Multi-objective optimization based on improved non-dominated sorting genetic algorithm II for tolerance allocation of auto-body parts

Abstract

The auto-body is usually composed of compliant sheet metals. The assembly variation is inevitable in the process of auto-body assembly. A reasonable tolerance allocation method for compliant sheet metal body parts is an important means to control assembly variation and improve the quality of automotive assembly. In this article, on the basis of tolerance–deviation model and tolerance–cost model, a multi-objective optimization model for tolerance allocation of auto-body is established. Part tolerance and manufacturing cost are regarded as design variables in the model. Then, orthogonal design and cumulative sorting strategy are proposed to modify the non-dominated sorting genetic algorithm II. Finally, the process of tolerance allocation is demonstrated through an example of the rear lamp bracket assembly. The results show that the improved non-dominated sorting genetic algorithm II algorithm has obvious optimization effect.

Keywords

Tolerance allocation non-dominated sorting genetic algorithm II sheet metal parts multi-objective optimization tolerance–cost model

Introduction

Auto-body is one of the four major components of an automobile. The quality of auto-body has an important impact on vehicle performance and appearance. However, there is a conflict between quality requirements and manufacturing costs. The variation level of the compliant sheet metal body parts is the key factor affecting the conflict. Therefore, it is of great significance to find a reasonable tolerance allocation method to solve the contradiction between quality and cost.

Tolerance allocation is the process of assigning the known assembly tolerance to the part tolerance according to certain criteria. The traditional method of tolerance allocation is to fit out the relation curve between tolerance and cost by a set of empirical tolerance–cost data, and then get the tolerance–cost numerical model. Typical models are exponential model,¹ reciprocal model² and polynomial model,³ and so on. All these models are based on the lowest cost for tolerance allocation. It is very hard to balance the quality and cost.

In recent years, many studies on multi-objective tolerance allocation have been done. These studies mainly focused on two aspects: the establishment of tolerance allocation model and the selection of optimization algorithm.

Heish⁴ put forward a flexible cost–tolerance optimization model by considering the manufacturing cost of the components, the process capability of the manufacturing operations, and the quality loss of products. Li et al.⁵ established multilevel optimization for tolerance allocation. Both quality- and cost-driven tolerance allocation problems are formulated. Cui and Zhang⁶ presented an integration optimization model of tolerance and maintenance for multi-station sheet metal assembly. Based on the variation propagation state space model, a quality loss model of multi-station assembly processes was built. Zheng et al.⁷ proposed an assembly tolerance allocation method using coalitional game theory in an attempt to find a trade-off between the assembly cost and the assembly quality. A multi-objective optimization model is introduced by using the reciprocal function and the worse case function as objective functions. Fan et al.⁸ set up a multi-objective tolerance allocation model based on quality loss model and cost–tolerance model. Balamurugan et al.⁹ suggested a model for optimal tolerance allocation by considering both tolerance cost and the present worth of quality loss such that the total manufacturing cost/loss is minimized.

Most of recent researches usually regard manufacturing cost and quality loss cost as objective functions. These studies mainly follow the rigid body assembly error theory, which is not suitable for the tolerance allocation of compliant sheet metal parts.

For different models, genetic algorithm, particle swarm algorithm and teaching-learning-based optimization (TLBO) algorithm, and so on are the main methods in current research to optimize the tolerance allocation problem. B Forouraghi¹⁰ used an enhanced particle swarm algorithm to simultaneously optimize product performance criteria, the manufacturing cost, and the stack-up tolerance. Sivakumar et al.¹¹ presented a new method based on elitist non-dominated sorting genetic algorithm II (NSGA-II) and differential evolution (DE) for the sensitivity-based conceptual design and the tolerance allocation for mechanical assemblies. Rao and More¹² developed a TLBO method for optimal selection of design and manufacturing tolerances. In order to achieve zero percentage rejection, Ramesh Kumar et al.¹³ proposed a new approach by implementing the non-traditional techniques: genetic algorithm, elitist NSGA-II, and DE for determining the optimum tolerance, zero percentage rejection, and manufacturing cost considering the varying quality loss constants. Among these optimization algorithms, NSGA-II is one of the most popular multi-objective optimization algorithms for its flexibility and extensiveness.

In this article, a multi-objective model for tolerance allocation of compliant sheet metals is discussed and the NSGA-II algorithm is introduced to solve the multi-objective problem. First, the assembly variation analysis model based on the power balance method is used to get the sensitivity matrix between parts and assembly. Then, the tolerance–deviation model and the tolerance–cost model are developed on the basis of the sensitivity matrix. After that, the multi-objective optimization model for tolerance allocation of the auto-body parts is established by considering the part tolerance and manufacturing cost as a multi-objective problem. Since the Pareto solution set and the convergence rate obtained by NSGA-II algorithm is not ideal, an improved NSGA-II algorithm is proposed by adopting orthogonal design and the cumulative sorting strategy.¹⁴ Finally, the proposed method is illustrated through a case study.

Assembly variation analysis model for sheet metal parts

The commonly used variation analysis methods for sheet metal part are influence coefficient method,¹⁵ static analysis method, and finite element method.¹⁶ Among these methods, the influence coefficient method is the most widely used method. This method assumes that the sheet metal part is in elastic deformation range and the stiffness matrix can be extracted by using finite element simulation software. Based on the influence coefficient method, the power balance principle is used in this article to set up the assembly variation analysis model. The sensitivity matrix between part and assembly can be obtained from the model. The assembly variation analysis model according to the power balance method¹⁷ is shown as formulas (1) and (2)

U = \frac{\sqrt{K_{1}}}{\sqrt{K_{1}} + \sqrt{K_{2}}} \cdot V_{1} + \frac{\sqrt{K_{2}}}{\sqrt{K_{1}} + \sqrt{K_{2}}} \cdot V_{2}

(1)

U = S_{1} \cdot V_{1} + S_{2} \cdot V_{2}

(2)

where $U$ is the assembly deviation vector; $V_{1}$ , $V_{2}$ is the deviation vector of part 1 and part 2, respectively; $K_{1}$ , $K_{2}$ is the stiffness matrix of part 1 and part 2, respectively; $S_{1}$ is the sensitivity matrix of part 1; and $S_{2}$ is the sensitivity matrix of part 2.

According to formulas (1) and (2), we can obtain the sensitivity matrix as follows

S_{1} = \frac{\sqrt{K_{1}}}{\sqrt{K_{1}} + \sqrt{K_{2}}}, S_{2} = \frac{\sqrt{K_{2}}}{\sqrt{K_{1}} + \sqrt{K_{2}}}

(3)

The stiffness matrix $K$ can be obtained by using finite element simulation software based on the influence coefficient method. If not consider other variation sources, the linear relationship between assembly deviation and part deviation can be written as

U = S_{1} \cdot V_{1} + S_{2} \cdot V_{2} = S \cdot V

(4)

where $S$ is the sensitivity matrix of the assembly output deviation to the part input deviation, and $V$ is parts deviation vector.

Multi-objective optimization of auto-body tolerance allocation

The assembly deviation has a great influence on the final assembly quality. The tolerance–deviation and the tolerance–cost models are introduced by considering the assembly deformation. On the basis of these models, the multi-objective optimization model for tolerance allocation is established by taking the manufacturing tolerance of parts as the design variable. The optimization model takes into account the manufacturing and process tolerance of the parts. It can be used to analyze the effects of variation propagation and manufacturing deviation on the tolerance allocation.

The tolerance–deviation relationship model

In section “Assembly variation analysis model for sheet metal parts,” the sensitivity matrix of part deviation and assembly deviation is obtained by using the power balance method, and the linear relationship between them is established as $U = S \cdot V$ . If suppose that the input deviation of the parts is random, the assembly quality can be described by the mean deviation and variance of the KPC (key product characteristic). Therefore, the linear equation can be expressed as an expectation¹⁸

E (U) = E (SV)

(5)

μ_{w} = S μ_{u}

(6)

where $μ_{w}$ is the mean deviation of assembly, and $μ_{u}$ is the mean deviation of part.

The covariance matrix of assembly variation can be defined as follows

Σ_{w} = S Σ_{u} S^{T}

(7)

where $Σ_{w}$ is the covariance matrix of the assembly deviation, $Σ_{u}$ is the covariance matrix of part deviation, and

Σ_{u} = [\begin{matrix} σ_{11}^{2} & \dots & σ_{1 n}^{2} \\ ⋱ & ⋰ \\ ⋮ & σ_{kk}^{2} & ⋮ \\ ⋰ & ⋱ \\ σ_{1 n}^{2} & σ_{nn}^{2} \end{matrix}]

(8)

The variation propagation in multi-station assembly system can be regarded as a discrete, linear system, so we can use the discrete state space method^19,20 to get the assembly system model. The dimension deviation of parts can be expressed as follows

X (k) = A (k) X (k - 1) + B (k) U (k)

(9)

And $A (k)$ , $B (k)$ is

A (k) = (S (k) - P (k) + I) (I + M (k))

(10)

B (k) = [- (S (k) - P (k) + I) \begin{matrix} M (k) & - (S (k) - P (k)) \end{matrix}]

(11)

where $X (k)$ is the state vector, $A (k)$ is the state matrix, $B (k)$ is the input matrix, $U (k)$ is the input vector, $S (k)$ is the sensitivity matrix, $P (k)$ is the deformation matrix, $M (k)$ is the relocation matrix, and $I$ is the identity matrix.

Assuming that the part deviation statistics are independent, that is, the covariance is zero, so the covariance relationship can be derived from the formulas (7) and (8)

Σ_{X_{k}} = A_{k} Σ_{X_{k - 1}} A_{k}^{T} + B_{k} Σ_{U_{k}} B_{k}^{T}

(12)

σ_{k}^{2} = diag (Σ_{k})

(13)

where $σ_{k}^{2}$ is the diagonal element of the covariance matrix $Σ_{k}$ .

Formula (13) contains the KPC variance of the part, so the tolerance–deviation relationship between KPC and KCC (key control characteristic) can be obtained.⁵ That is

t_{k}^{2} = A_{k}^{2} t_{k - 1}^{2} + B_{k}^{2} τ_{k}^{2}

(14)

t_{k} = F_{t} (t_{k - 1}, τ_{k}) = \sqrt{A_{k}^{2} t_{k - 1}^{2} + B_{k}^{2} τ_{k}^{2}}

(15)

where product tolerance $t$ is the tolerance corresponding to the KPC, and process tolerance $τ$ is the tolerance corresponding to the KCC.

In the article, the scope of the product tolerance is $0.01 mm \leq t \leq 2 mm$ , the scope of the process tolerance is $0.01 mm \leq τ \leq 2 / 3 mm$ .

The tolerance–cost relationship model

Reducing tolerance will increase the cost, while using large tolerance will lower the product quality and increase the product repair rate. Cost is divided into manufacturing cost and fixed cost. Manufacturing cost is related to the machining accuracy, fixture layout, and production specifications. Fixed cost includes clamping, loading and unloading, handling, and other operations. The most commonly used tolerance–cost model is exponential model and reciprocal model. In this article, the exponential model is applied to represent the cost of the sheet metal assembly system. For the n-dimensional product tolerances and m-dimensional process tolerances, the tolerance–cost relationship can be expressed as

c = F_{c} (t, τ) = \sum_{i = 1}^{n} e^{- 3 t_{i}} + \sum_{i = 1}^{m} e^{- 3 τ_{i}}

(16)

Multi-objective optimization model for tolerance allocation

The optimization model for tolerance allocation is developed based on the tolerance–deviation model and the tolerance–cost model. In this model, tolerance allocation takes into account both cost and quality, which is a multi-objective optimization problem. The model can be written as

Objective function

{\begin{matrix} min F_{c} (t, τ) = \sum_{i = 1}^{n} e^{- 3 t_{i}} + \sum_{i = 1}^{m} e^{- 3 τ_{i}} \\ min F_{t} (t_{k - 1}, τ_{k}) = \sqrt{A_{k}^{2} t_{k - 1}^{2} + B_{k}^{2} τ_{k}^{2}} \end{matrix}

(17)

Constraint conditions

Product tolerance : 0.01 mm < t_{i} < 2 mm

Process tolerance : 0.01 mm < τ_{i} < 2 / 3 mm

The optimization model consists of two objective functions $F_{c} (t, τ)$ and $F_{t} (t_{k - 1}, τ_{k})$ . Different from the single objective problem, the multi-objective problem may have multiple solutions. The solution set is usually called the Pareto solution set. The main purpose of the multi-objective optimization model is to make the tolerance allocation more reasonable, so as to achieve minimum cost and best quality.

Improved NSGA-II algorithm

Since the Pareto solution set and the convergence rate obtained by NSGA-II algorithm is not ideal, the NSGA-II algorithm is improved by using orthogonal design and the cumulative sorting strategy.

The initial population based on orthogonal design

The initial population is generated by the stochastic method in standard genetic algorithm, so the distribution of the initial population is not uniform and it is difficult to produce a good, representative individual. However, individuals obtained by the orthogonal design method can be more uniformly distributed in the solution space, which helps the NSGA-II algorithm to search the entire solution space and easy to bring good individuals.

As we know, the orthogonal design method is usually used for continuous space, while the problem to be solved in this article is discrete, so we need to quantify the continuous space. Each value in $X = {(x_{1}, x_{2}, \dots, x_{N}) | l_{i} \leq x_{i} \leq u_{i}}$ corresponds to each factor of the orthogonal design, so each individual has $N$ factors. The range of tolerance factor $x_{i}$ is quantified into $Q_{1}$ levels: $α_{i, 1}, α_{i, 2}, \dots, α_{i, Q_{1}}$ , where $Q_{1}$ is odd, $α_{i, j}$ is given by the following formula

α_{i, j} = {\begin{matrix} l_{i} & j = 1 \\ l_{i} + (j - 1) (\frac{u_{i} - l_{i}}{Q_{1} - 1}) & 2 \leq j \leq Q_{1} - 1 \\ u_{i} & j = Q_{1} \end{matrix}

(18)

The steps of generating the initial population by orthogonal design method can be referred to Chen and Cai.²¹

The cumulative sorting strategy

NSGA-II algorithm uses SBX (Simulated Binary Crossover) operator. The search ability of SBX operator is weak, which makes the Pareto sorting values cannot well reflect the density information of individuals. Therefore, the search ability of NSGA-II algorithm is decreased and the good individual may be eliminated.

In order to take into account the Pareto sorting values and density information of individuals, the cumulative sorting strategy is adopted in this article. The cumulative sorting method is as follows: first, each individual’s Pareto sorting value $p_{1}$ is obtained by hierarchical ordering strategy of NSGA-II algorithm. At the same time to evaluate each individual and calculate the dominating numbers $n_{p}$ of each individual, then the cumulative sorting value can be written as $p_{t} = p_{1} + n_{p}$ , where $p_{1}$ is sorting value of each individual, $n_{p}$ is dominating numbers of each individual, and $p_{t}$ is the cumulative sorting value.

The tolerance allocation based on the improved NSGA-II algorithm

Algorithm flow of the improved NSGA-II is shown in Figure 1. First, the initial population is generated by orthogonal design method and the non-dominated sorting and crowding distance calculation are carried out. After selection, crossover, mutation, and other basic genetic operations, the newly generated offspring population is combined with the parent population. Then, non-dominant sorting and crowding distance are calculated again. After that, the cumulative sorting method is applied to generate a new population. The new population is the same size as the initial population. Continue the loop until the condition is terminated.

Figure 1.

Flow chart of the improved NSGA-II algorithm.

Application

Case study description

Rear lamp bracket is a typical auto-body panel, it has the characteristics of space curved surface, asymmetric shape, and high surface quality. Assembly variation is easy to accumulate at the rear lamp bracket which affects the stability of the rear lamp and the appearance of the automobile. This article takes a rear lamp bracket assembly as an example to illustrate the tolerance allocation based on improved NSGA-II algorithm. The rear lamp bracket assembly is shown in Figure 2.

Figure 2.

Rear lamp bracket assembly.

The finite element software ABAQUS is used to simulate the assembly process of the rear lamp bracket. The mesh models are shown in Figure 3. The material of the two rear lamp brackets is low-carbon steel, which Young’s modulus is 210 GPa and Poisson ratio is 0.3. The thickness of the outer rear lamp bracket is 1 mm, and the thickness of the inner rear lamp bracket is 1.5 mm.

Figure 3.

Finite element analysis model of rear lamp bracket assembly: (a) the outer bracket, (b) the inner bracket, and (c) the rear lamp bracket assembly.

Key measurement points

The common method of evaluating the assembly quality is to select some key measurement points on the part and use the quality of these points to reflect the quality of the assembly. In the article, 12 key measurement points are selected to evaluate the assembly quality of rear lamp bracket assembly. In these 12 points, 6 points are arranged on the outer rear lamp bracket and 6 points are located on the inner rear lamp bracket. The 12 key measurement points are shown in Figure 4 and the coordinates of these points is shown in Table 1.

Figure 4.

Measurement points of rear lamp bracket assembly.

Table 1.

Coordinates of measurement points of rear lamp bracket assembly.

Points	Direction	Coordinates
		X	Y	Z
MP_01	X	5233.25	−739.00	1148.00
MP_02	X	5375.36	−778.00	854.00
MP_03	X	5286.33	−670.00	806.40
MP_04	X	5321.00	−667.00	893.00
MP_05	X	5360.00	−732.00	962.00
MP_06	X	5404.00	−619.00	884.00
MP_07	Y	5397.21	−538.40	1058.00
MP_08	X	5355.17	−548.00	1071.00
MP_09	Y	5209.44	−553.00	1078.00
MP_10	Z	5323.00	−396.00	1242.50
MP_11	Y	5259.48	−639.34	1229.00
MP_12	Y	5197.00	−635.88	1178.45

Assembly variation analysis based on power balance method

Assembly variation of the rear lamp bracket assembly is analyzed by using the power balance principle. The assembly variation analysis model is shown in formula (4).

Using finite element software ABQUAS, we simulate the assembly process of the rear lamp bracket. We can get the sensitivity matrix [S₁], [S₂]. The results are shown in Tables 2 and 3. Table 2 is the sensitivity matrix of the inner rear lamp bracket and Table 3 is that of the outer rear lamp bracket.

Table 2.

Sensitivity matrix of inner rear lamp bracket.

	W_P-01	W_P-02	W_P-03	W_P-04	W_P-05	W_P-06	W_P-07
MP-01	0.0416	0.0207	0.0207	0.0299	0.0413	0.0214	0.0287
MP-02	−0.0354	0.0075	0.0345	0.0514	0.0899	0.0412	0.1357
MP-03	0.2347	0.0681	0.0588	0.1265	−0.0104	0.024	−0.0641
MP-04	−0.0233	−0.0514	0.2279	1.1353	0.6871	−0.4256	0.5364
MP-05	−0.1065	−0.0222	0.2452	0.5037	0.5478	0.6336	0.1847
MP-06	−0.1365	0.0187	0.1454	−0.0374	0.9297	1.0884	0.0599
MP-07	0.0874	−0.0216	−0.0441	0.0856	−0.0274	−0.0187	−0.1587
MP-08	0.0455	−0.041	−0.0548	0.0321	−0.038	0.0119	−0.1124
MP-09	0.0097	−0.0346	−0.1112	−0.0206	−0.1259	−0.0013	−0.0732
MP-10	−0.0367	−0.0024	0.0844	0.0654	0.0782	0.0566	−0.0444
MP-11	0.2364	0.0267	−0.1113	0.0269	−0.0364	−0.0224	−0.0558
MP-12	0.0624	−0.0116	−0.1341	−0.0552	−0.0864	−0.066	−0.0674

Table 3.

Sensitivity matrix of outer rear lamp bracket.

	W_P-01	W_P-02	W_P-03	W_P-04	W_P-05	W_P-06	W_P-07
MP-01	−0.0024	0.0012	0.0006	−0.0008	0.002	0.0007	0.0056
MP-02	−0.0042	0.0024	−0.0019	0.0015	0.0024	0.0022	0.0245
MP-03	0.0148	−0.0097	0.0088	−0.0023	−0.0068	0.0034	−0.0414
MP-04	−0.0135	−0.0011	−0.0096	0.0411	0.01267	−0.0429	0.0135
MP-05	−0.0163	0.0031	−0.0055	0.0247	−0.0069	0.0667	0.0464
MP-06	−0.0297	0.0076	−0.0073	−0.0146	−0.0264	0.2421	0.0171
MP-07	0.0088	−0.007	−0.0026	0.0065	−0.0071	0.006	−0.0222
MP-08	0.0067	−0.0066	−0.0022	0.0032	−0.0076	0.0067	−0.0771
MP-09	0.0058	−0.0034	−0.0041	0.0072	−0.0037	0.0132	−0.0438
MP-10	−0.0021	−0.0025	0.0034	0.0024	−0.0065	0.0041	−0.0234
MP-11	0.0236	−0.0064	0.0055	0.0026	−0.0071	−0.0087	−0.0297
MP-12	0.0067	−0.0017	−0.0067	0.0006	−0.0038	0.0033	−0.0145

Optimization and results

The improved NSGA-II algorithm is applied to solve the multi-objective optimization model of the rear lamp bracket assembly. The relevant parameters are as follows: the crossover rate: 0.9, the mutation rate: 0.1, the maximum generation $gen = 100$ , and the population size $pop = 100$ .

The optimization model of the rear lamp bracket assembly is based on formula (17). The objective function and constraints are written to the MATLAB toolbox. After 100 generation iterations, the results are shown in Figure 5.

Figure 5.

Pareto set of NSGA-II and Improved NSGA-II.

Figure 5 shows the comparison of the Pareto solution set of the NSGA-II algorithm before and after improvement. Before improvement, the individual distribution is not uniform; cost individuals are mostly distributed in 0–1 and 3–5 intervals; there are many overlapping individuals. After improvement, the Pareto solution set is closer to the non-dominated front, and the distribution of the Pareto set is obviously improved. The results show that the improved NSGA-II algorithm can get better Pareto solution set. Based on the optimization results, we can make the products tolerance allocation according to the product quality requirements and manufacturing cost plan.

After getting the Pareto solution set, we need to figure out the most suitable solution from the solution set according to the actual production requirements. The methods commonly used to select the suitable solution are: AHP (Analytic Hierarchy Process) method, ELECTRE (Elimination Et Choice Translation Reality) selection method, and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method. In this article, the TOPSIS method is adopted to select the appropriate solution from the Pareto solution set.

Let sample space is $P$ , and $P = {X_{ij}}$ , population size $i = 1, 2, \dots, N$ , target number $j = 1, 2, \dots, M$ . The steps to select the optimal solution from Pareto solution set based on TOPSIS are as follows:

Step 1. To standardize the elements in $P$

Z_{ij} = \frac{X_{ij}}{(\sum X_{ij}^{2})}

(19)

Step 2. To determine the positive and the negative ideal solutions

Z_{i}^{+} = max (Z_{ij})

(20)

Z_{i}^{-} = min (Z_{ij})

(21)

Step 3. To calculate the distance between positive and negative ideal solutions

S_{i}^{+} = \sqrt{\sum_{j}^{M} \frac{{(Z_{ij} - Z_{i}^{+})}^{2}}{4}}

(22)

S_{i}^{-} = \sqrt{\sum_{j}^{M} \frac{{(Z_{ij} - Z_{i}^{-})}^{2}}{4}}

(23)

Step 4. To figure out the relative adjacent degree between target value and the ideal solution

C_{i}^{*} = \frac{S_{i}^{-}}{(S_{i}^{+} + S_{i}^{-})}

(24)

Step 5. To sort the Pareto solution sets by descending order according to $C_{i}^{*}$ and get the best solution.

In the article, tolerance and cost comprise target vector $P$ , the population size $N = 100$ , and the target number $M = 2$ . After the calculation according to above steps, the optimization result of tolerance allocation for rear lamp bracket can be obtained. We can see from Table 4, the optimal solution is No. 55. At this time, the assembly tolerance is 2.156544 and the cost is about 10.02359.

Table 4.

Optimization result of tolerance allocation.

Number	Tolerance	Cost	Sorting value
1	0.116158	14.3688	1
2	0.123681	14.30669	1
3	0.123796	14.25497	1
…	…	…	…
55	2.156544	10.02359	1
…	…	…	…
98	5.24673	8.424574	1
99	5.313259	8.39319	1
100	5.379694	8.403185	1

Conclusion

In the automobile manufacturing industry, quality and cost is always a pair of prominent contradictions. The traditional model which uses the lowest cost as a means to evaluate tolerance allocation will lead to the poor applicability of the allocation scheme. It is difficult to adapt to the development of globalization and the cooperation in different places. In this article, the tolerance–deviation model and the tolerance–cost model are discussed based on the sheet metal tolerance variation analysis model. Then, the multi-objective optimization model of tolerance allocation is obtained. The improved NSGA-II algorithm is used to optimize and control the multi-objective model. Finally, the effectiveness of the optimization model is verified by the case of rear lamp bracket assembly. The method presented in this article has certain theoretical and engineering application significance. It can be used to solve the multi-objective optimization problem of auto-body assembly.

Footnotes

Academic Editor: Xichun Luo

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Speckhart

. Calculation of tolerance based on a minimum cost approach. J Eng Ind: T ASME 1972; 94: 447–453.

Spotts

. Allocation of tolerances to minimize cost of assembly. J Eng Ind: T ASME 1973; 95: 762–764.

Dong

. Optimal process sequence identification and optimal process tolerance assignment in computer-aided process planning. Comput Ind 1991; 17: 19–32.

Heish

K-L

. The study of cost-tolerance model by incorporating process capability index into product lifecycle cost. Int J Adv Manuf Tech 2006; 28: 638–642.

Kokkolaras

Papalambros

et al . Product and process tolerance allocation in multistation compliant assembly using analytical target cascading. J Mech Des: T ASME 2008; 130: 091701-1–091701-9.

Cui

Zhang

. Tolerance allocation and maintenance optimal design for fixture in multi-station panel assembly process. Appl Mech Mater 2010; 34–35: 1039–1045.

Zheng

Jin

Lai

et al . Assembly tolerance allocation using a coalitional game method. Eng Optimiz 2011; 43: 763–778.

Fan

Jiao

et al . Optimization of assembly tolerance allocation for the front door of a car. J Guangxi Univ 2015; 40: 616–621.

Balamurugan

Saravanan

Dinesh Bahu

et al . Concurrent optimal allocation of geometric and process tolerances based on the present worth of quality loss using evolutionary optimisation techniques. Res Eng Des 2017; 28: 185–202.

10.

Forouraghi

. Optimal tolerance allocation using a multiobjective particle swarm optimizer. Int J Adv Manuf Tech 2009; 44: 710–724.

11.

Sivakumar

Balamurugan

Ramabalan

. Evolutionary sensitivity-based conceptual design and tolerance allocation for mechanical assemblies. Int J Adv Manuf Tech 2010; 48: 307–324.

12.

Rao

. Advanced optimal tolerance design of machine elements using teaching-learning-based optimization algorithm. Prod Manuf Res 2014; 2: 71–94.

13.

Ramesh Kumar

Padmanaban

Balamurugan

. Optimal tolerance allocation in a complex assembly using evolutionary algorithms. Int J Simul Model 2016; 15: 121–132.

14.

Chen

Xing

. The auto body tolerance allocation optimization based on improved NSGA-II algorithm. J Commun Netw 2015; 5: 13–17.

15.

Koren

. Stream of variation theory for automotive body assembly. CIRP Ann: Manuf Techn 1997; 46: 1–6.

16.

Hsieh

. Simulation and optimization of assembly processes involving flexible parts. Int J Vehicle Des 1997; 18: 455–465.

17.

Xing

Wang

. A new assembly variation analysis model based on the method of power balance for auto-body parts. Assembly Autom 2014; 34: 296–302.

18.

Zhang

. Study and application on tolerance allocation of compliant autobody panel multi-station assembly. Master’s Dissertation, Jilin University, Changchun, China, 2011, pp.31–33.

19.

Ding

Ceglarek

Shi

. Design evaluation of multi-station assembly processes by using state space approach. J Mech Des: T ASME 2002; 124: 408–418.

20.

Camelio

Ceglarek

. Modeling variation propagation of multi-station assembly systems with compliant parts. J Mech Des: T ASME 2004; 125: 673–681.

21.

Chen

Cai

. Application of improving orthogonal-based genetic algorithm in function optimization. Comput Eng Des 2008; 29: 413–418.