## 3 Product Gene Theory
### Product Gene Model for Dynamic Quality Control  
Biologically, a gene is a segment of DNA with specific genetic information. DNA consists of four nucleotides which include A, T, C, and G [34]. The sequence of these nucleotides determines the features of an organism. Therefore, a gene is also a group of specific nucleotide sequences which determines a set of specific characteristics [29]. Besides, genes can transmit genetic information to the next generation [5]. Similar to an organism, a processing surface is a basic unit to form its one product or one component. Correspondingly, a product gene unit describes the genetic information of a special surface which determines a set of features, the gene of one product or one part which consists of a set of gene units. Corresponding to biological genes, the relationship between biological gene and product gene for dynamic quality control of large equipment is shown in Figure 2.  
In addition, the gene of one product or one part is multi-level and interrelated, it can be explained by the descriptions as shown in Figures 3 and 4. GUq in Figure 4 represents gene unit \(q\). Different lines represent different relationships among the gene units, which include assembly relationship, influence relationship, and sequence relationship. Figure 2: Relationship between biological gene and product gene based on the processing surface. Considering the characteristics mentioned above, the definitions of product gene based on the processing surface are performed as follows:  
**Definition 2**. : _Product gene unit based on processing surface: In this study, we express the product gene of a surface with a model which consists of 6 sets as the expression (1). According to the corresponding relationship between product structure and product gene as shown in Figure 3, the gene of one product or one part can be expressed by expression (2)._  
\[GU=(L,C,P,I,R,PE) \tag{1}\]  
\[PG=\{GU(1),GU(2),\ldots,GU(q)\} \tag{2}\]  
_where \(L,C,P,I,R,\) and PE represent Label Information Set, Quality Characteristics Information Set, Processes Information Set, Quality Influence Factors Set, Interrelated Information Set and Quality Performance Information Set respectively._  
Figure 4: Interrelated relation schematic diagram of product gene units. Figure 3: Multi-level structure schematic diagram of the product gene. **Definition 3**. : _Label Information Set \(L\): It is used to describe the adsorption information of a processing surface and expressed by \(L=\) (UID, EC, PC, SC). Here, UID represents the unique code of gene unit, EC represents the code of equipment which is composed of the code of equipment name and equipment type, PC represents the code of part which is composed of the code of part name and part type, and SC represents the code of surface name and surface type._  
**Definition 4**. : _Quality Characteristics Information Set \(C\): \(C=\) (\(C_{1}\), \(C_{2}\), \(C_{3}\),..., \(C_{a}\)). It describes the size, precision, and feature properties of a processing surface. Here, the properties include length, width, radius, screw-pitch, roughness, hardness, and so on, of which each is represented by a \(C_{i}\)._  
**Definition 5**. : _Process Information Set \(P\): \(P=\) (\(P_{1}\), \(P_{2}\), \(P_{3}\),..., \(P_{b}\)). It describes the information of processing types and methods of a processing surface. Here, \(P_{i}\) consists of the allowance of one side and the processing mode of process \(i\), which is shown by \(P_{i}=\) (\(AW_{i}\), PMC\({}_{i}\))._  
**Definition 6**. : _Quality Influence Factors Set \(I\): \(I=\) (MAT, \(I_{1}\), \(I_{2}\), \(I_{3}\),..., \(I_{c}\)). Quality influencing factors mainly include operator, equipment, material, process, measurement, and environment (SMIE) [35]. Therefore, 5M1E is the basis of the quality influence factors set. In Quality Influence Factors Set, \(I_{i}=\) (\(CW_{i}\), TM\({}_{i}\), MA\({}_{i}\), ETEM\({}_{i}\), HMD\({}_{i}\), IM\({}_{i}\), TL\({}_{i}\), PTEM\({}_{i}\)) represents the information of influence quality of process \(i\).