Interval Data Uncertainty Research Articles

On the problem of calculating steady state modes of electric power systems under conditions of interval data uncertainty

Currently, there are several approaches for dealing with uncertain (inaccurate) data. The main ones are methods of probabilistic analysis, fuzzy set theory and interval analysis. In this paper, we consider the problem of calculating the parameters of steady-state modes of electric power systems with interval uncertainty of the initial data. The main reasons and the relevance of the use of interval methods for calculating the parameters of steady-state modes of networks of electrical systems are described. First, an interval calculation model is formulated, and then some interval iterative methods for solving nonlinear nodal equations of electrical networks are studied. Algorithms for the interval methods of Gauss-Seidel and Newton-Raphson are proposed for solving nonlinear nodal equations of steady state electrical networks. To demonstrate the level of efficiency of the developed algorithms, several test calculations were carried out with interval parameters formed through the middle and radius of the interval. The results of numerical calculations using these methods show that the Newton-Raphson method is superior to the Gauss-Seidel method in terms of the number of iterations and optimality (with a smaller width) of interval solutions.

Robust worst-practice interval DEA with non-discretionary factors

Traditionally, data envelopment analysis (DEA) evaluates the performance of decision-making units (DMUs) with the most favorable weights on the best practice frontier. In this regard, less emphasis is placed on non-performing or distressed DMUs. To identify the worst performers in risk-taking industries, the worst-practice frontier (WPF) DEA model has been proposed. However, the model does not assume evaluation in the condition that the environment is uncertain. In this paper, we examine the WPF-DEA from basics and further propose novel robust WPF-DEA models in the presence of interval data uncertainty and non-discretionary factors. The proposed approach is based on robust optimization where uncertain input and output data are constrained in an uncertainty set. We first discuss the applicability of worst-practice DEA models to a broad range of application domains and then consider the selection of worst-performing suppliers in supply chain decision analysis where some factors are unknown and not under varied discretion of management. Using the Monte-Carlo simulation, we compute the conformity of rankings in the interval efficiency as well as determine the price of robustness for selecting the worst-performing suppliers.

Improving logic-based Benders’ algorithms for solving min-max regret problems

This paper addresses a class of problems under interval data uncertainty, composed of min-max regret generalisations of classical 0-1 optimisation problems with interval costs. These problems are called robust-hard when their classical counterparts are already NP-hard. The state-of-the-art exact algorithms for interval 0-1 min-max regret problems in general work by solving a corresponding mixed integer linear programming formulation in a Benders’ decomposition fashion. Each of the possibly exponentially many Benders’ cuts is separated on the fly through the resolution of an instance of the classical 0-1 optimisation problem counterpart. Since these separation subproblems may be NP-hard, not all of them can be easily modelled by means of Linear Programming (LP), unless P = NP. In this work, we formally describe these algorithms through a logic-based Benders’ decomposition framework and assess the impact of three warm-start procedures. These procedures work by providing promising initial cuts and primal bounds through the resolution of a linearly relaxed model and an LP-based heuristic. Extensive computational experiments in solving two challenging robust-hard problems indicate that these procedures can highly improve the quality of the bounds obtained by the Benders’ framework within a limited execution time. Moreover, the simplicity and effectiveness of these speed-up procedures makes them an easily reproducible option when dealing with interval 0-1 min-max regret problems in general, especially the more challenging subclass of robust-hard problems

Data fitting problem under interval uncertainty in data

We consider the data fitting problem under uncertainty, which is not described by probabilistic laws, but is limited in magnitude and has an interval character, i.e., is expressed by the intervals of possible data values. The most general case is considered when the intervals represent the measurement results both in independent (predictor) variables and in the dependent (criterial) variables. The concepts of weak and strong compatibility of data and parameters of functional dependence are introduced. It is shown that the resulting formulations of problems are reduced to the study and estimation of various solution sets for an interval system of equations constructed from the processed data. We discuss in detail the strong compatibility of the parameters and data, as more practical, more adequate to the reality and possessing better theoretical properties. The estimates of the function parameters, obtained in view of the strong compatibility, have a polynomial computational complexity, are robust, almost always have finite variability, and are also only partially affected by the so-called Demidenko paradox. We also propose a computational technology for solving the problem of constructing a linear functional dependence under interval data uncertainty and take into account the requirement of strong compatibility. It is based on the application of the so-called recognizing functional of the problem solution set — a special mapping, which recognizes, by the sign of the values, whether a point belongs to the solution set and simultaneously provides a quantitative measure of this membership. The properties of the recognizing functional are discussed. The maximum point of this functional is taken as an estimate of the parameters of the functional dependency under construction, which ensures the best compatibility between the parameters and data (or their least discrepancy). Accordingly, the practical implementation of this approach, named «maximum compatibility method», is reduced to the computation of the unconditional maximum of the recognizing functional — a concave non-smooth function. A specific example of solving the data fitting problem for a linear function from measurement data with interval uncertainty is presented.

Multiobjective bilevel optimisation method to solve environmental-economic power generation and dispatch problem with interval data uncertainty

Multiobjective bilevel optimisation method to solve environmental-economic power generation and dispatch problem with interval data uncertainty

Multiobjective bilevel optimisation method to solve environmental-economic power generation and dispatch problem with interval data uncertainty

This paper presents a multiobjective bilevel programming (MOBLP) based interval goal programming (IVGP) method for modelling and solving to solve environmental-economic power generation and dispatch (EEPGD) problem using genetic algorithm (GA) in an inexact environment. In the proposed method, the objective functions to be optimised are represented as goals by introducing target intervals for the achievement of objective values in decision horizon. Then, using interval arithmetic rules, the defined planned – interval goals are converted into standard goals as in conventional goal programming (GP) to formulate minsum IVGP model of the problem. In the solution search process, a genetic algorithm (GA) computational scheme is employed in an interactive way to reach the solution of EEPGD problem. The effective use of the approach is illustrated through a case example of the standard IEEE 30-bus 6-generator test system. The model solution is also compared with the solutions obtained by using other approaches studied previously.

Robust global error bounds for uncertain linear inequality systems with applications

In this paper, we examine global error bounds for a linear inequality system in the face of data uncertainty by building upon recent developments in robust optimization, where the uncertain parameters are assumed to be in prescribed uncertainty sets. We present necessary and sufficient dual conditions for the existence of robust global error bounds. As a consequence, we obtain a robust Hoffman error bound in the case of commonly used interval data uncertainty, extending the well-known Hoffman's error bound. We then introduce the notion of radius of robust global error bound under interval data uncertainty and present a formula for finding the radius by examining the stability of robust global error bounds. It provides the radius of the largest interval uncertainty set in which the uncertain system admits a robust global error bound. As an immediate application to robust linear programming, we provide conditions for robust feasibility and give a formula for radius of robust feasibility of an uncertain linear program.

Characterizing robust local error bounds for linear inequality systems under data uncertainty

We present necessary and sufficient conditions for the existence of robust local error bounds for linear inequality system in the face of data uncertainty where the uncertain data belong to a prescribed compact uncertainty set. The robust local error bound for an uncertain system is defined in terms of the existence of local error bound for its robust counterpart, where the uncertain linear inequality is enforced for every possible value of the data in the uncertainty set. Some of these conditions are expressed using the projection of the origin and others are given by way of normal cones at the boundary points of the solution set. In the case of commonly used interval data uncertainty, we show that a qualification condition completely characterizes the existence of robust local error bounds.

Robust closed-loop supply chain network design for perishable goods in agile manufacturing under uncertainty

In this study, a general comprehensive model is proposed for strategic closed-loop supply chain network design under interval data uncertainty. The proposed model considers various assumptions such as multiple periods, multiple products, and multiple supply chain echelons as well as uncertain demand and purchasing cost. In addition, bill of materials for each product is considered via a new approach in management of forward and reverse flows of products for producing new products and reusing or disassembling returned products. Uncertainty of parameters in the proposed model is handled via an interval robust optimisation technique. The model assumptions are well matched with decision making environments of food and high-tech electronics manufacturing industries. The factors that make these two industries similar are time-dependent properties of products such as prices and warehousing lifetime period. The computational results of solving the proposed model via LINGO 8 demonstrate efficiency of the proposed model in dealing with uncertainty in an agile manufacturing context.