In this case, we want to distinguish which supplier has better process capability by the index Cpu and Cpl, so we apply the selection method proposed by Chou (1994).
The use of loss functions in quality assurance settings has grown with the introduction of Taguchi’s philosophy. The index Cpm incorporates with the variation of production items with respect to the target value and the specification limits preset in the factory. Huang and Lee (1995) proposed a mathematically complicated approximation method for selecting a subset of processes containing the best supplier from a given set of processes based on the index Cpm. 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.
Since we can not compare these two suppliers directly, we have to sample some products made by these two suppliers, then use the statistical analysis to realize which one has better process capability. Then we decide whether switch the present supplier or not. Before sampling, we have to decide how many sample sizes we should sample to achieve our objective power. And we use statistical simulation program, S-plus, to investigate the accuracy of the selection method.
Then build up Table7-19 to make users convenient to know the required sample size under an objective selection power.
4.1 Selection Power Analysis for Cpu and Cpl
4.1.1 Sample size required for designated selection power
Replacing the supplier will cause huge affection (no matter it’s visible or invisible). So the new supplier has to make sufficient information to prove that it is more capable. Otherwise we will not run risks of the disadvantage caused by wrong decision. We have to sample a required number of products made by these two suppliers to make a believable comparison under the designated selection power.
In order to satisfy the user’s need to distinguish which supplier has better process capability, we have to set two factors first, (1) the minimum of Cpu, Cpu0. In a purchasing contract, a minimum value of the PCI is usually specified.
Montgomery (2001) recommended the minimum quality requirements of Cpu and Cpl for processes runs under some designated capable conditions. In particular, 1.25 for existing processes, and 1.45 for new processes; 1.45 also for existing processes on safety, strength, or critical parameter, and 1.60 for new processes on safety, strength, or critical parameter. (2) the minimal difference of
C pu between these two suppliers, δ =Cpu2−Cpu1, then we can know how many sample sizes we should sample with determined power by the selection method. If
2
1 ˆ
ˆpu Cpu
C < and A<c then we conclude that the process capability of the new supplier better than that of the present supplier. By the way, it means that we have sufficient evidence to reject the null hypothesis H0:Cpu1 ≥Cpu2, otherwise we can not believe that the new supplier has better process capability to replace the present supplier. For the accuracy of this selection method, we use simulation program, S-plus, with 20,000 numbers to establish Tables1-4 which present the required sample to distinguish which supplier has better process capability under power condition = 0.95, and minimum of Cpu= 1.00, 1.25, 1.45, 1.60, the minimal difference of C pu between these two suppliers δ = 0.05(0.05)1.00 with power = 0.90, 0.95, 0.975, 0.99, the power here means the probability of rejecting the null hypothesis H0:Cpu1 ≥Cpu2 when Cpu1 <Cpu2 is true. And we make an example about the response time of LCD, the minimum of C = pu 1.00 and the minimal difference of Cpu between these two suppliers, δ =0.25 , the determined selection power = 0.95, then we can know we have to take 257 samples.
According to Table 7-14, which present the required sample to distinguish which supplier has better process capability under power condition = 0.95. And we find two phenomenon (1) Within fixed selection power, the larger the difference δ between two suppliers, the larger the required sample size. (2) With fixed δ and minimum of Cpu, the selection power increases, the required sample size increases. It’s only because when we want this selection analysis more
realizable, we most draw more products to avoid the variation of statistic estimation and risks of by wrong decision.
4.1.2 Phase I－Supplier Selection
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 a new supplier II (We want to avoid type I error happened. Since switching the supplier will cause a huge cost, over a span then we find it has been a great loss).
For the Phase I of Supplier Selection problem, the user should input the preset minimum requirement of Cpu values, and the minimal difference that must be differentiated between suppliers with designated selection power. The user may alternatively check Tables 7-14 for required sample size for selection power = 0.95, with designated selection power = 0.90, 0.95, 0.975, 0.99. In this case, we only need to compare the test statistic ˆ 1
Cpu and ˆ 2
Cpu , and the selection value A &c based on the test statistic and the required sample sizes.
4.1.3 Phase II－Magnitude Outperformed Detection
Because replacing the supplier will cause a huge cost, we have to compare process capability indices Cpu of these two suppliers. Although the process capability of the new supplier is better than that of the present supplier, the difference between these two suppliers may be too small to be noticed. At this situation, we may not decide to replace the present supplier, unless we can prove that there is a notable magnitude of the difference between these two suppliers.
This action of changing the supplier will be meaningful. So we further investigate the magnitude of the difference between these two suppliers in this stage.
Based on the selection method using the hypothesis test, we set a specified constant q , the notable magnitude of the difference between these two suppliers, and q>0, to realize the value of q , we will test H0 :Cpu1+q≥Cpu2 (the new supplier is not as capable as the present supplier with a magnitude, q ) versus
2 1
1:Cpu q Cpu
H + < (the new supplier is more capable than the present supplier with a magnitude, q ). By comparing these test statistics ˆ 1
Cpu , ˆ 2
Cpu , and the selection value A &c based on the test statistic and the required sample sizes. If the test apply to reject H (0 ˆ 1 ˆ 2
pu
pu q C
C + < and A<c), we can conclude that the new supplier is more capable than the present supplier at least a magnitude, q . In other words, We note that Cpu2 must be greater than the preset capability requirement, and Cpu2 =Cpu1 +q , where q = max{ q′ | test rejects
2
1 pu
pu q C
C + ≥ }. Then we decide to switch the present supplier to avoid waste such a huge exchanging cost.
4.2 Selection Power Analysis for Cpm
4.2.1 Sample size required for designated selection power
In practice, if a new supplier II wants to join competing the orders by claiming its capability better than the existing supplier I, then the new supplier II must furnish convincing information justifying the claim with prescribed level of confidence. Thus, the sample size required for designated selection power must be determined to collect actual data from the factories. The method, however, applies some approximating results and 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.
If the minimum requirement of Cpm values for two candidate processes,
0
Cpm , and the minimal difference δ =Cpm2 −Cpm1 are determined then the sample size required need to sample such that the suppliers must be differentiated with designated selection power. Thus, based on the proposed selection procedures, if If γˆ22 ≤ w×γˆ12 and γˆ12 > w×γˆ22 then we conclude that π2 is better supplier. Otherwise, we would believe that the existing supplier I is better than the new supplier II since we don’t have sufficient information to reject the null hypothesis. We investigate the selection method and accuracy analysis using simulation technique with simulated 10,000 numbers. For users’ convenience in applying our procedure in practice, we tabulate the sample size required for various designated selection power = 0.90, 0.95, 0.975, 0.99. The selection power is calculating the probability of rejecting the null hypothesis H0 :Cpm1 ≥Cpm2, while actually Cpm1 ≥Cpm2 is true, using simulation technique. Tables 1-4 summarize the sample size required for various capability requirements Cpm= 1.00, 1.33, 1.50, 1.67 and the difference δ = 0.05(0.05)1.00 under the
*
p -condition = 0.95, respectively. For example, if the capability requirement of suppliers Cpm is set to 1.00 and δ = 0.30, we would suggest to collect 151 samples to satisfy the designated selection power = 0.95.
We note that the sample size required is a function of Cpm, the difference δ between two suppliers and the designated selection power. From these tables, it can be seen that the larger the value of the difference δ between two suppliers, the smaller the sample size required for fixed selection power. For fixed δ and
Cpm, the sample size required increases as designated selection power increases.
This phenomenon can be explained easily, since the smaller of the difference and the larger designated selection power, the more collected sample is required to account for the smaller uncertainty in the estimation.
4.2.2 Phase I－Supplier Selection
In most applications, the supplier selection decisions would be solely based on the hypothesis testing comparing the two Cpm values, H0 :Cpm1 ≥Cpm2 versus H1:Cpm1 <Cpm2. If the test rejects the null hypothesis H0 :Cpm1 ≥Cpm2, then one has sufficient information to conclude that the new supplier II is superior to the original supplier I, and the decision of the replacement would be suggested.
For the Phase I of Supplier Selection problem, the practitioner should input the preset minimum requirement of Cpm values, and the minimal difference that must be differentiated between suppliers with designated selection power. The practitioner may alternatively check Tables 1-4 for sample size required for *p - condition = 0.95, with designated selection power = 0.90, 0.95, 0.975, 0.99. In this case one only need to compare the test statistic γˆi2, i=1.2, with the selection value w based on the selection procedure corresponding to the preset capability requirement and the required sample sizes.
4.2.3 Phase II－Magnitude Outperformed Detection
In Phase I of supplier selection problem, the supplier selection decisions would be solely based on the hypothesis testing comparing the two Cpm values without further investigating the magnitude of the difference between the two suppliers.
In other applications, the supplier selection decisions would be based on the hypothesis testing comparing the two Cpm values, H0 :Cpm1+q≥Cpm2, versus
2 1
2 :Cpm q Cpm
H + < , where q>0 is a specified constant. If the test rejects the null hypothesis H0 :Cpm1+q≥Cpm2 then one has sufficient information to conclude that supplier II is significantly better than supplier I by a magnitude of
q, and the replacement would then be made due to expensive cost for the supplier replacement. In this case one would have to compare the test statistic γˆi2, i=1,2, with the selection value w corresponding to the preset capability requirement for given sample and designated selection power, to ensure that the magnitude of the difference between the two suppliers exceeds q . We note that Cpm1 must be greater than the preset capability requirement, and Cpm2 =Cpm1+q, where q = max{ q′ | test rejects Cpm1+q′≥Cpm2}. The basic problem is checking whether or not the two suppliers meeting the preset capability requirement could be done by finding the lower confidence bounds on their process capabilities.