Enhance the Uncertainty Modeling Ability of Fuzzy Grey Cognitive Maps by General Grey Number

In real-life systems, people cannot get precise data. The data are represented in the forms of interval or multiple intervals in many cases. Most intelligent algorithms are designed for precise data in algorithm research. People always use a real number contained in the interval or multiple intervals as the candidate for the precise data. However, such a measure will lose lots of information contained in the interval or multiple intervals. Fuzzy Cognitive Map (FCM) is one of the famous intelligent algorithms. The Fuzzy Grey Cognitive Map (FGCM) was proposed to enable FCM to do interval computation. But FGCM can only deal with a single interval, as for multiple intervals, the FGCM is powerless. Thus, this paper aims to enhance the FGCM to make it can cope with multiple intervals. The paper introduces the general grey number and deduces the new activation functions according to Grey System Theory (GST) and Taylor series. Finally, an industrial process control problem is applied to verify the new algorithm. The results show that the new algorithm is not only compatible with the original FGCM and FCM, but can process more uncertain knowledge and data. In general, the new algorithm inherits most of FGCM’s characteristics and can cope with the data expressed by multiple intervals, which means it can be used in environments with more uncertain knowledge and data.


I. INTRODUCTION
Recently, coping with the uncertainty data and information is one of the main tasks of soft computing. Soft computing is different from the ''hard computing'', which need deterministic analytic techniques. Soft computing can deal with imprecision, uncertainty, partial truth, and approximations. The principal constituents of soft computing techniques are probabilistic reasoning, fuzzy logic, neuro-computing, genetic algorithms, belief networks, chaotic systems, as well as learning theory [1]. Fuzzy Cognitive Map (FCM) is one of the soft computing techniques [2]. Due to its simplicity, support of inconsistent knowledge, and circle causalities for knowledge modeling and inference [3], FCM has made many significant achievements in many research The associate editor coordinating the review of this manuscript and approving it for publication was Nagarajan Raghavan .
areas, including decision-support [4], [5], system simulation [6], system prediction [7], smart city [8], intrusion detection [9], and industrial process control [10]- [13] etc. FCM assigns fuzzy weights to topological graphs, so it is also treated as a fuzzy-neural system [14]. Unlike neural networks, FCM's nodes are explanatory, which gives engineers and data scientists convenience to conduct the what-if analysis of the system [15]. And an FCM can be easily built by exploiting the historical data or the experts' knowledge [16].
The data from real-life system always have uncertainty. People always use intervals to express the data with uncertainty, which leads to the emergence of grey system theory (GST) [17]. To make the use of such uncertainty data, many algorithms have been extended according to the GST, which include the FCM. The model is called Fuzzy Grey Cognitive Map [15] (FGCM), the FGCM can support the interval input, improving the uncertainty modeling ability of the FCM [18]. In this paper, the uncertainty modeling ability refers to the algorithm's ability to represent and process uncertain data. Although FGCM can process the uncertain data represented by intervals, the multiple intervals or discrete data are often encountered in real life, FGCM cannot deal with such data. Consider an occasion: a decision support model of an aircraft needs the data of the ground speed, but the ground speed obtained from the Inertial Navigation System (INS) is always different from the one calculated by the Global Positioning System (GPS). For example, the GPS gives a value falls in the data range of [215,220] because of the positioning error, but the INS gives a value falls in the data range of [200,210] caused by the cumulative error, if the decision support model is built as an FCM, a fuzzy number which falls into the union of [200,210] ∪ [215, 220] may be chosen as the input value because the FCM only supports a fuzzy number as its node value. Which number should be chosen as the input? Different numbers may cause totally different results, the pilot has to make a decision according to a number selected randomly in [200,210] ∪ [215, 220], which will involve many fateful consequences. If the decision support model is built as an FGCM, which interval should be chosen as the input of the FGCM? Some methods, like Dempster-Shafer (D-S) evidence theory [19], [20] would like to make a choice between them, but making a choice between [200,210] and [215, 220] will lose another interval's information, and it should be avoided to make a decision at the cost of losing partly information. Choosing [200,220] will add the uncertainty of the input data, which can also damage the precision of the decision-making. The best way is to make [200, 210] ∪ [215, 220] as the input of FGCM, in this way, no information is lost, and the pilot can make a decision by full information. However, FGCM cannot calculate with such data. The same predicament will be confronted more often when two or more experts were invited to determine the FGCM's weight.  35, 0.4] is more than 0.05 for sure. Actually, the average is not equal to the union, some information will be lost by averaging the intervals. A reasonable solution to cope with the multiple interval values is to introduce the general grey number into FGCM, change the inference process of the FGCM according to the arithmetic rules of the general grey number, which is the aim of this paper.
A general grey number is the union of interval grey numbers or real numbers [21]. For [22], [23] was proposed as the precursor of the general grey number. Then in 2012, the general grey number was clarified as the ''generalized grey number'' [24]. Almost simultaneously, the general grey number and their operations were summarized in [21]. After that, the general grey number can be used in many fields [25]- [27] because of its adequate representation and arithmetic operation system.
The FGCM uses the interval grey number as its basic element. Interval grey number is a special form of the general grey number, for example, [200,220] can be seen as an interval grey number. However, the general grey number runs different arithmetic operation systems with the interval grey number, which is the main obstacle to introduce the general grey number to the FGCM. To tackle the problem, this paper changed the data structure as well as the activation functions for FGCM. As the innovation is mathematical, one classical application of FGCM is enough to verify it. The new algorithm should be compatible with classical FGCM, and it can operate with the data in the form of the general grey number.
Thus, the paper's main contributions are: • Introducing the general grey number in the FGCM; • Formulating the FGCM's activation function in the form of the general grey number; • Validating the new models compatibility and availability by the industrial process control problem.
The FGCM using general grey number is a novel extension of the FCM, which combines GST and FCM more deeply than FGCM. When the data have more uncertainty, especially contains multiple intervals or discrete values, the FGCM using the general grey number should be applied.
The rest of the paper is organized as follows: Section II reviews the fundamentals of GST followed by the FGCM framework, introduces the general grey number; Section III shows how to apply the general grey number to FGCM and how the activation functions works; Section IV takes an industrial process control problem as an example to verify FGCM using general grey number; Section V analyzes the experiment result and computational complexity; finally, section VI concludes the characteristics of FGCM using general grey number and shows the future work.

II. PRELIMINARY KNOWLEDGE
Before turning to the proposed algorithm, the preliminary knowledge is described here.

A. GREY SYSTEM THEORY
Grey System Theory (GST) was proposed in the 1980s to deal with the uncertainty caused by incomplete information and VOLUME 8, 2020 inaccurate data. The GST has been successfully applied in the areas of environment [28], military [29], business [30], agriculture [31], geology [32], and meteorology [33].
In order to tackle the uncertainty of the system, representation is required firstly. The grey number is the basic component of the GST, which represents a crisp number falling into one or more ranges. The interval grey number is the most commonly used grey number, indicated as ⊗G ∈ [G, G], G G. If the lower bound is known but the upper bound is unknown, the interval grey number should be written as ⊗G ∈ [G, +∞); the same as the case if the upper bound is known but the lower bound is unknown, denoted as ⊗G ∈ (−∞, G]. If neither the upper bound nor lower bound is known, the grey number becomes a black number, indicated as ⊗G ∈ (−∞, +∞). No information can be operated from a black number. If a grey number's upper bound and lower bound are equal, then it is a white number, it means the complete certain information is known, indicated as ⊗G ∈ [G, G], G = G.
The interval grey numbers' operation rules are defined as below: The degree of greyness represents the level of uncertainty a grey number contains. For convenience, the degree of greyness will be called greyness in this paper. Given a grey number ⊗ generated from the domain , its greyness can be defined as: The µ() calculates the measure of the grey number or the domain. The greyness is valid in [0, 1], i.e., 0 g • (⊗) 1.
The higher the greyness is, the higher the uncertainty of the grey number. A white number's greyness is 0 and a black number's greyness is 1 obvious.

B. THE FUZZY GREY COGNITIVE MAP
The main idea of the FGCM is to use greyness to replace the fuzziness in the FCM, because a grey set can be specified to a fuzzy set [24]. The FGCM model keeps the same topological structure and the inference process as those in the FCM (Fig. 1). The nodes in FGCM represent the variables or concepts that have uncertainty. And grey weight (weight shown by grey numbers) can represent the uncertainty of the experts' suggestions or the incomplete data sets.
To carry out the iteration in the FGCM model, the node values and weight are often written as grey vectors and grey matrix. The grey vectors' and matrix's operation rules are the same with the real vectors and matrix. For example, Fig. 1 (right) shows a simple FGCM, its initial state vector is And the grey weight matrix is Like the FCM, the FGCM gets its new node values by where S(⊗) is the activation function to compress the node value into [0, 1] or [−1, 1], when the node value is in [0, 1], the activation function is when the node value falls in [−1, 1], the S(⊗) is the hyperbolic tangent: The positive parameter λ is used to control the steepness of the curve. The lager the λ is, the steeper the curve is.
The inference results reach a steady state, a limit cycle or chaotic results, which depends on the FGCM's structure [15].

C. THE GENERAL GREY NUMBER
This section introduces the general grey number's concept and arithmetic operations. Following Liu et al.'s work [21], we present some definitions and theorems are given here to provide a better understanding of general grey number [34]. They are fundamentals to enhance the FGCM's uncertainty modeling ability.
Definition 1: Assume ⊗ ∈ [a, a] is an interval grey number, with the distribution of the grey number ⊗ is unknown:

Definition 2: Assume ⊗ ∈ [a, a] is an interval grey number, with the distribution of the grey number ⊗ is already known, the kernel of the
Definition 3: Assume⊗ is the kernel of the grey number ⊗, g • is the greyness of the grey number, then the⊗ (g • ) is called the simplified form of the grey number.
For an interval grey number, the simplified form of it contains all information about itself. The simplified form of interval grey number can replace the interval grey number itself.
Definition 4: The interval grey number and white number are the basic elements of the general grey number.
Definition 5: A general grey number is defined as for all interval grey number are called the lower bound and upper bound of g ± . The general grey number is defined as the union of interval grey number or white number, it can express data have more uncertainty. The next step is to determine the operation of the general grey number. Before achieving this objective, define the kernel and greyness of the general grey number.
[a i , a i ] is a general grey number, its kernel is calculated bŷ If the distribution of the general grey number is already known, and the probability of g ± ∈ [a i , a i ], (i = 1, 2, · · · , n) is p i , and n i=1 p i = 1, p i > 0, i = 1, 2, · · · , n then the kernel is calculated bŷ µ() is the measure function in , then the greyness is defined as There are two axioms prepared for the operation rules of general grey number [21].
Axiom 1 (The Synthesis Axiom for Greyness): When adding or subtracting a set of n general grey numbers g 1 , g 2 , · · · , g n , the greyness g • of the operation result g ± can be calculated as follows: Axiom 2 (Axiom of Greyness Undiminished): When dividing or multiplying a set of n general grey numbers, the greyness of the operation result is no less than the maximum greyness of the operated general grey number in the set.
Based on these two axioms, the basic operating rules of the general grey number are summarized and proven in [35], and shown as follows: where where , (ĝ 1 = 0)

III. THE PROPOSED ALGORITHM
Comparing to the interval grey number, the general grey number can handle more uncertainties carried from the incomplete information. For the node value, two or more interval grey numbers can often be chosen as the input. If the FGCM is applied to such occasions, only one interval grey number can be used, or one uses the minimum (the greyness is minimum) interval grey number that contains all node values gotten before. For example, g ± ∈ [0.5, 0.6] [0.8, 0.9] is a general grey number. Its underlying white number may get its value from [0.5, 0.6] or [0.8, 0.9]. However, we cannot get its value within (0.6, 0.8). This is different from an interval [0.5, 0.9] where (0.6, 0.8) is a valid part for candidate values [24]. What's more, a general grey number allows candidate values in a discrete set, such as g ± ∈ {0.3, 0.4, 0.5}, which cannot be represented as an interval. Because of the conflict opinions raised by experts or the data from the environment, the weight and node values of the FGCM can also encounter multiple interval or discrete values cases. To make use of the general grey number in FGCM, the activation function of the general grey number's form is needed.

A. THE SIGMOID ACTIVATION FUNCTION IN THE FORM OF GENERAL GREY NUMBER
If the domain is [0, 1], then the sigmoid activation function is The result of S(g ± ) is a new general grey number, the kernel of theŜ(g ± ) is calculated aŝ the greyness is calculated as The proof of (29) and (30) is shown below: Proof: According to the Taylor's formula, according to the equation (27), the kernel of (g ± ) n isĝ n , the greyness is g • ; thus, the kernel of (−λ) n (g ± ) n n! is (−λ) nĝn n! , n = 1, 2, 3 · · · , the greyness is g • ; therefore, the kernel of e −λg ± isê according to axiom 1, the greyness of e −λg ± is According to the (20) and (24), the greyness of the S(g ± ) is And using (32), the kernel of S(g ± ) iŝ The proof is end.

B. THE HYPERBOLIC TANGENT ACTIVATION FUNCTION IN THE FORM OF GENERAL GREY NUMBER
If the domain is [−1, 1], then the hyperbolic tangent is often used as activation function: The kernel of tanh(g ± ) is the greyness is calculated as The proof of it is very similar to the sigmoid activation function's, so it will not be shown here. The figures of these two functions are shown in the Fig. 2 and 3. Like the FGCM's activation functions, the lager the λ is, the steeper the surface is.

Algorithm 1 FGCM Using General Grey Number
Data: the initial vector in the form of general grey number ⊗ C Result: the output vector in the form of general grey number while t < iteration_times do Like FCMs, the new algorithm iterates until the node values reach a steady point, a cycle contains several general grey numbers or chaotic results.
Compared with the FGCM, the new algorithm's data structure is general grey number whereas the FGCM's is interval grey number. And the activation functions' computing methods are different, the new algorithm's are (28), (29), (30) or (36), (37), (38), the kernel and greyness are calculated in order, but FGCMs use (12) or (13). The new model is more complex than original FGCM.
In next section, the validation experiments are conducted to show how the FGCM using the general grey number works.

IV. EXPERIMENTS AND RESULTS
The proposed algorithm can be used universally. And it is enough to validate it using a classical application of FCM or FGCM. We choose the industrial process control problem as an example to show how the general grey number works in FGCM.
To evaluate the performance of the proposed algorithm, three experiments are designed to test the compatibility and availability. In Section IV-A, we show the background of the industrial process control problem and how the FCM solves this problem, this FCM is the benchmark of the comparison experiments shown in Section IV-B. And then we build an original FGCM and FGCM using general grey number to prove the compatibility of the proposed algorithm in Section IV-B. In Section IV-C, two cases are specially designed to test the unique characteristics of the proposed algorithm, in which a single FCM or FGCM is not applicable. These two cases also proved that the proposed algorithm has a wider application domain than the FCM and FGCM. Finally, a more complexed situation where both nodes and weights values are two or more intervals is designed in Section IV-D to show the proposed algorithm's availability, at the same time, we exploit the original FGCM multiple times to solve this situation, the results range of the original FGCM is consistent with the proposed model, but so many original FGCM's runtime is too long to be tolerant.

A. THE INDUSTRIAL PROCESS CONTROL SYSTEM
The industrial process control problem was first discussed using FCM by C.D. Stylios in 1998 [36], [37]. Then, it was used in many validations of new FCM-relevant algorithms. In 2006, this problem was used as a case study to compare two unsupervised learning methods for fine-tuning fuzzy cognitive map causal links [38]. In 2011, Extended Great Deluge Algorithm (EGDA) has been considered as a training algorithm for FCM, the industrial process control problem was one of the case studies to validate the EGDA [39]. And the FGCM was applied to analyze the industrial process control problem in 2014 [13]. Therefore, the FGCM using general grey number is validated by the industrial process control problem here.  This industrial process control system aims to control the height and temperature of the liquid in 2 tanks in a suitable range. It contains two tanks, three valves, and a heating element, as shown in Fig. 4. V1 is the inlet valve, and its outlet valve (V2) is the inlet valve of tank 2. The heating element lies in tank 1, thus, the hot liquid heated in tank 1 can flow into the tank 2. V3 is the outlet valve of the tank 2. If the height of the liquid in tank 2 is too high, the valve 3 is needed to open.
The aims of such a control system are: where, the H i , (i = 1, 2) is the height of liquid in tank i, (i = 1, 2); the T i , (i = 1, 2) is the temperature of liquid in tank i, (i = 1, 2). Stylios and Salmeron have constructed the FCM and the FGCM models correspondingly. However, Salmeron applied a different set of weights with the Stylios' [13]. To make a convincing comparison, we use the original FCM built by Stylios as the benchmark. The FCM contains 8 nodes. Each node's definition [36] is shown below: The node values need to be normalized before processing by the FCM. For example, if the height of liquid in tank 1 is 20%, then the C1's value is 0.2. In practice, the normalized approach was selected by the engineers. If the C3 is 0, it means the valve 1 is totally closed. If the C3 is 1, it means the valve 1 is totally open. And the weight matrix is shown below. Next compatibility and availability experiments are conducted. The FGCM of this problem has been built in [40] and it was compatible with the original FCM we referred. To validate the compatibility, keep the weights be same with the FGCM, but transfer the weights into general grey number and build a new FGCM using general grey number. Given the same input vector, if the results are approximately equal, then the new model is compatible with FCM and FGCM.

B. COMPATIBILITY CHECK
As Salmeron has built the FGCM for this problem, we refer to the weight greyness in [40] and get a matrix of interval grey number. The weights are listed in Table 1 and transferred to the general grey numbers by Definition 1, 2 and 3.
Apply the weight above, the FGCM using general grey number is obtained; the topological graph is shown in Fig. 5, and the inputs are the same as FCM's. The only difference is the data structure. After 10 iterations, all three models: FCM, FGCM and FGCM using general grey number, reach a steady state, as shown in Table 2 and Fig. 6.
To make a comparison, we transfer the FGCM's outputs into the form of the general grey number, they are listed 163850 VOLUME 8, 2020  correspondingly in the fourth column (Table 2). Findings from Table 2 are summarized as following: • Both FGCM and FGCM using general grey number are compatible with the original FCM as their results always cover the FCM's results; • The kernels of the two FGCM's results are appropriately equal to the FCM's; • The greyness calculated by the FGCM using general grey number is appropriately equal to the ones calculated by FGCM.

C. THE UNIQUE CHARACTERISTICS OF THE PROPOSED ALGORITHM
Characteristics of FGCM using general grey number are explored here by setting the following 2 cases: the input contains the general grey number and the weights contain the general grey number. The FCM and the original FGCM cannot cope with these 2 cases, but the algorithm this paper proposed can tackle them.   [0.19, 0.21] ∪ [0.29, 0.31] is the input value, it is transferred into the form of general grey number 0.25 (0.04) using (15) and (17). Thus the initial vector is   of the FCM [41]. The simulation results are shown in Fig. 7, the model gets the same results with the FGCM using general grey number in Section IV-B.

2) THE WEIGHTS CONFRONT CONTRADICTORY EXPERTS' SUGGESTIONS
Now consider another case: one engineer believes that ⊗w 13 = [0.01, 0.25], another thinks ⊗w 13 = [0.4, 0.6] is more appropriate. According to the experience, the first engineer is more authoritative than the second, so the weight of the first engineer's weight is 0.6, and the weight of the second engineer's is 0.4. Use the (16) and (17), the ⊗w 13 should be 0.352 (0.1864) . In this way, both two of the engineer's advice is considered. So we update the weight matrix accordingly and use the initial vector (39) in the FGCM using general grey number. The nodes vector gets into a steady state at the sixth iteration. The steady state is shown in Table 3 and Fig. 8. Because the ⊗w 13 is the only changed weight, it can only influence the node 1 and node 3: the values of these two nodes have changed accordingly. Next, a more complex situation is designed to show the proposed algorithm can handle that most of the nodes and weights are in the form of multiple intervals.

D. MORE COMPLEX SITUATION
This experiment is designed to show the availability of the proposed algorithm and why we should use the FGCM using general grey number instead of simply run many different FGCMs when we encounter the multiple intervals data.
If most of nodes confront the the situation like section IV-C1, i.e., most nodes' input is multiple intervals or discrete data, as shown in Table 4. And most of the weights are made up by multiple intervals, as shown in Table 5. Using the proposed model, the results are show in Fig. 9 and Table 6.
If we want to use FGCM to solve the problem, we need to run n i=1 Cn i × n×n i=1,j=1 wn ij = 35831808 times. Cn i means the node C i contains Cn i intervals. If C i is a real number, Cn i = 1. n is the number of nodes. wn i is the number of intervals w ij contained, if w ij is a real number, wn i = 1. It is so time consuming that using many FGCMs to solve the cases that the data contains multiple intervals. Actually, under the platform of MacOS 10.15.5, using Python 3.7, the CPU is intel core i7, 2.3 GHz, the total runtime of all FGCMs is 110980.1357s. On the other hand, the FGCM using general grey number only needs to compute one time, and the runtime is 0.0056s.
The results of multiple FGCMs are overlapped with each other, after unioning the 35831808 intervals, the results are also summarized in Fig. 9 and Table 6.
Whether can we average the multiple intervals before running the FGCM? The answer is no. There is no meaning to average a set of interval grey numbers.

V. THE RESULTS ANALYSIS
In this section, a more thorough analysis of the experiments above is given.

A. COMPATIBILITY AND AVAILABILITY
In Section IV-B, we run the FGCM and the proposed algorithm independently. The FGCM is extended from the FCM directly according to [40], and the FGCM using general grey number is obtained from FGCM directly. The comparison fairness is ensured by: • The FCM, FGCM, and the proposed method share with the same structure.
• The inputs and weights of FGCM are extended from FCM directly according to [40].
• The inputs and weights of FGCM and proposed method are the same.
• The FCM, FGCM, and the proposed method iterates 10 times correspondingly.
• The FCM, FGCM, and the proposed method using the same type of activation function: sigmoid, and the parameter λ is set as 1 for all models.
• The only difference is the data structure.
The comparation results show the proposed method are compatible with the FGCM and FCM as expected: most of the results of the FGCM and the proposed method cover the results of FCM. And the kernel of the results of FGCM and the proposed algorithm is approximately equal to the FCMs.
The results of the proposed algorithm and the original FGCM have almost the same greyness, which ensures compatibility in the perspective of greyness. When FGCM processing the data, it calculates the upper bound and lower bound of the interval grey number, the greyness is calculated in an implicit and passive way. The proposed model calculates the greyness in an explicit and active way: first it calculates the kernel and then calculates the greyness directly. The uncertainty propagation of the FGCM using general grey number is another interesting problem deserving more explorations.
In Section IV-C and IV-D, we test if FGCM using general grey number can handle the multiple intervals in nodes, weights, and both of them. The results show that the proposed algorithm can cope with the occasions that the nodes and weights contain multiple intervals. A more complex case is conducted also, it shows that the FGCM using general grey number can handle the situation where both nodes and weights contain multiple intervals. But when the original FGCM encounter such occasions, it has to run many times, the time consumed cannot be tolerant.

B. THE COMPUTATIONAL COMPLEXITY
In this part, the computational complexity is analyzed. Consider a single FCM, FGCM, and FGCM using general grey number. Because their structures are same with each other, their computational complexity is the same, it is O(m×n×l), where m is the number of weights, n is the number of nodes, l is the number of iteration times.
Although a single FCM, FGCM, and FGCM using general grey number share the same time complexity, the runtime is different from each other. With the same m, n, and l, the proposed algorithm's runtime is longest, and the FGCM's is medium, the shortest is FCM's. The reason is that the arithmetical operations of general grey numbers are more complex than the operations of real numbers and interval grey numbers, especially for the addition. For the real number, it just needs an addition operation of float number. For the interval grey number, an addition operation needs 2 float addition operations. For the general grey number, an addition operation needs 2 abs() functions, 2 multiplications, 2 divisions, and three additions. So, with the same n, m, and l, the runtime of the proposed algorithm is longest.
We run the experiment in Section IV-B to show the runtime of each algorithm because the fairness is ensured in that experiment. The platform is MacOS 10.15.5, using Python 3.7, the CPU is intel core i7, 2.3 GHz. To eliminate the randomness of a single simulation, we run 10000 times experiments. The runtime distributions of each algorithm are shown in Fig. 10. It can be found that the runtime of the three algorithms obeys right-skewed distribution. The the medians of FCM's, FGCM's, and proposed algorithm's runtime are 0.000069, 0.003333, and 0.004359 correspondingly. We did a hypothesis-testing to confirm the runtimes order. Because the runtime data obey skewed distribution, Wilcoxon tests were used. The results show, in our simulation environment, the runtime of FGCM is longer than 46 times of FCM's, with the confidence of 99.96%; the runtime of FGCM using general grey number is longer than 1.297 times of the original FGCM, with the confidence of 99.93%.
However, when encountered the multiple interval input or weights, The program's time complexity becomes where Cn i means the node C i contains Cn i intervals, if C i is a real number, Cn i = 1, n is the number of nodes, wn i is the number of intervals w ij contained, if w ij is a real number, wn ij = 1. But apply the proposed algorithm, the program's time complexity is still O(m × n × l). Thus, in the perspective of runtime, when n i=1 Cn i × n×n i=1j=1 wn ij ≥ 1.297, i.e., once the multiple intervals encountered, the proposed model is more appropriate to be used.

VI. CONCLUSIONS AND FUTURE WORK
As a soft computing algorithm, FCM's uncertainty modeling ability has been improved by FGCM. But FGCM's uncertain modeling ability is limited due to the fact that it can only deal with an interval grey number. Thus, the aim of this paper is to enhance the FGCM's uncertainty modeling ability.
This paper introduces the general grey number into FGCM to make it have the ability to deal with multiple interval data. And two activation functions are deduced according to GST and Taylor series. In this way, the FGCM using general grey number is proposed. It transfers FGCM's interval grey number into the general grey number. The new algorithm's arithmetic operation and activation function are different from the FGCM. Finally, three experiments are conducted to validate the proposed algorithm of which the industrial process control problem is the background. The first experiment is employed to show the new model is compatible with FCM and FGCM. The second experiment shows the new model's characteristics which the original FCM and FGCM do not have. And the third experiment shows that the proposed algorithm can handle the more complex situation: both nodes and weights values are comprised of multiple intervals.
In general, the main finding of this paper are as follows: • The new model has a stronger uncertainty modeling ability since the general grey number is applied in the FGCM.
• Two activation functions in the form of the general grey number: sigmoid and tanh, make the new model can do the reasoning under the uncertainty represented by the general grey number.
• The reasoning ability and availability are proven by the industrial process control problem.
The new algorithm combines the new theoretical achievements of GST and FCM. The FCM provides the basic algorithm framework, and GST provides the representation and arithmetic operation method. The general grey number's characteristics make the FGCM can process multiple intervals, and the greyness will be kept during the inference. It can be used in the situation where exists more uncertainty, such as two or more sensors' data, experts' suggestions.
The proposed algorithm with general grey number increases the complexity of the single FGCM, but when encountered with the multiple interval input or weights, the time complexity keeps the same. The other limitation is that the results of the proposed method will be influenced by the extreme values, but this can be easily overcome by data cleaning or imputing in the data pre-processing.
Finally, there is still a lot of work to be undertaken in the future, such as the investigation of convergence property, the greyness propagation, the greyness's convergence, the applications of the FGCM using general grey number et al. And more theoretical and application researches should be conducted in order to construct an integrated theoretical system of FGCM using general grey number, i.e., the Fuzzy General Grey Cognitive Map (FGGCM).