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Full Fuzzy Fractional Programming Based on the Extension Principle
Abstract
We address the full fuzzy linear fractional programming problem with LR fuzzy numbers. Our goal is to revitalize a strict use of extension principle by employing it in all stages of our solution approach, thus deriving results that fully comply to it. Using the -cuts of the coefficients we present the linear optimization models that empirically derive the -cuts of the optimal objective fuzzy value, and discuss the optimization models able to derive the exact endpoints of the optimal objective values intervals. For initial maximization (minimization) problems the main issue is related to how to solve two stage min-max (max-min) problems to obtain the left (right) most endpoints. Our goals are as it follows: to obtain exact solutions to small-size problems; to obtain relevant information about solutions to large-scale problems that are in accordance to the extension principle; and to provide a procedure able to measure to which extent the solutions obtained by an approach to full fuzzy linear fractional programming comply to the extension principle. We illustrate the theoretical findings reporting numerical results, and including a relevant comparison to the results from the literature.KeywordsLinear fractional programmingLR fuzzy numbersExtension principle
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In this paper, we propose a solution approach to solving full fuzzy multiple objective linear fractional problems based on Zadeh’s extension principle. We adopt the idea of using triangular fuzzy numbers for the coefficients of the original problem and derive the shapes of the fuzzy variables with respect to the extension principle. The solution concept built in the novel approach strictly follows the basic arithmetic of fuzzy numbers, and the developed methodology contributes to correcting some inconsistencies in an existing approach from the recent literature. The solution we propose to the original problem is constructed out of the non-dominated points of crisp multiple objective linear fractional problems formed with feasible values of the fuzzy coefficients. The membership degree of each identified non-dominated point is computed with respect to the membership degrees of the coefficients involved. Our empirical results confirm and clearly illustrate the theoretical foundations
This article presents a fuzzy multi-objective linear fractional programming (FMOLFP) problem. The goal programming (GP) approach is used to solve the proposed problem. The LR (Left and Right) possibilistic variables are addressed to the suggested the fuzzy multi-objective linear fractional programming (FMOLFP) model to deal the uncertainty of the model parameters. An auxiliary model in which objective function is the distance between the pÀ ary aÀ optimal value restriction and pÀ ary fuzzy objective function is proposed. In the last, one solved example is given to illustrate and to support the validity of the suggested approach.
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.
In this paper, we discuss fully fuzzy linear fractional programming (FFLFP) problems under fuzzy nature with triangular fuzzy numbers. We recommend a simple method for the solution of FFLFP problems which preserves the fuzzy nature of the problem. The efficiency of the method proposed is illustrated with an example.
During the last decades, the art and science of fuzzy logic have witnessed significant developments and have found applications in many active areas, such as pattern recognition, classification, control systems, etc. A lot of research has demonstrated the ability of fuzzy logic in dealing with vague and uncertain linguistic information. For the purpose of representing human perception, fuzzy logic has been employed as an effective tool in intelligent decision making. Due to the emergence of various studies on fuzzy logic-based decision-making methods, it is necessary to make a comprehensive overview of published papers in this field and their applications. This paper covers a wide range of both theoretical and practical applications of fuzzy logic in decision making. It has been grouped into five parts: to explain the role of fuzzy logic in decision making, we first present some basic ideas underlying different types of fuzzy logic and the structure of the fuzzy logic system. Then, we make a review of evaluation methods, prediction methods, decision support algorithms, group decision-making methods based on fuzzy logic. Applications of these methods are further reviewed. Finally, some challenges and future trends are given from different perspectives. This paper illustrates that the combination of fuzzy logic and decision making method has an extensive research prospect. It can help researchers to identify the frontiers of fuzzy logic in the field of decision making.
In this paper, proposing a mathematical model with disjunctive constraint system, and providing approximate membership function shapes to the optimal values of the decision variables, we improve the solution approach to transportation problems with trapezoidal fuzzy parameters. We further extend the approach to solving transportation problems with intuitionistic fuzzy parameters; and compare the membership function shapes of the fuzzy solutions obtained by our approach to the fuzzy solutions to full fuzzy transportation problems yielded by approaches found in the literature.
Several methods currently exist for solving fuzzy linear fractional programming problems under nonnegative fuzzy variables. However, due to the limitation of these methods, they cannot be applied for solving fully fuzzy linear fractional programming (FFLFP) problems where all the variables and parameters are fuzzy numbers. So, this paper is planning to fill in this gap and in order to obtain the fuzzy optimal solution we propose a new efficient method for FFLFP problems utilized in daily life circumstances. This proposed method is based on crisp linear fractional programming and has a simple structure. To show the efficiency of our proposed method some numerical and real life problems have been illustrated. Keywords: Fully fuzzy linear fractional programming problem (FFLFPP); Triangular fuzzy numbers; Ranking function. MSC: 90C31, 91A35, 94D05.
Data envelopment analysis (DEA) is a prominent technique for evaluating relative efficiency of a set of entities called decision making units (DMUs) with homogeneous structures. In order to implement a comprehensive assessment, undesirable factors should be included in the efficiency analysis. The present study endeavors to propose a novel approach for solving DEA model in the presence of undesirable outputs in which all input/output data are represented by triangular fuzzy numbers. To this end, two virtual fuzzy DMUs called fuzzy ideal DMU (FIDMU) and fuzzy anti-ideal DMU (FADMU) are introduced into proposed fuzzy DEA framework. Then, a lexicographic approach is used to find the best and the worst fuzzy efficiencies of FIDMU and FADMU, respectively. Moreover, the resulting fuzzy efficiencies are used to measure the best and worst fuzzy relative efficiencies of DMUs to construct a fuzzy relative closeness index. To address the overall assessment, a new approach is proposed for ranking fuzzy relative closeness indexes based on which the DMUs are ranked. The developed framework greatly reduces the complexity of computation compared with commonly used existing methods in the literature. To validate the proposed methodology and proposed ranking method, a numerical example is illustrated and compared the results with an existing approach.
A wide variety of solution approaches to linear programming problems in fuzzy environment are proposed in the recent literature. For this study we consider a linear programming problem with fuzzy inequality constraints, and both coefficients and decision variables described by trapezoidal fuzzy numbers. Our solution approach takes into consideration decision maker’s acceptance degree of the violated fuzzy constraints.
We use the interval expectation of the trapezoidal fuzzy numbers to transform the original problem into an interval optimization problem. Then, using an order relation to rank the intervals; and handling the acceptance degree of the violated fuzzy constraints as a parameter in the optimization model, we analyze the Pareto optimal solutions to a parametric bi-objective linear programming problem. For a fixed value of the acceptance degree provided by the decision maker, but after a parametric analysis with respect to the parameter used for aggregating the two objectives, the decision maker becomes better informed about the nature of the problem he has to complete.
We illustrate our new solution approach using numerical examples found in the literature, and emphasize its advantages.
In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.
We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, -cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately.
This article presents an algorithm for solving fully fuzzy multi-objective linear fractional (FFMOLF) optimization problem. Some computational algorithms have been developed for the solution of fully fuzzy single-objective linear fractional optimization problems. Veeramani and Sumathi (Appl Math Model 40:6148–6164, 2016) pointed out that no algorithm is available for solving a single-objective fully fuzzy optimization problem. Das et al. (RAIRO-Oper Res 51:285–297, 2017) proposed a method for solving single-objective linear fractional programming problem using multi-objective programming. Moreover, it is the fact that no method/algorithm is available for solving a FFMOLF optimization problem. In this article, a fully fuzzy MOLF optimization problem is considered, where all the coefficients and variables are assumed to be the triangular fuzzy numbers (TFNs). So, we are proposing an algorithm for solving FFMOLF optimization problem with the help of the ranking function and the weighted approach. To validate the proposed fuzzy intelligent algorithm, three existing classical numerical problems are converted into FFMOLF optimization problem using approximate TFNs. Then, the proposed algorithm is applied in an asymmetric way. Since there is no algorithm available in the existing literature for solving this difficult problem, we compare the obtained efficient solutions with corresponding existing methods for deterministic problems.
In this paper, authors devoted to study a fully fuzzy fractional multi-objective transportation problem by using goal programming approach. Also trapezoidal membership functions are applied to each objective function and constraints to describe a each fuzzy goal. A numerical example is provided to illustrate the efficiency of the multi-objective proposed approach.
This paper presents optimality criteria for fuzzy-valued fractional multi-objective optimization problem. There are numerous optimality criteria which have been established for the deterministic fractional multi-objective optimization problems. Very few studies are available on the establishment of optimality criteria for fuzzy-valued multi-objective optimization problem. So, Karush–Kuhn–Tucker optimality criteria for fuzzy-valued fractional multi-objective problem are established by using Lagrange multipliers. First, the original problem is modified using the parametric approach of Dinkelbach into multi-objective non-fractional optimization problem, and then, the optimality conditions are established for the modified problem using the Hukuhara derivative. The established optimality criteria are verified by two numerical examples.
In this paper, we develop intuitionistic fuzzy data envelopment analysis (IFDEA) and dual IFDEA (DIFDEA) models based on - and -cuts. We determine intuitionistic fuzzy (IF) efficiencies based on - and -cuts. We develop an IF correlation coefficient (IFCC) between IF variables to validate the DIFDEA models. We propose an index ranking approach to rank the decision making units (DMUs). Also, we propose an approach to find the IF input–output targets which help to make inefficient DMUs as efficient DMUs in IF environment. Finally, an example and a health sector application are presented to illustrate and compare the proposed methods.
This paper deals with developing an efficient algorithm for solving the fully fuzzy linear fractional programming problem. To this end, we construct a new method which is obtained from combination of Charnes−Cooper scheme and the multi-objective linear programming problem. Furthermore, the application of the proposed method in real life problems is presented and this method is compared with some existing methods. The numerical experiments and comparative results presented promising results to find the fuzzy optimal solution.
The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.
In the present paper, we propose a new approach to solving the full fuzzy linear fractional programming problem. By this approach, we provide a tool for making good decisions in certain problems in which the goals may be modelled by linear fractional functions under linear constraints; and when only vague data are available. In order to evaluate the membership function of the fractional objective, we use the α-cut interval of a special class of fuzzy numbers, namely the fuzzy numbers obtained as sums of products of triangular fuzzy numbers with positive support. We derive the α-cut interval of the ratio of such fuzzy numbers, compute the exact membership function of the ratio, and introduce a way to evaluate the error that arises when the result is approximated by a triangular fuzzy number. We analyse the effect of this approximation on solving a full fuzzy linear fractional programming problem. We illustrate our approach by solving a special example – a decision-making problem in production planning.
In practice, some special LR fuzzy numbers, like the triangular fuzzy number, the Gaussian fuzzy number and the Cauchy fuzzy number, are widely used in many areas to deal with various vague information. With regard to these special LR fuzzy numbers, called regular LR fuzzy numbers in this paper, an operational law is proposed for fuzzy arithmetic, providing a novel approach to analytically and exactly calculating the inverse credibility distribution of some specific arithmetical operations based on the credibility measure. As an application of the operational law, an equivalent form of the expected value operator as well as a theorem for computing the expected value of strictly monotone functions is suggested. Finally, we utilize the operational law to construct a solution framework of fuzzy programming with parameters of regular LR fuzzy numbers, and such type of fuzzy programming problems can be handled by the operational law as the classic deterministic programming without any particular solving techniques.
The conventional extension principle was established on the Euclidean space and defined by considering the minimum or t-norm operator. The generalized extension principle proposed in this paper is established on the Hausdorff space and defined by considering an operator that is more general than the t-norm operator. On the other hand, based on the topological structure, we also discuss the properties of 0-level sets by considering the closure. Many interesting and useful equalities considering the 0-level sets will be obtained in this paper.
Mathematical programming is one of the areas to which fuzzy set theory has been applied extensively. Primarily based on Bellman and Zadeh's model of decision in fuzzy environments, models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming, in integer and fractional programming, and in dynamic programming. These models in turn have been used to offer computationally efficient approaches for solving vector maximum problems. This paper surveys major models and theories in this area and offers some indication on future developments which can be expected.
In this paper we aim to provide empirical solutions to a special class of full fuzzy linear fractional programming problems. We use trapezoidal fuzzy numbers to describe the parameters and derive empirical shape of the membership of the goal function optimal values of the problem. Our approach essentially follows the extension principle, and is based on solving crisp quadratic optimization problems. The model we propose treats in different ways, through two independent parameters, the objective function coefficients and coefficients in the constraints. To illustrate our theory, we solve a relevant instance and compare our numerical results with the numerical results recalled from the recent literature.
In this paper we show how a parametric discussion on the optimal objective values to two mathematical models involved in a fuzzy standard data envelopment analysis can provide an analytic description to the membership functions of the fuzzy efficiencies of the decision-making units. We recall the mathematical models under discussion from the literature, but we approach them from a novel perspective, thus providing an analytical alternative to the numerical methods used so far in the literature.
This study first surveys fuzzy linearization approaches for solving multi-objective linear fractional programming (MOLFP) problems. In particular, we review different existing methods dealing with fuzzy objectives on a crisp constraint set. Those methods transform the given MOLFP problem into a linear or a multi-objective linear programming (LP or MOLP) problem and obtain one efficient or weakly efficient solution of the main MOLFP problem. We show that one of these popular existing methods has shortcomings, and we modify it to be able to find efficient solutions. The main idea of LP-based methods is optimizing a weighted sum of numerators and the negative form of denominators of the given fractional objective function over the feasible set. We prove there is no weight region to guarantee the efficiency of the optimal solutions of such LP-based methods whenever the interior of the feasible set is nonempty. Moreover, MOLP-based methods obtain an equivalent MOLP problem to the main MOLFP problem using fuzzy set techniques. We prove MOLFP problems with a non-closed efficient set are not equivalent to MOLP ones whenever the equivalency mapping is continuous.
This paper proposes a fuzzy optimization problem whose objective function is Zadeh's extension of a function with respect to a parameter and the independent variable. Making use of a partial order relation, the extension of a function mapping each parameter to the corresponding local minimizing point is proven to provide the local smallest fuzzy value for the extended function. Deploying the usual order relation in the literature, the same point is proven to provide the minimal value for the function. Examples illustrate the results.
This study surveys the use of fuzzy numbers in classic optimization models, and its effects on making decisions. In a wide sense, mathematical programming is a collection of tools used in mathematical optimization to make good decisions. There are many sectors of economy that employ it. Finance and government, logistics and manufacturing, the distribution of the electrical power are worth to be first mentioned.
When real life problems are modeled mathematically, there is always a trade-off between model’s accuracy and complexity. By this survey, we aim to present in a concise form some mathematical models from the literature together with the methods to solve them. We will focus mainly on fuzzy fractional programming problems. We will also refer to but not describe in detail the multi-criteria decision-making problems involving fuzzy numbers and linear fractional programming models.
Linear ranking functions are often used to transform fuzzy multiobjective linear programming (MOLP) problems into crisp ones. The crisp MOLP problems are then solved by using classical methods (eg, weighted sum, epsilon‐constraint, etc), or fuzzy ones based on Bellman and Zadeh's decision‐making model. In this paper, we show that this transformation does not guarantee Pareto optimal fuzzy solutions for the original fuzzy problems. By using lexicographic ranking criteria, we propose a fuzzy epsilon‐constraint method that yields Pareto optimal fuzzy solutions of fuzzy variable and fully fuzzy MOLP problems, in which all parameters and decision variables take on LR fuzzy numbers. The proposed method is illustrated by means of three numerical examples, including a fully fuzzy multiobjective project crashing problem.
In the article, Veeramani and Sumathi [10] presented an interesting algorithm to solve a fully fuzzy linear fractional programming (FFLFP) problem with all parameters as well as decision variables as triangular fuzzy numbers. They transformed the FFLFP problem under consideration into a bi-objective linear programming (LP) problem, which is then converted into two crisp LP problems. In this paper, we show that they have used an inappropriate property for obtaining non-negative fuzzy optimal solution of the same problem which may lead to the erroneous results. Using a numerical example, we show that the optimal fuzzy solution derived from the existing model may not be non-negative. To overcome this shortcoming, a new constraint is added to the existing fuzzy model that ensures the fuzzy optimal solution of the same problem is a non-negative fuzzy number. Finally, the modified solution approach is extended for solving FFLFP problems with trapezoidal fuzzy parameters and illustrated with the help of a numerical example.
Mathematical programming has know a spectacular diversification in the last few decades. This process has happened both at the level of mathematical research and at the level of the applications generated by the solution methods that were created. To write a monograph dedicated to a certain domain of mathematical programming is, under such circumstances,especially difficult. In the present monograph we opt for the domain of fractional programming. Interest of this subject was generated by the fact that various optimization problems from engineering and economics consider the minimization of a ratio between physical and/or economical functions, for example cost/time, cost/volume,cost/profit, or other quantities that measure the efficiency of a system. For example, the productivity of industrial systems, defined as the ratio between the realized services in a system within a given period of time and the utilized resources, is used as one of the best indicators of the quality of their operation. Such problems, where the objective function appears as a ratio of functions, constitute fractional programming problem. Due to its importance in modeling various decision processes in management science, operational research, and economics, and also due to its frequent appearance in other problems that are not necessarily economical, such as information theory, numerical analysis, stochastic programming, decomposition algorithms for large linear systems, etc., the fractional programming method has received particular attention in the last three decades.
There are very few methods in literature to deal with fully fuzzy linear fractional programming (FFLFP) problems. In this paper, it is pointed out that in the existing methods (Journal of Applied Mathematics and Computing, 27 (2008), 227-242; Yugoslav Journal of Operations Research, 22 (2012), 41-50) for solving FFLFP problems, an inappropriate ranking property is used which may lead to the erroneous results. To resolve the flaw of the existing methods, a new method is proposed for solving FFLFP problems, and is illustrated with the help of a numerical problem.
An introduction to ratio optimization problems is provided which covers various applications as well as major theoretical and algorithmic developments. In addition to an extensive treatment of single-ratio fractional programming, three types of multi-ratio fractional programs are discussed: maximization of the smallest of several ratios, maximization of a sum of ratios and multi-objective fractional programs. Earlier as well as recent developments are discussed and open problems are identified. The article concludes with a comprehensive, up-to-date bibliography in fractional programming. Well over one thousand articles have appeared in more than thirty years of increasingly intensive research in fractional programming. The bibliography includes all references from the beginning until late 1993 to the extent they are known to the author at this time.
In this paper, we propose a method of solving the fully fuzzified linear fractional programming problems, where all the parameters
and variables are triangular fuzzy numbers. We transform the problem of maximizing a function with triangular fuzzy value
into a deterministic multiple objective linear fractional programming problem with quadratic constraints. We apply the extension
principle of Zadeh to add fuzzy numbers, an approximate version of the same principle to multiply and divide fuzzy numbers
and the Kerre’s method to evaluate a fuzzy constraint. The results obtained by Buckley and Feuring in 2000 applied to fractional
programming and disjunctive constraints are taken into consideration here. The method needs to add extra zero-one variables
for treating disjunctive constraints. In order to illustrate our method we consider a numerical example.
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Transportation models play an important role in logistics and supply chain management for reducing cost and improving service. This paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. The idea is based on the extension principle. A pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level α. From different values of α, the membership function of the objective value is constructed. Two different types of the fuzzy transportation problem are discussed: one with inequality constraints and the other with equality constraints. It is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem. Since the objective value is expressed by a membership function rather than by a crisp value, more information is provided for making decisions.
We try to provide a tentative assessment of the role of fuzzy sets in decision analysis. We discuss membership functions, aggregation operations, linguistic variables, fuzzy intervals and the valued preference relations they induce. The importance of the notion of bipolarity and the potential of qualitative evaluation methods are also pointed out. We take a critical standpoint on the state-of-the-art, in order to highlight the actual achievements and question what is often considered debatable by decision scientists observing the fuzzy decision analysis literature.
By decision-making in a fuzzy environment is meant a decision process in which the goals and/or the constraints, but not necessarily the system under control, are fuzzy in nature. This means that the goals and/or the constraints constitute classes of alternatives whose boundaries are not sharply defined.
An example of a fuzzy constraint is: “The cost of A should not be substantially higher than α,” where α is a specified constant. Similarly, an example of a fuzzy goal is: “x should be in the vicinity of x 0 ,” where x 0 is a constant. The italicized words are the sources of fuzziness in these examples.
Fuzzy goals and fuzzy constraints can be defined precisely as fuzzy sets in the space of alternatives. A fuzzy decision, then, may be viewed as an intersection of the given goals and constraints. A maximizing decision is defined as a point in the space of alternatives at which the membership function of a fuzzy decision attains its maximum value.
The use of these concepts is illustrated by examples involving multistage decision processes in which the system under control is either deterministic or stochastic. By using dynamic programming, the determination of a maximizing decision is reduced to the solution of a system of functional equations. A reverse-flow technique is described for the solution of a functional equation arising in connection with a decision process in which the termination time is defined implicitly by the condition that the process stops when the system under control enters a specified set of states in its state space.