Improvement of rock engineering system coding using fuzzy numbers

Abstract

A fuzzy expert semi-quantitative (FESQ) method is presented to analyze the parameters influencing rock mass cavability in block caving mines. In the proposed method, fuzzy numbers are used to allow the incorporation of uncertainty in coding the interaction matrix. Since in the classic ESQ method a fixed single value is defined as the interaction between two parameters, based on one expert view, it seems necessary to improve the method to be able to cover different views of different experts. To illustrate the application of the proposed method, first the parameters influencing the cavability in the block caving mines are selected, then a matrix of interaction between parameters are introduced using fuzzy numbers. Every fuzzy number has its own membership function designed based on experts’ views. Finally the dominant and subordinated parameters are determined and interpreted by fuzzy logic. The proposed fuzzy codding method is very useful in humanistic decision making systems with existence of uncertainty, vagueness and imprecision.

Keywords

ESQ codding fuzzy numbers cavability RES

1 Introduction

Rock engineering activities such as underground mining and tunneling disturb the initial in situ state of the rock masses and create a dynamic interaction between rock mass and structural parameters. The knowledge of the interaction mechanisms and interaction processes between the parameters of rock mass is important in understanding the behavior of rock mass [1]. By increasing the number of parameters, modeling the interaction mechanism and pathways by the conventional methods and analytical techniques becomes more difficult especially in the case of complex systems or systems with many factors with complex interrelationships. Hence, the need for new approaches for the analysis of complex processes is necessary for rock engineering. To address these limitations, the rock engineering systems (RES) approach was proposed by Hudson [2].

Rock engineering systems approach allows the simultaneous analysis of the relationship between the parameters of the rock mass, the site and its structure and assesses the effects of their interactions [2]. Many researchers have attempted to develop the RES method in various fields in rock mechanics such as [3 –10].

In the RES approach, the interactions between various parameters of the system are presented in a matrix form using a clockwise convention. The interaction matrix in rock engineering system is a key element to show the list of parameters which are used in rock engineering projects and the interaction between them. In this method, the influence parameters are located on the main diagonal of the interaction matrix and interaction with other parameters are located in the non-diagonal elements of the interaction matrix [11].

The precursor to the analysis must be a method of coding the interaction matrix in order to establish how the parameters influence each other via the mechanism. Several coding methods have been developed for this purpose, the most common being the “expert semi-quantitative” (ESQ) coding method. In the ESQ method, only a certain value (deterministic value) for the interaction parameters are considered while there are uncertainty in the parameter’s and relations between them [12]. Therefore, choosing a unique code cannot express the interactions accurately and completely. On the other hand, if several expert opinions are used for encoding the interaction matrix, eventually there is no choice except averaging the different codes. However, in many cases, the interaction is not fully known, and different experts have divers opinions about the interaction of two parameters on each other [13].

In this paper we thus propose a novel “Fuzzy ESQ” (FESQ) coding method to be used within the RES systems structure. In this novel method by using fuzzy numbers and their membership functions, uncertainties in the assignment of codes are dealt. For this purpose, a number of experts are asked to fill the interaction matrix of parameters influencing cavability. After that based on all points of view, a fuzzy number is designed for every element of the matrix. Some fuzzy algebraic operations are then done to interpret the parameters.

The assessment of parameters affecting the cavability of rock mass in block caving mines is considered to show the application of the proposed method. To fulfill this task, 14 parameters are considered as the main factors of the rock engineering system modeling. Eventually, it is shown how the RES method is employed in the fuzzy manner. The dominant and subordinated parameters and also the interactive parameters in the potential of rock mass cavability are finally introduced.

The rest of the paper is organized as follows. The rock engineering system and the ESQ method for codding the interaction matrix are discussed briefly in section 2. Some preliminaries in fuzzy numbers are explained in Section 3. Section 4 discusses the proposed fuzzy ESQ methodology. For a better illustration, the presented method is designed for analysis of parameters influencing the cavability of rock mass in block caving mines. Section 5 concludes the paper.

2 Rock engineering system (RES)

Given that an analytic approach is being developed, some methods should be designed to represent the total system behavior. Also, as it would be helpful to utilize the same presentational and analytic procedure for any circumstances or project, there is an obvious benefit in adopting a methodology that has a universal applicability.

The interaction matrix has been developed by Hudson (1992) for this purpose. The basic method of representing the relevant parameters, their interactions, and the rock mass/construction behavior is via the interaction matrix which is shown conceptually in Fig. 1 [2]. The subjects or parameters in question are placed along the leading diagonal (top left to bottom right) of the matrix. In Fig. 1, this is a 2 × 2 matrix where subject A is in the top left-hand box and subject B is in the bottom right-hand box. The influence of A on B is located in the top right-hand box; the influence of B on A is located in the bottom left-hand box. Thus the basic principle of the interaction matrix is to place the main subject or parameters along the leading diagonal and to consider the interactions in the off-diagonal boxes [2]. As an example of the interactions in a discontinuous rock mass, six binary interactions are shown in Fig. 2.

In this Figure, there are three basic subjects being considered: the rock structure, the rock stress and water flow. With just these six interactions shown in Fig. 2, it is clear that some methodological device is needed to be able to coherently consider the interactions and indeed any other interactions that may occur. Also, not only the natural processes that are involved in the rock mass interactions must be considered but also the effects of construction. This refers both to the disturbance induced by the construction process and to the presence of the new structure created by the construction [2].

2.1 Matrix coding

Some form of quantitative translation of observation and our engineering knowledge of the mechanism is required. This is achieved by coding the interaction matrices and studying the interaction intensity and dominance of each parameter. In Fig. 3, five matrix coding methods are shown [14]. Among these methods, the ESQ method is the most common approach for coding the interaction matrices [10].

2.2 The cause and effect plot

Once the matrix has been numerically coded, the sum of each row and each column is calculated. The sum of the row values is termed as the “cause” and the sum of the column is termed as the “effect”. Figure 4 shows the generation of the cause and effect co-ordinates. The main parameters, P_i, are listed along the leading diagonal with construction as the last box. From the construction of the matrix, it is clear that the rows passing through P_i represent the influence of P_i on all the other parameters in the system. Conversely, the column through P_i represents the influence of the other parameters, i.e. the rest of the system, on P_i [1].

The parameter intensity can be measured by the perpendicular distance of the parameter point from this line. The iconic representation is shown in Fig. 5, which illustrates parameters interaction intensity and parameters dominance characteristic. The two sets of 45° lines in the plot indicate contours of equal value of each of two characteristics. The parameter interaction intensity increases uniformly from zero to the maximum, the associated maximum possible parameter dominance value rises from zero to a maximum at 50% parameter interaction intensity and then reduces back to zero at the maximum parameter intensity value. The specific numerical values of the two characteristics are $(C + E) / \sqrt{2}$ and $(C - E) / \sqrt{2}$ as indicated in Fig. 5 [15].

3 Fuzzy numbers

A fuzzy number is an extension of a regular real number in the sense that it does not refer to one single value. It has some of possible values with their own weights between 0 and 1 which is named membership function. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line [16]. The calculations with fuzzy numbers allow the incorporation of uncertainty on parameters, properties, geometry, initial conditions, etc. Figure 6 shows the difference between a regular number with the value of 3 and a fuzzy number 3. In the crisp set theory, a number just have one value. For example the membership function of number 3 in crisp theory can be expressed as:

$μ_{number 3} = {\begin{matrix} 1 for x = 3 \\ 0 for x \neq 3 \end{matrix}$ (1)

And the membership function for the fuzzy number 3 which contain a closed neighborhood of number 3 can be defined as: $μ_{Fuzzy number 3} = {\begin{matrix} x - 2 for x \in [2 3] \\ \begin{matrix} 4 - x for x \in [3 4] \\ 0 otherwise \end{matrix} \end{matrix}$ (2)

Fuzzy sets can be further divided based on the type of membership function which describes them. The most common types are triangular, trapezoidal, and bell-shaped numbers. But they may also have an arbitrary membership function that fits a special situation. The definition of a fuzzy number is addressed in [17, 18] as follows:

Definition 1. A fuzzy number β is a mapping $β (x) : R \to [0 1]$ with the following properties:

•(Bounded support): there are two different values like a, b with a ≤ b called the endpoints of μ such that: ${\begin{matrix} μ (x) = 0 x \notin [a b] \\ μ (x) > 0 x \in [a b] \end{matrix}$ (3)

•(Normality): there are two different real number c, d with a ≤ c ≤ d ≤ b in sense that μ (x) =1 if and only if x ∈ [c, d]

•(Convexity): μ (x) is increasing in the interval [a, c] and decreasing in [d, b].

When a fuzzy number is designed, there are several operations which are done on fuzzy sets and numbers. The three main operations on fuzzy sets are complement, union and intersection. For calculation of the fuzzy sets union we use the S-norms. The most typical S-norm in fuzzy sets is the maximum function. In this paper the Yager class S-norm is used which is defined as [19]: $S_{ω} (μ_{a}, μ_{b}) = min [1, (μ_{a}^{ω} + μ_{b}^{ω})^{\frac{1}{ω}}]$ (4)where S_w is the S-norm of two different fuzzy sets with membership functions of μ_a, μ_b and ω is a parameter in the interval of (0, ∞). In addition, the T-norms are used to calculate the intersection of fuzzy sets. The most typical T-norms are minimum function and product. In this paper, the minimum function shown by min (o) is used as the T-norm.

The relations of fuzzy sets have also been introduced. The cartesian product of two fuzzy sets can be calculated using T-norms. In this paper, the cartesian product is calculated as follows: $μ_{a \times b} (x, y) = min (μ_{a} (x), μ_{b} (y))$ (5)

To obtain the most important values in a fuzzy set, α-cut of a fuzzy set is usually calculated. α-cut shows the values that have membership function more than α. It is simply defined in Definition 2.

Definition 2- An α-cut or α-level set of a fuzzy set A ⊆ X is an ordinary set A_α ⊆ X, such that: $A_{α} = {x | μ (x) \geq α}$ (6)

The algebraic operations are also determined for fuzzy numbers. The sum of two fuzzy numbers A and B with membership functions μ_a (x) , μ_b (x) can be calculated as follows: $μ_{A + B} (z) = sup_{x + y = z} min (μ_{a} (x), μ_{b} (y))$ (7)and similarly the subtracting of two fuzzy number A and B is defined as: $μ_{A - B} (z) = sup_{x - y = z} min (μ_{a} (x), μ_{b} (y))$ (8)

An example of summing and subtracting of two fuzzy numbers is displayed in Fig. 7.

In most of applications, after all fuzzy operations and calculation, we need a crisp value as an output. The nonfuzzy output is computed using defuzzifying methods. There are different ways to extract the crisp value of a fuzzy set. In this paper the center of gravity defuzzifier is used which is calculated as follows: $x^{*} = \frac{\int x μ (x) dx}{\int μ (x) dx}$ (9)where x^* is the crisp output of fuzzy set with membership function μ (x) Fig. 8 shows the center of gravity defuzzifier.

4 Fuzzy ESQ codding

To demonstrate the application of the fuzzy ESQ method, the assessment of effective parameters on the cavability of rock mass in block caving mines are considered as case study. In the following, a brief discretion about cavability of rock mass in block caving mine is explained.

4.1 Rock mass cavability

Block caving is a distinct caving method, applied mostly to large and massive ore bodies. Inherent to the method is low cost and high production capability, especially when it reaches full production [20].

The cavability of the rock mass is one of the fundamental issues for the current mining methods. Two principal techniques used in the industry to investigate the cavability of the rock mass: an empirical method and a method based on numerical analysis [21]. The cavability of a deposit defines the ability of the orebody and overlying rock mass to cave freely and spontaneously, once undercut to a sufficient dimension [22]. Providing indicative values of cavability has now become possible, although there is no general consensus regarding cavability criteria. Both large companies and academics, often independently, have carried out lots of research in the last decade or so, but a comprehensive formula is yet to be found [23]

Many parameters affect the cavability of rock mass. The selection of classification parameters mainly depends on its characteristics that include importance, accuracy and reliability. We selected two categories of parameters: induced factors and natural factors. The selection of parameters has been performed based on the literatures [21 , 24–27] and the experience gained from our own analysis of cavability in block caving mines. As a result, 13 key parameters that have the greatest impact on the cavability have been selected. These parameters are shown in Fig. 9.

4.2 ESQ fuzzy methodology

To deal with approximate nature of human reasoning in RES, fuzzy numbers are used to rank the interaction between the parameters. Instead of using crisp values of 0, 1, 2, 3, 4, five fuzzy sets are introduced for ranking the interaction. Some questionnaires are prepared and the twenty experts who are specialists in this field fill them by using one of these five fuzzy sets for every element of RES matrix. The fuzzy sets introduced for ranking is displayed in Fig. 10.

Now the goal is to calculate a fuzzy number with its own membership function based on the different experts’ point of views for every element in the matrix. To fulfill this goal and to cover all the specialists’ ideas, a union operation has been done on fuzzy sets. For every interaction value, the number of every fuzzy set which is voted by experts is considered and a weighted union operation is done using Yager class S-norm.

For instance for the effect of parameter P₁ (UCS) on the parameter P₂ (in-situ stress), 60 percent of experts voted on “Medium” interaction, 20 percent said it has “weak” interaction, and 10 percent voted on “Strong” interaction. Five percent of experts announced it has “No interaction” and similarly five percent of them voted on “Critical” interaction. Hence a union operation is done on the fuzzy sets with the weight which has been voted. For example a union between 5 fuzzy sets with the weights [0.05, 0.2, 0.6, 0.1, and 0.05] is performed for calculating the fuzzy number for effect of P₁ on P₂. The weighted fuzzy sets and the final computed fuzzy number are shown in Figs. 11 and 12. The Yager class S-norm has been performed using w = 1. As it is displayed in Figs. 11 and 12, the Yager class S-norm makes the final fuzzy membership function smoother than it has been done by maximum functionS-norm.

These operations are performed for all RES matrix elements and a fuzzy number is calculated for every element based on the experts’ views. The final fuzzy RES matrix is shown in Fig. 13.

After computing all the interaction values by fuzzy numbers, now it is necessary to calculate the cause and effect for every parameter. For calculating the cause value for parameter i, 13 fuzzy numbers in i-th row of the matrix should be summed. Similarly, the i-th column should be summed to obtain the effect for parameter i. For sum operation, Equation (7) is used. Finally the sum of cause and effect for every parameter should be computed. All the cause and effect parameters and the summation of them are displayed in Fig. 14. For better illustration, the crisp value equal to every fuzzy set is demonstrated by a fix orthogonal line in the Figures. The crisp output of every set is calculated using center of gravity defuzzifier Equation (9).

One of the most important graphs in the RES analysis is the cause-effect diagram (C–E plot). The position of parameters in this plot demonstrates how a parameter influences the overall system. Since the causes and effects for different parameters are fuzzy numbers, the position of parameters in C–E plot is not a fixed single point anymore and it is also a fuzzy set. There are several values for one parameter with its own membership function demonstrating the position of the parameter on the plot. For computing the C–E plot position for every parameter, Equation (5) is used. For instance, the positions of parameters P₇ and P₁₁ in the C–E plot are displayed in Figs. 15 and 16.

For dealing with vagueness and imprecision, it’s better to analyze the fuzzy value of cause and effect. To plot the histogram of summation and difference of causes and effects, an α-cut of every fuzzy set corresponding to summation and difference is considered. The α-cut is not a fuzzy set and shows the most important values in any fuzzy set. Since after algebraic operations the fuzzy sets are not normal any more, 70% of maximum of fuzzy membership function for every set is considered as α. The α-cut of fuzzy summation of cause and effect for every parameter is plotted in Fig. 17 and, the histogram of difference between cause and effect is displayed in Fig. 18. The defuzzified value of every set is shown by a circle. When the α-cut set has many members, it means that the corresponding fuzzy set has been wider and there are more vagueness about that parameter like P₇ in the Fig. 17. On the other hand, most of the experts have had close views about a parameter when the α-cut set has less member like P₄.

For better illustration and more simplicity, the nonfuzzy value of causes and effects are calculated using Equation (9) and the C–E graph of nonfuzzy values is plotted in Fig. 19.

As it is mentioned previously, the parameter with the maximum value of C–E is dominated on the system and the parameter with the minimum of C–E is subordinated by the system. So in the presented system based on fuzzy logic, both P₁ and P₄ should be considered as the most dominant parameters but with different membership values. The parameter P₁₄ is the most subordinate parameter. Figure. 19 also clarifies that P₂ is the most interactive parameter, which means small changes in parameter largely affect the overall system.

The example illustrates the application of the proposed fuzzy method. It is clear that the fuzzy method covers the different ideas of different experts. The paper presents a useful improvement to the ESQ method coding however the methodology still depends on the specialist opinions.

5 Conclusion

A fuzzy ESQ method for RES analysis is presented in this paper to allow the incorporation of uncertainty in coding the interaction matrix. The interaction intensity of parameters has been coded by fuzzy numbers. The membership function of every fuzzy number is computed according to the expert’s views. The values of cause and effect for every parameter are calculated using fuzzy operations. For better illustration, the proposed fuzzy method is used to analyze the parameters influencing the cavability in block caving mines. Fourteen effective parameters on the cavability of rock mass are selected and the fuzzy interaction matrix of cavability potential is created. The C–E diagram is plotted and the dominant, subordinate and interactive parameters are introduced based on their fuzzy values.

By using the presented fuzzy method the uncertainty and vagueness of human reasoning can be directly included in the analyses, and the effect of such uncertainties can be numerically expressed. This type of information is critically important in practice. For example in a field operation, a designer can identify the parameters that need special attention and consider them in designing a project.

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