## 3 Multi-Boundary Fuzzy LR  
The relationship between the RSSI and the distance (d) for the radio wave propagation is expressed as follows in a mathematical LR form:  
\[RSSI_{i}=A-Blog(d_{i}) \tag{1}\]  
where \(A\) and \(B\) are the relationship coefficients. The distance and the \(RSSI\) values express the regression coefficients with fuzzy numbers as follows due to the wave propagation uncertainty:  
\[RSSI^{\prime}_{i}=\dot{\mathrm{A}}-\dot{\mathrm{B}}log(d_{i}) \tag{2}\]  
The fuzzy membership functions of the fuzzy \(RSSI\) variables (\(RSSI^{\prime}\)) can be derived from the measurement uncertainties. The membership functions of fuzzy coefficients \(\dot{\mathrm{A}}\) and \(\dot{\mathrm{B}}\) can be evaluated using the fuzzy regression analysis based on the fuzzy extension principle [24; 25; 26]. The left-right (L-R) presentation of the fuzzy number provides a suitable means for representing the fuzzy coefficients. Let \(\dot{\mathrm{A}}\) be the coefficient expressed as follows:  
\[\dot{A}=f(C,L,R) \tag{3}\]  
where \(C\) is the central value, and \(L\) and \(R\) are the left and right spreads, respectively. The membership function \(\mu_{A}(x)\), of the triangular \(L\)-\(R\) fuzzy number is given by Equation (4). \[\mu_{A}(x)= \begin{cases}1-\frac{C-x}{L}for\ C-L\leq x\leq C,L>0\\ 1-\frac{x-C}{R}for\ C\leq x\leq C+R,R>0\end{cases}  \tag{4}\]Note that the maximum _l-r_ spreads are the boundary of the model that occurs at membership level 0.0. We incorporate the uncertainty not captured in the available measurement data sets by using the proposed MBFLR model and define a variable membership level \(\mu_{A}\) as a factor [27] for fitting the certainty captured by RSSI data, called the \"initial membership level\" (\(\mu_{AI}\)), to extend the L-R spreads for the uncertainty not captured at the lower membership levels (see Figure 1). According to this approach, each of the measured RSSI must be within the boundaries around the estimated regression curves at lower \(\mu_{AI}\) values. The spread of the membership function and the fuzziness of the regression variables can be controlled by specifying the \(\mu_{AI}\) level between 0 and 1. The \(\mu_{AI}\) value is useful in considering quantified uncertainties, such as the maximum or minimum RSSI, caused by the fading electromagnetic wave. Accordingly, the left and right spreads of the L-R fuzzy numbers A and B can be expressed as follows:  
\[C_{A}-L_{A}\ (1-\mu_{AI})\leq\ \dot{A}\ \leq C_{A}\ +R_{A}\ (1-\mu_{AI}) \tag{5}\]  
\[C_{B}-L_{B}\ (1-\mu_{AI})\leq\ \dot{B}\ \leq C_{B}\ +R_{B}\ (1-\mu_{AI}) \tag{6}\]  
The spread of the RSSI fuzzy number RSSI' obtained from the measurement uncertainty analysis is expressed as follows:  
\[C_{RSSI}-L_{RSSI}\leq C_{RSSI}\leq C_{RSSI}+R_{RSSI} \tag{7}\]  
Figure 1: Concept of MBFLR. Based on the uncertainty RSSI measurement, the fuzzy membership function of the output has a triangular shape (see Figure 1). The two corners of the triangular base represent the zero-membership level of fuzzy set A. For example, an initial membership level (\(\mu_{AI}\)) of 0.4 provides the lower bound of the fuzzy data curve 0.0(L) and 0.2(L) that will intersect with the left spread of the lower membership values 0.0 and 0.2, respectively. Similarly, the higher upper bound of the fuzzy rating curve 0.0(U) and 0.2(U) will intersect with the right spread of the lower membership values 0.0 and 0.2, respectively. Therefore, Equations (2) and (5)-(7) can be combined, such that the lower bound of the fuzzy data curve intersects at the boundary of the left spread, and the upper bound intersects at that of the right spread.