Optimizing Tolerance Allocation in the Remanufacturing Process of Used Electromechanical Products

Abstract

Optimizing tolerance allocation is crucial for balancing cost and performance in the remanufacturing of used electromechanical products. However, the traditional remanufacturing model of “individual part precision restoration + secondary machining trial assembly” lacks an integrated approach to tolerance planning in the design and manufacturing stages, leading to excessive fluctuations in cost and quality. To address this issue, a remanufacturing value-based tolerance allocation method is proposed, integrating remanufacturing value into the tolerance allocation process. First, a remanufacturing value quantification and evaluation indicator system was established at the failure surface layer (i.e., the remanufacturing processing surface) at the design stage and comprehensively considers the used part quality and enterprise processing capabilities. Quantification methods for each indicator were developed, and a comprehensive weighting strategy combining subjective enterprise standards and objective return quality adopted. Then, a multi-objective optimization model for remanufacturing tolerance allocation was established, targeting remanufacturing cost, quality loss, process stability, and corrected by the failure surface value. Finally, the beetle antennae search (BAS) algorithm was employed to determine the optimal solution. A case study on a used gearbox demonstrated that the proposed method significantly improves cost, quality loss, and process stability compared to the traditional remanufacturing approaches.

1. Introduction

With the rapid advancement of industrial technology, the cycle of upgrading and iteration for electromechanical products has been significantly shortened. These used products retain considerable residual value, and discarding them without further processing leads to resource wastage and environmental pollution. Remanufacturing processes restore used electromechanical products to an ‘as-new’ condition by repairing or replacing worn parts, thereby effectively preserving their residual value [1]. Remanufacturing offers substantial economic and environmental benefits compared to the manufacturing of new products and has emerged as a pivotal green manufacturing technology in contemporary industry [2,3].

Unlike new manufacturing, remanufactured electromechanical products exhibit significant variability in remanufacturing value across different failure surfaces (machining surfaces) due to uncertainties in component reuse methods, enterprise processing capabilities, and the quality of the used parts [4]. Traditional remanufacturing processes initially involve restoring individual used parts to precision, followed by trial assembly and secondary machining during the assembly stage. This approach results in excessive fluctuations in product quality and costs [5]. A primary cause of this issue is the lack of synchronization between the design and manufacturing stages [6]. During the design phase, tolerance redistribution is often not considered based on the varying remanufacturing values of different failure surfaces. Therefore, it is imperative to consider comprehensively the remanufacturing value of each failure surface during the design phase of remanufactured electromechanical products and to design a secondary tolerance allocation scheme accordingly. This approach is essential for further reduction of production costs and enhancement of product quality.

A rational assessment of the remanufacturing value of used electromechanical products is essential for determining the feasibility and approach to the remanufacturing of used parts. Numerous studies have concentrated on developing quantifiable evaluation metrics for the remanufacturing value of used components. For example, Fang et al. [7] introduced a quantification method for process indicators such as disassemblability and reparability of remanufactured products, leveraging the original CAD model. Omwando et al. [8] employed a two-tiered fuzzy computing approach to analyze both qualitative and quantitative indicators, and two control drives were subsequently compared in terms of the remanufacturing processes, economic feasibility, and resource utilization. Through this comparison, it was demonstrated that the quantification of remanufacturing value can enhance economic and technical decision-making for remanufactured products. Karaulova and Bashkite [9] utilized a life-cycle assessment (LCA) model to evaluate the remanufacturability of a used grab crane from technological, economic, and environmental standpoints, offering a holistic evaluation. Mark Ferguson et al. [10] proposed a tactical production-planning model, which introduced a quality grading method for remanufactured parts based on remanufacturing value, leading to increased remanufacturing profitability. Zhang et al. [11], using a boom cylinder of a concrete pump truck as an example, combined fuzzy evaluation with the analytic hierarchy process (AHP) to assess quantitatively the factors influencing the remanufacturability of used electromechanical products, taking into account the unique attributes of remanufacturing. While these studies effectively quantify remanufacturing value at the component level, a comprehensive quantification model for failure surfaces is still needed. Given that different failure surfaces of the same used part may have varying remanufacturing values, it is imperative to quantify the remanufacturing value of each failure surface during the secondary tolerance allocation design. This enables the assignment of differentiated tolerance precision levels.

Tolerance is a critical link between used parts and remanufactured products, underscoring the necessity for a rational formulation of a tolerance scheme, which is essential for the final cost and quality of remanufactured electromechanical products [12,13]. Unlike new product manufacturing, where a single optimal tolerance scheme can be universally applied across an entire batch, remanufactured products necessitate a more individualized approach due to their inherent uncertainties, requiring case-by-case determination of tolerance schemes. Liu et al. [14] developed a hierarchical assembly method for remanufactured assembly dimension chains based on the varying quality levels of used parts, employing process control methods, enabling the assembly of remanufactured engine pistons. They further optimized the tolerance allocation scheme for remanufactured assembly dimension chains by quantifying the uncertainty in the assembly process of remanufactured products using the entropy weight method, which was validated through a remanufactured engine assembly case study [15]. HU et al. [16] utilized a state–space model of used parts to select and match used parts, effectively controlling the assembly precision of remanufactured automobile engines. Liu [17] quantified the impact of uncertainty in used parts on the assembly quality of remanufactured products using an entropy model and proposed a tolerance design method for remanufactured assembly dimension chains, facilitating the successful remanufacturing and assembly of used gearboxes. These studies analyzed the impact of uncertainty on remanufacturing tolerance schemes from multiple perspectives, achieving effective part matching in terms of assembly precision. However, the differences in the remanufacturing value of failure surfaces and their effect on tolerance allocation have not yet been fully considered.

To address the limitations of existing research, this study proposes an innovative tolerance allocation method based on remanufacturing value at surface level. Unlike the conventional “part-level precision restoration + secondary machining trial assembly” model, this method quantifies remanufacturing value at the failure surface level by integrating individual part quality, overall assembly precision requirements, and enterprise processing capabilities. This strategy facilitates the allocation of higher machining precision to surfaces with greater remanufacturing value, thereby optimizing cost, reducing quality loss, and enhancing process stability.

In light of the previous work, one of the main factors that has hindered the further reduction of remanufacturing costs and the improvement of quality stability in used electromechanical products is the lack of effective methods to quantify the remanufacturing value of failure surfaces during the design stage and to use this information for secondary tolerance allocation design. To address this, a remanufacturing value quantification model was first established at the failure surface level to assess the value of each failure surface. Subsequently, a multi-objective optimization model for secondary tolerance allocation in remanufacturing was developed, targeting remanufacturing cost, quality loss, and process stability, with the remanufacturing value of failure surfaces incorporated as a correction factor. Finally, the beetle antennae search (BAS) algorithm was used to derive the optimal tolerance allocation scheme, ensuring compliance with machining capacity and assembly precision requirements.

This study aimed to establish a tolerance allocation method based on a remanufacturing value that aligns with the characteristics of the design and manufacturing stages of electromechanical product remanufacturing. The proposed method addresses the issues of high costs and quality fluctuations inherent in the traditional “individual part precision restoration + secondary machining trial assembly” model. By thoroughly considering uncertainties in the remanufacturing process, such as part quality, reassembly methods, and enterprise processing capabilities, the remanufacturing value of failure surfaces is quantified. This approach assigns higher machining precision to failure surfaces with greater remanufacturing value, thereby effectively reducing costs, minimizing quality loss, and improving process stability. This study provides a more tailored tolerance allocation strategy for remanufacturing used electromechanical products, laying a foundation for further enhancement of remanufacturing efficiency.

The remainder of this paper is structured as follows. In Section 2, the remanufacturing value index evaluation system and quantification method at the failure surface level are constructed, and based on this, a multi-objective optimization model for secondary tolerance allocation in the remanufacturing of used electromechanical products is constructed. In Section 3, the effectiveness and feasibility of the proposed method are validated through a remanufacturing case study of a gearbox. Finally, Section 4 summarizes the main conclusions of this paper and highlights future research directions.

2. Methodology

The consideration of remanufacturing value at the failure surface level during the design phase can render the final tolerance redistribution scheme more rational. The framework proposed in this paper for the redistribution of remanufacturing tolerance is bifurcated into two distinct stages, as depicted in Figure 1.

Stage1: Quantitative stage of remanufacturing value of each failure surface in the assembly dimension chain. First, a remanufacturing value evaluation system is established based on the return quality of the used parts; Then, the quantitative methods of each index are developed by using FCE (fuzzy comprehensive evaluation), ANN (artificial neural networks) and other methods; Finally, the remanufacture value of each failure surface is obtained by comprehensively considering the objective return quality and the subjective enterprise standard to confirm the index weight.

Stage2: Remanufacturing tolerance allocation scheme formulation and optimization stage. First, the multi-objective optimization model is established with the cost, quality, and process stability as the optimization objectives, the parts used strategy, closed-loop precision and enterprise processing capacity as constraints, and the remanufacturing value of the failure surface as the correction; Then, according to the characteristics of the optimization model, the BAS is used to obtain the optimal scheme of remanufacturing tolerance allocation.

2.1. Evaluation System of Remanufacture Value of Failure Surface

The remanufacturing value of the failure surface is jointly affected by the return quality of the used part and the failure surface itself. In terms of parts, the length of the remaining life of the parts will directly affect the stability of the parts during the second service, and the economic benefit determines the willingness of the company. In terms of failure surface, different return quality (such as damage degree of positioning benchmark, processing area, etc.) will bring different process difficulties, energy consumption, and pollutant emission changes to the failure surface with the change of processing accuracy. These factors will also lead to different remanufacturing values of the failure surface.

In conclusion, the remanufacturing value evaluation system for failure surfaces has been established, as illustrated in Figure 2. This evaluation indicator system is structured into two levels: the “part level” and the “failure surface level”, comprising five indicators. The economic indicator and remaining life indicator are determined by the overall condition of the remanufactured part, whereas the failure degree, process difficulty, and environmental benefit indicators are influenced by the specific conditions of the failure surface. In order to facilitate subsequent calculation, all indicators are adjusted to be benefit-oriented, meaning that the higher the quantified value of the indicator, the better is its performance.

2.1.1. Index Quantification

1.

Failure degree $F_i$

The failure degree of the failure surface not only affects the formulation of the remanufacturing process scheme [18], but also reflects the vulnerability of the failure surface when the product is in service. The higher the degree of damage, the greater is the vulnerability of the failure surface, indicating a higher conditional probability of failure under normal service conditions when subjected to initial events such as excessive operational loads, environmental factors (e.g., corrosion, temperature fluctuations), and accidental mechanical impacts. Obviously, under the same conditions, assigning a higher accuracy level to the failure surface, which is more vulnerable to damage, will cause the waste of processing resources.

To address the limitations of traditional failure degree evaluation methods, which classify failure as “high”, “medium”, or “low” and overly rely on the designer’s experience, leading to excessive subjectivity, FCE is used to quantify the failure degree of the failure surface. By refining fuzzy intervals, this approach reduces the influence of subjective judgment on the quantification results. After cleaning and testing, the damage volume of the failure surface can be obtained by the original CAD model [19]. Then, a series of quantitative equations are then used to map the damage volume to the score interval of [0–1].

A failure quantification standard for a used gearbox was taken as an example; the failure feature of failure surface can be extracted as shown in

Table 1. The failure quantification standard of used gearbox.

Failure Feature	Volumetric Damage Amount Interval	Quantified Score Equation	Quantified Score Interval [0, 1]
Wear	0	-----	1
	$0<x<3.0 {\mathrm{mm}}^3$	$f=1-{\displaystyle \frac{x-0}{3.0-0}}\times 0.5$	$(0.5,1]$
	$3.0 {\mathrm{mm}}^3\le x<6.0 {\mathrm{mm}}^3$	$f=0.5-{\displaystyle \frac{x-3.0}{6.0-3.0}}\times 0.5$	$[0,0.5)$
	$x\ge 6.0 {\mathrm{mm}}^3$	$0$	$0$
Corrosion	0	-----	1
	$0<x<2.6 {\mathrm{mm}}^3$	$f=1-{\displaystyle \frac{x-0}{2.6-0}}\times 0.5$	$(0.5,1]$
	$2.6 {\mathrm{mm}}^3\le x<5.2{\mathrm{mm}}^3$	$f=0.5-{\displaystyle \frac{x-2.6}{5.2-2.6}}\times 0.5$	$[0,0.5)$
	$x\ge 5.2 {\mathrm{mm}}^3$	$0$	$0$
Crack	0	-----	1
	$0<x<2.8 {\mathrm{mm}}^3$	$f=1-{\displaystyle \frac{x-0}{2.8-0}}\times 0.5$	$(0.5,1]$
	$2.8 {\mathrm{mm}}^3\le x<5.6 {\mathrm{mm}}^3$	$f=0.5-{\displaystyle \frac{x-2.8}{5.6-2.8}}\times 0.5$	$[0,0.5)$
	$x\ge 5.6 {\mathrm{mm}}^3$	$0$	$0$

.

2.

Economic benefits $E_{iN}$

The economic benefits determine whether a company is willing to engage in the remanufacturing of used parts. The final economic benefits of used parts after remanufacturing are determined by the price of the new parts, the procurement (or sale) cost of the used parts, and the remanufacturing processing costs. Therefore, when a used part N is determined to be remanufactured through the remanufacturability decision process [20], the remanufacturing economic benefit of N is shown as follows:

$$E_{iN}={\displaystyle \frac{C_{iNnew}-(C_{iNused}+C_{ijNre})}{C_{iNnew}}},$$

(1)

where $C_{iNnew}$ is the price of a new part for the remanufactured part N to which the failure surface i belongs; $C_{iNused}$ and $C_{ijNre}$ represent the procurement (or sale) cost of used part N and the historical average cost of used part N when using the remanufacturing process scheme j, respectively.

3.

Remaining life L_iN

The remaining life of a used part is a key indicator that affects the market acceptance of the remanufactured products. The failure surface of a part with a higher remaining life also has a higher remanufacturing value. By establishing the relationship between design data and service data of the used parts, the remaining life of the used parts N can be predicted as follows:

$$L_N=L_M-L_A,$$

(2)

where $L_N$ is the remaining life of the used parts; $L_M$ and $L_A$ are the average service life and actual service life of the used parts N, respectively. $L_A$ is a dynamic value that reflects the actual working time of the used parts in real service environment, and changes with variations in the service environment.

In this study, an artificial neural network (ANN) model is employed to predict the actual service life $L_A$ due to its strong self-learning and adaptive capabilities. By using the service condition date as the input and the actual service life as the output, a mapping relationship between the service environment and service life is established to predict the actual service life of the used parts in real operating conditions. The model is shown in Figure 3.

As shown in Figure 3, the ANN model employs a five-layer predictive structure. Initially, the Service Condition Data (Input Layer 1) of the used parts are utilized to predict the Actual Life. Subsequently, based on the predicted Actual Life, the model incorporates the Remanufacturing Process and Failure Degree to ultimately forecast the Remaining Life. The relevant training data are derived from the enterprise history database, using a gearbox gear shaft as an example, the parameters of the input and output layers are specified as follows:

Input Layer 1= {input power, operating speed, service time, operating temperature}

↓

Hidden Layer 1

↓

Output Layer 1 = {Actually Life}

Input Layer 2 = {Actually Life, remanufacturing process, failure degree}

↓

Hidden Layer 2

↓

Output Layer 2 = {Remaining Life}

After obtaining the remaining service life $L_N$, the quantification of the remaining life indicator for remanufactured part N, to which failure surface i belongs, is shown as follows:

$$L_{iN}={\displaystyle \frac{L_{iN}}{L_{iM}}},$$

(3)

It is worth noting that when the remanufactured part N has multiple failure surfaces needing tolerance design, the failure degree and remanufacturing process for each surface may differ, resulting in varying remaining life after remanufacturing. In this case, it should equal the remaining life of the failure surface with the shortest remaining life as follows:

$$L_{iN}=\min \left\{L_{iN}\right\}(i=1,2,\dots ),$$

(4)

4.

Process difficulty P_iN and Eco-benefits EC_iN

The performance of the failure surface in terms of processing difficulty, energy consumption, and pollutant emissions improves as the machining accuracy increases, but this performance varies across different failure surfaces [21]. For example, the location of the failure surface and the degree of damage to the positioning datum affect the difficulty of positioning detection, which in turn impacts processing difficulty; The area of failure surface ultimately affects the eco-efficiency in energy consumption and pollutant by affecting the processing time when the machining accuracy is improved. Moreover, the impact of machining accuracy on the process difficulty and eco-benefits of the failure surface is also constrained by the processing capacity of the enterprises.

The influence degree of failure surface machining accuracy on process difficulty and eco-benefits can be quantitatively represented by a value between 0 and 1. A higher value indicates less added impact on process difficulty, energy consumption, and pollution emissions when improving the unit process accuracy of the failure surface, resulting in better process difficulty and eco-benefit performance. The indicator of ease of process difficulty (P_iN)and eco-benefits (EC_iN) can be expressed by the fuzzy set $\Delta I\to$[significant impact, moderate impact, minor impact, minimal impact], and its corresponding evaluation value is $\left[\Delta I_1,\Delta I_2,\Delta I_3,\Delta I_4\right]$. Then, the membership function between PiN (ECiN) and unit process accuracy improvement of the failure surface can be established, as shown in Figure 4.

It is worth emphasizing that the fuzzy evaluation value set $\left[\Delta I_1,\Delta I_2,\Delta I_3,\Delta I_4\right]$ applies only to parts identified as suitable for remanufacturing. Following the preliminary remanufacturing feasibility assessment, the P_iN and E_Ci values for remanufactured parts are guaranteed to fall within a reasonable range. Consequently, extreme scenarios, such as excessively high process difficulty or inadequate environmental benefits, are effectively excluded during the fuzzy evaluation process.

As shown in Figure 4, the x-axis represents the process accuracy improvement preference of the failure surface which is quantified by experts, while the y-axis represents the membership between P_iN (EC_iN) and process accuracy improvement. $\left\{{pa}_1,{pa}_2,{pa}_3,{pa}_4\right\}$ are the process accuracy improvement preferences responding to $\left[\Delta I_4,\Delta I_3,\Delta I_2,\Delta I_1\right]$, which means the greater the value of $pa,$ the less the value of $\Delta I_i$. According to the previous quantified process accuracy improvement $pa$ and the scope of the process accuracy improvement $\left\{{pa}_1,{pa}_2,{pa}_3,{pa}_4\right\}$, the membership of process accuracy improvement $pa$ belonging to the evaluation value $\left[\Delta I_i,\Delta I_j\right]$ can be determined as $\left[\Delta I({pa}_i),\Delta I({pa}_j)\right]$ (i and j are adjacent integers), so the quantified value of the indicator of process difficulty (P_iN) and eco-benefits (EC_iN) can be calculated as follows:

$$T_2=\Delta I_i\times \Delta I({pa}_i)+\Delta I_j\times \Delta I({pa}_j),$$

(5)

2.1.2. Index Weight

After obtaining the quantitative value of each index, the next step is to obtain the weight of the index. For the remanufacturing tolerance redistribution of used electromechanical products, there is a difference between the return quality of used parts and the production standards of enterprises. The entropy weight method and AHP (analytic hierarchy process) are used to comprehensively formulate the weight of the index [22]. In which the entropy weight reflects the objective of the return quality differences between used parts, and AHP reflects the subjective production standards of the enterprises.

a.

Objective weight by entropy

Due to the uncertainty of the quality of the used parts, the remanufactured products lack the uniformity of the parts quality in new products, resulting in a relative evaluation of “better or worse” remanufacturing value indicators among failure surfaces (i.e., machining surfaces) within the target remanufactured assembly dimension chain. In this paper, the entropy weight model is used to measure the objective uncertainty of the remanufacturing value difference. The processes are as follows:

• Establishment of evaluation matrix. The number of failure surfaces is n, the number of indexes is m, according to Figure 2, the evaluation matrix of the failure surface remanufacturing value can be established as $E={\left\{e_{ij}\right\}}_{mn}$; $e_{ij}$ is the value of the ith failure surface in the jth index. The optimal set of remanufacturing value evaluations can be obtained as $E^{*}=\left\{e_1^{*},e_2^{*},\cdots ,e_j^{*}\right\}$, $e_j^{*}=\max \left\{e_{ij}\right\}$; then the formation of the standardization of the evaluation matrix $E^{\prime }$ is as follows:

  $$E^{\prime }={\left\{{e^{\prime }}_{ij}\right\}}_{mn},$$

  (6)

  where $e_{ij}^{\prime }={\displaystyle \frac{r_{ij}}{{\displaystyle \sum _j{r}_{ij}}}},j=1,2,\dots ,m;r_{ij}={\displaystyle \frac{e_{ij}}{e_j^{*}}}$.
• Calculation of entropy. For evaluation index j, the entropy value for the i-th failure surface is defined as follows:

  $$E_j=-K{\displaystyle \sum _{i=1}^n\frac{h_{ij}}{h_j}}\ln {\displaystyle \frac{h_{ij}}{h_j}},$$

  (7)

  where $h_j={\displaystyle \sum _{i=1}^n{h}_{ij}},j=1,2,\dots ,m;K={\displaystyle \frac{1}{\ln m}}$.
• Calculation of the entropy weight. Based on entropy, the entropy weight of the evaluation index j can be calculated as follows:

  $${\theta }_j={\displaystyle \frac{1-E_j}{{\displaystyle \sum _{j=1}^m\left(1-E_j\right)}}},$$

  (8)

b.

Subjective weight by AHP

The weight of AHP reflects the subjective importance of enterprises to different indexes in the remanufacture value evaluation system. AHP can determined the weight of the index through the pairwise comparison of the importance of each index. First, the importance of comparison of the pairwise index can be described as shown in

Table 2. The importance of comparison of pairwise index.

Relative Importance of Indicators	Quantitative Values
Equally important	1
Slightly more important	3
Moderately more important	5
Very important	7
Extremely important	9
Intermediate values	2/4/6/8

.

Then, the judgment matrix of importance between each index of the remanufacture value evaluation system is as follows, where is the importance of index i compared to index j:

$$U=\left\{\begin{array}{cccc}{u}_{11}& u_{12}& \cdots & u_{1j}\\ u_{21}& u_{22}& \cdots & u_{2j}\\ ⋮& ⋮& \cdots & ⋮\\ u_{i1}& u_{i2}& \cdots & u_{ij}\end{array}\right\}$$

Finally, the weight of evaluation index b in AHP is defined as follows:

$${\lambda }_j={\displaystyle \frac{{\overline{W}}_j}{{\displaystyle \sum _j{\overline{W}}_j}}},$$

(9)

where ${\overline{W}}_j=\sqrt[j]{M_i},M_i={\displaystyle \prod _{j=1}{u}_{ij}}(j=1,2,\dots ,6)$.

When the objective weight of the return quality and the subjective weight of the enterprise standard are obtained, respectively, the comprehensive weight of each evaluation index is shown as follows:

$${\omega }_j={\displaystyle \frac{{\lambda }_j\times {\theta }_j}{{\displaystyle \sum _{j=1}^5\left({\lambda }_j\times {\theta }_j\right)}}},$$

(10)

Then, the remanufacturing value of each failure surface can be standardized as below:

$${\beta }_i={\displaystyle \frac{{\displaystyle \sum _{j=1}^5{\omega }_j}{e}_{ij}}{\min \left({\displaystyle \sum _{j=1}^5{\omega }_j}{e}_{ij}\right)}},$$

(11)

2.2. Multi-Objective Optimization Model for Remanufacturing Tolerance Allocation

Once the remanufacturing value of the failure surface has been quantified, a multi-objective optimization model needs to be used to select an optimum tolerance redistribution scheme by comprehensively considering the information from the design, manufacturing, and assembly stage. Processing cost, quality loss, and process capability are considered as the variables in multi-objective functions, and the beetle antennae search algorithm (BAS) is employed to obtain the optimal scheme. Before introducing the objective functions, three points of assumptions are provided as follows:

• The dimension chain to be designed is in plane.
• The tolerance of the removal/additive process are symmetrically distributed.
• The failure surface to be designed needs the whole surface machining. For the failure surface of partial repair, it needs to be flush with the original plane without tolerance redistribution, and the tolerance must follow the original dimensions.

2.2.1. Objective Functions

• Tolerance-Cost Function

The process of remanufacturing for failure surface includes the removal process and additive process. The negative square cost model is used to construct the tolerance-cost function of the removal process as follows:

$$C_{ij-}=a_{0ij-}+{\displaystyle \frac{a_{1ij-}}{{\left(T_{ij-}\right)}^2}},$$

(12)

where $C_{ij-}$ is the cost of ith failure surface i using removal process j−; $a_{0ij-}$ and $a_{1ij-}$ are the fixed cost coefficient and tolerance cost coefficient, respectively; $T_{ij-}$ is the tolerance ith failure surface using removal process j−; the values of $a_{0ij-}$ and $a_{1ij-}$ can be obtained by fitting the statistical samples by the least square method.

The tolerance cost of the additive process is greatly affected by the equipment and parameters. According to reference [23], the tolerance satisfies the normal distribution when the equipment works with stable parameters. Therefore, the tolerance cost of the additive process in this paper is regarded as a fixed value related to the machining dimension, shown as follows:

$$C_{ij+}=a_{0ij+}=f\left(h_{ij+}\right),$$

(13)

where $C_{ij+}$ is the cost of the i-th failure surface using additive process j+; $a_{0ij+}$ is the fixed cost of process j+; $f\left(h_{ij+}\right)$ is the cost function related to the dimension recovery parameter $h_{ij+}$, which can be obtained by the original CAD model, and then $f\left(h_{ij+}\right)$ can be obtained by fitting the sample data.

To sum up, when the total number of removal processes and additive processes are a and b, respectively, the tolerance-cost function for the remanufacturing of used electromechanical products is shown as follows:

$$C={\displaystyle \sum _{i=1}^n\left({\displaystyle \sum _{j^{-}=1}^a{C}_{ij-}}+{\displaystyle \sum _{j^{+}=1}^b{C}_{ij+}}\right)},$$

(14)

2.

Tolerance-Quality Loss Function

According to the Taguchi theory, the deviation of the dimensions in the machining process will lead to quality loss of the mechanical product. In this paper, the exponential equation is used to construct the quality loss equation at the remanufacturing process level, shown as follows:

$$L=K_{ij}{(A_{ij0}-A_{ij})}^2,$$

(15)

$$K_{ij}={\displaystyle \frac{B_{ij}}{{\Delta }_{ij}^2}},$$

(16)

where $K_{ij}$ is the quality loss coefficient of the ith failure surface using process j; $A_{ij0}$ and $A_{ij}$ are the design dimension and actual machining dimension of the i-th failure surface using process j, respectively. When tolerances are symmetrically distributed, ${(A_{ij0}-A_{ij})}^2={\displaystyle \frac{{T_{ij}}^2}{4}}$, ${\Delta }_{ij}$ and $B_{ij}$ are the allowable limit dimension deviation of the ith failure surface using process j and the economic loss when the allowable deviation is exceeded.

Therefore, the total quality loss of the used electromechanical products is shown as follows:

$$L_{ij}={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^n\frac{K_{ij}{T_{ij}}^2}{4}}},$$

(17)

c.

Tolerance-Process Capability Function

The process capability index $C_p$ reflects whether the current process can stably produce products that meet the requirements. Maximizing $C_p$ within the specified range can maximize the process stability of the remanufacturing production under the tolerance design scheme and is a key factor in balancing the remanufacturing production costs and quality. The process capability of the failure surface i using process j is shown as follows:

$$C_{pij}={\displaystyle \frac{T_{ij}-{\epsilon }_0}{6\sqrt{{\sigma }_{ij}^2-(y^2-m^2)}}},$$

(18)

where ${\sigma }_{ij}$ represents the standard deviation of the dimensional deviation for failure surface i using process j; m and y represent the characteristic values of the current operation’s dimensional output and the target value, respectively, and ${\epsilon }_0$ represents the deviation of the distribution center relative to the design center. When the output operation exhibits a symmetrical tolerance distribution and the dimension center coincides with the design center, $C_{pij}$$={\displaystyle \frac{T_{ij}}{6{\sigma }_{ij}}}$. Under this condition, the total process capability for remanufacturing used electromechanical products is shown as follows:

$$D={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^n\frac{T_{ij}}{6{\sigma }_{ij}}}},$$

(19)

Combined with the remanufacturing value of failure surface, the objective function of the multi-objective optimization model is corrected as follows:

$$\left\{\begin{array}{l}C(T_{ij})={\displaystyle \sum _{i=1}^N{\frac{1}{\beta }}_i({\displaystyle \sum _{j=1}^{j-}{C}_{ij-}+}{\displaystyle \sum _{j=1}^{j+}{C}_{ij+})}}\hfill \\ L(T_{ij})={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^n{\beta }_i\frac{K_{ij}{T_{ij}}^2}{4}}}\hfill \\ D(T_{ij})={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^n{\beta }_i\frac{T_{ij}}{6{\sigma }_{ij}}}}\hfill \end{array}\right.,$$

(20)

2.2.2. Constraints Functions

• Constraints of closed-loop precision

The closed-loop dimension reflects the final assembly accuracy of the remanufactured product. Given the “as new” manufacturing standard, the assembly precision of remanufactured products must meet or exceed that of the original products. However, the uncertainty reuse mode of the used parts makes the control of closed-loop accuracy more complex; it is necessary to include this uncertainty in the integrated tolerance design of remanufacturing.

The parts of remanufactured products can be classified according to their reuse method into new parts, reused parts, and remanufactured parts. The tolerance of the new parts is determined by the specifications provided by the new product suppliers; for reused parts, the tolerance is considered zero due to their fixed dimensions; the tolerance for remanufactured parts, however, requires design and optimization.

Therefore, based on the uncertainty of part reused methods, the precision constraints of the closed-loop for remanufactured products are formulated as follows:

$${\displaystyle \sum {\xi }_i}{T}_i+{\displaystyle \sum {\xi }_{\mathrm{new} }}{T}_{\mathrm{new} }\le T_0,$$

(21)

where ${\xi }_i$ and ${\xi }_{new}$ are the tolerance transfer coefficient of the remanufacturing surface i and new parts, respectively; $T_i$ and $T_{new}$ are the tolerance of the remanufacturing surface i and new parts, respectively; $T_0$ represents the closed-loop dimension of the original assembly dimension chain.

2.

Constraints of processing capability

In the remanufacturing process, significant differences in processing capabilities exist among remanufacturing enterprises due to variations in equipment performance and personnel proficiency. Therefore, when designing tolerances, the processing capability constraints of the remanufacturing enterprise must be considered; the constraints of processing capability are shown as follows:

$$T_{i{j}^{-}\min }\le T_i\le T_{i{j}^{-}\max },$$

(22)

where $T_{ij\min }$ and $T_{ij\max }$ represent the minimum and maximum tolerances, respectively, that can be achieved for surface i using removal process j− under the current processing capability of the remanufacturing enterprise.

In summary, a multi-objective optimization model for remanufacturing the tolerance allocation of electromechanical products is established to minimize cost and quality loss while maximizing process stability. The model is constrained by closed-loop assembly accuracy, enterprise processing capabilities, and process stability requirements, as shown in the following equation:

$$\left\{\begin{array}{l}{f}_1(T_{ij})={\displaystyle \frac{C(T_{ij})-C_{min}}{C_{max}-C_{min}}}\\ f_2(T_{ij})={\displaystyle \frac{L(T_{ij})-L_{min}}{L_{max}-L_{min}}}\\ f_2(T_{ij})={\displaystyle \frac{D_{max}-D(T_{ij})}{D_{max}-D_{min}}}\end{array}\right.,$$

(23)

$$F{\left(T_{ij}\right)}_{min}={\omega }_1{f}_1\left(T_{ij}\right)+{\omega }_2{f}_2\left(T_{ij}\right)+{\omega }_3{f}_3\left(T_{ij}\right),$$

(24)

$$s.t.\left\{\begin{array}{l}{T}_{ijmin}\le T_{ij}\le T_{ijmax}\hfill \\ C_{pijmin}\le C_{pij}\le C_{pijmax}\hfill \\ {\displaystyle \sum _{i=1}^n{\zeta }_i}{T}_i+{\displaystyle \sum {\zeta }_{new}}{T}_{new}\le T_0\hfill \\ {\omega }_1+{\omega }_2+{\omega }_3=1\hfill \end{array}\right.,$$

(25)

2.2.3. Optimization Algorithm of BAS

In the tolerance design of remanufactured electromechanical products, it is essential to consider comprehensively the manufacturing costs, quality loss, and process capability. These objectives are constrained by design requirements, such as product precision, processing capability, and process stability, making it a typical multi-objective optimization problem. Traditional optimization algorithms typically convert multi-objective problems into single-objective problems before using predefined populations to optimize iteratively the objectives for an optimal solution. However, the need to filter the fitness values of multiple individuals in the population results in a relatively slow convergence speed.

To address the shortcomings of traditional multi-objective optimization algorithms, the beetle antennae search algorithm (BAS) is utilized to solve the multi-objective optimization model for remanufacturing tolerances. BAS is an intelligent algorithm developed by Professor Jiang Xiangyuan’s team at the China University of Petroleum in 2017 [24]. This algorithm simulates the foraging behavior of beetles in nature, using their antennae to search for scent information and adjusting their direction based on the intensity of the scents received, ultimately locating food through repeated iterations. When solving the tolerance optimization model, BAS can adjust its direction based solely on the relative fitness of individuals, without requiring gradient information or specific function forms; its global convergence properties have been thoroughly validated [25].

Compared to traditional multi-objective optimization algorithms, such as genetic algorithms (GA) and particle swarm optimization (PSO), BAS offers advantages including lower computational demands, faster search speeds, and a reduced likelihood of becoming trapped in local optima. As an efficient approximation search algorithm, the effectiveness of BAS has been validated across various engineering optimization problems [26].

Despite these advantages, BAS may encounter challenges in more complex scenarios outlined below.

1.

Handling Noise:

In noisy environments, additional preprocessing steps, such as data smoothing or averaging evaluations, may be necessary to mitigate the impact of noise on optimization results.

2.

Dynamic and Hybrid Strategies:

For problems with high-dimensional or highly complex objective functions, BAS may benefit from dynamic step-size adjustment or hybrid strategies that combine its lightweight structure with the global search capability of other algorithms.

Before employing BAS to solve the multi-objective optimization model, some simplifications are applied as follows:

• The beetle is simplified to a point mass, with the left and right antennae positioned on either side of the mass.
• The ratio of the movement step size $step$ to the distance between the antennae $d0$ is constant, that is, $step=c\times d0$.
• After each step of movement, the orientation of the beetle’s head is randomized.

Based on the above simplifications, the process of using BAS to solve the multi-objective optimization model for remanufacturing tolerance design is outlined as follows:

1.

Beetle position initialized.

A random solution is selected from the solution space of the optimization function F for remanufacturing tolerance design, with its coordinates serving as the initial position x of the beetle.

2.

Setting the antennae and the distance between left–right antennae.

Let $xl$ and $xr$ represent the points searched by the left and right antennae of the beetle, respectively, and let $d0$ denote the distance between the two antennae. Position $x$, $xl$, $xr$ on the same straight line, such that $d0=norm(xl-xr)$. According to the simplification condition (c), the beetle can orient in any direction, meaning that the direction vector of $xl$ towards $xr$ is arbitrary. A random vector $dir=(x_1,\dots ,x_i,\dots ,x_n)$ is used to represent this direction vector, and after normalization, $dir=dir/norm(dir)$. At this point, $dir=dir/norm(dir)$. When expressed in terms of the centroid, $xl$ and $xr$ can be represented as follows:

$$\left\{\begin{array}{c}xl=x+d0\times dir/2\\ xr=x-d0\times dir/2\end{array}\right.,$$

(26)

3.

Direction of Beetle Movement and Step Size

To solve the target function F, the scent values of the left and right antennae are calculated as: $fleft=f\left(xl\right) ; fright=f\left(xr\right)$, and then their magnitudes compared, after which the beetle moves using $step$ as the step size:

$$\left\{\begin{array}{c}x=x+step\times normal\left(xl-xr\right)\\ x=x-step\times normal\left(xl-xr\right)\end{array}\right.\hspace{1em}\begin{array}{c}fleft<frigh\\ fleft>fright\end{array},$$

(27)

After the beetle moves forward by a distance of step, the step size is updated as $step=step\times eta$. According to the literature [27], eta is set to 0.2.

The beetle stops moving when the precision requirements are met, or the specified number of steps is reached. At this point, the position of the beetle represents the optimal solution of the optimization model. The BAS solution process is illustrated in the accompanying Figure 5.

3. Case Study

As a typical electromechanical product, the gearbox is widely used in various engineering fields and serves as a significant source for remanufacturing. Its structural components include common part types such as housings, gears, and shafts, while its diverse service conditions lead to a wide range of failure types. A used gearbox model from an enterprise was selected as the research object. After prolonged service, this gearbox had experienced severe failures, rendering it unsuitable for further use and requiring remanufacturing.

After assessing the remanufacturability of each part of the gearbox, it was decided to replace the right bearing bush with a new part, the tolerance of the new right bearing bush is $6_{-0.050}^{+0.050}$ mm. The left bearing bush will be reused. The left housing, right housing, and gear shaft need remanufacturing, necessitating the design of remanufacturing tolerances. The closed-loop tolerance of the gearbox is $T_0\le 0.4 \mathrm{mm}$. The structure of the gearbox, the identifiers of the remanufacturing surfaces, and the assembly dimension chain are illustrated in Figure 6.

The failure surfaces which need tolerance design are A₁, A₃, A₄, A₅, A₈, A₉, and the tolerance transfer coefficient of the failure surfaces ${\xi }_i=1$ (i = 1, 3, …, 9). After remanufacturing the process design, the remanufacturing processes for each failure surface are shown in

Table 3. Remanufacturing process of each failure surface of the used gearbox.

Surface No.	Remanufacturing Process
$A_1$	rough turning→cold welding→fine turning
$A_3$	rough milling→bead welding→fine milling
$A_4$	bead welding→fine milling
$A_5$	rough turning→cold welding→fine turning
$A_8$	rough grinding→chromium plating→fine grinding
$A_9$	laser cladding→fine grinding

. The processing capability for the material removal processes is shown in

Table 4. Machining capability of removal processes.

	Equipment Models	Rough Machining/mm	Fine Machining /mm
Turning	CA6120	0.072–0.120	0.038–0.070
Milling	X5032	0.050–0.100	0.036–0.050
Grinding	M7350	0.054–0.072	0.018–0.052

, and the relevant cost parameters for each process are listed in

Table 5. The parameter of each remanufacturing process of the used gearbox.

	Process	${\mathit{a}}_0$/yuan	${\mathit{a}}_1$/yuan	$\mathit{\sigma }$/mm	${\mathit{K}}_{\mathit{i}\mathit{j}}$/yuan
$A_1$	rough turning	9.72	0.040	0.012	151
	cold welding	14.64
	fine turning	10.34	0.0120	0.007	256
$A_3$	rough milling	12.61	0.047	0.010	178
	bead welding	28.25
	fine milling	24.65	0.0103	0.006	232
$A_4$	bead welding	20.32
	fine milling	24.65	0.0105	0.006	773
$A_5$	rough turning	9.72	0.037	0.012	795
	cold welding	17.45
	fine turning	20.34	0.0120	0.007	780
$A_8$	rough grinding	11.91	0.044	0.009	700
	chromium plating	25.00
	fine grinding	23.31	0.0108	0.006	720
$A_9$	laser cladding	20.00
	fine grinding	23.31	0.0112	0.0060	745

. (Due to the confidentiality of cost data related to the company’s business, some data in the table were adjusted by ±20%). To ensure process stability, the enterprise has set a process capability threshold of $[C_{pijmin},C_{pijmax}]=[1.00,1.33]$.

3.1. Remanufacturing Value of the Failure Surface

3.1.1. Failure Degree of the Failure Surface

According to Section 2.1.2, the failure surface damage degree can be quantified based on the enterprise standards, and the damaged volume of the failure surface can be obtained from the original CAD model after cleaning and inspection. Taking failure surface A₁ as an example, the quantification standards for its damage degree are shown in Table 2. After inspection, the failure type of this surface was identified as crack failure, with a damaged volume of 2.69 mm³. Therefore, the damage degree of failure surface A₁ can be described as follows. The failure degree of other failure surface is shown in

Table 6. Quantification of failure degrees for gearbox failure surfaces.

Surface No.	Failure Type	Damaged Volume	${\mathit{F}}_{\mathit{i}}$
$A_1$	Impact crack	2.69 mm3	0.52
$A_3$	Fretting wear	1.68 mm3	0.73
$A_4$	Fretting wear	1.50 mm3	0.75
$A_5$	Fatigue crack	2.46 mm3	0.56
$A_8$	Fatigue wear	1.62 mm3	0.73
$A_9$	Chemical corrosion	2.24 mm3	0.57

$$F_1=\{\mathrm{Impact} \mathrm{crack}, 2.69 {\mathrm{mm}}^3\}=0.52$$

3.1.2. Economic Benefits of the Failure Surface

The price of new parts $C_{iNnew}$ and the procurement (or resale) price of used parts $C_{iNused}$ for gearbox remanufacturing parts were provided by the procurement department. The remanufacturing cost $C_{ijNre}$ was derived from the historical average cost of similar parts undergoing the same remanufacturing process, as supplied by the manufacturing department. Taking the gear shaft part as an example, $E_{iN}$ of the gear shaft = {$C_{iNnew}$ = 450, $C_{iNused}$ = 280, $C_{ijNre}$ (A8: chromium plating→fine grinding+ A9: laser cladding→fine grinding) = 48.5}

$$E_{iN}={\displaystyle \frac{480-(280+48.5)}{480}}=0.73$$

3.1.3. Remaining Life of the Failure Surface

The remaining life of the failure surface after remanufacturing is influenced by factors such as service condition data, failure degree, and remanufacturing processes. As shown in Figure 2, the remaining life indicator for the failure surface is quantified using ANN. Taking the gear shaft as an example, the training data for the ANN model are presented in

Table 7. The training data for the ANN model.

Dataset No.	Input Power (kW)	Operating Speed (r/min)	Service Time (h)	Operating Temperature (°C)	Actual Life (h)	Remanufacturing Process	Failure Degree	Remaining Life (h)
1	2.63	980	2400	75 ± 20	2657	Rough Grinding→Chromium Plating→Fine Grinding	0.73	8740
2	2.73	960	2200	73 ± 20	2657	Laser Cladding→Fine Grinding	0.57	9025
3	2.85	1000	2500	78 ± 20	2700	Rough Grinding→Nickel Plating→Fine Grinding	0.68	8900
4	3.00	1050	2600	80 ± 20	2750	Thermal Spraying→Fine Grinding	0.65	9100
5	2.50	950	2300	70 ± 20	2600	Laser Cladding→Polishing	0.55	9200
6	3.20	1100	2800	85 ± 20	2850	Welding Repair→Plasma Spraying→Fine Grinding	0.75	8600
7	2.70	980	2400	75 ± 20	2650	Surface Hardening→Fine Grinding	0.60	9000
…
23	2.55	920	2100	68 ± 20	2550	Rough Grinding→Electroless Plating→Polishing	0.70	8800
24	2.95	1020	2700	82 ± 20	2800	Laser Shock Peening→Fine Grinding	0.58	9150
25	3.10	1070	2900	88 ± 20	2950	Thermal Spraying→Surface Coating→Fine Grinding	0.62	9050
26	2.60	960	2250	73 ± 20	2625	Rough Grinding→Chromium Plating→Heat Treatment	0.72	8800
27	3.25	1120	2950	90 ± 20	3000	Cold Spraying→Fine Grinding	0.55	9300
28 #	2.80	1000	2500	77 ± 20	2700	Surface Hardening→Polishing	0.68	8900/8550 *
29 #	2.90	1050	2650	80 ± 20	2750	Plasma Spraying→Fine Grinding	0.60	9000/8693 *
30 #	2.75	970	2400	75 ± 20	2675	Laser Surface Melting→Fine Grinding	0.58	9100/9115 *

# represents the validation group; * represents the model-predicted values.

, where entries numbered 1–27 correspond to the training set, and entries numbered 28–30 correspond to the validation set. The results indicate that the neural network model achieves a prediction accuracy of over 96%, demonstrating high reliability and precision in its predictions.

The service condition data of the used gear box in Input Layer 1 are {input power: 2.63 KW, operating speed: 980 r/min, service time: 2400 h, operating temperature: 75 ± 20 °C}. Input Layer 2 includes:

A8 = {actual service time: 2657 h, remanufacturing process: rough grinding→chromium plating→fine grinding, failure degree: 0.73};

A9 = {actual service time: 2657 h, remanufacturing process: laser cladding→fine grinding, failure degree: 0.57};

The average service life L_M of this gear shaft is 9500 h. After training the ANN model, the remaining life of failure surfaces A8 and A9 following remanufacturing were determined to be 8740 h and 9025 h, respectively. According to Equation (3), the quantified remaining life indicator for the gear shaft is $L_M={\displaystyle \frac{8740h}{9500h}}=0.92$. The remaining life of each failure surface is shown in

Table 8. The remaining life of each failure surface of the used gearbox.

Surface No.	Remanufacturing Part	Remaining Life	LiN
$A_1$	left housing	0.92	0.92
$A_3$		0.95	0.92
$A_4$	right housing	0.85	0.80
$A_5$		0.80	0.80
$A_8$	gear shaft	0.74	0.71
$A_9$		0.71	0.71

.

3.1.4. Process Difficulty and Eco-Benefits of the Failure Surface

An expert group consisting of nine experts from the company’s design, manufacturing, and management departments was formed to quantify the process difficulty P_iN and eco-benefits EC_iN for each failure surface of the gearbox.

A fuzzy evaluation model was used to quantify the process difficulty value for failure surface A₁. The precision improvement potential interval for A₁ is pa = [0.0,0.3,0.6,0.9], corresponding to the inspection response interval $\Delta I=[0.2,0.5,0.7,1.0]$. Based on expert evaluation, the precision improvement preference for A₁ was determined to be 0.15. According to Equations (2)–(10), the process difficulty P_iN of A₁ is as follows:

$$P_{A1}=\Delta I_i\times \Delta I\left(p{a}_i\right)+\Delta I_j\times \Delta I\left(p{a}_j\right)=0.2\times {\displaystyle \frac{0.3-0.15}{0.3-0.0}}+0.5\times {\displaystyle \frac{0.15-0.0}{0.3-0.15}}=0.35$$

Similarly, the eco-benefit EC_iN of A₁ is pa = [0.9,0.6,0.3,0.0], the EC_iN of A₁ is as follows:

$$E{C}_{A1}=\Delta I_i\times \Delta I\left(p{a}_i\right)+\Delta I_j\times \Delta I\left(p{a}_j\right)=0.9\times {\displaystyle \frac{0.3-0.15}{0.3-0.0}}+0.6\times {\displaystyle \frac{0.15-0.0}{0.3-0.15}}=0.75$$

3.2. Remanufacturing Value of the Failure Surface

According to Section 3.1, the remanufacturing value index of each failure surface of the used gearbox is quantified as shown in

Table 9. The remaining life of each failure surface of the used gearbox.

Surface No.	${\mathit{F}}_{\mathit{N}}$	LiN	${\mathit{E}}_{\mathit{i}\mathit{N}}$	PiN	ECiN
$A_1$	0.52	0.92	0.48	0.35	0.75
$A_3$	0.73	0.92	0.48	0.52	0.45
$A_4$	0.75	0.80	0.52	0.54	0.44
$A_5$	0.56	0.80	0.52	0.37	0.71
$A_8$	0.73	0.71	0.73	0.42	0.64
$A_9$	0.57	0.71	0.73	0.44	0.61

. The subjective weights of each evaluation index are based on the enterprise standards (AHP), the objective weights for return quality (EWM), and the comprehensive weights as shown in

Table 10. The weight of each remanufacturing value index of the used gearbox.

	${\mathit{F}}_{\mathit{N}}$	LiN	${\mathit{E}}_{\mathit{i}\mathit{N}}$	PiN	ECiN
AHP	0.145	0.221	0.290	0.201	0.134
Entropy	0.191	0.150	0.239	0.195	0.228
Comprehensive	0.139	0.166	0.347	0.196	0.153

. The remanufacturing value of each failure surface is as shown in

Table 11. Remanufacturing value of each failure surface of the used gearbox.

Surface No.	${\mathit{A}}_1$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$	${\mathit{A}}_8$	${\mathit{A}}_9$
Remanufacturing value	1.01	1.04	1.03	1.00	1.14	1.10

.

3.3. Optimization by BAS

The beetle antennae search (BAS) algorithm was used to solve the model with the following parameter settings: initial beetle step length step = 1, ratio constant c = 2, and iteration count n = 50. The optimization objective weights specified by the enterprise are ${\omega }_1=0.5, {\omega }_2=0.2, {\omega }_3=0.3$. To demonstrate the applicability of the proposed algorithm in solving the tolerance allocation model, the genetic algorithm (GA) was also employed as a comparative method for simultaneous model solving. The convergence processes of the two algorithms are illustrated in Figure 7, and Table 11 displays the resulting tolerances for each machining surface along with the processing costs, quality loss, and process capability.

As illustrated in Figure 7, both the BAS and GA algorithms exhibit comparable initial convergence speeds, achieving rapid convergence during the early stages. However, as a multi-peak problem, the remanufacturing tolerance allocation multi-objective optimization model presents considerable challenges for global optimization. The BAS algorithm leverages its step-size advantage to effectively escape local optima, thereby yielding superior optimization results compared to the GA algorithm.

3.4. Results and Discussions

The final optimized tolerance scheme for gearbox remanufacturing is presented in

Table 12. Optimal tolerance scheme of the remanufactured gearbox.

Surface No.	$\mathbf{Process} ({\mathit{T}}_{\mathit{i}\mathit{j}}$/mm)	$\mathit{C}$/yuan	$\mathit{L}$/yuan	$\mathit{D}$
$A_1$	rough turning (0.117)→cold welding→fine turning (0.054)	189.2	81.1	13.41
$A_3$	rough milling (0.068)→bead welding→fine milling (0.044)
$A_4$	bead welding→fine milling (0.047)
$A_5$	rough turning (0.100)→cold welding→fine turning (0.050)
$A_8$	rough grinding (0.062)→chromium plating→fine grinding (0.031)
$A_9$	laser cladding→fine grinding (0.025)

. In this scheme, the closed-loop dimension T = 0.351 mm meets assembly precision requirements, eliminating the need for trial assembly and secondary machining in the assembling stage.

According to the enterprise-defined original tolerance scheme for this used gearbox, the final process restores dimensions and tolerances of the remanufactured parts to ensure assembly precision and success rate, while earlier processes adhere to a minimum processing cost principle for tolerances. Under this scheme, the remanufacturing cost, quality loss, and process capability of the gearbox are presented in

Table 13. Original tolerance scheme of the remanufactured gearbox.

Surface No.	$\mathbf{Process} ({\mathit{T}}_{\mathit{i}\mathit{j}}/mm)$	$\mathit{C}$/RMB	$\mathit{L}$/RMB	$\mathit{D}$
$A_1$	rough turning (0.120)→cold welding→fine turning (0.040)	212.5	94.5	12.8
$A_3$	rough milling (0.100)→bead welding→fine milling (0.040)
$A_4$	bead welding→fine milling (0.040)
$A_5$	rough turning (0.120)→cold welding→fine turning (0.040)
$A_8$	rough grinding (0.072)→chromium plating→fine grinding (0.020)
$A_9$	laser cladding→fine grinding (0.020)

.

Compared to the original scheme, the optimized tolerance scheme reduces costs by 11%, decreases quality loss by 14%, and improves process stability by 4.8%. Additionally, as the database continues to be updated and refined, and as expert evaluation systems become more mature, the additional costs associated with evaluation and algorithms will decrease rapidly. This will further amplify the cost advantages of the proposed method. These results demonstrate the effectiveness and practicality of the proposed method.

To further validate the robustness of the proposed approach, a sensitivity analysis was conducted to assess the impact of variations in the closed-loop dimensional tolerance on the key parameters, as shown in

Table 14. Sensitivity analysis of closed-loop tolerance on key parameters.

Initial Size	Closed-Loop	Cost	Quality Loss	Process Stability
Initial size	0.40 mm	189.2	81.1	13.41
Tolerance Expansion +10%	0.44 mm	166.5	89.2	14.75
Tolerance Expansion +20%	0.48 mm	147.6	98.1	16.09

.

The analysis shows that expanding tolerances by 10% to 20% can reduces costs by 12% to 22%, while quality loss increases by 10% to 21%, and process stability improves by 10% to 20%. These results highlight the trade-offs between cost, quality, and stability in tolerance allocation. Achieving the optimal balance requires careful consideration of these factors to meet both economic and quality objectives under varying conditions.

4. Summary and Conclusions

A remanufacturing value-based tolerance allocation method is proposed for remanufactured electromechanical products, encompassing a remanufacturing value evaluation index system, indicator quantification and weighting, and the optimization of the tolerance allocation scheme. First, a failure surface remanufacturing value quantification system was constructed, based on the overall return quality of the remanufactured parts and the specific conditions of the failure surfaces (i.e., remanufacturing processing surfaces). Then, models such as artificial neural network (ANN) and fuzzy comprehensive evaluation (FCE) were utilized to quantify the indicators according to their specific characteristics, while subjective and objective weights were determined using an integrated approach combining the analytic hierarchy process (AHP) and the entropy weight method (EWM). This ensures a balanced weighting approach, combining expert judgment with objective data variability. Finally, a multi-objective optimization model was established to optimize cost, quality loss, and process stability. The model incorporates constraints such as part recombination methods and closed-loop precision requirements, corrected by the remanufacturing value of the failure surfaces. The beetle antennae search (BAS) algorithm was used to solve the model, and a case study on a used gearbox validated the effectiveness of the proposed method. In general, the remanufacturing value-based tolerance allocation method demonstrates significant advantages over the traditional “individual part precision restoration + secondary trial assembly” remanufacturing mode. These advantages are evident in terms of reduced cost, minimized quality loss, and enhanced process stability. Unlike the universal tolerance schemes employed for new product batches, tolerance allocation for remanufactured products adopts a customized “case-by-case” approach. Specifically, it must account for uncertainties related to part recombination methods, part recovery quality, and the remanufacturing processes themselves. Higher precision tolerances are allocated to processing surfaces with greater remanufacturing value to maximize residual value. The BAS algorithm supports the generation of optimal tolerance schemes, thus minimizing costs, quality loss, and process instability.

The scalability of the proposed method was demonstrated to be applicable for both small- and medium-sized enterprises with limited resources and large manufacturing enterprises with advanced infrastructures. For small- and medium-sized enterprises, implementation can be initiated with lightweight solutions, such as exporting optimization results in standard formats (e.g., CSV or JSON) and integrating them manually into existing production workflows. Cloud-based platforms and open-source tools can be utilized for data analysis and optimization. During CAD model updates or failure surface scans, modeling can be restricted to critical machining surfaces to conserve computational resources, thereby minimizing infrastructure costs. Databases can be expanded and refined incrementally as production batches increase. For large enterprises, deeper integration can be achieved by embedding multi-objective optimization models and algorithms into MES systems, enabling real-time data acquisition, optimization, and feedback loops. ERP systems can be employed to supply precise cost and accuracy data to refine optimization objectives, while MES systems can automatically execute optimized tolerance schemes on production lines. Seamless integration can be facilitated through APIs and standardized data interfaces, ensuring efficient and scalable implementation across large-scale industrial environments.

Future research is recommended to extend the proposed method to more complex assembly dimension chains, such as parallel or hybrid chains, while incorporating additional process parameters, including environmental and energy considerations. Furthermore, adaptive integration strategies for ERP/MES systems, tailored to diverse industrial production scenarios, should be investigated to enhance the generality and automation of the proposed approach. In terms of algorithmic performance, although superior convergence characteristics can be achieved by the BAS algorithm by leveraging step-size adjustments to escape local optima under convergence speeds comparable to GA, its performance remains sensitive to parameter tuning—particularly the step size—and its robustness under high-dimensional or noisy conditions has yet to be thoroughly investigated. It is recommended that future research incorporate adaptive parameter adjustment mechanisms or explore hybridization with other algorithm to broaden its applicability in more complex scenarios.