**1.1 Intuitionistic fuzzy set**

Let X be a fixed set. An IFS *<sup>A</sup>*<sup>~</sup> in X is of the form *<sup>A</sup>*<sup>~</sup> <sup>¼</sup> <sup>&</sup>lt; *<sup>x</sup>*, *<sup>μ</sup>A*<sup>~</sup> ð Þ *<sup>x</sup>* , *vA*<sup>~</sup> ð Þ *<sup>x</sup>* <sup>&</sup>gt; : *<sup>x</sup>*<sup>∈</sup> *<sup>X</sup>* , where the *μA*<sup>~</sup> ð Þ *x* : *X* ! ½ � 0, 1 and *νA*<sup>~</sup> ð Þ *x* : *X* ! ½ � 0, 1 . This represents the degree of membership and of non membership respectively of the element *x*∈*X* to the set *A*~, which is a subset of the set X, for every element of *x*∈*X*, 0 ≤*μA*<sup>~</sup> ð Þþ *x vA*<sup>~</sup> ð Þ *x* ≤1 [22].

The value of πAð Þ¼ X 1 � μAð Þ� X vAð Þ X represents the degree of hesitation (or uncertainty) associated with the membership of elements x ɛ X in IFS A. This is known as the intuitionistic fuzzy index of A with respect to element x.

#### **1.2 Intuitionistic fuzzy number**

An IFN *A*~ is defined as follows [22]:

i. an intuitionistic fuzzy subset of the real line

ii. it is normal, i.e. there is any *<sup>x</sup>*<sup>0</sup> <sup>∈</sup>*<sup>R</sup>* such that *<sup>μ</sup>A*<sup>~</sup> ð Þ¼ *<sup>x</sup>* <sup>1</sup> *so vA*<sup>~</sup> ð Þ¼ *<sup>x</sup>* <sup>0</sup>

iii. a convex set for the membership function *μA*<sup>~</sup> ð Þ *x* i.e.

$$
\mu\_{\vec{A}}(\lambda \mathbf{x}\_1 + (1 - \lambda)\mathbf{x}\_2) \ge \min\left(\mu\_{\vec{A}}(\mathbf{x}\_1), \mu\_{\vec{A}}(\mathbf{x}\_2)\right) \forall \mathbf{x}\_1, \mathbf{x}\_2 \in R\lambda \in [0, 1],
$$

iv. a concave set for the non-membership function *vA*<sup>~</sup> ð Þ *x* i.e.

$$
v\_{\vec{A}}\left(\lambda\mathbf{x}\_1 + (1-\lambda)\mathbf{x}\_2\right) \le \max\left(v\_{\vec{A}}(\mathbf{x}\_1), v\_{\vec{A}}(\mathbf{x}\_2)\right) \forall \mathbf{x}\_1, \mathbf{x}\_2 \in R, \lambda \in [0, 1]
$$

A triangular intuitionistic fuzzy number *<sup>A</sup>*<sup>~</sup> <sup>¼</sup> *<sup>a</sup>*1, *<sup>a</sup>*2, *<sup>a</sup>*3; *<sup>a</sup>*<sup>0</sup> 1, *a*2, *a*<sup>0</sup> 3 is a subset of intuitionistic fuzzy set on the set of real number R whose membership and non membership are defined as follows:

**Figure 2.** *A triangular intuitionistic fuzzy number.*

*Evaluating the Organizational Hierarchy Using the IFSAW and TOPSIS Techniques DOI: http://dx.doi.org/10.5772/intechopen.95979*

$$\mu\_{\bar{A}}(\mathbf{x}) = \begin{cases} \frac{\mathbf{x} - a\_1}{a\_2 - a\_1}, & a\_1 < \mathbf{x} \le a\_2 \\\frac{a\_3 - \mathbf{x}}{a\_3 - a\_2}, & a\_2 < \mathbf{x} \le a\_3 \\\ 0, & otherwise \end{cases} \qquad v\_{\bar{A}}(\mathbf{x}) = \begin{cases} \frac{a\_2 - \mathbf{x}}{a\_2 - a\_1'}, & a\_1' < \mathbf{x} \le a\_2 \\\frac{\mathbf{x} - a\_2}{a\_3' - a\_2}, & a\_2 < \mathbf{x} \le a\_3' \\\ 1, & otherwise \end{cases}$$

Intuitionistic fuzzy set is widely recognised and is being studied and applied in various fields be it in science, psychology and other growing fields like consumer behaviour, advertising and communications where decision making is crucial (**Figure 2**).

In this work two methods of intuitionistic fuzzy sets viz. SAW (simple additive weight method) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) are used for ranking the various levels of employees in an organisation. The paper is organised as follows: Section 2 begins with the basic operations of intuitionistic fuzzy sets; Section 3 and 4 explain the intuitionistic fuzzy SAW algorithm and TOPSIS methodology which are used in the paper. Section 5 illustrates the procedure for evaluating the hierarchical level using the proposed algorithms. Section 6 is the final discussion and conclusion related to the evaluation procedure.

#### **2. Operations on intuitionistic fuzzy sets**

Let A and B are IFS s of the set X, then multiplication operator is defined as follows [19]:

$$\begin{aligned} A \otimes B &= \left[ \mu\_A(\mathbf{x}) \mu\_B(\mathbf{x})\_{V\_A}(\mathbf{x}) + \nu\_B(\mathbf{x}) - \nu\_A(\mathbf{x}) \nu\_B(\mathbf{x}), \mathbf{1} \\ &- \left\{ \mu\_A(\mathbf{x}) \, \mu\_B(\mathbf{x}) + (\nu\_A(\mathbf{x}) + \nu\_B(\mathbf{x}) - \nu\_A(\mathbf{x}) \nu\_B(\mathbf{x})) \right\} \end{aligned} \tag{1}$$

Let A = (μ, v) be an intuitionistic fuzzy number, a score function S of an intuitionistic fuzzy value can be represented as follows:

$$\mathbf{S(A)} = \mu \text{-v, S(A)} \, e[\cdot \mathbf{1}, \mathbf{1}] \tag{2}$$

If S (Ai) represents the largest among the values of {**S(**Ai**)**}, then the alternative Ai is the best choice.

#### **3. Intuitionistic fuzzy simple additive weighting algorithm**

This method is a simple additive weighting method developed by Hwang and Yoon [23]. According to this principle the first step ensures in obtaining a weighted sum of the performance ratings of each alternative under all attributes. Let A1, A2, A3, … , An be n alternatives which denotes the employee cadres. Let C1, C2, C3, … , Cm, be the criteria on the basis of which the evaluation is done. Further each criteria is assigned weight given by the decision makers and it is represented by a weighting vector W = {W1, W2, W3, … , Wn}, where W1, W2, W3, … , Wn are represented by intuitionistic fuzzy sets defined as follows:

$$\mathbf{W}\mathbf{j} = \left\{ \mu\_{\mathbf{w}}(\mathbf{x}\_{\mathbf{j}}), \mathbf{v}\_{\mathbf{w}}(\mathbf{x}\_{\mathbf{j}}), \pi\_{\mathbf{w}}(\mathbf{x}\_{\mathbf{j}}) \right\}, \text{where } \mathbf{j} = \mathbf{1}, 2, \dots, \mathbf{n}. \tag{3}$$

The procedure for Intuitionistic fuzzy SAW is being presented as follows:

**Step 1**: Construct an intuitionistic fuzzy decision matrix: *R* **= (***rij***)mxn** such that <sup>~</sup>*rij* <sup>¼</sup> *<sup>μ</sup>ij*, *<sup>ν</sup>ij*, *<sup>π</sup>ij* � �

$$
\tilde{R} = \begin{bmatrix}
\tilde{r}\_{11} & \tilde{r}\_{12} & \dots & \tilde{r}\_{1n} \\
\tilde{r}\_{21} & \tilde{r}\_{22} & \dots & \tilde{r}\_{2n} \\
\dots & \dots & \dots & \dots \\
\tilde{r}\_{m1} & \tilde{r}\_{m2} & \dots & \tilde{r}\_{mn}
\end{bmatrix}
$$

(i = 1,2, … ,m; j = 1,2, … ,n),. In ~*rij***,** *μij* indicates the degree that the alternative Ai satisfies Cj and *νij* indicates the degree that the alternative Ai does not satisfy the attribute Cj.

**Step 2:** This step entails performing the transformation by using Eq. (1) and obtain the total intuitionistic fuzzy scores V (Ai) for individual vendors. This is determined by the product of intuitionistic fuzzy weight vectors (W) and intuitionistic fuzzy rating matrix (R).

$$\mathbf{V(A\_i)} = \mathbf{R} \odot \mathbf{W} = \sum\_{i=1} \left[ \{ \mu\_{A\_i}(\mathbf{x}\_j), \nu\_{A\_i}(\mathbf{x}\_j), \pi\_{A\_i}(\mathbf{x}\_j) \} \otimes \{ \mu\_w(\mathbf{x}\_j), \nu\_w(\mathbf{x}\_j), \pi\_{w\_i}(\mathbf{x}\_j) \} \right] \tag{4}$$

**Step 3:** The third step is used for ranking the alternatives. Applying Eq. (**2**) a crisp score function S(A1),S(A2), … ,S(An) is calculated for the various alternatives. The largest value of S (Aj) among S(A1),S(A2), … ,S(An) represents the best alternative or vendor.

**Step 4:** This approach is compared with Jun Ye [24] on weighted correlation coefficient under intuitionistic fuzzy environment.
