Mathematical Programming Methods Research Articles

A two-stage real world serial batching scheduling problem: a case study

This paper concerns a real-life production management problem composed of two stages, which includes interdependent batch scheduling problems. Moreover, both stages require setup operations at the beginning and between the subsequent batches. From the optimisation point of view, this production management problem is complying with customers’ deadlines and hence minimising total lateness. To achieve this objective, the production planner must identify optimum or near-optimum batch schedules for both stages. Correspondingly, this paper aims to develop a methodology to manage the related production problem as accurately as possible. We formulate an optimisation model that employs the mathematical programming method in line with this. Afterwards, an algorithmic proposal based on the simulated annealing algorithm is also developed to solve the problem in realistic sizes. The computational capabilities of the developed model and the algorithm are evaluated on the randomly generated problem sets. A direct comparison between the mathematical model and the algorithm shows how efficiently the proposed algorithm solves real-world problems. Also, computational results indicate that the proposed algorithm satisfactorily solves the related real-world batch scheduling problem.

RECONSTRUCTION OF THE CORNEAL SURFACE OF THE HUMAN EYE USING A COMPUTATIONAL EVOLUTIONARY ALGORITHM. PRACTICAL APPLICATION IN NON-PATHOLOGICAL CASES

Increasingly, the use of geometric modelling techniques in Applied Ophthalmology is significant in the characterization of important pathologies of the cornea, such as Keratoconus. This article presents a novel method for the geometric reconstruction of the corneal surface from optical topography using a genetic algorithm. Traditionally, mathematical programming methods such as the least squares method have been used to obtain the coefficients of the corneal surface function, such as Navarro model or Zernike polynomials. This new method uses non-dominated multivariable genetic algorithm optimization to obtain the surface function coefficients from the point cloud obtained with corneal topographer device. Once the reconstruction is performed, the surface is represented using CAD software, and morphogeometric parameters are obtained. The experimental sample consisted in 33 healthy patients eyes, aged from 11 to 63, and without previous ocular surgeries or pathologies. Topographic data were obtained using a Scheimpflug Sirius tomographer (CSO, Italy). The computational optimization was executed under Matlab software environment (Mathworks, USA). The new method provides a lower mean squared error (MSE) than those obtained by the least squares or the nonlinear programming algorithms. Thus, the morphogeometric parameters obtained from the patient's corneas fit better, allowing for a better analysis of real clinical conditions.

Method of operating assessment of the state of network elements to ensure the quality of services in corporate high­speed multiservice communication networks

The article proposes intelligent methods and algorithms for operational assessment of the state of network elements to ensure indicators of the quality of communication services in corporate multi-service communication networks that function in a mode close to real time. A feature of corporate multi-service networks is the high dynamics of changes in their state. And the task of the automated network management system is to ensure the given quality of communication services. Thus, the relevance of this study is determined by the need to implement network management processes in a mode close to real time, with a given quality. The basis of the method of operational assessment of the state of network elements is the concept of using intelligent agents. In the proposed approach, intelligent agents are created as hierarchical fuzzy situational networks, in which management decisions, in contrast to known methods based on the application of reference situations, are produced on the basis of solving a hierarchical system of optimization tasks based on fuzzy mathematical programming methods. The main paradigm of their functioning is «situation – action». This makes it possible to significantly reduce the dimension of the task to be solved and obtain Pareto-optimal solutions for managing the quality of communication services. The intelligent agent of management solutions can be implemented automatically, provided that the network administrator delegates such capabilities to it. The method of estimating traffic parameters is based on the concept of conditional nonlinear Pareto-optimal filtering. Its essence is to apply a two-stage procedure; at the first stage, a forecast of the values of parameter estimates is made, but in the second, with the receipt of the following observations, their values are adjusted. The average relative error of traffic parameter estimation does not exceed 10%, which is sufficiently operational for managing the quality of communication services.

Entropy-metric estimation of the small data models with stochastic parameters

The formalization of dependencies between datasets, taking into account specific hypotheses about data properties, is a constantly relevant task, which is especially acute when it comes to small data. The aim of the study is to formalize the procedure for calculating optimal estimates of probability density functions of parameters of linear and nonlinear dynamic and static small data models, created taking into account specific hypotheses regarding the properties of the studied object. The research methodology includes probability theory and mathematical statistics, information theory, evaluation theory, and stochastic mathematical programming methods. The mathematical apparatus presented in the article is based on the principle of maximization of information entropy on sets determined as a result of a small number of censored measurements of “input" and “output" entities in the presence of noise. These data structures became the basis for the formalization of linear and nonlinear dynamic and static models of small data with stochastic parameters, which include both controlled and noise-oriented input and output measurement entities. For all variants of the above-mentioned small data models, the tasks of determining the optimal estimates of the probability density functions of the parameters were carried out. Formulated optimization problems are reduced to the forms canonical for the stochastic linear programming problem with probabilistic constraints.

Математическое моделирование, оптимизация структуры и режима работы оборудования конденсационных котлов

One of the priority areas of the development of science, technology, and engineering in the Russian Federation is energy saving issues. One of the promising areas to solve the problem of energy saving is to reduce waste heat losses of power plants. The designs of condensing heat exchangers used in industry and energy sector allow both to reduce waste heat losses and to significantly reduce moisture losses. Despite the substantial number of scientific publications on this issue and the positive experience of using the developed designs of condensing heat exchangers, most gas boiler houses and thermal power plants currently continue to operate without deep heat recovery units. To a great extent, it is due to the lack of the universal methods to calculate and optimize heat exchanger modes. Thus, to effective selection of the optimal structure and operating mode of the equipment, the development of mathematical models of power plants with condensing heat exchangers and software packages for their computer implementation is an urgent task. To design a model of a condensing boiler, equations of energy and mass balances are used. To solve the problem of optimal choice of structure and operating mode of the equipment, mathematical programming methods are used. A model and a method to solve the problem of choosing the optimal structure and operating mode of condensation heat exchangers have been developed. As a target optimization function, it is proposed to use the amount of fuel required to provide pre-set heat load. A computer program has been developed for optimal distribution of load between operating units. Analysis of the results obtained has showed an adequate description of real equipment model and the possibility to generate computer mode maps. Application of these maps allows significant savings of energy resources due to the optimal choice of mode and load distribution between operating equipment. The proposed approach allows us to formulate and solve inverse problems of diagnosing the state of condensing heat exchangers.

OPTIMIZING HARD QOS AND SECURITY WITH DISJOINT PATH ROUTING

Background. The combination of secure routing and hard QoS is a worthwhile topic that involves designing and implementing network protocols and systems that can provide high performance and robust protection for data flow due to shared goals. Secure QoS routing over disjoint paths is a challenging problem that requires balancing the trade-off between network security and bandwidth guarantees.
 Objective. This article investigates a mathematical model that can address secure QoS routing by formulating it as an optimization problem with a linear objective function and linear or bilinear constraints. The objective function aims to minimize the paths compromise probability, while the constraints ensure that the total bandwidth of the paths meets the QoS requirements.
 Methods. We use computer simulation of hard QoS and security with disjoint path routing. Also we use mathematical programming methods in order to describe secure QoS routing.
 Results. The article presents a numerical study of the model using different scenarios and parameters. The results show that the model can effectively provide secure QoS routing over disjoint paths with a high bandwidth guarantee level and a low compromise probability. The work analyses the sensitivity of the solutions to the QoS requirements and reveals that there is usually some margin in the bandwidth provision.
 Conclusions. The proposed model is a promising tool for secure QoS routing over disjoint paths in various network environments.

Full-waveform inversion using level set and cut elements for sharp-interface problems

This paper proposes a topology optimization procedure to reconstruct sharp interfaces of salt bodies in a background medium with known properties. The reconstruction technique uses the time-domain acoustic wave equation that provides the pressure field to minimize a data misfit objective function. The salt-background distinction is described by the level set-cut element approach proposed by Andreasen et al. (2020) [1]. This approach preserves the abrupt transition of properties across interfaces and provides accurate solutions to the problem. The nodal values of the level set function are parameterized as an explicit function of the design variables, and a “discretize (space)-then-derive”approach is implemented consistently with the discretized optimization problem. This setting allows for solving the minimization problem with mathematical programming methods; we use the conjugate gradient to update the level set shape-based reconstruction. A set of numerical examples demonstrate that the optimization algorithm is quite robust and is able to identify complex geometries of salt bodies.

Optimal dispatch of integrated energy system based on deep reinforcement learning

Optimized scheduling of integrated energy systems is of great significance for achieving multi-energy complementarity and economic operation of the system. However, the intermittency of renewable energy sources and the uncertainty of user energy demand cause random fluctuations in the supply and demand sides of the system. Traditional scheduling methods are difficult to accurately adapt to the dynamic changes of the actual environment. In view of the uncertainty associated with renewable energy and load in integrated energy systems, an optimal dispatch method based on deep reinforcement learning is proposed. This study first outlines the methodology of deep reinforcement learning and then presents an optimal dispatch model based on this approach. The model incorporates a well-designed state space, action space, and reward function. Next, the process of model solving using the Asynchronous Advantage Actor-Critic (A3C) algorithm is described. Finally, simulation results demonstrate that the proposed method can adaptively respond to the uncertainty of energy sources and loads, and achieve optimal performance comparable to that of traditional mathematical programming methods.

ПАРАМЕТРИЧЕСКИЙ МЕТОД РАСЧЕТА ТРЕУГОЛЬНЫХ ВЕСОВЫХ КОЭФФИЦИЕНТОВ В МАТРИЦЕ ПОПАРНЫХ СРАВНЕНИЙ

The article deals with development of the method for calculation of triangular weight coefficients in fuzzy problem of decision making. The study objective is to improve the accuracy of triangular weight coefficient determination. In order to achieve the above-mentioned objective, a set of methods of scientific and empirical investigation: analysis and splitting of the problem into two tasks, methods of mathematical programming and fuzzy logic, method of comparison. The required weight coefficients are represented in a parametrical form with unlike fuzzy parameters which allowed for two tasks to be separated and formalized: determination of modal values and the task to determine fuzziness parameters. Both tasks have been formulated in terms of non-linear mathematical programming. The proposed method differs from the known ones by using the parameters of triangular numbers fuzziness as optimized variables which are assumed to be different for the left and right borders in all weight coefficients.
 The proposed approach allowed for the accuracy of weight coefficients to be improved. The accuracy validation was conducted comparing with the known LLSM method on two matrices. The calculations have demonstrated that the integral absolute error of the proposed method more than twice smaller than that of the know method for each of the matrices.

An integrated model with a solution algorithm for improving an existing public bicycle sharing system

This study uses a time–space network flow technique combined with a mathematical programming method to construct a holistic model for improving an existing bicycle sharing system. The model considers bicycle rental station site selection, expansion of station capacity, expansion of bicycle fleet size, and bicycle deployment and relocation issues in the same integrated framework. The model can be formulated as an integer network flow problem with side constraints. Since the problem is NP-hard, we design a heuristic algorithm to solve the problem efficiently and effectively. Finally, numerical tests are conducted in order to perform a preliminary evaluation of the model and heuristic algorithm with data from an existing public bicycle sharing system. The results indicate that the model and heuristic algorithm could be helpful to planners to improve their bike-sharing system operations.

A machine learning based branch-cut-and-Benders for dock assignment and truck scheduling problem in cross-docks

In this work, we study the dock-door assignment and truck scheduling problem in cross-docks with an intraday planning horizon. This timeframe allows us to view the problem as a repeating operation with a frequency of 24 h. In practice, this repeating process follows a certain distribution, which is largely sustained even if we extend the horizon. While several modeling approaches and efficient solution algorithms have been proposed for various problem variations, the utilization of decomposition techniques in exact mathematical programming methods has been the most effective. Surprisingly, none of these techniques have taken advantage of the repeating patterns inherent in the problem. We start with a recently proposed compact model that is well-designed and can be exploited in a primal (Benders) decomposition technique, although it cannot be directly used to solve a practical-sized problem. We show that its modeling deficiencies can be fixed and propose a Benders decomposition framework together with several Alternative Objective Functions (AOFs) to generate customized Benders cuts, along with other valid inequalities that can be identified and separated. A classifier is trained to identify the most efficient AOF to use at different stages of the Benders iterations, to help avoid saturation of the master problem with dominated Benders cuts. Our extensive computational experiments confirm the significant performance improvement in the comparable decomposition framework.

Система непрерывного мониторинга технического состояния и оперативной диагностики паровой турбины

Power equipment deterioration occurs during operation, it causes loss in reliability and efficiency, unscheduled shutdowns, and accidents. Now predictive analytics is one of the promising directions in the field of power engineering which allows to control and analyze the technical condition of power equipment. The problems of localization of deviation of technological parameters and detection of anomaly in the operation of power equipment are consistently solved in the framework of predictive analytics. The problems of deviations localization and anomalies detection are solved by the methods of statistical modeling and the classification algorithms respectively. However, for steam turbines the localization of deviation and the detection of anomalies having slow-flowing character are a difficult problem. Therefore, the issue of development of a method for continuous monitoring of technical condition and maintenance diagnostics based on a mathematical model of the steam turbine section flow characteristics is worth noticing. The method allows us to consider the effect of changes in the open flow area of the individual sections of a steam turbine on the pressure distribution over the steam flow path. The model of a steam turbine has been developed within the matrix formalization methodology. The solution of the system of linear and nonlinear equations is carried out by methods of computational mathematics. The solution of the optimization problems of the steam flow path diagnostics is carried out by methods of mathematical programming. A mathematical model of the cogeneration steam turbine Т-250/305-23.5-DB and a method for continuous condition monitoring and maintenance diagnostics of steam turbine have been developed. It allows us to localize deviation and detect anomaly by recovery of the open flow area of the individual sections of a steam turbine based on the pressure distribution over the steam flow path. The results of the statistical analysis prove that a mathematical model of the cogeneration steam turbine Т-250/305-23.5-DB has been recognized adequate. The method for continuous condition monitoring and maintenance diagnostics of steam turbine has demonstrated consistency of the obtained results and ability to solve diagnostic problems in practice. The developed model and method can be used as a module in the development of a software package for predictive analytics of power equipment.

OPTIMIZATION METHODS IN CALCULATIONS OF MECHANISMS OF VEHICLES OF TRANSPORT

Purpose. The optimal design of vehicle mechanisms is a detailed review and evaluation of the scientific and technical literature and individual publications on the methodology for optimizing the inherent calculations of these types of mechanical structures. The main purpose of this review material is to consider a number of directions for designing mechanisms of means of transport, especially highlighting the use of optimization methods. Research in the field of designing mechanisms is carried out in two directions: by the synthesis method according to given provisions and by the method of mathematical programming. Methods. Methods for the synthesis of hinged-lever mechanisms can be divided into two groups. In the first one, methods of approximation of functions are used, in the second one, optimization methods. The search for the optimal solution in the second group means the simultaneous search for the correct formulation of the problem, in which the parameters of the mechanisms should be sought under conditions of the best satisfaction of the requirements of the technological process. The problems of optimal design of mechanisms most often continue to be formulated as problems of the best approximation of functions. Optimization synthesis is a relatively new direction in the synthesis of lever mechanisms. Optimization synthesis is carried out on the basis of nonlinear optimization methods. At present, the complexity and complexity of the problems that arise and require solution in the process of designing mechanisms have significantly increased. A certain amount of research has been carried out within the framework of the classical theory of finite and infinitely small plane displacements. In these works, the theory of spatial motion of a general form is developed in detail and applied to the problems of kinematic synthesis. Results. Since the simplex method is quite effective in solving problems of linear programming, it became the founder of a number of methods of sequential linearization or methods of secant planes for solving nonlinear programming. Practical significance. With an extensive list of modern literature and scientific publications on the synthesis of mechanisms and mathematical programming, a complete chronological review of the achievements of the last decade would be quite loud. Therefore, individual trends in the synthesis of mechanisms were only noted, focusing on the desire to obtain a “better” design, taking into account practical requirements. The synthesis of mechanisms is an area in which the application of mathematical programming methods is quite promising due to their free formulation. The constraints and requirements of engineering practical problems can often be taken into account in forms consistent with the data of modern methods. The main purpose of this study is to make widely known the achievements in the two fields with the hope that currently existing optimization tools will be applied to these synthesis problems and that, conversely, the optimization tools will be further improved as a result of research on the design of vehicle mechanisms. This will contribute to a better understanding and accelerated development of both mechanisms calculation methods and optimization methodology.

Random update particle swarm optimizer (RUPSO): A novel robust optimization algorithm

Achieving an optimal design, especially those with numerous variables, is a tedious process. It is regularly achieved by mathematical programming or Metaheuristic algorithm methods. A novel metaheuristic optimization algorithm is presented in this research. The proposed algorithm is based on the well-known Particle Swarm Optimization (PSO) algorithm, in which the updating process has been improved. The efficiency of the new method, named Random Update Particle Swarm Optimizer (RUPSO), has been verified for problems with continuous variables as well as problems with discrete variables. It is shown that RUPSO converges to better optimum solutions than PSO and other existing metaheuristic algorithms. It is for its capability of searching through the entire feasible search spaces and going over the local optima. Random updating process, which has been utilized in RUPSO, decreases the sensitivity of the algorithm and increases its stability in solving optimization problems. High stability of the method is also shown in the paper. The RUPSO algorithm finds the optimized answers with a very high convergence rate for problems with discrete or even continuous variables.

Optimization for Asphalt Pavement Maintenance Plans at Network Level: Integrating Maintenance Funds, Pavement Performance, Road Users, and Environment

As an infrastructure project, the cost and benefits of road maintenance should be measured through social costs. In order to fully consider user interests and make road maintenance decisions more reasonable, two types of user costs that may affect road maintenance benefits have been quantified through analysis of existing literature. At the same time, environmental issues have gradually become the focus of various industries, and in optimizing road maintenance decisions, the impact of environmental issues on decision making should also be considered. This article first analyzes and quantifies the user costs that affect the effectiveness of road maintenance. Secondly, based on the concept of sustainable development of roads, the optimization of road maintenance decisions is divided into two steps: the first step is to determine the minimum maintenance budget funds, and the second step is to determine the optimal plan. The specific optimization method is to use the 0-1 mathematical programming method to establish a network-level pavement maintenance decision optimization model based on a quantitative model. The most reasonable maintenance optimization plan is determined from four aspects: maintenance funds, maintenance performance improvement value, user benefit improvement, and reducing environmental impact. Finally, a provincial road network for a case study is selected. The applicability of the new model is verified through a case study. This study can help decision makers deal with asphalt pavement maintenance arrangements at the network level with four decision objectives: maintenance funds, pavement performance, road users, and environmental impacts.

Optimal routing and scheduling of unmanned aerial vehicles for delivery services

ABSTRACT In the logistics industry, unmanned aerial vehicles (UAVs) are mostly used for last-mile delivery, in combination with other types of vehicles. There is currently no operator in Taiwan that is completely reliant upon the usage of UAVs for cargo delivery services. Therefore, this study proposes a routing and scheduling model for UAVs by utilizing the network flow technique and mathematical programming methods. All advance requests must be satisfied, and the related operating constraints ensured in the model. The model aims to minimize the total operating cost. To effectively solve large problems that may occur in practice, this study develops a relax-and-fix heuristic. Numerical tests are conducted to preliminarily examine whether the model, coupled with the heuristic algorithm, could be applied in practice. The test results indicate that the proposed model and solution algorithm are effective and thus could be useful for UAV operators to perform delivery routing and scheduling.

IMPROVING THE EFFICIENCY OF 6–10 KV POWER GRIDS OF INDUSTRIAL ENTERPRISES THROUGH OPTIMAL REACTIVE POWER CONTROL

"The article deals with improving the efficiency of an industrial power system, which is determined by the level of power losses in its elements. One of the most important methods of increasing the efficiency of power supply systems and distribution networks that ensure the transmission and distribution of electricity is to reduce the amount of electrical energy that is expended in this process. Reducing power losses in electrical grids is achieved by reducing the reactive component of the electric current, i.e., by increasing the cos φ. The Electrical Installation Code recommends, first of all, using the synchronous motors available at the enterprise. It is reasonable to solve the problem of minimizing active power losses in the power grid using the method of mathematical modeling and mathematical programming. The article presents the main features of the criterion programming method for solving a class of problems with a positive degree of difficulty, which is quite well-known in the electric power industry for solving optimization problems. We propose the method of criterion programming to solve the problem of improving the efficiency of the power grid by optimal control of the operation of synchronous motors according to the reactive power mode. "

The measurement of profit, profitability, cost and revenue efficiency through data envelopment analysis: A comparison of models using BenchmarkingEconomicEfficiency.jl

We undertake a systematic comparison of existing models measuring and decomposing the economic efficiency of organizations. For this purpose we introduce the package BenchmarkingEconomicEfficiency.jl for the open-source Julia language including a set of functions to be used by scholars and professionals working in the fields of economics, management science, engineering, and operations research. Using mathematical programming methods known as Data Envelopment Analysis, the software develops code to decompose economic efficiency considering alternative definitions: profit, profitability, cost and revenue. Economic efficiency can be decomposed, multiplicative or additively, into a technical (productive) efficiency term and a residual term representing allocative (or price) efficiency. We include traditional decompositions like the radial efficiency measures associated with the input (cost) and output (revenue) approaches, as well as new ones corresponding to the Russell measures, the directional distance function, DDF (including novel extensions like the reverse DDF, modified DDF, or generalizations based on Hölder norms), the generalized distance function, and additive measures like the slack based measure, their weighted variants, etc. Moreover, regardless the underlying economic efficiency model, many of these technical inefficiency measures are available for calculation in a computer software for the first time. This article details the theoretical methods and the empirical implementation of the functions, comparing the obtained results using a common dataset on Taiwanese Banks.

Numerical study of the parametric optimization of the forced oscillation frequencies of the shell of a minimal surface on a trapezoidal contour under thermal and power loading

This research paper discusses various methods and approaches to the optimal design of structures. Methods for solving the optimization problem can be divided into two large groups. The first group includes methods that are based on the use of the necessary conditions for the extremes of the objective function. The second group consists of mathematical programming methods: linear, convex, dynamic programming, and random search. In mathematical terms, optimal design problems are optimization problems - the search for an extremum of the objective function and the values of the parameters at which the extremum is achieved. The choice of the optimality criterion is one of the main problems of optimal design. The most widely developed problems are those that have the optimization criterion of weight or volume of the structure while satisfying the conditions of strength, rigidity and stability.
 Optimal design problemsare also divided into three large groups. The first group is parametric optimization problems, which involve the optimization of one or more parameters, called design variables, to minimize or maximize the objective function. The second group is topological optimization, in which unnecessary material is discarded, where the Mises stress is zero, thereby minimizing the objective function. The third group is optimization of the shape of the object under study, when the shape corresponds to internal forces, the shell with the smallest area is modeled on a given cone (shells of minimal surfaces), as well as methods of applied geometry, where the surface shape is modeled for a certain load.
 To perform the parametric optimization of the forced vibrations of the shell of the minimum surface on a trapezoidal contour, the objective function is the weight of the spatial structure. The variables in the parametric optimization problem are the thickness of the finite elements from 1 to 100 mm. The structure constraint is imposed on the first forced oscillation frequency of 0.250 Hz. This type of problem is used to prevent resonance from process equipment that can affect the natural frequencies of the structure under external load.Subject of this study is an interesting applied problem for construction mechanics, as it is the first time to display the application of two types of optimization on one research object.
 The results of a numerical study of the parametric optimization of the minimum surface shell on a trapezoidal cage under thermal power loading. The parametric optimization helped to reduce the weight of the shell by 13.4%, which is 1810 kg of sheet steel. The first forced oscillation frequency meets the constraint of the optimization calculation. We constructed 10 forced vibration frequency shapes of the shell before and after optimization, and also presented the distribution of the shell thickness after the optimization calculation.

Mathematical programming methods for reducing inequity due to stock-outs of pharmaceuticals in low-resourced regions using a mobile pharmacy

Delivering essential pharmaceuticals to consumers in low- and middle-income countries (LMICs) is a complex global challenge that requires equitable solutions in the last mile of pharmaceutical supply chains. This paper proposes using a mobile pharmacy to alleviate the burden due to pharmaceutical stock-outs among rural communities from an equity lens. The Stock-out Severity Index (SSI) is used to assess inequity and is used as the equity objective in a mobile facility routing model. We develop two mathematical programming approaches for routing of the mobile pharmacy so as to minimize the mean absolute deviation of the SSI. The first is a mixed integer program (MIP) modeling approach, based on an approximation of the SSI. The second is an implicit enumeration approach that is implemented within a chronological decomposition heuristic. To help establish its applicability potential, the paper describes a case study that demonstrates how a mobile pharmacy can effectively reduce inequity in a section of rural Uganda. In the case study, costs associated with improving equity are evaluated. This paper finds that optimizing for equity only is associated with high operational costs, and demonstrates approaches that achieve equitable solutions though constrained cost increases.