Applications of Fuzzy Theory-Based Approaches in Tunnelling Geomechanics: a State-of-the-Art Review

Abstract

The first introduction of fuzzy theory in the nineteenth century created room for continuous research and application in various fields. Fuzzy set theory has been globally applied in geotechnical engineering, and research in this field continues to date. The fuzzy inference system is considered to be one of the most popular techniques adopted to resolve some of the geomechanical challenges faced in both surface and underground excavations. This paper unpacks fuzzy theory-based approaches in mine geomechanics with the aim of expanding the innovative application of the same approach specifically in tunnel geomechanics. This aim was achieved by conducting a review of recent successful and unsuccessful applications of fuzzy inference systems in underground excavations/tunnelling geomechanics. Indeed, this review has outlined some cardinal points associated with the ability of the technique to solve complex geomechanics problems. However, the success of the technique was accompanied by a few limitations associated with the methodology. Finally, a future outlook associated with the technique has been established.

1 Introduction

The majority of problems in geotechnical engineering are too complex to solve using conventional methods. This is because conventional methods such as rock mass classification are limited to the extent of their application. In most geotechnical problems, traditional descriptors such as the rock mass rating (RMR), rating index (for geotechnical parameters such as unconfined compressive strength in rock mass rating charts) and rock quality designation (RQD) are usually applied. Nevertheless, there are always subjective uncertainties associated with these conventional descriptors. For example, ambiguous words and phrases such as highly weathered, very blocky and large water inflow are used to describe geotechnical parameters that are rated using index values. When using the RQD approach, associated uncertainties result from core logging, core handling and alteration to the degree of fracturing [1].

Historically, the overlaying method was one of the techniques implemented to solve uncertainty. The method was first established by Ellison [2] with follow-up studies by Ellenberger [3] and Seegmiller [4]. The technique uses a popular concept of overlaying hazard maps to identify the most hazardous areas. Geological changes, mining factors and morphology are other ways to characterize hazardous areas. For example, areas with multiple geological structures can be considered more hazardous than areas with few or no geological structures. In other words, the overlaying technique is closely related to the so-called hazard map rating—a rating commonly applied in rock burst environments. An example of a hazard map can be seen in the study of Trifu and Suorinemi [5]. The following three factors were used to determine the ratings in this example: (1) degradation of the rock mass, (2) the availability of seismically active planes and (3) the presence and abundance of geological features. According to Huisman et al. [6], stress release, weathering and erosion of the exposed rock mass are the main causes of rock mass degradation. As such, the degradation of the rock mass has the potential to result in slope instability.

Hazard maps are developed using tracing maps and plan maps. Tracing maps consist of zones or shades of likelihood of hazard occurrences. Plan maps consist of geotechnical rating grids. Tracing maps are overlaid on plan maps, and subjective judgement is made based on zones with multiple overlaying geological parameters. One key drawback of the hazard map rating technique is that its accuracy has been found to be at an inadequate level of significance in various studies, such as those of Dhital [7], Stein et al. [8] and Gaspar-Escribano et al. [9]. The other common challenge is that hazard maps are based on incidents that occurred along the mining panel (e.g. [10]). Hazard maps do not consider laboratory testing and analytical results. In addition, each event at the mines occurred due to different factors, such as poor support design, poor support installation, ground conditions, the influence of geological features, the depth of mining and stresses acting around the excavation [11,12,13]. Consequently, it was found to be too complex to rely on the overlay method [10]. The complexity in the overlay method led to the development of a research approach known as fuzzy set theory. Fuzzy set theory (FST) was intended to address problems involving uncertainties, ambiguity and subjective judgement. The methodology adopted in FST to solve these problems is encapsulated in what is known as the fuzzy inference system [14]. A fuzzy inference system (FIS) is a form of artificial intelligence that enables one to mimic any system in ways that human beings think when solving problems.

The idea of fuzzy set theory contains fuzzy logic. In essence, fuzzy logic involves inferring new knowledge from existing data. Fuzzy logic is a computing methodology that is based on the degree of truth of a statement and is one of the well-known machine learning and artificial intelligence methods in data science. In fact, data are used to answer questions following either supervised or unsupervised learning, as shown in Fig. 1. The available data determine the machine learning algorithm category and the algorithm type utilized to solve the problem. According to Alloghani et al. [15], supervised and unsupervised learning vary due to the nature of available data and the objective of the output. Unsupervised learning involves many variables and unknown data, whereas supervised learning is characterized by a series of known variables. Therefore, FIS provides a method for encoding information that is based on experience into logical principles that a machine can comprehend. FIS is also known as the fuzzy logic controller or the fuzzy rule-based system (MathWorks [16]). In this review paper, the concept and use of FIS are further covered in depth.

Fig. 1

Machine learning flowchart (modified after [17])

The studies of Nguyen [1] and Nguyen and Ashworth [18] are the earliest known studies that outlined the successful application of the FIS in geomechanics. The first study proposed the application of FIS in decision-making processes associated with mining geomechanics problems. Here, Nguyen [1] introduced a technique to remedy uncertainties from previous conventional (i.e. rating index) and overlaying methods. This was done by incorporating expert knowledge, Bieniawski’s rock mass rating (RMR) system and Barton’s quality index (Q) as input parameters of the FIS-based model [1, 18]. Bellman and Zadeh [19] and Zadeh [20] advocated for the minimum to maximum operations for multiple-criteria modelling. Later in 1985, Nguyen and Ashworth were able to illustrate this technique by incorporating various methods and expertise as outlined above. Nguyen and Ashworth [18] concluded that elements of fuzzy set theory and operations may be suitable for application in mining geomechanics problems.

After conducting a critical review in various fields of application, geomechanics was deemed most suitable for fuzzy theory-based approaches. Scholars such as Hossein et al. [21], Mahdevari et al. [22], Mahdevari and Torabi [23] and Samimi Namin et al. [24] have shown evidence of the above statement. To date, FIS has been applied in rock mass classification, slope stability, tunnelling, foundation analysis, geotechnical project scheduling and cost planning, to name a few examples. The wide adoption of the FIS technique in solving geomechanics problems since its establishment is primarily ascribed to the accuracy and reliability of the produced results. Adoko et al. [25] exemplified the use of the FIS technique to solve geomechanical problems very well. This paper examines fuzzy theory-based approaches in mine geomechanics with the goal of advancing creative use of the same methodology, specifically in tunnel geomechanics. The application of fuzzy inference systems in geomechanical problems was therefore reviewed—the aim of which was to identify the most common technique used in solving geomechanical problems. This was achieved by examining the recent innovative application of fuzzy inference systems in mining geomechanics. Finally, this paper portrays the use and success of the FIS method in mine geomechanics with the purpose of outlining the recent updates and identifying the knowledge gap regarding the FIS technique.

2 Overview of Fuzzy Set Theory

The use of fuzzy reasoning and knowledge-based representation is broad when designing model algorithms in machine learning and artificial intelligence (AI). Fuzzy sets enable one to solve problems with uncertainties, ambiguity and/or incomplete information. Black [26], an American philosopher, first introduced the concept of fuzzy sets by conducting an exercise on vagueness in logical analysis. However, he was opposed by a traditional mathematician and did not continue with the concept. A few decades later, Zadeh [27] reintroduced the concept of fuzzy sets to address uncertainty. Zadeh [27] reasoned that there are many uncertainties that cannot be addressed using probability theory only, as it works based on the classic or crisp set. The section below unpacks the concept of fuzzy set theory, fuzzy logic and typical fuzzy inference techniques suggested.

2.1 The Concept of Fuzzy Set Theory

The term fuzzy simply means that something is unclear or vague (Oxford [28]). The concept of fuzzy sets is used in various disciplines, such as computer science, artificial intelligence, medicine and engineering decision-making theory. All these disciplines and others use a computer system to reason and decide like human beings. Most traditional reasoning mechanisms are crisp. In other words, the traditional reasoning mechanisms are represented by values (1) and (0), which are true or yes and false or no, respectively. However, fuzzy sets are different because they integrate vagueness by incorporating values between 1 and 0 to display the array of possibilities. Nguyen [1] described fuzzy sets as a system that totally depends on ambiguous and imprecise information in making appropriate decisions. It thus appears that fuzzy sets generalized the crisp set theory for them (fuzzy sets) to derive a reliable and reasonable solution. For instance, considering the example in Fig. 2, if one is to comment on the degree of weathering of a rock using a crisp set, the answer can only be yes (1) or no (0). In contrast, fuzzy sets will answer the same question in a range of possibilities. For instance, one can say the rock is extremely weathered (1), very weathered (0.75), moderately weathered (0.5), slightly weathered (0.25) or not weathered (0). Therefore, fuzzy set logic always has room for uncertainties. When incorrect measurements are taken or a model ignores certain effects, systematic uncertainties result and ultimately make it difficult to solve the problem at hand. Interestingly, fuzzy set theory has been reported by several authors (e.g. [25, 29]) to be capable of well representing not only systematic uncertainties but also the concepts of gradualness and bipolarity.

Fig. 2

Crisp set and fuzzy set illustrated (after [30])

Generally, a crisp set can be defined as a collection of elements with the same characteristics. In contrast, a crisp set can be mathematically represented in three ways: first, as a collection of all the elements (see Eq. 1); second, as a collection of elements such that the elements have the properties of P(x) (see Eq. 2); and last, as a characteristic function. In simple terms, the characteristic function representation defines whether an element is a member or a nonmember of the crisp set. Hence, crisp sets can only allow full membership functions or no membership function (see Eq. 3).

$$A={a}_1+{a}_2+{a}_3+\dots +{a}_n$$

(1)

$$A=\left[x\right|P(x)\Big],P: Properties$$

(2)

$${\mu}_a(x)=\left\{\begin{array}{c}1\ if\ x\ belongs\ to\ A\\ {}0\ if\ x\ does\ not\ belong\ to\ A\end{array}\right.$$

(3)

Fuzzy sets are elements with imprecise or vague boundaries and work based on the crisp set. Nevertheless, fuzzy sets can address imprecision and uncertainties. In other words, fuzzy sets are more general concepts of a crisp set. Engineering judgement involves uncertainties to some extent, hence the introduction of fuzzy set theory in rock engineering. When precise reasoning cannot be given due to uncertain input data, fuzzy logic devices are a means for reasoning. Algorithms in fuzzy logic use all the input data while solving a problem. Thus, the best decision viable is taken corresponding to the input elements. Mathematically, a fuzzy set can be expressed by Eq. (4). If a collection of objects is represented by x, then a fuzzy set à in x is a set of ordered pairs defined as follows [31]:

$$\overset{\sim }{A}=\left\{\left(x,{\mu}_{\overset{\sim }{A}}(x)\Big)|x\in X\right)\right\}$$

(4)

Fuzzy sets are characterized by membership functions (MF). MFs in fuzzy set theory are used to convert crisp sets into fuzzy sets. The MF of a fuzzy set is represented by ${\mu}_{\overset{\sim }{A}}(x)$ in the equation. The membership function generally represents x in the membership space, where x is known as the universe of discourse. The universe of discourse can be written as a collection of ordered pairs. Fuzzy rules are represented by fuzzy relations as follows [32, 33]:

$$R=A\to B$$

(5)

R can also be viewed as two fuzzy sets with two-dimensional membership functions that are mathematically expressed as follows [32]:

$${\mu}_R\left(x,y\right)=f\left({\mu}_A(x),{\mu}_B(y)\right)$$

(6)

Function f is known as the fuzzy implication function because it transforms the degree of membership of x in A and y in B to (x, y) in A × B.

The membership function value is the degree of belongingness, which describes the similarity of an element to a particular class. Therefore, the membership function value of a fuzzy set ranges from 0 to 1. Considering the example presented in Fig. 2, subjective judgement is applied when classifying the condition of the weathered rock. As such, the judgement will vary from person to person; hence, it is defined as a fuzzy set because there is no defined boundary. A fuzzy set is therefore represented as the degree of belongingness based on the probability and membership function. Probability is the frequency of likelihood that an element belongs to a class and membership function defining the similarity of that element to a class. The availability of an element speaks to probability, whereas the guarantee speaks to membership function.

It has been observed and proven that a fuzzy set is a simple extension of the concept of a crisp set. Crisp sets are elements with fixed or well-defined boundaries. A fuzzy set has been noted to allow partial membership, which means that an element can be partially true or partially false—hence the range between 1 and 0. However, the crisp set strictly allows full membership or no membership to every element of the universe of discourse [25, 34,35,36,37,38,39]. For the sake of argument, the fuzzy set appears to be completely different from the crisp set in terms of their membership [30, 31, 40]. This can be simply explained by looking at the characteristic function of crisp set A in crisp set theory. It appears that the set takes the value 0 or 1 based on the given element of the universal set belonging to A or not, while the fuzzy set has a membership function with a value ranging from 0 to 1 [40,41,42].

There are various types of membership function distributions in fuzzy sets. Nonetheless, the six common MFs are the singleton MF, triangular MF, trapezoidal MF, Gaussian MF, generalized bell-shaped MF and sigmoid MF. These names follow the shapes formed when plotting the membership functions on a Cartesian plane. The x-axis represents the input value, and the y-axis represents the corresponding fuzzy value. The singleton MF is simple and straightforward in that only one parameter {c} on the x-axis has a corresponding fuzzy value of 1. The rest of the parameters have a corresponding fuzzy value of 0, as shown in Fig. 3A.

Fig. 3

Singleton, triangular and trapezoidal membership functions. (A) Singleton membership function, (B) Triangular membership function, (C) Trapezoidal membership function (after [43])

The triangular MF is one of the most applied membership functions in fuzzy controller design [43]. To address the fuzziness of the input, triangular MF uses three parameters {a, b, c}, with which the base of the triangle is defined by a and c, whereas the height is defined by Ben Yahia et al. [44]. For each value of x, the membership function is described by μ_A(x) → [0, 1]. Parameters a and c are the lower and upper boundaries on the x-axis, respectively, and their corresponding fuzzy values are zero. Parameter b is the centre membership whose corresponding fuzzy value is 1 (see Fig. 3B). A trapezoidal MF is defined by four parameters {a, b, c, d}, as shown in Fig. 3C. The highest membership value that the element can take is described by span a–b. This means that if elements in the x-axis are between (a, b) and (c, d), the corresponding fuzzy value will be between 0 and 1. Triangular and trapezoidal MFs have been used extensively because of their straightforward mathematical formulas and computational effectiveness [43].

The Gaussian MF is characterized by two parameters {m, σ}, where m is the mean or centre of the Gaussian curve and σ is the standard deviation and spread of the curve (see Fig. 4A). The Gaussian curve is more similar to a general way of representing data distribution. Nonetheless, it is less preferred for fuzzification because of its mathematical complexity [43]. The Gaussian membership function has been used successfully in the estimation of the elastic constant of rocks [45].

Fig. 4

Gaussian, generalized bell shape and sigmoid membership functions. (A) Gaussian membership function, (B) Generalized bell shape membership function and (C) Sigmoid membership function (after [43])

The generalized bell-shape MF is characterized by three parameters {a, b, c}, as shown in Fig. 4B. The width, centre and slope of the curve are represented by a, c and b, respectively [43]. The sigmoid MF (Fig. 4C) is typically applied when classifying tasks in machine learning. It is used mainly in logistic reasoning and neural networks. The sigmoid MF is characterized by two parameters {a, c}, which are used to suppress the input and map it to fuzzy values between 0 and 1. The slope at the crossover point x = c is controlled by parameter a. Based on the sign of parameter a, the sigmoid MF is characteristically open right or left; hence, it is appropriate to represent very large or very negative concepts [43].

2.2 Fuzzy Logic

Fuzzy logic is a computing methodology that is based on the degree of truth of a statement. In fuzzy logic, the representation of values is not limited to true (1) or false (0), as it is in crisp set logic. However, it is now expressed by linguistic variables of true and false (1-0) [31]. A fuzzy inference system (FIS) is the main element of a fuzzy logic system. As previously mentioned, the FIS is also referred to as the fuzzy logic controller. This is because it deals with decision-making using the IF-THEN rule (If X is A, then Y is B).

The fuzzy logic process involves three major stages between input and output (Fig. 5). The first stage is the fuzzification stage, where crisp input data or values are converted into linguistic variables through membership functions of the knowledge base. Second, the fuzzy inference engine evaluates the degree of input membership using fuzzy rules. This stage resembles the human decision-making process. The third and last process is the defuzzification of the fuzzy output into crisp values [46].

Fig. 5

Fuzzy logic process (after [46])

Nguyen [1] used the FIS to solve geomechanical problems in underground excavations concerning RMR [25, 37]. The technique applied the so-called Bellman–Zadeh fuzzy aggregation scheme, which is preferred for synthesizing hazard indices for mining excavations. The Bellman–Zadeh fuzzy aggregation scheme was applied to evaluate the hazard index associated with mining tunnels. The same technique was also successfully implemented on rock mass classification from Bieniawski’s system through integration of expert knowledge between the two concepts. It was also well established in evaluating Barton’s quality index Q provided that the information on various contributing ratings is fuzzy. Adoko and Wu [37] highlighted that this technique is also well established in several categories of geomechanics. The geomechanics categories include rock mass classification, slope stability, tunnelling, foundation analysis, geotechnical project scheduling and cost planning.

2.3 Fuzzy Inference Techniques

Basic fuzzy set systems are presented in a group of ‘IF-THEN’ rules to express the correlation of input and output variables in the system [47]. Computational aspects of an FIS can have a single rule with a single antecedent. For example, IF (x) is large, while THEN (y) is small. The terms big and small are fuzzy and should therefore be expressed by membership functions [48]. FIS can also be represented by a single rule with multiple antecedents. Alternatively, FIS can have multiple rules with multiple antecedents. This means that for each antecedent, there will be a consequent, i.e. Rule X_n: If condition x, THEN restriction y. For example, considering the ground conditions in a tunnelling project, Rule 1 can be structured as follows: IF the rock mass is blocky THEN areal coverage (shotcrete) must be installed. Rule 2 can be as follows: if the rock is intact, then no support is needed. Rule 3: If the rock mass is jointed, THEN cable anchors should be installed. There can be more than one rule in an FIS; however, each rule must have its own condition and consequence.

According to Yazdani-Chamzini [49], several models have been developed to address linear and nonlinear behaviour systems. However, the most important ones include the Mamdani systems, the Takagi–Sugeno–Kang (TSK) systems and the Tsukamoto systems. These fuzzy inference systems differ with regard to their rule of application [50].

The Mamdani system can be expressed as follows [51]:

$$If\ {X}_1 is\ {A}_1, and\ {X}_2 is\ {A}_2\ THEN\ Y\ is\ B$$

(7)

In Eq. (7), both inputs (X₁ and X₂) are fuzzy sets or fuzzy numbers. The output (B) is also a fuzzy set. This inference system consists of two cases of two input Mamdani systems: the max–min inference method and the max production method.

In comparison, the Takagi–Sugeno–Kang (TSK) system is given by [52]:

$$If\ {X}_1 is\ {A}_1, and\ {X}_2 is\ {A}_2\ THEN\ Y\ is\ f\ \left({X}_1,{X}_2\right)$$

(8)

In the Sugeno system, both inputs (X₁ and X₂) are fuzzy sets or fuzzy numbers [53]. However, the output is a function of the inputs as specified by the inputs in the fuzzy rule. The rule has an output represented as a crisp function [52].

Finally, the Tsukamoto system can be defined as follows [54]:

$$If\ {X}_1 is\ {A}_1, and\ {X}_2 is\ {A}_2\ THEN\ Y\ is\ B$$

(9)

For Tsukamoto, both inputs (X₁ and X₂) are fuzzy sets or fuzzy numbers, and the output (B) is also a fuzzy set [55]. The only difference between the Mamdani system and the Tsukamoto system is that the membership function of the output fuzzy set (B) is a monotonic function. This means that each fuzzy rule is represented by a fuzzy set with a monotonic membership function. Monotonic functions are characterized by successive values that are either increasing, decreasing or constant [56].

Ali et al. [57] noted that the Mamdani, Sugeno and Tsukamoto fuzzy models are the three fuzzy inference systems that have been widely used. The authors further outlined that the differences among the three commonly used fuzzy inference systems lie in the aggregation and defuzzification procedures because of their fuzzy rules. According to Adoko and Wu [37], there are three conceptual elements of defining a fuzzy inference system, namely, the rule base, the database and the reasoning mechanism. The rule base forms part of fuzzy rules, whereas the parameters for input membership functions are established from the database, and the reasoning mechanism permits opportunity for judgement of output functions using fuzzy logic—and consequently making satisfactory conclusions therein.

3 Applications of Fuzzy Inference in Mine Geomechanics

Fuzzy inference systems are widely adopted to resolve some of the geomechanical challenges faced in both surface and underground excavations. This section explains the application and success of the FIS technique in mine geomechanics, with the purpose of outlining recent updates on the technique.

3.1 General View of FIS Methods Applied in Mining Engineering

Fuzzy set theory is globally applied in geotechnical engineering, with fuzzy inference systems being the most prominent resource employed [37]. This section demonstrates how fuzzy inference systems have evolved and grown in their application and techniques over the years. Two models are reviewed: the Mamdani FIS and the Sugeno FIS. These models were selected based on their popularity in geomechanics. In addition, the neuro-fuzzy system is reviewed because it is starting to gain popularity in geomechanics and tunnelling. Neuro-fuzzy systems are formed by coupling fuzzy systems with neural networks. They are mostly known as adaptive neuro-fuzzy inference systems (ANFISs).

3.1.1 Mamdani Fuzzy Inference System

The Mamdani FIS was initially introduced to create a control system. This was achieved by synthesizing crisp sets into linguistic control rules. Linguistic rules are constructed based on the expertise and experiences of human beings [51]. In this system, the output of each rule is a fuzzy set from the output membership function. The Mamdani FIS is more suitable for expert knowledge because it is more intuitive, and it understands the rule base easily. The final crisp output value is computed through defuzzification of the fuzzy output. Advantages of Mamdani FIS include intuitiveness, suitability for human input, understandable rule base and widespread acceptance [58]. The Mamdani fuzzy logic system has attracted interest in the field of engineering [59]. It is said to have also gained popularity because of its attractive features, such as simplicity, the ability to model complex and uncertainty problems effectively, decision-making and utilization of expert knowledge, which is common in the field of geomechanics and rock engineering [37, 49].

Application of the Mamdani FIS has gained popularity and has been used with success in rock slope stability assessment, burden prediction from rock geomechanical properties, tunnel convergence prediction, prediction of the blastability designation of rock, rock mass blastability, penetrability, diggability, rippability, excavatability and rock mass classification systems [59,60,61,62,63,64,65,66,67].

3.1.2 Takagi–Sugeno–Kang (TSK) Fuzzy Inference System

The Takagi–Sugeno–Kang FIS is also known as the Sugeno FIS. This system uses the singleton membership function for the output variable. The output is either a linear function or a constant of the input values. When comparing the defuzzification process of the Mamdani and Sugeno FISs, the Sugeno FIS is found to be more computationally efficient. This is because the Sugeno FIS uses the weighted average or weighted sum of a few datapoints rather than computing a centroid of a two-dimensional area [52]. The Sugeno FIS is advantageous because of its computational efficiency, suitability for mathematical analysis, ability to adapt and optimize techniques, ability to work well with linear techniques and ability to guarantee output surface continuity [58].

For example, the Sugeno FIS has been used by Matos et al. [68] to model the prediction of clean rock joint shear strength. The model was developed based on 44 shear strength tests carried out on different joints. The input variables for the model were the normal boundary stiffness, the initial normal stress acting on the joint, the joint roughness coefficient (JRC) value, the UCS, the basic friction angle and the shear displacement imposed on the joint. MATLAB® was used to implement the model with the Sugeno fuzzy rule controller. The consequents of the logical rules were zero-order constants that are defined based on the input variables. According to the combination of values assumed by the input variables, shear strength was the weighted average of the consequents. The model was then developed by membership functions and their parameters. The trapezoidal MF was used for the edges of the interval of each variable. Additionally, triangular MFs were used to complete the remaining values that were not covered in the trapezoidal MF. The results from the model proved that the predicted shear strength of clean rock joints fits the experimental data acceptably. In addition, the model also enhanced the characterization of the shear behaviour of discontinuities [68].

The Sugeno FIS has also been used in rock slope stability assessment by Chen et al. [69]. Jalalifar et al. [70] also applied the Sugeno FIS technique in conjunction with an artificial neural network to predict a rock engineering classification system such as rock mass rating (RMR). The blending of a fuzzy system and an artificial neural network to form a neuro-fuzzy system produces improved models. The neuro-fuzzy model by Jalalifar et al. [70] offered better prediction as opposed to traditional conventional modelling methodologies because the proposed weights technique was utilized in the process.

The history and development of fuzzy inference system applications in mining geomechanics is highlighted in Table 1.

Table 1 Fuzzy inference system techniques applied in mining geomechanics

Apart from basic fuzzy inference systems, there are advanced fuzzy inference systems that blend the basic fuzzy set system and soft techniques such as evolutionary computing and neural networks. Numerous software packages have been developed to model fuzzy inference systems. For example, MATLAB® has a built-in fuzzy toolbox and is one of the commonly used software packages in the industry [37]. The fuzzy logic toolbox in MATLAB® enables the user to solve problems with fuzzy logic. It is equipped with tools that allow the user to generate or edit fuzzy inference systems [79].

Generally, the basic fuzzy inference approach is adopted when solving geomechanical problems wherein both input and output data must be determined. For example, data can be collected through site observations and mapping. However, in cases where data are readily available from previous projects, a fuzzy neural network approach is used. Fuzzy systems have been coupled with neural networks, forming neuro-fuzzy systems that display the advantages of both methodologies [37]. An example of this combination is best explained by the adaptive neuro-fuzzy inference system (ANFIS) developed by Jang et al. [80].

ANFIS is said to work well with the Takagi–Sugeno–Kang (TSK) fuzzy inference system because it permits the formation of a fuzzy rule conclusion. Moreover, there is adjustment of the weighted linear combination of inputs and membership function parameters by either the backpropagation algorithm or a combination of the backpropagation algorithm with a least squares type of method ([81]; MathWorks Inc., 2010). In mine geomechanics, ANFIS has been used as follows: predicting liquefaction [82]; interpreting model footing response [83]; jointed rock mass deformation modulus [84]; potential of swelling in compacted soil [85]; mapping of landslide susceptibility [86]; slope stability assessment [69]; ANFIS model for rock burst prediction [34]; and prediction of blast-induced ground vibration [76].

The literature also revealed numerous methods of ANFIS application in tunnelling geomechanics, such as the modelling of tunnel boring machine performance [87]; stability analysis of tunnels under construction [88]; estimation of convergence in tunnelling [38]; and tunnelling risk assessment [49]. The above-listed applications prove that ANFIS is gaining popularity in the geomechanics space. A recent study conducted by Chen et al. [89] proposed the application of ANFIS to the structural safety evaluation of in-service tunnels. To evaluate the structural safety of concrete tunnels, indices and grading standards were initially created in consideration of the variables that affect the safety of in-service tunnels. A safety assessment system was then developed using ANFIS. This safety system for the tunnel was built after determining the indices and grading requirements for the structural safety evaluation of the tunnel based on the structural characteristics of the tunnel and a Chinese specification. From the results, it was concluded that the developed safety assessment system has a strong capacity for both learning and application. After learning, the system effectively emulated professionals in nonlinear fuzzy inference by employing the field-measured data of an in-service tunnel [89].

The application of ANFIS in tunnelling geomechanics has indeed evolved over time. The success of ANFIS models is undeniably influenced by the availability of field data to test the model. Recently, Parsajoo et al. [90] enhanced the tunnel boring machine (TBM) field penetration index of the ANFIS model. The researchers used the artificial bee colony (ABC) technique to optimize the parameters of the ANFIS membership functions. The key objective was to attain the lowest possible predicted error. To develop this model, data were gathered from more than 150 samples of the Queens water tunnel project. In the data, various TBM and rock mass properties were demonstrated. The ANFIS ABC approach was then contrasted with the fundamental ANFIS. The results from the ANFIS ABC model proved that average errors can be reduced by 3%. Meanwhile, the root mean square (RMS) and the coefficient of determination (R²) were 4.35% and 92.6%, respectively. It was discovered that, in comparison to the ANFIS model, the ANFIS ABC approach was useful to estimate TBM performance with fewer input variables and more accuracy [90]. Because ANFIS ABC was successful in forecasting the field penetration index, it is possible to use this model to solve additional tunnelling and mining engineering issues such as estimating rock strength and deformation.

3.2 Innovative Application of Fuzzy Inference Systems in Tunnel Geomechanics

The application of fuzzy inference systems in rock mechanics and engineering geology has grown rapidly over the years [91,92,93,94]. This is because of its ability to tolerate a wide range of uncertainties and to describe complex and multivariable nonlinear problems. This section highlights in themes the various examples in which fuzzy inference systems were used in the mining geomechanics space. The achievements and limitations are also highlighted in each example.

3.2.1 Slope Stability Analysis

Rock masses are frequently cut through to create tunnels for railroads and roadways. Often, the excavation of a tunnel may expose a rock slope that must be carefully monitored for slope stability. Fuzzy techniques have been established for slope stability analysis problems in surface excavations [37]. Supporting the above statement, it has been discovered that the study of Kacewicz [95] was the first attempt to apply fuzzy set theory in estimating the safety factor of Warsaw’s slope in Poland. The success of this study was governed by incorporating soil and rock mass parameters as fuzzy numbers. In addition, the slice method (Fellenius method) and principle of extension by Zadeh [96] were fully implemented to enable accurate prediction of the safety factor of the slope based on the intervals between the upper and lower limits. Finally, expert knowledge was also incorporated within the membership function, which then provided the opportunity to choose the ideal factor of safety.

The study by Kacewicz [95] has attracted several scholars to try and further the realism and possibility of this technique for predicting safety factors. One of the known follow-up studies was that of Juang et al. [71]. The authors revisited some of the existing slopes considering uncertainties based on the vertex method whereby uncertain soil parameters (expressed as fuzzy numbers) were discretized into a set of intervals. Nevertheless, a close look was considered; Juang et al. [71] were noted to reduce a problem into a series of interval analyses using only conventional mathematics. Therefore, this approach is purely deterministic rather than probabilistic, which allows the use of a computer-based program for slope stability analysis.

3.2.2 Stability Analysis of Tunnels Under Construction

The majority of civil and mining engineering works require information relating to the properties and behaviour of subsoil material. However, it is common in tunnel construction that there is limited or partial information on the constraints that have an effect on the construction. During the construction process, geological conditions usually change with depth, and these conditions are sometimes not satisfactorily established. To remedy such unanticipated situations, calibrated mathematical models have been used in conjunction with continuous monitoring to predict and analyse the behaviour and stability of a tunnel under construction. Nonetheless, the process is complex, whereas other techniques such as numerical, empirical, mathematical and artificial intelligence lack precision. These methods require knowledge and information that is uncertain during the design and construction phase of a tunnel. As a result, Luis Rangel et al. [88] developed an alternative strategy of analysing tunnel stability during the design and construction processes. The strategy was a hybrid—consisting of neural, neuro-fuzzy and analytical solutions [88]. The most important parameters considered for stability analysis in this technique include geostatic mean stress, ground shear strength, rock mass deformation modulus, concrete support strength, concrete support deformation modulus, displacement induced by the tunnel and stress in the support.

The system was designed such that it can reproduce the displacement induced at the periphery of the tunnel before and after support has been installed. Moreover, the system used a criterion that considers the dimensionless parameters based on the shear strength of the media. The technique prototype was tested using a database of 261 cases, where only 45 of these cases were real and the remaining 216 were synthetic. The results showed that the hybrid system can simultaneously compute the total and initial displacements, maximum stress acting on the lining and stability of the excavation as it advances without support. In addition, the prototype can be used in real time because of its ability to respond immediately. However, the system has some shortcomings since it does not consider factors such as the stress path during construction. Furthermore, the lack of real data in the database means that reality should be approximated. Nonetheless, more real data can be used to obtain better and more reliable outcomes using the neuro-fuzzy system [88].

3.2.3 Stress Analysis in Tunnels

During tunnel construction, redistribution of stresses in the periphery of the excavation is anticipated. Thus, the prediction of earth pressure balance is deemed important during tunnel construction. Elbaz et al. [77] developed an efficient multiobjective optimization model that integrates the adaptive neuro-fuzzy inference system (ANFIS) and genetic algorithm (GA) to forecast shield performance during tunnel construction. The integration of GA was introduced to enhance the level of accuracy of ANFIS. This hybrid model was tested in a real fired tunnel section in Guangzhou, China. Input parameters were classified into two categories, namely, geological parameters and operational parameters. However, for the hybrid model, three main input parameters were used: cutter head torque (CT), rotational speed screw rate (SC) and cutter head rotation speed (CR). The output for the model was set as the advance rate (AR). The researchers used the Takagi-Sugeno method to construct fuzzy rules because the method is highly reliable and can efficiently compute a synthetic system for fuzzy rules from the input to output database [77]. Moreover, ANFIS works well with the Sugeno FIS (MathWorks Inc., 2010).

The results from ANFIS and the hybrid model (ANFIS-GA) were compared, and it was observed that ANFIS-GA is more successful in predicting, with high accuracy, the advance rate that can be used to channel tunnel performance in the field. Sometimes, optimization of fuzzy calculation for a fuzzy inference system becomes inefficient when using standard optimization algorithms. To remedy this, GAs are used instead. As outlined in MathWorks Inc. (2010), GAs are more suitable for discontinuous, nondifferentiable, stochastic and highly nonlinear objective functions. Kalantary et al. [97] exemplified this application by investigating the correlation between the undrained shear strength (S_u) and the standard penetration test blow count (N_SPT) factor. Based on the above literature, one can conclude that hybrid models yield effective and reliable results when compared to pure fuzzy models.

3.2.4 Tunnel Convergence

Tunnels should be examined for any deterioration and convergence taking place with time. Understanding the type of deterioration process and effects on the tunnel, together with tunnel primary support and concrete lining degradation, is critical for consideration when designing tunnels. Displacement and convergence occur in the tunnel during tunnel construction. The convergence around a tunnel comprises two parts, namely, plastic and elastic parts [98]. Generally, elasticity is used when analysing the stress and strain distribution around an excavation. Hazrati Aghchai et al. [98] considered a circular tunnel under construction and analysed the elastic displacement around the tunnel. For a circular tunnel, Kirsch’s equation is used to determine the radial and tangential stresses around the tunnel. The rock mass around the tunnel was assumed to be homogenous, isotropic and in plane strain condition. In addition, nonhydrostatic in situ conditions were also considered in the model analysis. Although the model was a success, it did not consider the time factor, i.e. how long it would take for the tunnel to reach a certain displacement factor.

Adoko et al. [25] developed and implemented a model based on the Mamdani fuzzy system algorithm to predict the final ground convergence or closure at the vicinity of the tunnel. A critical review was performed to identify the most popular category in which the technique is well established in geomechanics. It was noted that the Mamdani technique gained extensive support and was predominantly explored when solving tunnelling convergence problems. As a result, most previous models famously known for solving tunnelling convergence were then eliminated because they could not incorporate the inherent subjective uncertainties associated with rock masses [23, 37, 99,100,101,102,103,104,105].

4 Application of Fuzzy Inference Systems in Tunnelling: Case Studies

This section provides discussions on recent studies that outline the successful and unsuccessful cases regarding the application of fuzzy sets in tunnelling geomechanics. The section also discusses the gap in knowledge regarding the application of FIS in geomechanics problems. Three case studies are presented in this section to demonstrate the successful application of the fuzzy inference system in tunnelling. These case studies were specifically chosen because they address convergence, safety levels and concrete lining cracks, which are major issues encountered in most tunnels worldwide.

4.1 Case Study 1: Estimation of the Convergence of a High-Speed Railway Tunnel in Weak Rock Using an Adaptive Neuro-Fuzzy Inference System (ANFIS) Approach

Tunnel construction has grown in countries such as China due to the demand for transportation and energy supply. Boidy [106] and Boidy et al. [107] are of the view that deterioration is bound to happen over time whenever a structure is built and operational. Tunnels suffer continuous loading from the surrounding rock mass after several years of operation, and this cannot be ignored. Subsequently, the levels of safety and serviceability drop [108]. Hence, continuous stability analysis of road tunnels is a significant task in tunnel safety. Adoko and Wu [38] adopted the New Australian Tunnelling Method (NATM) in estimating tunnel diameter convergence wherein systematic convergence measurements are applied in adjusting the design during construction, thus preventing fatal risks.

The study by Adoko and Wu [38] adopted the use of a new fuzzy model known as the adaptive neuro-fuzzy inference system (ANFIS), which can predict tunnel diameter convergence. The ANFIS established and fit for mathematical analysis (MathWorks Inc., 2010) is of the view that the Takagi-Sugeno FIS uses fewer rules than the Mamdani FIS.

The estimation of the convergence case study was conducted on high-speed railway tunnels wherein more than 1000 datasets were collected. These are Daguan tunnel No. 2 and Yaojia tunnel No. 1 in Hunan Province, China. The two tunnels were treated separately, i.e. the Daguan tunnel No. 2 dataset was used as training data, and the Yaojia tunnel No. 1 dataset was used for testing and validating the model. The dataset for each tunnel consisted of six inputs: the surrounding rock mass rating index, the ground engineering conditions rating index, the rock density, the tunnel overburden depth, the distance between the monitoring station and the tunnel heading face, and the time elapsed after the working face passed through the monitoring station. Conversely, the output parameters in the model consisted of the total convergence and convergence velocity of the crown sides of the tunnel. The ‘IF-THEN’ rule was applied in constructing the model. Gaussian MFs were considered because they are smooth and utilize nonzero at each point. Moreover, no measurements of convergence are required as input parameters other than parameters that influence tunnel deformation (tunnel geometry, rock mass properties, ground conditions, etc.). Therefore, the model is capable of convergence estimation.

The model combines fuzzy logic and artificial neuro-networks to determine membership functions. This approach was established to overcome the limitations of fuzzy logic and neural networks. For fuzzy systems, the identification of parameter membership functions is not efficient, especially when dealing with complex systems. Conversely, in neural networks, uncertainty lies in the determination of the ideal size and optimal architecture. For these reasons, ANFIS was introduced. According to Adoko and Wu [38], this system also gained popularity in algorithms employed in commercial software such as MATLAB® version R2010a. Two types of fuzzy inference systems have been prominent in the field of geotechnical engineering, as contested by Adoko and Wu [37]. These are the Mamdani type and the Takagi-Sugeno type.

The results from the study proved that the proposed model can forecast tunnel convergence with a high level of accuracy. The level of accuracy is evaluated by the root mean square error (RMSE) and the percentage of variance accounted for (VAF). Models that use the Takagi-Sugeno FIS, such as this one, show that there is enhancement in the level of accuracy for convergence velocity as opposed to models based on Mamdani fuzzy. The datasets for the Takagi-Sugeno FIS and Mamdani FIS models were 1000 and 135, respectively. The Mamdani model was built based on expert knowledge; hence, there was less dataset as opposed to the TSFIS model. The convergence velocity RMSE ranged between 0.4 and 2.05 mm/day, and the VAR ranged between 59.23 and 78.40% for the Mamdani model [25]. In contrast, RMSE and VAR for the Takagi-Sugeno FIS model range between 0.018 and 0.336 mm/day and 87.56 and 92%, respectively (see Table 2). For this reason, the convergence velocity was successfully and accurately estimated from the Takagi-Sugeno FIS model. To conclude this case study, the authors recommended that the proposed method be used in predicting tunnel convergences, but it should not be a substitute for in situ measurement of convergence [38]. In simple terms, the model can be used in conjunction with convergence monitoring programs for decision-making purposes.

Table 2 Performance of the prediction model [38]

4.2 Case Study 2: Fuzzy Evaluation Model for Highway Tunnel Safety in Karst Formations

Safety is a major concern in the design and construction of tunnels. Investigations are conducted in tunnels with the aim of detecting tunnel defects that affect tunnel safety and usability. This approach is a traditional decision method, as it is based on investigations and the detection of tunnel defects. The shortfall of this approach is that the results are not adequate enough, as they fail to create a calculation model for all tunnel defects. Moreover, there is minimal correlation shown between tunnel defects. Thus, the model is unable to access tunnel safety with accuracy. For these reasons, fuzzy theory was suggested to evaluate the highway tunnel safety in karst formations [109]. Research has been conducted from the standpoint of structural safety assessment with fuzzy theory. Nevertheless, there was no evidence of research conducted on the structural safety of in-service karst tunnels in this regard prior to that of Rao et al. [109]. The evaluation for highway tunnel safety in karst formations is linked to numerous complexities, ambiguities and uncertainties of qualitative and quantitative indices. Therefore, the proposed fuzzy theory approach to discern the complexity of the karst highway tunnel is the fuzzy comprehensive evaluation method [109].

The Hui-long-shan tunnel situated on the Shaogan Highway was built in 2010. It is one of the tunnels in 27 karst caves where flagstones were treated with mortar while being built [109]. The tunnel is 14.5 m wide and 1155 m and 1145 m long on the right and left sides, respectively. Rao et al. [109] explained that the karst area has been treated thoroughly. However, after the tunnel has endured repeated load changes as a result of tunnel lining dehiscence, the safety circumstances and potential risk of the tunnel remain unknown. The level of safety of a tunnel is influenced by factors such as tunnel design and/or construction flaws, cracks in the lining, support, and geological and hydrological surroundings. Moreover, the most alarming factors to be considered include deterioration of the surrounding rock with time, change in load distribution due to lining separation, lining deterioration due to water seepage, tunnel clearance deficiency and rapid deformation of tunnel reinforcement [109]. An investigation of tunnel lining conditions was conducted at the Hui-long-shan tunnel. Lining cracks, water seepage and breakage on the tunnel lining were discovered. Finally, ground penetrating radar (GPR) scanning was conducted to detect any deterioration within the tunnel lining and immediate surrounding rock mass. Accurately assessing the influence of these factors on tunnel safety is difficult. Thus, data relating to tunnel construction, service state, tunnel conditions, visual observation and testing of the lining state were first collected and analysed prior to fuzzy model evaluation.

Evaluation of complex systems in fuzzy theory must consider numerous fuzzy factors and factors in different layers. In simple terms, some of these factors depend on multiple other factors. The three-grade fuzzy comprehensive evaluation model was developed for tunnel safety, and the main parameters in the model were set up based on the hierarchical division of factors for tunnel safety. The hierarchical division consists of three levels, of which some of the factors are determined by several other factors. For example, in this case, leakage (level 1) depends on five factors in level 2. These are the leakage in the construction joint, leakage in cracks, leakage in the lining, function of the drainage system, subgrade water and ooze emitting from frost boiling. Furthermore, some of the second-level factors are also dependent on other factors in the third level. For example, leakage in cracks is influenced by leakage at the vault and leakage at the haunch of the tunnel. For the sake of argument, it is critical to adopt a multilevel fuzzy system to systematically assess practices with multiple factors at different levels, such as tunnel safety evaluation. For the fuzzy evaluation matrix, membership functions were applied when determining quantitative indices. After fuzzy evaluations, the results showed that the in-service karst tunnel is safe on the measuring section and leans towards general safety based on the principle of maximum membership. However, it is essential to monitor tunnel conditions and make any necessary repairs [109].

The fuzzy comprehensive evaluation method used in this study combined qualitative and quantitative indices, thus compensating for shortfalls in the conventional evaluation method based on empirical conclusions. To summarize this case study, the author stated that the membership parameters are significant for this model and should therefore be constantly optimized and improved in engineering applications. This will help to achieve accurate and consistent evaluation results.

4.3 Case Study 3: Evaluating the Cracks of Highway Tunnel Concrete Lining by Using a Fuzzy Inference System

Predicting cracks in a tunnel lining is difficult because different factors come into play. Such factors include and are not limited to the lining material strength, tunnel operational conditions, geological surroundings and stress field in the periphery of the tunnel. Cracks in civil engineering structures such as tunnels, bridges and dam walls are not accounted for. The mechanism by which these cracks transpire and the impairment they inflict on these structures is uncertain. The growth of technology and its application in various fields has positively impacted the research world. Studies show that the use of fuzzy systems to address uncertainties has grown over the years. Yousif [110] posits that scholars such as Rao et al. [109], Adoko and Wu [37], Monjezi et al. [111], Monjezi and Rezaei [66], Rezaei et al. [112] and Jalalifar et al. [70] demonstrate the successful establishment of fuzzy set theory in geomechanics, rock engineering and tunnelling. The aforementioned scholars developed models using different fuzzy techniques and assigned membership functions to fuzzy variables and consequently solved their respective geomechanical problems.

Yousif [110] conducted a study with the aim of designing a fuzzy inspection system to evaluate cracks in the concrete lining of a tunnel. This case study was conducted in northern Iraq at the Peshraw tunnel on the Sulaymaniyah highway. The tunnel is 2389 m long and was constructed in 2004. Yousif [110] based his research on Peshraw tunnel inspection crack data and a highway tunnel design manual with rules and principles for identifying maximum crack width. After inspecting the tunnel concrete lining, the identified cracks were classified into three categories, i.e. vertical, horizontal and diagonal cracks. In developing the fuzzy model, six inputs (concrete damage extent; load bear by reinforcement; load bear by concrete; damage variable, thickness of protection layer; and spacing and strain of reinforcement) and one output (crack width) were defined as membership variables. The field was divided into three sections, namely, minor (<0.8 mm), moderate (0.8–3.2 mm) and severe (>3.2 mm), based on the crack width. In the model, severe and minor fuzzy sets were modelled using trapezoidal membership functions, whereas moderate fuzzy sets were modelled using triangular membership functions.

Although the model was designed based on highway tunnel design specifications, it was evaluated using real field data at the Peshraw highway tunnel, and the results were compared to the inspection findings data results (Table 3). The crack width results obtained from the model are closely related to the field results of the measured crack widths. The average error for the FIS model in this example was 8.34% [110].

Table 3 Example of investigation crack data in the Peshraw tunnel vs. FIS results

Fuzzy model evaluation becomes comfortable when assessment parameters are based on actual evidence that reflects the field. Similarly, the Peshraw tunnel study model showed the comfortability of the fuzzy model. The author’s judgement of the results comparison was based on the maximum permit crack formula from the highway tunnel design specification manual, and the following conclusion remarks transpired: each membership function value is crucial in any fuzzy evaluation model; therefore, fuzzy evaluation models should be used more often in engineering to improve the reliability of the results [110].

4.4 Recent Case Studies on the Partial Successful Application of Fuzzy Inference System in Tunnelling Geomechanics

Tunnel convergence is closely related to tunnel deformation, which ultimately affects tunnel stability and serviceability [113]. For tunnels constructed by the New Australian Tunnelling Method (NATM), tunnel diameter convergence prediction is critical at an early stage since quick adjustments of the design can be prepared if necessary. Adoko et al. [25] conducted a study on a high-speed railway tunnel in Hunan, China. To develop the fuzzy model, 135 datasets were collected from the tunnel construction site. Data were collected using a convergence monitoring program. In addition, geotechnical investigation was conducted by means of field and laboratory testing. Digital extensometers were used to record tunnel deformation after the first lining was installed. The tunnel geometry, support conditions and geological surroundings are influential parameters for NATM behaviour. From these parameters, six input membership functions were defined for the fuzzy inference system, as shown in Fig. 6.

Fig. 6

Structure of the proposed fuzzy model (modified after [25])

A fuzzy model for the prediction of tunnel diameter convergence for a high-speed railway tunnel was set based on the Mamdani algorithm and triangular and trapezoid membership functions. Parameters influencing tunnel deformation, such as rock mass properties, ground engineering conditions and tunnel geometry, were defined in the Mamdani fuzzy inference model. The results from the model indicated the capability of tunnel diameter convergence prediction with good accuracy. However, the shortcoming of the model surfaced on the convergence velocity. Performance indices on the fuzzy model indicated convergence velocity as a concern since it did not execute with excellence, although it predicted the convergence velocity with impartial accuracy. The authors opened to further research with the aim of improving the fuzzy model capacity in detecting tunnel convergence velocity. A proposal was made for growing the dataset and ultimately defining ideal parameters for membership input into the fuzzy model [25].

The technicality of fuzzy set theory presented by the case studies confirms that fuzzy methods simplify the complexity of solving geotechnical problems as opposed to the traditional methodology. The benefit of adopting fuzzy systems in tunnelling is the ease and precision with which complicated issues may be resolved. Additionally, depending on the recently accessible data and suggested models, unknown information may be forecasted for future planning of improved tunnel designs. Tay and Lim [114] as well as Yazdani-Chamzini [49] asserted that expert knowledge and experience are beneficial since risk evaluation, prioritization and ranking are permitted in fuzzy models. Fuzzy models are usually empirical and should therefore be modified when adopted in a different environment or field of study. This shows that there is still room for improvement regarding the application of fuzzy theory in geotechnical engineering and tunnelling.

5 Conclusion and Future Works

Since the introduction of fuzzy set theory, numerous problems in various fields, including tunnelling and geomechanics, have been addressed. This does not, however, imply that all problems have been resolved. Rather, this implies that there is room for continuous research. Indeed, fuzzy set theory has gained support, popularity and continuous research on a global scale in solving geomechanical engineering problems. Fuzzy inference systems make it easier for tasks to be mechanized. Hence, there has been growth in the application of fuzzy inference systems (FISs) in many engineering disciplines to solve problems with uncertainties. This paper reviewed the application of FISs in solving geomechanical problems in underground excavation and tunnelling. The paper also gave some examples of the recent studies that outlined the successful and unsuccessful cases regarding the application of fuzzy sets in tunnelling geomechanics. Moreover, recent studies outlining various spaces of fuzzy inference application have surfaced. These include slope stability, tunnel convergence prediction, tunnel stability during construction and other rock mechanics, and geotechnical engineering problems.

The purpose of fuzzy set theory in the field of engineering is to model complex structures involved in engineering designs and decision-making with simplicity, reliability and accuracy. Continuous monitoring of engineering structures such as tunnels is critical for determining their stability. Fuzzy set techniques are indeed a great tool for analysing and estimating geotechnical aspects of the tunnel that have an impact on the safety and stability of the structure. The case studies presented in this review paper confirmed that the use of fuzzy models can result in more reliable and accurate results than traditional methodologies. Furthermore, the success of fuzzy models in application is influenced by datasets and the input membership functions. Thus, it is critical to have a vast dataset to execute fuzzy model output excellently.

Based on the critical review, it is confirmed that fuzzy set theory has been widely applied with success in geomechanics and tunnelling with a wide scope ranging from convergence, lining cracks, safety levels and other rock engineering-related problems in tunnelling. The Mamdani method and adaptive neuro-fuzzy inference systems have recently gained popularity and interest from numerous researchers. This is because of their simplicity and ability to solve complexity and uncertainty with accuracy and reliability. In conclusion, it is critical to understand that although fuzzy set theory is highly recommended and implemented recently, the effectiveness of its execution lies in how appropriate it is used. When developing fuzzy models to address any geomechanical problems, specifically for tunnels, the understanding of tunnel geometry, surrounding rock mass, support conditions and ground engineering conditions plays a vital role in implementing a successful fuzzy model. Expert knowledge is also commended for decision-making purposes because fuzzy models allow risk evaluation, prioritization and ranking.

The following areas of study are recommended for future research:

• Predicting the effects of fracturing on the stability of road tunnels.

• Developing a predictive stability analysis model for road tunnels using a fuzzy inference system.

• Developing deformation curves of the tunnel lining material through a fuzzy inference system.