Application of Uncertain AHP Method in Analyzing Travel Time Belief Reliability in Transportation Network

Abstract

Because predictions of transportation system reliability can provide useful information for intelligent transportation systems (ITS), evaluation of them might be viewed as a beneficial activity for reducing traffic congestion. This evaluation procedure could include some alternatives and criteria in a discrete decision space. To handle this evaluation process in an uncertain environment, a novel uncertain multi-criteria decision-making (MCDM) method is put forward in this paper. Considering the validity of uncertainty theory as a measure of epistemic uncertainty, we first introduce it into analytic hierarchy process (AHP) and provide the whole calculation procedure of the approach. The proposed approach is employed to evaluate regional travel time belief reliability in a case study. Additionally, a comparison is performed between the results of uncertain AHP and other MCDM methods to examine the efficiency of this method. These analyses show that uncertainty theory is particularly suited to be employed combination with the AHP method.

1. Introduction

The transportation system is critical to the proper operation of a city. Many experts and professors have dedicated their careers to relevant research over the years. Adoption of artificial intelligence (AI) technology, integration of multimodal systems, and the advent of new unknown elements such as COVID-19 have all had a significant impact on the transportation system. Simultaneously, safety has begun to be recognized as a crucial element influencing travel decisions. With the emergence of new problems and demands, it is critical to further investigate the transportation system’s performance.

The evaluation of the transportation system has been studied by many researchers during the past years to better understand the key factors that can increase overall transportation system performance [1,2]. It is well known that we are confronted with multiple criteria in transportation system evaluation procedures. As a result, multi-criteria decision-making (MCDM) methods are rapidly growing in related problems [3]. Taking Nantong urban expressway as an example, Jing et al. of Southeast University studied the scheme evaluation system of urban expressway by using analytic hierarchy process (AHP), fuzzy evaluation and grey relational analysis (GRA) [4]. In order to establish a safety index that could be implemented in a decision support system (DSS) for the railway transportation system, Sangiorgio et al. took into consideration the damage antecedents in both the railway infrastructure and the train equipment [5]. Senne et al. proposed an evaluation model based on AHP in terms of sustainability and integrated transportation [6]. A consistent combination of the ELECTRE TRI multi-criteria decision-sorting method and the DELPHI procedure was proposed to identify which urban public transport vehicles are acceptable [7]. It is undeniable that group decision making has evolved into a critical and vital component of MCDM. [8,9], and a group can better overcome the complexity of the problem [10,11]. In this study, we plan to analyze the structure of the transportation issue with the assistance of a group of experts, and decide how important the selected criteria are. As commonly asserted by its supporters, the benefits of AHP over other multi-criteria methodologies include its flexibility and intuitive appeal to experts [12].

As urbanization accelerates, traffic congestion has become a major constraint to urban development, negatively impacting the performance of the transportation system [13,14]. At the same time, statistics show that congestion levels are still worsening. We could conclude that one of the crucial actions that might contribute to the provision of improved transportation services is the evaluation of the transportation system with regard to significant congestion factors. While many other metrics (such as “level of service”) have been employed by the transportation industry to quantify congestion, travel time is a more direct indicator of how congestion impacts users [15]. Many audiences, both technical and nontechnical, use travel time to evaluate the efficacy of transportation systems. In summary, to acquire a better knowledge of regional traffic performance, we will focus more on travel time reliability evaluation in the transportation system.

However, experts usually have limited understanding of things and fail to have absolute confidence to make accurate judgments about the relative importance of factors expressed by Satty’s nine-point scale [16,17,18]. Therefore, this issue can affect the evaluation of transportation system reliability. Moreover, the evaluation procedure in AHP problems frequently contains uncertain data, which can be challenging for experts. According to studies, the transportation problem is always a typical uncertainty problem [13,14]. In addition to the system’s random uncertainty, we should note that the transportation system’s primary service target is people. The amount of traffic information, cognition of travelers, their goal, personal preferences, and other factors all contribute to the system’s uncertainty, which cannot be accurately measured by probability theory. However, when researching issues on travel time reliability, the existing epistemic uncertainty has not been taken into consideration by researchers deeply [19,20,21].

The fuzzy set theory, proposed by Zadeh in 1965, is the most widely used method for modeling the uncertainty of MCDM issues [22]. The introduction of fuzzy set theory to these MCDM problems has been widely applied in scientific and engineering domains [23]. However, fuzzy set theory still could not explain the actual existence of uncertain phenomena perfectly. Liu questioned the inconsistencies existing in fuzzy set theory, and put forward a novel axiomatic mathematical theory in 2007, which could better describe the uncertainty caused by subjective cognition [24]. Uncertainty theory has gradually grown into a relatively complete mathematical system over the years. It can more reasonably quantify inaccurate data information, which is mainly provided by experts in consideration of personal experience. Especially when there is neither historical data nor experimental data for reference, people have to rely on empirical data, then the uncertainty theory could be very useful and effective.

Many studies have been conducted with focusing on applications of uncertainty theory in problems in uncertain environments. Lv et al. applied the uncertainty theory to the vehicle scheduling problem and proposed an uncertain programming model for multiple distribution centers [25]. Song et al. proposed a new uncertain decision model to perform product configuration using redundancy and standardization in an uncertain environment [26]. Kang proposed the theoretical framework and the basic method of belief reliability to examine the epistemic uncertainty in reliability research, introducing uncertainty theory to reliability industry [27]. Although many financial and engineering industries employ uncertainty theory, there have been relatively few studies that have expanded on it or applied it to MCDM problems. In the current study, we intend to introduce the uncertainty theory to AHP to make up for the deficiency of ignoring epistemic uncertainty in the original method. In this extension, the preferences of experts on the criteria should be expressed as an uncertain valuable. The main contributions of this study are summarized as follows:

• We provide an extension of AHP method using uncertainty theory to quantify the subjective weights of criteria;
• The proposed uncertain AHP approach is applied to the evaluation of regional travel time belief reliability with congestion considerations, providing technical support to intelligent traffic systems (ITS) for intelligent decision making;
• The validity of introducing uncertainty theory into the MCDM methods to describe the existing uncertain information is verified.

The structure of this article is shown as follows: the definitions and theorems of uncertainty theory are introduced in Section 2, which provides a mathematical basis for the establishment of the evaluation procedure. In Section 3, we describe the overall calculation procedure of how to combine the uncertainty theory with the AHP method. In Section 4, we explain how to establish the evaluation model of the transportation system and provide a concrete case study to verify the validity and operability of the approach. Finally, the conclusions are presented in Section 5.

2. Preliminaries on Uncertainty Theory

The uncertain measure is a set function that satisfies the axioms of uncertainty theory [24]. Γ is a nonempty set (sometimes called a universal set) and $ℒ$ is a σ-algebra over Γ [24]. Each element Λ in $ℒ$ is called a measurable set, and the uncertain measure $ℳ$ is defined on the σ-algebra $ℒ$ [24]. That is, a number $ℳ\left\{\mathsf{\Lambda }\right\}$ is assigned to each event Λ to indicate the belief degree with which we believe Λ will happen [24].

Axiom 1.

(Normality Axiom [24])$ℳ\left\{\mathsf{\Gamma }\right\}=1$for the universal set Γ.

Axiom 2.

(Duality Axiom [24])$ℳ\left\{\mathsf{\Lambda }\right\}+ℳ\left\{{\mathsf{\Lambda }}^{\mathrm{c}}\right\}=1$for any event Λ.

Axiom 3.

(Subadditivity Axiom [24]) For every countable sequence of events${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2,\dots ,$we have

$$ℳ\left\{{\displaystyle \underset{i=1}{\overset{\infty }{\cup }}{\mathsf{\Lambda }}_i}\right\}\le {\displaystyle \sum _{i=1}^{\infty }ℳ\left\{{\mathsf{\Lambda }}_i\right\}}.$$

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Definition 1.

(Uncertain Measure [24]) The set function $ℳ$ is called an uncertain measure if it satisfies the normality, duality, and subadditivity axioms.

Axiom 4.

(Product Axiom [28]) Let (Γ_k, $ℒ$_k, $ℳ$_k) be uncertainty spaces for k = 1, 2,…The product uncertain measure $ℳ$ is an uncertain measure satisfying

$$ℳ\left\{{\displaystyle \prod }_{k=1}^{\infty }{\mathsf{\Lambda }}_k\right\}=\underset{k=1}{\overset{\infty }{{\displaystyle \mathsf{ʌ}}}}{ℳ}_k\left\{{\mathsf{\Lambda }}_{\mathrm{k}}\right\}$$

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where Λ_k are arbitrarily chosen events from $ℒ$_k for k = 1, 2, …, respectively.

The uncertain variable is used to represent quantities with uncertainty.

Definition 2.

(Uncertain Variable [24]) An uncertain variable is a function ξ from an uncertainty space (Γ,$ℒ$,$ℳ$) to the set of real numbers such that$\left\{\xi \in B\right\}$is an event for any Borel set B of real numbers.

Definition 3.

(Uncertainty Distribution [24]) The uncertainty distribution Φ of an uncertain variable ξ is defined by

$$\mathsf{\Phi }\left(x\right)=ℳ\left\{\xi \le x\right\}$$

(3)

for any real number x.

Definition 4.

(Zigzag Uncertainty Distribution [29]) An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution

$$\mathsf{\Phi }\left(x\right)=\{\begin{array}{ll}0,\hfill & ifx\le a\hfill \\ \frac{x-a}{2\left(b-a\right)},\hfill & ifa\le x\le b\hfill \\ \frac{\left(x+c-2b\right)}{2\left(c-b\right)},\hfill & ifb\le x\le c\hfill \\ 1,\hfill & ifx\ge c\hfill \end{array}$$

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denoted by 𝒵(a,b,c) where a, b, c are real numbers with a < b < c. See Figure 1.

Definition 5.

(Regular Uncertainty Distribution [29]) An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ(x) < 1, and

$$\underset{x\to -\infty }{\lim }\mathsf{\Phi }\left(x\right)=0,\underset{x\to +\infty }{\lim }\mathsf{\Phi }\left(x\right)=1.$$

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Definition 6.

(Inverse Uncertainty Distribution [29]) Let ξ be an uncertain variable with regular uncertainty distribution Φ(x). Then the inverse function ${\mathsf{\Phi }}^{-1}\left(\alpha \right)$ is called the inverse uncertainty distribution of ξ.

Definition 7.

(Expected Value [29]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by

$$E\left[\xi \right]={\int }_0^{+\infty }ℳ\left\{\xi \ge x\right\}\mathrm{d}x-{\int }_{-\infty }^0ℳ\left\{\xi \le x\right\}\mathrm{d}x$$

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provided that at least one of the two integrals is finite.

Theorem 1.

(Liu [29]) Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then

$$E\left[\xi \right]={\int }_0^1{\mathsf{\Phi }}^{-1}\left(\alpha \right)\mathrm{d}\alpha .$$

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Theorem 2.

(Operational Law [29]) Let${\xi }_1,{\xi }_2,\dots ,{\xi }_n$be independent uncertain variables with regular uncertainty distributions${\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,\dots ,{\mathsf{\Phi }}_n$, respectively. If$f\left(x_1,x_2,\dots ,x_n\right)$is continuous, strictly increasing with respect to$x_1,x_2,\dots ,x_m$and strictly decreasing with respect to$x_{m+1},x_{m+2},\dots ,x_n$, then

$$\xi =f\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)$$

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has an inverse uncertainty distribution

$${\mathsf{\Psi }}^{-1}\left(\alpha \right)=f\left({\mathsf{\Phi }}_1^{-1}\left(\alpha \right),\dots ,{\mathsf{\Phi }}_m^{-1}\left(\alpha \right),{\mathsf{\Phi }}_{m+1}^{-1}\left(1-\alpha \right),\dots ,{\mathsf{\Phi }}_n^{-1}\left(1-\alpha \right)\right).$$

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3. Uncertain Analytic Hierarchy Process

Analytic hierarchy process (AHP) is a particularly effective comprehensive evaluation method when dealing with some complex systems [30]. However, based on the essential characteristics of AHP, it is clear that AHP obviously very consistent with the assessors’ conduct. In summary, AHP heavily relies on personal experiences, knowledge, and intuition. As a result, it is critical that specialists provide absolute authoritative evaluation. It is also critical to better accurately measure the information provided by these specialists. However, when scaling preferences in classical AHP, the subjectivity of judgment is neglected.

Researchers have worked hard over the years to effectively evaluate the epistemic uncertainty present in the judgmental matrix. To describe uncertainty, experts have introduced theories such as fuzzy set theory, rough set theory, and gray system theory into AHP. However, certain inexplicable defects in these theories emerged throughout their practical use [24]. Furthermore, as previously stated, uncertainty theory can better describe epistemic uncertainty. As a result, this section attempts to incorporate uncertainty theory into AHP in order to compensate for the inadequacies of classic AHP.

3.1. Uncertain Scale for Preferences

On the basis of the relative scale measurement of classical AHP, we consider the intensity of importance as uncertain variables with a zigzag uncertainty distribution

$$ℳ\left({\xi }_{ij}\le x\right)={\mathsf{\Phi }}_{ij}\left(x\right)=\{\begin{array}{ll}0\hfill & \left(x\le a_{ij}\right)\hfill \\ \frac{x-a_{ij}}{2\left(b_{ij}-a_{ij}\right)}\hfill & \left(a_{ij}\le x\le b_{ij}\right)\hfill \\ \frac{\left(x+c_{ij}-2b_{ij}\right)}{2\left(c_{ij}-b_{ij}\right)}\hfill & \left(b_{ij}\le x\le c_{ij}\right)\hfill \\ 1\hfill & \left(x\ge c_{ij}\right)\hfill \end{array}$$

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denoted by $\mathcal{Z}\left(a_{ij},b_{ij},c_{ij}\right)$ where $a_{ij},b_{ij},c_{ij}$ are real numbers with $a_{ij}<b_{ij}<c_{ij}$.

That is, the number $ℳ\left({\xi }_{ij}\le x\right)$ is set to represent the belief degree with which we think it will happen that the intensity of importance of index i with respect to index j is lower than x. For the value of x, we can refer to the fundamental scale of 1–9 in the classical AHP. In contrast to the classic scaling method, in uncertain AHP, evaluators assign preference to any value from 1 to 9, while also providing belief degree one-to-one. At the same time, we believe that following zigzag uncertainty distribution is a suitable choice because it better reflects the statistical data and simplifies the calculation. The relative scale measurement is then obtained, as indicated in Table 1.

Table 1. Pairwise comparison scale ¹.

Evaluation	Explanation
$ℳ\left({\xi }_{ij}\le 1\right)=0$	We totally believe index i is preferred to index j.
$ℳ\left({\xi }_{ij}\le 1\right)=1$	We totally believe index j is preferred to index i.
$ℳ\left({\xi }_{ij}\le 3\right)=0$	We totally believe index i is moderately preferred to index j.
$ℳ\left({\xi }_{ij}\le 3\right)=1$	We do not believe index i is moderately preferred to index j, and we consider it to be of a lower priority.
$ℳ\left({\xi }_{ij}\le 5\right)=0$	We totally believe index i is strongly preferred to index j.
$ℳ\left({\xi }_{ij}\le 5\right)=1$	We do not believe index i is strongly preferred to index j, and we consider it to be of a lower priority.
$ℳ\left({\xi }_{ij}\le 7\right)=0$	We totally believe index i is very strongly preferred to index j.
$ℳ\left({\xi }_{ij}\le 7\right)=1$	We do not believe index i is very strongly preferred to index j, and we consider it to be of a lower priority.
$ℳ\left({\xi }_{ij}\le 9\right)=0$	We totally believe index i is extremely preferred to index j.
$ℳ\left({\xi }_{ij}\le 9\right)=1$	We do not believe index i is very extremely preferred to index j, and we consider it to be of a lower priority.
$0<ℳ\left({\xi }_{ij}\le x\right)<1$	Our belief level is somewhere between complete belief and complete disbelief.

¹ The values of x in Table 1 are chosen as 1, 3, 5, 7, and 9 only as examples for the reader’s reference, the actual values of x are shown in Table 2.

Table 1. Pairwise comparison scale ¹.

Evaluation	Explanation
$ℳ\left({\xi }_{ij}\le 1\right)=0$	We totally believe index i is preferred to index j.
$ℳ\left({\xi }_{ij}\le 1\right)=1$	We totally believe index j is preferred to index i.
$ℳ\left({\xi }_{ij}\le 3\right)=0$	We totally believe index i is moderately preferred to index j.
$ℳ\left({\xi }_{ij}\le 3\right)=1$	We do not believe index i is moderately preferred to index j, and we consider it to be of a lower priority.
$ℳ\left({\xi }_{ij}\le 5\right)=0$	We totally believe index i is strongly preferred to index j.
$ℳ\left({\xi }_{ij}\le 5\right)=1$	We do not believe index i is strongly preferred to index j, and we consider it to be of a lower priority.
$ℳ\left({\xi }_{ij}\le 7\right)=0$	We totally believe index i is very strongly preferred to index j.
$ℳ\left({\xi }_{ij}\le 7\right)=1$	We do not believe index i is very strongly preferred to index j, and we consider it to be of a lower priority.
$ℳ\left({\xi }_{ij}\le 9\right)=0$	We totally believe index i is extremely preferred to index j.
$ℳ\left({\xi }_{ij}\le 9\right)=1$	We do not believe index i is very extremely preferred to index j, and we consider it to be of a lower priority.
$0<ℳ\left({\xi }_{ij}\le x\right)<1$	Our belief level is somewhere between complete belief and complete disbelief.

¹ The values of x in Table 1 are chosen as 1, 3, 5, 7, and 9 only as examples for the reader’s reference, the actual values of x are shown in Table 2.

Table 2. The fundamental scale for x ¹.

x	Definition	Explanation
1	Equal importance	Two indexes contribute equally to the objective
3	Moderate importance of one over another	Experience and judgment strongly favor one index over another
5	Essential or strong importance	Experience and judgment strongly favor one index over another
7	Very strong importance	Experience and judgment strongly favor one index over another
9	Extreme importance	The evidence favoring one index over another is of tile highest possible order of affirmation
2,4,6,8	Intermediate values between the two adjacent judgments	When compromise is needed
Reciprocals	If index i has one of the above numbers assigned to it when compared with index j, then j has the reciprocal value when compared with i

¹ The definitions and explanations mentioned in Table 2 are all referenced in Satty’s article “How to make a decision: The analytic hierarchy process” [12].

Table 2. The fundamental scale for x ¹.

x	Definition	Explanation
1	Equal importance	Two indexes contribute equally to the objective
3	Moderate importance of one over another	Experience and judgment strongly favor one index over another
5	Essential or strong importance	Experience and judgment strongly favor one index over another
7	Very strong importance	Experience and judgment strongly favor one index over another
9	Extreme importance	The evidence favoring one index over another is of tile highest possible order of affirmation
2,4,6,8	Intermediate values between the two adjacent judgments	When compromise is needed
Reciprocals	If index i has one of the above numbers assigned to it when compared with index j, then j has the reciprocal value when compared with i

¹ The definitions and explanations mentioned in Table 2 are all referenced in Satty’s article “How to make a decision: The analytic hierarchy process” [12].

As previously noted, the preference, or the intensity of importance, is determined by the expertise and personal knowledge of decision makers. In order to turn abstract data into tangible data and quantify decision makers’ knowledge and expertise, this paper cites a questionnaire survey procedure [24]. To explain this procedure, an example, as shown below, is given.

Example 1.

The questionnaire survey procedure of evaluating the preference of index i to index j.

Q1: To what extent do you think that the preference of index i to index j is less than 1?

A1: 1%. (An experimental datum (1,0.01) is obtained.)

Q2: To what extent do you think that the preference${\xi }_{ij}$is less than 3?

A2: 40%. (An experimental datum (3,0.4) is obtained.)

Q3: To what extent do you think that the preference${\xi }_{ij}$is less than 5?

A3: 70%. (An experimental datum (5,0.7) is obtained.)

Q4: To what extent do you think that the preference${\xi }_{ij}$is less than 7?

A4: 99%. (An experimental datum (7,0.99) is obtained.)

Q5: So, to what extent do you think that the preference${\xi }_{ij}$is less than 6?

A5: 98%. (An experimental datum (6,0.98) is obtained.)

Q6: And then, to what extent do you think that the preference${\xi }_{ij}$is less than 4?

A6: 50%. (An experimental datum (4,0.5) is obtained.)

Six experimental data of the preference of index i to index j are obtained from the decision maker utilizing this questionnaire survey:

(1,0.01), (3,0.4), (4,0.5), (5,0.7), (6,0.98), (7,0.99).

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In the process of practical application, we can take as many values as possible near the points where the changes are sharp.

It is possible to use the principle of least squares to calculate the uncertainty distributions of uncertain variables Φ(x) [29].

Definition 8.

(Principle of Least Squares [29]) If the expert’s experimental data

$$\left(x_1,{\alpha }_1\right),\left(x_2,{\alpha }_2\right),\dots \left(x_n,{\alpha }_n\right)$$

(12)

are obtained, and the vertical direction is accepted, then we have

$$\min {\displaystyle \sum _{i=1}^n{\left(\mathsf{\Phi }\left(x_i|\theta \right)-{\alpha }_i\right)}^2}.$$

(13)

The optimal solution $\widehat{\theta }$ of Equation (13) is called the least squares estimate of θ, and then the least squares uncertainty distribution is $\mathsf{\Phi }\left(x_i|\widehat{\theta }\right)$ [29].

3.2. Steps for Applying Uncertain AHP

Generally speaking, four steps are involved:

• Building the hierarchical structure;
• Constructing the uncertain judgmental matrix;
• Calculating the relative importance factor;
• Calculating comprehensive evaluation scores and making decisions.

3.2.1. Building the Hierarchical Structure

First and foremost, specify the objective of the stated problem. The top layer, which symbolizes the overarching goal, was then defined. Second, for the criteria layer, choose the primary criteria affecting the previously indicated target. Based on this, the sub-criteria with the greatest influence on the target are chosen from the elements contained in each criterion. The criteria layer and the sub-criteria layer are both middle levels that are used to evaluate alternatives. Finally, the bottom layer contains the possibilities for achieving the goal and solving the problem. The hierarchical structure was then organized from top to bottom. So far, the structure of the complex problem has been presented in a simple and concise manner. The hierarchical structure is shown in Figure 2. O stands for the top layer. The criteria layer, which contains n criteria, is denoted as C₁, C₂, …, and C_n. The sub-criteria layer is denoted in the same way. The problems to be solved are different, and the corresponding middle layers are also different. A₁, A₂, …, and A_m are used to identify the m alternatives in the bottom layer.

3.2.2. Constructing the Uncertain Judgmental Matrix

Pairwise comparison of two elements produces an unclear judging matrix. Pairwise comparisons are used to assess the relative relevance of each criterion in relation to the goal and each sub-criterion in relation to each criterion. Domain experts of the assessment team must provide their judgment on the worth of one single pairwise comparison and related belief degree at a time through a questionnaire, using the scale shown in Table 1. The gathered data is then fitted using the least square approach. As a result, each relative importance’s uncertainty distribution is provided.

Finally, we may obtain the comparison result that is represented by matrix $O_{n\times n}={\left({\xi }_{ij}\right)}_{n\times n}$. The pairwise comparison judgmental matrix from the goal layer to the criteria layer is designated by $O_{n\times n}$. In the same way, we can also get the pairwise comparison judgmental matrixes from the criteria layer to the sub-criteria layer, which are denoted as ${\left(C_i\right)}_{k\times k},i=1,2,\dots ,n$.

Definition 9.

(Uncertain Judgmental Matrix) Let the relative importance${\xi }_{ij}$of the pairwise comparison be an uncertain variable that has a zigzag uncertainty distribution, that is${\xi }_{ij}\sim \mathcal{Z}\left(a_{ij},b_{ij},c_{ij}\right)$. After pairwise comparison of n objects, we can get a pairwise comparison judgmental matrix composed of$n\times n$pairwise comparisons, which is denoted as

$$O=\left[\begin{array}{cccc}{\xi }_{11}& {\xi }_{12}& \cdots & {\xi }_{1n}\\ {\xi }_{21}& {\xi }_{22}& \cdots & {\xi }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\xi }_{n1}& {\xi }_{n2}& \cdots & {\xi }_{nn}\end{array}\right],$$

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in which when i equals j,

$${\xi }_{ij}\sim \mathcal{Z}\left(1,1,1\right),$$

when i is not equal to j,

$$\{\begin{array}{l}{\xi }_{ij}\sim \mathcal{Z}\left(a_{ij},b_{ij},c_{ij}\right)\hfill \\ {\xi }_{ji}\sim \mathcal{Z}\left(a_{ji},b_{ji},c_{ji}\right)=\mathcal{Z}\left(1/c_{ij},1/b_{ij},1/a_{ij}\right).\hfill \end{array}$$

3.2.3. Calculating the Relative Importance Factor

The relative importance of each element in one layer to the element in the layer above could be derived once the judgmental matrix has been constructed.

Definition 10.

(Relative Importance Factor) For an upper-level goal/element, the relative importance of an element at this layer is called the relative importance factor, and the calculation formula is as follows. Supposing there are n factors for a target, the relative importance factor of the k-th factor is

$${\eta }_k={\displaystyle \sum _{j=1}^n{\xi }_{kj}}/{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{\xi }_{ij}}}$$

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where the${\xi }_{ij}$is the pairwise comparison among the corresponding uncertain judgmental matrix.

Assuming that the relative importance factors are independent of each other, the inverse distribution function of ${\eta }_k$ can be obtained as follows

$${\mathsf{\Psi }}_k^{-1}\left(\alpha \right)=\frac{{\displaystyle \sum _{j=1}^n{\mathsf{\Phi }}_{kj}^{-1}\left(\alpha \right)}}{{\displaystyle \sum _{j=1}^n{\mathsf{\Phi }}_{kj}^{-1}\left(\alpha \right)}+{\displaystyle \sum _{i\ne k}^n{\displaystyle \sum _{j=1}^n{\mathsf{\Phi }}_{ij}^{-1}\left(1-\alpha \right)}}},$$

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where the ${\mathsf{\Phi }}_{ij}^{-1}\left(\alpha \right)$ is the inverse uncertainty distribution of ${\xi }_{ij}$:

$${{\mathsf{\Phi }}_{ij}}^{-1}\left(\alpha \right)=\{\begin{array}{ll}\left(1-2\alpha \right)a_{ij}+2\alpha b_{ij}\hfill & \left(\alpha <0.5\right)\hfill \\ \left(2-2\alpha \right)b_{ij}+2\left(2\alpha -1\right)c_{ij}\hfill & \left(\alpha \ge 0.5\right).\hfill \end{array}$$

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Next, calculate the relative importance factor and its distribution function for each element and alternative in turn. The relative importance factors of the same level can form a vector:

$${W^{\prime }}_O={\left[{\eta }_1,{\eta }_2,\cdots ,{\eta }_n\right]}^{\mathrm{T}}.$$

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In order to facilitate the next calculation, we need to simplify the relative importance vector. An essential numerical characteristic of an uncertain variable is its expected value. The expected value also represents the value of an uncertain variable in the average sense. Based on Theorem 1, we can find the expectations for each relative importance factor, and then the vector could be simplified as:

$$W_O={\left[{{\eta }^{\prime }}_1,{{\eta }^{\prime }}_2,\cdots ,{{\eta }^{\prime }}_n\right]}^{\mathrm{T}},{{\eta }^{\prime }}_k=\frac{E\left[{\eta }_k\right]}{{\displaystyle \sum _{k=1}^{n}E\left[{\eta }_k\right]}}.$$

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The vector for the sub-criteria layer can be obtained as well:

$$W_{C_n}={\left[{{\eta }^{\prime }}_{n1},{{\eta }^{\prime }}_{n2},\cdots ,{{\eta }^{\prime }}_{nk}\right]}^{\mathrm{T}},{{\eta }^{\prime }}_{ni}=\frac{E\left[{\eta }_{ni}\right]}{{\displaystyle \sum _{i=1}^{k}E\left[{\eta }_{ni}\right]}}.$$

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Next, assume a total of p sub-criteria; we can obtain the weight value vector as:

$$W_{\mathrm{A}}={\left[e_1,e_2,\dots ,e_p\right]}^{\mathrm{T}}={\left[{{\eta }^{\prime }}_1{{\eta }^{\prime }}_{11},{{\eta }^{\prime }}_1{{\eta }^{\prime }}_{12},\dots ,{{\eta }^{\prime }}_n{{\eta }^{\prime }}_{n1},{{\eta }^{\prime }}_n{{\eta }^{\prime }}_{n2},\dots ,{{\eta }^{\prime }}_n{{\eta }^{\prime }}_{nk}\right]}^{\mathrm{T}}.$$

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3.2.4. Calculating Comprehensive Evaluation Scores and Making Decisions

For the p-th sub-criteria, we can get the vector composed of each preference factor of the bottom layer called the score vector: $W_{A_p}={\left[{\eta }_{1p},{\eta }_{2p},\cdots ,{\eta }_{mp}\right]}^{\mathrm{T}}$, and the inverse uncertainty distribution of ${\eta }_{kp}\left(k=1,2,\dots ,m\right)$ is ${\mathsf{\Psi }}_{kp}^{-1}\left(\alpha \right)$. Further merging the score vectors can form a score matrix:

$$W_S=\left[\begin{array}{cccc}{\eta }_{11}& {\eta }_{12}& \cdots & {\eta }_{1p}\\ {\eta }_{21}& {\eta }_{22}& \cdots & {\eta }_{2p}\\ ⋮& ⋮& ⋮& ⋮\\ {\eta }_{m1}& {\eta }_{m2}& \cdots & {\eta }_{mp}\end{array}\right].$$

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Finally, we can get the total score vector W of the bottom layer for the total goal:

$$W=W_S\cdot W_{\mathrm{A}}=\left[\begin{array}{cccc}{\eta }_{11}& {\eta }_{12}& \cdots & {\eta }_{1p}\\ {\eta }_{21}& {\eta }_{22}& \cdots & {\eta }_{2p}\\ ⋮& ⋮& ⋮& ⋮\\ {\eta }_{m1}& {\eta }_{m2}& \cdots & {\eta }_{mp}\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\\ ⋮\\ e_p\end{array}\right]={\left[w_1,w_2,\cdots ,w_m\right]}^{\mathrm{T}}$$

(23)

where the $w_i(i=1,2,\cdots m)$ is the total score of i-th alternative for the total goal.

4. Case Study

In this section, a case study is presented to demonstrate the procedure and verify the efficiency of the proposed uncertain AHP approach in evaluating regional travel time belief reliability. The case study is related to an open subject study focusing on the reliability evaluation of urban road traffic networks. The goal of this study was to reduce epistemic uncertainty during the original evaluation and generate more accurate reliability predictions so as to better manage and operate the transportation system. A group of three traffic experts (E₁ to E₃) from Beihang University was formed to perform this evaluation. They come from the School of Reliability and Systems Engineering and have worked for many years on the reliability of transportation systems.

4.1. Hierarchical Structure

Travel time belief time is chosen as the evaluation objective because it better quantifies the benefits of traffic management and operation activities than simple averages [31]. To facilitate the calculation, four alternatives defined as the level of reliability (A₁ to A₄) remain for further evaluations, such as excellent, good, average, and poor. In this case study, we quantify reliability R_B as poor when its value is greater than 0 and less than 0.25. By analogy, we can obtain

Table 3. The quantification of reliability levels.

The Level of Reliability	Quantified Values
Poor	0 ≤ RB ≤ 0.25
Average	0.25 ≤ RB ≤ 0.5
Good	0.5 ≤ RB ≤ 0.75
Excellent	0.75 ≤ RB ≤ 1

.

Because the interaction of all the sources of congestion produced unreliable travel times, the group of experts planned to start at the root of congestion. According to the FHWA “Traffic Congestion Reliability” reports, the hierarchical structure of the evaluation problem is depicted in Figure 3 [15]. In this structure, we have two criteria (recurring congestion and no-recurring congestion), and each of these criteria has related sub-criteria, which were carefully chosen by the experts (seven sub-criteria in total). The definitions of these criteria and sub-criteria are as follows:

Recurring congestion (C₁): Approximately half of the congestion is “recurring”, meaning it occurs on a daily basis. During peak travel periods, recurring congestion arises for one clear reason: there are just more vehicles than available road capacity [15].

• Bottlenecks (C₁₁): A bottleneck is defined as a place where a road narrows or where there is frequently heavy traffic, which causes vehicles to slow down or stop. This sub-criterion is described as a current estimate of the number of bottlenecks.
• Traffic signal timing and operations (C₁₂): This sub-criterion is defined as an estimate of the number of traffic light intersections where many vehicles converge at the current moment.
• Functional facilities (C₁₃): Functional facility buildings reflect the travel demand direction of travelers to a great extent. In daily life, people are faced with regular traffic needs such as commuting, school, and official business. This part of demand often appears every day, and its impact on the operation of the transportation system is relatively stable and lasting. The accompanying changes in traffic flow also have certain regularity. Therefore, buildings such as hospitals, schools, markets, subway stations, and so on are what we need to focus on. It is worth noting that the special functional buildings such as performance venues and tourist attractions cannot be ignored either. Studies have shown that the emergence of activities associated with these kinds of buildings often gathers huge traffic demands in a short time. Moreover, this temporary demand could occur concurrently with conventional travel demand.

No-recurring congestion (C₂): Aside from physical capacity, external events can have a significant impact on traffic flow. These non-recurring causes of traffic congestion include traffic incidents, work zones, bad weather, and special events [15]. The main result of these events is to “steal” physical capability from the roadway. Events such as adverse weather and public health issues may cause travelers to reevaluate their trip plans, which may influence traffic demand. The criterion itself is subject to a high degree of uncertainty, so we cannot form a firm perception of the occurrence of such incidents in pre-existing regions based on the historical data available. As a result, there is a non-negligible epistemic uncertainty in its assessment.

• Traffic incidents (C₂₁): Traffic incidents are unplanned roadway events that affect or impede the normal flow of traffic [32]. Traffic incidents increase the likelihood of secondary crashes and cause additional congestion. Incidents affect travel reliability, commerce, and transportation system performance. The transportation of dangerous goods also poses a threat to the traveling public. Slight carelessness may cause material losses or even casualties.
• Work zones (C₂₂): Aging and overcrowded roadways prompt a massive proportion of roadwork [33]. Every year, over 3,000 work zones have been established on more than 20% of U.S. roadways during peak construction seasons [34]. As a result of “daily changes in traffic patterns, narrowed rights-of-way, and other construction operations”, work zones may cause crashes [33,35]. Given this information, it is not surprising that researchers have paid attention to the risk of crashes occurring in work zones [36].
• Bad road weather (C₂₃): Severe weather conditions such as rain and flooding, snow and ice, low visibility, hurricanes, and high winds clearly have negative effects on travel time reliability, with travel time uncertainty increasing [37]. Tu discovered that travel times reliability decreased under bad weather, especially at greater inflow levels [38]. Maze et al. noted that severe winter storms increase the probability of being involved in a collision by up to 25 times, which is far higher than the risk raised by engaging in risky behaviors such as speeding or driving while intoxicated [39]. An extreme rainstorm easily floods roadways, resulting in traffic congestion. When faced with extremely bad weather, travelers also tend to change their travel plans. Thus, it can be seen that bad road weather is considered a criterion.
• Special events (C₂₄): Recently, public health issues such as COVID-19 have had a profound impact on global lifestyles and transportation [40]. These impacts typically include governmental precautions as well as individual decisions not to travel in order to decrease the danger of contamination [40,41]. As a result of these factors, global use of public transportation has decreased as well [42,43]. At the same time, the transmissibility of infectious diseases can affect an area for a short period of time and can cause all functional buildings in the area to lose their functionality.

During the structuring process, we discovered that the evaluation of travel time reliability is mostly performed for traffic changes with a rather short time horizon. This means that the expectations for timeliness and real-time evaluation results are relatively high. As a result, changes caused by financial criteria over a short period of time might be deemed nearly constant. Simultaneously, non-financial criteria such as the construction of work zones and functional facilities in the transportation system reflect, from a different perspective, the government’s financial considerations when carrying out transportation planning. As a result of referring to the structure established by some studies, we believe that the impact of specific financial criteria could be ignored in this issue for the time being [23,44,45].

4.2. Study Area and Data Collection

We selected the area near Beihang University for the study. The specific scope of the study is shown in Figure 4 and Figure 5 below, where the red dots in Figure 4 are traffic data detectors, and the blue lines are city roads. An organization collected travel speed data through these detectors from 27 July to 31 July 2020. The data collected allow experts to obtain information about the traffic situation over a three-day period and to assess the future traffic situation on this basis.

The specific data include detector information, roadway information, and speed information with the following specific feature values.

• Detector information: detection section serial number, starting and ending detector serial number, road name, road length, road direction, road class (e.g., 59567200000261, 525875 525859, North Fourth Ring Road Central Auxiliary, 232.931, 1, 4);
• Road information: road number, endpoint latitude and longitude coordinates (e.g., 20013930970, 115.75, 39.54681);
• Speed data: date, detector number, speed data (1 min collection interval).

4.3. Evaluation Procedure

The following steps and Figure 6 represent the process of evaluation of alternatives according to the structure and the proposed uncertain AHP approach mentioned above.

Step 1. According to the hierarchical structure of the problem, each expert of the group gives his judgment on the worth of one single pairwise comparison and related belief degree at a time through a questionnaire. Then, each relative importance’s uncertainty distribution is obtained through linear fitting. The collated data for the criteria and sub-criteria are represented in

Table 4. The collated data for criteria C₁ and C₂.

Professor	Belief Degree	x									Linear fit
		1	2	3	4	5	6	7	8	9
P1	$ℳ\left({\xi }_{21}\le x\right)$	0	0.24	0.72	0.92	0.98	1	1	1	1	𝒵 (1.1011,2.7865,7.4507)
P2		0	0.16	0.53	0.78	0.93	0.98	0.99	1	1
P3		0	0.2	0.57	0.86	0.91	0.94	1	1	1

,

Table 5. The collated data for criteria C₁₁, C_12, and C₁₃.

Weight	Professor	Belief Degree	x					Linear fit
			1	3	5	7	9
${\xi }_{12}^1$	P1	$ℳ\left({\xi }_{12}^1\le x\right)$	0	0.3	0.5	0.78	1	𝒵 (1.6889,5.0222,7.8218)
	P2		0	0.29	0.6	0.92	1
	P3		0	0	0.39	0.86	1
${\xi }_{13}^1$	P1	$ℳ\left({\xi }_{13}^1\le x\right)$	0	0.38	0.68	0.82	1	𝒵 (1.1439,3.3022,7.4447)
	P2		0	0.49	0.79	1	1
	P3		0	0.42	0.73	0.98	1
${\xi }_{23}^1$	P1	$ℳ\left({\xi }_{23}^1\le x\right)$	0	0.78	0.95	1	1	𝒵 (1,2.5544,5.1357)
	P2		0	0.65	0.94	0.99	1
	P3		0	0.5	1	1	1

and

Table 6. The collated data for criteria C₂₁, C₂₂, C_23, and C₂₄.

Weight	Professor	Belief Degree	x									Linear fit
			1	2	3	4	5	6	7	8	9
${\xi }_{12}^2$	P1	$ℳ\left({\xi }_{12}^2\le x\right)$	0	0.42	0.72	0.88	0.92	0.94	0.99	1	1	𝒵 (1,2.0274,5.49)
	P2		0	0.48	0.66	0.84	0.96	1	1	1	1
	P3		0	0.56	0.74	0.98	0.99	1	1	1	1
${\xi }_{31}^2$	P1	$ℳ\left({\xi }_{31}^2\le x\right)$	0	0.07	0.24	0.49	0.71	0.85	0.98	1	1	𝒵 (1.2687,4.0672,6.6768)
	P2		0	0.11	0.31	0.43	0.89	0.99	1	1	1
	P3		0	0	0.37	0.62	0.82	0.9	1	1	1
${\xi }_{14}^2$	P1	$ℳ\left({\xi }_{14}^2\le x\right)$	0	0.02	0.27	0.52	0.76	0.88	0.98	1	1	𝒵 (1.9635,4.0183,7.4081)
	P2		0	0.01	0.22	0.48	0.72	0.9	0.99	1	1
	P3		0	0	0.26	0.49	0.69	0.86	0.96	0.99	1
${\xi }_{32}^2$	P1	$ℳ\left({\xi }_{32}^2\le x\right)$	0	0	0	0.3	0.59	0.71	0.82	0.9	1	𝒵 (2.2749,4.9873,8.5974)
	P2		0	0	0.08	0.32	0.53	0.66	0.78	0.91	1
	P3		0	0	0.12	0.28	0.48	0.78	0.88	0.99	1
${\xi }_{24}^2$	P1	$ℳ\left({\xi }_{24}^2\le x\right)$	0	0	0.25	0.5	0.79	0.98	1	1	1	𝒵 (1.9885,4.0575,6.6836)
	P2		0	0	0.22	0.46	0.67	0.96	1	1	1
	P3		0	0	0.28	0.49	0.71	0.88	1	1	1
${\xi }_{34}^2$	P1	$ℳ\left({\xi }_{34}^2\le x\right)$	0	0	0	0.16	0.33	0.47	0.91	1	1	𝒵 (3.528,5.928,8.7465)
	P2		0	0	0	0.11	0.34	0.55	0.71	0.88	1
	P3		0	0	0	0.02	0.28	0.52	0.62	0.8	1

, respectively. At last, the uncertain judgmental matrix for the criteria and sub-criteria are constructed as shown in

Table 7. Uncertain matrix for criteria C₁ and C₂.

O	C1	C2
C1	${\xi }_{11}\sim \mathcal{Z}\left(1,1,1\right)$	${\xi }_{12}\sim \mathcal{Z}\left(0.13,0.36,0.91\right)$
C2	${\xi }_{21}\sim \mathcal{Z}\left(1.10,2.79,7.45\right)$	${\xi }_{22}\sim \mathcal{Z}\left(1,1,1\right)$

,

Table 8. Uncertain matrix for criteria C₁₁, C_12, and C₁₃.

C1	C11	C12	C13
C11	${\xi }_{11}^1\sim \mathcal{Z}\left(1,1,1\right)$	${\xi }_{12}^1\sim \mathcal{Z}\left(1.69,5.02,7.82\right)$	${\xi }_{13}^1\sim \mathcal{Z}\left(1.14,3.3,7.44\right)$
C12	${\xi }_{21}^1\sim \mathcal{Z}\left(0.13,0.2,0.59\right)$	${\xi }_{22}^1\sim \mathcal{Z}\left(1,1,1\right)$	${\xi }_{23}^1\sim \mathcal{Z}\left(1,2.55,5.14\right)$
C13	${\xi }_{31}^1\sim \mathcal{Z}\left(0.13,0.3,0.88\right)$	${\xi }_{32}^1\sim \mathcal{Z}\left(0.19,0.39,1\right)$	${\xi }_{33}^1\sim \mathcal{Z}\left(1,1,1\right)$

and

Table 9. Uncertain matrix for criteria C₂₁, C₂₂, C_23, and C₂₄.

C2	C21	C22	C23	C24
C21	${\xi }_{11}^2\sim \mathcal{Z}\left(1,1,1\right)$	${\xi }_{12}^2\sim \mathcal{Z}\left(1,2.03,5.49\right)$	${\xi }_{13}^2\sim \mathcal{Z}\left(0.15,0.25,0.79\right)$	${\xi }_{14}^2\sim \mathcal{Z}\left(1.96,4.02,7.41\right)$
C22	${\xi }_{21}^2\sim \mathcal{Z}\left(0.18,0.49,1\right)$	${\xi }_{22}^2\sim \mathcal{Z}\left(1,1,1\right)$	${\xi }_{23}^2\sim \mathcal{Z}\left(0.17,0.2,0.44\right)$	${\xi }_{24}^2\sim \mathcal{Z}\left(1.99,4.06,6.68\right)$
C23	${\xi }_{31}^2\sim \mathcal{Z}\left(1.27,4.07,6.68\right)$	${\xi }_{32}^2\sim \mathcal{Z}\left(2.27,4.99,8.6\right)$	${\xi }_{33}^2\sim \mathcal{Z}\left(1,1,1\right)$	${\xi }_{34}^2\sim \mathcal{Z}\left(3.53,5.93,8.75\right)$
C24	${\xi }_{41}^2\sim \mathcal{Z}\left(0.16,0.25,0.51\right)$	${\xi }_{42}^2\sim \mathcal{Z}\left(0.15,0.25,0.5\right)$	${\xi }_{43}^2\sim \mathcal{Z}\left(0.11,0.17,0.28\right)$	${\xi }_{44}^2\sim \mathcal{Z}\left(1,1,1\right)$

, respectively.

Step 2. According to Table 7, Table 8 and Table 9, Equations (18) and (19), and Theorem 1, the relative importance vectors are calculated. Equations (26), (30), and (35) show the results after the normalization of this step.

The inverse uncertainty distribution of η₁ and η₂ according to matrix A is shown respectively:

$$\begin{array}{ll}{\mathsf{\Psi }}_1^{-1}\left(\alpha \right)& =\frac{{\mathsf{\Phi }}_{11}^{-1}\left(\alpha \right)+{\mathsf{\Phi }}_{12}^{-1}\left(\alpha \right)}{{\mathsf{\Phi }}_{11}^{-1}\left(\alpha \right)+{\mathsf{\Phi }}_{12}^{-1}\left(\alpha \right)+{\mathsf{\Phi }}_{21}^{-1}\left(1-\alpha \right)+{\mathsf{\Phi }}_{22}^{-1}\left(1-\alpha \right)}\\ & =\{\begin{array}{ll}-(46\alpha +113)/(2576\alpha -1445)\hfill & \left(\alpha <0.5\right)\hfill \\ (492\alpha +18)/(154\alpha +566)\hfill & \left(\alpha \ge 0.5\right),\hfill \end{array}\end{array}$$

(24)

$$\begin{array}{ll}{\mathsf{\Psi }}_2^{-1}\left(\alpha \right)& =\frac{{\mathsf{\Phi }}_{21}^{-1}\left(\alpha \right)+{\mathsf{\Phi }}_{22}^{-1}\left(\alpha \right)}{{\mathsf{\Phi }}_{21}^{-1}\left(\alpha \right)+{\mathsf{\Phi }}_{22}^{-1}\left(\alpha \right)+{\mathsf{\Phi }}_{11}^{-1}\left(1-\alpha \right)+{\mathsf{\Phi }}_{12}^{-1}\left(1-\alpha \right)}\\ & =\{\begin{array}{ll}-(338\alpha +210)/(154\alpha -720)\hfill & \left(\alpha <0.5\right)\hfill \\ (2622\alpha -1290)/(2576\alpha -1131)\hfill & \left(\alpha \ge 0.5\right).\hfill \end{array}\end{array}$$

(25)

Then we can obtain the expected value of η_i based on Theorem 1. After normalization, we finally obtain the weight vector of the criteria layer as:

$$W_O=\left(0.3932,0.6068\right).$$

(26)

Moreover, the inverse uncertainty distribution of η₁₁, η_12, and η₁₃ according to matrix C₁ is shown respectively:

$${\mathsf{\Psi }}_1^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(1098\alpha +383)/(1658\alpha -2417)\hfill & \left(\alpha <0.5\right)\hfill \\ (4640\alpha -2652)/(4242\alpha -1909)\hfill & \left(\alpha \ge 0.5\right),\hfill \end{array}$$

(27)

$${\mathsf{\Psi }}_2^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(324\alpha +213)/(5130\alpha -3039)\hfill & \left(\alpha <0.5\right)\hfill \\ (1942\alpha -746)/(770\alpha +941)\hfill & \left(\alpha \ge 0.5\right),\hfill \end{array}$$

(28)

$${\mathsf{\Psi }}_3^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(74\alpha +132)/(6508\alpha -3316)\hfill & \left(\alpha <0.5\right)\hfill \\ -(814\alpha +24)/(608\alpha -2042)\hfill & \left(\alpha \ge 0.5\right).\hfill \end{array}$$

(29)

Then, we can obtain the weight vector of the sub-criteria layer as:

$$W_{C_1}=\left(0.3828,0.3291,0.2881\right).$$

(30)

Finally, the inverse uncertainty distribution of η₂₁, η₂₂, η_23, and η₂₄ according to matrix C₂ is shown respectively:

$${\mathsf{\Psi }}_1^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(638\alpha +411)/(9256\alpha -5417)\hfill & \left(\alpha <0.5\right)\hfill \\ (4416\alpha -2138)/(2300\alpha +1261)\hfill & \left(\alpha \ge 0.5\right),\hfill \end{array}$$

(31)

$${\mathsf{\Psi }}_2^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(482\alpha +334)/(11330\alpha -6144)\hfill & \left(\alpha <0.5\right)\hfill \\ (2498\alpha -1024)/(226\alpha +2608)\hfill & \left(\alpha \ge 0.5\right),\hfill \end{array}$$

(32)

$${\mathsf{\Psi }}_3^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(1584\alpha +807)/(5912\alpha -5483)\hfill & \left(\alpha <0.5\right)\hfill \\ (6814\alpha -4206)/(5644\alpha -2149)\hfill & \left(\alpha \ge 0.5\right),\hfill \end{array}$$

(33)

$${\mathsf{\Psi }}_4^{-1}\left(\alpha \right)=\{\begin{array}{ll}-(50\alpha +142)/(13678\alpha -6502)\hfill & \left(\alpha <0.5\right)\hfill \\ -(582\alpha +342)/(2122\alpha -4598)\hfill & \left(\alpha \ge 0.5\right).\hfill \end{array}$$

(34)

Then we can obtain the weight vector of the sub-criteria layer as:

$$W_{C_2}=\left(0.3157,0.25,0.2975,0.1368\right).$$

(35)

Step 3. According to Equations (26), (30), and (35), the weight value vectors are calculated.

$$W_A=\left(0.1505,0.1294,0.1133,0.1916,0.1517,0.1805,0.0830\right).$$

(36)

Step 4 and Step 5. For the seventh sub-criteria, we can get the score matrix S shown in

Table 10. The score matrix.

Level	Sub-criteria
	C11	C12	C13	C21	C22	C23	C24
Poor	𝒵 (0,0.2,0.4)	𝒵 (0.4,0.6,0.8)	𝒵 (0.2,0.4,0.6)	𝒵 (0,0.2,0.4)	𝒵 (0,0.2,0.4)	𝒵 (0.4,0.6,0.8)	𝒵 (0,0.2,0.4)
Average	𝒵 (0.4,0.6,0.8)	𝒵 (0.6,0.8,1)	𝒵 (0.6,0.8,1)	𝒵 (0.2,0.4,0.6)	𝒵 (0.2,0.4,0.6)	𝒵 (0.6,0.8,1)	𝒵 (0.2,0.4,0.6)
Good	𝒵 (0.6,0.8,1)	𝒵 (0.2,0.4,0.6)	𝒵 (0.4,0.6,0.8)	𝒵 (0.6,0.8,1)	𝒵 (0.6,0.8,1)	𝒵 (0.2,0.4,0.6)	𝒵 (0.6,0.8,1)
Excellent	𝒵 (0.2,0.4,0.6)	𝒵 (0,0.2,0.4)	𝒵 (0,0.2,0.4)	𝒵 (0.4,0.6,0.8)	𝒵 (0.4,0.6,0.8)	𝒵 (0,0.2,0.4)	𝒵 (0.4,0.6,0.8)

. Using the bad road weather criterion as an example, the weather conditions in the region during the statistical time were mostly cloudy and rainy; therefore, the experts concluded that the regional travel time reliability corresponds to the evaluation: average > poor > good > excellent. Then, we can get the total score vector W of the bottom layer for the total goal according to Equation (23). Based on the final score shown in

Table 11. The final total score vector.

wi	The Belief Reliability Level	Φi(x)	E(wi)
w1	Poor	𝒵 (0.108,0.56,1.52)	0.6869
w2	Average	𝒵 (0.399,1.92,3.6)	1.9598
w3	Good	𝒵 (0.453,2.32,4.16)	2.3133
w4	Excellent	𝒵 (0.201,0.8,1.92)	0.9302

, we consider the overall grade of travel time belief reliability in this study region to be: good, corresponding to a travel time belief reliability that can be quantified as 0.5 ≤ R_B ≤ 0.75.

4.4. Comparison and Discussion

For validating the results of the proposed approach, the problem is solved using the AHP, intuitionistic fuzzy analytic hierarchy process (IFAHP [17]), and neutrosophic fuzzy analytic hierarchy process (NF-AHP [46]) methods. The results are shown in

Table 12. Ranking results using different methods.

The Belief Reliability Level	Ranking Results
	AHP	IFAHP	NF-AHP	Uncertain AHP
Poor	4	4	4	4
Average	1	2	2	2
Good	2	1	1	1
Excellent	3	3	3	3

. We can see that the proposed method’s ranking result is confirmed by the latter two approaches, indicating that its result is valid. Moreover, the fact that the results of the AHP method and the uncertain AHP method differ suggests that the proposed approach is sufficiently effective in capturing the haziness and uncertainty.

The inspirations from the comparison results are summarized as follows:

• Despite using a different theory from fuzzy set theory, the method is similarly effective at representing the uncertainty contained in expert judgment;
• The concept of belief degree is simple to comprehend and allows experts to express their true attitudes and subjective preferences. However, the concepts of neutrosophic sets and intuitionistic fuzzy sets are also worth considering;
• The procedure of integrating the opinions of a group of experts has been simplified. Data gathered from multiple experts within a group could be merged and fitted together. Using the fitting method to determine the uncertain distribution of each matrix element is an effective strategy to reduce expert discrepancies and represent expert consistency, resulting in more reliable results;
• The proposed method lacks a consistency check. This is an issue that needs to be addressed further in future research.

5. Conclusions

The findings of reliability assessments could not only provide effective advice to travelers but also assist managers in keeping track of the road network’s operational conditions and where problems exist. Therefore, we developed a regional reliability evaluation approach to provide technical support to intelligent traffic systems for intelligent decision making. In this study, a novel uncertain MCDM approach was proposed to deal with this evaluation process in an uncertain environment. Considering the validity of uncertainty theory as a measure of epistemic uncertainty, we introduced it into AHP and provided the whole calculation procedure of the approach. To examine the efficiency of evaluation, we also made a comparison between the results of uncertain AHP with other MCDM methods. Uncertainty theory was proved to be particularly suited to be employed in combination with the AHP method. However, the proposed method still has limitations, and its further improvements could be summarized as follows:

• The proposed approach is significantly reliant on experts’ knowledge. As a result, in the process of uncertain decision making, we cannot rely solely on the subjective weights offered by experts. We could use objective data to improve the assessment’s accuracy;
• It is clear that in constructing the hierarchical structure, the depth of thinking in this article is not enough, and the final structure shown above is relatively rough and simple. More detailed criteria related to traffic congestion should be selected for evaluation;
• As more evaluation criteria are added, the number of pairwise comparisons grows, resulting in additional calculations. The uncertain MCDM approach should be simplified, for example, by considering the application of uncertainty theory for the step-wise weight assessment ratio analysis (SWARA) method [23];
• The effect of interrelations between criteria should be taken into consideration.

In conclusion, we have proposed a new theoretical method and proved its feasibility; however, future work will be studying further its application effects.