Fuzzy linear programming

By using the fuzzy set theory in [7] [19], the fuzzy optimization problem has been widely studied since 70s. The first method to solve the fuzzy linear programming problem in the method of [27], [4]. In [27], the main focus is the fuzzy goal and the application of the strategic theory to the fuzzy decision problem. However, the focus is on the concepts of mathematical programming and using the level set, and some classical results are generalized to include fuzzy constraints and objective functions. In [4], the fuzzy set theory is applied to the fuzzy linear programming problem, which proves how the fuzzy linear programming problem is solved without increasing the computation amount.

In this paper fuzzy linear programming is applied to profit maximization related machine scheduling problem where forecasted demand of product and profit are expressed by the fuzzy set notations. To obtain the optimal solution of such problem the conventional linear programming (LP) problem is formulated as fuzzy linear programming problem by replacing the crisp constraints and objective function by fuzzy set and then by simply solving a conventional LP problem which is equivalent to the formulated FLP problem. In the present approach a LP package is used to solve this problem.

The paper is organized in six sections. The introductory section is followed by presentation of some basic concepts necessary for the development of a mechan- ism for solving intuitionistic fuzzy linear programming problems. In this section, basic concept of Triangular Intuitionistic Fuzzy Number (TIFN) is described. In Section 3, we discuss fuzzy linear programming problem and introduce a new method analogous with it, to solve IFLPP when both the coefficient matrix of the constraints and cost coefficients are intuitionistic fuzzy in nature. In Section 4, there is a comparative study between some of the other optimization techniques with our proposed technique for solving an intuitionistic fuzzy linear program- ming problem. Section 5 concludes the present paper and refers to some prob- lems for further studies which is followed by a list of references in the last sec- tion.

Comparison of fuzzy numbers is considered one of the most important topics in fuzzy logic theory. The early and most important work in the field of comparing fuzzy numbers has been presented by Dubois and Prade [1]. A comparison between their work and other attempts that have been made in this area has been given by Bortolan and Degani [2]. On the other hand, the dominance possibility indices, which have been introduced by Dubois and Prade, were utilized in the field of fuzzy mathematical programming [3,4]and the field of stochastic fuzzy mathematical programming [5,6]. The approach used in these field was based on formulating a possibility function, whether in the case of trapezoidal fuzzy numbers or the case of triangular fuzzy numbers. In this paper, we are going to utilize Dubois and Prade’s dominance possibility and necessity indices, within a different approach, in the case of stochastic fuzzy linear programming problem. The dominance possibility and necessity, as well as the strict dominance possibility and necessity criteria, are utilized according to the chance-constrained method to transform the suggested problem to its deterministic-crisp equivalent. This approach helps avoiding any approximation that may exist due to comparing the inverse distribution function of fuzzy tolerance measures.

Abstract. Recently, Gasimov and Yenilmez proposed an approach for solving two kinds of fuzzy linear programming (FLP) problems. Through the approach, each FLP problem is first defuzzified into an equivalent crisp problem which is non-linear and even non-convex. Then, the crisp problem is solved by the use of the modified subgra- dient method. In this paper we will have another look at the earlier defuzzification process developed by Gasimov and Yenilmez in view of a perfectly acceptable remark in fuzzy contexts. Furthermore, it is shown that if the modified defuzzification process is used to solve FLP problems, some interesting results are appeared.

In this paper, a new method is proposed to find the fuzzy optimal solution of fully fuzzy linear programming (abbreviated to FFLP) problems. Also, we employ linear programming (LP) with equality constraints to find a non-negative fuzzy number vector ˜ x which satisfies ˜ A˜ x = ˜ b, where ˜ A is a fuzzy number matrix. Then we investigate the existence of a positive solution of fully fuzzy linear system (FFLS).

uncertainties the fuzzy concept has been introduced. Zimmermann categorized these uncertainties as fuzziness and proposed the linear programming problem formulation. Further fuzzy constraint coefficient in linear programming was considered by Fang and HU. Fuzziness can be categorized into two types namely ambiguity and vagueness. The description of uncertainty which is associated with linguistic information can be done by vagueness. Ambiguity is specified to the situation when the choice between two or more alternatives is left unspecified and the occurrence of the individual alternative is unknown payable to insufficiency in knowledge and tools. Ambiguity can be further classified based on the viewpoint of the ways the ambiguity comes form, that is either by preference based ambiguity or possibility based ambiguity. These ambiguities and uncertainties cannot be solved by traditional mathematical optimization techniques. To solve the problems with such ambiguities and uncertainties Fuzzy Optimization technique has been used. Fuzzy optimization solves the fuzzy model optimally using optimization techniques and tools by formulating fuzzy information in terms of their membership function. In fuzzy linear programming problem the constraint or objective function need not to be specified in precise like crisp terms. In such situations the fuzzy linear programming occurs. In general fuzzy linear programming the cost matrix or the coefficient matrix or the variable matrix or the column vector represents the fuzziness.

In section 2 have some basic definitions.In section 3 formulations of multi objective fuzzy linear programming prblam and raking formula are discussed. In section 4 the simplex method to solving multi objective fuzzy linear programming problem with numerical example. In section 5 conclusions are discussed

linear programming (FLP) proposed by Sanei- fard et al. [12]. Afterwards, many authors have considered various kinds of FLP problems and have proposed several approaches for solving these problems [2, 3, 4, 5, 7, 8, 9]. Fuzzy set theory, for instance, has been applied to many disciplines such as control theory, management science, mathematical modeling, and industrial applications. The concept of fuzzy linear pro- gramming (FLP) on a general level was first proposed by Saneifard et al [11]. Later on, other authors considered various types of FLP prob- lems and proposed several approaches to solve them. The fuzzy linear programming problem in which all parameters and variables are repre- sented by fuzzy numbers is known as fully fuzzy linear programing problem (FFLP) problems. Wang et al. [9] proposed a method to find the

three methods for solving three models of fuzzy integer linear programming based on the representation theorem and on fuzzy number ranking method. Nasseri et.al [14] proposed a new method for solving fuzzy linear programming problems in which he has used the fuzzy ranking method for converting the fuzzy objective function into crisp objective function. Lee and Li [12] discussed the comparison of fuzzy numbers. Amit Kumar et al. [3, 4] presented a new method for solving fuzzy linear programs with Trapezoidal fuzzy numbers. Cheng [8] used a centroid based distance method to rank fuzzy numbers in 1998.Kauffmann and Gupta [9] introduced to Fuzzy Arithmetic. Kolman and Hill [10] was introduced a FFLP problem. Chen [7] proposed the ranking trapezoidal fuzzy number using maximizing and minimizing set decomposition principle and sign distance. In this paper, some preliminaries are presented in section 2. Section 3 describes the proposed method with one numerical example and obtained results were discussed. Section 4 concludes the paper.

Fuzzy linear programming problem occur in many fields such as mathematical modeling, Control theory and Management sciences, etc. In this paper we focus on a kind of Linear Programming with fuzzy numbers and variables namely Fully Fuzzy Linear Programming (FFLP) problem, in which the constraints are in inequality forms. Then a new method is proposed to fine the fuzzy solution for solving (FFLP). Numerical examples are providing to illustrate the method.

for all and i ∈ I and k ∈ k . That is, the matrix equation (3) encompasses n x s simultaneous equations of the form (4). these equations are referred to as fuzzy relation equations.The set of all particular matrices of the form P that satisfy (1) is called its solution and denote the set of all solutions as

elimination consists in successive elimination of the unknowns. Each step transforms with the unknowns , , … to a in which one of the unknowns say does has been eliminated. The concept of fuzzy numbers and arithmetic operations with these numbers Zadeh, 1976). One of the major applications of fuzzy number arithmetic is treating linear systems and their parameters that are all partially

In this paper we present a Fuzzy linear Mathematical programming approach for optimal allocation of land under cultivation. Fuzzy Mathematical programming approach is more realistic and flexible optimal solution for the agricultural land cultivation problem. In this study we have discussed how to deal with decision making problems that are described by Fuzzy linear programming (Flp) models and formulated with the elements of uncertainty. This form of approximation can be convenient and sufficient for making good decisions.

One of the major difficulties to study such fuzzy linear programming problems with fuzzy coefficients is how to compare these fuzzy numbers. Thus an important issue of ranking of fuzzy numbers and its approximation method took considerable interest amongst the researchers. Some of the authors who made significant contributions in the area are Dubois and Prade[10], Heliperrn[15], Adrian[1,2]. This growing discipline attracted many authors to extend the theory of fuzzy sets to various application areas of industrial planning, production planning, agricultural production planning, economics etc. Atanossov[4, 5] extended the fuzzy set theory to intuitionistic fuzzy sets. This extended new set, named as intuitionistic fuzzy set, has a feature to accommodate hesitation factor of including an element in a fuzzy set apart

Atanassov[3,8-10] introduced Intuitionistic fuzzy set (IFS) which is one of the generalizations of fuzzy set theory characterized by a membership function, a non-membership function and a hesitancy function. In fuzzy sets the degree of acceptance is only considered but IFS is characterized by a membership function and a non-membership function so that the sum of both values is less than one. The concept of membership and non-membership was first considered by Angelov[1] in optimization problem and gave intuitionistic fuzzy approach to solve this. Luo.et al. [19] applied the inclusion degree of intuitionistic fuzzy set to multi criteria decision making problem. Pramanik et al.[12] solved a vector optimization problem using an intuitionistic fuzzy goal programming. A transportation model was solved by Jana et al.[7] using multi-objective intuitionistic fuzzy linear programming. Dey et al. [15] use Intuitionistic fuzzy optimization technique to optimize non-linear single objective two bar truss structural model.

We need to fractional linear programming in many real-world problems such as production and financial planning and institutional planning and return on investment, and others. Multi objective Linear fractional programming problems useful targets in production and financial planning and return on investment. Charnes and Cooper, Used variable transformation method to solve linear fractional programming problems [1]. Tantawy, Proposes a new method for solving linear fractional programming problems [2]. Jayalakshmi and Pandian, Proposed a new method namely, denominator objective restriction method for finding an optimal solution to linear fractional programming problems [3]. Moumita and De, Study of the fully fuzzy linear fractional programming problem using graded mean integration representation method [4]. Ezzati et al, Used a new algorithm to solve fully fuzzy linear programming problems using the multi objective linear programming problem [5]. Haifang et al, Solving a fully fuzzy linear programming problem through compromise programming [6].Pawlak, Used a rough set theory a new mathematical approach to imperfect knowledge [7]. Kryskiewice, Used a rough set theory to incomplete has found many interesting applications [8]. Pal, The rough set approach seems to be of fundamental importance to cognitive sciences, especially in the areas of machine learning, decision analysis, and expert systems [9]. Pawlak, Rough set theory, introduced by the author, expresses vagueness, not by means of membership, but employing a boundary region of a set. The theory of rough set deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations [10]. Tsumoto, Used the concept of lower and upper approximation in rough sets theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules [11]. Lu and Huang, The concept of rough interval will be introduced to represent dual uncertain information of many parameters, and the associated solution method will be presented to solve rough interval fuzzy linear programming problems [12].

Fuzzy linear programming has been studied extensively by numerous researchers, however, only few studies on fuzzy stochastic linear programming can be found up to date (Giri, Maiti and Maiti, 2014; Sakawa, Katagiri, and Matsu, 2014; Wang, and Watada, 2011). Various methods of fuzzy transformation and defuzzification methods have been proposed to defuzzify the optimization problem, such as max membership defuzzification method, centroid defuzzification method (CDM), weighted average method, mean max membership method, center of sums method, center of largest area method, first (or last) maxima, Yager’s robust ranking method, and GMIR. Although a lot of methods have been proposed to defuzzify the fuzziness in the problems, however up to now, limited studies have been made to search for the best methods among all the proposed methods. Mogharreban and DiLalla (2006) proposed that center of area is the best method of defuzzification, meanwhile Naaz, Alam and Biswas (2011) suggested that the center of gravity, bisector method, and mean of maxima methods were the three best defuzzification methods. There was a proposed method of defuzzification to defuzzify the generalized fuzzy numbers, which was GMIR method. However, no comparison between GMIR method and previously proposed method has been made.

Fuzzy linear programming is an application of fuzzy set theory in linear decision making problems and most of these problems are related to linear programming (LP) with fuzzy variables. In this paper, an approximate but convenient method for solving these problems with fuzzy non-negative technical coefficient and without using the ranking functions, is proposed. With the help of numerical examples, the method is illustrated.

complete solution set for the fuzzy linear programming problems using linear and nonlinear membership functions. Tang and Chang [29] developed a decision-making model and helped the DMs and researchers to understand the effect of multiple criteria decision-making on a capital budgeting investment, coefficients, and resources. They solved the problem based on the proposed method introduced by Gasimov and Yenimez [10]. Kiruthiga and Loganathan [16] reduced the Fuzzy MOLP problem to the corresponding ordinary one using the ranking function and hence solved it using the fuzzy programming technique. Hamadameen [11] proposed a technique for solving fuzzy MOLP problem in which the objective functions coefficients are triangular fuzzy numbers. Under uncertainty, vague, and imprecise of data, Garg [9] suggested an alternative approach for solving multi-objective reliability optimization problem. Chen et al. [5] applied fuzzy goal programming approach to treat equipment's purchasing problem for a flexible manufacturing cell.