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 thresholdgraph 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- valuedfuzzy 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-valuedfuzzythresholdgraph to model and solve this type of real problem.
Abstract. In this paper, we define irregular interval-valuedfuzzy graphs and their various classifications. Size of regular interval-valuedfuzzy graphs is derived. The relation between highly and neighbourly irregular interval-valuedfuzzy graphs are established. Some basic theorems related to the stated graphs have also been presented. Keywords: Interval-valuedfuzzy graphs, irregular interval-valuedfuzzy graphs, totally irregular interval-valuedfuzzygraph
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 . 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  presented a fuzzy ELECTRE method using the concept of the interval- valuedfuzzy set (IVFS) with unequal criteria weights, and the criteria values are considered as triangular interval-valuedfuzzy number, and also using triangular interval-valuedfuzzy number to distinguish the concordance and discordance sets, and
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. Fuzzygraph theory is successfully used in many problems, to handle the uncertainty that occurs in graph theory. An interval-valuedfuzzygraph is a generalized structure of a fuzzygraph 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–valuedfuzzy graphs such as neighbourly totally irregular interval- valuedfuzzygraph, highly irregular interval-valuedfuzzy graphs and highly totally irregular interval–valuedfuzzy graphs are introduced and investigated. A necessary and sufficient condition under which neighbourly irregular and highly irregular interval– valuedfuzzy graphs are equivalent is discussed.
17. N.Thillaigovindan and V.Chinnadurai, Interval-valuedfuzzy 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.
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 , 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- valuedfuzzygraph representation is better than a crisp graph version. Interval-valuedfuzzy graphs suitably represent every event.
In this paper we find the degree and classify it for Cartesian product of two intervalvalued 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.
Graph theory has so many applications in almost all real world problems. But since the world is full of uncertainty, fuzzygraph has a separate importance in many real life applications. The first definition of fuzzygraph was by Kaufmann  in 1973. But it was Rosenfeld  who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs as a generalization of Eulers graph theory in 1975. The works of Bhattacharya, Bhutani , Bhutani and Battou , Bhutani and Rosenfeld [12,13, 14], Mordeson , Mordeson and Nair [19,20], Mordeson and Peng , Sunitha and Vijayakumar [43-46], Nagoor Gani and Ahmed , Nagoor Gani and Malarvizhi , Nagoor Gani and Radha [24,25] form the foundation of all researches in fuzzygraph theory. In , Sunitha and Sunil Mathew made a very good survey of the researches done so far in fuzzygraph theory. Samanta and Pal introduced fuzzy tolerance graphs , fuzzy k-competion graphs and p-competition fuzzy graphs , fuzzythreshold graphs  and bipolar fuzzy hypergraphs .
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.
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 ; Chen, Lin, & Huang ; Kuo, Tzeng, & Huang ). 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.
In this paper, we introduced the concept of a quotient semigroup S/δ by an interval- valuedfuzzy congruence relation δ on a semigroup S, and present Homomorphism The- orems with respect to an interval-valuedfuzzy congruence relation. We also investigate idempotent-separating interval-valuedfuzzy congruence, a group interval-valuedfuzzy con- gruence on inverse semigroup and studied some important results.
Mi Jung Son  introduced intervalvaluedfuzzy soft set and defined some of its types. P. Rajarajeswari and P. Dhanalakshmi  developed interval-valuedfuzzy soft matrix theory. Zulqarnain. M and M. Saeed  defined some new types of intervalvaluedfuzzy 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  used interval-valuedfuzzy soft sets and Sanchez’s approach for medical diagnosis. In recent years many researchers [21-25] have been worked on applications of intervalvaluedfuzzy soft sets.
Interval-valuedfuzzy sets were introduced independently by Zadeh , Grattan-Guiness , Jahn , in the seventies, in the same year. An intervalvaluedfuzzy set (IVF) is defined by an interval-valued membership function. Jun.Y.B and Kin.K.H defined an intervalvaluedfuzzy R-subgroups of nearrings. Solairaju.A and Nagarajan.R defined the charactarization of intervalvalued Anti fuzzy Left h-ideals over Hemirings. M.G.Somasundara Moorthy and K. Arjunan have defined an intervalvaluedfuzzy subring of a ring under homomorphism. We introduce the concept of intervalvalued intuitionistic fuzzy subring of a ring and established some results.
Proof: Let µ e be an interval-valuedfuzzy 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-valuedfuzzy 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-valuedfuzzy 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
In real life scenario, we face so many uncertainties, in all walks of life. Zadeh’s classical concept of fuzzy set 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.
Intervalvaluedfuzzy sets were introduced independently by Zadeh , Grattan-Guiness , Jahn , in the seventies, in the same year. An intervalvaluedfuzzy set (IVF) is defined by an intervalvalued membership function. Jun.Y.B and Kin.K.H  defined an intervalvaluedfuzzy R- subgroups of nearrings. Solairaju. A and Nagarajan. R defined the charactarization of intervalvalued Anti fuzzy Left h-ideals over Hemirings. Azriel Rosenfeld  defined a fuzzy group. Asok Kumer Ray  defined a product of fuzzy subgroups.We introduce the concept of intervalvalued anti fuzzy subnearring of a nearring and established some results.
For the first time Zadeh (1965) introduced the concept of fuzzy sets and also Zadeh (1975) introduced the concept of an interval-valuedfuzzy 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-valuedfuzzy 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.
In this paper, we discuss various k-g inverses of k-regular interval-valuedfuzzy 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.