Table 1 Comparison of different optimization objectives

Considering in-plane variations

Minimize \(F\left(X\right)=\frac{1}{\sqrt{2{N}_{KPC}}}\sqrt{\sum_{i=1}^{{N}_{KPC}}[{{\sigma }^{2}\left(\delta {x}_{0}\right)}_{i}+{{\sigma }^{2}\left(\delta {y}_{0}\right)}_{i}]}\)

\({{\sigma }^{ }\left(\delta {x}_{0}\right)}_{i}, {{\sigma }^{ }\left(\delta {y}_{0}\right)}_{i}:\) The \(i\)th KPC’s variations in the x and y directions

\({N}_{KPC}\): Number of KPCs

Minimizing the pooled standard deviation of resultant errors at all KPCs

Minimizing variations caused by source variations and improving robustness

Cai [5]

Minimize \(F\left(X\right)=\sum_{i=1}^{m}ST {D}_{x}^{2}\left(i\right)+ST {D}_{z}^{2}\left(i\right)\)

\(ST {D}_{x}\), \(ST {D}_{y}^{ }:\) Standard deviations in the X and Z directions

Minimizing the summation of squared standard in-plane deviations

Masoumi et al. [10]

Minimize \({S}_{{\text{max}}}={\lambda }_{{\text{max}}}({D}^{{\text{T}}}D)\)

\(D:\) a sensitivity index reflects the relationship between the final variation and source variations

Minimizing the square of the 2-norm of sensitivity matrix

Minimizing the sensitivity of the part to source variations and improving robustness

Kim and Ding [11], Tian et al. [13], Huang et al. [14], Xie et al. [15]

Minimize \(F\left(X\right)=\sqrt{{\text {cond}}({S}^{{\text{T}}}S)}\)

\(S:\) a sensitivity index reflects the relationship between the final variation and source variations

Minimizing the square root of the condition number of the sensitivity matrix

Li et al. [12]

Considering out-of-plane deformations

Minimize \(F\left(X\right)=\sum_{i=1}^{m}{{w}_{i}(X)}^{2}\)

\({w}_{i}\left(X\right):\) the deformation perpendicular to the part surface at the \(i\) th node

Minimizing the sum of squares of nodal deformations

Minimizing deformations caused by forces and reducing assembly errors caused by deformations

Cai et al. [6]

Minimize \(F=\sum_{i=1}^{n}{u}_{i}\)

\({u}_{i}\): strain energy of the \(i\)th finite element

Minimizing the sum of strain energy of finite elements

Ahmad et al. [16−18], Bi et al. [19]

Minimize \(F={\text{max}}\left\{\frac{{e}^{i}}{{T}^{i}} \space \,{\text {for}}\space \,i=1,\cdots ,M\right\}\)

\({e}^{i}:\) profile error of the \(i\)th point

\({T}^{i}\): profile tolerance of the \(i\)th point

Minimizing the maximum ratio of error to tolerance

De Meter [20]

Minimize \(F={(\sum_{i=1}^{N}{\Delta }_{i})}^{{\text{T}}}(\sum_{i=1}^{N}{\Delta }_{i})\)

\({\Delta }_{i}:\) the rigid body motion at the \(i\)th fixturing point

Minimizing the total rigid body motion

Minimizing positioning errors caused by elastic deformations and improving positioning accuracy

Li and Melkote [21, 22]

Minimize \(H\left(x\right)=\sum_{i=1}^{{m}_{0}}{\varphi }_{i}(x)/{m}_{0}\)

\({\varphi }_{i}\left(x\right):\) the dimensional gap at node \(i\) along the interface between the compliant parts to be assembled

\({m}_{0}:\) the number of the nodes along the assembly interface between two parts

Minimizing the average dimensional gap along the interface between the compliant parts to be assembled

Reducing the assembly gap between two parts and improving the weld quality

Du et al. [23]

Minimize \(f=\frac{C}{{C}_{\text {max}}}+\sum_{k=1}^{K}p({C}_{puk})\)

\(C\): total cost of production

\({C}_{pu}:\) upper process capability index

\(p\left({C}_{puk}\right):\) a penalty function relative to \({C}_{pu}\) of the \(k\)th KPC

Minimizing the expense of production

Reducing the expense while meeting quality requirement

Aderiani et al. [24]