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Quality design method using process capability index based on Monte-Carlo method and real-coded genetic algorithm
International Journal of Production Economics 204 (2018) 358–364
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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Quality design method using process capability index based on Monte-Carlo method and real-coded genetic algorithm
T
Akimasa Otsuka∗, Fusaomi Nagata Sanyo-Onoda City University, 1-1-1 Daigaku-Dori, Sanyo-Onoda, Yamaguchi, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Statistical tolerance index Process capability index Monte-carlo simulation Real-coded genetic algorithm Product performance
Variability in the performance and quality of products is an important issue in production engineering. Quality variability in mechanical production is due to irregularity of parts dimensions caused by machining errors. The dimensions of each part are usually managed by conventional tolerance at the design stage. Tight tolerance values result in reduced performance variation along with an increase in the manufacturing cost. Therefore tolerancing, which is a downstream process in mechanical design, is important in a detailed design process. Although quality is usually controlled in the manufacturing stage, not only production strategy but also management strategy will change in a positive direction and manufacturing cost is also reduced if the quality is also controlled at the design stage. This is because the design stage is an upstream process in manufacturing. This paper focuses on quality control in the design stage, and proposes a novel design method of process capability, which can statistically control parts dimensions based on product performance. The method consists of a numerical method and a real-coded genetic algorithm. A case study was analysed to evaluate the effectiveness of the proposed method. The result showed that the proposed method suitably allocates the STI for each part so that the product satisfies the required product performance.
1. Introduction Actual dimensions of machined parts do not match the nominal dimensions specified at the design stage due to machining errors. Mechanical products consist of machined parts, and the product's performance depends on the functional dimension resulting from stack-up of the parts. Due to variability in the performance and quality of products, the dimensions of each part should be managed by conventional tolerance at the design stage. Tight tolerance values result in reduced performance variation along with increased manufacturing costs. Therefore, tolerancing, which is a downstream process in mechanical design, is an important factor in a detailed design process. Tolerancing methods are generally classified as worst-case or statistical. The worstcase method is traditionally used and involves easy calculations; stackup of parts' variations is modelled as the sum of the limit of the variation of each part. Although this method perfectly guarantees the interchangeability of parts, the specified tolerances tend to be tight. However, the statistical method allows the tolerances to be relaxed considering statistical distributions of the parts dimensions. The method is based on statistical rules to ensure its compatibility in mass production. Various studies have focused on statistical methods. For example, Skowronski and Turner (1997) examined Monte-Carlo ∗
variance reduction techniques, importance sampling, and correlation. They also proposed a method for using them in statistical tolerance synthesis. Zhang et al. (1998) proposed PCI (process capability index) based tolerance as a predetermined statistical tolerance zone. This tolerancing method can be used as an interface between design specification and statistical process control. Choi et al. (1999) applied a complex search method to ensure optimal allocation when tolerance limits were used and when Taguchi's quadratic loss function was defined. Li (2000) studied the relationship between unbalanced tolerance design and quality loss function. The study concluded that the optimal setting of the process mean, which minimizes the expected quality loss was obtained with respect to the asymmetrical ratio. Gao and Huang (2003) proposed an optimal tolerance balancing method for a nonlinear model. The method was based on process capability and validated through tests. Pillet (2003) detailed inertial tolerancing and compared it with traditional tolerancing methods. Pillet et al. (2005, 2015) proposed weighted inertia tolerance to obtain the best possible compromise between statistical and worst case tolerancing methods. The tolerance principle involved calculating the allowable range of the mean square deviation in relation to the target. Judic (2016) proposed a semiquadratic method known as “Process Tolerancing” and compared it with “Inertial Tolerancing.” Van Hoecke (2016) defined a tool risk
Corresponding author. Sanyo-Onoda City University, 1-1-1 Daigaku-Dori, Sanyo-Onoda, Yamaguchi, 756-0884, Japan. E-mail addresses: [email protected] (A. Otsuka), [email protected] (F. Nagata).
https://doi.org/10.1016/j.ijpe.2018.08.016 Received 7 March 2018; Received in revised form 11 July 2018; Accepted 12 August 2018 Available online 13 August 2018 0925-5273/ © 2018 Elsevier B.V. All rights reserved.
International Journal of Production Economics 204 (2018) 358–364
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distribution parameters of dimensional or geometrical errors of machined parts lot-by-lot. The STI is a useful tool to control performance and quality for mass produced products. However, it is not commonly used in current design processes due to its complexity.
caused by Gaussian hypothesis in statistical tolerancing, and related it to process capability indices. Although designers use the statistical method that considers a condition of product performance and manufacturing cost, the conditions are rounded to the conventional tolerances as a scalar value. The intention of the designer can be reflected onto actual products if more detailed information is added to the tolerances on the design drawing. Consequently, the design drawing allows designers to design a product with additional value. Fortunately, the statistical tolerance index (STI), which is a useful tolerance specification for mass production, has been standardized in ASME Y14.5. The STI has a limited process capability index. When the STI is specified in design drawings, manufacturing processes must satisfy the limitation of the STI under statistical process control. Although the STI may result an additional manufacturing cost, it is a beneficial trade-off. Because a product generally consists of several parts, there are two main problems when applying the STI to an actual design process: tolerance stack-up and tolerance allocation; these are the same as in conventional tolerancing. Before reasonably allocating the STI into parts' dimensions, the STI stack-up problem should be solved. Previous research has shown that the STI stack-up problem is known to be complex and difficult even if a product consists of only two parts. Srinivasan (1999) proved that a solution to the problem was generally represented by the Minkowski sum on the hyperplane of the mean and square of standard deviation. They provided an algebraic solution for the problem under the condition in which only limits of Cpk and Cc were specified. Based on their study, Otsuka and Nagata (2015) derived a more general algebraic solution for the problem and clarified its applicability condition. In addition, a numerical method using the MonteCarlo simulation has been developed to obtain the approximate solution for the STI stack-up problem, because the problem is difficult to solve algebraically. Otsuka and Nagata (2017) also proposed a design method of target dimensions and Cpm indices based on product performance and cost. This paper proposes the simultaneous design method of the target dimensions and four process capability indices, Cp, Cpk, Cc, and Cpm. The allocation method consists of the numerical method for the STI stack-up problem and a real-coded genetic algorithm using the UNDX crossover process (Kita et al., 2002; Ono et al., 2000). A case study was analysed to evaluate the effectiveness of the proposed method. The result shows that the proposed method suitably allocates the STI for each part so that the product satisfies certain conditions.
2.1. Process capability index (PCI) Machining processes in mass production must be controlled lot-bylot to prevent machining errors. Process capability indices (PCIs) are non-dimensional parameters that have been used for a long time to evaluate the machining process quantitatively; they are defined as follows,
Cp =
U−L 6σ
(1)
μ−L U−μ ⎞ , Cpk = min ⎛ 3σ ⎠ ⎝ 3σ
(2)
τ−μ μ−τ ⎞ Cc = max ⎛ , ⎝τ − L U − τ ⎠
(3)
Cpm =
U−L 6 σ 2 + (μ − τ )2
(4)
where L, U, μ, σ and τ are the lower limit of size, upper limit of size, process mean, process standard deviation and target dimension, respectively. T is the tolerance defined by the difference of U and L. When PCIs are limited within certain specified values such as Cp ≥ p, Cpk ≥ k, Cc ≤ c, and Cpm ≥ m where p, k, c, and m are design parameters, the process must be controlled to maintain its capability within each parameter range. As the PCIs are defined by the process mean and/or standard deviation, the process is assumed to be controlled under statistical process control. When several STIs are simultaneously specified on a dimension, the allowable range of the mean and standard deviation tends to be complex. A population parameter zone (PPZ) is usually used to visually represent the area. 2.2. Population parameter zone The PPZ represents an allowable range on the μ - σ plane when STIs are specified. Fig. 2 shows examples of PPZ when multiple STIs are specified. The horizontal axis is the mean value and the vertical axis is the standard deviation value. The shaded area is the allowable range of the mean and the standard deviation. The solid boundary of the range is the allowable limit. When STIs are specified for each part dimension, the dimension has a corresponding PPZ. During the design process, designers need to decide suitable values and types of STI for each part dimension based on the required function, product performance, and cost of the final product. The decision process is equivalent to tolerance allocation. The STI stack-up problem must be solved before discussing the STI allocation problem, because the allocation problem is an inverse of the stack-up problem.
2. Statistical tolerance index (STI) STI is a specification that uses a process capability index as an additional indicator for a manufacturing process with a conventional tolerance. STIs can be specified by adding “ST” into a hexagon. Fig. 1 (a) shows a design drawing in which conventional tolerances are specified. Fig. 1 (b) shows a design drawing in which conventional tolerances and STIs are specified. The specified dimensions, tolerances, and STIs in Fig. 1 are for illustrative purposes only. STI is applicable to both dimension and geometrical tolerances, so that STI can limit the
30.0±0.05
30.0±0.05
ST
A
A A
A (a) conventional tolerance
ST
(b) statistical tolerance index
Fig. 1. Example design drawings. 359
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function, the functional dimension of the assembly X is given as follows. n
X=
Cp
Cpk
i=1
Cc
U
U
L
(6)
If the parts' dimensions are independent of each other as in the model, the mean and the standard deviation of the functional dimension of mass produced assemblies, μX and σX, can be calculated as follows.
Cpm
Cc
L
∑ xi
n
Fig. 2. Examples of population parameter zone and allowable area when STI are specified.
μX =
3. Stack-up analysis of STI
σX =
∑ μi i=1
(7)
n
∑ σi2 i=1
The functionality and performance of the final product usually depend on a functional dimension of assembly consisting of several parts. In the design process of the final product, the constraints of the functional dimension are decided based on customer and supplier demands. Subsequently, designers must allocate the constraints to parts' tolerances and specify them on the design drawing as shown in Fig. 1. The allocation process is carried out based on a solution of a tolerance stackup problem. When designing STIs, the stack-up problem should be solved first. As shown in Fig. 3, the problem can be converted into a synthesis of each part's PPZ.
The assumption of a normal distribution is not required because equations (7) and (8) are valid regardless of the distributions of the parts' dimensions. When the STIs are specified on each part's dimension with conventional tolerance, the PPZ of each part can be generated on the μi - σi planes. Accordingly, the PPZ of the functional dimension is calculated on the μi - σi plane, which is the solution to the STI stack-up problem. The boundary of the PPZ is an important element because it is the allowable limit of the mean and standard deviation. 3.2. Probabilistic solution to STI stack-up using the Monte-Carlo simulation
3.1. Modelling of stack-up analysis of STI
A method using the Monte-Carlo simulation is proposed in order to solve general STI stack-up problems. Each boundary on the PPZ of parts is the worst-case mean and standard deviation. Therefore, a boundary on assembly is obtained as the worst-case stack-up of them. The term “worst-case” denotes the worst situation between the mean and standard deviation. In the first procedure, uniform random numbers of means of each part are generated within the range limited by Cc. Minimum standard deviation of each PPZ is calculated based on the random number and other specified STIs, such as Cp, Cpk, and Cpm. The mean and standard deviation of assembly are then calculated using equations (7) and (8) using the random number and the minimum standard deviation, respectively. A point cloud is obtained by repeating these processes. The PPZ boundary of assembly can be estimated based on the point cloud. The method is applicable if the function h in equation (5) is linear, and if the parts dimensions are independent of each other. Because the Monte-Carlo simulation is a probabilistic method, there is an approximation error in the solution. The accuracy of the solution may depend on the number of trials of the Monte-Carlo simulation. Fig. 4 shows the procedure of the method.
Several existing studies have focused on the stack-up problem for conventional tolerance using the worst-case and statistical methods. However, there is little research about the problem of STI stack-up. Srinivasan (1999) proved that a solution to the problem was represented by the Minkowski sum on the μ - σ2 plane. They also demonstrated an algebraic solution for the STI stack-up problem when Cpk and Cc were specified and when the values of k and c for each part were the same. However, this solution is not of practical use due to a lack of a degree of design freedom. Furthermore, the calculation details of the Minkowski sum were not shown in that report. Otsuka and Nagata (2015) proposed a numerical method using the Monte-Carlo simulation for the problem of STI stack-up, and the method can be applied in more general situation. In this study, the method is used when solving the STI stack-up problem. In general, a functional dimension X consisting of n parts can be written as follows,
X = h (x1, x2, ..., x n )
(5)
where i is the parts identifier and n is the number of parts. As h can be linear or nonlinear, in this study, h is assumed to be linear. Fig. 3 also shows an assembly model consisting of three parts. When h is a linear
9.0±0.24 ST Cpk
1.32 Cc
4.0±0.1 ST Cp
4. Allocation method of the STI Designing the process capability limit of each part, that is the STI allocation problem, is a nonlinear and global optimization problem, because the problem is inverse to that of the STI stack-up. The differential information cannot be used in an STI allocation problem because the stack-up solution is obtained via the Monte-Carlo simulation. Furthermore, it is known whether search space is convex or nonconvex. Therefore, a genetic algorithm, which has been often used not only in tolerance allocation but also in many optimization problems, is applied. Numerous global optimization methods have been developed and applied to tolerance allocation problems of actual products. Metaheuristic methods are popular, especially in tolerance allocation problem. For example, Chen (2001) proposed a tolerance allocation method using simulated annealing and a neural network. Yang and Naikan (2003) proposed an algorithm for optimum allocation of tolerances using a neural network. Walter et al. (2015) presented a methodology for the least cost tolerance allocation of systems with time-variant deviations using particle swarm optimization. Chen and Fischer (2000) researched nonlinearly constrained tolerance allocation
0.85 1.50 Cc 0.95 Cpm 0.33
7.0±0.2 ST Cpk
(8)
1.32 Cpm 0.44
Stack up problem
= 20.0±0.54 ST
Fig. 3. Representation of the stack-up problem using PPZ. 360
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2 Part 1
1. Generate a uniform random number * within allowable value of part1, which is calculated based on specified Cc value. 2. Calculate all allowable standard deviations corresponding to specified Cp, Cpk, Cpm.
2 2 ( * , *)
3. Select the minimum standard deviation and name it as * .
L
U
*
4. Point ( *, *) is on a boundary of a certain STI.
Part n
Part 2 ( * , *)
( *, *)
Assembly
( *, *)
5. Repeat the procedure mentioned above for all parts and obtain the point ( *, *) .
6. Calculate * and * based on statistical rule. 7. Plot ( * , * ) on the
plane.
8. Return to 1. and iterate as many as possible. 9. Set of points is obtained, and an approximate solution can be obtained as a convex-hull envelope. Fig. 4. Flowchart of the proposed method for solving the stack-up problem of the STI.
x-y plane: Design variable space
problems and concluded that genetic algorithms performed well for the problem in their study. Noorul et al. (2005) proposed a procedure using a genetic algorithm for gear train and overrunning clutch assemblies and compared it with conventional techniques. Based on these studies, a genetic algorithm, in particular a real-encoded genetic algorithm is applied to the STI allocation problem in this study. In tolerance allocation, a relational model between cost and tolerance is used and tolerances are usually determined considering costminimization. For example, Walter and Wartzack (2013) presented a methodology for the least cost tolerance allocation of systems with time dependent motion behavior. If tolerances are design parameters, then the cost-tolerance relationship, which is often used in tolerancing, can be applied. However, the relationship between cost and process capability indices, in particular PPZ, is not thoroughly discussed in this field. Therefore, in this study, the size of the allowable area on the PPZ is set as an evaluation function because manufacturing cost decreases as the area becomes larger. Allowable limits of the process capability indexes, p, k, c, m, and target value τ are set as the design values.
Unit vectors
Color map: Two dimensional normal distribution. (Children will be generated based on the probability.) Fig. 5. Schematic of UNDX in the case of two design parameters. n
4.1. Real-coded genetic algorithm
n
C1 = m + ∑kparam z k ek , C2 = m + ∑kparam z k ek =1 =1 m = (P1 + P2)/2
A real-coded genetic algorithm is applied to find the optimal solution due to the large number of design parameters in the STI allocation problem. In a real-coded genetic algorithm, the crossover process is important because it affects the convergence of the optimal solution. In this study, the unimodal normal distribution crossover (UNDX) is applied as the crossover process in the genetic algorithm (Kita et al., 2002). The crossover operator outperformed other crossover operators in several benchmarking tests and was successfully applied to the design of the lens systems (Ono et al., 2000). In the UNDX, each chromosome is a real vector as a representation in the design variable space. Fig. 5 shows the schematic of the UNDX in the case of two design parameters, which is the simplest case. In the UNDX, when the number of design parameters is nparam, children vectors C1 and C2 are generated as follows,
z k ≅ N (0, sk2) (k = 1, …., nparam) s1 = αd1, sk = βd2/ nparam e1 =
(P2 − P1) P2 − P1
, ei ⊥ ej, (i, j = 1, …, n param ; i ≠ j )
d1 = P2 − P1 , d2 = (P3 − P1)·e2
(9)
where P1, P2 and P3 are parent vectors of Parent 1, 2 and 3, respectively. d1 is the distance between Parents 1 and 2, d2 is the distance between Parent 3 from the line connecting Parents 1 and 2. α and β are constants. In this study, the constants α and β are set to be 0.5 and 0.35, respectively, according to recommendation in the original UNDX report (Ono et al., 1999, 2000; Kita et al., 2002). In the UNDX, vector P3 is used to calculate d2, then the parameter sk, which is the standard deviation of normal random numbers zk, is decided. If nparam is bigger, C1 and C2 are generated away from the line connecting P1 and P2. To avoid this situation, sk is divided by the square root of nparam. Fig. 6 shows a complete flowchart of the proposed method to design STI. 361
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PPZ of X
Evaluation
500 generations?
Yes
Satisfying yield requirement ?
S.D.
Performance
Calculate the stack-up of statistical tolerance indices
Initial configuration
No Elitism and Selection
Mean
Satisfying performance requirement ?
Performance and PPZ of X
Crossover (UNDX) Mutation
Calculate evaluated value
Return evaluated value
ST Cp
.... Cpk
....
ST Cp
.... Cpk
....
ST Cp
.... Cpk
....
End Fig. 6. Flowchart of proposed method to design STI for each part.
4.2. Product performance In this study, the STI is designed based on the product's performance. The performance of product Y depends on the functional dimension described as Y = f(X). The functional dimension of the accepted product is distributed within a total tolerance zone between the upper size limit UX and the lower size limit LX. A representative value of each lot is defined as the estimated value of the product performance, as follows,
EY =
∫L
UX
X
f (X ) N (μX , σX ) dX
PPZ of xi
Specifications on drawing
Fig. 7. Design flow of the STI based on product performance.
X x1 x2 x3
(10)
where function N denotes a normal distribution, LX and UX are the lower and upper limits of the functional dimension, respectively, and EY indicates the customer demand for product performance in the STI design process. Once the minimum EY value is set based on customer demand, the demand can be directly projected onto PPZ (μX - σX plane) as an allowable area, as shown in Fig. 7. The STI is designed to calculate each design parameter so that the stack-up of the STI does not infringe on the allowable area. Furthermore, the design parameters should be designed with a more relaxed STI value in terms of manufacturing cost. When more relaxed STI values are set to each part, the allowable area on the μX -σX plane becomes larger. Therefore, the inverse value of the allowable area is set as an objective function in the genetic algorithm and the objective function should be small.
x5
x4
x6
Fig. 8. A virtual product model used in the case study.
Table 1 Basic parameters of the virtual product. (Unit mm).
5. Case study A case study was analysed using a virtual assembly, which consisted of six linearly stacked-up parts, as shown in Fig. 8. The assembly is the model of a rotating element used in various mechanical products such as cars, machine tools, home appliances, and so on. The virtual product is an academic model and is a representative model of linearly stacked-up products. Even if the number of parts increases, the proposed method can be applied. However, it takes longer time to converge. The basic parameters of parts' dimensions, tolerances, lower and upper limit sizes are assumed, as shown in Table 1. In this case, the functional dimension X was the same as in equation (6) with n = 6. The performance function of the product was assumed to be equation (11) as shown in Fig. 9 (a), which has an asymmetrical nominal-the-best characteristic. Although it would be more preferable to use an actual performance function, it does not affect the effectiveness of the proposed method. Fig. 9 (b) shows the EY values projected on the μX -σX plane.
i
1
2
3
4
5
6
X
xi Ui Li
40 40.6 39.4
6 6.3 5.7
12 12.4 11.6
78 78.8 77.2
24 24.5 23.5
40 40.6 39.4
200 203.2 196.8
Y = 20 X − 140 − 0.2(X − 198)3 − 130
(11)
Fig. 10 shows chromosome structure, which consists of pi, ki, ci, mi and τi. Each gene is a real number. In this study, assembly consists of six parts, so that the size of chromosome is 30. Parameter tuning is important in the genetic algorithm in terms of the convergence performance and avoiding local minimum. However, the sensitivity of the parameters to the objective function is not known analytically even for such a simple model. In this study, the parameters of the genetic algorithm were manually tuned through several trial-anderror iterations. The search space of each design parameter and the parameters of the genetic algorithm are listed in Table 2. 362
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25
1.0 (mm)
EY
X
Y
Table 2 Parameters of the real-coded genetic algorithm.
20
0 196.8
203.2
0 196.8
X (mm)
200 μX (mm)
(a)
(b)
200
0
203.2
Fig. 9. Product performance characteristic and EY values on PPZ in the case study.
(12)
Furthermore, the initial parameter setting was also determined through several trial-and-error iterations. Real values of p, k, c, and m of all initial individuals are randomly set considering severe situations, such as pi = 2.0–2.5, ki = 2.0–2.5, ci = 0.3–0.5, mi = 1.1–1.6, respectively. Conversely, target dimensions τi are randomly set near the center of each tolerance range. Optimization was performed under these conditions.
p k c m τ
c1 ~ c6
90% 1% 60 4 1.0 ≤ pi ≤ 2.5 1.0 ≤ ki ≤ 2.5 0.3 ≤ ci ≤ 1.0 0.1 ≤ mi ≤ 1.0 Li ≤ τi ≤ Ui
x1
x2
x3
x4
x5
x6
1.031 1.000 0.300 0.723 40.06
1.881 1.797 1.000 0.274 5.91
1.843 1.938 0.418 0.753 11.66
1.000 1.001 0.892 0.239 77.49
1.000 1.000 0.987 0.261 23.87
1.015 1.099 0.866 0.235 39.78
1.0
Generation 1 100 200 500
(mm)
3 2
X
Objective function
Design parameters obtained by the optimization are rounded to the forth decimal place and are listed in Table 3. Fig. 11 (a) shows the convergence curve for 500 generations. The result shows that the optimal solution is well converged. Fig. 11 (b) shows the historical allowable areas during the evolution process. In Fig. 11(b), each line shows a limit of the allowable area at generations 1, 100, 200, and 500. The figure also shows that allowable area showed an incremental trend. This result indicates that the initial solution is a strict STI, but the final optimal solution is a relaxed STI. Fig. 12 shows that the allowable area at the 500th generation overlaps the one in Fig. 9 (b). This result shows that product performance always satisfies equation (12) if the STI is specified on the drawing based on the optimal solution. Fig. 13 shows a design drawing that includes specified dimensions, tolerances and the design parameters rounded to the second decimal place. To guarantee performance demand, the parameters p, k, and m should be rounded up, and parameter c should be rounded down. Target in the figure dimensions are omitted, and it can be specified if it is required. Some parameters in the figure are redundant. For example, in the parameters of part 3, Cp ≥ 1.9 and Cpk ≥ 2.0 are specified. In this case, the former constraint is perfectly included in the latter one. Therefore, it is preferable to remove the Cp specification to reduce redundancy in actual design drawing. Finally, it is concluded that the proposed method can help suitably design the STI and target values for each part. Until now, there has not been a design method of process capability indices. The proposed method makes it possible for designers to design process capability indices based on customer demand for product performance and manufacturing cost. Compared with existing tolerancing methods, the proposed method has novelty and advantage with regard to considering both product performance and manufacturing cost. It can be used as a decision-making support tool not only in designing and manufacturing stages but also in the overall management of the
k1 ~ k6
Crossover rate Mutation rate Number of Individual Number of elite Search space of p Search space of k Search space of c Search space of m Search space of τ
4
6. Result and discussion
p1 ~ p6
500
Table 3 Optimized design parameters.
Assuming required performance EY to be larger than 20 as follow.
EY ≥ 20
Generation
1 0
100
200
300
400
500
0 196.8
200
Generation
μX (mm)
(a)
(b)
203.2
Fig. 11. Convergence curve and historical results of the allowable area during optimization.
1.0
(mm)
20
X
EY
0 196.8
200
203.2
0
μX (mm) Fig. 12. Product performance EY and allowable area according to optimized parameters.
company. For example, based on the design drawing, the company or designer can decide whether to outsource a part or create it in their factory. However, the proposed method has some limitations. It can be
m1 ~ m6
Number of design parameter is 6 × 5 = 30.
363
1
~
Fig. 10. Chromosome structure used in UNDX. 6
International Journal of Production Economics 204 (2018) 358–364
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40±0.6 ST Cp 1.1 Cpk
1.0 Cc
6±0.3 ST Cp 1.9 Cpk
1.8 Cc
12±0.4 ST Cp 1.9 Cpk
78±0.8 ST Cp 1.0 Cpk
0.3 Cpm 0.8
This work was supported by JSPS KAKENHI Grant Number 26870774 and 17K18029.
1.0 Cpm 0.3
2.0 Cc
1.1 Cc
Acknowledgments
0.4 Cpm 0.8
References Chen, M.C., 2001. Tolerance synthesis by neural learning and nonlinear programming. Int. J. Prod. Econ. 70 (1), 55–65. Chen, T.C., Fischer, G.W., 2000. A GA-based search method for the tolerance allocation problem. Artif. Intell. Eng. 14 (2), 133–141. Choi, H.G.R., Park, M.H., Salisbury, E., 1999. Optimal tolerance allocation with loss function. J. Manuf. Sci. Eng. 122 (3), 529–535. Gao, Y., Huang, M., 2003. Optimal process tolerance balancing based on process capabilities. Int. J. Adv. Manuf. Technol. 21 (7), 501–507. Judic, J.M., 2016. Process Tolerancing: a new approach to better integrate the truth of the processes in tolerance analysis and synthesis. Procedia CIRP 43, 244–249. Kita, H., Ono, I., Kobayashi, S., 2002. Theoretical analysis of the unimodal normal distribution crossover for real-coded genetic algorithms. Trans. Soc. Instrum. Control Eng. 2 (1), 187–194. Li, M.H.C., 2000. Quality loss function based manufacturing process setting models for unbalanced tolerance design. Int. J. Adv. Manuf. Technol. 16 (1), 39–45. Noorul, H.A., Sivakumar, K., Saravanan, R., Muthiah, V., 2005. Tolerance design optimization of machine elements using genetic algorithm. Int. J. Adv. Manuf. Technol. 25 (3–4), 385–391. Ono, I., Satoh, H., Kobayashi, S., 1999. A real-coded genetic algorithm for function optimization using the unimodal normal distribution crossover. J. Jpn. Soc. Artif. Intell. 14 (6), 1146–1155 (In Japanese). Ono, I., Kobayashi, S., Yoshida, K., 2000. Optimal lens design by real-coded genetic algorithms using UNDX. Comput. Meth. Appl. Mech. Eng. 186 (2–4), 483–497. Otsuka, A., Nagata, F., 2015. Stack up analysis of statistical tolerance indices for linear function model using Monte Carlo simulation. In: Proceedings of the 20th International Conference on Engineering Design, pp. 143–152 vol. 6(1). Otsuka, A., Nagata, F., 2017. Design method of cpm-index based on product performance and manufacturing cost. Comput. Ind. Eng. 113, 921–927. Pillet, M., 2003. Inertial tolerancing in the case of assembled products. In: Recent Advances in Integrated Design and Manufacturing in Mechanical Engineering. Springer, Dordrecht, pp. 85–94. Pillet, M., Duret, D., Sergent, A., 2005. Weighted inertial tolerancing. Qual. Eng. 17 (4), 687–692. Pillet, M., Maire, J.L., Hernandez, P., Vincent, R., 2015. Tolerancement inertiel calcul optimal. In: 11e Cong. Int. De Genie Industriel-CIGI2015, (In French). Skowronski, V.J., Turner, J.U., 1997. Using Monte-Carlo variance reduction in statistical tolerance synthesis. Comput. Aided Des. 29 (1), 63–69. Srinivasan, V., 1999. In: Paul, D.J.R. (Ed.), Dimensioning and Tolerancing Handbook. McGraw-Hill, New York (chapter 8). Van Hoecke, A., 2016. Tool risk setting in statistical tolerancing and its management in verification, in order to optimize customer's and supplier's risk. Procedia CIRP 43, 250–255. Walter, M., Wartzack, S., 2013. Statistical Tolerance-cost-optimization of Systems in Motion Taking into Account Different Kinds of Deviations, Smart Product Engineering. Springer, Verlag Berlin Heidelberg, pp. 705–714. Walter, M.S.J., Spruegel, T.C., Wartzack, S., 2015. Least cost tolerance allocation for systems with time-variant deviations. Procedia CIRP 27, 1–9. Yang, C.C., Naikan, V.N.A., 2003. Optimum design of component tolerances of assemblies using constraint networks. Int. J. Prod. Econ. 84 (2), 149–163. Zhang, Y., Low, Y.S., Fang, X.D., 1998. PCI-based tolerance as an interface between design specifications and statistical quality control. Comput. Ind. Eng. 35 (1), 201–204.
0.8 Cpm 0.3
24±0.5 ST Cp 1.0 Cpk
1.0 Cc
40±0.6 ST Cp 1.1 Cpk
0.9 Cpm 0.3
1.1 Cc
0.8 Cpm 0.3
Fig. 13. Design drawing of the virtual product model with conventional tolerance and STI.
used under the conditions in which tolerance limits and performance function are known and parts are linearly stacked. 7. Conclusion This paper proposes an STI design method, which is an inverse solution to the stack-up problem. The proposed method consists of the Monte-Carlo simulation and a real-coded genetic algorithm. The efficiency of the proposed method was evaluated via a case study using a virtual product model. The STI was suitably designed and optimized so that the product satisfied required product performance. As the proposed method considers both product performance and manufacturing cost, it can be used as a decision-making support tool not only in the designing and manufacturing stages but also in the overall management of the company. As future work, the proposed method is planned to be extended to geometrical tolerance, and also to be researched in terms of acceptance sampling and failure rate. Furthermore, in this method, cost function in optimization is defined as the inverse value of the allowable area and the area is uniformly evaluated. In fact, the bottom side of the area cannot be achieved due to the limits of machining accuracy. Therefore, more suitable evaluated function should be researched.
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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Quality design method using process capability index based on Monte-Carlo method and real-coded genetic algorithm
T
Akimasa Otsuka∗, Fusaomi Nagata Sanyo-Onoda City University, 1-1-1 Daigaku-Dori, Sanyo-Onoda, Yamaguchi, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Statistical tolerance index Process capability index Monte-carlo simulation Real-coded genetic algorithm Product performance
Variability in the performance and quality of products is an important issue in production engineering. Quality variability in mechanical production is due to irregularity of parts dimensions caused by machining errors. The dimensions of each part are usually managed by conventional tolerance at the design stage. Tight tolerance values result in reduced performance variation along with an increase in the manufacturing cost. Therefore tolerancing, which is a downstream process in mechanical design, is important in a detailed design process. Although quality is usually controlled in the manufacturing stage, not only production strategy but also management strategy will change in a positive direction and manufacturing cost is also reduced if the quality is also controlled at the design stage. This is because the design stage is an upstream process in manufacturing. This paper focuses on quality control in the design stage, and proposes a novel design method of process capability, which can statistically control parts dimensions based on product performance. The method consists of a numerical method and a real-coded genetic algorithm. A case study was analysed to evaluate the effectiveness of the proposed method. The result showed that the proposed method suitably allocates the STI for each part so that the product satisfies the required product performance.
1. Introduction Actual dimensions of machined parts do not match the nominal dimensions specified at the design stage due to machining errors. Mechanical products consist of machined parts, and the product's performance depends on the functional dimension resulting from stack-up of the parts. Due to variability in the performance and quality of products, the dimensions of each part should be managed by conventional tolerance at the design stage. Tight tolerance values result in reduced performance variation along with increased manufacturing costs. Therefore, tolerancing, which is a downstream process in mechanical design, is an important factor in a detailed design process. Tolerancing methods are generally classified as worst-case or statistical. The worstcase method is traditionally used and involves easy calculations; stackup of parts' variations is modelled as the sum of the limit of the variation of each part. Although this method perfectly guarantees the interchangeability of parts, the specified tolerances tend to be tight. However, the statistical method allows the tolerances to be relaxed considering statistical distributions of the parts dimensions. The method is based on statistical rules to ensure its compatibility in mass production. Various studies have focused on statistical methods. For example, Skowronski and Turner (1997) examined Monte-Carlo ∗
variance reduction techniques, importance sampling, and correlation. They also proposed a method for using them in statistical tolerance synthesis. Zhang et al. (1998) proposed PCI (process capability index) based tolerance as a predetermined statistical tolerance zone. This tolerancing method can be used as an interface between design specification and statistical process control. Choi et al. (1999) applied a complex search method to ensure optimal allocation when tolerance limits were used and when Taguchi's quadratic loss function was defined. Li (2000) studied the relationship between unbalanced tolerance design and quality loss function. The study concluded that the optimal setting of the process mean, which minimizes the expected quality loss was obtained with respect to the asymmetrical ratio. Gao and Huang (2003) proposed an optimal tolerance balancing method for a nonlinear model. The method was based on process capability and validated through tests. Pillet (2003) detailed inertial tolerancing and compared it with traditional tolerancing methods. Pillet et al. (2005, 2015) proposed weighted inertia tolerance to obtain the best possible compromise between statistical and worst case tolerancing methods. The tolerance principle involved calculating the allowable range of the mean square deviation in relation to the target. Judic (2016) proposed a semiquadratic method known as “Process Tolerancing” and compared it with “Inertial Tolerancing.” Van Hoecke (2016) defined a tool risk
Corresponding author. Sanyo-Onoda City University, 1-1-1 Daigaku-Dori, Sanyo-Onoda, Yamaguchi, 756-0884, Japan. E-mail addresses: [email protected] (A. Otsuka), [email protected] (F. Nagata).
https://doi.org/10.1016/j.ijpe.2018.08.016 Received 7 March 2018; Received in revised form 11 July 2018; Accepted 12 August 2018 Available online 13 August 2018 0925-5273/ © 2018 Elsevier B.V. All rights reserved.
International Journal of Production Economics 204 (2018) 358–364
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distribution parameters of dimensional or geometrical errors of machined parts lot-by-lot. The STI is a useful tool to control performance and quality for mass produced products. However, it is not commonly used in current design processes due to its complexity.
caused by Gaussian hypothesis in statistical tolerancing, and related it to process capability indices. Although designers use the statistical method that considers a condition of product performance and manufacturing cost, the conditions are rounded to the conventional tolerances as a scalar value. The intention of the designer can be reflected onto actual products if more detailed information is added to the tolerances on the design drawing. Consequently, the design drawing allows designers to design a product with additional value. Fortunately, the statistical tolerance index (STI), which is a useful tolerance specification for mass production, has been standardized in ASME Y14.5. The STI has a limited process capability index. When the STI is specified in design drawings, manufacturing processes must satisfy the limitation of the STI under statistical process control. Although the STI may result an additional manufacturing cost, it is a beneficial trade-off. Because a product generally consists of several parts, there are two main problems when applying the STI to an actual design process: tolerance stack-up and tolerance allocation; these are the same as in conventional tolerancing. Before reasonably allocating the STI into parts' dimensions, the STI stack-up problem should be solved. Previous research has shown that the STI stack-up problem is known to be complex and difficult even if a product consists of only two parts. Srinivasan (1999) proved that a solution to the problem was generally represented by the Minkowski sum on the hyperplane of the mean and square of standard deviation. They provided an algebraic solution for the problem under the condition in which only limits of Cpk and Cc were specified. Based on their study, Otsuka and Nagata (2015) derived a more general algebraic solution for the problem and clarified its applicability condition. In addition, a numerical method using the MonteCarlo simulation has been developed to obtain the approximate solution for the STI stack-up problem, because the problem is difficult to solve algebraically. Otsuka and Nagata (2017) also proposed a design method of target dimensions and Cpm indices based on product performance and cost. This paper proposes the simultaneous design method of the target dimensions and four process capability indices, Cp, Cpk, Cc, and Cpm. The allocation method consists of the numerical method for the STI stack-up problem and a real-coded genetic algorithm using the UNDX crossover process (Kita et al., 2002; Ono et al., 2000). A case study was analysed to evaluate the effectiveness of the proposed method. The result shows that the proposed method suitably allocates the STI for each part so that the product satisfies certain conditions.
2.1. Process capability index (PCI) Machining processes in mass production must be controlled lot-bylot to prevent machining errors. Process capability indices (PCIs) are non-dimensional parameters that have been used for a long time to evaluate the machining process quantitatively; they are defined as follows,
Cp =
U−L 6σ
(1)
μ−L U−μ ⎞ , Cpk = min ⎛ 3σ ⎠ ⎝ 3σ
(2)
τ−μ μ−τ ⎞ Cc = max ⎛ , ⎝τ − L U − τ ⎠
(3)
Cpm =
U−L 6 σ 2 + (μ − τ )2
(4)
where L, U, μ, σ and τ are the lower limit of size, upper limit of size, process mean, process standard deviation and target dimension, respectively. T is the tolerance defined by the difference of U and L. When PCIs are limited within certain specified values such as Cp ≥ p, Cpk ≥ k, Cc ≤ c, and Cpm ≥ m where p, k, c, and m are design parameters, the process must be controlled to maintain its capability within each parameter range. As the PCIs are defined by the process mean and/or standard deviation, the process is assumed to be controlled under statistical process control. When several STIs are simultaneously specified on a dimension, the allowable range of the mean and standard deviation tends to be complex. A population parameter zone (PPZ) is usually used to visually represent the area. 2.2. Population parameter zone The PPZ represents an allowable range on the μ - σ plane when STIs are specified. Fig. 2 shows examples of PPZ when multiple STIs are specified. The horizontal axis is the mean value and the vertical axis is the standard deviation value. The shaded area is the allowable range of the mean and the standard deviation. The solid boundary of the range is the allowable limit. When STIs are specified for each part dimension, the dimension has a corresponding PPZ. During the design process, designers need to decide suitable values and types of STI for each part dimension based on the required function, product performance, and cost of the final product. The decision process is equivalent to tolerance allocation. The STI stack-up problem must be solved before discussing the STI allocation problem, because the allocation problem is an inverse of the stack-up problem.
2. Statistical tolerance index (STI) STI is a specification that uses a process capability index as an additional indicator for a manufacturing process with a conventional tolerance. STIs can be specified by adding “ST” into a hexagon. Fig. 1 (a) shows a design drawing in which conventional tolerances are specified. Fig. 1 (b) shows a design drawing in which conventional tolerances and STIs are specified. The specified dimensions, tolerances, and STIs in Fig. 1 are for illustrative purposes only. STI is applicable to both dimension and geometrical tolerances, so that STI can limit the
30.0±0.05
30.0±0.05
ST
A
A A
A (a) conventional tolerance
ST
(b) statistical tolerance index
Fig. 1. Example design drawings. 359
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function, the functional dimension of the assembly X is given as follows. n
X=
Cp
Cpk
i=1
Cc
U
U
L
(6)
If the parts' dimensions are independent of each other as in the model, the mean and the standard deviation of the functional dimension of mass produced assemblies, μX and σX, can be calculated as follows.
Cpm
Cc
L
∑ xi
n
Fig. 2. Examples of population parameter zone and allowable area when STI are specified.
μX =
3. Stack-up analysis of STI
σX =
∑ μi i=1
(7)
n
∑ σi2 i=1
The functionality and performance of the final product usually depend on a functional dimension of assembly consisting of several parts. In the design process of the final product, the constraints of the functional dimension are decided based on customer and supplier demands. Subsequently, designers must allocate the constraints to parts' tolerances and specify them on the design drawing as shown in Fig. 1. The allocation process is carried out based on a solution of a tolerance stackup problem. When designing STIs, the stack-up problem should be solved first. As shown in Fig. 3, the problem can be converted into a synthesis of each part's PPZ.
The assumption of a normal distribution is not required because equations (7) and (8) are valid regardless of the distributions of the parts' dimensions. When the STIs are specified on each part's dimension with conventional tolerance, the PPZ of each part can be generated on the μi - σi planes. Accordingly, the PPZ of the functional dimension is calculated on the μi - σi plane, which is the solution to the STI stack-up problem. The boundary of the PPZ is an important element because it is the allowable limit of the mean and standard deviation. 3.2. Probabilistic solution to STI stack-up using the Monte-Carlo simulation
3.1. Modelling of stack-up analysis of STI
A method using the Monte-Carlo simulation is proposed in order to solve general STI stack-up problems. Each boundary on the PPZ of parts is the worst-case mean and standard deviation. Therefore, a boundary on assembly is obtained as the worst-case stack-up of them. The term “worst-case” denotes the worst situation between the mean and standard deviation. In the first procedure, uniform random numbers of means of each part are generated within the range limited by Cc. Minimum standard deviation of each PPZ is calculated based on the random number and other specified STIs, such as Cp, Cpk, and Cpm. The mean and standard deviation of assembly are then calculated using equations (7) and (8) using the random number and the minimum standard deviation, respectively. A point cloud is obtained by repeating these processes. The PPZ boundary of assembly can be estimated based on the point cloud. The method is applicable if the function h in equation (5) is linear, and if the parts dimensions are independent of each other. Because the Monte-Carlo simulation is a probabilistic method, there is an approximation error in the solution. The accuracy of the solution may depend on the number of trials of the Monte-Carlo simulation. Fig. 4 shows the procedure of the method.
Several existing studies have focused on the stack-up problem for conventional tolerance using the worst-case and statistical methods. However, there is little research about the problem of STI stack-up. Srinivasan (1999) proved that a solution to the problem was represented by the Minkowski sum on the μ - σ2 plane. They also demonstrated an algebraic solution for the STI stack-up problem when Cpk and Cc were specified and when the values of k and c for each part were the same. However, this solution is not of practical use due to a lack of a degree of design freedom. Furthermore, the calculation details of the Minkowski sum were not shown in that report. Otsuka and Nagata (2015) proposed a numerical method using the Monte-Carlo simulation for the problem of STI stack-up, and the method can be applied in more general situation. In this study, the method is used when solving the STI stack-up problem. In general, a functional dimension X consisting of n parts can be written as follows,
X = h (x1, x2, ..., x n )
(5)
where i is the parts identifier and n is the number of parts. As h can be linear or nonlinear, in this study, h is assumed to be linear. Fig. 3 also shows an assembly model consisting of three parts. When h is a linear
9.0±0.24 ST Cpk
1.32 Cc
4.0±0.1 ST Cp
4. Allocation method of the STI Designing the process capability limit of each part, that is the STI allocation problem, is a nonlinear and global optimization problem, because the problem is inverse to that of the STI stack-up. The differential information cannot be used in an STI allocation problem because the stack-up solution is obtained via the Monte-Carlo simulation. Furthermore, it is known whether search space is convex or nonconvex. Therefore, a genetic algorithm, which has been often used not only in tolerance allocation but also in many optimization problems, is applied. Numerous global optimization methods have been developed and applied to tolerance allocation problems of actual products. Metaheuristic methods are popular, especially in tolerance allocation problem. For example, Chen (2001) proposed a tolerance allocation method using simulated annealing and a neural network. Yang and Naikan (2003) proposed an algorithm for optimum allocation of tolerances using a neural network. Walter et al. (2015) presented a methodology for the least cost tolerance allocation of systems with time-variant deviations using particle swarm optimization. Chen and Fischer (2000) researched nonlinearly constrained tolerance allocation
0.85 1.50 Cc 0.95 Cpm 0.33
7.0±0.2 ST Cpk
(8)
1.32 Cpm 0.44
Stack up problem
= 20.0±0.54 ST
Fig. 3. Representation of the stack-up problem using PPZ. 360
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2 Part 1
1. Generate a uniform random number * within allowable value of part1, which is calculated based on specified Cc value. 2. Calculate all allowable standard deviations corresponding to specified Cp, Cpk, Cpm.
2 2 ( * , *)
3. Select the minimum standard deviation and name it as * .
L
U
*
4. Point ( *, *) is on a boundary of a certain STI.
Part n
Part 2 ( * , *)
( *, *)
Assembly
( *, *)
5. Repeat the procedure mentioned above for all parts and obtain the point ( *, *) .
6. Calculate * and * based on statistical rule. 7. Plot ( * , * ) on the
plane.
8. Return to 1. and iterate as many as possible. 9. Set of points is obtained, and an approximate solution can be obtained as a convex-hull envelope. Fig. 4. Flowchart of the proposed method for solving the stack-up problem of the STI.
x-y plane: Design variable space
problems and concluded that genetic algorithms performed well for the problem in their study. Noorul et al. (2005) proposed a procedure using a genetic algorithm for gear train and overrunning clutch assemblies and compared it with conventional techniques. Based on these studies, a genetic algorithm, in particular a real-encoded genetic algorithm is applied to the STI allocation problem in this study. In tolerance allocation, a relational model between cost and tolerance is used and tolerances are usually determined considering costminimization. For example, Walter and Wartzack (2013) presented a methodology for the least cost tolerance allocation of systems with time dependent motion behavior. If tolerances are design parameters, then the cost-tolerance relationship, which is often used in tolerancing, can be applied. However, the relationship between cost and process capability indices, in particular PPZ, is not thoroughly discussed in this field. Therefore, in this study, the size of the allowable area on the PPZ is set as an evaluation function because manufacturing cost decreases as the area becomes larger. Allowable limits of the process capability indexes, p, k, c, m, and target value τ are set as the design values.
Unit vectors
Color map: Two dimensional normal distribution. (Children will be generated based on the probability.) Fig. 5. Schematic of UNDX in the case of two design parameters. n
4.1. Real-coded genetic algorithm
n
C1 = m + ∑kparam z k ek , C2 = m + ∑kparam z k ek =1 =1 m = (P1 + P2)/2
A real-coded genetic algorithm is applied to find the optimal solution due to the large number of design parameters in the STI allocation problem. In a real-coded genetic algorithm, the crossover process is important because it affects the convergence of the optimal solution. In this study, the unimodal normal distribution crossover (UNDX) is applied as the crossover process in the genetic algorithm (Kita et al., 2002). The crossover operator outperformed other crossover operators in several benchmarking tests and was successfully applied to the design of the lens systems (Ono et al., 2000). In the UNDX, each chromosome is a real vector as a representation in the design variable space. Fig. 5 shows the schematic of the UNDX in the case of two design parameters, which is the simplest case. In the UNDX, when the number of design parameters is nparam, children vectors C1 and C2 are generated as follows,
z k ≅ N (0, sk2) (k = 1, …., nparam) s1 = αd1, sk = βd2/ nparam e1 =
(P2 − P1) P2 − P1
, ei ⊥ ej, (i, j = 1, …, n param ; i ≠ j )
d1 = P2 − P1 , d2 = (P3 − P1)·e2
(9)
where P1, P2 and P3 are parent vectors of Parent 1, 2 and 3, respectively. d1 is the distance between Parents 1 and 2, d2 is the distance between Parent 3 from the line connecting Parents 1 and 2. α and β are constants. In this study, the constants α and β are set to be 0.5 and 0.35, respectively, according to recommendation in the original UNDX report (Ono et al., 1999, 2000; Kita et al., 2002). In the UNDX, vector P3 is used to calculate d2, then the parameter sk, which is the standard deviation of normal random numbers zk, is decided. If nparam is bigger, C1 and C2 are generated away from the line connecting P1 and P2. To avoid this situation, sk is divided by the square root of nparam. Fig. 6 shows a complete flowchart of the proposed method to design STI. 361
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PPZ of X
Evaluation
500 generations?
Yes
Satisfying yield requirement ?
S.D.
Performance
Calculate the stack-up of statistical tolerance indices
Initial configuration
No Elitism and Selection
Mean
Satisfying performance requirement ?
Performance and PPZ of X
Crossover (UNDX) Mutation
Calculate evaluated value
Return evaluated value
ST Cp
.... Cpk
....
ST Cp
.... Cpk
....
ST Cp
.... Cpk
....
End Fig. 6. Flowchart of proposed method to design STI for each part.
4.2. Product performance In this study, the STI is designed based on the product's performance. The performance of product Y depends on the functional dimension described as Y = f(X). The functional dimension of the accepted product is distributed within a total tolerance zone between the upper size limit UX and the lower size limit LX. A representative value of each lot is defined as the estimated value of the product performance, as follows,
EY =
∫L
UX
X
f (X ) N (μX , σX ) dX
PPZ of xi
Specifications on drawing
Fig. 7. Design flow of the STI based on product performance.
X x1 x2 x3
(10)
where function N denotes a normal distribution, LX and UX are the lower and upper limits of the functional dimension, respectively, and EY indicates the customer demand for product performance in the STI design process. Once the minimum EY value is set based on customer demand, the demand can be directly projected onto PPZ (μX - σX plane) as an allowable area, as shown in Fig. 7. The STI is designed to calculate each design parameter so that the stack-up of the STI does not infringe on the allowable area. Furthermore, the design parameters should be designed with a more relaxed STI value in terms of manufacturing cost. When more relaxed STI values are set to each part, the allowable area on the μX -σX plane becomes larger. Therefore, the inverse value of the allowable area is set as an objective function in the genetic algorithm and the objective function should be small.
x5
x4
x6
Fig. 8. A virtual product model used in the case study.
Table 1 Basic parameters of the virtual product. (Unit mm).
5. Case study A case study was analysed using a virtual assembly, which consisted of six linearly stacked-up parts, as shown in Fig. 8. The assembly is the model of a rotating element used in various mechanical products such as cars, machine tools, home appliances, and so on. The virtual product is an academic model and is a representative model of linearly stacked-up products. Even if the number of parts increases, the proposed method can be applied. However, it takes longer time to converge. The basic parameters of parts' dimensions, tolerances, lower and upper limit sizes are assumed, as shown in Table 1. In this case, the functional dimension X was the same as in equation (6) with n = 6. The performance function of the product was assumed to be equation (11) as shown in Fig. 9 (a), which has an asymmetrical nominal-the-best characteristic. Although it would be more preferable to use an actual performance function, it does not affect the effectiveness of the proposed method. Fig. 9 (b) shows the EY values projected on the μX -σX plane.
i
1
2
3
4
5
6
X
xi Ui Li
40 40.6 39.4
6 6.3 5.7
12 12.4 11.6
78 78.8 77.2
24 24.5 23.5
40 40.6 39.4
200 203.2 196.8
Y = 20 X − 140 − 0.2(X − 198)3 − 130
(11)
Fig. 10 shows chromosome structure, which consists of pi, ki, ci, mi and τi. Each gene is a real number. In this study, assembly consists of six parts, so that the size of chromosome is 30. Parameter tuning is important in the genetic algorithm in terms of the convergence performance and avoiding local minimum. However, the sensitivity of the parameters to the objective function is not known analytically even for such a simple model. In this study, the parameters of the genetic algorithm were manually tuned through several trial-anderror iterations. The search space of each design parameter and the parameters of the genetic algorithm are listed in Table 2. 362
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25
1.0 (mm)
EY
X
Y
Table 2 Parameters of the real-coded genetic algorithm.
20
0 196.8
203.2
0 196.8
X (mm)
200 μX (mm)
(a)
(b)
200
0
203.2
Fig. 9. Product performance characteristic and EY values on PPZ in the case study.
(12)
Furthermore, the initial parameter setting was also determined through several trial-and-error iterations. Real values of p, k, c, and m of all initial individuals are randomly set considering severe situations, such as pi = 2.0–2.5, ki = 2.0–2.5, ci = 0.3–0.5, mi = 1.1–1.6, respectively. Conversely, target dimensions τi are randomly set near the center of each tolerance range. Optimization was performed under these conditions.
p k c m τ
c1 ~ c6
90% 1% 60 4 1.0 ≤ pi ≤ 2.5 1.0 ≤ ki ≤ 2.5 0.3 ≤ ci ≤ 1.0 0.1 ≤ mi ≤ 1.0 Li ≤ τi ≤ Ui
x1
x2
x3
x4
x5
x6
1.031 1.000 0.300 0.723 40.06
1.881 1.797 1.000 0.274 5.91
1.843 1.938 0.418 0.753 11.66
1.000 1.001 0.892 0.239 77.49
1.000 1.000 0.987 0.261 23.87
1.015 1.099 0.866 0.235 39.78
1.0
Generation 1 100 200 500
(mm)
3 2
X
Objective function
Design parameters obtained by the optimization are rounded to the forth decimal place and are listed in Table 3. Fig. 11 (a) shows the convergence curve for 500 generations. The result shows that the optimal solution is well converged. Fig. 11 (b) shows the historical allowable areas during the evolution process. In Fig. 11(b), each line shows a limit of the allowable area at generations 1, 100, 200, and 500. The figure also shows that allowable area showed an incremental trend. This result indicates that the initial solution is a strict STI, but the final optimal solution is a relaxed STI. Fig. 12 shows that the allowable area at the 500th generation overlaps the one in Fig. 9 (b). This result shows that product performance always satisfies equation (12) if the STI is specified on the drawing based on the optimal solution. Fig. 13 shows a design drawing that includes specified dimensions, tolerances and the design parameters rounded to the second decimal place. To guarantee performance demand, the parameters p, k, and m should be rounded up, and parameter c should be rounded down. Target in the figure dimensions are omitted, and it can be specified if it is required. Some parameters in the figure are redundant. For example, in the parameters of part 3, Cp ≥ 1.9 and Cpk ≥ 2.0 are specified. In this case, the former constraint is perfectly included in the latter one. Therefore, it is preferable to remove the Cp specification to reduce redundancy in actual design drawing. Finally, it is concluded that the proposed method can help suitably design the STI and target values for each part. Until now, there has not been a design method of process capability indices. The proposed method makes it possible for designers to design process capability indices based on customer demand for product performance and manufacturing cost. Compared with existing tolerancing methods, the proposed method has novelty and advantage with regard to considering both product performance and manufacturing cost. It can be used as a decision-making support tool not only in designing and manufacturing stages but also in the overall management of the
k1 ~ k6
Crossover rate Mutation rate Number of Individual Number of elite Search space of p Search space of k Search space of c Search space of m Search space of τ
4
6. Result and discussion
p1 ~ p6
500
Table 3 Optimized design parameters.
Assuming required performance EY to be larger than 20 as follow.
EY ≥ 20
Generation
1 0
100
200
300
400
500
0 196.8
200
Generation
μX (mm)
(a)
(b)
203.2
Fig. 11. Convergence curve and historical results of the allowable area during optimization.
1.0
(mm)
20
X
EY
0 196.8
200
203.2
0
μX (mm) Fig. 12. Product performance EY and allowable area according to optimized parameters.
company. For example, based on the design drawing, the company or designer can decide whether to outsource a part or create it in their factory. However, the proposed method has some limitations. It can be
m1 ~ m6
Number of design parameter is 6 × 5 = 30.
363
1
~
Fig. 10. Chromosome structure used in UNDX. 6
International Journal of Production Economics 204 (2018) 358–364
A. Otsuka, F. Nagata
40±0.6 ST Cp 1.1 Cpk
1.0 Cc
6±0.3 ST Cp 1.9 Cpk
1.8 Cc
12±0.4 ST Cp 1.9 Cpk
78±0.8 ST Cp 1.0 Cpk
0.3 Cpm 0.8
This work was supported by JSPS KAKENHI Grant Number 26870774 and 17K18029.
1.0 Cpm 0.3
2.0 Cc
1.1 Cc
Acknowledgments
0.4 Cpm 0.8
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0.8 Cpm 0.3
24±0.5 ST Cp 1.0 Cpk
1.0 Cc
40±0.6 ST Cp 1.1 Cpk
0.9 Cpm 0.3
1.1 Cc
0.8 Cpm 0.3
Fig. 13. Design drawing of the virtual product model with conventional tolerance and STI.
used under the conditions in which tolerance limits and performance function are known and parts are linearly stacked. 7. Conclusion This paper proposes an STI design method, which is an inverse solution to the stack-up problem. The proposed method consists of the Monte-Carlo simulation and a real-coded genetic algorithm. The efficiency of the proposed method was evaluated via a case study using a virtual product model. The STI was suitably designed and optimized so that the product satisfied required product performance. As the proposed method considers both product performance and manufacturing cost, it can be used as a decision-making support tool not only in the designing and manufacturing stages but also in the overall management of the company. As future work, the proposed method is planned to be extended to geometrical tolerance, and also to be researched in terms of acceptance sampling and failure rate. Furthermore, in this method, cost function in optimization is defined as the inverse value of the allowable area and the area is uniformly evaluated. In fact, the bottom side of the area cannot be achieved due to the limits of machining accuracy. Therefore, more suitable evaluated function should be researched.
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