ESTIMATED INCAPABILITY INDEX: RELIABILITY AND
DECISION MAKING WITH SAMPLE INFORMATION
W. L. PEARN1AND G. H. LIN2∗
1Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, Republic of China 2Department of Communication Engineering, National Penghu Institute of Technology, Taiwan, Republic of China
SUMMARY
The process incapability index Cpp, which provides an uncontaminated separation between information concerning the process precision and process accuracy, has been proposed to measure process performance for industry applications. In this paper, we investigate the reliability of the natural estimator computationally, based on theα-level confidence relative error for various sample sizes. We also develop a decision-making procedure for judging if the process satisfies the preset quality requirement. The investigation is useful to the practitioners in determining the sample sizes required in their applications for the decisions reliable to the desired level. Copyright 2002 John Wiley & Sons, Ltd.
KEY WORDS: process incapability index; non-central chi-square distribution;α-level confidence relative error 1. THE INCAPABILITY INDEXCpp
A process incapability indexCpp, providing numerical measures on process performance, has been proposed by Greenwich and Jahr-Schaffrath [1]. The indexCpp is a simple transformation ofCpm∗ , a general form of the capability indexCpmconsidered by Chan et al. [2], which provides an uncontaminated separation between information concerning the process precision and the process accuracy. The index Cpp is defined as the following: Cpp= σ D 2 + µ − T D 2
where µ is the process mean, σ is the process standard deviation, D = min{(USL − T )/3, (T −
LSL)/3}, USL and LSL are the upper and the lower
specification limits, andT is the target value. If we define Cip = (σ/D)2 and Cia = [(µ − T )/D]2, then Cpp can be expressed as Cpp = Cip + Cia. The indexCipmeasures the process variation relative to the specification tolerance, which reflects process precision. The indexCiameasures the relative process departure, which reflects process accuracy. We note that the mathematical relationships Cip = 1/(Cp)2 and Cia = 9(1 − Ca)2 can be established, where
Cp and Ca are two basic process capability indices ∗Correspondence to: G. H. Lin, Department of Communication Engineering, National Penghu Institute of Technology, Taiwan 880, Republic of China.
considered by Kane [3] and Pearn et al. [4]. Thus,
Cpp= Cip+ Cia= 1/(Cp)2+ 9(1 − Ca)2. 2. ESTIMATION OFCpp
To estimate the process incapability (a combined mea-sure of process imprecision and process inaccuracy), we consider the natural estimator ˆCpp defined as the following, where ¯X =ni=1Xi/n, which also can be written as a function of Cip ˆCpp = 1 n n i=1 (Xi− ¯X)2 D2 + ( ¯X − T )2 D2 =Cip n n ˆCpp Cip = Cip n n i=1 (Xi − T )2 σ2
If the process characteristic is normally distributed, then the estimator ˆCppis distributed as[Cip/n]χn2(δ), whereχn2(δ) is a non-central chi-square distribution with n degrees of freedom and non-centrality parameterδ = n(µ − T )2/σ2= nC
ia/Cip. Therefore, the probability density function (PDF) of ˆCpp can be easily derived and expressed as the following, fory > 0, which can also be rewritten as a function ofCipand
Cia h(y)=∞ k=0 [(ny)/(2C ip)]k+n/2exp[−(ny)/(2Cip)] y(k + n/2) ×(δ/2)(k + 1)kexp(−δ/2) Received 15 February 2001
5 4 2 3 1 5 4 3 2 1 (a) (b) 5 4 3 2 1 5 4
3
2 1 (c) (d)
Figure 1. PDF plot of ˆCpp for (a)Cia = 5.06, Cip = 1.00, 0.56, 0.44, 0.36, 0.25, (b) Cia = 2.25, Cip = 1.00, 0.56, 0.44, 0.36, 0.25,
(c)Cia= 0.56, Cip= 1.00, 0.56, 0.44, 0.36, 0.25 and (d) Cia= 0.00, Cip= 1.00, 0.56, 0.44, 0.36, 0.25 =∞ k=0 [(ny)/(2C ip)]k+n/2exp[−(ny)/(2Cip)] y(k + n/2) ×[nCia/(2Cip)]kexp[−nCia/(2Cip)] (k + 1)
In Figures 1(a)–(d), we plot the PDF of ˆCpp for
n = 20, Cia= 5.06, 2.25, 0.56 and 0.00, respectively, with commonly used values ofCip = 1.00 (curve 1), 0.56 (curve 2), 0.44 (curve 3), 0.36 (curve 4), and 0.25 (curve 5). As can be seen from the figures, for a fixed value ofCiathe variance of ˆCpp decreases as the value of Cip increases and for a fixed value of
Cip the variance of ˆCpp decreases as the value ofCia decreases.
We note that those Cip values are equivalent to the widely used capability requirements,Cp = 1.00, 1.33, 1.50, 1.67 and 2.00. For industry applications, a process is called ‘incapable’ ifCip> 1.00 (equivalent to Cp < 1.00) and is called ‘capable’ if 0.56 <
Cip ≤ 1.00 (equivalent to 1.00 ≤ Cp < 1.33). A process is called ‘satisfactory’ if 0.44 < Cip ≤ 0.56 (equivalent to 1.33 ≤ Cp < 1.50), called ‘good’ if 0.36 < Cip≤ 0.44 (equivalent to 1.50 ≤ Cp< 1.67), called ‘excellent’ if 0.25 < Cip≤ 0.36 (equivalent to 1.67 ≤ Cp< 2.00) and is called ‘super’ if Cip≤ 0.25 (equivalent to Cp 2.00). On the other hand, the values ofCiaare equivalent toCa = 0.25, 0.50, 0.75 and 1.00 respectively. We note that if the process is perfectly centered, thenCia = 0.00 (or equivalently
Ca= 1.00) (see Pearn et al. [4]).
If the process characteristic follows the normal distribution, Pearn and Lin [5] showed that ˆCppis the MLE, and the UMVUE of Cpp. They also showed that ˆCpp is consistent,√n( ˆCpp − Cpp) converges to
N(0, 2CipCia + 2CipCpp) in distribution and ˆCpp is asymptotically efficient. Thus, the estimator ˆCpp has all the desired statistical properties and using ˆCpp to estimate process incapability would be reasonable.
OPTIONS REPLACE PAGESIZE = 58 LINESIZE = 78 NODATA; DATA CRE;
CIA = ;
CIP1 = ; CIP2 = ; CIP3 = ; CIP4 = ; CIP5 = ; DO N = 10 TO 200 BY 10;
D1 = N*CIA/CIP1; D2 = N*CIA/CIP2;
D3 = N*CIA/CIP3; D4 = N*CIA/CIP4; D5 = N*CIA/CIP5; A1 = CIP/(CIA + CIP1)*(1/N)*CINV(0.025, N, D1); B1 = CIP/(CIA + CIP1)*(1/N)*CINV(0.975, N, D1); C1 = MAX(ABS(A1 - 1), ABS(B1 - 1)); A2 = CIP/(CIA + CIP2)*(1/N)*CINV(0.025, N, D2); B2 = CIP/(CIA + CIP2)*(1/N)*CINV(0.975, N, D2); C2 = MAX(ABS(A2 - 1), ABS(B2 - 1)); A3 = CIP/(CIA + CIP3)*(1/N)*CINV(0.025, N, D3); B3 = CIP/(CIA + CIP3)*(1/N)*CINV(0.975, N, D3); C3 = MAX(ABS(A3 - 1), ABS(B3 - 1)); A4 = CIP/(CIA + CIP4)*(1/N)*CINV(0.025, N, D4); B4 = CIP/(CIA + CIP4)*(1/N)*CINV(0.975, N, D4); C4 = MAX(ABS(A4 - 1), ABS(B4 - 1)); A5 = CIP/(CIA + CIP5)*(1/N)*CINV(0.025, N, D5); B5 = CIP/(CIA + CIP5)*(1/N)*CINV(0.975, N, D5); C5 = MAX(ABS(A5 - 1), ABS(B5 - 1)); OUTPUT; END; FORMAT C1 C2 C3 C4 C5 6.4; PROC PRINT DATA = CRE; VAR N C1 C2 C3 C4 C5; RUN;
Figure 2.
3. RELIABILITY ANALYSIS
To evaluate the reliability of the estimator ˆCpp, we consider the measurement criteria called the
α-level confidence relative error, which is defined
as CREα( ˆCpp) = maxα{| ˆCpp − Cpp|/Cpp} = maxα|( ˆCpp/Cpp) − 1| = maxα{|Lα/2− 1|, |U1−α/2− 1|}, where Lα/2 and U1−α/2 satisfy the probability equation P{Lα/2 ≤ ˆCpp/Cpp ≤ U1−α/2} = 1 − α, which can be obtained as follows
P{Lα/2≤ ˆCpp/Cpp ≤ U1−α/2} = 1 − α = P Lα/2≤ C ip n n ˆC pp Cip (Cip+ Cia)−1 ≤ U1−α/2 = P Lα/2≤ C ip n(Cip+ Cia) χ2 n(δ) ≤ U1−α/2 where once again χn2(δ) is a non-central chi-square distribution with n degrees of freedom and non-centrality parameterδ = n(µ − T )2/σ2 = nCia/Cip.
Therefore, the bounds U1−α/2, and Lα/2, may be obtained as: U1−α/2= C ip n(Cip+ Cia) χ2 n,1−α/2(δ) Lα/2= C ip n(Cip+ Cia) χ2 n,α/2(δ)
Thus, CREα( ˆCpp) = c presents that with at least
(1−α) confidence the relative deviation (relative error)
of ˆCpp, maxα{| ˆCpp−Cpp|/Cpp} = maxα{|( ˆCpp/Cpp)− 1|} = maxα{|U1−α/2 − 1|, |Lα/2 − 1|}, will be no greater than c. The (1 − α)% confidence relative error of ˆCpp can be calculated using the SAS computer program (see Figure2). Tables1–3display CREα( ˆCpp) values with α = 0.05, 0.025 and 0.01, for
Cip= 1.00, 0.56, 0.44, 0.36 and 0.25, and Cia= 5.06, 2.25, 0.56 and 0.00, withn = 10(10)200.
For example, forCip = 0.25 and Cia = 5.06, with
α = 0.05 and n = 150, we have CREα( ˆCpp) = 0.0695, which indicates that with at least 95% confidence the obtained ˆCpp value will be within 6.95% of the trueCpp value. Thus, for the described
Table 1. CREα( ˆCpp) for (a) Cia= 5.06, α = 0.05, various Cip,n = 10(10)200; (b) Cia= 2.25; (c) Cia= 0.56; and (d) Cia= 0.00 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 (a) (c) 10 0.5258 0.4091 0.3669 0.3322 0.2787 10 0.9591 0.8765 0.8331 0.7903 0.7100 20 0.3630 0.2838 0.2550 0.2312 0.1944 20 0.6507 0.5968 0.5682 0.5399 0.4867 30 0.2932 0.2297 0.2066 0.1874 0.1578 30 0.5209 0.4786 0.4561 0.4338 0.3917 40 0.2522 0.1979 0.1781 0.1616 0.1361 40 0.4457 0.4100 0.3909 0.3720 0.3362 50 0.2246 0.1764 0.1587 0.1441 0.1215 50 0.3953 0.3639 0.3472 0.3305 0.2989 60 0.2043 0.1606 0.1446 0.1313 0.1107 60 0.3586 0.3303 0.3152 0.3002 0.2716 70 0.1887 0.1483 0.1336 0.1213 0.1023 70 0.3303 0.3045 0.2906 0.2768 0.2505 80 0.1761 0.1385 0.1248 0.1133 0.0956 80 0.3078 0.2838 0.2709 0.2581 0.2337 90 0.1657 0.1304 0.1175 0.1067 0.0900 90 0.2892 0.2667 0.2547 0.2427 0.2198 100 0.1570 0.1236 0.1113 0.1012 0.0853 100 0.2736 0.2524 0.2410 0.2297 0.2081 110 0.1495 0.1177 0.1061 0.0964 0.0813 110 0.2602 0.2401 0.2293 0.2186 0.1980 120 0.1430 0.1126 0.1015 0.0922 0.0778 120 0.2486 0.2294 0.2191 0.2089 0.1893 130 0.1372 0.1081 0.0974 0.0885 0.0747 130 0.2383 0.2201 0.2102 0.2004 0.1816 140 0.1321 0.1041 0.0938 0.0853 0.0720 140 0.2293 0.2117 0.2023 0.1928 0.1748 150 0.1275 0.1005 0.0906 0.0823 0.0695 150 0.2212 0.2043 0.1952 0.1861 0.1687 160 0.1234 0.0972 0.0876 0.0797 0.0673 160 0.2138 0.1975 0.1887 0.1799 0.1632 170 0.1196 0.0943 0.0850 0.0773 0.0652 170 0.2072 0.1914 0.1829 0.1744 0.1582 180 0.1162 0.0916 0.0826 0.0751 0.0634 180 0.2011 0.1858 0.1776 0.1693 0.1536 190 0.1130 0.0891 0.0803 0.0730 0.0617 190 0.1955 0.1807 0.1727 0.1647 0.1494 200 0.1101 0.0868 0.0783 0.0712 0.0601 200 0.1904 0.1760 0.1682 0.1604 0.1455 (b) (d) 10 0.7100 0.5781 0.5258 0.4810 0.4091 10 1.0483 1.0483 1.0483 1.0483 1.0483 20 0.4867 0.3983 0.3630 0.3327 0.2838 20 0.7085 0.7085 0.7085 0.7085 0.7085 30 0.3917 0.3214 0.2932 0.2689 0.2297 30 0.5660 0.5660 0.5660 0.5660 0.5660 40 0.3362 0.2763 0.2522 0.2315 0.1979 40 0.4836 0.4836 0.4836 0.4836 0.4836 50 0.2989 0.2459 0.2246 0.2062 0.1764 50 0.4284 0.4284 0.4284 0.4284 0.4284 60 0.2716 0.2236 0.2043 0.1876 0.1606 60 0.3883 0.3883 0.3883 0.3883 0.3883 70 0.2505 0.2065 0.1887 0.1733 0.1483 70 0.3575 0.3575 0.3575 0.3575 0.3575 80 0.2337 0.1927 0.1761 0.1618 0.1385 80 0.3329 0.3329 0.3329 0.3329 0.3329 90 0.2198 0.1813 0.1657 0.1523 0.1304 90 0.3127 0.3127 0.3127 0.3127 0.3127 100 0.2081 0.1717 0.1570 0.1442 0.1236 100 0.2956 0.2956 0.2956 0.2956 0.2956 110 0.1980 0.1635 0.1495 0.1374 0.1177 110 0.2811 0.2811 0.2811 0.2811 0.2811 120 0.1893 0.1563 0.1430 0.1314 0.1126 120 0.2684 0.2684 0.2684 0.2684 0.2684 130 0.1816 0.1500 0.1372 0.1261 0.1081 130 0.2573 0.2573 0.2573 0.2573 0.2573 140 0.1748 0.1444 0.1321 0.1214 0.1041 140 0.2475 0.2475 0.2475 0.2475 0.2475 150 0.1687 0.1394 0.1275 0.1172 0.1005 150 0.2387 0.2387 0.2387 0.2387 0.2387 160 0.1632 0.1349 0.1234 0.1134 0.0972 160 0.2308 0.2308 0.2308 0.2308 0.2308 170 0.1582 0.1307 0.1196 0.1100 0.0943 170 0.2235 0.2235 0.2235 0.2235 0.2235 180 0.1536 0.1270 0.1162 0.1068 0.0916 180 0.2169 0.2169 0.2169 0.2169 0.2169 190 0.1494 0.1235 0.1130 0.1039 0.0891 190 0.2108 0.2108 0.2108 0.2108 0.2108 200 0.1455 0.1203 0.1101 0.1012 0.0868 200 0.2053 0.2053 0.2053 0.2053 0.2053
condition, if the calculated value ˆCpp= 1.27, then the true value ofCppwould be between 1.27/(1 + 6.95%) and 1.27/(1 − 6.95%) (with at least 95% confidence). We note that the α-level confidence relative error CREα( ˆCpp), which is obtained from the same approach as used for finding the confidence interval, provides the practitioners with more direct and easily understood information than the confidence interval approach regarding the accuracy of their estimations
and suggests a clear range on the true value of the process performance measure using the indexCpp.
4. A DECISION MAKING PROCEDURE Under the usual normality assumption,n ˆCpp/(Cpp −
Cia) is distributed as χn2(δ), a non-central chi-square distribution with n degrees of freedom and non-centrality parameter δ = n(µ − T )2/σ2 =
Table 2. CREα( ˆCpp) for (a) Cia= 5.06, α = 0.025, various Cip,n = 10(10)200; (b) Cia= 2.25; (c) Cia= 0.56; and (d) Cia= 0.00 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 (a) (c) 10 0.6138 0.4754 0.4258 0.3850 0.3223 10 1.1410 1.0382 0.9846 0.9323 0.8348 20 0.4214 0.3284 0.2947 0.2670 0.2241 20 0.7659 0.7001 0.6656 0.6316 0.5679 30 0.3395 0.2652 0.2383 0.2160 0.1816 30 0.6102 0.5591 0.5321 0.5055 0.4554 40 0.2916 0.2282 0.2052 0.1861 0.1566 40 0.5205 0.4777 0.4549 0.4325 0.3901 50 0.2593 0.2032 0.1828 0.1658 0.1396 50 0.4607 0.4232 0.4033 0.3836 0.3463 60 0.2357 0.1849 0.1663 0.1510 0.1272 60 0.4172 0.3836 0.3657 0.3479 0.3143 70 0.2175 0.1707 0.1536 0.1395 0.1175 70 0.3839 0.3532 0.3368 0.3205 0.2897 80 0.2029 0.1594 0.1434 0.1303 0.1098 80 0.3573 0.3289 0.3137 0.2986 0.2701 90 0.1909 0.1500 0.1350 0.1226 0.1034 90 0.3354 0.3089 0.2947 0.2806 0.2538 100 0.1808 0.1421 0.1279 0.1162 0.0980 100 0.3171 0.2921 0.2788 0.2655 0.2402 110 0.1721 0.1353 0.1218 0.1107 0.0933 110 0.3014 0.2778 0.2651 0.2525 0.2285 120 0.1645 0.1294 0.1165 0.1059 0.0893 120 0.2878 0.2653 0.2532 0.2412 0.2184 130 0.1579 0.1242 0.1119 0.1016 0.0857 130 0.2758 0.2543 0.2428 0.2313 0.2095 140 0.1519 0.1196 0.1077 0.0979 0.0825 140 0.2652 0.2446 0.2335 0.2225 0.2015 150 0.1466 0.1154 0.1040 0.0945 0.0797 150 0.2557 0.2359 0.2253 0.2146 0.1944 160 0.1494 0.1117 0.1006 0.0914 0.0771 160 0.2472 0.2281 0.2178 0.2075 0.1880 170 0.1375 0.1083 0.0976 0.0887 0.0748 170 0.2394 0.2209 0.2110 0.2011 0.1822 180 0.1335 0.1052 0.0948 0.0861 0.0727 180 0.2323 0.2144 0.2048 0.1952 0.1769 190 0.1299 0.1023 0.0922 0.0838 0.0707 190 0.2258 0.2085 0.1991 0.1898 0.1720 200 0.1265 0.0997 0.0898 0.0816 0.0689 200 0.2198 0.2030 0.1939 0.1848 0.1675 (b) (d) 10 0.8348 0.6762 0.6138 0.5605 0.4754 10 1.2558 1.2558 1.2558 1.2558 1.2558 20 0.5679 0.4631 0.4214 0.3857 0.3284 20 0.8381 0.8381 0.8381 0.8381 0.8381 30 0.4554 0.3725 0.3395 0.3110 0.2652 30 0.6656 0.6656 0.6656 0.6656 0.6656 40 0.3901 0.3197 0.2916 0.2673 0.2282 40 0.5666 0.5666 0.5666 0.5666 0.5666 50 0.3463 0.2842 0.2593 0.2379 0.2032 50 0.5008 0.5008 0.5008 0.5008 0.5008 60 0.3143 0.2583 0.2357 0.2163 0.1849 60 0.4531 0.4531 0.4531 0.4531 0.4531 70 0.2897 0.2382 0.2175 0.1996 0.1707 70 0.4165 0.4165 0.4165 0.4165 0.4165 80 0.2701 0.2222 0.2029 0.1863 0.1594 80 0.3874 0.3874 0.3874 0.3874 0.3874 90 0.2538 0.2090 0.1909 0.1753 0.1500 90 0.3634 0.3634 0.3634 0.3634 0.3634 100 0.2402 0.1979 0.1808 0.1660 0.1421 100 0.3434 0.3434 0.3434 0.3434 0.3434 110 0.2285 0.1883 0.1721 0.1580 0.1353 110 0.3263 0.3263 0.3263 0.3263 0.3263 120 0.2184 0.1800 0.1645 0.1511 0.1294 120 0.3114 0.3114 0.3114 0.3114 0.3114 130 0.2095 0.1727 0.1579 0.1450 0.1242 130 0.2984 0.2984 0.2984 0.2984 0.2984 140 0.2015 0.1662 0.1519 0.1396 0.1196 140 0.2869 0.2869 0.2869 0.2869 0.2869 150 0.1944 0.1604 0.1466 0.1347 0.1154 150 0.2765 0.2765 0.2765 0.2765 0.2765 160 0.1880 0.1552 0.1419 0.1304 0.1117 160 0.2672 0.2672 0.2672 0.2672 0.2672 170 0.1822 0.1504 0.1375 0.1264 0.1083 170 0.2587 0.2587 0.2587 0.2587 0.2587 180 0.1769 0.1460 0.1335 0.1227 0.1052 180 0.2510 0.2510 0.2510 0.2510 0.2510 190 0.1720 0.1420 0.1299 0.1194 0.1023 190 0.2439 0.2439 0.2439 0.2439 0.2439 200 0.1675 0.1383 0.1265 0.1163 0.0997 200 0.2374 0.2374 0.2374 0.2374 0.2374
nCia/(Cpp− Cia). Let cobe a statistic calculated from the sample data satisfying P{Cpp ≤ co} = 1−α. Then,
cois a 100(1−α)% upper confidence limit for Cpp. We note that P{Cpp ≤ co} = P{Cpp− Cia≤ co− Cia} = P{1/(Cpp− Cia) ≥ 1/(co− Cia)} = P{n ˆCpp/(Cpp− Cia) ≥ n ˆCpp/(co− Cia)} = P{χ2 n(δ) ≥ n ˆCpp/(co− Cia)} = 1 − α Therefore, n ˆCpp/(co − Cia) = χn,α2 (δ), where χ2
n,α(δ) is the (lower) αth percentile of the χn2(δ) distribution. A 100(1 − α)% upper confidence limit on Cpp can be written in terms of ˆCpp as
Table 3. CREα( ˆCpp) for (a) Cia= 5.06, α = 0.01, various Cip,n = 10(10)200; (b) Cia= 2.25; (c) Cia= 0.56; and (d) Cia= 0.00 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 (a) (c) 10 0.7217 0.5564 0.4974 0.4491 0.3751 10 1.3693 1.2399 1.1734 1.1087 0.9892 20 0.4924 0.3823 0.3427 0.3101 0.2599 20 0.9087 0.8278 0.7856 0.7444 0.6674 30 0.3955 0.3081 0.2765 0.2505 0.2103 30 0.7201 0.6579 0.6253 0.5933 0.5332 40 0.3391 0.2648 0.2378 0.2155 0.1811 40 0.6123 0.5604 0.5331 0.5062 0.4557 50 0.3013 0.2355 0.2116 0.1919 0.1614 50 0.5406 0.4955 0.4717 0.4482 0.4039 60 0.2736 0.2141 0.1925 0.1746 0.1469 60 0.4888 0.4485 0.4271 0.4060 0.3662 70 0.2523 0.1976 0.1777 0.1612 0.1357 70 0.4492 0.4124 0.3929 0.3736 0.3372 80 0.2352 0.1844 0.1659 0.1505 0.1267 80 0.4176 0.3837 0.3656 0.3478 0.3140 90 0.2212 0.1735 0.1561 0.1417 0.1193 90 0.3917 0.3601 0.3432 0.3266 0.2950 100 0.2094 0.1643 0.1478 0.1342 0.1131 100 0.3700 0.3403 0.3244 0.3087 0.2790 110 0.1992 0.1564 0.1407 0.1278 0.1077 110 0.3514 0.3233 0.3084 0.2935 0.2653 120 0.1904 0.1495 0.1346 0.1222 0.1030 120 0.3354 0.3087 0.2944 0.2803 0.2534 130 0.1827 0.1435 0.1292 0.1173 0.0989 130 0.3213 0.2958 0.2822 0.2686 0.2430 140 0.1758 0.1381 0.1244 0.1129 0.0952 140 0.3088 0.2844 0.2713 0.2583 0.2337 150 0.1696 0.1333 0.1200 0.1090 0.0919 150 0.2976 0.2741 0.2616 0.2491 0.2254 160 0.1640 0.1290 0.1161 0.1055 0.0889 160 0.2875 0.2649 0.2528 0.2408 0.2179 170 0.1590 0.1250 0.1126 0.1023 0.0862 170 0.2784 0.2566 0.2449 0.2332 0.2111 180 0.1544 0.1214 0.1093 0.0993 0.0838 180 0.2701 0.2489 0.2376 0.2263 0.2049 190 0.1501 0.1181 0.1064 0.0966 0.0815 190 0.2625 0.2420 0.2310 0.2200 0.1992 200 0.1462 0.1150 0.1036 0.0941 0.0794 200 0.2554 0.2355 0.2248 0.2142 0.1940 (b) (d) 10 0.9892 0.7968 0.7217 0.6578 0.5564 10 1.5188 1.5188 1.5188 1.5188 1.5188 20 0.6674 0.5420 0.4924 0.4501 0.3823 20 0.9999 0.9999 0.9999 0.9999 0.9999 30 0.5332 0.4346 0.3955 0.3620 0.3081 30 0.7891 0.7892 0.7891 0.7892 0.7892 40 0.4557 0.3723 0.3391 0.3106 0.2648 40 0.6692 0.6692 0.6692 0.6692 0.6692 50 0.4039 0.3305 0.3013 0.2761 0.2355 50 0.5898 0.5898 0.5898 0.5898 0.5898 60 0.3662 0.3001 0.2736 0.2508 0.2141 60 0.5325 0.5325 0.5325 0.5325 0.5325 70 0.3372 0.2766 0.2523 0.2314 0.1976 70 0.4887 0.4887 0.4887 0.4887 0.4887 80 0.3140 0.2578 0.2352 0.2158 0.1844 80 0.4540 0.4540 0.4540 0.4540 0.4540 90 0.2950 0.2423 0.2212 0.2030 0.1735 90 0.4256 0.4256 0.4256 0.4256 0.4256 100 0.2790 0.2293 0.2094 0.1921 0.1643 100 0.4017 0.4017 0.4017 0.4017 0.4017 110 0.2653 0.2182 0.1992 0.1829 0.1564 110 0.3814 0.3814 0.3814 0.3814 0.3814 120 0.2534 0.2085 0.1904 0.1748 0.1495 120 0.3637 0.3637 0.3637 0.3637 0.3637 130 0.2430 0.2000 0.1827 0.1677 0.1435 130 0.3483 0.3483 0.3483 0.3483 0.3483 140 0.2337 0.1924 0.1758 0.1614 0.1381 140 0.3346 0.3346 0.3346 0.3346 0.3346 150 0.2254 0.1856 0.1696 0.1558 0.1333 150 0.3224 0.3224 0.3224 0.3224 0.3224 160 0.2179 0.1795 0.1640 0.1507 0.1290 160 0.3114 0.3114 0.3114 0.3114 0.3114 170 0.2111 0.1740 0.1590 0.1460 0.1250 170 0.3014 0.3014 0.3014 0.3014 0.3014 180 0.2049 0.1689 0.1544 0.1418 0.1214 180 0.2923 0.2923 0.2923 0.2923 0.2923 190 0.1992 0.1642 0.1501 0.1379 0.1181 190 0.2840 0.2840 0.2840 0.2840 0.2840 200 0.1940 0.1599 0.1462 0.1343 0.1150 200 0.2763 0.2763 0.2763 0.2763 0.2763
unknown. In practice, we can use the UMVUE, ˜Cia= [( ¯X −T )/D]2−(Sn−1)2/(nD2), recommended by Pearn and Lin [5] to estimate Cia (see also Greenwich and Jahr-Schaffrath [1]) where Sn−1 = [ni=1(Xi − ¯X)2/(n − 1)]1/2 is the conventional estimator of the process standard deviationσ. We note that δ = n(µ − T )2/σ2, will also be estimated as
ˆδ = n( ¯X − T )2/(S
n−1)2. Thus, if ˆCpp≤ χn,α2 (ˆδ)(C − ˜Cia)/n, where C is the recommended maximum value,
then we claim that the process satisfies the quality requirement for at least 100(1 − α)% of the time.
A simple procedure, based on the recommended maximum valueC, for judging whether the process satisfies the preset quality requirement is presented in the following. The SAS software package can be used for generating the valuesχn,α2 (ˆδ)(C − ˜Cia)/n using the command CINV(α, n, ˆδ), with input of the process information, USL, LSL,T , C, α-risk, ¯X, Sn−12 , and ˜Cia.
Procedure
Step 1. Decide the quality requirement C (normally
set to 1.00, 0.56, 0.44, 0.36 and 0.25) and theα-risk (normally set to 0.01, 0.025 or 0.05), the chance of wrongly concluding an incapable process as capable.
Step 2. Calculate the values of ˆδ = n( ¯X − T )2/s2, ˜Cia and ˆCppfrom the sample data.
Step 3. Conclude that the process satisfies the
quality requirement if the ˆCpp value is less than
χ2
n,α(ˆδ)(C − ˜Cia)/n. Otherwise, we do not have enough information to conclude that the process is capable.
For example, consider a manufacturing process with an upper specification limit USL = 20, a lower specification LSL = 10, a target value T = 15, capability requirement is set to C = 1.00. For the calculated sample mean ¯X = 14.5 and sample varianceS2
n−1 = 2.0, under the risk α = 0.05, we calculateD = min{(USL − T )/3, (T − LSL)/3} and
ˆδ = n( ¯X − T )2/s2, ˜C
ia. Running the SAS program with those input data, we obtain the critical value
χ2
n,α(ˆδ)(C − ˜Cia)/n = 0.7246 for n = 50. Thus, if the calculated value ˆCpp < 0.7246, then we conclude that the process satisfies the quality requirement.
REFERENCES
1. Greenwich M, Jahr-Schaffrath BL. A process incapability index. International Journal of Quality and Reliability Management 1995; 12(4):58–71.
2. Chan LK, Cheng SW, Spiring FA. A new measure of process capability:Cpm. Journal of Quality Technology 1988;
20(3):162–175.
3. Kane VE. Process capability indices. Journal of Quality Technology 1986; 18(1):41–52.
4. Pearn WL, Lin GH, Chen KS. Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory and Methods 1998; 27(4):985–1000.
5. Pearn WL, Lin GH. On the reliability of the estimated incapability index. Quality and Reliability Engineering International 2001; 17:279–290.
Authors’ biographies:
Dr Wen Lea Pearn is a Professor of Operations
Research and Quality Management in the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, Republic of China. He received his PhD degree from the University of Maryland at College Park, MD, USA. He worked for AT&T Bell Laboratories at Switch Network Control and Process Quality Centers.
Dr Gu Hong Lin received his PhD degree in Quality
Man-agement from the National Chiao Tung University, Taiwan, Republic of China. Currently, he is an Associate Professor in the Department of Communication Engineering, National Penghu Institute of Technology, Penghu, Taiwan, Republic of China.
