Fuzzy Inference Systems-based Approaches in Geotechnical Engineering- a Review Amoussou Coffi Adoko PhD Student, ASCE Student Member, China University of Geosciences (Wuhan) Email:
[email protected]
Li Wu * Professor, Faculty of Engineering, China University of Geosciences (Wuhan) Email:
[email protected]; tel: +8613871159076 *Corresponding author
ABSTRACT In general, fuzzy set theory (FST) is used in geotechnical engineering to handle uncertain data and subjective information. The techniques employed can be classified into four categories: basic fuzzy inference (fuzzy set/logic); advanced fuzzy inference (combining other soft computing techniques), fuzzy probability theory and Klisiński fuzzy plasticity methods. The principal objective of the present paper is to review the applications based on fuzzy inference techniques as they represent the most used FST approaches recently. First, it identifies and describes the different ways of application within topics such as slope stability, rock engineering, foundation and tunneling. It is observed that the adaptive neuro-fuzzy inference system (ANFIS) has been in use over the past ten years, especially in rock engineering related topics. Finally it is suggested that further research need to be done in order to develop a rigorous methodology or criteria to identify the characteristics of the inference system (e.g. type of systems, defuzzification methods and assignment of membership functions to fuzzy variables) that solve best a given geotechnical problem. A proper use of FST as a research tool in geotechnical problems associated with uncertainties can help in improving the accuracy of the results.
KEYWORDS:
Geotechnical engineering, fuzzy set theory, fuzzy inference systems
INTRODUCTION Over the past decades, Fuzzy Set Theory (FST) has been used in geotechnical engineering problems to cope with uncertain data due to lack of precision, incompleteness, vagueness and randomness of the information as well as incorporating subjective judgment from expects into problems analysis. In the first applications, dealing properly with subjectivity in geomechanics - 1543 -
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decision processes, motivated Nguyen and Ashworth (1985) to apply FST in rock engineering classification system while Kacewicz (1987) employed it for slope stability analysis in order to overcome some limitations of probability theory which provides the necessary tools to solve only problems associated with randomness but cannot solve properly epistemic uncertainties. Since then, FST has been proved to be a useful tool for modeling imprecision of the data and has been applied successfully in various geotechnical problems. Introduced by Zadeh (1965), FST provides the means for representing epistemic uncertainty using set theory and describes the concept of gradualness and bipolarity (Dubois and Prade, 2010). A fuzzy set is an extension of the concept of a crisp set. While a crisp set only allows full membership or no membership to every element of a universe of discourse, a fuzzy set allows for partial membership. Basically fuzzy set theory includes fuzzy variables (type-1 fuzzy set and type-2 fuzzy set or higher order fuzzy set) or fuzzy functions, fuzzy logic, fuzzy inference system, fuzzy probability, and hybrid fuzzy set (combination of other soft computing techniques such as neural network and genetic algorithm). Fuzzy inference system models define relationships between input and output variables of a system by using linguistic labels (i.e., fuzzy sets) in a collection of IF-THEN rules, Mandani and Takagi-Sugeno systems being the most commonly used. Ample details on the FST can be found in many literatures, for example in (Zimmermann, 1999; Celikyilmaz &Turksen, 2009). Since early 80’s where the first applications of FST in geotechnical engineering appeared , it has been developing intensively and currently it is employed in wide variety of problems for instance, slope stability, rock engineering, tunneling, project management, and even constitutive relation of geomaterials. The literature reveals that researchers used this theory in different ways in order to deal with the level of complexity associated to the problems. In some cases fuzzy set basically was needed while in other cases advanced methods were implemented. Due to its advantages of handling epistemic uncertainty, it has been useful as predictive models, in situation where getting complete or updated data (statistical distribution parameters for example) over a period was unrealistic or impossible. Many researchers employed FST to overcome some shortcoming of traditional method of dealing with imprecise data such as probability theory (Kacewicz, 1987; Dodagoudar & Venkatachalam, 2000) and classical approach of prediction (multi regression analysis) (Yagiz & Gökceoğlu, 2010). Moreover, the possibility to use expert knowledge with a fuzzy reasoning approach to enhance some geotechnical problems analysis makes FST to be considered as powerful tool of describing uncertain data. Finally it offers the platform to conceive many fuzzy geotechnical problems based on Zadeh’s original intuition e.g. fuzzy set plasticity theory and fuzzy damage theory (Klisinski, 1988; Mishnaevsky & Schmauder, 1996; Min & Phan, 2010). Also, FST raised debate among researchers, mostly logicians and mathematicians. It is believed that fuzzy logic is useless and inconvincible (Lindley, 1994; Pelletier, 2004). Nevertheless its applications in many fields including geotechnical engineering are increasing. Without engaging in controversial aspects FST, the present paper intends to provide a deep insight of the use of Fuzzy Inference Systems (FISs) in order to optimize its applications. It aims at identifying the different methods and tools of FISs applications in geotechnical engineering and discusses some further research questions.
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REVIEW METHODOLOGY With the purpose of achieving one of the important objectives of this paper, the review methodology consists mainly in bibliography search which is one of the common research methods for a review article (Dereli et al., 2011). Our literature retrieval process was basically done through computer search on online database namely IEEE Xplore, SpringerLink, ISI-Web of Science, ScienceDirect and Engineering Village as well as direct search in Google. Using the descriptors “fuzzy set theory” along with some geotechnical engineering such as slope stability, rock engineering, foundation, constitutive models and tunneling, we retrieved more than 850 papers including non relevant or repetitive articles. Then we did a refinement process and only papers concerning fuzzy inference systems application with relevance in geotechnical engineering were considered. After multiple review of every single article, the best candidates suitable for reference were chosen. Within this selection, special attention was taken to analyze the technique of fuzzy inference systems employed. In addition, a general classification according to the used FST methods and the type of problem solved was made. This allows us to present, as findings, first a bibliometric analysis in order to highlight the main trends of research on the topic, and secondly to produce a concise literature review showing the state of the art of the Fuzzy Inference System (FIS) in use in geotechnical engineering.
BIBLIOMETRIC ANALYSIS Bibliometric analysis refers to a research method which helps on producing the state of the art for a given research topic. It is commonly in use and describe trend of publications by employing quantitative analysis and statistics. The methodology used for bibliometric analysis, includes the same approach of the literature review, with the same database and was made at the initial stages of literature review process where getting general statistics about the use of Fuzzy Set Theory (FST) in geotechnical engineering, has been our concern. Globally speaking, it has been found that in geotechnical engineering FST is widely applied. One of the most employed FST resources is fuzzy inference systems (FIS). Fig.1 compares the number of publication based on FIS and other fuzzy techniques over the past 3 decades 80’s, 90’s and 00’s. It was observed that there was an increasing acceptance of FISs applications. However it should be noticed that our information source was restricted to ScienceDirect database indexing important journals that publish papers on this area such as Fuzzy Sets and Systems, Computers and Geotechnics, Engineering Geology, Expert Systems with Applications, Tunnelling and Underground Space Technology and Computers & Geosciences.
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FIS
Other methods
Number of papers
160 140 135
120
130 119
100 80
89
123
83
60 40 20 8
7
1
6
16
7
0 1980's
1990's
2000's
2010-2011 Years
Figure 1: Distribution of publications according to the techniques (FST, FIS and others) used from 1980-2011. Table1 provides a distribution of the reference papers that have been reviewed according to the FIS techniques employed so far. It is shown that applications based on Mandani systems, basic fuzzy set/logic and hybrid system using neural network are the most used. The most popular topics are related to rock engineering. It is due to their simplicity and effectiveness to convey uncertain information and capability to provide best tools for problems classification, decisionmaking and expert knowledge utilization which are very common in rock engineering. Table 1: Distribution of the main references according to the fuzzy inference techniques employed. Type of fuzzy inference systems Basic fuzzy set/logic
Number of papers 13
Mandani systems
18
Seguno-Tagaki systems
5
Topics Slope stability, foundation, rock mass classification, geotechnical project scheduling and cost planning determination geotechnical parameters, prediction of soil uniaxial compressive strength, fuzzyfication of Chen plastic model of concrete and rock sawability. Rock mass blastability, penetrability, diggability, rippability, excavability; rock mass classification systems, prediction of flyrock in mining surface, burden, rock fragmentation, backbreak in open-pit blasting and TBMs thrust and torque requirement prediction of maximum charge per delay in surface mining, impact hammer performance, constitutive modeling, swelling potential of compacted soils; rock engineering classification system and rock slope stability assessment
Vol.16 [2011], Bund. M Hybrid system using Neural network (ANFIS and other)
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Hybrid system using Genetic Algorithms
5
1547 constitutive modeling of undrained response of sand mixtures, angle of shearing resistance of soils, tunnel boring machine performance modeling, liquefaction prediction, footing response modeling, modulus of deformation of jointed rock masses, slake durability of shaly rock and landslide susceptibility mapping Optimum design of dynamic compaction of soil, slope stability and decision making in geotechnical engineering.
APPLICATION OF FUZZY SET THEORY IN GEOTECHNICAL ENGINEERING: LITERATURE REVIEW Classification of the fuzzy set theory techniques in use in geotechnical engineering Even though the classification of FST applications in geotechnical engineering is not the principal objective of this paper, it can help to understand how FST has been applied so far. Our classification is subjective (others authors may classify in another way), and is based on some criteria that have been set. It should be noticed that it may contain some categorization fallacies but the key idea about this classification is to provide concise information about the applications. The criteria are based on theory and methods used namely fuzzy set (membership functions), fuzzy logic, fuzzy inference system, fuzzy probability measure, and fuzzy plasticity as well as the combination with other soft computing i.e. neural network and genetic algorithm. Table 1: A general classification of fuzzy set technique in use in geotechnical engineering Fuzzy techniques Basic fuzzy set/fuzzy logic and inference
Advanced fuzzy inference systems
Example of applications Rock mass classification (Nguyen & Ashworth, 1985), slope stability (Kacewicz, 1987; Juan et al. (1998), sawability classification of building stones (Tutmez et al., 2007) and risk assessment for rock stability (Wang et al., 2011). Mandani type systems
Sugeno type systems Systems using Neural Network
Systems using Genetic Algorithm Hybrid formulation
A new Mandani-based model to predict burden from rock geomechanical properties (Monjezi & Rezaei, 2011) and Mandani fuzzy inference model prediction of the blastability designation of rock (Azimi et al., 2010) Rock engineering classification system (Jalalifar et al., 2011), rock slope stability assessment (Chen et al., 2011). Constitutive modeling of undrained response of sand mixtures (Calabar et al., 2010) and prediction of maximum charge per delay in surface mining (Alipour & Ashtiani, 2011). Slope stability(Zhang & Lin 2006; Xue et al.2007)
Soft computing techniques based model of the angle of shearing resistance of soils (Kayadelen et al., 2009)
Vol.16 [2011], Bund. M Fuzzy probability theory Fuzzy plasticity theory
1548 Slope reliability (Dodagoudar & Venkatachalam, 2000) Cyclic constitutive modeling (klisinski, 1988), rock fragmentation (Mishnaevsky & Schmauder, 1996) and soil-water hysteresis model for unsaturated sands (Min & Phan, 2010)
Basic fuzzy set/fuzzy logic and inference This group utilizes the basic theory of fuzzy set: membership function, α-cut concept and fuzzy relation. This represents the simplest way to apply fuzzy set theory. The objective is to analyze the propagation of uncertainty in a deterministic model or system where the inputs contain fuzzy set properties. Typically, parameters in a deterministic model are assumed to be fuzzy numbers. By doing so, operations on fuzzy numbers are needed in order to compute all calculation for the given deterministic model. Techniques and operations include solving interval arithmetic or interval expression as a constrained global optimization problem after α-cut transformation (Degrauwe, 2007) or using Zadeh (1975) extension principles as well as fuzzy logic or reasoning (the minmax aggregation, etc…) to get the final result. Optimization methods for fuzzy numbers calculation for this category can be vertex method Juang et al. (1998). The literature showed that this category of fuzzy technique has started been used in use geotechnical engineering by early 80’s. As matter of fact, Nguyen (1985) and Nguyen and Ashworth (1985) researched rock mass classification with the help of the min-max aggregation operation developed by Bellman and Zadeh (1970) for multi-criteria decision modeling to select the most likely rock mass class. They used the Bieniawski’s Rock Mass Rating RMR system and Barton’s Quality index Q where the inputs were computed with fuzzy techniques. The approach consisted firstly, in assigning membership function 𝜇 (i represents the class and j the criterion of classification) to the inputs; secondly a fuzzy binary relation table constituted by the 𝜇 for every i and j. Then min-max aggregation procedure is applied to derive the rock mass class. This approach does have some shortcoming; mainly the fuzzy min-max operation is suitable for systems with non-interactive criteria (Dubois and Prade, 1980) which is not the case in rock mass classification systems where classification criteria are weighted differently. For this reason, other fuzzy rock mass classification methodologies based on the fuzzy weighted average algorithm and the so called “refined rating”, were proposed respectively by Juang and Lee’s (1990) and Habibagahi and Katebi’s (1996). Another area of earlier application belonging to this category of fuzzy techniques is slope stability analysis. Kacewicz (1987) applied successfully fuzzy set theory to estimate the factor of safety of Warsaw’s slope in Poland. The main soil and rock masse parameters (weight, water pore pressure, friction angle, and cohesion) were treated as fuzzy numbers. With the method of slices (Fellenius method) and Zadeh principle of extension (1975), the factor of safety was determined in terms of intervals with upper y lower limit. The subjective point of view (expert knowledge) was incorporated into the membership function which allowed choosing the ideal factor of safety. Similarly Juan et al. (1998), analyzed some existing slopes considering uncertainties based on the so called vertex method, anchored in the α-cut concept, in which uncertain soil parameters were discretized and expressed as fuzzy numbers then put into a set of
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intervals. The main difference here is that the problem is reduced to a series of intervals analyses and use only conventional mathematics. This approach is deterministic rather than probabilistic, that enabled them to use a PC-based computer program for slope stability analysis (PCSTABL) together with the vertex method. Fetz et al. (1999), based on interval analysis on α-cut set, devoted a research work to application of fuzzy models in geotechnical engineering by analyzing a mat foundation (raft) with fuzzy finite-element computation, a fuzzy dynamic system (a fuzzy differential equation was solved as example) and uncertainty in geotechnical project scheduling as well as cost planning while Möller et al. (2001), presented the solution techniques for fuzzy finite element method with the help of α-level optimization. Other applications include the determination of characteristic values of measured geotechnical parameters (Nawari and Liang, 2000), the prediction of the uniaxial compressive strength of Ankara agglomerates by mean of membership function Gökceoğlu (2002), fuzzyfication of Chen plastic model of concrete (Kruis and Stemberk, 2005), sawability classification of building stones (Tutmez et al., 2007), the selection of an appropriate excavation construction (Pan, 2009) and risk evaluation of surrounding rock stability (Wang et al,. 2011). As it is discussed, all these applications utilize basic fuzzy set and fuzzy logic methods. Those employing advanced fuzzy inference techniques will be presented in the coming section.
Advanced fuzzy inference This category represents the most comprehensive fuzzy set theory in use in geotechnical engineering. Besides the basics, it utilizes resources of Fuzzy Inference System (FIS), along with other soft techniques e.g. neural network and evolutionary computing. Recently this technique has been used intensively and the results showed that it is suitable to analyze some rock mass geomechanical properties like diggability (Iphar and Goktan, 2006), rippability (Basarir et al., 2007), excavability (Khademi et al., 2010), or penetrability (Hoseinie et al., 2009) which is referred to a classification of a rock mass in terms of how easy it can be dug, ripped, excavated, or penetrated. As a general observation, the Mamdani Fuzzy model is often used in geotechnical problems because of its simplicity and effectiveness to handle linguistic variables; even though other FISs are available namely the Takagi-Sugeno-Kang fuzzy (TSK) model, the Tsukamoto fuzzy model and the Singleton fuzzy model (Azimi et al., 2010). Basically, rule base, database and reasoning mechanism are three conceptual elements of a FIS. The fuzzy rules constitute the rule base and the database determines the membership functions associated with the inputs parameters to be used in the rule base while the reasoning mechanism provides the platform to derive an adequate conclusion (output) by using fuzzy logic. At this stage the extraction of a crisp set from a fuzzy set, called defuzzification is performed. Defuzzification methods include centroid of area (COA), bisector of area (BOA), mean of maximum (MOM), smallest of maximum (SOM), and largest of maximum (LOM) (Khademi et al., 2010). A schematic illustration of the fuzzy inference model is shown (fig. 2). Many software packages are available to model FIS; one of the most commonly used being the fuzzy toolbox of Matlab. The literature revealed countless application of this way of using fuzzy set theory. Khademi et al. (2010) developed a model to predict excavability of rock mass (how easy a rock mass can
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be excavated) according to the Rock Mass Excavability RME classification. The model utilized seven inputs parameters: uniaxial compressive strength, drilling rate index, homogeneity of discontinuities, number of joints per meters, orientation of discontinuities, stand-up time and groundwater inflow; five (5) classes of rock; theoretically there are 18,750 (5x5x2x5x3x5x5) possible if-then rules for the FIS however 9,300 of them were excluded for being in contradiction of the rock nature, and the defuzzification method was the centroid of area (COA). The result indicated that this fuzzy model solved the problem of sharp transition boundaries between two adjacent rock classes and can be used for rating-based rock engineering classification systems.
Figure 2: Schematic illustration of the fuzzy inference method Also, Azimi et al. (2010), applied FIS to rock mass blastability designation (BD) classification systems. They developed a methodology in terms of “Effective Rules’’ for the construction of the if-then rules to reduce the number of rules and then efficiently solve fuzzy inference systems with a large number of fuzzy rules (which is computer consuming). The approach includes the method of matrix intersection in Rock Engineering System and the maximum and minimum numbers of effective rules for a crisp input set. Similarly Yagiz and Gökceoğlu (2010), analyzed prediction of rock mass brittleness for underground rock excavation and design by applying FIS and nonlinear regression analysis. For the fuzzy inference model, the inputs were unit weight, uniaxial compressive strength, Brazilian tensile strength and the output was the brittleness index. It was shown that even though the nonlinear multiple regression models performed better, the fuzzy model exhibited a high performance. Moreover, many other researchers employed successfully the FIS in geomechanics related problems with the same approach. For example, the prediction of Geological Strength Index (GSI) (Sonmez et al., 2003), deformation modulus of rock masses (Gökceoğlu & Zorlu 2004), assessment of failure susceptibility of soil slopes using fuzzy logic (Saboya Jr. et al., 2006), sinkhole occurrences over abandoned mines (Deb & Choi, 2006), swell and shrink factor affecting earthwork optimization of highways (Goktepe et al., 2008), analysis and prediction of rock fragmentation due to blasting (Monjezi et al., 2009), identification of expansive soils and assessment of expansion potential (Reddy et al., 2009), integration of fuzzy expert systems and discrete event simulation in construction engineering (Shaheen et al., 2009), constitutive
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modeling (Zhang & Zhang, 2010), building damage risk assessment on mining terrains (Malinowska, 2011), development of a fuzzy model to predict flyrock in surface mining (Rezaei et al., 2011), prediction of TBMs thrust and torque requirement (Acaroğlu, 2011) and burden prediction from rock geomechanical properties (Monjezi & Rezaei, 2011). So far the general approach concerns a class of problems where having inputs data, an output has to be determined. It consists in mapping inputs characteristics to a single-valued output or an output related decision-making via fuzzy rules which have been predetermined. But sometimes, in geotechnical routine, both input and output data are already available from past projects or records; and the fuzzy inference system that describe best the data has to be found and then can be used as prediction model. In such cases, fuzzy neural network approach like the Adaptive Neuro-Fuzzy Inference System (ANFIS) may be employed where the system properties (i.e. fuzzy set and fuzzy rules) are determined by learning machine capable of acquiring knowledge from data. ANFIS works with Takagi-Sugeno-Kang type fuzzy inference system in which the conclusion of a fuzzy rule is constituted, weighted linear combination of the inputs and the membership function parameters are adjusted using either a backpropagation algorithm alone or in combination with a least squares type of method (Sugeno and Kang, 1988; The MathWork, 2010). A literature survey indicates that it has contributed a lot to some complex geotechnical problem. For example, Chen et al. (2011) developed an ANFIS based model for slope stability assessment. The methodology includes the database preparation where they collected 53 set of data of epimetamorphic rock slopes; the input vector (each data set) consisted of 5 parameters bulk density γ, height H, inclination β, cohesion c and internal friction angle ϕ; the output was the stability state where 0 denoted failure and 1 denoted stability. The data were divided into two parts. The first part consisting of 41 data pairs was used as training data while the second part (remaining 12 sets of data) served as the prediction test. The results showed that the model can accurately predict rock slope in complex conditions. Many other applications based on the same principle and fuzzy neural network includes: tunnel boring machine performance modeling (Alvarez Grima et al., 2000) liquefaction prediction (Rahman and Wang, 2002), interpretation of a model footing response (Provenzano et al., 2004), modulus of deformation of jointed rock masses (Gökceoğlu et al., 2004), slake durability study of shaly rock and its predictions (Singh et al.,2005), tunnel stability analysis during construction (Rangel et al., 2005), swelling potential of compacted soils (Kayadelen et al., 2009), modeling of the angle of shearing resistance of soils using soft computing systems (Kayadelen et al., 2009), neuro-fuzzy based constitutive modeling of undrained response of Leighton Buzzard Sand mixtures (Calabar et al., 2010), landslide susceptibility mapping in Klang valley, Malaysia (Sezer et al., 2011) and fuzzy modeling approaches for the prediction of maximum charge per delay in surface mining (Alipour and Ashtiani 2011). At times, the optimization of the fuzzy calculation for a FIS cannot be done efficiently with standard optimization algorithms. In this case, Genetic Algorithm (GA) reveals been useful mainly when the objective function is discontinuous, non differentiable, stochastic, or highly nonlinear (The MathWork, 2010). Pasdarpour et al., (2009), presented a genetic algorithm methodology using fuzzy system for the optimum design of dynamic compaction of soil. The input parameters for the fuzzy system were tamper weight, height of tamper drop, print spacing, tamper radius, number of impact and soil layer geotechnical properties. Then GA optimization technique was employed to identify the maximum output which was the depth of low degree of
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improvement. Results proved that the presented approach augmented the depth of improvement about 20-30% in comparison to traditional design and was able to optimize dynamic compaction design. Similarly, Kalantary et al., (2009), investigated correlation between two soils test parameters, the undrained shear SU and the Standard Penetration Test blow count NSPT by using genetic algorithm and neural network (GMDH type). Finally, in order to achieve optimal prediction, combination of Genetic Algorithm, Neural Network, and Fuzzy Inference System or hybrid models were used with satisfactory results for decision making task in geotechnical engineering. Some applications based on this combination include slope stability with improved genetic algorithm (Zhang & Lin, 2006), slope stability based on genetic algorithm and fuzzy neural network (Xue et al., 2007), evolutionary fuzzy neural inference system for decision making in geotechnical engineering (Cheng et al., 2008).
REMARKS, DISCUSSION AND IMPLICATION FOR FURTHER RESEARCH Remarks and discussion As limitation of our paper, for the literature review, our references cover mainly, articles with international peer review indexed in IEEE Xplore, SpringerLink, ISI-Web of Science, ScienceDirect and Engineering Village databases. It doesn’t take into consideration all publications on the topic; instead we have selected the most representative papers which suffice to reach our goal. Our main objective was to present to readers a concise deep insight of how models based on fuzzy inference systems have been implemented so far in geotechnical engineering. Consequently readers are invited to be cautious about the interpretation of our results because it may happen that some techniques in use are not covered in the present paper. However we believe that this paper presents the principal trends of fuzzy inference techniques in geotechnical engineering. As far as remarks are concerned, we found that type-2 fuzzy set (fuzzy set whose membership function is also a fuzzy set) seemed not in use. The applications are based on type-1 fuzzy set or simply fuzzy set. Another remark is related to a proper use of fuzzy mathematics resources. Mostly, researchers in geotechnical engineering application context, use indifferently the notions of fuzzy sets and fuzzy numbers which seem to express the same idea. Then, parameters are assumed to be fuzzy set or fuzzy number. Actually from a strict mathematical point of view, fuzzy set and fuzzy number don’t define the same thing (Dubois and Prade, 2010). In addition, the literature reveals that fuzzy logic and fuzzy inference are sometimes questionable and are useless (Cheeseman, 1985; Lindley, 1994; Pelletier 1994 & 2004). Some researchers reported confusion on fuzzy set application for instance the existence of several views of membership function representing different concepts and the interpretation of any gradual notion as membership function even though it has nothing to do with a set (i.e. the case of fuzzy number widely in use in geotechnical engineering application) (Dubois & Prade, 1997). Nevertheless, it was proved that there is a need for fuzzy logic and is very useful as modeling tools (Zadeh 2002, Entemann, 2002). Without entering into the mathematical foundation of fuzzy set theory, it is important to bear in mind the basic idea behind this theory so that it can be used
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properly as a tool. Dubois and Prade (2010) suggested that a clear distinction should be made between ontic fuzzy set (conjunctive) and epistemic fuzzy set (disjunctive) because it will determine which mathematical tool should be used like fuzzy random variable, fuzzy system and fuzzy optimization.
Further research questions Due to its benefits, FST can be incessantly used for many applications such as classification task (rock mass classification, soil classification), selection purpose based on multi criteria and construction management based on the different techniques discussed above. However, for an optimal use of FST, further research should be done in order to fill the gaps described previously related to its application. In fuzzy inference related problems, the logic behind the popular choice of Mandani-type over other types of inference system, in the context of geotechnical engineering reported by many authors (e.g. Khademi. 2009), needs to be investigated deeply. Other system like Seguno-type might be suitable for some type of problems. Also the choice of defuzzification method used in many researches, seems to be standard. For example the centroid of area method is preferred over other methods. In addition, the assignment of membership function to the input variables is one of the key steps of FIS. Independently of the method of construction (either arbitrarily based on the user’s experience or designed using machine learning methods) the membership function should convey effectively the imprecise information of the input. For these reasons, we suggest that more research should be undertaken and aimed at finding the best methodology that will employ adequately the three conceptual elements of a FIS (rule base, database and reasoning mechanism) according to the type of problem to be solved. Other soft computing techniques (e.g. neural network and genetic algorithm) provide more resources and can be more adopted in further research improving the capacity of fuzzy inference system. Finally, the limitations and deficiencies of FST application in geotechnical context should be researched because they are unknown while FST is adopted more and more.
CONCLUSION To sum up, Fuzzy Inference Systems (FISs) have been applied in geotechnical engineering research covering a wide range of topics such as rock engineering, soil classification, constitutive modeling geotechnical site investigation, foundation, dams, settlements, slope stability, tunneling operation, construction management, tunnel stability and other geomechanical problems. Mandani type of FISs and ANFIS are the most common inference systems in use recently while the basic type of inference have been employed mainly during the 80’s and 90’s. Also the applications of artificial intelligence methods are increasing recently. Also it was shown that the use of FISs in geotechnical engineering was twofold. First epistemic uncertainty (lack of information, updated data unavailable and impreciseness) were handled successfully as well as expert knowledge and linguistic variables which were very important for some decision-making process. Secondly, fuzzy systems models were proved to be a good tool for prediction mainly when neural network or genetic algorithms were combined with.
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Even though FIS has a lot of merits, its main weakness is associated with whether it is properly used or not. We hope that the results of this review will highlight the importance of FISs applications in geotechnical engineering, provide valuable information about the current state of the art and finally, will stimulate more research interest of academicians and practitioners in geotechnical engineering.
ACKNOWLEDGEMENT The authors appreciate the support of China University of Geosciences (Wuhan)
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