Multi-Attribute Decision-Making Method in Preventive Maintenance of Asphalt Pavement Based on Optimized Triangular Fuzzy Number

Abstract

By choosing the right pavement maintenance plan, we can reduce resource utilization, reduce environmental pollution, and extend road life, which is important for improving social sustainability. At present, the selection of road maintenance programs mostly adopts multiple attribute decision-making (MADA), in particular, the analytic hierarchy process (AHP) is often used. However, this method needs to use expert scoring data, which leads to strong subjectivity and poor reliability. Therefore, it reduces the science of road maintenance scheme selection. In order to reduce the subjectivity of the score and obtain a more suitable road maintenance scheme, this paper applies a multi-criteria decision-making method that characterizes attribute information by triangular fuzzy numbers (TFN) in the discrete decision space. Firstly, we invite experts to score the importance of the selection of pavement preventive maintenance technical solutions with respect to the indicators affecting the selection of solutions. Secondly, the two indicators of similarity and reliability are used to quantitatively evaluate the indicators and programs, respectively. Finally, we compare the weighted programs according to the overall possibility degree of each program. In actual cases, the overall possibility degree of each scheme obtained by this method is 1.0002–0.0477, and the optimal solution is fog sealing technology. The decision-making model applied in this paper can be considered in multiple dimensions, which can scientifically reduce the subjectivity of expert scoring. The best maintenance plan can also be quickly obtained through the simple calculation method in this paper.

1. Introduction

The pavement system plays a vital role in the national economy and people’s daily lives, but due to external forces and environmental factors, the durability of the pavement will be reduced over time, and varying degrees of damage will occur [1]. To ensure the desired performance of highways, they need regular maintenance. In addition to maintaining the sustainability of pavement work, the environmental impacts and costs incurred in maintenance work can equally affect the sustainability of society. Efficient and accurate maintenance strategies can reduce the carbon emissions of the transportation industry, while also saving maintenance costs. The traditional pavement maintenance method is mainly corrective maintenance, but this method misses the best time for maintenance, and the pavement deterioration will become more serious, leading to an increase in maintenance costs, carbon emissions, and energy consumption [2]. The main reason for this is to maintain the pavement in a cost-effective manner. Therefore, in order to maximize the benefits of maintenance with the most economical solutions, Canada was the first to propose the concept of “pavement management” in the 1960s and established the OPAC system. Nowadays, developed countries such as the United States and Europe have also formed a complete framework structure for road maintenance management decision-making systems of “road network evaluation, performance prediction, demand analysis, maintenance project ranking, and optimization decision-making”, and some developing countries in Africa, South America and Southeast Asia have also begun to establish and implement their own national conditions with the help of the World Bank, the International Road Association and other institutions. Some developing countries in Africa, South America, and South East Asia, with the help of the World Bank, the International Road Association, and other organizations, have also begun to establish and implement road maintenance management systems in line with their national conditions [3].

However, how to better balance the budget shortfall and maintenance benefits has been continuously explored, and there are a number of excellent research results, such as: a stochastic optimization model applied from two perspectives: the number of good pavement segments and the effect of global warming [4]; a multi-stage stochastic mixed integer planning model that takes into account both budget and pavement deterioration uncertainties [5]; an integer linear programming method using pavement performance as an indicator [6]; MADA using the best-worst method (BWM) and gray correlation analysis (GRA) [7]; combining dynamic programming with traditional genetic algorithms for budget allocation planning of roads [8]; designing models that use a multi-classification machine learning algorithm to predict the type of pavement treatment [9]; maintenance planning based on funding and resources, considering risks and failures [10]; a multi-stage stochastic planning approach that considers parameter uncertainty in pavement maintenance [11] etc. However, the above models consider fewer factors and are less reliable. So some scholars increase the consideration of influencing factors, not only considering the maintenance quality [12] and maintenance cost [13], but also considering resource consumption [14] and human toxicity (HT) [15], in an attempt to obtain more scientific program decision-making results from multiple dimensions.

Decision-making for pavement maintenance programs considering multiple factors is a MADA problem that can be solved by transforming the problem into a 0–1 single objective optimization problem [16], expert opinion survey method [17], theoretical approach of rough set [18,19], the best-worst method (BWM) [20], base-criterion method (BCM) [21], halo effect using the convolutional neural networks (HECON) [22], whale optimization algorithm (WOA) [23], etc., among these methods, the most applied in pavement M&R optimization is the hierarchical analysis method (AHP) [24]. AHP is a MADA analysis method combining qualitative and quantitative analysis. It has been developed for more than 30 years, and many researchers are familiar with it. This method is relatively simple and can quickly and easily calculate the best road maintenance plan. And its decision-making process is quantifiable, and the thinking is clear. It can calculate the comprehensive weight of each indicator while considering various factors [25], and it is especially helpful for the evaluation of multi-objective and multi-criteria system analysis [26].

In the field of road maintenance decision-making, some scientists combine multi-attribute utility theory (FAHP-MAUT) with fuzzy analytic hierarchy processes to prioritize maintenance activities [27]. Some scientists have also used AHP-based dual evaluation from engineering and economic aspects [28]. In order to select the best maintenance activities more scientifically, scientists have constructed a hierarchical analysis program with pavement performance as the first level [29]. These methods generally need to use more subjective scores, such as expert scoring, as a basis for judgment, but few people take into account the subjectivity of reducing the scoring of experts in the field of road maintenance decision-making. For these empirical data, how to make them more scientific and then more accurately determine which program is better has always been our goal.

In reality, many decision-making problems are fuzzy and complex. There are phenomena in which the evaluation value of the program, the attribute weight, and the expert’s cognition of the decision-making problem cannot be completely determined or are difficult to express in quantitative numerical form; therefore, the fuzzy set is proposed. With the wide application of fuzzy sets, the fuzzy fault tree analysis method combined with a fault tree is developed [30]. A new model combining three-piece convex hazard rate function (HRF) and fuzzy set [31], a real-time gain-scheduling mechanism for discrete-time Takagi–Sugeno fuzzy systems [32], a fuzzy-based organization’s maintenance support potential level assessment method [33], and so on. Among these methods, the TFN can effectively represent information that is difficult to describe with accurate values and can also be flexibly converted to other fuzzy numbers to solve related problems in many fields. Therefore, it has been widely studied by many experts and scholars. Ali [34] introduced a triangular fuzzy membership function in the predictive fuzzy system; Temelcan [35] developed an algorithm for solving the FRLP problem with fuzzy rough numbers in the form of triangular ambiguity. In solving linear programming problems, Ammar [36] suggested an algorithm for solving triangular fuzzy rough integer programming problems; Ghoushchi [37] constructed a method to solve fully fuzzy linear programming based on alpha-cut theory and modified TFN. In addition, scholars have combined fuzzy dispersion measures [38], gray degrees [39], etc. with TFN and innovated TFN. However, only focusing on one aspect that is conducive to decision-making or measuring uncertainty will ignore the impact of decision-making information reliability on decision-making. Simply solving the weight from the dimension of the program also lacks consideration of the difference in the evaluation of different programs under the condition of each attribute, which is easy to cause problems such as low discrimination of decision results, insufficient persuasion, and even inconsistency with the actual situation, which is not conducive to making the best evaluation decision for the decision-making program.

In this paper, the optimized TFN-MADA method is used to process the expert scoring data. From the two aspects of the indicator and program, the similarity difference and uncertainty between each information are obtained, respectively, and then the weight is obtained. Finally, the program is ranked according to the total possibility degree of the sum of pairwise comparisons between programs.

By applying the method presented in this paper, the subjective degree of the score data can be further reduced, the weight of the more scientific evaluation index can be obtained, the decision-making suggestions of the pavement maintenance scheme can be obtained, and then the maintenance scheme that is more suitable for road maintenance can be selected. This method presented in this paper can also be used to improve the multi-attribute decision-making methods that require expert scoring data, such as AHP, which can make these methods more scientific and accurate. The method of this paper is relatively simple, the original data is easy to obtain, and the data demand is small. It is suitable for the selection of preliminary or emergency maintenance schemes, and it is also suitable for roads with missing data.

2. Optimized TFN-MADA Method

TFN-MADA is a multi-criteria decision-making method applicable to discrete decision-making that can accurately and reasonably qualitatively analyze objective things on the basis of subjective thinking and understanding in line with human fuzzy characteristics. However, the traditional TFN-MADA method only uses indicator dimension as the basis for solving the indicator weights, ignoring the reliability of the decision-making information and the evaluation of the differences between different programs under different indicators. As a result, the selection of the program has the defects of high homogeneity, low authority, and even deviation from reality, which affect the evaluation and selection of the final program.

Based on the above, this paper applies an optimization TFN-MADA method [40], first using the principle of departure maximization to determine the indicator similarity, then using the principle of entropy weighting method to evaluate the randomness of the evaluation of each indicator for different programs, then obtaining the reliability-based attribute weights of homogeneous difference indicators and random indicators, assigning the weights to the evaluation indicators, and finally combining with the theory of TFN likelihood degree comparison relationship. Finally, the relationship matrix of two-by-two comparisons between technical programs is obtained, and the programs are ranked and selected from the best to the worst according to their overall possibility degree. The specific ideas are shown in Figure 1.

2.1. Triangular Fuzzy Numbers

Definition 1. 

A TFN $\tilde{z}=\left[z^L,z^M,z^U\right]$ is a fuzzy set defined on the set R of real numbers, whose membership function is defined as follows: where $z^L,z^M and z^U$ are called the lower bound, the mode, and the upper bound of the TFN $\tilde{z}=\left[z^L,z^M,z^U\right]$, respectively, and $z^L\le z^M\le z^U$. If $z^L\ge 0$, then the TFN $\tilde{z}=\left[z^L,z^M,z^U\right]$ is called a positive TFN. If $z^U\le 0$, then the TFN $\tilde{z}=\left[z^L,z^M,z^U\right]$ is called a negative TFN. Equation (1) is its membership function [41]:

$${\mathsf{\phi }}_{\tilde{\mathrm{z}}}(\mathrm{a})=\left\{\begin{array}{c}\frac{\mathrm{a}-{\mathrm{z}}^{\mathrm{L}}}{{\mathrm{z}}^{\mathrm{M}}-{\mathrm{z}}^{\mathrm{L}}},\hspace{1em}{\mathrm{z}}^{\mathrm{L}}\le a\le {\mathrm{z}}^{\mathrm{M}}\\ \frac{{\mathrm{z}}^{\mathrm{U}}-\mathrm{a}}{{\mathrm{z}}^{\mathrm{U}}-{\mathrm{z}}^{\mathrm{M}}},\hspace{1em}{\mathrm{z}}^{\mathrm{M}}\le a\le {\mathrm{z}}^{\mathrm{U}}\\ 0\hspace{1em} ,\hspace{1em}\hspace{1em}\mathrm{others}\end{array}\right.$$

(1)

Because the triangular fuzzy number in this article is the score of the indicator, so $z^L\ge 0$.

Because this article applies triangular fuzzy numbers to expert scoring, all the values involved are positive, so there are simplified operation rules [42] as shown in Equations (2)–(4):

$${\tilde{\mathrm{z}}}_2=\left[{\mathrm{z}}_1^{\mathrm{L}}+{\mathrm{z}}_2^{\mathrm{L}},{\mathrm{z}}_1^{\mathrm{M}}+{\mathrm{z}}_1^{\mathrm{M}},{\mathrm{z}}_1^{\mathrm{U}}+{\mathrm{z}}_1^{\mathrm{U}}\right]$$

(2)

$${\tilde{\mathrm{z}}}_1{\tilde{\mathrm{z}}}_2=\left[{\mathrm{z}}_1^{\mathrm{L}}{\mathrm{z}}_2^{\mathrm{L}}, {\mathrm{z}}_1^{\mathrm{M}}{\mathrm{z}}_1^{\mathrm{M}},{\mathrm{z}}_1^{\mathrm{U}}{\mathrm{z}}_1^{\mathrm{U}}\right]$$

(3)

$$\mathsf{\alpha }\tilde{\mathrm{z}}=\left[\mathsf{\alpha }{\mathrm{z}}^{\mathrm{L}}, \mathsf{\alpha }{\mathrm{z}}^{\mathrm{M}},\mathsf{\alpha }{\mathrm{z}}^{\mathrm{U}}\right]$$

(4)

2.2. Basic Model

Set $\mathrm{X}=\left\{{\mathrm{X}}_1,{\mathrm{X}}_2,\cdots ,{\mathrm{X}}_{\mathrm{i}}\right\}$ is the program set, and $\mathrm{Y}=\left\{{\mathrm{Y}}_1,{\mathrm{Y}}_2,\cdots ,{\mathrm{Y}}_{\mathrm{j}}\right\}$ is the indicator set. According to the TFN $\left({\tilde{\mathrm{Z}}}_{\mathrm{i}1},{\tilde{\mathrm{Z}}}_{\mathrm{i}2},\cdots ,{\tilde{\mathrm{Z}}}_{\mathrm{i}\mathrm{j}}\right)$ column vector form, the evaluation of the program set is made based on the indicator set. Since different evaluation indicators are mostly of different magnitudes and units of magnitude, the fuzzy numbers are normalized in order to eliminate the effect of magnitude between indicators, as shown in Equations (5) and (6):

$$\mathrm{When} \mathrm{it} \mathrm{is} \mathrm{a} \mathrm{benefit-based} \mathrm{indicator}, \left\{\begin{array}{c}{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}=\frac{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}}{\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{U}}}\\ {\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}=\frac{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}}{\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}}\\ {\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{N}}=\frac{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{U}}}{\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}}\end{array}\right.$$

(5)

$$\mathrm{When} \mathrm{it} \mathrm{is} \mathrm{a} \mathrm{cost-based} \mathrm{indicator}, \left\{\begin{array}{c}{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}=\frac{\frac{1}{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{U}}}}{\sum _{\mathrm{i}=1}^{\mathrm{m}}\frac{1}{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}}}\\ {\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}=\frac{\frac{1}{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}}}{\sum _{\mathrm{i}=1}^{\mathrm{m}}\frac{1}{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}}}\\ {\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{N}}=\frac{\frac{1}{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}}}{\sum _{\mathrm{i}=1}^{\mathrm{m}}\frac{1}{{\mathrm{Z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{U}}}}\end{array}\right.$$

(6)

Letting ${\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}}=\left[{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}},{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}},{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{U}}\right],0<{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}\le {\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{M}}\le {\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{N}}<1$, and according to Equation (7), the normalized decision matrix of triangular fuzzy number multiple attribute decision-making is obtained.

$$\begin{gathered} \mathrm{Z}={\left[{\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}}\right]}_{\mathrm{m}\times \mathrm{n}}=\left[\begin{array}{ccc}\begin{array}{cc}{\tilde{\mathrm{z}}}_{11}& {\tilde{\mathrm{z}}}_{12}\\ {\tilde{\mathrm{z}}}_{13}& {\tilde{\mathrm{z}}}_{14}\end{array}& \cdots & \begin{array}{c}{\tilde{\mathrm{z}}}_{1\mathrm{n}}\\ {\tilde{\mathrm{z}}}_{2\mathrm{n}}\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}{\tilde{\mathrm{z}}}_{\mathrm{m}1}& {\tilde{\mathrm{z}}}_{\mathrm{m}2}\end{array}& \cdots & {\tilde{\mathrm{z}}}_{\mathrm{m}\mathrm{n}}\end{array}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left[\begin{array}{ccc}\begin{array}{cc}\left[{\mathrm{z}}_{11}^{\mathrm{L}},{\mathrm{z}}_{11}^{\mathrm{M}},{\mathrm{z}}_{11}^{\mathrm{N}}\right]& \left[{\mathrm{z}}_{12}^{\mathrm{L}},{\mathrm{z}}_{12}^{\mathrm{M}},{\mathrm{z}}_{12}^{\mathrm{N}}\right]\\ \left[{\mathrm{z}}_{21}^{\mathrm{L}},{\mathrm{z}}_{21}^{\mathrm{M}},{\mathrm{z}}_{21}^{\mathrm{N}}\right]& \left[{\mathrm{z}}_{22}^{\mathrm{L}},{\mathrm{z}}_{22}^{\mathrm{M}},{\mathrm{z}}_{22}^{\mathrm{N}}\right]\end{array}& \cdots & \begin{array}{c}\left[{\mathrm{z}}_{1\mathrm{n}}^{\mathrm{L}},{\mathrm{z}}_{1\mathrm{n}}^{\mathrm{M}},{\mathrm{z}}_{1\mathrm{n}}^{\mathrm{N}}\right]\\ \left[{\mathrm{z}}_{2\mathrm{n}}^{\mathrm{L}},{\mathrm{z}}_{2\mathrm{n}}^{\mathrm{M}},{\mathrm{z}}_{2\mathrm{n}}^{\mathrm{N}}\right]\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}\left[{\mathrm{z}}_{\mathrm{m}1}^{\mathrm{L}},{\mathrm{z}}_{\mathrm{m}1}^{\mathrm{M}},{\mathrm{z}}_{\mathrm{m}1}^{\mathrm{N}}\right]& \left[{\mathrm{z}}_{\mathrm{m}2}^{\mathrm{L}},{\mathrm{z}}_{\mathrm{m}2}^{\mathrm{M}},{\mathrm{z}}_{\mathrm{m}2}^{\mathrm{N}}\right]\end{array}& \cdots & \left[{\mathrm{z}}_{\mathrm{m}\mathrm{n}}^{\mathrm{L}},{\mathrm{z}}_{\mathrm{m}\mathrm{n}}^{\mathrm{M}},{\mathrm{z}}_{\mathrm{m}\mathrm{n}}^{\mathrm{N}}\right]\end{array}\right] \end{gathered}$$

(7)

2.3. Similarity

In TFN-MADA problems, there is often a tendency to give higher attribute weights to indicators with higher reliability. Aiming at the shortcomings of the traditional TFN-MADA method, the decision-making method is optimized and improved by comprehensively considering the similarity between the decision indicators and the reliability of the decision-making program.

For any two canonical TFN ${\tilde{\mathrm{z}}}_1=\left[{\mathrm{z}}^{\mathrm{L}},{\mathrm{z}}^{\mathrm{M}},{\mathrm{z}}^{\mathrm{U}}\right]$ and ${\tilde{\mathrm{z}}}_2=\left[{\mathrm{z}}^{\mathrm{L}},{\mathrm{z}}^{\mathrm{M}},{\mathrm{z}}^{\mathrm{U}}\right]$, there is a similarity calculation in Equation (8) [43]:

$$\mathrm{c}\left({\tilde{\mathrm{z}}}_1,{\tilde{\mathrm{z}}}_2\right)=1-\frac{\left|{\mathrm{z}}_1^{\mathrm{L}}-{\mathrm{z}}_2^{\mathrm{L}}\right|+\left|{\mathrm{z}}_1^{\mathrm{M}}-{\mathrm{z}}_2^{\mathrm{M}}\right|+\left|{\mathrm{z}}_1^{\mathrm{U}}-{\mathrm{z}}_2^{\mathrm{U}}\right|}{3}$$

(8)

It can be obtained from the similarity of program X_i to the j th indicator Y_j with other programs calculated by Equation (9):

$${\mathrm{c}}_{\mathrm{i}}\left({\mathrm{Y}}_{\mathrm{j}}\right)={\sum }_{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)\hspace{1em}\hspace{1em}\left(1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n},1\le \mathrm{k}\le \mathrm{m}\right)$$

(9)

So for the j th metric Y_j, the total similarity between all schemes and other schemes can be calculated by Equation (10):

$$\mathrm{c}\left({\mathrm{Y}}_{\mathrm{j}}\right)={\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\sum }_{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)\hspace{1em}\hspace{1em}\left(1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n},1\le \mathrm{k}\le \mathrm{m}\right)$$

(10)

If $0\le {\mathrm{C}}_{\mathrm{j}}\le$1, and $\sum _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{j}}^2=1$, from the deviation maximization decision method, it can be seen that: the greater the total similarity under a certain indicator, the smaller the impact of this indicator on the choice of the program, C_j is also smaller; similarly, the smaller the total similarity under a certain indicator, the greater the impact of this indicator on the choice of the program, C_j is also bigger, and more consideration should be given to this indicator in decision-making. To sum up, the total similarity of all indicators to all programs should be maximized under the effect of C_j, in other words, the obtained C_j must make the reciprocal of the total similarity of all indicators to all programs maximum under the action of C_j.

Constructing a similarity deviation maximization model as shown in Equation (11):

$$\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{G} ({\mathrm{C}}_{\mathrm{j}})={\sum }_{\mathrm{j}=1}^{\mathrm{n}}\frac{{\mathrm{C}}_{\mathrm{j}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\sum }_{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)}$$

(11)

Constructing the Lagrangian function as shown in Equation (12):

$$\mathrm{L}\left({\mathrm{C}}_{\mathrm{j}}, \mathsf{\lambda }\right)={\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\sum }_{\mathrm{k}=1}^{\mathrm{m}}\frac{{\mathrm{C}}_{\mathrm{j}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\sum }_{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)}+\mathsf{\lambda }\left({\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{j}}^2-1\right)$$

(12)

Defining Equation (13):

$$\left\{\begin{array}{c}\frac{\partial \mathrm{L}}{\partial {\mathrm{C}}_{\mathrm{j}}}=\sum _{\mathrm{i}=1}^{\mathrm{m}}\sum _{\mathrm{k}=1}^{\mathrm{m}}\frac{1}{\sum _{\mathrm{i}=1}^{\mathrm{m}}\sum _{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)}+2\mathsf{\lambda }{\mathrm{C}}_{\mathrm{j}}=0\\ \frac{\partial \mathrm{L}}{\partial \mathsf{\lambda }}=\sum _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{j}}^2-1=0\end{array}\right.$$

(13)

The solution is Equation (14):

$${\mathrm{C}}_{\mathrm{j}}^{\mathrm{\prime }}=\frac{1/\sum _{\mathrm{i}=1}^{\mathrm{m}}\sum _{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)}{\sqrt{\sum _{\mathrm{j}=1}^{\mathrm{n}}\left[1/{\left(\sum _{\mathrm{i}=1}^{\mathrm{m}}\sum _{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)\right)}^2\right]}}$$

(14)

Making ${\mathrm{C}}_{\mathrm{j}}=\frac{{\mathrm{C}}_{\mathrm{j}}^{\mathrm{\prime }}}{\sum _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{j}}^{\mathrm{\prime }}}$, which will be ${\mathrm{C}}_{\mathrm{j}}^{\mathrm{\prime }}$ normalized, the similarity of the indicators calculation Equation (15) can be obtained:

$${\mathrm{C}}_{\mathrm{j}}=\frac{1/\sum _{\mathrm{i}=1}^{\mathrm{m}}\sum _{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)}{\sum _{\mathrm{j}=1}^{\mathrm{n}}\left(1/\sum _{\mathrm{i}=1}^{\mathrm{m}}\sum _{\mathrm{k}=1}^{\mathrm{m}}\mathrm{c}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}},{\tilde{\mathrm{z}}}_{\mathrm{k}\mathrm{j}}\right)\right)}\hspace{1em}\hspace{1em}\left(1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n},1\le \mathrm{k}\le \mathrm{m}\right)$$

(15)

The higher the sum of the similarity of a decision program to other decision programs under a given indicator, the lower the influence of the indicator on the decision of the program, and the smaller the similarity C_j; conversely, the larger the similarity C_j. Therefore, the total similarity of all indicators to all decision options should be minimized under the effect of C_j.

2.4. Reliability

Fuzzy entropy can be used to quantify the amount of information in a fuzzy set and can therefore be used to measure the reliability of a program. The smaller the entropy value of the program, the more information it contains, the higher the reliability, and the more it can support the decision-maker. The traditional method of calculating TFN fuzzy entropy usually uses the average area method, but this will lead to the loss of fuzzy information.

Therefore, in this paper, the entropy function calculation equation with unknown variable b is defined as Equation (16):

$$\mathrm{e}\left(\mathrm{b}\right)=4\mathrm{b}(1-\mathrm{b})$$

(16)

The entropy function for the TFN $\tilde{z}=\left[z^L,z^M,z^U\right]$ is calculated by Equation (17):

$$\mathrm{e}\left({\mathsf{\phi }}_{\tilde{\mathrm{z}}}\left(\mathrm{a}\right)\right)=\left\{\begin{array}{c}4\left(\frac{\mathrm{a}-{\mathrm{z}}^{\mathrm{L}}}{{\mathrm{z}}^{\mathrm{M}}-{\mathrm{z}}^{\mathrm{L}}}\right)\left(1-\frac{\mathrm{a}-{\mathrm{z}}^{\mathrm{L}}}{{\mathrm{z}}^{\mathrm{M}}-{\mathrm{z}}^{\mathrm{L}}}\right),\hspace{1em}{\mathrm{z}}^{\mathrm{L}}\le a\le {\mathrm{z}}^{\mathrm{M}}\\ 4\left(\frac{{\mathrm{z}}^{\mathrm{U}}-\mathrm{a}}{{\mathrm{z}}^{\mathrm{U}}-{\mathrm{z}}^{\mathrm{M}}}\right)\left(1-\frac{{\mathrm{z}}^{\mathrm{U}}-\mathrm{a}}{{\mathrm{z}}^{\mathrm{U}}-{\mathrm{z}}^{\mathrm{M}}}\right),\hspace{1em}{\mathrm{z}}^{\mathrm{M}}\le a\le {\mathrm{z}}^{\mathrm{U}}\\ \hspace{1em} 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\hspace{1em}\mathrm{others}\end{array}\right.$$

(17)

The entropy of the $\tilde{\mathrm{z}}$ is:

$$\begin{array}{cc}E\left(\tilde{z}\right)\hfill & =\int e\left({\phi }_{\tilde{z}}\left(a\right)\right)s\left(a\right)da\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(s\left(a\right)=C\right)\hfill \\ & ={\int }_{z^L}^{z^M}\frac{4\left[-a^2+\left(z^L+z^M\right)a-z^L{z}^M\right]}{{\left(z^M-z^L\right)}^2}s\left(a\right)da+{\int }_{z^L}^{z^M}\frac{4\left[-a^2+\left(z^L+z^M\right)a-z^L{z}^M\right]}{{\left(z^M-z^L\right)}^2}s\left(a\right)da\hfill \\ & =\frac{2c\left(z_{ij}^U-z_{ij}^L\right)}{3}\hfill \end{array}$$

Therefore, in the normalization matrix, the equation for calculating the entropy of indicator Y_j in program X_i is Equation (18):

$${\mathrm{E}}_{\mathrm{i}\mathrm{j}}\left({\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}}\right)=\frac{2\mathrm{c}\left({\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{U}}-{\mathrm{z}}_{\mathrm{i}\mathrm{j}}^{\mathrm{L}}\right)}{3},1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n}$$

(18)

The calculation method for the sum of the entropies of all metrics for the i th program X_i is Equation (19):

$${\mathrm{E}}_{\mathrm{i}}={\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{E}}_{\mathrm{i}\mathrm{j}}\left({\tilde{\mathrm{r}}}_{\mathrm{i}\mathrm{j}}\right)$$

(19)

The calculation equation for the reliability R_ij of indicator Y_j in program X_i is Equation (20):

$${\mathrm{R}}_{\mathrm{i}\mathrm{j}}=\frac{{\mathrm{E}}_{\mathrm{i}\mathrm{j}}\left({\tilde{\mathrm{r}}}_{\mathrm{i}\mathrm{j}}\right)}{{\mathrm{E}}_{\mathrm{i}}},1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n}$$

(20)

The smaller R_ij is, the lower the uncertainty of indicator C_j in program X_i, and the higher the reliability; conversely, the lower the reliability.

Since C_ij and R_ij are based on the program and indicators for consideration of reliability and confidence, respectively, the two can be superimposed on each other to respond more comprehensively to the impact of indicators on the program selection of the weight. Therefore, the weight can be calculated according to Equation (21):

$${\mathsf{\omega }}_{\mathrm{i}\mathrm{j}}={\mathrm{C}}_{\mathrm{j}}\left(1-{\mathrm{R}}_{\mathrm{i}\mathrm{j}}\right),1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n}$$

(21)

2.5. Decision Sequencing

Multiplying the normalized initial scoring decision matrix by the weights ${\mathsf{\omega }}_{\mathrm{i}\mathrm{j}}$, the weighted comprehensive value of each program X_i is calculated using Equation (22):

$${\tilde{\mathrm{b}}}_{\mathrm{i}}={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{{\mathsf{\omega }}_{\mathrm{i}\mathrm{j}}·\tilde{\mathrm{r}}}_{\mathrm{i}\mathrm{j}}$$

(22)

The equation for calculating the possibility degree of TFN is defined as Equation (23) [44]:

$$\begin{gathered} \mathrm{p}\left({\tilde{\mathrm{z}}}_1\ge {\tilde{\mathrm{z}}}_2\right)=\mathsf{\rho }\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{z}}_1^{\mathrm{M}}-{\mathrm{z}}_2^{\mathrm{L}},0\right)-\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{z}}_1^{\mathrm{L}}-{\mathrm{z}}_2^{\mathrm{M}},0\right)}{{\mathrm{l}}_{{\tilde{\mathrm{z}}}_1}^{\left(2\right)}+{\mathrm{l}}_{{\tilde{\mathrm{z}}}_2}^{\left(2\right)}}+(1-\mathsf{\rho })\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{z}}_1^{\mathrm{U}}-{\mathrm{z}}_2^{\mathrm{M}},0\right)-\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{z}}_1^{\mathrm{M}}-{\mathrm{z}}_2^{\mathrm{U}},0\right)}{{\mathrm{l}}_{{\tilde{\mathrm{z}}}_1}^{\left(1\right)}+{\mathrm{l}}_{{\tilde{\mathrm{z}}}_2}^{\left(1\right)}} \\ {\mathrm{l}}_{{\tilde{\mathrm{z}}}_1}^{\left(1\right)}={\mathrm{z}}_1^{\mathrm{U}}-{\mathrm{z}}_1^{\mathrm{M}},{\mathrm{l}}_{{\tilde{\mathrm{z}}}_2}^{\left(1\right)}={\mathrm{z}}_2^{\mathrm{U}}-{\mathrm{z}}_2^{\mathrm{M}},{\mathrm{l}}_{{\tilde{\mathrm{z}}}_1}^{\left(2\right)}={\mathrm{z}}_1^{\mathrm{M}}-{\mathrm{z}}_1^{\mathrm{L}},{\mathrm{l}}_{{\tilde{\mathrm{z}}}_2}^{\left(2\right)}={\mathrm{z}}_2^{\mathrm{M}}-{\mathrm{z}}_2^{\mathrm{L}} \end{gathered}$$

(23)

where $\mathsf{\rho }$ is the decision maker’s preference for risk, when $\mathsf{\rho }>0.5$ is positive utility, the decision maker is risk-loving; when $\mathsf{\rho }=0.5$ is risk-neutral, the decision maker is neither risk-loving nor risk-averse; when $\mathsf{\rho }<0.5$ is negative utility, the decision maker is risk-averse.

By comparing the weighted comprehensive values of each program, the comparative possibility degree of program X_i is better than that of scheme X_j, the equation is Equation (24).

$$\mathrm{p}\left({\mathrm{X}}_{\mathrm{i}}\succ {\mathrm{X}}_{\mathrm{k}}\right)=\mathrm{p}\left({\tilde{\mathrm{b}}}_{\mathrm{i}}\ge {\tilde{\mathrm{b}}}_{\mathrm{k}}\right),\hspace{1em}1\le \mathrm{i}\le \mathrm{m},\hspace{1em}1\le \mathrm{k}\le \mathrm{m}$$

(24)

Thus, the overall comparative possibility degree of scheme X_i is calculated by Equation (25):

$$\mathsf{\psi }\left({\mathrm{X}}_{\mathrm{i}}^{\succ }\right)=\frac{1}{\mathrm{m}-1}{\sum }_{\mathrm{k}\ne \mathrm{i}}^{\mathrm{m}}\mathrm{p}\left({\mathrm{X}}_{\mathrm{i}}\succ {\mathrm{X}}_{\mathrm{k}}\right),1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{k}\le \mathrm{m}$$

(25)

According to the overall comparative possibility degree $\mathsf{\psi }\left({\mathrm{X}}_{\mathrm{i}}^{\succ }\right)$, the programs are sorted to select the optimal scheme. The optimization idea of TFN is shown in Figure 2.

3. Evaluation Indicators

Evaluation indicators for road maintenance programs can be categorized by their nature into two main types: benefit-based indicators and cost-based indicators. The larger the benefit-based indicators, the better the program; the smaller the cost-based indicators, the better the program.

3.1. Benefit-Based Indicators

3.1.1. Material Strength

Different maintenance programs require different raw materials for maintenance, such as: asphalt, mixes of asphalt and other materials, recycling material, etc. Different materials have different strengths and different levels of road performance enhancement.

3.1.2. Pavement Performance

Pavement performance conditions have a direct impact on traffic and user costs [45]. According to the 2011 Pavement Disease Identification Manual issued by the Minnesota Department of Transportation’s Office of Materials and Road Research, this state evaluates the overall performance of pavements using the Pavement Quality Index (PQI_Min) and calculation methods as shown in Equation (26), which is a combination of the Pavement Ride Quality Index (RQI) and the Pavement Rating Index (SR) [46].

$${\mathrm{P}\mathrm{Q}\mathrm{I}}_{\mathrm{M}\mathrm{i}\mathrm{n}}=\sqrt{\mathrm{R}\mathrm{Q}\mathrm{I}\times \mathrm{S}\mathrm{R}}$$

(26)

3.1.3. Pavement Life

Pavement durability is a measure of the ability of a pavement to consistently maintain good serviceability over time, and is usually expressed in terms of pavement service life, which is the time elapsed for pavement performance to exceed a minimum threshold after a rehabilitation treatment [47].

3.1.4. Comfort

Vehicle vibration is an important factor that affects comfort and poses a threat to health due to prolonged vibration of the body during long distance travel. In this paper, the comfort level is mainly considered as the Road Travel Quality Index (RQI) [48], and its calculation methods are shown in Equation (27):

$${\mathrm{R}\mathrm{Q}\mathrm{I}}_{\mathrm{M}\mathrm{i}\mathrm{n}}=5.6972-2.104\sqrt{\mathrm{I}\mathrm{R}\mathrm{I}}$$

(27)

The RQI score in the U.S. specification is divided into 5 levels from 0 to 5 in order of poor, worse, medium, good, and excellent, and the higher the score, the better the pavement smoothness performance.

3.1.5. Pavement Skid Resistance

During the service life of a road, the skid resistance of asphalt pavement gradually deteriorates due to the combined effects of vehicle loads, rainfall, temperature, and other factors, and its performance decreases more slowly in the early stages and rapidly in the later stages [49], which has a great impact on road safety. Referring to the Technical Specification for Highway Asphalt Pavement Maintenance (JTG 5142-2019) [50], this paper mainly considers the pavement Skid Resistance Index (SRI).

3.1.6. Pavement Aesthetics

Pavement cracks, rutting, surface gumming, misalignment, etc. will affect the aesthetics of the pavement, which is not conducive to improving the overall image of the city.

3.2. Cost-Based Indicators

3.2.1. Engineering Costs

Project costs include new construction costs (e.g., the original gravel road surface is changed to asphalt), reconstruction costs (major and medium repair costs), daily maintenance costs, residual value of the road surface, and vehicle operating costs [46].

3.2.2. Traffic Disruption

Traffic disruptions need to be considered in the context of the duration of the disruption, the magnitude of the impact, and the number of people affected [51]. Includes additional vehicle operating costs, travel time conversion costs, and traffic accident costs due to traffic disruptions caused by road maintenance [52].

3.2.3. Carbon Emissions

China is on track to achieve “peak carbon” and “carbon neutrality” by 2030 and 2060, respectively, which has prompted environmental managers and policymakers to focus on the carbon emissions of products and services in general. A significant portion of greenhouse gas emissions come from our extensive paved road infrastructure system [53]. The calculation method for carbon emissions in this paper is shown in Equation (28):

$$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n} \mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n} \mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\times \mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l} \mathrm{o}\mathrm{f} \mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}$$

(28)

Among them, the carbon emission factors include greenhouse gases generated by energy consumption, material production processes, and the work of machinery or equipment. Activity levels include energy consumption, the amount of consumables, and the workload of machinery or equipment [54].

3.2.4. Noise Pollution

Studies have found that noise pollution not only causes great harm to the body, but also has a strong hidden nature, and people who are subjected to noise hazards for a long time are physically and psychologically harmed [55]. Nowadays, noise pollution has become one of the four major environmental problems worldwide. The European Environment Agency (EEA) recently released the first EU noise assessment report, saying that more than 125 million Europeans are exposed to traffic noise pollution that exceeds the legal guideline level, resulting in up to 10,000 premature deaths per year.

4. Road Maintenance Case

4.1. Overview of the Project

In this study, the road section in Jiangsu Province, which was built in 2010, was selected as the research object. This road surface is in good technical condition and has not been overhauled. The total length of the two-way road from the starting point to the end point is 96 km. The road section is tested in 2021, and relevant data are in

Table 1. Indicators of pavement condition.

Norm	Mileage/km			Score (of Student’s Work)	Make One’s Judgment
	Excellent	Very Much	Total
Pavement Damage Condition Index PCI	90	6	96	97.89	first class
Road Travel Quality Index RQI	96	0	96	94.34	excellent
Rutting Depth Index RDI	88	8	96	92.42	excellent
Slip Resistance Index (SRI)	32	64	96	89.19	excellent

:

The overall pavement technical condition index PQI of the target road section was 95.34, with a high level of road condition, with pavement breakage, smoothness, and rutting maintained at an excellent level and skid resistance maintained at a good level. Among them, cracking totaled 3.31 m², strip repair 6125.4 m, and block repair 67.95 m².

4.2. Case Calculations

• Step 1: Initial data

At present, a variety of more mature preventive maintenance techniques have been developed, the main ones being: grouting or sealing, micro-surfacing, thin slurry sealing, gravel sealing, composite sealing, fog sealing, thin-layer overlay, and in situ thermal regeneration. The program was generated setting $\mathrm{X}=\left\{{\mathrm{X}}_1,{\mathrm{X}}_2,\cdots ,{\mathrm{X}}_8\right\}$ in turn.

It was found that the factors affecting the selection of pavement maintenance programs can be categorized into benefit and cost indicators, as shown in

Table 2. Programmatic choice impact indicators.

	Indicator
Benefit-based indicators	Material strength Y1
	Pavement Performance Y2
	Pavement life Y3
	Comfort Y4
	Anti-skid road surface Y5
	Pavement aesthetics Y6
Cost-based indicators	Project cost Y7
	Traffic Disruption Y8
	Carbon emissions Y9
	Noise pollution Y10

. The indicator was generated setting $\mathrm{Y}=\left\{{\mathrm{Y}}_1,{\mathrm{Y}}_2,\cdots ,{\mathrm{Y}}_{10}\right\}$ in turn.

For this research, an online and paper version of the questionnaire was developed to collect the perceptions of different people on the importance of each indicator for each program choice, with the importance scoring scores presented in the form of TFN. The questionnaire is shown in Appendix A. The score should be between 0 and 1, where 0 means that the indicator is completely unimportant for the selection of the program, and 1 means that the indicator completely determines the selection of the program. The questionnaire was distributed to a total of 10 experts from different professional positions, including 30% of relevant researchers, 30% of highway maintenance engineers, 20% of professors from universities, and 20% of road designers. Only background information was provided to them, and expert opinions were consulted anonymously. Each expert scored each program based on his or her own opinions and experience and tried to maintain their independence. The final effective recovery rate of the questionnaire was 80%. The upper limit, lower limit, and median of each score of the questionnaire were averaged to obtain the initial decision matrix.

• Step 2: Normalization Processing

According to the scores of each expert in the returned questionnaire, the initial TFN decision matrix of benefit-type indicators and cost-type indicators is obtained after summarizing and analyzing. According to Equations (5) and (6), each attribute indicator is normalized, and the normalized TFN decision matrix D₁ and D₂ of benefit-type indicators and cost-type indicators are obtained, as shown in

Table 3. Benefit-based indicator normalization matrix D₁ (×10⁻¹).

D1	Y1	Y2	Y3	Y4	Y5	Y6
X1	[1.199, 1.276, 1.338]	[1.241, 1.330, 1.422]	[1.497, 1.567, 1.636]	[1.447, 1.548, 1.642]	[0.556, 0.529, 0.731]	[1.029, 1.109, 1.166]
X2	[1.243, 1.306, 1.400]	[1.064, 1.131, 1.216]	[1.241, 1.338, 1.400]	[1.184, 1.339, 1.496]	[1.349, 1.454, 1.644]	[1.260, 1.347, 1.445]
X3	[1.272, 1.350, 1.446]	[1.182, 1.269, 1.343]	[1.327, 1.426, 1.527]	[1.283, 1.409, 1.551]	[0.913, 1.101,1.324]	[1.214, 1.300, 1.379]
X4	[1.228, 1.306, 1.385]	[1.108, 1.193, 1.295]	[1.344, 1.444, 1.564]	[1.316, 1.426, 1.551]	[0.913, 1.101, 1.233]	[1.244, 1.331, 1.445]
X5	[1.127, 1.231, 1.308]	[1.167, 1.239, 1.311]	[0.629, 0.687, 0.782]	[0.625, 0.713, 0.858]	[1.230, 1.451, 1.644]	[1.275, 1.347, 1.429]
X6	[1.199, 1.276, 1.338]	[1.226, 1.315, 1.406]	[1.548, 1.620, 1.691]	[1.431, 1.600, 1.770]	[0.397, 0.529, 0.685]	[0.814, 0.887, 0.969]
X7	[1.040, 1.098, 1.185]	[1.152, 1.223, 1.311]	[0.578, 0.634, 0.709]	[0.543, 0.661, 0.785]	[2.421, 2.731, 2.922]	[1.321, 1.395, 1.494]
X8	[1.084, 1.157, 1.246]	[1.211, 1.300, 1.390]	[1.190, 1.285, 1.382]	[1.184, 1.304, 1.442]	[0.913, 1.101, 1.324]	[1.198, 1.284, 1.363]

and

Table 4. Cost-based indicator normalization matrix D₂ (×10⁻¹).

D2	Y7	Y8	Y9	Y10
X1	[1.575, 1.873, 2.269]	[1.285, 1.432, 1.585]	[1.027, 1.088, 1.180]	[1.675, 1.824, 1.994]
X2	[1.002, 1.150, 1.378]	[1.304, 1.411, 1.500]	[1.228, 1.297, 1.414]	[1.100, 1.164, 1.241]
X3	[1.002, 1.150, 1.285]	[1.087, 1.174, 1.255]	[1.135, 1.257, 1.372]	[1.129, 1.183, 1.241]
X4	[0.973, 1.150, 1.285]	[1.126, 1.195, 1.277]	[1.089, 1.201, 1.331]	[1.072, 1.144, 1.200]
X5	[0.705, 0.808, 0.919]	[1.126, 1.195, 1.309]	[1.290, 1.442, 1.548]	[1.139, 1.213, 1.292]
X6	[1.575, 1.873, 2.269]	[1.186, 1.256, 1.351]	[1.174, 1.241, 1.305]	[1.139, 1.203, 1.272]
X7	[0.516, 0.573, 0.652]	[1.067, 1.143, 1.213]	[1.228, 1.322, 1.439]	[1.053, 1.105, 1.180]
X8	[1.224, 1.424, 1.609]	[1.107, 1.195, 1.277]	[1.058, 1.152, 1.247]	[1.100, 1.164, 1.211]

.

• Step 3: Numerical calculation

The similarity of indicators C_j from Equations (8)–(15) is calculated:

$$\begin{gathered} {\mathrm{C}}_1=0.0984,{\mathrm{C}}_2=0.0978,{\mathrm{C}}_3=0.1009,{\mathrm{C}}_4=0.1009,{\mathrm{C}}_5=0.1040, \\ {\mathrm{C}}_6=0.0991,{\mathrm{C}}_7=0.1022,{\mathrm{C}}_8=0.0984,{\mathrm{C}}_9=0.0984,{\mathrm{C}}_{10}=0.0997. \end{gathered}$$

The reliability ${\mathrm{R}}_{\mathrm{i}\mathrm{j}}\left(1\le \mathrm{i}\le 8,1\le \mathrm{j}\le 10\right)$ of the indicator $\mathrm{Y}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,\cdots ,{\mathrm{y}}_{10}\right\}$ set in the program $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_8\right\}$ is calculated by Equations (18)–(20). We can see the result as shown in

Table 5. Reliability R_ij (×10⁻¹).

Rij	Y1	Y2	Y3	Y4	Y5	Y6	Y7	Y8	Y9	Y10
X1	0.573	0.746	0.573	0.801	0.721	0.561	2.853	1.232	0.628	1.312
X2	0.729	0.701	0.736	1.444	1.368	0.854	1.743	0.910	0.861	0.653
X3	0.798	0.736	0.915	1.232	1.886	0.757	1.301	0.771	1.087	0.516
X4	0.731	0.870	1.024	1.093	1.483	0.933	1.448	0.703	1.121	0.592
X5	0.869	0.690	0.733	1.114	1.983	0.740	1.027	0.876	1.236	0.733
X6	0.589	0.760	0.602	1.432	1.217	0.653	2.934	0.697	0.551	0.564
X7	0.738	0.807	0.662	1.225	2.542	0.875	0.693	0.738	1.073	0.647
X8	0.729	0.804	0.864	1.161	1.850	0.743	1.735	0.763	0.851	0.500

.

• Step 4: Weighted post-processing

The reliability-based weights ${\mathsf{\omega }}_{\mathrm{i}\mathrm{j}}\left(1\le \mathrm{i}\le 8,1\le \mathrm{j}\le 10\right)$ of each indicator are derived according to Equation (21) and then calculated as the weighted composite value ${\tilde{\mathrm{b}}}_{\mathrm{i}}\left(1\le \mathrm{i}\le 8\right)$ of each program $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_8\right\}$ according to Equation (22). The results are as follows:

$$\begin{gathered} {\tilde{\mathrm{b}}}_1=\left[0.1116,0.1205,0.1324\right] \\ {\tilde{\mathrm{b}}}_2=\left[0.1078,0.1164,0.1270\right] \\ {\tilde{\mathrm{b}}}_3=\left[0.1040,0.1136,0.1234\right] \\ {\tilde{\mathrm{b}}}_4=\left[0.1028,0.1124,0.1220\right] \\ {\tilde{\mathrm{b}}}_5=\left[0.0924,0.1014,0.1110\right] \\ {\tilde{\mathrm{b}}}_6=\left[0.1043,0.1138,0.1245\right] \\ {\tilde{\mathrm{b}}}_7=\left[0.0960,0.1044,0.1133\right] \\ {\tilde{\mathrm{b}}}_8=\left[0.1014,0.1111,0.1211\right] \end{gathered}$$

• Step 5: Relationship Matrix

According to (23) to (24), a pairwise comparison of the measured values of the possibility degree of the weighted comprehensive value of the program is carried out. This case stipulates $\mathsf{\rho }=0.5$. The comparison possibility degree relation matrix ${\mathrm{P}}_{8\times 8}=\mathrm{p}{\left({\mathrm{X}}_{\mathrm{i}}\succ {\mathrm{X}}_{\mathrm{k}}\right)}_{8\times 8}$ is constructed as follows:

$${\mathrm{P}}_{8\times 8}=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}0.5000& 0.7184\\ 0.2816& 0.5000\end{array}& \begin{array}{ccc}0.8792& 0.9435& 1.0000\\ 0.6691& 0.7351& 1.0000\end{array}\\ \begin{array}{cc}0.1209& 0.3309\\ 0.0565& 0.2650\end{array}& \begin{array}{ccc}0.5000& 0.5648& 1.0000\\ 0.4353& 0.5000& 1.0000\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}0.8517& 1.0000\\ 0.6441& 1.0000\end{array}& \begin{array}{c}0.9863\\ 0.7958\end{array}\\ \begin{array}{cc}0.4776& 0.9889\\ 0.4140& 0.9313\end{array}& \begin{array}{c}0.6267\\ 0.5630\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{cc}0.0000& 0.0000\\ 0.1483& 0.3559\end{array}& \begin{array}{ccc}0.0000& 0.0000& 0.5000\\ 0.5224& 0.5860& 1.0000\end{array}\\ \begin{array}{cc}0.0000& 0.0000\\ 0.0137& 0.2043\end{array}& \begin{array}{ccc}0.0111& 0.0688& 0.6665\\ 0.3734& 0.4370& 1.0000\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0.0000\\ 3.2448\\ 0.0028\end{array}& \begin{array}{cc}0.3336& 0.0000\\ 0.9972& 0.6466\\ 0.5000& 0.1411\end{array}\\ 0.3535& \begin{array}{cc}0.8590& 0.5000\end{array}\end{array}\end{array}\right]$$

The overall comparative possibility degree of each scenario is calculated according to Equation (25). The calculation results are shown in Figure 3:

$$\begin{gathered} \mathsf{\psi }\left({\mathrm{X}}_1^{\succ }\right)=0.9113 \\ \mathsf{\psi }\left({\mathrm{X}}_2^{\succ }\right)=0.7322 \\ \mathsf{\psi }\left({\mathrm{X}}_3^{\succ }\right)=0.5871 \\ \mathsf{\psi }\left({\mathrm{X}}_4^{\succ }\right)=0.5236 \\ \mathsf{\psi }\left({\mathrm{X}}_5^{\succ }\right)=0.0477 \\ \mathsf{\psi }\left({\mathrm{X}}_6^{\succ }\right)=1.0002 \\ \mathsf{\psi }\left({\mathrm{X}}_7^{\succ }\right)=0.1272 \\ \mathsf{\psi }\left({\mathrm{X}}_8^{\succ }\right)=0.4630 \end{gathered}$$

• Step 6: Output of the optimal solution

Sorting $\mathsf{\psi }\left({\mathrm{X}}_{\mathrm{i}}^{\succ }\right)\left(1<\mathrm{i}<8\right)$, the programs can be obtained from the best to the worst as ${{\mathrm{X}}_6>\mathrm{X}}_1>{\mathrm{X}}_2>{\mathrm{X}}_3>{\mathrm{X}}_4>{\mathrm{X}}_8>{\mathrm{X}}_7>{\mathrm{X}}_5$. So, the optimal solution is the fog sealing technology X₆.

4.3. Case Validation

From the reference [24], the service life and unit cost of each preventive maintenance program are shown in

Table 6. Preventive maintenance program life and unit cost table.

Xi	X1	X2	X3	X4	X5	X6	X7	X8
Unit cost (yuan)	10	21.5	17.5	16.5	55	7.5	57.5	24
Service life (year)	1.5	2.5	2.5	2	3.5	1.5	5	2

as follows:

Based on the service life and unit cost of each program, considering the time value of funds, the ‘Equivalent Annual Cost Method’ [56] is used to evaluate each program as Equation (29).

$$\mathrm{E}\mathrm{A}\mathrm{C}\left({\mathrm{X}}_{\mathrm{i}}\right)={\mathrm{C}}_{\mathrm{i}}\frac{{\mathrm{i}}_{{\mathrm{C}}_{\mathrm{i}}}{\left(1+{\mathrm{i}}_{{\mathrm{C}}_{\mathrm{i}}}\right)}^{{\mathrm{n}}_{\mathrm{i}}}}{{\left(1+{\mathrm{i}}_{{\mathrm{C}}_{\mathrm{i}}}\right)}^{{\mathrm{n}}_{\mathrm{i}}}-1}$$

(29)

Among them, the discount rate i is 8% according to Methods and Parameters for Economic Evaluation of Construction Projects (Third Edition); C_i is the unit cost (yuan) of the conservation program X_i; and n is the service life (year) of the conservation program. The calculation results for each program are in

Table 7. The calculation results of the two models.

Xi	X1	X2	X3	X4	X5	X6	X7	X8
Equivalent annual cost method	7.3366	9.8280	7.9995	9.2515	18.6299	5.5025	14.3964	13.4567
Overall comparative possibility degree	0.9113	0.7322	0.5871	0.5236	0.0477	1.0002	0.1272	0.4630

as follows:

Since the overall comparative possibility degree is a benefit-based indicator, i.e., the higher the program, the better, and the equivalent annual cost is a cost-based indicator, i.e., the lower the program, the better, as can be seen from Figure 4, in the calculation results of the two methods, the fog sealing layer maintenance program is the optimal choice, and the composite sealing layer and the thin layer overlay are both inferior choices. However, the results of the ordering of the program’s superiority and inferiority are not completely consistent; the ordering in this paper is X₂ > X₃ > X₄, while in [56] we got the order of X₃ > X₄ > X₂. The reason for this difference is mainly because the reference [56] only comes from the pavement life and engineering cost considerations, while this article’s considerations are more factors. Relevant studies show that the micro-surface treatment technology originated from the thin slurry sealing technology and, at the same time, is superior to the thin slurry sealing technology, which has the advantages of being economical and quick and is favorable for paving [57]. The thin-slurry sealing layer is less capable of coping with reflection cracks [58]. The maintenance time after gravel sealing is longer [59]. The maintenance time after gravel sealing is longer. Because this case is located in Jiangsu Province, given the rainfall and precipitation concentration, the road base are easy to erode by water. In terms of waterproofing performance, the thin slurry sealing layer seepage coefficient is basically less than 10 mL/min, with better waterproofing and water sealing effects than the gravel sealing layer [60,61]. In summary, this paper concludes that the road maintenance system is better than gravel sealer. All in all, the ranking of the road maintenance program obtained in this article is more scientific and more applicable to the case in the paper. Therefore, the method of optimization TFN-MADA in this article has certain scientific and practical feasibility in the application of the decision-making direction of road maintenance programs.

5. Conclusions

(1)

This paper applies the optimization TFN-MADA method, based on expert scoring data, to analyze the indicators and maintenance programs affecting the preventive maintenance of pavements, applies the principles of deviation maximization and entropy weighting to consider and analyze the indicators and the program perspectives, respectively, and scientifically selects the preferred maintenance program.

(2)

The validity of the model was verified by the case study, and the preferred order of maintenance programs for the target road section was obtained: fog sealing technology > joint sealing technology > micro-surfacing technology > slurry sealing technology > crushed stone sealing technology > hot in-place recycling technology > thin layer covering technology > composite seal overlay techniques.

(3)

Because the fog sealing layer technology has convenient construction, fast traffic, good economic benefits [62], more friendly to the environment [63] etc., it is more reasonable to obtain a higher score in this case; because the composite sealing layer and sealing cover technology adopt two technologies, their economic cost is significantly higher than other preventive maintenance technologies [64], so their score is lower in this case.

(4)

The decision-making method of small data pavement maintenance applied in this paper can quickly and scientifically select the preliminary maintenance scheme at the decision-making site, and can also select the scheme without the influence of data loss.

(5)

The small-data pavement maintenance decision-making methodology applied in this research allows for rapid, scientific, and reliable selection of preliminary maintenance programs at the decision-making site.

(6)

Because the decision-making method in this study is to deal with the expert score, and then sort the scheme, the score of the expert score has a great influence on the decision-making result. However, the number of expert score samples in this case is limited. Therefore, in the future, a database of scores can be constructed, and algorithms such as deep learning can be used to further optimize the initial triangular fuzzy number decision matrix, making the selection of preventive maintenance schemes more accurate.

(7)

Due to the limited data in this case, which is a road in Jiangsu, China, the extreme weather, slope, natural disasters, and other indicators are not considered. In the future, when using this method, it is necessary to improve the index set according to the local situation. This paper mainly discusses the road maintenance schemes commonly used in Jiangsu and does not study maintenance schemes such as ultra-thin wear layers. In the future, more practical cases need to be accumulated, and more maintenance schemes need to be considered.