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.

Show more
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.

Show more
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.

Show more
15 Read more

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.

Show more
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.

Show more
12 Read more

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.

Show more
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

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.

Show more
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

13 Read more

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

Show more
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.

Show more
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].

Show more
22 Read more

36 Read more

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.

Show more
14 Read more