For any integer k ≥ 1 and any graph G = (V, E) with minimum degree at least k − 1, we define a function f: V → {0, 1, 2} as a Roman k-tuple dominating function on G if for any vertex v with f(v) = 0 there exist at least k and for any vertex v with f(v) ̸ = 0 at least k − 1 vertices w in its neighborhood with f(w) = 2. The minimum weight of a Roman k-tuple dominating function f on G is called the Roman k-tuple domination number of the graph where the weight of f is f(V ) = ∑ v∈ V f(v). In this paper, we initiate to study the Roman k-tuple domination number of a graph, by giving some tight bounds for the Roman k-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman k-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. [1] in 2004 for the Roman domination number.

Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number g ´k, t (G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper for a given graph G with minimum degree at least k, we find some sharp lower and upper bounds on the k-tuple total domination number of the m-Mycieleskian graph mm (G) of G in terms on k and g´ k, t (G).Specially we give the sharp bounds g´k, t (G) +1 and g´k, t (G) +k for g´ k, t (m1 (G)), and characterize graphs with g´ k, t (m1 (G)) = g´ k, t (G) +1.

For any integer k³1, a set S of vertices in a graph G=(V,E) is a k-tuple total dominating set of G if any vertex of G is adjacent to at least k vertices in S, and any vertex of V-S is adjacent to at least k vertices in V-S. The minimum number of vertices of such a set in G we call the k-tuple total restrained domination number of G. The maximum number of classes of a partition of V such that its all classes are k-tuple total restrained dominating sets in G we call the k-tuple total restrained domatic number of G.In this paper, we give some sharp bounds for the k-tuple total restrained domination number of a graph, and also calculate it for some of the known graphs. Next, we mainly present basic properties of the k-tuple total restrained domatic number of a graph.

For an integer k ≥,2, a Roman k-tuple dominating function, (or just RkDF), in a graph G is a function f: V (G)→, {0,1,2} satisfying the condition that every vertex u for which f(u) ≠,0 is adjacent to at least k vertices v for which f(v) = 2, and every vertex u for which f(u) 6= 0 is adjacent to at least k-1 vertices v for which f(v) = 2. The Roman k-tuple domination number of G is the minimum weight of an RkDF in G. In this note we settle two problems posed in [Roman k-tuple Domination in Graphs, Iranian J. Math. Sci. Inform. 15 (2020), 101-115].

In real-world decision-making problems, experts often prefer to express their views, regarding problem parameters, in a natural language rather than precise numerical form. Linguistic representation models have been widely used to solve many decision-making problems with qualitative information. Game theory has been found successful applications in a wide range of areas. This paper presents an extensive study of matrix games with qualitative payoffs. The work uses 2-tuple intuitionistic fuzzy linguistic values (2-TIFLVs) to represent the payoffs of the matrix game. We develop the mathematical formulation and concepts of the solutions for matrix games with payoffs represented by 2-TIFLVs. Paper also shows that matrix games with payoffs of 2-TIFLVs have solutions that can be obtained by transforming the matrix game in a pair of linear/nonlinear programming problems. Finally, a real-life numerical is given to illustrate the developed method.

In this paper, we investigate the multiple attribute decision making (MADM) problems with 2-tuple intuitionistic fuzzy linguistic information. Then, we utilize arithmetic and geometric operations to develop some 2-tuple intuitionistic fuzzy linguistic aggregation operators. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the 2-tuple intuitionistic fuzzy linguistic MADM problems. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Because of the complexity of decision-making environment, the uncertainty of fuzziness and the uncertainty of grey maybe coexist in the problems of multi-attribute group decision making. In this paper, we study the problems of multi-attribute group decision making with hybrid grey attribute data (the precise values, interval numbers and linguistic fuzzy variables coexist, and each attribute value has a certain grey degree), and present a new grey hybrid multi-attribute group decision making method based on grey linguistic 2-tuple. Concretely, the concept of grey linguistic 2-tuple is defined based on the traditional linguistic 2-tuple, and the transformation methods of transforming the precise values, interval numbers and linguistic fuzzy variables into the grey linguistic 2-tuples are presented respectively. Further, a new grey linguistic 2-tuple weighted averaging (GLTWA) operator is presented to aggregate multiple decision makers' individual decision information into comprehensive decision information, and then a ranking method based on grey 2-tuple correlation degree is presented to rank all alternatives and to select the winners. An application decision making example of supplier selection is also given to validate the method developed and to highlight the implementation, practicality and effectiveness of the presented method.