Selective Assembly to Minimize Surplus Components by Shifting the Process Mean

SELECTIVE ASSEMBLY TO MINIMIZE SURPLUS COMPONENTS BY SHIFTING THE PROCESS MEAN

Shun Matsuura 1 and Nobuo Shinozaki 2

1 Research Fellow of the Japan Society for the Promotion of Science, Keio University, 3-14-1, Hiyoshi, Kohoku, Yokohama 223-8522, Japan [email protected] 2 Keio University, 3-14-1, Hiyoshi, Kohoku, Yokohama 223-8522, Japan [email protected]

Abstract We consider a product assembled from two components when the quality characteristic is the clearance between the matching components (or the sum of the dimensions of the matching components). Some variation is inevitable in any production process, and the quality of the assembled product is dependent on the dimensional variability of the component parts. A product is rejected when its clearance is outside a given specification range. Random assembly of the matching components may lead to an unacceptably large number of rejected products. In such a situation, selective assembly should be effective for reducing it. In this approach, the components are sorted into several classes according to their dimensions, and the product is assembled by randomly selecting matching components from the corresponding classes. Many previous works are focused on equal width and equal probability partitioning schemes. When there is large difference between the variances of the two component dimensions, equal width partitioning will result in large number of surplus components and equal probability partitioning will result in some rejected products. Some authors have proposed the method of manufacturing the component with smaller variance at two (or more) shifted means to cope with this difficulty. However, determination of optimal mean shift has not been addressed. This paper treats with the problem of determining optimal mean shift in equal width and equal probability partitioning schemes. Some numerical results are given which show that using optimal mean shift considerably reduces the number of surplus components in equal width partitioning and may enable us to manufacture all products within the specification range in equal probability partitioning. We also show some merits and demerits of equal width and equal probability partitioning schemes.

Keywords: Clearance specification, Match gauging, Optimization, Statistical process control, Tolerance limit

INTRODUCTION

We consider a product assembled from two components when the quality characteristic is the difference of the relevant dimensions of the matching components (i.e., clearance). Note that although we use the clearance as the assembly dimension of interest, our discussion is equally valid in the case in which it is the sum of the dimensions of the matching components. Some variation is inevitable in any production process, and the quality of the assembled product is dependent on the dimensional variability of the component parts. The clearance of any product should be within a given specification range. A product is rejected when its clearance is outside the specification. Random assembly of the matching components may lead to an unacceptably large number of rejected products. In such a situation, selective assembly should be effective for reducing it. In this approach, any component with its dimension outside specified limits of the dimensional distribution is rejected, and the remaining components are sorted into several classes according to their dimensions. The product is then assembled by randomly selecting matching components from the corresponding classes. This approach enables the assembly of high-precision products from relatively low-precision components, which may lead to a cost reduction compared to the alternative of manufacturing the respective components at a higher level of precision. A piston and cylinder assembly (Figure 1) is an example from an automobile industry in Japan. A matching pair of piston and cylinder is chosen to satisfy a given clearance specification. If the clearance is too small, the oil film between the cylinder wall and piston becomes too thin and piston scuffing occurs. If the clearance is too large, the piston inclines in the cylinder and abnormal noise occurs. Random assembly of pistons and cylinders lead to an unacceptably large number of products which do not satisfy the clearance specification. Thus, the automobile industry has used selective assembly. Pistons and cylinders are sorted into several classes according to their outer and inner diameters, respectively. The smaller pistons are matched with the smaller cylinders and the bigger pistons with the bigger cylinders. Other applications of selective assembly include a pin and bushing assembly, a hole and shaft matching, and a camshaft, valve, and valve lifter assembly.

1 Outer diameter of piston

Inner diameter of cylinder

Figure 1. Piston and Cylinder Assembly

A great deal of research and development effort has been devoted to the subject of selective assembly. Kwon et al. (1999) studied optimal partitioning of the distributions of the two component dimensions to minimize expected squared error loss, assuming that the two component dimensions had the same normal distribution. Mease et al. (2004) extensively studied some optimal partitioning strategies for several loss functions and distributions. In particular, they developed optimal partitioning for minimizing the expected squared error loss for the case in which the two component dimensions are identically distributed. They gave equations for optimal partition limits, and showed that the solution to them is unique when the density function of the dimensional distribution is strongly unimodal. Matsuura and Shinozaki (2007) extended the results to the case in which measurement error is present. Matsuura and Shinozaki (2009) studied optimal partitioning to maximize profit, including optimal choice of truncation points of the dimensional distributions, in the presence and absence of measurement error. These papers are focused on optimal partitioning of the dimensional distributions, and did not take account of the tolerance constraint on the clearance. In selective assembly when the dimensional variability of the two components are approximately equal and a tolerance constraint on the clearance is given, equal width partitioning scheme, studied by Pugh (1986) for instance, enables us to manufacture all products to satisfy the clearance specification. However, it is not unusual that the variances of the two component dimensions are different. If it is the case and we use equal width partitioning scheme, large number of surplus components are expected due to the difference between the numbers of components in the corresponding classes. On the other hand, Fang and Zhang (1995) studied equal probability partitioning scheme in which the corresponding classes have equal probability, that is, we have no surplus component. They proposed an algorithm for equal probability partitioning to manufacture all products within the specification range. However, equal probability partition to manufacture all products within the specification range does not exist when the difference between the variances of the two component dimensions is large or the clearance specification is tight. In the unequal variances case, some authors proposed the method of manufacturing the component with smaller variance at two (or more) shifted means and mixing them. Although Matsuura and Shinozaki (2008) studied optimal mean shift which minimizes clearance variation, they did not take account of the tolerance constraint on the clearance. On the other hand, although Kannan et al. (1997) and Kannan and Jayabalan (2002) took account of the clearance specification, they did not discuss optimal mean shift which minimizes the number of surplus components. This paper studies optimal mean shift in equal width and equal probability partitioning schemes when the component with smaller variance is manufactured at two shifted means and a tolerance constraint on the clearance is given. We give models, notation, and assumptions in Section 2. In Sections 3 and 4, we describe equal width and equal probability partitioning schemes and discuss optimal mean shift in both the schemes. Section 5 gives some numerical results which show that using optimal mean shift considerably reduces the number of surplus components in equal width partitioning and may enable us to manufacture all products within the specification range for tighter clearance specification in equal probability partitioning. We also show some merits and demerits of equal width and equal probability partitioning schemes. The final section gives concluding remarks.

2 MODELS, NOTATION, AND ASSUMPTIONS

Let X and U denote the dimensions of the components with larger and smaller variance, respectively. The clearance X- U must be within the specification range [C- D, C + D ] , where C is a given target clearance and D is a given tolerance limit. Suppose that the process mean of U can be adjusted. Then we let C = 0 without loss of generality. We also suppose that X and U are normally distributed and we let t =SD[ U ] / SD [ X ] ( 0< t 1 ). Throughout this paper, we assume E[ X ]= 0 and SD[ X ]= 1 without loss of generality. Then, X is distributed as N(0, 1) . Let F(x ) denote the cumulative distribution function of N(0, 1) . The component with smaller variance is manufactured at two shifted means, and we let b be the two means. Then, the dimension of the component with smaller variance, denoted by Y , follow the mixture distribution of U s with means b . Its cumulative distribution function is expressed as 1 � 骣y+ b 骣 y b FY ( y| b ) =睚 F琪 + F 琪 . 2 铪桫t 桫 t We also let D be the common specification limits for the two component dimensions X and Y . A component with its dimension in the intervals (-� - D ] and (D, ) is rejected before to the assembly process. Let F and G denote the cumulative distribution functions of X and Y after truncation at D , respectively. They are expressed as F(x ) - F ( - D ) F( x ) = ,- D＃ x D , F(D ) - F ( - D ) F( y| b ) - F ( - D | b ) G( y| b ) = Y Y FY( D| b ) - F Y ( - D | b ) 1{F (y- b ) + F ( y + b )} - 1 { F (-D - b ) + F ( - D + b )} =2t t 2 t t ,-D＃ y D . 1D- b D + b 1 - D - b - D + b 2{F (t ) + F ( t )} - 2 { F ( t ) + F ( t )} Let n denote the number of classes. The respective partition limits for X and Y are denoted by

(x1, x 2 , … ,) xn- 1 and (y1, y 2 , … ,) yn- 1 . In selective assembly, the components X�( xi-1 x i ] are matched with the components Y�( yi-1 y i ] as shown in Figure 2.

Component X

reject reject

-D x x ・・・ x D x D 1 2 n-1 n-1

matching matching matching

Component Y

reject reject

-D y y ・・・ y D y D 1 2 n-1 n-1 Figure 2. Selective Assembly

3 We let X i and Yi denote the truncated random variables of X and Y defined on (xi-1, x i ] and (yi-1, y i ] , respectively, where x0= y 0 = - D and xn= y n = D . The probabilities of the i th class of the components X and Y , F( xi )- F ( x i-1 ) and G( yi| b ) - G ( y i-1 | b ) , are denoted by pi and qi . In Sections 3 and 4, we describe equal width and equal probability partitioning schemes and discuss optimal mean shift in both the schemes.

EQUAL WIDTH PARTITIONING SCHEME

Equal width partitioning scheme partitions the distributions so that all classes have equal widths. The number of classes needed to satisfy the clearance specification is 轾2D n = . 犏D We note that 犏轾x denotes the minimum integer which is not smaller than x . The partition limits for X and Y are given as

(x1, x 2 , … ,) xn-( 1 = y 1 … , y) 2 , ( , y n - 1 = - D + 2 w ,- … D) + w , , D - w , 2D where w = n . We easily see that

|Xi - Y i |＃ w D, i =1 , 2 , … , n . Thus, all products satisfy the clearance specification. However, some surplus components result from the difference between pi and qi . The probability that a component is used for assembly is n n

R( b )=邋 min( pi , q i ) = min{ F ( x i ) - F ( x i-1 ) , G ( y i | b ) - G ( y i - 1 | b )} . i=1 i = 1 We easily see that R(0)= 1 holds when t =1. Thus, we see that we do not need to shift the process mean and have no surplus component when the two component dimensions have the same variance. However, in the unequal variances case (t <1), b = 0 does not necessarily maximize R( b ) and we should choose optimal mean shift b* which maximizes R( b ) , that is, b* =arg max R ( b ) . b 0

We give some properties of b* and R( b* ) .

Proposition 3.1. (The lower limit of R( b ) .) For any D and b , D R( b )� min( f ( x )| g ( x , b )) dx -D where f ( ) and g(讄 b ) denote the derivative functions of F( ) and G(讄 b ) , respectively.

Proposition 3.2. (The lower limit of R( b* ) .) Let D b† =arg max min( f ( x ) , g ( x | b )) dx . b 0 -D Then, for any D , D R( b* )吵 R ( b† ) max min( f ( x ), g ( x | b )) dx . b 0 -D

From Proposition 3.2, we see that it is guaranteed that R( b* ) is not smaller than D max min(f ( x ), g ( x | b )) dx for any D . We notice that b† and b 0 -D

4 D max min(f ( x ), g ( x | b )) dx do not depend on D . Therefore, if b 0 -D D max min(f ( x ), g ( x | b )) dx is sufficiently large, it may be an effective method that we use b† b 0 -D without regard to the value of D , compared to the alternative of computing b* which depends on D . In Section 5, we will give some numerical results which show that using optimal mean shift b* considerably increases R( b ) , especially for the case in which t is small.

EQUAL PROBABILITY PARTITIONING SCHEME

Equal probability partitioning is the partitioning in which the corresponding classes have equal probability, that is, no surplus component exists. The goal is to find the number of classes n and the sets of partition limits

(x1, x 2 , … ,) xn- 1 and (y1, y 2 , … ,) yn- 1 which satisfy the following conditions:

xi- y i-1 � , = i ,1 , 2 , … , n (1)

xi-1 - y i � D, = i ,1 , 2 , … , n (2)

pi= q i , i =1 , 2 , … , n . (3)

We note that if the conditions (1) and (2) are satisfied, then all products satisfy the clearance specification. Fang and Zhang (1995) gave the following algorithm for finding equal probability partitioning which satisfies the conditions (1)-(3). We note that F -1 ( ) and G-1(讄 b ) denote the inverse functions of F( ) and G(讄 b ) , respectively.

-1 1. If F(- D + D )� G ( + D D| b ) , then put x1 = - D + D and y1 = G( F ( - D + D ) | b ) . If -1 F(- D + D ) > G ( - D + D | b ) , then put y1 = - D + D and x1 = F( G ( - D + D | b )) .

2. We repeat the following for i =2 , 3 , … , sequentially unless D- xi � and D- yi � hold. If -1 F( yi+ D )� G ( x i D| b ) , then put xi+1 = y i + D and yi+1 = G( F ( y i + D ) | b ) , and if -1 F( yi+ D ) > G ( x i + D | b ) , then put yi+1 = x i + D and xi+1 = F( G ( x i + D | b )) .

3. If D- xi � and D- yi � holds for some i , then put xi+1= y i + 1 = D and n= i +1 and finish the algorithm.

However equal probability partitioning which satisfies the conditions (1)-(3) does not exist under a condition as discussed in the following proposition.

Proposition 4.1. The number of classes n and the partition limits (x1, x 2 , … ,) xn- 1 and (y1, y 2 , … ,) yn- 1 which satisfy the conditions (1)-(3) exist if and only if D >sup |F-1 ( u ) - G - 1 ( u | b ) | . 0

We see from this proposition that, letting d (b )= sup | F-1 ( u ) - G - 1 ( u | b ) |, 0 d (b ) . Thus, if we choose optimal mean shift b* which minimizes d (b ) , that is, b*=arg min sup | F -1 ( u ) - G - 1 ( u | b ) | , b 0 { } 0

5 equal probability partitioning which satisfies the conditions (1)-(3) no matter how small D is. We also give the following proposition.

Proposition 4.2. Suppose that D >sup |F-1 ( u ) - G - 1 ( u | b ) | . 0

From this proposition, we see that the algorithm minimizes the number of classes in the set of equal probability partitions which satisfy the conditions (1)-(3).

NUMERICAL RESULTS

In this section, we give some numerical results on equal width and equal probability partitioning schemes for D = 3 and some values of D . Tables 1-3 compare R(0) and R( b* ) for D =2 , 1 , 0 . 6 in equal width partitioning scheme. From these tables, we see that using optimal mean shift results in considerable improvement on R( b ) . In other words, using optimal mean shift considerably reduces the number of surplus components compared with not shifting. Especially for the case in which t is small, the improvement is quite substantial. For example, although about 30% of components are surplus without shift for D = 2 and t =0 . 3 , we have no surplus component by manufacturing the component with smaller variance at two means �0 8559 . Table 4 compares d (0) and d (b* ) for t =1 , 0 . 8 , 0 . 5 , 0 . 3 in equal probability partitioning scheme. We see that using optimal mean shift enables us to manufacture all products within the specification range for tighter tolerance limit D in equal probability partitioning compared to not shifting. For example, when t =0 . 5 , although equal probability partitioning which satisfies the conditions (1)-(3) exists only for D >1 . 2832 without shift, it exists for D >0 . 3741 by manufacturing the component with smaller variance at two means �1 0194 . Next we give a numerical example to compare equal width and equal probability partitioning schemes. We let t =0 . 5 and D =0 . 6 . From Table 2, if we use equal width partitioning scheme and b* =0 . 6516 , then we need 10 classes (the partition limits for X and Y are (- 2 . 4 ,- 1 . 8 ,- 1 . 2 ,- 0 . 6 , 0 , 0 . 6 , 1 . 2 , 1 . 8 , 2 . 4) ) and about 10% of components are surplus. On the other hand, although equal probability partitioning which satisfies the conditions (1)-(3) does not exist since d (0)= 1 . 2832 > 0 . 6 = D if we use b = 0 , it exists since D =0 . 6 > 0 . 3741 = d (b* ) if we use b* =1 . 0194 . Table 5 is the result obtained using the algorithm for equal probability partitioning when we use b* =1 . 0194 . From this table, we see that equal probability partitioning reduces surplus components from 10% to 0% by increasing number of classes from 10 to 17 compared with equal width partitioning.

Table 1. R(0) , b* and R( b* ) for t =0 . 8 in equal width partitioning D n R(0) b* R( b* ) 2 3 0.8957 0.5586 1 1 6 0.8957 0.5586 0.9925 0.6 10 0.9054 0.5418 0.9893

6 Table 2. R(0) , b* and R( b* ) for t =0 . 5 in equal width partitioning D n R(0) b* R( b* ) 2 3 0.7300 0.7595 1 1 6 0.7300 0.7595 0.9636 0.6 10 0.6829 0.6516 0.9084

Table 3. R(0) , b* and R( b* ) for t =0 . 3 in equal width partitioning D n R(0) b* R( b* ) 2 3 0.6854 0.8559 1 1 6 0.6854 0.8559 0.9572 0.6 10 0.4982 0.6356 0.8019

Table 4. d (0) , b* and d (b* ) in equal probability partitioning t d (0) b* d (b* ) 1 0 0.8 0.4371 0.6269 0.0503 0.5 1.2832 1.0194 0.3741 0.3 1.9157 1.2378 0.7382

Table 5. Equal probability partitioning under D =0 . 6 when X: N(0, 1) and Y: N(� 1, 0194 . (0 5)2 ) Class Partition limits for X Partition limits for Y Clearance Probability min max min max min max % 1 -3.000 -2.774 -3.000 -2.400 -0.600 0.226 0.14 2 -2.774 -2.481 -2.400 -2.174 -0.600 -0.081 0.38 3 -2.481 -2.005 -2.174 -1.881 -0.600 0.169 1.60 4 -2.005 -1.281 -1.881 -1.444 -0.561 0.600 7.78 5 -1.281 -0.844 -1.444 -1.150 -0.131 0.600 9.96 6 -0.844 -0.550 -1.150 -0.917 0.073 0.600 9.21 7 -0.550 -0.317 -0.917 -0.682 0.132 0.600 8.47 8 -0.317 -0.082 -0.682 -0.281 -0.036 0.600 9.21 9 -0.082 0.198 -0.281 0.518 -0.600 0.478 11.12 10 0.198 0.423 0.518 0.798 -0.600 -0.095 8.57 11 0.423 0.677 0.798 1.023 -0.600 -0.121 8.71 12 0.677 1.026 1.023 1.277 -0.600 0.002 9.69 13 1.026 1.576 1.277 1.626 -0.600 0.299 9.53 14 1.576 2.226 1.626 2.013 -0.437 0.600 4.46 15 2.226 2.613 2.013 2.266 -0.041 0.600 0.86 16 2.613 2.866 2.266 2.504 0.110 0.600 0.24 17 2.866 3.000 2.504 3.000 -0.134 0.496 0.07

CONCLUSION

In selective assembly when the component with smaller variance is manufactured at two shifted means, this paper has studied determining optimal mean shift in equal width and equal probability partitioning schemes. Some numerical results have shown that using optimal mean shift considerably reduces the number of surplus components in equal width partitioning and enables us to manufacture all products within the specification range for tighter clearance specification in equal probability partitioning compared to not shifting. It has also been shown that by using equal

7 probability partitioning we have no surplus component but need larger number of classes compared with equal width partitioning.

ACKNOWLEDGEMENTS

This research was supported by Grant-in-Aid for JSPS Fellows, 20 381.

REFERENCES

Fang, X. D. and Zhang, Y. (1995), “A new algorithm for minimizing the surplus parts in selective assembly,” Computers and Industrial Engineering, Vol.28, pp. 341-350. Kannan, S. M., Jayabalan, V. and Ganesan, S. (1997), “Process design to control the mismatch in selective assembly by shifting the process mean,” Proceedings of International Conference on Quality Engineering and Management, P.S.G. College of Technology, Coimbatore, South India, pp. 85-91. Kannan, S. M. and Jayabalan, V. (2002), “Manufacturing mean design for selective assembly to minimize surplus parts,” Proceedings of International Conference on Quality and Reliability, I.C.Q.R. RMIT University, Melbourne, Australia, pp. 259- 264. Kwon, H. M., Kim, K. J. and Chandra, M. J. (1999), “An economic selective assembly procedure for two mating components with equal variance,” Naval Research Logistics, Vol.46, pp. 809-821. Matsuura, S. and Shinozaki, N. (2007), “Optimal binning strategies under squared error loss in selective assembly with measurement error,” Communications in Statistics–Theory and Methods, Vol.36, No.16, pp. 2863-2876. Matsuura, S. and Shinozaki, N. (2008), “Selective assembly to minimize clearance variation by shifting the process mean when two components have dissimilar variances (in Japanese),” Journal of the Japanese Society for Quality Control, Vol.38, No.2, pp. 73- 81. Matsuura, S. and Shinozaki, N. (2009), “Selective assembly for maximizing profit in the presence and absence of measurement error,” in Lenz, H. J. and Wirlich, P. T. (Ed.), Frontiers in Statistical Quality Control 9, Physica Verlag, Heidelberg (to appear). Mease, D., Nair, V. N. and Sudjianto, A. (2004), “Selective assembly in manufacturing: statistical issues and optimal binning strategies,” Technometrics, Vol.46, pp. 165-175. Pugh, G. A. (1986), “Partitioning for selective assembly,” Computers and Industrial Engineering, Vol.11, pp. 175-179.