In this paper, we present a Modified Revised Simplex Method for minimizing **fuzzy** transportation problem of a **Triangular** **Fuzzy** **Numbers**. In which the supplies and demands are **triangular** **fuzzy** **numbers**. A parametric approach is used to obtain a **fuzzy** basic feasible solution with the help of Revised Simplex Method. We get optimal solution of **fuzzy** transportation problem in which number of constraints equal to number of occupied cells.

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Quadratic Programming (QP) is a mathematical modeling technique de- signed to optimize the usage of limited resources and has been widely applied to solve real world problems. In conventional quadratic programming model the parameters are known constants. However in many practical situations, it is not reasonable to require that the constraints or the objective function in quadratic programming problems be specified in precise, crisp terms. In such situations, it is desirable to use some type of **Fuzzy** Quadratic Programming (FQP) problem. In this paper a new approach is proposed to derive the **fuzzy** objective value of **fuzzy** quadratic programming problem, where the con- straints coefficients and the right-hand sides are all **triangular** **fuzzy** **numbers**. The proposed method is solved using MATLAB TM toolbox and the numerical

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Abstract. This study presents an approximate approach for ranking **fuzzy** **triangular** **numbers** based on the distance method of a **fuzzy** **triangular** number and its area. The total approximate is determined by convex combining of **fuzzy** **triangular** number’s relative and its area that based on decision maker’s optimistic perspectives. The proposed approach is simple in terms of computational efforts and is efficient in ranking a large quantity of **fuzzy** **triangular** **numbers**. By a group of examples in [2] demonstrate the accuracy and applicability of the proposed approach. Finally we construct a new ranking system for **fuzzy** **triangular** number which is very realistic and also matching our intuition as the crisp ranking system on R.

Robust ranking method has been used for rank- ing the **fuzzy** **numbers**. The Hungarian (One’s assignment) method has been applied to solve the **fuzzy** assignment problem. An illustrative numerical example is provided to demonstrate the effectiveness of the **fuzzy** assignment prob- lems.

In general, the theory of IFS is the generalization of **fuzzy** sets. Therefore it is expected that, IFS could be used to stimulate human decision-making process and any activities requiring human expertise and knowledge which are inevitably imprecise or not totally reliable. In this paper, we have introduced a more general definition of **triangular** **fuzzy** number in intuitionistic **fuzzy** environment such that the degree of satisfaction and rejection are so considered that the sum of both values is always less than one. Basic arithmetic operations and -cut sets are defined for this types of **fuzzy** **numbers**. Also a GTIFN is approximated to a nearest interval number by using cut sets. Finally, average ranking index is proposed to find inequality relations between two GTIFNs. This approach is very simple and easy to apply in real life problems. Next, we approximate a GTIFN to a nearest interval number.

Abstract. In this paper the polynomial interpolation of **triangular** **fuzzy** number is discussed. First general form of the polynomial with **fuzzy** coefficients is proposed. The hermite interpolation method is studied with **triangular** **fuzzy** number an example is provided to illustrate the algorithm.

(2013) presented some types and properties of type-2 **triangular** **fuzzy** matrices. The paper 2 of this paper, we recall the definition of type-2 **triangular** **fuzzy** number and some 2 **triangular** **fuzzy** **numbers**. In section-3, we review the definition of type-2 **triangular** **fuzzy** matrices (T2TFM) -4, we define adjoint of T2TFMs. In section-5, we derive some more special properties 6, relevant numerical examples are presented. Finally in section

In this research, we develop a set of **Fuzzy** Intention Product Design System. The system is capable of breaking the restrictions of time and space. Through the **Fuzzy** Analytic Hierarchy Process (FAHP), the **triangular** **fuzzy** **numbers** are put into the Pairwise Comparison Matrix to prevent the **fuzzy** problems during the process of conducting criteria measurement and judgment. By means of FAHP, the AHP Exact Value will be substituted by the Interval Value. And the experts will make strategic decisions with humanity when evaluating problems and give a weight value to the measure factors.

Abstract. Priority queuing models have a wide range of application in computer network system. In this paper the performance measures of **fuzzy** priority queuing model are computed using L- R method. L -R method is convenient and flexible compared to other methods. Numerical illustration is given to check the validity of the proposed method. Keywords: **Fuzzy** queue, priority discipline performance measure, L-R method and **triangular** **fuzzy** **numbers**.

Once the hierarchy is established, the **fuzzy** pair wise comparison takes place. The experts compare all the criteria on the same level of the hierarchy. A pair wise comparison is performed by using **Fuzzy** linguistic terms in the scale of 0 – 10 described by the **Triangular** **Fuzzy** **Numbers** in the Table 1.1. In Buckley's method, the element of the negative judgment is treated as an inverse and reversed order of the **fuzzy** number of the corresponding positive judgment. Thus it requires not only a rigorous manipulation in the construction of reciprocal matrix but also due to transitivity the result becomes inconsistent. Again to reflect pessimistic, most likely and optimistic decision making environment, **triangular** **fuzzy** **numbers** with minimum value, most plausible value & maximum value are considered.

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The conventional AHP method is incapable of handling the uncertainty and vagueness involved in the mapping of one’s preference to an exact number or ratio The major difficulty with classical AHP is its inability in mapping human judgments .In recent years it has been observed the due to confusion in decision makers mind probable deviations should be integrated to the decision making process in **Fuzzy**-AHP, pair wise comparisons are done using **fuzzy** linguistic preference scale. For simplicity the reciprocal **fuzzy** **numbers** are replaced by individual **Triangular** **Fuzzy** **Numbers** in the pair wise comparison matrix.

A project is combination of interrelated activities which must be executed in a certain order before the entire task can be completed. An activity in a project is usually viewed as a job requiring time and resources for its completion. The Critical Path Method (CPM) plays a vital role in planning, scheduling and controlling the complex projects consisting of number of work contents. The successful implementation of CPM requires the availability of clear determined parameters (time duration, cost etc) for each activity. But in reality, due to uncertainty of information as well as the variation of management scenario, it is often difficult to obtain the exact activity time estimates. Under such situations it is highly impossible to formulate the mathematical model through the classical traditional methods. Therefore the **fuzzy** set theory proposed by Zadeh [7] can play a significant role in this kind of problems to handle the ambiguity about the time duration of deeds in a project network. Atanassov [5], [6] extended the **fuzzy** sets to the theory intuitionistic **fuzzy** sets in which both the degree of belonging and degree of non-belonging are considered. Several authors such as Takahashi et al [8], Sophia Porchelvi and Sudha [14] to [16] Error! Reference source not found. , Kiran Yadav et al [1] and Nagoor Gani et al [9] have studied **fuzzy** Critical Path Method in Intuitionistic **Fuzzy** Environment. Sophia Porchelvi and Sudha have studied Intuitionistic **Fuzzy** Critical Path in a Network using a new ranking method. De. P.K and Amita Bhinchar [12], [13] discussed **fuzzy** critical path analysis by a ranking method. Jayagowri and Geetharamani [11] obtained intuitionistic **fuzzy** critical path by using metric distance ranking method. Elizabeth and Sujatha [18] presented two different algorithms to obtain the critical path in a **fuzzy** network problem involving **triangular** intuitionistic **fuzzy** **numbers** and **triangular** **fuzzy** **numbers**.

In this paper decagonal **Fuzzy** number has been newly introduced and the alpha cut operations of arithmetic function principles using addition, subtraction multiplication and division has been fully modified with some conditions and has been explained with numerical examples. In a particular case of the growth rate in bacteria which consists of ten points is difficult to solve using trapezoidal or **triangular** **fuzzy** **numbers**, therefore decagonal **fuzzy** **numbers** plays a vital role in solving the problem. It also helps us to solve many optimization problems in future which has ten parameters as in the above case.

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**Fuzzy** Game has been applied in many fields such as Operations Research, Control Theory and Management Sciences etc. In this paper, we consider some operations of **triangular** **fuzzy** **numbers** and a solution of **Fuzzy** Game Problem with **triangular** **fuzzy** **numbers**. The Solution of such **Fuzzy** games with pure strategies by minimax-maximin principle is discussed.

Abstract— Achieving an accomplished e-commerce depends on high quality websites which are preferred by prospective customers. Assessing e-commerce website quality can be considered as a multicriteria decision making problem because of its complex structure including qualitative and quantitative factors. In this study, we propose a new methodology based on **fuzzy** analytic hierarchy process. The proposed methodology enables multiple decision makers on evaluation and uses **triangular** **fuzzy** scale that includes both positive and negative **fuzzy** **numbers**. The methodology includes simple mathematical calculations, and it yields **triangular** **fuzzy** **numbers** of alternatives’ weights. At the last step, obtained alternatives’ weights are ranked by integral values method. In the empirical study, three e-commerce websites, which are the most famous in Turkey, are evaluated by the methodology. The findings of this study shows that proposed methodology can tackle the e-commerce website quality assessing, effectively.

In this research, the linguistic variables such as **triangular** **fuzzy** **numbers** are used for the concepts of importance and necessity, and a new approach is proposed to enhance the decision maker’s ability to reduce ambiguity in decision issues. The results can illustrate the impact of taking into account the concept of necessity in the decision-making process. Notably, G-**numbers** can be a combination of various planning and decision-making issues.

Ranking of **fuzzy** **numbers** plays a very significant role in linguistic multi-criteria decision making problems. Several **fuzzy** ranking methods have been proposed since 1976. The linguistic terms are represented quantitatively using **fuzzy** sets and then **fuzzy** optimal alternative is calculated which gives the relative merit of each alternative. S. Abbasbandy and T.Hajjari [1] in 2009 proposed a new method based on the left and right spreads at some – levels and defined magnitude of **fuzzy** **numbers**. Ranking is done based on this magnitude. S. Abbasbandy and B. Asady [2] in 2006 proposed sign distance method by considering a **fuzzy** origin and then calculating distance with respect to the origin. If and is the origin, then distance is defined as

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In this work , the **fuzzy** transportation problems using interval valued **triangular** **fuzzy** **numbers** have been discussed the initial basic feasible solution of the same transportation problem is obtained by different methods such as North West Corner rule , North East Corner rule , Least cost Method and Best Candidate Method.

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The purpose of the critical path method (CPM) is to identify the critical activities in the critical path of an activity network. In the real world for many projects we have to use human judgment for estimating the duration of activities. However, the unknowns or vagueness about the time duration for activities in network planning, has led to the development of **fuzzy** CPM. A way to deal with this imprecise data is to employ the concept of fuzziness, where the vague activity times can be represented by **fuzzy** sets. In this paper a new method based on **fuzzy** theory is developed to solve the project scheduling problem under **fuzzy** environment. Assuming that the duration of activities are **triangular** **fuzzy** **numbers**, in this method we compute total float time of each activity and **fuzzy** critical path without computing forward and backward pass calculations. Through a numerical example, calculation steps in this method and the results are illustrated. Compare with other **fuzzy** critical method the proposed method is simple, fast and effective to find total float time of each activity and **fuzzy** critical path in a **fuzzy** project network.

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Abstract. In this paper, we formulate a transportation problem in which sources, destinations and costs are different types of **fuzzy** **numbers**. We used real, **fuzzy** and intuitionistic **fuzzy** **numbers** are employed to get the optimal solution. Mixed intuitionistic **fuzzy** BCM is used to find the optimal solution in terms of **triangular** intuitionistic **fuzzy** **numbers**. The method is illustrated by a numerical examples.