Let G = (V(G), E(G)) be a connected graph and d(f, y) denotes the distance between edge f and vertex y, which is defined as). An edge metric generator with minimum number of vertices is called an edge metric basis for graph G and the cardinality of an edge metric basis is called the edge metric dimension represented by edim(G). In this paper, we study the edge metric dimension of flower graph f n×3 and also calculate the edge metric dimension of the prism related graphs D n and D t n . It has been concluded that the edge metric dimension of D n is bounded, while of f n×3 and D t n is unbounded.

Probabilistic interval-valued hesitant fuzzy sets (PIVHFSs) are an extension of interval-valued hesitant fuzzy sets (IVHFSs) in which each hesitant interval value is considered along with its occurrence probability. These assigned probabilities give more details about the level of agreeness or disagreeness. PIVHFSs describe the belonging degrees in the form of interval along with probabilities and thereby provide more information and can help the decision makers (DMs) to obtain precise, rational, and consistent decision consequences than IVHFSs, as the correspondence of unpredictability and inaccuracy broadly presents in real life problems due to which experts are confused to assign the weights to the criteria. In order to cope with this problem, we construct the linear programming (LP) methodology to find the exact values of the weights for the criteria. Furthermore these weights are employed in the aggregation operators of PIVHFSs recently developed. Finally, the LP methodology and the actions are then applied on a certain multiple criteria decision making (MCDM) problem and a comparative analysis is given at the end.

Picture fuzzy sets (PFSs) are comparatively a new extension of fuzzy sets which describe the human opinions that has more answers like acceptance, rejection, neutral and desist, which cannot be correctly presented in fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). The PFSs are categorized by three objects, the degree of belonging, the degree of neutral belonging and the degree of non- belonging such that the total of these three degrees must not be more than one. So far, there is no such work presented in the literature which deals with unknown weights of criteria based on PFSs. In the present work, we have developed a linear programming (LP) model to find the exact weights from the given constraints of weights for the criteria and construct a modified distance based on similarity measure between picture fuzzy sets. Then we have utilized this similarity measure to achieve the best option in the multiple criteria decision making (MCDM) problem. Lastly, two practical examples for the selection of alternatives are presented to compare the obtained results with the existing similarity measures.

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The notion of bipolar fuzzy sets (B p FSs) has got much attention from the experts or decision-makers (DMs). B p FSs have ample information in the form of two degrees called the positive belonging degree (P v BD) and a negative belonging degree (N v BD). In this article, we introduced the concept of bipolar picture fuzzy sets (BP c FSs) by connecting the concepts of B p FSs and picture fuzzy sets (P c FSs). Firstly, we presented the concept, operational rules, score, and accuracy functions of BP c FSs. Secondly, a distance measure is formulated for the BP c FSs and then implemented for the extension of TOPSIS. Thirdly, a multiple criteria decision making (MCDM) model is proposed to handle the uncertain MCDM problems. Lastly, a practical example related to the sum of money's investment is exemplified to validate and effectiveness of the proposed model.