Fuzzy Inference System
Fuzzy logic is a form of manyvalued logic and it deals with reasoning that is approximate rather than fixed and exact. The nature of uncertainty in a slope design is a very important that should considered. Fuzzy set theory was developed specially to deal with uncertainties that are nonrandom in nature.
The fuzzy set was first introduced in 1965 by Lofti Zadeh as a mathematical way to represent linguistic vagueness . It can be considered as a generalization of classical set theory. In a classical set, an element belongs to or does not belong to a set. That is, the membership of an element is crisp (0, 1), and an ‘‘A’’ crisp set of real objects are described by a unique membership function such as X_{A }in fig.3.1 (a).
Fig. 3.1 (a) Crisp set and (b) Fuzzy set
Contrary, a fuzzy set is a generalization of an ordinary set which assign the degree of membership for each element to range over the unit interval between 0 and 1 as shown in fig. 3.1(b). In addition, fuzzy set theory can be used for developing rulebased models which combine physical insights, expert knowledge and numerical data in a transparent way that closely resembles the real world. An element of the variable can be a member of the fuzzy set through a membership function that can take values in the range from 0 to 1. Membership functions (MF) can either be chosen by the user arbitrarily based on the user’s experience or can also be designed using machine learning methods (e.g., artificial neural networks, genetic algorithms, etc.). There are different shapes of membership functions; triangular, trapezoidal, piecewiselinear, Gaussian, bell shaped, etc. The fuzzy rules provide a system for describing complex (uncertain, vague) systems by relating input and output parameters using linguistic variables. A fuzzy if–then rule assumes the form ‘‘if x is A then y is B,’’ where A and B are linguistic values defined by fuzzy sets on universes of discourse X and Y, respectively.
Fuzzy inference is the process of formulating an input fuzzy set map to an output fuzzy set using fuzzy logic. In fact, the core section of a fuzzy system is the FIS part, which combines the facts obtained from the fuzzification with the rule base and conducts the fuzzy reasoning process. Generally, the basic structure of a FIS consists of three conceptual components, rule base, database, and reasoning mechanism. A rule base contains a selection of fuzzy rules and a database defines the membership functions used in the fuzzy rules. A reasoning mechanism performs the fuzzy reasoning based on the rules and given facts to derive a reasonable output or conclusion. There are several FISs that have been employed in various applications. The most commonly used include:
· Mamdani Fuzzy Model;
· TakagiSugenoKang fuzzy (TSK) model;
· Tsukamoto fuzzy model;
· Singleton fuzzy model.
The differences between these FISs lie in the consequents of their fuzzy rules, and thus their aggregation and defuzzification procedures differ accordingly. Defuzzification is a process of reducing an aggregated (or clipped) fuzzy set into a crisp number, presumably the most representative value of that fuzzy set interval. There are two methods which are generally used for defuzzification i.e. Centre of area (Centroid) method and Ranking index method.
The Mamdani Fuzzy model is often used in geotechnical problems because of its simplicity and effectiveness to handle linguistic variables. 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.
Fuzzy logic provides an inference structure that enables the human reasoning capabilities to be applied to artificial knowledgebased systems. Fuzzy logic provides a means for converting linguistic strategy into control actions and thus offers a highlevel computation.Fuzzy logic provides mathematical strength to the emulation of certain perceptual and linguistic attributes associated with human cognition, whereas the science of neural networks provides a new computing tool with learning and adaptation capabilities. The theory of fuzzy logic provides an inference mechanism under cognitive uncertainty, computational neural networks offer exciting advantages such as learning, adaptation, fault tolerance, parallelism, and generalization.
Table 1: Distribution of the main references according to the fuzzy inference techniques employed.
Type of fuzzy set/logic 
Number of papers 
Topics 
Basic fuzzy set/logic 
13 
Slope stability, foundation, rock mass classification, geotechnical project scheduling and cost planning determination geotechnical parameters, prediction of soil uniaxial compressive strength, fizzyfication of Chen plastic model of concrete and rock sawability. 
Mandani systems 
18 
Rock mass blastability, penetrability, diggability, rippability, excavability; rock mass classification systems, prediction of flyrock in mining surface, burden, rock fragmentation, backbreak in openpit blasting and TBMs thrust and torque requirement 
SegunoTagaki systems 
5 
Prediction of maximum charge par delay in surface mining, impact hammer performance, constitutive modeling, swelling potential of compacted soils, rock engineering classification system and rock slope stability assessment 
Hybrid system using Neural network (ANFIS and other) 
12 
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 landside susceptibility mapping 
Hybrid system using Genetic Algorithms 
5 
Optimum design of dynamic compaction of soil, slope stability and decision making in geotechnical engineering. 
Table 1: A general classification of fuzzy set technique in use in geotechnical engineering
Fuzzy techniques 

Example of applications 
Basic fuzzy set/fuzzy logic and inference 

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). 
Advanced fuzzy inference systems 
Mandani type systems 
A new Mandanibased model to predict burden from rock geomechanical propertics (Monjezi & Rezaei, 2011) and Mandani fuzzy inference model prediction of the blastability designation of rock (Azimi et al., 2010) 

Sugeno type systems 
Rock engineering classification system (Jalalifar et al., 2011), rock slope stability assessment (Chen et al., 2011) 

Systems using Neural Network 
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). 

Systems using Genetic Algorithm 
Slope stability (Zhang & Lin 2006; Xue et al. 2007) 

Hybrid formulation 
Soft computing techniques based model of the angle of shearing resistance of soils (Kayadelen et al., 2009) 
Fuzzy probability theory 

Slope reliability (Dodagoudar & Venkatachalam,2000) 
Fuzzy plasticity theory 

Cyclic constitutive modeling (klisinski,1988), rock fragmentation (Mishnaevsky & Schmauder, 1996) and soilwater hysteresis model for unsaturated sands (Min & Phan, 2010) 
Neurofuzzy inference systems have been used in many areas in civil engineering applications. A stability assessment model for epimetamorphic rock slopes has been developed by using Adaptive NeuroFuzzy Inference System (ANFIS) for its capacity of dynamic nonlinear analyses. the inference system is employed to predict the stability of the slope by choosing bulk density γ, the height H, the inclination β, the shear strength parameters c and ϕ, of the slope as inputs, while the stability state as output.
In order to forecast the factor of safety (FS) or the status of stability (S) in the case of rock or soil slopes, the factors that influence FS and S have to be determined. The input layer data consists of six input parameters that perfect stability in the case of failure. The output layer is composed of a single output parameter, either the factor of safety FS, or the status of stability.
This fuzzy methodology enables the engineer to investigate the effect of parameter uncertainty on the computed stability of a slope in systematic way. The resulting factor of safety fuzzy set provides more information than does a single, fixed factor of safety value as obtained from conventional methods. With this approach the variation and possible range of factor of safety values which reflect uncertainty in the input parameters can be determined.
Shangguan et.al.(2010) has simulated Probabilistic Neural Networks Forecasting the stability of Slope. Unit weight, Cohesion, Internal friction angle, Slope angle, Slope height and Pore pressure used as input parameter and factor of safety is simulated as output parameter. Then he also compare between the estimated and practical states of slope safety with different methods.
ANN Among all data, we 80% used for the training and remaining for validating the prediction capability The dataset must covers a wide spectrum of soil and seismic parameters. When preparing input data for a particular site, of primary importance is the recognition of the conditions which caused the slope to become unstable and the processes which triggered that movement. From the results presented here, it can be observed that the neural network results are considerably close to value calculated by Bishop’s classical method. In all cases, it is over 92% and in most cases, it is over 95%.
ANN Several important parameters, including total stress, effective stress, angle of slope, coefficient of cohesion, internal friction angle, and horizontal coefficient of earthquake, were used as the input parameters, while the slope stability was the output parameter. The results are compared with the classical methods of limit equilibrium to check the ANN model’s validity.
The application of fuzzy set theory is used to SMR classification by incorporating fuzzy sets and assesses the stability of rock slope. The Mamdani fuzzy algorithm may be used to construct the if–then rules for evaluating rock slope stability.
Thus, the FSMR method provides a better assessment of slope stability than other slope stability classification systems and can also predict of rock slope failure. It is also noted that engineering judgment is required in the stability analysis, and fuzzy set theory has also been found to be a useful tool for rock engineers and engineering geologists who study rock slope stability.
Fuzzy Slope engineering is an important branch in geotechnical engineering. Slope engineering is a complicated systematic engineering. A lot of Engineering is related to slope stability, such as mining, road and bridge, water conservancy and structure engineering, etc. Its stabilization directly concerns the safety of engineering. With economical development and largescale construction cause, it takes up more and more important place.Based on reviews of evaluation methods of slope stability, the main research work conducted in this paper is as follows:
(1)Considering the uncertain problems of stability analysis which have the characteristics of random and fuzziness, the author uses the maximum membership degree principle to analyze and evaluate the slope stability. Ridge distribution in effect factor of quantity and trapezium distribution in the effect factor of ration are applied here to construct membership function. The gradation analysis method is used here to determine the proportion of importance of each effect factor. The method of two class synthesis assessment is adopted to analyze the stability of slope.
(2)There are eleven effect factors chosen to analyze fuzzily the slope stability. We selected angle of cut slope, state of underwater, angle between surface of cut slope and major structure plane, efflorescence, etc. as major factors effect slop stability. The slop stability is assessed by each factor.
(3)Based on one concrete engineering case, the method of fuzzy analysis is examined, and this result demonstrates that eleven membership functions, constructed by the author, are reasonable. So the proportion of importance is reasonable. The membership functions and the distribution of the proportion of importance can also be applied to analyze the stability of similar slopes.
(4)Put the judgment of fuzzy comprehensive evaluation as the input of neural network by MATLAB. We transport out the final judgment through the neural network that possess learning ability.