A Single-Valued Pentagonal Neutrosophic Framework for Optimal Cloud Service Selection: Integrating FUCOM,MEREC and TODIM

Abstract

With the advancement and expansion of cloud computing services, selecting a proper cloud service is regarded as a challenging endeavor, since there is fierce competition in the industry for delivering a quality experience to its customers. Several Multi-Criteria Decision Making (MCDM) procedures have been adopted by decision makers over the past years to identify the best cloud services, however these approaches suffer from rank reversal and inconsistent results. For addressing these challenges, we introduced a framework integrating the Full Consistency Method (FUCOM) and Method based on the removal effects of criteria (MEREC) for weight estimation and TOmada de Decisao Interativa Multicriterio (TODIM) for ranking the alternatives in a pentagonal neutrosophic environment to determine the optimal cloud service. Comparative study and sensitivity analysis are done to check the efficacy and robustness of our model. The results demonstrate that, in comparison to current methods, the suggested model performs better and produces more consistent results.

Keywords

MCDM cloud service selection decision making pentagonal neutrosophic sets integrated weight TODIM

1. Introduction

Cloud computing has been proved to immensely benefit the IT industry in recent years by providing optimized resources and on-the-go services via the internet (Alam, 2022; Mudassar et al., 2022; Zhao & Rabiei, 2023). In today's rapidly changing world, businesses and organizations increasingly rely on cloud services to meet their technological needs. With the assistance of today's modern internet technology, the cloud computing paradigm can provide its consumers flexible and substantial services. The traditional cloud computing paradigm delivers services based on software, platform, and infrastructure, dependent on the needs of the user. These assist users by providing on-demand software applications, facilitating platform and ecosystem to develop applications according to the user's need and providing resources (virtual storage, cloud servers, and automation) for carrying out computational tasks over the internet (Tomar et al., 2023). Since cloud computing has reshaped corporate organizations of all sizes by delivering on-demand services, reduced costs, scalable, and elastic assistance, the majority of businesses are now outsourcing their day-to-day operations to cloud-based computing (Hazra et al., 2022).

With the rapidly changing business landscape, the selection of a Cloud Service Provider (CSP) is crucial for organizations looking to leverage the benefits of remote cloud services. Numerous Companies implemented cloud computing to facilitate latest web-based services while lowering the infrastructure costs. Design and engineering professionals must consider different heterogeneous conditions with criteria having intricate dependencies when selecting cloud services, which appears as an impossible task to accomplish manually, posing a significant challenge. To ensure the best fit for their necessities, consumers need to consider several factors based on the requirements of the QoS (Quality of Service) when selecting a CSP (Hayyolalam & Kazem, 2018). In such cases, it is vital to examine the distinct characteristics that describe the diverse cloud services offered by various CSPs. These factors include pricing models, data security and governance, user feedback and software stack improvement, responsibility for infrastructure management, security and privacy concerns, availability of technical support and customer service, scalability and performance, and integration with existing systems (Siegel & Perdue, 2012). Consumers must carefully evaluate and prioritize these factors to accurately identify the CSP that best aligns with their specific needs and objectives. By doing so, businesses can optimize their resources, improve their operational efficiency, and enhance their overall performance. Therefore, the primary challenge is to identify an efficient and reliable cloud service that meets and exceeds customer expectations while upholding a high standard of accuracy, consistency, and reliability. This makes Cloud Service Selection (CSS) a noteworthy and fascinating research problem. The interaction of various alternatives, QoS criteria, and decision makers’ perspectives makes the CSS problem suitable to consider as an MCDM problem. (Youssef, 2020).

1.1 Limitations

In recent years, one of the most promising fields, known as MCDM, has evolved with efficient and robust decision making models that can tackle real-world challenges. Despite the fact that researchers have used MCDM to efficiently construct a trust factor in selecting the optimum CSP, there are still certain limitations and constraints that need to be addressed.

As a result, several limitations in the existing literatures must be emphasized on:

−
The majority of studies depend heavily on subjective or objective weight evaluation. The decision-makers’ proficiency in determining weights from qualitative and linguistic data is prioritized in the subjective weighting strategy. It is simple to understand, but there is a possibility of biasness toward particular choices, which could lead to ambiguous findings due to personal preferences. The objective weighting strategy, on the other hand, makes decisions based on available facts and mathematical calculations. It does not favor the decision maker's own preferences, yet it may occasionally produce conflicting outcomes in relation to the real data (Deepa et al., 2019).
−
Since there is still unpredictability in group decision making, several decision making models implied techniques to address fuzziness. However, there are still discrepancies because randomness cannot be completely eliminated using fuzzy approaches because, in fuzzy approaches, the decision expert gives his judgment of a linguistic term in membership values only but there may be certain cases where indeterminacy prevails along with membership and non-membership values while giving the judgment.
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The decision making model should put more emphasis on handling erroneous, imprecise, and fake QoS criteria published by CSPs.
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The decision-making model should be adaptable enough to accept a wide range of criteria and cloud alternatives while still producing accurate outcomes.

1.2 Motivation

Considering the above mentioned issues in this domain of CSS and from the review of existing literatures in the following section (Section 2), certain questions arises regarding improvement of service quality for an organization, specifically how randomness can be dealt more efficiently. To what extent can we achieve optimized results? How to achieve consistent weights to determine the relative importance or priority of different criteria which are crucial for decision making. In light of the aforementioned concerns, we thus present a novel method for selecting desired cloud services by integrating weighting techniques which aggregates both subjective as well as objective weight estimation that takes into account both quantitative and qualitative properties. To deal with uncertain and vague data, our approach uses Single-Valued Pentagonal Neutrosophic (SVPN) method which is based on pentagonal neutrosophic sets (Das & Chakraborty, 2021). We obtained the objective weights using MEREC (Keshavarz-Ghorabaee et al., 2021), a modern objective weighing technique that takes into account the removal effects of the criteria. The subjective weights are computed by FUCOM (Pamučar et al., 2018), which is based on pairwise comparisons and requires just n-1 criteria comparisons. For the final ranking of the alternatives, the TODIM (Gomes, 2009) technique is employed, which takes the integrated weights obtained by FUCOM and MEREC, and provides the outcome based on the evaluations. Insofar as we are aware, no other researcher has investigated the benefits associated with combining the MEREC and FUCOM methodologies, hence, this study is the first to apply this integrated appraoch. Moreover, their assessments are then further examined by another MCDM methodology, TODIM, to get the final rankings. Also to our best knowledge no researcher considered SVPN decision making approach in CSS problem.

1.3 Contribution

The prime contribution of our study are as follows:

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A novel hybrid approach for selection of cloud service providers is proposed based on QoS.
−
We utilized the single valued pentagonal neutrosophic numbers to assist in dealing with imprecision and achieving optimal decisions.
−
Our model uses integrated weighting approach to generate the weighting scores for robust results.
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A robust ranking mechanism based on prospect theory is incorporated to generate the final alternative rankings.
−
The consistency of the proposed method is tested with sensitivity analysis and comparing with existing methods.

The remaining paper is structured as follows: In Section 2, the related works are discussed. Preliminaries are mentioned in Section 3. In Section 4 the proposed hybrid model is depicted. A case study along with comparative analysis is presented in Section 5, which is followed by conclusion at Section 6.
2. Related Works

The popularity of cloud computing has skyrocketed as a method of delivering services over the internet. As a subscription service, it facilitates on-demand access to computational resources, allowing customers to dynamically access and use external resources on a subscription basis. Due to its popularity, a lot of research has been done on this domain of selecting cloud services based on QoS (Nath et al., 2022; Monika & Sangwan, 2022; Kumar et al., 2017; Sidhu & Singh, 2017). In (Sun et al., 2013), the authors proposed a method based on Analytic Hierarchy Process (AHP) to take into account user's qualitative preferences and convert them into quantitative values. Their approach targeted multi users having different priorities in the shared cloud services. The authors in (Karim et al., 2013) introduced a novel approach for CSS that is based upon mapping of user's QoS requirements. The mapping is done between the requirements in SaaS layer with the alternatives in the IaaS layer for which they designed a model to address complex QoS specifications. The authors addressed customer satisfaction in (Ding et al., 2017), where they proposed a novel method for rank prediction in personalized CSS. They focused their research on assessing similarity rankings and predicting cloud services while taking into account aspects such as data scarcity and varying customer requirements. A cloud broker based architecture is proposed by the authors of (Nawaz et al., 2018) to tackle cloud selection problem addressing the issue of changing user preferences. Their architecture consists of Markov chain to track the patterns of client's preferences and use these patterns along with Best Worst Method (BWM) to select and rank appropriate services. Their approach displayed better consistency in the results as compared to AHP. The authors of (Jatoth et al., 2019) presented a hybrid MCDM model consisting Grey TOPSIS and AHP to rank cloud services using quantified QoS. Here the AHP technique is used to determine criteria weights which would be fitted to grey TOPIS. The grey numbers are used to address the uncertainties in the CSS. Their model is comprised of 4 entities in cloud: consumer, broker, service repository and service providers. They considered seven CSPs in their case study to find the optimal service using their SELCLOUD model. The authors of (Kumar et al., 2021) proposed a novel framework named CCS-OSSR for selecting optimal cloud services. Their model is integration of BWM and TOPSIS. Implementation of Fuzzy AHP and TOPSIS for predicting ranks of cloud services can be seen in (Kumar et al., 2022). In (Sidhu & Singh, 2019) authors used PROMETHEE for selecting trustworthy cloud based data servers. The method employs the standard parameters for analyzing the selection measure of reliable cloud data servers. The authors of (Wang et al., 2020) utilized Fuzzy TODIM based on alpha level sets to consider Decision Makers behavior in Decision Making process.

An essential component of MCDM is estimating the weights given to the various criteria that are used to evaluate alternatives. These weights have a substantial impact on the final decision outcome. Numerous methods have been devised for calculating the weight of the criteria, which may be subjective, objective or hybrid. The authors of (Popović et al., 2022) designed a model incorporating a novel grey full-consistency method (FUCOM-G) integrated with SWOT analysis. With their approach they tried to optimize the logistic operations and functioning of supply-chain process. In (Demir et al., 2022), to deal with sustainability issues in urban mobility plans, the authors designed a model consisting FUCOM and CoCoSo in fuzzy environment. A combination of FUCOM and ADAM method can be seen in (Andrejić et al., 2023) to address the issue of selecting optimal distribution channel in businesses. Here FUCOM generates the criteria weights and ADAM performs the final rankings. The authors of (Simic et al., 2022) proposed a hybrid framework to enhance group decision-making in PLqROFSs configuration. The framework is a combination of power average operator, Archimedean operator and FUCOM. To address the issue of prioritizing the sustainable policies to mitigate the effects of changing climate in urban transportation, a hybrid model consisting MEREC and MARCOS using type-2 neutrosophic numbers was introduced in (Mishra et al., 2022). To rank global low-carbon tourism strategies using fourteen sustainability indicators considering social, economic, and environmental factors, (Chaurasiya & Jain, 2023) presents an integrated decision-making framework using neutrosophic approach to address conflicting indicators and sustainability concerns. Their model is an integration of MEREC, generalized Dombi operators and MULTIMOORA. (Zamani-Sabzi et al., 2016) describes a hybrid MCDM model that combines integrated weighting methods MEREC and SWARA with a ranking method CORPAS under Pythagorean fuzzy environment. Their goal was to identify the best banking management system for increasing efficiency in the banking sector.

While numerous MCDM techniques have been developed to address the CSS problem, fuzzy techniques and their variations play a vital role to deal uncertainties. In (Sun et al., 2016), a fuzzy ontology based decision making model was developed to map relations between the database objects to relate the services in the cloud. Their model consists of fuzzy AHP to generate the criteria weights and fuzzy TOPSIS to provide the final rankings. A Cloud Service Selection with CSSCI framework was introduced by the authors of (Sun et al., 2019) which aggregates non-linearity mappings among the criteria by using fuzzy estimation in conjunction with the Choquet Integral. To solve the issue of uncertainty in CSP data, the authors of (Gireesha et al., 2020) introduced an approach for CSP selection for choosing trusty CSPs by applying IIVIFS-WASPAS approach. Here they incorporated integrated weight estimation technique and present a novel preference-attitudinal score as well as accuracy function based on decision maker's opinion. An approach to select optimal cloud vendor can be seen in (Krishankumar et al., 2020) where they applied an aggregating operator for aggregating the Intuitionistic Fuzzy (IF) weights in the first stage and in the later stage they perform ranking by extending the VIKOR under the IF environment. The authors of (Otay and Yıldız, 2020) applied Pythagorean fuzzy sets along with AHP and VIKOR to solve the issue of selecting the appropriate cloud computing services. The Pythagorean fuzzy sets are considered an extension of the IF sets. There are CSS researches which have used neutrosophic decision making because it can deal with uncertainties much more efficiently than fuzzy and intuitionistic fuzzy sets. In (Ghorui et al., 2023) a pentagonal intuitionistic fuzzy number based model is developed embedding AHP and TOPSIS for optimal selection of cloud service providers. (Tiwari & Kumar, 2022) investigates a CSS framework in a neutrosophic environment while taking into account both the functional and non-functional needs of cloud customers. Here the authors integrated single valued neutrosophic sets along with modified TOPSIS to choose the suitable service provider. In (Attya et al., 2023) the authors built a model using Modified Generative Adversarial Network and a Neutrosophic MCDM ranking algorithm. A robust modified TOPSIS based method for CSS is developed by (Tiwari et al., 2023) which is a generalized TOPSIS that can process different QoS data including neutrosophic ones. The pentagonal neutrosophic system in (Nath et al., 2023) is an example of a generalization of neutrosophic sets. In order to satisfy the decision maker and arrive at the best solution, the authors in this case used the idea of pentagonal single-valued neutrosophic numbers.

One of the prominent MCDM model TODIM acronym for TOmada de Decisão Interativa e Multicritério is built around the concept of prospect theory which deals with the psychological aspects of the decision maker where it demonstrates each alternative's dominance over others by forming certain multi-attribute functions (Kahneman & Tversky, 2013). Based on existing literatures numerous studies have employed the traditional TODIM approach. Notably, the TODIM method has evolved beyond its original numerical framework to encompass diverse fuzzy sets. Moreover, it has been combined with other decision-making methodologies to enhance its applicability and effectiveness in different contexts. To address MAGDM challenges in stock investment, (Wu et al., 2022) introduces the Intuitionistic Fuzzy CPT-TODIM (IF-CPT-TODIM) method, which leverages IFSs for enhanced decision-making. The authors of (Nguyen et al., 2023) addressed the challenges in portfolio selection concerning financial performance of any firms. Their research proposes a multidimensional financial evaluation index system integrating generalized TODIM with entropy weighting that tailored for stock investment over a long-term period. For selecting the optimal service platforms for cloud databases (Ranjan et al., 2023) proposed an integrated model comprising TODIM and probability uncertain linguistic. TODIM exceeds other methods like MARCOS (Stević et al., 2020), VIKOR (Chiu et al., 2013), and TOPSIS due to its explicit consideration of risk, subjective factors, and behavioral aspects in decision-making. VIKOR and TOPSIS are more susceptible to rank reversal. The flexibility, transparency, and empirical validation of TODIM make it a reliable choice across diverse decision contexts. Thus it has been extended to SVPN setting and implemented in our decision model for criteria ranking.

The literature reviewed above depicted the use of a variety of MCDM methods, each with its advantages and disadvantages. However, the fact remains that strong decision models that can handle ambiguity and inconsistency while yielding improved outcomes are still required. Also, increasing the number of criteria or alternatives complicates the decision-making process, resulting in inconsistent results. Thus, it becomes difficult to revisit all of the comparative results if inconsistent results are encountered. Considering the aforementioned issues, we created a novel integrated cloud service selection model that combines the strengths of both subjective and objective weighting techniques, as well as extending the model to an SVPN setting for the first time in the literature.

3. Preliminaries

This section presents the preliminaries necessary to develop our model.

3.1 Single-Valued Pentagonal Neutrosophic Sets (SVPNS)

Many researchers have used the concept of Single-Valued Neutrosophic Sets (SVNS) in their studies of MCDM because it is far more efficient in dealing with uncertainties and indeterminacy than prior methods (Ali et al., 2022; Başhan et al., 2020; Chai et al., 2021; Luo et al., 2022). Several variations of SVNS have emerged with time which includes variations as triangular, trapezoidal and pentagonal neutrosophic sets (Abdel-Basset et al., 2018; Das & Chakraborty, 2021; Ye, 2015). Some definitions and notions used for neutrosophic sets are as follows:

Definition 1.
(Smarandache, 1999) A neutrosophic set $\tilde{F} \in X$ is defined by:
$\begin{aligned} \tilde{F} = {(x, μ_{\tilde{F}} (x), σ_{\tilde{F}} (x), ϑ_{F} (x)) | x \in X} \end{aligned}$
where X is an universal set and the functions $μ_{\tilde{F}} (x)$ stands for membership, $σ_{\tilde{F}} (x)$ stands for indeterminacy and $ϑ_{\tilde{F}} (x)$ stands for non-membership. The limits are as follows:
$\begin{aligned} μ_{\tilde{F}} (x), σ_{\tilde{F}} (x), ϑ_{\tilde{F}} (x) : X \to] 0^{-}, 1^{+} [ \end{aligned}$
such that $0^{-} \leq μ_{\tilde{F}} (x) + σ_{\tilde{F}} (x) + ϑ_{\tilde{F}} (x) \leq 3^{+}$
Definition 2.
(Wang et al., 2010) A single-valued neutrosophic set $\tilde{F} \in X$ is defined by:
$\begin{aligned} \tilde{P} = {(x, μ_{\tilde{F}} (x), σ_{\tilde{F}} (x), ϑ_{F} (x)) | x \in X} \end{aligned}$
where X is an universal set and the functions $μ_{\tilde{F}} (x)$ stands for membership, $σ_{\tilde{F}} (x)$ stands for indeterminacy and $ϑ_{\tilde{F}} (x)$ stands for non- membership. The limits are as follows:
$\begin{aligned} μ_{\tilde{F}} (x), σ_{\tilde{F}} (x), ϑ_{\tilde{F}} (x) : X \to [0, 1] \end{aligned}$
such that $0 \leq μ_{\tilde{F}} (x) + σ_{\tilde{F}} (x) + ϑ_{\tilde{F}} (x) \leq 3$
Definition 3.
(Chakraborty et al., 2019) Let $j = [(j_{1}, j_{2}, j_{3}, j_{4}, j_{5}); μ_{\tilde{j}}]$ , $j * = [(j_{1} , j_{2} , j_{3} , j_{4} , j_{5} ); σ_{\tilde{j}}]$ and $j^{+} = [(j_{1}^{+}, j_{2}^{+}, j_{3}^{+}, j_{4}^{+}, j_{5}^{+}); ϑ_{\tilde{j}}]$ , then a SVPN number be defined as, such that $\tilde{j} \in R$ , where $R$ is a set of real numbers and $μ_{\tilde{j}}, σ_{\tilde{j}}, ϑ_{\tilde{j}} \in [0, 1]$ . Then the membership function $μ_{\tilde{j}} (x) : R \to [0, μ]$ , indeterminacy function $σ_{\tilde{j}} (x) : R \to [σ, 1]$ and non-membership function $ϑ_{\tilde{j}} (x) : R \to [ϑ, 1]$ of $\bar{j}$ is given by:

$\begin{aligned} μ_{\tilde{j}} (x) & = {\begin{array}{lc} μ_{\tilde{j l 1}} (x) & j_{1} \leq x \leq j_{2} \\ μ_{\tilde{j l 2}} (x) & j_{2} \leq x \leq j_{3} \\ μ & j_{3} \\ μ_{\tilde{j u 1}} (x) & j_{3} \leq x \leq j_{4} \\ μ_{\tilde{j u 2}} (x) & j_{4} \leq x \leq j_{5} \\ 0 & o t h e r w i s e \end{array} \end{aligned}$

$\begin{aligned} σ_{\tilde{j }} (x) & = {\begin{array}{lc} σ_{\tilde{j * l 1}} (x) & j_{1} * \leq x \leq j_{2} * \\ σ_{\tilde{j * l 2}} (x) & j_{2} * \leq x \leq j \\ σ & j_{3} * \\ σ_{\tilde{j * u 1}} (x) & j_{3} * \leq x \leq j_{4} * \\ σ_{\tilde{j * u 2}} (x) & j_{4} * \leq x \leq j_{5} * \\ 0 & o t h e r w i s e \end{array} \end{aligned}$

$\begin{aligned} ϑ_{\tilde{j^{+}}} (x) & = {\begin{array}{lc} ϑ_{\tilde{j^{+} l 1}} (x) & j_{1}^{+} \leq x \leq j_{2}^{+} \\ ϑ_{\tilde{j^{+} l 2}} (x) & j_{2}^{+} \leq x \leq j_{3}^{+} \\ ϑ & j_{3}^{+} \\ ϑ_{\tilde{j^{+} u 1}} (x) & j_{3}^{+} \leq x \leq j_{4}^{+} \\ ϑ_{\tilde{j^{+} u 2}} (x) & j_{4}^{+} \leq x \leq j_{5}^{+} \\ 0 & o t h e r w i s e \end{array} \end{aligned}$

Figure 1. represents linear pentagonal neutrosophic number where the black line represents membership function and, blue and red lines represents the indeterminacy and non-membership functions respectively. Here, the range of $τ$ is between $0 \leq τ \leq 1$ and if the value of $τ$ is 0 or 1 then the pentagonal number will convert to a triangular neutrosophic number.

Figure 1.
A Linear Pentagonal Neutrosophic Representation (Chakraborty et al., 2019).

The representation of the linguistic terms to rate the alternatives based on criteria can be observed in Table 1.

Table 1.
Linguistic Expression for Pentagonal Neutrosophic Numbers.

Linguistic Expressions Pentagonal Neutrosophic Numbers

Very Low (VL) (0,0.1,0.2,0.3,0.4; $μ, σ, ϑ$ )

Low (L) (0.1,0.2,0.3,0.4,0.5; $μ, σ, ϑ$ )

Medium Low (ML) (0.2,0.3,0.4,0.5,0.6; $μ, σ, ϑ$ )

Fair (F) (0.3,0.4,0.5,0.6,0.7; $μ, σ, ϑ$ )

Medium High (MH) (0.4,0.5,0.6,0.7,0.8; $μ, σ, ϑ$ )

High (H) (0.5,0.6,0.7,0.8,0.9; $μ, σ, ϑ$ )

Very High (VH) (0.6,0.7,0.8,0.9,1; $μ, σ, ϑ$ )

3.2 Generating Crisp Values from SVPN Number

Linguistic Expressions	Pentagonal Neutrosophic Numbers
Very Low (VL)	(0,0.1,0.2,0.3,0.4; $μ, σ, ϑ$ )
Low (L)	(0.1,0.2,0.3,0.4,0.5; $μ, σ, ϑ$ )
Medium Low (ML)	(0.2,0.3,0.4,0.5,0.6; $μ, σ, ϑ$ )
Fair (F)	(0.3,0.4,0.5,0.6,0.7; $μ, σ, ϑ$ )
Medium High (MH)	(0.4,0.5,0.6,0.7,0.8; $μ, σ, ϑ$ )
High (H)	(0.5,0.6,0.7,0.8,0.9; $μ, σ, ϑ$ )
Very High (VH)	(0.6,0.7,0.8,0.9,1; $μ, σ, ϑ$ )

For converting neutrosophic numbers into definite values the score and accuracy function is implied. From (Krishankumar et al., 2020) the score and accuracy function generation approach is considered to transform SVPN number ${\tilde{k}}_{P N} = (k_{1}, k_{2}, k_{3}, k_{4}, k_{5}; μ, σ, ϑ)$ which is defined as follows:

− Score function: The score function for ${\tilde{k}}_{P N}$ is scaled as

\begin{aligned} {\tilde{S c}}_{P N} = \frac{1}{15} (k_{1} + k_{2} + k_{3} + k_{4} + k_{5}) x {2 + μ - σ - ϑ} \end{aligned}

(1)

− Accuracy function: The accuracy function is given as

\begin{aligned} {\tilde{A c}}_{P N} = \frac{1}{15} (k_{1} + k_{2} + k_{3} + k_{4} + k_{5}) x {2 + μ - σ} \end{aligned}

(2)

3.3 SVPN-FUCOM for Subjective Weight Estimation

In 2018, Pamu $\overset{˘}{c}$ ar et al. (Pamučar et al., 2018) presented the new MCDM subjective model FUCOM. It validates the results by performing a Deviation from Maximum Consistency (DMC) and is predicated on the idea of pairwise comparison of the criteria. With only a few binary comparisons needed (n-1) to estimate the weights, as compared to methods like AHP and BWM which require n(n-1)/2 and 2n-3 pairwise comparisons respectively, FUCOM is a far simpler and feasible approach than the current methods for defining criteria weight (Pamucar et al., 2021). Also comparing to other subjective methods like LBWA (Žižović & Pamucar, 2019) and DIBR (Tešić et al., 2022), FUCOM is much more adaptable to complex decision environments while being transparent and providing interpretable results, and for this reason it's a popular weight estimating approach considered by many researchers. This study expands the classical FUCOM in SVPN setting.

The algorithmic steps of SVPN-FUCOM are listed below:

Step 1: Establishing decision criteria. Assume there are n (j = 1,2,..,n) criteria for evaluation denoted by the set ${\bar{C}}_{j} = {C_{1}, C_{2}, \dots, C_{n}} .$

Step 2: Ranking of decision criteria. The criteria are ranked by the DMs based upon their preference of importance. The criterion with highest weight coefficient is assigned the first rank and so on till the criterion with the least weight coefficient receives the last rank. The criteria ranking obtained is ${\bar{C}}_{j (i)} > {\bar{C}}_{j (2)} > \dots > {\bar{C}}_{j (q)}$ where q is the observed criterion rank. The equality operator “=” is used instead of “>” if two or more criteria are the same.

Step 3: Criteria comparisons with SVPNs. Using the linguistic expressions from Table 1, all the criteria are compared with one another with respect to the criteria having first rank (Very High). For all of the criteria given in Step 2, SVPN criteria significance ( ${\bar{w}}_{C_{j (q)}}$ ) is acquired. As the first criterion is self-compared ( ${\bar{w}}_{C_{j (q)}} = E S$ ), the remaining n-1 criteria must be compared. Therefore, the pentagonal neutrosophic comparative significance $({\bar{φ}}_{q / (q + 1)})$ is accomplished by using Eq. (3).

\begin{aligned} {\bar{φ}}_{q / (q + 1)} = \frac{{\bar{w}}_{C_{j (q + 1)}}}{{\bar{w}}_{C_{j (q)}}} = \frac{(w_{C_{j (q + 1)}}^{L}, w_{C_{j (q + 1)}}^{L M}, w_{C_{j (q + 1)}}^{M}, w_{C_{j (q + 1)}}^{U M}, w_{C_{j (q + 1)}}^{U})}{(w_{C_{j (q)}}^{L}, w_{C_{j (q)}}^{L M}, w_{C_{j (q)}}^{M}, w_{C_{j (q)}}^{U M}, w_{C_{j (q)}}^{U})} \end{aligned}

(3)

Thus, utilizing Eq. (4), a SVPN vector with the comparative importance of evaluation criteria is achieved.

\begin{aligned} \bar{Φ} = ({\bar{φ}}_{1 / 2}, {\bar{φ}}_{2 / 3}, \dots, {\bar{φ}}_{q / (q + 1)}) \end{aligned}

(4)

where

{\bar{φ}}_{q / (q + 1)}

signifies the importance of

C_{j (q)}

rank with respect to

C_{j (q + 1)}

rank.

Step 4. Estimation of Optimal weights. Here the final SVPN criteria weight coefficients are generated $({\bar{w}}_{1}, {\bar{w}}_{2}, \dots, {\bar{w}}_{n})^{T}$ . The final weight coefficient values should satisfy the following two conditions:

\begin{aligned} \frac{{\bar{w}}_{q}}{{\bar{w}}_{q + 1}} & = {\bar{φ}}_{q / (q + 1)} \end{aligned}

(5)

\begin{aligned} \frac{{\bar{w}}_{q}}{{\bar{w}}_{q + 2}} & = {\bar{φ}}_{q / (q + 1)} \otimes {\bar{φ}}_{q / (q + 1) (q + 2)} \end{aligned}

(6)

To fulfill the conditions, the weight coefficients $({\bar{w}}_{1}, {\bar{w}}_{2}, \dots, {\bar{w}}_{n})^{T}$ must satisfy $| \frac{{\bar{w}}_{q}}{{\bar{w}}_{q + 1}} = {\bar{φ}}_{q / (q + 1)} | \leq χ$ and $| \frac{{\bar{w}}_{q}}{{\bar{w}}_{q + 2}} = {\bar{φ}}_{q / (q + 1)} \otimes {\bar{φ}}_{q / (q + 1) (q + 2)} | \leq χ$ states to minimize the value of $χ$ . With the defined requirements, Eq. (7) can be used to generate the optimal SVPN values of the weight coefficients $({\bar{w}}_{1}, {\bar{w}}_{2}, \dots, {\bar{w}}_{n})^{T}$ of the evaluation criteria.

\begin{aligned} min χ \end{aligned}

s.t.

\begin{aligned} {\begin{array}{ll} | \frac{{\bar{w}}_{q}}{{\bar{w}}_{q + 1}} = {\bar{φ}}_{q / (q + 1)} | \leq χ, & \forall_{j} \\ | \frac{{\bar{w}}_{q}}{{\bar{w}}_{q + 2}} = {\bar{φ}}_{q / (q + 1)} \otimes {\bar{φ}}_{q / (q + 1) (q + 2)} | \leq χ, & \forall_{j} \\ \sum_{j = 1}^{n} w_{j} = 1, & \forall_{j}, \\ w_{j}^{L} \leq w_{j}^{L M} \leq w_{j}^{M} \leq w_{j}^{U M} \leq w_{j}^{U}, \\ w_{j}^{L} \geq 0, & \forall_{j} \\ j = 1, 2, \dots, n \end{array} \end{aligned}

(7)

where

{\bar{w}}_{j} = w_{j}^{L}, w_{j}^{L M}, w_{j}^{M}, w_{j}^{U M}, w_{j}^{U}

and

{\bar{φ}}_{q / (q + 1)} = φ_{q / (q + 1)}^{L}, φ_{q / (q + 1)}^{L M}, φ_{q / (q + 1),}^{M} φ_{q / (q + 1)}^{U M}, φ_{q / (q + 1)}^{U}

3.4 SVPN-MEREC for Objective Weight Estimation

The MEREC approach, developed by M. Keshavarz-Ghorabaee et al. (Keshavarz-Ghorabaee et al., 2021), considers removal of effects mechanism for obtaining objective weights. Compared to methods like WENSLO (Pamucar et al., 2023) and LOPCOW (Ecer & Pamucar, 2022), its more efficient in incorporating multiple expert perspectives, foster consensus and ensure comprehensive consideration of diverse viewpoints. It is transparent and much more adaptable in case of addition of criteria or alternatives. We expand the classical MEREC approach with SVPN setting. Here let m denotes the number of cloud services (alternative), n represents the criteria of cloud services and k be the decision makers. As the study considers group decision making for better judgment, let us consider $a = {a_{1}, a_{2}, \dots, a_{m}}$ , $c = {c_{1}, c_{2}, \dots, c_{n}}$ and $e = {e_{1}, e_{2}, \dots, e_{k}}$ be finite sets of alternatives, criteria and decision makers respectively. The procedural steps for SVPN-MEREC are as follows:

Step 1: Constructing the initial SVPN decision matrices $\tilde{M} = [{\tilde{f}}_{i j}]_{m x n}$ . Table 1 provides a pentagonal neutrosophic linguistic scale for defining the decision matrices at initial stage. The initial decision matrix is given as:

Step 2: Performing normalization of the SVPN decision matrix denoted by ${\tilde{V}}_{i j}$ which is required to maintain the balance between the magnitude and dimensions of the attributes. It is a necessary step to maintain an equilibrium between the beneficial and non-beneficial criteria. Let $c^{+}$ and $c^{-}$ be the sets representing benefit and non-benefit criteria respectively, then the following equation can be used for normalization:

\begin{aligned} {\tilde{V}}_{i j} = {\begin{cases} \frac{m i n_{k} [{\tilde{S c}}_{P N} ({\tilde{f}}_{i j})]}{{\tilde{S c}}_{P N} ({\tilde{f}}_{i j})}, & j \in c^{+} \\ \frac{{\tilde{S c}}_{P N} ({\tilde{f}}_{i j})}{m a x_{k} [{\tilde{S c}}_{P N} ({\tilde{f}}_{i j})]}, & j \in c^{-} \end{cases}, i, j = 1, \dots, m; 1, \dots, n \end{aligned}

(9)

Where ${\tilde{V}}_{i j}$ denotes the normalized decision evaluation for the alternatives with respect to criteria.

Step 3: Calculating the overall performance of the alternatives using a logarithmic non-linear function where the minimum value of ${\tilde{V}}_{i j}$ signifies better results in performance. This performance measure $Q_{i}$ can be calculated as:

\begin{aligned} Q_{i} = \ln (1 + (\frac{1}{n} \sum_{j} | \ln ({\tilde{V}}_{i j}) |)), i = 1, \dots, m \end{aligned}

(10)

Step 4: Determining performance of individual alternative by removing criteria of each alternative separately. Let $Q_{i j}^{'}$ denote the performance measure of k^th alternative by removal of j^th criteria.

\begin{aligned} Q_{i j}^{'} = \ln (1 + (\frac{1}{n} \sum_{k, k \neq j} | \ln ({\tilde{V}}_{i j}) |)), i = 1, \dots, m \end{aligned}

(11)

Step 5: Computing the aggregation of the absolute deviations where the criteria's removal effect is calculated based on the values obtained from previous two steps. The value of $G_{j}$ is computed as:

\begin{aligned} G_{j} = \sum_{i = 1}^{m} | Q_{i j}^{'} - Q_{i} | \end{aligned}

(12)

Step 6: Calculating the final weights using the formula:

\begin{aligned} w_{j} = \frac{G_{j}}{\sum_{j} G_{j}}, j = 1, \dots, n \end{aligned}

(13)

4. Proposed Framework

The CSS framework proposed is a hybrid and robust approach to select optimal cloud services among the set of services in a peantagonal neutrosophic environment. The algorithmic process is divided into two phases. In the first phase, based on the decisions from a group of decision makers, the subjective and objective weights are calculated which are further integrated to obtain the final weights. Then these weights are further utilized by SVPN-TODIM to generate the final ranking of the cloud services. The flow diagram of the proposed model is depicted in Figure 2.

Figure 2.

Schematic representation of the proposed method.

The steps for the proposed method is given below:

Phase 1: Creation of initial decision matrix and criteria weight estimation.

Step 1. The initial step is to input data. This step is further divided as: a)

Based on selection of certain CSPs, a group of decision makers are requested to provide their opinion. They provide subjective opinions in the form of ranking or scores as response. Additionally, they also offer the alternatives’ objective criteria values. Let there be a set consisting of p decision makers as $d m = {d m_{1}, d m_{2}, \dots, d m_{p}}$ , q alternatives as $a l = {a l_{1}, a l_{2}, \dots, a l_{q}}$ and s criteria as $c r = {c r_{1}, c r_{2}, \dots, c r_{s}}$ with (p,q,s) $\geq 2$ .

The decision makers are provided a series of questionnaires that consists of ranking of alternatives from 1 to n and a 7-point linguistic conversion table to set the linguistic value ranging from Very Low to Very High.

Each decision maker lays down their opinions in the form of subjective and objective decision matrices similar to (7) where they provide their decisions for each alternatives with respect to the criteria. All the matrices of the decision makers are then averaged to obtain the final decision matrix.

Step 2. Calculation of subjective and objective weights using SVPN-FUCOM and SVPN-MEREC respectively and integrating them using the following formula:

\begin{aligned} I = δ S + (1 - δ) O \end{aligned}

(14)

where S and O stands for subjective and objective weights respectively and value of

δ

is in the range 0 to 1.

Phase 2: Providing ranks to alternative by taking the weights generated at Phase 1.

Step 3. Here the TODIM method is used to provide the final rankings of the cloud services. Here we extend the traditional TODIM into SVPN-TODIM. Suppose $a = {a_{1}, a_{2}, \dots, a_{m}}$ be a finite set of alternatives and $c = {c_{1}, c_{2}, \dots, c_{n}}$ be a finite set of criteria. Let $I = {l_{1}, l_{2}, \dots, l_{n}}$ be a finite set of the integrated weight of the attributes obtained from the previous phase where $I_{k} \in [0, 1]$ and $\sum_{k = 1}^{n} I_{k} = 1$ . $\tilde{Q} = (q_{i j})_{m x\; n} = (k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, μ_{i j,} σ_{i j,} ϑ_{i j,})_{m x\; n}$ be a SVPN matrix provided by the decision maker where $[< μ_{i j,} σ_{i j,} ϑ_{i j} > \in 0, 1]$ , $0 \leq μ_{i j} + σ_{i j} + ϑ_{i j} \leq 3$ and $i = 1, \dots, m, j = 1, \dots, n$ .

Step 4. Construct the normalized decision matrix $\tilde{Q^{N}} = (q_{i j}^{N})_{m x\; n}$ by applying the following normalization rule:

\begin{aligned} \tilde{Q^{N}} = {\begin{array}{ll} q_{i j}, & i f c_{j}^{+} \\ n e g (q_{i j}), & i f c_{j}^{-} \end{array} \end{aligned}

(15)

where

c_{j}^{+}

represents the benefit criteria and

c_{j}^{-}

represents the cost criteria

Step 5. Next the relative weight of individual criteria $c_{j}$ are calculated as:

\begin{aligned} W_{J R} = \frac{W_{J}}{W_{R}}; J, R = 1, \dots, n \end{aligned}

(16)

where the weight of each criteria denoted as

W_{J}

is divided by the highest weight

W_{R} = m a x {W_{J} | J = 1, \dots, n}

;

0 \leq W_{J R} \leq 1.

Step 6. With the help of eq. (15), the dominance score of alternative $a_{i}$ over individual alternative $a_{k}$ based on criteria $c_{j}$ can be evaluated with the following equation:

\begin{aligned} \emptyset_{j} (a_{i}, a_{k}) = {\begin{array}{ll} \sqrt{\frac{(q_{i j} - q_{k j}) W_{J R}}{\sum_{j = 1}^{n} W_{J R}}} & f o r (q_{i j} - q_{k j}) > 0 \\ 0 & f o r (q_{i j} - q_{k j}) = 0 \\ - \frac{1}{θ} \sqrt{\frac{(q_{i j} - q_{k j}) (\sum_{j = 1}^{n} W_{J R})}{W_{J R}}}, & f o r (q_{i j} - q_{k j}) < 0 \end{array} \end{aligned}

(17)

Step 7. Determine each alternative's degree of dominance ( $a_{i}$ ) over the remaining alternatives $a_{k}$ (k = 1,…,n) using:

\begin{aligned} \emptyset_{j} (a_{i}, a_{k}) = \sum_{j = 1}^{n} \emptyset_{j} (a_{i}, a_{k}) \end{aligned}

(18)

Step 8. Thus the overall dominance degree of individual $a_{i}$ can be computed using the following expression:

\begin{aligned} φ (a_{i}) = \frac{\sum_{p = 1}^{n} \emptyset_{j} (a_{i}, a_{k}) - m i n_{i} {\sum_{p = 1}^{n} \emptyset_{j} (a_{i}, a_{k})}}{m a x_{i} {\sum_{p = 1}^{n} \emptyset_{j} (a_{i}, a_{k})} - m i n_{i} {\sum_{p = 1}^{n} \emptyset_{j} (a_{i}, a_{k})}} \end{aligned}

(19)

The more $φ (a_{i})$ indicates better $a_{i}$ in the above expression.

5. Case Study

To validate our model, we performed an experiment with a real cloud dataset to select the best cloud services and evaluate the efficiency of the proposed method. Sensitivity analysis, performance analysis and ranking analysis are also performed to test the model's robustness and validate it against other models.

5.1 Experimental Analysis for Ranking Cloud Services

For this experiment, a real cloud service dataset QWS (Al-Masri & Mahmoud, 2008) is used with LINGO 18.0 and MATLAB 2020b software to implement the proposed approach. We considered six QoS parameters (Response Time (CR1), Availability (CR2), Throughput (CR3), Reliability (CR4), Compliance (CR5), and Latency (CR6), as well as five service providers (interop2 (A1), skynode (A2), IP2Geo (A3), MyAmazonSecure (A4), and GoogleSearchService (A5)). Here the beneficial criteria are Availability, Throughput, Reliability and Compliance whereas non-beneficial criteria are response time and latency. Furthermore, no sub-criteria were evaluated in our case study. Table 2 elaborates the parameters of QoS with their definition.

Table 2.
Descriptions of QoS Parameters.

S.No. QoS parameters Definitions Effect

1 Response Time The duration between submitting a request and receiving a response Non-Beneficial

2 Availability Ratio of successful to total invocations Beneficial

3 Throughput Total Invocations for a specific Time Frame Beneficial

4 Reliability Ratio of error messages to all messages received Beneficial

5 Compliance The degree to which a document conforms to the WSDL specification Beneficial

6 Latency Duration of time taken by server to handle a specific request Non-beneficial

S.No.	QoS parameters	Definitions	Effect
1	Response Time	The duration between submitting a request and receiving a response	Non-Beneficial
2	Availability	Ratio of successful to total invocations	Beneficial
3	Throughput	Total Invocations for a specific Time Frame	Beneficial
4	Reliability	Ratio of error messages to all messages received	Beneficial
5	Compliance	The degree to which a document conforms to the WSDL specification	Beneficial
6	Latency	Duration of time taken by server to handle a specific request	Non-beneficial

For better understanding, the steps involved in ranking cloud services using the SVPN algorithm are explained below along with numerical calculations:

Step 1. Formation of initial decision matrix with QoS criteria

Table 3. exhibits the initial decision matrix of the service providers in relation to the criteria.

Step 2. Converting the initial decision matrix into linguistic terms

Table 3.

Initial Decision Matrix of CSP with QoS Criteria.

CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	107.57	80	1.7	67	78	18.21
Skynode	80.55	94	13.6	60	89	26.19
IP2Geo	130.25	79	2.4	73	78	93.75
MyAmazonSecure	107.57	89	2.2	73	100	5.71
GoogleSearch Service	172.67	87	6.8	73	78	14

The decision matrix is created by the cloud specialists using their expertise. Expert ratings on linguistic terms are used to build the decision matrix. We considered a group decision making approach having four decision experts to provide their ratings displayed in Table 4.

Step 3. Converting the decision matrix to SVPN set

Table 4.

Linguistic Term Based Decision Matrix.

DM1
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	F	F	H	VH	ML	F
Skynode	VL	MH	VH	H	MH	VL
IP2Geo	L	VH	H	MH	VH	F
MyAmazonSecure	L	H	VH	MH	MH	L
GoogleSearch Service	VL	H	VH	VH	F	VL
DM2
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	ML	VH	H	VH	H	ML
Skynode	VL	F	VH	MH	MH	ML
IP2Geo	VL	H	VH	MH	VH	F
MyAmazonSecure	L	H	MH	H	F	F
GoogleSearch Service	L	MH	VH	F	F	VL
DM3
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	L	F	VH	VH	MH	F
Skynode	F	MH	MH	H	VH	L
IP2Geo	F	MH	H	H	VH	F
MyAmazonSecure	VL	H	F	H	F	VL
GoogleSearch Service	F	MH	H	H	MH	F
DM4
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	VL	MH	MH	VH	H	VL
Skynode	F	VH	H	H	VH	F
IP2Geo	L	MH	H	VH	MH	F
MyAmazonSecure	VL	H	VH	H	F	L
GoogleSearch Service	L	VH	MH	MH	MH	F

By translating each linguistic term in the decision matrix into a SVPN value, a mapping function that is displayed in Table 1. is used to compute the neutrosophic set decision matrix. Every linguistic term is substituted with its matching SVPN value by the mapping function. In our experiment the value of $μ_{\tilde{F}} (x)$ is 1 and, value of both $σ_{\tilde{F}} (x) and ϑ_{\tilde{F}} (x)$ are 0. The converted linguistic data can be seen in Table 5.

Table 5.

Single-Valued Pentagonal Neutrosophic Decision Matrix.

DM 1
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	(0.3,0.4,0.5,0.6,0.7;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.2,0.3,0.4,0.5,0.6;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)
Skynode	(0,0.1,0.2,0.3,0.4;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0,0.1,0.2,0.3,0.4;1,0,0)
IP2Geo	(0.1,0.2,0.3,0.4,0.5;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)
MyAmazon Secure	(0.1,0.2,0.3,0.4,0.5;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.1,0.2,0.3,0.4,0.5;1,0,0)
GoogleSearch Service	(0,0.1,0.2,0.3,0.4;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)	(0,0.1,0.2,0.3,0.4;1,0,0)
................
DM4
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	(0,0.1,0.2,0.3,0.4;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0,0.1,0.2,0.3,0.4;1,0,0)
Skynode	(0.3,0.4,0.5,0.6,0.7;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.6,0.7,0.8,0.,1;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)
IP2Geo	(0.1,0.2,0.3,0.4,0.5;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)
MyAmazon Secure	(0,0.1,0.2,0.3,0.4;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.5,0.6,0.7,0.8,0.9;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)	(0.1,0.2,0.3,0.4,0.5;1,0,0)
GoogleSearch Service	(0.1,0.2,0.3,0.4,0.5;1,0,0	(0.6,0.7,0.8,0.9,1;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.4,0.5,0.6,0.7,0.8;1,0,0)	(0.3,0.4,0.5,0.6,0.7;1,0,0)

The values in Table 5 are converted into crisp values using the score function for each DM, and the average of the results is calculated to create a decision table, as shown in Table 6. In the case of Interop2 for CR1, the SVPN number is converted into a crisp value for DM1 as shown below:

\begin{aligned} {\tilde{S c}}_{P N} = \frac{1}{15} (0.3 + 0.4 + 0.5 + 0.6 + 0.7) x {2 + 1 - 0 - 0} = 0.500 \end{aligned}

Table 6.

Normalized SVPN Decision Matrix.

CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	0.350	0.600	0.700	0.800	0.600	0.400
Skynode	0.350	0.625	0.725	0.675	0.700	0.350
IP2Geo	0.325	0.675	0.725	0.675	0.750	0.500
MyAmazonSecure	0.250	0.700	0.675	0.675	0.525	0.325
GoogleSearchService	0.325	0.675	0.725	0.650	0.550	0.350

Similarly ${\tilde{S c}}_{P N}$ for DM2, DM3 and DM4 are 0.400, 0.300 and 0.200 respectively.

The average of all the values is 0.350

Step 4. Determination of Integrated weights

This step involves calculating the subjective and objective weights, which are subsequently transformed into the final optimal weights by applying Eq. (14). Table 7. shows the integrated final weights. It is to be ensured that the summation of weights should be equal to 1 in TODIM. To calculate integrated weight for CR1 we can write:

\begin{aligned} I = 0.5 * 0.143 + (1 - 0.5) * 0.130 = 0.136 \end{aligned}

Table 7.

Final Integrated Weights Against the Attributes.

Weights	CR1	CR2	CR3	CR4	CR5	CR6
SVPN-FUCOM	0.143	0.132	0.191	0.131	0.177	0.231
SVPN-MEREC	0.130	0.115	0.068	0.088	0.226	0.373
Integrated-Weight	0.136	0.124	0.129	0.109	0.202	0.302

Similarly, all the other weights were calculated

Step 5. Normalization of SVPN decision matrix and calculation of relative weights.

The normalization process is achieved based on beneficial and non-beneficial criteria by following Eq. (15) which is represented in Table 8. After that the relative weights are generated using Eq. (16) which can be seen in Table 9.

Step 6. Computing the dominance degree of alternative

Table 8.

Normalized SVPN Decision Matrix.

CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	0.180	0.183	0.197	0.230	0.192	0.188
Skynode	0.180	0.191	0.204	0.194	0.224	0.215
IP2Geo	0.194	0.206	0.204	0.194	0.240	0.150
MyAmazonSecure	0.252	0.214	0.190	0.194	0.168	0.232
GoogleSearchService	0.194	0.206	0.204	0.187	0.176	0.215

Table 9.

Relative Weights of Individual Criteria.

CSP	CR1	CR2	CR3	CR4	CR5	CR6
Weight	0.278	0.407	0.525	0.358	0.670	1.000

The dominance degree of each alternative can be constructed using the Eq. (17) which is represented in Table 10.

Step 7. Finding each alternative's degree of dominance over the remaining alternatives

Table 10.

Dominance Degree of Alternatives.

CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	0	0	0	0	0	0
Skynode	−0.286	−0.246	0.000	0.063	−0.393	−0.295
IP2Geo	−0.286	−0.427	−0.292	0.063	−0.482	0.108
MyAmazon Secure	0.025	−0.493	−0.667	0.063	0.070	−0.375
GoogleSearch Service	−0.286	−0.427	−0.292	0.069	0.058	−0.295
...............
CSP	CR1	CR2	CR3	CR4	CR5	CR6
Interop2	0.039	0.061	0.030	−0.625	−0.278	0.091
Skynode	0.039	0.050	0.000	−0.255	−0.482	0.000
IP2Geo	0	0	0	−0.255	−0.556	0.141
MyAmazon Secure	−0.599	−0.246	−0.405	−0.255	−0.197	−0.231
GoogleSearch Service	0	0	0	0	0	0

The dominance degree ( $a_{i}$ ) of each alternative with respect to other alternatives $a_{k}$ is calculated using Eq. (18), as shown in Table 11.

Step 8. Computing the overall dominance degree

Table 11.

Dominance Degree ( $a_{i}$ ) of Each Alternative with Respect to Other Alternatives ( $a_{k}$ ).

CSP	A1	A2	A3	A4	A5
Interop2	0	−0.333	−0.698	−0.863	−0.688
Skynode	−1.080	0	−0.321	−0.611	−0.645
IP2Geo	−1.348	−0.887	0	−0.625	−0.670
MyAmazon Secure	−1.616	−1.419	−1.505	0	−1.476
GoogleSearch Service	−1.205	−0.622	−0.314	−2.389	0

The final overall dominance degree can be calculated using equation (19):

\begin{aligned} φ (A 1) = 0 φ (A 2) = 0.695, φ (A 3) = 0.843, φ (A 4) = 1, φ (A 5) = 0.619 \end{aligned}

Thus, the final CSP ranking can be specified as $A 4 > A 3 > A 2 > A 5 > A 1$ where $A 4$ (MyAmazonSecure) is considered the most efficient Alternative.

5.2 Sensitivity and Ranking Analysis

For proper evaluation of the proposed method, sensitivity analysis is performed to test the efficacy of the model in various situations. This is necessary for tracking the model's behavior to see if it can maintain the optimal result if certain alternatives are added, removed, or criteria are swapped, as this may result in a rank reversal problem. An optimal ranking criteria can become non-optimal in a rank reversal problem if certain alternatives are added or removed. In this case, we performed sensitivity analysis in the following ways:

5.2.1 Case 1(By removing alternative)

To begin, a sensitivity analysis is carried out by removing a cloud service from the existing cloud service database. Five experiments are run to test the framework's consistency after a service is removed. Interop2 is removed from the first experiment, and the rankings of the remaining services are calculated. The dominance degree of each alternative is calculated in every experiment that is shown in table 12. The ranks of the services can be observed in Figure 3. It is to observe in the figure that MyAmazonSecure remains the best service until it is removed. Also, ranks of the remaining services remains the same in each experiment until the optimal service changes. Thus it concludes that the proposed method is consistent against rank reversal on removal of alternatives.

Table 12.
Dominance Degrees of Services for Different Experiments for Case 1.

CSP Exp1 Exp2 Exp3 Exp4 Exp5

Interop2 0 0 0 0 0

Skynode 0.253 0 0.732 0.867 0.950

IP2Geo 0.727 0.654 0 1.000 1.000

MyAmazon Secure 1.000 1.000 1.000 0 0.955

GoogleSearch Service 0 0.501 0.515 0.733 0

CSP	Exp1	Exp2	Exp3	Exp4	Exp5
Interop2	0	0	0	0	0
Skynode	0.253	0	0.732	0.867	0.950
IP2Geo	0.727	0.654	0	1.000	1.000
MyAmazon Secure	1.000	1.000	1.000	0	0.955
GoogleSearch Service	0	0.501	0.515	0.733	0

Figure 3.

Ranks of services at different experiments in case 1.

5.2.2 Case 2 (by Adding Alternative)

Here the sensitivity analysis is carried out by adding cloud service to the initial cloud service database. To begin with, Interop2 and Skynode were the initial two services that were considered while performing the first experiment. Later on other services were added in the subsequent experiments. Table 13. shows the dominance degree and Figure 4. displays ranks of the services for every experiment. From Table 13. we get the idea that the dominance degree of the best cloud service remains consistent till a new optimal service is added to the cloud database. Similarly, the same observations of the service ranking can be seen in Figure 4. Hence its proved that the proposed method is again safe from rank reversal when new cloud alternative is added.

Figure 4.

Ranks of Services at Different Experiments in Case 2.

Table 13.

Dominance Degrees of Services for Different Experiments for Case 2.

CSP	Exp1	Exp2	Exp3	Exp4
Interop2	0.000	0.000	0.000	0.000
Skynode	1.000	0.877	0.716	0.695
IP2Geo		1.000	0.813	0.843
MyAmazonSecure			1.000	1.000
GoogleSearchService				0.619

5.2.3 Case 3 (by Switching Criteria)

We changed the level of weights based on the QoS attribute to see how the final results changed. If the ranking of CSPs appears unaltered in the majority of cases, the system is considered robust; otherwise, it is considered sensitive. For our experiment, we swapped the criteria weights one at a time and ran 15 test runs to see what happened when we got different ( $a_{i}$ ) values. For example, CR1-CR2 represents the weight exchange between CR1 and CR2. The sensitivity analysis results are shown in Figure 5, where it can be seen that A4 is the best CSP, followed by A5. As a result, our proposed framework is not sensitive to changing criterion weigh

Figure 5.

Sensitivity analysis based on criteria swap ping.

5.3 Comparative Analysis

The outcome of the experimental analysis is compared with existing CSS frameworks based on MCDM to verify the accuracy and resilience of the framework in the face of the rank reversal issue. We compared the proposed model with BWM-TOPSIS (Kumar et al., 2021), Fuzzy AHP-TOPSIS (Kumar et al., 2022), Fuzzy TODIM (Wang et al., 2020) and Neutrosophic CODAS (Simic et al., 2022).

5.3.1 Generalized Comparison

An identical dataset was used for comparison, and ranking similarity was observed, with the majority of the rankings being similar to the proposed technique, demonstrating the consistency of the proposed method. The comparative results can be seen in Figure 6. Crisp methods such as AHP and BWM have rank reversal issues that fuzzy methods can address (Başhan et al., 2020; Luo et al., 2022). However, because fuzzy approaches only consider membership values, they cannot fully address any ambiguity that may exist. For which the neutrosophic approach employed by (Simic et al., 2022) is far more robust and consistent. The proposed method is built on a single-valued pentagonal neutrosophic framework, which is far more robust and capable of dealing with indeterminacy because it takes into account pentagonal fuzzy numbers, which are a generalization of triangular and trapezoidal fuzzy numbers. Furthermore, the preceding methods used only subjective or objective weighting techniques to generate weights, whereas our approach uses an integrated weighting method, which provides more stable and optimal weights by integrating the two weight estimating types and adding an extra layer of robustness to the overall framework.

Figure 6.

A Comparison of Different Methods for Ranking Services with the Proposed Method.

5.3.2 Spearman's Correlation Test on Ranks

We employed the Spearman correlation test to evaluate the consistency among criteria ranks obtained by five methods. This test, known for its efficiency in measuring agreement between ordinal data arrays, was chosen due to its non-parametric nature and computational simplicity (Žižović & Pamucar, 2019). The following equation is used to calculate the Spearman's rank correlation coefficient (q):

\begin{aligned} q = 1 - \frac{6 \sum D_{i}^{2}}{n (n^{2} - 1)} \end{aligned}

(20)

where

q = Spearman's rank correlation coefficient

$D_{i}$ = Difference between rank s

n = Total observations

Table 14. presents the Spearman rank-order correlation between each pair of methods, along with the average correlation score for each method and Figure 7. depicts the same results in graphical form. The results show that the Fuzzy TODIM approach has the most consistent criteria rank compared to other sets of criteria rankings. Furthermore, the proposed method's rank consistency is nearly identical to the Fuzzy TODIM approach and has a high level of correlation with the other methods in the comparison.

Table 14.

Average Correlation Scores for Spearman's Correlation Coefficient.

	Fuzzy AHP+ TOPSIS	BWM-TOPSIS	Neutrosophic CODAS	Fuzzy TODIM	Proposed Method
Fuzzy AHP + TOPSIS	1.000	0.900	0.400	0.900	0.700
BWM-TOPSIS	0.900	1.000	0.200	0.800	0.600
Neutrosophic CODAS	0.400	0.200	1.000	0.700	0.900
Fuzzy TODIM	0.900	0.800	0.700	1.000	0.900
Proposed Method	0.700	0.600	0.900	0.900	1.000
Average	0.780	0.700	0.640	0.860	0.820

Figure 7.

Graphical depiction of the average rank orders of Spearman's rank correlation.

6. Conclusion

The CSS problem is considered as one of the most challenging problem today which this study addresses by introducing a novel approach. Here we proposed a novel framework which combines three prominent methods viz. FUCOM, MEREC and TODIM in single-valued pentagonal neutrosophic environment. The single-valued pentagonal neutrosophic sets are capable to handle the indeterminacy and vagueness more efficiently than other fuzzy methods as they enable concurrent application of the truth and falsity membership functions. Along with that the indeterminacy membership function allows the decision makers to convey how confident they are about the level of satisfaction and dissatisfaction in the decision making process. The proposed method is divided into two phases. In the initial phase, the objective weights are produced from the QoS data supplied by the QWS dataset, while the subjective weights are initially derived from the expert's preferences based on the various QoS criteria. SVPN-FUCOM and SVPN-MEREC are then used to estimate the subjective and objective weights, which are further integrated to produce the final optimal weights. In the next phase, these weights are inputted to SVPN-TODIM to generate the final optimal rankings of the cloud services. A real life dataset was used to perform the experimental analysis of the proposed method where five cloud services (interop2, skynode, IP2Geo, MyAmazonSecure, and GoogleSearchService) and 6 QoS criteria (Response Time, Availability, Throughput, Reliability, Compliance and Latency) were taken from the dataset. The results from the experimental analysis showed the efficacy of the proposed framework. Among the selected cloud services MyAmazonSecure achieved the first rank outperforming the others. The consistency and reliability of the proposed framework are analyzed by conducting sensitivity analysis in three different cases, which revealed that our model is stable in case of rank reversal phenomenon. Our model was also compared to some existing models where the results showed similarity in ranking of cloud services thus proving that the proposed model can handle vague and imprecise data more efficiently. Our model demonstrates a strong correlation with the other methods using the Spearman's Rank correlation test as well. Though our proposed hybrid method offers advantages by combining multiple techniques to improve decision quality yet their remains some challenges. Ensuring seamless integration and coherence among the various components of hybrid methods can be challenging. Furthermore, while our study focused on single-valued neutrosophic sets, there are many other types of fuzzy sets that can be implied (spherical, hesitant, picture, plithogenic, and so on). The final rankings from this study are based on the opinions and assessments of five decision makers. Different ranking results may emerge from the upcoming analysis based on the perspectives of various experts. Considering the limitations, the model can be extended in future with different combination of MCDM techniques to enhance it further.

Footnotes

ORCID iDs

Satyabrata Nath

Purnendu Das

Pradip Debnath

Nurulla Mansur Barbhuiya

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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A Single-Valued Pentagonal Neutrosophic Framework for Optimal Cloud Service Selection: Integrating FUCOM,MEREC and TODIM

Abstract

Keywords

1. Introduction

1.1 Limitations

1.3 Contribution

3. Preliminaries

3.1 Single-Valued Pentagonal Neutrosophic Sets (SVPNS)

5.1 Experimental Analysis for Ranking Cloud Services

5.2.1 Case 1(By removing alternative)

Table 12. Dominance Degrees of Services for Different Experiments for Case 1. CSP Exp1 Exp2 Exp3 Exp4 Exp5 Interop2 0 0 0 0 0 Skynode 0.253 0 0.732 0.867 0.950 IP2Geo 0.727 0.654 0 1.000 1.000 MyAmazon Secure 1.000 1.000 1.000 0 0.955 GoogleSearch Service 0 0.501 0.515 0.733 0

5.3.1 Generalized Comparison

Footnotes

ORCID iDs

Funding

Declaration of Conflicting Interests

References

Table 12.
Dominance Degrees of Services for Different Experiments for Case 1.

CSP Exp1 Exp2 Exp3 Exp4 Exp5

Interop2 0 0 0 0 0

Skynode 0.253 0 0.732 0.867 0.950

IP2Geo 0.727 0.654 0 1.000 1.000

MyAmazon Secure 1.000 1.000 1.000 0 0.955

GoogleSearch Service 0 0.501 0.515 0.733 0