製程能力指標於供應商決策之應用

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(1)國立交通大學工業工程與管理學系碩士班. 碩士論文. 製程能力指標於供應商決策之應用 The Application for Supplier Selection Problem Based on Process Capability Indices. 研. 究. 生：林涵琦. 指導教授：彭文理博士. i.

(2) 誌謝研究所的生涯最重要的兩點:好老師加好同學,很幸運的我都有了. 彭文理老師一世英明,碰到如此頑劣學生如我,也是用鼓勵代替責罵,耐心地一路指導過來,相信老師教學的功力和修行這兩年也因我而增加不少, 鍾媽媽兩年來對我的關愛,真的是令我感覺又是溫暖又是感動,我最愛鍾媽媽了.感謝彭德保老師在論文口試過程中縝密的指導,讓學生的論文內容能夠更加嚴謹. 感謝實驗室的好夥伴們,阿妮是最好的姊妹,聰明又堅強,大哥在課業和生活上的教導,改正了很多我做人處世的觀念,小哥在我系模考超爛時安慰我,排程作業寫不出來也罩我,阿參會陪我一起熬夜趕投影片陪我散布,小桃紅讓我記得,該唸英文了,芭樂在我報告被電後會給我建議,在我面臨困惑時也給我很多意見,帥遠….簡直是研究所生活的＂重＂心啊,讓我時時刻刻警惕自己的身材有無變形,跟大家一起鎖上門打 AOK 的日子是最令人開心的,讓寫論文漫長的時間頓時縮減許多,學長建緯跟英仲,從我尚未入學就一直關照,尤其在論文寫作上給了許多的意見,于婷學姊更是如同大姊般的照顧大家,學弟們一群寶,乖乖寫論文吧你們,能夠在生管實驗室跟大家過這兩年真的是很開心的呢.還有,熊波波先生,謝謝你對我的疼愛與耐心包容,我們要一直很幸福喔. 最後謝謝爸爸辛苦養我這個敗家女讀完了碩士,哥哥和惟惟也十分的獨立,讓我能在新竹安心唸書工作,還能在我月光光時援助我,我永遠是你們的敗家女. 謹將此論文與榮耀獻給我最思念的媽媽. ii.

(3) 製程能力指標於供應商決策之應用研究生：林涵琦. 指導教授：彭文理博士. 國立交通大學工業工程與管理學系碩士班. 摘要製程能力指標（Probability Capability Indices）是藉由一個指標的數值來衡量製程的能力與績效。在本篇論文中共應用了單邊指標 C pu ， C pl 以及雙邊指標 Cpm 分別來分辨兩家供應商製程能力，分別利用了兩個決策法則（1）Chou 在 1994 利用所提出，建立了三個單邊檢定來比較兩個相互競爭的供應商的製程能力（2）Huang and Lee 在 1995 年基於指標 Cpm 所提出的數學逼近的法則，主要功能在於由一群候選的供應商中選出一組包含有最佳供應商的集合，本研究並應用了上述的決策方法分別建立了一個實用的決策程序供使用者能用作於供應商決策時使用，由於我們無法直接地對兩個供應商作比較，我們必須分別從兩個供應商的產品進行抽樣，並使用統計分析來了解何者具有較佳的製程能力，即可決定是否要更換現有的供應商。為了證明本研究的可靠性，我們利用了模擬工具做了準確度分析，了解在欲達到的目標檢定力之下，所必須抽取的樣本數目為何。而兩階層的決策程序首先能選出較佳的供應商，再進一步的分別求出兩供應商製程能力的差距。最後，本研究應用了實際的例子，分別是：STN-LCD， TFT-LCD 及汽車玻璃三種產品製程的樣本，套用本研究的決策程序來做供應商的選擇。. iii.

(4) The Application for Supplier Selection Problem Based on Process Capability Indices Student：Han-Chi Lin. Advisor：Dr. Wen-Lea Pearn. Institute of Industrial Engineering and Management National Chiao Tung University Abstract Process capability indices (PCIs) have been used in the manufacturing industry to provide quantitative measures on process potential and performance. In this paper, we obtain the unilateral index C pu , C pl and bilateral index Cpm to distinguish which supplier has better process capability, so we apply the selection method proposed by Chou (1994) developed three one-sided tests for comparing two process capability indices to choose 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. We implement this method, and develop a practical step-by-step procedure for practitioners to use in making supplier selection decisions. Since we can’t 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. To make our research realizable, we make an accuracy analysis by building tables to make users convenient to know the required sample size under an objective selection power. Accuracy of the selection method is investigated using simulation technique. The accuracy study provides useful information about the sample size required for designated selection power. A two- phase selection procedure is developed to select better supplier and further examine the magnitude of the difference between the two suppliers. Finally, we also investigate a real-world case on the STN-LCD (Super Twisted Nematic Liquid Crystal Display) ,TFT-LCD (Thin Film Transistor Liquid Crystal Display) and automobile window manufacturing process, and apply the selection procedure using actual data collected from the factories, to reach a decision in supplier selections.. iv.

(5) Contents 摘要.............................................................................................................................. iii Abstract .........................................................................................................................iv Contents .........................................................................................................................v List of table ..................................................................................................................vii List of figure .............................................................................................................. viii Notations .......................................................................................................................ix 1. Introduction................................................................................................................1 2. Literature Review.......................................................................................................1 2.1 Literature Review.............................................................................................1 2.2 Distribution of the PCIs ...................................................................................4 2.2.1 Distribution of Estimated C pu and C pl .............................................4 2.2.2 Distribution of the Estimated C pm .......................................................5 3. Selection Method .......................................................................................................7 3.1 Selection Method .............................................................................................7 3.1.1 Selection Method of C pu and C pl .....................................................7 3.1.2 Selection Method of C pm .....................................................................9 3.2 Selection Procedure .......................................................................................11 3.2.1 Selection Procedure of C pu and C pl ...............................................11 3.2.2 Selection Procedure of C pm ...............................................................12 4. Accuracy Analysis ...................................................................................................13 4.1 Selection Power Analysis for C pu and C pl ................................................14 4.1.1 Sample size required for designated selection power .........................14 4.1.2 Phase I－Supplier Selection ...............................................................15 4.1.3 Phase II－Magnitude Outperformed Detection ..................................15 4.2 Selection Power Analysis for C pm ................................................................16 4.2.1 Sample size required for designated selection power .........................16 4.2.2 Phase I－Supplier Selection ...............................................................17 4.2.3 Phase II－Magnitude Outperformed Detection ..................................17 5. Example ...................................................................................................................18 5.1 Application Example TFT-LCD.....................................................................18 5.1.1 Data Analysis and Supplier Selection .................................................19 5.1.2 Phase I－Supplier Selection ...............................................................20 5.1.3 Phase II－Magnitude Outperformed Detection ..................................21 5.2 Application Example Automobile Windows..................................................21 5.2.1 Data Analysis and Supplier Selection .................................................23 5.2.2 Phase I－Supplier Selection ...............................................................24 5.2.3 Phase II－Magnitude Outperformed Detection ..................................24 5.3 Application Example STN-LCD ...................................................................25 5.3.1 Data Analysis and Supplier Selection .................................................26 v.

(6) 5.3.2 Phase I－Supplier Selection ...............................................................27 5.3.3 Phase II－Magnitude Outperformed Detection ..................................28 6. Conclusion ...............................................................................................................28 References....................................................................................................................30 Appendix A. The sample sizes information.................................................................32 Appendix B. The sample data for application .............................................................38. vi.

(7) List of table Table 1. The calculated sample statistics for two suppliers.(Cpu)............................20 Table 2. Magnitude outperformed detection of selection procedure. (Cpu) ...........21 Table 3. The calculated sample statistics for two suppliers.(Cpl) ............................24 Table 4. Magnitude outperformed detection of selection procedure. (Cpl)............25 Table 5. The calculated sample statistics for two suppliers.(Cpm) ...........................27 Table 6. Magnitude outperformed detection of selection procedure. (Cpm) ..........28 Table 7. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpu1 = 1.00, Cpu2 = 1.05(0.05)2.00.............................................................32 Table 8. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpu1 = 1.25, Cpu2 = 1.30(0.05)2.25.............................................................32 Table 9. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpu1 = 1.45, Cpu2 = 1.50(0.05)2.45.............................................................33 Table 10. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpu1 = 1.60, Cpu2 = 1.65(0.05)2.60. ..................................................33 Table 11. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpl1 = 1.00, Cpl2 = 1.05(0.05)2.00. ...................................................34 Table 12. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpl1 = 1.25, Cpl2 = 1.30(0.05)2.25. ...................................................34 Table 13. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpl1 = 1.45, Cpl2 = 1.50(0.05)2.45. ...................................................35 Table 14. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpl1 = 1.60, Cpl2 = 1.65(0.05)2.60. ...................................................35 Table 15. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpm1 = 1.00, Cpm2 = 1.05(0.05)2.00..................................................36 Table 16. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpm1 = 1.33, Cpm2 = 1.38(0.05)2.33..................................................36 Table 17. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpm1 = 1.50, Cpm2 = 1.55(0.05)2.50..................................................37 Table 18. Sample size required for power = 0.90, 0.95, 0.975, 0.99 under p* = 0.95, with Cpm1 = 1.67, Cpm2 = 1.72(0.05)2.67. ...............................................37 Table 19. Sample data collected from suppliers (using Cpm) ...................................38 Table 20. Sample data collected from suppliers for (using Cpu) ..............................39 Table 21. Sample data collected from suppliers (using Cpl) ....................................41. vii.

(8) List of figure Figure 1. The electronic field effect of the liquid crystal ............................................19 Figure 2. The picture tube theorem..............................................................................19 Figure 3. Normal probability plot for response time data of Supplier I. .....................20 Figure 4. Normal probability plot for response time data of Supplier II. ....................20 Figure 5. Histogram for supplier I. ..............................................................................20 Figure 6. Histogram for supplier II. .............................................................................20 Figure 7. The structure of the sandwich glass..............................................................22 Figure 8. Automobile’s front window..........................................................................22 Figure 9. Automobile's side window ...........................................................................22 Figure 10. Normal probability plot for thickness data of Supplier I............................23 Figure 11. Normal probability plot for thickness data of Supplier II. .........................23 Figure 12. Histogram for supplier I. ............................................................................24 Figure 13. Histogram for supplier II. ...........................................................................24 Figure 16. Normal probability plot for thickness data of Supplier I............................27 Figure 17. Normal probability plot for thickness data of Supplier II. .........................27 Figure 18. Histogram for supplier I. ............................................................................27 Figure 19. Histogram for supplier II. ...........................................................................27. viii.

(9) Notations T ：target value LSL ：the lower specification limits preset by the process engineers USL ：the lower specification limits preset by the process engineers π i ： supplier i , for i = 1, 2 X ij ：the measurements of samples independently drawn from supplier i , for i = 1, 2 n ：the number of the sample size drawn from supplier 1(for case Cpu,Cpl) m ：the number of the sample size drawn from supplier 2(for case Cpu,Cpl) ni ：the number of the sample size drawn from supplier i = 1, 2 (for case Cpm) µi ：the population mean of supplier i , for i = 1, 2 σ i2 ：the population variation of supplier i , for i = 1, 2 xi ：the sample mean calculated from data of supplier i , for i = 1, 2 Si ：the sample variation calculated from data of supplier i , for i = 1, 2 Yi 2 ：the MLE of σ i2 γ 2 ：the average loss of group γî2 ：the unbiased estimator of the average loss of a group of supplier i , for i = 1, 2 2 γ [i ] ：the ordered γ 2 γˆ[2i ] ：the ordered γî2 π (i ) ：the population associated with γ [2i ] C pu ：the UMVUE of Cpu C pl ：the UMVUE of Cpl Cˆ pm ( CCS ) ：an estimator of Cpm Cˆ pm ( B ) ：the MLE of Cpm C pu 0 ：the minimal requirement of Cpu values for two candidate processes C pl 0 ：the minimal requirement of Cpl values for two candidate processes C pm 0 ：the minimal requirement of Cpm values for two candidate processes δ ：the minimal difference of PCIs between these two suppliers A ：the likelihood ratio test statistics c ：the critical value w ：the weight number used to decide the range of a subset including the best supplier p * ：the least probability of a correct selection, 0.5 < p* < 1 q ：the notable magnitude of the difference between these two suppliers. ix.

(10) 1. Introduction. Process capability indices (PCIs), the purpose of which is to provide numerical measures of whether the ability of a manufacturing process meets a predetermined level of production tolerance or not, have received considerable research attention and increased usage in process assessments and purchasing decisions in the automotive industry during last decade.. In this paper, we obtain the unilateral index C pu , C pl and bilateral index Cpm to distinguish which supplier has better process capability. For this purpose, we apply the selection method proposed by Chou (1994) developing three one-sided tests to select between competing processes that which is more capable. Using the hypothesis test to find the larger C pu , C pl . 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. Under the circumstance, to search the larger Cpm which are used to provide unitless measure of the process performance is equivalent to look for the smaller γ 2 . 2. Literature Review 2.1 Literature Review Process capability indices have been used in the manufacturing industry to provide quantitative measures on process potential and performance. Including C p , C pu , C pl , C pk , C pm , C pmk , (see Kane (1986), Chan et al. (1988), Boyles (1991) and Pearn et al. (1992)). While C p , C pk , C pm and C pmk are appropriate measures for processes with two-sided specifications (which require both LSL and USL ), C pu and C pl have been designed specifically for processes with one-sided specification limit (which require only LSL or USL ). Those indices are effective tools for process capability analysis and quality assurance, and the formula for those indices are easy to understand and straightforward to apply. The Cp index was developed by Kane (1986), which considers the overall process variability relative to the manufacturing tolerance to measure process precision (product consistency). Due to simplicity of the design, Cp cannot reflect the tendency of process centering (targeting). Cp =. USL − LSL . 6σ. 1.

(11) The index C pu measures the capability of a smaller-the-better process with an upper specification limit USL , whereas the index C pl measures the capability of a larger-the-better process with a lower specification limit LSL . Pearn and Chen (2001) develop a similar procedure using these one-sided capability indices C pu and C pl to test whether practitioners’ processes meet the capability requirement. And set a convenient table display the critical value for various α -risk, sample sizes n and the desired quality condition. Further, Pearn and Lin provide the information of p-value required for making decisions. When the process mean is off center of the specification, the index C pk results in that one specification limit (the one closer to the process mean). And two calculations, C pu and C pl , have to be computed. In other words, the C pk index is the minimum of C pu and C pl . The index C pk defined as:. ⎧USL − µ µ − LSL ⎫ C pk = min ⎨ , ⎬ 3σ ⎭ ⎩ 3σ where USL is the upper specification limit, LSL is the lower specification limit, µ is the process mean and σ is the process standard deviation, and T is the target value. The index Cpk was developed because Cp does not adequately deal with cases where process mean µ is not centered. However, Cpk alone still cannot provide adequate measure of process centering. That is, a large value of Cpk does not really tell us anything about the location of the mean in the tolerance interval. When a process is centered, C p and C pk will be the same number, therefore, C pk is preferred because it’s not dependent on the process being centered. The index C pk takes the process mean into consideration but it can fail to distinguish between on-target processes from off-target processes (Pearn et al.(1992)). “Although the process capability indices C p and C pk are widely used to provide useful measures of process potential and performance. These indices don’t adequately address the issue of process centering” (Boyles(1991)). In other words, they are not related to the cost of failing to meet customer desires. To overcome this deficiency, several indices have been proposed that include the deviations from the target value when assessing the capability of a particular process. Hsiang and Taguchi (1985) considered am extension of C p to address the issue directly. And it also named C pm by Chan et al. (1988). Process capability 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 Cpm is defined as:. Cpm =. USL − LSL 6 σ 2 + ( µ − T )2. [. =. USL − LSL 6γ. ]. we note γ 2 = σ 2 + ( µ − T ) 2 = E ( X − T ) 2 to be the major part of the denominator of C pm , which incorporates two variation components: (i) variation 2.

(12) to the process mean and (ii) deviation of the process mean from the target. Since E ( X − T ) 2 was the expected loss where we have noted that the loss of a characteristic X missing the target is often assumed to be well approximated by 2 the symmetric squared error loss function, loss( X ) = ( X − T ) . Hence, the capability index C pm is a loss-based index.. [. ]. For on-target processes, the value of Cpm index reaches its maximum, implying that the process capability runs under the desired condition. On the other hand, the smaller value of Cpm means the higher expected loss and the poorer process capability. Therefore, the index Cpm is considered to be more sensitive than Cp and Cpk in reflecting the process loss. Boyles (1991) has provided a definitive analysis of Cpm and its usefulness in measuring process centering. He notes that both Cpk and Cpm coincide with Cp when µ = T and decrease as µ moves away from T. However, Cpk < 0 for µ > USL or µ < LSL , whereas Cpm of process with µ − T = ∆ > 0 is strictly bounded above by the Cp value of a process with σ = ∆ . In the initial stage of production setting, 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, service and so on, which need to be considered in selecting the best suppliers. Several selection rules have been proposed for selecting the means or variances in analysis of variance (see Gibbons, Olkin and Sobel (1977), Gupta and Panchapakesan (1979), Gupta and Huang (1981) for more details). PCIs are useful management tools, particularly in the manufacturing industry, which provide common quantitative measures on manufacturing capability and production quality. In the situation of the manufacturing process being control, we assume that the quality characteristic X is normally distributed, USL and LSL are usually fixed and determined in advance, the larger Cp is equivalent to looking for the smallest σ 2 . Tseng and Wu (1991) considered the problem of selecting the best manufacturing process from k available manufacturing processes based on the precision index Cp and a modified likelihood ratio (MLR) selection rule is proposed. Chou (1994) developed three one-sided tests (Cp , CPU , CPL ) for comparing two process capability indices to choose between competing processes when the sample sizes are equal. 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.. Since we couldn’t 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 to switch the present supplier or not. To make our research realizable, we make an accuracy analysis by building tables to make users convenient to know the required sample size under an objective selection 3.

(13) power. Accuracy of the selection method is investigated using simulation technique. The accuracy study provides useful information about the sample size required for designated selection power. A two- phase selection procedure is developed to select better supplier and further examine the magnitude of the difference between the two suppliers. Finally, we also investigate a real-world case on the STN-LCD (Super Twisted Nematic Liquid Crystal Display), TFT-LCD (Thin Film Transistor Liquid Crystal Display) and automobile window manufacturing process, and apply the selection procedure using actual data collected from the factories, to reach a decision in supplier selections. 2.2 Distribution of the PCIs In this paper, we obtain the unilateral index C pu , C pl and bilateral index Cpm to distinguish which supplier has better process capability. The formula for these indices are easy to understand and straightforward to apply. In practice, sample data must be collected to calculate these indices. Therefore, a great degree of uncertainty may most practitioners simply look at the value of the estimators calculated from the sample data, then make a conclusion on whether their processes meet the preset capability requirement. This approach is highly unreliable since sampling errors are ignored. Thus, we then introduce the distributional properties of the estimated index C pu , C pl , Cpm and the unbiased estimator of loss function, γˆ 2 , is considered. 2.2.1 Distribution of Estimated C pu and C pl C pu and C pl have been designed particularly for processes with one-sided specifications (which require only the upper or the lower specification limit). Chou and Owen (1989) considered the natural estimators of C pu and C pl , Cˆ pu and Cˆ pl , which are defined as the following: USL − X X − LSL , Cˆ pl = , Cˆ pu = 3S 3S. where X = ∑i =1 xi / n , S 2 = ∑i =1 ( xi − x ) 2 /(n − 1) , USL and LSL are the upper and the lower specification limits preset by the process engineers or product designers. Under normality assumption, Chou and Owen (1989) show that the estimator Cˆ pu is distributed as cn tn −1 (δ ) , where cn = (3 n ) −1 , and tn −1 (δ ) is a non-central t distribution with ( n − 1) degrees of freedom and non-centrality parameter δ = 3 nC pu , the same distribution of Cˆ pl (with δ = 3 nC pl ). But both Cˆ pu and Cˆ pl are unbiased. Pearn and Chen (2001) added the correction factor 1/ 2 bn −1 = [2 /( n − 1)] Γ[( n − 1) / 2] / Γ[( n − 2) / 2] to correct the natural estimators of ~ ~ C pu and C pl , and obtain these unbiased estimators, C pu and C pl , which are n. n. 4.

(14) defined as the following: b (USL − X ) ~ , C pu = bn −1Cˆ pu = n −1 3S b ( X − LSL) ~ , C pl = bn −1Cˆ pl = n −1 3S. ~ ~ then we have E (C pu ) = C pu , and E (C pl ) = C pl , since bn −1 < 1 , then ~ ~ Var (C pu ) < Var (Cˆ pu ) and Var (C pl ) < Var (Cˆ pl ) . Since both estimators depend only ~ and on the sufficient and complete statistics ( X , S 2 ) of ( µ ,σ 2 ) and C pu ~ C pl are UMVUEs of C pu and C pl . The r-th moment and the variance of C pu are as the following:. [ ]. ~ E C pu. r. =. (Γ[(n − 1) / 2]) r −1 Γ[(n − 1 − r ) / 2] E (Z ) r , r r (3 n ) (Γ[(n − 2) / 2]). ⎧ Γ[(n − 1) / 2]Γ[(n − 3) / 2] ⎫ ~ − 1⎬ C pu Var C pu = ⎨ (Γ[(n − 2) / 2]) 2 ⎩ ⎭. [ ]. [ ]. 2. +. 1 Γ[(n − 1) / 2]Γ[(n − 3) / 2] 9n (Γ[(n − 2) / 2]) 2. ~ where Z = n (USL − X ) / σ , it’s easy to verify that E (C pu ) = C pu . The results of ~ the r-th moment, the expected value and the variance of estimator C pl are the same. And Pearn and Lin use the UMVUEs of C pu and C pl to calculate the critical values and the p-value for making decisions. ~ ~ Further the PDF (probability density function) of C pu and C pl was be obtained as: 2 ⎧ ⎞ ⎤ ⎫⎪ ⎛ 3x ny ∞ 3 n /(n − 1) ⋅ 2 −n / 2 ⎪ 1 ⎡⎢ ( n −2 ) / 2 f ( x) = y + ⎜⎜ exp ⎨− × y − δ ⎟⎟ ⎥ ⎬dy 2 bn −1 π Γ[( n − 1) / 2] ∫0 b n 1 − ⎢ ⎠ ⎥⎦ ⎪⎭ ⎝ n −1 ⎪⎩ ⎣. where bn −1 = [2 /( n − 1)] Γ[( n − 1) / 2] / Γ[( n − 2) / 2] δ = 3 nC pl ). 1/ 2. and. δ = 3 nC pu. (or. 2.2.2 Distribution of the Estimated C pm Since the process mean µ and the process variance σ 2 must be estimated from the sample. Thus, the estimated index Cˆ pm is obtained by replacing µ and σ 2 by their estimators. Chan, Cheng, and Spring (1988) and Boyles (1991) proposed two different estimators of Cpm respectively defined as the following: 5.

(15) Cˆ pm(CCS ) =. d 3 s2 + ( x − T ) 2. and. Cˆ pm( B ) =. d 3 sn2 + ( x − T ) 2. ,. where d = (USL − LSL)/2 is the half width of the specification interval, x = ∑ in=1 xi / n, s2 = ∑ in=1 ( xi − x ) 2 /(n − 1) and sn2 = ∑ in=1 ( xi − x ) 2 / n. In fact, the two estimators, Cˆ pm(CCS ) and Cˆ pm( B ) , are asymptotical equivalent. Assuming that the process data are normally distributed and T = M , Chan, Cheng, and Spring (1988) derived the probability density function of Cˆ pm(CCS ) = Y as fY ( y) =. ⎡ 1⎛ a ⎞ ⎤ ∞ λ j (a / y2 ) n /2 + j −1 exp λ , − + ⎢ ⎜ ⎟⎥ ∑ 2 2j 2 n/2 −1 y3 ⎢⎣ 2 ⎝ y ⎠ ⎥⎦ j =1 j ! Γ(n /2 + j )2. a. y>0.. 2 (1 + λ / n)(n − 1) and λ = n( µ − T ) 2 /σ 2 . Experts in statistical where a = Cpm distributions will easily recognize that Cˆ pm(CCS ) can be shown to be functions of the inverse moments of a non-central chi-square distribution. An alternative equivalent formula was provided by Pearn, Kotz and Johnson (1992).. The distributional properties of Cˆ pm(CCS ) are intractable for asymmetrical specifications ( (USL + LSL)/2 ≠ T ). When the case of (USL + LSL)/2 = T , Cˆ pm(CCS ) is a biased estimator of Cpm , but is asymptotically unbiased. Detailed descriptions and proofs of the properties of Cˆ pm(CCS ) are given in Chan, Cheng, and Spring (1988). On the other hand, Boyles (1991) considered that it would be more appropriate to replace the factor n − 1 by n in the denominator since the term γˆ( B ) = sn2 + ( x − T ) 2 in the denominator of Cˆ pm( B ) is the uniformly minimum variance unbiased estimator (UMVUE) of the term σ 2 + ( µ − T ) 2 . We note that x and sn2 are the maximum likelihood estimators (MLEs) of µ and σ 2 , respectively. Hence, the estimated Cˆ pm( B ) is also the MLE of Cpm . The approach by simply looking at the calculated values of the estimated indices and then make a conclusion on whether the given process is capable, is highly unreliable as the sampling errors have been ignored. As the use of the capability indices grows more widespread, users are becoming educated and sensitive to the impact of the estimators and their sampling distributions on constructing confidence intervals and performing hypothesis testing. Under the assumption of normality, Kotz and Johnson (1993) obtained the r-th moment, and calculated the first two moments, the mean, and the variance of Cˆ pm . Cheng (1994) has developed a hypothesis testing procedure where tables of the approximate p-values were provided for some commonly used capability requirements, using the natural estimator of Cpm . The practitioners can use the obtained results to determining if their process satisfies the targeted quality condition. But Cheng’s approach requires further estimation of the distribution characteristic ( µ − T )/σ when calculating the p-values, which introduces additional sampling errors thus making the decisions made less reliable. Zimmer and Hubele (1997) provided tables of exact percentiles for the sampling distribution of the estimator Cˆ pm . Zimmer, Hubele and Zimmer (2001) 6.

(16) proposed a graphical procedure to obtain exact confidence intervals for Cpm , where the parameter ( µ − T )/σ is assumed to be a known constant. On the other hand, using the method similar to that presented in Vännman (1997), Pearn and Lin (2002) obtained an exact form of the cumulative distribution function of Cˆ pm . Under the assumption of normality, the cumulative distribution function of Cˆ pm can be expressed in terms of a mixture of the chi-square distribution and the normal distribution: FCˆ (x) = 1- ∫ 0b pm. n /(3 x ). ⎛ b2 n. ⎞ − t 2 ⎟ ⎡⎣φ (t + ξ n ) + φ (t − ξ n ) ⎤⎦ dt, ⎝ 9x ⎠. G ⎜. 2. (1). C pl 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 , χ n2−1 , 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 C pu and C pl , 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 C pu and C pl Chou (1994) developed three one-sided tests for comparing two process capability indices ( C p , C pu , C pl ) to choose between competing processes when the sample sizes are equal. Based on the hypothesis testing comparing the two C pu values, H 0 : C pu1 ≥ C pu 2 versus H 1 : C pu1 < C pu 2 . If the test rejects the null hypothesis H 0 : C pu1 ≥ C pu 2 , 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 X 11 , X 12 ,..., X 1n and X 21 , X 22 ,..., X 2 m be the measurements of two samples independently drawn from two suppliers π i following the normal 7.

(17) distributions N ( µi , σ i2 ) , for i = 1,2 . In practice, the number of the sample size n , m ( n = m ) should be decided first based on C pu 0 , 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 Cˆ pui can be calculated. Nevertheless, the estimator Cˆ pui 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 − X ij following the normal distributions N (U − µi , σ i2 ) , for i = 1,2 . The sample mean and the sample standard deviation, Oi and Si , are calculated from supplier i , for i = 1,2 . Then the test could be transferred to test H 0 : o1 / σ 1 ≥ o2 / σ 2 versus H 1 : o1 / σ 1 < o2 / σ 2 . And the 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:. O1 / S1 < O2 / S 2 and A < c it is equivalent to. Cˆ pu1 < Cˆ pu 2 and A < c Using the likelihood test, the test statistic A given as:. ⎤ ⎡ 2Y1Y2 A=⎢ ⎥ 2 2 1/ 2 2 2 1/ 2 ⎣ (O1 + 2Y1 ) (O2 + 2Y2 ) − W1W2 ⎦. [. n. ]. which is equivalent to ⎡ 2 A= ⎢ 2 1 / 2 2 1/ 2 ⎢⎣ ( aCˆ pu1 + 2) ( aCˆ pu 2 + 2) − aCˆ pu1Cˆ pu 2. [. ⎤ ⎥ ⎥⎦. n. ]. where Yi 2 = ( n − 1) Si2 / n , a = 9n / (n − 1) Under the process measurements follow a normal distribution. Cˆ pu has a pdf. proportional to a non-central t distribution. Since A is a function of Cˆ pu1 and Cˆ pu 2 , it’s difficult to determine the distribution of A . Hence it’s impossible to find c such that Pr{A < c | H 0 } equal an appropriate value of α . Using this fact, we can show that − 2 ln A has an approximate chi-square distribution with one degree under H 0 is true. Then we can find the critical value of the test, c , as follows: ⎡ χ 2 (1 − 2α ) ⎤ c ≈ exp ⎢ − 1 ⎥ , 0 < c < 1. 2 ⎣ ⎦. 8.

(18) 3.1.2 Selection Method of C pm Huang and Lee (1995) considered the supplier selection problem based on the index C pm , 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 γ 2 , 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 C pm is equivalent to looking for the smallest γ 2 . The selection rule of Huang and Lee (1995) is that retain the population i in the selected subset if and only if γî2 ≤ w × min1i ≠≤ jj ≤k γˆ 2j , 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 π i be the population with mean µ i and variance σ i2 , i = 1,2 , and X i1 , X i 2 ,..., X ini are the independent random samples from π i , i = 1,2 . When the populations are ranked in terms of γî2 , our interest is to select the better process with smaller value γ 2 . We denote a correct selection as CS, and assume that the ordered γ 2 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 γî2 of γ i2 , i = 1,2 , and γî2 is defined as follows: ni. γî2 =. ∑(x j =1. S i2 =. ij. − T )2 =. ni. ( ni − 1) S i2 + ni ( xi − T ) 2 , ni. 1 1 ni ( xij − xi ) 2 , xi = ∑ ni − 1 j =1 ni. ni. ∑x j =1. ij. ,. For cases with two candidate processes, comparing Cˆ pm1 and Cˆ pm 2 is equivalent to compare γˆ12 and γˆ22 . Hence, by the result of Pearn, Kotz and Johnson (1992),. σ 12. 2. ⎛µ −T ⎞ ⎟⎟ . γˆ ~ χ (λi ) , λi = ni ⎜⎜ 1 ni ⎝ σ1 ⎠ 2 i. 2 ni. 9.

(19) where χ n2i (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 γˆ 2 . The selection rule R is that: Consider π i as the better supplier if and only if γî2 ≤ w × γˆ 2j and γˆ 2j > w × γî2 , i = 1,2 and i ≠ j . For satisfying the p * -condition, then ⎧⎪ 1 ⎛⎜ 1 1 ⎞⎟ νˆ[ k ] ⎫⎪ w1 = exp ⎨− 2 L1 + − ⎬ νˆ[1] ⎜⎝ νˆ[1] νˆ[ k ] ⎟⎠ νˆ[1] ⎪⎭ ⎪⎩ and , ⎧⎪ 1 ⎛⎜ 1 1 ⎞⎟ νˆ[ k ] ⎫⎪ w2 = exp ⎨− 2 L2 + − ⎬ νˆ[1] ⎜⎝ νˆ[1] νˆ[ k ] ⎟⎠ νˆ[1] ⎪⎭ ⎪⎩. . Choose the value of w which is larger than 1 and choose the value as small as possible, so w = min{w1 , w2 }, if w1 > 1 and w2 > 1 w = w1 , if w1 > 1 and w2 ≤ 1 , w = w2 , if w2 > 1 and w1 ≤ 1 ;. where. L1 =. − d2 + k −1. d1 = a ∑. i =1. k −1. d 2 = b∑. − d 2 − d 22 − 4 d 1 d 3 d 22 − 4 d 1 d 3 , , L2 = 2d1 2d1 ai + ak ai. ak ⎞ ⎟ , ⎟ ⎠. ak ab ⎛⎜ k − 1 + ∑ a1 a * ⎜⎝ i = 1. ai + ak ai. a k ⎞ ⎛ k −1 ⎟⎜ ∑ ⎟ ⎜ i =1 ⎠⎝. 1+. i =1. d3 =. b ⎛ ⎜∑ 4 a * ⎜⎝ i = 1 2. 2. ⎛ a ⎞ a 2 ⎛⎜ k − 1 ⎜⎜ 1 + k ⎟⎟ + ∑ a 1 ⎠ a * ⎜⎝ i = 1 ⎝. k −1. k −1. a * = 0 .5 − a ∑. i =1. ak a1. ⎞ ⎟, ⎟ ⎠. ,. 2. ⎞ ⎟ − ln( p * 2 k − 1 2 a * ), ⎟ ⎠ ak 1 , b = − 0 . 513277 , a = − 0 . 085514 ,aj = νˆ[ j ] ai ak ai. ⎛ x −T ( n i + λî ) 2 ˆ νî = , λ i = n i ⎜⎜ i ( n i + 2 λî ) ⎝ Si where νî is used to estimate ν i ,. 2. ⎞ ⎟⎟ , ⎠ i = 1,2 , and ordered νî are denoted by. νˆ[1] ≤ νˆ[ 2 ] .. 10.

(20) 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 C pu and C pl 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 C pu and C pl . Step1. Determine the specification limits USL . Check the appropriate Table1-4 to find the corresponding n based on C pu 0 , 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 Cˆ pui , i = 1,2 and the value of a ⎡ 1 n ⎤ 1 n xi = ∑ xij , Si = ⎢ ( xij − xi ) 2 ⎥ ∑ n j ⎣n −1 j ⎦. 1/ 2. USL − xi 9n C pui = , a= 3S i n −1 Step3. Calculate the value of A and c . n. ⎤ ⎡ 2 A=⎢ ⎥ , 2 1/ 2 2 1/ 2 2 2 ⎢⎣ (aCˆ pu1 + 2) (aCˆ pu 2 + 2) − aCˆ pu1Cˆ pu 2 ⎥⎦. {. }. c = exp − χ 12 (1 − 2α ) / 2. Step4. Use the decision rule to conclude which supplier is better: If Cˆ < Cˆ and A < c then we conclude that that π pu1. pu 2. supplier. Otherwise, we conclude π 2 is better supplier.. 1. is better. 11.

(21) 3.2.2 Selection Procedure of C pm 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 specification limits USL , LSL , target value T , the probability p * , and the constants a = −0.085514 , b = −0.513277 . Step 2: Calculate the sample mean xi , sample standard deviation S i , the value of γî2 , i = 1,2 . ni. γˆ =. ∑ (x j =1. 2 i. S i2 =. ij. − T )2. ni. =. ( n i − 1 ) S i2 + n i ( x i − T ) 2 , ni. 1 1 ni ( xij − xi ) 2 , xi = ∑ ni − 1 j =1 ni. ni. ∑x j =1. ij. ,. Step 3: Calculate the value of λî ,νî , a j ( a j = 1 / νˆ[i ] ) , and a * , i = 1,2 k −1. a * = 0 .5 − a ∑ i =1. 1 ak ,aj = , b = − 0 . 513277 , a = − 0 . 085514 νˆ[ j ] ai. ⎛ x −T ( n + λî ) 2 ˆ νî = i , λ i = n i ⎜⎜ i ( n i + 2 λî ) ⎝ Si. 2. ⎞ ⎟⎟ , ⎠. Step 4: Calculate d 1 , d 2 , d 3 , and the value of L1 , L2 .. L1 =. − d2 +. − d 2 − d 22 − 4 d 1 d 3 d 22 − 4 d 1 d 3 , , L2 = 2d1 2d1 2. ⎛ a k ⎞ a 2 ⎛⎜ k − 1 ⎜ ⎟+ d1 = a ∑ ⎜1 + ∑ a 1 ⎟⎠ a * ⎜⎝ i = 1 i =1 ⎝. ai + ak ai. ak ⎞ ⎟ , ⎟ ⎠. ak ab ⎛⎜ k − 1 + ∑ a1 a * ⎜⎝ i = 1. ai + ak ai. a k ⎞⎛ k −1 ⎟⎜ ∑ ⎟⎜ i =1 ⎠⎝. k −1. k −1. d 2 = b∑. 1+. i =1. b 2 ⎛ k −1 ⎜∑ d3 = 4 a * ⎜⎝ i = 1. ak a1. ⎞ ⎟, ⎟ ⎠. 2. ak ai. ⎞ ⎟ − ln( p * 2 k − 1 2 a * ), ⎟ ⎠. Step 5: Calculate the value of w. 12.

(22) w = min{w1 , w2 }, if w1 > 1 and w2 > 1 w = w1 , if w1 > 1 and w2 ≤ 1 , w = w2 , if w2 > 1 and w1 ≤ 1 ;. ⎧⎪ 1 ⎛⎜ 1 1 ⎞⎟ νˆ[ k ] ⎫⎪ w1 = exp ⎨− 2 L1 + − ⎬ νˆ[1] ⎜⎝ νˆ[1] νˆ[ k ] ⎟⎠ νˆ[1] ⎪⎭ ⎪⎩ . and ⎧⎪ 1 ⎛⎜ 1 1 ⎞⎟ νˆ[ k ] ⎫⎪ + − w2 = exp ⎨− 2 L2 ⎬ νˆ[1] ⎜⎝ νˆ[1] νˆ[ k ] ⎟⎠ νˆ[1] ⎪⎭ ⎪⎩. 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 π 1 is more capable. If γˆ22 ≤ w × γˆ12 and γˆ12 > w × γˆ22 then we conclude that the process of π 2 is more capable. If γˆ12 ≤ w × γˆ22 and γˆ22 ≤ w × γˆ12 , we doesn’t have enough information to make supplier selection.. 4. Accuracy Analysis In this case, we want to distinguish which supplier has better process capability by the index C pu and C pl , 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 γ 2 , 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. 13.

(23) 4.1 Selection Power Analysis for C pu and C pl 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 C pu , C pu 0 . In a purchasing contract, a minimum value of the PCI is usually specified. Montgomery (2001) recommended the minimum quality requirements of C pu and C pl 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 = C pu 2 − C pu1 , then we can know how many sample sizes we should sample with determined power by the selection method. If Cˆ pu1 < Cˆ pu 2 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 H 0 : C pu1 ≥ C pu 2 , 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 C pu = 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 H 0 : C pu1 ≥ C pu 2 when C pu1 < C pu 2 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 C pu 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 C pu , the selection power increases, the required sample size increases. It’s only because when we want this selection analysis more 14.

(24) 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 C pu values, H 0 : C pu1 ≥ C pu 2 versus H 1 : C pu1 < C pu 2 If the test rejects the null hypothesis H 0 : C pu1 ≥ C pu 2 , 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 C pu 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 Cˆ pu1 and Cˆ pu 2 , 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 C pu 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 H 0 : C pu1 + q ≥ C pu 2 (the new supplier is not as capable as the present supplier with a magnitude, q ) versus H 1 : C pu1 + q < C pu 2 (the new supplier is more capable than the present supplier with a magnitude, q ). By comparing these test statistics Cˆ pu1 , Cˆ pu 2 , and the selection value A & c based on the test statistic and the required sample sizes. If the test apply to reject H 0 ( Cˆ pu1 + q < Cˆ pu 2 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 C pu 2 must be greater than the preset capability requirement, and C pu 2 = C pu1 + q , where q = max{ q′ | test rejects 15.

(25) C pu1 + q ≥ C pu 2 }. Then we decide to switch the present supplier to avoid waste such a huge exchanging cost.. 4.2 Selection Power Analysis for C pm 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 C pm values for two candidate processes, C pm 0 , and the minimal difference δ = C pm 2 − C pm1 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 π 2 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 H 0 : C pm1 ≥ C pm 2 , while actually C pm1 ≥ C pm 2 is true, using simulation technique. Tables 1-4 summarize the sample size required for various capability requirements C pm = 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 C pm 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 C pm , 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 C pm , 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.. 16.

(26) 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 C pm values, H 0 : C pm1 ≥ C pm 2 versus H 1 : C pm1 < C pm 2 . If the test rejects the null hypothesis H 0 : C pm1 ≥ C pm 2 , 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 C pm 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 γî2 , 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 C pm values, H 0 : C pm1 + q ≥ C pm 2 , versus H 2 : C pm1 + q < C pm 2 , where q > 0 is a specified constant. If the test rejects the null hypothesis H 0 : C pm1 + q ≥ C pm 2 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 γî2 , 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 C pm1 must be greater than the preset capability requirement, and C pm 2 = C pm1 + q , where q = max{ q′ | test rejects C pm1 + q′ ≥ C pm 2 }. 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.. 17.

(27) 5. Example 5.1 Application Example TFT-LCD LCD (liquid crystal display) is the technology used for displays in notebook and other smaller computers. Like light-emitting diode (LED) and gas-plasma technologies, LCDs allow displays to be thinner than cathode ray tube (CRT) technology. LCDs consume much less power than LED and gas-display displays because they work on the principle of blocking light rather than emitting it. To achieve the color on a pixel in an LCD panel, a current is applied to the crystals at that pixel to change the state of the crystals. Response times refer to the amount of time it takes for the crystals in the panel to move from an on to off state. A rising response time refers to the amount of time it takes to turn on the crystals and the falling time is the amount of time it takes for the crystals to move from an on to off state. Rising times tend to be very fast on LCDs, but the falling time tends to be much slower. This tends to cause a slight ghosting effect on bright moving images on black backgrounds. Simply to say, it’s the time takes for pixels to come up (become lit) and come down (become dark). The lower the response time, the less of a ghosting effect there will be on the screen. The electronic field effect of the liquid crystal is displayed in Figure 1, when the electronic field which between the electrode started to driving, it will attract to the electronic field works to make the liquid crystal turn its direction. And the optics effects will be produced. The picture tube theorem is showed in the Figure 2, An LCD is made with either a passive matrix or an active matrix display grid. The active matrix LCD is also known as a thin film transistor (TFT) display. The passive matrix LCD has a grid of conductors with pixels located at each intersection in the grid. A current is sent across two conductors on the grid to control the light for any pixel. An active matrix has a transistor located at each pixel intersection, requiring less current to control the luminance of a pixel.The current in an active matrix display can be switched on and off more frequently, improving the screen refresh time. Electrode. Electrode. LC. LC. Electrode. Turn off the electronic field. Electrode. Turn on the electronic field. 18.

(28) Figure 1. The electronic field effect of the liquid crystal. LC. Back Light. Vertical Polarizer Plate. Electrode. Electrode. Horizontal Polarizer Plate. Glass. LC. Back Light. Vertical Polarizer Plate. Electrode. Electrode. Horizontal Polarizer Plate. Glass. Figure 2. The picture tube theorem To illustrate which has better process capability between the two suppliers, we presented a case study on TFT-LCD manufacturing processes, which located on the Science-Based Industrial Park in Taiwan. These factories manufacture carious types of the LCD. For a particular model of the TFT-LCD investigated, the upper specification limit, USL of the response time is set to 20ms（ms， milliseconds）. If the characteristic data does not fall under the USL , the performance of the TFT-LCD will be discounted. We will use the software “LaCie calibration probe” to do the variable set of the LCD, then calculating the time takes for pixels to come up and come down.. 5.1.1 Data Analysis and Supplier Selection Before doing the data analysis, we set two factors first, (1) the minimum of C pu (2) the minimal difference of C pu between these two suppliers, δ = C pu 2 − C pu1 , then we can know how many sample sizes we should sample with determined power by the selection method. In this example, we set the minimum of C pu =1.00 and the minimal difference of C pu between these two suppliers, δ = 0.25 , the determined selection power = 0.95, then we can know we have to take 257 samples by checking Table 1. Then we present the data drew from these two suppliers in Table. In order to affirm these data as normal distributed, we show the distribution of these data in Figure 3-4. And we set these data to be a histogram in Figure 5-6.. 19.

(29) 20.0 19.5. 19.5. data2. data1. 19.0. 19.0. 18.5. 18.5. 18.0 18.0. -3. -2. -1. 0 1 Normal Distribution. 2. 3. Figure 3. Normal probability plot for response time data of Supplier I.. -3. -1. 0 1 Normal Distribution. 2. 3. Figure 4. Normal probability plot for response time data of Supplier II.. 1.2. 1.2. 0.8. 0.8. 0.4. 0.4. 0.0. -2. 0.0 18.118.218.4 18.518.718.819.0 19.119.319.419.6 19.719.8 data1. 18.1 18.2 18.4 18.5 18.7 18.8 18.9 19.1 19.2 19.4 19.5 19.7 19.8 data2. Figure 5. Histogram for supplier I.. Figure 6. Histogram for supplier II.. 5.1.2 Phase I－Supplier Selection We will test H 0 : C pu1 ≥ C pu 2 versus H 1 : C pu1 < C pu 2 by comparing these test statistics Cˆ pu1 , Cˆ pu 2 , and the selection value A & c based on the test statistic and the required sample sizes. If Cˆ pu1 < Cˆ pu 2 and A < c then we conclude that the process capability of the new supplier better than that of the present supplier. The calculated sample statistics for two suppliers are summarized in Table1. Table 1. The calculated sample statistics for two suppliers.(Cpu) Population. X. S. Cˆ pu. I. 19.00094 0.3072499 1.083872. II. 18.97955 0.2724119 1.248655. 20.

(30) Based on the selection method, the values Cˆ pu1 = 1.083872 and Cˆ pu 2 = 1.248655 . In this case one only need to compare the test statistic Cˆ pu1 and Cˆ pu 2 , by A = 0.1102599 and c = 0.2585227 , the outcome presents Cˆ pu1 < Cˆ pu 2 and A < c , then we conclude that the process of this new supplier is capable.. 5.1.3 Phase II－Magnitude Outperformed Detection To realize the lower bound value of the magnitude, h, we will test H 0 : C pu1 + h ≥ C pu 2 versus H 1 : C pu1 + h < C pu 2 . By comparing these test statistics Cˆ , Cˆ , and the selection value A & c based on the test statistic and the pu1. pu 2. required sample sizes. From the estimation of Phase I, we list the obtained selection values A and c and the decision based on the selection procedure for h = 0.01, 0.03(0.001)0.035 in Table 2. Therefore, from the analysis of magnitude outperformed detection based on sample statistics, the magnitude of the difference between the two suppliers is h = 0.034. By the way, we can conclude that the new supplier is more capable than the present supplier at least a magnitude, h=0.034. Table 2. Magnitude outperformed detection of selection procedure. (Cpu) 1.118872 Cˆ 1.093872 1.113872 1.116872 1.117872 pu1. Cˆ pu 2. 1.248655. 1.248655. 1.248655. 1.248655. 1.248655. h. 0.01. 0.03. 0.033. 0.034. 0.035. A. 0.1449597 0.2361393 0.2523842 0.2579458. 0.2635801. c. 0.2585227 0.2585227 0.2585227 0.2585227. 0.2585227. Decision. Reject Ho Reject Ho Reject Ho Reject Ho Don’t Reject Ho. 5.2 Application Example Automobile Windows Up to now, the number of registered vehicle (including the intercity bus, truck, car and wagon) has tended to 18,215,069. Thus, with the growing number of vehicle, there is a need for automobile windows. For the safety, the automobile window always be the sandwich glass. The sandwich glass inserted with the special membrane (PVB film) between two pieces of tempered glass was dealt with by the high pressure of high temperature. The structure of the sandwich was displayed in Figure 7. After the glass is broken, chip can still be glued together, it is a kind of safe type glass. The sandwich glass can absorb the ultraviolet ray in the sunlight effectively; protect the personal safety in maximum.. 21.

(31) Tempered glass. PVB film. Figure 7. The structure of the sandwich glass. Tempered glass is commonly used with various applications in our real life. Especially, it is used in automobile's side windows, front windows ( displayed in Figure 8, 9 ). There are some characteristic of the tempered glass: (1) the strength against still-mode impact resistance is three to five times over that of regular glass. (2) Resilient to sudden temperature drop with its heat endurance much superior than common glass. (3) When broken, its fragments differ from usual pointy shards but rather in curd configuration, which greatly reduces the impact of cuts. Based on these characteristics of the tempered glass, we have to temper the glass to avoid the dangers coming with the broken glass in some special occasions, like the automobile window, the microwave oven and so on. It is a high-impact glass with its broken fragments in curds featuring an optimal performance in safety.. Figure 8. Automobile’s front window. Figure 9. Automobile's side window. Tempered glass is derived by heating the raw glass sheeting to a temperature of near-melting point, with an evenly distributed cool air for rapid cooling to form a surface hardening process in order to overcome physical expandability found in glass. The outer surface is quickly cooled for a reinforced characteristic, which is known as tempered glass. In order to keep the high optical quality, we have to ask the thickness of the tempered glass at least 0.5mm. , no any distortion, wave and other defects on the surface due to it’s treatment temperature lower then the thermo tempered glass, so easy to laminating fabrication. Too thin tempered glass will result in danger when it broken (more break pattern) and increasing the difficult when it be processed and can’t suffer the outside force impact To illustrate which has better capability between the two suppliers, we 22.

(32) present a case study on the automobile window manufacturing process, which located on the Tafa industrial region in Taiwan. These factories manufacture various types of the tempered glass. For the particular model of the automobile window investigated, the lower specification limit, LSL of an automobile side window’s thickness is set to be 0.5mm. And we use thickness gauge to inspect the inspection for thickness. If the characteristic data does not fall over the tolerance LSL , the safety of the automobile window will be discounted.. 5.2.1 Data Analysis and Supplier Selection Before doing the data analysis, we set two factors first, (1) the minimum of C pl (2) the minimal difference of C pl between these two suppliers, δ = C pl 2 − C pl1 , then we can know how many sample sizes we should sample with determined power by the selection method. In this example, we set the minimum of C pl =1.00 and the minimal difference of C pl between these two suppliers, δ = 0.25 , the determined selection power = 0.95, then we can know we have to take 257 samples by checking Table 1. Then we present the data drew from these two suppliers in Table. In order to affirm these data as normal distributed, we show the distribution of these data in Figure 10-11. And we set these data to be a histogram in Figure 12-13. 0.60 0.58. 0.58. 0.56. data2. data1. 0.56. 0.54. 0.54 0.52 0.52 0.50 0.50 -3. -2. -1. 0 1 Normal Distribution. 2. 3. Figure 10. Normal probability plot for thickness data of Supplier I.. -3. -2. -1. 0 1 Normal Distribution. 2. 3. Figure 11. Normal probability plot for thickness data of Supplier II.. 23.

(33) 30 20. 15. 20. 10 10 5. 0. 0 0.51 0.52 0.52 0.53 0.54 0.55 0.55 0.56 0.57 0.58 0.58 0.59 0.60. 0.51 0.52 0.52 0.53 0.54 0.55 0.55 0.56 0.57 0.58 0.58 0.59 0.60. data1. data2. Figure 12. Histogram for supplier I.. Figure 13. Histogram for supplier II.. 5.2.2 Phase I－Supplier Selection We will test H 0 : C pl1 ≥ C pl 2 versus H 1 : C pl1 < C pl 2 by comparing these test statistics Cˆ pl1 , Cˆ pl 2 , and the selection value A & c based on the test statistic and the required sample sizes. If Cˆ pl1 < Cˆ pl 2 and A < c then we conclude that the process capability of the new supplier better than that of the present supplier. The calculated sample statistics for two suppliers are summarized in Table 3. Table 3. The calculated sample statistics for two suppliers.(Cpl) X Population S Cˆ pl. I. 0.5487296 0.01592503 1.019979. II. 0.548967 0.01335757 1.221954. Based on the selection method, the values Cˆ pl1 = 1.019979 and Cˆ pl 2 = 1.221954 . In this case one only need to compare the test statistic Cˆ pl1 and Cˆ pl 2 , by A = 0.02891871 and c = 0.2585227 , the outcome presents Cˆ pl1 < Cˆ pl 2 and A < c , then we conclude that the process of this new supplier is capable.. 5.2.3 Phase II－Magnitude Outperformed Detection To realize the lower bound value of the magnitude, q , we will test H 0 : C pl1 + q ≥ C pl 2 versus H 1 : C pl1 + q < C pl 2 . By comparing these test statistics Cˆ pl1 , Cˆ pl 2 , and the selection value A & c based on the test statistic and the required sample sizes. From the estimation of Phase I, we list the obtained selection values A and c and the decision based on the selection procedure for h = 0.01, 0.05, 0.07(0.001)0.074 in Table 4. 24.

(34) Therefore, from the analysis of magnitude outperformed detection based on sample statistics, the magnitude of the difference between the two suppliers is q = 0.034. By the way, we can conclude that the new supplier is more capable than the present supplier at least a magnitude, q =0.074. Table 4. Magnitude outperformed detection of selection procedure. (Cpl). Cˆ pl1 Cˆ. 1.029979. 1.069979. 1.089979. 1.090979. 1.091979. 1.092979. 1.093979. 1.221954. 1.221954. 1.221954. 1.221954. 1.221954. 1.221954. 1.221954. q. 0.01. 0.05. 0.07. 0.071. 0.072. 0.073. 0.074. pl 2. A. 0.04169824 0.1447203 0.2377967 0.2432623 0.2488044 0.2544226. 0.2601165. c. 0.2585227 0.2585227 0.2585227 0.2585227 0.2585227 0.2585227. 0.2585227. Decision Reject Ho Reject Ho Reject Ho Reject Ho Reject Ho Reject Ho Don’t Reject Ho. 5.3 Application Example STN-LCD Liquid crystals have been employed for display applications with various configurations. Most of the displays produced recently involve the use of either Twisted Nematic (TN) or Super Twisted Nematic (STN) liquid crystals, the technology of the STN display was introduced recently to improve the performance of LCD without using the TFT. A larger twist angle results in a significantly larger electro-optical distortion. This leads to a substantial improvement in the contract and viewing angles over TN displays. The STN-LCD products are popularly used in making the PDAs, notebook personal computers, word processors, and other peripherals. A typical assembly drawing for the STN-LCD product is depicted in Figure 14 and the custom glass and modules of the STN-LCD product is displayed in Figure 15.. Figure 12. An assembly drawing for the STN-LCD product.. Figure 2. The custom glass and modules of the STN-LCD product.. 25.