# Integrated Design of Process-Tolerance for Remanufacturing Based on Failure Feature
## 2 Materials and Methods
### Multi-Objective Function Construction
#### 2.3.1 Objective Functions  
Cost Function  
The cost of IDP-T for remanufacturing includes the cost of machine tools, cutting tools and additional materials; remanufacturing process of failure surface can be divided into removal processes and additive processes. When there are \(n\) processes in the remanufacturing of failure surface \(i\), the total cost is shown in Equation (5). \[C_{i}=\sum_{j^{-}=1}^{a}C_{ij^{-}}+\sum_{j^{+}=1}^{b}C_{ij^{+}}(a+b=n) \tag{5}\]  
where \(C_{ij^{-}}\) and \(C_{ij^{+}}\) are the cost of removal process and additive process of surface \(i\) using process \(j\), respectively; \(a\) and \(b\) are the numbers of removal process and additive process of surface \(i\), respectively. The cost of removal process \(C_{ij^{-}}\) will increase with the improvement of the machining accuracy. The negative square model is used to construct the tolerance-cost function of the removal process, as in Equation (6). \[C_{ij^{-}}=a_{0 ij^{-}}+\frac{a_{1 ij^{-}}}{\left(T_{ij^{-}}\right)^{2}} \tag{6}\]  
where \(a_{0 ij^{-}}\) and \(a_{1 ij^{-}}\) are the fixed cost coefficient and tolerance cost coefficient of surface \(i\) using removal process \(j\), respectively; \(T_{ij^{-}}\) is the tolerance of surface \(i\) using removal process \(j\). The values of \(a_{0 ij^{-}}\) and \(a_{1 ij^{-}}\) can be obtained by fitting the enterprise cost-tolerance statistical samples by the least square method. For additive processes, different equipment and parameters will lead to greater difference in precision. According to literature [22; 23], for the same equipment, when the main parameters remain stable, the tolerance is relatively fixed in unit thickness and meets the normal distribution. Therefore, when economic parameters are adopted, the additive process cost can be regarded as a fixed value related to the thickness of the additive, as shown in Equation (7). \[C_{ij^{+}}=a_{0 ij^{+}}=f\left(h_{ij^{+}}\right) \tag{7}\]  
where \(a_{0 ij^{+}}\) and \(f\left(h_{ij^{+}}\right)\) are fixed cost, and cost functions are related to the thickness of the additive when dimensions are restored under economic processing parameters, respectively. When the used product, having \(N\) failure surface needs remanufacturing tolerance design, the total cost is shown in Equation (8). \[C=\sum_{i=1}^{N}\left(\sum_{j=-1}^{a}C_{ij^{-}}+\sum_{j^{+}=1}^{b}C_{ij^{+}}\right) \tag{8}\]Quality Loss Function  
Quality loss is an economic loss due to a less than expected performance of the product. According to Taguchi theory, size deviation of the parts will cause quality loss and ultimately affect the subsequent processing. Exponential equations are used to construct the quality loss function of remanufacturing processes, shown as Equations (9) and (10). \[L=K_{ij}(A_{0 ij}-A_{ij})^{2} \tag{9}\]  
\[K_{ij}=\frac{B_{ij}}{\Delta_{ij}^{2}} \tag{10}\]  
where \(K_{ij}\) is the quality loss coefficient of surface \(i\) using process \(j\); \(A_{0 ij}\) and \(A_{ij}\) are the original design dimension and remanufacturing actual machining dimension of surface \(i\) using process \(j\), respectively; \(B_{ij}\) and \(\Delta_{ij}\) are the lost cost by out of tolerance and allowance tolerance of surface \(i\) using process \(j\), respectively. In process level, the physical meaning of quality loss is the economic loss of subsequent process and service caused by the tolerance of the surface \(i\) using process \(j\). When the tolerance of removal process \(T_{ij^{-}}\) is symmetrically distributed, the quality loss of surface \(i\) using process \(j\) is shown as Equation (11). \[L_{ij^{-}}=K_{ij^{-}}\left(\frac{T_{ij^{-}}^{2}}{4}\right) \tag{11}\]  
For the additive process, quality loss varies greatly with different equipment; the tolerance is regarded as a fixed value related to the thickness of the additive. Therefore, the quality loss of additive processes is shown in Equations (12) and (13).