A Hybrid Model for Fitness Influencer Competency Evaluation Framework

By inergency On Feb 3, 2024

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3.1. Bayesian BWM Technique

The BWM method was proposed by Rezaei in 2015 as a novel pairwise comparison weighting method that effectively improves the limitations of traditional AHP studies by solving the problems of excessive comparisons and unstable consistency [40]. In the BWM data survey phase, structured expert questionnaires are designed to compare these criteria with others on a 9-scale, and two sets of vectors (Best-to-Others and Others-to-Worst vectors) are obtained by pairwise comparisons to identify the best and worst criteria to help decision-makers make more accurate evaluations [41]. The BWM approach has been widely applied in various industrial evaluation projects, such as risk evaluation [42], blockchain [43], and smart cities [44]. However, traditional BWM uses the simplest arithmetic average to integrate the opinions of multiple experts, and if the experts disagree during the evaluation process, the evaluation value obtained by using the arithmetic average can no longer express the real situation [45]. To overcome the limitations of traditional BWM, Mohammadi and Rezaei [16] proposed a new method to optimize the traditional BWM approach, called Bayesian BWM, which is calculated by the concept of probability distribution when integrating group evaluation information to obtain the best set of criteria group weights. The basic criterion for weight generation in MCDM is that the sum of weights is 1, and each weight is greater than the equivalent of 0. From the concept of probability, the criterion can be considered a random event, and the possibility of the criterion occurring is the generation of weight. Therefore, the model is constructed as a probabilistic model. Therefore, it is reasonable to use BWM as the basis for constructing a probabilistic model. Existing studies have widely applied Bayesian BWM to solve the problem of evaluating weights, including school performance evaluation [45], electricity retailers [46], and airport resilience evaluation [41].

This study uses the suite software provided by Mohammadi and Rezaei [16] to perform Bayesian BWM-related calculations. The description and brief steps of Bayesian BWM for this study are as follows:

A literature review and expert group discussions are used to identify criteria $c_{i} = \{c_{1}, c_{2}, \dots, c_{n}\}$ for evaluating the competencies of $n$ fitness influencers. These criteria can be assigned to six dimensions to form a hierarchical evaluation framework.

Determine the most important (or best) and least important (or worst) criteria from among the n criteria.

The expert evaluation uses a scale of 9, which is set from 1 to 9 to present the importance of the most important criteria relative to other criteria. A scale of 1 indicates equal importance compared to the most important criteria, while a scale of 9 indicates absolute importance compared to other criteria. The greater the difference in scale, the greater the difference in relative importance. The BO vector is expressed as

$A_{B i} = (a_{B 1}, a_{B 2}, \dots, a_{B i}, \dots, a_{B n})$ The importance of the criterion is the most important criterion, which is denoted by $a_{B i}$ .

This step is similar to Step 3, where the experts evaluate the importance of other criteria compared to the least important criterion. The OW vector is expressed as

$A_{i W} = (a_{1 W}, a_{2 W}, \dots, a_{i W}, \dots, a_{n W}) T$ where $a_{i W}$ represents the relative importance of other criterion $i$ compared to the least important criterion $W$ . $a_{B B} = 1$ and $a_{W W} = 1$ are required due to the equal importance of self-comparisons.

The probability model of polynomial distribution is constructed by

A_{B i}

and

A_{i W}

, so the probability function of polynomial distribution of

A_{i W}

is as Equation (1).

$P (A_{i W}| w_{i}) = \frac{(\sum_{i = 1}^{n} a_{i W})!}{\prod_{i = 1}^{n} a_{i W}!} \prod_{i = 1}^{n} w_{i}^{a_{i W}}$

(1)

w_{i}

is the probability distribution of weights, and the probability of

w_{i}

and

a_{i W}

is proportional, so Equation (2) can be obtained. The weight probability

w_{W}

of the least important criterion is shown in Equations (3) and (4) and can be obtained by combining Equations (2) and (3).

$w_{i} \propto \frac{a_{i W}}{\sum_{i = 1}^{n} a_{i W}}, \forall i = 1, 2, \dots, n$

(2)

$w_{W} \propto \frac{a_{W W}}{\sum_{i = 1}^{n} a_{i W}} = \frac{1}{\sum_{i = 1}^{n} a_{i W}}$

(3)

$\frac{w_{i}}{w_{W}} \propto a_{i W}, \forall i = 1, 2, \dots, n$

(4)

In addition, the weight probability of the most important criterion is shown in Equation (5).

$\frac{1}{w_{B}} \propto \frac{a_{B B}}{\sum_{i = 1}^{n} a_{B i}} = \frac{1}{\sum_{i = 1}^{n} a_{B i}} \Rightarrow \frac{w_{B}}{w_{i}} \propto a_{B i}, \forall i = 1, 2, \dots, n$

(5)

The model is constructed using Dirichlet’s probability distribution to obtain the optimal weight value

w_{i}

, with Equation (6). as its probability function.

$D i r (w_{i}| α) = \frac{1}{B (α)} \prod_{i = 1}^{n} w_{i}^{α_{i} - 1}$

(6)

$α$ is the parameter of the vector, and usually the value is set to 1. $w_{i} \geq 0$ and $\sum w_{i} = 1$ are required to comply with the concept of MCDM.

The Bayesian BWM is a soft computing method that takes into account the survey data of several experts and integrates them to obtain a set of optimal group weights $w_{i}^{a g g}$ . The steps are as follows:

There are

j

experts

j = 1, 2, \dots, J

in the expert group, and the weight of the individual criterion is

w_{i}^{j}

after the experts’ evaluation, and the group weight

w_{i}^{a g g}

is obtained by integrating all of

w_{i}^{j}

. The BO and OW vectors of the first expert to the Jth expert are denoted by

A_{B i}^{1 : J}

and

A_{i W}^{1 : J}

. These vectors are used to construct the joint probability distribution function of the group decision as in Equation (7).

$P (w_{i}^{a g g}, w_{i}^{1 : J} |A_{B i}^{1 : J}, A_{i W}^{1 : J})$

(7)

The optimal weight

w_{i}^{j}

of each expert is obtained based on the

A_{B i}

and

A_{i W}

vectors of each expert, while the optimal weight

w_{i}^{a g g}

of the expert group is determined by

w_{i}^{j}

. The Bayesian-level model is constructed based on the Bayesian iterative operations, which means that the

A_{B i}

and

A_{i W}

vectors of the experts generate

w_{i}^{j}

, and the new group optimal weight

w_{i}^{a g g}

is computed on a rolling basis after the evaluation data of multiple experts are added one after another. Considering that the variables are independent of one another, the joint probability of the Bayesian model is shown in Equation (8).

$P (w_{i}^{a g g}, w_{i}^{1 : J} |A_{B i}^{1 : J}, A_{i W}^{1 : J}) \propto P (A_{B i}^{1 : J}, A_{i W}^{1 : J} |w_{i}^{a g g}, w_{i}^{1 : J}) P (w_{i}^{a g g}, w_{i}^{1 : J})$

(8)

Equation (8) can be further deduced as follows.

$P (A_{B i}^{1 : J}, A_{i W}^{1 : J} |w_{i}^{a g g}, w_{i}^{1 : J}) P (w_{i}^{a g g}, w_{i}^{1 : J}) = P (w_{i}^{a g g}) \prod_{j = 1}^{J} P (A_{i W}^{j} |w_{i}^{j}) P (A_{B i}^{j} |w_{i}^{j}) P (w_{i}^{j} |w_{i}^{a g g})$

(9)

From Equation (9), the corresponding probability function can be found by specifying the statistical distribution of each variable. The distributions of

A_{B i}^{j} |w_{i}^{j}

and

A_{i W}^{j} |w_{i}^{j}

are shown in Equation (10).

$A_{B}^{j} |w_{i}^{j} ~ m u l t i n o m i a l (\frac{1}{w_{i}^{j}}), \forall_{j} = 1, 2, \dots, J;$

$A_{i W}^{j} |w_{i}^{j} ~ m u l t i n o m i a l (w_{i}^{j}), \forall_{j} = 1, 2, \dots, J$

(10)

w_{i}^{j}

under the condition

w_{i}^{a g g}

can be constructed as a Dirichlet distribution as shown in Equation (11).

$w_{i}^{j} |w_{i}^{a g g} ~ D i r (γ \times w_{i}^{a g g}), \forall_{j} = 1, 2, \dots, J$

(11)

The mean value of the Dirichlet distribution is, and the non-negative parameter is $γ$ .

The

w_{i}^{j}

must be in the proximity of

w_{i}^{a g g}

since it is the mean of the distribution, the proximity is determined by the parameter

γ

, and the distribution of the parameter

γ

obeys gamma distribution as in Equation (12).

$γ ~ g a m m a (a, b)$

(12)

The shape and scale parameters of the gamma distribution are a and b, respectively. Finally, the optimal group weight

w_{i}^{a g g}

obeys the Dirichlet distribution as in Equation (13).

$w_{i}^{a g g} ~ D i r (α)$

(13)

The parameter $α$ is set to 1.

After constructing the probability distribution of all variables, the optimal group weight $w_{i}^{a g g}$ is obtained by simulating the experiment $p$ times through Markov-chain Monte Carlo (MCMC) technology.

3.2. Modified TOPSIS-AL Technique

TOPSIS is one of the popular MADM methods used in recent years for evaluating performance and ranking alternatives. In this method, the Positive and Negative Ideal Solutions (PIS and NIS) are identified among the combinations of alternatives, and the distance between each alternative and the PIS and NIS is calculated to obtain the relative position of each alternative. The best choice is the alternative closest to the PIS and furthest from the NIS. The TOPSIS method is easy to understand, simple to compute, and has solved many different problems [47,48]. The concept of aspiration level is introduced in this study as TOPSIS-AL. Whereas the original TOPSIS defined the current best alternative as the most desirable solution, TOPSIS-AL defined the aspiration level as PIS and the opposite worst value as NIS. The steps of TOPSIS-AL in this study are as follows:

There are k fitness influencers

A_{p} = \{A_{1}, A_{2}, \dots, A_{k}\}

and h criteria

c_{f} = \{c_{1}, c_{2}, \dots, c_{h}\}

(in the construction of the performance matrix, the vertical axis of the matrix is the fitness influencer

A_{p}

, and the horizontal axis is the criterion

c_{f}

). The evaluation value

d_{p f}

represents the performance of fitness influencer p under criterion f, as in Equation (14).

$D = {[d_{p f}]}_{k \times h} = {[\begin{matrix} d_{11} & d_{12} & \dots & d_{1 f} & \dots & d_{1 h} \\ d_{21} & d_{22} & \dots & d_{2 f} & \dots & d_{2 h} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ d_{p 1} & d_{p 2} & \dots & d_{p f} & \dots & d_{p h} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ d_{k 1} & t_{k 2} & \dots & d_{k f} & \dots & d_{k h} \end{matrix}]}_{k \times h}, p = 1, 2, \dots, k; f = 1, 2, \dots, h$

(14)

To have a uniform unit for all the obtained evaluation criteria and to allow all the performance values in the matrix to converge to a value range between 0 and 1, the normalization method is used to obtain the matrix

D^{*}

(Equation (15)). The conventional normalization method is to take the alternative with the best performance under each criterion as the denominator, which will lead to the situation of “picking the best apple from a bucket of rotten apples”. Therefore, the concept of aspiration level is introduced in the study to modify the normalization equation, as shown in Equation (16).

$D^{*} = {[d_{p f}^{*}]}_{k \times h}$

(15)

$d_{p f}^{*} = \frac{d_{p f}}{{d_{f}}^{a s p i r e}}$

(16)

Considering the different importance of criteria, the weight obtained in Bayesian BWM is multiplied by the normalized performance matrix to obtain the weighted normalized performance matrix, as shown in Equation (17).

$D^{* *} = (w_{i}^{a g g}) \cdot (D^{*})$

(17)

Based on the concept of aspiration level, after matrix normalization, PIS and NIS should be 1 and 0. Therefore, after considering the weights, the PIS and NIS of the system can be obtained, as in Equations (18) and (19).

$PIS = (z_{1}^{+}, z_{2}^{+}, \dots, z_{n}^{+}) = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})$

(18)

$NIS = (z_{1}^{-}, z_{2}^{-}, \dots, z_{n}^{-}) = (0, 0, \dots, 0)$

(19)

In this paper, the Euclidean distances are used to define the separation of fitness influencer p from the PIS and NIS, as in Equations (20) and (21).

$S_{p}^{+} = \sqrt{\sum_{f = 1}^{h} {(z_{f}^{+} - d_{p f}^{* *})}^{2}}$

(20)

$S_{p}^{-} = \sqrt{\sum_{f = 1}^{h} {(d_{p f}^{* *} - z_{f}^{-})}^{2}}$

(21)

The closeness coefficient (CC_p) is proposed by Kuo [48], which improves many shortcomings of conventional TOPSIS to obtain more reliable ranking results, as shown in Equation (22). The new ranking index has a better judgment basis, with the value range of CC_p ranging from −1 to 1 and the sum of CC_p being 0.

$C C_{p} = \frac{w^{+} S_{p}^{-}}{\sum_{p = 1}^{k} S_{p}^{-}} - \frac{w^{-} S_{p}^{+}}{\sum_{p = 1}^{k} S_{p}^{+}}$

(22)

where w⁺ and w⁻ represent the relative importance of PIS and NIS, respectively. Since w⁺ + w⁻ = 1, how much w⁺ and w⁻ are set will affect each other. In the absence of special circumstances, such as a particularly optimistic or pessimistic bias, w⁺ and w⁻ are both set to 0.5.

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