Mathematical Programming Research Articles

Formation of a heterogeneous group of UAVS with a reasonable number of false and real drones

The subject of scientific research is the joint use of false and real unmanned aerial vehicles (UAVs) as part of a heterogeneous group of UAVs to perform military tasks. This article aimsto determine the appropriate number of false and real drones (UAVs) as part of a heterogeneous group of UAVs to ensure that a certain number of real UAVs can fly to targets with the aim of further reliable target destruction. The scientific task is to develop a methodology for determining the appropriate number of false drones in a heterogeneous group of UAVs, considering the diversity of UAVs included in the UAV group. To achieve the goals of scientificresearch, partial scientific tasks were solved. The joint use of false drones as part of a UAV group to defeat targets with a given degree of damage was formalized. The formalization was carried out taking into account two possible cases of use: a) when the enemy has a sufficient number of means to destroy the entire group of UAVs; b) when the enemy has an insufficient number of means to destroy the entire UAV group. A mathematical model for determining the optimal composition of false and real drones (UAVs) as parts of a heterogeneous group of UAVs has been developed, which will allow to fulfill the task of defeating enemy targets with the desired degree of reliability. A program code has been developed that simplifies the mathematical calculations in the presented mathematical model and allows it to be used in the process of making an appropriate military decision. An algorithm to find the numbers of real and false UAVs in a heterogeneous group of UAVs is proposed. The obtained formulas and algorithms were verified by computer simulation using the Monte Carlo method. Methods. The mathematical model is based on combinatorial methods of probability theory. Programming for calculating analytical formulas and computer modeling of the Monte Carlo method was carried out based on the R computer language. The following results were obtained. A multifunctional algorithm is presented: on one hand, its application makes it possible to determine the optimal number of false UAVs in a heterogeneous group of UAVs to ensure that the required number of real UAVs reach the target, and on the other hand, to determine the predicted loss level of real UAVs in a heterogeneous group of UAVs when using a certain number of false drones.Conclusions. The availability of the developed mathematical model, algorithm, and program code makes it possible to predict the possible results of the combat use of heterogeneous groups of UAVs based on the initial parameters and to substantiate recommendations for a possible composition of such groups.

Industry 4.0 Ready: Transforming STEM Laboratory Courses – Continuous Distillation as Example

AbstractLaboratory courses are key to science, technology, engineering, and mathematics (STEM) programs in higher education, but often lack the flexibility for distance learning and inclusiveness for all students. To address these challenges, TU Bergakademie Freiberg introduced remote laboratories not only for international degree programs and the “Disti4Study” application, which provides interactive visualization of industrial thermal separation processes. This paper focuses on the development and didactic challenges associated with the “Continuous Distillation Laboratory”. It illustrates how rethinking the technical and didactic frameworks of traditional laboratories can align with new learning objectives, such as those related to Industry 4.0, including the Internet of Things and data management.

The classified mathematical papers of A. A. Albert: a glimpse into the application of mathematics to cryptologic problems during the 1950s and 1960s

Abraham Adrian Albert “A cubed” (1905–1972) was an algebraist best known academically for his study of associative and non-associative algebras. By the 1940s he had also developed an interest in cryptology and its relationship to mathematics. Throughout the following two decades, Albert would participate in several mathematics programs to study and advise on cryptanalysis. In particular, Albert’s work with the National Security Agency (NSA) set a framework for the development of cryptology as a mathematical discipline. This research aims to present a timeline for and the breadth of Albert’s promotion of cryptology in the post-WWII American mathematical community and to examine, for the first time, reports Albert authored under contract with the NSA. In doing so, we hope to better understand the relationship between the NSA and academic mathematicians at the time of writing. The reports, released in 2020 by the NSA in response to a 2010 Freedom of Information Act request, have been arranged by the author into eleven collections based on content. Each collection will be introduced with their ideas and results given in a roughly chronological order. These collections span ideas in group theory, field theory, matrix algebra, and geometry, most often with application to cryptological systems.

Solving Mathematical Programs with Complementarity Constraints Arising in Nonsmooth Optimal Control

AbstractThis paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call . The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes’ relaxation (SIAM J. Optim. 11, 918–936, 2001) with (Math. Program. 106, 25–57, 2006) as NLP solver manages to solve 73.8% of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.

Applying a 2D Variational Rigid Block Modeling Method to Rubble Stone Masonry Walls Considering Uncertainty in Material Properties

ABSTRACT The computational cost of detailed micro-modeling of masonry structures can be minimized with rigid block analysis methods, where a variational formulation is particularly interesting as it allows for nonlinear pushover analysis. This work investigates the accuracy of this modeling method in predicting the response of rubble stone masonry walls in shear compression tests. To this end, we extend the mathematical programming problem by including cohesion, compressive strength, and tensile strength in the formulation and implement an improved pushover procedure, which guards compatibility of the displaced shape and crack patterns at the intersection of elastic and rigid contact models. Additionally, we reduce model uncertainty, introduced through sampling material properties with distributions estimated based on results from tests on mortar masonry joints and on mortar specimens, by correlating the predicted crack patterns with the observed ones. The findings show that (i) model falsification based on crack pattern is an effective method for reducing the parameter uncertainty of stone masonry walls; and that (ii) the improved pushover procedure predicts the pre- and post-peak response of stone masonry walls very well with analysis times of less than 20 minutes for 2D models where stones and mortar are modeled explicitly.

Advancements in Multi-Objective Mathematical Programming Models and Solution Approaches for Sustainable Waste Management Practices in Vehicle Routing Problem: A Systematic Literature Review

In recent years, the integration of sustainable principles into the Vehicle Routing Problem (VRP) for efficient waste management has gained significant attention. This paper provides an overview of advancements in multi-objective mathematical programming (MP) models and the optimization techniques for sustainable waste management practices within the VRP framework. Sustainable waste management practices in the context of the VRP involve optimizing routes and schedules aim to reduce the environmental impact of waste collection. It investigates the integration of multi-objectives with sustainability criteria, including social, economic, and environmental considerations. Furthermore, it surveys various optimization methodologies applied in waste management to enhance efficiency and sustainability practices. The Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement was employed as a formal systematic review guideline for data gathering. Data was obtained from studies conducted between 2017 and 2023. The analysis included a total of 18 papers from online sources and databases. Based on the literature review, it has been observed that the majority of the research focused on employing Mixed-Integer Linear Programming (MILP) models to tackle the optimization problem for waste management. The literature survey demonstrated significant contributions in computational performance, cost-effectiveness, operational efficiency, and utilization of real-world data by improving the solution approaches, aiming to prioritize economic, social, and environmentally friendly solutions. The paper aims to serve as a convenient reference for researchers and practitioners seeking information on using the MP model and solution approaches to optimize VRP for waste management.

Enhancing Distribution Networks with Optimal BESS Sitting and Operation: A Weekly Horizon Optimization Approach

The optimal sitting and operation of Battery Energy Storage Systems (BESS) plays a key role in energy transition and sustainability. This paper presents an optimization framework based on a Multi-period Optimal Power Flow (MOPF) for the optimal sitting and operation of BESS alongside PV in active distribution grids. The model was implemented in AMPL (A Mathematical Programming Language) and solved using the Knitro solver to minimize power losses over one week, divided into hourly intervals. To demonstrate the applicability of the proposed model, various analyses were conducted on a benchmark 33-bus distribution network considering 1, 2 and 3 BESS. Along with the reduction in power losses of up to 17.95%, 26% and 29%, respectively. In all cases, there was an improvement in the voltage profile and a more uniform generation curve at the substation. An additional study showed that operating over a one-week horizon results in an energy gain of 1.08 MWh per day compared to single daily operations. The findings suggest that the proposed model for optimal sitting and operation of BESS in the presence of Renewable Energy Sources (RES) applies to real-world scenarios.

The Value of Drilling—A Chance-Constrained Optimization Approach

Managing uncertainty is a core challenge in mine planning. Mine planners often represent various planning variables, such as equipment performance and geological parameters, as random variables due to inherent uncertainties. This paper looks at geological uncertainty and its impact on mine planning. Some traditional approaches to manage this uncertainty include using conditional simulations or mathematical programming in the planning process. Drilling additional holes, despite its cost, is a common method to reduce uncertainty using additional samples to reduce deposit variance. In this paper, we first outline an ore blending optimization model which uses chance-constrained programming to manage property limit risk when selecting the order of ore feed into a processing facility. In coal mining, in tactical planning horizons, the order of coal seam removal is usually predetermined, allowing a blending model to ensure optimal feed properties. Using chance-constrained programming allows us to blend the uncertainties from geological models to maximize plant output while adhering to property constraints. We use the chance-constrained blending model to determine the value of additional information from infill drilling. The model prioritizes drilling locations that reduce uncertainty and improve blending outcomes. A case study on a coking coal mine in Queensland, Australia, demonstrates the model’s application, highlighting significant improvements in blending by reducing the variance of high-quality blocks. The study concludes that targeting high-quality blocks for variance reduction can better accommodate lower-quality material, offering a more valuable approach than the traditional focus of reducing uncertainty in low-quality blocks. This approach provides insights for improving mine planning strategies and showcases the potential of chance constraints in optimizing ore blending under uncertainty.

Solving a bilevel strategic offering model for a generation company in a hydrothermal energy market: A new convexification approach

This paper proposes a bilevel strategic offering (SO) model suitable for a strategic company (SC) that operates in a hydro-dominated day-ahead energy market. The upper-level model of the proposed SO model maximizes the SC’s profits, while the lower-level model embodies a hydrothermal market clearing (MC). This MC incorporates detailed hydro and thermal constraints for units of both the SC and non-strategic companies (NSC), in addition to the network constraints. The proposed solution approach entails replacing the lower-level model with its corresponding primal and dual constraints, along with a strong duality equality constraint. This leads to a non-convex mathematical programming with equilibrium constraints (MPEC) model, which is then reformulated to feature only a single bilinear term in its structure. Finally, the recast MPEC model is convexified around an initial point and solved iteratively until a convergence criteria is achieved. The model and solution approach proposed are evaluated on an adapted version of the IEEE 24-bus system, exploring scenarios where the SC acts strategically or non-strategically (perfect competition).

Augmented multi-agent algorithm utilizing intelligent search range detection heuristic to solve assembly line sequencing problem: A case study in the truck industry

Car sequencing is a widely adopted approach to implicitly enhance the short-term balancing of mixed-model assembly lines in practical applications. In this research, we investigate this problem in the context of a real-world case taken from a truck final assembly line. Our study considers cross-ratio constraints that address limitations related to dependent features of the products. Additionally, to manage the distribution of violation errors in cases where predefined congestion level thresholds (soft rules) exist, we propose a piecewise linear penalty function. To tackle these considerations, we formulate a mathematical programming model aimed at minimizing the total weighted penalty incurred from violating both independent and dependent sequencing rules. As the problem falls into the NP-hard category, we develop a multi-start parallel local search algorithm based on multi-agent approach to efficiently solve it. The algorithm is augmented by incorporating a self-adaptive diversification mechanism and a novel intelligent search range detection heuristic. The proposed heuristic enables the searching agents to intelligently explore the solution space rather than relying on a purely random search. The numerical experiments are conducted using 30 real-world instances from three truck assembly lines to validate the quality of the proposed algorithm. The comparison of the results indicates that our algorithm outperforms state-of-the-art algorithms. Furthermore, computational experiments demonstrate the effectiveness of each proposed mechanism in enhancing the algorithm's performance.

Breaking the bias: perspectives of a WiSTEM2D programme inspiring female undergraduates for a technical future

PurposeThis is an account of student experiences with the 2020/2021 Johnson & Johnson (J&J) Women in Science, Technology, Engineering, Mathematics, Manufacturing and Design (WiSTEM2D) intervention programme at the University of Limerick (UL).Design/methodology/approachThe WiSTEM2D programme supports female undergraduate students undertaking science, technology, engineering, mathematics, manufacturing and design (STEM2D) courses, helping them develop and grow in their degrees, research and beyond.FindingsDuring the programme, data were gathered from students in the form of qualitative information, which was used to analyse perceptions of the programme. This analysis revealed that students found the WiSTEM2D programme beneficial to them and that it improved their confidence levels, career knowledge and overall networking abilities.Originality/valueWiSTEM2D is one of several initiatives globally that are attempting to empower and encourage women to be active in STEM.

A Mathematical Programming Model for Minimizing Energy Consumption on a Selective Laser Melting Machine

The scheduling problem in additive manufacturing is receiving increasing attention; however, few have considered the effect of scheduling decisions on machine energy consumption. This research focuses on the nesting and scheduling problem of a single selective laser melting (SLM) machine to reduce total energy consumption. Based on an energy consumption model, a nesting and scheduling problem is formulated, and a mixed integer linear programming model is proposed. This model simultaneously determines part-to-batch assignments, part placement in the batch, and the choice of build orientation to reduce the total energy consumption of the SLM machine. The energy-saving potential of the model is validated through numerical experiments. Additionally, the effect of the number of alternative build orientations on energy consumption is explored.

Towards efficient solutions for vehicle routing problems for oxygen supply chains

This work investigates the integrated production–inventory-routing problem (PIRP) for liquid oxygen supply chains consisting of multiple plants, customers, and heterogeneous vehicles. Solving such a problem is challenging, especially when dealing with large-scale industrial scales. A two-level procedure is adopted that determines decisions regarding production, inventory, and product allocation by simplifying the routing component in the first level. Then, the routing decisions are considered in the lower level for which 3-index mathematical programming formulations are presented. To address the combinatorial complexity of the lower-level decisions, we propose alternative multi-trip heterogeneous vehicle routing problem (MTHVRP) formulations together with a column generation-based solution strategy. A set of test instances and a real-world case study demonstrate the applicability and performance of the formulations and solution methods.

System-of-Systems Resilience Analysis and Design Using Bayesian and Dynamic Bayesian Networks

A System-of-Systems (SoS) is characterized both by independence and by inter-dependency. This inter-dependency, while allowing an SoS to achieve its objectives, also means that failures can cascade throughout the SoS. An SoS needs to be resilient to deal with the impact of complex internal and external environments. In this paper, we propose a resilience analysis method of an SoS based on a hierarchy structure. Firstly, we establish a hierarchy structure, which is ranked from high to low as capability level, activity level and system level. Then, Bayesian Networks (BNs) and Dynamic Bayesian Networks (DBNs) are used to analyze the resilience of the SoS. A resilience-based system importance metric is introduced, which is used in the budget allocation optimization problem during the development domain of an SoS. This paper proposes a mathematical programming model aimed at optimizing SoS resilience by optimally using budget to the subsystem. The application of the proposed approach is demonstrated using a case study: a Next Generation Air Transportation setting. The study results provide evidence that the proposed inter-dependency analysis based on Bayesian theory and the SoS resilience design approach can assist SoS system engineers in increasing expected SoS resilience during the development domain.

A Bi-Objective Model for the Multi-Period Inventory-Based Reverse Logistics Network: A Case Study from an Automobile Component Distribution Network

Supply chain management and distribution network design has attracted the attention of many researchers in recent years. The timely satisfaction of customer demands leads to reducing costs, improving service levels, and increasing customer satisfaction. For this purpose, in this research, the mathematical programming models for a two-level distribution network including central warehouses, regional warehouses, and customers are designed so that several products with definite demands in multiple periods are distributed from central warehouses to customers. In this problem, two objective functions are considered. The first objective function seeks to minimize the costs of establishment, transportation, inventory, and shortage, and the second objective function attempts to maximize the satisfaction level corresponding with the supply rate of different goods for numerous customers. The presented models include the basic model, inventory-based model, multi-period inventory-based model, and multi-period inventory-based reverse logistics model. The validation and applicability of the proposed models were demonstrated by implementation in a real case study of the automobile industry. The LINGO software 20.0 was used to solve the models. The results show that incorporating the inventory management policies into the basic model and converting from a single-period to a multi-period reverse logistics model will significantly increase company profitability and customer satisfaction.

Urban Air Mobility for Last-Mile Transportation: A Review

Urban air mobility (UAM) is a revolutionary approach to transportation in densely populated cities. UAM involves using small, highly automated aircraft to transport passengers and goods at lower altitudes within urban and suburban areas, aiming to transform how people and parcels move within these environments. On average, UAM can reduce travel times by 30% to 40% for point-to-point journeys, with even greater reductions of 40% to 50% in major cities in the United States and China, compared to land transport. UAM includes advanced airborne transportation options like electric vertical takeoff and landing (eVTOL) aircraft and unmanned aerial vehicles (UAVs or drones). These technologies offer the potential to ease traffic congestion, decrease greenhouse gas emissions, and substantially cut travel times in urban areas. Studying the applications of eVTOLs and UAVs in parcel delivery and passenger transportation poses intricate challenges when examined through the lens of operations research (OR). By OR approaches, we mean mathematical programming, models, and solution methods addressing eVTOL- and UAV-aided parcel/people transportation problems. Despite the academic and practical importance, there is no review paper on eVTOL- and UAV-based optimization problems in the UAM sector. The present paper, applying a systematic literature review, develops a classification scheme for these problems, dividing them into routing and scheduling of eVTOLs and UAVs, infrastructure planning, safety and security, and the trade-off between efficiency and sustainability. The OR methodologies and the characteristics of the solution methods proposed for each problem are discussed. Finally, the study gaps and future research directions are presented alongside the concluding remarks.

A robust optimisation approach for the placement of forest fire suppression resources

AbstractThis research develops an initial attack plan for combating forest fires in any wildland areas susceptible to fire outbreaks. To be eligible for such a plan, the landscape must have been previously mapped and modelled concerning spatial and topographic data and fuel levels. Thus, when ignition occurs, one can predict the expected fire behaviour in terms of spread direction and rate of spread. With such information, decisions can be taken on where and when to position the suppression resources. This paper extends a recent contribution to this subject, generalising each node's resource requirement, allowing a more precise modelling of non‐homogeneous landscapes. Moreover, we treat the cases where the estimated number of resources may not be sufficient to deal with the fire intensity, which becomes revealed only at the fire scene. In such cases, additional resources may be needed to contain the fire effectively. This worst‐case approach is modelled with the support of the robust optimisation paradigm. We propose a deterministic mathematical programming model, a robust optimisation counterpart, and a robust tabu search (RoTS) algorithm. We adapt instances from the literature, which are optimally solved by a commercial solver and used for assessing the quality of the RoTS. The proposed algorithm could optimally solve 94 of 96 instances. Finally, we conducted a Monte Carlo simulation as part of a risk analysis assessment of the generated solutions.

MPCC: strong stability of weakly nondegenerate S-stationary points

ABSTRACT In this paper, we consider the class of mathematical programmes with complementarity constraints (MPCC). Specifically, we focus on strong stability of M- and S-stationary points for MPCC. Kojima introduced this concept for standard nonlinear optimization problems. It refers to several well-posedness properties of the underlying problem. Besides its topological definition, the challenge is to state an algebraic characterization of strong stability. We obtain such a description for S-stationary points whose components of Lagrange vectors corresponding to bi-active constraints do not mutually vanish. We call these points weakly nondegenerate. Moreover, we show that a particular constraint qualification is necessary for strong stability.

Модификация модели прогнозирования финансово-экономического состояния для задачи управления развитием медицинских учреждений

The article analyzes the model of forecasting the financial and economic condition, considers the possibility of its transformation for managing the development of medical institutions, and provides an appropriate modification using methods of discriminant analysis and mathematical programming. The formulated modification of the model has a clear mathematical justification, which allows, on the one hand, to prove its validity, and on the other hand constructively shows the possibility of its use, taking into account the specifics of the medical industry. The purpose of the work. The purpose of this work is to study changes in the forecasting model of the financial and economic condition, taking into account the specifics of the industry, as well as to evaluate the application of methods of discriminant analysis, reducing them to a mathematical programming problem, the limitations of which are reduced to systems of linear inequalities obtained by building maximally joint subsystems. Materials and methods. It is proposed to use the method of discriminant analysis with extensible precedent sets. It is assumed that in order to improve the quality of solving this problem, expert judgments about objects that were not initially included in the training sample are involved. At the same time, the goal is to determine the best objects to involve them in the learning material in order to maximize the solution of the problem. To formalize the model based on the predicted values, we will encrypt all available data into a vector of states. The considered methods show their effectiveness. The results of the study. During the modification of the economic and financial condition model, a conclusion is made about the linearization of the model, which is also associated with the differentiation operation. The possibility of using neural networks is also demonstrated for the case of differentiation of complex functions, and in the case of convex programming, for marginal estimates obtained using the directional differentiation operation. Conclusion. The results obtained allow us to assert that the use of strict mathematical tools, a modified model of the economic and financial condition for the medical industry is easy to use for development management, since it can be implemented as some component of a computer program (neural network) into which the administrator enters data, and at the output receives a processed result.

Health Equity Considerations in Cost-Effectiveness Analysis: Insights from an Umbrella Review.

Cost-effectiveness analyses (CEA) are important in healthcare decision-making and resource allocation; however, expanding the scope of CEAs beyond the traditional clinicoeconomic concepts to also include value elements such as health equity has attracted much interest in recent years. This umbrella review aimed to synthesize evidence on how equity concepts have been considered in modified types of CEAs. Publicly available articles in MEDLINE were searched on January 25, 2024, to identify systematic reviews (SLRs) published in English since 2013 that incorporate health equity considerations in CEAs. Title/abstract, full-text article screening and data extraction were conducted by a single reviewer and validated by a second reviewer. Results were qualitatively synthesized to identify common themes. Eight SLRs were included. Distributional CEAs (DCEA), equity-based weighting, extended CEA (ECEA), mathematical programming and multi-criteria decision analysis (MCDA) were the most discussed approaches. A lack of consensus on the best approach for incorporating health equity into CEAs was highlighted, as these approaches are not currently consistently used in decision-making. Important limitations included scarcity of robust data to inform health equity indices, bias associated with commonly used health outcome metrics and the challenge of accounting for additional contextual factors such as fairness and opportunity costs. Proposals to expand CEAs to address equity issues come with challenges due to data unavailability, methods complexity, and decision-makers unfamiliarity with these approaches. Our review indicates that extended and distributional CEAs can support decision-making by capturing the impact of inequity on the clinical and cost-effectiveness assessment of treatments, although future modeling should account for additional contextual factors such as fairness and opportunity costs. Recommendations for actions moving forward include standardization of data collection for outcomes related to equity and familiarity with methodologies to account for the complexities of integrating health equity considerations in CEAs.