**2.2 Characterization of a composite of pure aluminum and graphene nanoplatelets**

Physical and mechanical tests on the fabricated composite were used to classify the pure aluminum GNPs composite. These experiments were carried out on wellprepared specimens prepared according to ASTM Standards such as ASTM E8 & ASTM A370 for Tensile Test and Impact test, to investigate the impact of reinforcements in pure aluminum for aircraft applications.

## **3. Decision methods**

A multi-attribute decision-making problem is represented by a matrix X, which contains n alternatives and m criteria.

$$\mathbf{X} = \begin{pmatrix} \mathbf{X\_{11}} & \mathbf{X\_{12}} & \mathbf{X\_{13}} & \cdots & \cdots & \cdots & \mathbf{X\_{1m}} \\ \mathbf{X\_{21}} & \mathbf{X\_{22}} & \mathbf{X\_{23}} & \cdots & \cdots & \cdots & \mathbf{X\_{2m}} \\ \mathbf{X\_{31}} & \mathbf{X\_{32}} & \mathbf{X\_{33}} & \cdots & \cdots & \cdots & \mathbf{X\_{3m}} \\ \mathbf{X\_{41}} & \mathbf{X\_{42}} & \mathbf{X\_{43}} & \cdots & \cdots & \cdots & \mathbf{X\_{4m}} \\ \hline \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \mathbf{X\_{n1}} & \mathbf{X\_{n2}} & \mathbf{X\_{n3}} & \cdots & \cdots & \cdots & \mathbf{X\_{nm}} \end{pmatrix} \tag{1}$$

where Xij is the ith alternative's output on the jth criterion. In MADM methods, the weight value (wj) for each criterion must be determined such that the number of all criterion weights equals to one. The entropy approach is used to calculate these weights.

#### **3.1 Evaluation based on distance from average solution (EDAS) method**

The EDAS method is a distance-based technique that employs positive and negative distances from the average solution. The options are listed in ascending order. The procedural steps for n alternative composite and m parameters suggested [27, 28] are as follows:

$$\text{Let } [\mathbf{X}]\_{\mathbf{n}\times\mathbf{m}} \text{ be the decision} - \text{making matrix} \tag{2}$$

**Step 1**. The average solution (AVj).

$$\mathbf{AV}\_{\mathbf{j}} = \sum\_{i=1}^{n} \mathbf{X}\_{\mathbf{ij}} / \mathbf{n} \tag{3}$$

**Step 2**. The positive distance from average (PDA) for beneficial and non-beneficial criterion.

$$\begin{aligned} \text{If } \mathbf{j}^{\text{th}} \text{ criterion is beneficial,} \\ \text{PDA}\_{\text{ij}} &= \frac{\max \left( \mathbf{0}, \left( \mathbf{X}\_{\text{ij}} - \mathbf{A} \mathbf{V}\_{\text{j}} \right) \right)}{\mathbf{A} \mathbf{V}\_{\text{j}}} \\ \text{And if } \mathbf{j}^{\text{th}} \text{ criterion is non-bineficial,} \\ \text{PDA}\_{\text{ij}} &= \frac{\max \left( \mathbf{0}, \left( \mathbf{A} \mathbf{V}\_{\text{j}} - \mathbf{X}\_{\text{ij}} \right) \right)}{\mathbf{A} \mathbf{V}\_{\text{j}}} \end{aligned} \tag{4}$$

*Selection of Optimal Material from Stir Cast Aluminum Graphene Nano Platelets… DOI: http://dx.doi.org/10.5772/intechopen.100478*

**Step 3.** The negative distance from average (NDA) for beneficial and non-beneficial criterion.

$$\begin{aligned} \text{If } \mathbf{j}^{\text{th}} \text{ criterion is beneficial,} \\ \text{NDA}\_{\text{ij}} &= \frac{\max \left( \mathbf{0}, (\mathbf{A} \mathbf{V}\_{\text{j}} - \mathbf{X}\_{\text{ij}}) \right)}{\mathbf{A} \mathbf{V}\_{\text{j}}} \\ \text{And if } \mathbf{j}^{\text{th}} \text{ criterion is non-bineficial,} \\ \text{NDA}\_{\text{ij}} &= \frac{\max \left( \mathbf{0}, (\mathbf{X}\_{\text{ij}} - \mathbf{A} \mathbf{V}\_{\text{j}}) \right)}{\mathbf{A} \mathbf{V}\_{\text{j}}} \end{aligned} \tag{5}$$

**Step 4.** Eq. (6) calculates the positive and the negative weighted sums.

$$\text{SP}\_{\text{i}} = \sum\_{\text{j}=1}^{\text{m}} \left( \text{W}\_{\text{j}} \* \text{PDA}\_{\text{ij}} \right) \tag{6}$$

$$\text{SNN}\_{\text{i}} = \sum\_{\text{j=1}}^{\text{m}} \left( \text{W}\_{\text{j}} \* \text{NDA}\_{\text{i}\textsuperscript{\text{j}}} \right) \tag{7}$$

Wj = weight of jth criterion.

**Step 5.** Normalized values of SPi and SNi.

$$\text{NSP}\_{\text{i}} = \frac{\text{SPi}}{\max \, i \, \text{SPi}} \tag{8}$$

$$\text{NSNi} = \frac{\text{SNi}}{\max i \text{SNi}} \tag{9}$$

**Step 6.** The appraisal score (ASi) for all alternatives is calculated using Eqs. (8) and (9).

$$\mathbf{AS}\_{\mathbf{i}} = \mathbf{0}.\mathbf{5} \ast (\mathbf{N}\mathbf{S}\mathbf{i} + \mathbf{N}\mathbf{S}\mathbf{N}\_{\mathbf{i}}) \tag{10}$$

Where 0 ≤ ASi ≤ 1.

The alternative with the outstanding appraisal score is chosen as the best, among the other selective alternatives.

#### **3.2 VIse Kriterijumska Optimizacija kompromisno Resenje (VIKOR) method**

According to the VIKOR technique given by Rao [29] and Chatterjee et al. [30] for material selection, the compromise may be common for resolving the dispute, and the practicable option may be nearest to the ideal solution. The options are primarily assessed based on all of the characteristics taken into account.

**Step 1.** Determine the decisive criteria and select the best alternatives based on those criteria.

**Step 1: (a)** In the decision matrix, determine the best, (xij) max and the worst, (xij) min values of all the criteria.

$$\mathbf{X}\_{\mathbf{i}}^{+} = \max\_{\mathbf{i}} \left( \mathbf{X}\_{\mathbf{i}\mathbf{j}} \right) \left[ \mathbf{J} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{m} \right] \tag{11}$$

$$\mathbf{X}\_{\mathbf{i}}^{-} = \min\_{\mathbf{i}} \left( \mathbf{X}\_{\mathbf{i}\mathbf{j}} \right) \left[ \mathbf{J} = \mathbf{1}, 2, \dots, \mathbf{m} \right] \tag{12}$$

**Step 2:** Compute the values of utility measure (Si) and regret measure (Ri).

$$\mathbf{S}\_{\mathbf{i}} = \sum\_{\mathbf{j}=1}^{m} \left( \mathbf{W}\_{\mathbf{j}}^{\*} \left( \left( \mathbf{X}\_{\mathbf{i}}^{+} - \mathbf{X}\_{\mathbf{i}\mathbf{j}} \right) / \left( \mathbf{X}\_{\mathbf{i}}^{+} - \mathbf{X}\_{\mathbf{i}}^{-} \right) \right) \right) \tag{13}$$

$$\mathbf{R}\_{\mathrm{i}} = \max\_{\mathbf{j}} \left( \mathbf{W}\_{\mathrm{j}}^{\*} \left( \left( \mathbf{X}\_{\mathrm{i}}^{+} - \mathbf{X}\_{\mathrm{i}\natural} \right) / \left( \mathbf{X}\_{\mathrm{i}}^{+} - \mathbf{X}\_{\mathrm{i}}^{-} \right) \right) \right) \tag{14}$$

**Step 3:** Find the values of S\* , S�, *R\**, R�.

$$\mathbf{S}\* = \min\_{\mathbf{i}} \left( \mathbf{S}\_{\mathbf{i}} \right) \\ \mathbf{S}^{-} = \max\_{\mathbf{i}} \left( \mathbf{S}\_{\mathbf{i}} \right) \tag{15}$$

R ∗ ¼ min <sup>i</sup>ð Þ Ri R� ¼ max <sup>i</sup>ð Þ Ri (16)

**Step 4.** Compute the value of Q <sup>i</sup> for j = 1, 2, … , m.

$$\mathbf{Q}\_{i} = \mathbf{N} \* \left( (\mathbf{S}\_{i} - \mathbf{S} \*) / (\mathbf{S}^{-} - \mathbf{S} \*) \right) + (\mathbf{1} - \mathbf{N}) \* \left( (R\_{i} - R \*) / (R^{-} - R \*) \right) \tag{17}$$

The alternatives are ranked in the ascending order for the Q <sup>i</sup> values. The one having the lower value of Q <sup>i</sup> is considered as the best alternative.
