A systematic top–down approach for the identification and decomposition of product key characteristics

Abstract

Methodologies for the identification of key characteristics have been widely applied in quality management through the selection of critical dimensions and the measurement of variations. However, methods for both the identification and decomposition of key characteristics have not yet been developed, and more research is still required. In answering this need, a systematic top–down decomposition approach for the identification of key characteristics is proposed. The methodology of the identification and decomposition of key characteristics can be divided into two steps: first is the construction of candidate characteristics, and second is the identification of key characteristics. These steps are based on precise mathematical definitions. Initially, the necessary information for the construction of the candidate characteristics is obtained from analysis based on assembly-oriented graph, and that information is then conveyed utilizing feature adjacency matrix. A concept for a propagation chain is then proposed, and a search algorithm for an auto-generating propagation chain is obtained through feature adjacency matrix. The degree of influence of the candidate characteristics on the key characteristics is then defined. A formula is derived that can calculate the degree of influence by utilizing a variation model. Finally, the process of the identification of the key characteristics is achieved according to the relative degrees of influence. An aircraft boarding gate is presented in order to validate the proposed methodology. Two key characteristics of the aircraft boarding gate are identified, and the results indicate the methodology’s feasibility.

Introduction

The ever increasing levels of industrial competition are forcing enterprises to satisfy the demands for improved quality and more competitive costs for their products. To enhance the quality while reducing the costs of their products to the greatest extent, key characteristics (KCs) or tolerances of their products must be selected for rigorous monitoring and control. However, for complex products such as automobiles and aircraft, identifying these KCs from thousands of different parts is an extremely difficult task. In addressing this problem, the methodologies for the identification of KCs have been of great significance in both academia and industry during recent decades. ¹

KCs include products, subassemblies, parts, and process features that, when varying from their nominal values, significantly impact either one or more of the final cost, the performance, or the safety of a product. ² KCs can also be divided into product key characteristic (PKCs) and process KCs, depending on whether they are relevant to a product or a process.^1,3 This article’s research has focused on the identification of PKCs, which can help designers to narrow their focus to a smaller range of characteristics having the most significant influence on the product’s function. This is an effective approach that not only improves product quality and optimizes enterprise resources but also may be the basis for further applications of KCs, such as tolerance analysis and tolerance synthesis. Past research on the identification and decomposition of KCs has focused chiefly on either delivery of KCs or prioritization of KCs.

In the delivery of KCs, Mantripragada and Whitney ⁴ and Whitney ⁵ defined KCs as assembly-level dimensions related by design intention, and the delivery of those KCs was through a datum flow chain (DFC). A DFC is a type of directed acyclic graph that incorporates heterogeneous product and assembly process information such as functional requirements, shape characteristics, constraints, and tolerances. The DFC could formulate the KCs’ delivery through the processes of product decomposition and assembly. However, the method did not offer a mathematical definition for the DFC, so it was only suitable for the analysis at a logic level. Mathieu and Marguet ⁶ adopted the assembly-oriented graph (AOG) methodology to formulate an assembly model. Through analysis of the AOG, a propagation chain (PC), which was an error accumulation route of the KCs, was developed. This method was actually an extension of the DFC methodology. Dantan et al. ⁷ analyzed the delivery relationship between PKCs and process KCs, and offered an information model to convey this relationship. This method was a type of qualitative method based on engineering knowledge. In the above-mentioned research of the delivery of KCs, factors influencing KCs could be obtained, thus providing the basis for the identification of KCs. However, these types of methods have shortcomings. First, the lack of strict mathematical descriptions hinders their applications. Second, few methods are developed to further identify KCs from the factors.

To prioritize KCs, Thornton, ² Lee and Thornton, ³ and Thornton ⁸ proposed a mathematical model to determine the relative importance of the KCs, which provides helpful guidance in the identification of KCs quantitatively. However, the goal of that article was to analyze how to reduce the variation of KCs, and it did not discuss how to obtain the KCs. Chakhunashvili et al. ⁹ and Johansson et al. ¹⁰ described a statistic-based engineering method utilizing variation mode and effect analysis (VMEA) in which the influencing factors of PKCs were first obtained, and then the importance of the influencing factors was determined by calculating risk priorities. Key component identification methods from literature^11,12,15 that have been applied in industry were similar to the VMEA methodology. The aforementioned methodologies introduced quantitative methods for prioritizing KCs. The process of identifying and decomposing KCs may be more systematic and accurate based on the advantages utilized by these methods.

A more systematic methodology of identifying KCs is still needed due to the limitations of the methods developed by the previously mentioned researchers. The solution presented in this article is a top–down approach for identifying and decomposing KCs systematically. Section “Preliminary background and comprehensive conceptual solution” presents preliminary background and this article’s comprehensive conceptual solution, in two major steps, while sections “The construction of CCs” and “The identification of KCs” describe in detail the processes of these two steps. Section “Case study” presents an industrial example to illustrate the feasibility of the proposed methods, and section “Conclusion” presents the final summary.

Preliminary background and comprehensive conceptual solution

KCs that have maximum influence on performance should be determined according to customer requirements or production requirements, during the initial stages of product design. These KCs are commonly called top-level KCs. A product composed of many parts is usually divided into several units according to predetermined rules. The top-level KCs will be decomposed into sub-level KCs during the process of product decomposition and will eventually form a hierarchical tree. An illustrated diagram of the KCs’ structural tree is shown in Figure 1, where $K{C}_{i,j}$ represents the jth KC in the ith level. KCs in Figure 1 can be categorized into three types. The first type is the customer requirements, which is at the top level, such as $K{C}_{0,1}$ and $K{C}_{0,2}$ . It is assumed that the KCs at the top level are known. The aim of this article is to complete the decomposition of these known KCs. The second type of KCs are those at the assembly level, such as $K{C}_{1,1}$ , which can be further decomposed into the third type of KCs at the parts level, such as $K{C}_{n,1}$ , which cannot be decomposed. As shown above, the identification and decomposition of KCs are in fact a systematic process. In this article, we have divided the process into two main steps. The first step is the construction of candidate characteristics (CCs) and the second step is the identification of KCs.

Figure 1.

A product’s KCs’ structural tree.

CCs refer to the influencing factors of the KCs to be decomposed, which actually determines the extent of the identification of KCs. Let $y$ be a KC to be decomposed, and let $X$ be the set of CCs for this KC. Then, the construction of CCs can be expressed as follows

$$y\stackrel{\mathrm{Influenced}\mathrm{by}}{\to }X=\{\begin{array}{cccc}\hfill x_0,\hfill & \hfill x_1,\hfill & \hfill \begin{array}{cc}\hfill \cdots ,\hfill & \hfill x_k\hfill \end{array},\cdots ,\hfill & \hfill x_p\hfill \end{array}\}$$

(1)

where $x_k(k=1,2,\cdots ,p)$ is the kth CC in $X$ and $p$ is the number of CCs.

The identification of KCs refers to the process of identifying several KCs from the CCs. Let $Y$ be the set of KCs in the new level. Then, the identification of KCs can be expressed using equation (2)

$$\begin{array}{c}X\stackrel{\mathrm{Identify}}{\to }Y=\{\begin{array}{cccc}\hfill y_0,\hfill & \hfill y_1,\hfill & \hfill \begin{array}{cc}\hfill \cdots ,\hfill & \hfill y_k\hfill \end{array},\cdots ,\hfill & \hfill y_q\hfill \end{array}\},\hfill \\ \text{where}Y\subseteq X\hfill \end{array}$$

(2)

where $y_k(k=1,2,\cdots ,q)$ denotes the kth KCs and $q$ is the number of KCs. If the KCs in equation (2) can be decomposed, then repeat steps 1 and 2 until the KCs’ structural tree is generated.

The construction of CCs

As expressed by equation (1), CCs should first be generated for the identification of KCs. In this section, two types of information for the construction of CCs are extracted based on AOG. A feature adjacency matrix (FAM) is then used to present the information, and finally the CCs are generated by an automatic procedure.

FAM

A complex product is composed of many parts through specific assembly sequences to satisfy or achieve KCs. The assembly model for the product contains the dimensions or tolerances which have influence on KCs. Thus, the essential information for the identification and decomposition of KCs can be extracted from the assembly model. An AOG ⁶ is a graph that conveys assembly information utilizing circles, arcs, and dotted lines. It is an intuitive tool for conveying geometric information, assembly relationships, dimensions, and assembly sequence. The essential information for the construction of CCs will be obtained more efficiently from an analysis of the AOG.

A simple assembly and its top-level KC (the distance between features $f_8$ and $f_{10}$ ) are illustrated in Figure 2(a). The goal of decomposition of this KC is to find sub-level KCs, which are in Parts 1–4. Four parts of the assembly and some of their features, illustrated in Figure 2(b), are used to construct the AOG illustrated in Figure 2(c). The red curve designated as a PC, which will be discussed in section “PC,” shows that the KCs are satisfied by an accumulation of parts. The accumulation, or propagation, is divided into two procedures, which are the propagation inside a part and the propagation between parts. As shown in Figure 2(c), $d_{1(2)}$ is the dimension between $f_1$ and $f_2$ , $d_{2(3)}$ is the dimension between $f_2$ and $f_3$ , $m_{3(9)}$ is the mate between $f_3$ and $f_9$ , and $d_{9(10)}$ is the dimension between $f_9$ and $f_{10}$ . When $d_{1(2)}$ and $d_{2(3)}$ contain some variations, the related three features ( $f_1$ , $f_2$ , and $f_3$ ) will deviate from their nominal positions. Similarly, the variations in $d_{9(10)}$ will also cause the related features ( $f_9$ and $f_{10}$ ) to deviate from their nominal positions. However, these variations are not independent. The variations in Part 1 will propagate to Part 3 through the mate, $m_{3(9)}$ . Thus, it can be seen that $f_{10}$ is influenced by $d_{1(2)}$ , $d_{2(3)}$ and $d_{9(10)}$ . Utilizing the same analytic procedure, it can be concluded that $f_8$ is influenced by $d_{4(5)}$ , $d_{5(6)}$ , and $d_{7(8)}$ . Dimensions, including $d_{1(2)}$ , $d_{2(3)}$ , $d_{9(10)}$ , $d_{4(5)}$ , $d_{5(6)}$ , and $d_{7(8)}$ are CCs. Thus, for a complex product, two aspects of information are needed for the construction of CCs. One aspect is the dimension between the features, and the other is the mate between the parts, which is also represented by features. In this article, features are considered as the basic information unit in an assembly model. Based on graph theory, the two aspects of information can be expressed as follows

$$G=(F,R)$$

(3)

$$F=\{f_1,f_2,\cdots ,f_{n_1}\}$$

(4)

$$R=\{r_1,r_2,\cdots ,r_{n_2}\}$$

(5)

$$r_j=\{f_i,f_k,d_{i(k)}\}\mathrm{or}{r}_j=\{f_i,f_k,m_{i(k)}\}$$

(6)

In these equations, $F=\{f_1,f_2,\cdots ,f_{n_1}\}$ is a set of features and $n_1$ is the number of features; $R=\{r_1,r_2,\cdots ,r_{n_2}\}$ is a set of relationships between features and $n_2$ is the number of relationships; and $r_j$ is the jth relationship which has two possible values. When $r_j=\{f_i,f_k,d_{i(k)}\}$ , then there exists a dimension $d_{i(k)}$ between $f_i$ and $f_k$ ; if $r_j=\{f_i,f_k,m_{i(k)}\}$ , then there exists a mate $m_{i(k)}$ between $f_i$ and $f_k$ . Let $i=1,2,\cdots ,n_1$ ; $k=1,2,\cdots ,n_1$ ; and $j=1,2,\cdots ,n_2$ . The information from equation (3) can be represented by the data structure FAM in equation (7)

$$A={(a_{i(j)})}_{n_1\times n_1}=\begin{array}{cc}\begin{array}{cccc}{f}_1& f_2& \cdots & f_{n1}\end{array}& \\ \left[\begin{array}{cccc}{a}_{1(1)}& a_{1(2)}& \cdots & a_{1(n_1)}\\ a_{2(1)}& a_{2(1)}& \cdots & a_{2(n_1)}\\ ⋮& ⋮& ⋮& ⋮\\ a_{n_1(1)}& a_{n_1(2)}& \cdots & a_{n_1(n_1)}\end{array}\right]& \begin{array}{c}{f}_1\\ f_2\\ ⋮\\ f_{n_1}\end{array}\end{array}$$

(7)

where $A$ is a matrix with the size $n_1\times n_1$ ; $a_{i(j)}$ ( $i=1,2,\cdots ,n_1$ , $j=1,2,\cdots ,n_1$ ) is the element of $A$ denoting the relationship between $f_i$ and $f_j$ . If $a_{i(j)}=d_{i(j)}$ , then there exists a dimension $d_{i(j)}$ between $f_i$ and $f_j$ . If $a_{i(j)}=m_{i(j)}$ , then there exists a mate $m_{i(j)}$ between $f_i$ and $f_j$ . If $a_{i(j)}=0$ , then there is no relationship between $f_i$ and $f_j$ . FAM provides the necessary information for the construction of CCs. In the next subsection, the search algorithm for the construction of CCs will be presented based on FAM.

Figure 2.

An assembly of a product and its AOG: (a) a simple assembly, (b) parts and some of their features, and (c) AOG of the assembly.

PC

As shown in Figure 2(c), a PC is a chain of features, which reflects the routes of variations propagating to a KC. According to section “Preliminary background and comprehensive conceptual solution,” dimensions influencing KCs are actually CCs. Therefore, CCs will be generated with the construction of PC. The mathematical expression of a PC is defined as follows

$$G_L=(F_L,R_L)$$

(8)

where $F_L(F_L\subseteq F)$ is a set of features and $R_L(R_L\subseteq R)$ is a set of relationships. Obviously, the PC is a subset of equation (3).

A procedure for generating a PC is shown in Figure 3. It is divided into three major steps.

Figure 3.

Procedure for automatically generating a PC.

Key step 1

Read FAM and determine start feature $f_s$ and end feature $f_e$ according to the KCs to be decomposed, such as $f_8$ and $f_{10}$ in Figure 2.

Key step 2

Let $f$ equals $f_s$ , then search $f_x$ that has a dimensional or mating relationship with $f$ . If $f_x$ equals $f_e$ , go to key step 3. If not, update $f$ and repeat key step 2.

Key step 3

Form a PC according to the results of key step 1 and key step 2. Then, CCs will be obtained through a combination of PC and FAM.

The identification of KCs

The goal of the identification of KCs is to find sub-KCs in the CCs. Without loss of generality, let $Y={[\begin{array}{cccc}\hfill y_1\hfill & \hfill y_2\hfill & \hfill \cdots \hfill & \hfill y_m\hfill \end{array}]}^T$ ( $m$ is the number of KCs) be the KCs in a specific level that is to be decomposed, in Figure 1. According to the method proposed in section “The construction of CCs,” each element of $Y$ has its own CCs. Let $X_k={[\begin{array}{cccc}\hfill x_1^k\hfill & \hfill x_2^k\hfill & \hfill \cdots \hfill & \hfill x_{n_k}^k\hfill \end{array}]}^T$ be the CCs of $y_k$ , where $n_k$ is the number of CCs in $X_k$ , $k=1,2,\cdots ,m$ . For a complex product, a CC of a part or component may influence different KCs. With this basis, all the CCs are regarded as a set $X=\bigcup _{k=1}^m{X}_k={[\begin{array}{cccc}\hfill x_1\hfill & \hfill x_2\hfill & \hfill \cdots \hfill & \hfill x_n\hfill \end{array}]}^T$ ( $n$ is the number of the CCs). $Y$ and $X$ correspond to the KC and CC levels, as shown in Figure 4, respectively. It is important to note that not all CCs are relevant to all KCs. The relationships between KCs and CCs are represented by equation (14), which will be discussed later.

Figure 4.

Hierarchical relationship between KCs and CCs.

The information shown in Figure 4 actually provides an object analysis for the identification of KCs, which consists of three steps. First, the degree of influence of CCs on KCs is defined based on the Taguchi loss function. ¹³ A variation model is then presented to derive the degree of influence for a specific CC. Finally, the identification of KCs is accomplished according to the degree of influence.

Analysis of the degrees of influence of a CC

Any variations of KCs may have a significant influence on the quality of the product. The Taguchi loss function can quantitatively describe the influence. As illustrated in Figure 4, when $y_i$ deviates from its targeted value, the quality loss caused by the deviation can be expressed by equation (9) ¹³

$$L_i=k_i({\sigma }_i^2+{\delta }_i^2)$$

(9)

where $L_i$ represents the loss of $y_i$ ; $k_i=B_i/{(\mathrm{TU}-\mathrm{TL}/2)}^2$ is the proportionality constant, $B_i$ is the failure loss when $y_i$ exceeds the tolerance, and $\mathrm{TU}$ and $\mathrm{TL}$ are the upper and lower tolerances, respectively; ${\sigma }_i^2$ is the standard deviation (SD); ${\delta }_i={\mu }_i-{\tau }_i$ is the mean shift; and ${\mu }_i$ and ${\tau }_i$ are the expectation value and targeted value, respectively.

The variations of CCs will propagate to KCs during the assembly process. To analyze the influence of $x_i$ on $y_i$ in Figure 4, the loss sensitivities of $x_j$ on $y_i$ are defined as follows

$$\frac{\partial L_i}{\partial {\sigma }_j}\text{and}\frac{\partial L_i}{\partial {\delta }_j}$$

(10)

In actual production, the SD ${\sigma }_j$ and the mean shift ${\delta }_j$ may be different from their target values, under the influences of multiple process parameters. Assume ${\sigma }_j$ changes from ${\sigma }_j$ to $\sigma {\prime }_j$ , and ${\delta }_j$ changes from ${\delta }_j$ to $\delta {\prime }_j$ ( $\sigma {\prime }_j$ and $\delta {\prime }_j$ can be obtained from the process capability data or from industry experience). The quality loss changes of $y_i$ caused by this situation can then be presented as follows

$$\Delta L_{\mathrm{ij}}=\underset{{\sigma }_j}{\overset{\sigma {\prime }_j}{\int }}\frac{\partial L_i}{\partial {\sigma }_j}d{\sigma }_j+\underset{{\delta }_j}{\overset{\delta {\prime }_j}{\int }}\frac{\partial L_i}{\partial {\delta }_j}d{\mu }_j$$

(11)

where $\Delta L_{\mathrm{ij}}$ is the degree of influence of $x_j$ on $y_i$ . Equation (11) establishes a relationship between the CC variations and the KCs’ loss of quality. The relative importance of CCs can be determined by the degrees of influence. For simplicity, new KCs are identified according to the relative importance of the CCs.

Variation model

In this section, the relationship between the CCs’ variations and the KCs’ variations will be analyzed to calculate the degrees of influence of the CCs. As shown in Figure 4, equation (12), the relationship between the deviations in KCs to the deviations in CCs is represented as a linear equation by using Taylor expansions. It should be noted that $\Delta x_i$ is typically very small, and ${(\Delta x_i)}^2$ is not considered in equation (12)

$$\Delta y_i=\frac{\partial y_i}{\partial x_1}\Delta x_1+\frac{\partial y_i}{\partial x_2}\Delta x_2+\cdots +\frac{\partial y_i}{\partial x_n}\Delta x_n$$

(12)

where $\partial y_i/\partial x_k$ is the variation transfer coefficient, which quantifies the sensitivity of $y_i(i=1,2,\cdots ,m)$ to $x_k(k=1,2,\cdots ,n)$ . In general, the variation transfer coefficient is determined by a product’s structure. Based on equation (12), the relationship between $\Delta Y$ and $\Delta X$ can be derived

$$\Delta Y=\phi \Delta X$$

(13)

where $\Delta Y={\left[\begin{array}{cccc}\hfill \Delta y_1\hfill & \hfill \Delta y_2\hfill & \hfill \cdots \hfill & \hfill \Delta y_m\hfill \end{array}\right]}^T$ and $\Delta X={[\begin{array}{cccc}\hfill \Delta x_1\hfill & \hfill \Delta x_2\hfill & \hfill \cdots \hfill & \hfill \Delta x_n\hfill \end{array}]}^T$ ; $\phi$ is the variation transfer coefficient matrix as expressed in equation (14)

$$\phi =\left[\begin{array}{cccc}\hfill \frac{\partial y_1}{\partial x_1}\hfill & \hfill \frac{\partial y_1}{\partial x_2}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial y_1}{\partial x_n}\hfill \\ \multicolumn{1}{c}{\frac{\partial y_2}{\partial x_1}}& \hfill \frac{\partial y_2}{\partial x_2}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial y_2}{\partial x_n}\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill \cdots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{\frac{\partial y_m}{\partial x_1}}& \hfill \frac{\partial y_m}{\partial x_2}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial y_m}{\partial x_n}\hfill \end{array}\right]$$

(14)

It is assumed that all CCs adhere to normal distribution and are mutually independent. The statistical relationship between KCs and CCs can then be expressed as equations (15) and (16)

$${\sigma }_i^2={\left(\frac{\partial y_i}{\partial x_1}\right)}^2{\sigma }_1^2+{\left(\frac{\partial y_i}{\partial x_2}\right)}^2{\sigma }_2^2+\cdots +{\left(\frac{\partial y_i}{\partial x_n}\right)}^2{\sigma }_n^2$$

(15)

$${\delta }_i=\frac{\partial y_i}{\partial x_1}{\delta }_1+\frac{\partial y_i}{\partial x_2}{\delta }_2+\cdots +\frac{\partial y_i}{\partial x_n}{\delta }_n$$

(16)

Define $\lambda$ by squaring equation (14)

$$\lambda =\left[\begin{array}{cccc}\hfill {\left(\frac{\partial y_1}{\partial x_1}\right)}^2\hfill & \hfill {\left(\frac{\partial y_1}{\partial x_2}\right)}^2\hfill & \hfill \cdots \hfill & \hfill {\left(\frac{\partial y_1}{\partial x_n}\right)}^2\hfill \\ \multicolumn{1}{c}{{\left(\frac{\partial y_2}{\partial x_1}\right)}^2}& \hfill {\left(\frac{\partial y_2}{\partial x_2}\right)}^2\hfill & \hfill \cdots \hfill & \hfill {\left(\frac{\partial y_2}{\partial x_n}\right)}^2\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill \cdots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{{\left(\frac{\partial y_m}{\partial x_1}\right)}^2}& \hfill {\left(\frac{\partial y_m}{\partial x_2}\right)}^2\hfill & \hfill \cdots \hfill & \hfill {\left(\frac{\partial y_m}{\partial x_n}\right)}^2\hfill \end{array}\right]$$

(17)

Then, combining equations (13)–(17), the variations between KCs and CCs can be expressed as equation (18)

$${\sigma }_i^2=\sum _{j=1}^n{\lambda }_{\mathrm{ij}}{\sigma }_j^2,{\delta }_i=\sum _{j=1}^n{\phi }_{\mathrm{ij}}{\delta }_j$$

(18)

where ${\sigma }_i^2$ and ${\delta }_i(i=1,2,\cdots ,m)$ denote the SD and the mean shift of $y_i$ , respectively; ${\sigma }_j^2$ and ${\delta }_j(j=1,2,\cdots ,n)$ denote the SD and the mean shift of $x_j$ , respectively; and ${\lambda }_{\mathrm{ij}}={\phi }_{\mathrm{ij}}^2$ . Finally, combining with equations (9)–(11) and (18), the degree of influence of $x_j$ on $y_i$ can be derived as

$$\Delta L_{\mathrm{ij}}=k_i({\lambda }_{\mathrm{ij}}(\sigma {\prime }_j^2-{\sigma }_j^2)+{\phi }_{\mathrm{ij}}^2(\delta {\prime }_j^2-{\delta }_j^2))$$

(19)

The total degrees of influence of $x_j$ on all the KCs is derived as follows

$$\Delta L_j=\sum _{i=1}^m{k}_i({\lambda }_{\mathrm{ij}}({\sigma \prime }_j^2-{\sigma }_j^2)+{\phi }_{\mathrm{ij}}^2({\delta \prime }_j^2-{\delta }_j^2))$$

(20)

In this article, new KCs are selected by prioritizing the degrees of influence. The number of new KCs can be determined according to the actual situations. Generally, two or three will be selected for simple parts. More KCs are likely to be selected for complex parts. In the application, it is difficult to get an accurate value for the proportionality constant ( $k_i$ ) in equation (20). A standardized proportionality constant ( $k_i$ ) is proposed as the basis of the KCs’ identification, based on the relative size of the CCs’ degrees of influence on the KCs ¹⁴

$$k_i=\frac{4}{{(T{U}_i-T{L}_i)}^2}$$

(21)

The degree of influence after standardization is represented as follows

$$\Delta L_j=\sum _{i=1}^m\frac{4}{{(T{U}_i-T{L}_i)}^2}({\lambda }_{\mathrm{ij}}({\sigma \prime }_j^2-{\sigma }_j^2)+{{\phi }^2}_{\mathrm{ij}}({\delta \prime }_j^2-{\delta }_j^2))$$

(22)

Case study

In this section, an aircraft boarding gate will be used to verify the feasibility of the proposed method. The example is introduced first and the FAM is generated second, to represent dimensions and mattings of the boarding gate. Third, the CCs are obtained based on the FAM, and finally, the degrees of influence of the CCs are derived and analyzed.

Illustration of the case study

As shown in Figure 5, the contour requirements of the boarding gate are ensured by several measuring points on the skin, such as the red points, $M{P}_1$ – $M{P}_5$ . The distances between the measuring points and the point, $O$ , are defined as top-level KCs, such as $K{C}_{0,1}$ – $K{C}_{0,5}$ . The boarding gate contacts the fuselage at the yellow point in Figure 5. These contact points are distributed near the measuring points, such as $C{P}_1$ – $C{P}_5$ . The structure of the contact point is shown in the top left of Figure 5. Considering the similarity of $K{C}_{0,1}-K{C}_{0,5}$ is selected to illustrate the proposed method.

Figure 5.

An aircraft boarding gate and its KCs.

The construction of CCs

Two types of information are needed for the construction of CCs according to section “The construction of CCs”; one is the dimension, and the other is the mate. The procedure for obtaining the two types of information is divided into two steps. In the first step, the features of the parts are obtained. Five parts and their features are obtained, as illustrated in Figure 6. Part 2 is taken as an example with its five major features, represented as $f_{11}$ , $f_{12}$ , $f_{13}$ , $f_{14}$ , and $f_{15}$ . Second, the dimensions and mates are obtained through a combination of features. For example, $f_{12}$ and $f_{14}$ are used to represent the width of Part 2, and $f_{14}$ and $f_{18}$ are used to represent the mating between Parts 1 and 2.

Figure 6.

Related parts and features.

As shown in Table 1, the Relationship column is used to represent the dimensions and mates. For example, the dimension between $f_1$ and $f_3$ is 10 cm. The mate between $f_5$ and $f_{12}$ is M1. It should be noted that $f_{23}$ is a design datum of the boarding gate, and some dimensions are presented from this feature. However, $f_{23}$ does not belong to any part, so it is not illustrated in Figure 6.

Table 1.

Dimensions and mates of the boarding gate.

Feature	Feature	Relationship	Feature	Feature	Relationship
f 1	f 3	10	f 11	f 14	91.5
f2	f 4	34.5	f 13	f 14	3
f 3	f 5	0	f 12	f 14	91.5
f 4	f 6	5	f 16	f 17	80
f 3	f 10	18.7	f 10	f 16	M4
f 3	f 9	18	f 18	f 19	2.5
f 3	f 7	15	f 14	f 18	M5
f 3	f 8	20	f 20	f 21	9
f 5	f 12	M1	f 17	f 18	M6
f 4	f 15	M2	f 1	f 20	M7
f 2	f 22	M3	f 23	f 11	1267.7
f 23	f 16	1286.3	f 23	f 21	1286.7
f 23	f 19	1327.4

The FAM for the boarding gate is generated by applying equation (7) to Table 1

$$A=\begin{array}{cc}\begin{array}{cccccccccc}{f}_1& f_2& f_3& \cdots & f_{15}& f_{16}& \cdots & f_{21}& f_{22}& f_{23}\end{array}& \\ \left[\begin{array}{cccccccccc}0& 0& 10& \cdots & 0& 0& \cdots & 0& 0& 0\\ 0& 0& 0& \cdots & 0& 0& \cdots & 0& M3& 0\\ 10& 0& 0& \cdots & 0& 0& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& \ddots & 0& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0& \cdots & 0& 0& 0\\ 0& 0& 0& \cdots & 0& 0& \cdots & 0& 0& 1286.3\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0& \cdots & 0& 0& 1286.7\\ 0& M3& 0& \cdots & 0& 0& \cdots & 0& 0& 0\\ 0& 0& 0& \cdots & 0& 1286.3& \cdots & 1286.7& 0& 0\end{array}\right]& \begin{array}{c}{f}_1\\ f_2\\ f_3\\ ⋮\\ f_{15}\\ f_{16}\\ ⋮\\ f_{21}\\ f_{22}\\ f_{23}\end{array}\end{array}$$

(23)

Finally, let $f_{23}$ and $f_{19}$ be the start and end features, respectively. A PC is generated by the algorithm presented in Figure 3. The PC is shown as follows

$$f_{23}\to f_{21}\to f_{20}\to f_1\to f_3\to f_5\to f_{12}\to f_{14}\to f_{18}\to f_{19}$$

(24)

$f_{21}\to f_{20}\to f_1\to f_3$ is simplified to: $f_{21}\to f_3$ . The relationships between the features in the PC are expressed as follows: $r_1=\{\begin{array}{ccc}\hfill f_{23},\hfill & \hfill f_{21},\hfill & \hfill 1,286.7\hfill \end{array}\}$ , $r_2=\{\begin{array}{ccc}\hfill f_{21},\hfill & \hfill f_3,\hfill & \hfill 19\hfill \end{array}\}$ , $r_3=\{\begin{array}{ccc}\hfill f_3,\hfill & \hfill f_5,\hfill & \hfill 0\hfill \end{array}\}$ , $r_4=\{\begin{array}{ccc}\hfill f_5,\hfill & \hfill f_{12},\hfill & \hfill M1\hfill \end{array}\}$ , $r_5=\{\begin{array}{ccc}\hfill f_{12},\hfill & \hfill f_{14},\hfill & \hfill 91.5\hfill \end{array}\}$ , $r_6=\{\begin{array}{ccc}\hfill f_{14},\hfill & \hfill f_{18},\hfill & \hfill M5\hfill \end{array}\}$ , $r_7=\{\begin{array}{ccc}\hfill f_{18},\hfill & \hfill f_{19},\hfill & \hfill 2.5\hfill \end{array}\}$ . Therefore, $r_1$ , $r_2$ , $r_5$ , and $r_7$ are the CCs, which are expressed as $l_{1,1}$ , $l_{1,2}$ , $l_{1,3}$ , and $l_{1,4}$ , as illustrated in Figure 7.

Figure 7.

CCs of $L_{0,2}$ .

The identification of KCs

Figure 7 shows a dimensional view of the boarding gate. The KC is indicated by $L_{0,2}$ , which is affected by a combination of $l_{1,1}$ , $l_{1,2}$ , $l_{1,3}$ , and $l_{1,4}$ . It also shows the quantitative relationship between the KC and the CCs represented by equation (25) as

$$L_{0,2}=l_{1,1}\cos {\alpha }_1-l_{1,2}\cos {\alpha }_2+l_{1,3}+l_{1,4}$$

(25)

The $\phi$ matrix is derived by applying equation (14) to equation (25)

$$\phi ={[\begin{array}{cccc}\hfill \cos {\alpha }_1\hfill & \hfill -\cos {\alpha }_2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}]}^T$$

(26)

Squaring the matrix elements in $\phi$ yields the $\lambda$ matrix

$$\lambda ={\left[\begin{array}{cccc}\hfill {(\cos {\alpha }_1)}^2\hfill & \hfill {(-\cos {\alpha }_2)}^2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]}^T$$

(27)

Table 2 contains the illustrative sample values used for the analysis. Nominal values of KC and CCs are presented as follows: $L_{0,2}=1,327.4$ , $l_{1,1}=1,286.7$ , $l_{1,2}=19$ , $l_{1,3}=91.5$ , $l_{1,4}=2.5$ , ${\alpha }_1=15°$ , and ${\alpha }_2=60°$ . The tolerances for $L_{0,2}$ are $\pm 1$ . Application of these data to equation (22), the degrees of influence of the CCs is derived, as shown in the column titled Degrees of influence in Table 2.

Table 2.

Parameters and results.

CC	Target		Actual		Degrees of influence/ ${10}^{-2}$	Rank
	${\sigma }_j$	${\delta }_j$	$\sigma {\prime }_j$	$\delta {\prime }_j$
$l_{1,1}$	0.20	0	0.25	0.01	2.11	1
$l_{1,2}$	0.04	0	0.07	0.004	0.08	4
$l_{1,3}$	0.05	0	0.10	0.04	0.91	2
$l_{1,4}$	0.04	0	0.06	0.006	0.21	3

CC: candidate characteristic.

Table 2 shows that $l_{1,1}$ has the highest degree of influence, which is distinguished from the others because $l_{1,1}$ is the most critical CC to $L_{0,2}$ , and its initial variation is large. The most attention should thus be allocated to $l_{1,1}$ , and it should be controlled by a large proportion of the company’s available resources. Table 2 shows that $l_{1,2}$ has the lowest degree of influence and does not significantly contribute to $L_{0,2}$ , thus the least attention should be allocated to it. CCs, $l_{1,3}$ and $l_{1,4}$ , have medium degrees of influence, thus in actual production, they should only be monitored occasionally. When their variations become unstable, they should be controlled. $l_{1,1}$ may be the sub-KC, and thus it should be further decomposed. However, as it is treated similarly to the aforementioned processes, so the process will not be repeated here.

Conclusion

This article puts forward a type of top–down approach for the identification and decomposition of KCs. The feasibility of the proposed method is verified by an industrial example. Utilizing this method, the identification and decomposition of the KCs are divided into two steps: the construction of CCs and the identification of KCs. In the first step, the CCs are obtained, which are the basis of the identification of KCs. In the second step, a formula is derived to calculate the degree of influence of the CCs. New KCs are identified according to the relative size of their degrees of influence. Both these steps are based on strict mathematical definitions.

The proposed methodology in this article has good applications in industry. However, there still exist some shortcomings. First, the proposed method is based on an assembly model and does not consider any process factors from actual production. Therefore, this method can only be applied to the identification of PKCs. Second, the identification of the KCs is based on the relative size of the degrees of influence. This could lead to multiple selections or the omission of a KC. Therefore, subsequent research should focus on the integration of the proposed methodology with current computer-aided design/computer-aided manufacturing (CAD/CAM) systems and the identification of process KCs.