scholarly journals Universal method for solving optimization problems under the conditions of uncertainty in the initial data

2021 ◽  
Vol 1 (4 (109)) ◽  
pp. 46-53
Author(s):  
Lev Raskin ◽  
Oksana Sira ◽  
Larysa Sukhomlyn ◽  
Yurii Parfeniuk
Keyword(s):  
Mathematical Programming ◽  
Objective Function ◽  
Membership Function ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Original Data ◽  
Structural Basis ◽  
Mathematical Programming Problem ◽  
Universal Method ◽  
Analytical Expression

This paper proposes a method to solve a mathematical programming problem under the conditions of uncertainty in the original data. The structural basis of the proposed method for solving optimization problems under the conditions of uncertainty is the function of criterion value distribution, which depends on the type of uncertainty and the values of the problem’s uncertain variables. In the case where independent variables are random values, this function then is the conventional theoretical-probabilistic density of the distribution of the random criterion value; if the variables are fuzzy numbers, it is then a membership function of the fuzzy criterion value. The proposed method, for the case where uncertainty is described in the terms of a fuzzy set theory, is implemented using the following two-step procedure. In the first stage, using the membership functions of the fuzzy values of criterion parameters, the values for these parameters are set to be equal to the modal, which are fitted in the analytical expression for the objective function. The resulting deterministic problem is solved. The second stage implies solving the problem by minimizing the comprehensive criterion, which is built as follows. By using an analytical expression for the objective function, as well as the membership function of the problem’s fuzzy parameters, applying the rules for operations over fuzzy numbers, one finds a membership function of the criterion’s fuzzy value. Next, one calculates a measure of the compactness of the resulting membership function of the fuzzy value of the problem’s objective function whose numerical value defines the first component of the integrated criterion. The second component is the rate of deviation of the desired solution to the problem from the previously received modal one. Absolutely similarly designed is the computational procedure for the case where uncertainty is described in the terms of a probability theory. Thus, the proposed method for solving optimization problems is universal in relation to the nature of the uncertainty in the original data. An important advantage of the proposed method is the ability to use it when solving any problem of mathematical programming under the conditions of fuzzily assigned original data, regardless of its nature, structure, and type

scholarly journals Fresa: A Plant Propagation Algorithm for Black-Box Single and Multiple Objective Optimization

2021 ◽  
Vol 2 (4) ◽  
Keyword(s):  
Mathematical Programming ◽  
Objective Function ◽  
Optimization Problems ◽  
Population Based ◽  
Black Box ◽  
Multiple Objective ◽  
Multiple Objective Optimization ◽  
Plant Propagation ◽  
Propagation Algorithm ◽  
Multiple Dispatch

Fresa implements a nature inspired plant propagation algorithm for the solution of single and multiple objective optimization problems. The method is population based and evolutionary. Treating the objective function as a black box, the implementation is able to solve problems exhibiting behaviour that is challenging for mathematical programming methods. Fresa is easily adapted to new problems which may benefit from bespoke representations of solutions by taking advantage of the dynamic typing and multiple dispatch capabilities of the Julia language. Further, the support for threads in Julia enables an efficient implementation on multi-core computers.

scholarly journals Approach to optimizing charging infrastructure of autonomous trolleybuses for urban routes

Informatics ◽  
2021 ◽  
Vol 18 (4) ◽  
pp. 79-95
Author(s):  
М. Ya. Kovalyov ◽  
B. M. Rozin ◽  
I. A. Shaternik
Keyword(s):  
Mathematical Programming ◽  
Service Life ◽  
Objective Function ◽  
Heuristic Method ◽  
Cost Effective ◽  
Linear Constraints ◽  
Mathematical Programming Problem ◽  
Electric Transport ◽  
Charging Infrastructure ◽  
Charging Stations

P u r p o s e s.  When designing a system of urban electric transport that charges while driving, including autonomous trolleybuses with batteries of increased capacity, it is important to optimize the charging infrastructure for a fleet of such vehicles. The charging infrastructure of the dedicated routes consists of overhead wire sections along the routes and stationary charging stations of a given type at the terminal stops of the routes. It is designed to ensure the movement of trolleybuses and restore the charge of their batteries, consumed in the sections of autonomous running.The aim of the study is to create models and methods for developing cost-effective solutions for charging infrastructure, ensuring the functioning of the autonomous trolleybus fleet, respecting a number of specific conditions. Conditions include ensuring a specified range of autonomous trolleybus running at a given rate of energy consumption on routes, a guaranteed service life of their batteries, as well as preventing the discharge of batteries below a critical level under various operating modes during their service life.M e t ho d s. Methods of set theory, graph theory and linear approximation are used.Re s u l t s. A mathematical model has been developed for the optimization problem of the charging infrastructure of the autonomous trolleybus fleet. The total reduced annual costs for the charging infrastructure are selected as the objective function. The model is formulated as a mathematical programming problem with a quadratic objective function and linear constraints.Co n c l u s i o n. To solve the formulated problem of mathematical programming, standard solvers such as IBM ILOG CPLEX can be used, as well as, taking into account its computational complexity, the heuristic method of "swarm of particles".  The solution to the problem is to select the configuration of the location of the overhead wire sections on the routes and the durations of charging the trolleybuses at the terminal stops, which determine the corresponding number of stationary charging stations at these stops.

A General Framework for Decomposition Analysis in Optimal Design

1993 ◽  
Author(s):  
Terrance C. Wagner ◽  
Panos Y. Papalambros
Keyword(s):  
Optimal Design ◽  
Mathematical Programming ◽  
Decomposition Method ◽  
General Framework ◽  
Optimization Problems ◽  
Undirected Graph ◽  
Decomposition Analysis ◽  
Decomposition Methods ◽  
Graph Representation ◽  
Mathematical Programming Problem

Abstract Numerous methods exist in the literature for decomposing mathematical programming (MP) problems. The question for the designer wishing to utilize any of these methods is, what (if any) structure exists in a particular problem, and what (if any) decomposition method(s) may be appropriate for the problem at hand. The paper develops a formal methodology, termed Decomposition Analysis, to answer this question. Decomposition of a mathematical programming problem requires identification of linking variables or functions which effect independent optimization problems. Examination of prevalent methods reveals various structures in an MP problem which determine appropriate decomposition methods for a particular problem. An undirected graph representation of the MP problem facilitates rigorous identification of the desired structures that allow decomposition. The representation is the foundation of the methodology to analyze any particular MP problem for its decomposability.

scholarly journals Fractional two-stage transshipment problem under uncertainty: application of the extension principle approach

2021 ◽  
Author(s):  
Harish Garg ◽  
Ali Mahmoodirad ◽  
Sadegh Niroomand
Keyword(s):  
Objective Function ◽  
Membership Function ◽  
Solution Method ◽  
Fuzzy Numbers ◽  
Objective Function Value ◽  
Transformation Method ◽  
Extension Principle ◽  
Two Stage ◽  
Objective Value

AbstractIn this paper, a fuzzy fractional two-stage transshipment problem where all the parameters are represented by fuzzy numbers is studied. The problem uses the ratio of costs divided by benefits as the objective function. A solution method which employs the extension principle is used to find the fuzzy objective value of the problem. For this purpose, the fuzzy fractional two-stage transshipment problem is decomposed into two sub-problems where each of them is tackled individually using various $$\alpha$$ α levels to obtain the fuzzy objective function value and its associated membership function. To deal with the nonlinearity of the objective function the Charnes–Cooper transformation method is embedded to the proposed approach. The superior efficiency of the presented formulation and the proposed solution method is examined over a numerical example as well as a case study comparing to the literature.

scholarly journals Determination of the Optimal Values of the Parameters of Linear Corrective Devices of the Automatic Control System by the Integral Quadratic Criterion

2021 ◽  
pp. 55-74
Author(s):  
Badri Gvasalia ◽  
Keyword(s):  
Control System ◽  
Mathematical Programming ◽  
Automatic Control ◽  
Objective Function ◽  
Automatic Control System ◽  
Random Search ◽  
Extreme Points ◽  
Mathematical Programming Problem ◽  
Nonlinear Mathematical Programming ◽  
Optimal Values

When designing an automatic control system (ACS), it is important to determine the optimal values of the parameters of the correcting device according to any of the criteria. Recently, more and more publications have appeared on the use of the method of nonlinear mathematical programming for solving problems of synthesis of an automated control system. The article discusses a method for determining the optimal parameters of linear correcting devices according to the quadratic integral criterion. The novelty is the presentation of the above mentioned problem in the form of a nonlinear mathematical programming problem. To find a multiparametric, multiextremal, i.e. complex, objective function, a random search method is used. Also, a class of automatic control systems having one extreme objective function is highlighted. For this case, the theorems are proved, and the formula for determining extreme points is given in an analytical form. Numerical examples are also considered. Programs have been developed for implementing the corresponding algorithms on a computer in VBA language. The graphs of the corresponding transients are given.

scholarly journals Lagrange duality and saddle point optimality conditions for semi-infinite mathematical programming problems with equilibrium constraints

2019 ◽  
Vol 29 (4) ◽  
pp. 433-448
Author(s):  
Kunwar Singh ◽  
J.K. Maurya ◽  
S.K. Mishra
Keyword(s):  
Mathematical Programming ◽  
Saddle Point ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Mathematical Programming Problem ◽  
Dual Model ◽  
Equilibrium Constraints ◽  
Convexity Assumptions ◽  
Mathematical Programming Problems

In this paper, we consider a special class of optimization problems which contains infinitely many inequality constraints and finitely many complementarity constraints known as the semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC). We propose Lagrange type dual model for the SIMPEC and obtain their duality results using convexity assumptions. Further, we discuss the saddle point optimality conditions for the SIMPEC. Some examples are given to illustrate the obtained results.

scholarly journals Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1272
Author(s):  
Bogdana Stanojević ◽  
Milan Stanojević ◽  
Sorin Nădăban
Keyword(s):  
Mathematical Programming ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Extension Principle ◽  
Fuzzy Arithmetic ◽  
Advantages And Disadvantages ◽  
Fuzzy Environment ◽  
Basic Ideas ◽  
Mathematical Programming Problems

Optimization problems in the fuzzy environment are widely studied in the literature. We restrict our attention to mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers. Since the review of the recent literature on mathematical programming in the fuzzy environment shows that the extension principle is widely present through the fuzzy arithmetic but much less involved in the foundations of the solution concepts, we believe that efforts to rehabilitate the idea of following the extension principle when deriving relevant fuzzy descriptions to optimal solutions are highly needed. This paper identifies the current position and role of the extension principle in solving mathematical programming problems that involve fuzzy numbers in their models, highlighting the indispensability of the extension principle in approaching this class of problems. After presenting the basic ideas in fuzzy optimization, underlying the advantages and disadvantages of different solution approaches, we review the main methodologies yielding solutions that elude the extension principle, and then compare them to those that follow it. We also suggest research directions focusing on using the extension principle in all stages of the optimization process.

Convex Grey Optimization

2019 ◽  
Vol 53 (1) ◽  
pp. 339-349
Author(s):  
Surafel Luleseged Tilahun
Keyword(s):  
Objective Function ◽  
Optimization Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Probability Distributions ◽  
Optimal Solution ◽  
Decision Variables ◽  
Constraint Set ◽  
Grey Number ◽  
Real Scenario

Many optimization problems are formulated from a real scenario involving incomplete information due to uncertainty in reality. The uncertainties can be expressed with appropriate probability distributions or fuzzy numbers with a membership function, if enough information can be accessed for the construction of either the probability density function or the membership of the fuzzy numbers. However, in some cases there may not be enough information for that and grey numbers need to be used. A grey number is an interval number to represent the value of a quantity. Its exact value or the likelihood is not known but the maximum and/or the minimum possible values are. Applications in space exploration, robotics and engineering can be mentioned which involves such a scenario. An optimization problem is called a grey optimization problem if it involves a grey number in the objective function and/or constraint set. Unlike its wide applications, not much research is done in the field. Hence, in this paper, a convex grey optimization problem will be discussed. It will be shown that an optimal solution for a convex grey optimization problem is a grey number where the lower and upper limit are computed by solving the problem in an optimistic and pessimistic way. The optimistic way is when the decision maker counts the grey numbers as decision variables and optimize the objective function for all the decision variables whereas the pessimistic way is solving a minimax or maximin problem over the decision variables and over the grey numbers.

An Objective Approach for Constructing a Membership Function Based on Fuzzy Harvda-Charvat Entropy and Mathematical Programming

2016 ◽  
Vol 20 (4) ◽  
pp. 535-542
Author(s):  
Takashi Hasuike ◽  
◽  
Hideki Katagiri ◽  
Keyword(s):  
Mathematical Programming ◽  
Decision Maker ◽  
Membership Function ◽  
Piecewise Linear ◽  
Natural Extension ◽  
Interval Estimation ◽  
Optimal Solution ◽  
Mathematical Programming Problem ◽  
Partial Linear ◽  
Intermediate Value

This paper proposes an objective approach to the construction of an appropriate membership function that extends to our previous studies. It is important to set a membership function with subjectivity and objectivity to obtain a reasonable optimal solution that complies with the decision maker’s feelings in real-world decision making. To ensure objectivity and subjectivity of the obtained membership function, an entropy-based approach based on mathematical programming is integrated into the interval estimation considered by the decision maker. Fuzzy Harvda-Charvat entropy, which is a natural extension of fuzzy Shannon entropy, is introduced as general entropy with fuzziness. The main steps of our proposed approach are to set intervals with membership values 0 and 1 to enable a decision maker to judge confidently, and to solve the proposed mathematical programming problem strictly using nonlinear programming. In this paper, the given membership function is assumed to be a piecewise linear membership function as an approximation of nonlinear functions, and each intermediate value of partial linear function is optimally obtained.

scholarly journals Optimum Design of Reinforced Cylindrical Shells Under Combined Axial Compression and Internal Pressure

2021 ◽  
Vol 24 (2) ◽  
pp. 50-58
Author(s):  
Heorhii V. Filatov ◽  
Keyword(s):  
Mathematical Programming ◽  
Internal Pressure ◽  
Objective Function ◽  
Programming Problem ◽  
Random Search ◽  
Cylindrical Shells ◽  
Mathematical Programming Problem ◽  
Stability Conditions ◽  
Search Area ◽  
Critical Stresses

This paper discusses the use of the random search method for the optimal design of single-layered rib-reinforced cylindrical shells under combined axial compression and internal pressure with account taken of the elastic-plastic material behavior. The optimality criterion is the minimum shell volume. The search area for the optimal solution in the space of the parameters being optimized is limited by the strength and stability conditions of the shell. When assessing stability, the discrete rib arrangement is taken into account. In addition to the strength and stability conditions of the shell, the feasible space is subjected to the imposition of constraints on the geometric dimensions of the structural elements being optimized. The difficulty in formulating a mathematical programming problem is that the critical stresses arising in optimally-compressed rib-reinforced cylindrical shells are a function of not only the skin and reinforcement parameters, but also the number of half-waves in the circumferential and meridional directions that are formed due to buckling. In turn, the number of these half-waves depends on the variable shell parameters. Consequently, the search area becomes non-stationary, and when formulating a mathematical programming problem, it is necessary to provide for the need to minimize the critical stress function with respect to the integer wave formation parameters at each search procedure step. In this regard, a method is proposed for solving the problem of optimally designing rib-reinforced shells, using a random search algorithm whose learning is carried out not only depending on the objective function increment, but also on the increment of critical stresses at each extremum search step. The aim of this paper is to demonstrate a technique for optimizing this kind of shells, in which a special search-system learning algorithm is used, which consists in the fact that two problems of mathematical programming are simultaneously solved: that of minimizing the weight objective function and that of minimizing the critical stresses of shell buckling. The proposed technique is illustrated with a numerical example.

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