approximation area comparison method.

The term "Metaverse," first used in the 1992 Neal Stephenson book Snow Crash, is a relatively recent addition to technology experts' and scientists' everyday language. In Greek, "Meta" refers to a condition of existence that is higher or more sublime than what is now understood or experienced. Other levels of awareness or perception may be investigated and developed. A virtual environment that replaces the real world is called the "Metaverse" [1], [2].

It offers a complete Virtual Reality (VR) experience with well-known characters, objects, user interfaces, and social networks. [3] and [4] scholarly contributions support the definition in question, which states that the Metaverse possesses several essential characteristics, including a pervasive presence, absorbed verisimilitude, and the ability to grow in terms of player count, intricate settings, and a vast range of interactive possibilities. The Metaverse has evolved as an essential route for brand marketing, given the increasing virtual interaction of consumers during the pandemic. The younger generation is projected to show more interest in digital ownership after the epidemic. People's engagement in the Metaverse is as meaningful as how they interact with the world [5], [6]. The best possible realization of the Metaverse depends on high-speed internet access. The VR platform "Second Life" serves as an instructive example. It was established before cell phones were widely used, and its popularity declined partly because it did not allow instantaneous mobile interactions. Technologies that enable sophisticated interactions with people, digital objects, and settings are VR and augmented reality. The book presents the Metaverse as a virtual world, an internet-based cosmos that combines software agents and avatars to create augmented reality [7], [8]. The term "Metaverse" refers to a network of permanent, socially linked immersive platforms allowing smooth communication and dynamic interactions with virtual items. It depicts a Three-Dimensional (3D) depiction of a simulation that interacts with the natural world via a variety of applications [9], [10]. Multi-Criteria Decision-Making (MCDM) is one of the best tools for effective Metaverse platform selection to tackle such complicated challenges. In essence, MCDA came from operations research, which uses a variety of approaches but also has a humorous rational base from other fields of study [11], [12]. MCDA methods are widely used in both public and commercial sectors to inform choices on resource management in agriculture, immigration, education, transportation, investment, environment, defense, and health care, among other topics[13]–[15]. This study aims to propose an MCDM methodology named Multi-Attributive Border Approximation area Comparison (MABAC) for the section best Metaverse platform under different criteria.

This section introduces the steps of the MABAC method. The MABAC method ranks the alternatives and selects the optimal option [16]–[18]. Fig. 1 shows the steps of the MABAC method.

Step 1. Build the decision matrix.

The experts and decision-makers build the decision matrix between criteria and alternatives. The experts used the crisp values between 1 and 9. The decision matrix is built using Eq. (1).

a11 ⋯ a1n

A = [ ⋮ ⋱ ⋮ ] , i = 1,2, … , m;j = 1,2, … , n. (1)

am1 ⋯ amn Step 2. Normalize the decision matrix.

The decision matrix is normalized for beneficial and non-beneficial criteria by using Eqs. (2) and (3).

aij − ai

yij = , i = 1,2, … , m, j = 1,2, … , n. (2)

ai − ai

aij − ai

y = , i = 1,2, … , m, j = 1,2, … , n. (3)

ij − +

ai − ai

where ai = max(a1, a2, … am) and ai = min(a1, a2, … am).

Step 3. Compute the criteria weights.

The weight criteria weights are computed using the average method. The experts and criteria are evaluated the criteria, and then the mean of their evaluation is computed to obtain the criteria weights.

Step 4. Compute the weighted normalized decision matrix.

The criteria weights are multiplied by the normalization matrix to obtain the weighted normalized decision matrix using Eq. (4).

tij = wj + yijwj, i = 1,2, … , m, j = 1,2, … , n. (4)

Step 5. Compute the Border Approximation Area matrix (BAPA).

The BAPA is computed by using Eq. (5).

BAPA = (∏ t ) , j = 1,2, … , n. (5)

j ij

Step 6. Compute the distance from the BAPA.

The distance from the BAPA is computed using Eq. (6).

dij = tij − BAPAj , i = 1,2, … , m. (6)

Step 7. Compute the total distance.

The total distance from the BAPAj is computed by using Eq. (7).

Fi = ∑ dij, i = 1, … , m. (7)

Step 8. Rank the alternatives.

The alternatives are ranked based on the total distance of Fi.

This section introduces the results of the MCDM methodology to select the optimal Metaverse platform among various criteria and alternatives. The experts and decision-makers evaluated the nine criteria and 15 platforms. Fig. 2 shows the list of criteria.

Step 1. Build the decision matrix.

The experts and decision-makers build the decision matrix between criteria and alternatives. The experts used the crisp values between 1 and 9. The decision matrix is built using Eq. (1).

Step 2. Normalize the decision matrix.

The decision matrix is normalized for beneficial and non-beneficial criteria by using Eqs. (2) and (3) as shown in Table 1.

Step 3. Compute the criteria weights.

The weight criteria weights are computed using the average method. The experts and criteria are evaluated the criteria, and then the mean of their evaluation is computed to obtain the criteria weights. Fig. 3 shows the criteria weights.

Step 4. Compute the weighted normalized decision matrix.

The criteria weights are multiplied by the normalization matrix to obtain the weighted normalized decision matrix using Eq. (4), as shown in Table 2.

The BAPA is computed by using Eq. (5).

MPC4 MPC5 MPC6 MPC7 MPC8 MPC9

0.188482 0.100972 0.141361 0.233358 0.262528 0.263276 0.125654 0.078534 0.125654 0.194465 0.262528 0.335079 0.157068 0.100972 0.172775 0.252805 0.222139 0.335079 0.172775 0.112191 0.094241 0.213912 0.201945 0.311144 0.109948 0.089753 0.109948 0.194465 0.141361 0.215408 0.125654 0.112191 0.157068 0.136126 0.282723 0.167539 0.172775 0.123411 0.094241 0.155572 0.222139 0.191473 0.219895 0.078534 0.188482 0.213912 0.161556 0.239342 0.204188 0.100972 0.094241 0.272251 0.201945 0.28721 0.157068 0.157068 0.109948 0.252805 0.141361 0.215408 0.172775 0.112191 0.157068 0.194465 0.282723 0.311144 0.141361 0.078534 0.141361 0.252805 0.222139 0.263276 0.172775 0.100972 0.141361 0.272251 0.161556 0.335079 0.172775 0.100972 0.141361 0.213912 0.141361 0.239342 0.204188 0.13463 0.125654 0.194465 0.222139 0.167539

Step 6. Compute the distance from the BAPA.

The distance from the BAPA is computed using Eq. (6). As shown in Table 3.

Step 7. Compute the total distance.

The total distance from the BAPAj is computed using Eq. (7), as shown in Fig. 4.

Step 8. Rank the alternatives.

The alternatives are ranked based on the total distance of Fi as shown in Fig. 5. Alternative 1 is the best, and alternative 5 is the worst.

This section shows the rank of alternatives under different cases in criteria weights. We proposed ten cases in criteria weights. Fig. 6 shows the ten instances in criteria weights. Fig. 7 shows the rank of alternatives under different cases. The results show the rank of other options is stable.

In this section, we compare the MCDM methodology with other MCDM methods, such as TOPSIS, VIKOR, EDAS, and COPRAS methods, to show the effectiveness of the proposed MCDM methodology. Fig. 8 shows the rank of alternatives under comparative analysis. The results show the proposed method is robust compared with other MCDM methods.

The Metaverse platforms are evaluated, and the optimal one is selected among various criteria. The MCDM methodology is used to deal with multiple criteria. The MABAC method is used as an MCDM methodology to rank the alternatives. The nine criteria and 15 alternatives are used in this study. The experts and decision-makers evaluate the requirements and options based on their opinions. Then, this study used a crisp value from 1 to 9 to assess the criteria and alternatives to build the decision matrix. The decision matrix is normalized based on the positive and negative criteria. This study used all positive requirements. The criteria weights are computed by using the mean method. The criteria weights multiply the normalization matrix to show the weighted normalization decision matrix. The results show that alternatives 13 and 5 are the best.

No specific funding was received for this study.

The authors declare no conflicts of interest regarding the publication of this paper.