Mathematical Programming Problem Research Articles

Кількісне оцінювання функціональної ефективності мережевих структур за аналізом чутливості оптимізаційної моделі до зовнішніх впливів

The real network organizational structure is constantly under the influence of external influences, which can cause unexpected consequences due to the cascading spread of relevant disturbances directed at critically important parameters. Parametric sensitivity analysis is the latest computational procedure, the results of which provide the researcher with an idea and quantitative assessment of the appropriate response of the structure in the form of an analytical platform as a composition and interaction of scenario analysis, cascade process development and optimization modeling technology. The main task is timely adaptation to expected changes in business analytics, oriented to optimal resources distribution. The generalized minimum cost problem is a typical example of widespread problems of optimal distribution and use of limited, in particular, energy resources, of any nature and purpose, using the scientific management and analysis of decisions, in its network model attention is paid to the potentials of nodes. The maximum flow problem is an example of the active and effective application of methods and models of flows optimization, where the processes and objects used a network organization, the energy systems and complexes investigated, where arc parameters are critical in its model. In the article, for these classic optimization problems, mathematical and spreadsheet models were built on concrete examples. Direct and dual mathematical programming problems were solved, the following stages were carried out - the organization of a computer experiment and the construction of an analytical platform for evaluating the behavior of the network structure under the influence of external influences and disturbances as their consequences. To take into account the specific conditions of real objects with a network structure at the stage of modification, the model supplemented with appropriate restrictions and correction of input data sets. The obtained results should be useful for planning and management personnel and top-management for discussion and decision-making regarding, for example, the optimal placement of energy equipment, the weighted distribution of energy flows in the "source-sink" system or the appointment of executors for work, automated project management using project networks, etc. Keywords: optimization modeling, maximum network flow, minimum cost flows, network organizational structures, decision making, spreadsheet modeling and analytics.

THE PENALTY FUNKCIONS METHOD AND OPTIMALITY CONDITONS

The presented work is devoted to one of the most important areas of applied mathematics, namely mathematical programming. The following general mathematical programming problem is considered where are Lipschitz, and is closed set. The purpose of the work is to obtain optimality conditions for the problem posed using the method of penalty functions. The obtained necessary conditions are expressed in the language of subdifferentials, which are a generalization of the method of indefinite Lagrange coefficients for nonsmooth optimization problems.

Формирование оптимального кредитного портфеля с использованием нечётко-множественного подхода

The problem of profit maximization from lending to a financial organization is considered. A set of loan con-tracts forms a loan portfolio, which makes it possible to use the portfolio theory in the study of the designated problem. We will consider the formation of a loan portfolio as the formation of an investment portfolio. The problem arises of redistributing the funds of the investment portfolio in order to minimize its possible risk and maximize profitability. This problem in classical formulation can be solved according to Markowitz risk diversification. Thus, the problem of risk minimization and profitability maximization from lending to a financial institu-tion can be represented as a fuzzy mathematical programming problem, and more specifically, as a fuzzy linear programming problem. The resulting problem is solved by considering discrete α-levels and is reduced to a set of linear programming problems constructed for each of them. A clear expected optimal solution can be obtained by applying the defuzzification method, which allows you to convert a fuzzy set into a crisp number.

Hydromechanical Transmission IC2OC: Component Sizing and Optimization

The IC2OC transmission is a continuous transmission whose layout can change from simple IC to simple OC configuration and vice versa. It was proposed to cover a wider range of vehicle speeds without adding gears. Its sizing can lead to higher efficiencies than those of the IC and OC layouts. Therefore, this work deals with the sizing methodologies of this transmission. Two methodologies are proposed and discussed: the first uses the functional and constitutive equations of the transmission; the second is based on a mathematical programming problem. Both methodologies start from the choice of the full mechanical point speeds. The comparison between the two methods is carried out on the transmission of a commercially available 230 kW reach stacker. The comparison shows that the functional method, leaner and faster, can provide results very close to those obtained with the heavy and time-consuming optimization, provided that the values of the two full mechanical point speeds are the optimal ones for the two basic transmissions taken individually.

Planning Courses for Student Success at the American College of Greece

We are concerned with the personalized student course plan (PSCP) problem of optimizing the plan of courses students at the American College of Greece will need to take to complete their studies. We model the constraints set forth by the institution so that we guarantee the validity of all produced plans. We formulate several different objectives to optimize the resulting plan, including the fastest completion time, course difficulty balance, and maximization of the expected student grade point average given the student’s performance in passed courses. All resulting problems are mixed-integer linear programming problems with a number of binary variables, that is, the max number of terms times the number of courses available for the student to take. The resulting mathematical programming problem is solvable in less than 10 seconds on a modern commercial off-the-shelf PC, whereas the manual process used to take more than one hour of advising time for every student and, as measured by the objectives set forth, resulted in suboptimal schedules. History: This paper was refereed.

Risk management in software development in the field of information protection

In the context of increasing digitalization of sea and river transport, the immersion of the entire transport infrastructure in the digital environment, the volume of information flows serving the transport process is increasing, and the requirements for information protection are increasing in conditions of an aggressive external information environment. In the digital environment, a software product must not only serve the document flow of transport chains, but also ensure its confidentiality. Automation of transport facilities (ships, platforms, berths, warehouses) management increases their potential vulnerability from unauthorized access to control systems; the latter must be taken into account in servicing programs, increasing their size (for example, there is a need to fragment coherent information blocks and their alternative routing). This, in turn, increases the risk of errors in the software products themselves and significantly complicates their structure. Risks of failures (including confidentiality violations) during the software operation that implements information exchange can entail significant material and reputational losses for the developer. If the developer is legally and functionally involved in the industry, such losses can and should be considered among the general range of risks in water transport. An important aspect of risk management, which until recently was practically not considered in a quantitative aspect within the framework of mathematical models, is the joint consideration of losses when a risk situation occurs and the developer’s costs associated with reducing the likelihood of such occurrences; the latter may entail a reduction in the total expected losses, formalized as the corresponding mathematical expectation. As a result, risk management can be formulated in terms of mathematical programming problems with different (discrete or continuous) sets of constraints and with different properties of objective functions.

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

A new gray optimization model for disposing or re-commercializing unsold goods in reverse logistics networks with a cross-docking center

PurposeThis paper investigates a problem in a reverse logistics (RLs) network to decide whether to dispose of unsold goods in primary stores or re-commercialize them in outlet centers. By deducting the costs associated with each policy from its revenue, this study aims to maximize the profit from managing unsold goods.Design/methodology/approachA new mixed-integer linear programming model has been developed to address the problem, which considers the selling prices of products in primary and secondary stores and the costs of transportation, cross-docking and returning unwanted items. As a result of uncertain nature of the cost and time parameters, gray numbers are used to deal with it. In addition, an innovative uncertain solution approach for gray programming problems is presented that considers objective function satisfaction level as an indicator of optimism.FindingsAccording to the results, higher costs, including transportation, cross-docking and return costs, make sending goods to outlet centers unprofitable and more goods are disposed of in primary stores. Prices in primary and secondary stores heavily influence the number of discarded goods. Higher prices in primary stores result in more disposed of goods, while higher prices in secondary stores result in fewer. As a result of the proposed method, the objective function satisfaction level can be viewed as a measure of optimism.Originality/valueAn integral contribution of this study is developing a new mixed-integer linear programming model for selecting the appropriate goods for re-commercialization and choosing the best outlet center based on the products' price and total profit. Another novelty of the proposed model is considering the matching percentage of boxes with secondary stores’ desired product lists and the probability of returning goods due to non-compliance with delivery dates. Moreover, a new uncertain solution approach is developed to solve mathematical programming problems with gray parameters.

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

This study considers equilibrium problems, focusing on identifying finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set. This comprehensive concept encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of the existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios.

CLUSTERING-BASED OBJECTS STATE DIAGNOSTICS IN CONDITIONS OF FUZZY SOURCE DATA

У The problem of distribution of a set of objects, the state of which is determined by a set of controlled parameters, into a set of subsets of objects maximally homogeneous in their properties is considered. Relevance of the problem and important advantage of the clustering procedure: when its implementing it is possible to reduce the initial difficult problem of high dimensionality objects analysing to the solution of a number of simpler problems of lower dimensionality. This circumstance acquires additional attractiveness and importance if the initial data of the problem contain uncertainty, for example, are vaguely defined. Research object is the procedure of partitioning a set of objects into clusters under conditions of uncertainty. In this regard, the purpose of the study is to develop a method for solving the problem of clustering in conditions where the initial data on objects controlled parametersthe values contain uncertainty. The method of solving the problem is based on clustering procedure mathematical model development, containing analytical expressions for the criterion of its effectiveness, written in the form of a twice fractionally quadratic function. The impossibility of mathematical programming problem direct solution initiated the development of a heuristic algorithm for its solution. As a result, an iterative method was obtained and applied to solve the clustering problem under conditions of fuzzy initial data. The developed computational procedure is based on a reasonable system of rules for performing operations on fuzzy numbers. The situations when the belonging functions of problem fuzzy parameters are defined on infinite or compact media are considered. The developed system of rules allows to correctly perform operations in the metric of fuzzy defined states between clustering objects. The proposed method is easily generalised to the case when the uncertainty of the initial data is hierarchical.

A solution method for stochastic multilevel programming problems. A systematic sampling evolutionary approach

Stochastic multilevel programming is a mathematical programming problem with some given number of hierarchical levels of decentralized decision makers and having some kind of randomness properties in the problem definition. The introduction of some randomness property in its hierarchical structure makes stochastic multilevel problems computationally challenging and expensive. In this article, a systematic sampling evolutionary method is adapted to solve the problem. The solution procedure is based on realization of the random variables and systematic partitioning of each hierarchical level’s decision space for searching an optimal reaction. The search goes sequentially upwards starting from the bottom up through the top hierarchical level problem. The existence of solution and convergence of the solution procedure is shown. The solution procedure is implemented and tested on some selected deterministic test problems from literature. Moreover, the proposed algorithm can be used to solve stochastic multilevel programming problems with additional complexity in their problem definition.

On the Problem of Maximizing the Probability of Successful Passing of a Time-Limited Test

—The problem of finding the optimal sequence of performing a set of tasks in a timelimited test is considered. That is, a task group is allocated for mandatory initial execution in the test, the remaining tasks are performed during the remaining time until the end of the test. For each correctly completed task of the test, the subject is awarded a certain number of points. The proposed criterion is the probability that the total number of points scored for the test exceeds a certain level, which is a fixed parameter, while simultaneously fulfilling the time limit of the test. Random parameters are the user’s response time to each test task. The correctness of the user’s answer to the task is modeled by a random variable with a Bernoulli distribution. The resulting stochastic bilinear programming problem boils down to a deterministic integer problem of mathematical programming.

Optimizing Chance Constraint Multiple-Objective Fractional Mathematical Programming Problem Involving Dependent Random Variable

This manuscript suggests a methodology to solve chance constraint multiple-objective linear fractional mathematical programming problem in which the parameters are dependent random variables to each other. The proposed problem is formulated by taking few of the parameters as continuous dependent random variables. The proposed model cannot be solved directly by using existing methodology. Thus in order to solve the proposed model, an equivalent deterministic model is derived. The procedure to solve the proposed model is accomplished in two main steps. Initially, the proposed multiple-objective chance constraint linear fractional mathematical problem is transformed to deterministic equivalent multiple-objective linear fractional mathematical programming by the help of chance constrained method. In the second step, multiple-objective functions, which consist of fractional functions is solved by using lexicographic programming approach. Finally, an example is mentioned to illustrate the methodology.

Optimality conditions for nonconvex mathematical programming problems using weak subdifferentials and augmented normal cones

Optimality conditions for nonconvex mathematical programming problems using weak subdifferentials and augmented normal cones

Methodology of delivering space parts, tools and accessories to a near Earth orbital station on the basis of the reliability factor

The main aspects of the deployment of multi-module orbital manned stations are considered. The methodology for planning the delivery of spare parts, tools and accessories used in planning of transport operations to the International Space Station, is analyzed. It is shown that its application is related to the problem of mathematical programming with a large number of variables, which complicates the process of finding the optimal planning option in terms of economic efficiency. Transport operations to national manned orbital stations are carried out less frequently as compared to the International Space Station due to the enormous costs of spacecraft launches taking into account possible emergency situations. An approach to improving the methodology for planning the delivery of spare parts, tools and accessories for possible use on national manned orbital stations is proposed. It consists in taking into account the reliability of on-board equipment. The advantage of the improved methodology is simplification of planning transport operations without the use of mathematical programming. In addition, the improved methodology is clear, simple and can be implemented step by step.

Modern algorithm of linear and quadratic programming in optimization and problems of deformation of structures of variable structure in contact conditions

Various mathematical problems, in which the goal of finding the extremum of a functional is set, belong to the problems of mathematical programming and optimization problems. Practically directed problems of finding the optimal solution are extremely numerous in economics, management, technology, and others. They are related to increasing production efficiency, reducing resource costs, improving design solutions and technological processes, reducing mass, dimensions, etc. Among them, an important role is given to the methods of limiting the maximum stresses caused by external loads. Solving such problems begins with mathematical formalization. Constructive, economic or technological indicators are chosen as variation parameters. The search for the best solution is reduced to the selection of a set of parameters that provide a stationary value of the objective function. Extreme problems of practical orientation contain equality-inequality constraints in mathematical models. In improving the technical characteristics of machines, a significant role belongs to engineering and technical workers, who find optimal options at the design stage. At the same time, an essential element of the design process is the modeling of the determining processes in structures, taking into account the main influencing factors and behavior scenarios. Optimization is an important area of applied mathematics that provides effective tools for such modeling. Universal Algorithm is proposed in work [3] – a numerous scheme for solving quadratic programming (QP) problems for calculating the optimal point of a wide range of applied problems. At the same time, the linear programming (LP) problem is considered as a partial case of the (QP) problem. That is, in the universal algorithm for setting 2 optimization problems, they are formalized in a single and convenient form of symmetric matrix dependence, which makes it possible to build a single effective algorithm based on matrix algebra operations. In particular, it allows you to consider the practical tasks of calculating VAT in constructions of a variable structure consisting of separate parts connected by one-way connections. The main goal of this work is to analyze the behavior of the algorithm when increasing the number of constraints of the inequality type, to refine the computational scheme, and to formulate conclusions. Two model problems are considered as examples of the algorithm. This is a classic "transport" problem of LP and the behavior of a model of a bridge structure with one-way connections in cables under variations of wind loads. The number of ropes has been increased to 20, and the limits of one-way connections to 40.

Fuzzy Programming Approach to Solve Multi-Objective Fully Fuzzy Transportation Problem

The aim of this study is presenting the solution methodology of multi-objective fuzzy transportation problem with fuzzy decision variables, where all the input parameters and decisions variables of the programming problems are assumed to be triangular fuzzy number and triangular fuzzy decision variables respectively. Moreover the objectives under considerations are minimization of cost of transportation and minimization of shipping time under fuzzy environment. The fuzziness of the objective functions and the fuzzy constraints of the programming problem are defuzzified using the ranking function and the equality property between two fuzzy numbers, respectively. The consequent crisp multi-objective fuzzy transportation problem is tackled by employing fuzzy mathematical programming approach. Finally fuzzy decision is made after solving the resultant mathematical programming problem using LINGO(Schrage and LINDO Systems (1997)) software. Illustrative numerical example is presented in support of the proposed methodology.

Adaptive Subgradient Methods for Mathematical Programming Problems with Quasiconvex Functions

Adaptive Subgradient Methods for Mathematical Programming Problems with Quasiconvex Functions

UPPER BOUND FOR THE COMPETITIVE FACILITY LOCATION PROBLEM WITH DEMAND UNCERTAINTY

We consider a competitive facility location problem with two competing parties operating in a situation of uncertain demand scenario. The problem to find the best solutions for the parties is formulated as a discrete bi-level mathematical programming problem. In the paper, we suggest a procedure to compute an upper bound for the objective function on subsets. The procedure could be employed in implicit enumeration schemes capable to compute an optimal solution for the problem under study. Within the procedure, additional constraints iteratively augment the high-point relaxation of the initial bi-level problem, what strengthens the relaxation and improves the upper bound’s quality. New procedure to generate such cuts allows to construct the strongest cuts without enumerating the parameters encoding them.

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.