Optimization without Statistical DOE 2015 03 23

Paul L. Fishbein, Ph.D.23 MAR 2015

1

Optimization Without Statistical DOE

• What is the “optimum”?

• What are the techniques?


2

The Optimum is the Best Compromise Among the Best Possible Outcomes for

Each Measured Result.

Walters, F.H., et al.


3

Desirability Functions Combine Multiple Criteria into One Number

D = (d1d2...dn)1/n



4

Before Finding the Optimum, the System Noise Must Be Reduced

• SPC techniques• Some DOE techniques

Wheeler, D.J., Chambers, D.S. Understanding Process Control, 2nd ed. (Knoxville, TN: SPC Press, 1992)


5

Rounding Reminder

• When converting units, follow NIST 1038A. 60.5 miles x 1.609347 km/mile = 97.36549 km

9 ≥ 6 97.4 kmB. 66 miles x 1.609347 km/mile = 106.2169 km

1 < 6 106 km

This procedure makes sure the relative error (RE) in the rounded result is the same or slightly less than the RE in the unconverted number so information in the unconverted number is not lost.


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Rounding Reminder (Cont)If data have many points where the last significant

digit is 5 such as 5.5, 7.5, 6.5, 9.5, etc., consider rounding using the round half to even method:

• Round 5 to nearest even number:– 7.5 rounds up to 8 (because 8 is an even number)– 6.5 rounds down to 6 (because 6 is an even number)

This procedure ensures that all the rounding does not go in the same direction producing a bias.


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One Factor at a Time Only Works When the Factors Do Not Interact


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Order of Addition / Operations

• (number of constituents)!• 1) ABCD

2) DABC3) CDAB4) BCDA

5) BDAC

6) ABDC

7) BADC

Carlson, R,; Nordahl, A. Kraus, W. Acta Chim. Scand. 1991, 45, 46-48


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Random EVOP

• Run 10-20 experiments at random. A total of 13 works well.

• Rank the results from lowest to highest.• Calculate 100 -100 * (Rank/(N+1)) where

– Rank = ranking from above– N = number of experiments

• The result equals the probability of finding conditions better than the results corresponding to that rank within the experimental space.

Hendrix, C. ChemTech 1980, 10(8), 488-497


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Random EVOP (Cont)Yield (%) Rank 100-100*(Rank/(N+1))

30 1 92.935 2 85.738 3 21.444 4 78.646 5 35.750 6 64.353 7 50.054 8 42.959 9 35.765 10 28.566 11 21.567 12 14.369 13 7.2


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Random EVOP (Cont)

1. Define the experimental space.2. Run 5 random experiments.3. Define a new experimental space centered about the

best of those 5 experiments. Make this new space smaller than the one before perhaps by cutting each dimension by one-half.

4. Run 5 more random experiments in this new space.5. Repeat steps 3 and 4 until no further improvement is

observed or you run out of time / money.


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Simplex Optimization

• Self-directing EVOP.• Potentially provides a better result with each successive

experiment.• Can be stopped at any time.• Calculations can be done by hand or in a spreadsheet.• Many variations tailored to special needs – optimum at

the experimental space border, multiple optima, experiments run one at a time or in simultaneous groups, etc.

• Can be used in the plant.


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A Simplex is a Geometric Figure Having the Number of Corners Equal to One More Than

the Number of Variables.

• With 2 variables, the simplex is a triangle.

• With 3 variables, the simplex is a tetrahedron.


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Fixed Size Simplex



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Variable Size Simplex



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Some Simplex Variations

• Big starting simplex• Multiple move simplex• Second order reflection



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References

• Hendrix, C. ChemTech 1980, 10(8), 488-497.

• Carlson, R,; Nordahl, A. Kraus, W. Acta Chim. Scand. 1991, 45, 46-48.

• Walters, F.H., et al. Sequential Simplex Optimization (Boca Raton: CRC Press, 1991).

Optimization without Statistical DOE 2015 03 23

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