Suppose that there are ATMs that are linked with bank branches. Each ATM has a **threshold** limit amount. This limit sets the maximum withdrawals per day. The goal is to determine a replenishment schedule for allocating cash inventory at bank branches to service a preassigned subset of ATMs. The problem can be modelled as a **threshold** **graph** since each ATM has a **threshold** of transactions. In reality, each ATM can have a different withdrawal limit. These withdrawal limits can be represented by an **interval**- **valued** **fuzzy** set. The **threshold** limit can be set such that the branches can replenish the ATMs without ever hampering the ﬂow of transactions in each ATM. Motivated by this example, we investigate use of the **interval**-**valued** **fuzzy** **threshold** **graph** to model and solve this type of real problem.

Show more
Abstract. In this paper, we define irregular **interval**-**valued** **fuzzy** graphs and their various classifications. Size of regular **interval**-**valued** **fuzzy** graphs is derived. The relation between highly and neighbourly irregular **interval**-**valued** **fuzzy** graphs are established. Some basic theorems related to the stated graphs have also been presented. Keywords: **Interval**-**valued** **fuzzy** graphs, irregular **interval**-**valued** **fuzzy** graphs, totally irregular **interval**-**valued** **fuzzy** **graph**

Show more
11 Read more

At present, **graph** theoretical concepts are highly utilized by computer science applications. Especially in research areas of computer science including data mining, image segmentation, clustering, image capturing networking, for example, a data structure can be designed in the form of tree which in turn utilized vertices and edges. Similarly modeling of network topologies can be done using **graph** concepts.

The Elimination and Choice Translating Reality (ELECTRE) method is one of the outranking relation methods and it was first introduced by Roy [3]. The **threshold** values in the classical ELECTRE method are playing an importance role to filtering alternatives, and different **threshold** values produce different filtering results. As we known that the evaluation data in classical ELECTRE method are almost exact values that can affect the **threshold** values. Moreover, in real world cases, exact values could be difficult to be precisely determined since analysts’ judgments are often vague; for these reasons, we can find some studies [4,5,8] developed the ELECTRE method with type 2 **fuzzy** data. Vahdani and Hadipour [4] presented a **fuzzy** ELECTRE method using the concept of the **interval**- **valued** **fuzzy** set (IVFS) with unequal criteria weights, and the criteria values are considered as triangular **interval**-**valued** **fuzzy** number, and also using triangular **interval**-**valued** **fuzzy** number to distinguish the concordance and discordance sets, and

Show more
of all closed sub-intervals of the **interval** [0, 1], [-1, 0] be the set of all closed sub-intervals of the **interval** [-1, 0] and elements of these sets are denoted by uppercase letters. If μ0C [0, 1] or K [-1, 0] then it can be represented as μ = [μ L , μ u ] where μ L and μ u are the lower and upper

Abstract. Concepts of **graph** theory are applied in many areas of computer science including image segmentation, data mining, clustering, image capturing and networking. **Fuzzy** **graph** theory is successfully used in many problems, to handle the uncertainty that occurs in **graph** theory. An **interval**-**valued** **fuzzy** **graph** is a generalized structure of a **fuzzy** **graph** that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using **fuzzy** graphs. In this paper, new concepts of irregular **interval**–**valued** **fuzzy** graphs such as neighbourly totally irregular **interval**- **valued** **fuzzy** **graph**, highly irregular **interval**-**valued** **fuzzy** graphs and highly totally irregular **interval**–**valued** **fuzzy** graphs are introduced and investigated. A necessary and sufficient condition under which neighbourly irregular and highly irregular **interval**– **valued** **fuzzy** graphs are equivalent is discussed.

Show more
13 Read more

17. N.Thillaigovindan and V.Chinnadurai, **Interval**-**valued** **fuzzy** generalized bi-ideals, Proceedings of the National Conference on Algebra, **Graph** theory and their Applications, Department of Mathematics, Manonmaniam Sundaranar University, Narosa, (2009) 85-98.

12 Read more

In our daily life, the colouring of a **graph** is the most significant component of research in optimization technology and is used for various applications, viz. administrative sciences, wiring printed circuits, resource allocation [4], arrangement problems, and so on. These problems are represented by proper crisp graphs and are analysed by colouring these graphs. In the usual **graph** colouring problem, nodes receive the minimum number of colours such that two adjacent nodes do not have the same colour. A few studies discuss this point [6,7,11,14] . An **interval**- **valued** **fuzzy** **graph** representation is better than a crisp **graph** version. **Interval**-**valued** **fuzzy** graphs suitably represent every event.

Show more
11 Read more

In this paper we find the degree and classify it for Cartesian product of two **interval** **valued** intuitionistic **fuzzy** graphs. We can also extend it to other product like Strong product, tensor product, lexicographic product, etc. We may implement this concept to find the strength of the product of two algorithms which is also useful to solve the problem containing combinatorics. It is useful to the areas including geometry, algebra, number theory, topology, operations research, and computer science.

10 Read more

15 Read more

sets (FS) ([1, 2, 3]). According to the IVIFS definition, M A (x), N A (x) and H A (x) are intervals, where M A (x) denotes the range of support party, N A (x) denotes the range of opposition party, and H A (x) denotes the range of absent party. Moreover, the inferior of M A (x) (INF(M A (x))) is the firm support party of event A, the inferior of N A (x) (INF(N A (x))) is the firm opposition party of event A, the inferior of H A (x) (INF(H A (x))) is the firm absent party of event A, the superior of H A (x) (SUP(H A (x))) is the maximum absent party of event A, and SUP(H A (x))-INF(H A (x)) denotes the convertible absent part, where INF(M A (x))+ INF(N A (x))+SUP(H A (x))=1. Atanassov has divided the convertible absent part into two parts: SUP (M A (x))-INF (M A (x)) being the absent party which can be converted into support party, and SUP (N A (x))-INF (N A (x)) being the absent party which can be converted into the opposition party, where SUP (M A (x))-INF (M A (x))+ SUP (N A (x))-INF (N A (x)) = SUP(H A (x))-INF(H A (x)). Thus, Atanassov’s IVIFS is based on point estimation, which means that these intervals can be regarded as the estimation result of an experiment . However, the proportions of the absent party converted to the support party and to the opposition party may not be constants. For example, SUP (M A (x))-INF (M A (x)) is a constant for one experiment, but it may be a different constant for any other case. Thus, according to **interval** estimation, we provide a novel GIVIFS model to meet real need.

Show more
The single most important decision faced by management when dealing with multiple objectives is the selection of an appropriate solution, which optimizes the proposed criteria simultaneously. Therefore, it is hardly surprising that much of the literature on operations research focuses on the Multiple Objective Programming Problems. Modeling real world problems with crisp values under many conditions is inadequate because human judgment and preference are often ambiguous and cannot be estimated with exact numerical values (Chen [1]; Chen, Lin, & Huang [2]; Kuo, Tzeng, & Huang [3]). There are ways to rank competitive alternatives but ranking competing alternatives in terms of their overall performance with respect to some criterions in **fuzzy** environment is possible by the use of **fuzzy** TOPSIS methodology.

Show more
12 Read more

In this paper, we introduced the concept of a quotient semigroup S/δ by an **interval**- **valued** **fuzzy** congruence relation δ on a semigroup S, and present Homomorphism The- orems with respect to an **interval**-**valued** **fuzzy** congruence relation. We also investigate idempotent-separating **interval**-**valued** **fuzzy** congruence, a group **interval**-**valued** **fuzzy** con- gruence on inverse semigroup and studied some important results.

12 Read more

Mi Jung Son [15] introduced **interval** **valued** **fuzzy** soft set and defined some of its types. P. Rajarajeswari and P. Dhanalakshmi [16] developed **interval**-**valued** **fuzzy** soft matrix theory. Zulqarnain. M and M. Saeed [17] defined some new types of **interval** **valued** **fuzzy** soft matrix and gave an application of IVFSM in a decision making problem. Anjan Mukherjee and Sadhan Sarkar [18, 19] introduced Similarity measures for **interval**-**valued** intuitionistic **fuzzy** soft sets and gave applications in medical diagnosis problems. B. Chetia and P. K. Das [20] used **interval**-**valued** **fuzzy** soft sets and Sanchez’s approach for medical diagnosis. In recent years many researchers [21-25] have been worked on applications of **interval** **valued** **fuzzy** soft sets.

Show more
Proof: Let µ e be an **interval**-**valued** **fuzzy** right ideal of S and [ α, β ] ∈ Im µ. By Theorem 3.1, e U ( µ, e [ α, β ]) is a ternary subsemiring. Let x ∈ U ( µ, e [ α, β ]) and y, z ∈ S. Then µ e ( xyz ) ≥ µ e ( x ) ≥ [ α, β ] . Thus µ e ( xyz ) ≥ [ α, β ] , then xyz ∈ U ( µ, e [ α, β ]) . Hence U ( µ, e [ α, β ]) is a right ideal of S. Conversely, let U ( µ, e [ α, β ]) be a right ideal for all [ α, β ] ∈ Im µ. By Theorem 3.1, e µ e is an **interval**-**valued** **fuzzy** ternary subsemiring. If there exist x, y, z ∈ S such that µ e ( xyz ) < [ α, β ] = µ e ( x ) , then x ∈ U ( µ, e [ α, β ]) and y, z ∈ S with xyz ∈ / U ( µ, e [ α, β ]) . This contradicts that U ( µ, e [ α, β ]) is a right ideal. Hence µ e ( xyz ) ≥ µ e ( x ) . Therefore µ e is an **interval**-**valued** **fuzzy** right ideal of S. Example 3.2. Let S be a ternary semiring consists of non-positive integers with usual addition and ternary multiplication. Let

Show more
In real life scenario, we face so many uncertainties, in all walks of life. Zadeh’s classical concept of **fuzzy** set[20] is strong to deal with such type of problems. Since the initiation of **fuzzy** set theory, there are suggestions for higher order **fuzzy** sets for different applications in many fields. Among higher **fuzzy** sets intuitionistic **fuzzy** set introduced by Atanassov [1,2,3] have been found to be very useful and applicable.

12 Read more

11 Read more

For the first time Zadeh (1965) introduced the concept of **fuzzy** sets and also Zadeh (1975) introduced the concept of an **interval**-**valued** **fuzzy** sets, which is an extension of the concept of **fuzzy** set. Atanassov and Gargov, 1989 introduced the notion of **interval**-**valued** intuitionistic **fuzzy** sets, which is a generalization of both intuitionistic **fuzzy** sets and **interval**-**valued** **fuzzy** sets. On other hand, Satyanarayana et al., (2012) applied the concept of **interval**-**valued** intuitionistic **fuzzy** ideals. In this paper we introduce the notion of **interval**-**valued** intuitionistic **fuzzy** homomorphism of BF-algebras and investigate some interesting properties.

Show more
10 Read more

In this paper, we discuss various k-g inverses of k-regular **interval**-**valued** **fuzzy** matrices. In section 2, some basic definitions and results needed are given. In section 3, characterization of various k-g inverses of k-regular IVFM are determined.