Clustering and selection of knowledge meshes of knowledgeable manufacturing systems based on decomposition of fuzzy relational data

Abstract

This article presents a study on the clustering and selection of knowledge meshes in the knowledgeable manufacturing system that transforms all types of advanced manufacturing modes into corresponding knowledge meshes and selects the best combination of knowledge meshes to satisfy enterprise requirements. The appropriate knowledge mesh for enterprises not only includes the matching of knowledge mesh function but also that of performance perfection and structure. Thus, the similarity degree of knowledge meshes, whose properties are proved to relate to the operations of knowledge meshes, is constructed from the functional matching, the perfection degree and the layer number of lowest-layer knowledge point. The similarity values, taken as cluster data, are used to construct the fuzzy relational matrix to compress the high-dimensional feature space. The decomposition of the matrix is transformed into an optimization problem solved by the gradient method. The knowledge meshes with higher membership degree in each class are taken as reference knowledge meshes to identify user’s requirements exactly. The comparison of target knowledge mesh with reference knowledge meshes definitely narrows down the knowledge mesh selection to a certain type. Based on the above, the knowledge mesh clustering and selection method is exemplified. The results show that the proposed method works well in narrowing the search range and clarifying user requirements.

Keywords

Knowledgeable manufacturing system knowledge mesh fuzzy relational clustering similarity matrix-decomposed

Introduction

With the development of advanced manufacturing technologies, various advanced manufacturing modes have emerged. They are predicated on certain requirements of enterprises, so each one has its advantage. Because of the diversity of user requirements, these modes are basically independent with their own limitations. It is likely that some enterprises may adopt only one type of advanced manufacturing mode while others may employ a combination of several. In fact, an advanced manufacturing mode can be taken as a kind of advanced manufacturing knowledge. If all types of complementary advanced manufacturing modes are transformed into their corresponding advanced manufacturing knowledge in the advanced manufacturing system, the enterprise can be allowed to select the most appropriate combination of advanced manufacturing modes or the best mode for operation. Therefore, the concept of knowledgeable manufacturing system (KMS) was proposed in 2000¹ and has been paid more and more attention to since then.^2–4

In KMS, all types of advanced manufacturing modes are transformed into corresponding knowledge meshes (KMs) and stored in the KM base of KMS.⁵ New KMs are obtained by the operations of KM multiple set.^6,7 Once a KM multiple set operation expression is obtained, a new KM can be inferred from the expression by the developed KM-based inference engine and transformed into its corresponding software of KMS automatically by the developed automatic program construction software. Then the enterprises can choose the most appropriate combination of the modes or the best when necessary. Related studies have been conducted on the concepts, systematic structure and primary characters of KMS.¹ A KM-based knowledge representation method for advanced manufacturing modes and a new approach to self-reconfiguration of KMS were proposed by Yan.⁵ A topological structure and simplification of KM were discussed by Yang and Yan.⁸ The automatic construction and optimization of the KM based on the user’s function requirements were studied by Yan and Xue.^7,9 The fuzzy classification and searching for KM based on information granular were further explored by Yang and Yan.¹⁰ But the selection method of single KM in the optimization of KM multiple set operation expression has not yet been explored.⁷ And the fuzzy classification and searching method for KM was used for the high-level users who can describe their specific needs clearly.¹⁰ However, there are many various KMs and reconfigured KMs in the KM base. These KMs have both advantages and disadvantages, and certain parts overlap each other. The similar KMs complicate the selecting of KM. Thus, selecting KM from KM base for all users is the problem to be solved in this article.

Various methods have been proposed for the reference model. Hwang et al.¹¹ proposed a neutral reference model that consists of a neutral skeleton model and an external reference model. It can effectively share and propagate engineering change information in a distributed collaborative design environment in which different computer-aided design systems are used by participating companies. Sharifi et al.¹² presented a conceptual agile manufacturing model based on a wide-ranging review of agile manufacture and manufacturing architectures and described a framework for analyzing and developing a company’ s agile characteristics. Kong et al.¹³ introduced a pervasive representation model as a basis to realize computer automatic processing for modular product development because partitioning, development, fabrication and management of construction blocks of modular products still rely heavily on manual labors of sophisticate designers. Oztemel and Tekez¹⁴ introduced a reference model called reference model for intelligent integrated manufacturing system to promote the adaptation of integrated manufacturing system. However, these reference models are not aimed at the reference model of KMS. In KM base, each KM is stored separately and any calling of KM’s elements is realized by KM multiple set operation. If some typical KMs could be taken as reference models, KMs can be better selected and optimized.

Clustering algorithms have been used extensively for pattern recognition,^15–17 data mining,^18–21 information compression²² and fuzzy modeling.^23–26 Traditional hard clustering puts each object into exactly one cluster, while fuzzy clustering allows for partial membership of objects to cluster. Therefore, fuzzy clustering is among the most widely employed due to its effectiveness, simplicity and computational efficiency. Since Zadeh^27,28 proposed fuzzy sets using the idea of partial membership and described by fuzzy membership functions, many fuzzy clustering methods have been proposed and studied by researchers, among which are fuzzy c-means clustering,²⁹ fuzzy relational clustering,³⁰ fuzzy subspace clustering,³¹ fuzzy kernel clustering with outliers,³² entropy-based fuzzy clustering³³ and vector fuzzy c-means.³⁴ However, KM is a new knowledge representation for advanced manufacturing mode. Each KM contains lots of data and inherent information. Yang and Yan¹⁰ proposed a dynamic clustering of KMs by adjusting the clustering number according to user requirements to simplify the next self-reconfiguration operations. However, the feature space under no user demands is high dimensional, under which the distribution of KMs goes sparse. Thus, considering the relevance and complexity of KMs, it is necessary to develop a new KM clustering method for optimal selection of KMs.

A new method for KM selection based on fuzzy relational clustering is proposed, whose clustering data are the similarity degrees between KMs. The reasonable similarity is an important factor of mining KMs’ relations. The perfection degree, matching degree and similarity of KM are defined in turn in terms of performance, function and structure. Their properties are discussed. The sample-class relation is constructed by decomposing fuzzy relational matrix. As KMs having higher membership in each class with the target KM give a good reference, they are compared to determine the most similar class. Users tend to select KMs in a certain class, which narrows the scope of their choice. The test example has verified the validity of the proposed method for KMs selection.

The rest of the article is organized as follows: section “Similarity measure of KMs” presents the similarity model of KM, which includes the introduction of KM, the definitions of KM matching degree, perfection degree and similarity degree as well as related properties and computation steps. In section “KM fuzzy clustering and selection,” the decomposition of fuzzy relational matrix by gradient method is discussed, and a new selection method for KM is proposed, and their implementation steps are listed. Section “Empirical analysis” provides an empirical study to verify the proposed methods. Final conclusions and remarks are given in section “Conclusion.”

Similarity measure of KMs

KM

An advanced manufacturing system is the concrete realization of a manufacturing mode, and it consists of a number of relatively independent modules linked to each other. Any sub-module of a module or even sub-sub-modules can also be viewed as this. Setting the module as a knowledge point, the advanced manufacturing mode is supposed to consist of the points and the relationships between them. Then a network structure can be formed, whereby an advanced manufacturing mode can be transformed into a KM.⁵ The KM definition is as follows:

Definition 1

The KM can be defined as a six-tuple

KM = {P, R, M, F_{P}, F_{R}, F_{M}}

where

$P = {p_{1}, p_{2}, \dots, p_{m}}$ is a finite knowledge point set, which consists of m knowledge points.

$R \subseteq P \times P$ , $R = {r_{1}, r_{2}, \dots, r_{n}}$ is a finite complexity relationship set composed of n complexity relationships (containing affiliations and messages). Complexity relationships reflect inheritances and message exchanges between father and son knowledge points. There is one-to-one correspondence between a complexity relationship and the corresponding pair of father and son knowledge points. Except for the root knowledge point, each knowledge point is affiliated at least to one father knowledge point through a complexity relationship.

$M \subset P \times P$ , $M = {m_{1}, m_{2}, \dots, m_{k}}$ is a finite message relationship set that consists of k message relationships between knowledge points, that is, all relationships except complexity relationships between father and son knowledge points. $R \cap M = ϕ$ , $ϕ$ is a null set. Message relationships reflect message exchanges between knowledge points except those between father and son knowledge points.

$F_{P} = {F_{p 1}, F_{p 2}, \dots, F_{pm}}$ is a finite set that is made up of all functions defined on the set P. $\forall F_{pi} \subset F_{P}$ and i = 1, …, m, $F_{pi}$ is a finite set containing all functions of knowledge point $p_{i}$ and is a subset of F_P .

$F_{R} = {F_{r 1}, F_{r 2}, \dots, F_{rn}}$ is a finite set that consists of all inherited flows and message flows defined on the set R. $\forall F_{rj} \subset F_{R}$ and j = 1, …, n, $F_{rj}$ is a finite set of all inherited flows and message flows of the complexity relationship $r_{j}$ and a subset of $F_{R}$ as well. Inherited flows are message flows inherited by son knowledge points, which exist between father knowledge points and other knowledge points.

$F_{M} = {F_{m 1}, F_{m 2}, \dots, F_{mk}}$ is a finite set composed of all message flows defined on the set M. $\forall F_{ml} \subset F_{M}$ and l = 1, …, k, $F_{ml}$ is a finite set made up of all message flows of the message relationship $m_{l}$ and is a subset of $F_{M}$ .

As shown in Definition 1, a KM is a large set including knowledge points, message relationships, message flows and functions of knowledge points. Any KM corresponds to a set of functional topological space generated by knowledge points under the different functional environments.⁸ The reconfigured KMs are continually constructed by the topological bases of functional topological space. These topological bases are in fact the lowest-layer knowledge points of KM, that is, the smallest function modules in actual system. Once a new KM determines the lowest-layer knowledge points, the other elements of KM and corresponding operations will be called to reconfigure the KM. If the root knowledge point is taken as the first-layer knowledge point, its son knowledge point connecting the root knowledge point through a complexity relationship can be regarded as the second-layer knowledge point, and the son knowledge point of son knowledge point as the third-layer knowledge point, which connects the root knowledge point through two complexity relationships. To compare two KMs, the lowest-layer knowledge point of KM and its layer number are defined as follows.

Definition 2

The lowest-layer knowledge point of KM is the knowledge point that has no son knowledge point in KM and connects the root knowledge point through at least $α - 1$ complexity relationships. The positive integer $α$ is defined as the layer number of the lowest-layer knowledge point.

Because the lowest-layer knowledge point of KM plays an important role in the construction of KMs, the similarity model is based on its characteristics.

Similarity model

To cluster and select KMs, it is necessary to discuss the similarity of KM. Yan and Xue⁷ regard the fuzzy function–satisfaction degree as similarity, while Ding et al.³⁵ take the number of the same functions as such. However, a KM is an abstract model of a manufacturing mode in KMS. The KM similarity is not only the match of lowest-layer knowledge points’ function but also that of its satisfaction degree. And the structure of KM also influences its similarity. The matching degree of KMs’ function is defined as follows.

Definition 3

Suppose the lowest-layer knowledge point sets of KMs $V$ and $W$ are $P_{V} = {p_{v_{1}}, p_{v_{2}}, \dots, p_{v_{m}}}$ and $P_{W} = {p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ . The functional number of knowledge point $p_{v_{i}}$ is $l_{v_{i}}$ ( $i = 1, 2, \dots, m$ ) and that of knowledge point $p_{w_{j}}$ is $l_{w_{j}}$ ( $j = 1, 2, \dots, n$ ). The number of the functions that are the same as those of knowledge point $p_{v_{i}}$ is $l_{v_{i}, W}$ in $P_{W}$ and the number of the functions that are the same as those of knowledge point $p_{w_{j}}$ is $l_{w_{j}, V}$ in $P_{V}$ . Then, $f (P_{V}, P_{W}) = (\sum_{i = 1}^{m} l_{v_{i}, W} + \sum_{j = 1}^{n} l_{w_{j}, V}) / (\sum_{i = 1}^{m} l_{v_{i}} + \sum_{j = 1}^{n} l_{w_{j}})$ on $P_{V} \times P_{W}$ can be defined as the matching degree of KMs $V$ and $W$ . As $\sum_{i = 1}^{m} l_{v_{i}, W} = \sum_{j = 1}^{n} l_{w_{j}, V}$ , it can be simplified as

f (P_{V}, P_{W}) = \frac{2 \sum_{i = 1}^{m} l_{v_{i}, W}}{\sum_{i = 1}^{m} l_{v_{i}} + \sum_{j = 1}^{n} l_{w_{j}}}

It is obvious that $0 \leq f (P_{V}, P_{W}) \leq 1$ . When KMs $V$ and $W$ have the identical functions, $f (P_{V}, P_{W}) = 1$ holds; while not, $f (P_{V}, P_{W}) = 0$ holds and $f (P_{V}, P_{V}) = 1$ . When the lowest-layer knowledge point’s functional number of KMs $V$ and $W$ is fixed, the matching degree is an increasing function of the same functional number of KMs $V$ and $W$ . And $f (P_{V}, P_{W}) = f (P_{W}, P_{V})$ . The following properties of matching degree further show the similarity relation between KMs.

Property 1

If the functions of KM $V$ include all those of KM $W$ and the remaining ones of $V$ which are different from Ws are also different from those of $X$ , then $f (P_{V}, P_{X}) \leq f (P_{W}, P_{X})$ holds.

Suppose $P_{V + W}$ is the lowest-layer knowledge point’s set of the union of KMs $V$ and $W$ . If $V$ and $W$ share no identical function and KM $X$ includes all those of $V$ , $f (P_{W}, P_{X}) \leq f (P_{V + W}, P_{X})$ holds.

Suppose $P_{V \cap W}$ is the lowest-layer knowledge point’s set of the intersection of KMs $V$ and $W$ . If the functions of KM $X$ include both Vs and Ws, $f (P_{V \cap W}, P_{X}) \leq \min {f (P_{V}, P_{X}), f (P_{W}, P_{X})}$ holds.

Proof

Suppose the lowest-layer knowledge points of KMs $V$ , $W$ and $X$ are $P_{V} = {p_{v_{1}}, p_{v_{2}}, \dots, p_{v_{m}}}$ , $P_{W} = {p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ and $P_{X} = {p_{x_{1}}, p_{x_{2}}, \dots, p_{x_{r}}}$ , respectively. The functional numbers of corresponding knowledge points are $l_{v_{i}}$ , $l_{w_{j}}$ and $l_{x_{k}}$ , and the numbers of corresponding identical functions are $l_{v_{i}, X}, l_{x_{k}, V}, l_{w_{j}, X}$ and $l_{x_{k}, W}$ .

1. The total functional number of KMs $V$ and $X$ equals that of KMs $W$ and $X$ , that is $\sum_{i = 1}^{m} l_{v_{i}, X} + \sum_{k = 1}^{r} l_{x_{k}, V} = \sum_{j = 1}^{n} l_{w_{j}, X} + \sum_{k = 1}^{r} l_{x_{k}, W}$ . As $\sum_{i = 1}^{m} l_{v_{i}} \geq \sum_{j = 1}^{n} l_{w_{j}}$ , there exists

\begin{matrix} f (P_{V}, P_{X}) & = \frac{\sum_{i = 1}^{m} l_{v_{i}, X} + \sum_{k = 1}^{r} l_{x_{k}, V}}{\sum_{i = 1}^{m} l_{v_{i}} + \sum_{k = 1}^{r} l_{x_{k}}} \\ \leq \frac{\sum_{j = 1}^{n} l_{w_{j}, X} + \sum_{k = 1}^{r} l_{x_{k}, W}}{\sum_{j = 1}^{n} l_{w_{j}} + \sum_{k = 1}^{r} l_{x_{k}}} = f (P_{W}, P_{X}) \end{matrix}

(1)

2. $P_{V + W} = {p_{v_{1}}, p_{v_{2}}, \dots, p_{v_{m}}, p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ . Then, $\sum_{i = 1}^{m} l_{v_{i}, X} = \sum_{k = 1}^{r} l_{x_{k}, V} = \sum_{i = 1}^{m} l_{v_{i}}$ .

\begin{matrix} f (P_{W}, P_{X}) & = \frac{\sum_{j = 1}^{n} l_{w_{j}, X} + \sum_{k = 1}^{r} l_{x_{k}, W}}{\sum_{j = 1}^{n} l_{w_{j}} + \sum_{k = 1}^{r} l_{x_{k}}} \\ \leq \frac{\sum_{i = 1}^{m} l_{v_{i}, X} + \sum_{j = 1}^{n} l_{w_{j}, X} + \sum_{k = 1}^{r} l_{x_{k}, W}}{\sum_{i = 1}^{m} l_{v_{i}} + \sum_{j = 1}^{n} l_{w_{j}} + \sum_{k = 1}^{r} l_{x_{k}}} \\ \leq \frac{\sum_{i = 1}^{m} l_{v_{i}, X} + \sum_{j = 1}^{n} l_{w_{j}, X} + \sum_{k = 1}^{r} l_{x_{k}, W} + \sum_{k = 1}^{r} l_{x_{k}, V}}{\sum_{i = 1}^{m} l_{v_{i}} + \sum_{j = 1}^{n} l_{w_{j}} + \sum_{k = 1}^{r} l_{x_{k}}} \\ = f (P_{V + W}, P_{X}) \end{matrix}

(2)

3. Suppose $P_{V \cap W} = {p_{s_{1}}, p_{s_{2}}, \dots, p_{s_{t}}}$ . Then, $\sum_{h = 1}^{t} l_{s_{h}} = \sum_{h = 1}^{t} l_{s_{h}, X} \leq min {\sum_{i = 1}^{m} l_{v_{i}, X}, \sum_{j = 1}^{n} l_{w_{j}, X}}$ and $\sum_{i = 1}^{m} l_{v_{i}} = \sum_{i = 1}^{m} l_{v_{i}, X}$ , $\sum_{j = 1}^{n} l_{w_{j}} = \sum_{j = 1}^{n} l_{w_{j}, X}$ . And

\begin{matrix} f (P_{V \cap W}, P_{X}) & = \frac{\sum_{h = 1}^{t} l_{s_{h}, X} + \sum_{k = 1}^{r} l_{x_{k}, V \cap W}}{\sum_{h = 1}^{t} l_{s_{h}} + \sum_{k = 1}^{r} l_{x_{k}}} \\ \leq \frac{\sum_{i = 1}^{m} l_{v_{i}, X} + \sum_{k = 1}^{r} l_{x_{k}, V \cap W}}{\sum_{i = 1}^{m} l_{v_{i}} + \sum_{k = 1}^{r} l_{x_{k}}} \\ \leq \frac{\sum_{i = 1}^{m} l_{v_{i}, X} + \sum_{k = 1}^{r} l_{x_{k}, V}}{\sum_{i = 1}^{m} l_{v_{i}} + \sum_{k = 1}^{r} l_{x_{k}}} = f (P_{V}, P_{X}) \end{matrix}

(3)

Similarly, $f (P_{V \cap W}, P_{X}) \leq f (P_{W}, P_{X})$ . Then, $f (P_{V \cap W}, P_{X}) \leq min {f (P_{V}, P_{X}), f (P_{W}, P_{X})}$ .

Property 1 shows that matching degree of union is not necessarily higher than that of subset, as presented in Property 1 (1). However, the former may be higher than the latter under the conditions of Property 1 (2). Similarly, the matching degree of intersection is not necessarily lower than that of the original set. But the former may be lower than the latter under the preconditions of Property 1 (3).

The above shows that Definition 1 reflects the functional similarity between KMs. However, for user, the KM that has user functional requirements is necessary but not enough. How much a KM implements functions, that is, the satisfaction degree of user for these functions, is the same important. Some users hope to improve functions to meet the further development, while others only need the basic functions to save the cost. In addition, as we know, the representation of knowledge point is the same while the corresponding contents in practical system may vary a little. For example, for knowledge points “c” in Figures 3 and 4, which have the functions of production management, their son knowledge points have not be the same in KMs $W_{1}$ and $W_{2}$ . So it is necessary to compare this kind of difference between KMs and define it as the perfection degree of KM. If the matching degree is regarded as the functional comparison, the perfection degree is taken as the performance comparison. However, the degree cannot be defined simply by perfection or imperfection or the values 0 and 1. Using a fuzzy set for the definition is a more practical choice.

Definition 4

Suppose the lowest-layer knowledge point’s set of KM $W$ is $P_{W} = {p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ . A fuzzy set on $P_{W}$ is defined. $μ (p_{w_{j}}) : P_{W} \to [0, 1], p_{w_{j}} \in P_{W}$ . $μ (p_{w_{j}})$ is the functional perfection degree of knowledge point $p_{w_{j}}$ and simplified as $μ_{w_{j}}$ . $μ_{W} = (μ_{w_{1}}, μ_{w_{2}}, \dots, μ_{w_{n}})$ is the functional perfection degree of KM $W$ . $μ_{w_{j}} = 0$ denotes that the element $p_{w_{j}}$ does not exist in KM. The larger the $p_{w_{j}}$ value, the more perfect its function.

According to Definition 2, there is one-to-one correspondence between each KM and its fuzzy set $μ$ . It is assumed that each KM corresponds to one fuzzy vector $μ$ , $μ \in {[0, 1]}^{n}$ . The functional perfection degree $μ$ , which is obtained by some experts dividing the levels of functional perfection degree and using fuzzy analysis method to evaluate KMs, is also stored in KM base. It is the inherent information of KM and reflects the comparison of KM’s quality in contrast to the matching degree.

Definitions 1 and 2 reflect the similarity from the two aspects of functional quality and quantity. Even if the matching degree and perfection degree of two KM are completely the same, their corresponding structures of KM may differ. The layer number of knowledge point can reflect the KM’s structure, which is then introduced into the definition of KM’s similarity. Taking function, performance and structure into account, Definition 5 is set up as follows.

Definition 5

The similarity degree of KMs $V$ and $W$ is

\begin{matrix} sim (V, W) = \\ \frac{\sum_{i = 1}^{m} {\bar{α}}_{v_{i}, W} \cdot l_{v_{i}, W} \cdot {\bar{μ}}_{v_{i}, W} + \sum_{j = 1}^{n} {\bar{α}}_{w_{j}, V} \cdot l_{w_{j}, V} \cdot {\bar{μ}}_{w_{j}, V}}{\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} + \sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}}} \end{matrix}

(4)

The lowest-layer knowledge point sets of $V$ and $W$ are $P_{V} = {p_{v_{1}}, p_{v_{2}}, \dots, p_{v_{m}}}$ and $P_{W} = {p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ , respectively. $μ_{v_{i}}$ , $l_{v_{i}}$ and $α_{v_{i}}$ represent the functional perfection degree, functional number and layer number of knowledge point $p_{v_{i}}$ , and $μ_{w_{j}}$ , $l_{w_{j}}$ and $α_{w_{j}}$ are the functional perfection degree, functional number and layer number of knowledge point $p_{w_{j}}$ , respectively. $l_{v_{i}, W}$ is the number of the same functions as those of knowledge point $p_{v_{i}}$ in $P_{W}$ . $α_{v_{i}, W}$ and $μ_{v_{i}, W}$ are the layer number and functional perfection degree of the knowledge points relating to the same functions. $l_{w_{j}, V}$ is the number of the same functions as those of knowledge point $p_{w_{j}}$ in $P_{V}$ . $α_{w_{j}, V}$ and $μ_{w_{j}, V}$ are the layer number and functional perfection degree of knowledge points relating to the same functions. ${\bar{α}}_{v_{i}, W} = \min {α_{v_{i}, W}, α_{v_{i}}}$ , ${\bar{α}}_{w_{j}, V} = \min {α_{w_{j}, V}, α_{w_{j}}}$ , ${\bar{μ}}_{v_{i}, W} = \min {μ_{v_{i}, W}, μ_{v_{i}}}$ and ${\bar{μ}}_{w_{j}, V} = \min {μ_{w_{j}, V}, μ_{w_{j}}}$ .

If the knowledge point having the same function as knowledge point $p_{v_{i}}$ is not the only one, divide $l_{v_{i}, W}$ into sum formula to make each item correspond to the only knowledge point in $P_{W}$ . Then operate it. For example, if $p_{v_{i}}$ shares the same functions with $p_{w_{1}}$ and $p_{w_{2}}$ of $P_{W}$ , then ${\bar{α}}_{v_{i}, W} \cdot l_{v_{i}, W} \cdot {\bar{μ}}_{v_{i}, W} = {\bar{α}}_{v_{i 1}, W} \cdot l_{v_{i 1}, W} \cdot {\bar{μ}}_{v_{i 1}, W} + {\bar{α}}_{v_{i 2},_{W}} \cdot l_{v_{i 2}, W} \cdot {\bar{μ}}_{v_{i 2}, W}$ , where ${\bar{α}}_{v_{i 1}, W} = \min {α_{w_{1}}, α_{v_{i}}}$ ; ${\bar{α}}_{v_{i 2}, W} = \min {α_{w_{2}}, α_{v_{i}}}$ ; $l_{v_{i 1}, W}$ is the same functional number of $p_{w_{1}}$ and $p_{v_{i}}$ ; $l_{v_{i 2}, W}$ is the same functional number of $p_{w_{2}}$ and $p_{v_{i}}$ ; ${\bar{μ}}_{v_{i 1}, W} = \min {μ_{w_{1}}, μ_{v_{i}}}$ ; ${\bar{μ}}_{v_{i 2}, W} = \min {μ_{w_{2}}, μ_{v_{i}}}$ . $l_{v_{i}, W}$ can be separated, that is, the knowledge points having the same function in $P_{V}$ and $P_{W}$ can be viewed as one-to-one. Then, $\sum_{i = 1}^{m} {\bar{α}}_{v_{i}, W} \cdot l_{v_{i}, W} \cdot {\bar{μ}}_{v_{i}, W} = \sum_{j = 1}^{n} {\bar{α}}_{w_{j}, V} \cdot l_{w_{j}, V} \cdot {\bar{μ}}_{w_{j}, V}$ , and equation (4) can be simplified into

sim (V, W) = \frac{2 \sum_{i = 1}^{m} {\bar{α}}_{v_{i}, W} \cdot l_{v_{i}, W} \cdot {\bar{μ}}_{v_{i}, W}}{\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} + \sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}}}

(5)

It is obvious that $0 \leq sim (V, W) \leq 1$ . When KMs $V$ and $W$ share identical functions, and the functional perfection degree and layer number of knowledge point corresponding to those functions are also the same, $sim (V, W) = 1$ holds. When KMs $V$ and $W$ share no identical function, $sim (V, W) = 0$ holds, and $sim (V, V) = 1$ . And $sim (V, W) = sim (W, V)$ . The properties of similarity degree between KMs are similar to Property 1.

Property 2

When the lowest-layer knowledge point’s functional number of KMs $V$ and $W$ is fixed, $sim (V, W)$ is an increasing function of the same functional number of KMs $V$ and $W$ as well as that of the match degree of corresponding knowledge point layer numbers and that of corresponding knowledge point functional perfection degrees.

KM $V$ includes all functions of KM $W$ . The functional perfection degrees and layer numbers of lowest-layer knowledge points corresponding to the same functions are identical in KMs $V$ and $W$ . And those of $V$ , which are different from those of W, are also different from those of X. Then, $sim (V, X) \leq sim (W, X)$ holds.

KMs $V$ and $W$ share no identical function. KM $X$ includes all functions of KM $V$ . The functional perfection degrees and layer numbers of lowest-layer knowledge points corresponding to the same functions of KMs $V$ and $X$ are the same. Then, $sim (W, X) \leq sim (V + W, X)$ holds.

The functions of KM $X$ include those of both V and W. The functional perfection degrees and layer numbers of knowledge points corresponding to the same functions of KMs $V$ and $X$ are the same, and so are those of $V \cap W$ and $X$ . Then, $sim (V \cap W, X) \leq \min {sim (V, X), sim (W, X)}$ holds.

Proof

Suppose the lowest-layer knowledge point sets of KMs $V$ , $W$ , $X$ and $V \cap W$ are $P_{V} = {p_{v_{1}}, p_{v_{2}}, \dots, p_{v_{m}}}$ , $P_{W} = {p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ , $P_{X} = {p_{x_{1}}, p_{x_{2}}, \dots, p_{x_{r}}}$ and $P_{V \cap W} = {p_{s_{1}}, p_{s_{2}}, \dots, p_{s_{t}}}$ . The lowest-layer knowledge point functional perfection degrees are $μ_{v_{i}}$ , $μ_{w_{j}}$ , $μ_{x_{k}}$ and $μ_{s_{h}}$ , the functional numbers are $l_{v_{i}}, l_{w_{j}}, l_{x_{k}}$ and $l_{s_{h}}$ , and the corresponding lowest-layer knowledge point layer numbers are $α_{v_{i}}, α_{w_{j}}, α_{x_{k}}$ and $α_{s_{h}}$ . The functional perfection degrees of lowest-layer knowledge points having identical functions are $μ_{v_{i}, W}$ , $μ_{x_{k}, V}$ , $μ_{x_{k}, W}$ , $μ_{v_{i}, X}$ and $μ_{s_{h}, X}$ , the same functional numbers are $l_{v_{i}, W}, l_{v_{i}, X}, l_{x_{k}, V}, l_{x_{k}, W}$ and $l_{s_{h}, X}$ , and the corresponding layer numbers are $α_{v_{i}, W}$ , $α_{x_{k}, V}$ , $α_{x_{k}, W}$ , $α_{v_{i}, X}$ and $α_{s_{h}, X}$ , as defined in Definition 5. Let ${\bar{α}}_{v_{i}, W} = \min {α_{v_{i}, W}, α_{v_{i}}}$ , ${\bar{α}}_{x_{k}, V} = \min {α_{x_{k}, V}, α_{x_{k}}}$ , ${\bar{α}}_{x_{k}, W} = \min {α_{x_{k}, W}, α_{x_{k}}}$ , ${\bar{α}}_{v_{i}, X} = \min {α_{v_{i}, X}, α_{v_{i}}}$ , ${\bar{α}}_{s_{h}, X} = \min {α_{s_{h}, X}, α_{s_{h}}}$ , ${\bar{μ}}_{v_{i}, W} = \min {μ_{v_{i}, W}, μ_{v_{i}}}$ , ${\bar{μ}}_{x_{k}, V} = \min {μ_{x_{k}, V}, μ_{x_{k}}}$ , ${\bar{μ}}_{x_{k}, W} = \min {μ_{x_{k}, W}, μ_{x_{k}}}$ , ${\bar{μ}}_{v_{i}, X} = \min {μ_{v_{i}, X}, μ_{v_{i}}}$ and ${\bar{μ}}_{s_{h}, X} = \min {μ_{s_{h}, X}, μ_{s_{h}}}$ . Then, the following is the proof of formulas (1)–(4):

1. When the lowest-layer knowledge point’s functional number of KMs $V$ and $W$ is fixed, it means that $\sum_{i}^{m} l_{v_{i}} + \sum_{j = 1}^{n} l_{w_{j}}$ is a constant in equation (5).

(a) If $α_{v_{i}}, μ_{v_{i}}, α_{W_{j}}$ and $μ_{W_{j}}$ are fixed, $sim (V, W)$ in equation (5) can be viewed as the function of $l_{v_{i}, W}$ . It is clear that $sim (V, W)$ is an increasing function of $l_{v_{i}, W}$ .

(b) $\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}}$ is divided into two parts $\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}} = \sum_{i}^{m} α_{v_{i}, W} l_{v_{i}, W} μ_{v_{i}, W} + \sum_{j'} α_{w_{j'}} l_{w_{j'}} μ_{w_{j'}}$ , where $\sum_{i}^{m} α_{v_{i}, W} l_{v_{i}, W} μ_{v_{i}, W}$ denotes the sum of values concerning identical functions between KMs $V$ and $W$ , and $\sum_{j'} α_{w_{j'}} l_{w_{j'}} μ_{w_{j'}}$ denotes others. Since the knowledge points having the same functions in $V$ and $W$ can be viewed as one-to-one in terms of Definition 5, there exists

\begin{matrix} sim (V, W) \\ = \frac{2 \sum_{i = 1}^{m} \min {α_{v_{i}}, α_{v_{i}, W}} \cdot l_{v_{i}, W} \cdot \min {μ_{v_{i}}, μ_{v_{i}, W}}}{\sum_{i = 1}^{m} (α_{v_{i}} l_{v_{i}} μ_{v_{i}} + α_{v_{i}, W} l_{v_{i}, W} μ_{v_{i}, W}) + \sum_{j'} α_{w_{j'}} l_{w_{j'}} μ_{w_{j'}}} \end{matrix}

(6)

When $l_{v_{i}}, μ_{v_{i}}, l_{w_{j}}, μ_{w_{j}} and l_{v_{i}, W}$ is fixed, $sim (V, W)$ can just be taken as the function of $α_{v_{i}, W}$ . Then, two cases should be discussed:

Case 1. If $α_{v_{i}} \leq α_{v_{i}, W}$ , ${\bar{α}}_{v_{i}, W} = \min {α_{v_{i}, W}, α_{v_{i}}} = α_{v_{i}}$ . Then, the smaller $α_{v_{i}, W}$ , the larger $sim (V, W)$ .

Case 2. If $α_{v_{i}} > α_{v_{i}, W}$ , ${\bar{α}}_{v_{i}, W} = \min {α_{v_{i}, W}, α_{v_{i}}} = α_{v_{i}, W}$ . When $α_{v_{i}, W}$ turns larger, the numerator and denominator of equation (6) increase by the same value. Then, $sim (V, W)$ turns larger by inequality $(b + c) / (a + c) \geq b / a, a \geq b > 0, c > 0$ .

It has been proved that the smaller the $| α_{v_{i}, W} - α_{v_{i}} |$ value, the larger the $sim (V, W)$ , that is, the more the KMs $V$ and $W$ layer numbers match, the larger the $sim (V, W)$ . Thus, $sim (V, W)$ has been proved to be an increasing function of the match degree of corresponding knowledge point layer numbers.

(c) Similarly, $sim (V, W)$ is an increasing function of the matching degree of corresponding knowledge point perfection degrees by equation (6).

2. KM $V$ includes all functions of KM $W$ and the functions of $V$ which are different from those of W are also different from those of X. It means that the same functional number of KMs $V$ and $X$ is equal to that of KMs $W$ and $X$ , that is, $\sum_{k = 1}^{r} l_{x_{k}, V} = \sum_{k = 1}^{r} l_{x_{k}, W}$ . The functional perfection degrees and layer numbers of KM V lowest-layer knowledge points sharing identical functions with KM $X$ are the same as those of KM W lowest-layer knowledge points sharing identical functions with KM $X$ , that is, $μ_{x_{k}, V} = μ_{x_{k}, W}$ , and $α_{x_{k}, V} = α_{x_{k}, W}$ . Then, $κ_{x_{k}, V} = κ_{x_{k}, W}$ , ${\bar{α}}_{x_{k}, V} = {\bar{α}}_{x_{k}, W}$ and $\sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V} = \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, W} l_{x_{k}, W} {\bar{μ}}_{x_{k}, W}$ . And $\sum_{i = 1}^{m} l_{v_{i}} \geq \sum_{j = 1}^{n} l_{w_{j}}$ . Then

\begin{matrix} sim (V, X) & = \frac{2 \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V}}{\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ \leq \frac{2 \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, W} l_{x_{k}, W} {\bar{μ}}_{x_{k}, W}}{\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ = sim (W, X) \end{matrix}

(7)

3. Since KMs $V$ and $W$ share no identical function and KM $X$ includes all functions of KM $V$ , $l_{v_{i}} = l_{v_{i}, X}$ holds. The functional perfection degrees and layer numbers of lowest-layer knowledge points corresponding to the same functions of KMs $V$ and $X$ are the same, that is, $μ_{v_{i}} = μ_{v_{i}, X} = {\bar{μ}}_{v_{i}, X}$ and $α_{v_{i}} = α_{v_{i}, X} = {\bar{α}}_{v_{i}, X}$ . Then, $\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} = \sum_{i = 1}^{m} {\bar{α}}_{v_{i}, X} l_{v_{i}, X} {\bar{μ}}_{v_{i}, X} = \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V}$ . And

\begin{matrix} sim (W, X) = \\ \frac{2 \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, W} l_{x_{k}, W} {\bar{μ}}_{x_{k}, W}}{\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ \leq \frac{2 \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, W} l_{x_{k}, W} {\bar{μ}}_{x_{k}, W} + \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V}}{\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}} + \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V}} \\ = \frac{2 \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, W} l_{x_{k}, W} {\bar{μ}}_{x_{k}, W} + \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V}}{\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}} + \sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}}} \\ \leq \frac{2 (\sum_{k = 1}^{r} {\bar{α}}_{x_{k}, W} l_{x_{k}, W} {\bar{μ}}_{x_{k}, W} + \sum_{k = 1}^{r} {\bar{α}}_{x_{k}, V} l_{x_{k}, V} {\bar{μ}}_{x_{k}, V})}{\sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}} + \sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}}} \\ = sim (V + W, X) \end{matrix}

(8)

4. The functions of KM $X$ include those of both V and W, that is, $l_{v_{i}} = l_{v_{i}, X}$ . Then, the functions of KM $X$ include those of $V \cap W$ , that is, $l_{s_{h}} = l_{s_{h}, X}$ . The functional perfection degrees and layer numbers of knowledge points corresponding to the same functions of KMs $V$ and $X$ are the same, and so are those of $V \cap W$ and $X$ , that is, $α_{v_{i}} = α_{v_{i}, X} = {\bar{α}}_{v_{i}, X}$ , $μ_{v_{i}} = μ_{v_{i}, X} = κ_{v_{i}, X}$ , $α_{s_{h}} = α_{s_{h}, X} = {\bar{α}}_{s_{h}, X}$ and $μ_{s_{h}} = μ_{s_{h}, X} = κ_{s_{h}, X}$ . $\sum_{h = 1}^{t} α_{s_{h}} l_{s_{h}} μ_{s_{h}} = \sum_{h = 1}^{t} {\bar{α}}_{s_{h}, X} l_{s_{h}, X} {\bar{μ}}_{s_{h}, X} \leq \sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} = \sum_{i = 1}^{m} {\bar{α}}_{v_{i}, X} l_{v_{i}, X} {\bar{μ}}_{v_{i}, X}$ . Then

\begin{matrix} sim (V \cap W, X) = \\ \frac{2 \sum_{h = 1}^{t} {\bar{α}}_{s_{h}, X} l_{s_{h}, X} {\bar{μ}}_{s_{h}, X}}{\sum_{h = 1}^{t} α_{s_{h}} l_{s_{h}} μ_{s_{h}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ = \frac{\sum_{h = 1}^{t} α_{s_{h}} l_{s_{h}} μ_{s_{h}} + \sum_{h = 1}^{t} {\bar{α}}_{s_{h}, X} l_{s_{h}, X} {\bar{μ}}_{s_{h}, X}}{\sum_{h = 1}^{t} α_{s_{h}} l_{s_{h}} μ_{s_{h}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ \leq \frac{2 \sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}}}{\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ = \frac{2 \sum_{h = 1}^{t} {\bar{α}}_{v_{i}, X} l_{v_{i}, X} {\bar{μ}}_{v_{i}, X}}{\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} + \sum_{k = 1}^{r} α_{x_{k}} l_{x_{k}} μ_{x_{k}}} \\ = sim (V, X) \end{matrix}

(9)

Similarly, $sim (V \cap W, X) \leq sim (W, X)}$ . So $sim (V \cap W, X) \leq \min {sim (V, X), sim (W, X)}$ .

Property 2 indicates that similarity of KMs relates to the functional number, the matching degree of perfection degree and layer number of knowledge points having identical functions. The greater the number of knowledge points having identical functions, the more similar the KMs, and the smaller the dissimilarity of the perfection degree and layer number of corresponding knowledge point, the more similar the KMs. Property 2 also suggests that the similarity degree of union is not necessarily higher than that of subset, but the former may be higher than the latter under the preconditions of Property 2 (3). In a real system, although the combined KMs may complement each other, they are not necessarily the best choice for user, the reason being that some KM combinations may lower the similarity degree. Similarly, the similarity degree of intersection is not necessarily lower than that of the original set. But the former may be lower than the latter under the preconditions of Property 2 (4). Therefore, when KMS has secured the desired KMs through KM operations, the impacts caused by the operations should be considered in the similarity.

Computation of KM similarity degree

Steps to compute similarity degree are as follows:

Step 1. Suppose the lowest-layer knowledge point sets of KMs $V$ and $W$ are $P_{V} = {p_{v_{1}}, p_{v_{2}}, \dots, p_{v_{m}}}$ and $P_{W} = {p_{w_{1}}, p_{w_{2}}, \dots, p_{w_{n}}}$ . Read their functional perfection degrees $μ_{W} = (μ_{w_{1}}, μ_{w_{2}}, \dots, μ_{w_{n}})$ and $μ_{V} = (μ_{v_{1}}, μ_{v_{2}}, \dots, μ_{v_{m}})$ from KM base.

Step 2. Keep searching for father knowledge points of a lowest-layer knowledge point till the root knowledge point is found. Obtain the layer numbers $α_{v_{i}}$ and $α_{w_{j}}$ of the lowest-layer knowledge points $p_{v_{i}}$ and $p_{w_{j}}$ .

Step 3. Compare the knowledge points in different KMs. Loop each function of knowledge point $p_{v_{i}}$ ’s function set in KM $V$ to match that of $p_{w_{j}}$ ’s in KM $W$ . Obtain the functional number $l_{v_{i}}$ for the knowledge point $p_{v_{i}}$ and $l_{w_{j}}$ for $p_{w_{j}}$ as well as the same functional number $l_{v_{i}, W}$ , the layer number $α_{v_{i}, W}$ and the functional perfection degree $μ_{v_{i}, W}$ for the knowledge points relating to the same functions.

Step 4. Following Steps 1–3 and Definition 5, obtain the similarity degree $sim (V, W)$ for KMs $V$ and $W$ .

KM fuzzy clustering and selection

Decomposition of fuzzy relational matrix

Suppose that there are KMs $V_{1}, V_{2}, \dots, V_{N}$ in the KM base. Compare them to obtain the $N (N - 1) / 2$ similarity value by equation (4). They make up a $N \times N$ matrix termed as fuzzy relational matrix. To obtain the KM–class relation, it is necessary to decompose the fuzzy relational matrix.

For this, suppose fuzzy relational matrix $R = [r_{ij}], i, j = 1, 2, \dots, N$ , where $r_{ij} = sim (V_{i}, V_{j})$ . $R$ has reflexivity and symmetry properties, that is, $r_{ii} = 1$ and $r_{ij} = r_{ji}$ . And suppose the matrix $G = [g_{ij}], i = 1, 2, \dots, N, j = 1, 2, \dots, c, c < N$ , where $c$ is the class number. And $G^{T}$ denotes the transposition of $G$ . If $G$ can satisfy $R = G \circ G^{T}$ , where “∘” refers to the composition operator of relations, it indicates that a KM–class relation is obtainable by decomposition of $R$ .

The operator is s-t convolution of fuzzy set. Let $t (x, y) = xy$ and $S (x, y) = x + y - xy$ . Then the decomposition of $R$ can be transformed into $G = [g_{ij}]$ to satisfy $r_{i j} = S_{k = 1}^{c} (t (g_{i k}, g_{j k}))$ . As an accurate solution of $G$ is hard to acquire, it is transformed into an optimization problem, that is, seeking an approximate $G$ to minimize $Q = ∥ R - G \circ G^{T} ∥^{2}$ , that is

M i n_{G} Q = M i n_{G} {\sum_{i = 1}^{N} \sum_{j = 1}^{N} {[r_{i j} - S_{k = 1}^{c} (t (g_{i k}, g_{j k}))]}^{2}}

(10)

$G$ is calculated using the gradient method. $G$ is constantly updated by $G = G - β \cdot \nabla_{G} Q$ , that is

g_{ik} = g_{ik} - β \cdot \frac{\partial Q}{\partial g_{ik}}

(11)

where $β (> 0)$ denotes the learning rate. The iterative process of $G$ starts from a random matrix, whose element values are taken from 0 to 1. The detailed calculation of gradient is

\frac{\partial Q}{\partial g_{a b}} = - 2 \sum_{i = 1}^{N} \sum_{j = 1}^{N} [r_{i j} - S_{k = 1}^{c} (t (g_{i k}, g_{j k}))] \frac{\partial}{\partial g_{a b}} (S_{k = 1}^{c} (t (g_{i k}, g_{j k})))

(12)

where the derivative is divided into four cases:

When $i \neq a, j \neq a$ , the partial derivative of $S_{k = 1}^{c} (t (g_{i k}, g_{j k}))$ for $g_{ab}$ is 0.

If $i = a, j \neq a$ and $A = S_{k = 1, k \neq b}^{c} (t (g_{a k}, g_{j k}))$ , then $\partial / \partial g_{a b} (S_{k = 1}^{c} (t (g_{i k}, g_{j k}))) = \partial / \partial g_{a b} (A + g_{a b} g_{j b} - A g_{a b} g_{j b}) = g_{j b} (1 - A) .$

If $i \neq a, j = a$ and $B = S_{k = 1, k \neq b}^{c} (t (g_{i k}, g_{a k}))$ , then $\partial / \partial g_{a b} (S_{k = 1}^{c} (t (g_{i k}, g_{j k}))) = g_{i b} (1 - B)$ .

If $i = a, j = a$ and $C = S_{k = 1, k \neq b}^{c} (t (g_{a k}, g_{a k}))$ , then $\partial / \partial g_{a b} (S_{k = 1}^{c} (t (g_{i k}, g_{j k}))) = 2 g_{a b} (1 - C)$ .

Selection of KMs

Each KM in the KM base is affiliated with a class by the above method and maximum membership principle. The KMs having the highest membership are taken as reference KMs for user, especially for the low-quality user who may fail to identify his specific needs clearly. For high-level users, these reference KMs are not necessarily of the best choice, but can narrow the range of choice.

Suppose that the target KM $\bar{W}$ satisfies high-level user’s needs. The lowest-layer knowledge point sets of $\bar{W}$ is $P_{\bar{W}} = {p_{{\bar{w}}_{1}}, p_{{\bar{w}}_{2}}, \dots, p_{{\bar{w}}_{m}}}$ . First calculate the similarity of the target KM and the KM having the highest membership in each class. The larger the value, the more similar the KMs. Then, compare all KMs of this class with the target KM. The KM having the greatest similarity is the user’s choice.

As maximizing user’s needs is the requirement of selection, calculating similarity according to user’s needs is different from comparing KMs in the KM base. Modify equation (4) into equation (13), where $α_{{\bar{w}}_{i}, W}$ , $l_{{\bar{w}}_{i}, W}$ and $μ_{{\bar{w}}_{i}, W}$ in the denominator are the layer number, the same functional number and the functional perfection degree of knowledge point in KM $W$

\begin{matrix} sim (\bar{W}, W) = \\ \frac{\sum_{i = 1}^{m} {\bar{α}}_{{\bar{w}}_{i}, W} \cdot l_{{\bar{w}}_{i}, W} \cdot {\bar{μ}}_{{\bar{w}}_{i}, W} + \sum_{j = 1}^{n} {\bar{α}}_{w_{j}, \bar{W}} \cdot l_{w_{j}, \bar{W}} \cdot {\bar{μ}}_{w_{j}, \bar{W}}}{\sum_{i = 1}^{m} α_{{\bar{w}}_{i}} l_{{\bar{w}}_{i}} μ_{{\bar{w}}_{i}} + \sum_{i = 1}^{m} α_{{\bar{w}}_{i}, W} l_{{\bar{w}}_{i}, W} μ_{{\bar{w}}_{i}, W}} \end{matrix}

(13)

Fuzzy clustering realization and KM selection

The process goes as follows:

Step 1. Suppose three are $N$ KMs $V_{1}, V_{2}, \dots, V_{N}$ in KM base. Using equation (4), obtain the similarity degree $r_{ij} = sim (V_{i}, V_{j})$ of two KMs as fuzzy clustering data. Construct the fuzzy relational matrix $R = [r_{ij}], i, j = 1, 2, \dots, N$ .

Step 2. Decompose $R$ and make $G = [g_{ij}], i = 1, 2, \dots, N, j = 1, 2, \dots, c$ , to satisfy $R = G \circ G^{T}$ , where “ $\circ$ ” is s-t convolution and $c$ is the class number. Construct the optimization objective function as equation (10).

Step 3. Calculate $G$ by the gradient method. Iteration of $G$ starts from a matrix of random assignment. Set the learning rate $β$ and the number of iterations. Obtain the membership relationships $G$ of KMs by equations (11) and (12).

Step 4. The KM having the highest membership in each class is taken as reference KM.

Step 5. The user’s function requirements are layered by modules and sub-modules to map the target KM $\bar{W}$ . Using equation (13), calculate the similarity degree of target KM and reference KMs to determine from which class to select. Compare the KMs in the specified class with the target KM. The KM having the highest similarity degree is the best KM for user.

Empirical analysis

Suppose that there are 15 KMs $W_{1}, W_{2}, \dots, W_{15}$ in the KM base. Now let us cluster them and select reference KMs for enterprise.

Calculation of similarity degree

As similarity degree calculation needs complete data of each KM, here only that of KMs $W_{1}$ and $W_{2}$ is presented as an example due to the limit of space. Suppose that there are two simplified management information systems of vehicle factory, as shown in Figures 1 and 2. They are, respectively, represented by KMs $W_{1}$ and $W_{2}$ , as shown in Figures 3 and 4. For simplicity, in formal representation of the two KMs, only knowledge point set $P$ , complexity relationship set $R$ and message relationship set $M$ are set, while the finite set $F_{P}$ of all functions on set $P$ , finite set $F_{R}$ of all inherited flows and message flows on set $R$ and finite set $F_{M}$ of all message flows on set $M$ are omitted.

The lowest-layer knowledge point sets of $W_{1}$ and $W_{2}$ are $P_{W_{1}} = b 1, b 2, c 1, c 2, c 3, d 1, d 2$ and $P_{W_{2}} = b 1, b 2, cc 1, c 2, c 3, e 1, e 2$ , respectively. The layer number is 3. Their functions are as follows: b1 → recorded information, monthly and quarterly statistics; b2 → voucher inquiry, voucher entry and voucher delete; c1 → complete status of production and equipment fault; cc1 → complete status of production, equipment fault and raw material scrap; c2 → process analysis, statement of stamping plan and document of stamping process; c3 → parts demand planning, complete status of production and raw material used; d1 → check plan, check card, product quality and check information; d2 → incoming goods inspection, key procedure check and factory inspection; e1 → equipment cards, technical data, affiliated fittings and special equipment; e2 → equipment out, equipment deployment, equipment seal and equipment idle. The perfection degrees of KMs $W_{1}$ and $W_{2}$ are $μ_{W_{1}} = {0.8638, 0.7634, 0.4178, 0.5550, 0.6204, 0.3010, 0.1567}$ and $μ_{W_{2}} = {0.8011, 0.7472, 0.3011, 0.4099, 0.4582, 0.2057, 0.1257}$ , respectively. By equation (5), the similarity degree of $W_{1}$ and $W_{2}$ is

\begin{matrix} sim (W_{1}, W_{2}) & = (2 \cdot 3 \cdot 0.8011 + 2 \cdot 3 \cdot 0.7472 + 2 \cdot 2 \cdot 0.3011 + 2 \cdot 3 \cdot 0.4099 + 2 \cdot 3 \cdot 0.4582) \\ / [(3 \cdot 0.8011 + 3 \cdot 0.7472 + 3 \cdot 0.3011 + 3 \cdot 0.4099 + 3 \cdot 0.4582 + 4 \cdot 0.2057 + 4 \cdot 0.1257) \\ + (3 \cdot 0.8638 + 3 \cdot 0.7634 + 2 \cdot 0.4178 + 3 \cdot 0.5550 + 3 \cdot 0.6204 + 4 \cdot 0.3010 + 3 \cdot 0.1567)] \\ = 0.7858 \end{matrix}

(14)

The functional perfection degrees of 15 KMs are listed in Table 1. And all KMs are normalized into the KMs having the same lowest-layer knowledge points by regarding the missing lowest-layer knowledge points as knowledge points of - perfection degree. Data are optional upon certain rules. The financial management of $W_{1}, W_{2}, W_{3}, W_{8}$ , the production and supply management of $W_{4}, W_{6}, W_{7}, W_{11}$ , the quality, equipment and warehouse management of $W_{9}, W_{10}, W_{13}, W_{14}, W_{15}$ have higher perfection degrees. But all aspects of $W_{5}$ have lower values and those of $W_{12}$ are not prominent either.

Table 1.

Functional perfection degree of KMs.

		W ₁	W ₂	W ₃	W ₄	W ₅	W ₆	W ₇	W ₈	W ₉	W ₁₀	W ₁₁	W ₁₂	W ₁₃	W ₁₄	W ₁₅
Financial management	Enter-in-ledger report	0.9638	0.8011	0.8571	0.2461	0.0669	0.1560	0.4579	0.9093	0.2875	0.1875	0.1697	0.6326	0.0000	0.0000	0.2277
	Credentials management	0.8634	0.8472	0.8887	0.0000	0.0326	0.3413	0.0000	0.8525	0.2875	0.2875	0.1195	0.8784	0.0000	0.2530	0.0000
	Purchase audit	0.8955	0.9638	0.8571	0.2356	0.1123	0.5623	0.0000	0.8784	0.5369	0.2356	0.5241	0.7523	0.0000	0.0000	0.0000
Production management	Production monitoring	0.4178	0.3011	0.2087	0.8995	0.0784	0.8277	0.8963	0.2842	0.2625	0.2625	0.8440	0.6530	0.2940	0.3659	0.2124
	Planning scheduling	0.5550	0.4099	0.0000	0.9440	0.0530	0.8423	0.8963	0.1540	0.2625	0.3625	0.8940	0.7659	0.3989	0.2274	0.3491
	Material management	0.6204	0.4582	0.3091	0.9940	0.0659	0.9124	0.8205	0.1093	0.3776	0.2364	0.8989	0.6274	0.3684	0.1281	0.1349
Quality management	Measure management	0.0000	0.0000	0.1255	0.1989	0.0274	0.1491	0.3697	0.2326	0.8413	0.8495	0.3525	0.2697	0.8427	0.8660	0.8511
	Quality inspection	0.2010	0.0000	0.1251	0.2684	0.0281	0.2349	0.3195	0.1784	0.7277	0.7963	0.2842	0.7195	0.8881	0.8277	0.9444
	Process control	0.1568	0.0000	0.2404	0.2427	0.0660	0.2511	0.1440	0.1530	0.7423	0.8963	0.3540	0.6440	0.9017	0.9365	0.9673
Equipment management	Equipment file	0.0000	0.1057	0.1912	0.2881	0.0277	0.2444	0.1940	0.1659	0.9124	0.8205	0.0000	0.6940	0.8426	0.9319	0.8020
	Technique state	0.0000	0.1057	0.1375	0.1017	0.0365	0.2673	0.1989	0.1274	0.9491	0.9010	0.0000	0.7989	0.8697	0.9285	0.8361
	Pre-repair pre-examine	0.2625	0.2625	0.8440	0.1251	0.2684	0.0281	0.2349	0.3195	0.8989	0.9566	0.0000	0.5674	0.8956	0.9563	0.8467
Supply management	Basic data management	0.2625	0.3625	0.8940	0.9562	0.3697	0.9623	0.8556	0.1784	0.7277	0.1784	0.8987	0.5896	0.2625	0.2625	0.8440
	Material customization	0.3776	0.2364	0.8989	0.8995	0.3195	0.8277	0.9563	0.1530	0.7423	0.1530	0.8823	0.7896	0.2625	0.3625	0.8940
	Quota customization	0.8413	0.8495	0.3525	0.9440	0.1440	0.8423	0.9745	0.1784	0.7277	0.1784	0.9128	0.6895	0.3776	0.2364	0.8989
Warehouse management	Document management	0.7277	0.7963	0.2842	0.1251	0.2684	0.0281	0.2349	0.3195	0.8653	0.9512	0.2625	0.6398	0.8569	0.9746	0.8759
Warehouse management	Production warehouse	0.7423	0.8963	0.3540	0.6124	0.1989	0.0000	0.9124	0.3569	0.8787	0.9356	0.3625	0.8522	0.9658	0.9258	0.8999

To simplify the calculation process of fuzzy relational matrix, suppose that the layer numbers of lowest-layer knowledge point are the same and all matching degrees are 1. And $m \geq n$ . Then equation (5) is simplified as

\begin{matrix} sim (V, W) & = \frac{2 \sum_{i = 1}^{m} {\bar{α}}_{v_{i}, W} \cdot l_{v_{i}, W} \cdot {\bar{μ}}_{v_{i}, W}}{\sum_{i = 1}^{m} α_{v_{i}} l_{v_{i}} μ_{v_{i}} + \sum_{j = 1}^{n} α_{w_{j}} l_{w_{j}} μ_{w_{j}}} \\ = \frac{2 \sum_{i = 1}^{m} {\bar{μ}}_{v_{i}, W}}{\sum_{i = 1}^{m} (μ_{v_{i}} + μ_{w_{i}})} \end{matrix}

(15)

Obtain the fuzzy relational matrix of 15 KMs by equation (15) as shown in Table 2. The lower part of fuzzy relational matrix is given because of symmetry.

Table 2.

Fuzzy relational matrix of KMs.

	W ₁	W ₂	W ₃	W ₄	W ₅	W ₆	W ₇	W ₈	W ₉	W ₁₀	W ₁₁	W ₁₂	W ₁₃	W ₁₄	W ₁₅
W ₁	1.0000
W ₂	0.8880	1.0000
W ₃	0.6820	0.6798	1.0000
W ₄	0.5983	0.5388	0.5793	1.0000
W ₅	0.3898	0.4071	0.4338	0.3601	1.0000
W ₆	0.5920	0.5295	0.6018	0.8522	0.3079	1.0000
W ₇	0.6225	0.5867	0.5775	0.8857	0.3672	0.7714	1.0000
W ₈	0.7393	0.7313	0.7698	0.4319	0.4682	0.4524	0.4552	1.0000
W ₉	0.5786	0.5960	0.6189	0.5803	0.3280	0.5711	0.6117	0.4826	1.0000
W ₁₀	0.4886	0.5123	0.4735	0.4351	0.3182	0.3904	0.4721	0.4884	0.8565	1.0000
W ₁₁	0.6226	0.5352	0.5960	0.8754	0.3679	0.8664	0.8329	0.4589	0.5886	0.4144	1.0000
W ₁₂	0.7376	0.7021	0.6835	0.6678	0.3152	0.6699	0.7011	0.6016	0.8164	0.7004	0.6687	1.0000
W ₁₃	0.4794	0.4998	0.4456	0.4590	0.3195	0.3838	0.5228	0.3993	0.8428	0.9096	0.4265	0.6944	1.0000
W ₁₄	0.4600	0.4598	0.4481	0.4107	0.3309	0.3648	0.4749	0.4294	0.8438	0.9229	0.3917	0.6769	0.9197	1.0000
W ₁₅	0.4868	0.5049	0.5469	0.5918	0.3167	0.5060	0.6464	0.3801	0.8861	0.8290	0.5568	0.7385	0.8641	0.8403	1.0000

Clustering and selection of KMs

The fuzzy relational matrix is decomposed using MATLAB 7.5. The initial value of $G$ is the matrix of random assignment. Let $c = 3$ and $β = 0.07$ . Iterate 500 times. The results are given in Table 3. $W_{4}, W_{6}, W_{7}, W_{11}$ belong to the first class, $W_{9}, W_{10}, W_{13}, W_{14}, W_{15}$ to the second class, and $W_{1}, W_{2}, W_{3}, W_{8}$ to the third class. These KMs have higher membership of ownership class (>0.9). The results show that $c = 3$ is a reasonable number of class. In comparison with Table 1, the KMs of the first class have higher perfection degree in production and supply management, but lower in other aspects. Those of the second have higher perfection degree in quality, equipment and warehouse management. And the third class has higher perfection degree in financial management. The clustering results match the characteristics of Table 1 exactly.

Table 3.

Membership of KMs’ class.

	W ₁	W ₂	W ₃	W ₄	W ₅	W ₆	W ₇	W ₈	W ₉	W ₁₀	W ₁₁	W ₁₂	W ₁₃	W ₁₄	W ₁₅
First class	0.5080	0.4003	0.5252	1.0000	0.2499	0.9798	0.9615	0.2714	0.5228	0.2503	0.9971	0.6490	0.2432	0.2093	0.4533
Second class	0.1635	0.2211	0.1576	0.1794	0.0000	0.0891	0.2985	0.0506	0.9599	0.9867	0.1288	0.6984	0.9983	0.9908	0.9582
Third class	0.9037	0.9183	0.8398	0.2517	0.5712	0.2507	0.2784	0.9532	0.3845	0.3745	0.2811	0.5827	0.3564	0.3479	0.3227

On the other hand, KMs in the KM base are automatically classified in Table 3 without any user participation. The KMs having higher membership in class, such as $W_{4}, W_{8}, W_{10}$ , can be taken as reference KMs. For the high-quality user who can clearly describe specific needs, his target KM is $\bar{W}$ , its layer number being the same as that of existing KMs’ lowest-layer knowledge point. Then, equation (13) is simplified as $sim (\bar{W}, W) = 2 \sum_{i = 1}^{m} {\bar{μ}}_{{\bar{w}}_{i}} / \sum_{i = 1}^{m} (μ_{{\bar{w}}_{i}} + μ_{W ({\bar{w}}_{i})})$ . Suppose that the functional perfection degree of $\bar{W}$ for each function in Table 1 is {0.2000, 0.4000, 0.0000, 0.9000, 0.9500, 0.9500, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.9657, 0.9475, 0.9256, 0.2700, 0.5795}. Then, $sim (\bar{W}, W_{4}) = 0.9453$ , $sim (\bar{W}, W_{8}) = 0.4316$ and $sim (\bar{W}, W_{10}) = 0.4982$ . According to the principle of maximum membership, $\bar{W}$ belongs to the first class. The KMs of the first class include $W_{4}, W_{6}, W_{7}, W_{11}$ . Calculating the similarity degree yields $sim (\bar{W}, W_{4}) = 0.9453$ , $sim (\bar{W}, W_{6}) = 0.8949$ , $sim (\bar{W}, W_{7}) = 0.9020$ and $sim (\bar{W}, W_{11}) = 0.9367$ . Thus, $W_{4}$ is the best KM for user.

Conclusion

KMS realizes the optimal implementation of advanced manufacturing modes by KM. Meanwhile, more and more KMs are stored in the KM base, intensifying the difficulty of selection by user, especially low-level user. To solve this problem, the clustering and selection methods of KMs based on decomposition of fuzzy relational data are proposed.

In terms of the KM representation, there exist certain inevitable overlapping parts among KMs. So in this article, the reference KMs gained by clustering are selected to clarify the relationships of KMs, thus narrowing the scope of user’s choice. For typical reference KMs, the similarity model between KMs plays an essential role. Starting from the representation of KM, the matching degree is a quantitative comparison of KM, while the perfection degree is a performance comparison. The layer number is an influence factor of KM’s structure. Based on the above, the similarity function is constructed. As proven by a number of tests, similarity degree varies with the changes in self-reconfiguration operations and can well reflect the KM’s characteristics.

The similarity values between KMs as clustering data convert the original clustering space into that constituted by similarity and overcome the sparse distribution of high-dimensional sample sets. The decomposition of fuzzy relational matrix yields a KM–class relationship enabling user to select KMs in the class rather than to choose from all KMs. The KMs having higher membership are taken as reference KMs for low-level users. Meanwhile, they narrow the scope of selection for high-level users. It is a new idea that KMs are selected by mining the relationships among them. The reference KMs do help user identify his demands for proper choice.

Although the simulation has verified the effectiveness of the proposed method, further exploration into the acquisition of perfection and matching degrees is a must, and that makes the focus of our future study. In addition, for other possible fuzzy clustering methods, classification number and validity of clustering are worthy to be studied as well.

Footnotes

Acknowledgements

The authors thank the two anonymous referees and Professor Li Lu for their valuable comments and suggestions.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by a key program of National Natural Science Foundation of China under Grant No. 60934008 and Postdoctoral Foundation of Jiangsu Province under Grant No. 1107010145.

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