8. Comparisons of algorithms

Sun Woong Kim, Steven G. Heeringa and Peter W. Solenberger

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Using the four controlled selection problems given in Section 2, we present some results from the two methods using d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2C@ and d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE1@ in the new algorithm, and compare the solutions for these two methods to solutions generated under the algorithms previously described by Jessen (1970), Jessen (1978), Causey et al. (1985), Huang and Lin (1998), and Winkler (2001). The solutions from the two methods using d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2C@ and d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE1@ were obtained by implementing the SOCSLP, running on the version 9.2 of SAS/OR (2008). Solutions for the algorithm of Sitter and Skinner (1994) using LP were also obtained using PROC LP of the version 9.2 of SAS/OR (2008). Solutions for the other methods are the results as they appeared in the original papers.

The answers to two questions help us compare the algorithms: 1) Are the solutions from the new methods different from those of the previous algorithms described in Section 5? 2) Do the solutions from the new methods give higher probabilities of selection for optimum arrays compared to those generated using the previous methods?

Prior to the comparison of the algorithms, we need to take a look at the results in Table 8.1 obtained from the two methods. In the table, the method using d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2C@ and the one using d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE1@ are denoted by N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacaaIYaaabeaaaaa@39BD@ and N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacqGHEisPaeqaaaaa@3A72@ , respectively. Since when calculated by d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2C@ ( d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE1@ ), the arrays with the same distance value are in the same group, there would be different groups for all possible arrays (see Remark 6.2). Let G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4raaaa@38CE@ denote the number of the different groups. Also, let O F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4taiaadAeaaaa@39A1@ be the actual value of the objective function (6.5) or (6.6) and t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iDaaaa@38FB@ the actual number of T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam ivaaaa@38DB@ , the number of transitions, introduced in Section 6.3. They are all obtained from the SOCSLP, and t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iDaaaa@38FB@ especially indicates the number of iterations in phase 1 and 2 of the PROC LP in the software.

Table 8.1
Results with the new methods
Table summary
This table displays the results of Results with the new methods Problem 2.1, Problem 2.2, Problem 2.3 and Problem 2.4 (appearing as column headers).
  Problem 2.1 Problem 2.2 Problem 2.3 Problem 2.4
N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacaaIYaaabeaaaaa@3BE0@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacqGHEisPaeqaaaaa@3C95@ N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacaaIYaaabeaaaaa@3BE0@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacqGHEisPaeqaaaaa@3C95@ N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacaaIYaaabeaaaaa@3BE0@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacqGHEisPaeqaaaaa@3C95@ N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacaaIYaaabeaaaaa@3BE0@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacqGHEisPaeqaaaaa@3C95@
G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4raaaa@3AF1@ 4 3 9 2 6 2 157 14
O F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4taiaadAeaaaa@3BC4@ 1.336 0.62 1.689 0.64 1.582 0.72 1.661 0.701
t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iDaaaa@3B1E@ 2 2 8 6 18 15 43 41

As seen in the table, most values of G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4raaaa@38CE@ are much smaller than L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam itaaaa@38D3@ , the number of all possible arrays given in Table 6.1, except for the case of the large value of “157” for Problem 2.4, which arises simply due to the fact that the a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yyamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3AF1@ are given to three decimal places. When using d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2D@ , the values of O F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4taiaadAeaaaa@39A1@ range between 1 and 2, while they are always less than 1, when using d * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE2@ . Most values of t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iDaaaa@38FB@ do not reach the 95% CI of T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam ivaaaa@38DB@ shown at the bottom of Table 6.1. Thus, the actual computational demands are less than those expected in the theory.

The solutions from different algorithms for the first three problems are presented in order in Table 8.2 through Table 8.4. Results for Problem 2.4 are simply described below. (The table of solutions to this problem is available on request.) In Table 8.2, the method of Sitter and Skinner (1994), Jessen’s (1970) method 2 and method 3 are denoted by S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4uaiaadofaaaa@39B2@ , J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam Osaiaaikdaaaa@398D@ and J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam Osaiaaiodaaaa@398E@ , respectively. The solutions for J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam Osaiaaikdaaaa@398D@ and J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam Osaiaaiodaaaa@398E@ in the table are from Jessen (1970, page 782). The table shows that all methods except Jessen’s (1970) method 3 yield the same solution for the 3  x  3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaa aaaaaaa8qacaaIZaGaaeiiaiaabIhacaqGGaGaaG4maaaa@3BDC@ array Problem 2.1. In the common solutions, the probability of selection for the optimum arrays, denoted by B k B p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaaca WGWbWaaeWaaeaacaWGcbWaaSbaaSqaaiaadUgaaeqaaaGccaGLOaGa ayzkaaaaleaacaWGcbWaaSbaaWqaaiaadUgaaeqaaSGaeyicI48efv 3ySLgzgjxyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbbiab=fa8cnaa BaaameaacqGHEisPaeqaaaWcbeqdcqGHris5aaaa@4DCB@ , is 0.5.

Table 8.2
Comparison of solutions to Problem 2.1
Table summary
This table displays the results of Comparison of solutions to Problem 2.1. The information is grouped by B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbmqaaeGaciGaaiaabeqaam aaeaqbaaGcbaGaemOqai0aaSbaaSqaaiabdUgaRbqabaaaaa@3F83@ (appearing as row headers), p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmaabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3E90@ (appearing as column headers).
B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbmqaaeGaciGaaiaabeqaam aaeaqbaaGcbaGaemOqai0aaSbaaSqaaiabdUgaRbqabaaaaa@3F83@ p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmaabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3E90@
N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIYaaabeaaaaa@39C5@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacqGHEisPaeqaaaaa@3A7A@ S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaado faaaa@39BA@ J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaaik daaaa@3995@ J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaaio daaaa@3996@
0 1 1 1 0 1 1 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmWaaa qaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaa baGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaaaaaa@3C9D@ 0.2 0.2 0.2 0.2 0.1
1 0 1 1 1 0 0 1 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmWaaa qaaiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaa baGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaaaaaa@3C9D@ Note * 0.5 0.5 0.5 0.5 0.4
1 1 0 0 1 1 1 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmWaaa qaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXaaa baGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGymaaaaaaa@3C9D@ 0.3 0.3 0.3 0.3 0.2
0 1 1 1 1 0 1 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmWaaa qaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaa baGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGymaaaaaaa@3C9D@         0.1
1 0 1 0 1 1 1 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmWaaa qaaiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIXaaa baGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaaaaaa@3C9D@         0.1
1 1 0 1 0 1 0 1 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmWaaa qaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaigdaaeaacaaIWaaa baGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaaaaaa@3C9D@         0.1
Total 1.0 1.0 1.0 1.0 1.0
Total Note  0.5 0.5 0.5 0.5 0.4

In Table 8.3, Jessen’s (1978) method is denoted by J S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam Osaiaadofaaaa@39A9@ . The solution for J S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam Osaiaadofaaaa@39A9@ in the table is from Jessen (1978, pages 375-376). As shown in the table, the new methods using d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2C@ and d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE1@ have the same solution for the Problem 2.2 4 × 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG inaiabgEna0kaaisdaaaa@3B95@ array; however only one-half of the arrays in those solutions overlap with the arrays in the solutions from the methods of Sitter and Skinner (1994) and Jessen (1978). Also, the Sitter and Skinner and Jessen methods provide a lower probability of 0.6 to optimum arrays, whereas the new methods allocate the higher probability of 0.8 to the arrays.

Table 8.3
Comparison of solutions to Problem 2.2
Table summary
This table displays the results of Comparison of solutions to Problem 2.2. The information is grouped by B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbmqaaeGaciGaaiaabeqaam aaeaqbaaGcbaGaemOqai0aaSbaaSqaaiabdUgaRbqabaaaaa@3F83@ (appearing as row headers), p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmaabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3E90@ (appearing as column headers).
B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbmqaaeGaciGaaiaabeqaam aaeaqbaaGcbaGaemOqai0aaSbaaSqaaiabdUgaRbqabaaaaa@3F83@ p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmaabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3E90@
N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIYaaabeaaaaa@39C5@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacqGHEisPaeqaaaaa@3A7A@ S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaado faaaa@39BA@ J S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaado faaaa@39B1@
0 0 1 1 0 1 0 1 1 1 0 0 1 0 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaaaaaa@41BE@ 0.2 0.2    
0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGym aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWa aabaGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaaaaaa@41BE@ Note * 0.2 0.2 0.4 0.2
0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGym aaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWa aabaGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaaaaaa@41BE@ Note * 0.2 0.2    
0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGym aaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigdaaeaacaaIWa aabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaicda aeaacaaIXaaaaaaa@41BE@ Note * 0.4 0.4 0.2 0.4
0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGim aaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaigdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIXaaaaaaa@41BE@     0.2  
0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGym aaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWa aabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaigda aeaacaaIWaaaaaaa@41BE@     0.2  
0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGym aaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaaaaaa@41BE@       0.2
0 0 1 1 0 1 0 1 1 0 1 0 1 1 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabqabaa aaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaigdaaeaacaaIWa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaaaaaa@41BE@       0.2
Total 1.0 1.0 1.0 1.0
Total Note  0.8 0.8 0.6 0.6

Problem 2.3, with 141 possible arrays, is considerably larger than the above two problems. The solutions to this problem under the five methods are compared in Table 8.4. In the table, the methods of Causey et al. (1985) and Huang and Lin (1998) are denoted by C A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4qaiaadgeaaaa@3990@ and H U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam isaiaadwfaaaa@39A9@ , respectively. The solutions for C A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4qaiaadgeaaaa@3990@ and H U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam isaiaadwfaaaa@39A9@ in the table are from Causey et al. (1985, page 906) and Huang and Lin (1998, Figure 3), respectively.

Table 8.4
Comparison of solutions to Problem 2.3
Table summary
This table displays the results of Comparison of solutions to Problem 2.3. The information is grouped by B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbmqaaeGaciGaaiaabeqaam aaeaqbaaGcbaGaemOqai0aaSbaaSqaaiabdUgaRbqabaaaaa@3F83@ (appearing as row headers), p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmaabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3E90@ (appearing as column headers).
B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbmqaaeGaciGaaiaabeqaam aaeaqbaaGcbaGaemOqai0aaSbaaSqaaiabdUgaRbqabaaaaa@3F83@ p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmaabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3E90@
N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIYaaabeaaaaa@39C5@ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacqGHEisPaeqaaaaa@3A7A@ S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaado faaaa@39BA@ C A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaadg eaaaa@3998@ H U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaadw faaaa@39B1@
0 2 0 1 0 1 0 0 0 2 0 0 1 1 0 0 1 0 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@ 0.2 0.2 0.2    
0 2 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@ 0.1 0.2 0.03    
0 2 0 1 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@ 0.1        
0 2 0 2 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@ 0.1        
0 2 0 2 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@ 0.1        
1 2 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIXaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@ Note * 0.1   0.08    
1 2 0 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIXaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@ Note * 0.3 0.4 0.2 0.4 0.4
0 2 0 2 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@   0.2      
0 2 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGymaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@     0.11    
0 2 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@     0.03    
0 2 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@     0.03    
0 2 0 2 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@     0.09    
0 2 0 2 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@     0.08    
0 2 0 2 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@     0.03    
1 2 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIXaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@     0.06    
1 2 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIXaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@     0.06    
0 2 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 0 0 1 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@       0.2  
0 2 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaigda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@       0.2 0.2
0 2 0 2 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@       0.2  
0 2 0 1 0 1 0 0 0 2 0 0 1 1 0 0 0 0 0 1 0 0 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaaaa@479B@         0.2
0 2 0 2 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabGWaaa aaaeaacaaIWaaabaGaaGOmaaqaaiaaicdaaeaacaaIYaaabaGaaGim aaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaaaa@479B@         0.2
Total 1.0 1.0 1.0 1.0 1.0
Total Note  0.4 0.4 0.28 0.4 0.4

We note that all these methods provide different solutions, and about half of the arrays overlap between the new methods and the method of Sitter and Skinner (1994). Moreover, the solutions from the methods of Causey et al. (1985) and Huang and Lin (1998) are quite unlike the solution from the method using d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE1@ . The method using d 2 * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiaaikdaaeaacaGGQaaaaaaa@3D2D@ and Sitter and Skinner’s method distribute the probabilities of selection to two optimum arrays, whereas the other three methods just allocate the probability to only one optimum array. Sitter and Skinner’s method appears to be less effective in selecting optimum arrays since their method gives the probability of 0.28 to those, while the others give the higher probability of 0.4.

The solutions to Problem 2.4, which is the largest of the given problems, are compared under the four methods ( N 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacaaIYaaabeaaaaa@39BD@ , N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OtamaaBaaaleaacqGHEisPaeqaaaaa@3A72@ , S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4uaiaadofaaaa@39B2@ , and Winkler’s (2001) method). Only two arrays, including one optimum, overlap in the solutions, and the two new methods give the same probabilities (0.127 and 0.483) to those arrays. Even when comparing the method using d * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaia iMykW7caWGKbWaa0baaSqaaiabg6HiLcqaaiaacQcaaaaaaa@3DE2@ with the methods of Sitter and Skinner (1994) and Winkler (2001), their solutions are very different. Also, the new methods give the same probability of selection of 0.483 to the optimum array, whereas the other previous methods give the lower probabilities of 0.385 and 0.104, respectively.

In summary, it seems that the new methods successfully achieve S1 and S2 of optimal solutions. Note that the new methods consistently give higher probabilities of selection for optimum arrays and that the totals of those probabilities are always the same. The solutions from the new methods are very different from those obtained using previous methods, when the controlled selection problems are not small. This implies that the solutions from the previous methods may be far from optimal under criteria S1 and S2 (R1 and R2).

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