Intuitionistic Connection Cloud Model Based on Rough Set for Evaluation of the Shrinkage–Swelling of Untreated and Lime-Treated Expansive Clays

Abstract

The evaluation of the shrinkage–swelling characteristic of expansive clay is of great significance, but it is a complex problem since the evaluation process involves numerous uncertain factors, such as randomness, non-subordination, and hesitation uncertainties. Here, an intuitionistic connection cloud model has been proposed to address this issue. First, an evaluation index system is established. According to the reliability of interval-valued evaluation indexes, the corresponding cloud numerical characteristic parameters are specified based on the membership interval generated by the intuitionistic fuzzy principle. Moreover, the improved conditional information entropy based on rough set theory is utilized to assign the index weight. Subsequently, combined with the weight, the intuitionistic connection degree of the sample to the classification standard is determined to identify the shrinkage–swelling grade. Finally, a case study on the shrinkage–swelling grade of untreated and lime-treated expansive clays in Hefei Xinqiao International Airport was performed to illustrate the validity and reliability of the model. The results show that the proposed model is reasonable and feasible for the evaluation of the shrinkage–swelling grade of untreated and lime-treated expansive clays.

1. Introduction

Expansive clay with strong hydrophilic and considerable shrinkage–swelling potential is a special soil that will directly affect the stability and safety of the building foundation and may cause serious damage to the structure [1]. So, to improve the strength of expansive clay, lime treatment is used to deal with the above problems in engineering practice. Sometimes, bad construction quality control and neglect of shrinkage may also result in engineering hazards and huge economic losses. Consequently, to explore the appropriate treatment or modification methods and ensure the safety and stability of the engineering structure founded on the expansive clay, it is important to appropriately estimate the shrinkage–swelling grade of untreated and lime-treated expansive clays [2]. However, the problem has not yet been solved because expansive clay in different regions shows specific physical features and engineering behaviors due to its various compositions and environment. Therefore, it is very necessary to develop an evaluation method to enhance the accuracy and reliability of the evaluation of the shrinkage–swelling problem via uncertainty and certainty information and human behavioral characteristics.

To accurately assess the shrinkage–swelling grade, some scientists have introduced uncertainty analysis theories—for example, extension theory [3], fuzzy mathematics [4], grey theory [5], rough set theory [6], neural network [7], and support vector machine [8] to depict the uncertainty of evaluation factors. Although these methods have made some substantial progress, they still have some defects in their corresponding application processes: extension theory is an effective method by which to solve the incompatibility problem in the real world, but it does not consider the ambiguity of the evaluation object; while the fuzzy mathematical membership function is difficult to determine in practical application; the grey theory has low accuracy for the evaluation of information dispersion samples; the rough set theory eliminates some important evaluation factors in the process of attribute simplification; and machine learning methods, including artificial neural networks and support vector machines, cannot describe the random ambiguity of the interaction between the influencing factors. To address these shortcomings, Wang et al. presented classification methods, coupling set pair analysis theory with triangular fuzzy numbers or a cloud model to classify the degree of shrinkage and expansion for expansive clay [9,10]. They can express the transition between certainty and uncertainty of evaluation indicators. Compared with the above methods, the cloud model can effectively deal with the fuzziness and randomness of the index in the infinite interval, but it ignores the limited interval of the index value distribution and cannot deal with the correlation between multiple evaluation indexes. Recently, Wang et al. [11] proposed a connection cloud model to overcome the above defect of the traditional cloud model to a certain extent. But it still does not consider the hesitant property of the connection degree, which may result in evaluation information loss; for instance, the connection degree is a crisp value of 1 when the index value is the expectation of the connection cloud. This is not consistent with the actual state and may not reflect the hesitant characteristics of the connection degree. Accordingly, some researchers introduced the intuitionistic normal cloud to analyze uncertain problems [12,13]. However, thus far, little attention has been paid to the classification associated with weight considering the importance and information of the index under multiple uncertainties.

As we know, proper index weighting is the fundamental procedure of evaluation. Objective weighting methods, e.g., the entropy method, the maximum deviation method, and the TOPSIS method, are commonly utilized to assign evaluation index weight. They can fully make use of information but cannot embody the importance of the evaluation index. To overcome this shortcoming, Chen and Huang [14] applied the advantage of the rough set to determine the importance and weight of attributes. This method makes full use of the objective information of sample data but ignores the cases where the weight of the index is 0. However, each index has a certain effect on the shrinkage–swelling property, so its influence should be considered in the evaluation procedure. Based on the discussion discussed above, index weight determination also needs further investigation because previous objective weighting methods mainly focused on index information; few focus on the importance of index information. Thus, there is a need for the development of suitable methods for the shrinkage–swelling classification of untreated and lime-treated expansive clays.

In summary, expansive clay in different regions exhibits specific engineering behaviors due to its various compositions and environment. So, the evaluation of the shrinkage–swelling characteristic of expansive clay inevitably encounters hesitant uncertainties. However, existing uncertainty analysis methods still lack unified consideration of fuzzy, hesitation, and randomness uncertainties. This may result in difficulties in obtaining a precise description of uncertainty when the measured values are on the mean value. In addition, previous objective weighting methods mainly focused on index information; few focused on the importance of index information. Therefore, the objective of our work is to provide a new means of simultaneously depicting the membership, non-membership, and hesitation degree for the evaluation of the shrinkage–swelling grade associated with index information importance. The proposed method can take into account the uncertainty of the hesitant characteristics of evaluation indicators to achieve the goal of improving the reliability and accuracy of the actual evaluation of the shrinkage–swelling grade.

Given that the previous classifications of shrinkage–swelling grades rarely considered the description of the hesitation degree in various types of uncertainty indicators and their weighting, existing evaluation methods still lack unified consideration of fuzzy, hesitation, and randomness uncertainties. To simultaneously depict the membership, non-membership, and hesitation degree responding to classification standards associated with index information importance, it is very necessary to develop an evaluation method under multiple uncertain environments. Therefore, based on the above considerations, to improve the flexibility and practicability of uncertainty processing, this paper attempts to use a method of coupling the concept of the intuitionistic fuzzy set and the connection cloud model and the rough set and proposes a new intuitionistic connection cloud model to dialectically analyze and evaluate the shrinkage–swelling grade based on the description of membership, non-membership, and hesitation characteristics of interval-valued indicators of classification standards. At the same time, this study introduces an improved conditional information entropy weighting method based on a rough set to enhance the reliability of the evaluation results.

2. Theory

2.1. Intuitionistic Fuzzy Set

At first, Atanassov [15] introduced the concept of intuitionistic fuzzy sets (IFS) as a generalization of fuzzy sets. The IFS is the extension of fuzzy sets and gives both the membership degree and the non-membership degree of an element belonging to a set. So, the IFS has the characteristics of comprehensive delicacy and flexible description of the uncertainty system and overcomes the defect of the fuzzy set, which can only describe the “this is also another” and cannot describe the “not one is not the other”.

Definition 1 ([12,15]).

Let X be a conclusive domain. If x corresponds to two mappings, $u\to \left[0,1\right]$ and $v\to \left[0,1\right]$, with the condition $0\le u\left(x\right)+\nu \left(x\right)\le 1,\hspace{0.33em}\forall x\in X$, an IFS A in X is defined as:

$$A=\left\{\left.〈x,u\left(x\right),\nu \left(x\right)〉\right|x\in X\right\},$$

(1)

$$\pi (x)=1-u\left(x\right)-\nu \left(x\right),$$

(2)

where u(x), ν(x), and π(x) denote the membership function, non-membership function, and hesitancy function of element x belonging to the intuitionistic fuzzy set A, respectively. π(x) represents the hesitancy degree for the object x to A.

Unlike the traditional fuzzy set, which his only characterized by a membership function, the non-membership function and hesitancy function in the IFS are used to depict uncertain information; that is, the intuitionistic fuzzy sets possess the capability of dealing with uncertain information through membership degrees, non-membership degrees, and hesitation degrees. Hence, it is more reasonable and stable in simultaneously depicting the degree of support, opposition, and neutrality of the object x for a particular state through u(x), ν(x), and π(x), respectively. So, IFSs can provide additional options for describing the properties of things and have been widely applied in various fields. This also provides a new concept for solving the problem that the connection cloud cannot describe the hesitant characteristics of fuzzy random concepts. Here, the information is coupled with IFS to make up for the above defect in the connection cloud model.

2.2. Intuitionistic Connection Cloud Model

The normal cloud model can transform qualitative concepts into quantitative data; thus, it is widely used in the analysis of uncertainty problems. However, it still has some defects. First, the cloud droplets mapped by the normal cloud generator are established in an infinite interval, which makes it difficult to describe the actual distribution of the evaluation index in the finite interval. Second, the actual distribution of the evaluation index sometimes struggles to meet the assumption of normal distribution. Finally, the common interaction among uncertainties in the classification problem is ignored, so the conversion trend of the evaluation results at the grading threshold cannot be effectively reflected. These shortcomings limit the application scope of the normal cloud and may cause deviations between the results and the actual situation. To make up for these defects of the normal cloud, Wang et al. improved the normal cloud model by using the connection number theory and proposed a new connection cloud model [10]. Connection number theory developed from the principle of set pair analysis can analyze uncertain problems from the perspectives of identity, discrepancy, and contrary aspects.

Definition 2.

Let P be a quantitative domain and Q be a qualitative concept of P. If x ∈ P is a random event of concept Q and satisfies x~N(Ex, (En′)²), En′~N(En, He²), then the distribution of x in the universe P is called the connection cloud, and each x is a simulated cloud droplet. Similar to the certainty in the normal cloud, the membership degree of the cloud droplet x in the connection cloud is called the connection degree y(x), and its mathematic model is as follows [11]:

$$y(x)=\exp \left(-\frac{9}{2}{\left|\frac{x-Ex}{3E{n}^{\prime }}\right|}^k\right),$$

(3)

where En′ denotes random numbers following normal distribution based on entropy En and hyper entropy He; and k is the order of the connection cloud. When k = 2, the connection cloud will degenerate into a normal cloud.

The connection cloud model considers the actual distribution characteristics of evaluation indexes and describes the relationship between certainty and uncertainty relationships of information, the relationship between indexes and grades, and the conversion trend between adjacent grades. These advantages enable the connection cloud to be widely used in various fields. However, like normal clouds, it still does not deal well with the problem of non-subordination and the hesitation of fuzzy concepts. It can be seen from Equation (3) that when the value of the simulated cloud droplet x is equal to the mathematical expectation Ex of the cloud, its connection degree is equal to 1, which does not fully reflect the hesitant characteristics of some fuzzy random concepts [10]; that is, in some cases, the membership of any element of the discussed region cannot be accurately determined, and it is therefore impossible to find the elements that entirely belong to the ambiguous and uncertain classification set. The above drawbacks can be addressed by intuitionistic fuzzy sets, which allow membership to oscillate randomly in an interval rather than being fixed on a crisp number. Therefore, this paper proposes a new intuitionistic connection cloud model using intuitionistic fuzzy sets.

Definition 3.

Define x (x ∈ P) as a one-time implementation of Q in the region P. To achieve this, the connection degree is no longer equal to a fixed value of 1 corresponding to x = Ex but a uniform random number for the connection cloud model. The membership degree u and non-membership degree v with the condition $0\le u+v\le 1$ are introduced to express the hesitancy degree of the connection degree. Combined with the numerical characteristic parameters of the connection cloud model, an intuitionistic connection degree z(x) is given as

$$z(x)=\beta \cdot \exp \left(-\frac{9}{2}{\left|\frac{x-Ex}{3E{n}^{\prime }}\right|}^k\right),$$

(4)

where β is a random number satisfying uniform distribution U [u, 1 − v]. If $u=1,\hspace{0.33em}\nu =0$, the intuitionistic connection cloud degenerates into a connection cloud. As denoted above, the cloud drop generation algorithm for intuitionistic connection cloud is depicted as follows:

(1)

Initialization of numbers of cloud drops and the lower limitations of the possible values of the membership degree and non-membership degree corresponding to Ex.

(2)

Generation of random numbers En′ from the normal distribution with entropy En and hyper entropy He.

(3)

Generation of random numbers x_m satisfying the normal distribution of expectation Ex and standard deviation En′.

(4)

Generation of random numbers β following the uniform distribution of lower limitation u and upper limitation 1 − v.

(5)

Specification of the intuitionistic connection degree of cloud drops through Equation (4). A corresponding drop is obtained as (x_m, z(x_m)).

(6)

Repeat 2 to 5 steps until all cloud drops are obtained.

To illustrate the advantages of an intuitionistic connection cloud, Figure 1 uses an example to reflect the theoretical optimization process from the normal cloud to the connection cloud and then to the intuitionistic connection cloud. The classification standard interval of the known value to be evaluated is [6,12]. The numerical characteristics of the normal cloud and the connection cloud are (Ex, En, He) = (9.0, 3.0, 0.01) and (Ex, En, He, ξ, k) = (9.0, 3.0, 0.01, 9.0, 1.70), and the corresponding cloud map is obtained after generating 1000 cloud droplets. However, at the classification thresholds x = 6 and 12, there is both the possibility of belonging to the adjacent grades and the possibility of not belonging to the adjacent grades; the membership degree should be assigned to 0.5 at these two points. It can be seen from Figure 1 that the connection degree of the two boundary values 6 and 12 in the connection cloud is equal to 0.5, but the degree of certainty in the normal cloud is not equal to 0.5. Therefore, the connection cloud truly reflects the certainty and uncertainty at the thresholds and its conversion trend, which effectively compensates for the defects of the normal cloud. On the other hand, the membership at the Ex of the connection cloud and norm model is a crisp value 1. This cannot express the hesitation properties. To better deal with the non-membership and hesitation properties of some fuzzy concepts [12], the intuitionistic cloud model incorporates the concept of the hesitancy degree. In Figure 1, let β~U [0.9, 1]; according to definition 3, the intuitionistic connection degree in the intuitionistic connection cloud is no longer the crisp value of 1 when x takes the mathematical expectation value of 9; instead, it fluctuates in the range of 0.9~1.0. Therefore, compared with the connection cloud and normal cloud, the intuitionistic connection cloud can more effectively capture uncertainty and realize the objective description of the degree of non-membership and hesitation.

3. Evaluation Model Based on Intuitionistic Connection Cloud Model

Basic Principle and Evaluation Process

The basic principle of the evaluation model based on the intuitionistic connection cloud model is as follows. First, select the evaluation indicators and classification standards to construct the evaluation index system. Then, specify the numerical characteristic parameters of the intuitionistic connection cloud and generate the intuitionistic connection cloud. Next, investigate the weight assigned via the improved information entropy method based on the rough set and determine the intuitionistic connection degree of the sample to each grade. Finally, identify the shrinkage–swelling grade of the sample according to the principle of maximum membership degree. The corresponding process is shown in Figure 2. And detailed steps are listed as follows.

Step 1: Set up an evaluation index system of the shrinkage–swelling grade. The shrinkage–swelling mechanism of untreated and lime-treated expansive clays is complex, and there are many influencing factors. According to the references [10,16] and the Chinese National Standards GB/50112 [17], five kinds of indexes, including the liquid limit C₁, total shrinkage–swelling rate C₂, plasticity index C₃, natural water content C₄, and free expansion rate C₅, are selected as evaluation indicators, and the shrinkage–swelling property is divided into four grades: extreme high (I); high (II); moderate (III); and low (IV). The corresponding classification standard [16] is listed in

Table 1. Classification standard for the shrinkage–swelling grade [16].

Grade	Liquid Limit C1/%	Total Shrinkage–Swelling Rate C2/%	Plasticity Index C3	Natural Water Content C4/%	Free Expansion Rate C5/%
Extreme High, I	55~100	6~100	35~100	0~15	85~200
High, II	50~55	4~6	25~35	15~25	70~85
Moderate, III	45~50	2~4	18~25	25~35	55~70
Low, IV	40~45	−2~2	0~18	35~100	0~55

.

Step 2: Determine the weight of the evaluation index via the improved information entropy method based on the rough set. Since the rough set does not require any prior knowledge outside the dataset of the decision problem itself, it can effectively avoid human subjective errors [18] but ignore the cases where the weight of the index is 0. As we know, each index has a certain effect on the shrinkage–swelling property of soil, so their influence cannot be ignored in the evaluation process. To make full use of the objective information of sample data and take into account the impact of various attributes, this paper introduces an improved conditional information entropy I(D|C) method to distinguish index importance; that is, the corresponding weight w of the evaluation index is as follows [18]:

$$w\left(c_i\right)=\frac{sig\left(c_i\right)+I\left(D|c_i\right)}{{\displaystyle \sum _{b\in C}\left[sig\left(b\right)+I\left(D|b\right)\right]}},$$

(5)

$$I\left(D|C\right)={\displaystyle \sum _{i=1}^m\frac{{\left|C_i\right|}^2}{{\left|U\right|}^2}}{\displaystyle \sum _{j=1}^n\frac{\left|D_j\cap C_i\right|}{\left|C_i\right|}}\left(1-\frac{\left|D_j\cap C_i\right|}{\left|C_i\right|}\right),$$

(6)

$$sig\left(c\right)=I\left(D|C-\left\{c\right\}\right)-I\left(D|C\right)+\frac{\left({\displaystyle \sum _{a\in C}\left|U|\left\{a\right\}\right|-{\displaystyle \sum _{a\in C-\left\{c\right\}}\left|U|\left\{a\right\}\right|}}\right)}{{\displaystyle \sum _{a\in C}\left|U|\left\{a\right\}\right|}},$$

(7)

where w(c_i) is the weight of index I; I(D|C) is the conditional information entropy D to C; and sig(c) represents the importance of evaluation index c, $\forall c\in C$. The domain U is divided into two parts according to the condition attribute C and the decision attribute D, and U|C = {C₁, C₂, ⋯, C_m}, U|D = {D₁, D₂, ⋯, D_n}.

Step 3: Determination of membership limitation. The connection cloud model can generally deal with the cases where the sample data is a crisp value, while data are often nearly continuous interval data for actual uncertainty problems. In this case, the connection cloud model often only takes the sample mean for analysis, ignoring the integrity and volatility of the sample, and does not make full use of all sample information. This limitation will inevitably lead to the loss of the original sample information, thus affecting the accuracy of the results. Unfortunately, the decision-making interval in the traditional evaluation model based on the intuitionistic normal cloud model is often subjectively determined based on expert rating scores. So, it is not consistent with the idea that the engineers hope to evaluate it based on the objective measured data; this also limits its application scope in practical engineering to some extent. On the other hand, the accuracy of engineering geological parameters obtained via statistics based on multiple measured data should depend on the sample size and the discreteness of the data itself. In general, the trend of data variation and the relative deviation rate reduces with the increasing number of samples, while the accuracy of the parameters increases. Therefore, based on the principle of data variability in statistics, this paper introduces the concept of reliability to replace the “confidence level interval”, completely relying on human subjective evaluation in the traditional intuitionistic cloud model. The calculation model is given as

$$p=1-\frac{\delta \cdot t_{\alpha }}{\sqrt{K-1}},$$

(8)

where p denotes reliability; δ is the coefficient of data variation; α represents the significance level (also known as the risk level in engineering); t_α is the confidence coefficient; and K is the sample number.

The above method can avoid information loss and distortion and effectively ensure the rationality and applicability of the evaluation results. After obtaining the reliability p, let the lower limitation u = p. For the upper membership limitation of 1 − v, the traditional intuitionistic cloud rarely has 1 − v = 1 in practical applications because the sample interval value is determined via artificial scoring, and the accuracy of human subjective judgment struggles to reach 100%. However, the interval of the sample values in the proposed model is determined based on measured data, which avoids subjective scoring, so that the randomness of sample value and non-membership v is close to 0. Since the upper and lower limitations of membership need to be specified for each sample, the corresponding membership limitation matrix can be formed:

$$M=\left[\begin{array}{ccc}\left[u_{11},1-v_{11}\right]& \cdots & \left[u_{1m},1-v_{1m}\right]\\ ⋮& \ddots & ⋮\\ \left[u_{h1},1-v_{h1}\right]& \cdots & \left[u_{hm},1-v_{hm}\right]\end{array}\right],$$

(9)

where [$u_{hm},\hspace{0.33em}1-v_{hm}$] is the membership interval of the index m of sample h. To specify the connection degree of the sample to each grade, combined with the index weight vector w, the aggregated membership interval vector R is obtained as follows:

$$R=M{w}^{\prime },$$

(10)

where w′ is the transpose matrix of the matrix w = [w₁, …, w_m].

Step 4: Calculate the numerical characteristics of the intuitionistic connection cloud, generate N cloud droplets in a finite interval, and then specify the intuitionistic connection degree. Since the computational efficiency of the one-dimensional cloud model may decrease with the increasing number of indicators and samples, the multi-dimensional cloud is used here to deal with the discussed issue. The connection degree of cloud droplets $x^i\left(x_1^i, x_2^i, \cdots , x_m^i\right)$ is

$$y\left\{x^i(x_1^i, x_2^i, \cdots , x_m^i\left.)\right\}\right.=\exp \left(-\frac{9}{2}{\displaystyle \sum _{j=1}^m{\left|\frac{x_j^i-E{x}_j^i}{3E{n}_j^{i’}}\right|}^{k_j^i}}\right),$$

(11)

$$E{x}_j^i=\frac{C_{\min j}^i+C_{\max j}^i}{2},$$

(12)

$$E{n}_j^i=\frac{{\xi }_j^i}{3},$$

(13)

$$H{e}_j^i=\eta ,$$

(14)

$$k_j^i=\frac{\ln (\frac{\ln 4}{9})}{\ln \left|\frac{{\zeta }_j^i - E{x}_j^i}{3E{n}_j^{i’}}\right|},$$

(15)

where $E{x}_j^i$, $E{n}_j^i$, $H{e}_j^i$, and $k_j^i$ are the expectation, entropy, hyper entropy, and order of grade i (i = 1, 2, …, n) of the evaluation index j (j = 1, 2, …, m), respectively; $C_{\max j}^i$ and $C_{\min j}^i$ denote the interval upper and lower limitations of grade i of index j; η is an ambiguity-corrected parameter (here, η = 0.01); ${\zeta }_j^i$ represents the width of the left or right half branch of the connection cloud for grade i of the evaluation index j; and ${\zeta }_j^i$ is $C_{\max j}^i$ or $C_{\min j}^i$

Step 5: Calculate the multidimensional intuitionistic connection degree of the sample and determine the shrinkage–swelling grade. Let the h-th (h = 1, 2, …, P) sample to be tested be $x_h=\left(x_{h1}, x_{h2}, \cdots , x_{hm}\right)$. Combined with the weight, the multidimensional intuitionistic connection degree $z_i^h$ of the sample belonging to grade i can be calculated according to the following Equation (16), and its classification L_h is determined according to the maximum membership principle.

$$z_i^h(x_h)=\beta \cdot \exp \left(-\frac{9}{2}{\displaystyle \sum _{j=1}^m{w}_j{\left|\frac{x_{hj}^i-E{x}_j^i}{3E{n^{\prime }}_j^i}\right|}^{k_j^i}}\right)$$

(16)

$$L_h=\max (z_1^h, z_2^h, \cdots , z_n^h)$$

(17)

4. Materials and Results

4.1. Data and Model Implementation

To verify the reliability and rationality of the model proposed, in this paper, a series of sample data of untreated and lime-treated expansive clays was obtained from the foundation treatment area of Hefei Xinqiao International Airport [10]. The measured values of samples are listed in

Table 2. Measured data of samples.

Case	Soil Type	Liquid Limit C1/%	Total Shrinkage–Swelling Rate C2/%	Plasticity Index C3	Natural Water Content C4/%	Free Expansion Rate C5/%
A	Untreated expansive clay	[44.7,48.2]	[0.26,0.44]	[23.6,24.8]	[20.4,26.5]	[44.3,62.3]
B	4% lime-treated expansive clay	[45.6,46.7]	[−0.76,−0.33]	[20.3,21.5]	[16.3,22.7]	[14.0,18.0]
C	6% lime-treated expansive clay	[43.2,43.4]	[−0.73,−0.27]	[15.3,16.0]	[16.5,22.5]	[6.0,11.0]
D	7% lime-treated expansive clay	[43.4,50.0]	[−0.15,−0.20]	[15.0,23.0]	[14.0,23.0]	[0.8,9.3]
E	8% lime-treated expansive clay	[43.3,43.7]	[−0.763,−0.33]	[13.5,16.0]	[16.0,22.5]	[2.5,4.5]
F	10% lime-treated expansive clay	[44.3,44.8]	[−1.08,−0.61]	[12.3,16.0]	[16.2,22.8]	[0.0,0.0]

. Each parameter in Table 2 was obtained based on at least six sets of experiments. The data in Table 2 are taken from the measured data in the engineering implementation design stage; i.e., the significance level α was assigned 0.05 based on the engineering practice experience, and t_α was obtained at 1.645 according to the t distribution table. The corresponding reliability of the evaluation index of each case was obtained by Equation (8). However, it is difficult to calculate the coefficient of variation δ and the reliability through Equation (8) when the mean value is 0; i.e., for the case discussed above, with the sampling numbers K, the theory of hypothesis testing is used to calculate the reliability of the index, i.e.,

$$p=\exp \left[\ln \left(1-C\right)/K\right].$$

(18)

Here, the confidence probability C = 0.95 and K = 6; thus, p = 0.6070. The obtained reliability of sample data is listed in

Table 3. Reliability of sample data.

Samples	C1/%	C2/%	C3	C4/%	C5/%
Undisturbed expansive clay	0.9976	0.8410	0.9832	0.9346	0.9089
4% lime-treated expansive clay	0.9933	0.7849	0.9849	0.9078	0.9372
6% lime-treated expansive clay	0.9988	0.7937	0.9876	0.9271	0.8413
7% lime-treated expansive clay	0.9563	0.9181	0.8455	0.9000	0.5214
8% lime-treated expansive clay	0.9975	0.8322	0.9437	0.9275	0.8266
10% lime-treated expansive clay	0.9968	0.8361	0.9875	0.9057	0.6070

. Then, according to the above discussion step 3, a membership limitation interval with the limitations of reliability was built to generate the corresponding matrix as follows:

$$M={\left[\begin{array}{ccccc}\left[0.9976,1\right]& \left[0.8410,1\right]& \left[0.9832,1\right]& \left[0.9346,1\right]& \left[0.9089,1\right]\\ \left[0.9933,1\right]& \left[0.7849,1\right]& \left[0.9849,1\right]& \left[0.9078,1\right]& \left[0.9372,1\right]\\ \left[0.9988,1\right]& \left[0.7937,1\right]& \left[0.9876,1\right]& \left[0.9271,1\right]& \left[0.8413,1\right]\\ \left[0.9563,1\right]& \left[0.9181,1\right]& \left[0.8455,1\right]& \left[0.9000,1\right]& \left[0.5214,1\right]\\ \left[0.9975,1\right]& \left[0.8322,1\right]& \left[0.9437,1\right]& \left[0.9275,1\right]& \left[0.8266,1\right]\\ \left[0.9968,1\right]& \left[0.8361,1\right]& \left[0.9875,1\right]& \left[0.9057,1\right]& \left[0.6070,1\right]\end{array}\right]}_{.}$$

Based on the classification standard interval in Table 1, the grades for measured index values were identified. According to the discussed step 2, the index weight vector was assigned as w = [0.1912, 0.2132, 0.1912, 0.2132, 0.1912]. Subsequently, the integrated membership interval was obtained as shown in

Table 4. The integrated membership interval of samples.

Case	Case A	Case B	Case C	Case D	Case E	Case F
Integrated interval	[0.9311,1]	[0.9183,1]	[0.9075,1]	[0.8318,1]	[0.9044,1]	[0.8668,1]

. From Equations (12)–(15), the numerical characteristic values of the connection cloud for the j-th indicator at grade i can be determined. For instance, the obtained numerical characteristics of the connection cloud for grade I are listed in

Table 5. Numerical characteristics of connection cloud for grade I.

Index	Ex	EnL	EnR	He	aL	aR	kL	kR
C1	77.5	9.167	33.333	0.01	27.5	100	9.3217	1.2540
C2	53.0	16.333	33.333	0.01	49.0	100	44.8877	2.4775
C3	67.5	14.167	33.333	0.01	42.5	100	6.9729	1.6643
C4	7.5	8.333	5.833	0.01	25.0	17.5	1.5537	2.2077
C5	142.5	24.167	66.667	0.01	72.5	200	8.0698	1.5006

.

Based on the numerical characteristics of all indicators at each grade, 2500 cloud droplets are generated according to Equations (11)–(16) to simulate multi-dimensional connection clouds and intuitionistic connection clouds for each grade. Based on the indicators of liquid limit and plasticity index, the two-dimensional connection cloud and intuitionistic connection cloud were generated for comparative analysis, as shown in Figure 3. The one-dimensional clouds, as shown in Figure 3c–f, are the projections of the two-dimensional cloud in the directions of liquid limit or the plasticity index.

Taking case A as an example by which to illustrate the evaluation procedures, the random number generated from the uniform distribution U [u, 1 − ν], weight values, and the corresponding numerical characteristics parameters of each grade and each index value were substituted into the Equation (16), and the intuitionistic connection degrees z(I) = 0.0000, z(II) = 0.0204, z(III) = 0.3543, and z(IV) = 0.0936 were obtained. According to the principle of maximum membership degree, the shrinkage–swelling grade was specified as III. Since the generation of uniform random numbers is for randomness, considering the concept of the Monte-Carlo simulation [19], step 5 was repeated 500 times to obtain 500 intuitionistic connection degrees, and their mean was taken as the optimal estimation of the overall true value μ to reduce the influence of random errors on the classification results.

4.2. Results and Comparison

The evaluation results and comparisons with other methods are shown in

Table 6. Evaluation results and comparisons.

Case	z(x)				Proposed Model	Connection Cloud Model	Normal Cloud Model [10]	Connection Expectation Method [10]	Chinese National Standards GB/50112 [17]
	Grade I	Grade II	Grade III	Grade IV
A	0.0000	0.0204	0.3543	0.0936	III	IV	III	III	III
B	0.0000	0.0002	0.0145	0.0570	IV	IV	III	IV	IV
C	0.0000	0.0000	0.0038	0.0947	IV	IV	IV	IV	IV
D	0.0000	0.0000	0.0036	0.0414	IV	IV	IV	IV	IV
E	0.0000	0.0000	0.0020	0.0876	IV	IV	IV	IV	IV
F	0.0000	0.0000	0.0013	0.0888	IV	IV	IV	IV	IV

. It can be seen from Table 6 that the evaluation results of the proposed model are completely consistent with those of specifications and are almost consistent with other methods. This indicates that the intuitionistic connection cloud model based on an improved rough set used to assess the shrinkage–swelling grade is effective and feasible.

The results of the comparative analysis also show that the accuracy of the proposed model reached 100%, while in the connection cloud model and normal cloud model, individual sample grade misjudgment occurred. For example, among the five index values of case A, C₁ and C₃ were in grade III, C₂ was in grade IV, C₄ was between grade II and III, and C₅ was between grade III and IV; so, it was more reasonable to specify the shrinkage–swelling grade as grade III. Additionally, the intuitionistic connection degree takes into account the reliability of samples. This is conducive to coping with complex problems with numerous uncertain information. At the same time, the improved information entropy weighting method based on the rough set does not rely on any prior knowledge and can assign weight via objective data simultaneously considering the importance of the index. In general, an intuitionistic connection cloud model with universality and interpretability can deal with the interval-valued index containing multiple uncertainties and further characterize the hesitant characteristics of the problem relative to the traditional cloud models.

These results also suggest that the application of the specification method may rely too much on the reliability of the parameters and cannot describe the uncertainty of information acquisition and cognitive uncertainty. However, the evaluation of the shrinkage–swelling grade will encounter subjective ambiguities, which is not conducive to the application of field engineers. In addition, various categories of uncertainty, such as fuzziness, randomness, and hesitation, may be inevitably encountered in evaluations, so the analysis of shrinkage–swelling grade cannot rely only on methods based on a single type of uncertainty. So, the proposed method coupled with IFS and the connection cloud model can therefore overcome the defects of the previous evaluation method using uncertainty analysis theory to some extent.

Existing research shows that shrinkage–swelling features of untreated and lime-treated expansive clays are influenced by the liquid limit, the total expansion rate, plasticity index, natural moisture content, free expansion rate, and decision-maker hesitation information factors, so the actual evaluation of the shrinkage–swelling grade is a complex problem affected by the cross, blending, and dynamic uncertainty indexes. However, engineers often take the free expansion rate as the classification standard of the shrinkage–swelling grade for the expansive clay; that is, the foundation deformation obtained from the expansion rate measured under the pressure of 50 kPa based on the Chinese National Standards GB/50112 is used to specify shrinkage–swelling grade. Thus, the specification method with only a single type of information has certain limitations. Case studies and comparisons with the other methods in Table 6 indicate that the proposed method, based on an intuitionistic connection cloud model, can obtain more accuracy and reliability of classification relative to the other methods. On the other hand, the application result of the proposed model is only based on examples. Currently, this may make it difficult to provide concise equations or tables similar to specifications for engineers; so, to establish a concise formula form, a large number of engineering practices need to be investigated and summarized. In addition, the parameter values in the case study are relatively small compared with the standard range. These parameters, shown in Table 2, are the measured values of untreated and improved expansive clay in the foundation treatment area of Hefei Xinqiao International Airport. Although the parameter index values have some limitations, this does not affect the validation of the model’s application. The data for the case study are centrally distributed in a certain interval relative to those of the classification standard, as shown in Table 1, which are the possible range of parameters. This just reflects the characteristics of expansive clay as a special soil; that is, its engineering characteristics are regional. Therefore, in order to verify the reliability of the proposed method, it is necessary to conduct more verification analyses with a larger parameter fluctuation range only, especially for untreated expansive clay.

5. Discussion

5.1. Description of Interval-Valued Index

As is known to all, to improve the reliability of sample index values and reduce random errors, engineers often determine index value via repeated sampling, so the measured index value is an interval value. Measured values of free expansion rate C₅ for each sample were taken here to illustrate this problem. Figure 4 illustrates the statistical characteristics of measured values obtained from six sets of measured data. It can be seen that the standard deviation for the untreated expansive clay is the largest, while the coefficient of variation for 7% lime-treated expansive clay is the largest. Obviously, data processing using the mean or endpoint value was inappropriate. However, the connection cloud model and normal cloud model often use the average value of the evaluation indicator to process the data; the connection expectation model always depicts the indicator using the endpoint values of the actual interval value. Evidently, these values do not take into account all the characteristics of the data, and the corresponding result may be different from the actual information of the samples.

5.2. Superiority of the Proposed Model

The membership degree of the cloud model changes continuously in the interval [0, 1]. The sample of connection degree closer to 1 has a greater possibility of belonging to the given grade, and the sample of connection degree closer to 0 presents a greater possibility that the sample does not belong to the given grade. Moreover, membership 0.5 is the mean value of the membership interval [0, 1], which is the boundary point between belonging and not belonging; it is the fuzziest point. For example, as seen in Table 1, it is obvious that the threshold value 35 of index C₃ between grade I and grade II is subordinate to both grade I and grade II, the degree of belonging to grade I and grade II is 0.5, and the ambiguity at this point is the highest. So, it is more likely that this point belongs to grade I and grade II. It can be seen from Figure 3e that the membership y(III) and y(IV) at this point are both 0.5 in the connection cloud model; so, the connection cloud model can well describe the transformation trend between adjacent grades. However, in the connection cloud model, if the indicator value x is the expected value, then y(x) = 1, which is an accurate representation and ignores the hesitant nature. In contrast, the intuitionistic connection cloud not only retains the advantages of connection cloud but also overcomes the above shortcoming such that the vertex of the intuitionistic connection degree is no longer a crisp value of 1 but fluctuates in an interval slightly less than 1. It is seen from Table 3 that the minimum reliability of the plasticity index is about 0.85, and the maximum intuitionistic connection degree is a random number uniformly generated between 0.85 and 1. Figure 3f is a demonstration of the above random uniform implementation. Thus, the intuitionistic connection cloud model can depict relationships between the interval-valued indexes and classification standards from membership, non-membership, and hesitation aspects, and can deal with multiple uncertainties without information loss or distortion relative to the normal cloud model and connection cloud model.

The evaluation of the shrinkage–swelling grade is a complex problem with uncertainty indicators. The case study shows that the intuitionistic connection model can overcome the shortcomings of the conventional cloud models and has a better description of uncertainties. The model proposed here can significantly enhance reliability because it can consider hesitation characteristics of the interval-valued index. It has the following benefits over other methods.

(1)

Compared with the normal cloud model, the intuitionistic connection cloud model offers a useful basis for the depiction of the certain and uncertain relationships between the interval-valued indexes with classification standards from three perspectives, namely, identity, discrepancy, and contrary aspects. Additionally, it overcomes the disadvantage of the requirement of normal distribution.

(2)

The proposed model without information loss or distortion can simultaneously express ambiguity, randomness, and hesitation; so, it has the capability to depict high uncertainty in the evaluation of the shrinkage–swelling grade of untreated and lime-treated expansive clays, enhancing the description of the evaluation result “not one is not the other”.

(3)

The weights assigned according to the conditional information entropy method based on the rough set can effectively find the index importance. This weighting method can not only avoid prior knowledge and a large number of data requirements but also reduce the errors caused by human factors.

6. Conclusions

The evaluation of the shrinkage–swelling grade of untreated and lime-treated expansive clays involves numerous uncertain evaluation factors and human cognition. However, the traditional evaluation methods mostly rely on single uncertainty assumptions or data distribution and cannot also simultaneously depict the hesitation and randomness of the index. Therefore, there is a necessity to develop a novel model to improve the accuracy and reliability of evaluation results. The intuitionistic connection cloud model based on improved conditional information entropy weighting was introduced herein to analyze the shrinkage–swelling grade of expansive clay and lime-treated expansive clay with consideration of the hesitation, fuzziness, and randomness of the indicator in an infinite interval. A case study was furthermore conducted to verify the feasibility and validity of the proposed model. The main conclusions are drawn as follows.

(1)

The case study indicates that the intuitionistic connection cloud model used to evaluate the shrinkage–swelling grade is effective and feasible. And it can not only reflect the conversion situation at the classification threshold but also capture the hesitation characteristics at the evaluation interval, especially the central expectation point. The intuitionistic connection cloud model can depict hesitant characteristics of the fuzzy and random index, and the results obtained from the proposed model without valid information loss or distortion are more accurate and reliable relative to those of the traditional cloud models.

(2)

The weighting method based on conditional information entropy and the rough set greatly reduces the subjective deviation. It not only overcomes the problem of the weight of a few indexes potentially taking 0 or being unable to be assigned via the traditional objective weighting method but also considers the importance of indicators.

(3)

Different from the method of taking the mean value of traditional cloud models, the model proposed here expresses the measured value with an interval and determines the membership interval of the sample according to its reliability to replace the confidence level interval artificially determined in the intuitionistic normal cloud. So, the proposed model can explore the true characteristics of the original data to the greatest extent and promote an evaluation process and a final result closer to reality.

Although the case study verifies the effective application of the proposed model to the shrinkage–swelling evaluation, the intuitionistic connection cloud model is a useful tool with which to depict the hesitation and randomness of information simultaneously. However, to make evaluation results more reasonable and credible, the question of how to better couple two types of weights still needs further investigation.