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; majoriza-

tion; Schur-convexity, Kantorovich cost function.

1 Introduction

The common theme of discrete matching problems is to match elements of one finite set

S1 to elements of another finite set S2 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-to-

subset matching.

∗This work is in honor of Professor Alan G. Hawkes on his 75th birthday.†Lirongcui@bit.edu.cn, kongdejing12345@163.com, School of Management and Economics, Bei-

jing Institute of Technology, Beijing, 100081,China. Supported by the NSF of China under grant 71371031.‡lih@math.wsu.edu, Department of Mathematics, Washington State University, Pullman, WA 99164,

U.S.A. Supported by NSF grant DMS 1007556.

1

The early work in matching appeared in the 1940s (see, e.g., [4]), and the research has

accelerated in observational studies since the 1970s 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 [18, 19, 21], economics [6, 23],

sociology [8, 15], epidemiology and medicine [13, 1], 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 [16, 17], where S1 and S2 are usually multi-dimensional Euclidean

spaces (or general Hilbert spaces). Given two probability distributions f1 and f2 on S1 and

S2 respectively, the goal is to find an optimal matching (or transport) map from S1 to S2so that certain cost functional of f1 and f2 is minimized. Such an optimal transport exists

under mild model assumptions, and finding it explicitly can be done in the situations such

as S1 = S2 = R.

In this paper we develop a discrete matching method based on the theory of majorization

[14] and apply it to quality assurance. The simplest setup of our problem is described as

follows. Let S1 = {x1, . . . , xn} and S2 = {y1, . . . , yn} denote two finite sets with the same

size. If xi < yj for all 1 ≤ i, j ≤ n, then find a permutation π(·) of {1, . . . , n} so as to

minimize the total variance

dπ(S1,S2) :=1

n

n∑i=1

[(yπ(i) − xi)−

1

n

n∑k=1

(yπ(k) − xk)]2, (1.1)

subject to∑n

i=1(yi − xi) = 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 1.1. Due to manufacturing variability,

the measurements, x1, . . . , xn, of inside parts are represented by a random variable X, and

the measurements, y1, . . . , yn, of outside parts are represented by a random variable Y . We

assume that P (X ≤ Y ) = 1 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. 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 (1.1), 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., [11, 12]) and

2

Figure 1.1: The values of the diameters Φ1 and Φ2 for inside (axle) and outside (sleeve) parts

are x1, . . . , xn and y1, . . . , yn, 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 (1.1) 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 opti-

mal matching problems with multiple objective functions, that are often arising from reli-

ability modeling and quality management. In terms of discrete matching in the design of

observational studies, our optimal matching problem (1.1) 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 match-

ing problems consist of a large class of metrics, including the Mahalanobis distance (1.1)

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 persis-

tent 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 [14, 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 = (x1, . . . , xn) ∈ Rn, x(k) denotes the kth smallest among components

x1, . . . , xn, and x[k] denotes the kth largest among components x1, . . . , xn, 1 ≤ k ≤ n.

Definition 2.1. Let x = (x1, . . . , xn),y = (y1, . . . , yn) ∈ Rn.

1. x is said to be majorized by y, denoted as x ≺ y, if

k∑i=1

x(i) ≥k∑i=1

y(i), k = 1, . . . , n− 1, andn∑i=1

x(i) =n∑i=1

y(i).

2. x is said to be weakly majorized by y, denoted as x ≺w y, if

k∑i=1

x[i] ≤k∑i=1

y[i], k = 1, . . . , n.

For example, (1, 2, 3) ≺ (2, 0, 4) and (1, 2, 3) ≺w (2, 1, 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 ∈ Rd. For any non-negative vector x = (x1, . . . , xn) ∈ Rn,

(x, x, . . . , x) ≺ (x1, x2, . . . , xn) ≺( n∑i=1

xi, 0, . . . , 0), (2.1)

where x =∑n

i=1 xi/n denotes the average.

Definition 2.2. A real-valued function φ defined on a set D ⊆ Rd is said to be Schur-convex

on D if x ≺ y =⇒ φ(x) ≤ φ(y).

If a symmetric function φ defined on an open subset D is differentiable, then φ is Schur-

convex if and only if the partial derivative ∂φ(x)/∂xk satisfies Schur’s condition:

(xi − xj)(∂φ(x)

∂xi− ∂φ(x)

∂xj

)≥ 0, for all i 6= j. (2.2)

4

For example, the function φ(x) =∑n

i=1 xpi is Schur-convex for any p ≥ 2. This, together

with (2.1), imply that for any vector (x1, . . . , xn),( 1

n

n∑i=1

xi

)p≤ 1

n

n∑i=1

xpi , 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 [14] 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., [14, 10]. The following

properties of majorization and Schur-convexity can be found in [14].

Theorem 2.3. Let x = (x1, . . . , xn),y = (y1, . . . , yn) ∈ Rn.

1. x ≺w y ⇐⇒ φ(x) ≤ φ(y) for all Schur-convex and non-decreasing functions φ.

2. x ≺ y if and only if there exist a finite number (say m ≥ 1) of vectors x(i), i = 1, . . . ,m,

such that

x = x(1) ≺ x(2) ≺ · · · ≺ x(m) = y

where x(i) and x(i+1) differ in two coordinates only, and all x(i)s are of the following

form

(z1, . . . , zk−1, zk+∆, zk+1, . . . , zl−1, zl−∆, zl+1, . . . , zn), 1 ≤ 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 z1 ≥ z2 ≥ · · · ≥ zn and ∆ ≥ 0,

φ(z1, . . . , zk−1, zk+∆, zk+1, . . . , zl−1, zl−∆, zl+1, . . . , zn) is non-decreasing in ∆, (2.4)

where 1 ≤ k < l ≤ n.

4. For any convex function g(·) and any increasing Schur-convex function φ(·),

ψ(x) = φ(g(x1), . . . , g(xn))

is also Schur-convex. In particular, the function∑n

i=1 g(xi) and max{g(x1), . . . , g(xn)}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 = (x1, . . . , xn),y = (y1, . . . , yn) ∈ Rn, such that x(n) ≤ y(1), then

φ(g(y(1) − x(1)), . . . , g(y(n) − x(n))) ≤ φ(g(y1 − x1), . . . , g(yn − xn))

≤ φ(g(y(1) − x(n)), . . . , g(y(n) − x(1)))

for any convex function g(·) and any increasing Schur-convex function φ(·).

Proof: Since φ(·) is symmetric, we assume without loss of generality that yis are already

arranged in the increasing order; that is, y1 ≤ y2 ≤ · · · ≤ yn. In light of Theorem 2.3 (4), we

need to show that(y1 − x(1), . . . , yn − x(n)

)≺(y1 − x1, . . . , yn − xn

)≺(y1 − x(n), . . . , yn − x(1)

). (2.5)

1. To prove the first inequality in (2.5), we write, without loss of generality, the compo-

nents of(y1 − x1, . . . , yn − xn

)in terms of order statistics x(i)s:(

y1 − x1, . . . , yn − xn)

=(y1 − x(k), . . . , yl − x(1), . . .

), k, l ≥ 1.

Obviously,

y1 − x(k) ≤ min{y1 − x(1), yl − x(k)} ≤ max{y1 − x(1), yl − x(k)} ≤ yl − x(1)

which imply that (y1 − x(1), yl − x(k)

)≺(y1 − x(k), yl − x(1)

).

Keeping all other components i 6= 1, i 6= l the same, we perform the first Robin Hood

transform on components 1 and l:

x(1) :=(y1−x(1), . . . , yl−x(k), . . .

)≺(y1−x(k), . . . , yl−x(1), . . .

)=(y1−x1, . . . , yn−xn

).

Note that the first component of x(1) is the same as the first component of(y1 −

x(1), . . . , yn − x(n)). Starting from the second component of x(1), repeat similar Robin

Hood transforms on remaining components, leading to a sequence of Robin Hood

transforms:(y1 − x(1), . . . , yn − x(n)

)= x(n−1) ≺ · · · ≺ x(2) ≺ x(1) ≺

(y1 − x1, . . . , yn − xn

)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) :=(y1 − x1, . . . , yn − xn

)=(y1 − x(k), . . . , yl − x(n), . . .

), k, l ≤ n.

Obviously,

y1 − x(n) ≤ min{y1 − x(k), yl − x(n)} ≤ max{y1 − x(k), yl − x(n)} ≤ yl − x(k)

which imply that (y1 − x(k), yl − x(n)

)≺(y1 − x(n), yl − x(k)

).

Keeping all other components i 6= 1, i 6= l the same, we perform the first reverse Robin

Hood transform on components 1 and l:(y1−x1, . . . , yn−xn

)=(y1−x(k), . . . , yl−x(n), . . .

)≺(y1−x(n), . . . , yl−x(k), . . .

)=: y(1).

Note that the first component of y(1) is the same as the first component of(y1 −

x(n), . . . , yn− x(1)). Starting from the second component of y(1), repeat similar reverse

Robin Hood transforms on remaining components, to get

y(2) :=(y1 − x(n), y2 − x(n−1), . . .

),

repeat again and again on remaining components, which lead to a sequence of reverse

Robin Hood transforms:(y1 − x1, . . . , yn − xn

)≺ y(1) ≺ y(2) ≺ · · · ≺ y(n−1) =

(y1 − x(n), . . . , yn − x(1)

)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. For

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=1(yi − xi) is fixed, leading naturally

to majorization. If the sum is not fixed (e.g., xis and yis may be drawn from larger

batches), then the weak majorization can be used.

7

Corollary 2.6. Let x = (x1, . . . , xn),y = (y1, . . . , yn) ∈ Rn, such that x(n) ≤ y(1). Define

the Kantorovich cost function:

Kg(x,y) := min{ 1

n

n∑i=1

g(yπ(i) − xi) : all permutations π(·) of {1, . . . , n}}

(2.6)

where g(·) is a convex function. Then it follows from Theorem 2.4 that

Kg(x,y) =1

n

n∑i=1

g(y(i) − x(i)).

The Kantorovich cost function (2.6) is a discrete version of Kantorovich’s function used

in the Monge-Kantorovich mass transportation problem [16, 17].

Example 2.7. Let x = (x1, . . . , xn),y = (y1, . . . , yn) ∈ Rn, such that x(n) ≤ y(1). Define the

variance of the difference vector (y1 − x1, . . . , yn − xn) as

Var(y − x) =1

n

n∑i=1

[(yi − xi)− (y − x)

]2, where y − x =

1

n

n∑i=1

(yi − xi).

It follows from Corollary 2.6 that Var(y − x) achieves the minimum

1

n

n∑i=1

[(y(i) − x(i))− (y − x)

]2when 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 2.1).

Figure 2.1: The values of the diameters Φ1 and Φj, 2 ≤ j ≤ m + 1, for inside (axle) and

outside (sleeve) parts are xi and yim−m+1, . . . , yim, respectively.

8

Consider x = (x1, . . . , xn) ∈ Rn,y = (y1, . . . , ynm) ∈ Rnm, such that x(n) ≤ y(1). The goal

is to match xi to a subset of m components of y so as to minimize the following objective

functions: find a permutation π(·) on {1, . . . , nm} such that

min1

nm

n∑i=1

m∑j=1

g(yπ(im−m+j) − xi), where g(·) is convex, and (2.7)

min1

nm

n∑i=1

m∑j=1

[yπ(im−m+j) − yi]2 =

1

n

n∑i=1

{ 1

m

m∑j=1

[yπ(im−m+j) − yi]2}

(2.8)

where yi = 1m

∑mj=1 yπ(im−m+j), 1 ≤ i ≤ n. Since xi is fixed for the subset {im − m +

1, . . . , im}, the second objective function can be written as

1

nm

n∑i=1

m∑j=1

[yπ(im−m+j) − yi]2 =

1

n

n∑i=1

{ 1

m

m∑j=1

[(yπ(im−m+j) − xi)− (yi − xi)

]2}.

That is, the second objective function describes the average of variances of matching differ-

ences with subsets.

Proposition 2.8. Assume without loss of generality that the components of x are arranged

in the increasing order: x1 ≤ x2 ≤ · · · ≤ xn. The optimal matching (2.7) and (2.8) can be

achieved via the permutation

yπ(im−m+j) = y(im−m+j), i = 1, . . . , n, j = 1, . . . ,m. (2.9)

Proof: We enlarge x as follows,

x∗ = (x1, . . . , x1︸ ︷︷ ︸m

, x2, . . . , x2︸ ︷︷ ︸m

, . . . , xn, . . . , xn︸ ︷︷ ︸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

n∑i=1

{ 1

m

m∑j=1

[yπ(im−m+j) − yi]2}

=n∑i=1

{ 1

m

m∑j=1

y2π(im−m+j) − yi2}

=1

m

n∑i=1

m∑j=1

y2π(im−m+j) −1

m2

n∑i=1

[ m∑j=1

yπ(im−m+j)

]2Since

∑ni=1

∑mj=1 y

2π(im−m+j) is invariant under any permutation π(·), minimizing (2.8) boils

down to maximizing∑n

i=1

[∑mj=1 yπ(im−m+j)

]2.

9

Since∑n

i=1 z2i is Schur-convex in (z1, . . . , zn), we have, for any permutation π(·),( m∑j=1

yπ(j), . . . ,

m∑j=1

yπ(nm−m+j)

)≺( m∑j=1

y(j), . . . ,

m∑j=1

y(nm−m+j)

)

=⇒n∑i=1

[ m∑j=1

yπ(im−m+j)

]2≤

n∑i=1

[ m∑j=1

y(im−m+j)

]2which 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.

3 Numerical Examples and Invariant Optimal Match-

ing Strategies

As mentioned in Sections 1 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 3.1. 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 3.1. The optimal matching

criterion is the variance (see Example 2.7):

min Var(y − x) =1

n

n∑i=1

[(yπ(i) − xi)− (y − x)

]2,

where y − x = 1n

∑ni=1(yπ(i)−xi). 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

10

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 3.1: 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 16/8! to obtain

the optimal matching. We may also encounter the worst matching with probability 16/8!.

The total gap variance in the optimal matching is 6.9038 × 10−7 and the largest total gap

variance is 1.1609× 10−5 . �

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

11

matching criteria are the total variance and the sum of variances with subsets:

min1

nm

n∑i=1

m∑j=1

(yπ(im−m+j) − xi − y − x

)2, and

min1

nm

n∑i=1

m∑j=1

[yπ(im−m+j) − yi]2

where y − x = 1nm

∑ni=1

∑mj=1(yπ(im−m+j) − xi) and yi = 1

m

∑mj=1 yπ(im−m+j), 1 ≤ 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:(x2

y2, y17, y11, y13

),

(x1

y1, y16, y5, y26

),

(x6

y35, y3, y29, y34

),

(x9

y6, y19, y25, y32

),

(x3

y21, y14, y7, y23

),

(x8

y22, y31, y20, y4

),

(x5

y33, y30, y10, y27

),

(x4

y28, y15, y24, y12

),

(x7

y9, y36, y8, y18

),

where xis are arranged in the increasing order. This is a 1-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

12

Figure 3.1: 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 3.1). Consider the

data sets X = {(x1, y1), (x2, y2), . . . , (xn, yn)}, U = {u1, . . . , unm1} and V = {v1, . . . , vnm2}.The goal is to match {x1, . . . , xn} to U and match {y1, . . . , yn} to V ; that is, to find a

permutation π(·) on {1, . . . , nm1} and a permutation τ(·) on {1, . . . , nm2} so as to minimize

min1

nm1

n∑i=1

m1∑j=1

(uπ(im1−m1+j) − xi − u − x

)2; (3.1)

min1

nm1

n∑i=1

m1∑j=1

[uπ(im1−m1+j) − ui]2; (3.2)

min1

nm2

n∑i=1

m2∑j=1

(vτ(im2−m2+j) − yi − v − y

)2; (3.3)

min1

nm2

n∑i=1

m2∑j=1

[vτ(im2−m2+j) − vi]2, (3.4)

where u − x = 1nm1

∑ni=1

∑m1

j=1(uπ(im1−m1+j)−xi) and ui = 1m1

∑m1

j=1 uπ(im1−m1+j), 1 ≤ i ≤ n,

and v − y = 1nm2

∑ni=1

∑m2

j=1(vτ(im2−m2+j) − yi) and vi = 1m2

∑m2

j=1 vτ(im2−m2+j), 1 ≤ i ≤ n.

Without loss of generality, we assume that x1 ≤ x2 ≤ · · · ≤ xn, and correspondingly

(y1, y2, . . . , yn) = (y(ρ(1)), y(ρ(2)), . . . , y(ρ(n))), where ρ(·) is the permutation on {1, . . . , n} that

maps i to the index of the ρ(i)th smallest among y1, . . . , yn. Using Proposition 2.8, the

permutation

uπ(im1−m1+j) = u(im1−m1+j), 1 ≤ i ≤ n, 1 ≤ j ≤ m1,

13

minimizes (3.1) and (3.2), and the permutation

vτ(im2−m2+j) = v(ρ(i)m2−m2+j), 1 ≤ i ≤ n, 1 ≤ j ≤ m2,

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,(x3; y3

u2, u1; v10, v7, v2

),

(x2; y2

u5, u3; v9, v11, v5

),

(x4; y4

u6, u7; v1, v4, v3

),

(x1; y1

u4, u8; v6, v8, v12

).

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 1-to-2

matching followed by a 1-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

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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, 18, 19, 21],

our method can be applied to a wide class of optimal criteria metrics, including the Maha-

lanobis 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 manage-

ment. 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 sup-

ported partly by the NSF of China under grant 71371031 and NSF grant DMS 1007556. The

authors would like to thank two anonymous referees and editor for their valuable suggestions

on the improvements of the paper.

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