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Abstract

Intersection traffic congestion evaluation is essential for effective intelligent transportation system planning, and an objective and precise assessment of traffic congestion is vital to ensure the smooth circulation of traffic. Multiple criteria decision-making is a method for evaluating the degree of traffic congestion. A hybrid multiple criteria decision-making method integrating the fuzzy analytic hierarchy process, techniques for order preference by similarity to an ideal solution, and gray correlation techniques are presented in this work. The proposed method applied fuzzy analytic hierarchy process to determine the weight of the evaluation index; subsequently, gray correlation techniques for order preference by similarity to an ideal solution were integrated to construct the hybrid decision-making method. A case study of traffic congestion at intersections with five evaluation indexes verified the effectiveness of the hybrid method. The evaluation results of the different methods show that the proposed method overcomes the one-sidedness of analytical hierarchy process–techniques for order preference by similarity to an ideal solution and analytical hierarchy process–gray correlation. Thus, the proposed hybrid decision-making model provides a more accurate and reliable method for evaluating the degree of traffic congestion.

Keywords

Fuzzy analytic hierarchy process gray correlation techniques for order preference by similarity to an ideal solution multi-objective decision-making intelligent transportation system congestion evaluation decision analysis

Introduction

Because of the rapid development of the economy and the acceleration of urbanization, motor vehicle ownership and road traffic volume have dramatically increased. The demands from traffic on urban road infrastructure are becoming heavier because traffic congestion incidents more frequently occur.^1,2 Traffic congestion at the intersection has seriously affected people’s daily travel activities and bring about environmental pollution and waste of resources.³ On the contrary, convenient and smooth traffic at the intersection not only can greatly improve people’s work efficiency but also can improve the image of the city. It can be seen that research about traffic congestion at the intersection is necessary and significant. Traffic congestion at the intersection is complicated and changeable with many evaluation criterions. In other words, it is impossible to accurately assess the traffic congestion at the intersection depending on one criteria. Thus, multiple criteria decision-making (MCDM) method was introduced to study the problem of traffic congestion.^4–6 Based on this, a scientific system was established to evaluate intersection congestion.

MCDM has been applied to a variety of traffic congestion evaluations.⁷ Analytical hierarchy process (AHP) is a simple, flexible, and practical MCDM, which breaks down complex problems into several progressive levels and determines weight by comparing indices.⁸ Techniques for order preference by similarity to an ideal solution (TOPSIS) is a classical method to MCDM problems; evaluation schemes are sorted according to the distance relationship among data sequences. However, combining TOPSIS and AHP offers more favorable results; in this combination, TOPSIS is used for design selection, and AHP is applied to calculate weights for the concerned criteria.^9,10

However, AHP, TOPSIS, and gray relational analysis (GRA) all have their own respective defects. For example, although TOPSIS sorts evaluation schemes according to the distance to the ideal solution and negative ideal solution, the distance between evaluation schemes and the ideal or negative ideal solution may be equal. Furthermore, TOPSIS cannot reflect variation tendencies in the data sequence. Gray correlation (GC)^11,12 takes the similarity of curves as a “yardstick” to reflect the change trends in the data sequence. GRA applies especially to poor information, but defects are evident in the overall assessment of evaluation schemes. In other words, the GC degree between evaluation alternatives and the ideal one may be equal. Therefore, neither TOPSIS nor GC is suitable for evaluating intersection congestion.

To more accurately and objectively evaluate traffic congestion, we proposed a new method that integrates fuzzy analytic hierarchy process (FAHP), TOPSIS, and GC to evaluate traffic congestion. Specifically, the FAHP is used to determine the preference weights of the evaluation index, and GRA subsequently is introduced to improve TOPSIS and combine Euclidean distance with the GC degree for the assessment of intersection traffic congestion. Overall, this hybrid method provides more reliable evaluation results. In this study, a novel method for assessing traffic congestion was determined by considering the weights of evaluation criteria, the location relationship among data sequences, and their situation changes. This article provides traffic management departments with constructive suggestions related to intersection congestion. Compared with previous studies, this article offers the following two noteworthy contributions:

A hybrid MCDM approach integrating FAHP, GC, and TOPSIS, called FGT, is proposed to evaluate intersection traffic congestion, making full use of quantitative analysis and the weight allocation features of FAHP and the selection abilities of GC and TOPSIS.

By comparing the proposed approach with the FAHP-TOPSIS and FAHP-GC methods, this research validates its effectiveness and feasibility in evaluating intersection traffic congestion.

The remainder of this article is organized as follows. Section “Literature review” describes the existing evaluation methods involving FAHP, TOPSIS, and GC. In section “Existing evaluation methods,” the novel evaluation method that combines FAHP, TOPSIS, and GRA is outlined. In section “The FGT method,” the method is applied to the evaluation of traffic congestion, and the effectiveness of this hybrid method is proven. Finally, the conclusions are presented in section “Assessment example.”

Literature review

MCDM provides a good ideal for multi-standard and complex traffic problems. Traffic system is a very complex task involving multiple criteria related to economic, environmental, and socio-political issues. MCDM techniques actually assist decision makers to choose the most reasonable program by assessing such problems. As far as we know, many methods exist for evaluating traffic problems: for example, the AHP,^13–16 the FAHP,^17–19 TOPSIS,²⁰ AHP-TOPSIS,²¹ AHP-evidential reasoning,^22,23 GRA, Decision-Making and Evaluation Laboratory (DEMATEL), and the AHP-TOPSIS-GC.^24,25

At present, many scholars have done an in-depth study of traffic problems, and most of researches focus on MCDM techniques. For instance, Sadi-Nezhad and Damghani²⁶ adopt fuzzy TOPSIS method to assess the traffic police centers’ performance. Mohajeri and Amin²⁷ use AHP to select the railway station site. De Luca²⁸ uses AHP to handle the public engagement in strategic transportation planning. Awasthi et al.²⁹ adopt fuzzy TOPSIS to handle the Metro transportation problem. Arslan³⁰ uses a fuzzy AHP to evaluate transportation projects. Abbas³¹ adopts MCDM techniques to assess such problems, most of which were related to transportation science. More detailed works refer to related reviews.³² Moreover, MCDM approach has been applied to other traffic-related areas, such as airline industry, electrified vehicles, and logistic center, which is shown in Table 1.

Table 1.

Application of MCDM approach in traffic and related areas.

Method	Area	Method	Area
AHP	Sustainable transport strategies,^33,34 railways,³⁵ transport system,³⁶ airport³⁷	TOPSIS	Public transit service³⁸
Fuzzy AHP	Public transport service,³⁹ airline industry,⁴⁰ aircraft⁴¹	Fuzzy TOPSIS	Airline industry⁴²
AHP and SMARTER	Transport infrastructure projects⁴³	TOPSIS and COPRAS, SAW	Gas terminal location⁴⁴
Fuzzy AHP, Choquet integral and trapezoidal fuzzy sets	Rail transit⁴⁵	Fuzzy TOPSIS and Fuzzy AHP	3PL transportation,⁴⁶ seaport⁴⁷
MCA	Spatial infrastructure projects⁴⁸	VIKOR and interval type-2 fuzzy sets	Rail transit⁴⁹
Fuzzy ELECTRE III	Airline industry⁵⁰	Fuzzy VIKOR	Transportation firm⁵¹

MCDM: multiple criteria decision-making; AHP: analytical hierarchy process; TOPSIS: techniques for order preference by similarity to an ideal solution; SMARTER: simple multi-attribute rating technique exploiting ranks; MCA: multi-criteria analysis; COPRAS: complex proportional assessment; SAW: simple additive weighting; VIKOR: vlsekriter-ijumska optimizacija i kompro resenje in serbian, meaning multi-criteria optimization and compromise solution; ELECTRE III: Elimination and choice translating reality III.

The review of the literature indicates that a large number of evaluation methods are applied to traffic system. But, very few people focus on the assessment of traffic congestion at the intersection. In order to fill the gap of the traffic system, in this article, the hybrid MCDM method integrating fuzzy AHP, TOPSIS, and GC is put forward for the assessment of traffic congestion at the intersection. FAHP is used to determine the weight vector of traffic index structure, which is established based on traffic system characteristics, that is, delay indicator, queuing indicator, saturation index, number of stops, and occupancy index. GC-TOPSIS is applied to obtain the final ranking of traffic congestion at the intersection.

Existing evaluation methods

The weight of each evaluation index in evaluating schemes is not the same. We introduced FAHP to determine the weights of indices more objectively and accurately in the TOPSIS and GC evaluation method.

The evaluation scheme was ordered using TOPSIS and GC, according to the Euclidean distance and the similarity between data sequences, respectively. Their basic evaluation steps are described as follows.

FAHP method

AHP was introduced by Saaty in the early 1970s;⁵² however, its method for determining the weight is defective and complicated. Therefore, a MCDM method, FAHP^18,19 was developed. At present, FAHP is applied to the evaluation of design schemes⁵³ and involves the following steps:

Step 1. Construct the fuzzy judgment matrix

A certain element of the above grade is used as the evaluation criteria to compare two metrics at the corresponding level and determine the matrix element to construct a fuzzy judgment matrix

$A = {(a_{i j})}_{n \times n} = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}]$ (1)

where $n$ represents the number of elements associated with the upper criterion. Notably, $a_{ij} = (l, m, n)$ represents the triangular fuzzy number, with $n$ representing the lower limit of the triangular fuzzy number, $l$ representing the upper limit of the triangular fuzzy number, and $m$ representing the median point of the triangular fuzzy number; moreover, both $l$ and $n$ represent a range of values of the relative importance of the two metrics. Selection of the relative importance on the basis of the fuzzy scale is shown in Table 2:

Table 2.

Fuzzy scale values and their meanings.

Numerical scale	Definition	Explanation
1	Equal importance	Two factors have the same importance
3	Slightly more important	Compared the two elements, the former is more important than the latter slightly
5	Obviously important	Compared the two elements, the former is more important than the latter obviously
7	Very strong important	An activity is strongly favored, and its dominance is demonstrated in practice
9	Absolute importance	One indicator have absolute advantage compared with another
2, 4, 6 and 8	The middle of the adjacent judgment value	Importance degree between the above-mentioned numerical
Reciprocals (1/a_ij)	The degree of importance that activity i is compared to activity j becomes the reciprocal when j is compared to i

To ensure the accuracy and reliability of the final fuzzy matrix, the geometric mean technique was used to determine the final fuzzy matrix using formula (2)

$\begin{matrix} a_{ij} & = (a_{ij}^{1} \otimes a_{ij}^{2} \otimes \dots \otimes a_{ij}^{n}) \\ = [{(l_{ij}^{1} \times l_{ij}^{2} \times \dots \times l_{ij}^{n})}^{\frac{1}{k}}, {(m_{ij}^{1} \times m_{ij}^{2} \times \dots \times m_{ij}^{n})}^{\frac{1}{k}}, \\ {(n_{ij}^{1} \times n_{ij}^{2} \times \dots \times n_{ij}^{n})}^{\frac{1}{k}}] \end{matrix}$ (2)

where $a_{ij}$ is the element of the final fuzzy matrix; $k$ is the number of fuzzy matrices determined by experts; and $l$ , $m$ , and $n$ are the upper limit, median point, and lower limit of the triangular fuzzy number, respectively. The other elements of the final fuzzy matrix were then calculated using the same calculation procedure.

Step 2. Construct the fuzzy weights

To calculate the fuzzy weights of each indicator, the calculation process can be described as follows

$b_{i} = {(a_{i 1} \otimes \dots \otimes a_{ij} \otimes \dots \otimes a_{in})}^{\frac{1}{n}}$ (3)

Next, to obtain the fuzzy weight of each indicator, the following procedure is used

$W_{i} = {(b_{1} \otimes \dots \otimes b_{i} \otimes \dots \otimes b_{n})}^{- 1}$ (4)

where $W_{i}$ is the fuzzy weight of each index.

Step 3. Construct the standard weights

Standard weights denoted as BNP_wi must be calculated to weigh each index; here, we used the following method

$BN P_{W_{i}} = \frac{[({U_{W}}_{i} - {L_{W}}_{i}) + ({M_{W}}_{i} - {L_{W}}_{i})]}{3} + {L_{W}}_{i}$ (5)

TOPSIS method

TOPSIS, proposed by Hwang and Yoon in 1981, is an effective method for solving MCDM problems.⁵⁴ A finite number of evaluation schemes can be sorted according to their distance to the ideal solution using TOPSIS, the results of which present the evaluative quality in the existing schemes. The ideal solution is the optimal solution that can be envisaged, wherein each attribute achieves the optimal value—in other words, the ideal solution maximizes the benefit or minimizes the cost. By contrast, the solution that maximizes the cost or minimizes the benefit is called the negative ideal solution.⁵⁵ The calculation steps of TOPSIS are as follows:

Step 1. Construct the initial decision matrix

The structure of the initial decision matrix can be described using

$X = \begin{matrix} \begin{matrix} E_{1} & E_{2} & \dots & E_{n} \end{matrix} \\ \begin{matrix} D_{1} \\ D_{2} \\ ⋮ \\ D_{m} \end{matrix} & [\begin{matrix} X_{11} & X_{12} & \dots & X_{1 n} \\ X_{21} & X_{22} & \dots & X_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ X_{m 1} & X_{m 2} & \dots & X_{mn} \end{matrix}] \end{matrix}$ (6)

where ${D_{i}}$ are the design alternatives, $i \in {1, 2, \dots, m}$ and ${E_{j}}$ are the attributes of evaluation objects, $j \in {1, 2, \dots, n}$ .

Step 2. Establish a standardized matrix by normalizing the initial decision matrix

The structure of the standardized matrix can then be described using

$Y = {[y_{ij}]}_{m \times n}$ (7)

For the index where “the greater it is, the more desirable it is,” $y_{ij}$ of the standardized matrix is calculated as

$y_{ij} = \frac{X_{ij}}{max_{i} X_{ij}}, i \in {1, 2, \dots, m}, j \in {1, 2, \dots, n}$ (8)

For the index where “the smaller it is, the more desirable it is,” $y_{ij}$ of the standardized matrix is calculated as

$y_{ij} = \frac{min_{j} X_{ij}}{X_{ij}}, i \in {1, 2, \dots, m}, j \in {1, 2, \dots, n}$ (9)

Step 3. Calculate the standardized matrix Q combined with the weight of each index determined using FAHP

Specifically

$Q = ω^{T} Y = [\begin{matrix} ω'_{1} y_{11} & ω'_{2} y_{12} & \dots & ω'_{m} y_{1 m} \\ ω'_{1} y_{21} & ω'_{2} y_{22} & \dots & ω'_{m} y_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ω'_{1} y_{n 1} & ω'_{2} y_{n 2} & \dots & ω'_{m} y_{nm} \end{matrix}]$ (10)

Step 4. Select the optimal and least optimal values for each attribute to determine the ideal and negative ideal solutions

The ideal solution and the negative ideal solution are as follows

$Q_{j}^{+} = {max_{1 \leq i \leq n} ({{Q_{i}}_{j}}_{i = 1}^{n}) | j \in J^{+}, min_{1 \leq i \leq n} ({{Q_{i}}_{j}}_{i = 1}^{n}) | j \in J^{-}}$ (11)

$Q_{j}^{-} = {min_{1 \leq i \leq n} ({{Q_{i}}_{j}}_{i = 1}^{n}) | j \in J^{+}, max_{1 \leq i \leq n} ({{Q_{i}}_{j}}_{i = 1}^{n}) | j \in J^{-}}$ (12)

where $J^{+}$ denotes the positive indicators for “the greater it is, the more desirable it is,” and $J^{-}$ denotes the negative indicators for “the smaller it is, the more desirable it is.”

Step 5. Calculate the Euclidean distance for each scheme to the ideal and negative ideal solutions

The calculation methods are as follows

$S_{i}^{+} = \sqrt{{\sum_{j = 1}^{n} [Q_{ij} - Q_{j}^{+}]}^{2}}, (j = 1, 2, \dots, m)$ (13)

$S_{i}^{-} = \sqrt{{\sum_{j = 1}^{n} [Q_{ij} - Q_{j}^{-}]}^{2}}, (j = 1, 2, \dots, m)$ (14)

Step 6. Calculate the relative similarity degree, $F_{i}$ , of each scheme, $D_{i}$ , based on the Euclidean distance

The application of the formula is as follows

$F_{i} = \frac{S_{i}^{-}}{S_{i}^{+} + S_{i}^{-}}, (i = 1, 2, \dots, m)$ (15)

where $F_{i}$ is a value less than 1. Notably, the more favorable the value is, the greater the design scheme is.

Thus, the TOPSIS analysis model evaluates the benefits and drawbacks of each solution on the basis of the Euclidean distance between each scheme and the ideal scheme. However, the Euclidean distance of two schemes may be identical; in this situation, we cannot determine the degree of superiority of these two schemes. Moreover, the distance to both the ideal and negative ideal solutions may be equal. This disadvantage of TOPSIS is outlined in Figure 1.

Figure 1.

Case of equal Euclidean distance during TOPSIS analysis.

Points $D$ and $E$ represent two decision-making schemes $A$ represents the ideal solution, $B$ represents the negative ideal solution, $H$ and $F$ represent two random decision-making schemes, and $C$ represents the center of the circle. If the straight line $AB$ is perpendicular to $DE$ , the distance of decision-making schemes $D$ and $E$ to the ideal solution and the negative solution $A$ and $B$ is equal, and TOPSIS cannot evaluate the advantages and disadvantages of the two schemes. Thus, in these instances, TOPSIS is a defective method of analysis.

GC method

GC analysis is a key part of the GC system, and it also is an effective method for the mining of data internal rule. GC analysis is based on analyzing the geometry of data sequences and the related degrees of curve geometry.^56–58 Each scheme can be sorted according to the degree of similarity of the data sequence: the closer the curve shape, the bigger the GC. We can then interpret the principle of GC analysis intuitively.

First, the following four factors are assumed to be present in a data sequence

$\begin{matrix} a^{(0)} = (a_{1}^{(0)}, a_{2}^{(0)}, \dots, a_{n}^{(0)}) \\ a^{(1)} = (a_{1}^{(1)}, a_{2}^{(1)}, \dots, a_{n}^{(1)}) \\ a^{(2)} = (a_{1}^{(2)}, a_{2}^{(2)}, \dots, a_{n}^{(2)}) \\ a^{(3)} = (a_{1}^{(3)}, a_{2}^{(3)}, \dots, a_{n}^{(3)}) \end{matrix}$

where $a^{(0)}$ is the main factor or the reference sequence, and $a^{(1)}$ , $a^{(2)}$ , $a^{(3)}$ are the comparative factors or sequences (Figure 2).

Figure 2.

Comparison of data sequence curves.

Curve (0) is closest to curve (1); that is, they possess the most similar geometric shapes. We considered $a^{(0)}$ and $a^{(1)}$ to have the maximum GC degree under the definition of GC; because the difference in the geometrical shape of curves (0) and (2) is greater than that of curve (1), the GC degree of $a^{(0)}$ and $a^{(2)}$ is smaller than $a^{(1)}$ . By contrast, because the difference in the geometrical shape of curves (0) and (3) is the biggest, the GC degree of $a^{(0)}$ and $a^{(3)}$ is smallest. Overall, the closer the geometric shape is to the curve, the bigger the GC degree is; moreover, the decision scheme is closer to the ideal scheme. GC involves the following steps:

Step 1. Calculate the GC coefficient between the ith scheme and the ideal solution for the jth index according to the weighted normalized matrix of TOPSIS and the ideal solution

The equation is

$r_{ij}^{+} = \frac{min_{i} min_{j} | Q_{j}^{+} - Q_{ij} | + ρ max_{i} max_{j} | Q_{j}^{+} - Q_{ij} |}{| Q_{j}^{+} - Q_{ij} | + ρ max_{i} max_{j} | Q_{j}^{+} - Q_{ij} |}$ (16)

where $ρ$ is the distinguishing coefficient and $ρ \in [0, 1]$ . We selected $ρ = 0.5$ for this study.

Next, the GC coefficient matrix between each scheme and the ideal solution is found

$R^{+} = [\begin{matrix} r_{11}^{+} & r_{12}^{+} & \dots & r_{1 m}^{+} \\ r_{21}^{+} & r_{22}^{+} & \dots & r_{2 m}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ r_{n 1}^{+} & r_{n 2}^{+} & \dots & r_{nm}^{+} \end{matrix}]$ (17)

Finally, the GC degree $R_{i}^{+}$ , between each scheme and the ideal solution on the basis of the GC coefficient matrix, is calculated

$R_{i}^{+} = \frac{1}{m} \sum_{j = 1}^{m} r_{ij}^{+}, (i \in {1, 2, \dots, m})$ (18)

Step 2. Calculate the GC coefficient between the ith scheme and the negative ideal solution for the jth index on the basis of the weighted normalized matrix

$r_{ij}^{-} = \frac{min_{i} min_{j} | Q_{j}^{-} - Q_{ij} | + ρ max_{i} max_{j} | Q_{j}^{-} - Q_{ij} |}{| Q_{j}^{-} - Q_{ij} | + ρ max_{i} max_{j} | Q_{j}^{-} - Q_{ij} |}$ (19)

Next, the GC coefficient matrix is calculated using formula (20)

$R^{-} = [\begin{matrix} r_{11}^{-} & r_{12}^{-} & \dots & r_{1 m}^{-} \\ r_{21}^{-} & r_{22}^{-} & \dots & r_{2 m}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ r_{n 1}^{-} & r_{n 2}^{-} & \dots & r_{nm}^{-} \end{matrix}]$ (20)

Finally, the GC degree $R_{i}^{-}$ , between each scheme and the negative ideal solution on the basis of the GC coefficient matrix, is calculated

$R_{i}^{-} = \frac{1}{m} \sum_{j = 1}^{m} r_{ij}^{-}, (i \in {1, 2, \dots, m})$ (21)

Step 3. Calculate the GC relative closeness, $C_{i}$ , of each scheme on the basis of Steps 1 and 2. Then, sort each scheme according to the GC relative closeness

$C_{i} = \frac{R_{i}^{-}}{R_{i}^{+} + R_{i}^{-}}, (i = 1, 2, \dots, m)$ (22)

The FGT method

To compensate for the drawbacks of the TOPSIS method, and to more objectively and accurately evaluate the benefits and drawbacks of each scheme, this article proposes a novel evaluation method called the FGT method, which combines the FAHP, GC, and TOPSIS. The FGT method fully employs fuzzy theory to determine the weight of each evaluation index, as well as the GC and TOPSIS methods to sort the evaluation schemes. The following steps outline the FGT process:

Step 1. Calculate the weight of each evaluation index in the evaluation scheme using FAHP;

Step 2. Construct the initial decision matrix;

Step 3. Establish standardized matrix $Y = [y_{ij}]$ by normalizing the initial decision matrix;

Step 4. Calculate standardized matrix $Q$ combined with the weight of each index determined through FAHP;

Step 5. Determine the ideal and negative ideal solutions according to the value of each evaluation index in each evaluation scheme;

Step 6. Calculate the Euclidean distance, $S_{i}^{+}$ , between the ideal solution and each evaluation scheme; simultaneously, calculate the Euclidean distance, $S_{i}^{-}$ , between the negative ideal solution and each evaluation scheme;

Step 7. On the basis of the standardized matrix (Step 4) and the ideal solution (Step 5), calculate the GC degree $R_{i}^{+}$ using the GC method. Then, calculate the GC degree $R_{i}^{-}$ using the GC method on the basis of the standardized matrix (Step 4) and the negative ideal solution (Step 5);

Step 8. Apply the dimensionless method to normalize $S_{i}^{+}$ , $S_{i}^{-}$ , $R_{i}^{+}$ , $R_{i}^{-}$

$G_{i}^{'} = \frac{G_{i}}{max_{1 \leq i \leq m} G_{i}}$ (23)

where $G_{i}$ denotes $S_{i}^{+}$ , $S_{i}^{-}$ , $R_{i}^{+}$ , $R_{i}^{-}$ .

Step 9. Improve the reliability of the evaluation results by integrating the GC degrees $R_{i}^{+}$ , $R_{i}^{-}$ and Euclidean distances $S_{i}^{+}$ , $S_{i}^{-}$ through the simple weighted method. The greater the $S_{i}^{-}$ and $R_{i}^{+}$ are, the more effective is the design scheme. The calculation method is described as follows

$U_{i}^{+} = h \times S_{i}^{-} + (1 - h) \times R_{i}^{+}, (i \in {1, 2, \dots, m})$ (24)

$U_{i}^{-} = h \times S_{i}^{+} + (1 - h) \times R_{i}^{-}, (i \in {1, 2, \dots, m})$ (25)

where $h$ reflects the preference of a decision maker regarding location relations.

Step 10. Calculate the comprehensive closeness index for each evaluation scheme to determine the benefits and drawbacks of each assessment program

$L_{i} = \frac{U_{i}^{+}}{U_{i}^{-} + U_{i}^{+}}, (i = {1, 2, \dots, m})$ (26)

where $L_{i}$ is the basis for ranking the benefits and drawbacks of each evaluation scheme in the proposed hybrid evaluation method. Notably, the $L_{i}$ value must be between 0 and 1, and the greater that $L_{i}$ is, the more effective is the evaluation scheme.

The calculation process based on the description of the hybrid evaluation method is detailed in Figure 3.

Figure 3.

Flowchart of the hybrid assessment method.

Assessment example

In this section, our hybrid evaluation method is applied to evaluate traffic congestion, and its effectiveness is reviewed.

Determining the evaluation index for degree of traffic congestion

To quantitatively consider the impact of traffic flow on the overall operation of the intersection and ensure that the evaluation results are scientific, comprehensive, and objective, the establishment of evaluation indicators should be scientific, objective, and comprehensive, as well as fully reflect the degree of traffic congestion. Depending on the situation at the intersection, this method uses five criteria as the evaluation index.

Delay indicator

Intersection delay is calculated by dividing the total additional travel time of the driver, passenger, or pedestrian by the effect of the control measures and of other road users by the flow through the corresponding cross section of the road.⁵⁹ It is a key performance index of intermittent traffic flow. This indicator determines the time loss caused by vehicles or pedestrians passing through intersections. The average delay is the most commonly used evaluation indicator of signalized intersections; overall, a greater delay value is indicative of a more crowded intersection.

Queuing indicator

Queue length is a useful index for evaluating the rationality of the design length of intersection entrances and the crowded conditions of intersections. Average queue length refers to the average length of the waiting queue of vehicles at each entranceway prior to the intersection^60,61 and can directly reflect crowding at the intersection. Under normal circumstances, a longer average queue length is indicative of a greater amount of traffic.

Saturation index

Saturation is the ratio of actual traffic volume of a lane group at an intersection to its capacity.⁶² Notably, total saturation refers to the saturation value reached by the phase with the highest degree of saturation, rather than the sum of phase saturation. Overall, a greater total saturation value is indicative of a greater amount of traffic.

Number of stops (parking rate)

Parking rate refers to the ratio of the delay time of deceleration–acceleration time. The average number of stops at the intersection (parking rate) can directly reflect the travel state of the vehicles at an intersection. Overall, a higher parking rate is indicative of a greater amount of traffic.

Occupancy index

Finally, time-share occupancy refers to the ratio of time that a car passes or stays in an entire observation period.⁶³ Time-share occupancy can be used to reflect the traffic flow of the corresponding imported lane.

Using these five indicators, evaluation indexes of traffic congestion can be clearly displayed (Figure 4).

Figure 4.

Evaluation index of traffic congestion at intersections.

Determining the weights of five evaluation indexes of traffic congestion degree using the FAHP method

The five traffic congestion indicators were calculated using the FAHP method according to the following steps:

Step 1. Nine experts assessed the relative importance of the five indicators of traffic congestion. On the basis of the assessment results and the definitions listed in Table 1, we established a fuzzy judgment matrix for each expert concerning the evaluation index system for the degree of traffic congestion. We then calculated the comprehensive fuzzy judgment matrix according to Formula (2) on the basis of the fuzzy judgment evaluation results of the nine experts. The comprehensive fuzzy judgment matrix $A$ is shown as follows

$A = \begin{matrix} \begin{matrix} P 1 & P 2 & P 3 & P 4 & P 5 \end{matrix} \\ \begin{matrix} P 1 \\ P 2 \\ P 3 \\ P 4 \\ P 5 \end{matrix} & [\begin{matrix} (1.00, 1.00, 1.00) & (1.00, 2.00, 3.00) & (2.29, 3.30, 4.31) & (4.00, 5.00, 6.00) & (5.65, 6.65, 7.65) \\ (0.33, 0.50, 1.00) & (1.00, 1.00, 1.00) & (1.00, 2.00, 3.00) & (2.62, 3.63, 4.64) & (3.91, 4.93, 5.94) \\ (0.23, 0.30, 0.44) & (0.33, 0.50, 1.00) & (1.00, 1.00, 1.00) & (1.00, 2.00, 3.00) & (6.29, 3.30, 4.31) \\ (0.17, 0.2, 0.25) & (0.22, 0.28, 0.38) & (0.33, 0.50, 1.00) & (1.00, 1.00, 1.00) & (1.26, 2.29, 3.30) \\ (0.13, 0.15, 0.18) & (0.17, 0.20, 0.26) & (0.23, 0.30, 0.44) & (0.30, 0.44, 0.79) & (1.00, 1.00, 1.00) \end{matrix}] \end{matrix}$

Step 2. To obtain the weight of each evaluation index of the traffic congestion degree, $b_{i}$ was calculated on the basis of fuzzy judgment matrix $A$ and formula (3). The calculation results are summarized in Table 3.

Step 3. We subsequently calculated the fuzzy weight of each evaluation index of the degree of traffic congestion, referring to the calculation results of b_i and formula (4). The results were as follows

$\begin{matrix} W_{1} = (0.325, 0.441, 0.718) \\ W_{2} = (0.205, 0.267, 0.484) \\ W_{3} = (0.112, 0.150, 0.284) \\ W_{4} = (0.069, 0.087, 0.160) \\ W_{5} = (0.043, 0.050, 0.089) \end{matrix}$

Step 4. Finally, the best non-fuzzy performance (BNP) values of the fuzzy weights of each evaluation index were calculated using formula (5), and the weights and values were sorted. These results are presented in Tables 4 and 5.

Table 3.

Calculated results of b_i.

b _i	b ₁	b ₂	b ₃	b ₄	b ₅
Calculation results	(2.20, 2.94, 3.59)	(1.28, 1.78, 2.42)	(0.70, 1, 1.42)	(0.43, 0.58, 0.80)	(0.27, 0.33, 0.44)

Table 4.

BNP values of $W_{i}$ .

Fuzzy weight of each index	W ₁	W ₂	W ₃	W ₄	W ₅
BNP value of W_i	0.504	0.319	0.182	0.105	0.060

BNP: best non-fuzzy performance.

Table 5.

Weights of each evaluation index.

Evaluation indexes	Weights	BNP value	Rank
Average delay	(0.352, 0.441, 0.718)	0.504	5
Average queue length	(0.205, 0.267, 0.484)	0.319	4
Average number of stops	(0.112, 0.150, 0.284)	0.182	3
Total saturation	(0.069, 0.087, 0.160)	0.105	2
Average occupancy	(0.043, 0.050, 0.089)	0.060	1

BNP: best non-fuzzy performance.

Evaluation of traffic congestion degree

Step 1. We organized the results of the intersection traffic survey and expert assessments by listing the traffic evaluation index values for nine time periods in one day (Table 6).

Step 2. Next, we established the initial decision matrix, which is detailed in Table 7.

Step 3. The standardized matrix $Y$ was calculated by normalizing the initial decision matrix according to formula (9). These results are shown in Table 8.

Step 4. Subsequently, the standardized matrix $Q$ , combined with the weight of each index, was calculated (Table 9), and the ideal solution $Q_{j}^{+}$ and the negative ideal solution $Q_{j}^{-}$ were obtained (Table 10).

Step 5. We also calculated the Euclidean distances $S_{i}^{+}$ and $S_{i}^{-}$ using the information in Table 10 and formulas (13) and (14)

$\begin{matrix} S_{i}^{+} = {0.4281, 0.1515, 0.0037, 0.0550, 0.0820, 0.0178, 0.0148, 0.1112, 0.1689} \\ S_{i}^{-} = {0.0000, 0.2830, 0.4257, 0.4574, 0.3744, 0.4149, 0.4188, 0.3536, 0.3107} \end{matrix}$

Table 6.

Value of traffic evaluation index.

	Time period	P1	P2	P3	P4	P5
1	0:00–5:00	18.55	0.88	0.53	10.67	0.35
2	5:00–7:00	26.37	2.94	0.62	29.02	2.97
3	7:00–9:00	39.59	27.61	0.69	102.33	34.77
4	9:00–11:00	32.20	21.46	0.69	116.55	26.22
5	11:00–13:00	29.43	16.69	0.69	107.02	23.5
6	13:00–16:00	37.02	26.12	0.67	118.37	29.75
7	16:00–18:00	37.26	37.49	0.70	103.69	28.18
8	18:00–20:00	27.03	11.64	0.66	90.67	12.32
9	20:00–23:00	23.41	6.05	0.62	58.53	7.65

Table 7.

Initial decision matrix.

	P1	P2	P3	P4	P5
1	18.55	0.88	0.53	10.67	0.35
2	26.37	2.94	0.62	29.02	2.97
3	39.59	27.61	0.69	102.33	34.77
4	32.20	21.46	0.69	116.55	26.22
5	29.43	16.69	0.69	107.02	23.5
6	37.02	26.12	0.67	118.37	29.75
7	37.26	37.49	0.70	103.69	28.18
8	27.03	11.64	0.66	90.67	12.32
9	23.41	6.05	0.62	58.53	7.65

Table 8.

Standardized matrix.

	P1	P2	P3	P4	P5
1	1	1	1	1	1
2	0.7035	0.2993	0.8548	0.3677	0.1178
3	0.4686	0.0319	0.7681	0.1043	0.010
4	0.5761	0.0410	0.7681	0.0915	0.0133
5	0.6303	0.0527	0.7681	0.0997	0.0149
6	0.5011	0.0337	0.7910	0.0901	0.0118
7	0.4979	0.0235	0.7571	0.1029	0.0124
8	0.6863	0.0756	0.8030	0.1177	0.0284
9	0.7924	0.1455	0.8548	0.1823	0.0456

Table 9.

Standardized matrix $Q$ combined with the weight of each index.

	P1	P2	P3	P4	P5
1	0.5040	0.3190	0.1820	0.1050	0.0600
2	0.3546	0.0955	0.1556	0.0386	0.0071
3	0.2362	0.0102	0.1398	0.0110	0.0006
4	0.2904	0.0131	0.1398	0.0097	0.0008
5	0.3177	0.0168	0.1398	0.0105	0.0009
6	0.2526	0.0108	0.1440	0.0095	0.0007
7	0.2509	0.0075	0.1378	0.0108	0.0007
8	0.3459	0.0241	0.1461	0.0124	0.0017
9	0.3994	0.0464	0.1556	0.0191	0.0027

Table 10.

Ideal solution and negative ideal solution of the standardized matrix $Q$ .

	P1	P2	P3	P4	P5
$Q_{j}^{+}$	0.2362	0.0075	0.1378	0.0095	0.0006
$Q_{j}^{-}$	0.5040	0.3190	0.1820	0.1050	0.0600

Subsequently, the GC coefficient matrices $R^{+}$ , $R^{-}$ were obtained according to formulas (17) and (20), and the GC degrees $R_{i}^{+}$ , $R_{i}^{-}$ between each scheme and the ideal and negative ideal solutions were calculated

$\begin{matrix} R_{i}^{+} = {0.5648, 0.7814, 0.9944, 0.9406, 0.9181, 0.9690, 0.9810, 0.8829, 0.8229} \\ R_{i}^{-} = {1.0000, 0.6448, 0.5673, 0.5782, 0.5859, 0.5733, 0.5681, 0.6019, 0.6385} \end{matrix}$

Step 6. Finally, the dimensionless method was applied to normalize $S_{i}^{+}$ , $S_{i}^{-}$ , $R_{i}^{+}$ , $R_{i}^{-}$ according to formula (23), where $h = 0.5$ , to obtain the comprehensive closeness index

$L_{i} = {0.2212, 0.5844, 0.7702, 0.7336, 0.6914, 0.7537, 0.7594, 0.6584, 0.5933}$

Comparison with existing methods

To verify the accuracy of the proposed hybrid evaluation method, the results of the FGT were then compared with the results of the FAHP-GC and FAHP-TOPSIS methods. The weights that were determined using FAHP were subsequently applied to the three evaluation methods. The evaluation result of FAHP-GC was obtained using formulas (24) and (25), where $h = 0$ ; by contrast, the evaluation result of FAHP-TOPSIS was obtained using formulas (24) and (25) where $h = 1$ . The evaluation results of all three evaluation methods are outlined in Table 11.

Table 11.

Closeness index of the three methods.

Scheme	FGT		FAHP-GC		FAHP-TOPSIS
	L_i	Rank	C_i	Rank	F_i	Rank
1	0.221	9	0.361	9	0	8
2	0.584	8	0.548	8	0.650	7
3	0.770	1	0.637	1	0.991	1
4	0.734	4	0.619	4	0.893	4
5	0.691	5	0.610	5	0.820	5
6	0.754	3	0.628	3	0.959	3
7	0.759	2	0.633	2	0.966	2
8	0.658	6	0.595	6	0.761	6
9	0.593	7	0.563	7	0.650	7

FAHP-GC: fuzzy analytic hierarchy process–gray correlation; FAHP-TOPSIS: fuzzy analytic hierarchy process–techniques for order preference by similarity to an ideal solution; FGT: FAHP, GC, and TOPSIS.

Table 11 indicates that the evaluation results of the three methods are consistent for the degree of traffic congestion. The validity and accuracy of our proposed method are confirmed through data comparison. Moreover, the data clearly indicate that the FAHP-TOPSIS evaluation method cannot effectively compare the two time periods of traffic congestion when there are two sets of equal data. However, the proposed method is a comprehensive evaluation method that is based on Euclidean distance and data sequence curve similarities. This evaluation method overcomes the shortcomings of FAHP-TOPSIS and FAHP-GC, rendering the evaluation results more accurate and reliable. In summary, the proposed method is feasible and effective for evaluating traffic congestion at intersections.

Conclusion

Traffic decision-making plays a significant role in urban public transport planning; in this article, traffic congestion can be effectively evaluated by a hybrid MCDM method combining the FAHP, TOPSIS, and GC techniques. Obviously, the MCDM method combining FAHP and TOPSIS has proved to be defective when evaluating the traffic congestion degree because FAHP-TOPSIS cannot accurately evaluate two schemes in Table 10 that have equal distance from the ideal solution. Similarly, FAHP-GC mainly focuses on situation changes among data sequences. It may not be precise enough and thus leads to some misleading conclusions without considering their comprehensive features.

In this work, the ranking results are compared with commonly used MCDM methods involving TOPSIS, GC, and FAHP. The results demonstrates that the proposed method can be used to solve real-life MCDM problems in urban public transport planning, and the method can provide an accurate evaluation of the alternatives and offer a more reasonable selection. The main novel elements of the proposed method are summarized below. First, the proposed method overcomes the shortcomings of the FAHP-TOPSIS and FAHP-GC methods, and it can be used as an accurate and effective multi-attribute scheme decision-making method. Especially, FAHP is determined as the technique of weight assignment, thus reducing the influence of subjective preference. Second, FGT is a useful and reliable tool for evaluation of traffic congestion degree based on the results of an empirical case, comparison analysis. A suitable hierarchical structure of each criterion considering delay indicator, queuing indicator, saturation index, parking rate, and occupancy index is built for intersection traffic congestion evaluation, and the reasonable weight for each criterion is derived. Third, this method can be used to guide decision makers in making better decisions when constructing a best intersection and traffic congestion control. Moreover, the proposed approach can be applied to other fields, for example, selection of suppliers and plant location selection.

With the advancement of science and technology, and the development of multi-attribute theories, we predict that new theories and methods can be identified and integrated to assess traffic congestion. In particular, we suggest that uncertainty theory should be considered in multi-attribute decision-making.⁶⁴

Footnotes

Academic Editor: Liping Jiang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported in part by the Zhejiang Provincial Natural Science Foundation of China (grant no. LR14E050003),the National Natural Science Foundation of China (grant no. 51405075),and the Funds for International Cooperation and Exchange of the National Natural Science Foundation of China (grant no. 51561125002). Sincere appreciation is extended to the reviewers of this paper for their helpful comments.

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Evaluation of traffic congestion degree: An integrated approach

Abstract

Keywords

Introduction

Literature review

Existing evaluation methods

FAHP method

TOPSIS method

GC method

The FGT method

Assessment example

Determining the evaluation index for degree of traffic congestion

Delay indicator

Queuing indicator

Saturation index

Number of stops (parking rate)

Occupancy index

Determining the weights of five evaluation indexes of traffic congestion degree using the FAHP method

Evaluation of traffic congestion degree

Comparison with existing methods

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References