1 Introduction

The quality of the product is crucial to every production process. An assembly is often composed of more than two components. The quality of the parts used in the process has an impact on the functionality of the assembled product [1]. Tolerance affects component quality as it determines how well two components fit together. Because the components are made with tighter tolerances, precise assembly is better adapted to functional needs [2]. Due to variances in component size, manufacturers are still having problems manufacturing precise assemblies. The rejection of excess components owing to dimensional differences raises assembly costs. Regardless, selective assembly is a successful strategy for manufacturing precise assemblies at a lower cost of manufacture. Lower manufacturing costs result from eliminating or reducing secondary procedures used in the fabrication of wide-tolerance components [3, 4]. Typically, components are organized into bins during selective assembly using techniques like the equal area method, the equal width method, or the uniform tolerance method. Assemblies are created by selecting and combining components from designated bins in a random manner, following specific requirements [5, 6]. Prior research works had mainly aimed at reducing issues such as variations in clearance and surplus components. They preferred the individual methods namely equal/unequal area method, equal/unequal width method, and equal/unequal number of bins method for grou** components. Still, no research papers focus on the use of the combination of equal area and equal width approaches in conjunction with equal bin numbers while producing linear and/or non-linear assemblies with high success rate. The aforementioned method allows for a large number of bin combinations, making this challenging situation an NP-hard problem [7]. The "Literature Review" section helps readers understand the current study by discussing and reviewing prior works that are pertinent to the proposed study. The Literature Review section of the research paper examines the various evolutionary algorithmic methods that have been used in the past to address selective assembly problems. The Problem Overview section gives a general description of the research project. The Case Study section includes in-depth information about the instance case, which in this case is the assembly of ball bearings. The Solution Methodology section describes the strategy used to address problems with selective assembly generated by the SSO algorithm. The Results and Discussion section presents and analyzes the computational findings, and the Conclusion section summarizes the key findings and conclusions.

2 Literature Review

Kern [1] studied the challenges and roadblocks in predicting and controlling production variability during the product design and process phases. He provided solutions by recommending tools and techniques to overcome these obstacles and also developed equations to eliminate clearance variations for various selective assembly methods. By constructing components from various combinations of selected groups, Kannan et al. [2] introduced a new selective assembly approach that uses a genetic algorithm (GA) to produce the minimum clearance variation. They used a radial assembly to test their method. In order to remove excess components during assembly, Siva Kumar et al. [3] proposed an approach based on the use of GA for choosing the best combination of component groups. The effectiveness of their suggested approach was demonstrated using a gearbox shaft assembly.

In order to reduce surplus components during the manufacture of piston and piston ring assemblies, Asha et al. [4] presented a novel technique. In order to determine the best combination of the suggested strategies, they employed a non-dominated sorting GA. By using a particle swarm optimization approach, Kannan et al. [5] were able to minimize assembly variance by about 80% and determine the optimal group combinations for mating component assembly. With the intention of reducing clearance variance and minimizing excess components, Kannan et al. [6] have put forth a novel selective assembly technique for skewness-prone components. For each clearance change, they calculated the number of components in the selection group mixtures using GA.

A new selective assembly technique based on GA was introduced by Wang et al. [7] for getting rid of clearance differences in gear assembly while ensuring that there are no extra parts. By employing the identical width strategy to segregate the parts with smaller measurement differences, Matsuura and Shinozaki [8] established an optimal manufacturing mean design to reduce the number of extra components. Raj et al. [9] suggested using a GA to select the best mixture of all components for minimizing dimensional variances. The number of components dictated by this method represents the length of the chromosomes.

Yue et al. [10] used a Genetic Algorithm to discover the optimal combination of the groups that were selected to create a hole and shaft assembly with the least degree of clearance variance. The clearance variation was 30 µm based on the proposed method. Babu and Asha [11] used a technique based on an artificial immune system for selecting the best mixture of groups that has the lowest loss value within the range of specification and the lowest variation in assembly tolerance.

Selective assembly employing machines with infinite buffer capacity has been the subject of extensive investigation. However, many assembly processes in the real world rely on faulty machinery and limited buffers. A 2-part assembly method with uncertain Bernoulli machines and constrained storage was researched by Ju and Li [12]. They obtained great accuracy in their performance evaluation by employing a two-level decomposition approach. By lowering component variance, Xu et al. [13] suggested a unique method for improving hard disk drive production profits. In order to eliminate deficient components before assembly and choose matching pairings of components, they developed theories of discarding and binning.

A genetic algorithm-based selective assembly method was established by Lu and Fei [14] to increase assembly success rates by reducing surplus components. To do this, they employed a special 2D chromosomal structure and a genetic algorithm. The suggested approach worked better with assemblies that have many dimension chains. To assess assembly loss, Babu and Asha [15] suggested an equal interval-based Taguchi loss function. To determine the best selection group combination, they applied an updated sheep flock heredity algorithm that took into consideration of clearance fluctuation and assembly loss value. When building assemblies using a selective assembly approach, Ju et al. [16] took into account the use of untrustworthy machines and limited buffers. The production procedures for powertrain assembly in the automobile sector were used as an example. They developed a 2-level disintegration approach to assess the performance while assuming dependability models for Bernoulli machines.

Chu et al. [17] introduced an approach that meets the stringent backlash criteria of RV reducers. They utilized a genetic algorithm to address this challenge. In a similar vein, Asha and Babu [18] attempted to minimize excess components and clearance variation by adopting a meta-heuristic approach. Aderiani et al. [19] developed a multi-stage technique for selective assembly that effectively minimized clearance variation without the need for extra components. They employed the GA approach to optimize the selection of component groups, considering the complete dimensional distribution of the parts. Their method outperformed previous approaches by achieving a 20% reduction in variation. In a different approach, Hui et al. [20] assessed the quality of linear axis assembly in a machine tool by using a data-driven modeling technique. They developed a model that utilizes a genetic algorithm and a synthetic minority oversampling approach to measure assembly quality. Their data-driven modeling technique proved to be effective. **ng et al. [21] attempted to optimize the selective assembly of holes and shafts using relative entropy and dynamic programming. Jeevanantham et al. [22] used a genetic algorithm to minimize the surplus parts in selective assembly. Filipovich et al. [23] experimented with the model to make the maximum number of rod-piton groups. Rajamani [24] tried an assembly installation operation selective tool to effectively reduce the assembly variation in aerospace assemblies. Mahalingam et al. [25] used the moth-flame optimization technique to introduce an equal area amongst uneven bin numbers to increase the number of successful ball bearing assemblies. Lin and Huang [26], Tran et al. [27], Zhang et al. [28], Li et al. [29], and Ramesh et al. [30] dealt with the different manufacturing-related problems and solved them using multi-response optimization approaches.

Over time, researchers have approached the challenges of selective assembly through various methodologies. However, after a comprehensive review of related literature, a new problem setting has been identified. In the following section of the paper, we will delve into this novel problem and explore potential solutions.

3 Problem Overview

Manufacturing industries often face the challenge of producing an exact number of assemblies due to the presence of excess components resulting from tolerance variations and distribution. To solve this problem, components are divided into bins according to their tolerances in order to create as many assemblies as possible by randomly choosing components from the bins and putting them together. The key elements that impact the cost of assembly production by removing extra parts are the assembly specification, the manufacturing tolerance of individual components, and the bins. Techniques such as equal area and equal width are used to divide the components into bins based on their normal distribution of tolerances. However, figuring out the best pair of bins for varied assembly standards and the quantity of closer assemblies for each bin number from the manufactured components takes time. This study aims to provide an innovative approach for increasing assembly success rates by eliminating extra components.

4 EEE Method

Selective assembly, which divides components into various bins with respect to the variation in tolerance, has been the subject of several studies in the past to increase the success rate of manufacturing assemblies. The distribution of parts in terms of tolerance variation was previously classified using an equal number of bins based on the equal width/area of the distribution. But in 2015, Lu and Fei [14] organized the parts by classifying them into bins with regards to the unequal group sizes of particular ball bearing assembly components. 81.3 percent was the rate of success for this method, and the gap between it and earlier publications' success rates was only 0.63 percent. It is suggested to investigate the impact of categorizing the components with equal bin numbers using both the equal area and equal width approaches in order to further boost the success rate. Therefore, this study develops a unique approach referred to as Equal Area-Equal Width-Equal Bin Numbers (EEE) to categorize the components into bins in a single stage. This method takes into account the combination of normal and skewed distributions of components. The number of parts of the prescribed components that are available in each bin, as well as the total number of parts of the components that need to be assembled, define how many times a bin number will occur. The length of bin combinations is determined by the highest bin number in which the components are categorized. The study provides a step-by-step methodology for calculating the percentage of assembly success rates.

Step no. 1: Arrange the quantity of pieces produced (Ni) for each assembly component in increasing order according to the size.

Step no.2: Set the number of bins (NoB). It is equal for all the components.

Step no.3: Assign the method namely equal area/equal width for each component to categorize the respective parts into bins. Identify the parts and its numbers that lie into each bin of the respective components as per the method assigned. The number of parts of ith component to be grouped in jth bin (Pij) is calculated based on two cases.

Case 1—Equal area method: As per this method the Pij is calculated using Eq. 1.

$${P}_{ij}=\frac{{N}_{i}}{NoB}$$
(1)

Case 2—Equal width method: As per this method, the bin width (Wi) is calculated using Eq. 2.

$${W}_{i}=\frac{D{max}_{i}-{Dmin}_{i}}{NoB}$$
(2)

where, Dmaxi & Dmini is the maximum and minimum dimensions of a part in the pool of component i. Based on the calculated value of Wi, the number of parts of ith component are grouped into different bins.

Step no.4: Fix the arbitrary constant (K) for this work as ‘0’.

Step no.5: Determine length (L) of the grou** of bins for the ith component using Eq. (3).

$${L}_{i}=(NoC*NoB)-K$$
(3)

where, NoC – Total number of components.

Step no.6: Calculate the repetition of jth bin numbers occurred in the Li (Rij) using Eq. (4).

$${R}_{ij}=L*\frac{{P}_{ij}}{{N}_{i}}$$
(4)

Step no.7: Determine the combination of bins for each component by taking into account its frequency. As an instance, if component A is separated into 3 bins, 111,222,333 is the bin combination and the number of bins in the combination will be 9. Therefore, there are 9! possible permutations. One example of a combination would be 131,231,223.

Step no.8: Step numbers 1 to 8 are repeated for each component.

Step no.9: The assembly procedure consists of selecting a part (k) at random from the relevant bin specified in the bin combination (d) until the smallest value of Pij is reached. Equation 5 depicts the calculation of assembly clearance (Ld). It is confirmed using the manufacturer's given lower limit (LL) and upper limit (UL) values as prescribed in Eq. 6. This decides whether the assembly is approved or refused. Excess parts will be used if the identical bin number occurs in the remaining places of the bin combination. The maximum number of potential assemblies (Yd) may be calculated using Eq. 7, which is based on the minimal number of pieces [min(Pij)] present in the bins indicated in each location of the bin combination.

$${L}_{dk}=\sum_{i=1}^{NoC}\pm {D}_{ij=dk}$$
(5)

Yk – Assembly success / failure index

$${Y}_{k}=1 if LL\le {L}_{dk}\le UL else 0$$
(6)
$${NoY}_{d}=\sum_{k=1}^{\text{min}({P}_{ij})}{Y}_{k}$$
(7)

Step no.10: The above procedure is applied to the entire length of the bin combination and the total number of assemblies (T) produced is calculated using Eq. 8.

$$T=\sum_{d=1}^{\text{L}}{NoY}_{d}$$
(8)

Step no.11: Determine the success rate (SR) as a percentage using Eq. 9.

$$SR=100*\frac{T}{N}$$
(9)

The next section discusses the algorithmic approach utilised in this study.

5 Algorithmic Approach

If the number of bins for the assembly components is set, determining the success rate for assembly can be done easily using either the equal area or equal width method. However, in this study, both methods are used simultaneously when dividing the parts of each component into bins to observe the effect on the success rate percentage. When analyzing the numerous permutations and combinations of approaches for separating the portions of each component into bins, the task might be called an NP-hard problem. To propose a solution, an algorithmic method is required. The Salp Swarm Optimization (SSO) algorithm was chosen for this study because it is described by Mirjalili [31], Laith et al. [32] Castelli et al. [33], Ehteram et al. [34], Abualigah et al. [35], Abd El-sattar et al. [36], Alkoffash et al. [37], Bhattacharjee et al. [38], and Ahmed et al. [39] as having the benefits of having few setup factors, being simple to apply, and having quick convergence. Furthermore, the performance of the SSO algorithm is compared to the results of standard algorithms such as Antlion Optimization (ALO) and Genetic algorithms. Figure 1 depicts the pseudocodes for various methods. The factors and their values are addressed in Table 1.

Fig. 1
figure 1

Pseudocode of Salp Swarm Optimization Algorithm, Antlion Optimization Algorithm and Genetic Algorithm

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Table 1 Factors and their values
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6 Numerical Illustration

This section provides a numerical example to explain how to calculate the percentage of ball bearing assembly (BBA) success rate. The instance stated by Lu and Fei [14] is employed for numerical demonstration. The assembly's components and specifications are shown below.

$$Diameter\;of\;outer\;race, A={50}_{-0.000}^{+0.012} mm$$
$$Diameter\;of\;inner\;race, B={35}_{-0.012}^{+0.000} mm$$
$$Diameter\;of\;ball,C={7.5}_{-0.006}^{+0.000} mm$$

The clearance range will remain between 18 and 24 microns. The dimensional tolerance of the components is obtained directly from Lu and Fei [14]. Figure 2 depicts the dimensional characteristics of the BBA used in this investigation.

Fig. 2
figure 2

BBA (Lu and Fei [14])

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This study considers the Selective Assembly approach (SlM) with four levels of bins and two levels [either Equal Area (EA-Level 1) or Equal Width (EW-Level 2) approach] for the assembly of three components A, B, and C. Table 2 clearly demonstrates this.

Table 2 Parameters and its levels
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The use of full factorial design ensures thorough exploration of parameter effects, robustness to unexpected interactions, statistical efficiency, generalizability of results, and ease of interpretation, making it a suitable choice for experimental studies. The full factorial design is considered for establishing the combinations of parameters to conduct the tests. Hence, 32 (4 × 23) combinations are possible. Table 3 shows the experimental design.

Table 3 Experimental design
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As an instance (E.No. 2), for the three bins, the grouped component’s dimensional tolerances (in microns) are presented in Fig. 7. In this instance, the parts of components A and B are grouped into 3 bins based on the equal width method. Whereas the equal area method is being followed for grou** the parts of component B into 3 bins. The number of parts of components grouped in each bin is presented in Fig. 8. Similarly, the grou** of parts of components with respect to the number of bins 4, 5, and 6 is identified.

As discussed earlier, the calculation of the assembly success rate is most difficult when variations occur in the number of bins. Further, two different methods, namely the uniform area and uniform width methods, are being used while grou** the parts of each component into bins in view of visualizing the corresponding enhancement in the percentage of assembly success rate. The issue is deemed NP-hard in terms of the possible random combinations of varying the grou** technique and the number of bins. As a result, the Salp Swarm Optimization (SSO) method is employed to evaluate the optimal bin combinations. Tables 12, 13, and 14 show the findings of the optimal bin combinations for all the components. The number of assemblies to be prepared is determined using the step-by-step approach outlined in Sect. 4. The results of a test run are given in Fig. 9. Totally, 15 test runs were carried out, and the results are given in Table 4. The stop** criterion while running the salp swarm optimization algorithm for a run is considered as 100 iterations, and/or the repetition of same results for the 50 consecutive iterations.

Table 4 Results of test runs
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The significance of the results obtained through the test runs for all 32 experiments is evaluated through the normality test using Minitab software. For instance, the summary report of experiment no. 1 is given in Fig. 3. Further, the normality test results of all the test runs, considering all 32 experiments, are presented in Table 5.

Fig. 3
figure 3

Normality test results for experiment no.1

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Table 5 Results of evaluation of statistical significance through normality test of all test runs
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From the above table, based on the P-value, it is clearly evident that the test results are normally distributed and the data are statistically significant. Table 6 shows the results of the analysis of variance (ANOVA) on the SSO algorithm findings for the value of the number of assemblies built. According to the ANOVA analysis, the P value of the regression model is less than 0.05 for the number of assemblies made.

Table 6 ANOVA analysis for no. of assemblies obtained through SSO algorithm
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The pareto chart investigates the significance of the number of bins for each component as well as ways for distributing component parts into various bins in order to increase the number of assemblies. Figure 4a and b show the Pareto chart and residual plot, respectively. These numbers clearly show the impact of the aforementioned parameters on the number of successful assemblies.

Fig. 4
figure 4

Output of SSO algorithm and the statistical analyzes

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Further, the efficacy of the proposed SSO algorithm is compared with the outcomes of the GA and ALO algorithms. A nonparametric test called the Friedman test resembles the parametric 2-way analysis (Nagarajan et al. [40]) of variance, and it is used for determining the substantial differences in the behavior of the algorithms used in this work. The Friedman test results are presented in Table 7.

Table 7 Best NoA for different algorithms
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The maximum mean rank value is obtained using the SSO algorithm compared with the other two. It shows the effectiveness of using the SSO algorithm. The result of the ANOVA for the Friedman test is given in Table 8. It is concluded from Table 8 that the result is significant since the probability value is less than 0.05. Further, the statistical analysis of the results obtained through the SSO, GA, and ALO algorithms is presented in Fig. 5. Since the P value is less than 0.05 in all three cases, the obtained results are statistically significant. The percentage success rate of making assembly (SR) through the SSO algorithm is calculated [as per Eq. (9)] as 92%.

Table 8 ANOVA for friedman test
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Fig. 5
figure 5

Statistical analysis on the results of a SSO, b GA, and c ALO algorithms

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Furthermore, the paired T test is used with Minitab software to estimate the statistical significance of the results of the algorithms utilized. The results are given in Tables 9, 10, and 11. The T-value in a paired T-test is a statistical test to determine whether there is a significant difference between two paired algorithms. The T-value is calculated by dividing the difference between the means of the two samples by the standard error of the difference. The P value in a paired T-test is a measure of the probability of obtaining the observed difference between the means of the two algorithms if there is actually no difference between the algorithms. The P value is calculated by comparing the T-value to the cumulative distribution function of the T-distribution. As listed in Table 11, the P value of less than 0.05 is typically considered to be statistically significant, which means that there is less than a 5% chance of obtaining the observed difference between the means if there is actually no difference between the algorithms.

Table 9 Descriptive Statistics
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Table 10 Estimation for Paired Difference
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Table 11 Testing of significant difference
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Convergence plot given in Fig. 6 can be used to better explain the results comparison of different algorithms based on the following three criteria.

  1. a)

    The slopes of the convergence plot: SSO algorithm with the steeper slope converges with maximum number of assemblies (46 assemblies) more quickly than the GA and ALO algorithms (42 assemblies) with the shallower slope. This means that the SSO algorithm with the steeper slope is more likely to find a better solution in a shorter amount of time.

  2. b)

    The time to convergence: The time to convergence is the number of iterations it takes for the algorithm to reach maximum number of assemblies. The SSO algorithm (23rd iteration) converged with little amount of time and less computational effort than the GA and ALO algorithms (41st and 61st iteration respectively).

  3. c)

    The final error: The final error is the error between the algorithm's final solution and the desired value. The SSO algorithm has the lower final error value of 4 and hence, it is more accurate than the GA and ALO algorithms (final error value is 8).

Fig. 6
figure 6

Convergence plot–SSO vs GA vs ALO algorithms

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7 Conclusions

In the course of this research, a unique novel approach that will henceforth be referred to as the Equal Area-Equal Width-Equal Bin Numbers (EEE) method was devised with the intention of increasing the quantity of ball bearing assemblies that could be produced by combining the component parts of three distinct types (A, B, and C). A full factorial design was considered for conducting the experiments. The following conclusions were drawn out of these experimental results.

  1. 1.

    Considering the outcomes of all 32 experiments, it was found that the normality test results of all the test runs were normally distributed and the data were statistically significant.

  2. 2.

    The Salp Swarm Optimization (SSO) algorithm was used for the evaluation of the best bin combinations in order to identify the possibility of making the maximum number of assemblies. The percentage success rate of making assembly (SR) through the SSO algorithm was calculated as 92%. When measured against the findings of previous literature, the percentage increase in the assembly success rate comes out to 13.16.

  3. 3.

    In order to prove the efficiency of the proposed EEE approach, the results obtained from the SSO algorithm, the GA methodology, and the ALO algorithm were compared with one another. The SSO algorithms outperformed others.

  4. 4.

    The results of nonparametric test called the Friedman test were used to determine the substantial differences in the behavior of the algorithms used in this work.

  5. 5.

    A paired T-test was carried out to identify the statistical significance between the SSO and GA & ALO algorithms. According to the findings of the tests, the number of ball bearing assemblies increased significantly when the SSO algorithm was used rather than GA and ALO. This is because the SSO algorithm has a very small number of setup parameters, allows for accurate estimate, maintains population variety, is simple to comprehend and put into practice, and converges quite quickly.

  6. 6.

    The convergence plot offered more evidence supporting the hypothesis.

Hence, the unique EEE approach suggested in this study could be more beneficial for mass production industries that seek high assembly success rates. The suggested work may be extended to apply for other complex assemblies by considering unequal bin numbers, along with equal areas and widths. The machine learning approach may be used instead of using regression models for predicting the number of successful assemblies.