Continuous Linear Programming Research Articles

Efficient and robust optimal design for quantile regression based on linear programming

When informing decisions with experimental data, it is often necessary to quantify the distribution tails of uncertain system responses using limited data. To maximize the information content of the data, one is naturally led to use experimental design. However, common design techniques minimize global statistics such as the average estimation or prediction variance. Novel methods for optimal experimental design that target distribution tails are developed. To achieve this, pre-asymptotic estimates of the data uncertainty are produced via an upper bound on a prescribed quantile, computed using quantile regression. Two optimal design problems are formulated: (i) Minimize the variance of the upper bound; and (ii) Minimize the Conditional Value-at-Risk of the upper bound. Additionally, each design problem is augmented with an added cardinality constraint to bound the number of experiments. These optimal design problems are reduced to continuous and mixed-integer linear programming problems. Consequently, the proposed methods are extremely efficient, even when applied to large datasets. The application of the proposed design formulation is demonstrated through a sensor placement problem in direct field acoustic testing.

Optimizing satellite ground station facilities scheduling for RSGS: a novel model and algorithm

ABSTRACT As the rapid increasing of the number of the satellites, the number of facilities of the satellite ground stations is becoming insufficient and the conflicts of serving satellite data missions exist. To make full use of the facilities of the China Remote Sensing Satellite Ground Station (RSGS), scheduling optimization of these facilities was studied in this paper, which was not well-solved by the original method in the operation system of RSGS. The problem is formulated as a continuous time-based mixed integer linear programming (MILP) model. A decomposition optimization algorithm is proposed to solve the large-scale problem, which is difficult to solve using traditional methods. In the first step, the particle swarm optimization (PSO) algorithm is used to optimize the multi-ground station relay tasks, and the multi-ground station optimization problem is transformed into several single ground station optimization problems. Next, in two additional steps, optimization of the facilities at a single ground station is performed to further reduce the size of the model. The proposed method has been applied to operations systems of RSGS, and the approach presented in this article has demonstrated stable performance and sufficient efficiency online, resulting in the automation of facility scheduling.

Mathematical formulations for project scheduling problems with categorical and hierarchical skills

In this paper, we present six extensions to the multi-skilled resource-constrained project scheduling problem (MSRCPSP) by introducing hierarchical levels of skills. These hierarchical skills can impact the MSRCPSP in multiple different ways. This paper studies efficiency differences, cost differences, quality differences and more. For each of these problems we propose and analyse seven continuous and time-indexed (mixed-)integer linear programming formulations. A modular artificial dataset is generated that assembles instances of the presented problems as well as combinations of these problems. In the computational experiments, we solve these instances using the proposed mathematical formulations with the CPLEX solver. Finally, we compare the results of the different formulations for the resource-constrained project scheduling problems with hierarchical levels of skills in order to explain their inherent similarities and differences.

A METHOD FOR SOLVING THE CANONICAL PROBLEM OF TRANSPORT LOGISTICS IN CONDITIONS OF UNCERTAINTY

Subject.The canonical task of transport logistics in the distributed system "suppliers - consumers" is considered. Goal. Development of an accurate algorithm for solving this problem according to the probabilistic criterion in the assumption of the random nature of transportation costs has been done. Tasks. 1. Development of an exact method for solving the problem of finding a plan that minimizes the total cost of transportation in conditions when their costs are given by their distribution densities. 2. Development of a method for solving the problem when the distribution density of the cost of transportation is not known. Methods. A computational scheme for solving the problem is proposed, which is implemented by an iterative procedure for sequential improvement of the transportation plan. The convergence of this procedure is proved. In order to accelerate the convergence of the computational procedure to the exact solution, an alternative method is proposed based on the solution of a nontrivial problem of fractional nonlinear programming. The method reduces the original complex problem to solving a sequence of simpler problems. The original problem is supplemented by considering a situation that is important for practice when, in the conditions of a small sample of initial data, there is no possibility of obtaining adequate analytical descriptions for the distribution densities of the random costs of transportation. To solve the problem in this case, a minimax method is proposed for finding the best transportation plan in the most unfavorable situation, when the distribution densities of the random cost of transportation are the worst. To find such densities, the modern mathematical apparatus of continuous linear programming was used. Results. A mathematical model and a method for solving the problem of transport logistics in conditions of uncertainty of the initial data are proposed. The desired plan is achieved using the solution of the fractional nonlinear programming problem. Conclusions: The problem of forming a transportation plan is considered, provided that their costs are random values. Also, a method for solving the problem of optimization of transportation for a situation where the density of distribution of random cost cannot be correctly determined is considered.

Optimal operation planning of distribution systems under uncertainty in load modeling by using decentralized energy resources

The new elements connected in electrical distribution systems increase the complexity of grids as well as planning and operating. The benefits of classical techniques, such as Conservation Voltage Reduction (CVR), combined with a coordinated operation of distribution elements, can contribute to increase efficiency and to reduce energy consumption of the distribution systems.This paper proposes an optimization model for a day-ahead operation of an unbalanced distribution system with photovoltaic distributed generation (DG) penetration. The CVR technique will be considered in deterministic, stochastic and robust approaches, considering the uncertainty of its parameters in the load modeling. The proposed optimization model aims to minimize energy consumption at the substation considering the operation of traditional elements, such as On Load Tap Changers (OLTC) at substation and capacitor banks (CB), but also modern elements, such as intelligent photovoltaic inverters.In the proposed approach, the mathematical problem, initially modelled as a mixed-integer non-linear programming problem, was converted into a continuous linear programming problem. The results were obtained using a modified IEEE123-bus test system indicating good results for utilities to save energy and reduce power demand in three optimization approaches: deterministic, stochastic and robust, and the last one proved to be better for being conservative for voltage violations and lower computational burden.

DEVELOPMENT OF METHODS FOR SUPPLY MANAGEMENT IN TRANSPORTATION NETWORKS UNDER CONDITIONS OF UNCERTAINTY OF TRANSPORTATION COST VALUES

The problem of transport management in a distributed logistics system "suppliers – consumers" is considered. Under the assumption of a random nature of transportation costs, an exact algorithm for solving this problem by a probabilistic criterion has been developed. This algorithm is implemented by an iterative procedure for sequential improvement of the transportation plan. The rate of convergence of a computational procedure to an exact solution depends significantly on the dimension of the problem and is unacceptably low in real problems. In this regard, an alternative method is proposed, based on reducing the original problem to solving a nontrivial problem of fractional-nonlinear programming. A method for solving this problem has been developed and substantiated. The corresponding computational algorithm reduces the fractional-nonlinear model to the quadratic one. The resulting problem is solved by known methods. Further, the original problem is supplemented by considering a situation that is important for practice, when in the conditions of a small sample of initial data there is no possibility of obtaining adequate analytical descriptions for the distribution densities of the random costs of transportation. In this case, the available volume of statistical material is sufficient only to estimate the first two moments of unknown distribution densities. For this marginal case, a minimax method for finding the transportation plan is proposed. The first step is to solve the problem of determining the worst distribution density with the given values of the first two moments. In the second step, the transportation plan is found, which is the best in this most unfavorable situation, when the distribution densities of the random cost of transportation are the worst. To find such densities, let’s use the modern mathematical apparatus of continuous linear programming

A GRASP-based algorithm for solving the emergency room physician scheduling problem

This paper addresses a physician scheduling problem in an Emergency Room (ER) requiring a long-term work calendar to allocate work days and types of shift among all the doctors. The mathematical model is created without simplifications, using the real calendar, including holidays. This precludes the possibility of cyclic-type solutions, and involves numerous and varied constraints (demand, workload, ergonomics, fairness, etc.). An effective solution to this very difficult practical problem cannot be obtained, for large instances, with exact solution methods. We formulate a mathematical representation of a real-world ER physician scheduling problem featuring a hybrid algorithm combining continuous linear programming with a greedy randomized adaptive search procedure (GRASP). Linear programming is used to model a general physician-demand covering problem, where the solution is used to guide the construction phase of the GRASP, to obtain initial full schedules for subsequent improvement by iterative application of Variable Neighborhood Descent Search (VNDS) and Network Flow Optimization (NFO). A computational study shows the superiority of our approach over the Integer Linear Programming method in a set of instances of varying size and difficulty inspired by a real setting. The methodology is embedded in a software tool for generating one-year-ahead physician schedules for a local ER. These solutions, which are now in use, outperform the manually-created schedules used previously.

FACTORING MANAGEMENT ON THE BASIS OF FORMATION OF OPTIMIZING FACTORING PORTFOLIO OF TRADE ENTERPRISES

Abstract. The article is devoted to the study of the effectiveness of factoring management as an innovative product based on the formation of the optimization factoring portfolio of trade enterprises, which will serve as a basis for management decisions on the implementation of this service in business. As a result of the research a number of conclusions and methodical recommendations of practical direction are formulated. The formation of the optimization factoring portfolio is based on the principles of the CAPM portfolio investment model proposed by W. Sharp. The conditions for the formation of portfolio investments, which are reflected in the price model of the capital market, were adapted to the formation of the factoring portfolio. The key conditions of the factoring agreement are: limits on credit risk and a given level of income, that is the company must choose a portfolio with the maximum profitability, which is achieved by minimizing the cost of this service; seek to increase the funding limit for debtors; to remove from the portfolio of debtors of the enterprise with a high risk of non-repayment of debt. We propose to determine the debtors of the enterprise that will form the optimization factoring portfolio through the variables x1 – xn. When x = 1, the debtor is added to the factoring portfolio, and if x = 0, then it is necessary to exclude the debtor from the portfolio. In the language of factoring regression agreements, you can use a continuous linear programming problem in which the variable x will be in the range from 0 to 1, that is funding limits for debtors can be broken. The target function of the factoring portfolio of commercial enterprises is proposed both for «invoice-discounting» (only receivables financing) and for «full factoring», that is full-fledged factoring (the use of additional related factoring services). The profitability of the enterprise from factoring operations depends on the number of debtors in the factoring portfolio. The greater the number of debtors in the portfolio, the higher the profitability of the company from factoring. However, the company needs to consider debtors with high credit risk in order to minimize the non-repayment of receivables. Therefore, debtors with the highest credit risk of factoring should be excluded from the factoring portfolio. To assess the economic efficiency of factoring management of trade enterprises on the basis of the formed optimization factoring portfolios, we propose to use the Cronbach & Gleser method. The proposed proposals for the management of factoring of commercial enterprises, which is based on the formation of factoring portfolios, will regulate debt relations between enterprises with the involvement of third parties, that is, to prevent the occurrence of negative phenomena in the debt relations between commercial enterprises in the future, as well as will help maximize profits in the studied business entities. Further research will focus on adapting the factoring portfolio of commercial enterprises to changes in the value of financial resources under the influence of transformational changes in the economy. Keywords: factoring portfolio, trading enterprises, transformational changes, factoring management, debt relations between enterprises, profit maximization. JEL Classification С44, С65, С61, F65 Formulas: 11; fig.: 0; tabl.: 0; bibl.: 20.

Development of modern models and methods of the theory of statistical hypothesis testing

Typical problems of the theory of statistical hypothesis testing are considered. All these problems belong to the same object area and are formulated in a single system of axioms and assumptions using a common linguistic thesaurus. However, different approaches are used to solve each of these problems and a unique solution method is developed. In this regard, the work proposes a unified methodological approach for formulating and solving these problems. The mathematical basis of the approach is the theory of continuous linear programming (CLP), which generalizes the known mathematical apparatus of linear programming for the continuous case. The mathematical apparatus of CLP allows passing from a two-point description of the solution of the problem in the form {0; 1} to a continuous one on the segment [0; 1]. Theorems justifying the solution of problems in terms of CLP are proved. The problems of testing a simple hypothesis against several equivalent or unequal alternatives are considered. To solve all these problems, a continuous function is introduced that specifies a randomized decision rule leading to continuous linear programming models. As a result, it becomes possible to expand the range of analytically solved problems of the theory of statistical hypothesis testing. In particular, the problem of making a decision based on the maximum power criterion with a fixed type I error probability, with a constraint on the average risk, the problem of testing a simple hypothesis against several alternatives for given type II error probabilities. The method for solving problems of statistical hypothesis testing for the case when more than one observed controlled parameter is used to identify the state is proposed

APPLICATION PROBABILISTIC ECONOMIC MATHEMATICAL MODELS FOR OPTIMIZATION PLANNING OF AN IRREDUCIBLE STOCK AND IMPROVING THE EFFICIENCY OF LOGISTICS SUPPLY MANAGEMENT OF THE STATE DEFENCE FORCES WITHOUT REDUSING THE LEVEL OF RELIABILITY

Предметом вивчення в статті є логістичне забезпечення сил оборони держави. Метою дослiдження є розробка ймовірнісної економіко-математичної моделі управління запасами для планування незнижувального запасу засобів матеріально-технічного забезпечення сил оборони держави. Завдання дослідження: сформулювати задачу визначення розміру незнижувального (резервного) запасу матеріально-технічних засобів за умови, що максимально можливе значення ймовірності його вичерпання протягом визначеного періоду не перевищить граничного значення у термінах економіко-математичної моделі Уілсона; шляхом статистичної обробки даних спостережень дій сил оборони держави при виконанні ними завдань за призначенням представити задачу визначення оптимального незнижувального запасу за видами витрат засобів матеріально-технічного забезпечення без зниження рівня надійності як задачу перевірки багатоальтернативних статистичних гіпотез; на підставі розв’язання задачі континуального лінійного програмування визначити оптимальний незнижувального запасу відповідно характеру витрат з метою підвищення ефективності управління логістичним забезпеченням сил оборони держави засобами матеріально-технічного забезпечення. Методологічною основою дослідження стали загальнонаукові та спеціальні методи наукового пізнання. Отримані такі результати . Витрати запасів матеріально-технічних засобів при виконанні силами оборони держави завдань за призначенням протягом певного періоду часу є нормально розподіленою випадковою величиною з математичним очікуванням та середнєквадратичним відхиленням, обрахованими за результатами статистичної обробки даних. Задача визначення оптимального незнижувального запасу за видами витрат матеріально-технічних засобів без зниження рівня надійності сформульована як задача перевірки багатоальтернативних статистичних гіпотез та розв’язана як задача континуального лінійного програмування. Висновки. Визначення розміру незнижувального (резервного) запасу матеріально-технічних засобів підрозділів сил оборони держави при виконанні ними завдань за призначенням за умови, що максимально можливе значення ймовірності його вичерпання протягом визначеного періоду не перевищить граничного значення доцільно сформулювати у термінах економіко-математичної моделі Уілсона. Припущення про те, що витрати запасів матеріально-технічних засобів при виконанні силами оборони держави при виконанні завдань за призначенням, є випадковою величиною, підкореною нормальному закону розподілу, дозволить скористатися відомим апаратом теорії імовірності та перевірки багато альтернативних статистичних гіпотез та робити розрахунки за даними існуючої статистичної вибірки. Статистична обробка даних, отриманих при виконанні силами оборони держави завдань за призначенням дає змогу отримати вихідні дані та обрахувати прогнозовані оптимальні ймовірності розподілу витрат матеріально-технічних засобів відповідно до поточного, середнього, капітального ремонту та безповоротних втрат.

A two-level vertex-searching global algorithm framework for bilevel linear fractional programming problems

ABSTRACT We propose a new two-level vertex-searching algorithm framework that finds a global optimal solution to the continuous bilevel linear fractional programming problem over a compact polyhedron, in which both the upper and the lower objectives are linear fractional. Our solution method adopts the vertex-searching approach on the polyhedron, and the search space is determined by the set of candidates of optimal base to the lower level problem. In order to search base, a modified enumerative scheme, that is a new upper bound filter scheme inserted into the classical enumerative scheme, is proposed. The main solution procedure is designed on solving a sequence of upper and lower level mathematical programs; instead of a single-level problem reformulation approach, which is popularly and widely used in literature. An extension on general upper level objective functions such as quasiconvex/quasiconcave for the proposed vertex-searching approach is discussed. Numerical experiments show that our algorithm leads us to a global optimum. We conclude that our proposed algorithm framework has the simplest solution procedure and has potential efficiency advantages, which may reduce the complexity of enumerative schemes for medium or large-scale problems, while comparing with existing global algorithms such as the Kth-best algorithm (a two-level vertex-searching algorithm) and the single-level duality-based reformulation algorithm (a single-level vertex-searching algorithm).

A multi-user multi-operator computing pricing method for Internet of things based on bi-level optimization

The Internet of mobile things is a promising paradigm that generates, stores, and processes amount of real-time data to render rich services for mobile users. Along with the increase of mobile devices in the field of Internet of things, more and more intelligent applications, such as face recognition and virtual reality, have emerged. These applications typically consume large amounts of computing and energy resources. However, due to the physical size limitations of Internet of things terminals, their computing capacity and power are limited, where users’ needs for application processing delay and power consumption cannot be met. Therefore, the concept of edge cloud computing has been proposed, which enhances the computing capacity of Internet of things terminals by offloading user tasks to edge servers for computation. When there are multiple operators, it is important to understand how users choose an operator to perform computation and how operators can reasonably price the computing capacity to meet their own interests. Therefore, we study the computation pricing and user decision-making problems of Internet of things under multi-user and multi-operator scenarios. The problem is divided into three phases and modeled as a two-level optimization problem. While an operator’s goal is to minimize the loss of his interests, the user’s goal is to minimize the computation cost (energy consumption and price). First, since the lower-level user decision-making problem is an integer linear programming problem, we transform it into an equivalent continuous linear programming problem by relaxation. Second, we transform the bi-level optimization problem into an equivalent single-level optimization problem by substituting the lower problem’s Karush–Kuhn–Tucker conditions into an upper problem. Finally, we use a spatial branch and bound algorithm to solve the problem. Experimental results show that the proposed algorithm can effectively maintain the benefits of both operators and users in the field of Internet of things.

DYNAMIC PROBLEM OF FORMATION OF SECURITIES PORTFOLIO UNDER UNCERTAINTY CONDITIONS

The analysis of known methods for solving the problem of forming a portfolio of securities in the face of uncertainty is carried out. Traditionally, the problem is solved under the assumption that for each type of asset, the values of the main statistical characteristics of the random value of their profitability (mathematical expectation and variance) are known. At the same time, the variance of portfolio returns, which is minimized, is used as a criterion for portfolio optimization. Two alternative approaches to solving the formulated problem are proposed. The first of them provides a decision on the criterion of the probability that the random total portfolio return will not be lower than the given. It is assumed that the random return for each type of asset is distributed normally and the statistical characteristics of the respective densities are known. The original problem is reduced to the problem of maximizing the quadratic fractional criterion in the presence of linear constraints. To solve this non-standard optimization problem, a special iterative algorithm is proposed that implements the procedure for sequential improvement of the plan. The method converges and the computational procedure for obtaining a solution can be stopped by any of the standard criteria. The second approach considers the possibility of solving the problem under the assumption that the distribution densities of random asset returns are not known, however, based on the results of preliminary statistical processing of the initial data, estimates of the values of the main numerical characteristics for each of the assets are obtained. To solve the problem, a new mathematical apparatus is used – continuous linear programming, which is a generalization of ordinary linear programming to the case when the task variables are continuous. This method, in the considered problem, is based on solving an auxiliary problem: finding the worst-case distribution density of a random total portfolio return at which this total return does not reach an acceptable threshold with maximum probability. Now the main minimax problem is being solved: the formation of the best portfolio in the worst conditions. The resulting computational scheme leads to the problem of quadratic mathematical programming in the presence of linear constraints. Next, a method is proposed for solving the problem of forming a portfolio of securities, taking into account the real dynamics of the value of assets. The problem that arises in this case is formulated and solved in terms of the general theory of control, using the Riccati equation.

Method of hill tunneling via weighted simplex centroid for continuous piecewise linear programming

This paper works on a heuristic algorithm with determinacy for the global optimization of continuous piecewise linear (CPWL) programming. CPWL is widely applied since it can be equivalently transformed into D.C. programming, and further, concave optimization over a polyhedron. Considering that the super-level sets of concave piecewise linear functions are polyhedra, we propose the hill tunneling via simplex centroid (HTSC) algorithm, which is able to escape a local optimum to the other side of its contour surface by cutting across the super-level set. The searching path for the hill tunneling is established by using the centroid of a constructed simplex. In the numerical experiments, the proposed HTSC algorithm is compared with CPLEX and the hill detouring (HD) method, which shows its superior performance on the numerical efficiency and the global search capability.

A simplex-type algorithm for continuous linear programs with constant coefficients

We consider continuous linear programs over a continuous finite time horizon $T$, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions, where we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In previous papers we have shown that these problems possess optimal strongly dual solutions. We also have presented a detailed description of optimal solutions and have defined a combinatorial analogue to basic solutions of standard LP. In this paper we present an algorithm which solves this class of problems in a finite bounded number of steps, using an analogue of the simplex method, in the space of measures.

Analyzing Node-Weighted Oblivious Matching Problem via Continuous LP with Jump Discontinuity

We prove the first non-trivial performance ratio strictly above 0.5 for the weighted Ranking algorithm on the oblivious matching problem where nodes in a general graph can have arbitrary weights. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the weighted and the unweighted versions of the problem. Using a new class of continuous linear programming (LP), we prove that the ratio for the weighted case is at least 0.501512, and we improve the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). Unlike previous continuous LP, in which the primal solution must be continuous everywhere, our new continuous LP framework allows the monotone component of the primal function to have jump discontinuities, and the other primal components to take non-conventional forms, such as the Dirac δ function.

An effective method for optimization of continuous and discrete parameters of heat and power plants

Modern thermal power plants are complex technological systems. Therefore, making informed decisions when studying them requires the use of mathematical modeling and nonlinear optimization methods for plant parameter. The most complex task is to solve a mixed optimization problem wherein a part of optimization parameters vary continuously, and the other can take only discrete (integer) values. An effective method is developed to solve a thermal power plant optimization problem with continuous and discrete parameters. The method suggests an iterative procedure for solving continuous nonlinear programming problems and discrete-continuous linear programming problems. For each iteration, we add new constraints obtained by linearizing nonlinear inequality constraints and the objective function of the initial problem to the system of inequality constraints of linear problem. The effectiveness of the proposed method is exemplified by the optimization of a combined cycle power plant with a mixture of working media (the STIG scheme). Design characteristics of heating surface of a waste heat recovery boiler represent the discrete parameters to be optimized. The research demonstrates a considerable reduction in computational effort compared to the branch and bound method.

Ranking on Arbitrary Graphs: Rematch via Continuous Linear Programming

Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are es...

Linear programming problems on time scales

In this work, we study linear programming problems on time scales. This approach unifies discrete and continuous linear programming models and extends them to other cases ?in between?. After a brief introduction to time scales, we formulate the primal as well as the dual time scales linear programming models. Next, we establish and prove the weak duality theorem and the optimality conditions theorem for arbitrary time scales, while the strong duality theorem is established for isolated time scales. Finally, examples are given in order to illustrate the effectiveness of the presented results.

Hill tunneling method via peak subpoints for continuous piecewise linear programming

Hill tunneling method via peak subpoints for continuous piecewise linear programming