**4. Principle of TOPSIS for decision making with intuitionistic fuzzy set**

TOPSIS methodology is proposed by [25]. The fundamental principle underlying this theory is that the alternative which is chosen entails that it has the least distance from the positive ideal- solution (i.e. alternative) and its distance is the farthest from the negative ideal- solution (i.e. alternative).

Suppose there exists n decision making alternatives given by the set *A* = {*A*1*, A*2*,. .., An*} from which a most preferred alternative is to be selected. These are assessed based on m attributes, both quantitative and qualitative. The set of all attributes is denoted by *X* = {*x*1*, x*2*,. .., xm*}. The ratings of different alternatives Aj on attributes xi are expressed with intuitionistic fuzzy sets *Fij* <sup>¼</sup> *<sup>μ</sup>ij*, *<sup>ν</sup>ij* � � where *<sup>μ</sup>ij* <sup>∈</sup>½ � 0, 1 , *<sup>ν</sup>ij* <sup>∈</sup>½ � 0, 1 and 0 ≤*μij* þ *νij* ≤1. Thus, the ratings of any alternatives Aj on all m attributes xi are expressed with intuitionistic fuzzy vector *μ*1*<sup>j</sup>*, *ν*1*<sup>j</sup>* D E, *<sup>μ</sup>*2*<sup>j</sup>* , *ν*2*<sup>j</sup>* DD E, *::* … , *<sup>μ</sup>mj*, *<sup>ν</sup>mj* � � D E *<sup>T</sup>*.

The intuitionistic fuzzy decision matrix is represented as *<sup>F</sup>* <sup>¼</sup> *<sup>μ</sup>ij*, *<sup>ν</sup>ij* � � � � *mxn*

$$= \begin{bmatrix} (\mu\_{11}, \nu\_{11}) & (\mu\_{12}, \nu\_{12}) & \dots & \dots & (\mu\_{1n}, \nu\_{1n}) \dots \\ (\mu\_{21}, \nu\_{21}) & (\mu\_{22}, \nu\_{22}) & \dots & (\mu\_{2n}, \nu\_{2n}) \\ & \dots & \dots & \dots & \dots \\ (\mu\_{m1}, \nu\_{m1} & (\mu\_{m2}, \nu\_{m2}) & \dots & (\mu\_{mn}, \nu\_{mn}) \end{bmatrix} \tag{5}$$

It is assumed that the weights *ω<sup>i</sup>* of the attributes *xi* ∈*X* are real numbers known a priori i.e. the weight vector *<sup>ω</sup>* <sup>¼</sup> ð Þ *<sup>ω</sup>*1, *<sup>ω</sup>*2,*ω*3, … *<sup>ω</sup><sup>m</sup> <sup>T</sup>* of attributes are known.

Since the weights of the attributes are not precisely defined therefore they are treated as intuitionistic fuzzy sets i.e. the weight of each factor is expressed with the *Evaluating the Organizational Hierarchy Using the IFSAW and TOPSIS Techniques DOI: http://dx.doi.org/10.5772/intechopen.95979*

intuitionistic fuzzy set *ω<sup>i</sup>* ¼ *xi*, *αi*, *β<sup>i</sup>* f g h i where *α<sup>i</sup>* ∈ ½ � 0, 1 and *β<sup>i</sup>* ∈½ � 0, 1 are respectively the degree of membership and non membership respectively of the attribute *xi* ∈ *X*. Usually *ω<sup>i</sup>* ¼ *xi*, *αi*, *β<sup>i</sup>* f g h i is denoted by *ω<sup>i</sup>* ¼ *αi*, *β<sup>i</sup>* h i in short. The weight of all attributes is concisely expressed in the vector format as follows:

$$\begin{aligned} \boldsymbol{\alpha} &= \begin{pmatrix} \alpha\_1, \alpha\_2, \alpha\_3, \dots \alpha\_m \end{pmatrix}^T \\ &= \begin{pmatrix} \langle \alpha\_1, \beta\_1 \rangle, \langle \alpha\_2, \beta\_2 \rangle, \dots \langle \alpha\_m, \beta\_m \rangle \end{pmatrix}^T \end{aligned} \tag{6}$$

#### **4.1 Principle and process of TOPSIS**

The entire methodology can be summarized as follows:


$$F = \left( \left< \mu\_{\vec{\eta}}, \nu\_{\vec{\eta}} \right> \right)\_{m \ge n} \tag{7}$$


$$
\begin{aligned}
\left<\mu\_{\vec{\imath}\vec{\jmath}},\nu\_{\vec{\imath}\vec{\jmath}}\right> &= aF\_{\vec{\imath}} \\ &= \left<\alpha\_i,\beta\_i\right>\left<\mu\_{\vec{\imath}\vec{\jmath}},\nu\_{\vec{\imath}\vec{\jmath}}\right> \\ &= \left<\alpha\_i\mu\_{\vec{\imath}\vec{\jmath}},\beta\_i+\nu\_{\vec{\imath}\vec{\jmath}}-\beta\_i\nu\_{\vec{\imath}\vec{\jmath}}\right>
\end{aligned}
\tag{8}
$$

5.For calculating the intuitionistic fuzzy positive ideal –solution and intuitionistic fuzzy negative ideal –solution the following formulas are obtained

$$\begin{aligned} A^{+} &= \left( \langle \mu\_1^{+}, \nu\_1^{+} \rangle, \langle \mu\_2^{+}, \nu\_2^{+} \rangle, \dots \langle \mu\_m^{+}, \nu\_m^{+} \rangle \right)^T \\ A^{-} &= \left( \langle \mu\_1^{-}, \nu\_1^{-} \rangle, \langle \mu\_2^{-}, \nu\_2^{-} \rangle, \dots \langle \mu\_m^{-}, \nu\_m^{-} \rangle \right)^T \end{aligned} \tag{9}$$
 
$$where \ \mu\_i^{+} = \max \mathbf{1}\_{\leq j \leq n} \left\{ \mu\_{\hat{\boldsymbol{\eta}}} \right\} \ \nu\_i^{+} = \min \mathbf{1}\_{1 \leq j \leq n} \left\{ \nu\_{\hat{\boldsymbol{\eta}}} \right\} \tag{10}$$
 
$$\mu\_i^{-} = \min\_{1 \leq j \leq n} \left\{ \mu\_{\hat{\boldsymbol{\eta}}} \right\} \ \nu\_i^{-} = \max\_{1 \leq j \leq n} \left\{ \nu\_{\hat{\boldsymbol{\eta}}} \right\}$$

6.The Euclidean distances of the various alternatives *A <sup>j</sup>*(j = 1,2, … n) from the intuitionistic fuzzy positive ideal and intuitionistic fuzzy negative ideal solution are computed using the following equations

$$\mathbf{D}(\mathbf{A}\_{\mathrm{j}},\mathbf{A}^{+}) = \sqrt{\frac{1}{2} \left( \sum\_{i=1}^{m} \left[ \left( \mu\_{\mathrm{ij}} - \mu\_{\mathrm{i}}^{+} \right)^{2} + \left( \nu\_{\mathrm{ij}} - \nu\_{\mathrm{i}}^{+} \right)^{2} + \left( \pi\_{\mathrm{ij}} - \pi\_{\mathrm{i}}^{+} \right)^{2} \right] \right)} \tag{11}$$

$$\mathbf{D}(\mathbf{A}\_{\mathbf{j}},\mathbf{A}^{-}) = \sqrt{\frac{1}{2} \left( \sum\_{i=1}^{m} \left[ \left(\mu\_{\mathbf{i}\mathbf{j}} - \mu\_{\mathbf{i}}^{-}\right)^{2} + \left(\nu\_{\mathbf{i}\mathbf{j}} - \nu\_{\mathbf{i}}^{-}\right)^{2} + \left(\pi\_{\mathbf{i}\mathbf{j}} - \pi\_{\mathbf{i}}^{-}\right)^{2} \right]} \right)} \tag{12}$$

7.Thereafter the relative closeness degree *λ <sup>j</sup>* of the alternatives Aj (j = 1,2 … ,n) to the intuitionistic fuzzy positive ideal solution are obtained from

$$\lambda\_j = \frac{D(A\_j, A^-)}{D(A\_j, A^+) + D(A\_j, A^-)}, j = 1, 2, \dots, n \tag{13}$$

8.Lastly determine the ranking order of the alternatives Aj (j = 1,2 … n) according to the non increasing order of the relative closeness degrees *λ <sup>j</sup>* and the best alternative from A.

Using the two approaches the different level of workers in the organisation are assessed. For a better understanding of the situation an example is worked out below:

### **5. Numerical example**

The example is illustrated as below:

An organization has employed six decision making criteria in order to select the most effective hierarchical level in an organization based on the following criterions.


The hierarchical levels of an organization were broadly restricted to four and were compared based on the six decision making criteria (as indicated in **Table 1**).


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

The employees are rated (A, B, C) based on the judgement provided by experts in the organisation .


The intuitionistic fuzzy decision matrix has been constructed as below (**Table 2**):

#### **Table 2.**

*Intuitionistic fuzzy decision matrix.*

The weights for the criteria are as below:


**Table 3.** *Weights of the criteria.*

The total intuitionistic fuzzy score V(HLi) for each hierarchical level is calculated as follows:

V HL ð Þ¼ <sup>1</sup> ½ð Þ *:*7*; :*1*; :*2 ∗ ð Þ *:*2*; :*4*; :*4 � þ ½ð Þ *:*5*; :*3*; :*2 ∗ ð Þ *:*2*; :*2*; :*6 � þ ½ � ð Þ *:*8*; :*1*; :*1 ∗ ð Þ *:*1*; :*5*; :*4 þ½ð Þ *:*7*; :*2*; :*1 ∗ ð*:*5, *:*3, *:*2Þ� þ ½ð Þ *:*5*; :*3*; :*2 ∗ ð Þ *:*3*; :*4*; :*3 � þ ½ � ð Þ *:*8*; :*1*; :*1 ∗ ð Þ *:*2*; :*4*; :*4 V HL ð Þ¼ <sup>1</sup> <sup>½</sup>*:*<sup>7</sup> <sup>∗</sup> *:*2*; :*<sup>1</sup> <sup>þ</sup> *:*4‐*:*<sup>1</sup> <sup>∗</sup> *:*4*;* <sup>1</sup>‐ð Þ *:*<sup>7</sup> <sup>∗</sup> *:*<sup>2</sup> <sup>þ</sup> *:*<sup>1</sup> <sup>þ</sup> *:*4‐*:*<sup>1</sup> <sup>∗</sup> *:*<sup>4</sup> �þ½*:*<sup>5</sup> <sup>∗</sup> *:*2, *:*<sup>3</sup> <sup>þ</sup> *:*2‐ *:*<sup>3</sup> <sup>∗</sup> *:*2, 1‐ð*:*<sup>5</sup> <sup>∗</sup> *:*<sup>2</sup> <sup>þ</sup> *:*<sup>3</sup> <sup>þ</sup> *:*2‐*:*<sup>3</sup> <sup>∗</sup> *:*2Þ� þ ½ � *:*<sup>8</sup> <sup>∗</sup> *:*1*; :*<sup>1</sup> <sup>þ</sup> *:*5‐*:*<sup>1</sup> <sup>∗</sup> *:*5*;* <sup>1</sup>‐ð Þ *:*<sup>8</sup> <sup>∗</sup> *:*<sup>1</sup> <sup>þ</sup> *:*<sup>1</sup> <sup>þ</sup> *:*5‐*:*<sup>1</sup> <sup>∗</sup> *:*<sup>5</sup> þ½*:*<sup>7</sup> <sup>∗</sup> *:*5, *:*<sup>2</sup> <sup>þ</sup> *:*3‐*:*<sup>2</sup> <sup>∗</sup> *:*3, 1‐ð*:*<sup>7</sup> <sup>∗</sup> *:*<sup>5</sup> <sup>þ</sup> *:*<sup>2</sup> <sup>þ</sup> *:*3‐*:*<sup>2</sup> <sup>∗</sup> *:*3Þ� þ ½ � *:*<sup>5</sup> <sup>∗</sup> *:*3*; :*<sup>3</sup> <sup>þ</sup> *:*4‐*:*<sup>3</sup> <sup>∗</sup> *:*4*;* <sup>1</sup>‐ð Þ *:*<sup>5</sup> <sup>∗</sup> *:*<sup>3</sup> <sup>þ</sup> *:*<sup>3</sup> <sup>þ</sup> *:*4‐*:*<sup>3</sup> <sup>∗</sup> *:*<sup>4</sup> þ½ � *:*<sup>8</sup> <sup>∗</sup> *:*2*; :*<sup>1</sup> <sup>þ</sup> *:*4‐*:*<sup>1</sup> <sup>∗</sup> *:*4*;* <sup>1</sup>‐ð Þ *:*<sup>8</sup> <sup>∗</sup> *:*<sup>2</sup> <sup>þ</sup> *:*<sup>1</sup> <sup>þ</sup> *:*4‐*:*<sup>1</sup> <sup>∗</sup> *:*<sup>4</sup> V HL ð Þ¼ <sup>1</sup> ½ð Þþ *:*14, *:*46, *:*4 ð Þþ *:*1, *:*44, *:*46 ð Þþ *:*08, *:*55, *:*37 ð Þþ *:*35, *:*44, *:*21 ð Þþ *:*15, *:*58, *:*27 ð Þ *:*16, *:*46, *:*38 � V HL ð Þ¼ <sup>1</sup> ½ � 0*:*98, *:*013, *:*007

Similarly, the intuitionistic fuzzy scores for other hierarchical levels are calculated as:

$$\begin{aligned} \mathrm{V(HL\_2)} &= [\mathbf{0.99}, \mathbf{0.09}, \mathbf{0.01}] \\ \mathrm{V(HL\_3)} &= [\mathbf{0.82}, \mathbf{0.02}, \mathbf{.178}] \\ \mathrm{V(HL\_4)} &= [\mathbf{0.6}, \mathbf{0.28}, \mathbf{.372}] \end{aligned}$$

The score functions for each hierarchical level calculated using Eq. (2) stands as follows:

$$\mathbf{S(HL\_1)} = \mathbf{0.98-013} = \mathbf{0.967}$$

$$\mathbf{S(HL\_2)} = \mathbf{0.99-009} = \mathbf{0.981}$$

$$\mathbf{S(HL\_3)} = \mathbf{0.82-0.002} = \mathbf{0.818}$$

$$\mathbf{S(HL\_4)} = \mathbf{0.6-0.028} = \mathbf{0.572}$$

The hierarchical level with the largest score function value is HL2 i.e. the middle management.

The ranking order is as below:

$$\text{HL}\_2 > \text{HL}\_1 > \text{HL}\_3 > \text{HL}\_4$$

The ranking order for the hierarchical levels is in agreement with Jun Ye [24] result on weighted correlation coefficient under intuitionistic fuzzy environment. **The TOPSIS methodology**


The weights for the criteria are as mentioned in **Table 3**. The weighted IF decision matrix is obtained as:


$$\mathbf{A}^+ = \{ (0.\mathbf{35}, 0.04), (0.4\mathbf{5}, 0.02), (0.2\mathbf{5}, 0.02), (0.2\mathbf{1}, 0.08) \}.$$

$$\mathbf{A}^{-} = \left\{ \left( \mathbf{0.08, 0.12} \right), \left( \mathbf{0.06, 0.80} \right), \left( \mathbf{0.03, 0.25} \right), \left( \mathbf{0.03, 0.16} \right) \right\}$$

$$\mathbf{D}\_1(\mathbf{1}, \mathbf{A}^{+}) = \frac{1}{2} \begin{bmatrix} \left( 0.14 - 0.35 \right)^2 + \left( 0.04 - 0.04 \right)^2 + \left( 0.08 - 0.61 \right)^2 + \left( 0.14 - 0.45 \right)^2 + \\ \left( \left( 0.04 - 0.45 \right)^2 + \left( 0.04 - 0.02 \right)^2 + \left( 0.08 - 0.53 \right)^2 + \left( 0.10 - 0.25 \right)^2 + \left( 0.04 - 0.02 \right)^2 \end{bmatrix}$$

$$+ \left( 0.16 - 0.73 \right)^2 + \left( 0.10 - 0.21 \right)^2 + \left( 0.16 - 0.08 \right)^2 + \left( 0.04 - 0.71 \right)^2$$

$$= \frac{1}{2} \begin{bmatrix} 0.0441 + 0 + 0.2809 + 0.0961 + 0.0004 + 0.2025 + 0.0225 + 0.0004 + 0.3249 + 0.0121 \end{bmatrix}^{1/2}$$

$$= \frac{1}{2} \sqrt{1.4892}$$

$$= \frac{1}{2} \times 1.19966$$

$$= 0.59983$$

Similarly the other measures are calculated as follows:

$$\mathbf{D}(2,\mathbf{A}^+) = \mathbf{0.6251}$$

$$\mathbf{D}(3,\mathbf{A}^+) = \mathbf{0.6462}$$

$$\mathbf{D}(4,\mathbf{A}^+) = \mathbf{0.5925}$$

$$\mathbf{D}(5,\mathbf{A}^+) = \mathbf{0.80475}$$

$$\mathbf{D}(6,\mathbf{A}^+) = \mathbf{0.67749}$$

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

Also

$$\begin{aligned} \mathbf{D}(\mathbf{1},\mathbf{A}^{-}) &= \frac{1}{2} \begin{bmatrix} (0.14-0.08)^2 + (0.04-0.12)^2 + (0.08-0.80)^2 + (0.14-0.06)^2 + (0.14-0.06)^2 \\ (0.04-0.80)^2 + (0.08-0.14)^2 + (0.10-0.03)^2 + (0.04-0.25)^2 \\ + (0.16-0.53)^2 + (0.10-0.03)^2 + (0.16-0.16)^2 + (0.04-0.81)^2 \end{bmatrix} \\ &= \frac{1}{2} \begin{bmatrix} 0.0036 + 0 + 0.0064 + 0.5184 + 0.0064 + 0.5776 + 0.0036 + 0.0049 + 0 + 0.5929 \\ + 0.0064 + 0.4489 \end{bmatrix} \\ &= \frac{1}{2} \sqrt{17138} \\ &= 0.6545 \\ &= 0.6545 \\ &\mathbf{D}(3,\mathbf{A}^-) = 0.64033 \\ &\mathbf{D}(4,\mathbf{A}^-) = 0.57621 \\ &\mathbf{D}(5,\mathbf{A}^-) = 0.619394 \\ &\mathbf{D}(6,\mathbf{A}^-) = \mathbf{0.710} \end{aligned}$$

Now the relative closeness degree *λ <sup>j</sup>* of the alternatives Aj (j = 1,2 … ,n) to the intuitionistic fuzzy positive ideal solution are obtained from

$$\lambda\_j = \frac{D(A\_j, A^-)}{D(A\_j, A^+) + D(A\_j, A^-)}, j = 1, 2, \dots, n$$

$$\lambda\_1 = \frac{0.6545}{0.6545 + 0.59983} = 0.52179$$

$$\lambda\_2 = \frac{0.6900}{0.6900 + 0.6251} = 0.5246$$

$$\lambda\_3 = \frac{0.64033}{0.64033 + 0.6462} = 0.4977$$

$$\lambda\_4 = \frac{0.57621}{0.57621 + 0.5925} = 0.4930$$

Lastly the ranking order of the alternatives Aj (j = 1,2 … n) according to the non increasing order of the relative closeness degrees *λ <sup>j</sup>* is as follows:

#### HL2 > HL1> HL4> HL3

To obtain an overall result of the two methods for finding the effectiveness of the employees the average of the two methods is sought. This is shown in the following table as below:

