Matching via Majorization for Consistency of Product Quality Lirong Cui Dejing Kong Haijun Li Abstract A new matching method is introduced in this paper to match attributes of parts in order to ensure consistent quality of products that are assembled from matched parts. The method yields invariant optimal matching that depends only on ranking of attributes of assembling parts where the optimal matching criteria consist of a large class of metrics including the Kantorovich cost function for matched pairs. Our method is non-parametric and based on the theory of majorization that reinforces a general rank-invariant matching strategy that small matches with small and large matches with large for variance reduction. Using this majorization-based matching method several specific multi-part matching problems are investigated in order to illustrate its wide applicability in quality assurance. Key words and phrases: Invariant optimal solutions; matching pattern; majorization; Schur-convexity Kantorovich cost function. Introduction The common theme of discrete matching problems is to match elements of one finite set S to elements of another finite set S 2 with some cost associated to matched elements and the goal is to find best matching strategies for minimizing the total associated cost of all possible matched elements. Matching is usually one-to-one but can also be a subset-tosubset matching. This work is in honor of Professor Alan G. Hawkes on his 75th birthday. Lirongcui@bit.edu.cn kongdejing2345@63.com School of Management and Economics Beijing Institute of Technology Beijing 0008China. Supported by the NSF of China under grant 73703. lih@math.wsu.edu Department of Mathematics Washington State University Pullman WA 9964 U.S.A. Supported by NSF grant DMS 007556.
The early work in matching appeared in the 940s see e.g. [4] and the research has accelerated in observational studies since the 970s after the paper by Cochran and Rubin [2] and the paper by Rubin [20]. Optimal matching has now been widely studied in various contexts; for example in design of observational studies [8 9 2] economics [6 23] sociology [8 5] epidemiology and medicine [3 ] and in political science [7] to mention just a few. Optimal matching has been also studied as design problems in optimizing network flows see e.g. [5] and the reference therein. The continuous matching problem can be formulated as the Monge-Kantorovich mass transportation problem [6 7] where S and S 2 are usually multi-dimensional Euclidean spaces or general Hilbert spaces. Given two probability distributions f and f 2 on S and S 2 respectively the goal is to find an optimal matching or transport map from S to S 2 so that certain cost functional of f and f 2 is minimized. Such an optimal transport exists under mild model assumptions and finding it explicitly can be done in the situations such as S = S 2 = R. In this paper we develop a discrete matching method based on the theory of majorization [4] and apply it to quality assurance. The simplest setup of our problem is described as follows. Let S = {x... x n } and S 2 = {y... y n } denote two finite sets with the same size. If x i < y j for all i j n then find a permutation π of {... n} so as to minimize the total variance d π S S 2 := n i= [ y πi x i n 2 y πk x k ]. subject to n i= y i x i = c being a constant. The problem was motivated from optimal assembling of parts. For example an axle s diameter is an important attribute and its sleeve has a large hole size. In an assembling process two parts are equipped with each other; one is inside and another is outside see Figure.. Due to manufacturing variability the measurements x... x n of inside parts are represented by a random variable X and the measurements y... y n of outside parts are represented by a random variable Y. We assume that P X Y = because of conforming regularity. In practice parts are matched randomly in an assembling process which cannot make the dimension gaps of paired parts consistently minimal; that is the variance of these gaps is rather large. k= The dimension gap between two paired parts is crucial to ensure the quality of products since the larger dimension gaps could result in shorter lifetimes for products due to various factors such as friction and vibration. An important issue is as formulated in. to match the inside and outside parts properly to make the product quality highly consistent in a batch of products. Quality management has plenty of contents such as sampling plans see e.g. [ 2] and 2
Figure.: The values of the diameters Φ and Φ 2 for inside axle and outside sleeve parts are x... x n and y... y n respectively. quality controls see e.g. [9] etc. Delivering consistency of product quality has been always an important issue in quality management but it has not been studied deeply specially in quantitative analysis. Consistent product quality not only makes efficient use of materials but also facilitates effective management of production system operation. In today s digital era detailed data related to parts or sub-systems can be shared by many different departments such as design and planning manufacturing quality assurance etc. The shared product data in manufacturing provide valuable information on quality management of assembling parts. Based on part manufacturing information the optimal matching strategies for assembling parts are called for to safeguard consistency of assembled product quality. In this paper we shall present and make more precise a heuristic invariant optimal matching rule: small matches with small and large matches with large to ensure the consistency of product or system quality. Specifically we show in this paper how the one-to-one matching problem. can be solved explicitly via majorization see Section 2 for the definition. We also show this majorization-based matching method can be used to solve general subset-to-subset optimal matching problems with multiple objective functions that are often arising from reliability modeling and quality management. In terms of discrete matching in the design of observational studies our optimal matching problem. can be thought of as an optimal matching with a single covariate for each treatment and control individual. In contrast to optimal matching in observational studies the optimal matching criteria used in our matching problems consist of a large class of metrics including the Mahalanobis distance. and Kantorovich cost function for matched pairs. Our majorization-based optimal matching can be proceeded by successive applications of a finite number of local matchings for persistent variance reductions and so the method can be applied to matching for large batches of products that may have various matching constraints. Theory of majorization is elegant and mathematically powerful [4 3] and our majorization-based method can be extended 3
to continuous optimal matching problems that are arising from e.g. shape recognition or data compression. To the best of our knowledge this is the first paper that applies optimal matching to quality assurance. The rest of the paper is organized as follows. Section 2 presents main comparison results for the majorization-based matching. Section 3 discusses optimal matching solutions for several multi-part product assemblings. The remarks in Section 4 conclude the paper. 2 Optimal Matching via Majorization Various notions of majorization describe the fundamental phenomena of spread out. For any real vector x = x... x n R n x k denotes the kth smallest among components x... x n and x [k] denotes the kth largest among components x... x n k n. Definition 2.. Let x = x... x n y = y... y n R n.. x is said to be majorized by y denoted as x y if k x i i= k y i k =... n and i= x i = i= y i. i= 2. x is said to be weakly majorized by y denoted as x w y if k x [i] i= k y [i] k =... n. i= For example 2 3 2 0 4 and 2 3 w 2 4. That is if x y then y is more spread out than x does. Obviously x w y if and only if x z and z y component-wise for some z R d. For any non-negative vector x = x... x n R n x x... x x x 2... x n x i 0... 0 2. where x = n i= x i/n denotes the average. Definition 2.2. A real-valued function φ defined on a set D R d is said to be Schur-convex on D if x y = φx φy. If a symmetric function φ defined on an open subset D is differentiable then φ is Schurconvex if and only if the partial derivative φx/ x k satisfies Schur s condition: φx x i x j φx 0 for all i j. 2.2 x i x j 4 i=
For example the function φx = n i= xp i is Schur-convex for any p 2. This together with 2. imply that for any vector x... x n n i= x i p n i= x p i for any p 2. As illustrated in this example the notion of majorization provides a powerful method for deriving various inequalities. A most comprehensive treatment on majorization and Schur convexity is the monograph [4] by Marshall Olkin and Arnold. A theory of majorization on partially ordered sets was developed in [22] and stochastic versions of majorization and Schur-convexity have also been introduced in the literature; see e.g. [4 0]. The following properties of majorization and Schur-convexity can be found in [4]. Theorem 2.3. Let x = x... x n y = y... y n R n.. x w y φx φy for all Schur-convex and non-decreasing functions φ. 2. x y if and only if there exist a finite number say m of vectors x i i =... m such that x = x x 2 x m = y where x i and x i+ differ in two coordinates only and all x i s are of the following form z... z k z k + z k+... z l z l z l+... z n k < l n R. 2.3 3. A symmetric function φ defined on an open set D is Schur-convex if and only if for any z z 2 z n and 0 φz... z k z k + z k+... z l z l z l+... z n is non-decreasing in 2.4 where k < l n. 4. For any convex function g and any increasing Schur-convex function φ ψx = φgx... gx n is also Schur-convex. In particular the function n i= gx i and max{gx... gx n } are Schur-convex for any convex function g. 5
The transform 2.3 is known as the Robin Hood transform that allocates positive weight from a larger component to a smaller component so as to make the vector less spread out. Such a transform is powerful and essentially reduces derivation of any majorization-based inequality to a two-dimensional problem. The Robin Hood transform on partial ordered sets has been also developed [22]. Theorem 2.4. If x = x... x n y = y... y n R n such that x n y then φgy x... gy n x n φgy x... gy n x n φgy x n... gy n x for any convex function g and any increasing Schur-convex function φ. Proof: Since φ is symmetric we assume without loss of generality that y i s are already arranged in the increasing order; that is y y 2 y n. In light of Theorem 2.3 4 we need to show that y x... y n x n y x... y n x n y x n... y n x. 2.5. To prove the first inequality in 2.5 we write without loss of generality the components of y x... y n x n in terms of order statistics xi s: y x... y n x n = y x k... y l x... k l. Obviously y x k min{y x y l x k } max{y x y l x k } y l x which imply that y x y l x k y x k y l x. Keeping all other components i i l the same we perform the first Robin Hood transform on components and l: x := y x... y l x k... y x k... y l x... = y x... y n x n. Note that the first component of x is the same as the first component of y x... y n x n. Starting from the second component of x repeat similar Robin Hood transforms on remaining components leading to a sequence of Robin Hood transforms: y x... y n x n = x n x 2 x y x... y n x n and the first inequality of 2.5 follows. 6
2. To prove the second inequality of 2.5 we use the reverse Robin Hood transforms to make the vectors more spread out. Write y 0 := y x... y n x n = y x k... y l x n... k l n. Obviously y x n min{y x k y l x n } max{y x k y l x n } y l x k which imply that y x k y l x n y x n y l x k. Keeping all other components i i l the same we perform the first reverse Robin Hood transform on components and l: y x... y n x n = y x k... y l x n... y x n... y l x k... =: y. Note that the first component of y is the same as the first component of y x n... y n x. Starting from the second component of y repeat similar reverse Robin Hood transforms on remaining components to get y 2 := y x n y 2 x n... repeat again and again on remaining components which lead to a sequence of reverse Robin Hood transforms: y x... y n x n y y 2 y n = y x n... y n x and the second inequality of 2.5 follows. Remark 2.5. Theorem 2.4 presents a general result for optimal matching and more importantly local Robin Hood transforms used in the proof is very powerful. example if there are some constraints on matching vectors x and y then inequalities can be established by performing Robin Hood transforms within the region defined by the constraints. In matching problems n i= y i x i is fixed leading naturally to majorization. If the sum is not fixed e.g. x i s and y i s may be drawn from larger batches then the weak majorization can be used. For 7
Corollary 2.6. Let x = x... x n y = y... y n R n such that x n y. Define the Kantorovich cost function: { } K g x y := min gy πi x i : all permutations π of {... n} n i= where g is a convex function. Then it follows from Theorem 2.4 that K g x y = gy i x i. n i= 2.6 The Kantorovich cost function 2.6 is a discrete version of Kantorovich s function used in the Monge-Kantorovich mass transportation problem [6 7]. Example 2.7. Let x = x... x n y = y... y n R n such that x n y. Define the variance of the difference vector y x... y n x n as Vary x = [ ] 2 y i x i y x where y x = n n i= It follows from Corollary 2.6 that Vary x achieves the minimum [ ] 2 y i x i y x n i= y i x i. when the ith smallest x i matches the ith smallest y i ; that is optimal matching occurs when small matches small and large matches large. For subset-to-subset matching the key issue is on specifications of optimal matching criteria and it often has more than one objective functions. For example optimal matching can be achieved by minimizing some cost function of all matched subsets as well as variances within subsets. We illustrate our marjorization-based matching method in the following one-to-subset matching problem see Figure 2.. i= Figure 2.: The values of the diameters Φ and Φ j 2 j m + for inside axle and outside sleeve parts are x i and y im m+... y im respectively. 8
Consider x = x... x n R n y = y... y nm R nm such that x n y. The goal is to match x i to a subset of m components of y so as to minimize the following objective functions: find a permutation π on {... nm} such that min min nm nm i= i= m gy πim m+j x i where g is convex and 2.7 j= m [y πim m+j ȳ i ] 2 = n j= { m i= m } [y πim m+j ȳ i ] 2 j= 2.8 where ȳ i = m m j= y πim m+j i n. Since x i is fixed for the subset {im m +... im} the second objective function can be written as nm i= m [y πim m+j ȳ i ] 2 = n j= { m i= m [ yπim m+j x i ȳ i x i ] } 2. j= That is the second objective function describes the average of variances of matching differences with subsets. Proposition 2.8. Assume without loss of generality that the components of x are arranged in the increasing order: x x 2 x n. The optimal matching 2.7 and 2.8 can be achieved via the permutation Proof: We enlarge x as follows y πim m+j = y im m+j i =... n j =... m. 2.9 x = x... x }{{} x 2... x 2... x }{{} n... x n. }{{} m m m It then follows from Corollary 2.6 that the permutation 2.9 minimizes 2.7. To show the permutation 2.9 also minimizes 2.8 consider { m i= m } [y πim m+j ȳ i ] 2 j= { m } = y 2 2 πim m+j ȳ m i i= j= = m yπim m+j 2 [ m ] 2 y m m 2 πim m+j i= Since n m i= j= y2 πim m+j is invariant under any permutation π minimizing 2.8 boils down to maximizing n [ m i= j= y ] 2. πim m+j j= i= j= 9
Since n i= z2 i is Schur-convex in z... z n we have for any permutation π m y πj... j= = m m y πnm m+j y j... j= [ m ] 2 y πim m+j i= j= j= m y nm m+j j= [ m ] 2 y im m+j which implies that 2.8 is minimized via the permutation 2.9. It is worth mentioning that the permutation 2.9 is just one of n! permutations that minimizes 2.8 but it is the only one that also minimizes 2.7. Note that the objective function 2.8 can be made more general but using variance as an optimal criterion is a common practice in quality assurance. i= j= 3 Numerical Examples and Invariant Optimal Matching Strategies As mentioned in Sections and 2 invariant optimal matching strategies explore ranking of matching elements and their solutions do not depend on specific values of these elements. Using the method we developed in Section 2 the optimal solutions can be constructed explicitly using majorization. It is worth mentioning that in manufacturing practice there can be many types of optimal matching; for example matching with two types of parts can be one-to-one one-to-subset subset-to-subset and matching with more than two types of parts can be one-to-one-to-one one-to-one-to-subset subset-to-subset-to-subset etc. All these matching problems require various optimal criteria to ensure consistency precision and compatibility of materials. We begin with two illustrative numerical examples on one-to-one and one-to-subset matching before discussing a more complex matching problem. Example 3.. A two-part product consists of an axle and its sleeve and the size data of 8 products for both parts and their rankings are given in Table 3.. The optimal matching criterion is the variance see Example 2.7: min Vary x = n i= [ y πi x i y x] 2 where y x = n n i= y πi x i. This is a one-to-one matching problem in which the total variance of dimension gaps of matched parts is the optimal criterion. The optimal matching 0
is obtained using Corollary 2.6 and is listed in Table 3.2. Note that the optimal matching is not unique in this example because some parts have identical matching sizes but all optimal solutions subscribe our invariant optimal matching strategy that small matches with small and large matches with large. Table 3.: The size data are shown in the top table and the ranked data in the increasing order are shown in the bottom table. If we match both parts randomly we only have a chance with probability 6/8! to obtain the optimal matching. We may also encounter the worst matching with probability 6/8!. The total gap variance in the optimal matching is 6.9038 0 7 and the largest total gap variance is.609 0 5. Table 3.2: Optimal matching with gaps Example 3.2. A two-part product consists of an axle and four sleeves in different positions of the axle and the size data for axles and sleeves are given in Table 3.3. The optimal
matching criteria are the total variance and the sum of variances with subsets: min min nm nm i= i= m yπim m+j x i y x 2 and j= m [y πim m+j ȳ i ] 2 j= where y x = nm n i= m j= y πim m+j x i and ȳ i = m m j= y πim m+j i n. After sorting in the increasing order data sets of axles and sleeves respectively we obtain the optimal matching using Proposition 2.8 as follows: x 2 y 2 y 7 y y 3 x 8 y 22 y 3 y 20 y 4 x y y 6 y 5 y 26 x 5 y 33 y 30 y 0 y 27 x 6 y 35 y 3 y 29 y 34 x 4 y 28 y 5 y 24 y 2 x 9 y 6 y 9 y 25 y 32 x 7 y 9 y 36 y 8 y 8 x 3 y 2 y 4 y 7 y 23 where x i s are arranged in the increasing order. This is a -to-4 matching problem with two objective optimal criteria but as long as objective functions are Schur-convex our matching method see Theorem 2.4 yields the invariant optimal matching strategy that small matches with small and large matches with large. Table 3.3: Size data for a one-to-subset matching 2
Figure 3.: Subset-to-subset optimal sequential matching Our majorization-based match method can be applied to sequential optimal matching. We illustrate this using a two-dimensional matching problem see Figure 3.. Consider the data sets X = {x y x 2 y 2... x n y n } U = {u... u nm } and V = {v... v nm2 }. The goal is to match {x... x n } to U and match {y... y n } to V; that is to find a permutation π on {... nm } and a permutation τ on {... nm 2 } so as to minimize min min min min nm nm nm 2 nm 2 m uπim m +j x i u x 2 ; 3. i= j= m [u πim m +j ū i ] 2 ; 3.2 i= j= m 2 vτim2 m 2 +j y i v y 2 ; 3.3 i= j= m 2 [v τim2 m 2 +j v i ] 2 3.4 i= j= where u x = n m nm i= and v y = nm 2 n i= j= u πim m +j x i and ū i = m m j= u πim m +j i n m2 j= v τim 2 m 2 +j y i and v i = m2 m 2 j= v τim 2 m 2 +j i n. Without loss of generality we assume that x x 2 x n and correspondingly y y 2... y n = y ρ y ρ2... y ρn where ρ is the permutation on {... n} that maps i to the index of the ρith smallest among y... y n. permutation u πim m +j = u im m +j i n j m Using Proposition 2.8 the 3
minimizes 3. and 3.2 and the permutation v τim2 m 2 +j = v ρim2 m 2 +j i n j m 2 minimizes 3.3 and 3.4. Example 3.3. A multi-part product consists of a twined axle and two different sleeves and the first axle needs to match the two sleeves of the first type and the second axle needs to match the three sleeves of the second type. The size data for axles and sleeves are given in Table 3.4. The optimal matching strategy is given below x 3 ; y 3 x 2 ; y 2 x 4 ; y 4 x ; y. u 2 u ; v 0 v 7 v 2 u 5 u 3 ; v 9 v v 5 u 6 u 7 ; v v 4 v 3 u 4 u 8 ; v 6 v 8 v 2 Note that the multi-dimensional optimal matching is a special case of optimal matching on partially ordered sets. This problem can be viewed as a sequencial matching with a -to-2 matching followed by a -to-3 matching with multiple objective optimal criteria. Again as long as objective functions are Schur-convex our matching method see Theorem 2.4 yields the invariant optimal matching strategy that small matches with small and large matches with large. Table 3.4: Subset-to-subset optimal sequential matching 4
4 Concluding Remarks The matching problems presented in this paper were motivated from a consulting problem in quality management and the goal is to match non-overlapping subsets of parts of one type to non-overlapping subsets of parts of another type so as to minimize the total matching variance and the sum of variances within subsets. We develop a majorization-based method to solve this problem and find the optimal solutions explicitly by constructing a sequence of pair-wise local matching operations for persistent variance reductions. In contrast to optimal matching problems in observational studies [2 20 8 9 2] our method can be applied to a wide class of optimal criteria metrics including the Mahalanobis distance and Kantorovich cost functionals. Our method focuses on the structural properties of matching such as ranking of matched elements with the aim of developing the majorization-based method for continuous matching problems arising from quality management. The majorization-based matching method developed in this paper sheds new light on the optimal matching heuristic that small matches with small and large matches with large for a wide class of objective functions. Our future research includes majorization-based optimal matching for partially ordered data sets and its application to quality assurance. Acknowledgement This work is in honor of Professor Alan G. Hawkes on his 75th birthday. This work was supported partly by the NSF of China under grant 73703 and NSF grant DMS 007556. The authors would like to thank two anonymous referees and editor for their valuable suggestions on the improvements of the paper. References [] Brookhart M. A. Schneeweiss S. Rothman K. J. Glynn R. J. Avorn J. and Sturmer T. 2006: Variable selection for propensity score models. American Journal of Epidemiology 63 49-56. [2] Cochran W. G. and Rubin D. B. 973: Controlling bias in observational studies: A review. Sankhya Ser. A 35 47-446. [3] Egozcue M. and Wong W. K. 200: Gains from diversification on convex combinations: A majorization and stochastic dominance approach. European Journal of Operational Research 200: 893-900. 5
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