Selecting cloud database services provider through multi-attribute group decision making: a probabilistic uncertainty linguistics TODIM model

Abstract

Cloud database services platforms (CDSP) have aided in both the generation of new ideas and the reduction of costs. Choosing the finest CDSP is a critical part of enterprise management for helping businesses respond to increasing market pressure and client demand. This study tried to solve this problem and proposed the integration concept of the TODIM (TOmada de Decisão Interativa e Multicritério) and the probability uncertain linguistic (PUL), which is a typical decision-making technique focused on the theory of prospects (PT-PUL-TODIM). The concept of probability uncertainty linguistic term sets (PULTS) is discussed. The cumulative weight of probabilistic double hierarchy linguistic knowledge is then determined by using the following cumulative prospect theory. Based on the experts’ opinions, complex expressions with experience in the Vietnam market were collected and further transformed into a PULTS matrix. However, the CRiteria Importance Through Inter-Criteria Correlation (CRITIC) technique is implemented to calculate the objective criteria weights. The PT-PUL-TODIM optimization approach is developed and adopted for the CDSP. Finally, we compared other methods and conducted a sensitivity analysis to compare the PT-PUL-TODIM approach with other ways to assess the validity, practicality, and usefulness of the technique, and the comparison revealed the merits and limitations of the model.

KEYWORDS:

• Multiple attribute group decision making
• probability uncertainty linguistic TODIM
• theory of prospects
• cloud database service providers’ selection
• CRITIC

JEL Classifications:

• C42
• C63
• D80

1. Introduction

Zadesh (1965) was the first to mention fuzzy sets (FS) [1]. Novel fuzzy set concepts, including intuitionistic fuzzy sets (IFS) [2], picture fuzzy sets (PFS) [3] and Pythagorean fuzzy sets (PyFS), have been developed one after the other [4,5]. According to Liu et al. [6], a new concept of normal wiggly hesitant fuzzy linguistic term set (NWHFLTS) has been a more significant instrument for assisting us in locating additional appropriate data. For probabilistic linguistic word sets, [7] developed a new probability linguistic term set (PLTS). Mo [8] combined PLTS with D number theory. Under PLTS, Du and Liu [9] created multi-Muirhead average operators. Under PLTS, He et al. [10] proposed the generalized Dice similarity metric. Lin et al. [11] define a new set of linguistic terms called probabilistic uncertain linguistic terms set (PULTS). The probabilistic uncertain linguistic (PUL) and the grey relational projection (GRP) approach were defined by [12]. Aisaiti et al. [13] developed the PULTS preference relation. He et al. [14] developed the Evaluation based on Distance from Average Solution (EDAS) within the context of PULTS. X. Gou et al. [15] developed an MCDM method named the probabilistic double hierarchy linguistic alternative queuing method (PDHL-AQM), where the decision-making result is intuitive by a directed graph or a 0–1 precedence relationship matrix, which applied the PDHL-AQM to solve a practical MCDM problem involving the real economy development evaluation under the perspective of economic financialization. Krishankumar et al. [16] presented a new big data-driven decision model with data in the form of complex expressions, which were transformed into a holistic decision matrix by adopting probabilistic linguistic information and based on the distance from average solution (EDAS) approach is extended to probabilistic linguistic information for the rational ranking of cloud vendors selection for a healthcare center in India. Ramadass et al. [17] proposed the PROMETHEE–Borda method is extended to PLTS for the evaluation of cloud vendors from probabilistic linguistic information with unknown/partial weight values. Feng et al. [18] developed a QUALItative FLEXible (QUALIFLEX) multiple criteria model based on PULTS. Wei et al. [19] examined the Multi-Attributive Border Approximation area Comparison (MABAC) technique in a PULTS setting. Bashir et al. [20] developed new PULTS procedures, metrics and operators.

The TOmada de Decisão Interativa e Multicritério (TODIM) refers to interactive multi-criteria decision-making [21–24], MABAC means multi-attributive border approximating areas comparisons technique [19,25], the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method [26–30], the Evaluation based on Distance from Average Solution (EDAS) typically, these strategies are employed to address the multi-attribute decision-making (MADM) or (MAGDM) difficulties [14,24,31–33]. The cosine similarity-based DHHFL-ELECTRE II method was proposed to solve MADM problem in the double hierarchy hesitant fuzzy linguistic environment in the performance evaluation of financial logistics enterprises [34]. The TODIM technique is different from these well-known systems due to its diverse treatment of profits and losses. According to He et al. [24], TODIM could be utilized to identify the optimum river-water transfer strategy [35] performed an additional application research based on the TODIM methodology for the site location of low-speed wind turbines. To avoid multicollinearity, [22,36,37] devised the TODIM approach in a fuzzy Pythagorean environment with interval values, [38] enhanced the standard TODIM technique's applicability in a green supplier chosen using probabilistic hesitant fuzzy information. In a fuzzy T-spherical environment [39] employed the TODIM approach. Luo and Liang (2021) evaluated the quality of cleaning products using the TODIM approach. The TODIM technique was defined by [40] as an interval-valued intuitionistic fuzzy set (IVIFSs). Zhao et al. [41] developed a behavioural linguistics distribution model of MCGDM using the TODIM approach. Arya and Kumar [28] used an image fuzzy set to merge the TODIM and VIKOR methods (PFS). Luo et al. [7] suggested a TOPSIS-TODIM hybrid model in their investigation of the fuzzy set of linguistic pictures created from PFS. However, the newly proposed intuitionistic fuzzy statistical variance method and finally, ranking of cloud vendor is done using newly proposed three-way VIKOR method under intuitionistic fuzzy environment which introduces neutral category along with cost and benefit for better understanding the nature of criteria [42].

According to a survey of existing studies, there have been many studies applying the classic TODIM approach. In fact, the TODIM approach has been expanded from the numerical value to various fuzzy sets, and it has been integrated with other decision methods. Based on Li et al. [43] addressed multiple-attribute group decision-making challenges utilizing interval intuitionistic fuzzy set theory, a technique analysis of existing TODIM is discussed, in addition to an application that focuses on the facility location of airport terminals to illustrate the method. Irvanizam et al. [44] proposed based on the TODIM technique employing MCDM was described across an assessment for finding solutions and a smartphone selection problem. The proposed design utilized triangle fuzzy set theory to adjust the linguistic terms of the set of criteria weights and applied therefore the average score of Beta distribution to convert the triangular fuzzy values into crisp values. When evaluating the degree of dominance for every alternative pairing focused on loss and gain, the TODIM technique may identify the best smartphone depending on the recommendations of the decision maker [45]. Extending the concept of evidence theory, a ranking model under probabilistic hesitant fuzzy set (PHFS) was proposed, which was a robust form of the hesitant fuzzy set that correlates occurrence probabilities with numerous membership grades, providing flexibility to experts during preference elicitation and facilitating the right management of uncertainty. Irvanizam et al. [46] suggested an expanded TODIM decision-making technique towards multiple-attribute decision-making (MADM) issues in a linguistics context applying dual-connection numbers (DCNs). The weights of attributes are determined by using CRITIC in the non-linear space. Following this, cloud vendors are ranked in a personalized fashion using the proposed algorithm that encompasses the WASPAS procedure and rank fusion schemes [47]. The extensive method incorporates fuzzy linguistics that the scores of alternatives and criteria for all these are expressed as triangular fuzzy numbers (TFNs) to convey uncertainty information. A cadre selection problem is used as an example to demonstrate the conformance and validity of the extended TODIM technique and to compare it with other methods. The compromise between the strengths of the prospect theory with the TODIM approach; scenario research involving the NSSP decision-making problems was provided to highlight the value of the proposed techniques [37]. The application of CPT-SVN-TODIM method was developed by [48], which is used in the assessment of medical emergency management. The reliability of the CPT-SVN-TODIM method is confirmed by comparing it with some other methods. However, limited investigations have been conducted to modify the TODIM approach by incorporating unique theoretical ideas into the TODIM model [38,41,49]. PT was proposed, which altered the factors influencing choice outcomes from probability and final resources to weighting and potential losses [37,50]. Integration of PT and TODIM with PULTS to provide any type of decision maker with a robust psychology explanation and accurate actual data. Using the PT-PUL-TODIM technique to identify cloud database service providers (CDSPs) becomes extremely beneficial for business managerial decisions.

Based on the above motivation, the innovations and the primary contributions of this research within the prospect theory focused TODIM proposed model developed within the perspective of PUL to MADM and MAGDM provide as follows:

1. Give a weight-determining method to obtain the weight vector of criteria, which is the important element in the process of decision-making using the PT-PUL-TODIM.

2. Develop the CRITIC to calculate the objective weight of attributes under PULTSs, which to deal with MAGDM problem for PT-PUL-TODIM model is proposed.

3. Apply the PT-PUL-TODIM technique to solve a practical MAGDM problem involving the CDSPs. Additionally, some comparative analyses are made to show the advantages and reasonableness of the PT-PUL-TODIM.

4. A case study relate to the evaluation critical factors that impact the selecting of cloud database providers is conducted with the model constructed based on PT-PUL-TODIM. The decision result can provide some reference values for the enterprises to make decisions and also to adjust products and services for increasing clients’ satisfaction.

5. The proposed model in this paper is compared with the existing models.

The following are the article's primary sections: basic theoretical information on PULTS is introduced in Section 1. In Section 2, the prospect theory, which represents the main idea behind the new method, is explained in more detail. The most important part of this work is Section 3, which discusses the article's proposed approach. The CDSPs are the focus of the example in Section 4. Finally, to determine whether it satisfies the specified technique’s application effect, in Section 5, we selected different current techniques in the probability uncertain linguistic environment and evaluated by comparing the others with the PT-PUL-TODIM technique. A flow chart is drawn to show the main work of this paper in Figure 1.

Figure 1. The flow chart of the proposed TODIM method.

2. Preliminary knowledge

In this section, we start to explain several fundamentals concerning the probability uncertain linguistic term set (PULTS).

Theorem 1:

[51]: $V=\{v_{-3}=\text{ extremely bad, }{v}_{-2}=\text{very bad},v_{-1}=\text{bad},v_0=\text{medium},v_1=\text{great},v_2=\text{very great},v_3=\text{extremely great} }$ is an illustration of the common LTS $V=\left\{v_{\vartheta }\mid \vartheta =-\varpi ,\cdots ,-2,-1,0,1,2,\cdots ,\varpi \right\}$ whereby each characteristic $v_{\vartheta }$ represents for a linguistic. The classification method $TF$ could assist us to replace the definition $v_{\vartheta }$ onto the evident $\hat{t}$. (1) $TF:\left[v\mathrm{\_}\left(-\varpi \right),v\mathrm{\_}\varpi \right]\to \left[0,1\right],TF\left(v\mathrm{\_}\vartheta \right)=\left(\vartheta +\varpi \right)/2\varpi =t^{\wedge }.$(1) In contrast, the function $T{F}^{-1}$ restores the sharp $\hat{t}$ to the linguistic term $v_{\vartheta }$ in the reverse direction. (2) $T{F}^{-1}:\left[0,1\right]\to \left[v_{-\varpi },v_{\varpi }\right],T{F}^{-1}\left(\hat{t}\right)=v_{\left(2\hat{t}-1\right)\varpi }=v_{\vartheta }.$(2) Following the development of the hesitant fuzzy term sets (HFTS) and probability linguistic term sets (PLTS), Lin et al. (2018) established the probability uncertainty linguistic term sets (PULTS).

Theorem 2:

[52]: According to the $\text{LTS}V=\left\{v_{\vartheta }\mid \vartheta =-\varpi ;\cdots ;-2;-1;0;1;2;\cdots ;\varpi \right\}$, as PULTS are formulated with: (3) $PU\left(\hat{\pi }\right)=\left\{\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]\left({\hat{\pi }}^{\left(m\right)}\right)\mid {\hat{\pi }}^{\left(m\right)}\ge 0;m=1;2;\cdots ;\mathrm{\#}PU\left(\hat{\pi }\right);\sum _{m=1}^{\mathrm{\#}PU\left(\pi \right)}{\hat{\pi }}^{\left(m\right)}\le 1\right\}$(3) where ${\hat{L}}^{\left(m\right)}\text{and}{\hat{U}}^{\left(m\right)}$ are the lowest and highest points of uncertain linguistic term (ULT) $\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]$, respectively $({\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\in V\text{as}\text{well}\text{as}{\hat{L}}^{\left(m\right)}\le {\hat{U}}^{\left(m\right)})$. In addition, ${\hat{\pi }}^{\left(m\right)}$ represents the corresponding probability of ULT $\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]$, as well as the overall quantity of ULT in PULTS $PU\left(\hat{\pi }\right)$ is # $PU\left(\hat{\pi }\right)$.

Specifically, it is required to synthesize the original PULTS if there is an exclusion or crossing connection between various separate ULTs in the equation $PU\left(\hat{\pi }\right)=\left\{\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]\left({\hat{\pi }}^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}PU\left(\hat{\pi }\right)\right\}$. For purposes of inclusion, the more comprehensive ULT is further segmented, such as in the following: $\left[v_0,v_2\right]\left(0.6\right)$ are divided into $\left[v_0,v_1\right]\left(0.3\right)\text{and}\left[v_1,v_2\right]\left(0.3\right)\text{in}PU\left(\hat{\pi }\right)=\left\{\left[v_0,v_2\right]\left(0.6\right),\left[v_0,v_1\right]\left(0.4\right)\right\}$ so that $PU\left(\hat{\pi }\right)=\left\{\left[v_0,v_2\right]\left(0.6\right),\left[v_0,v_1\right]\left(0.4\right)\right\}$ For crossovers, relatively similar part is differentiated out of existing PULTSs, for illustration, $PU\left(\hat{\pi }\right)=\left\{\left[v_{-1},v_1\right]\left(0.6\right),\left[v_0,v_2\right]\left(0.4\right)\right\}$ is turned into $PU\left(\hat{\pi }\right)=\left\{\left[v_{-1},v_0\right]\left(0.3\right),\left[v_0,v_1\right]\left(0.5\right),\left[v_1,v_2\right]\left(0.2\right)\right\}$.

Theorem 3:

[11]: According to the ${\overline{\pi }}^{\left(m\right)}={\hat{\pi }}^{\left(m\right)}/\sum _{m=1}^{\ast PU\left(\hat{\pi }\right)}{\hat{\pi }}^{\left(m\right)}$, the PULTS $PU\left(\hat{\pi }\right)=\left\{\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]\left({\hat{\pi }}^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}PU\left(\hat{\pi }\right)\right\}$ compatible with standardization easel $\stackrel{-}{P}\stackrel{-}{U}\left(\stackrel{-}{\pi }\right)=\left\{\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]\left({\stackrel{-}{\pi }}^{\left(m\right)}\right)\mid {\stackrel{-}{\pi }}^{\left(m\right)}\ge 0;m=1,2,\cdots ,\mathrm{\#}PU\left(\hat{\pi }\right);\sum _{m=1}^{\mathrm{\#}PU\left(\hat{\pi }\right)}{\stackrel{-}{\pi }}^{\left(m\right)}=1\right\}$.

Theorem 4:

[11]: In facilitating the computation of PULTS, we generally perform the following $\mathrm{\#}P{U}_1\left({\hat{\pi }}_1\right)-\mathrm{\#}P{U}_2\left({\hat{\pi }}_2\right)$ minimal ULTs with zero probability from PULTS $P{U}_2\left({\hat{\pi }}_2\right)$ is added in PULTS $P{U}_2\left({\hat{\pi }}_2\right)$.

Theorem 5:

[11]: The expectation value $EX\left(PU\left(\hat{\pi }\right)\right)$ and deviation value $DE\left(PU\left(\hat{\pi }\right)\right)$ of PULTS $PU\left(\hat{\pi }\right)=\left\{\left[{\hat{L}}^{\left(m\right)},{\hat{U}}^{\left(m\right)}\right]\left({\hat{\pi }}^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}PU\left(\hat{\pi }\right)\right\}$ are respectively defined by the following equations (4) and (5).

Moreover, there are following rules for any two PULTSs: $\begin{array}{rl}P{U}_1\left({\hat{\pi }}_1\right)& =\{\left[{\hat{L}}_1^{\left(m\right)},{\hat{U}}_1^{\left(m\right)}\right]\left({\hat{\pi }}_1^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}P{U}_1\left({\hat{\pi }}_1\right)\}\text{ and}\\ P{U}_2\left({\hat{\pi }}_2\right)& =\left\{\left[{\hat{L}}_2^{\left(m\right)},{\hat{U}}_2^{\left(m\right)}\right]\left({\hat{\pi }}_2^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}P{U}_2\left({\hat{\pi }}_2\right)\right\}\end{array}$

First, when $\text{EX}\left(P{U}_1\left({\hat{\pi }}_1\right)\right)>\text{EX}\left(P{U}_2\left({\hat{\pi }}_2\right)\right)$, we can directly acquire that $P{U}_1\left({\hat{\pi }}_1\right)>P{U}_2\left({\hat{\pi }}_2\right)$. Second, when $EX\left(P{U}_1\left({\hat{\pi }}_1\right)\right)=EX\left(P{U}_2\left({\hat{\pi }}_2\right)\right)$ and $DE\left(P{U}_1\left({\hat{\pi }}_1\right)\right)>DE(\phantom{{\hat{\pi }}_2}P{U}_2\left({\hat{\pi }}_2\right))$, we can also get the identical conclusion that $P{U}_1\left({\hat{\pi }}_1\right)>P{U}_2\left({\hat{\pi }}_2\right)$. Finally, if and only if $\text{EX}\left(P{U}_1\left({\hat{\pi }}_1\right)\right)=EX\left(P{U}_2\left({\hat{\pi }}_2\right)\right)$ and $DE\left(P{U}_1\left({\hat{\pi }}_1\right)\right)=DE\left(P{U}_2\left({\hat{\pi }}_2\right)\right),P{U}_1\left({\hat{\pi }}_1\right)=P{U}_2\left({\hat{\pi }}_2\right)$ appears. (4) $\begin{array}{rl}EX\left(PU\left(\hat{\pi }\right)\right)& =\frac{\sum _{m=1}^{\mathrm{\#}PU\left(\hat{\pi }\right)}\left(\frac{TF\left({\hat{L}}^{\left(m\right)}\right)\cdot {\hat{\pi }}^{\left(m\right)}+TF\left({\hat{U}}^{\left(m\right)}\right)\cdot {\hat{\pi }}^{\left(m\right)}}{2}\right)}{\sum _{m=1}^{\mathrm{\#}PU\left(\hat{\pi }\right)}{\hat{\pi }}^{\left(t\right)}}\end{array}$(4) (5) $\begin{array}{rl}DE\left(PU\left(\hat{\pi }\right)\right)& =\frac{\sqrt{\sum _{m=1}^{\mathrm{\#}PU\left(\hat{\pi }\right)}{\left(\frac{TF\left({\hat{L}}^{\left(m\right)}\right)\cdot {\hat{\pi }}^{\left(m\right)}+TF\left({\hat{U}}^{\left(m\right)}\right)\cdot {\hat{\pi }}^{\left(m\right)}}{2}-EX\left(PU\left(\hat{\pi }\right)\right)\right)}^2}}{\sum _{m=1}^{\mathrm{\#}PU\left(\hat{\pi }\right)}{\hat{\pi }}^{\left(t\right)}}\end{array}$(5)

Theorem 6:

[53]: If we assume that $P{U}_1\left({\hat{\pi }}_1\right)=\{\left[{\hat{L}}_1^{\left(m\right)},{\hat{U}}_1^{\left(m\right)}\right]\left({\hat{\pi }}_1^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}P{U}_1\left({\hat{\pi }}_1\right)\phantom{{\hat{U}}_1^{\left(m\right)}}\}$ and $P{U}_2\left({\hat{\pi }}_2\right)=\left\{\left[{\hat{L}}_2^{\left(m\right)},{\hat{U}}_2^{\left(m\right)}\right]\left({\hat{\pi }}_2^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}P{U}_2\left({\hat{\pi }}_2\right)\right\}$ are both PULTS, therefore the formula (6) is the Hamming distance expansion. (#PU $\left({\hat{\pi }}_1\right)=\mathrm{\#}P{U}_2\left({\hat{\pi }}_2\right)=\mathrm{\#}PU)$ (6) $d\left(P{U}_1\left({\hat{\pi }}_1\right),P{U}_2\left({\hat{\pi }}_2\right)\right)=\frac{1}{2\mathrm{\#}PU}\sum _{m=1}^{\mathrm{\#}PU}\left(\left|\begin{array}{c}\left|TF\left({\hat{L}}_1^{\left(m\right)}\right)\cdot {\hat{\pi }}_1^{\left(m\right)}-TF\left({\hat{L}}_2^{\left(m\right)}\right)\cdot {\hat{\pi }}_2^{\left(m\right)}\right|\\ \left|TF\left({\hat{U}}_1^{\left(m\right)}\right)\cdot {\hat{\pi }}_1^{\left(m\right)}-TF\left({\hat{U}}_2^{\left(m\right)}\right)\cdot {\hat{\pi }}_2^{\left(m\right)}\right|\end{array}\right|\right)$(6)

3. The PT-TODIM technique with PULTS for MAGDM

3.1 The theory of prospects

This concept also explains how individuals regularly adopt unreasonable or inconsistent decisions [54]. Even though prospect theory has been utilized to explain decision-making in economics, law, political science and medical fields, its applications to interpreting decision in the implementation of CDSP has not been explored [55]. The perspective that a decision maker’s choice can be influenced by two factors, advantages, but also loss, and decision weights, is demonstrated by prospect theory (PT), which illustrates this viewpoint as demonstration. This distinctive approach was defined by [50] through the utilization of the following equations (7)–(8), the prospects operate $\tilde{P}\left(w\right)$, the values function $\tilde{C}\left(w_s\right)$ and calculate the weights perform $\tilde{G}\left(h_s\right)$ as follows as Equation (9): (7) $\begin{array}{rl}\tilde{P}\left(w\right)& =\sum _{s=1}^f\tilde{C}\left(w_s\right)\cdot \tilde{G}\left(h_s\right)\end{array}$(7) (8) $\begin{array}{rl}\tilde{C}\left(w_s\right)& =\{\begin{array}{ll}{\left(w_s-w_0\right)}^{\gamma },& \text{ if }{w}_s\ge w_0\\ -\kappa {\left(w_0-w_s\right)}^{\xi },& \text{ if }{w}_s<w_0\end{array}\end{array}$(8) (9) $\begin{array}{rl}\tilde{G}\left(h_s\right)& =\{\begin{array}{l}{\displaystyle \frac{h_s^x}{{\left(h_s^x+{\left(1-h_s\right)}^x\right)}^{\frac{1}{x}}}},\text{ if }{w}_s\ge w_0\\ {\displaystyle \frac{h_u^z}{{\left(h_u^z+{\left(1-h_s\right)}^z\right)}^{\frac{1}{z}}}},\text{ if }{w}_s<w_0\end{array}\end{array}$(9)

For the value function $\tilde{C}\left(w_s\right)$, the value $w_s-w_0$ indicates gains for decision-makers if the real value $w_s$ is not smaller than all the chosen normal point $w_0$. If not, its value $w_0-w_s$ will result in losses for decision makers. During actual decision-making, decision-makers with distinct types of personalities have varying psychological evaluations of profit and losses. Therefore, the variables $k,j\text{and}s$ and in equation (8) represent the decision makers’ psychology. In summary, the bigger overall acceptable risk of the decision maker, the higher the value of $\text{A}$ relative to $\text{B}$ and $k<1$. Throughout most situations, the decision maker is risk adverse, which corresponds to $k>1$ including $k\le s$.

The adjustment of the probabilities that have been skewed as a result of the psychology of the decision makers is represented by the weight function $\tilde{G}\left(h_s\right)$. According to Kahneman and Tversky (1979) research, the psychology of decision makers can have an impact on how such individuals think about the objective probability that an event will take effect. As a result, to reach a decision that is more precise, it is essential to adjust the value of the subjective probability according to the weights function $\tilde{G}\left(h_s\right)$. The curve of the weights function $\tilde{G}\left(h_s\right)$ is represented by the values of $x\text{and}z$.

3.2 The integration of PT-PUL-TODIM technique

This research aims to develop an innovative PULTS of MAGDM concept based on the integration of the theory of prospects and TODIM methods. In this part, we will describe how this new model actually functions. There is pertinent background information in the following. $A=\left\{A_1,A_2,\cdots ,A_{\tau }\right\},C=\left\{C_1,C_2,\cdots ,C_{\theta }\right\}\text{and}\mathrm{\wp }=\left\{{\mathrm{\wp }}_1,{\mathrm{\wp }}_2,\cdots ,{\mathrm{\wp }}_{\delta }\right\}$ represent sets of options, attributes and decision-makers, in which the attribute weight values are fully unidentified. In addition, decision-makers use the uncertainty linguistic term sets (ULTS) to articulate their perspectives that are compiled in $\delta$ ULTS decision matrix: $\begin{array}{rl}{\hat{U}}^{\left(e\right)}& ={\left(\left[{\hat{L}}_{\text{ι}ϵ}^{\left(e\right)},{\hat{U}}_{\text{ι}ϵ}^{\left(e\right)}\right]\right)}_{\tau \times \theta }\\ & \times ({\hat{L}}_{\text{ι}ϵ}^{\left(e\right)},{\hat{U}}_{\text{ι}ϵ}^{\left(e\right)}\in V;{\hat{L}}_{\text{ι}ϵ}^{\left(e\right)}\le {\hat{U}}_{\text{ι}ϵ}^{\left(e\right)};\text{ι}=1,2,\cdots ,\tau ;ϵ=1,2,\cdots ,\theta .;e=1,2,\cdots ,\delta .)\end{array}$Consequently, the following is the detailed procedure:

Step 1: Changing the negative attributes must be transformed into a positive one for ensuring information accuracy. Specifically, if such a value of the negative attributes is $\left[v_a,v_b\right]$, then it should be transformed into $\left[v_{-b},v_{-a}\right]$.

Step 2: Using the persistent ULTS decision matrix and Theorem 3, the pretreatment and reorganized PULTS matrices may be obtained as $\tilde{K}=(P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right){)}_{\delta \tau }={\left(\{\left[{\hat{L}}_{\text{ι}ϵ}^{\left(e\right)},{\hat{U}}_{\text{ι}ϵ}^{\left(e\right)}\right]{\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}|m=1,2,\dots ,P{U}_{\text{ι}ϵ}(\hat{\pi }\left)\right\}\right)}_{\tau \times \theta }$. Significantly, ${\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}$ represents the potential of ULTS $\left[{\hat{L}}_{\text{ι}ϵ}^{\left(m\right)},{\hat{U}}_{\text{ι}ϵ}^{\left(m\right)}\right]$ showing in alternatives ${\text{A}}_{\iota }$ within attributes ${\text{C}}_{ϵ}$.

Step 3: Determine the weight value using the CRITIC method [56]. Criteria Importance Through Inter-Criteria Correlation (CRITIC), which was proposed by Diakoulaki, Mavrotas, and Papayannakis (1995), will be discussed in this part to determine the objective weights of attributes. The precise computation algorithms for this combined weight methodology are provided below.

1. Construct the probabilistic uncertain linguistic correlation coefficient matrix $PC{C}_{ϵ\alpha }$ by calculating the correlation coefficient between attributes using Equation (10)

   (10) $\begin{array}{rl}PC{C}_{ϵ\alpha }& =\frac{\left(\begin{array}{l}{\displaystyle \left(\sum _{m=1}^{\mathrm{\#}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)}\frac{\begin{array}{c}\left(TF\left({\hat{L}}_{\text{ι}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}-TF\left({\hat{L}}_{ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{ϵ}^{\left(m\right)}\right)\\ {\displaystyle +\left(TF\left({\hat{U}}_{\text{ι}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}-TF\left({\hat{U}}_{ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{ϵ}^{\left(m\right)}\right)}\end{array}}{2}\right)}\\ {\displaystyle \left(\sum _{m=1}^{\mathrm{\#}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)}\frac{\begin{array}{c}\left(TF\left({\hat{L}}_{\text{ι}\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}\alpha }^{\left(m\right)}-TF\left({\hat{L}}_{ϵ\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\alpha }^{\left(m\right)}\right)\\ {\displaystyle +\left(TF\left({\hat{U}}_{\text{ι}\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}\alpha }^{\left(m\right)}-TF\left({\hat{U}}_{\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\alpha }^{\left(m\right)}\right)}\end{array}}{2}\right)}\end{array}\right)}{\left(\begin{array}{l}{\displaystyle \sqrt{\sum _{t=1}^{\tau }\left(\sum _{m=1}^{\mathrm{\#}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)}\frac{\begin{array}{c}\left(TF\left({\hat{L}}_{\text{ι}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}-TF\left({\hat{L}}_{ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{ϵ}^{\left(m\right)}\right)\\ {\displaystyle +\left(TF\left({\hat{U}}_{\text{ι}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}-TF\left({\hat{U}}_{ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{ϵ}^{\left(m\right)}\right)}\end{array}}{2}\right)}}\\ {\displaystyle \sqrt{\sum _{t=1}^{\tau }\left(\sum _{m=1}^{\mathrm{\#}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)}\frac{\begin{array}{c}\left(TF\left({\hat{L}}_{\text{ι}\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}\alpha }^{\left(m\right)}-TF\left({\hat{L}}_{ϵ\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\alpha }^{\left(m\right)}\right)\\ {\displaystyle +\left(TF\left({\hat{U}}_{\text{ι}\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}\alpha }^{\left(m\right)}-TF\left({\hat{U}}_{\alpha }^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\alpha }^{\left(m\right)}\right)}\end{array}}{2}\frac{}{}\right)}}\end{array}\right)}\\ & {\displaystyle \times ϵ,\alpha =0,1,2,\dots ,\theta }\end{array}$(10) where $\begin{array}{rl}\hat{P}{\hat{U}}_{ϵ}\left(\hat{\pi }\right)& =\left\{\left[{\hat{L}}_{ϵ}^{\left(m\right)},{\hat{U}}_{ϵ}^{\left(m\right)}\right]\left({\hat{\pi }}_{ϵ}^{\left(m\right)}\right)=\left[\frac{1}{\tau }\sum _{\text{ι}=1}^{\tau }{\hat{L}}_{\text{ι}ϵ}^{\left(m\right)},\frac{1}{\tau }\sum _{\text{ι}=1}^{\tau }{\hat{U}}_{\text{ι}ϵ}^{\left(m\right)}\right]\left(\frac{1}{\tau }\sum _{i=1}^{\tau }{\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}\right)\right\}\text{and}\\ \hat{P}{\hat{U}}_{\alpha }\left(\hat{\pi }\right)& =\left\{\left[{\hat{L}}_{\alpha }^{\left(m\right)},{\hat{U}}_{\alpha }^{\left(m\right)}\right]\left({\hat{\pi }}_{\alpha }^{\left(m\right)}\right)=\left[\frac{1}{\tau }\sum _{l=1}^{\tau }{\hat{L}}_{\text{ι}\alpha }^{\left(m\right)},\frac{1}{\tau }\sum _{\text{ι}=1}^{\tau }{\hat{U}}_{\text{ι}\alpha }^{\left(m\right)}\right]\left(\frac{1}{\tau }\sum _{l=1}^{\tau }{\hat{\pi }}_{\text{ι}\alpha }^{\left(m\right)}\right)\right\}.\end{array}$

2. Determine the attribute's probabilistic uncertain linguistic standard deviation $PS{D}_{ϵ}$ using Equation (11): (11) $\begin{array}{rl}PS{D}_{ϵ}& {\displaystyle =\sqrt{\frac{\sum _{t=1}^{\tau }\left(\sum _{m=1}^{\mathrm{\#}P{U}_{\text{l}ϵ}\left(\hat{\pi }\right)}\frac{\begin{array}{c}\left(TF\left({\hat{L}}_{\text{l}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{l}ϵ}^{\left(m\right)}-TF\left({\hat{L}}_{ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{ϵ}^{\left(m\right)}\right)\\ +\left(TF\left({\hat{U}}_{\text{l}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{l}ϵ}^{\left(m\right)}-TF\left({\hat{U}}_{ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{ϵ}^{\left(m\right)}\right)\end{array}}{2}\right)}{\tau -1}}}\\ & \times ϵ=1,2,\dots ,\theta .\end{array}$(11)

3. As in Equations (12) and (13), compute the objective weight vector of attribute $y=(y_1,y_2,\cdots ,y_{\theta }{)}^T(y_{ϵ}\ge 0\text{and}\sum _{ϵ=1}^{\theta }{y}_{ϵ}=1)$. (12) $\begin{array}{rl}{I}_{ϵ}& =PS{D}_{ϵ}\cdot \sum _{\alpha =1}^{\theta }\left(1-PC{C}_{\alpha }\right),ϵ=1,2,\cdots ,\theta .\end{array}$(12) (13) $\begin{array}{rl}{y}_{ϵ}& =\frac{I_{ϵ}}{\sum _{ϵ=1}^{\theta }{I}_{ϵ}},ϵ=1,2,\cdots ,\theta \text{. }\end{array}$(13)

Step 4: According to the deviations of attribute probabilities in PT, Equations (14) and (15), which combine the actual weight with the other weight matrices to exactly the right attribute weight, and finally the rectified relative significance ${\ell }_{l\sigma ϵ}^{\ast }\left(y_{ϵ}\right)$ is calculated. (14) $\begin{array}{rl}{\ell }_{\text{ι}\sigma ϵ}\left(y_{ϵ}\right)& =\{\begin{array}{l}{\left(y_{ϵ}\right)}^x/{\left({\left(y_{ϵ}\right)}^x+{\left(1-y_{ϵ}\right)}^x\right)}^{\frac{1}{x}},P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)\ge P{U}_{\sigma ϵ}\left(\hat{\pi }\right)\\ {\left(y_{ϵ}\right)}^z/{\left({\left(y_{ϵ}\right)}^z+{\left(1-y_{ϵ}\right)}^z\right)}^{\frac{1}{z}},P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)<P{U}_{\sigma ϵ}\left(\hat{\pi }\right)\end{array}\end{array}$(14) (15) $\begin{array}{rl}{\ell }_{\text{ι}\sigma ϵ}\left(y_{ϵ}\right)& =\frac{{\ell }_{\text{ι}\sigma ϵ}\left(y_{ϵ}\right)}{\underset{ϵ}{\text{max}}\left\{{\ell }_{\text{ι}\sigma ϵ}\left(y_{ϵ}\right)\right\}}\text{ι},\sigma =1,2,\dots ,\tau ;ϵ=1,2,..,\theta .\end{array}$(15) Step 5: Determine the degrees of relative dominance denoted by the symbol ${\tilde{D}}_{ϵ}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)\left(\text{ι},\sigma =1,2,\cdots ,\tau ;ϵ=1,2,\cdots ,\theta \right)$ using Equation (17). Rectified relative significance ${\ell }_{\text{ι}\sigma ϵ}^{\ast }\left(y_{ϵ}\right)$ is one of the prerequisites for the relative dominance degree $d_{ϵ}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)$, which is acquired using Equation (16), for another prerequisite is the distance ${\tilde{D}}_{ϵ}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)$. However, the values of the relative domination degrees that correlate to the same attributes could be preserved within the same matrices (16) $\begin{array}{rl}& \begin{array}{l}{\tilde{D}}_{ϵ}\left(ϵ=1,2,\cdots ,\theta \right)\\ \begin{array}{c}{d}_{ϵ}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)={\displaystyle \frac{1}{2\cdot \mathrm{\#}PU}\sum _{m=1}^{\mathrm{\#}PU}\left(\begin{array}{c}\left|TF\left({\hat{L}}_{\text{ι}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}-TF\left({\hat{L}}_{\sigma ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\sigma ϵ}^{\left(m\right)}\right|\\ +\left|TF\left({\hat{U}}_{\text{ι}ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\text{ι}ϵ}^{\left(m\right)}-TF\left({\hat{U}}_{\sigma ϵ}^{\left(m\right)}\right)\cdot {\hat{\pi }}_{\sigma ϵ}^{\left(m\right)}\right|\\ \end{array}\right)}\\ \text{ι},\sigma =1,2,\cdots ,\tau ;ϵ=1,2,\cdots ,\theta \end{array}\end{array}\end{array}$(16) (17) $\begin{array}{rl}& {\tilde{D}}_{ϵ}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)=\{\begin{array}{ll}{\displaystyle \frac{{\ell }_{\text{ι}\sigma ϵ}^{\ast }\left(y_{ϵ}\right)\cdot {\left(d_{ϵ}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)\right)}^j}{\sum _{ϵ=1}^{\theta }{\ell }_{1\sigma ϵ}^{\ast }\left(y_{ϵ}\right)}}& ,\text{if}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)>P{U}_{\sigma ϵ}\left(\hat{\pi }\right)\\ 0& ,\text{if}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)=P{U}_{\sigma ϵ}\left(\hat{\pi }\right)\\ -k\cdot {\displaystyle \frac{\left(\sum _{ϵ=1}^{\theta }{\ell }_{1\sigma ϵ}^{\ast }\left(y_{ϵ}\right)\right)\cdot {\left(d_{ϵ}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)\right)}^s}{{\ell }_{\text{ι}\sigma ϵ}^{\ast }\left(y_{ϵ}\right)},}& \text{ if}P{U}_{\text{ι}ϵ}\left(\hat{\pi }\right)<P{U}_{\sigma ϵ}\left(\hat{\pi }\right)\end{array}\end{array}$(17) where all $k,j\text{and}s$ are parameters. (18) $\begin{array}{rl}& \begin{array}{cccc}{\text{A}}_1& {\text{A}}_2& \dots & {\text{A}}_{\tau }\end{array}\\ & {\tilde{D}}_{ϵ}=({\tilde{D}}_{ϵ}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right){)}_{\tau \times \tau }=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ ⋮\\ {\text{A}}_{\tau }\end{array}\left(\begin{array}{cccc}0& {\tilde{D}}_{ϵ}\left({\text{A}}_1,{\text{A}}_2\right)& \cdots & {\tilde{D}}_{ϵ}\left({\text{A}}_1,{\text{A}}_{\tau }\right)\\ {\tilde{D}}_{ϵ}\left({\text{A}}_2,{\text{A}}_1\right)& 0& \cdots & {\tilde{D}}_{ϵ}\left({\text{A}}_2,{\text{A}}_{\tau }\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{D}}_{ϵ}\left({\text{A}}_{\tau },{\text{A}}_1\right)& {\tilde{D}}_{ϵ}\left({\text{A}}_{\tau },{\text{A}}_2\right)& \cdots & 0\end{array}\right)\end{array}$(18) Step 6: Using Equation (19), sum the relative value of domination degrees ${\tilde{D}}_{ϵ}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)$ for various attributes to get the total dominating degrees $\tilde{D}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)\left(1,\sigma =1,2,\cdots ,\tau \right)$. Gather the results into matrices $\tilde{D}$ one at a time with Equation (20). (19) $\begin{array}{rl}\tilde{D}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)& =\sum _{ϵ=1_{ϵ}}^{\theta }\tilde{D}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)1,\sigma =1,2,\cdots ,\tau \end{array}$(19) (20) $\begin{array}{rl}& \begin{array}{cccc}{\text{A}}_1& {\text{A}}_2& \dots & {\text{A}}_{\tau }\end{array}\\ \tilde{D}& =(\tilde{D}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right){)}_{\tau \times \tau }=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ ⋮\\ {\text{A}}_{\tau }\end{array}\left(\begin{array}{cccc}0& \tilde{D}\left({\text{A}}_1,{\text{A}}_2\right)& \cdots & \tilde{D}\left({\text{A}}_1,{\text{A}}_{\tau }\right)\\ \tilde{D}\left({\text{A}}_2,{\text{A}}_1\right)& 0& \cdots & \tilde{D}\left({\text{A}}_2,{\text{A}}_{\tau }\right)\\ ⋮& ⋮& \ddots & ⋮\\ \tilde{D}\left({\text{A}}_{\tau },{\text{A}}_1\right)& \tilde{D}\left({\text{A}}_{\tau },{\text{A}}_2\right)& \cdots & 0\end{array}\right)\end{array}$(20) Step 7: Using Equation (21), we obtain the standardized general dominating degrees $\tilde{N}\left({\text{A}}_{\text{ι}}\right)\left(\text{ι}=1,2,\cdots ,\tau \right)$ that serve as the ultimate criterion for identifying the optimum path. In general, the higher value of $\tilde{N}\left({\text{A}}_{\text{ι}}\right)$, the better option. (21) $\begin{array}{ll}& \tilde{N}\left({\text{A}}_{\text{ι}}\right)={\displaystyle \frac{\sum _{\sigma =1}^{\tau }\tilde{D}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)-\underset{\text{ι}}{\text{min}}\left\{\sum _{\sigma =1}^{\tau }\tilde{D}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)\right\}}{\underset{\text{ι}}{\text{max}}\left\{\sum _{\sigma =1}^{\tau }\tilde{D}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)\right\}-\underset{\text{ι}}{\text{min}}\left\{\sum _{\sigma =1}^{\tau }\tilde{D}\left({\text{A}}_{\text{ι}},{\text{A}}_{\sigma }\right)\right\}}}\\ & \text{ι}=1,2,\cdots ,\tau \end{array}$(21)

4. Case study and sensitivity analysis

4.1. Case study of CDSPs selection in Vietnam

The enterprise aiming a complete digital transformation must plan for the use of a cloud database. The adoption of technology will provide them with greater flexibility and agility while also lowering risk and operational expenses. There is no generally applicable solution. Each enterprise must first set priorities. Only then can a cloud database service provider that matches its requirements be found. For each enterprise, it is critical to adopt cloud database services for data storage, and increasingly, analytics and computation. This implies that enterprises are essentially outsourcing a great deal of anxiety that comes with storing and maintaining large amounts of data. Space, power utilization, networking infrastructure, and security become problems for cloud database service providers, who are generally well-equipped to cope with them. Another significant advantage of employing cloud database solutions is that enterprises may be highly scalable. Most provide plans that allow them to start small and gradually increase the amount of storage capacity available as their demand develops.

Our investigation was started at the beginning of 2022; at this step of the suggested model, several round-table discussions with members of the board were conducted. The study's entry requirements and decision options were created with consultation from an expert panel. The panel members were chosen from a pool of talented capable professionals, with multiple criteria taken into consideration. Researchers discovered that expertise in the cloud database services industry was the most important attribute. The second condition was advanced-level professional experience (at least ten years as a senior executive or company owner in the field), and the third requirement was member of the executive of a corporation. If necessary, this is the first exhaustive and extensive evaluation of expert opinion over a lengthy period. All of these experts participated in a face-to-face discussion, where they were first asked open-ended questions and then tasked with compiling a list of the necessary criteria and alternatives for evaluating the impact variables associated with CDSP adoption. After collecting these lists, redundant selection criteria and alternatives were deleted, leaving the final selection criteria and alternatives. The membership of the board of experts is provided in Table 1.

Table 1. The board of experts and their detail

Expert $\left(e\right)$	Responsibility	Years of experience	Education level	Major	Association
1	Company Owner	13	Master	IT Management	Member of Board
2	Senior Executive	15	Master	IT Management and Business Administration	Vice President
3	Professional FinTech expert	16	Master	IT Management and Business Administration	Member of Board
4	Company Owner	18	Master	System Administration and Business Administration	Member of Board
5	Senior Executive	20	Master	Cybersecurity Management and Computer Networking	Vice President

Additionally, providers all provide additional services that can meet enterprises’ analytics and data visualization demands without their valuable data ever leaving the safety of the cloud. Selecting the best cloud database services provider (CDSP) from the market is a crucial enterprise management decision-making activity, in accordance with the benefit-and-return principle. The evaluation of CDSPs usually is based on the following four aspects: (1) ${\text{C}}_1$ is the improving agility and innovation; (2) ${\text{C}}_2$ is the lower cost; (3) ${\text{C}}_3$ is the faster time to market; (4) ${\text{C}}_4$ is the reducing risks. In addition, panel members ${\mathrm{\wp }}_e\left(e=1,2,3,4,5\right)$ provided relevant validation data to five CDSPs ${\text{A}}_{\text{ι}}\left(\text{ι}=1,2,3,4,5\right)$, as shown in Tables 2–6. $V=\left\{\begin{array}{rl}{v}_{-3}& =\text{extremely bad }\left(\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{B}}\right),v_{-2}=\text{very bad}\left(\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}}\right),v_{-1}=\text{bad }\left(\underset{\sim }{\mathrm{B}}\right),v_0\\ & =\text{medium}\left(\underset{\sim }{\mathrm{M}}\right),v_1=\text{great }\left(\text{G}\right),v_2=\text{very great}\left(\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right),v_3\\ & =\text{extremely great}\left(\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right)\end{array}\right\}$The example that follows illustrates how PT-PUL-TODIM can be used in CDSP selection.

Table 2. The ULT matrix ${\tilde{U}}^{\left(1\right)}$_.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$

Table 3. The ULT matrix ${\tilde{U}}^{\left(2\right)}$_.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$

Table 4. The ULT matrix ${\tilde{U}}^{\left(3\right)}$_.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$

Table 5. The ULT matrix ${\tilde{U}}^{\left(4\right)}$_.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$

Table 6. The ULT matrix ${\tilde{U}}^{\left(5\right)}$_.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$

Step 1: Tables 7–11 contain the outcomes of the transformation, which is to make the negative attribute into a positive one.

Table 7. The standard ULT matrices with ${\tilde{U}}^{\left(1\right)}$_.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$

Table 8. The standard ULT matrices with ${\tilde{U}}^{\left(2\right)}$.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$

Table 9. The standard ULT matrices with ${\tilde{U}}^{\left(3\right)}$.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$

Table 10. The standard ULT matrices with ${\tilde{U}}^{\left(4\right)}$

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$

Table 11. The standard ULT matrices with ${\tilde{U}}^{\left(5\right)}$.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{B}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_2$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_3$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{E}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$
${\text{A}}_4$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{B}},\underset{\sim }{\mathrm{M}}\right]$
${\text{A}}_5$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{M}},\underset{\sim }{\mathrm{G}}\right]$	$\left[\underset{\sim }{\mathrm{G}},\underset{\sim }{\mathrm{V}}\underset{\sim }{\mathrm{G}}\right]$

Step 2: Using the persistent ULTS decision matrix and Theorem 3, we obtain the pretreatment and reorganized PULTS matrices as

$\tilde{K}={\left(P{U}_{1ϵ}\left(\hat{\pi }\right)\right)}_{\delta \times \tau }={\left(\left\{\left[{\hat{L}}_{\iota ϵ}^{\left(m\right)},{\hat{U}}_{\text{ι}ϵ}^{\left(m\right)}\right]\left({\hat{\pi }}_{\iota ϵ}^{\left(m\right)}\right)\mid m=1,2,\cdots ,\mathrm{\#}P{U}_{\iota ϵ}\left(\hat{\pi }\right)\right\}\right)}_{\tau \times \theta }$, as shown in Table 12.

Table 12. PULTS matrices with $\tilde{K}$.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\text{A}}_1$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.6\right),\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.3\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.1\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.6\right),\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.3\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.1\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.6\right),\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.3\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.1\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.6\right),\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.3\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.1\right)\end{array}\right\}$
${\text{A}}_2$	$\left\{\begin{array}{c}\left[{\text{v}}_1,{\text{v}}_2\right]\left(0\right),\left[{\text{v}}_1,{\text{v}}_2\right]\left(0.9\right),\\ \left[{\text{v}}_2,{\text{v}}_3\right]\left(0.1\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.2\right),\left[{\text{v}}_1,{\text{v}}_2\right]\left(0.6\right),\\ \left[{\text{v}}_2,{\text{v}}_3\right]\left(0.2\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.1\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.3\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.6\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_1,{\text{v}}_2\right]\left(0\right),\left[{\text{v}}_1,{\text{v}}_2\right]\left(0.7\right),\\ \left[{\text{v}}_2,{\text{v}}_3\right]\left(0.3\right)\end{array}\right\}$
${\text{A}}_3$	$\left\{\begin{array}{c}\left[{\text{v}}_0,{\text{v}}_1\right]\left(0\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.6\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.2\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.6\right),\left[{\text{v}}_1,{\text{v}}_2\right]\left(0.3\right),\\ \left[{\text{v}}_2,{\text{v}}_3\right]\left(0.1\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.2\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.2\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.6\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.2\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.6\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.2\right)\end{array}\right\}$
${\text{A}}_4$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.2\right),\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.6\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.2\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.1\right),\left[{\text{v}}_0,{\text{v}}_3\right]\left(0.3\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.6\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.1\right),\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.5\right),\\ \left[{\text{v}}_0,{\text{v}}_1\right]\left(0.4\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-3},{\text{v}}_{-2}\right]\left(0.1\right),\left[{\text{v}}_{-2},{\text{v}}_{-1}\right]\left(0.5\right),\\ \left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.4\right)\end{array}\right\}$
${\text{A}}_5$	$\left\{\begin{array}{c}\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.1\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.7\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.2\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.4\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.4\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.2\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_{-1},{\text{v}}_0\right]\left(0.2\right),\left[{\text{v}}_0,{\text{v}}_1\right]\left(0.6\right),\\ \left[{\text{v}}_1,{\text{v}}_2\right]\left(0.2\right)\end{array}\right\}$	$\left\{\begin{array}{c}\left[{\text{v}}_1,{\text{v}}_2\right]\left(0\right),\left[{\text{v}}_1,{\text{v}}_2\right]\left(0.6\right),\\ \left[{\text{v}}_2,{\text{v}}_3\right]\left(0.4\right)\end{array}\right\}$

Step 3: As with Equations (10)–(13), CRITIC technique is adopted like a method for finding the objective weight vector of attribute $y=(y_1,y_2,y_3,y_4{)}^T=(0.1805,0.3064,0.2056,0.3076{)}^T.$Step 4: According to the deviations of attribute probabilities in PT, Equations (14) and (15) combine the actual weight with the other weight matrices to exactly the right attribute weight, and Table 13 presents finally the rectified relative significance ${\ell }_{l\sigma ϵ}^{\ast }\left(y_{ϵ}\right)$ is calculated (Note that the value of constants $x=0.61\text{and}z=0.69$ in Equation (14) were being taken from Kahneman's (1992) observational evidence).

Table 13. Rectified relative significance.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$		${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
${\ell }_{12}^{\ast }$	$0.7271$	$0.9976$	$0.7946$	1	${\ell }_{34}^{\ast }$	$0.7478$	1	$0.7965$	$0.9717$
${\ell }_{13}^{\ast }$	$0.7271$	$0.9976$	$0.7946$	1	${\ell }_{35}^{\ast }$	$0.7460$	$0.9675$	$0.7946$	1
${\ell }_{14}^{\ast }$	$0.7289$	1	$0.7965$	$0.9717$	${\ell }_{41}^{\ast }$	$0.7460$	$0.9675$	$0.7856$	1
${\ell }_{15}^{\ast }$	$0.7271$	$0.9976$	$0.7946$	1	${\ell }_{42}^{\ast }$	$0.7271$	$0.9976$	$0.7856$	1
${\ell }_{21}^{\ast }$	$0.7695$	$0.9980$	$0.8104$	1	${\ell }_{43}^{\ast }$	$0.7271$	$0.9675$	$0.7856$	1
${\ell }_{23}^{\ast }$	$0.7695$	$0.9980$	$0.8197$	1	${\ell }_{45}^{\ast }$	$0.7271$	$0.9675$	$0.7856$	1
${\ell }_{24}^{\ast }$	$0.7695$	$0.9980$	$0.8197$	1	${\ell }_{51}^{\ast }$	$0.7695$	$0.9980$	$0.8104$	1
${\ell }_{25}^{\ast }$	$0.7460$	$0.9675$	$0.7946$	1	${\ell }_{52}^{\ast }$	$0.7289$	1	$0.7875$	$0.9717$
${\ell }_{31}^{\ast }$	$0.7695$	$0.9980$	$0.8104$	1	${\ell }_{53}^{\ast }$	$0.7289$	1	$0.7875$	$0.9717$
${\ell }_{32}^{\ast }$	$0.7271$	$0.9976$	$0.7856$	1	${\ell }_{54}^{\ast }$	$0.7478$	1	$0.7965$	$0.9717$

Step 5: Compute the relative dominance degree ${\tilde{D}}_{ϵ}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)(1,\sigma =1,2,3,4,5;ϵ=1,2,3,4)$ based on Equation (17), the results can be found in Table 9. In fact, Table 14 shows the distance $d_{ϵ}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)$ calculated using Equation (16). By then, all values of the relative domination level that correlate to the same attributes can be recorded in the same matrices ${\tilde{D}}_{ϵ}\left(ϵ=1,2,3,4\right)$. (Note that parameter values estimated $j=0.88,s=0.88\text{and}k=2.25$ in Equation (17) were obtained by [57]). $\begin{array}{rl}& \begin{array}{ccccc}{\text{A}}_1& {\text{A}}_2& {\text{A}}_3& {\text{A}}_4& {\text{A}}_5\end{array}\\ {\tilde{D}}_1& =({\tilde{D}}_1\left({\text{A}}_{\iota },{\text{A}}_{\sigma }\right){)}_{5\times 5}=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ {\text{A}}_3\\ {\text{A}}_4\\ {\text{A}}_5\end{array}\left(\begin{array}{ccccc}0& -3.1523& -2.7065& -1.3533& -2.1843\\ 0.0623& 0& 0.0469& 0.0443& 0.0334\\ 0.0535& -2.3758& 0& 0.0308& 0.0239\\ 0.0267& -2.2446& -1.5575& 0& -0.9936\\ 0.0431& -1.6836& -1.2090& 0.0196& 0\end{array}\right)\\ & \begin{array}{ccccc}{\text{A}}_1& {\text{A}}_2& {\text{A}}_3& {\text{A}}_4& {\text{A}}_5\end{array}\\ {\tilde{D}}_2& =({\tilde{D}}_2\left({\text{A}}_{\iota },{\text{A}}_{\sigma }\right){)}_{5\times 5}=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ {\text{A}}_3\\ {\text{A}}_4\\ {\text{A}}_5\end{array}\left(\begin{array}{ccccc}0& -1.7361& -2.0196& -1.0878& -1.3478\\ 0.0610& 0& 0.0625& 0.0691& 0.0364\\ 0.0710& -1.7792& 0& -2.2442& 0.0310\\ 0.0382& -1.9677& 0.0789& 0& 0.0558\\ 0.0474& -1.0346& -0.8812& -1.5868& 0\end{array}\right)\\ & \begin{array}{ccccc}{\text{A}}_1& {\text{A}}_2& {\text{A}}_3& {\text{A}}_4& {\text{A}}_5\end{array}\\ {\tilde{D}}_3& =({\tilde{D}}_3\left({\text{A}}_{\iota },{\text{A}}_{\sigma }\right){)}_{5\times 5}=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ {\text{A}}_3\\ {\text{A}}_4\\ {\text{A}}_5\end{array}\left(\begin{array}{ccccc}0& 0.0548& 0.0521& 0.0415& 0.0467\\ -2.4101& 0& 0.0115& 0.0272& 0.0482\\ -2.2919& -0.5040& 0& 0.0356& 0.0495\\ -1.8259& -1.1949& -1.5678& 0& -1.2495\\ -2.0528& -2.1198& -2.1798& 0.0284& 0\end{array}\right)\\ & \begin{array}{ccccc}{\text{A}}_1& {\text{A}}_2& {\text{A}}_3& {\text{A}}_4& {\text{A}}_5\end{array}\\ {\tilde{D}}_4& =({\tilde{D}}_4\left({\text{A}}_{\iota },{\text{A}}_{\sigma }\right){)}_{5\times 5}=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ {\text{A}}_3\\ {\text{A}}_4\\ {\text{A}}_5\end{array}\left(\begin{array}{ccccc}0& -1.6363& -0.5672& 0.0384& -1.6843\\ 0.0578& 0& 0.0421& 0.0593& 0.6203\\ 0.0200& -1.1925& 0& 0.0346& -1.2415\\ -1.0884& -1.6799& -0.9816& 0& -1.7127\\ 0.0594& 0.0219& 0.0438& 0.0604& 0\end{array}\right)\end{array}$

Table 14. Distance between each two alternatives.

	${\text{C}}_1$	${\text{C}}_2$	${\text{C}}_3$	${\text{C}}_4$
$d\left({\text{A}}_1,{\text{A}}_2\right)$	0.2445	0.1778	0.2001	0.1666
$d\left({\text{A}}_1,{\text{A}}_3\right)$	0.2056	0.2111	0.1888	0.0500
$d\left({\text{A}}_1,{\text{A}}_4\right)$	0.0944	0.1056	0.1444	0.1056
$d\left({\text{A}}_1,{\text{A}}_5\right)$	0.1612	0.1334	0.1668	0.1723
$d\left({\text{A}}_2,{\text{A}}_3\right)$	0.1778	0.1833	0.0333	0.1167
$d\left({\text{A}}_2,{\text{A}}_4\right)$	0.1167	0.2056	0.0889	0.1722
$d\left({\text{A}}_2,{\text{A}}_5\right)$	0.1223	0.1001	0.1723	0.0556
$d\left({\text{A}}_3,{\text{A}}_4\right)$	0.1111	0.2389	0.1222	0.0944
$d\left({\text{A}}_3,{\text{A}}_5\right)$	0.0833	0.0833	0.1778	0.1222
$d\left({\text{A}}_4,{\text{A}}_5\right)$	0.0667	0.1611	0.0944	0.1778

Step 6: Using Equation (19), sum the value of the relative domination degrees ${\tilde{D}}_{ϵ}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)$ for various attributes to get the total dominating degrees $\tilde{D}\left({\text{A}}_1,{\text{A}}_{\sigma }\right)(1,\sigma =1,2,3,4,5)$. Gather the results into matrices $\tilde{D}$ one at a time. $\begin{array}{rl}& \begin{array}{ccccc}{\text{A}}_1& {\text{A}}_2& {\text{A}}_3& {\text{A}}_4& {\text{A}}_5\end{array}\\ \tilde{D}& =(\tilde{D}\left({\text{A}}_{\iota },{\text{A}}_{\sigma }\right){)}_{5\times 5}=\begin{array}{c}{A}_1\\ {\text{A}}_2\\ {\text{A}}_3\\ {\text{A}}_4\\ {\text{A}}_5\end{array}\left(\begin{array}{ccccc}0& -6.4700& -5.2412& -2.3612& -5.1697\\ -2.2291& 0& 0.1630& 0.1999& -0.5023\\ -2.1474& -5.8515& 0& -2.1431& -1.1371\\ -2.8494& -7.0871& -4.0281& 0& -3.9001\\ -1.9029& -4.8261& -4.2263& -1.4783& 0\end{array}\right)\end{array}$Step 7: Using Equation (21), obtain the standardized general dominating degrees $\tilde{N}\left({\text{A}}_{\text{ι}}\right)\left(\text{ι}=1,2,3,4,5\right)$ that serve as the ultimate criterion for identifying the optimum path. In general, the higher value of $\tilde{N}\left({\text{A}}_{\text{ι}}\right)$, the better option. $\begin{array}{ll}& \tilde{N}\left({\text{A}}_1\right)=0,\tilde{N}\left({\text{A}}_2\right)=1,\tilde{N}\left({\text{A}}_3\right)=0.4699,\tilde{N}\left({\text{A}}_4\right)=0.0809,\tilde{N}\left({\text{A}}_5\right)=0.4040,\\ & {\text{A}}_2>{\text{A}}_3>{\text{A}}_5>{\text{A}}_4>{\text{A}}_1\end{array}$

4.2. Sensitivity analysis

To demonstrate that the results obtained by this technique are stable and resistant to modification by the vector of weighting factors. Then, using the Sensitivity Analyzer App, we generated 100 sets of random numbers with 5 values and a sum of one as the values of the weights for the criteria. In addition, a single of the criteria carries the most weight in every set. As depicted in Figure 2, the proposed technique has been used again to compute the results, and the mean scores of 100 sets of weights have been obtained. The overall average is proportional to the alternative ordering, so the alternative ranking can be derived from Figure 2. Still, we can observe that alternative A2 has always been superior to all other alternatives and is compatible with the optimal solution we determined in Section 4.1. Accordingly, the method we developed has a high level of stability, and its findings are robust to changes in criterion values. In addition, it demonstrates further that outcomes of the TODIM approach in MAGDM are reliable and not susceptible to external interference; consequently, much more precise results can be obtained when compared to other ranking methods.

Figure 2. The score of alternatives in different criterion sets.

5. Solving the case with some existing techniques and comparative analysis

5.1. Solving the case with some existing techniques

5.1.1 Solving the case by the PUL-EDAS and PUL-GAR

In this part, PUL-EDAS [14] and PUL-GAR [19] have been chosen and calculated using the same starting data. Table 15 shows the results of the final calculation. The theory of prospects and PUL-TODIM technique and another two approaches all generate the same consistently acceptable outcome, demonstrating the predictability of the new PT-PUL-TODIM model's decision-making outcomes. The PT-PUL-TODIM model indicates the proposed variation is observed between alternative solutions. Furthermore, from the point of view of model ideology, our proposed technique completely accommodates the contribution of DM's psychology to decision results.

Table 15. The ranking result of based on PUL-EDAS and PUL-GAR techniques.

	PUL-EDAS	PUL-GAR
${\text{A}}_1$	$0.0478$	$0.4477$
${\text{A}}_2$	1	$0.6209$
${\text{A}}_3$	$0.5894$	$0.5410$
${\text{A}}_4$	$0.1476$	$0.3934$
${\text{A}}_5$	$0.6801$	$0.5979$
Ranking	${\text{A}}_2>{\text{A}}_5>{\text{A}}_3>{\text{A}}_4>{\text{A}}_1$	${\text{A}}_2>{\text{A}}_5>{\text{A}}_3>{\text{A}}_1>{\text{A}}_4$

5.1.2 Solving the case by the PDHL-VIKOR method

We try to solve the case by the PDHL-VIKOR method [42,58]. Since the following method may be affected by the values of the criteria weights, we use the maximum deviation method [59] to determine the weight information first.

We get the following optimization model to determine the weight vector of criteria: (22) $\{\begin{array}{rl}maxd\left(w\right)& =\sum _{j=1}^n{w}_j{d}_j=\frac{1}{m-1}\sum _{j=1}^n\sum _{i=1}^m\sum _{k=1,k\ne i}^m{w}_j\sqrt{{\left(S\left(I_{ij}\left(p\right)\right)-S\left(I_{kj}\left(p\right)\right)\right)}^2}\\ w_j& \ge 0,\sum _{j=1}^n{w}_j^2=1,j=1,2,\dots ,n\end{array}$(22) According to the Lagrange function, the formula above in Equation (22) can be solved and the weight of each criterion can be obtained by (23) $w_j=\frac{\sum _{j=1}^m\sum _{k=1,k+i}^m\sqrt{{\left(S\left(I_{ij}\left(p\right)\right)-S\left(I_{kj}\left(p\right)\right)\right)}^2}}{\sum _{j=1}^n\sum _{i=1}^m\sum _{i=1,k+i}^m\sqrt{{\left(S\left(I_{ij}\left(p\right)\right)-S\left(I_{kj}\left(p\right)\right)\right)}^2}},j=1,2,\dots ,n$(23) Thus, using Equation (23) we get the vector of the criteria weights. The three measures of the VIKOR method are calculated. Therefore, the ranking of alternatives based on three measures is shown in Table 16.

Table 16. The numerical results and rank derived by the PDHL-VIKOR.

	PDHLGU	rank	PDHLIK	rank	PDHLC	rank
${\text{A}}_1$	0.9227	5	0.3655	5	$1.0000$	5
${\text{A}}_2$	$-$1.033	1	$-$0.1681	1	$0.0000$	1
${\text{A}}_3$	$-$0.3113	3	0.0684	3	$0.4092$	3
${\text{A}}_4$	0.7220	4	0.2335	4	$0.8263$	4
${\text{A}}_5$	$-$0.6576	2	$-$0.0857	2	$0.1764$	2

From the result based on the PDHL-VIKOR method, the ranking of the five alternatives can be obtained as ${\text{A}}_2>{\text{A}}_5>{\text{A}}_3>{\text{A}}_4>{\text{A}}_1$ and the most best CDSP is ${\text{A}}_2$.

5.2 Comparative analysis

The PT-PUL-TODIM is developed given the following benefits: probabilistic uncertainty linguistic terms are close to managers’ and businesses’ decision-marking linguistic phrases. A new PULTS for the MAGDM approach focused on the integration of the theory of prospects and TODIM methods to examine the significance levels and probability distribution of linguistic terms. In other words, decision-makers often hesitate among many linguistic terms and have varying preferences. The primary and differentiating concept of the PT-PUL-TODIM technique is matched among multi-criteria methods with minimal implementation complexity and high predictive power, given that it is founded on a value function that adheres more closely to prospect theory.

Additionally, based on the PDHL-VIKOR method, we conclude that alternative A2 is the compromise solution, and we cannot obtain the compromise solution when we employ the traditional VIKOR method. By deleting the second hierarchy linguistic terms and the probability information, respectively, the optimal alternative is A2 based on the existing methods in probabilistic linguistic environment [35,39,60,61]. Although the optimal alternative of all methods discussed above is the same as A2, their final rankings are different as shown in Table 17.

Table 17. Ranking results based on different methods.

Methods	Results
The proposed method	${\text{A}}_2>{\text{A}}_3>{\text{A}}_5>{\text{A}}_4>{\text{A}}_1$
PUL-EDAS	${\text{A}}_2>{\text{A}}_5>{\text{A}}_3>{\text{A}}_4>{\text{A}}_1$
PUL-GAR	${\text{A}}_2>{\text{A}}_5>{\text{A}}_3>{\text{A}}_1>{\text{A}}_4$
PDHL-VIKOR	${\text{A}}_2>{\text{A}}_5>{\text{A}}_3>{\text{A}}_4>{\text{A}}_1$

Considering the lack of knowledge or the loss of assessment data, some experts do not provide any evaluation information for an attribute. In the framework of the language phrase, the decision-making challenge with missing evaluation values can be transformed into a probability problem. It can be used to explain unknown and uncertain values rather than replace them with null values, which is crucial for collaborative decision-making and large data decision making.

5.3. Discussion

From the preceding analysis, it is evident that the order determined by the PT-PUL-TODIM approach differs slightly from that determined by PUL-EDAS, PUL-GAR and PDHL-VIKOR. These three approaches provide the same optimal alternative, which is A2. This demonstrates that the strategy we suggested is acceptable and effective for this study. The cloud database service providers in Vietnam can be a solid ground for developing best practice guides regarding digital transformation and digital economics, which have an immediate need for a set of effective decision-making models based on selecting CDSP evaluation challenges. There is now the PT-PUL-TODEM to provide a better choice strategy for MAGDM scenarios, which is significant for clients and enterprises. Information from the surveys was conducted to determine and rank the criteria for success supported by the experts’ opinions and a review of the relevant literature. Five criteria from the available CDSP evaluation literature are employed in this regard. Five experts in the field of cloud database services information for evaluating sustainable CDSP. The analysis results found that improving agility and innovation, lower cost, faster time to market, and reducing risks which were impacted by selecting cloud database service provider followed, and the ranking of alternatives is followed by A2 > A3 > A5 > A4 > A1.

The core concept of each technique may result in fine distinctions between negative and positive characteristics, but the concentration is on MADM in the actual number. In addition, the initial weight vector of the attributes is regarded as the ultimate weight vector. Integration of the information entropy and EDAS method under PULTS provides an integrated model, in which information entropy is used for deriving the priority weights of each attribute and EDAS with PULTS is used to determine the final ranking of all alternatives. In addition, the PUL-GAR technique, which combines weight information, emphasizes only the degree of distance near the frontier approximate solution region for all attributes. The primary objective of PUL-GAR is to simplify the computation of PULTS through an effort to accomplish extra precise numerical results. PUL-GAR becomes a simple yet reliable mathematical approach that may be integrated with many other techniques; the prospective profits and losses are regarded to ensure that the overall ranking outcome may become detailed. VIKOR is a decision-making technique for selecting a compromise solution that is closest to the positive ideal solution and obtaining the compromise solution [7,28,62]. Gou et al. [29] introduced the theory of a probabilistic double hierarchy linguistic term set and then presented the PDHL–VIKOR approach, which was the expanded form of VIKOR in a PDHL environment. Particularly, the PDHL-VIKOR is more exhaustive since it considers the links between each alternative and both positive and negative ideal solutions. Our suggested PT-PUL-TODIM technique takes psychological aspects that influence the behaviour of decision markers into greater account. In addition, the comparison between the suggested technique and early standardization and distance measurement reveals disparities. In the meantime, we approach the model approach construction differently. The theory of He et al. [14] is limited in its scope of applicability and understanding of random dominance. Therefore, we utilized the cumulative prospect theory which is studied by Wei et al. [19] to overcome the above limitation.

6. Conclusion and future research directions

Our proposed technique in this study does have an important improvement over existing techniques regarding the general development of the concept logic structure. The merits of the PT-PUL-TODIM method established in this article include: (1) the proposed model is an expansion of the traditional TODIM approach; (2) the method improves the way MAGDM problems are solved; (3) the model generates an original PUL evaluation model; (4) the method is effectively implemented for the selection of cloud database service providers with novel data transformation, data imputation and decision making; (5) the sensitivity analysis of the significance of the criteria’s and reveals the robustness of the model. To ensure the application effect, the proposed technique was compared with the three present approaches. The method integrates the benefits of the PT as well as the TODIM technique, which ensures that the DM's psychology is considered throughout the entire operation for processing decision-related data. A case study on the CDSP selection problem was presented to demonstrate the effectiveness of the offered techniques. Although there are minor differences in the overall score of the options, these variations are likely attributable to the distinctive assessment criteria of the various methodologies.

Despite the fact that the study has positive aspects, several limitations highlight the potential for future research. In the near future, our team will continue to focus on MADM and MAGDM, where novel innovation and design will be explored. First, the developed algorithms and methodologies may apply to several additional advantageous MAGDM problems. Using the following concepts [63–65], the suggested methodologies could be expanded or included in many more fuzzy configurations to address group decision-making problems more effectively. Second, we will investigate further CDSP concerns, such as the inability to perform implementation system activities with the highest level of customer satisfaction [62,66–68].

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

Informed consent

This article does not contain any studies with human participants performed by any of the authors.

Acknowledgments

This work was supported by Posts and Telecommunications Institute of Technology, Vietnam (PTIT).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

This work was supported by Posts and Telecommunications Institute of Technology, Vietnam (PTIT).