A Performance Evaluation Method for Long and Steep Uphill Sections of Heavy-Haul Railway Lines

Abstract

Any system for evaluating the safety service performance of heavy-haul railway lines must effectively reflect the real-time service status of the line. The working conditions of heavy-load lines are complex and diverse, particularly on uphill sections. Existing evaluation systems struggle to accurately reflect the service conditions of long and steep uphill sections bearing heavy loads, posing a significant threat to the safe operation of these lines. To address this problem, we propose a new method for evaluating the safety service performance of long and steep uphill sections of heavy-haul railway lines by establishing a scoring system based on the Analytic Hierarchy Process (AHP). First, damage indicators for heavy-haul lines are categorized into three groups: track geometry status indicators, track structure status indicators, and track traffic status indicators. Using data from existing heavy-haul lines and maintenance experiences, we determine a score deduction standard, classifying lines into four levels based on their safety service quality. Next, we establish a coefficient table for the service performance of long and steep uphill sections after the corresponding scores are deducted. Using data for the length and elevation grade of the actual uphill section, we adjust the deducted scores of the track structure status indicators, enhancing the evaluation system’s accuracy in describing the working conditions. Finally, we verify the stability of the entire system by conducting a sensitivity analysis of the indicator evaluation results using the One-At-a-Time (OAT) method. This method fills a critical gap in the safe operation and maintenance of heavy-haul railways and provides a safety guarantee for the operation of long uphill sections of heavy-haul railways.

1. Introduction

As an effective way to improve railway freight capacity, the use of heavy-haul railway transportation has garnered attention across the world. In countries like the United States, Brazil and Australia, which have large territories and rich resources, and where large volumes of bulk cargoes (e.g., coals and ores) need to be transported, heavy-haul railway transportation has been especially vigorously developed. However, heavy-haul trains also operate on many mixed passenger and freight main lines in Europe. In China, the Shenmu–Shuozhou section of the Baotou–Shenmu Railway is a heavy-haul section of the line that plays an important role in the delivery of goods between the Inner Mongolia Autonomous Region and the coalfields in Shaanxi Province.

The application of inspection technology can be seen as one of the key measures used to ensure the safe operation of trains on heavy-haul railway lines [1]. Xiong Longhui et al. [2] summarized methods with the potential to achieve rapid and accurate detection and evaluation of rail defects; these methods included ultrasonic flaw detection, electromagnetic flaw detection, and visual flaw detection, among other comprehensive techniques. Data on rail wear, rail damage, rail repair and replacement, and line operation were used. Xie et al. [3] proposed a long-term, real-time online monitoring of heavy railway track defects. He et al. [4] proposed a RUL prediction method for heavy-haul railway lines based on an improved deep-spiking residual neural network. Shang Peipei et al. [5] analyzed a development law for outer-side rail wear on curved sections of line and damage to rails on straight sections relative to cumulative total passing mass; they then established a prediction model and provided recommended values for replacement cycles on a heavy-haul section of the Shuozhou–Huanghua Railway; Yin Duanquan et al. [6] aimed to eliminate the problem of rail end cracks in turnout areas of heavy-haul railways. Using macroscopic and microscopic observation of morphology, they analyzed types of cracks and their causes, measured damaged rails and joint bars, and observed fracture surfaces. In addition, they inspected the metallographic structures of fractures, identified the physical and chemical properties of damaged rails, and proposed a number of remediation measures; David F.N. Oliveira et al. [7] used autoencoders with two unsupervised anomaly detection models and produced balanced results for different scenarios, demonstrating the ability of such models to carry out autonomous detection on railway lines. Finally, based on the maintenance status of equipment on the Shuozhou–Huanghua Railway line, etc., Wang Feng et al. [8] discussed a repair cycle applicable to this equipment using theoretical analysis, indoor experiments, and field tests. Chen Xianmai et al. [9] simulated the concrete sleeper structure using parallel-bonded spherical elements to model the transfer of train loads. By verifying the vertical stiffness and lateral resistance of the sleepers and comparing them with experimental data, they confirmed the reliability of the model.

In previous studies, the analytic hierarchy process (AHP) has been widely used for evaluation purposes. Sandeep Panchal et al. [10] conducted a risk assessment using AHP. By considering the different causes of landslides on roads, they produced a landslide hazard rating map. Emre Demir et al. [11] used AHP to conduct a decision-making study of three bus routes in Antalya, Turkey. Several important criteria for evaluating economy, comfort, environment, and safety were considered, and the most efficient bus routes in the city were recommended. Diogo Silva Costa et al. [12] proposed a multi-criteria decision-making technique combining AHP and TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) to optimize a process automation plan for a software-automated robot. They also applied this proposed method for the selection of automation processes in real-life scenarios.

While previous studies have analyzed track life on heavy-haul lines and suggested repairs, none have provided a comprehensive evaluation method or management system for long and steep uphill sections of the railway. Furthermore, the focuses of these studies varied, and they were mostly theoretical and simulative in nature, falling short of meeting the specific requirements for performance evaluation of heavy-haul uphill lines.

In this study, using contemporary data from the Shenmu–Shuozhou section of the Baotou–Shenmu Railway, we aimed to analyze the operational rules and characteristics of heavy-haul trains on long and steep uphill sections. Our goal was to develop a new evaluation strategy for these sections using the Analytic Hierarchy Process (AHP). This strategy employs the One-At-a-Time (OAT) method to conduct a sensitivity analysis of evaluation indicators and verify the reliability of the evaluation results, ensuring that detection and maintenance capabilities are well-matched.

The overall flow of the algorithm proposed in this paper is illustrated in Figure 1. The main contributions of this work can be summarized as follows: we have established a heavy-haul railway evaluation system model suitable for long and steep uphill conditions. This model includes multiple evaluation indicators such as track quality index change rate, rail system, total freight volume, track slope, and length. The model is structured into three hierarchical levels to comprehensively describe the evaluation process.

• An evaluation system model suitable for long and steep uphill sections of heavy-haul railway lines has been established. This model includes multiple evaluation indicators such as track quality index change rate, rail system, total freight volume, track slope, and track length. For descriptive purposes, the model is divided into three hierarchical structures.
• A score deduction coefficient table for long and steep uphill sections of heavy-haul railway lines is proposed, with characteristic indicators of uphill slope conditions extracted. By combining the two principal characteristic indicators—slope grade and slope length—the mismatch problem between specific working conditions and the score deduction table at different levels is resolved.
• The One-At-a-Time (OAT) method is employed to conduct sensitivity analysis of established index weights at each level, verifying the percentage fluctuations of lines at different levels by adjusting weight values through up-and-down modifications. Experimental results indicate that the system model is not sensitive to fluctuations in the weight values of the criterion and index layers, demonstrating good stability and reliability. Compared with previously reported methods for determining scoring standards, the evaluation system proposed in this paper more accurately reflects the service status of long and steep uphill sections of railway lines.

2. Analysis of Working Conditions of Long and Steep Uphill Sections

The terrain of the entire Shenmu–Shuozhou railway line is complex. Bridges, tunnels, and culverts comprise 30% of its length, and many bridges and tunnels are connected consecutively. There are also long and steep uphill sections, with a maximum grade of 12‰. These are the harshest sections for heavy-haul trains, and they affect the transportation capacity of the Shenmu–Shuozhou railway line.

Compared with straight-line locomotion on level ground, the power requirements of a train are higher when it runs on long and steep uphill sections, and they rise still further with increasing grades and lengths of slopes. The Shenmu–Shuozhou railway line contains dense sublines and a number of long and steep uphill sections. Most of the uphill sections of the Baotou–Shenmu Railway serve heavy-haul trains, which require more power for uphill running. On these sections of the line, wheel-rail contact conditions are more demanding, resulting in increased wear to rail surfaces. Under such conditions, the driving safety of heavy-load trains can be affected.

In the present study, to establish an applicable evaluation method for uphill sections on the Shenmu–Shuozhou railway line, it was necessary to first carry out a targeted analysis of the working conditions of uphill sections of heavy-haul railways.

Heavy-haul trains require more traction and braking power when running on long and steep uphill slopes than when running on flat and straight lines, and traction and braking times also increase. Ref. [13] This impacts the condition of the line, as the surface of the rail becomes worn and damaged, with consequent implications for operational safety.

When a heavy-haul train runs uphill with a heavy load, the rail surface is under great pressure and is exposed to the risk of wear. The larger traction force required for uphill running increases the force to which the rail surface is subjected, resulting in increased wear to the rail, which eventually reduces its service life. Such a situation has implications for railway safety and transportation efficiency; maintenance and replacement costs may also impacted, and the stability and speed of trains themselves may be affected.

Total transportation volume is another key factor to be considered when assessing uphill sections of a line. When total transportation volume rises, wear to rail surfaces caused by heavy-haul trains passing through uphill sections also increases. Such wear means that trains require still-higher traction and braking forces, resulting in further deterioration in the condition of lines.

Therefore, when analyzing working conditions on long and steep uphill sections of heavy-haul railway lines, it is necessary to focus on the combined influence of vertical wear upon the rail and total transportation volume when determining the service life of the line.

3. Establishment of an Evaluation Model System for Long and Steep Uphill Sections

In this section, we describe our construction of a Track Service Performance Index (TSI, $T_{SPI}$) to assess the performance status of railway lines, and we introduce a hierarchical model for long and steep uphill sections of heavy-haul lines.

3.1. Determination of Hierarchy Structure

Inspection data for long and steep uphill railway lines bearing heavy-haul trains are mainly of four types: track geometry data, track structure data, track traffic freight volume data, and track slope data. Track geometry data primarily consists of horizontal curvature, vertical curvature, left and right steering, left and right heights, etc. Track structure data mainly includes inspection data concerning rail profile structures, rail top- and side-surface wear, sleepers, fastenings, etc. In the present study, to fully reflect the quality level of the line under this working condition and to make full use of inspection data, a three-category, three-layer heavy-haul railway $T_{SPI}$ system was established. The three categories were Track Geometry Index $T_{GI}$, Track Structure Index $T_{SI}$, and Track Freight Volume Index $T_{FVI}$. The three layers were the target layer, the criterion layer, and the indicator layer.

3.2. Selection of Evaluation Indicators

3.2.1. Selection and Calculation Method of $T_{GI}$

Track geometry refers to the geometric state of each part of the track, and it reflects the smoothness of the track. Sound-track geometry is an important factor in the maintenance of train safety. In the present study, to reflect and evaluate track geometry in a comprehensive manner, the existing data were combined with indicators related to heavy-haul railways, and various indicators were selected, including the track quality index $T_{QI}$, track quality index change rate $R_{TQI}$ (reflecting, in unit time, the difference between the measured $T_{QI}$ and that of the previous timepoint $T_{QI}^{\prime }$), geometric irregularity deviation value exceeding limit $C$, and dynamic deviation $T$. Based on actual test data, scores were deducted from the original values of these indicators. The calculation formula was as follows:

$${\begin{array}{c}T\end{array}}_{GI}=100-G\left(T_{QI},R_{TQI},C,T\right)$$

(1)

where $G\left(\right)$ is the score deduction function for $T_{GI}$.

3.2.2. Selection and Calculation Method of $T_{SI}$

The structural defects of heavy-haul railway lines mainly arise from the rail system (rail surface wear, joint damage, etc.), the fastener system (missing fasteners, fastener damage, non-qualified fasteners, etc.) and the ballast bed system (sleeper failure rate, mud-pumping, slurry, sand overflow, etc.). In light of this, we used a rail condition index ($R_g$), a joint part condition index ($J_e$), and a trackbed condition index ($T_{ea}$) to evaluate track structure status.

According to the score deduction standard, scores were deducted from the original values of these three indicators, using the following calculation formula:

$${\begin{array}{c}T\end{array}}_{SI}=100-S\left(R_{CI},J_{PCI},T_{BCI}\right)$$

(2)

where, $S\left(\right)$ is the score deduction function for $T_{FVI}$.

3.2.3. Selection and Calculation Method of $T_{SI}$

The track freight volume index refers to the total transportation volume passing on a unit line section of the track. The transportation volume $T_V$ is adopted. This indicator reflects the actual transportation intensity that the line bears. The calculation formula is

$${\begin{array}{c}T\end{array}}_{FVI}=100-F\left(T_V\right)$$

(3)

where $F\left(\right)$ is the score deduction function for $T_{FVI}$.

3.3. Determination of Indicator Score Deduction Criteria

Currently, the maintenance of heavy-haul railway lines is carried out unit by unit. Lines and turnouts are first divided into units; then, these units are sorted into different levels according to the score deduction; finally, the location and method of maintenance and repair are determined according to the level. The basic principle of unit-by-unit repair is also applied in the maintenance of long and steep uphill sections.

When calculating score deductions, different score deduction criteria should be applied for long and steep uphill lines and for general straight lines. In the present study, score deduction criteria were determined by considering the different levels of section units and the different indicators of long and steep uphill sections, in line with the following: “China TG/GW102-2019 Rules for Repair of General-speed Railway Lines” [14]; “Implementation Method of Equipment Management for Baotou–Shenmu Railway Maintenance Depot”; and “Technical Specification for Comprehensive Maintenance of Heavy-haul Railway Lines”. We also incorporated the actual operation and maintenance conditions of long and steep uphill lines. The deduction criteria and deducted scores for each part are shown in

Table A1. Score deduction criteria for geometric status evaluation indicators of long and steep uphill sections of the line.

Indicators		Criteria For Score Deduction	Unit	Deducted Score
Geometric Irregularity Deviation Value Exceeding Limit	Height (mm)	8	Position	25
		12		50
		20		100
	Steering (mm)	8		25
		10		50
		16		100
	Twist of Track (mm)	8		25
		10		50
		14		100
	Cross Level (mm)	8		25
		12		50
		18		100
	Gauge (mm)	+8		25
		−6
		+12		50
		−8
		+20		100
		−10
	Gauge Change Rate (%)	2		25
		2.5		50
Track Quality Index (TQI)		$\mathrm{T}\mathrm{Q}\mathrm{I}\ge$15	Per Unit	50
Change Rate of Track Quality Index (RTQI)		5%$<R_{TQI}\le$15%		20
		$15\%<R_{TQI}\le$25%		40
		$R_{TQI}>$25%		60
Vibration Acceleration of Carbody	Lateral Acceleration (m/s2)	>1.0	Position	25
		>1.5		50
		>2.0		100
	Longitudinal Acceleration (m/s2)	>0.6		25
		>0.9		50
		>1.5		100

,

Table A3. Score deduction criteria for evaluation indicators of the structural status of long and steep uphill sections of the line.

Indicators		Criteria For Score Deduction	Unit	Deducted Score
Rail System	Rail Head Nucleus Flaw	Yes	Position	10
	Crack of Rail Web	Yes		10
	Transverse Crack at Rail Bottom	Yes		10
	Longitudinal Crack at Rail Bottom	Yes		10
	Transverse Crack of Screw Hole	Yes		10
	Longitudinal Crack of Screw Hole	Yes		10
	Tread Scratch	Yes		10
	Fish Scale Pattern	Yes		10
	Spalling	Over 15 mm in length And over 3 mm in depth		10 + 5i
		Over 25 mm in length And over 3 mm in depth		20 + 10i
	Rail Corrugation	Trough depth over 0.3 mm		10 + 5i
	Rail Vertical Wear	Over 5 mm		50 + 5i
		Over 10 mm		100 + i
	Rail Side Wear	Over 16 mm		10
		Over 21 mm		10
	Rail Joint Insufficient Gap	Over 1 mm		10
	Gage Line or Rail End	Over 1 mm		10
	Rail Light Band Abnormal	Yes		10
Joint Part System	Fastener Missing	Yes		20
	Fastener Skew	Yes		20
	Fastener Buried	Yes		20
	Rubber Gasket Skew	Yes		20
Ballast Bed System	Ballast Bed Mud-pumping	Yes	Every 10 m Extension	20
	Ballast Bed Unclean Rate	Over 25%	Every 100 m Extension	20
	Sleeper Failure Rate	Over 4%	Per Unit	100

and

Table A4. Score deduction criteria for freight volume evaluation indicators of long and steep uphill sections of the line.

Indicators	Criteria For Score Deduction	Unit	Deducted Score
Total Passing Freight Volume	700 Mt	Unit	50 + 5i
	1500 Mt		100

in the Appendix A.

When considering uphill working conditions, the complex nature of the operating environment makes it difficult for the two important indicators, i.e., slope grade and slope length, to be well used in any evaluation system for the line concerned. In the present study, we used actual vehicle inspection data for trains presently running on the long and steep uphill sections of the Shenmu–Shuozhou railway line, transportation records for previous years, the equipment management and implementation method used by the Shenmu–Shuozhou Railway Engineering Department, and comprehensive maintenance technical specifications for heavy-haul railway lines, as well as historical data and the personal experience of experts, to establish a coefficient table for the long and steep uphill sections, as shown in the attached

Table A2. Score deduction coefficient i for long and steep uphill sections of the line.

	Grade	$0\mathit{‰}<{\mathit{T}}_{\mathit{G}}\le 4\mathit{‰}$	$4\mathit{‰}<{\mathit{T}}_{\mathit{G}}\le 8\mathit{‰}$	$8\mathit{‰}<{\mathit{T}}_{\mathit{G}}\le 12\mathit{‰}$
Length
$0 \mathrm{M}<T_L\le 250 \mathrm{M}$		1	2	3
$250 \mathrm{M}<T_L\le 500 \mathrm{M}$		2	3	4
$500 \mathrm{M}<T_L\le 750 \mathrm{M}$		3	4	5
$750 \mathrm{M}<T_L\le 1000 \mathrm{M}$		4	5	6
$T_L\ge 1000 \mathrm{M}$		5	6	7

. By looking up values in the table, an evaluation system for rating uphill sections of various grades and lengths could be implemented quickly and effectively. In the table, $T_G$ (in ‰) is the track grade, $T_L$ (in m) indicates the track length, and i indicates the score deduction coefficient of the slope.

3.4. Establishment of Scoring System $T_{SPI}$ for Long and Steep Uphill Sections

The scoring system $T_{SPI}$ for long and steep uphill sections of heavy-haul railway lines is illustrated in Figure 2.

The score deduction function of $T_{GI}$ was expressed as follows:

$$G\left(T_{QI},R_{TQI},C,T\right)={\displaystyle \sum _{i=1}^4{w}_{1i}{u}_{1i}}$$

(4)

where $w_{1i}$ is the weight of the ith item of $T_{GI}$; and $i=1,2,3,4$; $u_{1i}$ is the deducted score of the ith item of $T_{GI}$. The value range of $u_{1i}$ is 0–100. When $u_{1i}$ is greater than 100, it is assigned a value of 100.

The score deduction function of $T_{SI}$ was expressed as follows:

$$S\left(R_{CI},J_{PCI},T_{BCI}\right)={\displaystyle \sum _{j=1}^3{w}_{2j}{u}_{2j}}$$

(5)

where $w_{2j}$ is the weight of the jth item of $T_{SI}$; and $u_{2j}$ is the deducted value of the jth item of $T_{SI}$. The value range of $u_{2j}$ is 0–100. When $u_{2j}$ is greater than 100, it is assigned a value of 100.

The deduction function of $T_{FVI}$ was expressed as follows:

$$F\left(T_V\right)=w_{31}{u}_{31}$$

(6)

where $w_{31}$ is the weight of the first item of $F_{FVI}$; and $u_{31}$ is the deducted value of the first item of $F_{FVI}$.

The calculation formula of $T_{SPI}$ was expressed as follows:

$$T_{SPI}=w_1{T}_{GI}+w_2{T}_{SI}+w_3{T}_{FVI}$$

(7)

where $w_1$ is the weight of $T_{GI}$, $w_2$ is the weight of $T_{SI}$; $w_3$ is the weight of $T_{FVI}$’ and $w_1+w_2+w_3=1$.

3.5. Classification of Line Quality

According to the calculated $T_{SPI}$, line sections are divided into four quality levels with descending scores, i.e., I, II, III, and IV, corresponding to excellent, qualified, repairable, and disposable, respectively. The classification standard for line section unit quality levels is given in

Table 1. Standard of line unit quality level for long and steep uphill.

Level I	Level II	Level III	Level IV
$T_{SPI}\ge 70$	$70\ge T_{SPI}\ge 50$	$50\ge T_{SPI}\ge 20$	$T_{SPI}\le 20$

.

4. Determination of Evaluation Index Weights

In this section, we describe our design of evaluation index weights based on the hierarchical model for long and steep uphill sections of heavy-haul lines, as shown in Figure 2.

A judgment matrix was constructed to directly determine the accuracy and scientific value of the evaluation results. This involved a pairwise comparison of each factor by experts under the same evaluation to assess the relative importance of the two factors and thereby determine the scale definition value of the index. To this end, we used the commonly used assignment principle with the 1–9 scale method proposed by Professor T.L. Saaty, and specific definitions are listed in

Table 2. Definition of 1–9 scales of judgment matrix elements.

$\mathbf{Definition} (\mathbf{Element} {\mathbf{X}}_{\mathbf{i}}$$\mathbf{to} \mathbf{Element} {\mathbf{X}}_{\mathbf{j}}$)	$\mathbf{Judgment} \mathbf{Scale} ({\mathbf{r}}_{\mathbf{i}\mathbf{j}}$$\mathbf{Value}, {\mathbf{r}}_{\mathbf{i}\mathbf{j}}={\mathbf{X}}_{\mathbf{i}}/{\mathbf{X}}_{\mathbf{j}}$)
Equally important	1
Slightly Important	3
Important	5
Strongly important	7
Extremely important	9
Middle value of two adjacent judgment	2, 4, 6, 8
Scale value of the ratio of $X_i$ to $X_j$ is $r_{ij}=1/r_{ji}$	Reciprocal value of the judgment scale

.

After the definition of the scale value for the indicator was determined, the indicators of each level were calculated. The sum-product method was used, and the calculation process was as follows:

• The judgment matrix was determined as A = (a_ij)_n×n follows;
• The elements in $A$ were normalized by column so that:

  $$\overline{a_{ij}}=a_{ij}/{\displaystyle \sum _{k=1}^n{a}_{kj}}, i,j=1,2,\cdots ,n$$

  (8)
• After normalization, the values of each column in the same row were added so that:

  $$\widetilde{w_i}={\displaystyle \sum _{j=1}^n\overline{a_{ij}}}, i,j=1,2,\cdots ,n$$

  (9)
• By dividing the resulting vector of the addition operation by $n$, a weight vector was obtained as follows:

  $$w_i=\widetilde{w_i}/n$$

  (10)
• The maximum characteristic root was now determined thus:

  $${\lambda }_{\max }={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{{(Aw)}_i}{w_i}}$$

  (11)

  where ${(Aw)}_i$ is the ith component of the vector $Aw$.

After the weights were calculated, it was necessary to check the consistency of the calculated weights as follows:

• Calculation of consistency index $CI$:

  $$CI={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}}$$

  (12)
• Calculation of consistency ratio $CR$:

  $$CR={\displaystyle \frac{CI}{RI}}$$

  (13)

  where $RI$ is the random consistency index. Specific values are shown in

  Table 3. Random consistency index.

  Order	3	4	5	6	7
  RI	0.58	0.89	1.12	1.24	1.32

  .

Finally, $CR$ was used to determine whether the judgment matrix was consistent. Generally, when $CR<0.1$, the judgment matrix may be considered consistent, and the eigenvector of $A$ can be used as the weight vector. When $CR>0.1$, the judgment matrix $A$ needs to be tuned until the consistency requirements are met.

By repeating the steps above, the weights of each layer were calculated.

4.1. Determination of Criterion Layer Weights

Based on the expert experiences and the questionnaire, the judgment matrix of the criterion layer was obtained, as shown in

Table 4. Criterion layer judgment matrix.

w	Geometry	Structure	Freight Volume
Geometry	1	1/7	1/5
Structure	7	1	3
Track Freight Volume	5	1/3	1

. Using the above given AHP calculation steps, the weights of criterion layer indicators were found to be as follows: $w_1$ = 7.377%, $w_2$ = 64.339%, and $w_3$ = 28.284%. In addition, the consistency ratio $C_{\mathrm{R}}$ was determined to be 0.063, fulfilling the above-stated requirement.

4.2. Determination of Weights of $T_{GI}$

Based on the expert experiences and the questionnaire, the judgment matrix of $T_{GI}$ was obtained, as shown in

Table 5. Judgment matrix of $T_{GI}$.

${\mathit{T}}_{\mathit{G}\mathit{I}}$	Geometric Irregularity Deviation Value Exceeding Limit	Quality Index	Change Rate	Acceleration
Geometric Irregularity Deviation Value Exceeding Limit	1	5	5	5
Quality Index	1/5	1	4	3
Change Rate	1/5	1/4	1	1
Acceleration	1/5	1/3	1	1

. Using the above given AHP calculation steps, the weights of indicators of $T_{GI}$ were found to be as follows: $w_{11}$ = 58.476%, $w_{12}$ = 23.513%, and $w_{13}$ = 8.847%, $w_{14}$ = 9.164%. In addition, the consistency ratio $C_{\mathrm{R}}$ was determined to be 0.079, fulfilling the above-stated requirement.

4.3. Determination of Weights of $T_{SI}$

Based on the expert experiences and the questionnaire results, the judgment matrix of $T_{SI}$ was obtained, as shown in

Table 6. Judgment matrix of $T_{SI}$.

TSI	Rail	Joint Part	Ballast Bed
Rail	1	7	4
Joint Part	1/7	1	1/3
Ballast Bed	1/4	3	1

. Using the AHP calculation steps presented above, the weights of indicators of $T_{SI}$ were found to be as follows: $w_{11}$ = 70.144%, $w_{12}$ = 8.532%, and $w_{13}$ = 21.324%. In addition, the consistency ratio $C_{\mathrm{R}}$ was determined to be 0.031, fulfilling the above-stated requirement.

5. Sensitivity Analysis of Indicator Weights

In order to verify whether the above evaluation system was reliable, it was necessary to validate the stability of the obtained results. In the present study, the OAT (one at a time) method was used to perform sensitivity analysis on the index weights of the criterion layer. This method evaluates the effects of changes in individual indicator weights on the results by changing the weight values of the indicators one by one. For our analysis, the index weight of the criterion layer was adjusted up and down by values of 10% of the original weight. At the same time, to ensure that the weight values were still summed to 1, the weight values of other indicators were adjusted proportionally. The ergonomic evaluation results were then recalculated to reflect the impact of weight changes upon them.

The formula of the index weight value after fluctuation $w_i^{\prime }$ is:

$$w_i^{\prime }=w_i\times (1\pm 10\%)$$

(14)

The formula for other indicator weights without fluctuation $w_j^{\prime }$ was as follows:

$$w_j^{\prime }=w_j\times {\displaystyle \frac{1-{\mathrm{w}}_i^{\prime }}{1-{\mathrm{w}}_i}}$$

(15)

In the above formulas, $w_i$ is the weight value of the main indicator that needs to be calculated with fluctuation, and $w_j$ is the weight value of other indicators at the same level as the main indicator.

5.1. Sensitivity Calculation

In order to reduce the influence of the subjectivity of expert scoring on the results and to verify the stability of the evaluation results, measurement data for the section of line between the Shendong and Sancha stations on the Shenmu–Shuozhou Railway were used for evaluation and inspection purposes. As with the OAT method described above, the key weight values of the criterion layer and the track structure status were adjusted by 10% up and down, respectively, for sensitivity analysis. Track geometry was also considered when using this evaluation system.

The sensitivity analysis of the criterion layer is shown in

Table 7. Sensitivity analysis of criterion layer.

Criterion Layer	Change Rate	Track Geometry	Track Structure	Track Freight Volume	Level I	Level II	Level III	Level IV
Track Geometry	−10%	6.639%	64.851%	28.509%	29.5455%	32.9545%	25.0000%	12.5000%
	+10%	8.115%	63.827%	28.059%	29.5455%	34.0909%	25.0000%	11.3636%
Track Structure	−10%	8.708%	57.905%	33.387%	26.136%	30.682%	30.682%	12.500%
	+10%	6.046%	70.773%	23.181%	29.545%	36.364%	23.864%	10.227%
Track Freight Volume	−10%	7.668%	66.876%	25.456%	29.5455%	36.3636%	23.8636%	10.2273%
	+10%	7.086%	61.802%	31.112%	26.136%	35.227%	26.136%	12.500%

, and the sensitivity analysis of the track structure state index layer is shown in

Table 8. Sensitivity analysis of track structure status index layer.

Criterion Layer	Change Rate	Rail	Joint Part	Ballast Bed	Level I	Level II	Level III	Level IV
Rail	−10%	63.130%	10.537%	26.334%	30.6818%	34.0909%	26.1364%	9.0909%
	+10%	77.158%	6.527%	16.314%	25.0000%	31.8182%	30.6818%	12.5000%
Joint Part	−10%	70.798%	7.679%	21.523%	29.5455%	32.9545%	25.0000%	12.5000%
	+10%	69.490%	9.385%	21.125%	29.5455%	34.0909%	25.0000%	11.3636%
Ballast Bed	−10%	72.045%	8.763%	19.192%	29.5455%	32.9545%	25.0000%	12.5000%
	+10%	68.243%	8.301%	23.456	29.5455%	34.0909%	26.1364%	10.2273%

.

5.2. Analysis of Evaluation Results

The sensitivity analysis results in Table 7 and Table 8 show that when the weight values of each indicator in the criterion layer and the track structure status layer fluctuate, the evaluation results change slightly. This indicates that the performance evaluation results obtained in the present study for a long and steep uphill section of a heavy-haul railway line are not sensitive to fluctuations in the index weight values for the criterion layer and the track structure status index layer. The evaluation results therefore exhibit good stability and reliability.

6. Validation of Results

The following quality scoring results were obtained by using part data selected from the uphill section of the line between Shendong and Sancha stations on the heavy-haul Shenmu–Shuozhou Railway. These were compared with data from the Baotou–Shenmu maintenance district to verify whether the results of our scoring method were accurate. The test results are shown in

Table 9. Comparison of overall evaluation scores and quality levels for each line unit.

Line Serial Number	Proposed Method By This Paper		Line Evaluation Method of Baotou–Shenmu Maintenance District
	Evaluation Score	Quality Level	Overall Evaluation Score	Quality Level
1	81.26	Qualified	79	Qualified
2	70.76	Qualified	69	Qualified
3	66.57	Qualified	63	Qualified
4	64.62	Qualified	58	Qualified
5	53.06	Qualified	49	Disqualified
6	32.08	Disqualified	25	Disqualified
7	14.75	Disqualified	12	Disqualified

.

From a comparison of the evaluation results presented in Table 9, it may be concluded that the results for quality status on the Baotou–Shenmu heavy-haul line obtained using the status evaluation system proposed in this paper are consistent or close to those of the Baotou–Shenmu maintenance district in most cases. Our system appropriately reflects the actual status of the entire line and evaluates the actual quality status of the heavy-haul line more comprehensively. This indicates that the method proposed by this paper effectively assesses the status of heavy haul lines. This method considers the track geometry, the track structure, and the freight volume.

In addition, by comparing the overall evaluation scores obtained using the two different methods (Table 9), it may be seen that the scores obtained using the evaluation system proposed in this paper are slightly higher than those obtained by the Baotou–Shenmu maintenance district. Using the method proposed in this paper, the results for Line 5 indicate that it is safe for continued operation, in contrast to the results produced by the Baotou–Shenmu maintenance district. In other words, the service lives of rails and other components are determined to be longer using the method proposed in this paper, with consequent potential savings in human, material, and financial costs.

7. Conclusions

(1) In the present study, we constructed a score deduction coefficient table for long and steep uphill sections of railway lines. Indicators for such sections were divided into three categories: track geometry, track structure, and track freight volume. Line tracks were subsequently divided into the four quality levels of I, II, III, and IV, indicating excellent, qualified, repairable, and disposable, respectively.

(2) The analytical hierarchy process (AHP) was used to comprehensively evaluate and calculate the three criteria indicators for uphill working conditions, obtain each indicator’s weight, and construct an evaluation system for working conditions on uphill sections of heavy-haul lines. The three weights of the criterion layer indicators were 7.377%, 64.339%, and 28.284%, respectively; the weights of the track geometry indicators were 58.476%, 23.513%, 8.847%, and 9.164%; and the weights of the track structure status indicators were 70.144%, 8.532%, and 21.324%, respectively.

(3) The OAT method was used to adjust the weights of key indicators in the system. When the weight value of a main indicator fluctuated by 10%, the maximum percentage change of each level was 3.409%, which verified the proposed system’s reliability and stability.

(4) Data for seven sections of the line were selected for evaluation using the proposed method, and our results were compared with those obtained by the Baotou–Shenmu Railway maintenance district. Using our method, an additional section of the line qualified as fit for continued operation. In conclusion, the method proposed in this paper can describe performance on uphill sections on heavy-haul lines more accurately, enabling the service life of corresponding track sections to be extended based on sound scientific knowledge.