A New Approach to Three Way Decision Through Spherical Double Hierarchy Linguistic Information

Branded apps offer both an essential tool for businesses and consumers to interact in real-time and share marketing messages as well as an innovative approach to business that promotes value co-creation. In order to explore the impact of branded apps on customers, this study provides a Three-way decision (TWDs) technique with decision-theoretic rough sets (DTRSs) for selecting branded apps under spherical double hierarchy linguistic term sets (SDHLTSs) information. SDHLTS is a combination of the first hierarchy linguistic term and the second hierarchy linguistic term that can more flexibly describe ambiguity and uncertainty in decision-making (DM) problems. To aggregate the SDHLTSs, we propose a series of aggregation operators and fundamental operational laws for SDHLTSs. A grey relational analysis is considered to evaluate a conditional probability, which improves decision-making. A step-wise algorithm of the TWDs method based on decision-theoretic rough sets (DTRSs) is given for SDHLTSs. Further, a proposed aggregation operator is used to compute the loss function, and the decision’s results are decided by the minimum-loss principle. Finally, a real-world case study of the brand experience of a branded app is considered to demonstrate the efficiency of the proposed methods.


I. INTRODUCTION
The use of multi-attribute decision marking is growing in tandem.With the world's rapid population growth, problems in medical decision-making due to increasing uncertainty and vagueness in real-world data.Classical tools did not address this type of uncertainty and ambiguity.In data, these real-life and media specialists are turning to modern tools to reduce the uncertainty in real-world data.Therefore Zadeh 1965 [1] the fuzzy set analyzed the uncertainty and ambiguity by only degree of membership (DM).Later, the fuzzy set filed to explain the uncertainty by only degree of membership (DM).Therefore Atanassov's [2] introduced the concept of The associate editor coordinating the review of this manuscript and approving it for publication was Vicente García-Díaz .
IFS by adding degree of non-membership (NDM) to an FS and created the so called intuitionstic fuzzy set (IFS) with satisfying the condition (DM ) + (NDM ) ≤ 1.The concept and results of IFS, as well as its application to DM problems.Yager [3], [4] extended the concept of IFS and modified a new theory called PyFS for such problems, subject to the constraint (DM ) 2  + (NDM ) 2  ≤ 1,proving that the theory has a better processing possibility for resolve the MCGDM problem.Therefore in Cuong [5] has developed the concept of a picture fuzzy set (PFS) define as (DM , NeDM and NDM ) represent with satisfying the condition (DM ) 2  + (NeDM ) 2 + (NDM ) 2 ≤ 1.While FS and IFS are becoming popular techniques, individuals are more familiar with using linguistic term sets (LTSs) in fact to convey their assessment data, like as ''very bad'', ''some what bad'', ''excellent'', etc.Therefore, LTSs can effectively link with difficult circumstances.This case, IFS theory has failed to describe the uncertainty in the daily life applications.To solve this problem, Zadeh [6] presented the linguistic variable (LV) firstly and then Xu [7] extended the discrete linguistic term set (LTS) to the continuous LTS.Then, based on the IFNs and LVs, Chen et al. [8] proposed linguistic intuitionistic fuzzy numbers (LIFNs).The degree of membership (DM) and degree of non-membership (NDM) of an LIFN are expressed by LVs.Although LIFNs can quantitatively process language evaluation information, its application range is also satisfying the condition (DM ) + (NDM ) of an LIFN must be less than the length of the LTS.In order to expand the scope of linguistic information expression, Garg [9] proposed linguistic Pythagorean fuzzy numbers (LPFNs), Liu and Liu [10] proposed linguistic q-rung orthopair fuzzy numbers (Lq-ROFNs).Although LqROFNs have a greater range of information expression capabilities than LPFNs and LIFNs, but Lq-ROFNs cannot express hesitant evaluation information.Therefore, for the evaluation of brand experience of branded app, it is necessary to find a new tool that can three way decision through spherical double hierarchy linguistic information.
Ashraf et al. [25] discovered by the linguistic picture fuzzy set (LPFS), where DM, NeDM, and NDM are represented by linguistic terms, and it also solves the MAGDM problems by utilizing the PFLNWAA and PFLNWGA aggregation operators.Jin et al. [11] introduced by the idea of linguistic spherical fuzzy aggregation operators(AOs) to help solve decision-making problems.
Since Zadeh [12] has proposed and discussed the relevance of computational frameworks to words (CW), many extended forms of LTSs [13] have been suggested and studied.Modelling of expert expression Gou et al. [14] presented the double hierarchy linguistic term sets (DHLSs), which consist of the first hierarchy and second hierarchy linguistic term sets.Some of the basic distance and similarity measures of the DHFLEs were proposed by Gou et al. [15].And more convenient and suitable for most DMs qualitative evaluation of project attributes through a single linguistic term set, DHLTSs express qualitative information more flexibly through complex linguistic expression [16].The linguistic generalized spherical fuzzy set was generalized from algebraic [17].They have different decision-making attitudes than DMs, making the decision more practical.So algebraic operators have some desirable properties.This explains the introduction of algebraic aggregation operations in this paper.
Various traditional decision-making procedures have been created in the literature, however, they just provide a ranking of the schemes and do not provide decision experts with precise suggestions for improvement.Three-way decisionmaking (TWDs) breaks through this limitation because the decision-making method conforms to people's thinking patterns.For that purpose, Yao [18], [19], [20] developed the TWD technique due to their unique ability to handle DM problems.Liu et al. [21] proposed the double-hierarchy linguistic term set (DHLTS), which was further extended to the HFLTS context as DHHFLTSs.Using the Bayesian process [22], [23], objects are intuitively separated into three disjoint regions.If an object is separated into positive, negative, or border regions, it indicates that the DMs should accept the object, reject the object, or delay the decision.Since TWDs necessarily correspond to human decision patterns, they have been used to a variety of fields, including medical treatment [26], [27], financial decision, [28] and activity rehabilitation [29].Numerous extended forms of fuzzy sets, such as fuzzy set, [32] triangular fuzzy number, [33] dual hesitant fuzzy set, [34], [35], [36] have been presented into the TWDs procedure to directly accessible the loss functions (LFs) in the TWDs.The advent of DHLTS heralds the arrival of a new tool for obtaining diagnostic facts on expression in TWDs.DHLTSs provide a new tool.For expressing assessment information in TWDs.As DMs evaluate projects attribute information, they may give the evaluation value through DHLTS.More intuitively, reducing the time to make a decision.DMs value prompt decisions.DHLTSs help DMs make better informed decisions, proving that they are useful and effective tools for DMs.
When designing the loss function matrix, [37] use the method of calculating LFs in combination with game theory.[38] invent an optimization problems through the study of the association between the threshold values and loss functions, and achieved the threshold values when solved the optimization problems.Reference [40] performed the multiple criteria experiment, given the new calculation techniques of LFs.However, DMs tend to evaluate loss functions in practise based on their own knowledge and experience, and this paper uses this method in conducting research.The several researchers have investigated the calculation of conditional probability, which is another crucial component of TWDs.Sun et al. [39] the entropy weight method was used to calculate attribute weights first, and weighted aggregation was used to calculate conditional probability.Given by [41] developed a decision theory rough sets (DTRSs) model and applied itÂăto TWDs.They used the grey relational analysis (GRA) method to compute the conditional probability [42].The states Ã and Ãc of the TWDs are indicated by positive ideal solution (PIS) and the negative ideal solution (NIS) separately.Wang et al. [43] calculated the conditional probability using two DM methods based on third-generation prospect theory.Reference [44] calculated the attribute weights through maximizing deviation technique [45] first, and after that achieved the conditional probability with the technique for order performance by similarity to an ideal solution (TOPSIS).From the above literature, we analyzed that there is no idea of using DHLTSs to solve TWDs, making problems depend on the SDHLTWA aggregation operator and providing details about the combination of the study of LSFSs and DHLTSs for handling uncertainty and fuzziness.From 600 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the above-mentioned goals, the motivation for study and main contributions of this work are as follows: Motivation of study: According to the literature review of LSFSs and DHLTSs.There is no concept of spherical double hierarchy linguistic term sets.A new approach to three way decision through spherical double hierarchy linguistic information.The main motivation of the article are given as follows.
(a) Gou et al. [14], [15] DHLTSs were developed by only considering positive membership degrees, but this approach has limitations due to the lack of negative and neutral membership degrees.Thus, we generalised DHLTSs by including negative and neutral membership degrees and developed spherical double hierarchy linguistic term sets (SDHLTSs).They are adaptive tools that allow decision experts to provide assessments in the form of SDHLTS.
(b) Li, Xang et al. [30] by investigating the DHLTSs, the LFs were established using Hamacher aggregation operators.We apply the idea to SDHLTSs as a way to calculate the LFs.
(c) Further, we developed the basic operational laws and the spherical double hierarchy linguistic weighted averaging (SDHLWA), spherical double hierarchy linguistic ordered weighted averaging (SDHLOWA), and spherical double hierarchy linguistic hybrid averaging (SDHLHA) operators to aggregate the LFs to account for various decision attitudes of decision experts, which improves the DM process.
(d) A few recent analyses on spherical double hierarchy linguistic information is insufficient, and about their properties, it is not fully investigated.It is necessary to introduce a novel concept such as spherical double hierarchy linguistic term sets (SDHLTSs) by extending the LSFSs and DHLTS to produce decision-relevant information more precisely.
(e) Finding unknown weight vectors for decision-makers or criteria is a critical issue.To address this issue, the entropy measure of SDHLTS-based is developed to obtain the weight of DMs and criteria.
(f) The calculation of the conditional probability by the GRA method takes into both the relationship between the relative positive ideal matrix(RPIM) and relative negative ideal matrix (RNIM).The above makes the decision results more objective.
(g) We applied the proposed methodology to TWDs for the brand experience of branded App to demonstrate the impact of three-way decision-making.
Contribution of study: In this current work, we create various types of SDHLTWA aggregation operators in the SDHLTS environment.The SDHLTS can undoubtedly be explicit.The uncertain subjective data in the most ideal manner, and SDHLTWA aggregation operators provide more versatility in the data aggregation process.The main contributions of the article are given as follows: (a)First, we define SDHLTSs with the help of LSFSs and DHLTSs.
(b) We introduce a new score and accuracy function for SDHLTS.
(c) We develop various types of SDHLTA aggregation operators for SDHLTS, such as the SDHLTWA aggregation operator, the SDHLTOWA aggregation operator, and the SDHLTHA aggregation operator to deal with group decisionmaking problems in which the attributes have interrelationships.(d) We present an MCGDM method for solving a numerical problem, and afterward Apply these TWDs to solve a numerical example.(e) Conditional probabilities are evaluated using the GRA method, taking into account the relationship between relatively positive and negative ideal solutions.
(f) The SDHLTSs-TOPSIS method and the SDHLTSs-GRA method are proposed based on TWDs, respectively.We use these TWDs to rank the alternatives for brand experience of branded apps and do some detailed comparative analysis based on their ranking results.
(g) We applied the proposed methodology to TWDs for the brand experience of branded apps to demonstrate the impact of three-way decision making.
The rest of the paper is arranged as follows: Sect II introduces the basic aspects of FHLS,SHLT, and DHLTS which will help later.Sect III introduces the novel notion of SDHLTSs and score functions.Sect IV contains distance measures and aggregation operators for SDHLTS such as (SDHLTWA) spherical double hierarchy linguistic weighted averaging aggregation operators, (SDHLOWA) spherical double hierarchy linguistic order weighted averaging aggregation operators and (SDHLHA) spherical double hierarchy linguistic hybrid averaging aggregation operators.Sect V discusses the algorithm for calculating the conditional probability using the GRA approach and a novel TWD model.Sect VI describes the application of the proposed method by solving a numerical example of the brand experience of branded apps to illustrate the feasibility of the proposed method.In Sect VII we compare the proposed method with the existing approach to demonstrate the applicability of our proposed method.Sect VIII explains the conclusion of the article.

II. PRELIMINARIES
We will introduce some basic literature on IFS, PFS, and linguistic picture term sets, linguistic spherical term sets, spherical double hierachy linguistic term sets in the section, which will be useful in later sections.
Operational Laws of Linguistic Spherical Fuzzy Set: are the two LSFNs [11], where .
Score and Accuracy Function For SFDHLS: be a spherical fuzzy double hierarchy linguistic set (i ∈ N) .The score Ŝ ĉ and accuracy (Ac) functions are defined as follows; (1) (2)

IV. SPHERICAL FUZZY DOUBLE HIERARCHY LINGUISTIC AGGREGATION OPERATORS
This part is dedicated to the analysis of SFDHLWA aggregation operators.The SFDHLOWA aggregate operator that weights the ordered position of the argument.The SFDHLHA aggregation operators are an important generalization of SFDHLWA and SFDHLOWA aggregation operators because they can weight ordered positions and the arguments themselves, and they exhibit basic properties of SFDHLWA, SFDHLOWA, and SFDHLHA aggregation operators.

SFDHLWA aggregation operators:
., n) be the collection of SFDHLVs and (ψ 1 , ψ 2 , . . .ψ n ) T represent weight vectors of given collection restricted to ψ i ≥ 0; n g=1 ψ i = 1.Then based on above operational laws spherical double hierarchy linguistic term weighted averaging (SFDHLWA) operator are defined as: Now, there are some basic properties which are discussed in detailed for the SFDHLWA aggregation operators that is idempotency, monotonicity and boundedness. Let ., n) be the family of SFDHLVs and represents the weight vectors of given collection restricted to ψ i ≥ 0; n g=1 ψ i = 1.Then its desirable properties are as follows: (1) (Idempotency): (3) (Boundedness): Suppose Ã− and SFDHLOWA aggregation operators can be defined as; Now, there are some basic properties which are discussed in detailed for the SFDHLOWA aggregation operators that is idempotency, monotonicity and boundedness.
Now, there are some basic properties which are discussed in detailed for the SFDHLHA aggregation operators that is idempotency, monotonicity and boundedness ., n) be the family of SFDHLVs and represents the weight vectors of given collection restricted to ψ i ≥ 0; n g=1 ψ i = 1.Then its desirable properties are as follows: (1) (Idempotency): (2) (Monotonicity): Suppose another colllection (3) (Boundedness): Suppose Ã− and Distance measure for spherical double hierarchy linguistic set: Ŝγ 2 ⟨℘η2 ⟩ be two spherical double hierarchy linguistic set.Then the distance measure between any two SDHLTSs for any℘ > 0 (∈ R) are defined as follows: Here we used the idea of Renyi entropy [57] and provide a novel entropy measure for SDHLTSs.
Entropy measure for spherical double hierarchy linguistic set: SDHLTSs entropy measure are defined as.

V. CONDITIONAL PROBABILITY BASED ON TOPSIS AND GRA METHOD
The decision in the form of SDHLTSs is determined in four phases.The following phases are used to find conditional probability based on TOPSIS and GRA method with SDHLTSs.PHASE 1.
In this phase, First construct the decision expert matrix in the form of SDHLTSs with all the unknown information about the weights of each expert matrixes.
(a) Construct the experts evaluation matrices where Ãℓ We find the weights of the expert matrix, because when the weights of the experts are hidden, it is difficult for the DM to get the correct result.Therefore, experts have the following steps to calculate the weights of the matrix.
(b) Construct the expert ideal matrix (E IM ) of normalized decision matrix Êℓ that is closer to all matrixes of each expert information is determined by applying the SFDHLWA operator; where (b1) Construct the expert right ideal matrix and expert left ideal matrix denoted by E RIM , E LIM defined as; where and where (b2) Calculate the distance measure denoted by RIM and E LIM .The using equation (11).

Phase11
(a) Construct the revised expert ideal matrix (R v E IM ) of normalized decision matrix Êℓ that is closer to all matrixes of each expert information is determined by applying the SFDHLWA operator; (b) Determine the Criteria weights ŵi first we calculate the score matrix of aggregated matrix then apply Renyi entropy [57] to determine entropy measure as follows: (c) Determine the weights of criterias as follows:

Phase III: Conditional Probability:
The TWD approach depend on two main things namely LF and conditional probability.To find the conditional probability first we defined SDHLTSs.Suppose Ã = ‫ג{‬ 1 , ‫ג‬ 2 , .., ‫ג‬ m } be set of alternatives and Č Ã = {ℸ 1 , ℸ 1 , .., ℸ n } conditional attribute in the form of SDHLTSs with unknown weights vectors.V = ℸ∈ Č Ã, V ℸ , V ℸ denotes a domain of the attribute T ,q: Ã × Č Ã → V denotes a function such that q(‫ג‬ i , T ) ∈ V T for everyT ∈ Č Ã, ‫ג‬ i ∈ Ã,where To provide an evaluation reports for each alternative based on conditional attribute there exist a e no of experts represented by a set Ê = Ê1 , Ê2 , .., Êe Then the expert evaluation matrix in the form of SDHLTS are represented by Ê * = ÃB ij m×n .
(I ) Determine the relative positive ideal matrix (R PIM ) and relative negative ideal matrix R NIM of SDHLTSs represented by . ., a − m are defined as and where ( j = 1, 2, .., n).When the TWDs, (R PIM ) and R NIM are added together, they are equal to the set of states, Ã and Ãc .
(II ) Grey coefficient for each alternative calculated from (R PIM ) and R NIM by utilizing following below equation.The grey coefficient for each alternative calculated from (R PIM ) is Similarly, the grey coefficient of each alternative calculated from R NIM is where and the identification coefficient ρ = 0.5,i = 1, 2, . . ., m and j = 1, 2, . . ., n.
Calculating the grey coefficient degree for each alternative from (R PIM ) and (R NIM ) are as follows and; (III ) Relative relational degree (RRD) represented by ℑ i .
(IV ) Where ℑ i is consider to be conditional probability of an object lie in state Ã, that is Such that 0 ≤ P r ( Ã/‫ג‬ i ) ≤ 1.

Phase IV: DECISION MAKING based on novel DHLDTRS Model with Spherical fuzzy set:
As presented in the above notion of SDHLTSs, we consist of two terms namely first hierarchy and second hierarchy linguistic term sets that can manage uncertainty and fuzziness better than a single term.We discussed the loss functions in TWDs using DHLEs, as well as how to construct a new DTRS model using DHLEs with spherical fuzzy settings.This model has two states such as, Ã, Ãc , which describe whether an element belongs to Ã or not, and three actions, {a P , a B a N }.Where a P , a B a N represent the action that is used to determine the objects ‫ג‬ i that is a P indicates a P belong to POS( Ã) positive region, a B indicates ‫ג‬ i belong to BND( Ã) boundary region and a N represent ‫ג‬ i is in NEG( Ã) negative region respectively.The states of set classify an object's over all stiuation, where as action represents judgement.Here we determined the loss function for SDHLTSs, which are given in table 1, we see that the determined LF are in the from spherical double hierarchy linguistic number (SDHLN), and ĥPP , ĥBP and ĥNP are the loss degrees generated by takings actions of a P ; a B and a N an for ‫ג‬ given state Ã, with DHLN settings.Similarly to the way, ĥPN , ĥBN and ĥNN represent the loss degrees produced by doing the identical actions on ‫ג‬ particular state Ãc .According to the definitions of SDHTN and DTRSs [36], [59], the morally permissible relationship is as follows: Such that, the loss degrees of inaccurate decision are more than the loss degrees of delayed decision, and both of these loss degrees are greater than the loss degrees of accurate decision.Conditional probabilities are an important component of Bayesian decision-making approaches [22], [23].P r ( Ã/‫ג‬ i ), P r ( Ãc ‫ג/‬ i ) represents the conditional probability belonging to Ã and Ãc respectively.All these belonging to real value subjected to P r ( Ã/‫ג‬ i ) + P r ( Ãc ‫ג/‬ i ) = 1.The expected loss for the corresponding action R r (a ‫ג|‬ i ) where( = A, B, N ) can be computed for a given object  i as follows: R r (a The minimum loss decision rules may be deduced from the results in [18] and [20] as follows.
represents that the action are acceptable.
represents the action are delayed.(c) Resolve ‫ג‬ i belong to BNG( Ã); if represents the action are rejected.The proposed method is graphically represented in Fig 1.
The TWDs Methodology: We propose a novel approach of TWDs in the context of SDHHLTSs in light of the afore mentioned findings.The three-way decision method is divided into four phases.
Phase I: First construct the decision experts matrix in the form of SDHLTSs with all unknown information about the weights of each decision experts matrixes.
Phase II: Construct the revised expert ideal matrix (R v E IM ) of normalized decision matrix Êℓ that is closer to all matrixes of each expert information is determined by applying the (SFDHLWA) aggregation operator and using the entropy measure method, we determine the criterion weights.As determined by using equation 19-20.Phase III: Calculate the (R PIM ) A + and the (R NIM ) A − using equation 21-21 for score function determined.
The conditional probability by using the GRA method, indicated by ℑ i and as determined by using equation 25-28.
Phase IV: Each action's expected loss can be aggregated using equation 29-30 in accordance with (SFDHLWA) aggregation operator.It is therefore possible to determine the score function of expected losses.Each actions expected loss can be aggregated using Formulas 33-35 in accordance with aggregation operators.It is therefore possible to determine the score function of expected losses.Finally, according to the decision rules, deduce the decision outcomes.

VI. NUMERICAL APPLICATION
The developed MAGDM method is inially demonstrated in this section with a numerical application involving Brand Experience of Branded App.Then in order to demonstrate the characteristics and advantages of the proposed technique, a comparison is made between it and another decisionmaking techniques that use SDHLTSs.
A. CASE STUDY:BRAND EXPERIENCE OF BRANDED APP Brand apps are not only an important platform for enterprises and customers to connect and deliver marketing messages in real time, but also a new business model that encourages the co-creation of value.This study develops a fuzzy multicriteria decision-making (FMCDM) analytical model to investigate the impact of brand apps on customers.
Mobile App and Branded App: On mobile devices, a free mobile app can be downloaded.Unlike traditional marketing tools, the mobile app integrates a variety of innovative contents and functions while also extending enterprise customer service.It promotes value co-creation between brands and customers through real-time online interaction and serves as a platform for communicating marketing messages.According to Newman et al. [46] employing applications to give value to customers provides a chance for many retailers to reestablish or strengthen their competitiveness.In the meantime, branded apps for mobile devices can help develop a distinct brand identity by using the brand's name, logo, or totem [47].
As a new marketing tool, mobile apps may help to increase brand loyalty and purchase intent, strengthen the customerbrand link, and increase overall sales by improving brand satisfaction [48].According to Stocchi et al. [49], this shift in communication approach is due in part to the strong influence of a good user experience through branded apps on brand loyalty and purchase intent.In encouraging brand-consumer interactions, such apps have an advantage over conventional advertising techniques.
Brand Experience of Branded App: Schmitt came up with the concept of the entire brand experience in 1999.He describes experiences as ''private events that occur in response to certain stimulation.They are usually the result of direct observation or participation in events real, fictional, or constructed and thus induced rather than self-generated.Brand experience is described as the customer experience, feelings, perceptions, and behavioral responses that result from brand-related stimuli all part of a brand's design, identity, packaging, communication, and environment [50].Lee and Kang [51] different dimensions of brand experience work together to provide the overall brand experience and help businesses build good rapport with consumers.Ambler et al. [52] the development of brand experience is the result of customers using a brand and discussing and collecting brand-related information or promotions.Nadzri et al. [53], clients having a brand experience even before encountering a brand.Customers are stimulated by multiple brands when browsing the items on the market.Therefore, marketers must not only focus on the functional attributes and efficacy of the product, but also understand consumer sentiment from the perspective of the overall brand experience.Kim and Yu [54] proposed that the overall brand experience of customers generated by interacting with a brand through a brand app [55], [56] consists of four important alternatives.
(a) Affective ‫ג‬ 1 : Intrinsic feelings about the brand; perceptions and attitudes toward events; subjective emotional experience, (b) Cognitive ‫ג‬ 2 : The process of using concepts, perceptions, judgments, and imagination to obtain brandrelated information: (c) Behavioral ‫ג‬ 3 : Behavioral response prompted by a brand that invigorates customers or makes them display a specific behavioral pattern: (d) Relational ‫ג‬ 4 : Formation of a certain relationship or connection with a brand.
In addition, the four brand experience of branded app response alternatives are assessed using four criteria.
(a) Emotions ℸ 1 .Using this app makes me feel joyful, absolutely thrilled, pleasant, and so on.(b3) The closeness indices is evaluated as follows: CI (1) CI (2) CI (3)  0.46857 0.63830 0.78490 (b4) Weight of expert matrix are evaluated as follows: ϒ (1) ϒ ( 2) ϒ (3)   0.25 0.336 0.414 610 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Phase II: (a): Compute the revised expert ideal matrix (R v E IM ) as shown in Table 10 as follows: (b) Utilizing Equation ( 12) the entropy measure associated to each attribute are calculated as follows:  represent as ℑ i , and conditional probability P r ( Ã/‫ג‬ i ) can be determined based on GRA that the object belong to the state Ã, as shown in Table 12.
Phase IV.
(a) Construct the loss functions matrix from the DM in the form of SDHLTNs as follows: (b) Based on SDHFLTA operational laws we derived the expected loss of each action by applying equation ( 30), (31) and (32) as follows: Graphically the expected loss function are represented in Fig 2: (c) Determine the decision result for each object further using the decision rules P, B and N based on the minimum 612 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.From above result we analyze that ‫ג‬ 1 , ‫ג‬ 4 are consider to be select and ‫ג‬ 2 , ‫ג‬ 3 assume to be rejected.

VII. COMPARISON SECTION
In this part, we compared the suggested approach's advantages and execution to the TOPSIS method and MADM method.

A. COMPARISON WITH TOPSIS METHOD
To calculate the conditional probability, we use the TOPSIS method by Liang et al. [44].As a result, this comparison takes into consideration the identical weights for R PIM and R PIM that we determined in our suggested technique.Then, we used the TOPSIS method to find out the conditional probability as shown in table 15.
The same LF as in table 13 is then used, and the expected loss is determined in table 16.

B. COMPARISION WITH TODIM METHOD
In this section, to demonstrate the effectiveness of the developed decision-making technique, we compare it with the existing TODIM technique [26].Therefore, this comparison is made by considering the same weights and evaluation matrixes of the decision experts as we have calculated in our proposed method.The detailed steps of the TODIM method are as follows.
Step 1. Determine the relative weight ω jŝ of the criteria ℸ j to ℸ ŝ by the given formula.
, where ω ŝ = max ω j , j = 1, 2, . . ., j Step 3. The overall dominance degree of the alternative ‫ג‬ i over each alternative ‫ג‬ ť according to the decision experts matrix is evaluated by the given formula.Step 5. Evaluate the overall value of the alternative ‫ג‬ i by the given equation as follows.

‫ג(‬
‫ג‬ i , ‫ג‬ ť Step 6. Rank all the alternatives by the overall values of ‫ג(‬ i ).The bigger ‫ג(‬ i ) is, the better the alternative.Below are the evaluation steps.
Step 1.As the weights of critria are ω j = (.07,.31,.29,.33)T .We calculated the relative weight ω jŝ of the criteria Cj to Cr are determined as.ω r = max {0.07, 0.31, 0.29, 0.33} = 0.33 ω jr = (0.212, 0.939, 0.878, 1.000) T Step 2-3.Calculate the dominance degree of the alternative ‫ג‬ i over each alternative ‫ג‬ ť with respect to the E ℓ under the criteria Cj, φ = 2.4, and the overall dominance degree of the alternative over each alternative is determined as.Step-6.To choose the best alternative by rank the values of ‫ג(‬ i ), the alternative with maximum value is the best choice.According to step 5, the ranking of ‫ג‬ i is ‫ג(‬ 1 ) > ‫ג(‬ 4 ) > ‫ג(‬ 2 ) > ‫ג(‬ 3 ) and it is clear that the best choice is ‫ג‬ 1 .
Hence from the overall value of the alternative ‫ג(‬ 1 ) > ‫ג(‬ 4 ) > ‫ג(‬ 2 ) > ‫ג(‬ 3 ), it is clear that the best alternative based on the TODIM method is ‫ג‬ 1 which are identical to that of the proposed method.Hence it is analyzed that our proposed method is efficient and practical to solve the and uncertainty in the DM problems.The graphical ranking alternative based on TOPSIS and TODIM Method are represented in Fig 3: We produced more results, ranks, and conditional probabilities using different methods and aggregation operators, and the best result is ‫ג‬ 1 , as shown in Tables 17 and 18. Topsis, BP, weighted aggregation and proposed method.Conditional probablities of existence methods is graphically represented in    TWDs model presented by Liang et al. [44].Conditional probabilities are estimated using the weighted aggregation method.The bidirectional projective (BP) [60] technique analyses the relationship between the scheme and the ideal solution, increasing the objectivity of conditional probability assessment.Conditional probabilities are estimated using the weighted aggregation operators [11].Using the same attribute weights, the conditional probability is calculated in Table 17 using other existing methods.Table 18 represents the ranking of alternatives based on conditional probability.As a result of Tables 17-18, we determined that the best result is ‫ג‬ 1 obtained from other current ways that are comparable to the proposed method, demonstrating the practicability of the suggested methods.

D. DISCUSSIONS
The loss created by actions in different states may be shown using the DTRS model, one of the TWD elements, and the action with the minimum loss principle.Conditional probability is one of the other elements that make up the TWD method.The conditional probabilities in most existing TWD models are provided directly by the DM [60], making selecting a result appear less difficult.We compute the weight of each attribute using the Renyi entropy and the RRD of the object given by the GRA method as the conditional probability.linguistic terms used to characterise the qualitative issue are more closely related to human expression habits.In the case of a single linguistic set, SDHLTS allows for more flexible expression of qualitative information.The appearance of SDHLTSs provided a novel method of transmitting evaluation information in TWDs.When DM evaluates project attribute data and can provide DHLTS with evaluation values more intuitively, which greatly reduces decision-making time.In the DHLT environment the proposed model is constructed.A new research direction is the TWD model based on the SDHLTS information system.As a result, it has a relatively high research value.In this paper, some desirable properties of the SDHLHWA operator are demonstrated, making decision-making problems more practical.
The main advantages of the proposed method are as follows: (1) The SDHLS, which consists of the FHLT and SHLT, can express the evaluation of DMs more flexibly in the TWD process.Therefore, for dealing withÂ decision-making problems, the SDHLS-based TWD method is a useful tool.
(2) The GRA method is used to calculate the conditional probability, which replaces hamming distance with a weighted grey relational degree as a distance measure to improve the TOPSIS model.Furthermore, the SDHL operator takes into account the different decision-making attitudes of DMs when aggregating LFs.They make the decision-making process more rational.

VIII. CONCLUSION
This study developed a research model with academic value based on the holistic brand experience theory by presenting the characteristics of a brand experience connected to branded apps, combining satisfaction and loyalty influence assessments, and evaluating the influence of each factor separately.Additionally, this research developed a TWD technique under SFSs and DHLTS that is adept at handling ambiguous and unclear information.We investigated the novel concept of SDHLTSs by extending the idea of DHLTSs to deal with uncertainty in the DM problem.Furthermore, we proposed SDHLWA, SDHLOWA, and SDHLHA aggregation operators.The basic desirable properties of the proposed operator were described in depth.The Renyi entropy measure was used for calculating the weights of the criteria.A GRA technique was utilized to calculate the conditional probability based on RRC, which replaces the hamming distance with the weighted GRC as a distance measure to improve the TOPSIS method.The aggregation of loss functions by the aggregation operators takes into account the different decision-making attitudes of DMs.TWDs were described in detail using the SFDHLT environment, taking into consideration evaluation values and loss functions.Finally, to show the practicability and effectiveness of the TWD method applied to real-world problems for selecting the best brand experience in branded apps.
In the future, research directions will include the aspects below: (a) Hamacher aggregation operators, (b) Yager aggregation operators, (c) Einstein aggregation operators (d) Dombi aggregation operators (e) It will also concentrate on incomplete information processing in decision-making application sectors such as online project recommendation, work, and production resumes, investment decision-making, online medical selection, disease prediction, etc.

FIGURE 1 .
FIGURE 1. Graphical presentation of proposed method.
(a) Resolve ‫ג‬ i belong to POS( Ã); if (b) Curiosityℸ 2 .This app arouses my curiosity.(c) Functional experience ℸ 3 .I want to use the functions (e.g., mobile shopping and information on new products) of the app again.(d) Self-identification ℸ 4 .I identify as a member of the brand community when I use the app.Due to the confedentiality of the information, only limited project details are presented.Assuming that the experts weight vectors and the criteria weights are totally unknown, the evaluated value of candidates while considering criteria.Now we applied the above problem to TWDs based on SDHLTSs setting.The step wise details are as follows.Phase 1.(a) Construct experts evaluation matrix in the from of double hierarchy spherical linguistic term sets, so the linguistic term set are denoted by S = {S 0 = medium, S 1 = low, S 2 = very low, S 3 = sightly low, S 4 = high low, S 5 = pretty low, S 6 = very high, S 7 = most high, S 8 = very most high } and ℘ = {℘ 0 = bad, ℘ 1 = fair-bad, ℘ 2 = verybad, ℘ 3 = a bit-bad, ℘ 4 =a bit-bad,℘ 5 = high,℘ 6 = pretty high, ℘ 7 = extremely high, ℘ 8 = definitly high } are defined on the basis of following set as follows: (b) Calculate the expert ideal matrix E IM as: (b1) Calculate the expert right ideal matrix E RIM as: (b1) Calculate the expert lift ideal matrix E LIM as: (b2) Utilizing Definition (11), the distance of E (e) ij to E IM , E RIM and E LIM as follows in Table-6 (D E IM ), (D E RIM ) and (D E LIM ) respectively.
C. USING EXISTING METHODS, CALCULATE CONDITIONAL PROBABILITYConditional probabilities, which are an important part of TWD, can also be used in ranking algorithms.In the 614VOLUME 12, 2024    Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE 1 .
(a).Description of acronyms used in this work.(b).Representation of variables used in this work.(c).Explanation of symbols used in this work.

TABLE 2 .
The actions, DHLT loss function in various states.

TABLE 9 .
(D E IM , D E RIM , D E LIM ).

TABLE 10 .
Revised expert ideal matrix (R v E IM ).

TABLE 16 .
Score function of expected losses.
According to decision experts weights evaluate the collective overall dominance degree of the alternative ‫ג‬ i of each ‫ג‬ ť as follows.

1
‫ג‬ i , ‫ג‬ ť = The collective overall dominance degree of the alternative ‫ג‬ i over each alternative ‫ג‬ ť are computed as follows.