Exponential Similarity Measure Based Selection of Cloud Service Provisioning in Cloud

This section provides a detailed analysis of the proposed exponential similarity metrics for provisioning the suitable cloud service provisioning process. The proposed model is designed to handle the multi-objective constraints and provide an appropriate framework for cloud providers. A multi-parameter-based approach is essential for selecting the optimal cloud service provider. Key factors such as security, technology, transparency, trustworthiness, business policies, reliability, performance, SLAs, and migration support must be considered to develop a structured selection framework. The CHP is introduced as a decision- making technique to handle complex, subjective choices. It employs pairwise comparisons to establish priorities, ensuring consistency in evaluation. The final phase of CHP integrates weight criteria and optional scores to compute a final score, synthesizing outcomes for optimal decision- making. It is proved with the following analysis.

Definition 1: Let A=(a_ij)_n×n be the square matrix that is mentioned as the positive reciprocal if a_ij>0, and $a_{i j}=\frac{1}{a_{j i}}$ is expressed as in Eq. (1):

$A=\left( \begin{matrix}   1 & {{a}_{12}} & ..{{a}_{1n}}  \\   \frac{1}{{{a}_{12}}} & 1 & ..{{a}_{2n}}  \\   \frac{1}{{{a}_{1n}}} & \frac{1}{{{a}_{2n}}} & ..1  \\\end{matrix} \right)$                      (1)

Definition 2: The reciprocal matrix A=(a_ij)_n×n is said to be as per the existing model is consistent perfectly when $a_{i j}=a_{i k} a_{k j} \forall i, j, k \in\{1,2,3, \ldots, n\}$.

Definition 3: The similarity measure among wo vectors v_i and v_jis denoted SM (v_i,v_j)→[0,1], whose domain is V². Here, V specifies the n-dimensional vector space and it holds the below valuables:

1. S(av_i, v_j)=1; $\forall \mathrm{v}_i \in V$. This has the reflexive property.
2. S(av_i, v_j)=0; $\forall \mathrm{v}_i,  \mathrm{v}_j \in V$, when v_i and v_j is not similar vectors to all;
3. SM(v_i, v_j)<SM(v_i, v_k), $\forall \mathrm{v}_i, \mathrm{v}_j, \mathrm{v}_k \in V$, when v_i is more like to v_k, thus it is same as v_j. Then, the similarity measure has the triangle property of metrics.

ESM: The objective optimal function is used to obtain the exponential similarity measures of the consistency index. The C* equals the Pairwise Comparison Matrix (PCM) order when the PCM is consistent. On the other hand, this relies on the ranges from 0 and 1. The consistency index of ESM makes the PCM's sizes available, i.e., $E S I=C^* / n$.

The matrix is consistent perfectly when the CCI is 1. The CCI's lower value represents that the PCM has a familiar constant. Commonly, if the CCI is less than 90%, the CCI needs to be more than 90%. Moreover, the PCM’s consistency is not improved using the existing cosine similarity method. Generally, the PCM's consistency needs to be enhanced using the process in the recursive technique for the entries of the revised judgment matrix till one does not attain the required accuracies for the consistency of the judgment matrix. The weighted geometric mean method or arithmetic mean is selected to enhance consistency.

2.1 Theoretical proof of exponential similarity measures

The memory-based collaborative filtering algorithm is proposed that accepts all similarity formulas for the user to service provisioning calculation. The selection process is predicted using any similarity measure by this algorithm for the specific service provider. This algorithm 1 compares the similarity measures for Root Mean Square Error (RMSE) outcomes for selection and user similarity. The selections are predicted with the help of user similarity. The similarity between service selection and users are identified using different measures. One of the approaches is the ESM which is used to compute the angle among two space vectors. Based on the study conducted by Esposito et al. [25], Pearson introduced the normalized factor of the similarity measure where Manhattan and Euclidean are the other similarity measures [26, 27]. The Manhattan distance is the sum of vertical and horizontal distances on the points between points. The theoretical similarity measures [28] like the replacements to compute the RMSE [29, 30] for the similarity function and every similarity measure’s results are compared.

The introduction of the ESM is used to measure the exponential distance that manages the vectors containing the empty elements. The given Eq. (2) describes the ESM as:

$Exp({{u}_{i}},{{u}_{j}})={{e}^{\frac{\sqrt{\sum\limits_{item=1}^{|j|}{{{({{u}_{i}}(item)-{{u}_{j}}(item))}^{2}}}}}{\varnothing \sqrt{f}}}}$                                    (2)

The both service similarity weight has considered for recommender list and weight of QoS parameters has utilized for ESM calculation. Here, $0 \leq \operatorname{Exp}\left(u_i, u_j\right) \leq 1$ equals the difference between maximal obtainable and minimal obtainable ratings. The user needs to provide and $|I|$ represent the items number. However, f means the elements' numbers that are rated using both users, and the similarity needs to be identified between intersections of filled elements. Figure 1 and Table 2 represent the similarities smoothly as smooth functions are computed to generate better coefficients like the similarity measure.

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Figure 1. Exponential similarity function

Table 2. Exponential similarity measure

The two vectors' distances need to be measured in the rating matrix such as way that the intersection of filled elements of two user’s vectors like filling the elements using both the user vectors to identify the similarity that is compared to find the similarities and the similarity measure's exponential natures generates the coefficient of similarity for the smooth function. The similarity measure has the issues that do not have the balanced nature that results from the various intersection of filled elements among vectors that create the biased similarity coefficients. It is represented in Figure 1, where user one and user three are considered the same users with a similarity coefficient. The two users are not assumed as completely identical since the two users have one filled element alone in general, and the user 3 has another filled element that is not intersected with user 1’s filled elements. Whereas, since many intersecting filled elements are nearer in value, user three and user 5 are the same users in the real world. Hence user 3's empty element is 0.89 when considering the similar-users are not the optimum rating. Another novel similarity measure is introduced to solve this issue the ESMs provide importance to the number of filled elements among two vectors. Eq. (3) is used in Figure 1. Since user three does not include the ratings of similar items like user one, which provides fewer intersecting elements, user one and user 3 are entirely dissimilar. The similarity measure of novel normalized exponential is represented by Eq. (3). Multi-criteria decision-making methods ensure accurate and consistent weighting, which is then used for selecting cloud service providers (see Table 3).

Table 3. Service providers

$\operatorname{Exp}-\operatorname{Normalized}\left(u_i, u_j\right)=\frac{e^{\frac{\sqrt{\sum_{\text{item}=1}^{|j|}\left(u_i(\text { item })-u_j(\text{ item})\right)^2}}{\emptyset \sqrt{f}}}}{\frac{\left|U_i\right|}{f}}$                        (3)

Here $\emptyset$ is equalized to the difference between maximal obtainable and minimal obtainable ratings (see Table 4 and Figure 2). The user needs to provide and $|I|$ represent the items number. However, $f$ means the elements' numbers that are rated using both users, and the similarity needs to be identified between intersections of filled elements. $\left|U_i\right|$ represents the number of filled elements for selecting the present vector user. The theoretical proof of the similarity measure is given below:

Theorem 1: Let $\widehat{\omega} *=(\widehat{\omega} * 1 . \hat{\omega} * 2 . \widehat{\omega} * 3 . \ldots, \widehat{\omega} * n)$ specifies the optimal solution, let C^* represents then,

$\widehat{\omega} *=\frac{\sum_j^n b_{i j}}{\sqrt{\sum_{k=1}^n\left(\sum_j^n b_{k j}^2\right)}} i=1,2,3 \ldots n$                         (4)

Table 4. RMSE based similarity measure

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Figure 2. Similarity measure comparison

The objective function with optimal value is expressed by Eq. (5):

${{C}^{*}}=\sqrt{\sum\limits_{k=1}^{n}{(\sum\limits_{j=1}^{n}{b_{ij}^{2}{{)}^{2}}}}}$                                        (5)

where, $\hat{\omega }*=\hat{\omega }*\sqrt{\sum\limits_{k=1}^{n}{\omega _{k}^{2}}}$, i=1,2,3, …, n.

Let, $\beta^*=\sqrt{\sum_{k=1}^n \omega_k^2} \geq 0$.

It leads in the form in Eq. (6) as,

$\omega _{i}^{*}=\hat{\omega }_{i}^{*}{{\beta }^{*}}$                                        (6)

The system of equations is used to solve similarity of exponential measure. Here, β* is the coefficient of weighted assignments, and the assignment weight is the partial exponential normalization that constrains the obtained weights for optimizing the similar exponential measure. The coefficient of weight assignment represents the reciprocal of sum of the distinct maximization optimum solution.

$\sum\limits_{i=1}^{n}{\omega _{i}^{*}=\sum\limits_{i=1}^{n}{{{{\hat{\omega }}}^{*}}{{\beta }^{*}}}}$                                     (7)

$\sum_{i=1}^n \omega_i^*=1$

$\beta^*=\frac{1}{\sum_{j=1}^n \widehat{\omega}_y^*}$                                       (8)

The adjusted priority vector is given by Eq. (10).

$\omega_i^*=\widehat{\omega}_i^* \beta^*=\widehat{\omega}_i^* \frac{1}{\sum_{j=1}^n \widehat{\omega}_y^*}$                                      (9)

Theorem 2: Let PCM $A=\left(a_{i j}\right)_{n^* n}$ is consistent perfectly where the matrix method is derived precisely with the optimal objective function value $C^*=n$, and the properties $\omega_j^*=$ $\frac{1}{\sum_{i=1}^n a_{i j}}$, where, $j=1,2,3, \ldots, n$. The PCM of matrix A intends to form the matrix as in Eq. (10):

$A=\left(\begin{array}{ccc}\frac{\omega_1}{\omega_1} & \frac{\omega_1}{\omega_1} & \frac{\omega_1}{\omega_1} \\ \vdots & \vdots & \vdots \\ \frac{\omega_n}{\omega_1} & \frac{\omega_n}{\omega_2} & \frac{\omega_n}{\omega_n}\end{array}\right)$                                      (10)

The similarity is attained with the priority vector $\omega$ and the j^th column vector with matrix A. It is expressed as in Eq. (11).

$\begin{gathered}C_j=\text { Exponential Similarity Matrix }\left(\omega, a_j\right) =\frac{\sum_{k=1}^n \omega_k a_{k j}}{\sqrt{\sum_{k=1}^n \omega_k^2} \sqrt{\sum_{k=1}^n a_{k j}^2}}\end{gathered}$                                     (11)

The values of $j$ are set as $1,2, \ldots, n$ where the priority vector $\omega=\left(\omega_1, \omega_2, \ldots, \omega_n\right) T$ and the PCM column entry is provided as $a_j=\left(a_{1 j}, \ldots, a_{n j}\right) T$. The corresponding PCM matrix value with the priority vector is, $a_{i j}=\frac{\omega_i}{\omega_j}$, where for all $j=1,2, \ldots, n$ as in Eq. (12):

$\begin{gathered}C_i=\text { Exponential Similarity Matrix }\left(\omega, a_j\right) =\frac{\sum_{k=1}^n \frac{\omega_k^2}{\omega_f}}{\sqrt{\sum_{k=1}^n \omega_k^2} \sqrt{\sum_{k=1}^n \frac{\omega_k^2}{\omega_j}}}=1\end{gathered}$                      (12)

The matrix is consistent ideally iff the similarity measure for the priority vector, and matrix-vector is equal to 1. Else, the matrix is not constant ideally. It helps to enhance the similarity measure among the priority vector and pairwise comparison matrix.

Figure 3 demonstrates the general architecture of the process for choosing a cloud service provider, illustrating a focus on users’ needs and preference details as a primary structure. All models begin with the identification of requirements which serves as the foundation of the entire thought process. Requirements typically consist of expectations: functional, technical, non-functional in terms of cost, security, performance, scale, and compliance needs. After this, users state their criteria preferences which delineate the ranking of parameters in-terms of importance for their cloud service. frameworks. The next step deals with the prioritization of criteria that is among the most crucial stages where each criterion is given relative focus in terms of importance. The multi-criteria decision-making approaches to criteria weighting because they guarantee accuracy and consistency in weighting allocation. Following the weighted criteria, the cloud service provider selection is carried out (Table 3). This entails assessing available CSPs against the defined criteria to ascertain where the specific user’s priorities and limitations meet the greatest value as the user’s cloud service provider. The model captures some feedback loops which are cycles whereby the user feedback or response is solicited after interested treatments have been applied. With this, the users can evaluate the effectiveness of their chosen provider and improve the selection process in the future. The feedback makes sure that the model captures something measurable and provides responsiveness to the user, technology, or fluctuations in service level changes. The Figure 3 highlights a self-directed and self-altering process in decision-making of a model where selection of the cloud service is not a singular action, but a repetitive loop driven by use and ever-evolving needs. This is excessive when attempting to improve accuracy in decision-making, increase cloud adoption satisfaction in the long run, and sustain transparency.

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Figure 3. Cloud service provider frameworks

2.2 Cloud selection process

The data needs to be analyzed using the CHP model. The company's business profile needs to be concerned with selecting the cloud criteria such as security, technologies, transparency, trustworthiness, business policies and governance, best industry standards, reliability and performance, migration support, and service level agreements. There are eight parameters (Figure 4) called service roadmap, technologies, business policies, governance, security, choosing the best industry standards and certifications, services partnership and services dependency, reliability and performance, service level agreements, business and company profile and migration support are represented in Table 5 for optimizing the cloud selection.

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Figure 4. Comparison of total similarity measure

Table 5. Total similarity measure based on cloud services