A New Development of FDOSM Based on a 2-Tuple Fuzzy Environment: Evaluation and Benchmark of Network Protocols as a Case Study

Abstract

Multicriteria decision-making (MCDM) is one of the most common methods used to select the best alternative from a set of available alternatives. Many methods in MCDM are presented in the academic literature, with the latest being the Fuzzy Decision by Opinion Score Method (FDOSM). The FDOSM can solve many challenges that are present in other MCDM methods. However, several problems still exist in the FDOSM and its extensions, such as uncertainty. One of the most significant problems in the use of the FDOSM is the loss of information during the conversion of a decision matrix into an opinion decision matrix. In this paper, the authors expanded the FDOSM into the 2-tuple-FDOSM to solve this problem. The methodology behind the development of the 2-tuple-FDOSM was presented. Within the methodology, definitions of the 2-tuple linguistic fuzzy method, which was used to solve the loss-of-information problem that is present in the FDSOM method, are presented. A network case study was used in the application of the 2-tuple-FDOSM. The final results show that the 2-tuple-FDOSM can be used to address the problem of loss of information. Finally, a comparison between the basic FDOSM, TOPSIS, and 2-tuple-FDOSM was presented.

1. Introduction

MCDM is a logical strategy used to arrange available options by preference to select the most favorable option. The objective of the decision-making process is to attain the most desirable goals with the fewest potential repercussions [1]. Decision-making becomes more complicated in the presence of uncertainty, insufficient knowledge, and situations involving numerous criteria evaluations. MCDM is the most widely used decision-making method [2]. Decision-making in the presence of numerous objectives or qualities is referred to as MCDM [3]. Multiple difficulties in selection and/or decision-making are frequently addressed using the MCDM technique. The main goal of MCDM is to assist decision-makers in selecting the best option and ranking it based on its efficacy by sorting the alternatives among the available options. To complete the ranking process, various options must be examined in order to rank the alternatives and choose the best one [4]. An MCDM problem is typically represented as a decision matrix. This decision matrix is an m × n two-dimensional matrix with m rows and n columns, with rows A₁, A₂,… A_m representing alternatives and columns C₁, C₂,… C_n representing criteria. For each C_j criterion, the DM matrix ranks the A₁ alternative [5,6,7]. Essentially, this necessitates an evaluation and assessment process involving quantitative and/or qualitative analysis by specialists (decision-makers) to determine the optimal alternative for each criterion. Using complicated mathematical computations, the objective function examines all of the criteria with respect to each choice in the DM matrix [8]. In addition, single or a group of decision-makers are included in this approach to subjectively evaluate choices regarding several performance criteria [9,10]. The mathematical approach and the human approach are the two basic methodologies used in the MCDM process. The first employs mathematical equations, such as the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) technique [11]. The Analytic Hierarchy Process (AHP) approach [11,12,13,14] is a method that falls into the second category which considers human preferences in computations. Each method has its own set of problems, such as issues regarding normalization [15,16], criteria weight [17], and distance measurement [18,19] in the mathematical approach. The human approach, on the other hand, faces the most significant difficulty (the inconsistency ratio [20,21,22,23]). Another issue that plagues MCDM methodologies (both mathematical and human-based approaches) is uncertainty and ambiguous information. Because they use linguistic words, decision-makers (experts) are unable to calculate weights in real numbers. As a result, the problems, which include this knowledge, become more difficult to solve. This problem has been addressed by several researchers [24,25,26,27]. Many studies in the academic literature have advised using fuzzy set numbers to deal with the problem of uncertainty and ambiguous data [28,29,30,31,32]. In 2020, the Fuzzy Decision by Opinion Score method (FDOSM) was proposed as a promising solution to overcome the challenges listed above [33]. The FDOSM, as with other MCDM approaches, attempts to aid the decision-maker in identifying the most promising alternative by taking many factors into account which are dependent on the decision-maker’s viewpoint [33].

The loss of information is a challenge that researchers using the FDOSM method face when solving any decision-making problem. This problem is generated due to the philosophy of the FDOSM when opinion matrices are combined. The main aim and contribution of this paper was to evolve the FDOSM into the 2-tuple-FDOSM to address the loss-of-information problem present in the basic FDOSM. This paper is organized as follows: in Section 2, the academic literature related to the FDOSM and the 2-tuple linguistic fuzzy method is presented. The methodology used in this paper is presented in Section 3. In Section 4, the network case study is presented. The final results relating to the 2-tuple-FDOSM and the comparative analysis are presented in Section 5. In addition to this, in Section 6, the comparative analysis is presented. Finally, in Section 7, the conclusion of this paper is reported.

2. Related Work

In the academic literature, many researchers have used the FDOSM or extended the FDSOM into another fuzzy environment to solve a multicriteria decision problem. Again, any decision-making problem is presented as a decision matrix, shown as follows:

$$DM=\begin{array}{c}\begin{array}{c}{A}_1\\ A_2\end{array}\\ \begin{array}{c}⋮\\ A_m\end{array}\end{array}\left[\begin{array}{cc}\begin{array}{cc}{x}_{11}& x_{12}\\ x_{21}& x_{22}\end{array}& \begin{array}{cc}\dots & x_{1n}\\ \dots & x_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ x_{m1}& x_{m2}\end{array}& \begin{array}{cc}⋮& ⋮\\ \dots & x_{mn}\end{array}\end{array}\right]$$

The main steps in the FDOSM are as follows [33]:

• Step 1: Create the decision matrix.
• Step 2: Select the ideal solution for each criterion (the ideal solution is one of the following: min, max, or critical value).
• Step 3: Create a reference comparison between the ideal solution and other values per criterion, according to the decision-maker’s opinion, to create the opinion matrix.
• Step 4: Convert the opinion matrix into triangular fuzzy numbers (TFNs).
• Step 5: Direct aggregation with the arithmetic mean.
• Step 6: Make the final decision (the lowest is the best).

In [34], the authors applied different scenarios to the basic form of the FDOSM and made comparisons between different aggregation operators. Additionally, in [35], the authors extended the FDOSM into a q-rung orthopair fuzzy method and integrated the FDOSM with another method (i.e., the fuzzy weighted zero inconsistency method). In [36], the authors extended the FDOSM into the Pythagorean fuzzy numbers. In another article [37], the authors extended the FDOSM into T-spherical fuzzy numbers. The authors in [30] extended the FDOSM into the interval type-2 trapezoidal fuzzy numbers. Additionally, in [38], the authors extended the FDOSM into the intuitionistic fuzzy numbers. In another article [39], the authors extended the FDOSM using Pythagorean fuzzy numbers. The authors of [40] extended the FDOSM using interval-valued Pythagorean fuzzy numbers. Finally, the authors of [41] extended the FDOSM using a Fermatean fuzzy environment. However, the FDOSM suffered from issues other than uncertainty, such as the problem of losing information, when opinion matrices were combined with the group-decision-making context. For example, suppose there are three decision-makers, and each decision-maker creates an opinion matrix from their own viewpoint. For alternative 1, the first decision-maker gives their opinion (with a slight difference), the second decision-maker gives their opinion (with a big difference), and the third decision-maker gives their opinion (with a huge difference). When the fuzzy numbers of the three opinions are combined (a slight difference, a big difference, and a huge difference) we do not achieve a result that fits with the decision-makers’ opinions. In this case, we lose some information in the combination process. Many researchers in the academic literature have mentioned that the 2-tuple fuzzy method is the best way of solving the problem of missing information in the combination process [42,43,44,45]. A 2-tuple is a linguistic phrase followed by a number. The fundamental advantage of this representation is that it is continuous in its domain, allowing it to express any information counts in the discourse universe. A computational technique is used in conjunction with this representation model to deal with 2-tuples without sacrificing information. Here, we used this fuzzy linguistic representation model in a decision-making problem, demonstrating that it is more accurate than earlier models [46].

A 2-tuple can be defined by two values, the first of which is a linguistic label and the second of which is a real number that represents the symbolic translation’s value [47].

A gap in the academic literature related to the FDOSM method was identified: the FDOSM has not been extended using the 2-tuple linguistic fuzzy method to solve the loss-of-information problem. Therefore, in this article, we presented a new extension of the FDOSM based on the 2-tuple fuzzy method, namely the 2-tuple-FDOSM, to address the problem of loss of information when combining opinion matrices.

3. Methodology

In this section, the methodology relating to the development of the 2-tuple-FDOSM is presented. Definitions of the 2-tuple linguistic fuzzy method, which was used to solve the loss-of-information problem in the FDSOM method, are presented.

3.1. Development 2-Tuple-FDOSM

In this section, we present the steps used in the extension of the FDOSM, namely the 2-tuple-FDOSM. In Figure 1, these steps are summarized. The steps are composed of two main stages. The first stage relates to a data-transformation unit, and the second stage relates to a data-processing unit.

Figure 1. The steps for 2-tuple-FDOSM.

3.1.1. Data-Transformation Unit

The decision matrix was transformed into an opinion matrix in two steps using the data-transformation unit.

Step 1: The following equation is used to select the ideal solution for each criterion used in the decision matrix (CWND, throughput, queue size, and packets lost). Equation (1) represents the selection step.

$$A^{*}=\left\{\left[\left(\underset{i}{\max }{v}_{ij}|j\in J\right),\left(\underset{i}{\min }{v}_{ij}|j\in J\right),\left(O{p}_{ij}\in I.J\right)|i=1,2,3,\dots ,m\right]\right\}$$

(1)

The $\underset{i}{\max }{v}_{ij}$ term denotes the typical value of the benefit MCDM criteria, the $\underset{i}{\min }{v}_{ij}$ term denotes the ideal solution of the cost MCDM criteria, and $O{p}_{ij}$ is the critical value when the ideal intermediate value lies between the $\underset{i}{\min }{v}_{ij}$ and $\underset{i}{\max }{v}_{ij}$. The critical value must be determined by the decision-maker.

Step 2: After identifying the ideal solution, by using a five-point Likert scale, the expert conducts a reference comparison (i.e., no difference, slight difference, difference, big difference, or huge difference) between the ideal solution and alternative values in the same criterion.

The following equation can be used to express this step:

$$O{p}_{Lang}=\left\{\left(\left(\underset{ij}{\tilde{v}}\otimes v_{ij}|j\in J\right).|i=1,2,3,\dots ,m\right)\right\}$$

(2)

The reference comparison is referred to as $\otimes$. The linguistic term opinion matrix, which is identified as follows, is the product of the data-transformation unit:

$$O{p}_{Lang}=\begin{array}{c}{A}_1\\ ⋮\\ A_m\end{array}\left[\begin{array}{ccc}o{p}_{11}& \cdots & o{p}_{1n}\\ ⋮& \ddots & ⋮\\ o{p}_{m1}& \cdots & o{p}_{mn}\end{array}\right]$$

(3)

The output of this unit is the opinion matrix according to the decision-maker’s preferences. This opinion matrix is the input for the next stage (i.e., data processing). In the following section, the data-processing unit steps are presented.

3.1.2. Data-Processing Unit

In this section, we show how we used the 2-tuple fuzzy method based on the following mathematical definitions to extend the FDOSM into the 2-tuple-FDOSM.

If a symbolic approach for aggregating linguistic information yields a value of $\beta \in \left[0,g\right]$, and $\beta \notin \left\{0,\dots ,g\right\}$, then an approximation function $\left({\mathrm{app}}_2(\cdot )\right)$ is employed to express the index of the result in $S$. Let $S=\left\{s_0,\dots ,s_g\right\}$ be a linguistic term set.

Definition 1.

Let $\beta$ be the result of a symbolic aggregation operation performed on the index values of a number of labels assessed in a linguistic term set S. $\beta \in \left[0,g\right]$, where $q+1$ is S’s the number of elements in a set or other grouping. A $\alpha$ symbolic translation is defined as $i=round\left(\beta \right)$, with $\alpha =\beta -i$ being two values such that $i\in \left[0,g\right]$ and $\alpha \in \left[-0.5,0.5\right)$. A linguistic term’s symbolic translation, $s_i$, is a value evaluated through $\left[-0.5,0.5\right)$ This supports the “variance of information” among information measurements $\beta \in \left[0,g\right]$ after a symbolic aggregation procedure, and the value that comes closest in $\left\{0,\dots ,g\right\}$ suggests the index of the nearest linguistic term in $S\left(i=round\left(\beta \right)\right)$.

This method expresses linguistic information as 2-tuples $\left(s_i,{\alpha }_i\right),s_i\in S$, and ${\alpha }_i\in \left[-0.5,0.5\right)$. The linguistic label center of the information is expressed by $s_i$; ${\alpha }_i$ is a numeric value which represents the value of the symbolic translation from the original outcome $\beta$ to the index label that is the nearest, $i$ in the linguistic term set $\left(s_i\right)$. For linguistic phrases and 2-tuples, as well as quantitative values and 2-tuples, this model specifies transformation functions [46].

Definition 2.

Given a linguistic term set$S=\left\{s_0,\dots ,s_g\right\}$and a value$\beta \in \left[0,g\right]$, the 2-tuple that reflects the equivalent information to$\beta$is computed using the following function, which indicates the output of a symbolic aggregation operation [46]:

$$\Delta :\hspace{1em}\left[0,g\right]\to S\times \left[-0.5,0.5\right)$$

(4)

$$\begin{array}{rr}\hfill \Delta \left(\beta \right)=\left(s_i,\alpha \right),& \hfill \mathit{with}\{\begin{array}{ll}{s}_i,\hfill & i=round\left(\beta \right)\hfill \\ \alpha =\beta -i,\hfill & \alpha \in \left[-0.5,0.5\right)\hfill \end{array} \end{array}$$

(5)

whereas round $(\cdot )$is the standard process of rounding, $s_i$is the nearest label for the index to “$\beta$”, and $\alpha$ is the worth of the symbolic translation.

Example: Consider a symbolic aggregation operation over labels in the range $S=\left\{s_0,s_1,s_2,s_3,s_4,s_5,s_6\right\}$,which acquires the value $\beta =2.8$ as a result; therefore; the information’s representation as a 2-tuple will be:

$$\mathsf{\Delta }\left(2.8\right)=\left(s_3,-0.2\right)$$

(6)

This is depicted graphically in Figure 2.

Figure 2. Illustration of a symbolic translation calculation.

Let $S=\left\{s_0,\dots ,s_g\right\}$ be a set of linguistic terms, and let $\left(s_i,\alpha \right)$ be a 2-tuple in Proposition 1. There is always a ${\mathsf{\Delta }}^{-1}$ function which produces the equivalent numerical value $\beta \in \left[0,g\right]\subset ℛ$ from a 2-tuple.

Proof. 

Consider the following function to observe how straightforward it is:

$$\begin{array}{rr}\hfill {\mathsf{\Delta }}^{-1}& \hfill :S\times \left[-0.5,0.5\right)\to \left[0,g\right]\end{array}$$

(7)

$$\begin{array}{rr}\hfill {\mathsf{\Delta }}^{-1}\left(s_i,\alpha \right)& \hfill =i+\alpha =\beta .\end{array}$$

(8)

□

Remark. 

It is clear from Definitions 2 and 3, as well as Proposition 1, that converting and adding a symbolic translation of a linguistic phrase to a linguistic 2-tuple entails adding a value of 0.

$$s_i\in S\Rightarrow \left(s_i,0\right)$$

(9)

There are many aggregation processes in the literature that allow us to mix data based on different criteria. The arithmetic mean is used for the purpose of aggregating the results because it prevents information loss [46].

Definition 3.

Given a set of 2-tuples$x=\left\{\left(r_1,{\alpha }_1\right),\dots ,\left(r_n,{\alpha }_n\right)\right\}$, the 2-tuple arithmetic mean${\overline{x}}^e$is calculated as

$${\overline{x}}^e=\Delta \left({\displaystyle \sum _{i=1}^n\frac{1}{n}{\Delta }^{-1}(r_i,{\alpha }_i)}\right)=\Delta \left(\frac{1}{n}{\displaystyle \sum _{i=1}^n{\beta }_i}\right)$$

(10)

We can calculate the mean of a set of linguistic values using the arithmetic mean for 2-tuples without losing any information [46].

4. Case Study

In 2009, Long-Term Evolution-Advanced (LTE-A) was described as the first concept of a successor to Long-Term Evolution (LTE) technology because the desire for high data rates in mobile networks was steadily increasing. LTE-A is a significant improvement compared to LTE technology. LTE-A is a high-speed wireless network for mobile phones that is divided into generational groups. It is an LTE-based enhancement. Many protocols are applied under LTE-A, and to evaluate and benchmark these protocols, MCDM needed to be applied due to issues including importance criteria, data variation, and multicriteria problems. In this case study, four evaluation criteria were used. The first criterion was the congestion control window (CWND), which is one of the variables used to determine the transmission process. The CWND assigned by the sender depends on the network status maintained in the predestination address [48]. The second criterion was the throughput of the average data rate based on the amount of data that successfully reached the destination over a specific data link. It also indicated the amount of bandwidth used [48]. The third criterion was the queue size, composed of permeability that is authorized through any link connection [48]. The final criterion used in this case study was the packet loss [48]. The benefit criteria in this case study were CWND, throughput, and queue size, and the cost criterion was the packet loss.

In Table 1, the decision matrix for this case study is shown.

Table 1. The decision matrix.

Alternatives	CWND	Throughput	Queue Size	Pkt Loss
A1	120	9,881,687	48,424.92	209.75
A2	130	9,912,326	56,199.3	81.75
A3	145	9,905,462	53,788.4	24
A4	200	10,120,778	57,674	27
A5	205	9,902,374	51,274.2	43.5
A6	212	10,023,750	51,180.7	64.5
A7	202	10,264,182	57,581.3	70.5
A8	225	10,106,678	52,895.4	94.5
A9	235	20,368,886	59,680.9	104

5. Results and Discussion

The results for the 2-tuple-FDOSM are presented in this section as follows:

5.1. The Result of the Opinion Matrix

The opinion matrix used in the evaluation and benchmarking of the LTE-A protocols is discussed in this section. This process was realized by converting the original decision matrix presented in Table 1 into the opinion decision matrix depicted in Table 2 by judging the three decision-makers’ preferences using a five-point Likert scale. The ideal solution was determined by the decision-maker as defined in Equation (1). It should be mentioned that the ideal solutions for each decision-maker are reported in Table 3. According to the FDOSM philosophy, the decision-maker selects the ideal solution depending on their own opinion, and the decision-maker is not forced to choose the max value for the benefit criteria and the min value for the cost criteria.

Table 2. The opinion matrices of the three decision-makers.

The Opinion Matrix of the First Decision-Maker
Alternatives	CWND	Throughput	Queue Size	Pkt Loss
A1	Huge Difference	Huge Difference	Huge Difference	Huge Difference
A2	Huge Difference	Big Difference	Slight Difference	Big Difference
A3	Big Difference	Big Difference	Difference	No Difference
A4	Difference	Difference	Slight Difference	Slight Difference
A5	Difference	Big Difference	Big Difference	Difference
A6	Difference	Difference	Big Difference	Difference
A7	Difference	Slight Difference	Slight Difference	Difference
A8	Slight Difference	Slight Difference	Difference	Big Difference
A9	No Difference	No Difference	No Difference	Big Difference
The Opinion Matrix of the Second Decision-Maker
Alternatives	CWND	Throughput	Queue Size	Pkt Loss
A1	Huge Difference	Slight Difference	No Difference	Huge Difference
A2	Huge Difference	No Difference	Huge Difference	Big Difference
A3	Big Difference	Slight Difference	Big Difference	Slight Difference
A4	No Difference	Difference	Huge Difference	No Difference
A5	Slight Difference	Slight Difference	Difference	Difference
A6	Slight Difference	Difference	Difference	Difference
A7	Slight Difference	Big Difference	Huge Difference	Difference
A8	Difference	Big Difference	Big Difference	Big Difference
A9	Difference	Big Difference	Huge Difference	Big Difference
The Opinion Matrix of the Third Decision-Maker
Alternatives	CWND	Throughput	Queue Size	Pkt Loss
A1	Huge Difference	Slight Difference	Big Difference	Huge Difference
A2	Huge Difference	Slight Difference	Slight Difference	Big Difference
A3	Huge Difference	Slight Difference	Difference	No Difference
A4	Big Difference	Slight Difference	Slight Difference	No Difference
A5	Big Difference	No Difference	Difference	Slight Difference
A6	Big Difference	Slight Difference	Difference	Difference
A7	Difference	No Difference	Slight Difference	Difference
A8	Slight Difference	No Difference	Difference	Big Difference
A9	No Difference	No Difference	No Difference	Big Difference

Table 3. The ideal solution for each decision-maker.

Decision-Makers	CWND	Throughput	Queue Size	Pkt Loss
Decision-maker 1	235	20,368,886	59,680.9	24
Decision-maker 2	200	9,912,326	48,424.92	27
Decision-maker 3	235	20,368,886	59,680.9	27

Therefore, to establish the opinion matrices of the decision-makers, reference comparisons were made between the ideal solution and other values of alternatives under the same criteria according to Equation (2). Table 2 presents the opinion decision matrices derived from three decisions makers’ preferences.

The opinion matrices created depended on the decision-makers’ opinions. In the table shown above, it can be seen that each decision-maker made a reference comparison between the ideal solution and other values in the same criterion from their own viewpoints to create the decision matrix.

In the following section, the final rank for the alternatives is reported.

5.2. The Final Rank

According to the philosophy of the FDOSM, opinion matrices are created using a five-point Likert scale. When these opinion matrices are aggregated to one opinion matrix, some information is lost. Therefore, we used 2-tuples to determine the final rank without losing information.

First of all, we transferred the opinion matrix into a fuzzy opinion matrix by converting the linguistic terms into numbers according to the fuzzy numbers that were used in the basic FDOSM [33]. The following table presents this information.

In accordance with the information presented in Table 4, the fuzzy opinion matrices for the three decision-makers are reported in Table 5.

Table 4. Conversion of the linguistic terms into fuzzy numbers.

Linguistic Terms	Fuzzy Numbers
No difference	(0, 0.1, 0.3)
Slight Difference	(0.1, 0.3, 0.5)
Difference	(0.3, 0.5, 0.75)
Big Difference	(0.5, 0.75, 0.9)
Huge Difference	(0.75, 0.9, 1)

Table 5. The fuzzy opinion matrices.

The Fuzzy Opinion Matrix for the First Decision-Maker
Alternatives	CWND			Throughput			Queue Size			Pkt Loss
A1	0.75	0.90	1.00	0.75	0.90	1.00	0.75	0.90	1.00	0.75	0.90	1.00
A2	0.75	0.90	1.00	0.50	0.75	0.90	0.10	0.30	0.50	0.50	0.75	0.90
A3	0.50	0.75	0.90	0.50	0.75	0.90	0.30	0.50	0.75	0.00	0.10	0.30
A4	0.30	0.50	0.75	0.30	0.50	0.75	0.10	0.30	0.50	0.10	0.30	0.50
A5	0.30	0.50	0.75	0.50	0.75	0.90	0.50	0.75	0.90	0.30	0.50	0.75
A6	0.30	0.50	0.75	0.30	0.50	0.75	0.50	0.75	0.90	0.30	0.50	0.75
A7	0.30	0.50	0.75	0.10	0.30	0.50	0.10	0.30	0.50	0.30	0.50	0.75
A8	0.10	0.30	0.50	0.10	0.30	0.50	0.30	0.50	0.75	0.50	0.75	0.90
A9	0.00	0.10	0.30	0.00	0.10	0.30	0.00	0.10	0.30	0.50	0.75	0.90
The Fuzzy Opinion Matrix for the Second Decision-Maker
Alternatives	CWND			Throughput			Queue Size			Pkt Loss
A1	0.75	0.90	1.00	0.10	0.30	0.50	0.00	0.10	0.30	0.75	0.90	1.00
A2	0.75	0.90	1.00	0.00	0.10	0.30	0.75	0.90	1.00	0.50	0.75	0.90
A3	0.50	0.75	0.90	0.10	0.30	0.50	0.50	0.75	0.90	0.10	0.30	0.50
A4	0.00	0.10	0.30	0.30	0.50	0.75	0.75	0.90	1.00	0.00	0.10	0.30
A5	0.10	0.30	0.50	0.10	0.30	0.50	0.30	0.50	0.75	0.30	0.50	0.75
A6	0.10	0.30	0.50	0.30	0.50	0.75	0.30	0.50	0.75	0.30	0.50	0.75
A7	0.10	0.30	0.50	0.50	0.75	0.90	0.75	0.90	1.00	0.30	0.50	0.75
A8	0.30	0.50	0.75	0.50	0.75	0.90	0.50	0.75	0.90	0.50	0.75	0.90
A9	0.30	0.50	0.75	0.50	0.75	0.90	0.75	0.90	1.00	0.50	0.75	0.90
The Fuzzy Opinion Matrix for the Third Decision-Maker
Alternatives	CWND			Throughput			Queue Size			Pkt Loss
A1	0.75	0.90	1.00	0.10	0.30	0.50	0.50	0.75	0.90	0.75	0.90	1.00
A2	0.75	0.90	1.00	0.10	0.30	0.50	0.10	0.30	0.50	0.50	0.75	0.90
A3	0.75	0.90	1.00	0.10	0.30	0.50	0.30	0.50	0.75	0.00	0.10	0.30
A4	0.50	0.75	0.90	0.10	0.30	0.50	0.10	0.30	0.50	0.00	0.10	0.30
A5	0.50	0.75	0.90	0.00	0.10	0.30	0.30	0.50	0.75	0.10	0.30	0.50
A6	0.50	0.75	0.90	0.10	0.30	0.50	0.30	0.50	0.75	0.30	0.50	0.75
A7	0.30	0.50	0.75	0.00	0.10	0.30	0.10	0.30	0.50	0.30	0.50	0.75
A8	0.10	0.30	0.50	0.00	0.10	0.30	0.30	0.50	0.75	0.50	0.75	0.90
A9	0.00	0.10	0.30	0.00	0.10	0.30	0.00	0.10	0.30	0.50	0.75	0.90

In the next step, according to [33], to create the group fuzzy opinion matrix, the internal aggregation for the fuzzy opinion matrices was performed using the following equation:

$$group opinion matrix=\left(min,aggregationoperators,max\right)$$

(11)

The arithmetic mean was used to aggregate the fuzzy opinion matrix and obtain the group numbered opinion matrix. In Table 6, the group numbered opinion matrix is shown.

Table 6. The group fuzzy opinion matrix.

Alternatives	CWND			Throughput			Queue Size			Pkt Loss
A1	0.75	0.9	1	0.1	0.5	1	0	0.583333	1	0.75	0.9	1
A2	0.75	0.9	1	0	0.383333	0.9	0.1	0.5	1	0.5	0.75	0.9
A3	0.5	0.8	1	0.1	0.45	0.9	0.3	0.583333	0.9	0	0.166667	0.5
A4	0	0.45	0.9	0.1	0.433333	0.75	0.1	0.5	1	0	0.166667	0.5
A5	0.1	0.516667	0.9	0	0.383333	0.9	0.3	0.583333	0.9	0.1	0.433333	0.75
A6	0.1	0.516667	0.9	0.1	0.433333	0.75	0.3	0.583333	0.9	0.3	0.5	0.75
A7	0.1	0.433333	0.75	0	0.383333	0.9	0.1	0.5	1	0.3	0.5	0.75
A8	0.1	0.366667	0.75	0	0.383333	0.9	0.3	0.583333	0.9	0.5	0.75	0.9
A9	0	0.233333	0.75	0	0.316667	0.9	0	0.366667	1	0.5	0.75	0.9

From the table displayed above, the loss of information when aggregating the fuzzy opinion matrix of the three decision-makers can clearly be seen. To solve the loss-of-information problem, the use of 2-tuples was the most suitable method. In accordance with the definitions that are presented in the methodology, the 2-tuple group opinion matrix is reported in Table 7. In addition, the fuzzy score was determined using the arithmetic mean in the following equation:

$$A_{m\left(x\right)=}\frac{{\displaystyle \sum }\left(a_f+a_m+a_l\right)\left(b_f+b_m+b_l\right)\left(c_f+c_m+c_l\right)\left(d_f+d_m+d_l\right) }{n}$$

(12)

where n is the number of evaluation criteria.

Table 7. The final result of 2-tuple-FDOSM.

Alternatives	Fuzzy Score			2-Tuple-FDOSM Score	Rank
A1	0.4	0.720833	1	BD, −0.3	9
A2	0.3375	0.633333	0.95	BD, −0.16	8
A3	0.225	0.5	0.825	DI	5
A4	0.05	0.3875	0.7875	SD, 0.08	1
A5	0.125	0.479167	0.8625	DI, −0.3	3
A6	0.2	0.508333	0.825	DI, 0.08	6
A7	0.125	0.454167	0.85	D, −0.5	2
A8	0.225	0.520833	0.8625	DI, 0.2	7
A9	0.125	0.416667	0.8875	DI, −0.09	4

According to the philosophy of the FDOSM, the best alternative is the closest to the no-difference linguistic term (the ideal solution). In Table 7, the best alternative is A₄, because it has the closest value to the no-difference linguistic term (the ideal solution). On the other hand, the worst alternative is A₁, because it has the furthest value from the ideal solution.

The basic FDOSM was applied [33] to an external group context on the same opinion matrix and the same fuzzy opinion matrix for the three decision-makers. Additionally, the arithmetic mean was used to determine the final score. The lowest value was the best alternative. The final result of the application of the basic FDOSM to a group is reported in Table 8.

Table 8. The final result of group basic FDOSM.

Alternatives	Score	Rank
A1	0.7097	9
A2	0.6194	8
A3	0.4958	5
A4	0.3958	1
A5	0.4806	4
A6	0.5139	6
A7	0.4597	3
A8	0.5139	6
A9	0.4222	2

Using the basic FDOSM, the same best and the worst alternatives that were determined using the 2-tuple-FDOSM were determined. However, we noticed some clear differences in the final rank for some alternatives (i.e., A₅, A₇, A₈, and A₉). These differences occurred because of the loss of information when using the basic FDOSM. On the other hand, the 2-tuple-FDOSM solved the problem in the basic FDOSM that caused alternatives to be given the same rank, i.e., A₆ and A₈ had the same rank. This problem occurred due to uncertainty. Therefore, it was shown that the 2-tuple-FDOSM is more flexible in dealing with uncertainty. Finally, in the following section, a comparative analysis between the 2-tuple-FDOSM and TOPSIS method is presented.

6. Comparative Analysis

In this section, we present a comparative analysis between the final rankings from the 2-tuple-FDOSM and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for the same case study, which was used in this paper. In accordance with [48], the TOPSIS method was applied to the same case study, and the final results are reported in Table 9.

Table 9. The final results for TOPSIS method.

Alternatives	Score	Rank
A1	0	9
A2	0.620835	7
A3	0.799564	3
A4	0.910848	1
A5	0.855029	2
A6	0.772874	4
A7	0.746926	5
A8	0.639203	6
A9	0.604535	8

According to the table displayed above, there are many differences in the positions of the alternatives. These differences occurred due to the issues that the TOPSIS method suffered from. The TOPSIS method addresses uncertainty but nevertheless displays the fundamental drawback found in the original TOPSIS method, which is the necessity for external sources to provide preference weights. The 2-tuple-FDOSM outperformed TOPSIS in terms of dealing with:

1—missing data. 2—immeasurable criteria. 3—the generation of criteria weight. 4—missing information and normalization (the process of unifying data from various scales and types in the decision matrix). 5—the ideal solution (the best value under the same criterion) and distance measurement (the distance between the ideal solution and other alternative solutions). 6—data that are ambiguous or unclear (fuzziness) [33]. In Table 10, it can be observed that the 2-tuple-FDOSM was found to be more resilient in comparison to the TOPSIS method.

Table 10. Comparison between 2-tuple-FDOSM and TOPSIS.

No.	Comparison Issue	2-Tuple-FDOSM	TOPSIS
1	Missing information	√	×
2	Immeasurable value	√	×
3	Weight	√	×
4	Normalization	√	×
5	Ideal solution and distance measurement	√	×
6	Ambiguous and vague information	√	√

Although TOPSIS only handled vague and ambiguous information (n = 1/6), the 2-tuple-FDOSM was able to solve all the previously described difficulties (n = 6/6).

7. Conclusions

The Fuzzy Decision by Opinion Score method (FDOSM) is one of the latest methods in the multicriteria decision-making approach. Many researchers have used the FDOSM to solve different MCDM problems, and have extended the FDOSM into another fuzzy environment. However, it has previously been shown that the FDOSM suffers from information loss when combined with opinion matrices. In accordance with the advantages that 2-tuples display in solving the information loss problem, in this article, we presented a new extension of the FDOSM with a 2-tuple environment, namely the 2-tuple-FDOSM. Here, through the methodology of this research, we presented the extension of the FDOSM into the 2-tuple-FDOSM. Additionally, we applied the 2-tuple-FDOSM to a network case study. The final results show that the 2-tuple-FDOSM addresses two main issues in the basic FDOSM; these issues are information loss during aggregation with opinion matrices, and an issue that caused alternatives to be given the same ranks, which occurred due to uncertainty. Finally, the 2-tuple-FDOSM was shown to be more flexible in dealing with uncertainty. In future work, researchers can apply the 2-tuple-FDOSM to solve problems that were solved with the basic FDOSM, and compare the final ranks to select the best solutions. In addition, researchers can extend the FDOSM into other fuzzy environments and compare the results with those of the basic FDOSM and 2-tuple-FDOSM.