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Optimize How-To¶

This walkthrough searches a fitted model for the settings that maximise one response while holding a second one inside a limit, and then checks the answer against the model it came from.

Optimize takes one goal, and everything else you care about becomes a constraint on another output, so this walkthrough declares two outputs on the same sixteen runs: Strength, as every other DOE walkthrough has it, and Warp, the distortion of the finished bond in millimetres, which is the response the constraint goes on.

The data¶

A three factor experiment on a bonding process, run twice through. Temp is in degrees C, Press in psi and Time in seconds. Strength is the peel strength of the finished bond and Warp is its distortion, so more Strength is better and less Warp is better.

Press Copy for Excel, then paste the table into a blank worksheet.

Temp	Press	Time	Strength	Warp
180	40	10	49.5	0.31
220	40	10	65.4	0.52
180	60	10	61.8	0.36
220	60	10	66.8	0.63
180	40	20	52.3	0.44
220	40	20	70.7	0.70
180	60	20	60.7	0.49
220	60	20	70.0	0.79
180	40	10	50.6	0.33
220	40	10	66.4	0.55
180	60	10	61.2	0.38
220	60	10	67.2	0.60
180	40	20	51.4	0.42
220	40	20	67.1	0.74
180	60	20	61.9	0.51
220	60	20	71.6	0.76

The first four columns are the same experiment the rest of the DOE walkthroughs use, so a Strength model built here matches the one on Run Regression How-To. The fifth column is the addition.

Steps¶

Put the data in Excel

Press Copy for Excel above the table, click cell A1 in a blank worksheet, and press Ctrl+V. The header row lands in row 1 and the 16 runs in rows 2 through 17.
Make a design sheet to hold it

QXL DOE New > Create Design > Special > Create Custom Design. Set the number of factors to 3 and the number of runs to 16, then set the number of outputs to 2.
Fill in the design sheet

Copy the five columns into the three factor columns and the two output columns. Name the factors Temp, Press and Time, and the outputs Strength and Warp.
Answer the two questions a custom design asks

Optimize is one of the three commands that triggers them. Leave Type: at Quantitative for all three factors, and move AB, AC and BC into Interactions in model.
Run the regression

QXL DOE New > Analyze Design > Run Regression. Optimize reads the model, not the runs, so there has to be a fitted model before it can search.
Optimize

QXL DOE New > Optimize.

Page 1, the factor ranges. Each input row carries Low and High, pre-filled with the settings the design was built with, and a Continuous check box. Leave all three at their design settings for the first run. Press Next >.

Page 2, the goal and the constraints. Under Set the optimization goal choose Strength and the aim Maximize. Leave Set the optimization constraints (optional) empty this first time. Press Optimize. The form hides while the search runs.

Page 3, the result. Optimized inputs lists each input against its Optimized value, and Outputs lists each response against the value the model predicts there.

Check the two models first¶

If your coefficients match these, the answers below will match too.

Term	Strength	Warp
Constant	62.1625	0.53312
Temp (A)	5.9875	0.12812
Press (B)	2.9875	0.03188
Time (C)	1.0500	0.07312
AB	-2.2375	0.00188
AC	0.6500	0.01312
BC	-0.1500	-0.00062
R-squared	0.9854	0.9907

All three main effects push Warp the same way they push Strength, which is what makes this a real trade rather than a free choice: the settings that give the strongest bond give the most distorted one. AB is the one term that does not, at -2.2375 on Strength against +0.00188 on Warp, and its Warp coefficient has a p value of 0.7108, so it is not a term to plan around.

The unconstrained answer, and how to check it¶

Optimize should return Temp 220, Press 60, Time 20, with a predicted Strength of 70.45.

You can check that by hand, and on this model you can check it exactly. The model holds main effects and two-factor interactions only, so it is multilinear: it has no peak in the middle of the range, and its largest value over the allowed box is always at one of the eight corners. There are only eight to evaluate:

Temp	Press	Time	Strength	Warp
180	40	10	50.40	0.314
180	40	20	51.50	0.436
180	60	10	61.15	0.376
180	60	20	61.65	0.494
220	40	10	65.55	0.541
220	40	20	69.25	0.714
220	60	10	67.35	0.609
220	60	20	70.45	0.781

The last row is the answer, and the Warp column is why the next section exists: the strongest corner is also the most distorted one, at 0.781 mm.

An answer that is not one of those eight corners, on this model, is worth a second look. Add a quadratic term or a centre point and that stops being true, because the surface then curves and its best point can sit inside the range.

Now add the constraint¶

Press < Back from page 3 to return to the goal page. Optimize again does not take you there: it re-runs the search from the setup you already gave it. Keep the goal at Maximize Strength, then under Set the optimization constraints (optional) press Add constraint, choose Warp, and give it an upper bound of 0.65.

Expect a Strength of about 68.08, around Temp 220, Press 59.8, Time 12.4, with Warp at 0.65.

Two things about that answer are worth noticing.

It is not a corner. The feasible region is now bounded by a surface rather than by the box, so its best point sits on that surface, part way along Time. This is the case where hand-checking the corners is no longer enough: the best feasible corner is 220, 60, 10 at a Strength of 67.35, and the search beats it by three quarters of a unit.

The numbers are approximate and the page says so. Two runs can return slightly different settings, both good, and a constraint is honoured as far as the search can manage rather than as an absolute bound. Read the reported Warp on page 3 against the 0.65 you asked for rather than assuming it was enforced.

The trade, in three lines¶

Warp allowed	Best Strength	At about
no constraint	70.45	220, 60, 20
0.70	about 68.98	220, 59.4, 15.4
0.65	about 68.08	220, 59.8, 12.4

Tightening Warp by 0.05 mm costs about 0.9 of Strength, and it is spent almost entirely on Time: Time falls from 15.4 to 12.4 while Press moves 0.4 and Temp does not move at all.

Time is what gets cut because Time is the worst buy. Divide each main effect's Strength coefficient by its Warp coefficient and you get how much Strength a unit of Warp buys through that factor:

Factor	Strength	Warp	Strength per unit of Warp
Temp	5.9875	0.12812	46.7
Press	2.9875	0.03188	93.7
Time	1.0500	0.07312	14.4

Press is the best buy and Time the worst, so the search holds Press and Temp near their maximums and spends the constraint on Time. That table is the useful output of an optimisation, more than any single row of the answer.

What to check before you act on the answer¶

The settings are a prediction, not a measurement. They come from the fitted model, so they are only as good as the fit. R-squared is 0.9854 on Strength and 0.9907 on Warp here, which is about as good as a designed experiment gets, and it is still a prediction.

The answer can be a combination no run used, and here it is: nothing was run at Time 12.4. That is the point of optimising a model rather than picking the best run, and it is also why confirming it with an actual run matters. The DOE Advisor suggests validation points for exactly this reason.

Check the answer sits inside the ranges you set, and that those ranges are settings the process can really hold.

Rounding, and why the answer may not be a round number¶

Clear Continuous on an input and the Rounding level: box appears. It is a power of ten: 0 rounds to whole numbers, 2 to two decimals, -2 to the nearest hundred. Leave it continuous and the search may return a setting with more decimals than you can dial in, which is what happened to Time above.

Set Time's rounding level to 0 and run the constrained search again to see the difference: the answer comes back at a whole number of seconds, and the Strength it reports is a little lower, because the search is choosing from fewer settings.

Things to try next¶

Press Optimize again on the result page without changing anything. It is a fresh search from the same setup, not a continuation, so a second press can return a different answer, and comparing the two tells you how flat the optimum is. On the unconstrained run it should come back to the same corner, because the best corner beats every other by more than a search would miss.
Narrow Temp to 180 through 200 on page 1 and run again. The answer changes completely, because the corner it wanted is no longer in the box.
Set the goal to Minimize Warp instead with no constraint. It should return 180, 40, 10, the opposite corner, at a Warp of 0.314.
Press Save to worksheet to write the result where you can keep it.