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Optimize¶

Optimize searches a fitted model for the factor settings that best meet a goal you set, while respecting any constraints you add on the other responses.

QXL DOE New > Optimize

It works on the model, not on the runs, so it can return a combination of settings that no run in the design used.

What it needs¶

A regression sheet. Optimize reads the fitted coefficients, so run the regression first.

The regression must not be in uncoded units. Optimize refuses to open on a sheet produced by Uncoded Coefficients and says Optimization cannot be performed for the regression in the uncoded units. Run the regression normally and optimize from that sheet instead.

What it does, and what to expect from the answer¶

Quantum XL does not combine your responses into a single desirability score. It optimizes one goal. Everything else you care about is entered as a constraint, and the search works within the ranges and levels you allowed for each input.

Three properties of the answer are worth knowing before you act on one:

It is a very good candidate, not a proven optimum. For a smooth model over a small region it will be effectively exact. For a rough surface at the edge of the design region, run it twice and see whether the answer moves.
Two runs can return slightly different settings, both good. That is expected and not a sign that something is wrong.
A constraint is honoured as far as the search can manage, but it is not an absolute bound. The reported answer can sit outside a constraint you set, so check the reported values against your constraints rather than assuming they were enforced.

Step 1: the region to search¶

The first page sets the range for each input. The defaults are the low and high values the design was built with, and narrowing them is how you keep the answer inside settings you can actually run.

Each input row carries Low and High boxes and a Continuous checkbox.

Clearing Continuous reveals a second box, Rounding level:, and that box is what decides the step the answer can land on. It is hidden while Continuous is ticked, and it must be a number: anything else gives Rounding level value must be numeric. Any decimal part you type is dropped, so the level is always a whole number.

The level is a power of ten, and it can be negative:

Rounding level	The answer is rounded to
0, the default	whole numbers, so 123.456 becomes 123
2	two decimal places, so 123.456 becomes 123.46
-2	the nearest hundred, so 123.456 becomes 100

For a categorical input the row offers a level list instead, with All levels as the default, so you can exclude a level rather than have it proposed.

Step 2: the goal and the constraints¶

The goal. Pick a response and then an aim from the goal menu:

Aim	What the search does
Maximize	make it as large as possible
Minimize	make it as small as possible
Set target	get it as close as possible to a value you give
Minimize DPM for all outputs.	work on the defects per million of every qualifying output at once, rather than on one response

Minimize DPM for all outputs. is the one aim that is not tied to a single response, and the menu offers it only when two or more outputs qualify for DPM on the same terms given below, a specification limit and a usable standard deviation. With one or none it is not in the list.

Otherwise there can be only one goal. Everything else you care about becomes a constraint.

What you can set a goal or constraint on. More than just the response value:

Quantity	Available for
The predicted response	a quantitative output
The predicted standard deviation	a quantitative output with a standard deviation model
Cpk	a quantitative output with at least one specification limit and a usable standard deviation
DPM, defects per million	a quantitative output with at least one specification limit and a usable standard deviation
The probability of a level	a binary or nominal output

Cpk and DPM are available as the goal quantity as well as as constraints. Both are computed from the predicted response and the predicted spread together, so unlike the response goals they take the standard deviation into account, and that is why a specification limit on its own is not enough to make them available: the output also needs a spread to compute them from, either a valid standard deviation model or a residual standard error from its own regression. An output with a specification limit but neither of those does not offer Cpk or DPM at all.

Constraints. Any number, on any of the quantities above. A typical setup maximizes a quantitative response while holding a failure probability below a limit and keeping a second response inside a band.

Running it, and reading the result¶

Optimize starts the search, and the dialog shows Optimizing, please wait…. while it runs.

The result page has two tables. Optimized inputs lists each Input name with its Optimized value. Outputs lists each Output with its Value, the prediction at those settings.

Two commands sit on that page:

Command	What it does
Optimize again	runs the search again from the same setup. Worth doing once, because two runs can return slightly different settings
Save to worksheet	writes the optimized inputs into the Set point column of the regression sheet the run came from, and stores the setup with the sheet. This is the only way the answer leaves the dialog

Saving also remembers the setup. The bounds, rounding and goal are written into a hidden area of the regression sheet, and the next time you open Optimize on that sheet those saved values are used in place of the defaults. A later run therefore starts from where the last one left off rather than from the full factor ranges.

Check the reported values against the constraints you set. As above, a constraint is not an absolute bound, so a reported answer can sit outside one.

Where Cpk and DPM come from¶

The capability figures beside the prediction treat the predicted mean and predicted standard deviation as the mean and standard deviation of a normal distribution:

\[ C_{pk} = \min\!\left(\frac{\text{USL} - \hat{y}}{3\hat{s}},\ \frac{\hat{y} - \text{LSL}}{3\hat{s}}\right) \]

\[ \text{DPM} = \left[\Phi\!\left(\frac{\text{LSL} - \hat{y}}{\hat{s}}\right) + 1 - \Phi\!\left(\frac{\text{USL} - \hat{y}}{\hat{s}}\right)\right] \times 10^{6} \]

A third figure, Cp written with a subscript p, sits above Cpk on the sheet and needs both limits:

\[ C_p = \frac{\text{USL} - \text{LSL}}{6\hat{s}} \]

With only one specification limit set, Cp is not reported, and Cpk and DPM use that limit's term alone. The tail areas are multiplied by one million with no bound applied to the result.

The same S-Hat substitution applies here. With no standard deviation model, or none left after terms have been removed, all three figures are computed from the response regression's standard error, so they stop varying with the set points.

The standard deviation used is either the predicted standard deviation at those settings or the standard error of the response model, and a standard deviation that comes out at zero or below is replaced by a small floor, so Cpk is always finite. Full details: Prediction Equation.