Forecasting the Number of Passengers on Hungarian Railway Routes Using a Similarity and Fuzzy Arithmetic-Based Inference Method

Abstract

In this study, we present a similarity and fuzzy arithmetic-based fuzzy inference method and show how effectively it can be used to forecast the number of passengers on a railway route. We introduce a novel fuzzy similarity measure that is derived from the so-called epsilon function, which may be viewed as an alternative to the exponential function. After demonstrating the most important properties of the new similarity measure, we construct a fuzzy inference method that is founded on arithmetic operations over triangular fuzzy numbers. This inference method utilizes the proposed similarity measure to derive weight values for the above-mentioned arithmetic operations. The motivation behind the proposed method is twofold. On the one hand, we aim to construct a method that is simple and easy to implement. On the other hand, we intend to ensure that this method meets the practical requirements for rail passenger forecasts. Using a real-life case study, we demonstrate how well our method can predict the expected number of passengers on a new railway route based on characteristics of this relation. With respect to the studied case, we may conclude that although the similarity and fuzzy arithmetic-based fuzzy inference system has only two adjustable parameters, it may be regarded as a viable alternative to Sugeno-type fuzzy inference systems with a much greater number of adjustable parameters tuned by various optimization techniques.

1. Introduction

Nowadays, when mobility plays a major role in the economy, in business and in private life, understanding the magnitude of the demand for a transport service has become an important factor for service providers. This is also the case for railway operators, for whom forecasting the expected number of passengers on a railway route that does not yet exist is of great importance. Accordingly, several methods have been developed to forecast rail passenger traffic. Some of them are mentioned here, although the list is not exhaustive.

There are some examples of fuzzy models used for short-term passenger demand prediction. A fuzzy model has been developed in a study to predict peak-time demand on the Beijing–Shanghai high-speed railway, which predicts peak cross sections during holiday seasons, helping rail dispatchers to minimize costs and the number of passengers who miss free capacity (see [1]). The application of this method has been validated in practice, showing that fuzzy forecasting has led to significant improvements compared to the previously used methods. There are also some examples of looking for passenger demand trends in a longer-term historical passenger flow database, and learning the characteristics of historical passenger flows can be used to predict future demands using regression techniques (see [2,3]). In other studies, models for air transport demand forecasting were developed by combining neural and fuzzy logic models, with a strong emphasis on the seasonality of transport demands (see [4,5]). A model based on fuzzy logic has also been developed in a study on the prioritization of transport investments, with the aim of prioritizing governmental investments that can have the greatest impact on the shift to public and sustainable transport modes (see [6]). There are also examples of research topics that deal with the human perception of public transport systems, with the aim of demonstrating the potential to increase public transportation ridership (see [7]). Many research articles deal with the implementation of artificial neural networks (ANNs) in transportation sciences and transportation forecasting. The findings and the results comparisons show that these applications have a better prediction ability than the standard methods; however, in most cases it is pointed out that there is more to come in the development and use of artificial neural networks for demand forecasting (see [8,9,10]). Recent studies on traffic forecasting suggest that combinations of existing travel demand forecasting methods can produce more accurate results than using different methods separately. Forecasting techniques ranging from regression models to artificial neural networks and support vector machines (SVMs) incorporating soft computing methods are discussed and compared in [11]. There are many examples where fuzzy forecasting models are combined with neural networks, combinations that can yield more accurate predictions, but the background to the functioning of neural networks might cause difficulties in some cases (see [12,13]). There are also examples of comparing different types of transport demand forecasting models and identifying factors that influence rail transport demand. Examples of such explanatory variables include average rail passenger travel distance, car ownership index, number of buses working in interurban routes per year, or a competition variable, expressed as the ratio of unit cost by bus to the unit cost by rail (see [14]). In some cases, Grey Relational Analysis is also adopted in transportation demand forecasting as well as other consumer demand models (see [15,16]). In other studies, the characteristics of total passenger demand, rail passenger demand and personal car demand are also discussed, with the possible modeling of the modal split (see [17,18]). Passenger demands can also be examined through capacity allocation and timetable planning models. There are examples for the development of timetable optimizations based on the characteristics of railway infrastructure and vehicles (see [19,20]).

The difficulties of standardizing different demand forecasting models is also highlighted, indicating the difficulties of choosing the right type of modeling methods that could be used in different phases of a transportation planning period. An article from recent years dealt with creating a standardized transport demand forecasting knowledge base from which the optimal forecasting method can be easily selected during each planning phase (see [21]).

In this study, we introduce a similarity and fuzzy arithmetic-based fuzzy inference method and demonstrate its effectiveness in predicting the expected number of future passengers on a railway connection. We present a new fuzzy similarity measure derived from the so-called epsilon function, which serves as an alternative to the exponential function (see [22,23]). After highlighting the key properties of this new similarity measure, we develop a fuzzy inference technique that employs arithmetic operations on triangular fuzzy numbers. This method uses the proposed similarity measure to determine weight values for the aforementioned operations. Our motivation for this approach is twofold: firstly, we aim to create a method that is straightforward and easy to apply; secondly, we aspire to ensure that this method fulfills the practical requirements for rail passenger forecasting. Through a real-world case study that uses data from the Hungarian State Railways (MÁV), we illustrate how effectively our method can estimate the average number of passengers for a new railway connection based on the specific characteristics of that connection. It is worth noting that while our similarity and fuzzy arithmetic-based inference system has only two adjustable parameters, it can serve as a practical alternative to Sugeno-type fuzzy inference systems, which typically have a much larger number of adjustable parameters optimized through various techniques such as the Adaptive Neuro Fuzzy Inference System (ANFIS) (see, e.g., [24,25]), genetic algorithm (GA) (see, e.g., [26,27,28]), particle swarm optimization (PSO) (see, e.g., [29]) and pattern-search (PS) method (see, e.g., [30]). Our model and the Sugeno-type fuzzy inference systems with the above-mentioned parameter optimization techniques were implemented in the MATLAB (R2024b) software package [31]. The respective MATLAB source files ‘SFAFIS_example.m’ and ‘FIS_similarity_and_fuzzy_arithmetic_class.m’, which are available at https://jonast.web.elte.hu/, include the example of our case study. With these two files, the results of our case study can be reproduced.

The remaining part of this paper is structured as follows. In Section 2, we provide an overview of the basic concepts we employ in our study. Here, we present the arithmetic operations that will be later utilized over triangular fuzzy numbers and describe a function that will be used to construct a similarity measure. In Section 3, we propose a new fuzzy similarity measure and demonstrate its main properties. A novel fuzzy inference method called the similarity and fuzzy arithmetic-based fuzzy inference method is then presented in  Section 4. In Section 5, by the means of a real-life case study, we demonstrate how this new method can be used to predict the number of passengers on a railway route. Here, we also compare our method with some much more sophisticated fuzzy and neuro-fuzzy methods. Lastly, we make some concluding remarks and outline possible directions for future research.

2. Preliminaries

Here, we will provide a brief overview of the concepts and notations that we will make use of later on. We will use the common notation $\mathbb{R}$ for the real line.

Definition 1

(cf. [32]). Let X be a non-empty set. We say that the function ${\mu }_A:X\to [0,1]$ is the membership function of the fuzzy set A on the universe X. For any $x\in X$, ${\mu }_A\left(x\right)\in [0,1]$ is the membership value of x in the fuzzy set A.

Definition 2

(cf. [33]). Let X be a non-empty set and let ${\mu }_A:X\to [0,1]$ be the membership function of the fuzzy set A on the universe X. For any $\alpha \in (0,1]$, we say that the ordinary set

$$A_{\alpha }=\{x\in X:{\mu }_A\left(x\right)\ge \alpha \}$$

(1)

is the α-cut of the fuzzy set A.

2.1. Some Arithmetic Operations on Triangular Fuzzy Numbers

We will construct a fuzzy arithmetic-based inference method in which we will utilize triangular fuzzy numbers. We will use the following definition of triangular fuzzy numbers.

Definition 3

(cf. [34]). We say that a fuzzy set A on the real line is a triangular fuzzy number with the parameters $a,b,c\in \mathbb{R}$, $a<b<c$, if its membership function ${\mu }_A(·;a,b,c):\mathbb{R}\to [0,1]$ is given by

$${\mu }_A(x;a,b,c)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}x\le a\hfill \\ {\displaystyle \frac{x-a}{b-a}},\hfill & \mathit{if}a<x\le b\hfill \\ {\displaystyle \frac{c-x}{c-b}},\hfill & \mathit{if}b<x\le c\hfill \\ 0,\hfill & \mathit{if}x>c.\hfill \end{array}\right.$$

(2)

From now on, we will use the notation $T(a,b,c)$ for a triangular fuzzy number with the real-valued parameters a, b and c, where $a<b<c$. That is, $A=T(a,b,c)$ means that A is a triangular fuzzy number with the parameters a, b and c, and so the membership function of A is the function ${\mu }_A(·;a,b,c):\mathbb{R}\to [0,1]$ given in Definition 3.

Remark 1.

It should be added that fuzzy numbers are a class of fuzzy sets (see, e.g., [34,35]). Namely, a fuzzy set A with the membership function ${\mu }_A:\mathbb{R}\to [0,1]$ is said to be a fuzzy number if

(a) 

A is normal, i.e., there exists a $x_0\in \mathbb{R}$, such that ${\mu }_A\left(x_0\right)=1$.

(b) 

A is fuzzy convex, i.e., for any $t\in [0,1]$ and $x,y\in \mathbb{R}$,

$${\mu }_A(tx+(1-t)y)\ge min\{{\mu }_A\left(x\right),{\mu }_A\left(y\right)\}.$$

(c) 

${\mu }_A$ is upper semi-continuous on $\mathbb{R}$.

(d) 

The support of A, i.e., the set $\{x\in \mathbb{R}:{\mu }_A\left(x\right)>0\}$, is bounded.

It can be verified that a fuzzy set with the membership function ${\mu }_A(·;a,b,c):\mathbb{R}\to [0,1]$ given in Equation (2) satisfies the criteria for a fuzzy number. This means that triangular fuzzy numbers are a class of fuzzy numbers.

It is well-known (see, e.g., [33,34]) that using the concept of $\alpha$-cut and interval arithmetic operations, the addition, subtraction, multiplication by scalar, multiplication and division operations over triangular fuzzy numbers can be defined so that any set of triangular fuzzy numbers is closed under these operations. Noting the definition of the $\alpha$-cut of a fuzzy set given in Definition 2, after direct calculation, we find that for any $\alpha \in (0,1]$, the $\alpha$-cut of the triangular fuzzy number $T(a,b,c)$, $T{(a,b,c)}_{\alpha }$, is the interval

$$T{(a,b,c)}_{\alpha }=\left[(1-\alpha )a+\alpha b,\alpha b+(1-\alpha )c\right],$$

(3)

where $a,b,c\in \mathbb{R}$ and $a<b<c$. Notice that in Definition 2, $\alpha \in (0,1]$; however, for $\alpha =0$, the $\alpha$-cut of $T(a,b,c)$ is the interval $[a,c]$. Hence, for triangular fuzzy numbers, we interpret the $\alpha$-cut for any $\alpha \in [0,1]$ as the interval given by Equation (3).

Now, we will show that the weighted arithmetic mean of triangular fuzzy numbers is a triangular fuzzy number whose parameters are the weighted arithmetic means of the corresponding parameters of the triangular fuzzy numbers in question. We will utilize the following operations over intervals.

Definition 4.

For an interval $[x,y]$ and a scalar $t\in \mathbb{R}$, $t>0$, the operation $t{\odot }_{\mathcal{I}}[x,y]$ is defined as

$$t{\odot }_{\mathcal{I}}[x,y]=[tx,ty].$$

(4)

Definition 5.

For the intervals $[x_1,y_1],[x_2,y_2],\dots ,[x_n,y_n]$, $n\in \mathbb{N}$, $n\ge 2$ the operation $\underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left[x_i,y_i\right]$ is defined as

$$\underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left[x_i,y_i\right]=\left[\sum _{i=1}^n{x}_i,\sum _{i=1}^n{y}_i\right].$$

(5)

Using the scalar multiplication operation given in Definition 4 and the summation operator given in Definition 5, we can state the following theorem.

Theorem 1.

Let $T(a_1,b_1,c_1),T(a_2,b_2,c_2),\dots ,T(a_n,b_n,c_n)$ be triangular fuzzy numbers, where $n\in \mathbb{N}$, $n\ge 2$, and $a_i<b_i<c_i$ for all $i\in \{1,2,\dots ,n\}$. Furthermore, let $w_1,w_2,\dots ,w_n\in [0,1]$ such that ${\sum }_{i=1}^n{w}_i=1$. Then, for any $\alpha \in [0,1]$,

$$\underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left(w_i{\odot }_{\mathcal{I}}T{\left(a_i,b_i,c_i\right)}_{\alpha }\right)=T{\left(\sum _{i=1}^n{w}_i{a}_i,\sum _{i=1}^n{w}_i{b}_i,\sum _{i=1}^n{w}_i{c}_i\right)}_{\alpha },$$

(6)

where $T{(p,q,r)}_{\alpha }$ denotes the α-cut of a triangular fuzzy number with the parameters $p,q,r\in \mathbb{R}$, $p<q<r$.

Proof. 

Let $\alpha \in [0,1]$. Based on Equation (3), the $\alpha$-cut of $T(a_i,b_i,c_i)$ is the interval

$$T{(a_i,b_i,c_i)}_{\alpha }=\left[(1-\alpha )a_i+\alpha b_i,\alpha b_i+(1-\alpha )c_i\right],$$

(7)

where $i\in \{1,2,\dots ,n\}$. Now, using Equation (7), and the scalar multiplication and addition operations over intervals given by Equation (4) and Equation (5), respectively, we find that

$$\begin{array}{cc}\hfill & \underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left(w_i{\odot }_{\mathcal{I}}T{\left(a_i,b_i,c_i\right)}_{\alpha }\right)=\underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left(w_i{\odot }_{\mathcal{I}}\left[(1-\alpha )a_i+\alpha b_i,\alpha b_i+(1-\alpha )c_i\right]\right)\hfill \\ \hfill =& \underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left[w_i\left((1-\alpha )a_i+\alpha b_i\right),w_i\left(\alpha b_i+(1-\alpha )c_i\right)\right]\hfill \\ \hfill =& \underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left[(1-\alpha )w_i{a}_i+\alpha w_i{b}_i,\alpha w_i{b}_i+(1-\alpha )w_i{c}_i\right]\hfill \\ \hfill =& \left[\sum _{i=1}^n\left((1-\alpha )w_i{a}_i+\alpha w_i{b}_i\right),\sum _{i=1}^n\left(\alpha w_i{b}_i+(1-\alpha )w_i{c}_i\right)\right]\hfill \\ \hfill =& \left[(1-\alpha )\left(\sum _{i=1}^n{w}_i{a}_i\right)+\alpha \left(\sum _{i=1}^n{w}_i{b}_i\right),\alpha \left(\sum _{i=1}^n{w}_i{b}_i\right)+(1-\alpha )\left(\sum _{i=1}^n{w}_i{c}_i\right)\right].\hfill \end{array}$$

(8)

Next, noting Equation (7) again, we see that

$$\begin{array}{cc}\hfill & \left[(1-\alpha )\left(\sum _{i=1}^n{w}_i{a}_i\right)+\alpha \left(\sum _{i=1}^n{w}_i{b}_i\right),\alpha \left(\sum _{i=1}^n{w}_i{b}_i\right)+(1-\alpha )\left(\sum _{i=1}^n{w}_i{c}_i\right)\right]\hfill \\ \hfill =& T{\left(\sum _{i=1}^n{w}_i{a}_i,\sum _{i=1}^n{w}_i{b}_i,\sum _{i=1}^n{w}_i{c}_i\right)}_{\alpha }.\hfill \end{array}$$

(9)

Hence, with Equations (8) and (9), we obtain

$$\underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left(w_i{\odot }_{\mathcal{I}}T{\left(a_i,b_i,c_i\right)}_{\alpha }\right)=T{\left(\sum _{i=1}^n{w}_i{a}_i,\sum _{i=1}^n{w}_i{b}_i,\sum _{i=1}^n{w}_i{c}_i\right)}_{\alpha }.$$

   □

We will make use of the following scalar multiplication and summation operations over triangular fuzzy numbers.

Definition 6.

For a triangular fuzzy number $T(a,b,c)$ and a $t\in \mathbb{R}$, $t>0$, the scalar multiplication operation $t\odot T(a,b,c)$ is defined as

$$t\odot T(a,b,c)=T(ta,tb,tc).$$

(10)

Definition 7.

For the triangular fuzzy numbers $T(a_1,b_1,c_1),T(a_2,b_2,c_2),\dots ,T(a_n,b_n,c_n)$, the summation operation $\underset{i=1}{\stackrel{n}{\oplus }}T(a_i,b_i,c_i)$ is defined as

$$\underset{i=1}{\stackrel{n}{\oplus }}T(a_i,b_i,c_i)=T\left(\sum _{i=1}^n{a}_i,\sum _{i=1}^n{b}_i,\sum _{i=1}^n{c}_i\right).$$

(11)

Exploiting Theorem 1 and using Definitions 6 and 7, we can state the following corollary.

Corollary 1.

Let $T(a_1,b_1,c_1),T(a_2,b_2,c_2),\dots ,T(a_n,b_n,c_n)$ be triangular fuzzy numbers, where $n\in \mathbb{N}$, $n\ge 2$, and $a_i<b_i<c_i$ for all $i\in \{1,2,\dots ,n\}$. Furthermore, let $w_1,w_2,\dots ,w_n\in [0,1]$ such that ${\sum }_{i=1}^n{w}_i=1$. Then, the following hold:

(a) 

The weighted arithmetic mean of $T(a_1,b_1,c_1),T(a_2,b_2,c_2),\dots ,T(a_n,b_n,c_n)$ with respect to the weights $w_1,w_2,\dots ,w_n$, i.e., $\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)$, is

$$\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)=T\left(\sum _{i=1}^n{w}_i{a}_i,\sum _{i=1}^n{w}_i{b}_i,\sum _{i=1}^n{w}_i{c}_i\right).$$

(12)

(b) 

For any $\alpha \in [0,1]$, the α-cut of the weighted arithmetic mean $\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)$, i.e.,  ${\left(\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)\right)}_{\alpha }$, is equal to the weighted arithmetic mean of the α-cuts $T{(a_1,b_1,c_1)}_{\alpha },T{(a_2,b_2,c_2)}_{\alpha },\dots ,T{(a_n,b_n,c_n)}_{\alpha }$ with respect to the weights $w_1,w_2,\dots ,w_n$. That is,

$${\left(\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)\right)}_{\alpha }=\underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left(w_i{\odot }_{\mathcal{I}}T{\left(a_i,b_i,c_i\right)}_{\alpha }\right).$$

(13)

Proof. 

(a).

Using Equations (10) and (11), we immediately get

$$\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)=\underset{i=1}{\stackrel{n}{\oplus }}T(w_i{a}_i,w_i{b}_i,w_i{c}_i)=T\left(\sum _{i=1}^n{w}_i{a}_i,\sum _{i=1}^n{w}_i{b}_i,\sum _{i=1}^n{w}_i{c}_i\right).$$

(b).

Noting Equations (6) and (12) and Theorem 1, we can write

$$\begin{array}{cc}\hfill & {\left(\underset{i=1}{\stackrel{n}{\oplus }}\left(w_i\odot T(a_i,b_i,c_i)\right)\right)}_{\alpha }=T{\left(\sum _{i=1}^n{w}_i{a}_i,\sum _{i=1}^n{w}_i{b}_i,\sum _{i=1}^n{w}_i{c}_i\right)}_{\alpha }\hfill \\ \hfill =& \underset{i=1}{{\stackrel{n}{\oplus }}_{\mathcal{I}}}\left(w_i{\odot }_{\mathcal{I}}T{\left(a_i,b_i,c_i\right)}_{\alpha }\right).\hfill \end{array}$$

   □

Note that the operators ${\odot }_{\mathcal{I}}$ and ${\oplus }_{\mathcal{I}}$ are defined over intervals, while ⊙ and ⊕ are defined over triangular fuzzy numbers. Corollary 1 (b) tells us that any $\alpha$-cut interval of the weighted arithmetic mean of some triangular fuzzy numbers is none other than the weighted arithmetic mean of the $\alpha$-cut intervals of the respective triangular fuzzy numbers. We will utilize this property of triangular fuzzy numbers in our fuzzy arithmetic-based inference method.

2.2. The Epsilon Function

We will use the so-called epsilon function, which was introduced by Dombi et al. in [22], to construct a fuzzy similarity measure. The epsilon function is defined as follows.

Definition 8

(cf. [22]). Let $\lambda \in \mathbb{R}$, $\lambda \ne 0$ and $\Delta \in \mathbb{R}$, $\Delta >0$. We say that the function ${\epsilon }_{\lambda ,\Delta }:(-\Delta ,+\Delta )\to {\mathbb{R}}^{+}$, which is given by

$${\epsilon }_{\lambda ,\Delta }\left(x\right)={\left({\displaystyle \frac{\Delta +x}{\Delta -x}}\right)}^{\lambda {\displaystyle \frac{\Delta }{2}}},$$

(14)

is an epsilon function with the parameters λ and Δ.

For more details on the epsilon function and its generalization, see [23]. With direct calculation, we find that

$${\epsilon }_{\lambda ,\Delta }\left(x\right){|}_{x=0}=1$$

(15)

and

$${\displaystyle \frac{\mathrm{d}{\epsilon }_{\lambda ,\Delta }\left(x\right)}{\mathrm{d}x}}{|}_{x=0}=\lambda {\displaystyle \frac{{\Delta }^2}{{\Delta }^2-x^2}}{\left({\displaystyle \frac{\Delta +x}{\Delta -x}}\right)}^{\lambda {\displaystyle \frac{\Delta }{2}}}{|}_{x=0}=\lambda {\displaystyle \frac{{\Delta }^2}{{\Delta }^2-x^2}}{\epsilon }_{\lambda ,\Delta }\left(x\right){|}_{x=0}=\lambda .$$

(16)

Since

$${\mathrm{e}}^{\lambda x}{|}_{x=0}=1$$

(17)

and

$${\displaystyle \frac{\mathrm{d}{\mathrm{e}}^{\lambda x}}{\mathrm{d}x}}{|}_{x=0}=\lambda ,$$

(18)

from Equations (15)–(18) we see that ${\epsilon }_{\lambda ,\Delta }\left(x\right)$ and ${\mathrm{e}}^{\lambda x}$ are identical to first order at $x=0$. Moreover, we have the following result.

Theorem 2

(cf. [22]). For any $x\in (-\Delta ,+\Delta )$,

$$\underset{\Delta \to \infty }{lim}{\epsilon }_{\lambda ,\Delta }\left(x\right)={\mathrm{e}}^{\lambda x}.$$

Based on Theorem 2 and the fact that ${\epsilon }_{\lambda ,\Delta }\left(x\right)$ and ${\mathrm{e}}^{\lambda x}$ are identical to first order at $x=0$, the epsilon function can be regarded as a good approximation of the exponential function. Exploiting this property of the epsilon function, we will use it to construct a similarity measure.

3. A Fuzzy Similarity Measure Derived from the Epsilon Function

It is well known that if $d:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ is a normalized distance measure in the vector space ${[0,1]}^n$ and $\lambda >0$, then for any vectors $\mathbf{x},\mathbf{y}\in {[0,1]}^n$,

$$s_{\lambda }^{*}(\mathbf{x},\mathbf{y})={\mathrm{e}}^{-\lambda d(\mathbf{x},\mathbf{y})}$$

(19)

may be viewed as a measure of similarity between the vectors $\mathbf{x}$ and $\mathbf{y}$. Noting Equation (19), we see that the similarity measure $s_{\lambda }^{*}:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ has the following elementary properties:

(a)

For any $\mathbf{x},\mathbf{y}\in {[0,1]}^n$, if $d(\mathbf{x},\mathbf{y})=0$, then $s_{\lambda }^{*}(\mathbf{x},\mathbf{y})=1$ (maximal similarity).

(b)

For any $\mathbf{x},\mathbf{y}\in {[0,1]}^n$, if $d(\mathbf{x},\mathbf{y})=1$, then $s_{\lambda }^{*}(\mathbf{x},\mathbf{y})={\mathrm{e}}^{-\lambda }$ (minimal similarity).

(c)

For any $\mathbf{x},\mathbf{y},{\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime }\in {[0,1]}^n$, $d({\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime })>d(\mathbf{x},\mathbf{y})$ implies $s_{\lambda }^{*}({\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime })<s_{\lambda }^{*}(\mathbf{x},\mathbf{y})$ (monotonocity).

Notice that if $\lambda$ is large, i.e., $\lambda \gg 0$, then ${\mathrm{e}}^{-\lambda }\approx 0$. That is, if $\lambda \gg 0$ and $d(\mathbf{x},\mathbf{y})=1$, then $s_{\lambda }^{*}(\mathbf{x},\mathbf{y})\approx 0$.

It is well-known that for any $\lambda >0$ and $x\in [0,1]$, the function $g_{\lambda }\left(x\right)={(1-x)}^{\lambda }$ is a good approximation of ${\mathrm{e}}^{-\lambda x}$. Furthermore, $g_{\lambda }\left(0\right)=1$, $g_{\lambda }\left(1\right)=0$ and $g_{\lambda }\left(x\right)$ and ${\mathrm{e}}^{-\lambda x}$ are identical to first order at $x=0$.

Now, let $\lambda >0$ and define $h_{\lambda }:[0,1]\to [0,1]$ as

$$h_{\lambda }\left(x\right):={\epsilon }_{-\lambda ,1}\left(x\right),x\in [0,1],$$

i.e., $h_{\lambda }(·)$ is an epsilon function on $[0,1]$ with the parameters $-\lambda$ and $\Delta =1$. Hence, noting Definition 8, $h_{\lambda }\left(x\right)$ is given by

$$h_{\lambda }\left(x\right)={\left({\displaystyle \frac{1+x}{1-x}}\right)}^{-{\displaystyle \frac{\lambda }{2}}}={\left({\displaystyle \frac{1-x}{1+x}}\right)}^{{\displaystyle \frac{\lambda }{2}}},x\in [0,1].$$

(20)

Taking into account the properties of the epsilon function, we readily find that for any $\lambda >0$, $h_{\lambda }\left(0\right)=1$, $h_{\lambda }\left(1\right)=0$ and $h_{\lambda }\left(x\right)$ and ${\mathrm{e}}^{-\lambda x}$ are identical to first order at $x=0$. Hence, we see that $g_{\lambda }(·)$ and $h_{\lambda }(·)$ have similar properties on $[0,1]$. The following theorem states that for any $\lambda >0$ and $x\in [0,1]$, $h_{\lambda }\left(x\right)$ is even a better approximation of ${\mathrm{e}}^{-\lambda x}$ than $g_{\lambda }\left(x\right)$.

Theorem 3.

For any $\lambda >0$ and $x\in [0,1]$, it holds that

$${(1-x)}^{\lambda }\le {\left({\displaystyle \frac{1-x}{1+x}}\right)}^{{\displaystyle \frac{\lambda }{2}}}\le {\mathrm{e}}^{-\lambda x}.$$

(21)

Proof. 

It is sufficient to show that

$${(1-x)}^2\le {\displaystyle \frac{1-x}{1+x}}\le {\mathrm{e}}^{-2x}$$

(22)

holds because noting that $\lambda >0$ and raising both members of Equation (22) to ${\displaystyle \frac{\lambda }{2}}$, we get Equation (21).

First, we will prove the left hand side inequality in Equation (22). Since $x\in [0,1]$, we have $1-x^2\le 1$, and as $1-x^2=(1-x)(1+x)$, we find that

$$(1-x)(1+x)\le 1,$$

from which, taking into account the fact that $1-x\ge 0$ and $1+x>0$,

$${(1-x)}^2\le {\displaystyle \frac{1-x}{1+x}}$$

follows.

Now, we will prove the right hand side inequality in Equation (22). With the Maclaurin series of ${\mathrm{e}}^x$, we have

$${\mathrm{e}}^x=\sum _{i=0}^{\infty }{\displaystyle \frac{x^i}{i!}},$$

and since $x\in [0,1]$, we see that

$${\mathrm{e}}^x\ge 1+x+{\displaystyle \frac{x^2}{2}},$$

from which

$${\mathrm{e}}^{-2x}\le {\displaystyle \frac{1}{{\left(1+x+\frac{x^2}{2}\right)}^2}}$$

(23)

follows. As $x\in [0,1]$, we also have

$${\displaystyle \frac{1}{{\left(1+x+\frac{x^2}{2}\right)}^2}}\le {\displaystyle \frac{1}{{\left(1+x\right)}^2}},$$

(24)

and so based on Equations (23) and (24), we find that

$${\displaystyle \frac{1}{{\left(1+x\right)}^2}}\ge {\mathrm{e}}^{-2x}$$

(25)

holds for any $x\in [0,1]$, and the equality in Equation (25) holds only if $x=0$. Now, define $f\left(x\right)$ as

$$f\left(x\right)={\mathrm{e}}^{-2x}-{\displaystyle \frac{1-x}{1+x}},x\in [0,1].$$

The first derivative of f is

$${\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}=2\left({\displaystyle \frac{1}{{\left(x+1\right)}^2}}-{\mathrm{e}}^{-2x}\right),$$

and taking into account the inequality given in Equation (25), we see that for any $x\in [0,1]$,

$${\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}\ge 0.$$

Since the equality in Equation (25) holds only if $x=0$, we find that for any $x\in (0,1]$,

$${\displaystyle \frac{\mathrm{d}f\left(x\right)}{\mathrm{d}x}}>0,$$

which means that $f(·)$ is a strictly increasing function on $(0,1]$. As $f\left(0\right)=0$ and $f(·)$ is a strictly increasing function on $(0,1]$, we see that for any $x\in [0,1]$, $f\left(x\right)\ge 0$. Therefore, for any $x\in [0,1]$, we have

$${\left({\displaystyle \frac{1-x}{1+x}}\right)}^{{\displaystyle \frac{\lambda }{2}}}\le {\mathrm{e}}^{-\lambda x}.$$

   □

Remark 2.

We should add that an immediate practical consequence of Theorem 3 is that if $\lambda \gg 0$, then $h_{\lambda }\left(x\right)\approx {\mathrm{e}}^{-\lambda x}$ on $[0,1]$.

Figure 1 shows some example plots of ${\mathrm{e}}^{-\lambda x}$ and its approximation by $g_{\lambda }\left(x\right)$ and the epsilon function $h_{\lambda }\left(x\right)={\epsilon }_{-\lambda ,1}\left(x\right)$ on $[0,1]$.

Figure 1. Example plots of ${\mathrm{e}}^{-\lambda x}$ and its approximation by $g_{\lambda }\left(x\right)={(1-x)}^{\lambda }$ and the epsilon function $h_{\lambda }\left(x\right)={\left({\displaystyle \frac{1-x}{1+x}}\right)}^{{\displaystyle \frac{\lambda }{2}}}$ on $[0,1]$.

Based on the above considerations, we define the epsilon-function-based normalized fuzzy similarity measure as follows.

Definition 9.

Let $d:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ be a normalized distance measure in the vector space ${[0,1]}^n$. We say that the function $s_{\lambda }:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ is an epsilon-function-based normalized fuzzy similarity measure with a parameter $\lambda >0$ in the vector space ${[0,1]}^n$, if for any $\mathbf{x},\mathbf{y}\in {[0,1]}^n$, $s_{\lambda }(\mathbf{x},\mathbf{y})$ is given by

$$s_{\lambda }(\mathbf{x},\mathbf{y})={\left({\displaystyle \frac{1-d(\mathbf{x},\mathbf{y})}{1+d(\mathbf{x},\mathbf{y})}}\right)}^{{\displaystyle \frac{\lambda }{2}}}.$$

(26)

Notice that $s_{\lambda }(\mathbf{x},\mathbf{y})=h_{\lambda }\left(d(\mathbf{x},\mathbf{y})\right)$, where $h_{\lambda }(·)$ is the epsilon function on $[0,1]$ given by Equation (20). The following lemma summarizes the main properties of the epsilon-function-based normalized fuzzy similarity measure given in Definition 9.

Lemma 1.

Let $d:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ be a normalized distance measure in the vector space ${[0,1]}^n$ and let $s_{\lambda }:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ be the corresponding epsilon-function-based normalized fuzzy similarity measure with a parameter $\lambda >0$ given by Equation (26). $s_{\lambda }(·)$ has the following properties:

(a) 

For any $\mathbf{x},\mathbf{y}\in {[0,1]}^n$, if $d(\mathbf{x},\mathbf{y})=0$, then $s_{\lambda }(\mathbf{x},\mathbf{y})=1$ (maximal similarity).

(b) 

For any $\mathbf{x},\mathbf{y}\in {[0,1]}^n$, if $d(\mathbf{x},\mathbf{y})=1$, then $s_{\lambda }(\mathbf{x},\mathbf{y})=0$ (minimal similarity).

(c) 

For any $\mathbf{x},\mathbf{y},{\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime }\in {[0,1]}^n$, $d({\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime })>d(\mathbf{x},\mathbf{y})$ implies $s_{\lambda }({\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime })<s_{\lambda }(\mathbf{x},\mathbf{y})$(monotonocity).

(d) 

If $\lambda \gg 0$, then for any $\mathbf{x},\mathbf{y}\in {[0,1]}^n$, $s_{\lambda }(\mathbf{x},\mathbf{y})\approx s_{\lambda }^{*}(\mathbf{x},\mathbf{y})$, where $s_{\lambda }^{*}(\mathbf{x},\mathbf{y})$ is given by Equation (19).

(e) 

If $d(\mathbf{x},\mathbf{y})\in (0,1)$, then

$$\underset{\lambda \to \infty }{lim}{s}_{\lambda }(\mathbf{x},\mathbf{y})=0.$$

(27)

Proof. 

Properties (a), (b) and (c) readily follow from the definition of $s_{\lambda }(·)$ given in Definition 9, while property (b) is a consequence of Theorem 3.    □

4. A Similarity and Fuzzy Arithmetic-Based Fuzzy Inference Method

In our model, we assume that the value of a dependent variable $y\in \mathbb{R}$ is a function of an n-dimensional normalized feature vector $\mathbf{x}\in {[0,1]}^n$ as follows:

$$y=f\left(\mathbf{x}\right)+\epsilon ,$$

(28)

where $n\in \mathbb{N}$, $n\ge 1$, f is an unknown mapping from ${[0,1]}^n$ to $\mathbb{R}$ and $\epsilon$ is a normally distributed random variable with mean of 0. Our goal is to build a fuzzy inference method to approximate function f.

4.1. Forming a Feature Vector–Linguistic Value of the Number of Passenger Pairs

We will construct a Similarity and Fuzzy Arithmetic-based Fuzzy Inference System (SFAFIS) to approximate the mapping f given in Equation (28). For this purpose, we will use a set of training data given by the ordered pairs

$$({\mathbf{x}}_1,y_1),({\mathbf{x}}_2,y_2),\dots ,({\mathbf{x}}_m,y_m),$$

(29)

where $m\in \mathbb{N}$, $m\ge 2$,

$${\mathbf{x}}_i=\left(x_{i,1},x_{i,2},\dots ,x_{i,n}\right)\in {[0,1]}^n,$$

$i=1,2,\dots ,m$, $n\in \mathbb{N}$, $n\ge 1$ and $y_i\in \mathbb{R}$, $y_i>0$. Here, the vector ${\mathbf{x}}_i$ and the scalar $y_i$ represent the normalized feature vector (explanatory or input vector) and the corresponding output in the ith observation, respectively, in the sample given by Equation (29).

Let

$$q_0=\underset{i=1,2,\dots ,m}{min}\left(y_i\right)\mathrm{and}{q}_1=\underset{i=1,2,\dots ,m}{max}\left(y_i\right),$$

and suppose that $y\in [q_0,q_1]$. Now, let $L_1,L_2,\dots ,L_{\ell }$ be the linguistic values like ‘very small’, ‘small’, ‘large’, ‘very large’, etc., of y where $\ell \in \mathbb{N}$, $\ell \ge 3$. This means that $L_1,L_2,\dots ,L_{\ell }$ are fuzzy sets on the interval $[q_0,q_1]$. Furthermore, let $q_{{\displaystyle \frac{k}{\ell -1}}}$ denote the ${\displaystyle \frac{k}{\ell -1}}$-quantile of the $y_1,y_2,\dots ,y_m$ values, where $k=1,2,\dots ,\ell -2$.

In our approach, the kth linguistic level $L_k$ of y is a triangular fuzzy number with the parameters $a_k,b_k,c_k\in \mathbb{R}$, $a_k<b_k<c_k$. That is,

$$L_k=T(a_k,b_k,c_k),$$

and the membership function ${\mu }_{L_k}(·;a_k,b_k,c_k):\mathbb{R}\to [0,1]$ of $L_k$ is given by

$${\mu }_{L_k}(y;a_k,b_k,c_k)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}y\le a_k\hfill \\ {\displaystyle \frac{y-a_k}{b_k-a_k}},\hfill & \mathrm{if}{a}_k<y\le b_k\hfill \\ {\displaystyle \frac{c_k-y}{c_k-b_k}},\hfill & \mathrm{if}{b}_k<y\le c_k\hfill \\ 0,\hfill & \mathrm{if}y>c_k\hfill \end{array}\right.$$

such that

if $k=1$, then

$$\begin{array}{cc}\hfill a_k& =q_0-\left(q_{{\displaystyle \frac{1}{\ell -1}}}-q_0\right)=2q_0-q_{{\displaystyle \frac{1}{\ell -1}}},\hfill \\ \hfill b_k& =q_0,\hfill \\ \hfill c_k& =q_{{\displaystyle \frac{1}{\ell -1}}},\hfill \end{array}$$

if $k\in \{2,\dots ,\ell -1\}$, then

$$\begin{array}{cc}\hfill a_k& =q_{{\displaystyle \frac{k-2}{\ell -1}}},\hfill \\ \hfill b_k& =q_{{\displaystyle \frac{k-1}{\ell -1}}},\hfill \\ \hfill c_k& =q_{{\displaystyle \frac{k}{\ell -1}}},\hfill \end{array}$$

if $k=\ell$, then

$$\begin{array}{cc}\hfill a_k& =q_{{\displaystyle \frac{\ell -2}{\ell -1}}},\hfill \\ \hfill b_k& =q_1,\hfill \\ \hfill c_k& =q_1+\left(q_1-q_{{\displaystyle \frac{\ell -2}{\ell -1}}}\right)=2q_1-q_{{\displaystyle \frac{\ell -2}{\ell -1}}}.\hfill \end{array}$$

Figure 2 shows an example of the plots of membership functions of the triangular fuzzy numbers (linguistic values) $L_1,L_2,\dots ,L_{\ell }$.

Figure 2. Example plots of the triangular fuzzy numbers (linguistic values) $L_1,L_2,\dots ,L_{\ell }$.

Next, for all $i=1,2,\dots ,m$, we transform the training data pair $({\mathbf{x}}_i,y_i)$ into the pair

$$\left({\mathbf{x}}_i,L_{k\left(i\right)}\right)=\left({\mathbf{x}}_i,T\left(a_{k\left(i\right)},b_{k\left(i\right)},c_{k\left(i\right)}\right)\right)$$

such that $k\left(i\right)\in \{1,2,\dots ,\ell \}$ is the smallest index for which the membership value ${\mu }_{L_{k\left(i\right)}}(y_i;a_{k\left(i\right)},b_{k\left(i\right)},c_{k\left(i\right)}))$ equals the maximum of the membership values

$${\mu }_{L_1}(y_i;a_1,b_1,c_1),{\mu }_{L_2}(y_i;a_2,b_2,c_2),\dots ,{\mu }_{L_{\ell }}(y_i;a_{\ell },b_{\ell },c_{\ell }).$$

That is, $k\left(i\right)$ is given as

$$k\left(i\right)=min\left\{r:r\in \{1,2,\dots ,l\},{\mu }_{L_r}(y_i;a_r,b_r,c_r)=\underset{t=1,\dots ,\ell }{max}\left({\mu }_{L_t}(y_i;a_t,b_t,c_t)\right)\right\}.$$

In other words, $k\left(i\right)\in \{1,2,\dots ,\ell \}$ is the smallest index for which, among the linguistic values $L_1,L_2,\dots ,L_{\ell }$, the linguistic value $L_{k\left(i\right)}$ best represents the $y_i$ crisp value.

Remark 3.

Alternatively, instead of the ${\displaystyle \frac{k}{\ell -1}}$-quantile, $q_{{\displaystyle \frac{k}{\ell -1}}}$ can be defined as

$$q_{{\displaystyle \frac{k}{\ell -1}}}={\displaystyle \frac{k}{\ell -1}},$$

where $k=1,2,\dots ,\ell -2$. In this case, we apply a grid partitioning, i.e., with $\ell -1$ equidistant points, we divide the range $[q_0,q_1]$ into ℓ consecutive intervals with the same Lebesgue measure (i.e., same length).

4.2. The Inference Mechanism

Suppose that $\mathbf{x}\in {[0,1]}^n$ is an input feature vector for which we aim to predict the value of y. Furthermore, let $s_{\lambda }:{[0,1]}^n\times {[0,1]}^n\to [0,1]$ be an epsilon-function-based normalized fuzzy similarity measure with a parameter $\lambda >0$ given in Definition 9. That is, for any $\mathbf{x},{\mathbf{x}}^{\prime }\in {[0,1]}^n$, $s_{\lambda }(\mathbf{x},{\mathbf{x}}^{\prime })$ characterizes how much vector $\mathbf{x}$ is similar to the vector ${\mathbf{x}}^{\prime }$. Now, for all $i=1,2,\dots ,m$, let $w_i$ be given by

$$w_i={\displaystyle \frac{s_{\lambda }(\mathbf{x},{\mathbf{x}}_i)}{{\sum }_{j=1}^m{s}_{\lambda }(\mathbf{x},{\mathbf{x}}_j)}}.$$

Notice that $w_i\in [0,1]$ and ${\sum }_{i=1}^n{w}_i=1$, and the weight $w_i$ may be regarded as the relative measure of how well the input feature vector $\mathbf{x}$ fits to the feature vector ${\mathbf{x}}_i$ of the ith training data pair $({\mathbf{x}}_i,y_i)$, where $i\in \{1,2,\dots ,m\}$.

Since

$$L_{k\left(i\right)}=T\left(a_{k\left(i\right)},b_{k\left(i\right)},c_{k\left(i\right)}\right),$$

for all $i=1,2,\dots ,m$, based on Corollary 1 (a), the weighted arithmetic mean of the triangular fuzzy numbers $L_{k\left(1\right)},L_{k\left(2\right)},\dots ,L_{k\left(m\right)}$ is the triangular fuzzy number $T(a,b,c)$, which is given by

$$\begin{array}{cc}\hfill T(a,b,c)& =\underset{i=1}{\stackrel{m}{\oplus }}\left(w_i\odot L_{k\left(i\right)}\right)=\underset{i=1}{\stackrel{m}{\oplus }}\left(w_i\odot T\left(a_{k\left(i\right)},b_{\left(k\right(i\left)\right)},c_{k\left(i\right)}\right)\right)\hfill \\ \hfill & =T\left(\sum _{i=1}^m{w}_i{a}_{k\left(i\right)},\sum _{i=1}^m{w}_i{b}_{k\left(i\right)},\sum _{i=1}^m{w}_i{c}_{k\left(i\right)}\right),\hfill \end{array}$$

(30)

where ⊙ and ⊕ are the scalar multiplication and summation operators over triangular fuzzy numbers given in Definition 6 and Definition 7, respectively. Here, we treat the triangular fuzzy number $Y=T(a,b,c)$ as the fuzzy output of our similarity and fuzzy arithmetic-based fuzzy inference method, where

$$a=\sum _{i=1}^m{w}_i{a}_{k\left(i\right)},b=\sum _{i=1}^m{w}_i{b}_{k\left(i\right)},c=\sum _{i=1}^m{w}_i{c}_{k\left(i\right)}.$$

With this approach, our method takes the linguistic value of $y_i$ (i.e., the triangular fuzzy number $L_{k\left(i\right)}$) into account in the aggregate fuzzy output Y as much as the input feature vector $\mathbf{x}$ fits to the ith feature vector ${\mathbf{x}}_i$. Hence, the defuzzified value of Y may be treated as a $\widehat{y}$ prediction of y for the feature vector $\mathbf{x}\in {[0,1]}^n$. Using the center of gravity defuzzyfication method, the $\widehat{y}$ prediction of y is

$$\widehat{y}={\displaystyle \frac{a+b+c}{3}}.$$

Figure 3 shows the schematic diagram of the similarity and fuzzy arithmetic-based fuzzy inference system (SFAFIS) that we have described so far.

Figure 3. Schematic diagram of the similarity and fuzzy arithmetic-based fuzzy inference system (SFAFIS).

4.3. Tuning the Hyperparameters

The similarity and fuzzy arithmetic-based fuzzy inference system presented above may be treated as a two-parameter function $f_{\lambda ,\ell }:{[0,1]}^n\to \mathbb{R}$ that models the relationship between ${\mathbf{x}}_i$ and $y_i$, i.e.,

$$y_i\approx f_{\lambda ,\ell }\left({\mathbf{x}}_i\right),$$

(31)

where $({\mathbf{x}}_i,y_i)$ is the ith training data pair, $i=1,2,\dots ,m$. Function $f_{\lambda ,\ell }$ has two parameters, a $\lambda >0$ and an $\ell \in \mathbb{N}$, $\ell \ge 3$, which may be viewed as the hyperparameters of our SFAFIS. Recall that $\lambda$ is the parameter of the epsilon-function-based normalized fuzzy similarity measure and ℓ is the number of linguistic levels utilized in the SFAFIS. The optimal values of $\lambda$ and ℓ, denoted by ${\lambda }_{\mathrm{opt}}$ and ${\ell }_{\mathrm{opt}}$, respectively, can be determined by solving the minimization problem

$$\sum _{i=1}^m{\left(y_i-f_{\lambda ,\ell }\left({\mathbf{x}}_i\right)\right)}^2\to min.$$

That is, ${\lambda }_{\mathrm{opt}}$ and ${\ell }_{\mathrm{opt}}$ are parameter values for which

$$\sum _{i=1}^m{\left(y_i-f_{{\lambda }_{\mathrm{opt}},{\ell }_{\mathrm{opt}}}\left({\mathbf{x}}_i\right)\right)}^2=\underset{\begin{array}{c}\lambda \in (0,\infty )\\ \ell \in \mathbb{N},\ell \ge 3\end{array}}{min}\left(\sum _{i=1}^m{\left(y_i-f_{\lambda ,\ell }\left({\mathbf{x}}_i\right)\right)}^2\right)$$

(32)

holds.

The quasi-optimal values of $\lambda$ and ℓ can be found using various numerical methods such as particle swarm optimization or genetic algorithms. In a case study, which we will present in the next section, we are going to demonstrate that, since there are just two tunable parameters in our SFAFIS, even a simple grid search method is an effective way for finding the quasi-optimal values of these parameters.

We should emphasize that although our inference system has only two tunable parameters, in practice, this may serve as a viable alternative to much more sophisticated multi-parametric modeling methods.

5. Case Study

In this case study, we demonstrate how our similarity and fuzzy arithmetic-based model can be applied in practice. The aim of this study was to predict the number of passengers on railway relations using real-life data. For this purpose, historical passenger traffic data of the Hungarian State Railways (MÁV) were collected.

In order to utilize the SFAFIS method, we had to identify those characteristics of the examined relations that mostly affect the number of passengers on these relations. That is, we had to identify the components of an input vector $\mathbf{x}$. Based on the findings of previous research articles (see, e.g., [14,36,37,38,39,40,41,42,43]) and preliminary studies of Hungarian State Railways, the following characteristics of railway routes were taken in to account as explanatory variables:

• $p_1$: Population of the the first city.
• $p_2$: Population of the the second city.
• l: Distance between the cities (km).
• $t_r$: Fastest possible travel time between the cities by railway (min).
• $t_b$: Fastest possible travel time between the cities by bus (min).
• $t_c$: Fastest possible travel time between the cities by car (min).
• $f_{IC}$: Number of average InterCity train departures between the cities.
• $f_b$: Number of average bus departures between the cities.
• $f_r$: Number of average train departures between the cities.
• $c_r$: Unit cost of transport by rail (per passenger-km) (HUF).
• $c_b$: Unit cost of transport by bus (per passenger-km) (HUF).
• $c_c$: Unit cost of transport by car (per passenger-km) (HUF).
• $Ico$: Car ownership index of the region.
• $d_r$: Distance of the railway station from the city centers (km).
• $d_b$: Distance of the bus station from the city centers (km).

Since the above fifteen input characteristics were taken into consideration, an input vector $\mathbf{x}\in {[0,1]}^{15}$ contained the normalized values of the above characteristics. The normalization of the inputs was performed using the well-known min–max normalization method on the data given in Appendix A.

For the sake of simplicity, we stipulated that the railway connections are examined independently, i.e., the number of passengers on different sections of the same route are not added up. Using the company’s sales database for the year 2022, we calculated the rounded value of the average monthly number of passengers for each examined connection. These are shown in column y in Appendix A. Here, we utilized data from 60 railway connections in Hungary. However, it turned out, that even with these real data, the data range is too large; therefore, more data would be required to accurately test the proposed model. Unfortunately, because of the size and usual passenger flows in Hungary, more real-world data could not be gathered; therefore, we selected the densest part of our database (i.e., connections with an average monthly passenger number between 800 and 3500) and generated more records based on the original data. First, we generated new records from the original ones such that each new record was derived as a weighted average of two original records with a randomly selected average ratio. Next, using the same approach, we generated records using the previously generated ones. In Appendix A, we summarize our database, which includes both original and generated data and has 81 records in total. The original records are marked by letters from A to Z, while generated data are marked by a combination of letters corresponding to the records that they were generated from. For example, record AB was generated from records A and B and record ABBC was generated from the records AB and BC.

We evaluated the performance of the SFAFIS method and compared it with some well-known methods using randomly selected training and test data sets from the sample S given in Appendix A. We performed the method evaluations repeatedly so that, in every iteration, the training data set $S_{\mathrm{tr}}$ and the test data set $S_{\mathrm{te}}$ were randomly selected from the sample S as

$$S_{\mathrm{tr}}=\left\{({\mathbf{x}}_{i_1}^{\left(\mathrm{tr}\right)},y_{i_1}^{\left(\mathrm{tr}\right)}),({\mathbf{x}}_{i_2}^{\left(\mathrm{tr}\right)},y_{i_2}^{\left(\mathrm{tr}\right)}),\dots ,({\mathbf{x}}_{i_{n_{\mathrm{tr}}}}^{\left(\mathrm{tr}\right)},y_{i_{n_{\mathrm{tr}}}}^{\left(\mathrm{tr}\right)})\right\},$$

$$S_{\mathrm{te}}=\left\{({\mathbf{x}}_{j_1}^{\left(\mathrm{te}\right)},y_{j_1}^{\left(\mathrm{te}\right)}),({\mathbf{x}}_{j_2}^{\left(\mathrm{te}\right)},y_{j_2}^{\left(\mathrm{te}\right)}),\dots ,({\mathbf{x}}_{j_{n_{\mathrm{te}}}}^{\left(\mathrm{te}\right)},y_{j_{n_{\mathrm{te}}}}^{\left(\mathrm{te}\right)})\right\},$$

such that $S_{\mathrm{tr}}\cap S_{\mathrm{te}}=\varnothing$ and $S_{\mathrm{tr}}\cup S_{\mathrm{te}}=S$, where

• $n_{\mathrm{tr}}$ and $n_{\mathrm{te}}$ are the number of training and test data pairs, respectively;
• ${\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)}$ and ${\mathbf{x}}_{j_v}^{\left(\mathrm{te}\right)}$ are the n-dimensional normalized feature vectors of the $i_u$th and $j_v$th railway connections in the training and test data sets, respectively (i.e., ${\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)},{\mathbf{x}}_{j_c}^{\left(\mathrm{te}\right)}\in {[0,1]}^n$);
• $y_{i_u}^{\left(\mathrm{tr}\right)}$ and $y_{j_v}^{\left(\mathrm{te}\right)}$ are the average number of passengers of the $i_u$th and $j_v$th railway relations in the training and test data sets, respectively;
• ${\widehat{y}}_{i_u}^{\left(\mathrm{tr}\right)}$ and ${\widehat{y}}_{j_v}^{\left(\mathrm{te}\right)}$ are the estimated (computed) values of $y_{i_u}^{\left(\mathrm{tr}\right)}$ and $y_{j_v}^{\left(\mathrm{te}\right)}$, respectively;
• $u=1,2,\dots ,n_{\mathrm{tr}}$ and $v=1,2,\dots ,n_{\mathrm{te}}$.

To characterize the goodness of the investigated methods, the following Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE) metrics were taken into account:

$${\mathrm{MAPE}}_{\mathrm{tr}}={\displaystyle \frac{100}{n_{\mathrm{tr}}}}\sum _{u=1}^{n_{\mathrm{tr}}}\left|{\displaystyle \frac{y_{i_u}^{\left(\mathrm{tr}\right)}-{\widehat{y}}_{i_u}^{\left(\mathrm{tr}\right)}}{y_{i_u}^{\left(\mathrm{tr}\right)}}}\right|,$$

(33)

$${\mathrm{MAPE}}_{\mathrm{te}}={\displaystyle \frac{100}{n_{\mathrm{te}}}}\sum _{v=1}^{n_{\mathrm{te}}}\left|{\displaystyle \frac{y_{j_v}^{\left(\mathrm{te}\right)}-{\widehat{y}}_{j_v}^{\left(\mathrm{te}\right)}}{y_{j_v}^{\left(\mathrm{te}\right)}}}\right|,$$

(34)

$${\mathrm{MAPE}}_{\mathrm{tr},\mathrm{te}}={\displaystyle \frac{100}{n_{\mathrm{tr}}+n_{\mathrm{te}}}}\left(\sum _{u=1}^{n_{\mathrm{tr}}}\left|{\displaystyle \frac{y_{i_u}^{\left(\mathrm{tr}\right)}-{\widehat{y}}_{i_u}^{\left(\mathrm{tr}\right)}}{y_{i_u}^{\left(\mathrm{tr}\right)}}}\right|+\sum _{v=1}^{n_{\mathrm{te}}}\left|{\displaystyle \frac{y_{j_v}^{\left(\mathrm{te}\right)}-{\widehat{y}}_{j_v}^{\left(\mathrm{te}\right)}}{y_{j_v}^{\left(\mathrm{te}\right)}}}\right|\right),$$

(35)

$${\mathrm{RMSE}}_{\mathrm{tr}}=\sqrt{{\displaystyle \frac{1}{n_{\mathrm{tr}}}}\sum _{u=1}^{n_{\mathrm{tr}}}{\left(y_{i_u}^{\left(\mathrm{tr}\right)}-{\widehat{y}}_{i_u}^{\left(\mathrm{tr}\right)}\right)}^2},$$

(36)

$${\mathrm{RMSE}}_{\mathrm{te}}=\sqrt{{\displaystyle \frac{1}{n_{\mathrm{te}}}}\sum _{v=1}^{n_{\mathrm{te}}}{\left(y_{j_v}^{\left(\mathrm{te}\right)}-{\widehat{y}}_{j_v}^{\left(\mathrm{te}\right)}\right)}^2},$$

(37)

$${\mathrm{RMSE}}_{\mathrm{tr},\mathrm{te}}=\sqrt{{\displaystyle \frac{1}{n_{\mathrm{tr}}+n_{\mathrm{te}}}}\left(\sum _{u=1}^{n_{\mathrm{tr}}}{\left(y_{i_u}^{\left(\mathrm{tr}\right)}-{\widehat{y}}_{i_u}^{\left(\mathrm{tr}\right)}\right)}^2+\sum _{v=1}^{n_{\mathrm{te}}}{\left(y_{j_v}^{\left(\mathrm{te}\right)}-{\widehat{y}}_{j_v}^{\left(\mathrm{te}\right)}\right)}^2\right)}.$$

(38)

Here, ${\mathrm{MAPE}}_{\mathrm{tr}}$ and ${\mathrm{MAPE}}_{\mathrm{te}}$ are the MAPE metrics for the training and test data sets, respectively, while ${\mathrm{MAPE}}_{\mathrm{tr},\mathrm{te}}$ is the MAPE value that takes into account both the training and test data sets. Similarly, ${\mathrm{RMSE}}_{\mathrm{tr}}$ and ${\mathrm{RMSE}}_{\mathrm{te}}$ are the RMSE metrics for the training and test data sets, respectively, while ${\mathrm{RMSE}}_{\mathrm{tr},\mathrm{te}}$ is the RMSE value that takes into consideration both the training and test data sets.

Notice that using the Equations (33), (34), (36) and (37), the ${\mathrm{MAPE}}_{\mathrm{tr},\mathrm{te}}$ and ${\mathrm{RMSE}}_{\mathrm{tr},\mathrm{te}}$ given in Equation (35) and Equation (38), respectively, can also be calculated as

$${\mathrm{MAPE}}_{\mathrm{tr},\mathrm{te}}={\displaystyle \frac{n_{\mathrm{tr}}{\mathrm{MAPE}}_{\mathrm{tr}}+n_{\mathrm{te}}{\mathrm{MAPE}}_{\mathrm{te}}}{n_{\mathrm{tr}}+n_{\mathrm{te}}}}$$

(39)

$${\mathrm{RMSE}}_{\mathrm{tr},\mathrm{te}}=\sqrt{{\displaystyle \frac{n_{\mathrm{tr}}{\mathrm{RMSE}}_{\mathrm{tr}}^2+n_{\mathrm{te}}{\mathrm{RMSE}}_{\mathrm{te}}^2}{n_{\mathrm{tr}}+n_{\mathrm{te}}}}}.$$

(40)

In every iteration, the number of randomly selected training data points was $n_{\mathrm{tr}}=66$, while the remaining $n_{\mathrm{te}}=15$ data points were utilized as test data.

For every randomly selected training data set, the proposed SFAFIS method was applied with $\lambda =30$, $\lambda =60$ and $\lambda =90$ parameter values. For each value of the $\lambda$ parameter, the quasi-optimal value of the number of linguistic variables, i.e., ${\ell }_{\mathrm{opt}}$, was determined by minimizing the ${\mathrm{RMSE}}_{\mathrm{tr}}^2$ quantity as described in Section 4.3, such that ${\ell }_{\mathrm{opt}}$ was searched for in the set $\{5,6,\dots ,200\}$. That is, for a given value of $\lambda$, ${\ell }_{\mathrm{opt}}$ is the value of parameter ℓ for which

$$\sum _{u=1}^{n_{\mathrm{tr}}}{\left(y_{i_u}^{\left(\mathrm{tr}\right)}-f_{\lambda ,{\ell }_{\mathrm{opt}}}\left({\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)}\right)\right)}^2=\underset{\begin{array}{c}\ell \in \{5,6,\dots ,200\}\end{array}}{min}\left(\sum _{u=1}^{n_{\mathrm{tr}}}{\left(y_{i_u}^{\left(\mathrm{tr}\right)}-f_{\lambda ,\ell }\left({\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)}\right)\right)}^2\right),$$

where $\left({\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)},y_{i_u}^{\left(\mathrm{tr}\right)}\right)$ is the $i_u$th data pair in the training data set and $f_{\lambda ,\ell }\left({\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)}\right)$ is the output of the SFAFIS method for input vector ${\mathbf{x}}_{i_u}^{\left(\mathrm{tr}\right)}$.

With the genfis method of the MATLAB (R2024b) software package [31], in every iteration, utilizing the training data, a Sugeno-type fuzzy inference system (FIS) was generated using the subtractive clustering (SC) algorithm. This Sugeno-type FIS had 15 inputs ($\mathbf{x}\in {[0,1]}^{15}$) with 8 Gaussian membership functions for each input, i.e., the number of parameters on the input side was $15\times 8\times 2=240$. The number of rules in the rule base, i.e., the number of clusters formed, was eight, and so the output side of the FIS consisted of eight linear functions of the inputs. That is, each of these linear functions had 16 parameters (number of dimensions in the input vectors plus one for the constant term in the linear function). Hence, the total number of tunable parameters in such a Sugeno-type FIS was $15\times 8\times 2+8\times 16=368$. These parameters were tuned (optimized) using the following techniques, along with a $k=5$-fold cross validation:

• Adaptive Neuro Fuzzy Inference System (ANFIS) (see, e.g., [24,25]);
• Genetic Algorithm (GA) (see, e.g., [26,27,28]);
• Particle Swarm Optimization (PSO) (see, e.g., [29]);
• Pattern-search (PS) method (see, e.g., [30]).

The parameter tuning (optimization) of the Sugeno-type FIS was performed using the tunefis method of the MATLAB (R2024b) software package. The modeling goodness of the SFAFIS method and that of the Sugeno-type FIS tuned by the above-listed methods were expressed in terms of MAPE and RMSE metrics. These goodness results are shown in Table 1, Table 2, Table 3 and Table 4. The MATLAB source codes that were used to generate the results in these tables are available at https://jonast.web.elte.hu/. The ‘SFAFIS_example.m’ file contains the example of our case study, and the ‘FIS_similarity_and_fuzzy_arithmetic_class.m’ file is the class file in which the method itself is implemented. Using these two files, the results of our case study can be reproduced. We should note that in the MATLAB implementation of SFAFIS, instead of the similarity measure $s_{\lambda }(\mathbf{x},\mathbf{y})$ given in Equation (26), the following modified variant was employed:

$$s_{\lambda }^{\prime }(\mathbf{x},\mathbf{y})={\left({\displaystyle \frac{1-d(\mathbf{x},\mathbf{y})}{1+d(\mathbf{x},\mathbf{y})}}\right)}^{\lambda },$$

i.e., $s_{\lambda }^{\prime }(\mathbf{x},\mathbf{y})=s_{2\lambda }(\mathbf{x},\mathbf{y})$.

Table 1. Modeling results of SFAFIS and the Sugeno-type FIS tuned using ANFIS.

Iteration	Method	${\mathbf{MAPE}}_{\mathbf{tr}}$	${\mathbf{MAPE}}_{\mathbf{te}}$	${\mathbf{MAPE}}_{\mathbf{tr},\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr}}$	${\mathbf{RMSE}}_{\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr},\mathbf{te}}$	ℓopt	$\mathit{\lambda }$
1	SFAFIS	0.2202	4.4183	0.99761	5.0363	135.89	58.654	123	30
		0.15377	4.7255	1.0004	4.213	139.69	60.233	128	60
		0.15324	4.8394	1.021	4.2055	141.68	61.087	128	90
	ANFIS	0.0014637	12.094	2.2407	0.04118	302.94	130.37	-	-
2	SFAFIS	0.22453	5.8265	1.2619	5.0409	126.06	54.438	119	30
		0.18751	5.186	1.1132	4.4291	130.86	56.454	119	60
		0.18602	4.9695	1.0719	4.4058	133.72	57.682	119	90
	ANFIS	0.0015201	9.7151	1.8003	0.08549	197.57	85.02	-	-
3	SFAFIS	0.32908	3.6451	0.94316	9.2391	72.864	32.446	101	30
		0.1394	2.8135	0.6346	3.8243	65.986	28.605	116	60
		0.13703	2.5159	0.57755	3.806	63.059	27.353	116	90
	ANFIS	0.0037684	6.3676	1.1823	0.097153	144.38	62.132	-	-
4	SFAFIS	0.22989	2.0713	0.57088	5.4742	69.623	30.366	129	30
		0.16468	1.4503	0.40276	3.833	52.779	22.974	129	60
		0.16229	1.3263	0.37785	3.8257	45.738	19.983	129	90
	ANFIS	0.0019279	4.5712	0.84808	0.11433	120.42	51.821	-	-
5	SFAFIS	0.32783	0.72639	0.40164	9.5702	15.558	10.929	125	30
		0.1617	0.72237	0.26553	4.1248	19.621	9.2282	125	60
		0.16172	0.76868	0.27412	4.1371	20.693	9.6562	125	90
	ANFIS	0.0016693	2.6455	0.49127	0.045098	68.991	29.689	-	-
6	SFAFIS	0.25164	5.5006	1.2237	5.8787	98.651	42.783	118	30
		0.1556	4.2795	0.91928	3.9032	79.832	34.535	118	60
		0.15255	3.8394	0.8353	3.8559	71.451	30.944	118	90
	ANFIS	0.0013355	5.9663	1.106	0.032256	90.34	38.876	-	-
7	SFAFIS	0.29038	3.0833	0.80759	8.4814	116.93	50.897	112	30
		0.17055	2.9978	0.69411	4.4023	114.11	49.267	132	60
		0.16441	3.1599	0.71913	4.2877	115.73	49.951	132	90
	ANFIS	0.0049698	4.4572	0.82945	0.51938	96.282	41.436	-	-
8	SFAFIS	0.3094	2.9724	0.80254	8.984	106.33	46.47	117	30
		0.1567	1.4212	0.39087	3.6034	47.162	20.554	129	60
		0.15741	0.98954	0.31151	3.6139	41.72	18.248	129	90
	ANFIS	0.0021185	8.9088	1.6515	0.066603	236.31	101.69	-	-
9	SFAFIS	0.30447	3.7847	0.94895	8.9396	65.489	29.315	113	30
		0.16889	1.4384	0.40398	4.0767	41.3	18.15	117	60
		0.15815	1.4636	0.39989	3.9599	43.411	19.02	117	90
	ANFIS	0.0013597	8.3017	1.5385	0.054377	146.21	62.918	-	-
10	SFAFIS	0.19831	4.9907	1.0858	5.0846	129.59	55.955	125	30
		0.16201	4.5909	0.98218	4.291	138.01	59.518	125	60
		0.16043	4.6778	0.99697	4.3038	139.36	60.095	125	90
	ANFIS	0.0014798	6.9232	1.2833	0.047309	140.56	60.487	-	-

Table 2. Modeling results of SFAFIS and the Sugeno-type FIS tuned using genetic algorithm (GA).

Iteration	Method	${\mathbf{MAPE}}_{\mathbf{tr}}$	${\mathbf{MAPE}}_{\mathbf{te}}$	${\mathbf{MAPE}}_{\mathbf{tr},\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr}}$	${\mathbf{RMSE}}_{\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr},\mathbf{te}}$	ℓopt	$\mathit{\lambda }$
1	SFAFIS	0.2202	4.4183	0.99761	5.0363	135.89	58.654	123	30
		0.15377	4.7255	1.0004	4.213	139.69	60.233	128	60
		0.15324	4.8394	1.021	4.2055	141.68	61.087	128	90
	GA	5.939 $\times {10}^{-13}$	54.011	10.002	1.2304 $\times {10}^{-11}$	1612.8	694.02	-	-
2	SFAFIS	0.31285	4.9046	1.1632	8.9279	124.97	54.379	118	30
		0.16985	3.5869	0.80263	4.5693	82.416	35.705	126	60
		0.15862	3.5916	0.79436	4.4574	87.145	37.717	126	90
	GA	8.2149 $\times {10}^{-13}$	30.979	5.7368	1.7463 $\times {10}^{-11}$	584.05	251.34	-	-
3	SFAFIS	0.31248	2.1682	0.65613	8.9226	60.615	27.3	119	30
		0.17735	1.7383	0.46642	4.4278	32.421	14.513	119	60
		0.17931	1.8002	0.47946	4.4923	34.058	15.207	119	90
	GA	3.9052 $\times {10}^{-13}$	37.424	6.9304	7.9338 $\times {10}^{-12}$	933.91	401.89	-	-
4	SFAFIS	0.266	3.1635	0.80258	8.4559	86.529	38.01	119	30
		0.14044	3.2805	0.72193	3.9928	108.97	47.032	128	60
		0.13474	3.5954	0.77561	3.9833	119.37	51.495	128	90
	GA	1.0777 $\times {10}^{-12}$	107.74	19.953	2.7237 $\times {10}^{-11}$	2169.8	933.72	-	-
5	SFAFIS	0.21337	3.796	0.87681	5.0801	177	76.305	118	30
		0.15085	4.1158	0.8851	4.0728	178.61	76.949	118	60
		0.14864	4.2597	0.90996	4.0587	188.58	81.236	118	90
	GA	2.3839 $\times {10}^{-13}$	23.942	4.4337	4.6876 $\times {10}^{-12}$	507.75	218.5	-	-
6	SFAFIS	0.20716	4.8887	1.0741	5.0419	120.99	52.264	129	30
		0.15365	4.9371	1.0395	4.305	131.88	56.884	128	60
		0.15182	4.9114	1.0332	4.2387	135.37	58.378	128	90
	GA	4.1628 $\times {10}^{-13}$	33.713	6.2431	7.9228 $\times {10}^{-12}$	748.46	322.09	-	-
7	SFAFIS	0.28009	2.9498	0.77448	8.277	55.694	25.104	117	30
		0.15527	1.8633	0.47157	4.0176	37.824	16.676	110	60
		0.15237	1.7295	0.44444	3.9751	32.959	14.63	132	90
	GA	1.0958 $\times {10}^{-13}$	48.42	8.9667	2.4363 $\times {10}^{-12}$	1099.8	473.27	-	-
8	SFAFIS	0.20807	6.7026	1.4108	4.7065	427.77	184.13	123	30
		0.16001	7.7727	1.5698	3.8988	473.47	203.78	123	60
		0.15815	7.88	1.5881	3.8891	477.16	205.37	123	90
	GA	1.3733 $\times {10}^{-13}$	20.04	3.7112	3.2496 $\times {10}^{-12}$	797.99	343.4	-	-
9	SFAFIS	0.29691	6.4588	1.438	8.9759	129.42	56.282	109	30
		0.1628	3.7829	0.83319	4.2356	73.225	31.742	132	60
		0.15557	3.1536	0.71077	4.1189	65.904	28.603	132	90
	GA	1.9898 $\times {10}^{-13}$	16.164	2.9932	5.7774 $\times {10}^{-12}$	476.88	205.22	-	-
10	SFAFIS	0.28708	1.2686	0.46884	8.1722	39.502	18.53	118	30
		0.15661	0.99323	0.31154	3.6767	40.177	17.605	129	60
		0.16178	1.0292	0.32242	3.7043	41.457	18.151	129	90
	GA	1.8214 $\times {10}^{-13}$	14.027	2.5976	4.2808 $\times {10}^{-12}$	413.89	178.11	-	-

Table 3. Modeling results of SFAFIS and the Sugeno-type FIS tuned using particle swarm optimization (PSO).

Iteration	Method	${\mathbf{MAPE}}_{\mathbf{tr}}$	${\mathbf{MAPE}}_{\mathbf{te}}$	${\mathbf{MAPE}}_{\mathbf{tr},\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr}}$	${\mathbf{RMSE}}_{\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr},\mathbf{te}}$	ℓopt	$\mathit{\lambda }$
1	SFAFIS	0.2202	4.4183	0.99761	5.0363	135.89	58.654	123	30
		0.15377	4.7255	1.0004	4.213	139.69	60.233	128	60
		0.15324	4.8394	1.021	4.2055	141.68	61.087	128	90
	PSO	5.939 $\times {10}^{-13}$	54.011	10.002	1.2304 $\times {10}^{-11}$	1612.8	694.02	-	-
2	SFAFIS	0.30661	2.6871	0.74744	9.3238	104.95	45.943	115	30
		0.16299	1.4458	0.40054	4.3656	49.901	21.832	132	60
		0.1574	1.173	0.34548	4.268	46.291	20.289	132	90
	PSO	4.1799 $\times {10}^{-13}$	32.744	6.0637	1.0934 $\times {10}^{-11}$	967.97	416.55	-	-
3	SFAFIS	0.18747	4.5946	1.0036	4.7889	110.26	47.645	121	30
		0.18007	4.511	0.9821	4.3273	122.34	52.792	105	60
		0.17921	4.2945	0.94131	4.3083	129.23	55.75	105	90
	PSO	1.0973 $\times {10}^{-13}$	8.6157	1.5955	2.0855 $\times {10}^{-12}$	183.57	78.995	-	-
4	SFAFIS	0.21829	9.7979	1.9923	4.9939	440.71	189.71	110	30
		0.16126	10.554	2.0858	4.1652	488.16	210.1	132	60
		0.15819	10.75	2.1196	4.1102	492.26	211.87	132	90
	PSO	6.0409 $\times {10}^{-13}$	38.055	7.0473	1.2908 $\times {10}^{-11}$	842.16	362.41	-	-
5	SFAFIS	0.19639	11.809	2.3468	4.5202	453.9	195.37	116	30
		0.1456	11.998	2.3404	3.5729	495.23	213.14	116	60
		0.1434	12.038	2.3461	3.5365	498.18	214.41	116	90
	PSO	2.7137 $\times {10}^{-13}$	48.339	8.9517	5.7643 $\times {10}^{-12}$	1072.2	461.42	-	-
6	SFAFIS	0.33054	3.5667	0.92982	9.8091	70.693	31.684	108	30
		0.16586	3.2494	0.73688	4.1939	74.449	32.261	126	60
		0.15436	3.6957	0.81016	4.0768	84	36.334	126	90
	PSO	1.7145 $\times {10}^{-13}$	18.289	3.3869	3.4361 $\times {10}^{-12}$	293.82	126.44	-	-
7	SFAFIS	0.28899	9.9587	2.0797	9.5004	431.5	185.89	126	30
		0.1397	9.593	1.8903	3.8897	475.58	204.69	126	60
		0.13334	9.4787	1.864	3.8658	479.45	206.35	126	90
	PSO	1.5571 $\times {10}^{-13}$	23.123	4.282	3.8223 $\times {10}^{-12}$	486.1	209.18	-	-
8	SFAFIS	0.19582	8.9888	1.8241	4.5347	441.01	189.82	109	30
		0.15987	8.5128	1.7067	4.129	484.7	208.62	132	60
		0.15786	8.5872	1.7188	4.1076	489.37	210.62	132	90
	PSO	3.6489 $\times {10}^{-13}$	31.322	5.8003	6.6528 $\times {10}^{-12}$	1599.4	688.29	-	-
9	SFAFIS	0.25163	6.8997	1.4827	7.7123	388.25	167.22	124	30
		0.14193	5.8837	1.2052	3.3971	461.81	198.75	124	60
		0.14254	6.0215	1.2312	3.4167	475.42	204.61	124	90
	PSO	4.8984 $\times {10}^{-13}$	58.248	10.787	8.6798 $\times {10}^{-12}$	1805.7	777.05	-	-
10	SFAFIS	0.19379	7.9628	1.6325	4.5137	394.07	169.63	132	30
		0.14739	8.1986	1.6384	4.0815	470.95	202.7	116	60
		0.14556	8.5111	1.6947	4.0623	486.14	209.23	116	90
	PSO	3.1618 $\times {10}^{-13}$	31.842	5.8967	8.6715 $\times {10}^{-12}$	1804	776.3	-	-

Table 4. Modeling results of SFAFIS and the Sugeno-type FIS tuned using pattern-search (PS) optimization.

Iteration	Method	${\mathbf{MAPE}}_{\mathbf{tr}}$	${\mathbf{MAPE}}_{\mathbf{te}}$	${\mathbf{MAPE}}_{\mathbf{tr},\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr}}$	${\mathbf{RMSE}}_{\mathbf{te}}$	${\mathbf{RMSE}}_{\mathbf{tr},\mathbf{te}}$	ℓopt	$\mathit{\lambda }$
1	SFAFIS	0.2202	4.4183	0.99761	5.0363	135.89	58.654	123	30
		0.15377	4.7255	1.0004	4.213	139.69	60.233	128	60
		0.15324	4.8394	1.021	4.2055	141.68	61.087	128	90
	PS	5.939 $\times {10}^{-13}$	54.011	10.002	1.2304 $\times {10}^{-11}$	1612.8	694.02	-	-
2	SFAFIS	0.23605	11.846	2.3861	5.8128	159.09	68.663	108	30
		0.15636	6.8386	1.3938	3.8409	121.46	52.384	119	60
		0.18602	4.9695	1.0719	4.4058	133.72	57.682	119	90
	PS	1.5614 $\times {10}^{-13}$	26.582	4.9227	3.8216 $\times {10}^{-12}$	482.6	207.68	-	-
3	SFAFIS	0.28364	8.512	1.8074	8.396	434.57	187.16	112	30
		0.1476	7.8513	1.5742	3.7915	474.26	204.12	126	60
		0.13703	2.5159	0.57755	3.806	63.059	27.353	116	90
	PS	1.5834 $\times {10}^{-13}$	41.438	7.6738	3.239 $\times {10}^{-12}$	782.9	336.91	-	-
4	SFAFIS	0.28301	8.1045	1.7314	8.5418	406.78	175.22	109	30
		0.15165	6.4616	1.3202	3.6838	464.64	199.97	132	60
		0.16229	1.3263	0.37785	3.8257	45.738	19.983	129	90
	PS	3.797 $\times {10}^{-13}$	73.306	13.575	7.2282 $\times {10}^{-12}$	1413.5	608.27	-	-
5	SFAFIS	0.29353	2.2379	0.65361	8.6	98.406	43.053	119	30
		0.15202	1.4259	0.38793	4.658	61.988	27.005	124	60
		0.16172	0.76868	0.27412	4.1371	20.693	9.6562	125	90
	PS	2.494 $\times {10}^{-13}$	52.274	9.6804	5.0776 $\times {10}^{-12}$	1196.8	515.02	-	-
6	SFAFIS	0.20946	4.5473	1.0128	4.9107	127.29	54.954	132	30
		0.16973	4.8916	1.0441	4.3344	140.26	60.484	132	60
		0.15255	3.8394	0.8353	3.8559	71.451	30.944	118	90
	PS	3.2409 $\times {10}^{-13}$	24.475	4.5324	5.7138 $\times {10}^{-12}$	926.2	398.57	-	-
7	SFAFIS	0.27993	2.9145	0.76781	8.0152	93.395	40.837	129	30
		0.13906	3.2011	0.7061	3.6172	111.41	48.054	128	60
		0.16441	3.1599	0.71913	4.2877	115.73	49.951	132	90
	PS	3.7447 $\times {10}^{-13}$	21.939	4.0628	7.1633 $\times {10}^{-12}$	542.97	233.66	-	-
8	SFAFIS	0.30154	4.6014	1.0978	8.8431	420.07	180.95	124	30
		0.17052	5.0451	1.0732	3.8546	459.5	197.77	123	60
		0.15741	0.98954	0.31151	3.6139	41.72	18.248	129	90
	PS	1.8569 $\times {10}^{-13}$	11.627	2.1531	3.8067 $\times {10}^{-12}$	265.96	114.45	-	-
9	SFAFIS	0.28664	4.8999	1.1409	8.684	117.96	51.363	117	30
		0.13164	2.2561	0.52507	3.2993	54.869	23.799	128	60
		0.15815	1.4636	0.39989	3.9599	43.411	19.02	117	90
	PS	2.6935 $\times {10}^{-13}$	31.185	5.7749	6.8908 $\times {10}^{-12}$	1237.4	532.5	-	-
10	SFAFIS	0.21633	5.5482	1.2037	5.1291	132.86	57.36	132	30
		0.16746	4.3179	0.93606	4.2094	133.92	57.754	132	60
		0.16043	4.6778	0.99697	4.3038	139.36	60.095	125	90
	PS	5.4371 $\times {10}^{-13}$	19.465	3.6046	1.1614 $\times {10}^{-11}$	402.97	173.41	-	-

Using the goodness results shown in Table 1, Table 2, Table 3 and Table 4, Mann–Whitney median tests were conducted to compare the modeling performance of the SFAFIS method with that of the Sugeno-type FIS tuned using the ANFIS, GA, PSO and PS methods. From the three SFAFIS metric values (for $\lambda =30,60$ and 90 parameter values) corresponding to each iteration in Table 1, Table 2, Table 3 and Table 4, the best one always was taken into account in the hypothesis tests. The results of the Mann–Whitney median tests are shown in Table 5. In this table, the notation $m_{\mathrm{M}}$ stands for the median of the corresponding metric for method M.

Table 5. Results of Mann–Whitney median tests for the modeling results of SFAFIS and the Sugeno-type FIS tuned using the ANFIS, GA, PSO and PS methods and the samples in Table 1, Table 2, Table 3 and Table 4.

Metric	Null and Alternative Hypotheses and p-Values
${\mathrm{MAPE}}_{\mathrm{tr}}$	${\mathrm{H}}_0$:	$m_{\mathrm{SFAFIS}}=m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PS}}$
	${\mathrm{H}}_1$:	$m_{\mathrm{SFAFIS}}>m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}>m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}>m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}>m_{\mathrm{PS}}$
	p-value:	0.000	0.000	0.000	0.000
${\mathrm{MAPE}}_{\mathrm{te}}$	${\mathrm{H}}_0$:	$m_{\mathrm{SFAFIS}}=m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PS}}$
	${\mathrm{H}}_1$:	$m_{\mathrm{SFAFIS}}<m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PS}}$
	p-value:	0.001	0.000	0.000	0.000
${\mathrm{MAPE}}_{\mathrm{tr},\mathrm{te}}$	${\mathrm{H}}_0$:	$m_{\mathrm{SFAFIS}}=m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PS}}$
	${\mathrm{H}}_1$:	$m_{\mathrm{SFAFIS}}<m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PS}}$
	p-value:	0.003	0.000	0.000	0.000
${\mathrm{RMSE}}_{\mathrm{tr}}$	${\mathrm{H}}_0$:	$m_{\mathrm{SFAFIS}}=m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PS}}$
	${\mathrm{H}}_1$:	$m_{\mathrm{SFAFIS}}>m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}>m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}>m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}>m_{\mathrm{PS}}$
	p-value:	0.000	0.000	0.000	0.000
${\mathrm{RMSE}}_{\mathrm{te}}$	${\mathrm{H}}_0$:	$m_{\mathrm{SFAFIS}}=m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PS}}$
	${\mathrm{H}}_1$:	$m_{\mathrm{SFAFIS}}<m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PS}}$
	p-value:	0.006	0.000	0.002	0.000
${\mathrm{RMSE}}_{\mathrm{tr},\mathrm{te}}$	${\mathrm{H}}_0$:	$m_{\mathrm{SFAFIS}}=m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}=m_{\mathrm{PS}}$
	${\mathrm{H}}_1$:	$m_{\mathrm{SFAFIS}}<m_{\mathrm{ANFIS}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{GA}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PSO}}$	$m_{\mathrm{SFAFIS}}<m_{\mathrm{PS}}$
	p-value:	0.006	0.000	0.003	0.000

Based on the results of the Mann–Whitney median tests shown in Table 5, with respect to the studied case, the following conclusions can be drawn:

• On the training data, the Sugeno-type fuzzy inference systems tuned by the ANFIS, GA, PSO and PS methods perform significantly better than the proposed SFAFIS method both in terms of MAPE and RMSE (see the Mann–Whitney test results for ${\mathrm{MAPE}}_{\mathrm{tr}}$ and ${\mathrm{RMSE}}_{\mathrm{tr}}$ in Table 5).
• On the test data, the proposed SFAFIS method performs significantly better than the Sugeno-type fuzzy inference systems tuned by the ANFIS, GA, PSO and PS methods both in terms of MAPE and RMSE (see the Mann–Whitney test results for ${\mathrm{MAPE}}_{\mathrm{te}}$ and ${\mathrm{RMSE}}_{\mathrm{te}}$ in Table 5).
• Taking into account the training and test data together, the proposed SFAFIS method performs significantly better than the Sugeno-type fuzzy inference systems tuned by the ANFIS, GA, PSO and PS methods both in terms of MAPE and RMSE (see the Mann–Whitney test results for ${\mathrm{MAPE}}_{\mathrm{te},\mathrm{tr}}$ and ${\mathrm{RMSE}}_{\mathrm{te},\mathrm{tr}}$ in Table 5).

Noting the modeling results shown in Table 1, Table 2, Table 3 and Table 4, in most of the cases, the ${\mathrm{MAPE}}_{\mathrm{te}}$ metric for the SFAFIS method is less than 5. This means that using the above-considered characteristics of a new railway connection, the SFAFIS method can predict the average number of passengers on this connection with an absolute relative error of less than 5%. This accuracy exceeds the practical expectations.

Table 6 shows the average training and inference times of the investigated methods examined on the same computer under the same conditions. In this table, the average inference time represents the mean value of the time needed to predict both the training and test output data for the iterations shown in Table 1, Table 2, Table 3 and Table 4. In every iteration, from the three SFAFIS methods (for $\lambda =30,60$ and 90 parameter values), the one with the best ${\mathrm{RMSE}}_{\mathrm{tr},\mathrm{te}}$ value was taken into account in computing the average inference time of the SFAFIS method.

Table 6. Training and inference times vs. method.

Method	Average Training Time (s)	Average Inference Time (s)
SFAFIS	2.288	0.018157
Sugeno-type FIS tuned by ANFIS	3.695	0.003318
Sugeno-type FIS tuned by GA	545.985	0.005004
Sugeno-type FIS tuned by PSO	310.319	0.003858
Sugeno-type FIS tuned by PS	1185.936	0.005838

Based on the average training and inference times shown in Table 6, we see that, for the data utilized in our case study, the average training time for the SFAFIS method was similar to that of the ANFIS method, and the average training times of these two methods (2.288 and 3.695 s) were considerably shorter than the average training times of the other three methods. At the same time, the average inference time for the SFAFIS method turned out to be longer than that of the other investigated methods. Here, we should emphasize that since we aim to predict the number of passengers on railway connections and do not need to do this real-time, neither the training time nor the inference time of the method used is critical.

Possibilities for Applying the SFAFIS Method to Other Railway Systems

As mentioned before, in our case study, fifteen characteristics of railway connections were taken into consideration, and so the input vector $\mathbf{x}\in {[0,1]}^{15}$ contained the normalized values of these characteristics. Here, we should note that these characteristics were identified based on previous research articles (see, e.g., [14,36,37,38,39,40,41,42,43]) and preliminary studies of Hungarian State Railways (MÁV). This means that we identified those characteristics of the examined railway connections that mostly affected the number of passengers on these connections. Hence, these characteristics are specific to the Hungarian railway system in the sense that these inputs are the most explanatory in the given system. At the same time, it is worth noting that the SFAFIS method presented in Section 4 can be applied in the same way to any another railway system, even if the input characteristics in that system are different. That is, once the input features of railway routes, and hence the explanatory variables, in a railway system are identified, and the historical number of passengers associated with each input vector is given, the SFAFIS method can be applied in the same way as described. The number of passengers can be determined from ticket sales data.

6. Conclusions and Plans for Future Research

In this study, we developed a new fuzzy inference method that utilizes a novel similarity measure and fuzzy arithmetic operations to model the relation between inputs and outputs. Using known input–output pairs, for a given input vector, we determine its similarity to all the known input vectors, and from these similarities, we derive weights. Using these weights, the output of our system is computed as the defuzzified value of the weighted average of triangular fuzzy numbers that represent the linguistic values of the outputs. Based on the results of a real-life case study, which we presented in Section 5, we may conclude that although the proposed method has only two adjustable parameters, its forecasting capability is comparable with that of Sugeno-type fuzzy inference systems, which have a much larger number of adjustable parameters.

In Section 3, we introduced the epsilon-function-based normalized fuzzy similarity measure and demonstrated its main properties. Later, we showed how this similarity measure can be used in the presented SFAFIS method. However, we did not study how the SFAFIS method would perform using other similarity measures. Since comparing the proposed similarity measure with other similarity measures is an interesting topic, we plan to study this in our future research.

We proposed the use of a simple grid search method to find the quasi-optimal values of the $\lambda$ and ℓ parameters of an SFAFIS system. However, as part of our future research, we plan to investigate how more effective optimization methods can be utilized for this purpose. In the proposed inference system, we perform arithmetic operations over triangular fuzzy numbers. We would also like to examine how our method can be adapted to other types of fuzzy numbers.

In the presented case study, we utilized passenger traffic data from the Hungarian State Railways (MÁV). This naturally raises the question of how the SFAFIS method can be applied to other transportation systems, such as high-speed rail networks, urban metro systems or air routes. We would therefore like to further explore this question in our future research.

In the wider field of transportation planning, in our future research, we would also like to investigate how the choice of transport mode can be modeled using fuzzy methods. Moreover, we are also motivated to answer the question of how and by what evaluation methods limited transport capacities can be allocated on a socioeconomic basis.