Decision-Making in Fuzzy Environment: A Survey

Hossam Kamal El-Din, Hossam E. Abd El Munim and Hani Mahdi

## Abstract

Multi-criteria decision-making (MCDM) is a crucial process in many business and management applications. The final decision is based upon the relative weights to the decision-making team. The analytic hierarchy process (AHP) has found to be one of the most successful approaches for evaluations of the weights and the importance of the criteria. However, most of the evaluated values are not so precise due to the fuzziness of the evaluating environment. This chapter surveys essentially the basic analytic hierarchy process and the fuzzy analytic hierarchy process (FAHP). It depicts through an example the steps for using the original analytic hierarchy process for two levels of criteria. Then, it uses the same example to explain the fuzzy approach in the evaluation. Finally, it compares both approaches.

Keywords: analytic hierarchy process (AHP), fuzzy analytic hierarchy process (FAHP), multi-criteria decision-making (MCDM), Chang's extent analysis

## 1. Introduction

Multi-criteria decision-making (MCDM) is a discipline that interacts with decisions to select the most optimal alternative with respect to multiple criteria for a specific goal. MCDM is well known for impartially solving problems of decisionmaking and for comparing the alternative comparatively to deduce the relative priority of the alternatives. Based on the relative priority value, the optimal alternative is defined and selected as a choice that can achieves the decision target.

Different MCDM techniques are in recent times broadly applied and used to resolve various decisions and predictive problems. These techniques are such as weighted sum model (WSM), weighted product model (WPM), analytic hierarchy process (AHP), technique for order preference by similarly to ideal solution (TOPSIS) and fuzzy AHP which is the fuzzification version of the AHP. Among these techniques, we will discuss analytical hierarchy process (AHP) and fuzzy analytical hierarchy process (FAHP) in this chapter.

The main objective of this chapter is to introduce a comparative analysis of analytic hierarchy process (AHP) developed by Saaty [1] and fuzzy analytic hierarchy process (FAHP) developed by Chang [2]. Both techniques will be introduced using a simple example for decision-making.

Saaty introduced an example for determining the type of the job that would be best for the person upon getting his/her PhD. This example was selected to cope with the original work of Saaty about AHP.

In the flow of the chapter, first the classical AHP and fuzzy AHP methods are introduced, then the summary of calculations are presented for AHP and fuzzy AHP as the next section. Finally, the chapter ends with comparison results, findings and comments about these methods.

## 2. Analytic hierarchy process (AHP)

The analytic hierarchy process (AHP) is developed by Saaty [1] as a multicriteria decision-making approach, which aids the decision maker to set relative priorities and to make the best decision. AHP has found to be one of the most successful approaches for evaluations the relative priorities of different criteria and for selection between alternatives. It gains recently high attention for many applications; see, for example, Ho and Ma [3]. AHP is especially suitable for complex decisions which involve the comparison of decision elements which are difficult to quantify. It is a technique for decision-making where there are a limited number of choices and these choices are characterized by a set of attributes (criteria). Each of these choices has different attributes' value.

zij : zji ¼ 1 (1)

(2)

Assume that the comparisons between the criteria A, B, C are as follows: A is moderately more important than B, A is extremely important than C, and B is moderately more important than C. According to Table 1, we will have A = 3B, A = 9C, and B = 3C. This is also means that B = (1/3)A, C = (1/9)A, and C = (1/3)B. The result of these pairwise comparisons is traditionally related to what we call the

2, 4, 6, 8 Intermediate value between adjacent scales

Step 4: This is the normalization step which consist of two parts. In the first part, normalization is carried out for each column entries according to the

> zij n <sup>i</sup>¼<sup>1</sup>zij

A BC

zij <sup>¼</sup> <sup>P</sup>

A 1 39 B 1/3 1 3 C 1/9 1/3 1

The normalization step for criteria: (a) summation of columns; and (b) dividing each cell by its columns'

The summation of the very simple example is shown in Figure 2a while the

pairwise comparison matrix as shown in Table 2.

Saaty's nine-point scale of pairwise comparison.

Intensity of importance Definition 1 Equally important

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

3 Moderately more important

5 Strongly important 7 Very strongly important 9 Extremely important

results of this normalization part are shown in Figure 2b.

The pairwise comparison matrix for the considered three criteria.

following equation:

Table 1.

Table 2.

Figure 2.

summation.

17

To explain the core of AHP we consider the very simple example for building these relative priorities between three items, although later we will consider the development of priorities using AHP for two levels of criteria and in fuzzy environment. The AHP procedure can be described in an algorithmic way in five steps which give finally the relative priorities between criteria. These steps will be explained using the very simple example as follows:

Step 1: Define the problem: let us say we have three criteria A, B, C and we want to know the relative priorities (importance) between these criteria to achieve a specific goal.

Step 2: Construct a simple decision hierarchy structure to emphasize the goal and the criteria as shown in Figure 1. Although, the goals are generally selecting one of different alternatives, here the goal is the simplest one that is generating the relative priorities between the criteria A, B, and C.

Step 3: Construct a set of pairwise comparison methods to all criteria.

A pairwise comparison is a process used to compare the criteria in pairs to judge which criterion is more important than the others using Saaty's nine-point scale of pairwise comparison as shown in Table 1.

In general, consider a matrix Z with n n matrix, where n is the number of evaluation criteria considered. Each entry zij of the matrix Z represents the importance of the ith criterion relative to the jth criterion. If zij > 1, then the ith criterion is more important than the jth criterion and in the otherwise, if zij < 1, then the ith criterion is less important than the jth criterion. If ith criterion and jth criterion have the same importance, then the entry zij is 1. The entries zij and zji satisfy the following constraint:

Figure 1. Simple decision hierarchical structure.


Table 1.

In the flow of the chapter, first the classical AHP and fuzzy AHP methods are introduced, then the summary of calculations are presented for AHP and fuzzy AHP as the next section. Finally, the chapter ends with comparison results, findings

The analytic hierarchy process (AHP) is developed by Saaty [1] as a multicriteria decision-making approach, which aids the decision maker to set relative priorities and to make the best decision. AHP has found to be one of the most successful approaches for evaluations the relative priorities of different criteria and for selection between alternatives. It gains recently high attention for many applications; see, for example, Ho and Ma [3]. AHP is especially suitable for complex decisions which involve the comparison of decision elements which are difficult to quantify. It is a technique for decision-making where there are a limited number of choices and these choices are characterized by a set of attributes (criteria). Each of

To explain the core of AHP we consider the very simple example for building these relative priorities between three items, although later we will consider the development of priorities using AHP for two levels of criteria and in fuzzy environment. The AHP procedure can be described in an algorithmic way in five steps which give finally the relative priorities between criteria. These steps will be

Step 1: Define the problem: let us say we have three criteria A, B, C and we want to know the relative priorities (importance) between these criteria to achieve a

Step 2: Construct a simple decision hierarchy structure to emphasize the goal and the criteria as shown in Figure 1. Although, the goals are generally selecting one of different alternatives, here the goal is the simplest one that is generating

A pairwise comparison is a process used to compare the criteria in pairs to judge which criterion is more important than the others using Saaty's nine-point scale of

In general, consider a matrix Z with n n matrix, where n is the number of evaluation criteria considered. Each entry zij of the matrix Z represents the importance of the ith criterion relative to the jth criterion. If zij > 1, then the ith criterion is more important than the jth criterion and in the otherwise, if zij < 1, then the ith criterion is less important than the jth criterion. If ith criterion and jth criterion have

the same importance, then the entry zij is 1. The entries zij and zji satisfy the

Step 3: Construct a set of pairwise comparison methods to all criteria.

and comments about these methods.

2. Analytic hierarchy process (AHP)

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these choices has different attributes' value.

pairwise comparison as shown in Table 1.

specific goal.

following constraint:

Simple decision hierarchical structure.

Figure 1.

16

explained using the very simple example as follows:

the relative priorities between the criteria A, B, and C.

Saaty's nine-point scale of pairwise comparison.

$$z\_{\vec{\imath}} \, . z\_{\vec{\jmath}} = \mathbf{1} \tag{1}$$

Assume that the comparisons between the criteria A, B, C are as follows: A is moderately more important than B, A is extremely important than C, and B is moderately more important than C. According to Table 1, we will have A = 3B, A = 9C, and B = 3C. This is also means that B = (1/3)A, C = (1/9)A, and C = (1/3)B. The result of these pairwise comparisons is traditionally related to what we call the pairwise comparison matrix as shown in Table 2.

Step 4: This is the normalization step which consist of two parts. In the first part, normalization is carried out for each column entries according to the following equation:

$$\overline{\mathbf{z}}\_{\text{ij}} = \frac{\mathbf{z}\_{\text{ij}}}{\sum\_{i=1}^{n} \mathbf{z}\_{\text{ij}}} \tag{2}$$

The summation of the very simple example is shown in Figure 2a while the results of this normalization part are shown in Figure 2b.


Table 2.

The pairwise comparison matrix for the considered three criteria.

Figure 2.

The normalization step for criteria: (a) summation of columns; and (b) dividing each cell by its columns' summation.

The second part, the weight wi of the criterion i is calculated by taking the average of the entries on each row of matrix Z. This results in the weight vector W

which in the very simple example becomes W ¼ 0:693 0:231 0:077 0 B@

$$w\_i = \frac{\sum\_{j=1}^{n} \overline{\mathbf{z}}\_{ji}}{\mathbf{n}} \tag{3}$$

To check the consistency, the values of λmax, CI, RI and CR which are:

(a) The new pairwise comparison matrix, and (b) the weight of each criterion.

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

3. Fuzzy analytic hierarchy process (FAHP)

calculated as A�<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>=u, <sup>1</sup>=m, <sup>1</sup>=<sup>l</sup> as shown in Table 4.

The addition of two fuzzy numbers is defined by:

And the multiplication of two fuzzy numbers is defined by:

be repeated.

Figure 3.

Chang [2].

criteria.

19

following example:

λmax ¼ 3:619,CI ¼ 0:3095, RI ¼ 0:58 and CR ¼ 0:534>0:1 (7)

This means that the evaluation of the matrix is inconsistent and all the comparisons of the elements are needed to be reconsidered and the previous steps need to

The conventional AHP is insufficient for dealing with fuzziness and uncertainty in multi-criteria decision-making (MCDM) because of inability of AHP to deal with the imprecision in the pairwise comparison process. Hence, the fuzzy AHP technique can be viewed as an advanced analytical method developed from the conventional AHP. The fuzzy AHP is proposed to find the uncertainty of AHP method. Different approaches are suggested as fuzzy AHP. The most two used methods for calculating the relative weights of the criteria are geometric means, which is proposed by Buckley [4], and the extent analysis methods which is proposed by

Fuzzy AHP (FAHP) has been shown successful in many applications [5–7]. The successfulness of FAHP attracts the researches to consider even different membership functions' form instead of using triangular membership functions to represents the fuzzy numbers which will be consider here after [8]. Also, in the following, we will consider only the extent analysis methods for calculating the relative weights of

The triangular number is denoted by three numbers A ¼ ð Þ l, m, u where "l" represents the lower value, "m" the medium value, and "u" the upper value, respectively lð Þ <sup>≤</sup> <sup>m</sup> <sup>≤</sup> <sup>u</sup> . The reciprocal triangular number is denoted by A�<sup>1</sup> and

The addition and the multiplications of two fuzzy numbers are explained by the

ð Þ l1, m1, u<sup>1</sup> ⊕ ð Þ¼ l2, m2, u<sup>2</sup> ð Þ l<sup>1</sup> þ l2, m<sup>1</sup> þ m2, u<sup>1</sup> þ u<sup>2</sup> (8)

ð Þ l1, m1, u<sup>1</sup> ⊗ ð Þ¼ l2, m2, u<sup>2</sup> ð Þ l1l2, m1m2, u1u<sup>2</sup> (9)

Consider two triangular fuzzy numbers A1 ¼ ð Þ l1, m1, u<sup>1</sup> and A<sup>2</sup> ¼ ð Þ l2, m2, u2:

1

CA

Step 5: The final step is a test to check for the consistency associated with the comparison matrix to examine the extent of consistency by using consistency ratio (CR) using the formula:

$$\text{Consistency ratio } (CR) = \frac{CI}{RI} \tag{4}$$

If CR < 0.1, then the pairwise comparison matrix Z is reasonable consistence otherwise it is inconsistence. Here, RI is a random matrix consistency index obtained through experiments using samples with large quantities. Random index (RI) values for the matrix of the order n = [1, 10] are shown in Table 3.

The consistency index (CI) indicates whether a decision maker provides the comparison of consistent values in a set of evaluations. CI is calculated using the formula:

$$CI = \frac{\lambda\_{\text{max}} - n}{(n - 1)} \tag{5}$$

The calculation of the CI demands to compute the normalized eigenvector of the matrix and the principal eigenvalue λmax of the matrix, which is obtained from summing the multiplication of the number of weights of all criteria in each column of the matrix with the eigenvector of the matrix.

$$
\lambda\_{\text{max}} = (\mathbf{1.44} \times \mathbf{0.693}) + (\mathbf{4.33} \times \mathbf{0.231}) + (\mathbf{13} \times \mathbf{0.077}) = \mathbf{3} \tag{6}
$$

For λmax = 3 and n = 3, then the value of CI = 0. The consistency here is ideal due to the fact that there is full consistency between the three pairwise comparisons, A = 3B, A = 9C, and B = 3C, which means any of these three equations can be deduced from the other two equations.

As CI = 0, and RI for three elements = 0.58, the CR = 0 < 0.1. This means that the evaluation of the matrix is consistent and all the comparisons of the elements are ideal (as CR = 0). This is the ideal case where the pairwise comparisons are perfect.

What happens if the ranking of the criteria is changed and the pairwise comparison matrix is reconstructed?

The new pairwise comparison matrix and the weight of each criterion are shown in Figure 3a and b, respectively.


Table 3.

Values of the random index (RI) for small problems.

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

The second part, the weight wi of the criterion i is calculated by taking the average of the entries on each row of matrix Z. This results in the weight vector W

wi ¼

P<sup>n</sup> <sup>j</sup>¼1zji n

Step 5: The final step is a test to check for the consistency associated with the comparison matrix to examine the extent of consistency by using consistency

Consistency ratio ð Þ¼ CR CI

If CR < 0.1, then the pairwise comparison matrix Z is reasonable consistence

The consistency index (CI) indicates whether a decision maker provides the comparison of consistent values in a set of evaluations. CI is calculated using the

CI <sup>¼</sup> <sup>λ</sup>max � <sup>n</sup>

matrix and the principal eigenvalue λmax of the matrix, which is obtained from summing the multiplication of the number of weights of all criteria in each column

The calculation of the CI demands to compute the normalized eigenvector of the

For λmax = 3 and n = 3, then the value of CI = 0. The consistency here is ideal due to the fact that there is full consistency between the three pairwise comparisons, A = 3B, A = 9C, and B = 3C, which means any of these three equations can be

As CI = 0, and RI for three elements = 0.58, the CR = 0 < 0.1. This means that the evaluation of the matrix is consistent and all the comparisons of the elements are ideal (as CR = 0). This is the ideal case where the pairwise comparisons are

What happens if the ranking of the criteria is changed and the pairwise com-

n 1 2 3 4 5 6 7 8 9 10 RI 0 0 0.58 0.9 1.12 1.2 1.32 1.41 1.45 1.49

The new pairwise comparison matrix and the weight of each criterion are shown

λmax ¼ ð1:44 � 0:693Þ þ ð4:33 � 0:231Þ þ ð Þ¼ 13 � 0:077 3 (6)

otherwise it is inconsistence. Here, RI is a random matrix consistency index obtained through experiments using samples with large quantities. Random index

(RI) values for the matrix of the order n = [1, 10] are shown in Table 3.

0:693 0:231 0:077 1

CA

RI (4)

ð Þ <sup>n</sup> � <sup>1</sup> (5)

(3)

0

B@

which in the very simple example becomes W ¼

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of the matrix with the eigenvector of the matrix.

deduced from the other two equations.

parison matrix is reconstructed?

in Figure 3a and b, respectively.

Values of the random index (RI) for small problems.

ratio (CR) using the formula:

formula:

perfect.

Table 3.

18

Figure 3. (a) The new pairwise comparison matrix, and (b) the weight of each criterion.

To check the consistency, the values of λmax, CI, RI and CR which are:

$$
\lambda\_{\text{max}} = \text{3.619, CI} = 0.3095, \text{RI} = 0.58 \text{ and CR} = 0.534 \text{>0.1} \tag{7}
$$

This means that the evaluation of the matrix is inconsistent and all the comparisons of the elements are needed to be reconsidered and the previous steps need to be repeated.

## 3. Fuzzy analytic hierarchy process (FAHP)

The conventional AHP is insufficient for dealing with fuzziness and uncertainty in multi-criteria decision-making (MCDM) because of inability of AHP to deal with the imprecision in the pairwise comparison process. Hence, the fuzzy AHP technique can be viewed as an advanced analytical method developed from the conventional AHP. The fuzzy AHP is proposed to find the uncertainty of AHP method. Different approaches are suggested as fuzzy AHP. The most two used methods for calculating the relative weights of the criteria are geometric means, which is proposed by Buckley [4], and the extent analysis methods which is proposed by Chang [2].

Fuzzy AHP (FAHP) has been shown successful in many applications [5–7]. The successfulness of FAHP attracts the researches to consider even different membership functions' form instead of using triangular membership functions to represents the fuzzy numbers which will be consider here after [8]. Also, in the following, we will consider only the extent analysis methods for calculating the relative weights of criteria.

The triangular number is denoted by three numbers A ¼ ð Þ l, m, u where "l" represents the lower value, "m" the medium value, and "u" the upper value, respectively lð Þ <sup>≤</sup> <sup>m</sup> <sup>≤</sup> <sup>u</sup> . The reciprocal triangular number is denoted by A�<sup>1</sup> and calculated as A�<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>=u, <sup>1</sup>=m, <sup>1</sup>=<sup>l</sup> as shown in Table 4.

The addition and the multiplications of two fuzzy numbers are explained by the following example:

Consider two triangular fuzzy numbers A1 ¼ ð Þ l1, m1, u<sup>1</sup> and A<sup>2</sup> ¼ ð Þ l2, m2, u2: The addition of two fuzzy numbers is defined by:

$$(l\_1, m\_1, u\_1) \oplus (l\_2, m\_2, u\_2) = (l\_1 + l\_2, m\_1 + m\_2, u\_1 + u\_2) \tag{8}$$

And the multiplication of two fuzzy numbers is defined by:

$$(l\_1, m\_1, \mu\_1) \otimes (l\_2, m\_2, \mu\_2) = (l\_1 l\_2, m\_1 m\_2, \mu\_1 \mu\_2) \tag{9}$$

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SA ¼ ð Þ 12, 13, 14 ⊗

S<sup>B</sup> ¼ ð Þ 3:25, 4:33, 5:5 ⊗

SC ¼ ð Þ 1:36, 1:44, 1:61 ⊗

V Mð Þ¼ <sup>1</sup> ≥ M2

shown in Figure 4. Accordingly, D is given by:

Linguistic variables for the importance weight of each criterion.

definition.

Table 5.

Vð Þ¼ M<sup>1</sup> ≥ M<sup>2</sup> 1 and

Figure 4.

21

1 <sup>21</sup>:<sup>11</sup> , <sup>1</sup>

The fuzzy pairwise comparison matrix for the considered three criteria.

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

> 1 <sup>21</sup>:<sup>11</sup> , <sup>1</sup>

1 <sup>21</sup>:<sup>11</sup> , <sup>1</sup>

> 8 >><

> >>:

D ¼ Vð Þ¼ M<sup>2</sup> ≥ M<sup>1</sup>

<sup>18</sup>:<sup>77</sup> , <sup>1</sup>

� �

1 if m<sup>1</sup> ≥ m<sup>2</sup> 0 if l<sup>2</sup> ≥u<sup>1</sup>

To compare M1 and M2 both possibilities Vð Þ M<sup>1</sup> ≥ M<sup>2</sup> and Vð Þ M<sup>2</sup> ≥ M<sup>1</sup> are needed. Considering Figure 4 as an example, we have (m<sup>1</sup> ≥ m2Þ which means that

where hgt is the highest intersection point, D is its value and d its ordinate as

u<sup>2</sup> � l<sup>1</sup> ð Þþ u<sup>2</sup> � m<sup>2</sup> ð Þ m<sup>1</sup> � l<sup>1</sup>

Vð Þ¼ M<sup>2</sup> ≥ M<sup>1</sup> hgt Mð Þ¼ <sup>1</sup> ∩ M<sup>2</sup> μ<sup>M</sup><sup>1</sup>

<sup>18</sup>:<sup>77</sup> , <sup>1</sup>

A (1, 1, 1) (2, 3, 4) (9, 9, 9) B (1/4, 1/3, 1/2) (1, 1, 1) (2, 3, 4) C (1/9, 1/9, 1/9) (1/4, 1/3, 1/2) (1, 1, 1)

> <sup>18</sup>:<sup>77</sup> , <sup>1</sup> 16:1

Step 5: compute the degree of possibility for each convex fuzzy number M1 and M2 that M1 ≥ M2 which will be denoted by Vð Þ M<sup>1</sup> ≥ M<sup>2</sup> defined by the following

> l<sup>2</sup> � u<sup>1</sup> ð Þ� m<sup>1</sup> � u<sup>1</sup> ð Þ m<sup>2</sup> � l<sup>2</sup>

<sup>16</sup>:<sup>61</sup> � � <sup>¼</sup> ð Þ <sup>0</sup>:568, <sup>0</sup>:693, <sup>0</sup>:<sup>842</sup> (12)

A BC

<sup>16</sup>:<sup>61</sup> � � <sup>¼</sup> ð Þ <sup>0</sup>:154, <sup>0</sup>:231, <sup>0</sup>:<sup>331</sup> (13)

¼ ð Þ 0:064, 0:077, 0:097 (14)

(15)

(17)

Otherwise

ð Þ¼ d D (16)

#### Table 4.

The scale of fuzzy AHP pairwise comparison.

We consider the same simple example to explain the core of fuzzy AHP for building these relative priorities. The following section outlines the Chang's extent analysis method on fuzzy AHP.

The fuzzy AHP procedure can be described in an algorithmic way in seven steps which give finally the relative priorities between criteria. These steps are:

Step 1: define the problem. This is the same example as mentioned before in AHP.

Step 2: develop the decision hierarchy like AHP step as mentioned before. Step 3: construct the fuzzy pairwise comparison matrices to all criteria. Here, the general form of the fuzzy pairwise comparison will be as follows:

$$\mathbf{Z} = \begin{pmatrix} \mathbf{z\_{ij}} \end{pmatrix}\_{\mathbf{n}\times\mathbf{n}} = \begin{bmatrix} (\mathbf{1},\mathbf{1},\mathbf{1}) & (\mathbf{l\_{12},\mathbf{m\_{12}},\mathbf{u\_{12}}) & \dots & (\mathbf{l\_{1n},\mathbf{m\_{1n}},\mathbf{u\_{1n}}) \\\\ (\mathbf{l\_{21},\mathbf{m\_{21}},\mathbf{u\_{21}})} & (\mathbf{1},\mathbf{1},\mathbf{1}) & \dots & (\mathbf{l\_{2n},\mathbf{m\_{2n}},\mathbf{u\_{2n}}) \\\\ \cdot & \cdot & \cdot & \cdot & \cdot \\\\ \cdot & \cdot & \cdot & \cdot & \cdot \\\\ (\mathbf{l\_{n1},\mathbf{m\_{n1}},\mathbf{u\_{n1}})} & (\mathbf{l\_{n2},\mathbf{m\_{n2}},\mathbf{u\_{n2}}) & \dots & (\mathbf{l\_{n}1,\mathbf{1})} \end{bmatrix} \tag{10}$$

where zij <sup>¼</sup> lij, mij, uij � �, zji <sup>¼</sup> zij �<sup>1</sup> <sup>¼</sup> <sup>1</sup>=uij, <sup>1</sup>=mij, <sup>1</sup>=lij � � for i, j <sup>¼</sup> <sup>1</sup>, …, n.

Using the linguistic scale for criteria and alternatives as shown in Table 10 to compare the criteria in pairs to judge which criterion is more important than the others. As we use the same comparisons between the criteria A, B, C, the fuzzy pairwise comparisons between these criteria can be expressed in the matrix form as shown in Table 5.

Step 4: calculate the value of fuzzy synthetic extent Si with respect to the ith criterion using the formula:

$$\mathbf{S}\_{i} = \left( \frac{\sum\_{j=1}^{n} l\_{i\bar{j}}}{\sum\_{i=1}^{n} \sum\_{j=1}^{n} u\_{i\bar{j}}}, \frac{\sum\_{j=1}^{n} m\_{i\bar{j}}}{\sum\_{i=1}^{n} \sum\_{j=1}^{n} m\_{i\bar{j}}}, \frac{\sum\_{j=1}^{n} u\_{i\bar{j}}}{\sum\_{i=1}^{n} \sum\_{j=1}^{n} l\_{i\bar{j}}} \right) \tag{11}$$

According to the previous example, the values of fuzzy synthetic extent are:

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736


Table 5.

We consider the same simple example to explain the core of fuzzy AHP for building these relative priorities. The following section outlines the Chang's extent

which give finally the relative priorities between criteria. These steps are:

Step 1: define the problem. This is the same example as mentioned

general form of the fuzzy pairwise comparison will be as follows:

Step 2: develop the decision hierarchy like AHP step as mentioned before. Step 3: construct the fuzzy pairwise comparison matrices to all criteria. Here, the

The fuzzy AHP procedure can be described in an algorithmic way in seven steps

ð Þ 1, 1, 1 ð Þ l12, m12, u12 … ð Þ l1n, m1n, u1n ð Þ l21, m21, u21 ð Þ 1, 1, 1 … ð Þ l2n, m2n, u2n : ::: : ::: ð Þ ln1, mn1, un1 ð Þ ln2, mn2, un2 … ð Þ 1, 1, 1

�<sup>1</sup> <sup>¼</sup> <sup>1</sup>=uij, <sup>1</sup>=mij, <sup>1</sup>=lij

Using the linguistic scale for criteria and alternatives as shown in Table 10 to compare the criteria in pairs to judge which criterion is more important than the others. As we use the same comparisons between the criteria A, B, C, the fuzzy pairwise comparisons between these criteria can be expressed in the matrix form as

Step 4: calculate the value of fuzzy synthetic extent Si with respect to the ith

P<sup>n</sup> <sup>j</sup>¼<sup>1</sup>mij

According to the previous example, the values of fuzzy synthetic extent are:

!

P<sup>n</sup> i¼1 P<sup>n</sup> <sup>j</sup>¼<sup>1</sup> mij ,

� � for i, j <sup>¼</sup> <sup>1</sup>, …, n.

Linguistic scale for importance Triangular fuzzy

1 and 3

3 and 5

5 and 7

7 and 9

Reciprocal

(1/3, 1/2, 1)

(1/5, 1/4, 1/3)

(1/7, 1/6, 1/5)

(1/9, 1/8, 1/7)

P<sup>n</sup> <sup>j</sup>¼<sup>1</sup>uij

P<sup>n</sup> i¼1 P<sup>n</sup> <sup>j</sup>¼<sup>1</sup> lij

(10)

(11)

analysis method on fuzzy AHP.

The scale of fuzzy AHP pairwise comparison.

before in AHP.

Table 4.

Crisp importance value

Triangular fuzzy numbers

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2 (1, 2, 3) Intermediate value between

4 (3, 4, 5) Intermediate value between

6 (5, 6, 7) Intermediate value between

8 (7, 8, 9) Intermediate value between

1 (1, 1, 1) Equally important (1, 1, 1)

3 (2, 3, 4) Moderately more important (1/4, 1/3, 1/2)

5 (4, 5, 6) Strongly important (1/6, 1/5, 1/4)

7 (6, 7, 8) Very strongly important (1/8, 1/7, 1/6)

9 (9, 9, 9) Extremely important (1/9, 1/9, 1/9)

Z ¼ zij

shown in Table 5.

20

� �:<sup>n</sup>�<sup>n</sup> <sup>¼</sup>

where zij ¼ lij, mij, uij

criterion using the formula:

Si ¼

� �, zji <sup>¼</sup> zij

P<sup>n</sup> <sup>j</sup>¼<sup>1</sup>lij

P<sup>n</sup> i¼1 P<sup>n</sup> <sup>j</sup>¼<sup>1</sup> uij , The fuzzy pairwise comparison matrix for the considered three criteria.

$$\mathbf{S\_A} = (12, 13, 14) \otimes \left(\frac{1}{21.11}, \frac{1}{18.77}, \frac{1}{16.61}\right) = (0.568, 0.693, 0.842) \tag{12}$$

$$\mathbf{S}\_{\text{B}} = (3.25, 4.33, 5.5) \otimes \left(\frac{1}{21.11}, \frac{1}{18.77}, \frac{1}{16.61}\right) = (0.154, 0.231, 0.331) \tag{13}$$

$$\mathbf{S}\_{\rm C} = (1.36, 1.44, 1.61) \otimes \left(\frac{1}{21.11}, \frac{1}{18.77}, \frac{1}{16.1}\right) = (0.064, 0.077, 0.097) \tag{14}$$

Step 5: compute the degree of possibility for each convex fuzzy number M1 and M2 that M1 ≥ M2 which will be denoted by Vð Þ M<sup>1</sup> ≥ M<sup>2</sup> defined by the following definition.

$$\mathbf{V}(\mathbf{M}\_1 \ge \mathbf{M}\_2) = \begin{cases} 1 \text{ if } m\_1 \ge m\_2 \\ 0 \text{ if } l\_2 \ge u\_1 \\ \frac{l\_2 - u\_1}{(m\_1 - u\_1) - (m\_2 - l\_2)} \text{ Otherwise} \end{cases} \tag{15}$$

To compare M1 and M2 both possibilities Vð Þ M<sup>1</sup> ≥ M<sup>2</sup> and Vð Þ M<sup>2</sup> ≥ M<sup>1</sup> are needed. Considering Figure 4 as an example, we have (m<sup>1</sup> ≥ m2Þ which means that Vð Þ¼ M<sup>1</sup> ≥ M<sup>2</sup> 1 and

$$\text{V}(M\_2 \ge M\_1) = \text{hgt}(M\_1 \cap M\_2) = \mu\_{M\_1}(d) = D \tag{16}$$

where hgt is the highest intersection point, D is its value and d its ordinate as shown in Figure 4. Accordingly, D is given by:

$$\mathcal{D} = \mathcal{V}(\mathcal{M}\_2 \ge \mathcal{M}\_1) = \frac{u\_2 - l\_1}{(u\_2 - m\_2) + (m\_1 - l\_1)} \tag{17}$$

Figure 4. Linguistic variables for the importance weight of each criterion.

Application of Decision Science in Business and Management

For two fuzzy numbers only two values of possibilities are needed. As the considered fuzzy numbers increase, the numbers of the needed calculated possibilities are increased non-linearly. To compare n fuzzy numbers, we need n nð Þ � 1 possible values. Consider the very simple example with the three criteria A, B, and C, the needed possibilities are:

$$\mathbf{V(M\_A \ge M\_B)} = \mathbf{1}, \mathbf{V(M\_A \ge M\_C)} = \mathbf{1} \tag{18}$$

Figure 5, apply AHP method and fuzzy AHP method and then compare the results between these two methods. As shown in Figure 5, the hierarchical structure consists of four levels. The first level (the top level) is the goal which is to determine the type of a suitable job, the second level is the criteria, the third level is the subcriteria and the fourth level is the alternative (the lowest level) which the person

According to Figure 5, 12 pairwise comparison matrices need to be stated: one for the criteria with respect to the goal, (flexibility, opportunity, security, reputation and salary), two for the sub-criteria which one of it is for the sub-criteria with respect to the flexibility (location, time and work), and the other is for the subcriteria with respect to the opportunity (entrepreneurial, salary potential and top level position). Nine comparison matrices for the four alternatives with respect to the criteria and the sub-criteria "the covering criteria" connected to the alternatives (domestic company, international company, college and state university). The covering criteria are: the first six are sub-criteria in the third level and the last three are criteria from the second level. As Saaty listed only three pairwise comparison matrices of 12, we listed the rest of pairwise comparison matrices to emphasize the example and show the result. Tables 6–8 indicate the pairwise comparison matrices

Table 9 shows the calculation of the global weight for sub-criteria with respect to its criterion by multiplying weight of each criterion to the weights of sub-criteria

After computing the relative weights of criteria and sub-criteria, the next step is

V ¼ S � W (29)

to compute relative weights of alternatives. Tables 10–18 indicate the pairwise comparison matrices for alternatives with respect to the covering criteria.

Once the weight vector of covering criteria W and the weight vector of the alternative S have been computed, the AHP obtains a vector V of global scores by

Finally, the alternative ranking is accomplished by ordering these global scores in a descending order. Table 19 shows the final weights of the alternatives with

will choose the kind of the job from these alternatives [10, 11].

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

for all criteria and sub-criteria.

that affect its criterion.

multiplying S and W as:

Figure 5. Best job decision [9].

23

$$\mathbf{V(M\_{B}\geq M\_{A})}=\mathbf{0},\ \mathbf{V(M\_{B}\geq M\_{C})}=\mathbf{1}\tag{19}$$

$$\mathbf{V}(\mathbf{M}\_{\mathbf{C}} \ge \mathbf{M}\_{\mathbf{A}}) = \mathbf{0}, \mathbf{V}(\mathbf{M}\_{\mathbf{C}} \ge \mathbf{M}\_{\mathbf{B}}) = \mathbf{0} \tag{20}$$

Step 6: the degree of possibility for a convex fuzzy number to be greater than (k) convex fuzzy numbers Mið Þ i ¼ 1, 2, …, k can be defined by the following equation:

$$\begin{aligned} \mathbf{V}(\mathbf{M}\_{\mathbf{i}} \ge \mathbf{M}\_{1}, \mathbf{M}\_{2}, \dots, \mathbf{M}\_{\mathbf{k}}) &= \mathbf{V}((\mathbf{M}\_{\mathbf{i}} \ge \mathbf{M}\_{1}) \text{ and } (\mathbf{M}\_{\mathbf{i}} \ge \mathbf{M}\_{2}) \text{ and } \dots (\mathbf{M}\_{\mathbf{i}} \ge \mathbf{M}\_{\mathbf{k}})) \\ = \min \mathbf{V}(\mathbf{M}\_{\mathbf{i}} \ge \mathbf{M}\_{\mathbf{k}}), (\mathbf{k} = \mathbf{1}, 2, \dots, n), (\mathbf{i} = \mathbf{1}, 2, \dots, n), \mathbf{k} \ne \mathbf{i} \end{aligned} \tag{21}$$

The minimum degrees of possibilities for criteria A, B, C are:

$$\mathbf{V(M\_A \ge M\_B, M\_C)} = \min \ (\mathbf{1}, \mathbf{1}) = \mathbf{1} \tag{22}$$

$$\mathbf{V}(\mathbf{M}\_{\rm B} \ge \mathbf{M}\_{\rm A}, \mathbf{M}\_{\rm C}) = \min \ (\mathbf{0}, \mathbf{1}) = \mathbf{0} \tag{23}$$

$$\mathbf{V(M\_C \ge M\_A, M\_B) = min\ (0, 0) = 0} \tag{24}$$

Step 7: the normalized weight vector W ¼ ð Þ w1, …, w<sup>2</sup> <sup>T</sup> of the fuzzy comparison matrix Z is:

Assuming ďð Þ¼ zi min V Mð Þ ð Þ <sup>i</sup> ≥ Mk: For kð ¼ 1, 2, …, n), k 6¼ i. Then the weight vector is given by:

$$\mathbf{W}' = (\mathbf{d}'(\mathbf{z}\_1), \mathbf{d}'(\mathbf{z}\_2), \dots, \mathbf{d}'(\mathbf{z}\_n))^T \tag{25}$$

Via normalization, the normalized weight vector is:

$$\mathbf{W} = (d(\mathbf{z}\_1), d(\mathbf{z}\_2), \dots, d(\mathbf{z}\_n))^T \tag{26}$$

Therefore, the weight vector for A, B and C is:

$$\mathbf{W}^\* = (\mathbf{1}, \mathbf{0}, \mathbf{0}) \tag{27}$$

And the normalized weight vector for A, B and C is:

$$\mathbf{W} = (\mathbf{1}, \mathbf{0}, \mathbf{0})\tag{28}$$

## 4. Examples of applications

The following example is proposed by Saaty about a simple decision for selecting a job [9] and it was selected to cope with the original work of Saaty about AHP. This example is a simple decision examined by someone to determine what kind of job would be best for him/her after getting his/her PhD. The goal is to determine the kind of job for which he/she is best suited as spelled out by the criteria. We will construct the pairwise comparison of criteria from the hierarchy structure shown in Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

For two fuzzy numbers only two values of possibilities are needed. As the considered fuzzy numbers increase, the numbers of the needed calculated possibilities are increased non-linearly. To compare n fuzzy numbers, we need n nð Þ � 1 possible values. Consider the very simple example with the three criteria A, B, and

Step 6: the degree of possibility for a convex fuzzy number to be greater

V Mð <sup>i</sup> ≥ M1; M2; …; MkÞ ¼ V Mð Þ ð Þ <sup>i</sup> ≥ M1 and Mð Þ <sup>i</sup> ≥ M2 and…ð Þ Mi ≥ Mk <sup>¼</sup> min V Mð Þ <sup>i</sup> <sup>≥</sup> Mk ,ð Þ <sup>k</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; …; <sup>n</sup> ,ð Þ <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; …; <sup>n</sup> , <sup>k</sup> 6¼ <sup>i</sup> (21)

The minimum degrees of possibilities for criteria A, B, C are:

Step 7: the normalized weight vector W ¼ ð Þ w1, …, w<sup>2</sup>

Via normalization, the normalized weight vector is:

And the normalized weight vector for A, B and C is:

Therefore, the weight vector for A, B and C is:

4. Examples of applications

22

For kð ¼ 1, 2, …, n), k 6¼ i. Then the weight vector is given by:

Assuming ďð Þ¼ zi min V Mð Þ ð Þ <sup>i</sup> ≥ Mk:

than (k) convex fuzzy numbers Mið Þ i ¼ 1, 2, …, k can be defined by the following

V Mð Þ¼ <sup>A</sup> ≥ MB 1, V Mð Þ¼ <sup>A</sup> ≥ MC 1 (18) V Mð Þ¼ <sup>B</sup> ≥ MA 0, V Mð Þ¼ <sup>B</sup> ≥ MC 1 (19) V Mð Þ¼ <sup>C</sup> ≥ MA 0, V Mð Þ¼ <sup>C</sup> ≥ MB 0 (20)

V Mð <sup>A</sup> ≥ MB, MCÞ ¼ min 1ð Þ¼ , 1 1 (22) V Mð <sup>B</sup> ≥ MA, MCÞ ¼ min 0ð Þ¼ , 1 0 (23) V Mð <sup>C</sup> ≥ MA, MBÞ ¼ min 0ð Þ¼ , 0 0 (24)

<sup>W</sup>` <sup>¼</sup> ð Þ <sup>ď</sup>ð Þ z1 , <sup>ď</sup>ð Þ z2 , …, <sup>ď</sup>ð Þ zn <sup>T</sup> (25)

<sup>W</sup> <sup>¼</sup> ð Þ <sup>d</sup>ð Þ z1 , dð Þ z2 , …, dð Þ zn <sup>T</sup> (26)

W` ¼ ð Þ 1, 0, 0 (27)

W ¼ ð Þ 1, 0, 0 (28)

The following example is proposed by Saaty about a simple decision for selecting a job [9] and it was selected to cope with the original work of Saaty about AHP. This example is a simple decision examined by someone to determine what kind of job would be best for him/her after getting his/her PhD. The goal is to determine the kind of job for which he/she is best suited as spelled out by the criteria. We will construct the pairwise comparison of criteria from the hierarchy structure shown in

<sup>T</sup> of the fuzzy comparison

C, the needed possibilities are:

Application of Decision Science in Business and Management

equation:

matrix Z is:

Figure 5, apply AHP method and fuzzy AHP method and then compare the results between these two methods. As shown in Figure 5, the hierarchical structure consists of four levels. The first level (the top level) is the goal which is to determine the type of a suitable job, the second level is the criteria, the third level is the subcriteria and the fourth level is the alternative (the lowest level) which the person will choose the kind of the job from these alternatives [10, 11].

According to Figure 5, 12 pairwise comparison matrices need to be stated: one for the criteria with respect to the goal, (flexibility, opportunity, security, reputation and salary), two for the sub-criteria which one of it is for the sub-criteria with respect to the flexibility (location, time and work), and the other is for the subcriteria with respect to the opportunity (entrepreneurial, salary potential and top level position). Nine comparison matrices for the four alternatives with respect to the criteria and the sub-criteria "the covering criteria" connected to the alternatives (domestic company, international company, college and state university). The covering criteria are: the first six are sub-criteria in the third level and the last three are criteria from the second level. As Saaty listed only three pairwise comparison matrices of 12, we listed the rest of pairwise comparison matrices to emphasize the example and show the result. Tables 6–8 indicate the pairwise comparison matrices for all criteria and sub-criteria.

Table 9 shows the calculation of the global weight for sub-criteria with respect to its criterion by multiplying weight of each criterion to the weights of sub-criteria that affect its criterion.

After computing the relative weights of criteria and sub-criteria, the next step is to compute relative weights of alternatives. Tables 10–18 indicate the pairwise comparison matrices for alternatives with respect to the covering criteria.

Once the weight vector of covering criteria W and the weight vector of the alternative S have been computed, the AHP obtains a vector V of global scores by multiplying S and W as:

$$\mathbf{V} = \mathbf{S} \cdot \mathbf{W} \tag{29}$$

Finally, the alternative ranking is accomplished by ordering these global scores in a descending order. Table 19 shows the final weights of the alternatives with

Figure 5. Best job decision [9].

## Application of Decision Science in Business and Management


## Table 6.

Pairwise comparison matrix of the main criteria with respect to the goal.


## Table 7.

Pairwise comparison matrix for the sub-criteria with respect to flexibility.


#### Table 8.

Pairwise comparison matrix for the sub-criteria with respect to opportunity.


#### Table 9.

Local weight and global weight for criteria and sub-criteria.


respect to the covering criteria. It is clear that the domestic company is the preferred candidate. The second candidate is the college, then the third candidate is the

Domestic Co 1 1/5 1/3 1/2 0.072 Int'l Co 5 1 3 5 0.54 College 3 1/3 1 2 0.172 State Univ. 4 1/5 1/2 1 0.162

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co 1 1/5 4 1/3 0.192 Int'l Co 5 1 2 1/4 0.263 College 1/4 1/2 1 2 0.202 State Univ. 3 4 1/2 1 0.343

Domestic Co 1 1/3 3 6 0.32 Int'l Co 3 1 4 2 0.45 College 1/3 1/4 1 2 0.123 State Univ. 1/6 1/2 1/2 1 0.107

Domestic Co 1 2 1/3 1/4 0.133 Int'l Co 1/2 1 1/6 1/2 0.092 College 3 6 1 2 0.483 State Univ. 4 2 1/2 1 0.292

Domestic Co 1 2 4 6 0.502 Int'l Co 1/2 1 3 4 0.3 College 1/4 1/3 1 2 0.124 State Univ. 1/6 1/4 1/2 1 0.074

Pairwise comparison matrix for the alternatives with respect to location.

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

Pairwise comparison matrix for the alternatives with respect to work.

Pairwise comparison matrix for the alternatives with respect to time.

Pairwise comparison matrix for the alternatives with respect to entrepreneurial.

Table 11.

Table 12.

Table 13.

Table 14.

Table 15.

25

international company and the last candidate is the state university.

Pairwise comparison matrix for the alternatives with respect to top level position.

#### Table 10.

Pairwise comparison matrix for the alternatives with respect to salary potential.

## Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736


## Table 11.

Flexibility Opportunities Security Reputation Salary Priorities

Location Time Work Priorities

Ent Sal-pot Top level pos Priorities

Local weight (SCL)

Time 0.221 0.0081 Work 0.686 0.0251

Salary potential 0.158 0.0194 Top level position 0.283 0.0348

Domestic Co Int'l Co College State Univ. Priorities

Global weight (CL SCL)

Flexibility 1 1/4 1/6 1/4 1/8 0.036 Opportunities 4 1 1/3 3 1/7 0.122 Security 6 3 1 4 1/2 0.262 Reputation 4 1/3 1/4 1 1/7 0.075 Salary 8 7 2 7 1 0.506

Location 1 1/3 1/6 0.091 Time 3 1 1/4 0.218 Work 6 4 1 0.691

Ent 1 2 5 0.557 Sal-pot 1/2 1 1/4 0.158 Top level pos 1/5 4 1 0.283

> Sub-criterion (SC)

Flexibility 0.0367 Location 0.093 0.0034

Opportunity 0.123 Entrepreneurial 0.557 0.0685

Domestic Co 1 4 3 6 0.555 Int'l Co 1/4 1 3 5 0.258 College 1/3 1/3 1 2 0.124 State Univ. 1/6 1/5 1/2 1 0.064

Pairwise comparison matrix for the alternatives with respect to salary potential.

Pairwise comparison matrix of the main criteria with respect to the goal.

Application of Decision Science in Business and Management

Pairwise comparison matrix for the sub-criteria with respect to flexibility.

Pairwise comparison matrix for the sub-criteria with respect to opportunity.

Local weight (CL)

Local weight and global weight for criteria and sub-criteria.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

24

Criterion (C)

Pairwise comparison matrix for the alternatives with respect to location.


#### Table 12.

Pairwise comparison matrix for the alternatives with respect to work.


#### Table 13.

Pairwise comparison matrix for the alternatives with respect to time.


#### Table 14.

Pairwise comparison matrix for the alternatives with respect to entrepreneurial.


#### Table 15.

Pairwise comparison matrix for the alternatives with respect to top level position.

respect to the covering criteria. It is clear that the domestic company is the preferred candidate. The second candidate is the college, then the third candidate is the international company and the last candidate is the state university.

## Application of Decision Science in Business and Management


As mentioned before, these results of a classical AHP are compared with the results of fuzzy AHP. Therefore, the evaluations are recalculated according to the fuzzy AHP on the same hierarchy structure. The 12 pairwise comparison matrices for all criteria, sub-criteria and alternatives are shown from Tables 20–32.

Table 23 shows the calculation of the global weight for sub-criteria with respect to its criterion by multiplying weight of each criterion to the weights of sub-criteria

Flexibility (1, 1, 1) (1/5, 1/4, 1/3) (1/7, 1/6, 1/5) (1/5, 1/4, 1/3) (1/9, 1/8, 1/7) 0 Opportunities (3, 4, 5) (1, 1, 1) (1/4, 1/3, 1/2) (2, 3, 4) (1/8, 1/7, 1/6) 0 Security (5, 6, 7) (2, 3, 4) (1, 1, 1) (3, 4, 5) (1/3, 1/2, 1) 0.237 Reputation (3, 4, 5) (1/4, 1/3, 1/2) (1/5, 1/4, 1/3) (1, 1, 1) (1/8, 1/7, 1/6) 0 Salary (7, 8, 9) (6, 7, 8) (1, 2, 3) (6, 7, 8) (1, 1, 1) 0.763

Flexibility Opportunities Security Reputation Salary Priorities

Location Time Work Priorities

Ent Sal-pot Top level pos Priorities

Local weight (SCL)

Time 0 0 Work 1 0

Salary potential 0 0 Top level position 0.35 0

Global weight (CL SCL)

Location (1, 1, 1) (1/4, 1/3, 1/2) (1/7, 1/6, 1/5) 0 Time (2, 3, 4) (1, 1, 1) (1/5, 1/4, 1/6) 0 Work (5, 6, 7) (3, 4, 5) (1, 1, 1) 1

Ent (1, 1, 1) (1, 2, 3) (4, 5, 6) 0.65 Sal-pot (1/3, 1/2, 1) (1, 1, 1) (1/5, 1/4, 1/3) 0 Top level pos (1/6, 1/5, 1/4) (3, 4, 5) (1, 1, 1) 0.35

Fuzzy pairwise comparison matrix for the sub-criteria with respect to flexibility.

Fuzzy pairwise comparison matrix of the main criteria with respect to the goal.

Fuzzy pairwise comparison matrix for the sub-criteria with respect to opportunity.

Sub-criterion (SC)

Flexibility 0 Location 0 0

Opportunity 0 Entrepreneurial 0.65 0

Local weight (CL)

Local weight and global weight for criteria and sub-criteria.

that affect its criterion.

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

Table 21.

Table 20.

Table 22.

Table 23.

27

Criterion (C)

## Table 16.

Pairwise comparison matrix for the alternatives with respect to security.


## Table 17.

Pairwise comparison matrix for the alternatives with respect to salary.


#### Table 18.

Pairwise comparison matrix for the alternatives with respect to reputation.


### Table 19.

Final weights of alternatives for AHP method.

## Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

As mentioned before, these results of a classical AHP are compared with the results of fuzzy AHP. Therefore, the evaluations are recalculated according to the fuzzy AHP on the same hierarchy structure. The 12 pairwise comparison matrices for all criteria, sub-criteria and alternatives are shown from Tables 20–32.

Table 23 shows the calculation of the global weight for sub-criteria with respect to its criterion by multiplying weight of each criterion to the weights of sub-criteria that affect its criterion.


Table 20.

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co 1 3 2 5 0.47 Int'l Co 1/3 1 1/3 1/2 0.104 College 1/2 3 1 3 0.29 State Univ. 1/5 2 1/3 1 0.129

Domestic Co 1 4 1/2 3 0.33 Int'l Co 1/4 1 1/6 1/2 0.077 College 2 6 1 2 0.462 State Univ. 1/3 1/2 1/2 1 0.13

Domestic Co 1 3 4 5 0.5 Int'l Co 1/3 1 1/2 5 0.197 College 1/4 2 1 2 0.19 State Univ. 1/5 1/5 1/2 1 0.072

Pairwise comparison matrix for the alternatives with respect to security.

Application of Decision Science in Business and Management

Pairwise comparison matrix for the alternatives with respect to salary.

Pairwise comparison matrix for the alternatives with respect to reputation.

Table 16.

Table 17.

Table 18.

Table 19.

26

Final weights of alternatives for AHP method.

Fuzzy pairwise comparison matrix of the main criteria with respect to the goal.


#### Table 21.

Fuzzy pairwise comparison matrix for the sub-criteria with respect to flexibility.


#### Table 22.

Fuzzy pairwise comparison matrix for the sub-criteria with respect to opportunity.


#### Table 23.

Local weight and global weight for criteria and sub-criteria.

## Application of Decision Science in Business and Management


#### Table 24.

Fuzzy pairwise comparison matrix for the alternatives with respect to salary potential.


#### Table 25.

Fuzzy pairwise comparison matrix for the alternatives with respect to location.


#### Table 26.

Fuzzy pairwise comparison matrix for the alternatives with respect to work.


#### Table 27.

Fuzzy pairwise comparison matrix for the alternatives with respect to time.


Once the weight vector of covering criteria W and the weight vector of the alternative S have been computed, the fuzzy AHP obtains a vector V of global

Domestic Co (1, 1, 1) (2, 3, 4) (3, 4, 5) (4, 5, 6) 0.649 Int'l Co (1/4, 1/3, 1/2) (1, 1, 1) (1/3, 1/2, 1) (4, 5, 6) 0.228 College (1/5, 1/4, 1/3) (1, 2, 3) (1, 1, 1) (1, 2, 3) 0.123 State Univ. (1/6, 1/5, 1/4) (1/6, 1/5, 1/4) (1/3, 1/2, 1) (1, 1, 1) 0

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co (1, 1, 1) (1/6, 1/5, 1/4) (1/4, 1/3, 1/2) (1/3, 1/2, 1) 0 Int'l Co (4, 5, 6) (1, 1, 1) (2, 3, 4) (4, 5, 6) 0.789 College (2, 3, 4) (1/4, 1/3, 1/2) (1, 1, 1) (1, 2, 3) 0.211 State Univ. (1, 2, 3) (1/6, 1/5, 1/4) (1/3, 1/2, 1) (1, 1, 1) 0

Fuzzy pairwise comparison matrix for the alternatives with respect to top level position.

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

Domestic Co (1, 1, 1) (3, 4, 5) (1/3, 1/2, 1) (2, 3, 4) 0.436 Int'l Co (1/5, 1/4, 1/3) (1, 1, 1) (1/7, 1/6, 1/5) (1/3, 1/2, 1) 0 College (1, 2, 3) (5, 6, 7) (1, 1, 1) (1, 2, 3) 0.564 State Univ. (1/4, 1/3, 1/2) (1/3, 1/2, 1) (1/3, 1/2, 1) (1, 1, 1) 0

Domestic Co (1, 1, 1) (2, 3, 4) (1, 2, 3) (4, 5, 6) 0.573 Int'l Co (1/4, 1/3, 1/2) (1, 1, 1) (1/4, 1/3, 1/2) (1/3, 1/2, 1) 0 College (1/3, 1/2, 1) (2, 3, 4) (1, 1, 1) (2, 3, 4) 0.394 State Univ. (1/6, 1/5, 1/4) (1, 2, 3) (1/4, 1/3, 1/2) (1, 1, 1) 0.033

Finally, the alternative ranking is accomplished by ordering these global scores in a descending order. Table 33 shows the final weights of the alternatives with respect to the covering criteria. It is clear that the college is the preferred candidate. The second candidate is the domestic company, then the third candidate is the state

university and the last candidate is international company.

Fuzzy pairwise comparison matrix for the alternatives with respect to salary.

Fuzzy pairwise comparison matrix for the alternatives with respect to reputation.

Fuzzy pairwise comparison matrix for the alternatives with respect to security.

V ¼ S � W (30)

scores by multiplying S and W as:

Table 31.

Table 32.

29

Table 29.

Table 30.

#### Table 28.

Fuzzy pairwise comparison matrix for the alternatives with respect to entrepreneurial.

### Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736


#### Table 29.

Fuzzy pairwise comparison matrix for the alternatives with respect to top level position.


#### Table 30.

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co (1, 1, 1) (3, 4, 5) (2, 3, 4) (5, 6, 7) 0.646 Int'l Co (1/5, 1/4, 1/3) (1, 1, 1) (2, 3, 4) (4, 5, 6) 0.354 College (1/4, 1/3, 1/2) (1/4, 1/3, 1/2) (1, 1, 1) (1, 2, 3) 0 State Univ. (1/7, 1/6, 1/5) (1/6, 1/5, 1/4) (1/3, 1/2, 1) (1, 1, 1) 0

Domestic Co (1, 1, 1) (1/6, 1/5, 1/4) (3, 4, 5) (1/4, 1/3, 1/2) 0 Int'l Co (4, 5, 6) (1, 1, 1) (1, 2, 3) (1/5, 1/4, 1/3) 0 College (1/5, 1/4, 1/3) (1/3, 1/2, 1) (1, 1, 1) (1, 2, 3) 0.493 State Univ. (2,3,4) (3, 4, 5) (1/3, 1/2, 1) (1, 1, 1) 0.507

Fuzzy pairwise comparison matrix for the alternatives with respect to salary potential.

Application of Decision Science in Business and Management

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co Int'l Co College State Univ. Priorities

Domestic Co (1, 1, 1) (1/4, 1/3, 1/2) (2, 3, 4) (5, 6, 7) 0.508 Int'l Co (2, 3, 4) (1, 1, 1) (3, 4, 5) (1, 2, 3) 0.492 College (1/4, 1/3, 1/2) (1/5, 1/4, 1/3) (1, 1, 1) (1, 2, 3) 0 State Univ. (1/7, 1/6, 1/5) (1/3, 1/2, 1) (1/3, 1/2, 1) (1, 1, 1) 0

Domestic Co (1, 1, 1) (1, 2, 3) (1/4, 1/3, 1/2) (1/5, 1/4, 1/3) 0 Int'l Co (1/3, 1/2, 1) (1, 1, 1) (1/7, 1/6, 1/5) (1/3, 1/2, 1) 0 College (2, 3, 4) (5, 6, 7) (1, 1, 1) (1, 2, 3) 0.626 State Univ. (3, 4, 5) (1, 2, 3) (1/3, 1/2, 1) (1, 1, 1) 0.374

Domestic Co (1, 1, 1) (1, 2, 3) (3, 4, 5) (5, 6, 7) 0.625 Int'l Co (1/3, 1/2, 1) (1, 1, 1) (2, 3, 4) (3, 4, 5) 0.375 College (1/5, 1/4, 1/3) (1/4, 1/3, 1/2) (1, 1, 1) (1, 2, 3) 0 State Univ. (1/7, 1/6, 1/5) (1/5, 1/4, 1/3) (1/3, 1/2, 1) (1, 1, 1) 0

Fuzzy pairwise comparison matrix for the alternatives with respect to work.

Fuzzy pairwise comparison matrix for the alternatives with respect to location.

Fuzzy pairwise comparison matrix for the alternatives with respect to time.

Fuzzy pairwise comparison matrix for the alternatives with respect to entrepreneurial.

Table 26.

Table 24.

Table 25.

Table 27.

Table 28.

28

Fuzzy pairwise comparison matrix for the alternatives with respect to security.


#### Table 31.

Fuzzy pairwise comparison matrix for the alternatives with respect to salary.


#### Table 32.

Fuzzy pairwise comparison matrix for the alternatives with respect to reputation.

Once the weight vector of covering criteria W and the weight vector of the alternative S have been computed, the fuzzy AHP obtains a vector V of global scores by multiplying S and W as:

$$\mathbf{V} = \mathbf{S} \cdot \mathbf{W} \tag{30}$$

Finally, the alternative ranking is accomplished by ordering these global scores in a descending order. Table 33 shows the final weights of the alternatives with respect to the covering criteria. It is clear that the college is the preferred candidate. The second candidate is the domestic company, then the third candidate is the state university and the last candidate is international company.


Table 33. Final weights of alternatives for fuzzy AHP method.

## 5. Conclusions

In this chapter, a comparative analysis of analytic hierarchy process and fuzzy analytic hierarchy process is presented using two levels of criteria example. The analytic hierarchy process method is mainly used in crisp values, the normalized weight of each alternative shows that domestic company has higher priority (0.381) than the other alternatives while the fuzzy analytic hierarchy process used in range values, the normalized weight of each alternative shows that college has higher priority (0.524) than the other alternatives.

The fuzzy analytic hierarchy process approach is preferred by decision makers than analytic hierarchy process approach because fuzzy analytic hierarchy process applies a range of values to incorporate the decision maker's uncertainly. It enhances the potential of the analytic hierarchy process for dealing with imprecise and uncertain human comparison judgments.

The example showed that weight values of some criteria, sub-criteria and alternatives in fuzzy analytic hierarchy process became zero, as shown in Tables 20–22, etc., which look odd as results, because normally all given criteria are used in pairwise comparisons and assumed to be evaluated to non-zero values. This is not a strange position because the decision makers may do not take into account one or more criteria for the evaluation even if these criteria are set in the hierarchy. Therefore, the fuzzy analytic hierarchy process approach provides to eliminate the unnecessary criterion or criteria if all of the decision makers assign "extremely important" value when compared with the other criteria and expresses the less important criteria.

Author details

31

Ain Shams University, Cairo, Egypt

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

provided the original work is properly cited.

Hossam Kamal El-Din, Hossam E. Abd El Munim and Hani Mahdi\* Computer and Systems Engineering Department, Faculty of Engineering,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: hani.mahdi@eng.asu.edu.eg

Decision-Making in Fuzzy Environment: A Survey DOI: http://dx.doi.org/10.5772/intechopen.88736

## Author details

5. Conclusions

Table 33.

important criteria.

30

priority (0.524) than the other alternatives.

Final weights of alternatives for fuzzy AHP method.

Application of Decision Science in Business and Management

and uncertain human comparison judgments.

In this chapter, a comparative analysis of analytic hierarchy process and fuzzy analytic hierarchy process is presented using two levels of criteria example. The analytic hierarchy process method is mainly used in crisp values, the normalized weight of each alternative shows that domestic company has higher priority (0.381) than the other alternatives while the fuzzy analytic hierarchy process used in range values, the normalized weight of each alternative shows that college has higher

The fuzzy analytic hierarchy process approach is preferred by decision makers than analytic hierarchy process approach because fuzzy analytic hierarchy process applies a range of values to incorporate the decision maker's uncertainly. It

enhances the potential of the analytic hierarchy process for dealing with imprecise

The example showed that weight values of some criteria, sub-criteria and alternatives in fuzzy analytic hierarchy process became zero, as shown in Tables 20–22, etc., which look odd as results, because normally all given criteria are used in pairwise comparisons and assumed to be evaluated to non-zero values. This is not a strange position because the decision makers may do not take into account one or more criteria for the evaluation even if these criteria are set in the hierarchy. Therefore, the fuzzy analytic hierarchy process approach provides to eliminate the unnecessary criterion or criteria if all of the decision makers assign "extremely important" value when compared with the other criteria and expresses the less

Hossam Kamal El-Din, Hossam E. Abd El Munim and Hani Mahdi\* Computer and Systems Engineering Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt

\*Address all correspondence to: hani.mahdi@eng.asu.edu.eg

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Saaty TL. The Analytic Hierarchy Process. New York: McGraw-Hill; 1980. p. 324

[2] Chang DY. Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research. 1996;95(3):649-655

[3] Ho W, Ma X. The state-of-the-art integrations and applications of the analytic hierarchy process. European Journal of Operational Research. 2018; 267(2):399-414

[4] Buckley JJ. Fuzzy hierarchical analysis. Fuzzy Sets and Systems. 1985; 17(3):233-247

[5] Kubler S, Robert J, Derigent W, Voisin A, Le Traon Y. A state-of the-art survey & testbed of fuzzy AHP (FAHP) applications. Expert Systems with Applications. 2016;65:398-422

[6] Nguyen PT, Vu NB, Van Nguyen L, Le LP, Vo KD. The application of fuzzy analytic hierarchy process (F-AHP) in engineering project management. In 2018 IEEE 5th International Conference on Engineering Technologies and Applied Sciences (ICETAS). IEEE. November, 2018. pp. 1-4

[7] Ge Z, Liu Y. Analytic hierarchy process based fuzzy decision fusion system for model prioritization and process monitoring application. IEEE Transactions on Industrial Informatics. 2018;15(1):357-365

[8] Prascevic N, Prascevic Z. Application of fuzzy AHP for ranking and selection of alternatives in construction project management. Journal of Civil Engineering and Management. 2017;23 (8):1123-1135

[9] Saaty TL. Decision making with the analytic hierarchy process. International Journal of Services Sciences. 2008;1(1): 83-98

[10] Saaty TL. How to make a decision: The analytic hierarchy process. European Journal of Operational Research. 1990;48(1):9-26

[11] Saaty TL, Vargas LG. Models, Methods, Concepts and Applications of the Analytic Hierarchy Process. Boston: Kluwer Academic Publishers; 2000

**33**

**Chapter 3**

**Abstract**

Business and Information System

eGovernment Service Practice: An

This chapter examines previous studies of alignment between business and information systems holistically in relation to the development of working associations among professionals from information system and business backgrounds in business organization and eGovernment sectors while investigating alignment research that permits the development and growth of information system, which is appropriate, within budget and on-time development. The process of alignment plays a key role in the construction of dependent associations among individuals from two different groups, and the progress of alignment could be enhanced by emerging an information system according to the investors' prospects. The chapter presents system theory to gather and analyze the data across the designated platforms. The outcomes classify that alignment among business and information system departments remains a priority and is of worry in different ways in diverse areas, which provides prospects for the

**Keywords:** theories, practices, goal modeling, IS alignment, working relationships,

The trend toward globalization of the business and eGovernment sector remains

undiminished and has produced philosophical renovations, both internal and external, as mostly business firms and eGovernment sectors seek to establish strong alignment in their value chain while attempting to hearth closer relations with their customers and commercial partners. In answer to, or anticipation of variations in their atmosphere, most of organizations and eGovernment sectors are deploying information system applications for this purpose, at a rising rate [1–4]. Thus, this has elevated a key question vital to the present business and eGovernment paradigm: how can an eGovernment and business organization truly justify its investments on information system in the context of donating to business organization and eGovernment performance, be it in terms of effectiveness, augmented market

share, output, or other pointers of structural usefulness?

Alignment Theories Built on

Holistic Literature Review

*Sulaiman Abdulaziz Alfadhel, Shaofeng Liu* 

*and Festus O. Oderanti*

forthcoming discussion and research.

system integration

**1. Introduction**

## **Chapter 3**

References

267(2):399-414

17(3):233-247

p. 324

[1] Saaty TL. The Analytic Hierarchy Process. New York: McGraw-Hill; 1980.

Application of Decision Science in Business and Management

[10] Saaty TL. How to make a decision:

The analytic hierarchy process. European Journal of Operational Research. 1990;48(1):9-26

[11] Saaty TL, Vargas LG. Models, Methods, Concepts and Applications of the Analytic Hierarchy Process. Boston: Kluwer Academic Publishers; 2000

[2] Chang DY. Applications of the extent

[3] Ho W, Ma X. The state-of-the-art integrations and applications of the analytic hierarchy process. European Journal of Operational Research. 2018;

[4] Buckley JJ. Fuzzy hierarchical analysis. Fuzzy Sets and Systems. 1985;

[5] Kubler S, Robert J, Derigent W, Voisin A, Le Traon Y. A state-of the-art survey & testbed of fuzzy AHP (FAHP) applications. Expert Systems with Applications. 2016;65:398-422

[6] Nguyen PT, Vu NB, Van Nguyen L, Le LP, Vo KD. The application of fuzzy analytic hierarchy process (F-AHP) in engineering project management. In 2018 IEEE 5th International Conference on Engineering Technologies and Applied Sciences (ICETAS). IEEE.

November, 2018. pp. 1-4

2018;15(1):357-365

(8):1123-1135

83-98

32

[7] Ge Z, Liu Y. Analytic hierarchy process based fuzzy decision fusion system for model prioritization and process monitoring application. IEEE Transactions on Industrial Informatics.

[8] Prascevic N, Prascevic Z. Application of fuzzy AHP for ranking and selection of alternatives in construction project

Engineering and Management. 2017;23

[9] Saaty TL. Decision making with the analytic hierarchy process. International Journal of Services Sciences. 2008;1(1):

management. Journal of Civil

analysis method on fuzzy AHP. European Journal of Operational Research. 1996;95(3):649-655
