In this paper, we consider some cosine similarity measures and distance measures between q‐rung orthopair fuzzy sets (q‐ROFSs). First, we define a cosine similarity measure and a Euclidean distance measure of q‐ROFSs, their properties are also studied. Considering that the cosine measure does not satisfy the axiom of similarity measure, then we propose a method to construct other similarity measures between q‐ROFSs based on the proposed cosine similarity and Euclidean distance measures, and it satisfies with the axiom of the similarity measure. Furthermore, we obtain a cosine distance measure between q‐ROFSs by using the relationship between the similarity and distance measures, then we extend technique for order of preference by similarity to the ideal solution method to the proposed cosine distance measure, which can deal with the related decision‐making problems not only from the point of view of geometry but also from the point of view of algebra. Finally, we give a practical example to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods.
In this paper, we first introduce the concept of q‐rung orthopair hesitant fuzzy set (q‐ROHFS) and discuss the operational laws between any two q‐ROHFSs. Then the distance measures between q‐ROHFSs are proposed based on the concept of “multiple fuzzy sets”, and we develop the TOPSIS method to the proposed distance measures. The proposed distance measures not only retain the preference information expressed by q‐ROHFSs, but also deal with the q‐rung orthopair hesitant fuzzy decision information more objectively, In fact, the method can avoid the loss and distortion of the information in actual decision‐making process. Furthermore, we give an illustrative example about the selection of energy projects to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods. Finally, we make the sensitivity analysis of the parameters in proposed distance measures about the selection of energy projects.
We present the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVIFOWCS) measure in this paper, which combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator. The main advantage of the IVIFOWCS measure provides a parameterized family of similarity measures, and the decision maker can use the IVIFOWCS measure to consider a lot of possibilities and select the aggregation operator in accordance with his interests. We have studied some of its main properties and particular cases such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity (IVIFOWACS) measure and the interval-valued intuitionistic fuzzy maximum cosine similarity (IVIFMAXCS) measure. The IVIFOWCS measure not only is a generalization of some similarity measure, but also it can deal with the correlation of different decision matrices for interval-valued intuitionistic fuzzy values. Furthermore, we present an application of IVIFOWCS measure to the group decision-making problem. Finally the existing similarity measures are compared with the IVIFOWCS measure by an illustrative example.
A method has been developed for determining qualitatively and quantitatively all commonly used soap germicides. It includes the halogenated salicylanilides, halogenated carbanilides and bisphenols both alone and in mixtures. Elimination of interfering substances is accomplished by extraction with dimethylformamide followed by evaporation to dryness. The residue is then taken up in both acidic alcohol and basic alcohol. The ultraviolet patterns obtained under these conditions are typical of the various germicides and are used to identify them.
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