= ∫ ⎜⎝ − ⎟ ⎣⎠ + + − ⎦ (1)
Cpl x>0, where b d σ= / , ξ =(µ−T)/σ , G( )⋅ is the cumulative distribution function of the chi-square distribution with degree of freedom n−1, χn2−1, and φ( )⋅ is the probability density function of the standard normal distribution N(0, 1). It is noted that we would obtain an identical equation if we substitute ξ by
ξ
− into equation (1) for fixed values of x and n.
3. Selection Method
3.1 Selection Method
Based on the distribution properties of the estimated PCIs Cpu and Cpl, Chou (1994) developed one-sided tests to select between competing processes that which is more capable. And Huang and Lee (1995) developed based on Cpm index a mathematically complicated approximation method for selecting a subset of processes containing the best supplier from a given set of processes.
3.1.1 Selection Method of Cpu and Cpl
Chou (1994) developed three one-sided tests for comparing two process capability indices (Cp, Cpu, Cpl) to choose between competing processes when the sample sizes are equal.
Based on the hypothesis testing comparing the two Cpu values,
2 1 0:Cpu Cpu
H ≥ versus H1:Cpu1<Cpu2. If the test rejects the null hypothesis
2 1 0:Cpu Cpu
H ≥ , then we have sufficient information to conclude that the new supplier II is better than the present supplier I, and we may switch to the new supplier II.
Let X11,X12,...,X1n and X21,X22,...,X2m be the measurements of two samples independently drawn from two suppliers πi following the normal
distributions N(µi,σi2), for i=1,2. In practice, the number of the sample size n, m(n=m) should be decided first based on Cpu0, δ , and the preset power. Using those Tables7-14, the practitioners may perform the capability testing without having to run the computer programs. The sample mean and the sample standard deviation, xi and Si, are calculated from supplier i , for i=1,2. The estimator
Cˆpui can be calculated.
Nevertheless, the estimator Cˆpui has distributions which are proportional to non-central t distribution. It is complicated to find the critical value of the test statistics to make a decision. Therefore Chou (1994) made a variable transformation that Oij =U −Xij following the normal distributions
) equality test of two coefficients of variation (publish by Miller & Karson (1977) ) could be used. By using the likelihood ratio test, the reject region was defined follows:
Using the likelihood test, the test statistic A given as:
[ ]
which is equivalent to
[ ]
Under the process measurements follow a normal distribution. Cˆpu has a pdf. proportional to a non-central t distribution. Since A is a function of ˆ 1
Cpu
and ˆ 2
Cpu , it’s difficult to determine the distribution of A . Hence it’s impossible to find c such that Pr
{
A<c|H0}
equal an appropriate value of α . Using this fact, we can show that −2lnA has an approximate chi-square distribution with one degree under H0 is true. Then we can find the critical value of the test, c,3.1.2 Selection Method of Cpm
Huang and Lee (1995) considered the supplier selection problem based on the index Cpm, and developed a rather complicated method for supplier selection applications. The method essentially compares the average loss of a group of candidate processes, and select a subset of these processes with small process loss
γ2, which with certain level of confidence containing the best process.
Due to the specification limits are usually fixed and determined in advance, searching the largest Cpm is equivalent to looking for the smallest γ2. The selection rule of Huang and Lee (1995) is that retain the population i in the selected subset if and only if γˆi2 ≤w×min1i≤≠jj≤kγˆ2j , where the value of w is determined by a function of parameters, which can be determined by calculating from collected samples. And we note that choose the value of w is larger than 1 and choose the value as small as possible.
The method, however, provides no indication on how one could further proceed with selecting the best population among those chosen subset of populations. We investigate this method for cases with two candidate processes.
Let πi be the population with mean µ i and variance σi2 , i=1,2 , and
ini
i
i X X
X1, 2,..., are the independent random samples from πi, i=1,2. When the populations are ranked in terms of γˆi2, our interest is to select the better process with smaller value γ2. We denote a correct selection as CS, and assume that the ordered γ2 as γ[21] <γ[22].
Let us denote π(i) as the population associated with γ , [i2] i=1,2. Then, the better population is π(i). We wish to define a procedure with selection rule R such that the probability of a correct selection is no less than a pre-assigned number p* and 0.5< p*<1 . That is, Pr(CS| )R ≥p* . We refer to this requirement as the p*-condition. The selection rule R based on the unbiased and consistent estimators γˆi2 of γi2, i=1,2, and γˆi2 is defined as follows:
For cases with two candidate processes, comparing ˆ 1
Cpm and ˆ 2
where χn2i
( )
λi is the non-central chi-squared distribution with degrees of freedom and non-centrality parameter λi.Selection rule R: Consider the problem of selecting two populations with the smaller γˆ2. The selection rule R is that: Consider πi as the better supplier if and only if γˆi2 ≤w×γˆ2j and γˆ2j >w×γˆi2, i=1,2 and i≠ . For satisfying the j
*
p -condition, then
⎪⎭ as possible, so
{
1, 2}
3.2 Selection Procedure
Chou (1994) developed one-sided tests for comparing two process capability indices to select between competing processes. And based on Cpm index a mathematically complicated approximation method is developed by Huang and Lee (1995) for selecting a subset of processes containing the best supplier from a given set of processes. After probing into these selection method, we develop the practical step-by-step procedure for practitioners to use in making supplier selection decisions. The main steps in tests are developed as:
3.2.1 Selection Procedure of Cpu and Cpl
To make users do this selection work conveniently, we summarized a selection procedure based the selection method proposed by Chou (1994) using the process capability index Cpu and Cpl.
Step1. Determine the specification limits USL . Check the appropriate Table1-4 to find the corresponding n based on Cpu0, δ , and the preset power, where n=m. Then input the sample data of size n,
m.
Step2. Calculate the sample mean xi, and sample standard deviation S , i the test statistic Cˆpui, i=1,2 and the value of a supplier. Otherwise, we conclude π2 is better supplier.
3.2.2 Selection Procedure of Cpm
Huang and Lee (1995) developed the mathematically complicated approximation method for dealing the selected problem. To make this method practical for in-plant applications, the selection procedure may be summarized and expand in our form as follows:
Step 1: Input the original sample data of size ni , i=1, 2 , set the
{
1, 2}
Step 6: Conclude which supplier is better using the following rule R:
If γˆ12 ≤ w×γˆ22 and γˆ22 > w×γˆ22 then we conclude that the process of π1 is more capable.
If γˆ22 ≤ w×γˆ12 and γˆ12 > w×γˆ22 then we conclude that the process of π2 is more capable.
Ifγˆ12 ≤ w×γˆ22and γˆ22 ≤ w×γˆ12 , we doesn’t have enough information to make supplier selection.