Open Access
 Issue EPJ Appl. Metamat. Volume 11, 2024 4 7 https://doi.org/10.1051/epjam/2024005 01 March 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 2 The RAM method

The steps for using the RAM method to rank alternatives are as follows [29].

Step 1: Construct a decision matrix with m rows and n columns, where m and n correspond to the number of alternatives to be ranked and the number of criteria for each alternative, respectively. Let xij represent the value of criterion j for alternative i, where j = 1 ÷ n and i = 1 ÷ m. The letters B and C are used to denote the respective criteria for benefit and cost.

Step 2: Normalize the data using formula (1).

(1)

Step 3: Calculate the normalized values, considering the weights of the criteria, according to (2).

(2)

where wj is the weight of the jth criterion.

Step 4: Calculate the total normalized score, taking into account the criteria weights using (3) and (4).

(3)

(4)

Step 5: Calculate the score for each alternative according to (5).

(5)

Step 6: Rank the alternatives in descending order based on their scores.

## 3 Criteria weight determination methods

Four criteria weight determination methods have been employed, including the Equal method, the Entropy method, the MEREC method, and the LOPCOW method. The Equal method is the simplest [30], while Entropy and MEREC are two recommended methods [31], and LOPCOW is considered the most recent approach [32].

When using the Equal weight method, all criteria have equal values [30].

To calculate criteria weights using the Entropy method, you need to apply the formulas sequentially from (6) to (8) [31].

(6)

(7)

(8)

Applying the formulas sequentially from (9) to (14) to calculate criteria weights using the MEREC method [31].

(9)

(10)

(11)

(12)

(13)

(14)

To calculate criteria weights using the LOPCOW method, you need to apply the formulas sequentially from (15) to (17) [32]. In (16), σ represents the standard deviation.

(15)

(16)

(16)

(17)

## 4 Selecting the optimal solution for flame retardant nanocomposite material fabrication

The selection of the best solution for manufacturing flame retardant materials is based on the results of a previous experimental study [33]. Seven solutions for manufacturing flame retardant materials by adding reinforcing agents have been implemented. Varying the percentage or type of reinforcing agent added to the original PVC material has created different solutions. To illustrate this, for instance, material “5ATH/PVC” means that 5% ATH was added to PVC. Another example is “10ZB/PVC,” which signifies the addition of 10% ZB to the PVC material. Table 1 displays information on eight types of flame retardant materials, with the first type being the original PVC material and the other seven being PVC materials with added reinforcing agents.

The pricing for each solution is determined using five parameters, which include tensile strength (C1), elongation at break (C2), elastic modulus (C3), total burning time (C4), and the limiting oxygen index to sustain combustion (C5). The physical properties (C1, C2, and C3) were measured according to ASTM D638 standards. Larger values of all three parameters are considered better (Type B criteria). Fire resistance (C4) was assessed according to ASTM D 3801 standards, and smaller values are preferable (Type C criteria). The limiting oxygen index (C5) is the minimum volume percentage of oxygen required to sustain the combustion of the material. This parameter is determined according to ASTM D 2863-97 standards, and larger values are better, indicating Type B criteria. All values of these parameters for various types of nanocomposite fire-resistant materials have been summarized in Table 1 [33].

Observing Table 1, we can see that the PVC base material has the highest values for both C1 and C2 among the eight options. In contrast, C3 and C5 have the highest values for the 15ATH/PVC material, and C4 has the smallest value, also belonging to this material type. This implies that there is no single material where all five parameters are the best. Determining a material choice requires considering the trade-off between all these parameters. This is an act of multi-criteria decision making (MCDM). To perform MCDM actions, the first step is to determine the weights for the five criteria (C1 to C5).

Formulas in Section 3 have been applied to calculate the weights for the criteria using four different methods, and the results have been consolidated in Table 2. We can observe that when using different methods, the weights of the criteria change significantly. The last row of this table demonstrates the percentage change in the weights of each criterion between the minimum and maximum values.

The application of the RAM method to determine the best solution is carried out in the following sequence. Normalized values have been calculated as per (1) and are summarized in Table 3.

The normalized values, taking into account the weights of the criteria, have been calculated as per (2). First, the set of criteria weights was computed using the Equal method. The computed results have been consolidated in Table 4.

The quantities S+i, S−i , RIi have been calculated using the respective formulas (3)(5). The results after calculation have been consolidated in Table 5. This table also lists the rankings of the options based on the values of RIi.

Thus, the ranking of options when the criteria weights are calculated using the Equal method has concluded. The ranking of options when the criteria weights are calculated using the remaining three methods (entropy, MEREC, LOPCOW) has also been conducted similarly. In Figure 1, the chart illustrates the rankings of various options when the weights of the criteria are determined using different methods.

According to the data in Figure 1, in all four different scenarios (corresponding to four different methods for determining weights), the 15ATH/PVC material consistently emerges as the best material, while PVC is consistently identified as the worst material. It's worth noting that the weights of the criteria vary significantly when calculated using different methods (refer to the last row of Tab. 2). However, this factor does not alter the determination of the best solution. This provides strong evidence that 15ATH/PVC is the best material among the eight surveyed. In other words, adding 15% ATH to the base PVC is the best solution for producing fire-resistant materials. Looking back at Table 1, we observe that the tensile strength (C1) and elongation at break (C2) of 15ATH/PVC are both inferior to the base PVC material. However, the reduction in these parameters for 15ATH/PVC compared to the base PVC is negligible, at 8.56% for C1 and 5.32% for C2. In contrast, two modulus parameters, elastic modulus (C3) and the limiting oxygen index for flame maintenance (C5) of 15ATH/PVC, are better than those of the base PVC. C3 of 15ATH/PVC has increased by 11.63% compared to the base PVC, while C5 of 15ATH/PVC has increased by 25.49% compared to the base PVC. Particularly noteworthy is the flame resistance (C4) of 15ATH/PVC, which at 0.5 (s), is only 1/15th of that of the base PVC. This indicates that 15ATH/PVC excels in quality compared to the base PVC.

To conclude the process of using MCDM methods to rank the options, sensitivity analysis should be performed [34, 35]. The Spearman rank correlation coefficient has also been utilized for sensitivity analysis [34, 35], and it is calculated using formula (18). In this formula, Di represents the rank differences of options in a specific scenario compared to another scenario.

(18)

Table 6 summarizes the Spearman rank correlation coefficient values when comparing scenarios with each other.

The smallest value of the Spearman rank correlation coefficient in Table 6 is 0.8333, indicating that the rankings of options change very little in different situations [36]. It's also essential to emphasize that when using different methods to determine the weights for the criteria, the weight values of the criteria vary significantly (refer to the last row of Tab. 2). This further clarifies the advantage of minimizing the phenomenon of rank reversal as proposed by the RAM method [31].

With all the results obtained from the analyses above, it's evident that when using the RAM method to rank options, the best approach is not dependent on the weights of the criteria. Adding 15% of the ATH reinforcement material to PVC is the optimal solution for manufacturing fire-resistant materials.

Table 1

Some types of fire-resistant nanocomposite materials [33].

Table 2

Weights of the criteria.

Table 3

Normalized values in the RAM method.

Table 4

Normalized values considering criteria weights.

Table 5

Some parameters in RAM and the ranking of options.

 Fig 1Ranking of options.
Table 6

Spearman rank correlation coefficients.

## 5. Conclusion

The addition of ATH and ZB reinforcement materials to the base PVC enhances the fire resistance of the compounds. The optimization of the optimal solution regarding the type and amount of reinforcement material has been performed in this study. RAM and four methods for determining criteria weights were concurrently employed to achieve this objective. This research demonstrates that the use of RAM in combination with four different methods for determining weights has led to a stable result, with the same material option being consistently considered the best. Specifically, in all scenarios, the option containing 15% aluminum hydroxide reinforcement material (15ATH/PVC) was rated the highest. This showcases the consistency and stability of the RAM method in selecting the optimal solution in this case. This result also confirms the advantages of the RAM method as proposed, highlighting its capability to balance between favorable and unfavorable criteria and mitigate the occurrence of rank reversal.

In the future, several tasks need to be addressed, including: the selection of nanocomposite materials considering even more criteria; optimizing the production process of 15ATH/PVC nanocomposite materials to ensure the best performance and cost efficiency; conducting tests to ensure the stability and safety of 15ATH/PVC materials in fire-resistant environments and other practical applications; further exploration of different reinforcement materials to enhance the quality and fire resistance of PVC-based nanocomposite materials.

## Funding

This research received no external funding.

## Conflicts of interest

The author have nothing to disclose.

## Data availability statement

Data sets generated during the current study are available from the corresponding author on reasonable request.

## Author contribution statement

I am the sole author of the paper.

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Cite this article as: Do Duc Trung, Using RAM method for optimal selection of flame retardant nanocomposite material fabrication solution. EPJ Appl. Metamat. 11, 4 (2024)

## All Tables

Table 1

Some types of fire-resistant nanocomposite materials [33].

Table 2

Weights of the criteria.

Table 3

Normalized values in the RAM method.

Table 4

Normalized values considering criteria weights.

Table 5

Some parameters in RAM and the ranking of options.

Table 6

Spearman rank correlation coefficients.

## All Figures

 Fig 1Ranking of options. In the text

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