Fuzzy Optimization Problem Research Articles

Optimality and duality for nonconvex fuzzy optimization using granular differentiability method

This paper is concerned with the optimality and duality for a class of nonconvex fuzzy optimization problems under granular differentiability. We first derive optimality conditions for fuzzy optimization problems with fuzzy inequality constraints based on granular differentiability and granular F-convexity. Then, Wolfe type and Mond-Weir type dual models corresponding to the primal problems are described, where their weak, strong, and strict converse dual theorems are explored to illustrate the connection between the optimal solutions of the primal and the dual models. Meanwhile, several examples are provided to support the corresponding theoretical results.

Fuzzy logic in the problems of determining the economic parameters of project implementation

The goal of this work is to forecast project execution terms and its main economic parameters using graph theory and fuzzy logic methods. The subject of research in the article is the method of calculating the main properties or parameters of the project, forecasting the terms of its implementation and the possibility of taking into account force majeure situations. The article discusses the task of finding the optimal plan for the project of creating an e-commerce site and calculating its main economic parameters. The work uses methods of the theory of network planning and management. Fuzzy logic methods are used to solve the fuzzy optimization problem. Graph theory methods, namely the CPM critical path method, are used to forecast project deadlines. The following results were obtained: the main economic parameters were calculated for two versions of the project to create an e-commerce site – sequential, when the team does not have many developers to implement it (or enough resources), and parallel execution of work, which allows you to optimize the execution time by involving additional workers. The cost of both projects was also calculated, which in the future can help managers draw conclusions regarding the implementation of one or another planning option for this type of project. For the first project, the project work plan will be completed in a time not exceeding 230 days, with a reliability of 30%. Or does not exceed 295 days with a reliability of 80% percent. For the second project, the project work plan will be completed in a time not exceeding 230 days, with a reliability of 30%. Or does not exceed 278 days with a reliability of 70% percent. Conclusions: the paper proposes a fuzzy mathematical model for finding the optimal plan and calculating the main economic parameters of the project of creating an e-commerce site with a fuzzy set of plans and a vaguely defined goal. The work also presents a method for solving this problem. The obtained results of the research are of great value for planning the project, for solving the question of the feasibility of its initiation, for forecasting the resources that will be needed for its implementation. These indicators are essential for improving processes and correct allocation of work, which can help strengthen competitiveness and increase project profits.

A Recurrent Neural Network Approach for Constrained Distributed Fuzzy Convex Optimization.

This article investigates a class of constrained distributed fuzzy convex optimization problems, where the objective function is the sum of a set of local fuzzy convex objective functions, and the constraints include partial order relation and closed convex set constraints. In undirected connected node communication network, each node only knows its own objective function and constraints, and the local objective function and partial order relation functions may be nonsmooth. To solve this problem, a recurrent neural network approach based on differential inclusion framework is proposed. The network model is constructed with the help of the idea of penalty function, and the estimation of penalty parameters in advance is eliminated. Through theoretical analysis, it is proven that the state solution of the network enters the feasible region in finite time and does not escape again, and finally reaches consensus at an optimal solution of the distributed fuzzy optimization problem. Furthermore, the stability and global convergence of the network do not depend on the selection of the initial state. A numerical example and an intelligent ship output power optimization problem are given to illustrate the feasibility and effectiveness of the proposed approach.

Using Genetic Algorithms and Core Values of Cooperative Games to Solve Fuzzy Multiobjective Optimization Problems

A new methodology for solving the fuzzy multiobjective optimization problems is proposed in this paper by considering the fusion of cooperative game theory and genetic algorithm. The original fuzzy multiobjective optimization problem needs to be transformed into a scalar optimization problem, which is a conventional optimization problem. Usually, the assignments of suitable coefficients to the corresponding scalar optimization problem are subjectively determined by the decision makers. However, these assignments may cause some biases by their subjectivity. Therefore, this paper proposes a mechanical procedure to avoid this subjective biases. We are going to formulate a cooperative game using the α-level functions of the multiple fuzzy objective functions. Under this setting, the suitable coefficients can be determined mechanically by involving the core values of the cooperative game, which is formulated using the multiple fuzzy objective functions. We shall prove that the optimal solutions of the transformed scalar optimization problem are indeed the nondominated solutions of fuzzy multiobjective optimization problem. Since the core-nondominated solutions will depend on the coefficients that are determined by the core values of cooperative game, there will be a lot of core-nondominated solutions that will also depend on the corresponding coefficients. In order to obtain the best core-nondominated solution, we shall invoke the genetic algorithms by evolving the coefficients.

A note on “A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem”

Moges et al. (Inf. Sci. 625 (2023) 457–475) proposed a method to find a Pareto-optimal solution (POS) of intuitionistic fuzzy multi-objective linear fractional optimization problem (MOLFOP). In this method, firstly, an intuitionistic fuzzy MOLFOP is transformed into a crisp MOLFOP with the help of a ranking function. Then, using the existing method (Fuzzy Sets Syst. 125 (2002) 335–342), the transformed crisp MOLFOP is transformed into a crisp multi-objective linear optimization problem (MOLOP). Finally, it is assumed that to find a POS of the considered intuitionistic fuzzy MOLFOP is equivalent to find a POS of the transformed crisp MOLOP. It is pertinent to mention that this assumption is valid only if the transformed crisp MOLFOP is equivalent to the considered intuitionistic fuzzy MOLFOP as well as the transformed crisp MOLOP is equivalent to the transformed crisp MOLFOP. In this note, a numerical example is considered to show that the transformed crisp MOLFOP is not equivalent to the considered intuitionistic fuzzy MOLFOP. Also, on the basis of the existing results (Fuzzy Sets Syst. 246 (2014) 156–159), it is pointed out that the transformed crisp MOLOP is not equivalent to the transformed crisp MOLFOP. Therefore, Moges et al.’s method is not valid and hence, the results of agricultural planning problem of Ethopia for cultivation of different crops, obtained by Moges et al. with their proposed method, are not correct.

A cooperative neural dynamic model for solving general convex nonlinear optimization problems with fuzzy parameters and an application in manufacturing systems

SummaryIn the presented study, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is calculated using a recurrent neural network (RNN) model. Since there is a few research for solving FNLOP by RNN's, we give a new approach to solve the problem. By reducing the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are given. Moreover, we use the KKT conditions into a RNN as an important tool to solve the problem. Besides, the global convergence properties and the Lyapunov stability of the dynamic model are studied in this study. In the final step, some illustrative examples are considered to establish the obtained results. Reported results are compared with some others network models.

APPLICATION OF FUZZY VECTOR OPTIMIZATION METHODS FOR DIET COMPILATION

The purpose of this work is to study an approach to solving the problem of fuzzy vector optimization for compiling a diet. It is necessary to develop a daily ration that provides human needs in nutrients and energy and is the best in terms of costs and weight. For this purpose the problem of choosing the best alternative from a given fuzzy set of alternatives is solved, while the quality of the alternative is evaluated using several partial efficiency cri- teria. The goal in the task is fuzzily defined. According to the Zadeh – Bellman idea, a fuzzy solution to a problem is the intersection of a fuzzy goal and a fuzzy set of alternatives. The paper considers the problem of fuzzy two-criteria optimization. Partial criteria that are minimized are the weight and cost of the daily ration. Fuzzy needs for nutrients and kilocalories are determined by fuzzy triangular numbers. For an approximate solution of the problem, an algorithm is proposed according to which a sequence of linear programming problems is solved. The computer program was created that implements the solution of this problem in the Wolfram Matematica package. The following results were obtained: a set of products, their weight and cost were determined for the daily diet. The degree of confidence that the found plan is optimal is determined. On the basis of the proposed mathe- matical model and the method of its solution, a software application allowing to choose food products, enter the necessary restrictions in a user- friendly form, and receive one or more options for the daily ration at the output was created. Such an application will be useful for nutritionists, ath- letes, doctors and other people who are concerned with the problems of healthy eating.

Compromise optimum allocation in neutrosophic multi-character survey under stratified random sampling using neutrosophic fuzzy programming

Survey sampling has wide range of applications in social and scientific investigation to draw inference about the unknown parameter of interest. In complex surveys, the sample information about the study variable cannot be expressed by a precise number under uncertain environment due fuzziness and indeterminacy. Therefore, this information is expressed by neutrosophic numbers rather than the classical numbers. The neutrosophic statistics, which is generalization of classical statistics, deals with the neutrosophic data that has some degree of indeterminacy and fuzziness. In this study, we investigate the compromise optimum allocation problem for estimating the population means of the neutrosophic study variables in a multi-character stratified random sampling under uncertain per unit measurement cost. We proposed the intuitionistic fuzzy cost function, modeling the fuzzy uncertainty in stratum per unit measurement cost. The compromise optimum allocation problem is formulated as a multi-objective intuitionistic fuzzy optimization problem. The solution methodology is suggested using neutrosophic fuzzy programming and intuitionistic fuzzy programming approaches. A numerical study includes the means estimation of atmospheric variables is presented to explore the real-life application, explain the mathematical formulation, and efficiency comparison with some existing methods. The results show that the suggested methods produce more precise estimates with less utilization of survey resources as compared to some existing methods. The Python is used for statistical analysis, graphical designing and numerical optimization problems are solved using GAMS.

About defuzzification methods influence on fuzzy traveling salesman problem’s solving

The article investigates the approach to using fuzzy numbers and the method of dynamic programming to find solutions to the traveling salesman problem, considering the fuzzy representation of time in real travel conditions. This allows for formulating a fuzzy optimization problem to find the best value of the objective function, which is determined by the amount of time required to travel between cities. The traveling salesman problem (TSP) is a classic problem of combinatorial optimization, which involves finding the shortest or fastest route among a set of cities. Fuzzy numbers are used to formalize the uncertainty and imprecision of input data, associated with the subjectivity in estimates of the duration of necessary travel intervals. For operating with fuzzy numbers, their transformation into a special form is proposed, and the formalization of the obtained fuzzy results into a crisp representation is carried out based on the center of gravity (CoG) method. A comparison of the results obtained based on solving the deterministic traveling salesman problem using defuzzified time distances and the defuzzification of the solution to the fuzzy traveling salesman problem was conducted. The results confirmed the dependency of the solution on the method of defuzzification. A program was developed that was used to compare the results of the traveling salesman problem using crisp and fuzzy numbers based on the dynamic method. A conclusion is drawn, indicating that the use of trapezoidal fuzzy numbers with the dynamic programming method leads to improved results of the problem compared to using crisp numbers based on the defuzzification of fuzzy distances. Methods of implementation and problematic areas of application of the computation results are presented and analyzed, demonstrating the constructiveness of the proposed approach for studying real processes.

New Approach to Solving Fuzzy Multiobjective Linear Fractional Optimization Problems

In this paper, an iterative approach based on the use of fuzzy parametric functions is proposed to find the best preferred optimal solution to a fuzzy multiobjective linear fractional optimization problem. From this approach, the decision-maker imposes tolerance values or termination conditions for each parametric objective function. Indeed, the fuzzy parametric values are computed iteratively, and each fuzzy fractional objective is transformed into a fuzzy non-fractional parametric function using these values of parameters. The core value of fuzzy numbers is used to transform the fuzzy multiobjective non-fractional problem into a deterministic multiobjective non-fractional problem, and the ε-constraint approach is employed to obtain a linear single objective optimization problem. Finally, by setting the value of parameter ε, the Dangtzig simplex method is used to obtain an optimal solution. Therefore, the number of solutions is equal to the number of used values, and the optimal solution is chosen according to the preference of the decision-maker. We have provided a didactic example to highlight the step of our approach and its numerical performances.

Enhanced Solutions to Intuitionistic Fuzzy Multiobjective Linear Fractional Optimization Problems via Lexicographic Method

Optimization involving fuzzy numbers generally, and intuitionistic fuzzy numbers particularly is more and more an essential part of any computational intelligence method; and might be of interest in modeling fuzzy control systems, or carrying out a fuzzy sensitivity analysis. The recent literature includes many research papers related to both theoretical modelling, and practical implementation of fuzzy decision support systems. Fuzzy optimization is one of the main part in such decisionmaking tools. A realistic solution to a fuzzy optimization problem is always desired, and similarly, a Pareto optimal solution to a multiple objective optimization problem is always a need. In this paper we analyze the shortcomings; and eliminate the weaknesses of a solution approach from the literature proposed by Moges et al. in 2023. Firstly, Moges et al. used an accuracy function to de-fuzzify the original intuitionistic fuzzy optimization problem; then linearized the obtained crisp problem; and finally involved fuzzy goals to solve the derived multiple objective linear problem. We present two faulty key points of their approach, namely the de-fuzzification and linearization steps; prove the inappropriateness of their results; and propose some improvements. The final aim in addressing the first key point is to clarify the border between defuzzifications made in accordance to true theoretical statements, and those made for modeling reasons. The methodological improvement we propose is related to the second key point and it assures that Pareto optimal solutions - highly required in multiple objective optimization - are obtained. To illustrate our point of view, we use a numerical example from the literature, and report better numerical results derived by the improved methodology.

On the minimal solution for max-product fuzzy relation inequalities

<p>Minimal solutions play a crucial role in constructing the complete solution set of the max-product fuzzy relation inequalities, as well as in solving the corresponding fuzzy relation optimization problems. In this work, we propose a sufficient and necessary condition for checking whether a given solution is minimal in the max-product system. Our proposed approach is useful for eliminating non-minimal solutions from the set of all quasi-minimal solutions. Our proposed checking approach helps reduce computational complexity when solving the max-product system or related optimization problems.</p>

Solving Fuzzy Convex Programming Problems via a Projection Neural Network Framework

In this paper, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is given using a projection recurrent neural network (RNN) scheme. Since there is a few research for resolving of FNLOP by RNNs, we describe a new framework to solve the problem. By reformulating the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are obtained. Moreover, we utilize the KKT conditions into a RNN as a capable tool to solve the problem. Besides, the global convergence and the Lyapunov stability of the neuro-dynamic model are established. In the final step, some simulation examples are stated to validate the obtained results. Reported results are compared with some other previous neural networks.

Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms

The fusion of evolutionary algorithms and the solution concepts of cooperative game theory is proposed in this paper to solve the fuzzy optimization problems. The original fuzzy optimization problem is transformed into a scalar optimization problem by assigning some suitable coefficients. The assignment of those coefficients is frequently determined by the decision-makers via their subjectivity, which may cause some biases. In order to avoid these subjective biases, a cooperative game is formulated by considering the α-level functions of the fuzzy objective function. Using the Shapley values of this formulated cooperative game, the suitable coefficients can be reasonably set up. Under these settings, the transformed scalar optimization problem is solved to obtain the nondominated solution, which will depend on the coefficients. In other words, we shall obtain a bunch of nondominated solutions depending on the coefficients. Finally, the evolutionary algorithms are invoked to find the best nondominated solution by evolving the coefficients.

Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator

Solving optimization problems in a fuzzy environment is an area widely addressed in the recent literature. De-fuzzification of data, construction of crisp more or less equivalent problems, unification of multiple objectives, and solving a single crisp optimization problem are the general descriptions of many procedures that approach fuzzy optimization problems. Such procedures are misleading (since relevant information is lost through de-fuzzyfication and aggregation of more objectives into a single one), but they are still dominant in the literature due to their simplicity. In this paper, we address the full fuzzy linear programming problem, and provide solutions in full accordance with the extension principle. The main contribution of this paper is in modeling the conjunction of the fuzzy sets using the “product” operator instead of “min” within the definition of the solution concept. Our theoretical findings show that using a generalized “min” operator within the extension principle assures thinner shapes to the derived fuzzy solutions compared to those available in the literature. Thinner shapes are always desirable, since such solutions provide the decision maker with more significant information.

New preference order relationships and their application to multiobjective interval and fuzzy interval optimization problems

Recently, a method to generate preference ordering relationships in interval space, which is based on a family of automorphisms in bidimensional Euclidean space, and the family of φ-interval optimization problems were introduced in the literature. In this article both the method to generate preference ordering relationships in interval space and the family of φ-interval optimization problems are generalized, providing frameworks for dealing with families of multiobjective interval and multiobjective fuzzy interval optimization problems called the family of φ-multiobjective interval optimization problems and the family of φ-multiobjective fuzzy interval optimization problems, respectively. It is shown that those multiobjective interval and multiobjective fuzzy interval optimization problems presented in the literature, whose formulations are based on a given preference order relationship, can be seen as particular cases of a φ-multiobjective interval optimization problem and of a φ-multiobjective fuzzy interval optimization problems, respectively. Some optimality conditions for both these two families of optimization problems are provided and the methods, concepts and results presented herein are illustrated by several examples.

Lagrangian dual theory and stability analysis for fuzzy optimization problems

This paper investigates the Lagrangian dual theory for a class of fuzzy optimization problems with equality and inequality constraints. To begin with, the Lagrangian dual problem corresponding to the primal fuzzy optimization problem is established by introducing the Lagrangian fuzzy-valued function, and the weak dual theorem is derived. Subsequently, a necessary and sufficient condition for the existence of saddle-points is formulated, which serves as the fundamental basis for proving the strong dual theorem. The stability of the optimization model when its constraints are perturbed by parameters is then analyzed. Finally, the developed Lagrangian dual theory is applied to support vector machines to address the binary classification problem of triangular fuzzy number datasets.

Optimization of time based fuzzy multi-objective reliability redundancy allocation problem for [formula omitted] system using tuning and neighborhood based fuzzy MOPSO algorithm

In this paper, a time based fuzzy multi objective reliability redundancy allocation problem (FMORRAP) is proposed for the xj−out−of−m series–parallel system. The main objective is to maximize the system reliability with minimization of system cost and system repairing cost by optimizing the number of redundant components at each stage. The objectives are achieved by satisfying entropy constraints with limited redundant components at each stage and same for the whole system. Here the uncertainty of reliability, cost and repairing cost of each component is maintained by using triangular fuzzy number (TFN). The decreasing factor of component reliability and cost follows the change in the length of radius for the inverse logarithmic spiral with respect to time. Similarly the increasing factor of component repairing cost follows the same for the logarithmic spiral. Tuning and Neighborhood based Fuzzy Multi-Objective Particle Swarm Optimization (TNF-MOPSO) algorithm is proposed to solve fuzzy multi-objective optimization problems (MOOP). The proposed problem and proposed algorithm are illustrated by using a bench mark problem. The proposed algorithm produced better membership of objective functions for most of the time values and high satisfaction level of reliability for all values of the time parameter. It also outperforms the standard approaches MOPSO and NF-MOPSO.

On optimality for fuzzy optimization problems with granular differentiable fuzzy objective functions

In recent years, there has been a growing interest in optimization problems with uncertainty. Fuzzy optimization is one of the approaches for investigation of such real-world extremum problems that contain uncertain data and, therefore, that are not well defined. Nonetheless, there is not enough discussion on the optimality conditions for optimal solutions (with regard to distinct fuzzy numbers) in (nonconvex) granular differentiable fuzzy optimization problems and methods for solving such uncertain extremum problems. Although the convexity notion is a very important property of optimization models, there are many real-world extremum problems and processes with uncertainty that cannot be modeled as convex fuzzy optimization problems. In this work, therefore, a new notion of granular differentiable generalized convexity is defined. Namely, the definition of granular differentiable F-convexity is introduced for fuzzy optimization problems and some properties of this concept are analyzed. Further, both the Fritz John necessary optimality conditions and, under the Guignard constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions are established for an optimal solution (with respect to distinct fuzzy numbers) in a fuzzy extremum problem with granular differentiable fuzzy-valued objective function and both inequality and equality constraints. As main applications of the introduced concept of granular differentiable F-convexity, the sufficiency of the derived Karush-Kuhn-Tucker necessary optimality conditions are proved and a new approach based on the so-called modified F-objective function method is proposed for solving the investigated fuzzy extremum problem.

Optimization Method of Photovoltaic Microgrid Energy Storage System Based on Price-based DR

Extreme events lead to an increasing number of power outages in the distribution network. It is currently the most effective method to restore power supply after distribution network failure to connect distributed photovoltaic to the distribution network in the form of microgrids. However, the randomness of distributed PV output and load is the biggest obstacle limiting its development. Therefore, an optimization method of photovoltaic microgrid energy storage system (ESS) based on price-based demand response (DR) is proposed in this paper. Firstly, based on the influence of the uncertainty of the time of use (TOU) and load on the price-based DR, a price-based DR model is built. Then, taking the minimum difference between photovoltaic output and load demand as the optimization objective, the optimal operation model of price-based DR based on the fuzzy chance constrained program (FCCP) is established, and an optimization model of photovoltaic microgrid ESS based on FCCP is obtained. Finally, the fuzzy chance optimization problem is converted into a deterministic optimization problem through constraint equivalent treatment. The proposed method is verified by numerical simulations.