1. Introduction

Smarandache [1] introduced a neutrosophic set by finding the term degree of indeterminacy from the logical point of view as an independent component to handle imprecise, indeterminate, and unpredictable information in real-world problems. Neutrosophic sets are defined by truth, indeterminacy, and false membership functions, which take values in the real standard interval. The abstraction of indeterminate value is explained in the neutrosophic environment [2]. The massive information only gives real-life application, which is incomplete and fuzzy. Many techniques have been adapted to manage such information, such as fuzzy theories and probability theories. However, the measure exemplifies potential effects in the neutrosophic environment that are unprejudiced or overtoned, which can be regulated as continuous, discrete, or mixed. For instance, the Viterbi algorithm based on a fuzzy environment gives positive properties, but when we consider the neutrosophic environment, it can achieve indeterminacy (neutral) properties, which are not applicable in fuzzy sets.

Smarandache [3] presented the concept of single-valued neutrosophic sets (SVN), assuming that truth, indeterminacy, and false memberships are in a single-value in order to overcome the limitations of neutrosophic sets. This concept addresses the neutrosophic traffic flow problems [4]. Moreover, some models are formed for accident situations [5], algebraic structures [6], and COVID-19 [7]. Recently, Chaw et al. [8] presented a decision-making method based on SVN by considering complex neutrosophic numbers and algebraic relations to determine the factors influencing the oil price. Several new types of distances and similarity measures are investigated by Chai et al. [9] and applied to pattern recognition and medical diagnosis problems. Wu and Fang [10] designed a multilevel evaluation framework to assess the teaching quality in higher education with the help of TOPSIS and SVN. Saber et al. [11] investigated a single-valued neutrosophic soft set to describe the topological structure. Wang et al. [12] presented the idea of interval-valued neutrosophic sets (IVN), which are more precise and flexible than SVN. The interval-values membership function of measured form the truth, indeterminacy, and falsity. Akram and Nasir [13] utilised the idea of IVN in the concept of graph theory and also examined line graph in [14]. In the aftermath, Siti Nurul Fitriah et al. [15] extended the interval-valued neutrosophic sets to examine the incidence graph structures and their operations. IVN has been used to develop a sustainable supplier selection platform [16] that allows decision makers to find a suitable supplier for their supply chain industries in any preordained period. The concept of redefined IVN is an extension of IVN that Uluçay [17] introduced in 2021, and further, he studied its algebraic operators. Ebadi Torkayesh et al. [18] developed a sustainable municipal waste management system model based on interval-valued neutrosophic sets and multidistance measures to measure the social indicators among Istanbul citizens.

Markov Chain (MC) is a model to predict the future depending on the present state. MC’s initial position state vector uses the next position of the state used in Monte Carlo problems [19]. Ponomarev et al. [20] explained the MC time distribution. Hunter [21] showed that the MC in mixed times and further determines the rate of convergence in the MC network problem. Garcia [22] illustrated that the MC is an uncertain situation. Mallak et al. [23] introduced max-min in the MC ergodic process. Gerencsér [24] explained the ranges in the MC using mixing time and also revealed that the connectivity graph of a MC is a cycle. Kou et al. [25] demonstrated the incidence of diseases using the MC. Chan et al. [26] explained the distribution probability in the MC. Adeleke et al. [27] define the academic score based on the MC.

García et al. [28] utilised matrix analysis to identify the behaviour of the fuzzy Markov chain (FMC). Garcia [22] dealt with the MC in the interval type-2 fuzzy set. Vajargah and Gharehdaghi [29] presented the membership value of the FMC in Faure and Kronecker sequences. The Transition Probability (TP) matrix is explained by the state of the matrix moving from one state to another state of a system. The fuzzy number is distributed by TP in a FMC [27]. Li and Xiu [30] built the FMC model based on fuzzy triangular numbers to identify the fuzzy transition probability matrix. Lei et al. [31] investigated a new forecasting algorithm based on combining the multi-aggregation prediction algorithm and the FMC model. FMC is used to stabilize nonlinear estimation of multidimensional [32]. Interval-value has been deliberate for neutrosophic probability (NP) and used to analyze the equilibrium of MC under IVN [33]. Nagarajan et al. [34] studied the long-run behaviour of the world financial year under the interval neutrosophic MC framework. Kuppuswami et al. [35] investigated the MC model based on neutrosophic numbers and, as an application, they predicted traffic volume.

The hidden Markov model is a probabilistic model under uncertainty conditions that can be applied to determine a representation sequence [36]. The fuzzy hidden Markov model is an efficient way of finding an optimized path among the states where uncertainty exists. Darong et al. [37] improved the initial value of the observation matrix of the hidden Markov model for the motor drive system of urban rail transit by considering a predictive neural network and an intuitionistic fuzzy environment. Zeng and Liu [38] investigated the type-2 fuzzy hidden Markov model, in which the membership function of each hidden state is modelled by Gaussian primary with an uncertain mean and standard deviation. Moreover, they have derived the operators based on the type-2 fuzzy Viterbi algorithm and the forward-backwards algorithm in order to study speech recognition. Recently, Nagarajan et al. [39] derived the aggregation operators and Frank triangular norms for the interval type-2 fuzzy hidden Markov model and, based on that and the Viterbi algorithm, established a decision-making process to choose the best medicine company.

The motivation of the present work is to consider neutrosophic single-valued and interval-valued on the hidden Markov model because the combination has not been considered so far in the literature. The hidden Markov model cannot find uncertain information. The fuzzy hidden Markov model cannot find uncertain information with the nonmembership function. The interval-valued fuzzy hidden Markov model cannot find out uncertainty information with nonmembership function. The intuitionistic hidden Markov model cannot find the information during the addition of membership and nonmembership degrees more significant than one. The interval-valued intuitionistic hidden Markov model cannot find the information when adding membership and nonmembership degrees greater than one. But the neutrosophic hidden Markov model effectively finds the optimal path between the states where vagueness exists. That is the cause of the neutrosophic hidden Markov model considered for this present work.

The structure of this paper is organized as follows: Section 2 contains the basic notions of neutrosophic sets, single-valued neutrosophic sets, interval-valued neutrosophic sets and their operations, and the neutrosophic hidden Markov chain. In Section 3, we examine childhood obesity in lockdown situation applications by using single and interval-valued neutrosophic sets. Section 4 establishes the comparative analysis of the given application with different hidden Markov models. In Section 5, we have provided the conclusion [30].

2. Preliminaries

2.1. Markov Chain

A Markov chain is a sequence of random variables with the following properties. For , is defined on the sample space and takes values from the finite set . Thus, . Also for and ,

The transition probabilities are independent of [30].

2.2. Fuzzy Markov Chain

A fuzzy stochastic process is said to be a fuzzy Markov chain if it satisfies the Markov property.where establish the state space of the process.

Here, are called the fuzzy probabilities of moving from state to state in one step. Hence, , where is the membership of the transition from state to state . The matrix is called the fuzzy transition probability matrix.

2.3. Neutrosophic Set

Consider the space consists of universal elements characterized by . The neutrosophic set is a phenomenon which has structure as , where the three grades of memberships are from of the element to the set , with the criterion as follows:

The functions, the truth, indeterminate, and falsity grades lie in real standard/nonstandard subsets of [15].

2.4. Single-Valued Neutrosophic Set (SVNS)

The space of objects contains global elements . A SVNS is represented by degrees of membership grades mentioned in Definition 2.1. For all , . An SVNS can be written as [12].

2.5. Interval -Valued Neutrosophic Set

Let be a space of objects with generic elements in is denoted by . An interval-valued neutrosophic set (IVNS) in is characterized by truth-membership function, , indeterminacy-membership function , and falsity-membership function . For each point , , , and an IVNS is defined by where, , j [1].

2.6. Neutrosophic Markov Chain

The NMC is a sequence of neutrosophic random variables with the property that the next neutrosophic state depends only on the current state.which is a neutrosophic mathematical system characterized as memory less [1, 40].

2.7. Operations on Interval-Valued Neutrosophic Numbers

Let and be two interval neutrosophic numbers then Addition: Multiplication: Multiplication Neutrosophic probability: Addition Neutrosophic probability [34]:

2.8. Interval Neutrosophic Markov Chain

An interval neutrosophic stochastic process is said to be an interval neutrosophic Markov chain if it satisfies the Markov property.where establish the state space of the process.

Here, are called the interval-valued neutrosophic probabilities of moving from state to state in one step. Hence, , where are the lower and upper truth-membership of the transition from state to state , respectively, are the lower and upper indeterminate membership of the transition from state to state , respectively, and are the lower and upper falsity-membership of the transition from state to state . The matrix is called the interval-valued neutrosophic transition probability matrix.

2.9. Neutrosophic Hidden Markov Model

The neutrosophic hidden Markov chain (NHMC) is a neutrosophic Markov chain , whose states are unobservable directly but observed through a sequence of observations generated by the state is conditional only on . The NHMC is similar to fuzzy hidden Markov chain [41], where the arithmetic operations are neutrosophic operations. The model is depicted in Figure 1.