Mathematical Programming Research Articles

ON THE SOLUTION UNIQUENESS IN PORTFOLIO OPTIMIZATION AND RISK ANALYSIS

In this paper, we consider the issue of solution uniqueness of the mean-deviation portfolio optimization problem and its inverse for asset returns distributed over a finite number of scenarios. Due to the asymmetry of returns, the risk is assessed by a general deviation measure introduced by Rockafellar et al. [(2006b) Optimality conditions in portfolio analysis with general deviation measures, Mathematical Programming 108, 515–540] instead of the standard deviation as in the classical Markowitz optimization problem. We demonstrate that, in general, one cannot expect the uniqueness of Pareto-optimal profit sharing in cooperative investment and the uniqueness of solutions in the mean-deviation Black–Litterman asset allocation model. For a large class of deviation measures, we provide a resolution of the above nonuniqueness issues based on the principle of law-invariance. We provide several examples illustrating the nonuniqueness and the law-invariant solution.

Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints Under Data Uncertainty

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC.

Electricity arbitrage for mobile energy storage in marginal pricing mechanism via bi-level programming

The rapid growth of renewable energy is integrated into the distribution system. The creation of both new market mechanisms and investment models has critical effects on the economics and security of the distribution market. Mobile energy storage has been used to increase the resilience of distribution grids due to their advantages in mobility and flexibility, which offer electricity arbitrage options for merchant investments. This paper presents a bi-level optimization framework based on location marginal pricing settlement of mobile energy storage financial rights revenue in active distribution systems. In the developed framework, the upper-level problem determines the optimal capacity, routing, and dispatching of mobile energy storage to maximize revenue in the liberalized electricity market. The lower-level problem performs the joint optimization of energy and reserve market clearing as well as renewable energy capacity optimization based on the alternating current optimal power flow model. The non-convexity of the line flow and location marginal pricing is linearized via second-order cone relaxation and Karush-Kuhn-Tucker. The bi-level programming is reformatted as mathematical programming with equilibrium constraints. Further, the revenue risk due to renewable energy uncertainty is measured via conditional value-at-risk. Finally, the effectiveness of the optimization framework and solution method is verified using the modified IEEE-33 test system. The results show that mobile energy storage promotes renewable energy and reduces distribution network costs by 2.3%.

Selection of aerodynamic characteristics and engine parameters of a maneuverable aircraft under epistemic uncertainty

At the preliminary stage of aircraft design, it is usually required to solve the problem of insufficient initial data for applying traditional algoristic-type mathematical programming models. In many cases, numerous input parameters cannot be accurately specified at the time of making design decisions. If the inaccuracy of parameters is not taken into account, the actual values of target functions may differ significantly from the calculated ones when solving optimization problems. In this regard, the issue of current interest is the development of algorithms to improve the reliability of design decisions under conditions of epistemological uncertainty when experts engage in the formation of initial data. The paper considers the problem of selecting aerodynamic characteristics and engine parameters of maneuverable aircraft under conditions of uncertainty associated with inaccuracy of expert data. The applied problem under consideration is further complicated by the necessity to include “black box” models in the algorithms being developed. The paper proposes algorithms that apply the uncertainty theory together with “black box” models that implement optimization calculation techniques derived from previous engineering practice of aircraft design. Using these algorithms, experts are able to set uncertain parameters whereby the lack of data is factored in using uncertainty distribution functions. In cases of monotonicity of uncertain parameters in target functions, application of the uncertainty theory provides a significant reduction in computational costs compared to the method of statistical modeling in optimization calculations. The paper presents the results of computational studies of the developed algorithms. A genetic algorithm (mathematical optimization solver) is used to optimize the search for design solutions. Pareto frontiers are obtained for different confidence levels enabling to make design decisions including values of aerodynamic characteristics and engine parameters of maneuverable aircraft.

A Decision Support Framework for Resilient and Sustainable Service Design

AbstractResilient and sustainable service design is essential for ensuring the longevity and effectiveness of service systems. However, existing literature often neglects key aspects such as articulating resilience attributes and integrating sustainability dimensions. This study proposes a decision support model for a “resilient-sustainable service design” that merges service design principles with resilient system attributes and organizational sustainability goals. The framework incorporates a multi-objective mathematical programming model and a multi-phased Quality Function Deployment (QFD) approach to derive Pareto optimal solutions using the Brute Force algorithm. Applied in the m-health service sector in Bangladesh, the study reveals significant challenges, including limited awareness of services and logistical inefficiencies. To address these issues, flexible strategies such as demand planning and service innovation are implemented. The findings have direct implications for the improvement of service delivery processes and underscore the importance of considering both resilience and sustainability. While focusing on Bangladesh’s m-health sector, the insights gained have broader relevance globally. The integration of resilience and sustainability principles into service design is crucial for addressing complex challenges across sectors and regions. Future research could involve longitudinal studies to capture evolving resilience strategies and explore resilient-sustainable service systems from a broader perspective. This entails examining various factors such as technological advancements and socio-economic dynamics shaping resilient and sustainable service ecosystems.

Revolutionizing Textile Manufacturing: Sustainable and Profitable Production by Integrating Industry 4.0, Activity-Based Costing, and the Theory of Constraints

The textile industry, a cornerstone of daily life and a highly globalized sector, faces significant environmental challenges due to its high water and energy consumption and extensive chemical usage. This study proposes a comprehensive green production planning and control model integrating Industry 4.0 concepts, activity-based costing (ABC), and the theory of constraints (TOC). The model utilizes mathematical programming to optimize product mix, maximize profitability, and minimize environmental impact. It leverages real-time sensing technologies and ERP systems to facilitate waste recovery, reduce carbon emissions, and achieve energy savings. Various carbon emission cost models, including continuous and discontinuous tax functions, are explored to balance corporate profitability with environmental sustainability. The findings demonstrate the model’s potential in optimizing resource utilization, reducing the environmental footprint, and enhancing profitability.

Knowledge-assisted hybrid optimization strategy of large-scale crude oil scheduling integrated production planning

As modern refinery crude oil scheduling scales up, traditional manual operations and independent optimization methods struggle with complexity and dynamics. This study proposes an empirically assisted integrated planning and scheduling optimization model.The integrated model is solved using a hybrid optimization algorithm combining mathematical programming and particle swarm optimization (MP/PSO). Long-term planning aims to minimize operational and transportation costs while maximizing refinery profits; short-term scheduling, based on initial long-term plans, aims to minimize unit switchovers. In the short-term scheduling phase, heuristic rules based on empirical operational knowledge generate a high-performing initial population to accelerate convergence. This strategy is crucial for enhancing refinery emergency response capabilities, ensuring stable operations, and improving economic benefits. Experimental results show that within a reasonable time frame, MP/PSO performs better than PSO and manual scheduling in large-scale crude oil scheduling scenarios.

Contemporary approaches in matheuristics an updated survey

AbstractMatheuristics are problem independent frameworks that use mathematical programming tools to obtain high quality heuristic solutions. They are structurally general enough to be applied to different problems with little adaptation to their abstract structure, so they can be considered as new or hybrid metaheuristics based on components derived from the mathematical model of the problems of interest. In this survey, we emphasize the mathematical tools and describe how they can be used to design heuristics. We focus on mixed-integer linear programming and report representative examples from the literature of how it has been used for effective heuristic optimization. References to contributions to matheuristics deriving from neighboring research areas such as Artificial Intelligence or Quantum Computing are also included. We conclude with some ideas for possible future developments. This paper extends an original version published in 4OR with new sections on CMSA, Incremental Core, AI hybrids and Quantum Heuristics, and includes references to several recent publications.

New Generalized Derivatives for Solving Variational Inequalities Using the Nonsmooth Newton Methods

AbstractVariational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python.

Uniting agricultural water management, economics, and policy for climate adaptation through a new assessment of water markets for arid regions

Flexible policies aimed at irrigated agriculture are essential to adapt to climate change. Despite the importance of this goal, little published work has conceptualized, formulated, developed, and applied an integrated optimization framework for irrigated agriculture to guide adaptation to climate-related water stress. This research addresses the question: how can water management plans for irrigated agriculture be designed to minimize economic losses caused by adapting to climate-induced water stress? The study answers this by developing an optimization approach that identifies water use patterns to minimize farm income losses during water shortages, considering three water shortage sharing programs. An optimization model, calibrated using positive mathematical programming, is applied to replicate historical land use while adapting to future water supplies that deviate from the historical pattern. The analysis focuses on two irrigated regions in North America’s Rio Grande Basin, illustrating land use, water use, and cropping patterns that minimize regional farm economic losses to shortages. These losses are assessed under three water-sharing strategies: intercrop and interdistrict trading (IIT), intercrop and intradistrict trading (IRT), and no trading (NT). The results demonstrate that IIT yields an average economic gain of $2.824 million per year, while IRT results in an average gain of $2.600 million per year compared to NT. These findings offer valuable insights for water managers, scientists, stakeholders, and policymakers tasked with developing irrigation management strategies in arid regions facing future water supply challenges. The methods developed and results shown here highlight a path forward, using scientific, economic, and policy innovations to strengthen agricultural livelihoods in regions facing uncertain water availability.

A Capstone Course for Secondary Math Educators Moving to College-Level Teaching

The University of Tennessee has a Master of Mathematics (M.M.) program for secondary educators interested in college-level teaching. The program includes a capstone course on topics related to college-level teaching and can be thought of as a kind of professional development which allows secondary educators to explore options for their careers. This paper outlines the design, structure, and implementation of the course, including that it incorporated findings from an active research project. We describe details of class design, activities, our reflections, and how the course can serve as a way for teaching ideas to flourish between high school and college educators.

Packing squares independently

Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications.

Mathematical programming and geotechnologies applied to allocation of forest fire detection towers

Mathematical programming and geotechnologies applied to allocation of forest fire detection towers

Heuristic algorithm for calculating the component composition of recombination gas without an adapted equation of state

Background. To conduct laboratory research of projects of gas enhanced oil recovery methods, a large volume of reservoir system is required. Often its replenishment by depth and/or separator samples is not economically feasible. Therefore, the task of fluid recombination from wellhead liquid sample and natural gas simulator arises. Objective. To calculate the component composition of recombination gas under the conditions of the task. Materials and methods. The heuristic calculation algorithm uses the tools of: modeling, inverse problems of physics, set theory, mathematical programming. Special focus is given to its empirical and theoretical justification. Results. The paper considers the limitations of the thermodynamic method. The conditions of the problem to be solved are outlined and justified. The applied algorithm of the solution is formulated and derived. The efficiency of the algorithm for oil systems is experimentally confirmed. Conclusions. The work will be of interest for specialists in the field of laboratory research projects of gas enhanced oil recovery methods. The issues of improving the accuracy of the applied algorithm and its further development are still under discussion.

Editorial for the Special Issue: "Novel Solutions and Novel Approaches in Operational Research"

Abstract This special issue of Business Systems Research (SI of the BSR) is being co-published by the Slovenian Society INFORMATIKA – Section for Operational Research (SSI -SOR). It focuses on recent advances in Operations Research and Management Science (OR / MS), with a particular emphasis on linking OR / MS with other areas of quantitative and qualitative methods in the context of a multidisciplinary framework. The ten papers that were chosen for this Special Issue of the BSR present advancements and new techniques (methodology) in the field of Operations Research (OR), as well as their application in a variety of fields, including risk management, mathematical programming, game theory, gravity, spatial analysis, logistics, circular economy, continuous improvement, sustainability, e-commerce, forecasting, Gaussian processes, linear regression, multi-layer perceptron, and machine learning.

Decision-Making Model to Support Agricultural Policies in Realizing Economic and Social Sustainability

Abstract Background Achieving economic and social sustainability is the goal of any policy when defining measures. We focus on the beef sector, where many challenges have arisen due to its structural characteristics, such as an unfavourable scale structure, high costs, low efficiency, and a low environmental footprint. This paper presents an example of the support provided by a mathematical programming model in the development of a Common Agricultural Policy Strategic Plan for the period 2023-2027. Methods/approach It is a model based on linear programming that allows such an ex-ante analysis by calculating production plans at the farm level and aggregating the results at the sector level. Objectives When defining the interventions, the question arose as to what the reform of the Common Agricultural Policy will bring and to what extent the sector should be supported in meeting these challenges. These were the concerns of agricultural policy that we sought to support by modelling different scenarios. Results The results show that the situation of the sector will worsen, especially for larger farms, but they also show the great importance of production-related payments to mitigate the negative trend. Conclusions The applied approach proves to be suitable for supporting the design of agricultural policy and achieving greater economic and social sustainability in the sector.

Choosing science and mathematics programs in college: practical and psychological arbiters in career-pathing

The objective of this exploratory qualitative research is to investigate the practical and psychological factors influencing students’ decisions to pursue science and mathematics-based courses for their career paths. The study will focus on a sample size of 25 second-year students in Calbayog City who have chosen to continue with their chosen course after completing one year of study. To collect data, the study will use one-on-one interviews, a qualitative research method. This method will allow for a thorough investigation of the participants’ experiences, viewpoints, and motives. Through one-on-one interviews, the researchers are able to acquire a thorough understanding of the factors that influence students’ decision-making processes. The findings of this study will provide important insights into the practical and psychological factors that impact students’ selections in science and mathematics courses. This information can be utilized to improve career advising programs and educational systems, as well as to guide students who are exploring comparable career options.

Augmented ɛ‐constraint‐based matheuristic methodology for Bi‐objective production scheduling problems

AbstractMatheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi‐objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ‐constraint, which transforms multi‐objective problems into single‐objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ‐constraint‐based matheuristic methodology (ɛ‐MH) is proposed to apply the idea of ɛ‐constraint to embedded MILP models, so that Pareto fronts obtained by meta‐heuristics can be further improved by solving a set of MILP models. Afterwards, four speed‐up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ‐MH. Finally, several real‐world bi‐objective scheduling problems are discussed to present potential applications for the proposed methodology.

Fast heuristics for the time‐constrained immobile server problem

AbstractWe propose easy‐to‐implement heuristics for time‐constrained applications of a problem referred to in the literature as the facility location problem with immobile servers, stochastic demand, and congestion, the service system design problem, or the immobile server problem (ISP). The problem is typically posed as one of allocating capacity to a set of M/M/1 queues to which customers with stochastic demand are assigned with the objective of minimizing a cost function composed of a fixed capacity‐acquisition cost, a variable customer‐assignment cost, and an expected‐waiting‐time cost. The expected‐waiting‐time cost results in a nonlinear term in the objective function of the standard binary programming formulation of the problem. Thus, the solution approaches proposed in the literature are either sophisticated linearization or relaxation schemes, or metaheuristics. In this study, we demonstrate that an ensemble of straightforward, greedy heuristics can rapidly find high‐quality solutions. In addition to filling a gap in the literature on ISP heuristics, new stopping criteria for an existing cutting plane algorithm are proposed and tested, and a new mixed‐integer linear model requiring no iterating algorithm is developed. In many cases, our heuristic approach finds solutions of the same or better quality than those found by exact methods implemented with expensive, state‐of‐the‐art mathematical programming software, in particular a commercial nonlinear mixed‐integer linear programming solver, given a five‐minute time limit.

Analyzing School Bus Electrification in Richmond, Virginia

School buses are an essential component of the transportation infrastructure, serving as a lifeline for students across the globe. However, the widespread use of diesel school buses has raised concerns about the health impact on millions of students exposed to harmful emissions daily. Recognizing this issue, school districts worldwide are urgently seeking cleaner energy alternatives. Electric school buses emerge as an environmentally friendly and sustainable option, fostering a healthier environment for both students and communities. However, school bus electrification faces the challenges of high upfront cost, cumbersome charging management, and constraints from power grids. To help school bus operators address those challenges, this study presents a data-driven analysis for school bus electrification. This study considered a real-world school bus system in Richmond, VA, and developed a mathematical programming model to analyze the system design, charging strategies, and charging load profiles for the electrification scenario. The study evaluated different charging strategies based on model outcomes, aiming to optimize efficiency and effectiveness. Ultimately, this research generated electric school bus charging demand profiles under various scenarios, shedding light on the feasibility and implications of transitioning to electric-powered school buses.