## 5 Numerical Illustration  
The proposed method is initially implemented in the existing problem (wheel mounting assembly) discussed by [PERSON] et al. (2013)[PERSON] et al. (2013) to show the method's effectiveness in case study 1. Later, it is implemented in knuckle joint assembly in case study 2. Case study 1: Wheel Mounting Assembly (WMA)  
The components of the wheel mounting assembly are given in the Supplemental File as Figure S1. The operations required to manufacture the components of the assembly are illustrated in Figure S1, starting from \(O1\) to \(O8\). The feasibility of performing the operations through the processes from P1 to P5 is given in Table S2. The infeasibility of performing the operation using the specific process is marked as \({}^{\prime}0^{\prime}\). The same way, the feasibility of using machines for the specific processes is also given in Table S2. Further, the cost and time function constants are represented in Table S3. The first stage of the proposed work, the procedure to determine the best machine for each process, is discussed below. Figure 3 represents the graphical representation of possible alternative processes and machines to carry out all the sub-stage operations to complete the manufacturing of the product. In the first stage, the best machine is selected for each process by implementing the heuristic approach. Figure 2: Flow chart of univariate search method. \(TC_{e11}\)--tolerance cost of \(e\)th discrete tolerance using process number 1 on machine number 1; \(MT_{e13}\)--machining time of \(e\)th discrete tolerance using process number 1 on machine number 3. As shown in Figure 4, it is clear that _dis-M_1 is less than _dis-M_3; therefore, machine 1 is considered the best machine for process 1 for achieving a discrete tolerance value of 0.01 mm. The sum of the difference between minimum discrete distance (\(min(dis_{ijk})\)) and discrete distance (\(dis_{ijk}\)) to that particular machine is calculated to select the best machine suitable for all discrete tolerances of the process tolerance. As shown in Table 1, it can be concluded that machine number 1 is the best machine for process number 1, shown graphically in Figure S2 (Supplementary File). Similarly, the best machine for the other processes is calculated and shown in Figure S2. It is clearly understood that machine numbers 2, 1, 3, and 4 are the best machines for process numbers 2, 3, 4, and 5, respectively. After implementing the heuristic approach, alternative machines are removed for each process; instead, the best machine is selected, which is shown in Figure 5.  
In the second stage, since the same allocated tolerances reported in [PERSON] et al. (2013) have been used for demonstration purposes, only univariate search method is implemented to get the best process, minimum tolerance cost, and minimum machining time for each sub-stage/operation of WMA. The univariate search method proposed in this work provides the best results in relatively less computation time than the existing method. The search space of the existing method proposed in [PERSON] et al. (2013) contains 403,200 combinations, whereas, in the proposed method, there are only 11 possible combinations in the search space. Equations (14) and (15) compute the possible combinations in the existing and proposed methods, respectively. Table 2 represents the 11 possible combinations obtained using the univariate search approach, and the first combination is shown as yellow shaded text in Figure 5.  
\[ncs_{e}=\prod_{i=1}^{no}(\sum_{j=1}^{np}nm_{ij}) \tag{14}\] \[=(2+2+2)(2+3+2)(2+2)(3+3)(2+2)(2+3)(3+2)(2+2)\] \[=6\times 7\times 4\times 6\times 4\times 5\times 5\times 4=403,200\]  
\[ncs_{u}=1+\sum_{i=1}^{no}np_{i}-no=1+3+3+2+2+2+2+2+2-8=11 \tag{15}\]  
Figure 4: Selection of the best machine based on discrete tolerance.