*2.2.2.6 Defecation disorder*

The author defines the universal value for the criteria for defecation disorder is [0,1] and divides it into 3 categories of fuzzy triangles, namely normal (N), difficult to do defecation (SB) and diarrhea (D). By using the concept of the Likert scale and the defuzzy method, Large of Maximum, **Table 6** is obtained as the weight of the criteria for BAB defects.

## *2.2.3 Determine the suitability rating of each alternative on each criterion*

Interview results from an expert (doctor) on **Table 7**. From the **Table 8**, the match rating value is obtained as follows:


**Table 5.** *Weight of headache.*

#### *Fuzzy Multi-Attribute Decision Making (FMADM) Application on Decision Support… DOI: http://dx.doi.org/10.5772/intechopen.94614*


**Table 6.**

*Defecation disorder weight.*


**Table 7.**

*Linguistics data.*


#### **Table 8.**

*Match rating value.*

The compatibility rating in this method is also called the decision matrix which will be normalized.

#### *2.2.4 The determination of the preference weight*

The determination of the preference weight is stated in **Table 9** as follows:

### *2.2.5 Normalization of the matrix*

$$R = \begin{bmatrix} 0.75 & 0.25 & 0 & 0.5 & 0.25 & 1\\ 0.75 & 1 & 1 & 0.5 & 0.5 & 0.75\\ 0.75 & 0.25 & 0 & 0 & 0.5 & 0.5\\ 1 & 0 & 0 & 0.75 & 0.75 & 0.75\\ 0.5 & 0 & 0 & 0.5 & 0.25 & 0.5\\ 1 & 0 & 0 & 0.5 & 0.75 & 1 \end{bmatrix} \tag{1}$$


**Table 9.**

*Preference weight (W).*

To find a matrix you can use the following formula:

$$r\_{ij} = \begin{cases} \frac{\mathcal{X}\_{ij}}{\max\limits\_{i} \mathcal{X}\_{ij}}, j \text{ is benefit attribute} \\\\ \frac{\min\limits\_{i} \mathcal{X}\_{ij}}{\mathcal{X}\_{ij}}, j \text{ is cost attribute} \end{cases} \tag{2}$$

*2.2.6 Finding preference values obtained from multiplication of weights W with normalized matrix R*

$$\mathbf{V}\_{\circ} = \sum\_{j=1}^{n} w\_{j} r\_{\circ j} \tag{3}$$

The results of the calculation are shown in **Table 10** as follows.

The highest value achieved by the second alternative (*V*2) is DBD so someone will be stated to suffer from DHF if they experience symptoms of high fever, spots (petheciae) very much, have experienced bleeding gums if they have entered a severe stage, rarely nausea, rarely headaches and have diarrhea, but to be sure to be able to use laboratory tests again.

In this case, SAW method is not appropriate if it is used to make a decision support system thus the author tries to use a method developed by Joo (2004) [6], namely the FMADM method with development or FDM.

#### *2.3 The FMADM method with SAW to diagnose a type of disease*

#### *2.3.1 Representation of the problem*

Consists of 3 stages, namely:

a. Objective Identification

The purpose of this decision is to determine or diagnose an illness that is suffered based on the initial symptoms experienced.

b. Identification of Criteria and Alternatives. The criteria used are still 6 types of diseases and 6 criteria (symptoms).


**Table 10.** *Preference value.* *Fuzzy Multi-Attribute Decision Making (FMADM) Application on Decision Support… DOI: http://dx.doi.org/10.5772/intechopen.94614*

c. The hierarchical structure that determines the disease is shown in the **Figure 1**.

## *2.3.2 Evaluation of Fuzzy Sets*

Consists of 4 stages, namely:

a. Selecting the set of ratings for the criteria weights. There are two things that must be done, namely determining the degree of importance and determining the degree of compatibility. T (importance) W = {*c*<sup>1</sup> = {N, DR, DS, DT, DST}, *c*<sup>2</sup> = {TA, DK, ABYK, BYK, SBYK}, *c*<sup>3</sup> = {TP, P}, *c*<sup>4</sup> =, *c*<sup>5</sup> = {TP, P, J, S}, *c*<sup>6</sup> = {NR, D, SB}}. T (match) S = {Very Low (SR), Low (R), Enough (C), High (T), Very High (ST)}.

The parameters of each level of interest are as follows:

$$\begin{aligned} \text{N} &= (\mathbf{0}, \mathbf{0}, \mathbf{0}.25), & \text{TA} &= (\mathbf{0}, \mathbf{0}, \mathbf{0}.25), \\ \text{DR} &= (\mathbf{0}, \mathbf{0}.25, \mathbf{0}.5), & \text{SDK} &= (\mathbf{0}, \mathbf{0}.25, \mathbf{0}.5), \\ \text{DS} &= (\mathbf{0}.25, \mathbf{0}.5, \mathbf{0}.75), & \text{ABYK} &= (\mathbf{0}.25, \mathbf{0}.5, \mathbf{0}.75), \\ \text{DT} &= (\mathbf{0}.5, \mathbf{0}.75, \mathbf{1}), & \text{BYK} &= (\mathbf{0}.5, \mathbf{0}.75, \mathbf{1}) \\ \text{DST} &= (\mathbf{0}.75, \mathbf{1}, \mathbf{1}), & \text{SBYK} &= (\mathbf{0}.75, \mathbf{1}, \mathbf{1}) \\ \text{TP} &= (\mathbf{0}, \mathbf{0}, \mathbf{1}), & \text{NR} &= (\mathbf{0}.25, \mathbf{0}.5, \mathbf{0}.75), \\ \text{P} &= (\mathbf{0}, \mathbf{1}, \mathbf{1}), & \text{D} &= (\mathbf{0}.5, \mathbf{0}.75, \mathbf{1}) \\ \text{J} &= (\mathbf{0}.25, \mathbf{0}.5, \mathbf{0}.75), & \text{SB} &= (\mathbf{0}.75, \mathbf{1}, \mathbf{1}) \\ \text{S} &= (\mathbf{0}.5, \mathbf{0}.75, \mathbf{1}), & & \\ \end{aligned}$$

The degree of compatibility of each decision criteria as follows:

$$\begin{aligned} \text{Very Low (SR)} &= (0, 0, 0.25), \\ \text{Low (R)} &= (0, 0.25, 0.5), \end{aligned}$$

**Figure 1.** *Hierarchy Structure.*

$$\begin{aligned} \text{Enough } (\mathsf{C}) &= (\mathsf{0}.\mathsf{25}, \mathsf{0}.\mathsf{5}, \mathsf{0}.\mathsf{75}), \\ \text{Height } (\mathsf{T}) &= (\mathsf{0}.\mathsf{5}, \mathsf{0}.\mathsf{75}, \mathsf{1}) \\ \text{Very High } (\mathsf{ST}) &= (\mathsf{0}.\mathsf{75}, \mathsf{1}, \mathsf{1}) \end{aligned}$$

Based on this, the degree of compatibility of each alternative is obtained to the decision criteria in **Table 11** and the branch of interest for the decision criteria in **Table 12**.

b. Aggregate the weight of criteria and the degree of compatibility of each alternative with its criteria, using the following equation:

$$Y\_i = \left(\frac{1}{k}\right) \sum\_{t=1}^{k} (o\_{it} a\_i) \tag{4}$$

$$Q\_i = \left(\frac{1}{k}\right) \sum\_{t=1}^{k} (p\_{it} b\_i) \tag{5}$$

$$Z\_i = \left(\frac{1}{k}\right) \sum\_{t=1}^{k} (q\_{it}c\_i) \tag{6}$$

The result is compatibility index obtained from the aggregation of the weight of the criteria and the degree of compatibility of each alternative with its criteria that's shown in **Table 13**.

#### *2.3.3 Selecting optimal alternatives*

Prioritizing decision alternatives based on aggregation results by substituting the fuzzy match index value into the following equation:

$$I\_T^a(F) = \left(\frac{\mathbf{1}}{2}\right) (ac + b + (\mathbf{1} - a)a) \tag{7}$$


**Table 11.**

*The degree of compatibility of each alternative to the decision criteria.*


**Table 12.**

*Branch of interest for decision criteria.*

*Fuzzy Multi-Attribute Decision Making (FMADM) Application on Decision Support… DOI: http://dx.doi.org/10.5772/intechopen.94614*


**Table 13.** *Compatibility index.*

By taking optimism degree (α), namely: α = 0 (not optimistic), α = 0.5 (optimistic) and α = 1 (very optimistic). The following results are obtained on **Table 14**.

Based on the results above, it can be seen that regardless of the degree of optimism, the alternative a2 is that DHF has the greatest value compared to other alternatives.
