**4. Methodology**

A hybrid model of the conventional AHP and fuzzy theory was utilized to conduct the LSM in this study. The AHP has shown good capacity in solving a multicriteria decision-making problem by incorporating expert knowledge into quantitative analysis. To reduce subjectifies involved in conventional AHP analysis, the fuzzy set theory was adopted to handle blurry sets or categories. Hence, the hybrid use of fuzzy sets and conventional AHPs effectively addresses the decision-making issues under multiple criteria. The theoretical background of the conventional AHP and fuzzy theory were briefly introduced in Sections 4.1 and 4.2, respectively. After that, the process of incorporating fuzzy theory into AHP was given in Section 4.3.

#### *4.1. The Theoretical Background of the Conventional AHP*

The AHP, originally developed by Saaty [38], has shown grea<sup>t</sup> potential for handling multicriteria decision-making (MCDM) issues. Implemented in GIS, the AHP has been successfully employed in LSM in many previous studies [13,29,31]. A detailed description of the AHP application steps in LSM was introduced by Van et al. [31] and can be summarized as follows:

Step 1: Dividing the decision problem into a hierarchical structure

In this step, a complex decision problem was decomposed into a hierarchical structure, including an "objective" level on the top, one or more "criterion" level(s) in the middle, and several decision alternatives at the bottom level. Although there are no universal rules to be followed in constructing such a hierarchy, it was suggested by Saaty [39] that the hierarchy be built based on the decision maker's knowledge and experience with the problem.

#### Step 2: Constructing the pairwise comparison matrix

In this step, a comparison matrix was constructed with each element indicating the pairwise comparison between all the decision elements. By asking the decision maker how important alternative A is compared to alternative B, the pairwise comparison results (relative importance) are usually rated using a linguistic variable, such as "Slightly Important", "Moderately Important", or "Extremely Important" (Table 2).


**Table 2.** Triangular fuzzy scale used in this study.

Step 3: Calculate the weights of each decision element and check its consistency

For each comparison matrix, the relative weights of each decision element were calculated using the eigenvalue method (or some other methods). Weights could be used only if consistency had been satisfied.

Step 4: Hierarchically aggregate weights from all "criterion" levels

In this step, the score of each alternative with respect to the final goal was calculated by aggregating the weights of decision elements' weights from all "criterion" levels.

The numerical intensity scale for the relative importance between two decision elements, proposed by Saaty [40], has been broadly used in the AHP. Table 2 shows that in this study, the importance of "Equally Important" to "Extremely Important" was scaled from 1 to 9. The numbers 2, 4, 6, and 8 were used to describe intermediate importance. Inverse importance was scaled using the reciprocals of the numbers from 1 to 1/9. The eigenvalue method was adopted to calculate the weights. In this regard, the consistency index (*CI*) was calculated as follows:

$$CI = \frac{\lambda\_{\text{max}} - n}{n - 1} \tag{3}$$

where λmax represents the largest eigenvalue of a matrix.

For evaluation of the *CI*, the term consistency ratio (*CR*) was introduced. The *CR* was defined as the ratio of a given *CI* and that of a randomly generated reciprocal matrix (*RI)*. Consistency is satisfied if *CR* < 0.1.

$$CR = \frac{CI}{RI} \tag{4}$$

#### *4.2. Triangular Fuzzy Number (TFN)*

In the practical application of the conventional AHP for LSM, the determination of the exact relative importance of two factors (A and B) is more difficult than to identify one factor as being more important to another. Given this, fuzzy theory was employed to extend the conventional AHP by scaling the experts' decisions as fuzzy numbers. Thus, assigning exact ratio values to pairwise comparison results was avoided. There are many types of fuzzy numbers. For this study, triangular fuzzynumbers(TFNs)wereused.ConceptsfortheTFN-AHParebrieflyintroducedinthefollowing.

 Let *M* be a TFN on *R*; then, its member function *x* ∈ *M* , μ*M*(*x*) : *R* → [0, 1] can be defined as follows:

$$\mu\_M(\mathbf{x}) \begin{cases} 0 & \mathbf{x} < a \text{ or } \mathbf{x} > c \\ (\mathbf{x} - a) / (b - a), & a \le \mathbf{x} \le b \\ (c - \mathbf{x}) / (c - b), & b \le \mathbf{x} \le c \end{cases} \tag{5}$$

where *a*, *b*, and *c* represent the left, modal, and right values of *M* , respectively (see Figure 4).

**Figure 4.** Illustration of the membership function of TFNs.

A TFN can be denoted by *M* = (*a*, *b*, *c*). Let *M* 1 = (*<sup>a</sup>*1, *b*1, *<sup>c</sup>*1) and *M* 2 = (*<sup>a</sup>*2, *b*2, *<sup>c</sup>*2) be two TFNs, where *a*1, *a*2 > 0, *b*1, *b*2 > 0 and *c*1, *c*1 > 0. The laws of the operations can be defined as follows:

*Land* **2020**, *9*, 535

• Summation of two TFNs:

$$
\overline{M}\_1 \oplus \overline{M}\_2 = (a\_1 + a\_2, b\_1 + b\_2, c\_1 + c\_2) \tag{6}
$$

• Subtraction of a TFN from another TFN:

$$M\_1 \Theta M\_2 = (a\_1 - a\_2, b\_1 - b\_2, c\_1 - c\_2) \tag{7}$$

• Multiplication of two TFNs:

$$
\overline{M}\_1 \otimes \overline{M}\_2 = (a\_1 \times a\_2, b\_1 \times b\_2, c\_1 \times c\_2) \tag{8}
$$

• Multiplication of a number and a TFN:

$$
\lambda \otimes \overline{M}\_1 = \left(\lambda \times a\_1, \lambda \times b\_1, \lambda \times \mathbf{c1}\right) \tag{9}
$$

• Division of a TFN by another TFN:

$$
\tilde{M}\_1 \bullet \tilde{M}\_2 = (a\_1, b\_1, c\_1) \bullet (a\_2, b\_2, c\_2) = (\frac{a\_1}{c\_2}, \frac{b\_1}{b\_2}, \frac{c\_1}{a\_2}) \tag{10}
$$

• Reciprocal of a TFN:

$$
\tilde{M}\_1^{-1} = (a\_1, b\_1, c\_1)^{-1} = (\frac{1}{c\_1}, \frac{1}{b\_1}, \frac{1}{a\_1}) \tag{11}
$$

#### *4.3. Integration of the AHP and TFN*

The integration of fuzzy sets with the AHP has shown grea<sup>t</sup> potential not only for use in LSM but also in many other multicriteria decision making processes, such as hospital location selection and tourist risk evaluation. Very reliable results have been obtained in these applications. The following sections will describe the TFN-AHP theory.

In TFN-AHP theory, experts' judgments are scaled using a TFN rather than a definite number. Then, the TFN comparison matrix is defined as follows:

$$
\widetilde{K} = (\widetilde{k}\_{\widetilde{i}j})\_{n \times n} = \begin{bmatrix}
\widetilde{1} & \widetilde{k}\_{12} & \dots & \widetilde{k}\_{1n} \\
\overline{k}\_{21} & \overline{1} & \dots & \overline{k}\_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\overline{k}\_{n1} & \overline{k}\_{n2} & \dots & \overline{1}
\end{bmatrix} = \begin{bmatrix}
\widetilde{1} & \overline{k}\_{12} & \dots & \overline{k}\_{1n} \\
\overline{k}\_{12}^{-1} & \overline{1} & \dots & \overline{k}\_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\overline{k}\_{1n}^{-1} & \overline{k}\_{2n}^{-1} & \dots & \overline{1}
\end{bmatrix} \tag{12}
$$

where *kij* = (*aij*, *bij*, *cij*) denotes the fuzzy comparison value of criterion *i* to criterion *j* and *kij*−<sup>1</sup> = ( 1*cij*, 1*bij*, 1*aij*) denotes the reciprocal value for *i*, *j* = 1, 2, ··· , *n* and *i* - *j*.

If such judgments are made by consulting more than one expert, the element value is calculated by the average of their decisions as follows:

$$k\_{ij} = \frac{1}{h} \otimes \left(k\_{ij}^1 + k\_{ij}^1 + \dots + k\_{ij}^1\right) \tag{13}$$

where *kpij* = [*lpij*, *mpij*, *upij*] *p* ∈ [1, 2, ··· , *h*] and *h* is the number of experts.

Consistent with Chang (1996) [41], the required steps to compute the weight vector for a TFN comparison matrix can be expressed using the following procedure:

*Land* **2020**, *9*, 535

First, calculate the fuzzy synthetic extent with respect to the *i*th alternative by normalization of the row sums of the fuzzy comparison matrix as follows:

$$\widetilde{S}\_i = \sum\_{j=1}^n \overline{k\_{ij}} \otimes \left[ \sum\_{l=1}^n \sum\_{j=1}^n \overline{k\_{lj}} \right]^{-1}, i = 1, 2, \cdots, n \tag{14}$$

Then, calculate weight vectors concerning each decision element under a certain criterion using the degree of possibility of *Mi* ≥ *<sup>M</sup>j*, which is defined as follows:

$$V\{\tilde{M}\_i \ge \tilde{M}\_j\} = \sup\_{y \ge x} \{ \min \{ \tilde{M}\_j(x), \tilde{M}\_i(y) \} \}\tag{15}$$

This equation can be equivalently expressed as follows:

$$V(\widetilde{M}\_i \ge \widetilde{M}\_j) = \begin{cases} 1 & b\_i \ge b\_j \\ \frac{c\_i - a\_j}{(c\_i - b\_i) + \left(b\_j - a\_j\right)} & a\_j \ge c\_i \\ 0 & \text{otherwise} \end{cases} \tag{16}$$

Finally, calculate the normalized vector of weights *W* = (*w*1, *<sup>w</sup>*2, ··· , *<sup>w</sup>n*) of the TFN comparison matrix *K* as follows: 

$$w\_{i} = \frac{V(\overline{M}\_{i} \ge \overline{M}\_{j} | j = 1, 2, \cdots, n; i \ne j)}{\sum\_{k=1}^{n} V(\overline{M}\_{i} \ge \overline{M}\_{j} | j = 1, 2, \cdots, n; i \ne j)}\tag{17}$$

The typical LSM is performed based on raster cells. For the TFN-AHP application in LSM, the criteria refer to a series of LCFs, whereas the alternatives refer to the raster cells within the study area. To perform the LSM using the TFH-AHP, a weighted linear combination (WLC) is conducted to calculate the LSI for each raster pixel as follows:

$$\text{LSI} = \sum\_{i=1}^{n} w\_i \cdot s\_i^{k(x,y)} \tag{18}$$

where *wi* is the weight of *i*th criterion (LCF) *i*, *s k*(*<sup>x</sup>*,*y*) *i* is the weight of the *k*th subcriteria (subclass for the LCF) in the *i*th criterion, and *k* is determined by the spatial location (*<sup>x</sup>*, *y*) of the raster cell.
