Example 24.2: Comparing DETMAX Algorithm to Sequential Algorithm

Details of the OPTEX Procedure

Example 24.2: Comparing DETMAX Algorithm to Sequential Algorithm

See OPTEX4 in the SAS/QC Sample Library

An automotive engineer wants to fit a quadratic model to fuel consumption data in order to find the values of the control variables that minimize fuel consumption (refer to Vance 1986). The three control variables and their possible settings are shown in the following table:

Variable	Values
AF	15	16	17	18
EGR	0.020	0.177	0.377	0.566	0.921	1.117
SA	10	16	22	28	34	40	46	52

Rather than run all 192 (4×6×8) combinations of these factors, the engineer would like to see whether the total number of runs can be reduced to 50 in an optimal fashion.

Since the factors have different numbers of levels, you can use the PLAN procedure (refer to the SAS/STAT User's Guide) to generate the full factorial set to serve as a candidate data set for the OPTEX procedure.

   proc plan;
      factors af=4 ordered egr=6 ordered sa=8 ordered
              / noprint;
      output out=a
             af  nvals=(15,16,17,18)
             egr nvals=(.020,.177,.377,.566,.921,1.117)
             sa  nvals=(10,16,22,28,34,40,46,52);
   run;

The DETMAX algorithm of Mitchell (1974a) is very commonly used for computer-generated optimal design. Although it is not the default search method for the OPTEX procedure, you can specify that it be used with the METHOD=DETMAX option in the GENERATE statement. For example, the following statements produce Output 24.2.1.

   proc optex data=a seed=61552;
      model af|egr|sa@2 af*af egr*egr sa*sa;
      generate n=50 method=detmax;
   run;

Output 24.2.1: Efficiencies with DETMAX Algorithm

The OPTEX Procedure

Design Number	D-Efficiency	A-Efficiency	G-Efficiency	Average Prediction Standard Error
1	46.4922	24.8987	95.2281	0.4202
2	46.4864	24.8562	95.5744	0.4205
3	46.4797	24.8830	95.3137	0.4203
4	46.4635	25.6461	94.8125	0.4175
5	46.4495	24.5376	95.5559	0.4237
6	46.4459	25.0749	94.8536	0.4197
7	46.4428	24.5111	95.3704	0.4240
8	46.4333	25.0321	95.1371	0.4199
9	46.4333	25.0321	95.1371	0.4199
10	46.4333	25.0321	95.1371	0.4199

The DETMAX search method can require considerable run time. For comparison, you can use the METHOD=SEQUENTIAL option in the GENERATE statement, as shown in the following statements, which produce Output 24.2.2.

   proc optex data=a seed=33805;
      model af|egr|sa@2 af*af egr*egr sa*sa;
      generate n=50 method=sequential;
   run;

Output 24.2.2: Efficiencies with Sequential Algorithm

The OPTEX Procedure

Design Number	D-Efficiency	A-Efficiency	G-Efficiency	Average Prediction Standard Error
1	46.4009	25.0472	93.8673	0.4200

In a fraction of the run time required by DETMAX, the sequential algorithm finds a design with a relative D-efficiency of 46.4009 / 46.4922 = 99.8% compared to the best design found by the DETMAX procedure and with better A-efficiency. As this demonstrates, if absolute D-optimality is not required, a faster, simpler search may be sufficient.

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