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 The RSREG Procedure

## Searching for Multiple Response Conditions

Suppose you want to find the factor setting that produces responses in a certain region. For example, you have the following data with two factors and three responses:

```   data a;
input x1 x2 y1 y2 y3;
datalines;
-1      -1         1.8 1.940  3.6398
-1       1         2.6 1.843  4.9123
1      -1         5.4 1.063  6.0128
1       1         0.7 1.639  2.3629
0       0         8.5 0.134  9.0910
0       0         3.0 0.545  3.7349
0       0         9.8 0.453 10.4412
0       0         4.1 1.117  5.0042
0       0         4.8 1.690  6.6245
0       0         5.9 1.165  6.9420
0       0         7.3 1.013  8.7442
0       0         9.3 1.179 10.2762
1.4142  0         3.9 0.945  5.0245
-1.4142  0         1.7 0.333  2.4041
0       1.4142    3.0 1.869  5.2695
0      -1.4142    5.7 0.099  5.4346
;
```

You want to find the values of x1 and x2 that maximize y1 subject to y2<2 and y3<y2+y1. The exact answer is not easy to obtain analytically, but you can obtain a practically feasible solution by checking conditions across a grid of values in the range of interest. First, append a grid of factor values to the observed data, with missing values for the responses.

```   data b;
set a end=eof;
output;
if eof then do;
y1=.;
y2=.;
y3=.;
do x1=-2 to 2 by .1;
do x2=-2 to 2 by .1;
output;
end;
end;
end;
run;
```

Next, use PROC RSREG to fit a response surface model to the data and to compute predicted values for both the observed data and the grid, putting the predicted values in a data set c.

```   proc rsreg data=b out=c;
model y1 y2 y3=x1 x2 / predict;
run;
```

Finally, find the subset of predicted values that satisfy the constraints, sort by the unconstrained variable, and display the top five predictions.

```   data d;
set c;
if y2<2;
if y3<y2+y1;

proc sort data=d;
by descending y1;
run;

data d; set d;
i = _n_;
proc print;
where (i <= 5);
run;
```

The final results are displayed in Figure 56.5. They indicate that optimal values of the factors are around 0.3 for x1 and around -0.5 for x2.

 Obs x1 x2 _TYPE_ y1 y2 y3 i 1 0.3 -0.5 PREDICT 6.92570 0.75784 7.60471 1 2 0.3 -0.6 PREDICT 6.91424 0.74174 7.54194 2 3 0.3 -0.4 PREDICT 6.91003 0.77870 7.64341 3 4 0.4 -0.6 PREDICT 6.90769 0.73357 7.51836 4 5 0.4 -0.5 PREDICT 6.90540 0.75135 7.56883 5

Figure 56.5: Top Five Predictions

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