Combinative-distance-based assessment approach for the evaluation of artificial intelligence cloud platforms using probabilistic linguistic hesitant fuzzy sets

Abstract

With the rapid expansion of artificial intelligence (AI) and machine learning, the evaluation of AI cloud platforms has become a critical research topic. Given the availability of many platforms, selecting the best AI cloud services that can satisfy the requirements and budget of an organization is crucial. Several solutions, each with its advantages and disadvantages, are available. In this study, a combinative-distance-based assessment approach was proposed in probabilistic linguistic hesitant fuzzy sets (PLHFSs) to accommodate the multiple characteristics of group decision-making. The original data were normalized using a standardized process that integrated numerous methodologies. Furthermore, under PLHFSs, the statistical variance approach was used to generate the weighted objective of the vector of assessment criteria. Finally, an AI cloud platform evaluation and comparison analysis case study was used to validate the feasibility of this method.

Keywords

Combinative-distance-based assessment (CODAS) method probabilistic linguistic hesitant fuzzy sets (PLHFSs)AI cloud platform evaluation multiple attribute decision-making (MADM)fuzzy environments

1 Introduction

Machine learning may be highly suitable for cloud computing applications. Cloud services enable the rapid provisioning, spin-down, shutdown, and activation of large-scale computing clusters with zero downtime, with data never leaving the servers or IT infrastructure of an organization. Cloud services can be scaled as companies expand. Regarding artificial intelligence (AI) initiatives, these are particularly advantageous. For instance, some projects may require training AI models on hundreds of terabytes of structured data, which may require weeks if these workloads are executed on premises rather than in a cloud environment with a large available hardware capacity. Cloud providers offer preset AI frameworks and toolkits; therefore, developers do not have to begin projects from scratch. These features ensure that AI development is efficient, accessible, and user friendly. Furthermore, tools and application programming interfaces are available in cloud services to assist managers with cost management, including monitoring service utilization across several departments, easily creating budgets for individual projects, and notifying developers of resource usage surges. Cloud AI platforms provide a frontier for businesses that want to explore AI but may not know where to begin [1]. With the increasing popularity of AI technologies, there is a growing need for evaluating cloud platforms that provide AI services. However, existing evaluation methods often cannot capture the uncertainty and hesitancy associated with decision-making in this context. Therefore, this study aims to bridge this research gap by proposing a combinative-distance-based assessment approach using probabilistic linguistic hesitant fuzzy sets. This innovative approach considers different evaluation criteria and incorporates uncertainties and hesitancies into decision-making processes, providing a more comprehensive and accurate assessment of AI cloud platforms. To achieve this, this study proposes a combinative-distance-based assessment approach, which utilizes probabilistic linguistic hesitant fuzzy sets (PLHFS) to model and evaluate the performance and quality of AI cloud platforms. PLHFS is a mathematical tool that can handle uncertainty and vagueness in subjective judgments. The motivation behind this approach is to provide a more comprehensive and accurate evaluation of AI cloud platforms by integrating multiple criteria and considering the uncertainty inherent in subjective judgments. Using PLHFS, the authors aim to capture the imprecision and hesitation involved in decision-making, which is particularly relevant in the evaluation of AI technologies where the outcomes may not be fully known or predictable. Overall, the motivation of this study is to contribute to the field of AI cloud platform evaluation by proposing a novel assessment approach that considers different criteria, uncertainty, and subjective judgments, ultimately providing better insights for decision-making and platform selection. The assessment of the critical functions of AI cloud platforms and AI cloud platform evaluation are critical research topics [2, 3]. The issues present in the evaluation of the AI cloud service platforms were previously studied using the multi-attribute decision-making (MADM) method [4, 5]. For instance, the TODIM method [6 –9], QUALIFLEX method [10, 11], MOORA method [12, 13], EDAS method [14 –17], MABAC method [18 –20], grey relational projection (GRP) method [21, 22], grey relational analysis (GRA) [23], TOPSIS method [24, 25], ELECTRE method [26 –28], combined compromise solution (CoCoSo) method [29, 30], simultaneous evaluation of criteria and alternatives (SECA) technique [28], and Copeland technique [29] are used in numerous situations for overcoming AI cloud platform (AICP) evaluation challenges. The combinative-distance-based assessment (CODAS) technique was established in [28]. In [30], the Euclidean distance was used as the standard measurement and the taxicab distance as the support measurement. In [31], the Archimedean Copula-based operations on q-rung probabilistic dual hesitant fuzzy (qRPDHF) components were proposed, and the important characteristics of these systems were investigated. New aggregation operators (AOs) may be created based on these operations, including the qRPDHF generalized Maclaurin symmetric mean (MSM), qRPDHF geometric generalized MSM, qRPDHF weighted generalized MSM, and the qRPDHF weighted generalized geometric generalized MSM operators. These aggregation operators are superior to the current operators on qRPDHF, and they are deployed in an empirical setting to select the best open-source LMS software to demonstrate the applicability of the suggested AOs. In this study, we introduce a novel assessment approach for evaluating AI cloud platforms. This approach combines the concepts of distance, combinative operators, and probabilistic linguistic hesitant fuzzy sets. By integrating these techniques, the proposed method provides a comprehensive evaluation framework for AI cloud platforms. This study introduces the concept of PLHFS to handle uncertainties and hesitations in decision-making. PLHFS extends fuzzy sets by incorporating linguistic terms and assigning probabilities to each linguistic term. This enables a more nuanced representation of uncertainty and hesitation in the evaluation process. The study employs combinative operators, such as the probabilistic linguistic hesitant fuzzy weighted average (PLHFWA) and probabilistic linguistic hesitant fuzzy weighted geometric (PLHWG) operators to aggregate the evaluation criteria. These operators consider both the importance weights and hesitation degrees associated with each criterion, providing a more accurate representation of decision-making. The proposed assessment approach uses a combinative-distance-based method to rank and select the best AI cloud platform. This method combines the PLHFWA and PLHWG operators with distance measures, enabling a more comprehensive evaluation that accounts for both the strengths and weaknesses of each platform. Table 1 presents a literature review of the practical implications of the CODAS approach.

Table 1
Literature relevant to the CODAS approach in fuzzy environments

The fuzzy environments Problems solving References

Fuzzy Sets (FSs) Evaluating the market segments [32]

Pythagorean FSs (PFSs) Information security risk analysis [33]

Interval-Valued Intuitionistic Fuzzy Sets Fintech success factors evaluation [29]

Interval-Valued Fuzzy Soft Sets Sustainable supplier selection [33]

Interval-Valued Pythagorean Fuzzy Sets Emergent decision [34]

Interval-Valued Hesitant Fuzzy Sets Low-speed wind farm projects [35]

Hesitant Fuzzy Linguistic Term Sets Adopting artificial intelligence technologies [3]

The Probabilistic double hierarchy linguistic sets Online food delivery platform evaluation [36]

The fuzzy environments	Problems solving	References
Fuzzy Sets (FSs)	Evaluating the market segments	[32]
Pythagorean FSs (PFSs)	Information security risk analysis	[33]
Interval-Valued Intuitionistic Fuzzy Sets	Fintech success factors evaluation	[29]
Interval-Valued Fuzzy Soft Sets	Sustainable supplier selection	[33]
Interval-Valued Pythagorean Fuzzy Sets	Emergent decision	[34]
Interval-Valued Hesitant Fuzzy Sets	Low-speed wind farm projects	[35]
Hesitant Fuzzy Linguistic Term Sets	Adopting artificial intelligence technologies	[3]
The Probabilistic double hierarchy linguistic sets	Online food delivery platform evaluation	[36]

As indicated by the previous discussion on the existing literature, an empirical study evaluating the AICP using the CODAS approach under PLHFSs is yet to be conducted. Therefore, the CODAS technique under PLHFSs is proposed to evaluate the AICP MADM. The main contributions of this study are as follows: The PLHF– CODAS approach was designed for MADM problems. A statistical variance technique based on PLHFSs was used to calculate the weight of each criterion. The Manhattan distance was established under PLHFSs.

The remainder of this paper is organized as follows: Section 2 presents the fundamental principles, and Section 3 presents the CODAS approach for MADM under PLHFSs. In the next section, the latest proposed approaches are applied to evaluate AI platform providers. A comparison of this method with other techniques is presented in Section 5. Finally, the limitations of this technique and anticipated outcomes of its implementation are discussed in Section 6. Figure 1 illustrates the proposed CODAS– PLHFS method.

Fig. 1

Proposed CODAS– PLHFS technique.

2 Preliminary knowledge

2.1 Probabilistic linguistic hesitant fuzzy sets

Theorem 1 Suppose that Y represents an unchanging set. The following equations identify the PLHFS M on Y: $M_{y} = {m_{y} (q_{y}^{n} ∣ l_{y}^{n}) ∣ q_{y}^{n}, l_{y}^{n}},$ (1) where the function $m_{y} (q_{y}^{n} ∣ l_{y}^{n}) = {q_{y}^{n} (l_{y}^{n})}$ refers to $q_{y}^{n} (l_{y}^{n})$ to denote the group of probabilistic hesitant fuzzy elements, $q_{y}^{n} \in R, 0 \leq q_{y}^{n} \leq 1, 0 \leq l_{y}^{n} \leq 1, n = 1, 2, \dots # m$ (where # m is a parameter value), $q_{y}^{n}$ represents the prospective membership degrees of the attributes, and $l_{y}^{n}$ is the amount of possible membership degrees, $\sum_{n = 1}^{# m} l_{y}^{n} = 1$ .

To achieve a comparable minimal or maximal degree of membership in qⁿ (lⁿ) , the studies cited in [38 –40] proposed the following novel normalization technique:

Theorem 2 [40, 41] Assume PHLFEs as $m_{y} (q_{y}^{n} ∣ l_{y}^{n}) = {q_{y}^{n} (l_{y}^{n})}$ , where $q_{y}^{n}$ refers to the n-th lowest component in $q_{y}^{n} (l_{y}^{n})$ . If d = max (# m_y), and d = # m_y, then the definitive type of standardisation is equivalent to the preparatory form. If d ≠ # m_y, then the standardisation procedure is as follows:

If DM involves risk-taking, then the following expression is obtained: $q_{y}^{n} (l_{y}^{n}) = {\begin{matrix} q_{y}^{1} (l_{y}^{1}), \dots q_{\underset{# m - 1}{y}}^{# m - 1} (l_{y}^{# m - 1}), q_{y}^{# m} (\frac{l_{y}^{# m}}{\underset{d - # m + 1}{d - # m + 1}}) \end{matrix}},$ (2)

If DM involves risk-averse, then the following expression is obtained: $q_{y}^{n} (l_{y}^{n}) = {q_{y}^{1} (\frac{l_{y}^{1}}{\underset{d - # m + 1}{d - # m + 1}}), q_{y}^{2} (l_{y}^{2}), \dots q_{y}^{# m} (l_{y}^{# m})},$ (3)

If DM involves risk-neutral, then the following expression is obtained: $\begin{matrix} q_{y}^{n} (l_{y}^{n}) \\ = {\begin{matrix} q_{y}^{1} (l_{y}^{1}), q_{\underset{\frac{# m}{2}}{y}}^{\frac{# m}{2}} (l_{y}^{\frac{# m}{2}}), q_{y}^{\frac{# m + 2}{2}} (\frac{l_{y}^{# m + 2 / 2}}{\underset{d - # m + 1}{d - # m + 1}}), \\ q_{y}^{2 d - # m + 4 / 2} (l_{\underset{\frac{# m - 2}{2}}{y}}^{2 d - # m + 4 / 2}), \dots q_{y}^{# m} (l_{y}^{# m}) \end{matrix}}, \end{matrix}$ (4) where # m denotes an even number and $\begin{matrix} q_{y}^{n} (l_{y}^{n}) \\ = {\begin{matrix} q_{y}^{1} (l_{y}^{1}), q_{\underset{\frac{# m - 1}{2}}{y}}^{\frac{# m - 1}{2}} (l_{y}^{\frac{# m - 1}{2}}), q_{y}^{\frac{# m + 1}{2}} (\frac{l_{y}^{# m + 1 / 2}}{\underset{d - # m + 1}{d - # m + 1}}), \\ q_{y}^{2 d - # m + 3 / 2} (l_{\underset{\frac{# m - 1}{2}}{y}}^{2 d - # m + 3 / 2}), \dots q_{y}^{# m} (l_{y}^{# m}) \end{matrix}}, \end{matrix}$ (5) where # m denotes an odd number.

Following the fundamental standardization stated previously, restrictions exist on the computation of probability when multiplying two components. Next, a normalization technique is used [42].

Theorem 3 [41, 43] Let $m_{y} (q_{y}^{n} ∣ l_{y}^{n}) = {q_{y}^{n} (l_{y}^{n})}, m_{1} (q_{1}^{a} ∣ l_{1}^{a}) = {q_{1}^{a} (l_{1}^{a})}$ and $f_{2} (q_{2}^{b} ∣ l_{2}^{b}) = {q_{2}^{b} (l_{2}^{b})}$ assuming several PHLFEs, n = 1, 2, ⋯ , # m, a = 1, 2, ⋯ , # m₁, b = 1, 2, ⋯ , # m₂, following that the ultimate normalisation procedure is described as follows:

Step 1 If $l_{1}^{1} < l_{2}^{1}$ , then $q_{1}^{1} (l_{1}^{1}) = q_{1}^{1} (l_{1}^{1})$ and $q_{2}^{1} (l_{2}^{1}) = q_{2}^{1} (l_{2}^{1})$ , otherwise, $q_{1}^{1} (l_{1}^{1}) = q_{1}^{1} (l_{2}^{1})$ and $q_{2}^{1} (l_{2}^{1}) = q_{2}^{1} (l_{2}^{1})$ .

Step 2 If $l_{1}^{1} < l_{2}^{1}$ and $l_{2}^{1} - l_{1}^{1} \leq l_{1}^{2}$ , then $q_{1}^{2} (l_{1}^{2}) = q_{1}^{2} (l_{2}^{1} - l_{1}^{1})$ and $q_{2}^{2} (l_{2}^{2}) = q_{2}^{1} (l_{2}^{1} - l_{1}^{1})$ . If $l_{1}^{1} < l_{2}^{1}$ and $l_{2}^{1} - l_{1}^{1} > l_{1}^{2}$ then $q_{1}^{2} (l_{1}^{2}) = q_{1}^{2} (l_{1}^{2})$ and $q_{2}^{2} (l_{2}^{2}) = q_{2}^{1} (l_{1}^{2})$ . If $l_{1}^{1} \geq l_{2}^{1}$ and $l_{1}^{1} - l_{2}^{1} \leq l_{2}^{2}$ , then $q_{1}^{2} (l_{1}^{2}) = q_{1}^{1} (l_{1}^{1} - l_{2}^{1})$ and $q_{2}^{2} (l_{2}^{2}) = q_{2}^{2} (l_{1}^{1} - l_{2}^{1})$ . If $l_{1}^{1} \geq l_{2}^{1}$ and $l_{1}^{1} - l_{2}^{1} > l_{2}^{2}$ , then $q_{1}^{2} (l_{1}^{2}) = q_{1}^{1} (l_{2}^{2})$ and $q_{2}^{2} (l_{2}^{2}) = q_{2}^{2} (l_{2}^{2})$ .

Step 3 If $l_{1}^{1} \geq l_{2}^{1}, l_{1}^{1} - l_{2}^{1} \leq l_{2}^{2}$ and $l_{2}^{1} \leq l_{2}^{2} - l_{1}^{1} + l_{2}^{1}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{2} (l_{1}^{2})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{2} (l_{1}^{2})$ . If $l_{1}^{1} \geq l_{2}^{1}, l_{1}^{1} - l_{2}^{1} \leq l_{2}^{2}$ and $l_{2}^{1} > l_{2}^{2} - l_{1}^{1} + l_{2}^{1}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{2} (l_{2}^{2} + l_{2}^{1} - l_{1}^{1})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{2} (l_{2}^{2} + l_{2}^{1} - l_{1}^{1})$ . If $l_{1}^{1} \geq l_{2}^{1}, l_{1}^{1} - l_{2}^{1} > l_{2}^{2}$ and $l_{1}^{2} \geq l_{2}^{2} + l_{2}^{3}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{2} (l_{2}^{3})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{3} (l_{2}^{3})$ . If $l_{1}^{1} \geq l_{2}^{1}, l_{1}^{1} - l_{2}^{1} > l_{2}^{2}$ and $l_{1}^{2} < l_{2}^{2} + l_{2}^{3}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{2} (l_{1}^{2} - l_{2}^{2})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{3} (l_{1}^{2} - l_{2}^{2})$ . If $l_{1}^{1} < l_{2}^{1}, l_{2}^{1} - l_{1}^{1} \leq l_{1}^{2}$ and $l_{1}^{2} + l_{1}^{1} \leq l_{2}^{2} + l_{2}^{1}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{2} (l_{1}^{2} - l_{2}^{1} + l_{1}^{1})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{2} (l_{1}^{2} - l_{2}^{1} + l_{1}^{1})$ .. If and $l_{2}^{1} + l_{1}^{2} \leq l_{1}^{3} + l_{1}^{1}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{3} (l_{2}^{1} - l_{1}^{1} - l_{1}^{2})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{1} (l_{2}^{1} - l_{1}^{1} - l_{1}^{2})$ . If and $l_{2}^{1} + l_{1}^{2} > l_{1}^{3} + l_{1}^{1}$ , then $q_{1}^{3} (l_{1}^{3}) = q_{1}^{3} (l_{1}^{3})$ and $q_{2}^{3} (l_{2}^{3}) = q_{2}^{1} (l_{1}^{3})$ ,

where $l_{1}^{1} + l_{1}^{2} + \dots l_{1}^{# a} = 1$ and $l_{2}^{1} + l_{2}^{2} + \dots l_{2}^{# b} =$ 1, # a = # b.

Theorem 4 [44] The score function of normalized $m_{y} (l_{y}^{n}) = {q_{y}^{n} (l_{y}^{n})}$ can be obtained as follows: $s (m_{y} (l_{y}^{n})) = \sum_{n = 1}^{# m} q_{y}^{n} l_{y}^{n},$ (6)

where # m represents the number of distinct degrees of membership, $q_{y}^{n}$ represents the n-th highest element, $l_{y}^{n}$ represents the possibility associated with the n-th highest membership degree.

Theorem 5 [44] Given a standardized PHLFE, $m_{y} (l_{y}^{n}) = {q_{y}^{n} (l_{y}^{n})}$ , Equation (7) provides the degree of deviation. $(q_{y}^{n} (l_{y}^{n})) = \sum_{n = 1}^{# m} l_{y}^{n} {[q_{y}^{n} - s (q_{y}^{n} (l_{y}^{n}))]}^{2} .$ (7)

Theorem 6 [45] Compare two components $m_{1} (ζ_{1}^{a}) = {q_{1}^{a} (ζ_{1}^{a})}$ and $m_{2} (b^{b}) = {q_{2}^{b} (h_{2}^{b})}$

$q_{1}^{a} (g_{1}^{a}) > q_{2}^{b} (b^{b})$ , if $s (q_{1}^{a} (h^{a})) > s (q_{2}^{b} (b^{b}))$ ;

$q_{1}^{a} (ζ_{1}^{a}) > q_{2}^{b} (ζ_{2}^{b})$ , if $s (q_{1}^{a} (ζ^{a})) > s (q_{2}^{b} (ζ_{2}^{b}))$ and $d (q_{1}^{a} (h_{1}^{a})) > d (q_{2}^{b} (ζ_{2}^{b}))$ ;

$q_{1}^{a} (b^{a}) = q_{2}^{b} (b^{b})$ , if $s (q_{1}^{a} (ζ^{a})) = s (q_{2}^{b} (ζ_{2}^{b}))$ and $d (q_{1}^{a} (h_{1}^{a})) = d (q_{2}^{b} (ζ_{2}^{b}))$ ;

$q_{1}^{a} (b_{1}^{a}) < q_{2}^{b} (b_{2}^{b})$ , if $s (q_{1}^{a} (b^{a})) = s (q_{2}^{b} (b_{2}^{b}))$ and $d (q_{1}^{a} (ζ_{1}^{a})) > d (q_{2}^{b} (b_{2}^{b}))$ .

Theorem 7 [40, 41] Given $m (l^{n}) = {q (l^{n})}, m_{1} (b^{a}) = {q_{1}^{a} (l_{1}^{a})}$ and $m_{2} (l_{2}^{b}) = {q_{2}^{b} (l^{b})}$ are two normalised PHLFEs, # m₁ = # m₂ = # m and h^a = h^b = lⁿ. $\begin{matrix} (1) m_{1}^{a} (l_{1}^{a}) \oplus m_{2}^{b} (g_{2}^{b}) \\ = \cup_{a = 1, \dots, * m_{1}, b = 1, \dots,; m_{2}} {(q_{1}^{a} + q_{2}^{b} - q_{1}^{a} q_{2}^{b}) ∣ l^{n}}; \end{matrix}$ (8)

$\begin{matrix} (2) m_{1}^{a} (l_{1}^{a}) \otimes m_{2}^{b} (l_{2}^{b}) \\ = \cup_{a = 1, \dots, # m_{1}, b = 1, \dots, # m_{2}} {q_{1}^{a} q_{2}^{b} ∣ l^{n}}; \end{matrix}$ (9)

$\begin{matrix} (3) (m)^{λ} \\ = \cup_{n = 1, 2, \dots, # m} {(q)^{λ} ∣ l^{n}}; \end{matrix}$ (10)

$\begin{matrix} (4) λ m \\ = \cup_{n = 1, 2, \dots, # m} {1 - (1 - q)^{λ} ∣ l^{n}} . \end{matrix}$ (11)

Theorem 8 [37, 46] Identified is the probabilistic hesitant fuzzy weighted averaging (PLHFWA) function: $\begin{matrix} PLHFWA (m_{1} (l_{1}^{a}), m_{2} (l_{2}^{b}), \dots, m_{f} (l_{f}^{n}) \otimes_{e = 1}^{f} (\leftarrow m_{e} t_{e}) \\ = \underset{{\bar{q}}_{1} \in {\bar{m}}_{1}, {\bar{q}}_{2} \in {\bar{m}}_{2}, \dots, {\bar{q}}_{f} \in {\bar{m}}_{f}}{\cup} {1 - \prod_{e = 1}^{f} (1 - q)^{t_{e}} l^{n}}, \end{matrix}$ (12) where t_e = (t₁, t₂, ⋯ , t_f) refers to the m_e weighted vectors and $\sum_{e = 1}^{f} t_{e} = 1, t_{e} \in [0, 1]$ .

Theorem 9 [37, 46] Definition of the probabilistic hesitant fuzzy weighted geometric (PLHFWG) function: $\begin{matrix} PHFWG (m_{1} ({l^{a}}_{1}), m_{2} ({l^{b}}_{2}), \dots, m_{f} ({l^{n}}_{f}) \otimes_{e = 1}^{f} (m_{e})^{t_{e}} \\ = \underset{{\bar{q}}_{1} \in {\bar{m}}_{1}, {\bar{q}}_{2} \in {\bar{m}}_{2}, \dots, {\bar{q}}_{f} \in {\bar{m}}_{f}}{\cup} {\prod_{e = 1}^{f} (q)^{t_{e}} (l^{n})}, \end{matrix}$ (13)

2.2 Determine the relative weights of the characteristics

In this section, the statistical variance technique using PLHFSs [47] was used to determine the goal weights. Based on [48], the PLHF statistical variance approach consists of the following steps:

The statistical variance technique was used to compute the variation in each value of the data relative to the mean: $R_{t} = \frac{1}{y} \sum_{r = 1}^{x} {(s (m_{rt}) - s (m_{rt}))}^{2}, t = 1, 2, \dots, y,$ (14)

where m_rt represents the mean PHEF value. The equation for computing the weight is as follows: $e_{t} = \frac{R_{t}}{\sum_{t = 1}^{y} R_{t}} .$ (15)

2.3 Euclidean and Taxicab distances for standard PHLFEs

The distance that was measured along the Euclidean axes [40, 41] was expressed by the following equation: $\begin{matrix} D (m_{a} (l_{a}), m_{b} (l_{b})) = \frac{1}{2} \\ (\sqrt{\frac{1}{# m} \sum_{n = 1}^{# m} {(q_{a}^{n} l_{a}^{n} - q_{b}^{n} l_{b}^{n})}^{2}} + \sqrt{\frac{1}{# m} \sum_{n = 1}^{# m} {(q_{a}^{n} - q_{b}^{n})}^{2} l_{a}^{n} l_{b}^{n}}) . \end{matrix}$ (16)

For standardized PHLFEs with an identical probability $l_{a}^{n} = l_{b}^{n}$ [38 –40], the distance between two points measured using Euclidean can be rewritten as follows: $D (m_{a} (l_{a}), m_{b} (l_{b})) = \sqrt{\frac{1}{# m} \sum_{n = 1}^{# m} {(q_{a}^{n} - q_{b}^{n})}^{2} {\overset{`}{l}}^{n}} {\overset{`}{l}}_{a}^{n} = {\overset{`}{l}}_{a}^{n} = {\overset{`}{l}}^{n} .$ (17)

Hermann Minkowski proposed the term taxicab distance between two vectors, also known as the Manhattan distance. This term is used in geometric distances to denote the true circumferential value of two points in standardized coordinates. This phenomenon was suggested by an extension of the Manhattan distance in a fuzzy environment [49]. This study established the Manhattan distance in conformity with the PLHFSs as follows: $d (m_{a} (l_{a}), m_{b} (l_{b})) = \sum_{n = 1}^{# m} | q_{a}^{n} l_{a}^{n} - q_{b}^{n} l_{b}^{n} | .$ (18)

The form of standardization is as stated below: $\begin{matrix} d (m_{a} (l_{a}), m_{b} (l_{b})) \\ = \sum_{n = 1}^{# m} ∣ {\overset{`}{q}}_{a}^{n} - {\overset{`}{q}}_{b}^{n} {\overset{`}{l}}^{n}, {\overset{`}{l}}_{a}^{n} = {\overset{`}{l}}_{a}^{n} = {\overset{`}{l}}^{n} . \end{matrix}$ (19)

3 DAS under probabilistic linguistic hesitant fuzzy for MADM

Setting the MADM decision matrix to $M^{k} = {[m_{rt}^{k} (l_{rt})]}_{x \times y}$ , where AICP_r ={ AICP₁, AICP₂, ⋯ AICP_x } denotes options, C_t ={ C₁, C₂, ⋯ C_y }, and k ={ k₁, k₂, ⋯ k_c } denotes the set of all DM. The approximate weights of the DM are $h_{k}, \sum_{k = 1}^{c} h_{k} = 1, (k = 1, 2, \dots c)$ . Furthermore, the weights for e_t ={ e₁, e₂, ⋯ e_y } are undefined. The specific phases of MADM in PLHSs using the expanded CODAS approach are presented below.

Step 1 Sndardize the decision matrix

Equation (20) transforms the negative criterion into a positive criterion as follows: ${\begin{matrix} m_{rt} (l_{rt}) = {q_{rt} (l_{rt})} \\ if the value of the element is positive \\ m_{rt} (l_{rt}) = {(1 - q_{rt}) (l_{rt})} \\ if if the value of the element is negative \end{matrix}$ (20)

Assuming that all experts are risk-takers, the decision matrix is standardized according to Theorems 2 and 3:

Step 2 Compute the combination of the decision matrix

The decision matrix of several DMs is consolidated into a single matrix ${\overset{`}{m}}_{rt} ({\overset{`}{l}}_{rt}) = {\overset{`}{q}}_{rt} ({\overset{`}{l}}_{rt})]_{x \times y}$ using the PLHFWA in Equation (21). $\begin{matrix} PHFWA ({\bar{M}}_{rt}^{1}, {\bar{M}}_{rt}^{2}, \dots, {\bar{M}}_{rt}^{c} = \otimes_{e = 1}^{f} ({\bar{M}}_{rt}^{k} h_{k}) = \\ \cup_{{\bar{q}}_{rt}^{1} \in {\bar{m}}_{rt}^{1}, {\bar{q}}_{rt}^{2} \in {\bar{m}}_{rt}^{2}, \dots, {\bar{q}}_{rt}^{c} \in {\bar{m}}_{rt}^{c}} {1 - \prod_{k = 1}^{c} (1 - {\bar{q}}_{rt}^{k})^{h_{k}} (l)} . \end{matrix} .$ (21)

Step 3 Equations (14) and (15) were used to compute the weight of each criterion.

Step 4 Establish weighted standardized decision matrices using Equation (22). ${\overset{`}{m}}_{rt} ({\overset{`}{l}}_{rt}) = {{\overset{`}{e}}_{t} {\overset{`}{q}}_{rt} ∣ {\overset{`}{l}}_{rt}} .$ (22)

Step 5 Calculate the scoring matrix. $s ({\overset{`}{m}}_{rt} ({\overset{`}{l}}_{rt})) = \sum_{n = 1}^{# m} {\overset{`}{q}}_{rt}^{n} {\overset{`}{l}}_{rt}^{n} .$ (23)

Step 6 Examine the negative ideal point (NIP) of options using Equation (24). ${\overset{`}{m}}_{t}^{-} ({\overset{`}{l}}_{t}) = \min_{r} s ({\overset{`}{m}}_{rt} ({\overset{`}{l}}_{rt})) .$ (24)

Step 7 Determine the Euclidean distance [43] and taxicab distance of options from the NIP using Equations (25) and (26). $D_{r} = \sqrt{\sum_{t = 1}^{y} \frac{1}{# m} \sum_{n = 1}^{# m} {({\overset{`}{q}}_{rt}^{n} - {\overset{`}{q}}_{t}^{n -})}^{2} {\overset{`}{l}}^{n}}, r = 1, 2, \dots, x;$ (25) $d_{r} = \sum_{t = 1}^{y} \sum_{n = 1}^{# m} ∣ {\overset{`}{q}}_{rt}^{n} - {\overset{`}{q}}_{t}^{n -} {\overset{`}{l}}^{n}, r = 1, 2, \dots, x .$ (26)

Step 8 Construct the relative’s assessment matrix using Equation (27) as follows: ${\overset{`}{m}}_{rk} = (D_{r} - D_{k}) + (φ (D_{r} - D_{k}) \times (d_{r} - d_{k})), r, k = 1, 2, \dots, x .,$ (27) where k∈ { 1, 2, ⋯ x } and φ (θ) is the exact value of the threshold function that is defined as follows: $φ (θ) = {\begin{matrix} 1 & if | θ | \geq τ \\ 0 & if | θ | < τ \end{matrix},$ (28)

Here, τ is an exact threshold value in the interval 0.01 --0.05, which implies that when the difference between the two options is less than τ (where τ = 0.03), this parameter contrasts the Manhattan distance.

Step 9 Compute the assessment value of each option using the following equation: $V_{r} = \sum_{k = 1}^{x} {\overset{`}{m}}_{rk}, r = 1, 2, \dots, x$ (29)

Step 10 Ordering the options available based on the overall score.

4 Case study

This study began at the end of 2022 when multiple round-table meetings with board executives were performed. An expert panel assisted in developing the eligibility criteria and potential decision options. Panel members were selected from a pool of talented and capable individuals based on various criteria. Researchers revealed that the most critical criterion was competence in the AI-cloud service market. The second most important criterion was advanced-level managerial expertise (at least ten years as a senior executive or company owner in the industry), and the third was membership in a corporation’s executive committee. This study is the first thorough and extended review of expert opinions over a long period of time. These experts participated in a face-to-face discussion in which open-ended questions we asked before producing a list of the criteria required and choices for evaluating impact factors connected with the AICP ranking. Following the compilation of the aforementioned lists, identical selection criteria and alternatives were removed, leaving only the final decision criteria and alternatives. A list of the members of the panel of professionals is presented in Table 2.

Table 2
Panel of DMs and specific expertise

DM(s) (e) Responsibilities Major Association Years of experience Level of education

1 Company Owner IT Management and Software Engineer Member of Board 19 PhD

2 AI Engineer IT Management and Big Data Analyst Member of Board 14 PhD

3 Professional Big Data Engineer/Architect IT Management and Data Scientist Member of Board 15 PhD

DM(s) (e)	Responsibilities	Major	Association	Years of experience	Level of education
1	Company Owner	IT Management and Software Engineer	Member of Board	19	PhD
2	AI Engineer	IT Management and Big Data Analyst	Member of Board	14	PhD
3	Professional Big Data Engineer/Architect	IT Management and Data Scientist	Member of Board	15	PhD

Table 3

First expert’s decision matrix K₁

Options	C ₁	C ₂	C ₃
AICP ₁	{0.3 (0.1) , 0.5 (0.2) , 0.6 (0.7)}	{0.5 (0.5) , 0.6 (0.5)}	{0.5 (0.3) , 0.2 (0.1) , 0.4 (0.6)}
AICP ₂	{0.4 (0.5) , 0.5 (0.5)}	{0.2 (0.1) , 0.4 (0.5) , 0.3 (0.4)}	{0.3 (0.6) , 0.4 (0.4)}
AICP ₃	{0.2 (0.3) , 0.4 (0.6) , 0.5 (0.1)}	{0.1 (0.7) , 0.4 (0.3)}	{0.2 (0.4) , 0.3 (0.6)}
AICP ₄	{0.4 (0.4) , 0.2 (0.6)}	{0.6 (0.1) , 0.3 (0.7) , 0.2 (0.2)}	{0.2 (0.2) , 0.1 (0.3) , 0.5 (0.5)}
AICP ₅	{0.3 (1)}	{0.3 (0.2) , 0.2 (0.8)}	{0.1 (0.4) , 0.4 (0.5) , 0.5 (0.1)}
Options	C ₄	C ₅	C ₆
AICP ₁	{0.5 (0.4) , 0.2 (0.2) , 0.6 (0.4)}	{0.5 (0.6) , 0.4 (0.4)}	{0.5 (0.6) , 0.6 (0.4)}
AICP ₂	{0.5 (0.3) , 0.3 (0.5) , 0.4 (0.2)}	{0.3 (0.4) , 0.4 (0.4) , 0.5 (0.2)}	{0.3 (0.2) , 0.5 (0.8)}
AICP ₃	{0.3 (0.2) , 0.2 (0.8)}	{0.2 (0.3) , 0.3 (0.4) , 0.4 (0.3)}	{0.1 (0.4) , 0.5 (0.2) , 0.3 (0.4)}
AICP ₄	{0.2 (0.4) , 0.3 (0.3) , 0.4 (0.3)}	{0.3 (0.4) , 0.4 (0.2) , 0.5 (0.4)}	{0.4 (0.5) , 0.3 (0.5)}
AICP ₅	{0.2 (0.2) , 0.3 (0.3) , 0.5 (0.5)}	{0.3 (0.3) , 0.4 (0.4) , 0.1 (0.3)}	{0.2 (0.4) , 0.4 (0.5) , 0.3 (0.1)}

Table 4

Second expert’s decision matrix K₂

Options	C ₁	C ₂	C ₃
AICP ₁	{0.6 (0.2) , 0.4 (0.7) , 0.5 (0.1)}	{0.3 (0.5) , 0.6 (0.5)}	{0.3 (0.5) , 0.8 (0.2) , 0.4 (0.3)}
AICP ₂	{0.4 (0.5) , 0.3 (0.5)}	{0.1 (0.6) , 0.5 (0.4)}	{0.2 (0.8) , 0.8 (0.2)}
AICP ₃	{0.3 (0.2) , 0.4 (0.6) , 0.1 (0.2)}	{0.7 (0.2) , 0.3 (0.8)}	{0.4 (0.4) , 0.2 (0.3) , 0.5 (0.3)}
AICP ₄	{0.4 (0.4) , 0.1 (0.4) , 0.3 (0.2)}	{0.4 (1)}	{0.3 (0.3) , 0.6 (0.2) , 0.2 (0.5)}
AICP ₅	{0.1 (0.2) , 0.3 (0.4) , 0.5 (0.4)}	{0.2 (0.3) , 0.3 (0.6) , 0.4 (0.1)}	{0.1 (0.5) , 0.4 (0.5)}
Options	C ₄	C ₅	C ₆
AICP ₁	{0.3 (0.1) , 0.5 (0.5) , 0.4 (0.4)}	{0.4 (0.5) , 0.5 (0.5)}	{0.5 (1)}
AICP ₂	{0.2 (0.5) , 0.4 (0.5)}	{0.3 (0.5) , 0.4 (0.3) , 0.2 (0.2)}	{0.3 (0.5) , 0.4 (0.2) , 0.5 (0.3)}
AICP ₃	{0.3 (0.4) , 0.7 (0.1) , 0.1 (0.5)}	{0.5 (0.4) , 0.3 (0.3) , 0.4 (0.3)}	{0.4 (0.6) , 0.2 (0.4)}
AICP ₄	{0.5 (0.2) , 0.4 (0.1) , 0.3 (0.7)}	{0.3 (0.6) , 0.4 (0.4)}	{0.6 (0.3) , 0.3 (0.7)}
AICP ₅	{0.2 (1)}	{0.1 (0.4) , 0.6 (0.6)}	{0.2 (0.1) , 0.3 (0.4) , 0.4 (0.5)}

The evaluation of essential AICP features not only benefits major organizations but also provides substantial client experiences and economic benefits. Almost all AI cloud providers offer complex deep data science, machine learning platforms, and API services. Hence, an AI cloud can save time and resources of enterprises. However, AI cloud evolution is incomplete until it transforms into highly innovative enterprises or, more precisely, into the digital businesses of the next generation. To establish such an atmosphere of limitless invention, the transition of AICPs should prioritize six critical functions. In addition to fueling innovation, the AI cloud may assist businesses in overcoming the most pressing difficulties of the upcoming year, such as rising energy costs and chip scarcity, which affect every aspect, including hardware purchases and business continuity. In the future, AI cloud providers and analytics organizations can enhance these services in the AICP. AI clouds and their subdomains are the primary drivers of the rapid expansion of technological ecosystems. Gartner defines AI cloud as the use of “advanced analysis and logic-based techniques” to replicate human intelligence. AI clouds offer various applications to individuals and businesses across industries. User organizations and their technology partners frequently design, customize, and operate AI cloud implementations in many prominent AICPs. Given the worldwide expansion of AI software, these systems are crucial. Gartner projected the market to reach $62.5 billion in 2022, a 21.3% rise from its value in 2021. International Data Corporation anticipates this market to reach $549.9 billion by 2025. The evaluation of AICP involves many critical functions such as C₁ intelligent automation, C₂ the ability to prototype and test applications quickly and cost effectively (cut down cost), C₃ increased and improved digital security, C₄ ready integration into the operational and security environment of other applications currently supported in the enterprise (deeper insights), C₅ capacity to scale from prototype to large-scale data and processing environments easily, and C₆ seamless data management. Furthermore, the weights of the three decision-makers were D_c = (0.25, 0.30, 0.45). Decision matrices of three DMs under PLHFSs.

Step 1 Standardize the decision matrix in Tables 5 to 7. Table 8

Step 2 Compute the combination of the decision matrix in Table 8.

Step 3 Compute the weight for each criterion using Equations (14) and (15). The results are e_t = {0.182, 0.218, 0.100, 0.166, 0.234}.

Step 4 Establish the weighted standardized decision matrices using Equation (22) in Table 10.

Step 5 Calculate the scoring matrices in Table 11.

Step 6 Examine the negative ideal point (NIP) of the options using Equation (24) in Table 12.

Step 7 Determine the Euclidean distance [43] and taxicab distance of the options from the NIP; the outputs are presented in Tables 13 and 14.

Step 8 Construct the relative’s assessment matrix in Table 15.

Table 5

First expert’s decision matrix K₃

Options	C ₁	C ₂	C ₃
AICP ₁	{0.6 (0.4) , 0.4 (0.6)}	{0.6 (0.2) , 0.5 (0.5) , 0.4 (0.3)}	{0.5 (0.7) , 0.6 (0.3)}
AICP ₂	{0.4 (0.2) , 0.3 (0.8)}	{0.3 (0.4) , 0.2 (0.5) , 0.4 (0.1)}	{0.2 (0.5) , 0.4 (0.5)}
AICP ₃	{0.2 (0.3) , 0.1 (05) , 0.4 (0.2)}	{0.4 (0.4) , 0.5 (0.1) , 0.3 (0.5)}	{0.6 (0.3) , 0.3 (0.6) , 0.1 (0.1)}
AICP ₄	{0.3 (1)}	{0.2 (0.4) , 0.3 (0.5) , 0.4 (0.1)}	{0.3 (0.1) , 0.4 (0.3) , 0.5 (0.6)}
AICP ₅	{0.4 (0.1) , 0.3 (0.7) , 0.4 (0.2)}	{0.3 (1)}	{0.3 (1)}
Options	C ₄	C ₅	C ₆
AICP ₁	{0.6 (0.3) , 0.4 (0.3) , 0.5 (0.4)}	{0.7 (0.1) , 0.6 (0.5) , 0.5 (0.4)}	{0.5 (0.1) , 0.7 (0.7) , 0.3 (0.2)}
AICP ₂	{0.4 (0.4) , 0.5 (0.1) , 0.3 (0.5)}	{0.3 (0.2) , 0.5 (0.5) , 0.4 (0.3)}	{0.2 (0.5) , 0.4 (0.3) , 0.5 (0.2)}
AICP ₃	{0.5 (0.3) , 0.7 (0.2) , 0.2 (0.5)}	{0.3 (0.7) , 0.6 (0.3)}	{0.4 (0.3) , 0.3 (0.5) , 0.6 (0.2)}
AICP ₄	{0.3 (0.5) , 0.4 (0.4) , 0.5 (0.1)}	{0.5 (0.2) , 0.4 (0.8)}	{0.4 (0.1) , 0.3 (0.4) , 0.5 (05)}
AICP ₅	{0.4 (0.5) , 0.2 (0.5)}	{0.2 (0.3) , 0.3 (0.7)}	{0.1 (0.4) , 0.5 (0.3) , 0.6 (0.3)}

Table 6

Sndardize the decision matrix with K₁ offered by the first expert

Options	C ₁	C ₂
AICP ₁	{0.3(0.1), 0.5(0.2), 0.6(0.2), 0.6(0.15), 0.6(0.15), 0.6(0.1), 0.6(0.1)}	{(0.5(0.1), 0.5(0.2), 0.5(0.2), 0.6(0.15), 0.6(0.15), 0.6(0.1), 0.6(0.1)}
AICP ₂	{0.4(0.1), 0.4(0.2), 0.4(0.2), 0.5(0.15), 0.5(0.15), 0.5(0.1), 0.5(0.1)}	{0.2(0.1), 0.4(0.2), 0.4(0.2), 0.3(0.15), 0.3(0.15), 0.3(0.1), 0.5(0.1)}
AICP ₃	{0.2(0.1), 0.2(0.2), 0.4(0.2), 0.4(0.15), 0.4(0.15), 0.5(0.1), 0.4(0.1)}	(l(0.l), 0.1(0.2), 0.1(0.2), 0.4(0.15), 0.4(0.15), 0.l(0.l), 0.l(0.l)}
AICP ₄	{0.4(0.1), 0.2(0.2), 0.2(0.2), 0.2(0.15), 0.2(0.15), 0.4(0.1), 0.2(0.1)}	{0.6(0.1), 0.3(0.2), 0.3(0.2), 0.3(0.15), 0.3(0.15), 0.2(0.1), 0.2(0.1)}
AICP ₅	{0.3(0.1), 0.3(0.2), 0.3(0.2), 0.3(0.15), 0.3(0.15), 0.3(0.1), 0.3(0.1)}	{1), 0.2(0.2), 0.2(0.2), 0.2(0.15), 0.2(0.15), 0.2(0.1), 0.3(0.1)}
Options	C ₃	C ₄
AICP ₁	{0.5(0.1), 0.5(0.2), 0.4(0.2), 0.4(0.15), 0.4(0.15), 0.4(0.1), 0.2(0.1)}	{0.2(0.1), 0.5(0.2), 0.5(0.2), 0.6(0.15), 0.6(0.15), 0.6(0.1), 0.2(0.1)}
AICP ₂	{0.3(0.1), 0.4(0.2), 0.4(0.2), 0.3(0.15), 0.3(0.15), 0.3(0.1), 0.3(0.1)}	(0.5(0.1), 0.5(0.2), 0.4(0.2), 0.3(0.15), 0.3(0.15), 0.3(0.1), 0.3(0.1)}
AICP ₃	{0.3(0.1), 0.2(0.2), 0.2(0.2), 0.3(0.15), 0.3(0.15), 0.3(0.1), 0.3(0.1)}	{0.3(0.1), 0.2(0.2), 0.2(0.2), 0.2(0.15), 0.2(0.15), 0.2(0.1), 0.3(0.1)}
AICP ₄	{0.1(0.1), 0.1(0.2), 0.2(0.2), 0.5(0.15), 0.5(0.15), 0.5(0.1), 0.5(0.1)}	{0.3(0.1), 0.3(0.2), 0.2(0.2), 0.4(0.15), 0.4(0.15), 0.2(0.1), 0.2(0.1)}
AICP ₅	{0.5(0.1), 0.1(0.2), 0.1(0.2), 0.4(0.15), 0.4(0.15), 0.4(0.1), 0.4(0.1)}	{0.3(0.1), 0.3(0.2), 0.2(0.2), 0.5(0.15), 0.5(0.15), 0.5(0.1), 0.5(0.1)}
Options	C ₅	C ₆
AICP ₁	{0.5(0.1), 0.5(0.2), 0.4(0.2), 0.5(0.15), 0.5(0.15), 0.4(0.1), 0.4(0.1)}	{0.5(0.1), 0.6(0.2), 0.6(0.2), 0.5(0.15), 0.5(0.15), 0.5(0.1), 0.5(0.1)}
AICP ₂	{0.5(0.1), 0.3(0.2), 0.2(0.2), 0.4(0.15), 0.4(0.15), 0.4(0.1), 0.5(0.1)}	{0.3(0.1), 0.5(0.2), 0.5(0.2), 0.5(0.15), 0.5(0.15), 0.5(0.1), 0.3(0.1)}
AICP ₃	{0.2(0.1), 0.2(0.2), 0.3(0.2), 0.4(0.15), 0.4(0.15), 0.3(0.1), 0.3(0.1)}	(0.5), 0.1(0.2), 0.1(0.2), 0.3(0.15), 0.3(0.15), 0.3(0.1), 0.5(0.1)}
AICP ₄	{0.4(0.1), 0.3(0.2), 0.3(0.2), 0.5(0.15), 0.5(0.15), 0.5(0.1), 0.4(0.1)}	{0.3(0.1), 0.3(0.2), 0.3(0.2), 0.4(0.15), 0.4(0.15), 0.4(0.1), 0.4(0.1)}
AICP ₅	{0.3(0.1), 0.3(0.2), 0.4(0.2), 0.1(0.15), 0.1(0.15), 0.4(0.1), 0.4(0.1)}	{0.3(0.1), 0.2(0.2), 0.2(0.2), 0.4(0.15), 0.4(0.15), 0.4(0.1), 0.4(0.1)}

Table 7

Standardize the decision of matrix with K₂ offered by the first expert

Options	C ₁	C ₂
AICP ₁	{0.4(0.1), 0.6(0.2), 0.4(0.2), 0.4(0.15), 0.4(0.15), 0.5(0.1), 0.4(0.1)}	{0.3(04), 03(0.2), 03(0.2), 0.6(0.15), 0.6(045), 0.6(04), 0.6(04)}
AICP ₂	{0.4(0.1), 0.3(0.2), 03(0.2), 0.4(0.15), 0.4(0.15), 03(0.1), 0.4(0.1)}	{0.1(04), 0.5(0.2), 03(0.2), 0.1(0.15), 0.1(0.15), 0.1(04), 0.4(04)}
AICP ₃	{0.4(0.1), 03(0.2), 0.1(0.2), 0.4(0.15), 0.4(0.15), 0.4(0.1), 0.4(0.1)}	{0.7(04), 03(0.2), 03(0.2), 0.3(0.15), 0.3(0.15), 0.7(04), 03(0.1)}
AICP ₄	{0.l(0.l), 03(0.2), 0.1(0.2), 0.4(0.15), 0.4(0.15), 0.l(0.l), 0.4(0.1)}	{0.4(04), 0.4(0.2), 0.4(0.2), 0.4(0.15), 0.4(0.15), 0.4(04), 0.4(04)}
AICP ₅	{03(0.1), 0.1(0.2), 03(0.2), 0.5(0.15), 0.5(0.15), 03(0.1), 0.5(04)}	{0.2(04), 0.2(0.2), 03(0.2), 0.3(0.15), 0.3(0.15), 0.3(04), 0.4(04)}
Options	C ₃	C ₄
AICP ₁	{0.8(04), 0.4(0.2), 0.4(0.2), 0.3(0.15), 0.3(0.15), 03(0.1), 0.3(04)}	{03(0.1), 0.4(0.2), 0.4(0.2), 0.5(0.15), 0.5(0.15), 0.5(04), 0.5(04)}
AICP ₂	{0.8(04), 0.2(0.2), 0.2(0.2), 0.2(0.15), 0.2(0.15), 0.2(04), 0.8(04)}	{0.2(04), 0.2(0.2), 0.2(0.2), 0.4(0.15), 0.4(0.15), 0.4(04), 0.4(04)}
AICP ₃	{0.2(04), 0.2(0.2), 0.4(0.2), 0.5(045), 0.5(045), 0.4(04), 0.4(04)}	{0.7(04), 03(0.2), 03(0.2), 0.1(045), 0.1(0.15), 0.1(04), 0.1(04)}
AICP ₄	{0.3(04), 03(0.2), 0.6(0.2), 0.2(0.15), 0.2(045), 0.2(04), 0.2(04)}	{0.4(04), 03(0.2), 03(0.2), 0.3(045), 0.3(0.15), 0.3(04), 03(0.1)}
AICP ₅	{0.1(04), 04(0.2), 04(0.2), 0.4(0.15), 0.4(045), 0.4(04), 0.4(04)}	{0.2(04), 0.2(0.2), 0.2(0.2), 0.2(0.15), 0.2(0.15), 0.2(04), 0.2(04)}
Options	C ₅	C ₆
AICP ₁	{0.4(04), 0.4(0.2), 0.4(0.2), 0.5(0.15), 0.5(045), 0.5(04), 0.4(04)}	{0.5(04), 03(0.2), 03(0.2), 0.5(045), 0.5(045), 0.5(04), 0.5(04)}
AICP ₂	{0.3(04), 03(0.2), 03(0.2), 0.4(0.15), 0.4(045), 0.2(04), 0.2(04)}	{0.5(04), 03(0.2), 0.4(0.2), 0.3(0.15), 0.3(0.15), 03(0.1), 03(0.1)}
AICP ₃	{0.3(04), 03(0.2), 0.5(0.2), 0.4(0.15), 0.4(0.15), 0.5(04), 0.5(04)}	{0.4(04), 0.4(0.2), 0.2(0.2), 0.4(0.15), 0.4(0.15), 0.2(04), 0.2(04)}
AICP ₄	{0.3(04), 0.4(0.2), 0.4(0.2), 0.3(0.15), 0.3(045), 0.3(04), 0.3(04)}	{0.3(04), 03(0.2), 03(0.2), 0.6(0.15), 0.6(045), 0.3(04), 03(0.1)}
AICP ₅	{0.6(04), 0.6(0.2), 04(0.2), 0.6(0.15), 0.1(0.15), 0.1(04), 0.1(04)}	{0.2(04), 03(0.2), 03(0.2), 0.4(045), 0.4(0.15), 0.4(04), 0.4(04)}

Table 8

Standardize the decision matrix with K₃ offered by the first expert

Options	C ₁	C ₂
AICP ₁	{0.6(0.1),0.6(0.2),0.4(0.2),0.4(0.15),0.4(0.15),0.6(0.1),0.4(0.1)}	{0.4(0.1),0.4(0.2),0.6(0.2),0.5(0.15),0.5(0.15),0.5(0.1),0.5(0.1)}
AICP ₂	{0.4(0.1),0.3(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.4(0.1),0.3(0.1)}	{0.4(0.1),0.3(0.2),0.3(0.2),0.2(0.15), 0.2(0.15),0.2(0.1),0.2(0.1)}
AICP ₃	{0.2(0.1),0.2(0.2),0.4(0.2),0.5(0.15),0.4(0.1),0.5(0.1)}	(0.1),0.4(0.2),0.4(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}
AICP ₄	{0.3(0.1),0.3(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}	{0.4(0.1),0.2(0.2),0.2(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}
AICP ₅	{0.4(0.1),0.4(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}	{0.3(0.1),0.3(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}
Options	C ₃	C ₄
AICP ₁	{0.5(0.1),0.5(0.2),0.5(0.2),0.6(0.15),0.6(0.15),0.5(0.1),0.5(0.1)}	{0.6(0.1),0.6(0.2),0.5(0.2),0.4(0.15),0.4(0.15),0.5(0.1),0.5(0.1)}
AICP ₂	{0.2(0.1),0.2(0.2),0.2(0.2),0.4(0.15),0.4(0.15),0.4(0.1),0.4(0.1)}	{0.5(0.1),0.4(0.2),0.4(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}
AICP ₃	{0.6(0.1),0.6(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.1(0.1)}	{0..5(0.2),0.7(0.2),0.2(0.15),0.2(0.15),0.2(0.1),0.2(0.1)}
AICP ₄	{0.4(0.1),0.4(0.2),0.5(0.2),0.5(0.15),0.5(0.15),0.5(0.1),0.3(0.1)}	{0.5(0.1),0.4(0.2),0.4(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}
AICP ₅	{0.3(0.1),0.3(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}	{0.2(0.1),0.2(0.2),0.2(0.2),0.4(0.15),0.4(0.15),0.4(0.1),0.4(0.1)}
Options	C ₅	C ₆
AICP ₁	{0.7(0.1),0.5(0.2),0.5(0.2),0.6(0.15),0.6(0.15),0.6(0.1),0.6(0.1)}	{0.5(0.1),0.3(0.2),0.7(0.2),0.7(0.15),0.7(0.15),0.7(0.1),0.7(0.1)}
AICP ₂	{0.4(0.1),0.4(0.2),0.3(0.2),0.5(0.15),0.5(0.15),0.5(0.1),0.5(0.1)}	{0.4(0.1),0.4(0.2),0.5(0.2),0.2(0.15),0.2(0.15),0.2(0.1),0.2(0.1)}
AICP ₃	{0.3(0.1),0.3(0.2),0.3(0.2),0.6(0.15),0.6(0.15),0.3(0.1),0.3(0.1)}	{0.4(0.1),0.4(0.2),0.6(0.2),0.3(0.15),0.3,0.3(0.1),0.3(0.1)}
AICP ₄	{0.5(0.1),0.4(0.2),0.4(0.2),0.4(0.15),0.4(0.15),0.4(0.1),0.5(0.1)}	{0.4(0.1),0.3(0.2),0.3(0.2),0.5(0.15),0.5(0.15),0.5(0.1),0.5(0.1)}
AICP ₅	{0.2(0.1),0.2(0.2),0.3(0.2),0.3(0.15),0.3(0.15),0.3(0.1),0.3(0.1)}	{0.5(0.1),0.5(0.2),0.1(0.2),0.6(0.15),0.6(0.15),0.1(0.1),0.1(0.1)}

Table 9

Combination of the decision matrix

Options	C ₁	C ₂
AICP ₁	$\begin{matrix} {0.444 (0.1), 0.572 (0.2), 0.469 (0.2), \\ 0.469 (0.15), 0.469 (0.15), 0.563 (0.1), \\ 0.469 (0.1)} \end{matrix}$	$\begin{matrix} {0.396 (0.1), 0.396 (0.2), 0.465 (0.2), \\ 0.572 (0.15), 0.572 (0.15), 0.572 (0.1), \\ 0.572 (0.1)} \end{matrix}$
	{0.400 (0.1) , 0.332 (0.2) , 0.332 (0.2) ,	{0.231 (0.1) , 0.416 (0.2) , 0.416 (0.2) ,
AICP ₂	$\begin{matrix} 0.405 (0.15), 0.405 (0.15), 0.396 (0.1), \\ 0.405 (0.1)} \end{matrix}$	$\begin{matrix} 0.194 (0.15), 0.194 (0.15), \\ 0.194 (0.1), 0.381 (0.1)} \end{matrix}$
AICP ₃	{0.287 (0.1) , 0.242 (0.2) , 0.294 (0.2) ,	{0.514 (0.1) , 0.279 (0.2) , 0.279 (0.2) ,
	0.432 (0.15) , 0.432 (0.15) , 0.432 (0.1) ,	0.332 (0.15) , 0.332 (0.15) ,
	0.432 (0.1)}	0.462 (0.1) , 0.245 (0.1)}
AICP ₄	{0.261 (0.1) , 0.271 (0.2) , 0.194 (0.2) ,	{0.469 (0.1) , 0.315 (0.2) , 0.315 (0.2) ,
	0.315 (0.15) , 0.315 (0.15) , 0.261 (0.1) ,	0.342 (0.15) , 0.342 (0.15) , 0.315 (0.1) ,
	0.315 (0.1)}	0.315 (0.1)}
AICP ₅	{0.332 (0.1) , 0.261 (0.2) , 0.300 (0.2),	{0.262 (0.1) , 0.231 (0.2) , 0.271 (0.2) ,
	0.388 (0.15) , 0.388 (0.15) , 0.300 (0.1),	0.271 (0.15) , 0.271 (0.15),0.271 (0.1) ,
	0.388 (0.1)}	0.342 (0.1)}
Options	C ₃	C ₄
AICP ₁	{0.653 (0.1) , 0.462 (0.2) , 0.432 (0.2) ,	{0.384 (0.1) , 0.497 (0.2) , 0.462 (0.2) ,
	0.435 (0.15) , 0.435 (0.15) , 0.396 (0.1) ,	0.506 (0.15) , 0.506 (0.15) , 0.532 (0.1) ,
	0.341 (0.1)}	0.424 (0.1)}
AICP ₂	{0.559 (0.1) , 0.266 (0.2) , 0.266 (0.2) ,	{0.397 (0.1) , 0.363 (0.2) , 0.327 (0.2) ,
	0.295 (0.15) , 0.295 (0.15) , 0.295 (0.1) ,	0.342 (0.15) , 0.342 (0.15) , 0.342 (0.1) ,
	0.595 (0.1)}	0.342 (0.1)}
AICP ₃	{0.376 (0.1) , 0.350 (0.2) , 0.315 (0.2) ,	{0.549 (0.1) , 0.341 (0.2) , 0.435 (0.2) ,
	0.388 (0.15) , 0.388 (0.15) , 0.342 (0.1) ,	0.161 (0.15) , 0.161 (0.15) , 0.161 (0.1) ,
	0.290 (0.1)}	0.194 (0.1)}
AICP ₄	{0.279 (0.1) , 0.279 (0.2) , 0.473 (0.2) ,	{0.405 (0.1) , 0.416 (0.2) , 0.304 (0.2) ,
	0.397 (0.15) , 0.397 (0.15) , 0.397 (0.1) ,	0.332 (0.15) , 0.332 (0.15) , 0.271 (0.1) ,
	0.332 (0.1)}	0.271 (0.1)}
AICP ₅	{0.300 (0.1) , 0.165 (0.2) , 0.165 (0.2) ,	{0.231 (0.1) , 0.231 (0.2) , 0.200 (0.2) ,
	0.372 (0.15) , 0.372 (0.15) , 0.372 (0.1) ,	0.363 (0.15) , 0.363 (0.15) , 0.363 (0.1) ,
	0.372 (0.1)}	0.363 (0.1)}
Options	C ₅	C ₆
AICP ₁	{0.539 (0.1) , 0.462 (0.2) , 0.432 (0.2) ,	{0.500 (0.1) , 0.483 (0.2) , 0.599 (0.2) ,
	0.532 (0.15) , 0.532 (0.15) , 0.506 (0.1) ,	0.571 (0.15) , 0.571 (0.15) , 0.571 (0.1) ,
	0.469 (0.1)}	0.571 (0.1)}
AICP ₂	{0.396 (0.1) , 0.332 (0.2) , 0.300 (0.2) ,	{0.416 (0.1) , 0.472 (0.2) , 0.462 (0.2) ,
	0.432 (0.15) , 0.432 (0.15) , 0.363 (0.1) ,	0.341 (0.15) , 0.341 (0.15) , 0.341 (0.1) ,
	0.397 (0.1)}	0.271 (0.1)}
AICP ₃	{0.271 (0.1) , 0.271 (0.2) , 0.388 (0.2) ,	{0.432 (0.1) , 0.322 (0.2) , 0.327 (0.2) ,
	0.469 (0.15) , 0.469 (0.15) , 0.388 (0.1) ,	0.342 (0.15) , 0.342 (0.15) , 0.262 (0.1) ,
	0.388 (0.1)}	0.332 (0.1)}
AICP ₄	{0.396 (0.1) , 0.372 (0.2) , 0.372 (0.2) ,	{0.332 (0.1) , 0.300 (0.2) , 0.300 (0.2) ,
	0.396 (0.15) , 0.396 (0.15) , 0.396 (0.1) ,	0.517 (0.15) , 0.517 (0.15) , 0.396 (0.1) ,
	0.396 (0.1)}	0.396 (0.1)}
AICP ₅	{0.418 (0.1) , 0.418 (0.2) , 0.261 (0.2) ,	{0.332 (0.1) , 0.341 (0.2) , 0.214 (0.2) ,
	0.397 (0.15) , 0.397 (0.15) , 0.261 (0.1) ,	0.469 (0.15) , 0.469 (0.15) , 0.322 (0.1) ,
	0.261 (0.1)}	0.322 (0.1)}

Table 10

Weighted standardized decision matrices

Options	Q ₁	Q ₂
AICP ₁	{0.101 (0.1) , 0.143 (0.2) , 0.109 (0.2) ,	{0.104 (0.1) , 0.104 (0.2) , 0.127 (0.2) ,
	0.109 (0.15) , 0.109 (0.15) , 0.140 (0.1) ,	0.169 (0.15) , 0.169 (0.15) ,
	0.109 (0.1)}	0.169 (0.1) , 0.169 (0.1)}
AICP ₂	{0.089 (0.1) , 0.071 (0.2) , 0.071 (0.2) ,	{0.056 (0.1) , 0.110 (0.2) , 0.110 (0.2) ,
	0.090 (0.15) , 0.090 (0.15) , 0.088 (0.1) ,	0.046 (0.15) , 0.046 (0.15) , 0.046 (0.1) ,
	0.090 (0.1)}	0.099 (0.1)}
AICP ₃	{0.060 (0.1) , 0.049 (0.2) , 0.062 (0.2) ,	{0.145 (0.1) , 0.069 (0.2) , 0.069 (0.2) ,
	0.098 (0.15) , 0.098 (0.15) , 0.098 (0.1) ,	0.084 (0.15) , 0.084 (0.15) , 0.126 (0.1) ,
	0.098 (0.1)}	0.0599 (0.1)}
AICP ₄	{0.054 (0.1) , 0.056 (0.2) , 0.039 (0.2) ,	{0.129 (0.1) , 0.079 (0.2) , 0.079 (0.2) ,
	0.067 (0.15) , 0.067 (0.15) , 0.054 (0.1) ,	0.087 (0.15) , 0.087 (0.15) , 0.079 (0.1) ,
	0.067 (0.1)}	0.079 (0.1)}
AICP ₅	{0.071 (0.1) , 0.054 (0.2) , 0.063 (0.2) ,	{0.064 (0.1) , 0.056 (0.2) , 0.067 (0.2) ,
	0.086 (0.15) , 0.086 (0.15) , 0.063 (0.1) ,	0.067 (0.15) , 0.067 (0.15) , 0.067 (0.1) ,
	0.086 (0.1)}	0.087 (0.1)}
Options	C ₃	C ₄
AICP ₁	{0.101 (0.1) , 0.060 (0.2) , 0.055 (0.2) ,	{0.077 (0.1) , 0.108 (0.2) , 0.098 (0.2) ,
	0.056 (0.15) , 0.056 (0.15) , 0.049 (0.1) ,	0.110 (0.15) , 0.110 (0.15) ,
	0.041 (0.1)}	0.118 (0.1) , 0.087 (0.1)}
AICP ₂	{0.079 (0.1) , 0.031 (0.2) , 0.031 (0.2) ,	{0.088 (0.1) , 0.079 (0.2) , 0.070 (0.2) ,
	0.034 (0.15) , 0.034 (0.15) , 0.034 (0.1) ,	0.073 (0.15) , 0.073 (0.15) ,
	0.087 (0.1)}	0.073 (0.1) , 0.073 (0.1)}
AICP ₃	{0.046 (0.1) , 0.042 (0.2) , 0.037 (0.2) ,	{0.135 (0.1) , 0.073 (0.2) , 0.099 (0.2) ,
	0.048 (0.15) , 0.048 (0.15) , 0.041 (0.1) ,	0.032 (0.15) , 0.032 (0.15) ,
	0.034 (0.1)}	0.032 (0.1) , 0.039 (0.1)}
AICP ₄	{0.032 (0.1) , 0.032 (0.2) , 0.062 (0.2) ,	{0.090 (0.1) , 0.093 (0.2) , 0.064 (0.2) ,
	0.049 (0.15) , 0.049 (0.15) , 0.049 (0.1) ,	0.071 (0.15) , 0.071 (0.15) ,
	0.040 (0.1)}	0.056 (0.1) , 0.056 (0.1)}
AICP ₅	{0.035 (0.1) , 0.018 (0.2) , 0.018 (0.2) ,	{0.047 (0.1) , 0.047 (0.2) , 0.040 (0.2) ,
	0.045 (0.15) , 0.045 (0.15) , 0.045 (0.1) ,	0.079 (0.15) , 0.079 (0.15) ,
	0.045 (0.1)}	0.079 (0.1) , 0.079 (0.1)}
Options	C ₅	C ₆
AICP ₁	{0.074 (0.1) , 0.060 (0.2) , 0.055 (0.2) ,	{0.150 (0.1) , 0.143 (0.2) , 0.193 (0.2) ,
	0.073 (0.15) , 0.073 (0.15) , 0.068 (0.1) ,	0.180 (0.15) , 0.180 (0.15) ,
	0.061 (0.1)}	0.180 (0.1) , 0.180 (0.1)}
AICP ₂	{0.049 (0.1) , 0.040 (0.2) , 0.035 (0.2) ,	{0.118 (0.1) , 0.139 (0.2) , 0.135 (0.2) ,
	0.055 (0.15) , 0.055 (0.15) , 0.044 (0.1) ,	0.093 (0.15) , 0.093 (0.15) ,
	0.049 (0.1)}	0.093 (0.1) , 0.072 (0.1)}
AICP ₃	{0.031 (0.1) , 0.031 (0.2) , 0.048 (0.2) ,	{0.124 (0.1) , 0.087 (0.2) , 0.089 (0.2) ,
	0.061 (0.15) , 0.061 (0.15) , 0.048 (0.1) ,	0.093 (0.15) , 0.093 (0.15) ,
	0.048 (0.1)}	0.069 (0.1) , 0.090 (0.1)}
AICP ₄	{0.049 (0.1) , 0.045 (0.2) , 0.045 (0.2) ,	{0.090 (0.1) , 0.080 (0.2) , 0.080 (0.2) ,
	0.049 (0.15) , 0.049 (0.15) , 0.049 (0.1) ,	0.157 (0.15) , 0.157 (0.15) ,
	0.049 (0.1)}	0.111 (0.1) , 0.111 (0.1)}
AICP ₅	{0.053 (0.1) , 0.053 (0.2) , 0.030 (0.2) ,	{0.090 (0.1) , 0.093 (0.2) , 0.055 (0.2) ,
	0.049 (0.15) , 0.049 (0.15) , 0.030 (0.1) ,	0.138 (0.15) , 0.138 (0.15) ,
	0.030 (0.1)}	0.087 (0.1) , 0.087 (0.1)}

Table 11

Scoring of the combined decision matrices

C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
AICP ₃	0.118	0.141	0.059	0.102	0.065	0.172
AICP ₄	0.082	0.078	0.043	0.075	0.046	0.111
AICP ₅	0.077	0.086	0.042	0.064	0.047	0.091
AICP ₄	0.056	0.086	0.046	0.073	0.048	0.110
AICP ₅	0.071	0.066	0.033	0.061	0.042	0.097

Table 12

NIP solution

C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
{0.054 (0.1)	{0.064 (0.1)	{0.035 (0.1)	{0.047 (0.1)	{0.053 (0.1)	{0.090 (0.1)
0.056 (0.2),	0.056 (0.2),	0.018 (0.2),	0.047 (0.2),	0.053 (0.2),	0.093 (0.2),
0.039 (0.2),	0.067 (0.2),	0.018 (0.2),	0.040 (0.2),	0.030 (0.2),	0.055 (0.2),
0.067 (0.15),	0.067 (0.15),	0.045 (0.15),	0.079 (0.15),	0.049 (0.15),	0.138 (0.15),
0.067 (0.15),	0.067 (0.15),	0.045 (0.15),	0.079 (0.15),	0.049 (0.15),	0.138 (0.15),
0.054 (0.100),	0.067 (0.100),	0.045 (0.100),	0.079 (0.100),	0.030 (0.100),	0.087 (0.100),
0.067 (0.100)}	0.087 (0.100)}	0.045 (0.100)}	0.079 (0.100)}	0.030 (0.100)}	0.087 (0.100)}

Table 13

Output of Euclidean distances

AICP ₁	AICP ₂	AICP ₃	AICP ₄	AICP ₅
0.145	0.074	0.078	0.050	0.017

Table 14

Output of Manhattan distances

AICP ₁	AICP ₂	AICP ₃	AICP ₄	AICP ₅
0.302	0.146	0.154	0.088	0.016

Table 15

Relative’s assessment matrix

	AICP ₁	AICP ₂	AICP ₃	AICP ₄	AICP ₅
AICP ₁	0.00	0.23	0.21	0.31	0.41
AICP ₂	-0.23	0.00	0.00	0.02	0.19
AICP ₃	-0.21	0.00	0.00	0.03	0.20
AICP ₄	-0.31	-0.02	-0.03	0.00	0.11
AICP ₅	-0.41	-0.19	-0.20	-0.11	0.00

Step 9 Compute the assessment value of every options as follows: $\begin{matrix} V_{1} = 1.16, V_{2} = - 0.02, V_{3} = 0.02, V_{4} = - 0.25, \\ V_{5} = - 0.90 . \end{matrix}$

Step 10 Rank the options available based on the overall score as follows: $\begin{matrix} r ({AICP}_{1}) ≻ r ({AICP}_{3}) \\ ≻ r ({AICP}_{2}) ≻ r ({AICP}_{4}) ≻ r ({AICP}_{5}) . \end{matrix}$

5 Solving the case study with several existing methods and comparative analysis

In this section, the PLHF–TODIM method [50], PLHFWA operator [37], PLHFWG operator [37], and PLHF–TOPSIS technique [51] are compared with the PLHF–CODAS technique.

5.1 Solving and comparing with PLHF-TODIM

The PLHF–TODIM technique [50] was applied for comparison with the proposed PLHF–CODAS technique. By combining the PLHF and TODIM techniques, this system provides a comprehensive and structured approach for decision-making. It enables decision-makers to consider multiple criteria, assign weights to each criterion, and assess the alternative criteria accordingly. The application of the PLHF-TODIM technique results in a ranking of alternatives, with higher-ranked alternatives being considered more favorable or preferred based on the evaluation of their attributes. Table 16 shows the results obtained by applying the PLHF–TODIM technique.

Table 16
Results of the PLHF–TODIM technique

Expression Outcomes

Relative weights $e_{t}^{}$ $e_{t}^{} = (0.777, 0.929, 0.427, 0.706, 0.427, 1.000)$

Overall degree ϑ (AICP_r) ϑ (AICP₁) = 1.000, ϑ (AICP₂) = 0.526, ϑ (AICP₃) = 0 . 517, ϑ (AICP₄) = 0.591, ϑ (AICP₅) = 0.0101

Outcome r (AICP₁) ≻ r (AICP₄) ≻ r (AICP₂) ≻ r (AICP₃) ≻ r (AICP₅)

	Expression	Outcomes
Relative weights	$e_{t}^{*}$	$e_{t}^{*} = (0.777, 0.929, 0.427, 0.706, 0.427, 1.000)$
Overall degree	ϑ (AICP_r)	ϑ (AICP₁) = 1.000, ϑ (AICP₂) = 0.526, ϑ (AICP₃) = 0 . 517, ϑ (AICP₄) = 0.591, ϑ (AICP₅) = 0.0101
Outcome		r (AICP₁) ≻ r (AICP₄) ≻ r (AICP₂) ≻ r (AICP₃) ≻ r (AICP₅)

5.2 Solving and comparing with the PLHF aggregation operators

Table 17 lists the results obtained by applying the PLHFWA function, which combines the probabilities and weights, to calculate an aggregated value. These results represent the overall consensus or combined information from the hesitant fuzzy set. It considers both the confidence in each possibility and relative importance assigned to each possibility, resulting in a comprehensive representation of the aggregated information. The specific calculation of the PLHFWA function may vary depending on the specific algorithm or method used. However, the general idea is to consider both the probabilities and weights to obtain a single aggregated value that reflects the overall consensus or combined information from the hesitant fuzzy set.

Table 17
Results of the PLHFWA operator

Options Overall values Score

AICP ₁ {{ 0.474 (0.1) , 0.481 (0.2) , 0.494 (0.2) , 0.527 (0.15) , 0.527 (0.15) , 0.542 (0.1) , 0.501 (0.1) } 0.505

AICP ₂ {0.389 (0.1) , 0.385 (0.2) , 0.374 (0.2) , 0.330 (0.15) , 0.330 (0.15) , 0.320 (0.1) , 0.383 (0.1)} 0.360

AICP ₃ {0.430 (0.1) , 0.300 (0.2) , 0.336 (0.2) , 0.350 (0.15) , 0.350 (0.15) , 0.349 (0.1) , 0.315 (0.1)} 0.341

AICP ₄ {0.366 (0.1) , 0.324 (0.2) , 0.313 (0.2) , 0.392 (0.15) , 0.392 (0.15) , 0.336 (0.1) , 0.338 (0.1)} 0.349

AICP ₅ {0.308 (0.1) , 0.278 (0.2) , 0.241 (0.2) , 0.380 (0.15) , 0.380 (0.15) , 0.314 (0.1) , 0.345 (0.1)} 0.315

Outcome r (AICP₁) ≻ r (AICP₂) ≻ r (AICP₄) ≻ r (AICP₃) ≻ r (AICP₅)

Options	Overall values	Score
AICP ₁	{{ 0.474 (0.1) , 0.481 (0.2) , 0.494 (0.2) , 0.527 (0.15) , 0.527 (0.15) , 0.542 (0.1) , 0.501 (0.1) }	0.505
AICP ₂	{0.389 (0.1) , 0.385 (0.2) , 0.374 (0.2) , 0.330 (0.15) , 0.330 (0.15) , 0.320 (0.1) , 0.383 (0.1)}	0.360
AICP ₃	{0.430 (0.1) , 0.300 (0.2) , 0.336 (0.2) , 0.350 (0.15) , 0.350 (0.15) , 0.349 (0.1) , 0.315 (0.1)}	0.341
AICP ₄	{0.366 (0.1) , 0.324 (0.2) , 0.313 (0.2) , 0.392 (0.15) , 0.392 (0.15) , 0.336 (0.1) , 0.338 (0.1)}	0.349
AICP ₅	{0.308 (0.1) , 0.278 (0.2) , 0.241 (0.2) , 0.380 (0.15) , 0.380 (0.15) , 0.314 (0.1) , 0.345 (0.1)}	0.315
Outcome	r (AICP₁) ≻ r (AICP₂) ≻ r (AICP₄) ≻ r (AICP₃) ≻ r (AICP₅)

In Table 18, the results obtained by applying the PLHFWG function are shown, which considers not only the values of the probabilistic hesitant fuzzy numbers but also their corresponding probabilities and weights. The weights reflect the relative importance or significance of each number in the calculation. To obtain the PLHFWG results, we first multiply each probabilistic hesitant fuzzy number by its corresponding weight. Subsequently, for each possible value within the range of the fuzzy number, the weighted product of that value and its corresponding probability are calculated. Finally, we obtain the geometric mean of all the weighted products calculated in the previous step to obtain the PLHFWG results.

Table 18

Results of the PLHFWG operator

Options	Overall values	Score
AICP ₁	{{ 0.461 (0.1) , 0.475 (0.2) , 0.486 (0.2) , 0.522 (0.15) , 0.522 (0.15) , 0.536 (0.1) , 0.489 (0.1) }	0.497
AICP ₂	{0.369 (0.1) , 0.376 (0.2) , 0.364 (0.2) , 0.314 (0.15) , 0.314 (0.15) , 0.308 (0.1) , 0.367 (0.1)}	0.347
AICP ₃	{0.408 (0.1) , 0.297 (0.2) , 0.329 (0.2) , 0.327 (0.15) , 0.327 (0.15) , 0.320 (0.1) , 0.299 (0.1)}	0.326
AICP ₄	{0.354 (0.1) , 0.319 (0.2) , 0.300 (0.2) , 0.380 (0.15) , 0.380 (0.15) , 0.328 (0.1) , 0.333 (0.1)}	0.339
AICP ₅	{0.301 (0.1) , 0.266 (0.2) , 0.236 (0.2) , 0.370 (0.15) , 0.370 (0.15) , 0.310 (0.1) , 0.342 (0.1)}	0.307
Outcome	r (AICP₁) ≻ r (AICP₂) ≻ r (AICP₄) ≻ r (AICP₃) ≻ r (AICP₅)

5.3 Solving and comparing with the PLHF–TOPSIS technique

HF–TOPSIS method [52] was compared with the PLHF–CODAS method. To analyze the results obtained using the PLHF-TOPSIS technique, we first need to identify the criteria used for evaluation. Next, we must assign weights to each criterion to reflect their relative importance. These weights can be determined based on the decision-maker’s preferences or by using an AHP. Once the criteria a weights are established, we evaluate each alternative against each criterion. This is usually performed using a scale or scoring system to evaluate the performance of each alternative. The resulting scores ce calculated based on data and information available for each criterion.

Using the Promethee II method, we determine the preference function for each alternative, which compares their performance against each criterion. This function considers both the weight of the criterion and difference in performance between alternatives.

After obtaining the preference functions, we implement TOPSIS technique to rank the alternatives. TOPSIS calculates the distance of each alternative from the ideal solution, which is determined based on the best performance for each criterion.he alternative obtained with the shortest distance to the ideal solution is considered the best-ranked alternative. Table 19 lists the computational results of the PLHF–TOPSIS technique.

Table 19
Results obtained using the PLHF–TOPSIS technique

Title Symbol Outcome

Distance $D_{r}^{+}$ $D_{r}^{+} = (0.00, 0.84, 0.92, 0.87, 1.09)$

$D_{r}^{-}$ $D_{r}^{-} = (1.16, 0.48, 0.26, 0.36, 0.22)$

Comparative proximity coefficients G ^* $G_{1}^{} = 1.00, G_{2}^{} = 0.46, G_{3}^{} = 0.32, G_{4}^{} = 0.39, G_{5}^{} = 0.27$

The outcome r (AICP₁) ≻ r (AICP₂) ≻ r (AICP₄) ≻ r (AICP₃) ≻ r (AICP*₅)

Title	Symbol	Outcome
Distance	$D_{r}^{+}$	$D_{r}^{+} = (0.00, 0.84, 0.92, 0.87, 1.09)$
	$D_{r}^{-}$	$D_{r}^{-} = (1.16, 0.48, 0.26, 0.36, 0.22)$
Comparative proximity coefficients	G ^*	$G_{1}^{} = 1.00, G_{2}^{} = 0.46, G_{3}^{} = 0.32, G_{4}^{} = 0.39, G_{5}^{*} = 0.27$
The outcome		r (AICP₁) ≻ r (AICP₂) ≻ r (AICP₄) ≻ r (AICP₃) ≻ r (AICP₅)

Given the unique characteristics of all three techniques, including PLHF-TODIM, the PLHF aggregation operators, and PLHF-TOPSIS, distinct results were obtained while processing the same original data. The results are presented below. Although the outcome of the proposed method differs from that of the other methods, the optimal solution is AICP₁.

5.4 Sensitivity analysis

We also aim to demonstrate that the results of the proposed system are stable and resistant to modification by the vector of the criterion weights. Using the Sensitivity Analyzer App, we generated 50 sets of numbers that were randomized with five possibilities, and the criteria weights were assigned such that their sum was unity. In addition, one criterion carries the most weight in each set. As depicted in Fig. 2, the proposed technique was applied, and the mean evaluations of 50 sets of weights were obtained. The mean score obtained correlates to the option ranking, and the alternative ranking can be derived from Fig. 2. Additionally, the alternative AICP₁ is shown to be always superior to all other alternatives and agrees with the optimal alternative determined in Section 4. Therefore, the proposed technique is highly stable, and any changes to the weight values of the criteria will not affect the precision of the results. In addition, this evaluation demonstrates that the outputs of the PLHF-CODAS technique for MADM are stable and not susceptible to external interference, producing more precise results than other ranking methods.

Fig. 2

Score of alternatives in different criterion sets.

6 Conclusion and future research directions

In this study, the performance of the proposed PLHF–CODAS technique was analyzed on the Euclidean and taxicab distances across the ideal negative solution. If the Euclidean distances between the two possibilities are comparable, the taxicab distance is used to contrast them based on the PLHF–CODAS principle. In this study, the PLHF–CODAS approach was used for MADM using PLHFSs. Specific examples of AICP evaluation and comparison analyses were used to validate the feasibility of this novel method. The following are the advantages of the expanded approach: The PLHF–CODAS technique was developed to resolve the probabilistic linguistic hesitant fuzzy MADM, in which the statistical variance approach was used to estimate the weight of every attribute under the PLHFSs. Several difference solving techniques and comparative analyses demonstrate the applicability of this extended method, which may be used in the evaluation of the critical functions of an AICP. The selection of an AICP may help businesses save time and money. Furthermore, the AI cloud can help businesses overcome critical difficulties in the future. Despite the advantage of this study, we encountered several limitations. This study began by evaluating the AICPs for a developing nation. Therefore, a sampling bias may be present. Additionally, future research may evaluate the performance indicators of the AICPs using a variety of performance metrics. Moreover, the outcome of the proposed method is identical to that of the other evaluated techniques, as the present study focused solely on the technique of surveying, which was used to evaluate and assess the AICPs based on the opinions of experts and available literature. Six indicators from the extant literature were used in this regard. The data for evaluating the indicators of the AICPs were gathered from three industry professionals. In the future, methods and algorithms that can be applied to additional real-world decision-making issues and can be used for other unpredictably unclear data will be presented.

Statements and declarations

Footnotes

Acknowledgments

This work was supported by Posts and Telecommunications Institute of Technology.

Funding

This work was supported by Posts and Telecommunications Institute of Technology.

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.

Data availability

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

Conflicts of interest

The authors declare that they have no conflict of interest.

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