized by selecting 17 CO2 sources from thermal power and combustion stations in the central EU region aggregating almost 3000 kg/s of CO2; these amounts are optimally allocated to 16 available sinks, which combine maximum solar capacity with the least distance to major ports. This study makes use of transport cost correlations, while the solar potential of the most promising solar cities is calculated in detail based on solar intensity databases and solar power plants cost assessment. Mixed-integer nonlinear programming (MINLP) techniques are incorporated in the optimization procedure under sinks capacity constraints. A major conclusion of this work is that for the mass flowrates considered and the sources locations chosen in the current study it is economically beneficial to transfer CO2 directly to the final sinks from the original sources rather than via intermediate sources. The MINLP formulation derives values of solar utilization cost of over €7B and transportation cost of over €2B per year covering transportation pipelines distances of 34,000 km: precise economic estimates of the proposed methodology regarding CO2 transportation and utilization might have significant implications on energy and environmental policy issues on European or global scale.

Robust Variable Selection with Optimality Guarantees for High-Dimensional Logistic Regression

High-dimensional classification studies have become widespread across various domains. The large dimensionality, coupled with the possible presence of data contamination, motivates the use of robust, sparse estimation methods to improve model interpretability and ensure the majority of observations agree with the underlying parametric model. In this study, we propose a robust and sparse estimator for logistic regression models, which simultaneously tackles the presence of outliers and/or irrelevant features. Specifically, we propose the use of L0-constraints and mixed-integer conic programming techniques to solve the underlying double combinatorial problem in a framework that allows one to pursue optimality guarantees. We use our proposal to investigate the main drivers of honey bee (Apis mellifera) loss through the annual winter loss survey data collected by the Pennsylvania State Beekeepers Association. Previous studies mainly focused on predictive performance, however our approach produces a more interpretable classification model and provides evidence for several outlying observations within the survey data. We compare our proposal with existing heuristic methods and non-robust procedures, demonstrating its effectiveness. In addition to the application to honey bee loss, we present a simulation study where our proposal outperforms other methods across most performance measures and settings.

An effective procedure for feature subset selection in logistic regression based on information criteria

In this paper, the problem of best subset selection in logistic regression is addressed. In particular, we take into account formulations of the problem resulting from the adoption of information criteria, such as AIC or BIC, as goodness-of-fit measures. There exist various methods to tackle this problem. Heuristic methods are computationally cheap, but are usually only able to find low quality solutions. Methods based on local optimization suffer from similar limitations as heuristic ones. On the other hand, methods based on mixed integer reformulations of the problem are much more effective, at the cost of higher computational requirements, that become unsustainable when the problem size grows. We thus propose a new approach, which combines mixed-integer programming and decomposition techniques in order to overcome the aforementioned scalability issues. We provide a theoretical characterization of the proposed algorithm properties. The results of a vast numerical experiment, performed on widely available datasets, show that the proposed method achieves the goal of outperforming state-of-the-art techniques.

Solving the nuclear dismantling project scheduling problem by combining mixed-integer and constraint programming techniques and metaheuristics

Scheduling of megaprojects is very challenging because of typical characteristics, such as expected long project durations, many activities with multiple modes, scarce resources, and investment decisions. Furthermore, each megaproject has additional specific characteristics to be considered. Since the number of nuclear dismantling projects is expected to increase considerably worldwide in the coming decades, we use this type of megaproject as an application case in this paper. Therefore, we consider the specific characteristics of constrained renewable and non-renewable resources, multiple modes, precedence relations with and without no-wait condition, and a cost minimisation objective. To reliably plan at minimum costs considering all relevant characteristics, scheduling methods can be applied. But the extensive literature review conducted did not reveal a scheduling method considering the special characteristics of nuclear dismantling projects. Consequently, we introduce a novel scheduling problem referred to as the nuclear dismantling project scheduling problem. Furthermore, we developed and implemented an effective metaheuristic to obtain feasible schedules for projects with about 300 activities. We tested our approach with real-life data of three different nuclear dismantling projects in Germany. On average, it took less than a second to find an initial feasible solution for our samples. This solution could be further improved using metaheuristic procedures and exact optimisation techniques such as mixed-integer programming and constraint programming. The computational study shows that utilising exact optimisation techniques is beneficial compared to standard metaheuristics. The main result is the development of an initial solution finding procedure and an adaptive large neighbourhood search with iterative destroy and recreate operations that is competitive with state-of-the-art methods of related problems. The described problem and findings can be transferred to other megaprojects.

A Prosumer Model Based on Smart Home Energy Management and Forecasting Techniques

This work presents an optimization framework based on mixed-integer programming techniques for a smart home’s optimal energy management. In particular, through a cost-minimization objective function, the developed approach determines the optimal day-ahead energy scheduling of all load types that can be either inelastic or can take part in demand response programs and the charging/discharging programs of an electric vehicle and energy storage. The underlying energy system can also interact with the power grid, exchanging electricity through sales and purchases. The smart home’s energy system also incorporates renewable energy sources in the form of wind and solar power, which generate electrical energy that can be either directly consumed for the home’s requirements, directed to the batteries for charging needs (storage, electric vehicles), or sold back to the power grid for acquiring revenues. Three short-term forecasting processes are implemented for real-time prices, photovoltaics, and wind generation. The forecasting model is built on the hybrid combination of the K-medoids algorithm and Elman neural network. K-medoids performs clustering of the training set and is used for input selection. The forecasting is held via the neural network. The results indicate that different renewables’ availability highly influences the optimal demand allocation, renewables-based energy allocation, and the charging–discharging cycle of the energy storage and electric vehicle.

Learn to relax: Integrating 0-1 integer linear programming with pseudo-Boolean conflict-driven search

Conflict-driven pseudo-Boolean solvers optimize 0-1 integer linear programs by extending the conflict-driven clause learning (CDCL) paradigm from SAT solving. Though pseudo-Boolean solvers have the potential to be exponentially more efficient than CDCL solvers in theory, in practice they can sometimes get hopelessly stuck even when the linear programming (LP) relaxation is infeasible over the reals. Inspired by mixed integer programming (MIP), we address this problem by interleaving incremental LP solving with cut generation within the conflict-driven pseudo-Boolean search. This hybrid approach, which for the first time combines MIP techniques with full-blown conflict analysis operating directly on linear inequalities using the cutting planes method, significantly improves performance on a wide range of benchmarks, approaching a “best-of-both-worlds” scenario between SAT-style conflict-driven search and MIP-style branch-and-cut.