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Picture fuzzy set is the most widely used tool to handle the uncertainty with the account of three membership degrees, namely, positive, negative, and neutral such that their sum is bound up to 1. It is the generalization of the existing intuitionistic fuzzy and fuzzy sets. This paper studies the interval probability problems of the picture fuzzy sets and their belief structure. The belief function is a vital tool to represent the uncertain information in a more effective manner. On the other hand, the Dempster–Shafer theory (DST) is used to combine the independent sources of evidence with the low conflict. Keeping the advantages of these, in the present paper, we present the concept of the evidence theory for the picture fuzzy set environment using DST. Under this, we define the concept of interval probability distribution and discuss its properties. Finally, an illustrative example related to the decision-making process is employed to illustrate the application of the presented work.

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In this paper, we propose the application of the concept of power domination integrity to an electric power network. A phasor measurement unit (PMU) is used to analyze and control the power system by measuring voltage phase in electrical nodes and transmission lines. Due to the high cost of PMUs, it is necessary to minimize the number of PMUs such that the depth of observability is ensured. Placing PMUs in a network can be formulated as a graph theoretic problem of finding the minimum number of nodes (PMUs) in a graph that has a maximum number of links with other nodes. To achieve this, the concept of domination in graph theory is applied to power networks by redefining "adjacency" of a vertex as an "observed" vertex. The power domination number identifies the number of PMUs to be placed. The proposed concept of power domination integrity gives not only the minimum number of PMUs but also identifies the optimal locations for PMU placement in an electric power network.

Hidden Markov model (HMM) has become increasingly popular in the last several years. Real-world problems such as prediction of web navigation are uncertain in nature; in this case, HMM is less appropriate i.e., we cannot assign certain probability values while in fuzzy set theory everything has elasticity. In addition to that, a theory of possibility on fuzzy sets has been developed to handle uncertainity. Thus, we propose a fuzzy hidden Markov chain (FHMC) on possibility space and solve three basic problems of classical HMM in our proposed model to overcome the ambiguous situation. Client's browsing behavior is an interesting aspect in web access. Analysis of this issue can be of great benefit in discovering user's behavior in this way we have applied our proposed model to our institution's website ( www.ssn.edu.in ) to identify how well a given model matches a given observation sequence, next to find the corresponding state sequence which is the best to explain the given observation sequence and then to attempt to optimize the model parameters so as to describe best how a given observation sequence comes about. The solution of these problems help us to know the authenticity of the website.