Literature Review

Process capability indices are convenient and powerful tools for measuring process performance proposed by several researchers such as Boyles (1991), Pearn

(1992), Kushler and Hurley (1992), Kotz and Johnson (1993), Vännman and Kotz (1995), Vännman (1997), Kotz and Lovelace (1998), Pearn

et al

et al (1998), Pearn

and Shu (2003) and references therein. By taking into consideration process location, process variation, and manufacturing specifications, those indices quantify process performance and reflect process consistency, process accuracy, process yield, and process loss. The process indices

C

_p,

C

_pk,

C

_pm and

C

_pmk take natural process tolerance, manufacturing specifications, process centering, and the target value of the process into consideration and take advantage of unitless measures. Those indices convey critical information regarding whether a process is capable for reproducing items satisfying the customer’s requirement. In practice, a minimal capability requirement would be preset by the customers/engineers. If the prescribed minimum capability fails to be met, one would conclude that the process is incapable. The first process capacity index appearing in the literature was the precision index

C

_p and defined as (see Juran (1974) and Kane (1986)):

6 USL LSL

C

= −

where is the upper specification limit, is the lower specification limit, and

USL LSL

σ is the process standard deviation. The index

C

_p measures process precision (process quality consistency) and does not consider whether the process is centered.

In order to reflect the deviations of the process mean from the target value, the index

C

pk was proposed. It considers process variation and location of process mean which is defined as:

min{ , } min ,

3 3 3

pk pu pl

d m

USL LSL

C C C

μ μ μ

σ σ σ

− − − −

⎧ ⎫

= = ⎨ ⎬=

⎩ ⎭ ,

where μ is the process mean,

d = ( USL LSL)/2 −

, and

m = ( USL LSL /2 + )

. However,

C

_pk alone still cannot provide adequate measure of process centering.

That is, a large

C

_pk does not really say anything about the location of the mean in the tolerance interval. To help account this, Hsiang and Taguchi (1985) introduced the index

C

_pm, which was also proposed independently by Chan (1988). The index is related to the idea of squared error loss, (where

T

is the target value), and this loss-based process capability index

et al

( ) ( )2

loss X

X T

−

C

pm, sometimes called Taguchi index. The index emphasizes on measuring the ability of the process to cluster around the target, which therefore reflects the degrees of the process targeting (centering). The index

C

_pm incorporates with the variation of production items with respect to the target value and the specification limits preset in the factory. The index

C

pm is defined as:

2 2

6 (

USL LSL

C

σ μ

T )

= −

+ −

Pearn

et al

. (1992) proposed an index called

C

_pmk, which combines the merits of the before three basic indices

C

_p,

C

_pk and

C

_pm. The index

C

_pmk has been defined as:

{ }

( )

min ,

pmk 3

USL LSL

C

T

μ μ

σ μ

− −

= + − .

The index

C

_pmk is more sensitive to the departure of the process mean μ from the target value

T

than the other three indices

C

_p,

C

_pk and

C

_pm.

Those indices are effective tools for process capability analysis and quality assurance. We could divide these indices into two categories according to the target value

T

. The first includes

C

_p and

C

_pk, which are independent of . Process loss incurred by the departure from the target is neglected. The second category includes

T

C

pm and

C

_pmk, which rectify the disadvantage by taking the target value into account.

The limitation on using those indices requires the assumption that the quality characteristic measurement must be obtaining from normal distributions. Somerville and Montgomery (1996) presented an extensive study to illustrate how poorly the normally based capability indices perform as a predictor of process fallout when the process is non-normally distributed. If the normally based capability indices are still used to deal with non-normal process data, the value of the capacity indices are incorrect and might misrepresent the actually product quality. Although new capacity indices have been developed for non-normal distributions, those indices are harder to compute and interpret, and are sensitive to data peculiarities such as bimodality or truncation. Moreover, those indices do not explicitly account for the manufacturing cost or customer’s loss. Process quality yield index is proposed to remedy these

disadvantages. ^q

Y

2.2. Quality Yield Y

and Relation Indices 2.2.1. Process Yield

Traditionally, process yield ^Y

is defined as the percentage of the processed product units passing the inspections. Units are inspected according to specification limits placed on various key product characteristics and sorted into two categories:

accepted (conforming items) and rejected (defectives). Process yield has long been the most common and standard criteria used in the manufacturing industries for judging process performance. For product units rejected during the inspection, additional costs would be incurred to the factory for scrapping or reworking. All passed product units are treated equally and accepted by the producer. No additional cost to the

factory is required. The definition of

Y

index is

( )

USL

LSL

Y = ∫ dF ^x

where and are the upper and the lower specification limits, respectively and

USL LSL

( )

F x

is the cumulative distribution function of measured characteristic

x . The

disadvantage of yield measure is that it does not distinguish the products that fall inside of the specification limits. Customers do notice unit-to-unit difference in these characteristics, especially if the variance is large and/or the mean is offset from the target.

2.2.2. Process Loss

To rectify this disadvantage, the quadratic loss function is considered to distinguish the products by increasing the penalty as departure from the target increases. However, the quadratic loss function itself does not provide comparison with the specification limits and depends on the unit of the characteristic. To address these issues, Johnson (1992) developed the relative expected loss for a symmetric

case as: ^e and is the half specification width. This measure has a direct relationship with

d = ( USL LSL − )/2

C

pm because

L

_e =(3

C

_pm)⁻². The disadvantage of the index is the difficulty in setting a standard for the index since it increases from zero to infinity. ^e

L

2.2.3. Quality Yield Y

The main idea of the quality yield index is that it penalizes yield for the variation of the product characteristics from its target. Ng and Tsui (1992) suggested it by connecting the proportion-conforming-based index

Y

and loss-function-based index . Unlike the yield index , the quality yield focuses on the ability of the process to cluster around the target by taking the relative loss within the specifications into consideration. It is different from the expected relative worth index defined by Johnson by truncating the deviation outside the specifications. With this truncation, will be between zero and one and thus has better interpretation. Then the index defined as:

While yield is the proportion of conforming products,

Y

_q can be interpreted as

the proportion of “perfect” products. By relating to the yield measure, which is familiar to engineers, it is much easier for the engineers to understand and accept this capacity measure. The advantage of the index over the index is that the value of the former goes from zero to one. Similarly to the yield index, the

Y

measure, the ideal value of is one, which provides the user a clear concept about the standard. Similar to yield

Y

, the index does not rely on the normality assumption. And it can be interpreted as the average degree of products reaching ‘on target’.

Y

L

Y

In recent years, several methods have been proposed about . Pearn

(2004a) proposed a reliable approach for measuring by converting the estimate into a lower confidence bound for process with a very low fraction of defectives. And Pearn (2005) further applied a nonparametric but computer intensive method called bootstrap to obtain a lower confidence bound on for capability testing purposes. In this paper, we would use the quality index to judge which supplier has a better process performance.

Y

et al

Y

et al

Y

2.3. Investigation in Supplier Selection

The decision-maker usually faces the problem of selecting the best manufacturing supplier from several available manufacturing suppliers. There are many factors, such as quality, cost, and service and so on, that need to be considered in selecting the best supplier. Production quality is one of the key factors in supplier evaluation. For this reason, several selection rules have been proposed for selecting the means or variance in analysis of variance by Gibbons .

et al (1977), Gupta and

Panchapakesan (1979), Gupta and Huang (1981) for more detail. Process capability indices are useful management tool, particularly in the manufacturing industry. Tseng and Wu (1991) considered the problem of selecting the best manufacturing process from available manufacturing processes based on the precision index

k C

_p and a modified likelihood ratio selection rule is proposed. Chou (1994) developed three one-sided tests (

C

_p,

C

_pu,

C

_pl ) for comparing two process capability indices in order to choose between competing process when the sample size are equal. Huang and Lee (1995) considered the supplier selection problem based on the index

C

_pm, and developed a mathematically complicated approximation method for selecting a subset of processes containing the best supplier from a given set of processes. Pearn (2004b) further provided useful information regarding the sample size regarding the sample size required for various designated selection power by using a simulation technique. A two-phase selection procedure was developed to select a better supplier and to examine the magnitude of the difference between the two suppliers. Chen and Chen (2004) offered four approximate confidence interval methods, one based on the statistical theory given in Boyles (1991) and three based on the bootstrap method, for selecting a better one of two suppliers. However, the method of comparing two suppliers in term of has not yet been discussed. In order to select a better supplier in process cabability, this article proposes the hypothesis testing for comparing the capability of two suppliers based on index.

et al

Y

3. Selection Method

在文檔中依據品質良率指標Yq應用複式抽樣方法於供應商選擇 (頁 12-16)

et al

et al (1998), Pearn

C

C

C

C

C

6

USL LSL

C

= −

USL LSL

C

C

d m

USL LSL

C C C

d = ( USL LSL)/2 −

m = ( USL LSL /2 + )

C

C

C

T

et al

loss X

X T

C

C

C

6 (

USL LSL

C

T )

= −

+ −

et al

C

C

C

C

C

{ }

( )

USL LSL

C

T

C

T

C

C

C

T

C

C

T

C

C

Y

2.2. Quality Yield Y

and Relation Indices 2.2.1. Process Yield

Traditionally, process yield Y

Y

( )

Y = ∫ dF x

USL LSL

( )

F x

x . The

2.2.2. Process Loss

d = ( USL LSL − )/2

C

L

C

L

2.2.3. Quality Yield Y

Y

Y

Y

Y

Traditionally, process yield ^Y

Y = ∫ dF ^x