J. Cent. South Univ. (2018) 25: 843-854

DOI: https://doi.org/10.1007/s11771-018-3788-6

Geological adaptability matching design of disc cutter using multicriteria decision making approaches

XIA Yi-min(夏毅敏)^{1, 2}, LIN Lai-kuang(林赉贶)^{1, 2}, WU Dun(吴遁)³, JIA Lian-hui(贾连辉)⁴, CHEN Zhuo(陈卓)^{1, 2}, BIAN Zhang-kuo(卞章括)^{1, 2}

1. State Key Laboratory of High Performance Complex Manufacturing, Central South University,Changsha 410083, China;

2. College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China;

3. China Railway 14th Construction Bureau Co., Ltd., Jinan 250014, China;

4. China Railway Engineering Equipment Group Co., Ltd., Zhengzhou 450016, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: Geological adaptability matching design of a disc cutter is the prerequisite of cutter head design for tunnel boring machines (TBMs) and plays an important role in improving the tunneling efficiency of TBMs. The main purpose of the cutter matching design is to evaluate the cutter performance and select the appropriate cutter size. In this paper, a novel evaluation method based on multicriteria decision making (MCDM) techniques was developed to help TBM designers in the process of determining the cutter size. The analytic hierarchy process (AHP) and matter element analysis were applied to obtaining the weights of the cutter evaluation criteria, and the fuzzy comprehensive evaluation and technique for order performance by similarity to ideal solution (TOPSIS) approaches were employed to determine the ranking of the cutters. A case application was offered to illustrate and validate the proposed method. The results of the project case demonstrate that this method is reasonable and feasible for disc cutter size selection in cutter head design.

Key words: tunnel boring machine (TBM); disc cutter; matching design; evaluation method; multicriteria decision making (MCDM)

Cite this article as: XIA Yi-min, LIN Lai-kuang, WU Dun, JIA Lian-hui, CHEN Zhuo, BIAN Zhang-kuo. Geological adaptability matching design of disc cutter using multicriteria decision making approaches [J]. Journal of Central South University, 2018, 25(4): 843–854. DOI: https://doi.org/10.1007/s11771-018-3788-6.

1 Introduction

Tunnel boring machines (TBMs) are widely used in tunnel construction worldwide due to their high construction efficiency, strong environmental adaptability, safety and reliability [1–4]. As a key component of the TBM cutter head, the disc cutter contacts and interacts with the rock directly, as shown in Figure 1. The disc cutter is the most efficient and popular cutting tool for rock breaking and is used frequently on different types of TBMs [5, 6].

With the development of TBM technology, the size of disc cutters has developed from the original 11 and 12 inch designs to the current 17, 18, 19, and 20 inch designs, with 21 inch disc cutters also having been studied and applied, as shown in Table 1 [7]. The disc cutter size not only determines the basic cutter structural parameters, such as the cutter ring diameter, cutter tip width and weight, but also affects the overall cutter performance. Different disc cutters sizes bring about obvious differences in their rock-cutter contact length, load rating, allowable wear volume and rock breaking capacity. The impact loads that the cutter ring can bear are also diverse, which results in significant differences in cutter service lifetimes. The ultimate load capacities of disc cutters change with their size owing to their bearing sizes. The economics of disc cutters with different sizes varies considerably due to the manufacturing costs. In addition, the cutter spacing and layout should be designed appropriately according to the chosen size of disc cutter. Furthermore, the structural parameters of the cutter housing should be aligned with the cutter size, which may influence the design and manufacturing of the entire cutter head. Therefore, the matching design of the disc cutter size based on the geological conditions of the construction site is the prerequisite for the cutter head design, which is important for the TBM manufacturing and construction units.

Figure 1 Schematic diagram of TBM

Table 1 Disc cutter sizes and load capacity development

Currently, several researchers have been working on cutter selection and design. ROBY et al [7] introduced recent improvements in the disc cutter components, including the cutter ring sizes, materials, lubricants, bearings, seals and cutter condition monitoring, and emphatically analyzed the distinctions of disc cutters of different sizes. ZHANG et al [8] preliminarily compared the rock breaking efficiency and wear resistance between 17 inch disc cutters and 19 inch disc cutters through project case studies. XIA et al [9] proposed a cutter selection method for shield machine based on fuzzy theory and applied it in practice. BILGIN et al [10] compared the performances of disc cutters and chisel tools used on the same TBM under difficult ground conditions and analyzed the effect of replacing the disc cutters with chisel tools on the TBM performance. SUN et al [11] studied the influence of the structural parameters of the cutter on its rock breaking capacity and obtained the optimized edge width and angle based on the mapping relationship. CHO et al [12] analyzed the rock cutting behavior under different cutter tip parameters based on LCM tests and three- dimensional numerical analysis. CHIAIA [13] studied rock breaking mechanisms using different types of cutters, including blunt rigid sphere, flat punch and circular cone. BALCI et al [14] compared the rock breaking performances of the V-type disc cutter and the CCS-type disc cutter through experimental and theoretical considerations. MARJI [15] analyzed crack coalescence during the cutting action of single and double type disc cutters by numerical methods and investigated the effects of specific cutter parameters on the load and specific energy. HUO et al [16] designed a new type of disc cutter under sliding support with an obvious vibration reduction. ZHANG et al [17] introduced the failure form of disc cutters and developed a new material for cutter ring which can improve cutter performance greatly. From the above, it can be found that the existing research of cutter selection and design mainly focused on the cutter types, cutter tip shapes, cutter tip number (single or double), bearing types, ring materials and cutter ring structural design. However, research on the selection and matching design of the cutter size is seriously insufficient. The existing research is mostly focused on the development of the disc cutter size or qualitative conclusions based on the construction experiences. For example, cutters with large sizes can be applied to hard rock conditions in view of the high bearing capacity and large wear volume, whereas a small-sized cutter with excellent economy and reliability can be employed in soft rock conditions. In the disc cutter matching design, there is no feasible theoretical method to evaluate and judge the influence of each factor on the cutter size selection, which is an urgent problem that needs to be solved immediately.

In this work, a novel cutter performance evaluation method was developed for cutter matching design. The fuzzy comprehensive evaluation, TOPSIS, AHP and matter element analysis were combined to evaluate the potential cutters and determine the optimal size of the disc cutter.

2 Determination of disc cutter evaluation criteria

In this study, the main purpose of the geological adaptability matching design for disc cutters is to evaluate the cutter performance and select the optimal cutter size among alternatives in the specific geological conditions. To perform cutter matching design, an evaluation model needs to be established, and the disc cutter performance evaluation criteria need to be determined. The factors affecting the disc cutter performance are diversity, multiplicity, relevance and fuzziness. The overall performance of the disc cutter depends on the correlation between factors and the adaptability to geology. As a complex fuzzy multicriteria decision making (MCDM) problem, a hierarchical evaluation model needs to be established to systematically express cutter performance evaluation criteria and then to select the best alternative with respect to criteria under consideration. Combining with quantitative and qualitative criteria, the disc cutter performance evaluation criteria are composed of the cutter structural performance, rock breaking performance, economy, geological adaptability, safety and reliability.

Since the wear properties and bearing capacity of the disc cutter are related to its structural parameters [18], the cutter ring diameter, weight, ultimate wear capacity and ultimate load capacity are selected as the cutter structural performance criteria. Since the rock breaking capacities of cutters with different sizes are substantially different, the cutter penetration capability, load rating and rock-cutter contact length are selected as the cutter rock breaking performance criteria. The economy, geological adaptability, safety and reliability of the disc cutter are selected as its performance qualitative criteria considering the cost of the cutter design and manufacturing, time and cost of cutter replacement, reliability of construction, and other aspects. The hierarchical performance evaluation model of the disc cutter, as depicted in Figure 2, can be divided into three levels. Level 1 represents the evaluation alternatives with different cutter sizes. The primary criteria are on the second level consisting of the cutter structural performance, rock breaking performance, economy, geological adaptability, safety and reliability. Level 3 comprises the secondary criteria, including the cutter ring diameter, weight, etc..

Figure 2 Hierarchical performance evaluation model of disc cutters

3 Evaluation and decision of disc cutter performance

Selecting appropriate evaluation approaches for the comprehensive decision of disc cutters with different sizes is the key to the cutter matching design. The commonly used comprehensive evaluation approaches include grey correlation analysis, fuzzy comprehensive evaluation, principal component analysis, TOPSIS (technique for order preference by similarity to an ideal solution) approach and approximation theory [19–22]. The fuzzy TOPSIS approach, which has been used for a variety of applications in fuzzy multicriteria decision making problems, is suitable for the disc cutter performance evaluation and decision [23].

A novel framework based on the combination of fuzzy comprehensive evaluation and fuzzy TOPSIS approach is presented to evaluate the cutter performance considering the multi-level structural characteristics of the evaluation model. Specifically, the fuzzy comprehensive evaluation approach is employed to evaluate the secondary criteria of cutter performance to form the fuzzy decision matrix. Then, the fuzzy TOPSIS approach is used to evaluate the cutter performance and determine the final order of alternatives so that the decision makers can choose the most suitable cutter size for TBMs.

3.1 Fuzzy comprehensive evaluation approach

The fuzzy comprehensive evaluation is an application of fuzzy mathematics which makes the comprehensive evaluation under the principle of fuzzy transformation and maximum membership degree [24, 25]. The fuzzy comprehensive evaluation procedure applied to the disc cutter evaluation can be described as follows:

Step 1: Determine the factor set of the evaluated object.

                       (1)

Step 2: Determine the evaluation comment set.

                       (2)

where m is the number of ranks.

Step 3: Establish the fuzzy relationship matrix R from U to V.

  (3)

where r_ij represents the fuzzy membership of factor u_i aiming at the comment v_j.

Step 4: Determine the vector for the factor weights.

                      (4)

Step 5: Obtain the comprehensive evaluation result vector.

                   (5)

where b_j is the membership degree value of the jth evaluation criterion with respect to each evaluation comment.

3.2 Fuzzy TOPSIS approach

The TOPSIS approach, introduced by HWANG et al [26], is based on the principle that the chosen alternatives should have the shortest distance from the ideal solution and the farthest distance from the negative ideal solution. To deal with the difficulty in determining precise ratings and weights for the criteria and alternatives, the fuzzy TOPSIS approach is proposed to solve real world problems in the fuzzy environment [27]. The merit of using a fuzzy approach is to determine the importance or preference of the criteria and alternatives using fuzzy numbers instead of crisp numbers, suited to the real world in a fuzzy environment [28]. The fuzzy TOPSIS approach procedure applied to the disc cutter performance evaluation can be described as follows:

Step 1: Determine the alternatives of disc cutter size and choose the linguistic ratings for alternatives with respect to the criteria.

The ratings of the alternatives from the decision makers are considered as linguistic terms in the fuzzy environment, as shown in Table 2. For instance, if the response of one decision maker for the preference rating of one alternative under consideration is fair (F), the corresponding triangular fuzzy number of that alternative should be (0.3, 0.5, 0.7).

Table 2 Linguistic ratings for alternatives

Step 2: Construct the fuzzy decision matrix.

Suppose that the number of alternatives is m and the number of criteria is n, then the fuzzy decision matrix can be represented by Eq. (6):

                   (6)

                      (7)

where x_ij is the aggregated rating of the ith alternative A_i with respect to the jth criterion C_j, and w_j represents the importance weight of criterion C_j.

Step 3: Establish the normalized matrix.

The disc cutter normalized fuzzy decision matrix is established based on the fuzzy decision matrix. Equations (8) and (9) are used to transform the different criteria scales into a comparable scale.

The larger, the better type:

                  (8)

The smaller, the better type:

                (9)

The normalization of the fuzzy decision matrix is represented by Eq. (10):

    (10)

whereis the normalized value of and

Step 4: Construct the weighted normalized fuzzy decision matrix.

By multiplying the weights of the criterion with the normalized fuzzy decision matrix the disc cutter weighted normalized fuzzy decision matrix is defined as:

     (11)

where

Step 5: Calculate the fuzzy positive ideal solution (FPIS A⁺) and the fuzzy negative ideal solution (FNIS A^–).

                  (12)

                  (13)

where and

Step 6: Calculate the separation of each alternative from the positive and negative ideal solution.

The distance and between each cutter size and the fuzzy ideal solution (A⁺ and A^–) can be calculated as follows:

            (14)

            (15)

The distance between two triangular fuzzy numbers and can be calculated by:

 (16)

where and

Step 7: Calculate the closeness coefficient and rank the order of each alternative.

The closeness coefficient (CC_i) representing the separation to the fuzzy ideal solutions can be calculated as follows:

                          (17)

The greater the value of CC_i, the better the performance of alternative A_i will be.

4 Determination of importance weights of evaluation criteria

The AHP is applied to obtaining the weights among the factors by establishing hierarchical structure models, which has been widely used to tackle MCDM problems [29]. However, the weights of the criteria determined by experts opinions are accompanied with subjectivity, one-sidedness and uncertainty, which may lead to unreliable judgments. To overcome this issue, the matter element analysis combined with AHP is used. Considering the closeness between the experts assessments and the objective value of the criteria, an expert validity matrix is established by each expert judgment from which the weights of the criteria can be accurately obtained [30]. The proposed method has following six steps:

Step 1: Construct the weight pairwise comparison matrix.

Several experienced experts in the field of TBM are invited to fill out a questionnaire and construct the weight pairwise comparison matrix of each criterion using the linguistic variables presented in Table 3.

Step 2: Determine the composite matter element matrix.

Suppose that the number of criteria is n, the count of experts is m, and Z_i is the pairwise comparison matrix provided by the ith expert,while θ_ij is the evaluated weight of the jth cutter performance evaluation criterion obtained from the consistent testing of the pairwise comparison matrix Z_i, then, the composite matter element matrix can be constructed as follows:

             (18)

Step 3: Determine the classical domain matter element matrix, joint domain matter element matrix and the evaluation matter element matrix.

              (19)

            (20)

                    (21)

where R_pj, R_0j, R^x_j are the joint domain matter element matrix, the classical domain matter element matrix and the evaluation matter element matrix of the jth criterion, respectively.

Table 3 Linguistic terms for evaluation criteria

Step 4: Determine the correlation function matter element matrix.

The correlation function matter element matrix consisting of the correlative degrees is defined as:

    (22)

where K(x_ij) is the correlative degree provided by the ith expert with respect to the jth cutter performance criterion, and it is defined as follows:

             (23)

In this equation:

(24)

 (25)

where and are the distances between x_ij and the classical domain x_0ji and the joint domain x_pj, respectively.

Step 5: Determine the expert validity matter element matrix.

The expert validity coefficient is calculated through the following equation:

              (26)

where k_i is the coefficient denoting the overall cognitive level of the ith expert on the performance criteria, and

The expert validity matter element matrix is constructed as below:

              (27)

Step 6: Obtain the weights of the cutter performance evaluation criteria.

The correction matter element matrix R_f is obtained by modifying the expert validity matrix as Eq. (28):

        (28)

where

The weight matter element matrix R_w for the cutter performance evaluation criteria can be constructed as follows:

                (29)

where w_j is the weight of the jth criterion and

5 Initialization and normalization of evaluation criteria

The evaluation criteria for the disc cutter performance evaluation model should be initialized and normalized. An expert investigation approach is used to acquire the linguistic ratings for the qualitative criteria based on the fuzzy TOPSIS approach. The qualitative criteria fuzzy initial matrix is constructed by transforming the expert judgments into triangular fuzzy numbers. There are three quantitative criteria initialization methods in this study. The disc cutter structural performance quantitative criteria can be obtained through access to the relevant information, the load rating in the cutter rock breaking performance criteria can be achieved through finite element analysis, and the cutter penetration capability and rock-cutter contact length can be calculated with the corresponding function [31].

The fuzzy TOPSIS approach is used to normalize the qualitative criteria, while the equations for the relative membership degree are utilized to normalize the quantitative criteria through Eqs. (30) and (31).

The larger, the better type:

                          (30)

The smaller, the better type:

                        (31)

According to the above research, the process of the disc cutter geological adaptability matching design method can be summarized as follows: Identify the disc cutter evaluation criteria, calculate the weights of the criteria, collect the evaluation data, construct the fuzzy decision matrix, comprehensively evaluate and make a decision, as shown in Figure 3.

Figure 3 Schematic diagram of geological adaptability matching design method for disc cutter

6 Project case

The geological adaptability matching design method is used to select the suitable cutter size for two TBMs in a diversion project located in northeast China. The length of the TBM construction is nearly 20 km, and a hard rock section is selected as the research subject in this study. The geologic formations in this stage consist of granite, tuff, sandstone and diorite rocks, and the main rock types along the tunnel alignment are II-type and III-type granites. The rock mass is relatively intact and the joints are generally undeveloped. The rock compressive strength is 90– 142 MPa, as measured through rock sampling, signifying hard rock.

6.1 Initialization of evaluation criteria

In view of the geological conditions, four commonly used cutter sizes were selected as the decision alternatives, including the 17-inch (A1), 18-inch (A2), 19-inch (A3), and 20-inch (A4). The quantitative criteria were initialized using the aforementioned initialization method, and the initial values of the quantitative criteria are shown in Table 4.

Table 4 Initial values of quantitative criteria

To attain the initial values of the qualitative criteria, TBM experts with a high degree of experience were invited to fill out a questionnaire survey, and the individual fuzzy decision matrix was established via the scale listed in Table 2. Then, the arithmetic means were computed to obtain the initial values of the criteria using Eq.(32):

                    (32)

where is the fuzzy weighted value of the ith alternative with respect to the jth criterion.

The initial values of the qualitative criteria are shown in Table 5.

Table 5 Initial values of qualitative criteria

6.2 Calculation of weights for evaluation criteria

After the evaluation criteria were initialized, the weights of the disc cutter performance evaluation criteria were calculated using the AHP and matter element analysis. In this study, eight TBM experts were invited to determine the rankings and weights of the criteria. To demonstrate the process of the proposed method, the weights of the cutter structural performance criteria were computed as an example. For instance, a questionnaire survey filled out by one of the experts is listed in Table 6. Afterwards, a pairwise comparison matrix (Z₁) was built from the expert comments based on the linguistic scale, as listed in Table 3.

Table 6 Evaluations from one of experts in linguistic variables and weights of criteria

The composite matter element matrix was formed by the consistency test of each individual pairwise comparison matrix as below:

The classical domain matter element matrix, joint domain matter element matrix and the evaluation matter element matrix were determined by Eqs. (19), (20) and (21), respectively. Then the correlation function matter element matrix was computed using Eqs. (22), (23), (24) and (25):

The expert validity matter element matrix was calculated as follows:

The weight matter element matrix for the structural performance evaluation criteria was obtained through Eqs. (28) and (29):

All of the mentioned weights for each level criteria were computed and summarized in Table 7.

Table 7 Weights of evaluation criteria

6.3 Evaluation and decision of cutter performance

To obtain the primary quantitative criteria, the fuzzy comprehensive evaluation was applied to evaluate the secondary criteria of the cutter performance. The relative membership degree values of secondary quantitative criteria were calculated by Eqs. (30) and (31), as shown in Table 8.

As an example, the fuzzy relationship matrix for the structural performance criteria was described as follows:

Table 8 Relative membership degree of quantitative criteria

The results of the comprehensive evaluation for the structural criteria were calculated using Eq.(5):

A similar calculation was performed for the rock breaking performacne ctriteria, and the evaluation results of the quantitative criteria are shown in Table 9.

Table 9 Evaluation results of quantitative criteria

The final fuzzy decision matrix of the primary criteria was constructed by integrating the evaluation results of the qualitative criteria (Table 5) and quantitative criteria (Table 9) after those values were transformed into triangular fuzzy numbers, and the decision matrix is depicted in Table 10. The normalized matrix was formed using Eqs. (8), (9) and (10). Afterwards, the weighted normalized fuzzy decision matrix was established by multiplying the weights of the primary criteria, obtained from the AHP and matter element analysis, with the normalized matrix as shown in Table 11.

The fuzzy positive and negative ideal solutions of each alternative for the criteria were calculated by Eqs. (12) and (13), and the results are shown in Table 12.

Table 10 Fuzzy decision matrix of primary criteria

Table 11 Weighted normalized fuzzy decision matrix

Table 12 Fuzzy positive and negative ideal solutions

The distances of each alternative from the fuzzy ideal solutions were calculated using Eqs. (14) and (15), and the results are listed in Table 13. The closeness coefficient of the four alternatives can be calculated using Eq.(17), as presented in Table 13. According to the CCi values, the ranking order of the four alternatives is depicted in Figure 4.

Table 13 Closeness coefficient of alternatives

Figure 4 Evaluation result of alternatives according to CCi values

According to the CCi values for the four alternatives, the ranking order of the cutter sizes is A3>A4>A1>A2, which means that the 19-inch cutter is the best and most suitable choice for the TBMs in the diversion project.

Based on the evaluation results for the primary criteria, the structural performance, rock breaking performance and geological adaptability of the alternatives A3 and A4 are obviously superior to those of A1 and A2, which means the cutter with a larger size provides a higher load rating and a larger allowable wear volume, making it more suitable for breaking rocks in hard rock tunneling conditions. The 19-inch disc cutter has the highest score for safety and reliability performance due to its wide and effective application. The 17-inch cutter is the winner for economy because of its lower manufacturing cost.

The 19-inch cutter was selected as the normal and gage disc cutter for the two TBMs applied to the diversion project based on the results of the cutter matching design method. So far these two TBMs have tunneled 8.5 km and 10.9 km with excellent performance and high construction efficiency. The application of the proposed method demonstrates its effectiveness and reliability, and it is expected that this method may serve as an assistance tool for TBM cutterhead design.

7 Conclusions

1) Geological adaptability matching design of disc cutters is the first step in the structural design of the TBM cutter head. According to the geological conditions, selecting the appropriate cutter size is an effective means to improve the rock-breaking efficiency and service life of disc cutters. In this paper, a disc cutter performance evaluation and decision method was presented for determining the cutter size based on a combination of multicriteria decision making approaches.

2) A hierarchical cutter performance evaluation model consisting of qualitative and quantitative criteria was presented considering the influencing factors of the cutter size selection. The AHP and matter element analysis approaches were used to obtain the weights of the cutter performance criteria. The fuzzy comprehensive evaluation and fuzzy TOPSIS approaches were employed to evaluate and prioritize the alternatives. The weights acquired from the AHP and matter element analysis were involved in the cutter performance evaluation, being used in the fuzzy comprehensive evaluation and fuzzy TOPSIS. The most suitable cutter size was determined on account of the closeness coefficient.

3) A case application was used to illustrate and validate the proposed method. The optimal size of the disc cutter was determined for two TBMs in a diversion project using the evaluation method. The construction results indicate that this approach may provide a scientific, effective and reliable measure for evaluating the disc cutter performance and determining the best cutter size in the design of a TBM.

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(Edited by HE Yun-bin)

中文导读

基于多准则决策的盘形滚刀地质适应性设计方法

摘要：盘形滚刀地质适应性设计是隧道掘进机(TBM)刀盘设计的重要环节，直接影响TBM的整体掘进效率。滚刀匹配设计的关键是通过滚刀性能评价选择适宜的滚刀尺寸。本文提出一种基于多准则决策的新型评价方法用以确定滚刀尺寸，以不同型号滚刀为设计对象，建立滚刀性能评价递阶层次模型，采用层次分析和物元理论计算滚刀性能指标权重值，并通过模糊综合评价法和模糊TOPSIS决策法对不同型号滚刀性能进行综合评判与决策。将滚刀地质适应性设计方法应用于某水利工程TBM刀盘正面滚刀设计，结果表明该方法有效可靠，可为刀盘设计过程中滚刀性能评价和匹配设计提供一定理论依据。

关键词：隧道掘进机；盘形滚刀；匹配设计；评价方法；多准则决策

Foundation item: Project(51475478) supported by the National Natural Science Foundation of China; Project(2013CB035401) supported by the National Basic Research Program of China; Project(2012AA041801) supported by the National High-tech Research and Development Program of China; Project(CX2014B058) supported by the Hunan Provincial Innovation Foundation for Postgraduate, China

Received date: 2016-11-28; Accepted date: 2017-05-31

Corresponding author: LIN Lai-kuang, PhD; Tel: +86-731-88876926; E-mail: linlaikuang34@163.com; ORCID: 0000-0001-9331- 3302