Jul 21, 2009 - Page 1 ... Robust Desirability Index optimization. The Distribution .... Balanced ratio of numerical and graphical optimization techniques. 10 / 35 ...
Multiobjective Optimization with Desirability Functions and Desirability Indices Dr. Heike Trautmann Statistics Faculty, TU Dortmund University
Drug Design Workshop, Leiden, 21st of July, 2009
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Content
1
The Desirability Index (DI) as a method for multicriteria optimization Desirability Functions Desirability Index
2
Robust Desirability Index optimization The Distribution of the DI
3
Desirabilities vs. Pareto-Optimality DFs as a tool for incorporating preferences in EMOA
4
Process Control as a new application field for the DI
5
References
2 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
MCO methods and expert knowledge Prior knowledge: Link criterion Specification of expert knowledge by means of link criterion before optimization → (Possibly) unambiguous solution Examples: Desirability index, weighted sum
Posterior knowledge: Multiobjective Optimization Optimization without usage of prior knowledge: → Set of optimal solutions → Selection of set of desired solutions after optimization based on expert knowledge Example: Pareto optimization 3 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Desirability Index Example: Production of Fruit Juice
1. Which factors influence the process quality ? X1 : Orange Juice (%), X2 : Pineapple Juice (%), X3 : Grapefruit J. (%)
2. By which factors can the process quality be measured? Y1
:
Content of Vitamin C (mg/l)
Y2
:
Overall content of fruit acid (g/l)
Yˆ1 = f1 (X1 , . . . , X3 ) Yˆ2 = f2 (X1 , . . . , X3 )
Y3
:
Relative Density of fruit ingredients
Yˆ3 = f3 (X1 , . . . , X3 )
3. How desirable are different values of the quality criteria? → Specification of Desirability Functions di (Yi ) 4 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
1.0 0.8 0.6
d2 0.4 0.2 200
300
0.0
100
400
3
4
5
6
7
Overall content of fruit acid (Y2)
8
0.4
d3
0.6
0.8
1.0
Content of Vitamin C (Y1)
0.2
0
0.0
0.0
0.2
0.4
d1
0.6
0.8
1.0
Desirability Index Example: Production of Fruit Juice
1.00
1.05
1.10
1.15
1.20
1.25
1.30
Relative Density of fruit ingredients (Y3)
5 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Desirability Index Example: Production of Fruit Juice 4. Combination into an overall unitless quality criterion: Desirability Index (DI)
3 Y D := ( di (Yi ))1/3 i=1
5. Maximization of the DI, i.e. overall process quality: As Yˆi = fi (X1 , . . . , X3 ) : 3 Y ˆ 1 , . . . , X3 ) = ( di [fi (X1 , . . . , X3 )])1/3 D(X i=1
X
opt
ˆ 1 , . . . , X3 ) = (60, 17, 23)0 = max D(X X1 ,...,X3
6. Process is set up based on optimal factor levels
6 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Desirability Index (DI) Optimization 1. Influence factors:
X1 , . . . , Xn
2. Quality criteria:
Y1 , . . . , Yk with Yi = fi (X1 , . . . , Xn , εi )
3. How desirable are different values of the quality criteria (DFs)? di (Yi )(i = 1, . . . , k),
d : R → [0, 1]
4. Combination into an overall unitless quality criterion (DI): D := f (d1 , . . . , dk ),
D : [0, 1]k → [0, 1]
5. Maximization of the DI as a function of influence factors: v u k uY k ˆ 1 , . . . , Xn ) = t max D(X di (fi (X1 , ..., Xn , 0)) X1 ,...,Xn
i=1 7 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Main types of DFs (I) Derringer/Suich (1980)
0.0 0.2 0.4 0.6 0.8 1.0
Harrington (1965)
ri = 1
0.0
0.4
di
0.8
li = 1
−3
−2
di (Yi0 )
=
Yi0
=
−1
0
1
2
exp(−|Yi0 |ni ) 2Yi − (USLi + LSLi ) USLi − LSLi
3
li = 0.2 LSLi
ri = 2.5 Y i Ti
0, ( Yi −LSLi )li , Ti −LSLi di (Yi ) = Yi −USLi ri (T ) , i −USLi 0,
USLi
Yi < LSLi LSLi ≤ Yi ≤ Ti Ti < Yi ≤ USLi Yi > USLi 8 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Main types of DFs (II) Harrington (1965)
Derringer/Suich (1980) ri = 0.5
0.8
0.8
●
di 0.4
di 0.4
●
●
ri = 3.5 Ti
Yi (1)
(1)
(2)
ri = 1
0.0
0.0
●
Yi
USLi
(2)
di (Yi ) = 1, 0(j) 0(j) Yi −USLi ri di (Yi ) = exp(− exp(−Yi )) ) , (T i −USLi 0(j) (j) 0, Yi = b0i + b1i Yi , j = 1, 2.
Value pairs (Yi , di ), (Yi , di )
Yi ≤ Ti Ti < Yi < USLi Yi ≥ USLi
9 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
The Desirability Index has become widely accepted 1. Different types of DIs: k Y Dg := ( di )1/k , i=1
D :=
k Y i=1
di ,
Dmin := min di , i=1,...,k
D := 1/k
k X
di
i=1
2. Applications: Mostly in chemistry, chemical and mechanical engineering Optimization of manufacturing-, production- or chemical processes: Examples: Optimization of a force balance in wind tunnel tests, Optimization of solid-state bioconversion of wheat straw, . . . 3. Remarks: Mostly Derringer-Suich DFs and the geometric mean as a DI Balanced ratio of numerical and graphical optimization techniques 10 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
The Desirability Index in noisy environments The DI is a random variable DI maximization using the models linking quality criteria to influence factors: classical approach: v u k uY k ˆ max D(X1 , . . . , Xn ) = t di (fi (X1 , ..., Xn , 0))
X1 ,...,Xn
i=1
ideal approach → Robust DI Optimization: v u k uY k max E [D(X1 , . . . , Xn )] = E t di (fi (X1 , ..., Xn , εi )) X1 ,...,Xn
with in general
i=1
εi
∼
N (0, σi2 ),
i = 1, . . . , k 11 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Example of Derringer/Suich (1980) Tire Tread Example Quality Criteria: Y1 : PICO Abrasion Index Y2 : 200 % - Modulus Y3 : Elongation at Break Y4 : Hardness Influence factors (phr: parts per hundred): X1 : (phr silica − 1.2)/0.5 X2 : (phr silane − 50)/10 X3 : (phr sulfur − 2.3)/0.5
12 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Tire Tread Example Mathematical models Yˆ1 (X )
=
139.1 + 16.5X1 + 17.9X2 + 10.9X3 − 4.0X12 − 3.5X22 − 1.6X32 +5.1X1 X2 + 7.1X1 X3 + 7.9X2 X3 ;
Yˆ2 (X )
=
σ ˆ1 = 5.6,
1261.1 + 268.2X1 + 246.5X2 + 139.5X3 − 83.6X12 − 124.8X22 +199.2X32 + 69.4X1 X2 + 94.1X1 X3 + 104.4X2 X3 ; σ ˆ2 = 328.7,
Yˆ3 (x) =
400.4 − 99.7X1 − 31.4X2 − 73.9X3 + 7.9X12 + 17.3X22 +0.4X32 + 8.8X1 X2 + 6.3X1 X3 + 1.3X2 X3 ;
Yˆ4 (X )
=
σ ˆ3 = 20.6,
68.9 − 1.4X1 + 4.3X2 + 1.6X3 + 1.6X12 + 0.1X22 − 0.3X32 −1.6X1 X2 + 0.1X1 X3 − 0.3X2 X3 ;
σ ˆ4 = 1.27
−1.633 ≤ Xi ≤ 1.633. 13 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Tire Tread Example Desirability Functions
Desirability 0.4 0.8
200 % Modulus
120
140 160 Value
0.0
0.0
Desirability 0.4 0.8
PICO Abrasion
1000
400
1300
Desirability 0.4 0.8
Hardness
500 Value
600
0.0
0.0
Desirability 0.4 0.8
Elong. at Break
1150 Value
60
65 70 Value
75 14 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Tire Tread Example Classical DI optimization (without noise) Maximization of geometric mean: max
X ∈[−1.633,1.633]
ˆ )= D(Y
4 Y
Yˆi
i=1
Optimum found: X Yˆ
=
(−0.05, 0.145, −0.868)
= (129.5, 1300, 465.7, 68) ˆ d(Y ) = (0.189, 1, 0.656, 0.932) ˆ = 0.583 D → acceptable desirability
15 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Tire Tread Example How robust is the optimum? 10000 realisations of quality criteria using optimal factor setting:
130 Y1
150
400
450 500 Y3
550
0 500
1500 Y2
2500
0.15
Density
0.00
0.010 0.000
Density
0.30
110
0.0006 0.0000
Density
0.06 0.03 0.00
Density
εi ∼ N (0, σi2 )
0.0012
Yi = fi (X1 , . . . , X4 ) + εi ,
64 66 68 70 72 Y4 16 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Tire Tread Example How robust is the optimum? Desirability Functions: 8 6 2
4
Density
2.0 1.0
2.0
0 0.0
0.2
0.4
0.6
0.0
0.4 d2
0.8
4
2.0
0.5
1.0
1.5
d1
0.6
3
0.5
2
0.4
Density
D
1
0.3
0
0.2
1.0
0.1
0.0
0.0
Density
0.0
Density
2.5
0.0
3.0
3.5
Density
3.0
Desirability Index:
0.0
0.4 d3
0.8
0.4
0.6 d4
0.8
1.0
17 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Tire Tread Example How robust is the optimum?
0.0
0.2
Fn(x) 0.4 0.6
0.8
1.0
Empirical distribution function of D
0.0
0.1
0.2
0.3 D
0.4
0.5
0.6
Summary: min(D)
=
0
q0.25 (D)
=
0.1097
med(D)
=
0.1872
mean(D)
=
0.1898
q0.75 (D) max(D)
= =
0.2633 0.5757
sd(D)
=
0.108
ˆ opt )) = 0.583 But: D(X → not realistic in the course of the process!! 18 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
The Desirability Index in noisy environments
Robust DI Optimization
⇓ ˆ Optimization of E (D) instead of D → Derivation of the distribution of the DI Yi = fi (x) + εi ,
εi ∼ N (0, σi2 )
⇒ Yi ∼ N (fi (x), σi2 ) ⇒ Distribution of DFs ⇒ Distribution of DI 19 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Distribution of the DI (Geometric Mean) (I) 1. One-Sided Specification: Density Function: fD (D) ≈ − √
1 1 · exp − ∗2 (log(−k · log(D)) − µ∗ )2 2σ 2π · σ ∗ · log(D) · D
Distribution Function: FD (D) ≈ 1 − Φ Qα
log(k) + log(− log(D)) − µ∗ σ∗
≈ exp(− exp(σ ∗ · z1−α − log(k) + µ∗ )) with
med(fD (D)) ≈ exp[− exp(µ∗ )/k] 20 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Distribution of the DI (Geometric Mean) (II) 2. Two-Sided Specification: Density Function fZ (z)
=
√ (z − µ˜1 − µ˜2 )2 2 (z σ˜2 2 − µ˜1 σ˜2 2 + µ˜2 σ˜1 2 ) erf · exp − q q √ 2(σ˜2 2 + σ˜1 2 ) 4 π(σ˜2 2 + σ˜1 2 ) σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2
(z − µ˜1 + µ˜2 )2 (z + µ˜1 − µ˜2 )2 (z σ˜2 2 − µ˜1 σ˜2 2 − µ˜2 σ˜1 2 ) (z σ˜2 2 + µ˜1 σ˜2 2 + µ˜2 σ˜1 2 ) erf erf q q + exp − √ √ 2(σ˜2 2 + σ˜1 2 ) 2(σ˜2 2 + σ˜1 2 ) σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2 σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2 (z + µ˜1 + µ˜2 )2 (z σ˜2 2 + µ˜1 σ˜2 2 − µ˜2 σ˜1 2 ) (z − µ˜1 − µ˜2 )2 (z σ˜1 2 + µ˜1 σ˜2 2 − µ˜2 σ˜1 2 ) erf erf + exp − q q + exp − √ √ 2(σ˜2 2 + σ˜1 2 ) 2(σ˜2 2 + σ˜1 2 ) σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2 σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2 2 2 2 (z − µ˜1 + µ˜2 )2 (z + µ˜1 − µ˜2 )2 (z σ˜1 2 − µ˜1 σ˜2 2 − µ˜2 σ˜1 2 ) (z σ˜1 + µ˜1 σ˜2 + µ˜2 σ˜1 ) erf − erf + exp − + exp q q √ √ 2(σ˜2 2 + σ˜1 2 ) 2(σ˜2 2 + σ˜1 2 ) σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2 σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2 (z σ˜1 2 − µ˜1 σ˜2 2 + µ˜2 σ˜1 2 ) (z + µ˜1 + µ˜2 )2 erf + exp − √ q 2 2 2(σ˜2 + σ˜1 ) σ˜2 σ˜1 2 σ˜2 2 + σ˜1 2
+ exp −
√ with erf (x) = 2 · Φ( 2x) − 1
(Gaussian Error Function). 21 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Visualization
3
3
(3)
1.0
2
2
fD(D)
(2)
fD(D)
(1) 1.5
fD(D)
2.0
2.5
4
4
3.0
5
5
One-Sided Specification
(4)
(5)
(6) 1
(7)
0.0
0.2
0.4
0.6
0.8
0
0
0.0
0.5
1
(8)
1.0
Wünschbarkeitsindex D
0.0
0.2
0.4
0.6
0.8
1.0
Wünschbarkeitsindex D
0.0
0.2
0.4
0.6
0.8
1.0
Wünschbarkeitsindex D
0.0
1.0
fD(D) 2.0
3.0
Two-Sided Specification
0.0
0.2
0.4
D
0.6
0.8
1.0
22 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Distribution of the DI (Minimum of the DFs) D := mini=1,...,k di
⇒
FD (D) = 1 −
k Y [1 − FDi (di )] i=1
1. One-Sided Specification:
k k X 1 log(− log(D)) − µ ˜j 1 ˜i Y φ log(− log(D)) − µ fD (D) = − Φ D · log(D) i=1 σ ˜i σ ˜i σ ˜j j=1
j6=i
2. Two-Sided Specification: fD (D) =
−
k X i=1
" # " #! (− log(D))1/ni (− log(D))1/ni − µ ˜i (− log(D))1/ni + µ ˜i φ +φ ni D log(D)˜ σi σ ˜i σ ˜i " # " #!! k Y (− log(D))1/nj − µ ˜j (− log(D))1/nj + µ ˜j · −1 + Φ +Φ σ ˜j σ ˜j j=1
j6=i 23 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Visualization
3.0
5
6
One-Sided Specification
4
2.5
5
(3)
3
4
2.0
fD(D)
3
fD(D)
(4)
(6) 2
fD(D)
(7) (5)
(2)
1.5
(1)
0.0
0.2
0.4
0.6
0.8
1.0
0
0
0.0
1
0.5
1
1.0
2
(8)
0.0
0.2
Wünschbarkeitsindex D
0.4
0.6
0.8
1.0
0.0
0.2
0.4
Wünschbarkeitsindex D
0.6
0.8
1.0
0.8
1.0
Wünschbarkeitsindex D
3
0.5 (5)
1
0.5
(4)
0
0.0
(6) 0.2
0.4
0.6
Wünschbarkeitsindex D
(8) (9)
2
1.0
(2)
0.0
1.0
fD(D)
1.5
(3)
fD(D)
4
(7) (1)
0.0
fD(D)
2.0
5
1.5
2.5
6
3.0
7
2.0
Two-Sided Specification
0.8
1.0
0.0
0.2
0.4
0.6
Wünschbarkeitsindex D
0.8
1.0
0.0
0.2
0.4
0.6
Wünschbarkeitsindex D
24 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Desirabilities vs. Pareto-Optimality Prior knowledge: Link criterion Specification of expert knowledge by means of link criterion before optimization → (Possibly) unambiguous solution Examples: Desirability index, weighted sum
Posterior knowledge: Multiobjective Optimization Optimization without usage of prior knowledge: → Set of optimal solutions → Selection of set of desired solutions after optimization based on expert knowledge Example: Pareto optimization 25 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Desirabilities vs. Pareto-Optimality Pareto-Optimality A realization of quality criteria (QC) Y = (Y1 , . . . , Yk )0 is said to be pareto-optimal if there is no other realization that keeps up the process quality regarding all criteria and improves at least one criterion. → The process cannot be improved upon without deteriorating at least one quality criterion.
A corresponding influence factor setting X = (X1 , . . . , Xn )0 then is pareto-optimal in factor space if the corresponding criteria vector Y is pareto-optimal in criteria space.
Problems: Determination of the Pareto-Set, Selection of optimal factor setting 26 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Pareto-Optimality of the optimal solution
Geometric Mean The optimized factor levels based on the DI are pareto-optimal The DI can be understood as a method for selecting a pareto-optimal solution from the Pareto-Set.
Minimum of DFs Pareto-Optimality of X opt is not guaranteed. But: easy to interprete The applying experts have to be aware of this “non-pareto-optimality“
27 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
DFs as a tool for incorporating preferences in EMOA
Desirability MCO problem: Minimize with
−d(Y ) = −d[f (X )] = −(d1 [f1 (X )], . . . , dk [fk (X )]) Y = (Y1 , . . . , Yk ) ∈ Y,
objectives
X = (X1 , . . . , Xn ) ∈ X ,
decision variables
di ∈ [0, 1]
i = 1, . . . , k,
desirability functions
fi ,
i = 1, . . . , k.
mathematical models
→ Focussing on relevant parts of the Pareto front becomes possible
28 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Binh-problem, Harrington DFs (d1 ,d2 )
30
40
50
d(Y) ●
10
20
0
10 20 30 40 50
Y
30
40
50
10 20 30 40 50
f1
●
0
10
30 Y
●
50
40 10
20
f2
30
40 0
0
10
20
f2
30
40 30
f2 20 10 0
●
0
50
Y
50
20
●
f1
0
●
10
●
0.0 0.2 0.4 0.6 0.8 1.0
●
0
d(Y) 0.0 0.2 0.4 0.6 0.8 1.0
●
50
d(Y) 0.0 0.2 0.4 0.6 0.8 1.0
NSGA-II, µ = 200, ]FE = 10000
0
10 20 30 40 50
f1
29 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
The Desirability Index for Process Control? DI was already used for Process Optimization → Ideal measure for evaluating the “degree of optimality“ over time Interpretation of Out-Of-Control Signals possible → Control limits for the DI can be transferred back to DFs Complexity reduction compared to separate univariate control charts for quality criteria
1.0
Exemplary Control Chart Out−Of−Control
0.8
UCL
0.6
T
0.4
LCL
0.2
Out−Of−Control
t
0
20
40
60
80
100 30 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Suitable Control Charts for the Desirability Index Problem: Most univariate control charts assume normality! → Adapted Single-Measurements-Control Chart : LCL/UCL = Q0.005 /Q0.995
resp.
LWL/UWL = Q0.025 /Q0.975 ,
0.2
0.4
D
0.6
UCL
LCL
0
50
100 t
150
200
Wünschbarkeitsindex (D) 0.2 0.4 0.6 0.8
0.8
Examples
UCL
LCL
0
50
100
150
200
31 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Extreme-Value-Chart for DI (Sample Size g ) Control Limits:
g LCL = Q(1− √ 0.99)/2 ,
Warning Limits:
g LWL = Q(1− √ 0.95)/2 ,
g UCL = Q(1+ √ 0.99)/2 g UWL = Q(1+ √ 0.95)/2
0.8
1.0
Example
D 0.6
UCL
0.4
LCL
1
2
3
t
4
5
6
32 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
Thank you very much!
33 / 35
Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
References Harrington, J. (1965): The desirability function, Industrial Quality Control 21 (10), pp. 494 - 498 Derringer, G.C. and Suich, D. (1980): Simultaneous optimization of several response variables, Journal of Quality Technology 12 (4), pp. 214 219 Trautmann, H. and Mehnen, J. (2008): Preference-Based Pareto-Optimization in Certain and Noisy Environments; Engineering Optimization 41, pp. 23-28 Mehnen, J., Trautmann, H. and Tiwari, A. (2007): Introducing User Preference Using Desirability Functions in Multi-Objective Evolutionary Optimisation of Noisy Processes. CEC 2007, IEEE Congress on Evolutionary Computation, pp. 2687-2694, Kay Chen Tan, Jian-Xin Xu (eds.), Singapore, 2007.
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Desirability Index
Robust DI Optimization
Desirabilities vs. Pareto-Optimality
DI Process Control
References
References Mehnen, J. and Trautmann, H. (2006): Integration of Expert’s Preferences in Pareto Optimization by Desirability Function Techniques; In: Proceedings of the 5th CIRP International Seminar on Intelligent Computation in Manufacturing Engineering (CIRP ICME ’06), Ischia, Italy, R. Teti (ed.), pp. 293-298 Trautmann, H. and Weihs, C. (2006): On the Distribution of the Desirability Index using Harrington’s Desirability Function, Metrika 63 (2), pp. 207-213 Trautmann, H. (2004a): Qualit¨atskontrolle in der Industrie anhand von Kontrollkarten f¨ ur W¨ unschbarkeitsindizes - Anwendungsfeld Lagerverwaltung; Dissertation at Dortmund University, http://hdl.handle.net/2003/2794 Trautmann, H. (2004b): The Desirability Index as an Instrument for Multivariate Process Control; Technical Report 43/04, SFB 475, Dortmund University.
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