A Novel Methodology for Simultaneous Minimization of Manufacturing Objectives in Tolerance Allocation of Complex Assembly

: Tolerance cost and machining time play crucial roles while performing tolerance allocation in complex assemblies. The aim of the proposed work is to minimize the above-said manufacturing objectives for allocating optimum tolerance to the components of complex assemblies, by considering the proper process and machine selections from the given alternatives. A novel methodology that provides a two-step solution is developed for this work. First, a heuristic approach is applied to determine the best machine for each process, and then a combined whale optimization algorithm with a univariate search method is used to allocate optimum tolerances with the best process selection for each sub-stage/operation. The efﬁciency of the proposed novel methodology is validated by solving two typical tolerance allocation problems of complex assemblies: a wheel mounting assembly and a knuckle joint assembly. Compared with previous approaches, the proposed methodology showed a considerable reduction in tolerance cost and machining time in relatively less computation time.


Introduction
All aspects of manufacturing such as machine investment cost, manufacturing cost, the functionality of the product, quality of manufacturing, and the reliability of the product directly connect with tolerance allocation. Hence, most of the researchers are still focusing on this research topic to improve manufacturing efficiency. It involves the allocation of critical dimensions of an assembly, known from the product's functional requirements. As per the literature, more equations are available based on the combinations of the component's tolerance values; however, few equations provide better results. Tolerance allocation requires discovering the best possible combination of the component's tolerances by considering the manufacturing objective(s) and the associated constraints. The researchers proposed different tolerance allocation strategies in different periods. The summary of those strategies is explained here. Further, the detailed comparison of the strategies is given in the Supplementary File as Table S1.
Heuristic Method (HM): The application of this method is seldom found in literature, since the usage of thumb rules, previous experiences, and standards [18,19] in solving the tolerance allocation problems. Nevertheless, the alternative methods, namely the branch and bound algorithm [20] and design of experiments [21], were used to identify the effective tolerance allocation models. Further, a new method was developed by integrating HM with Tabu search for optimal tolerance allocation and subsequent manufacturing cost reduction [22]. Armillotta (2020) [23] minimized the manufacturing cost of the mechanical assemblies by properly allocating the tolerances and choosing the right dimensional properties. Korbi et al. (2021) [24] proposed a computer-aided design model for analyzing the tolerance in manufacturing the mechanical assemblies.
Discrete Cost Function (DCF) and Continuous Cost Function (CCF) Models: The researchers have used different cost function models [25][26][27][28][29][30] in the various periods to evaluate the manufacturing cost. These models were classified as DCF and CCF based on their nature. However, the researchers mainly used the CCF model, since it yields closed-form solutions to the tolerance allocation problems. On the other hand, the DCF models [4,26,31] were not preferred due to the model fitting errors during the manual formulation. Further, several studies have been carried out by considering the objective function as the sum of quality loss (as per Taguchi quality loss concept) and manufacturing cost [1,8,[32][33][34][35][36][37][38][39][40][41].
Simple, Complex, and Non-Linear Assembly: The researchers used different tolerance allocation methods to obtain a solution concerning product type. For instance, the LMM was only used for simple products [1,8,42,43], which have only two mating components. Several works have been reported on finding the optimum allocated tolerance values for the components of complex assembly [44][45][46][47][48][49][50][51]. A limited number of authors have concentrated on non-linear assembly products that consist of more than two components [20,26,38].
Alternative Process Selection (APS): In practice, it is possible to produce components using more than one alternative process. It is necessary to select the proper process for the correct component to reduce the manufacturing cost. Some authors have considered alternative process selection (i.e., every combination of the process has a feasible tolerance range, and for a given process combination, the cost of machining is the function of the tolerance value) for optimum tolerance allocation of both simple and complex assemblies [4,10,13,15].   [52] dealt with the distribution of tolerance on the component dimension of a complex assembly with alternative process selection. Authors have attempted to reduce the manufacturing cost of a product using the Lagrange multiplier method for complex assemblies with the bottom curve follower method.
Alternative Machine Selection (AMS): It is possible to reduce the manufacturing cost of a product by choosing the suitable machine for the correct process. Several researchers have considered machine selection as one of the criteria for minimizing manufacturing costs in recent years. However, very few authors [9,17,34] have discussed alternative machine selection for optimum tolerance allocation, and even fewer studies [34] on minimizing both cost and machining time with process and machine selection for optimum allocation of tolerance have been reported in the literature.
From this literature review, it appears that no significant effort has been made to consider machine time as an objective in optimum tolerance allocation, even though machine time is a crucial manufacturing parameter. Therefore, in the present study, both tolerance cost and machining time are optimized. Realizing the complexity of the problem, a combined heuristic and univariate search method is introduced to select the best machine for each process and the optimum tolerance for each component using alternative process and machine selection, with available machine time as a constraint.

Problem Definition
The biggest challenge facing today's manufacturing companies is to reduce production costs while maintaining better quality and higher productivity. Tolerance allocation plays a vital role in achieving those goals. The production of a component involves selecting processes as well as selecting machines. These decisions directly influence the allocated tolerance values of components. The optimum allocated tolerance values govern the manufacturing costs and machining time of the product. Therefore, an operation may be possible with multiple alternatives, and as such, is treated as a non-polynomial hard problem.

Mathematical Formulation
The proposed work aims to simultaneously minimize tolerance cost and machining time, represented in Equation (1). Tolerance cost (TC i ) and machining time (MT ik ) are calculated using Equations (2) and (3). Equations (4) and (5), respectively, determine the critical dimension of the sub-assembly (Y) and its tolerance (t Y ). Machine engagement time (met) is estimated using Equation (6). The constraints considered in this work are expressed by Equations (7)- (9). The allocated tolerance (t i ) should be within the process limits, represented by Equation (7). The calculated sub-assembly tolerance within the given sub-assembly tolerance; this constraint is given by Equation (8). The total individual machine engagement time to manufacture the product should be less than the given available machine time (amt) represented in Equation (9). In addition, it is assumed that the following data are known well in advance before manufacturing the product: Objective function TC ijk Tolerance cost of ith sub-stage using jth process on kth machine MT ijk Machining time of ith sub-stage using jth process on kth machine t ijk Allocated tolerance of ith sub-stage in jth process and kth machine A j , B j Tolerance cost function constants for jth process X1 j , Y1 j Tolerance machining time function constants for jth process η jk Efficiency factor using jth process on kth machine Machine number index l Component number index met k kth machine engagement time amt k Available time of kth machine

Methodology
The proposed method consists of two stages: (i) selection of the best machine for each process by applying a heuristic approach; (ii) selection of the best process and optimum allocated tolerance for each component using combined whale optimization algorithm and univariate search method. In the first stage, the process tolerance is divided into nd number of discrete values using Equation (10) and the allocated tolerance (t ejk ) is calculated using Equation (11). The tolerance cost (TC ejk ) and machining time (MT ejk ) for t ejk are calculated using Equations (2) and (3), respectively. Nagarajan et al. (2018) explained that the distance method is used to combine the two different objective functions into a single one. For each discrete value, points are plotted on a graph where the x-axis and y-axis represent tolerance cost (TC ejk ) and machining time (MT ejk ), respectively. Assuming point (x1, y1) as the origin and point (x2, y2) as discrete tolerance cost and machining time, and substituting (x1, y1) as (0,0) and (x2, y2) as (TC ejk , MT ejk ) in Equation (12), then the distance equation becomes Equation (13). The detailed steps of the heuristic and univariate search methods are shown in Figures 1 and 2. The pseudocode for the combined whale optimization algorithm and a univariate search method is presented in Section 5.
where e is the index for discrete point of tolerance and takes from 1,2,3 . . . nd.

Numerical Illustration
The proposed method is initially implemented in the existing problem (wheel mounting assembly) discussed by Geetha et al. (2013) [34] 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 0 . 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. Using Equations (2), (3), (10), (11) and (13), discrete tolerance (td j ), tolerance (t ejk ), tolerance cost (TC ejk ), machining time (MT ejk ), and distance values (dis ejk ) are calculated for each machine and are shown in Table 1. Figure 4 is constructed for a discrete tolerance value of 0.01 mm (t ejk = 0.01 mm), assuming TC ejk on the x-axis and MT ejk on the y axis. The distance between the origin and the discrete point of the machine represents the critical factor that decides on selecting the best machine. The value of less distance is more likely to be the best machine for the process.  TC e11 -tolerance cost of eth discrete tolerance using process number 1 on machine number 1; MT e13 -machining time of eth discrete tolerance using process number 1 on machine number 3.
As shown in Figure 4, it is clear that dis-M1 is less than dis-M3; 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  (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. Using Equations (2) and (3), the tolerance cost and machining time are calculated for each operation with alternative processes and its corresponding best machine is shown in Figure 5. In all 11 combinations of alternative processes, the same tolerance mentioned in case 1 and case 2 by Geetha et al. (2013) is assumed for each operation. The alternative process and its corresponding machine-allocated tolerance of each sub-stage/operation and its tolerance cost and machining time are presented in Tables 2 and 3. The best result is shown as shaded text in both Tables 2 and 3. Figures 6 and 7 show a comparison of the tolerance cost and machining time for the existing problem presented in Geetha et al. (2013) and the proposed method; it can be seen that the proposed method works well, considering tolerance cost, machining time, and both as objective functions.  0.060784 0.061882 0.039412 0.041831 0.055569 0.070471 0.062745 0 Table 4 represents the % of savings in both tolerance cost and machining time as compared to the existing method and the proposed method. Case Study 2: Knuckle Joint Assembly In practice, the availability of machine time will restrict the selection of the machine for performing a process, which will influence the tolerance cost and machining time. Therefore, in this work, the available machine time is considered as a constraint in selecting the machine. The methodology is demonstrated using a knuckle joint assembly (Figure 8), consisting of six components performed in ten sub-stages. Two critical dimensions are considered for the proper functioning of the product. It is assumed that nine processes are performed using ten machines. Equations (16) and (17) are used to determine the critical dimension, and the tolerances of the critical dimensions are estimated using Equations (18)- (21). Table 5 shows the dimensions, sub-stages, tolerance symbols, and tolerance stake-up of the knuckle joint assembly.   Table 6 shows the cost and time function constants of each process of the knuckle joint assembly. The possible operation-process and process-machine feasibility matrix and the available time of the individual machines are presented in Tables 7 and 8, respectively.  Table 7. Possible operation-process and process-machine feasibility matrix of knuckle joint assembly. O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 P1 As per stage 1, the best machine for each process is determined using the heuristic method and is presented in Table 9 and Figure S3 (in Supplementary File). Table 9. Best machine for each process of the knuckle joint assembly.

Process Number
Best machine number 9 3 10 7 2 5 9 1 4 In the second stage, the whale optimization algorithm is implemented with a univariate search method. The optimum allocated tolerance for each sub-stage/operation, tolerance cost, machining time, and best process is obtained by assuming 100 whales, 100 iterations as stopping criteria with 20 runs. The parameters involved in the algorithm are given in Table 10. Further, the pseudocode of this algorithm is given in the Supplementary File. The convergence plot for tolerance cost and machining time for stopping criteria considered as 100 iterations are shown in Figure 9. The Pareto optimal solution for a sample run is shown in Figure 10. The tolerance cost and machining time for 31 runs are presented in Table 11. Out of these 31 runs, the best value is calculated using EDAS and CODAS multi-criteria decision-making techniques implemented by Adali & Tuş (2019) [53] The appraisal score (APS) and the assessment score (AS) obtained by implementing EDAS and CODAS method are presented in Table 11. Run number (R.No.) 11 has the highest APS value in EDAS, and run number 7 has the highest AS value in CODAS (Highlighted in Table 11). The optimum allocated tolerances of each sub-stage are presented for both methods and are listed in Table 12, along with the tolerance cost and machining time for the required sub-assembly tolerances of 0.55 mm and 0.22 mm. The optimum total tolerance cost and total machining time are presented in Table 13 for the given sub-assembly tolerance values with the constraint of available machine time.   Table 14 shows the machine engagement times of the individual machines to manufacture the knuckle joint assembly within the available machine time.
For supporting the proposed method, the statistical analysis for EDAS and CODAS methods are executed through Minitab software. The statistical analysis results and probability plots for both the methods are presented in Figure 11. The probability values are 0.329 and 0.231 for the EDAS and CODAS methods, respectively. Since the value of probability is greater than 0.005 in both cases, it is clearly understood that the results obtained in 31 runs are from normally distributed data.    11. Statistical analysis and probability plot for EDAS and CODAS methods.

Conclusions
Most previous studies on tolerance allocation problems concentrated on minimizing manufacturing costs, quality loss, or combining the two. Machining time, a vital manufacturing objective, has barely been contemplated. In this paper, the machining time was considered along with manufacturing cost in optimum tolerance allocation of complex assemblies, representing a more realistic product development scenario. Alternative machine and process selections with available machine time make this problem cumbersome and complicated. Therefore, a new methodology was developed that applies a heuristic approach and combines whale optimization algorithm with a univariate search method. The total manufacturing cost and machining time of 36.3 USD and 57.74 min reported in this paper for wheel mounting assembly is 2.73% and 15.35% less than the problem dealt with in case 1 by Geetha et al. (2013). Similarly in case 2, there was 6.22% and 20.32% of savings in the tolerance cost and machining time reported by implementing the proposed method. The results presented in this paper demonstrate that the proposed method can reduce tolerance cost and machining time simultaneously with less computation time. The proposed method is also suitable for solving two-and three-dimensional problems. As a further extension of this work, the operation sequence, machine sequence, or both may be considered with additional objectives such as total investment cost of machines, idle time of machines, idle cost of machines, and the number of machines required to manufacture the product.  Figure  S2: Selection of best machine for process number 1 to 5 to manufacture WMA, Figure S3: Selection of best machine for process number 1 to 9 to manufacture KJA, Table S1: Summary of literature survey,