about the distribution of derringer-suich type...

About the distribution of Derringer-Suich typedesirabilities

Detlef SteuerHelmut-Schmidt-Universitat Hamburg

Universitat der [email protected]

http://fawn.unibw-hamburg.de/steuer.html

11.3.2004

Detlef Steuer GfKl Dortmund 11.3.2004

Contents

• (Short) Introduction to MCO and desirabilities

• Definition of Derringer-Suich desirabilities

• Current practice using desirabilities

• Distribution of desirabilities

• Improving practice with “realistic desirabilities”

• Concluding remarks

1


The MCO Problem

• Example: design of a new cookie.

• Objectives: Not too hard to eat, but hard enough to not fall apart beforeconsumption.

• Problem: “Antagonistic” objectives must be optimised simultaneously.

• Natural aim: Trying to optimise overall quality.

• Need for compromise, need for expert knowledge!

• In general: Best object y = (y1, . . . , ym) ∈ Rm must be identified. Eachcomponent yi is of type (TV ) or (LB).

2


Desirabilities

• Introduced by Harrington 1965

• Idea: First transform all quality measures to a unitless scale

• Then “oranges and apples” can be compared, i.e. calculate some meanvalue for the different measures.

• Transformation is chosen according to expert knowledge, calculation ofmean allows some weighting of variables.

3


Derringer-Suich desirabillity functions

Derringer-Suich (1980) improved on Harrington using more flexible functionsDS-desirabilities for target value problems

dTVDS(y) :=

0, fur y < l

(y − lt− l )

βl, fur l ≤ y ≤ t(u− yu− t )

βr, fur t < z ≤ u0, fur u < y

DS-desirabilities for “the-larger-the-better” type problems

dLBDS(y) :=

0, fur y < l

(y − lt− l )

βl, fur l ≤ y ≤ t1, fur t < y

4


Unified notation:A Quintupel (l, t, u, βl, βr) defines a DS-desirability with:

If l < t < u ∈ R, βl, βr ∈ R+, then (l, t, u, βl, βr)(y) := dTVDS(y).

Ifu =∞, then alsoβr = 0 and (l, t,∞, βl, 0)(y) := dLBDS(y).

Desirability Index Two different ways:

(geometric mean)Q(y) := (m∏

i=1

di(yi))1m or

(maximin)Q(y) := mini=1,...,m

di(yi)

5


Desirability functions of the DS-type for some

parametersDS−desirability functions

target

desi

rabi

lity

LSL T USL

0.0

0.2

0.4

0.6

0.8

1.0

(LSL, T, USL, 1 ,1)(LSL, T, USL, 0.5, 2)(LSL ,T ∞ ,4 , 0)

6


The right question?

Current Practice. Is this what we want?

• Product y = (y1, . . . , ym) to be optimised, y depends on factor settingsx = (x1, . . . , xk), y = f(x) + ε, ε multivariate normal with diagonalcovarince.

• Define desirabilities di(yi) for each component.

• Performing experiments according some DOE

• Fit linear or quadratic response curves fi, i = 1, . . . ,m for the compo-nents.

• Perform numerical optimisation for Q(f(x)) over region of operability Oand estimate best factor setting xopt. (idealized desirabilities)

7


No!

This way we ignore the error and the non-linerarity of the desirabilities.

Today’s practice gives:

xopt := maxx∈O

Q(E(Y |x))

Simplified and probably wrong solution!

We really search:xopt := max

x∈OE(Q(Y |x))!

8


Distribution of Xopt may be multimodal

Model: y = x2 + ε, d = (−1, 0, 1, 0.1, 1), ε ∼ N(0, 0.1), f fully quadratic

Histogram of estimated optimum X (all cases)

X

Freq

uenc

y

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

050

100

150

200

opt

9


Q(y) may have complicated structure

Model: Y = x21 + x2

2, d = (−1, 0, 4, 1/2, 1/2)

1

Desirability index as function of factor space

desi

rabi

lity

X 2

X1

10


Why ignoring the error can be harmful

Simulation: Repeat optimisation from Derringer/Suich

• Four objectives y1 to y4, two of type (TV ), two of type (LB), threecontrollable variables x1 to x3.

• Central-composite design with 20 experiments

• Second-order models fi including all interactions.

• Repeatedly generate data using the estimated model.

• Each time record xopt. What happens?

11


Why ignoring the error can be harmful

0 2 4 6

01

23

45

6

Two-dimensional projections of 1000 estimated optima

Factor 1

Fact

or 2

XXX X X

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++ ++ +++++

Factor 1

Factor 2

Fact

or 3

Fact

or 3

About 5% of the estimated xopt have true desirability 0!

12


Incorporating the error

Distribution of desirability function values

Each di and Q are random variates!

Model: Y = f(x) + ε, ε ∼ N(0, σ2)

Random desirability defined as

d(x, ε) =

f(x) + ε− lt− l for l ≤ f(x) + ε < t;

u− f(x)− εu− t for t ≤ f(x) + ε < u;

0 else.

13


Distribution of d(x, ε) of type (l, t, u, 1, 1)

Fd(x,ε)(d) =

0 for d < 0;

Φ

(l + d · (t− l)− f(x)

σ

)+

1− Φ(u− d · (u− t)− f(x)

σ ) for d ∈ [0, 1];

1 for d > 1.

Density fd(x,ε) for d of type (l, t, r, 1, 1) and fixed x:

fd(x,ε)(d) =

0 for d 6∈ [0, 1);

Φ(l−f(x)σ ) + 1− Φ(u−f(x)

σ ) for d = 0 (failure rate);t−lσ ϕ(l+d·(t−l)−f(x)

σ ) + u−tσ ϕ(u−d·(u−t)−f(x)

σ ) for d ∈ (0, 1).

14


Density for a desirability (−1, 0, 1, 1, 1), withf(x) = y + ε, ε ∼ N(0, 0.5), E(f(x)) = 0.9

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

desirability

dens

ity

15


Expected value for a given x and d of type (l, t, r, 1, 1)

E(d(x, ε)) = Glf(x)− lt− l + gl

σ

t− l +Gru− f(x)

u− t − grσ

u− t with

Gl = Φ

„t− f(x)

σ

«− Φ

„l − f(x)

σ

«,

gl = ϕ

„l − f(x)

σ

«− ϕ

„t− f(x)

σ

«,

Gr = Φ

„u− f(x)

σ

«− Φ

„t− f(x)

σ

«,

gr = ϕ

„t− f(x)

σ

«− ϕ

„u− f(x)

σ

«.

16


Is this any better?

• Define realistic desirabilities using the now know distribution functionsand optimise Qreal(y) = (

∏m1 E(di(x, ε)))

1m

• Repeat optimisation from Derringer/Suich using Qreal

• Result using realistic instead of idealised desirabilities:Different estimation for xopt:

xidealopt = (−0.05, 0.145,−0.868) (Derringer/Suich)

xrealopt = (0.13, 0.50,−1.08) (realistic desirabilities)

• It’s better! True 10% relative improvement for values of Q:

Q(xrealopt ) = 0.44 and Q(xidealopt ) = 0.40

17


Better approach, but not simpler

• Optimisation is now looking for a “best” random variate.

• Order of distributions is needed.

• Concentrate on key features of distribution: Expected value? Failurerate? Mode?

• Only “some” factor setting can be compared:If F (d(x1, ε)) >st F (d(x2, ε)) (stochastically larger) then factor settingx1 is better than setting x2.

• New MCO!

18


Remarks

• For analytical expressions for the distributions the weights βl = βr = 1are crucial. For other exponents simulations are possible.

• The distribution of Q for the geometric must be simulated, because Fdare not in a family of stable distributions.

• For the maximin approach results for extrem value statistics may be used.

• The distribution of xopt seems completely out of reach for simple analyt-ical formulation. Finding xopt is a callibration problem of the many-to-many type and Q−1 is a not very nicely behaving function. Neverthelesssimulations are possible.

19

about the distribution of derringer-suich type...

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