Selection Method

Since we can not compare two suppliers directly, we have to sample some products made by two suppliers, and use some statistical analysis to compare which one has better process capability. Then we decide whether switch the present supplier or not. Let π_i be the pollution assumed to be normally distributed with mean μ_i and variance σ , _i²

i = 1,2

, and are the independent random samples from

1

,

2

,...,

i i ini

x x x

πi, . In most applications, if a new supplier#2 (S2) wants to compete for the orders by claiming that its capability is better than the existing supplier#1 (S1), then the new S2 must furnish convincing information justifying the claim with a prescribed level of confidence. Thus, the supplier selection decision would be based on the hypothesis testing comparing the two

values

1,2 i =

Y

q 0

:

_q1 _{q 2}

H Y ≥ Y

1

:

_q1 _{q 2}

H Y < Y

.

If the test rejects the null hypothesis

H Y

₀

:

_q1

≥ Y

_q2 , then one has sufficient information to conclude that the new S2 is superior to the original S1, and the decision of the replacement would be suggested. Equivalently, this test hypothesis problem can be rewritten as:

versus

0

:

_q2 _q1

0

H Y − Y ≤ H Y

₁

:

_q2

− Y

_q1

> 0

(difference testing) versus (ratio testing).

0

:

_q2

/

_q1

H Y Y ≤ 1 1

q1

2

q1

i 1

:

_q2

/

_q1

H Y Y >

Thus, if the lower confidence bound for the difference between two process capability indices is positive and then S2 has a better process capability than S1.

Otherwise, we do not have sufficient information to conclude that the S2 has a better process capability than S1. In this case, we would believe that is true, i.e. . Similar, if the lower confidence bound between two process capability indices is great than 1, then S2 has a better process capability than S1. Otherwise, if the lower confidence bound of the ratio statistic is less than 1, and then we would conclude that S1 has a better process capability than S2.

q2

Y − Y

0

:

_q2 _q1

0

H Y − Y ≤

q1 q

Y ≥ Y

2

/ Y

q

Y

Based on above reasons mentioned, we should know the information about the point estimator of . Ng and Tsui (1992) proposed a sample estimator based on a finite population of products. Suppose ^q

Y

1, 2,...,

i i in

x x x

denote the sample measurements of product characteristics. It follows that are estimated by collected sample data and can be defined as follows: ^q

Y

2 2

It is important to find a lower bound on the rather than just the sample point estimate. The index can be rewritten as follows (see Pearn ^q

Y

lower 100γ% confidence bound on

by calculating the lower 100

Y L

− e

Y

_q

Y

q γ₁% confidence

bound on

Y

and the upper 100γ₂% confidence bound on

L

_e (γ γ γ= × ). Pearn _{1 2} (2004a) obtain the 100γ% lower confidence bound for

and

simultaneously can be expressed as:

.

However, their investigations are all developed for evaluating whether a single supplier’s process conforms to a customer’s requirements. Due to the complexities of the sampling distributions of ˆ₂

Y

q ˆ₁

Y

q

− or , constructions of exact confidence intervals for and are difficult. ²

ˆq

3.2. Bootstrap Methodology

The bootstrap is the idea that in the absence of any other knowledge about a population, the distribution of values found in a random sample of size from the population is the best guide to the distribution in the population, introduced by Efron (1979, 1982). Franklin and Wasserman (1991) proposed an initial study of three bootstrap methods for obtaining confidence intervals for

n

C

pk when the process was normally distributed. Franklin and Wasserman (1992) also proposed an initial study of these bootstrap lower limits for

C

_p,

C

_pk , and

C

_pm. Chen and Tong (2003) obtained the

C

_pk1

− C

_pk2 confidence interval using bootstrap methods under a normal distribution of observation. We can find most of them concluded that the

performance such bootstrap limits for PCIs is quite satisfactory in the majority of these cases. It can be applied whenever the construction of confidence intervals for parameters using the standard statistical techniques becomes intractable. The simulation results performed in the bootstrap confidence limits were as well as the lower confidence limits applied by the parametric method in the normal process environment. Without using distribution frequency tables to compute approximate probability values, the bootstrap method generates a unique sampling distribution based on the actual sample rather than the analytic methods.

In the following four bootstrap confidence limits are employed to determine the lower confidence bounds of difference and ratio statistics and the results are used to select the better supplier of the two selections. For

n

₁=

n

₂ = , let two bootstrap

n

samples of size n drawn with replacement from two original samples be denoted by

{ x x

11^*

,

12^*

,..., x

1^*_n

} { x x

21^*

,

22^*

,..., x

2^*_n

}

. The bootstrap sample statistics , and are computed. There are possibly a total of such samples, the statistic is calculated for each of these, and the resulting empirical distribution is referred to as the bootstrap distribution of the statistic. Due to the overwhelming computation time, it is not of practical interest to choose such samples. Eforn and Tibshirani (1986) indicated that a roughly minimum of 1,000 bootstrap resamples is usually sufficient to compute reasonably accurate confidence interval estimates for population parameters. For accuracy purpose, we consider

*

ˆ1

Y

q

*

ˆ2

Y

q

n

ⁿ

n

n

3,000

B =

bootstrap resamples (rather than 1,000). Thus, we take

B = 3,000

bootstrap estimates

* *

ˆ ˆ 2

Y

q

θ = −

Y

ˆ ^*_q1

of

θ

= ( Y

_q2

− Y

_q1

)

* *

ˆ ˆ 2

Y

q

θ = /

Y

ˆ^*_q1

of

θ

= ( Y

_q2

/ Y

_q1

)

,

respectively and then ordered from smallest to the largest where =1,2,…..,

l B

( )^{* *}2

ˆ

_l

( Y ˆ

_q

θ

=

−

Y

ˆ )^*_{q1 ( )l}

or

θ

^ˆ

_{( )}^*_l

= ( Y ^ˆ

^*_q2 /

Y

ˆ^*_{q1 ( )})_l .

Four kinds of bootstrap confidence intervals can be derived, including the standard bootstrap confidence interval (SB), the percentile bootstrap confidence interval (PB), the biased corrected percentile bootstrap confidence interval (BCPB), and the bootstrap-t (BT) method introduced by Efron (1981) and Efron and Tibshiraniwill (1986) are conducted in this paper. The generic notations ˆ θ and θˆ^* will be used to denote the estimator of θ and the associated ordered bootstrap estimate.

Construction of a two-sided

100(1 2 )% −

α confidence limit will be described. We note that a lower

100(1 −

α

)%

confidence limit can be obtained by using only the lower limit. The formulation details for the four types of confidence intervals are displayed as follows.

[A] Standard Bootstrap (SB) Method

From the bootstrap estimates

B ˆ

_{( )}^*

θ , l

l = 1,2,..., , B

the sample average and the sample standard deviation can be obtained as:

*

1

ˆ B

θ

=

_{( )}^*

1 B

ˆ

l l

θ

∑

= ^{, * ( )* 2 1/ 2} 1

1 ˆ ˆ

1

B l l

S

^θ

B

θ θ

=

⎡ ⎡ *⎤ ⎤

=⎢⎣ −

∑

⎢⎣ − ⎥⎦ ⎥⎦ ^.

The quantity

S

_θ^* is an estimator of the standard deviation of ˆθ is approximately normal. Thus, the

100(1 2 )% −

α SB confidence interval for θ can be constructed as:

[θ

^ˆ

^*

− z S

_{α θ}^* , θ

^ˆ

^*

+ z S

_{α θ}^*],

where ˆθ is the estimated θ for the original sample, and z_α is the upper α quantile of the standard of the standard normal distribution.

[B] Percentile Bootstrap (PB) Method

From the ordered collection of θ ,

^ˆ

_{( )}^*_l

l = 1,2,..., , B

the α percentage and 1− percentage points are used to obtained the α

100(1 2 )% −

α PB confidence interval for θ

[θ

^ˆ

_{( )}^*_αB , ˆ₍^*_{(1 ) )}

αB

θ ₋ ].

[C] Biased-Corrected Percentile Bootstrap (BCPB) Method

While the percentile confidence interval is intuitively appealing it is possible that due to sampling errors, the bootstrap distribution may be biased. In other words, it is possible that bootstrap distribution may be shifted higher or lower than would be expected. A three steps procedure is suggested to correct for the possible bias by Efron (1982). First, we use procedure the ordered distribution of θˆ^* and calculate the probability 0⁼ θ

(ˆ*

p P ≤θˆ ) . Second, we compute the inverse of the cumulative ₀ distribution function of a standard normal based upon

p

₀ as

z

₀ =φ( )

p

₀ ,

(2 0 )

p

L =φ

z

−

z

_α

p

_U =φ(2

z

₀+

z

_α). Finally, by executing these steps we obtain the

100(1 2 )% −

α BCPB confidence interval

[ ˆ₍^* ₎

p BL

θ , ˆ₍^*

U )

θP B ].

[D] Bootstrap-t (BT) Method

By using bootstrapping to approximate the distribution of a statistic of the form ˆ ˆ

(θ θ− )/S_θ, the bootstrap approximation in this case is obtained by taking bootstrap samples from the original data values, calculating the corresponding estimates θˆ^* and their estimated standard error, and hence finding the bootstrapped T-values

T =

(θˆ^*- ˆθ )/ . The hope is then that the generated distribution will mimic the distribution of T. The

10

S

0(1 2 )% −

α BT confidence interval for θ may constitute as:

[θˆ^*−

t S

_α^* _θ^*_ˆ , θˆ^*−

t S

_α^* _θ^*_ˆ],

where

t

_α^* and

t

₁^*_−α are the upper α and 1− quantiles of the bootstrap α t-distribution respectively, i.e. by finding the values that satisfy the two equation

P((θˆ^*- ˆθ ) /

S

_θ^*>

t

_α^*)= and P((α θˆ^*- ˆθ ) /

S

_θ^*>

t

₁^*_−α)= − 1 α for the generated bootstrap estimates.

4. Performance Comparisons of Four Bootstrap Methods

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