Multi-Criteria-Optimisation and Desirability
Indices
Dipl.-Stat. Detlef Steuer
Universitat Dortmund
LS Computergestutzte Statistik
steuer@amadeus.statistik.uni-dortmund.de
March 1999
Abstract
The basic ideas of Desirability functions and indices are introduced
and compared to other methods of multivariate optimisation. It is
shown, that gradient based techniques are not in general appropriate
to perform the numerical optimisation for Desirability indices. The
problems are shown for direct modelling of Desirability indices. An
example is given to illustrate the sensitivity of estimated optimum
factor settings to modelling errors for individual targets.
Keywords: MCO, MCD, MCDA, Desirability Function, Desirability
Index, dimension reduction, numerical optimisation.
1 Introduction
When dealing with industrial production processes we often have to assess
quality of products. It turns out, that often more than one measured variable
or property has to be taken into account to describe "quality". Normally
quality not only should be measured, but also tried to improve.
The optimisation of quality will normally prove more dicult in the case of
two or more competing properties. Often it is not possible to improve one
of them without deteriorating one or more of the others. In the context of
optimisation properties are also called "targets".
A very common example can be found in the context of drug-design: A new
pill has to full two, often contradictory, requirements. For example rstly it
should be very well solvable in water, secondly it should not fall apart without
1
water. While experimenting, the designer realizes that improving solvability
deteriorates durability. Solvability and durability here are two competing
properties in drug-design. The problem with competing properties shows
up, if there exist two designs, one with better solvability, the other with
better durability. It becomes impossible to rank these alternatives without
knowledge about the relative importance of the variables. Weights of some
kind have to be introduced.
The described situation of multiple, partly contradictory objectives in op-
timisation is the starting point of a branch of Operations Research (OR),
called Multi-Criteria-Optimisation (MCO), Multi-Criteria-Decision-Making
(MCD) or Multi-Criteria-Decision-Analysis (MCDA). It is clear, that prob-
lems of this kind are not restricted to industrial production processes. They
appear wherever the "value" of a "decision" is measured in more than one
dimension. The word "product" will be used in this general interpretation
throughout the paper.
For this paper a product's multiple properties are assumed to be competing,
i.e. they can not be optimised at the same time. Otherwise all problems
would reduce to the one-dimensional case of optimisation, which is not of
interest here.
The aim of this work is to discuss some of the OR methods to handle the
situation of multiple objectives from a statistical point of view, with stress
on the so called desirability indices.
In Section 2 the formal context of multiple objectives from a statistical point
of view will be introduced, where the properties can not be set directly, as
it is often assumed in OR (within some restricted area). Instead properties
have to be understood as functions of some underlying controlled variables
(factors) which must be approximated via statistical modelling. Section 3
presents mathematical and practical concepts for MCO. The concept of "de-
sirability" is explained in Section 4. After that two examples for "desirability
functions" used in practice are given in Sections 5 and 6. In Section 7 we
show, that gradient based optimisation techniques are unsuitable for the
presented optimisation problem without further considerations. We discuss
some aspects of modelling "desirability indices" in Section 8 and give a short
conclusion in Section 9.
2 Formalism
In this section the formal setting of this paper is dened.
The focus here lies on a vector P 2 IR
Z
, which is dened by its coordi-
nates Y
i
(P ) = p
i
; i = 1 ; : : : ; Z. This vector P stands for a product with Z
2
properties, which describe its quality.
The properties Y
i
; i = 1 ; : : : ; Z;are functions of a nite number of factors
X
1
; X
2
; : : : ; X
F
and a stochastic error term . Therefore we get:
P =
0
B
B
B
@
Y
1
(P )
Y
2
(P )
.
.
.
Y
Z
(P )
1
C
C
C
A
with Y
i
(P ) = f
i
(X
1
; : : : ; X
F
;  ); i= 1 ; : : : ; Z:
This formula deserves some remarks:
First the same set of factors is used for all properties in this formula. This
is not a restriction, if we allow them to be purely formal parameters of the
f
i
. Second we note the error is not restricted to be additive!
Under these assumptions the objective in MCO in this statistical context may
be formulated as nding the setting X
opt
= ( X
1;opt
; : : : ; X
F;opt
), for which the
expectation E(Y
1
; Y
2
; : : : ; Y
Z
) is "best"!
Implicitly this calls for a ranking in IR
F
; F > 1. As is well known, no such
ranking exists. Therefore it can only be tried to get "close to a ranking" in
some sense. In the following the basic concepts of multivariate ranking will
be shown.
3 Mathematical concepts for MCO
In this section mathematical concepts and problems for ranking multivariate
observations will be presented. These considerations lead to requirements for
practical MCO-procedures.
3.1 Domination
A rst, very optimistic, approach to MCO is hoping for one "really best"
object, which means an object that is better than all alternatives in every of
the multiple targets. The idea of a ranking in IR
Z
is abandoned here. The
problem of ranking IR
Z
is reduced to nding the "maximum" in a given set
of alternatives using a coordinate wise "better" relation, symbolised here by
\>".
This concept is called domination and a formal denition is given below:
Denition 1 (Domination) Given two objects P
1;2
2 IR
Z
described by
the same properties Y
i
(P ); i = 1 ; : : : ; Z, it is said, that P
1
dominates P
2
(P
1
>> P
2
), if Y
i1
 Y
i2
8 i = 1 ;2; : : : ; Z and Y
j1
> Y
j2
for one 1  j  Z.
3
At the same time the domination of factor settings has been induced as
follows:
Denition 2 (Domination in factor space) Given two settings X
1
re-
sulting in product P
1
:= E(P (X
1
)) and X
2
resulting in product P
2
:= E(P (X
2
))
and the products described by the same properties as above, it is said, that
X
1
dominates X
2
(X
1
>> X
2
), if P
1
>> P
2
.
Formally this is a nice and clean mathematical concept. For practical use it
has an important drawback: There may not exist a dominating object. And
worse, as Z increases, it becomes more and more unlikely for an object to
dominate any other.
3.2 Pareto-Optimality
Having seen the probable non-existence of solutions using domination, the
less strict concept of "pareto-optimality" can be tried. It is dened as follows:
Denition 3 (Pareto-Optimality) Given a set M  IR
Z
of objects, an
object P 2 M is called pareto-optimum, if there is no object Q 2 M with
Q >> P .
Or less formal: If an object can not be improved in the coordinate wise
sense of domination, it is optimal in some way. Inversely, and perhaps more
important: If an object is not pareto-optimum, it should not be considered
as a solution of a MCO problem!
Pareto-Optimality in factor space can be dened analogous to domination in
factor space (see denition 2):
A factor setting X is pareto-optimum in factor space, if the corresponding
product P := E(X) is pareto-optimum in product-space.
This leads to a rst strict requirement for a MCO procedure: Any proposed
solution of a MCO problem must be pareto-optimal!
While the problem with domination is the missing guarantee for nding a
solution, the problem with pareto-optimality is -in general- the possibility of
many solutions for the MCO.
On the other hand the experimenter is looking for a unique, best solution,
so it has to be decided among the proposed solutions. To accomplish a
decision weights for the dierent targets must be introduced, representing
their relative importance. The nal decision will be highly problem specic.
Thus a second requirement for MCO procedures is found: it has to be possible
to take dierent importances of targets into account!
4
In a mathematical sense we are looking for a functional C : IR
Z
! IR, which
allows the ranking of all objects in IR in an appropriate, problem specic
way.
3.3 Approaches to MCO in OR
A lot of heuristic methods for solving MCO problems have been developed in
OR. The statistical properties of these algorithms often are not well known.
This is not too surprising, as these algorithms were not introduced with
focus on their statistical aspects. Instead they are constructed to generate
solutions in practice.
A short list of approaches symbolising dierent classes of MCO-solvers, could
contain the following entries:
 Pareto optimality, standing for the mathematical, not problem specic
approach,
 overlay plots, a graphical method,
 Prometee, representing the so called outranking procedures,
 desirability functions and -indices, which will be presented in detail in
this paper and
 (monetary) loss functions, a member of the utility function class.
Each of these entries has its specic advantages and disadvantages. Pareto
optimality has been considered in the section above. Overlay plots try to
simultaneously look on contour plots of all targets. This method fails with
more than three or four targets. Prometee uses pairwise comparison of all
possible experimental results for constructing a (partial pre-) order among
them. For each target Y a so called preference function P
Y
is dened, that
assigns a numerical value to the dierence of two experiments a and b in this
target: P
Y
(a; b) = P
Y
(ab). An expert dened weighted sum of these target
preferences serves as \preference index" (a; b) for comparison of two exper-
iments. The sum over all preference indices
P
b
(a; b) is called the \positive
outranking 
ow" for experiment a then. The sum
P
b
(b; a) is called the
\negative outranking 
ow" of experiment a. Based on these \
ows" a rank-
ing of all experiments can be produced. Prometee as a MCO tool is very

exible to use, but hard to interpret. Little is known about its robustness.
For monetary loss functions there is a problem in their concreteness: It is
in general not possible to specify them as exact as it would be required to
really interpret them as money value.
5
A more detailed overview about these techniques can be found in [4]. Desir-
ability functions will be explored in the next sections.
4 The Concept of Desirability
In this section we will describe the basic concept of desirability for MCO.
In the formal description of the MCO problem we nd Z dierent targets
to be optimised. Translated to real world problems it means that up to Z
dierent scales of measurement, i.e. times, lengths, weights and so on, have
to be forced to be commensurable. Some of the existing heuristics try to
ignore this problem. They assign a weight to each target and compare the
weighted units.
Desirability is dierent here. In a rst step every objective Y
i
; i = 1 ; : : : ; Z
gets translated by an individual, so called "desirability-function" into a unit-
less desirability-scale. The step of dening the desirability for every possible
outcome of Y
i
obviously becomes crucial to the process of multivariate op-
timisation. Naturally the statistician is unable to determine good or bad
weights! A close collaboration of statistician and expert is very important
when xing the desirability function for a target.
Which are useful functions for using as desirability-functions d? First we can
force them to take values in [0; 1]. The limitation to [0; 1] has a technical
justication, which will become obviously later on.
The most general denition of a desirability function now is the following:
Denition 4 (Desirability function) Any function d with
d : Domain of Y ! [0; 1]; Y ,! d(Y ):
is called desirability function (DF).
Not all of these functions may usefully serve as DFs. To full the require-
ments formulated in Section 3 it has to be required, that
1. d is 
exible enough to allow problem specic formulations. It should
be possible to give parameters LSL (lower specication limit), USL
(upper specication limit), T (target value) and, if needed, possibly
dierent weights 
l
and 
r
for deviations to the left respectively right
of T .
2. For target value problems d should increase monotonically in (1; T )
and decrease monotonically in (T;1). This guarantees the pareto-
optimality of any desirability-optimum solution of the optimisation
problem!
6
Perhaps the last property requires a little proof:
Lemma 1 (Proof: Desirability optimum points in factor space are
pareto-optimum.)
A desirability optimum X
opt
is a local optimum of the desirability-function
over factor space. Assume X
opt
is not a pareto-optimum point. Then there
exists Y
opt
in factor space, which gives a superior result and therefore higher
desirability. Contradiction! 2
Concrete examples of DFs used in practice are given in Sections 5 and 6.
After scaling all targets individually to a desirability scale, they have to be
combined to a single number, the overall desirability or desirability index
of the product P . The following gives the general formal denition for a
desirability index:
Denition 5 (Desirability Index) A desirability index DI is a function
D with
D : [0 ;1]
Z
! [0; 1]:
The obvious idea for a concrete formulation of D is to use some kind of mean
value. Often the geometric mean is used for D. It has the feature to assign
a DI of 0, if any of the individual DFs is 0. This is nicely interpretable: If
one of the product's properties is completely unacceptable, the product as
a whole is unacceptable. The use of the geometric mean also gives justi-
cation for choosing the interval [0; 1]. Individual desirabilities greater than
1 would allow to compensate shortcomings in some of the other properties.
Nevertheless other function are used also. Of these the maximin DI is most
important. It denes
D(P ) := max
X
min
i=1;::: ;Z
d
i
(X):
This formulation can be nicely interpreted also: A product is only as good
as it's worst property at the optimum factor setting.
In the statistical context the desirability index D may now be represented as a
function of the factors X
i1
; : : : ; X
iF
and the unknown errors 
i
; i = 1 ; : : : ; Z.
D(P ) := D(d
i
(Y
i
)
i=1;::: ;Z
)
= D(d
i
(f
i
(X
i1
; X
i2
; : : : ; X
iF
; 
i
))
i=1;::: ;Z
)
Each factor setting is evaluated by a number D 2 IR. The canonical ranking
in IR induces a ranking in IR
Z
and the "best" factor setting can easily be
7
identied. As a result the multi objective optimisation problem has been
turned into a response surface problem.
In literature about applications of DFs the most common forms of DFs are
those of Harrington and Derringer/Suich. Both will be presented here with
their formulation of desirability functions and indices for target value prob-
lems.
5 Desirability Function I
The concept and the name "desirability" were introduced by Edwin C. Har-
rington Jr. ([3]) in 1965.
For DFs Harrington used the following exponentials to handle target value
problems:
d
H
(Y ) := e
jY
0
j
n
; where Y
0
is an appropriate transformation of Y: (1)
Appropriate in the sense of Harrington is to choose Y
0
in a way, so that
d
H
(LSL) = d
H
(USL) = 1 =e. As a possible transformation he gives
Y
0
=
2Y  (USL + LSL)
USL LSL
:
Dened this way d
H
(Y ) is symmetric around the centre between LSL and
USL. The parameter n serves as an "deviation importance" parameter. Large
values of n result in 
at curves around the centre, thus punishing deviations
from the centre less hard than low values of n. Figure 1 shows two typical
DFs of this type, with parameters n = 4 and n = 1.
For combining all d
H
i
; i = 1 ;2; : : : ; Z Harrington preferred the geometric
mean. Written as a function in Y
i
; i = 1 ;2; : : : ; Z and nally in X
j
; j =
1; 2; : : : ; F we get the DI of Harrington as:
D
H
(P ) :=
Z
v
u
u
t
Z
Y
1
d
H
i
(Y
i
)
=
Z
v
u
u
t
Z
Y
1
d
H
i
(f
i
(X
1
; X
2
; : : : ; X
F
; 
i
)):
For optimisation purposes the Z-th root is unimportant, however for com-
paring the overall desirability with the per target desirabilities it is essential.
Harrington developed his DF with interpretation of the result in mind. In
his paper he even gave a scale for interpretation to use with his index. He
proposes to interpret
8
Figure 1: Desirability functions of Harrington-type for two dierent values
of n
specification
de
sir
ab
ilit
y
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
LSL USL
n=4
n=1
Desirability functions of  Harrington-type
de
sir
ab
ilit
y
1/e
 desirability 1 as "ultimate satisfaction" or "improvement beyond this
point has no value";
 desirability 0:8 1 as "excellent" or "well beyond anything available";
 desirability 0:630:8 as "good" or "slight improvement over industrial
quality";
 desirability 0:4 0:63 as "acceptable, but poor";
 desirability 0:3 0:4 as "borderline";
 desirability 0 0:3 as "unacceptable to completely unacceptable".
There are some disadvantages to Harrington's approach. The DFs are not too

exible, as for example they are always symmetric. Furthermore Harrington
has to use dierent exponential functions for maximisation (minimisation)
problems, thus hardening the comparability of multiple desirabilities.
9
On the other hand his functions have a big plus for being given in closed
form and being dierentiable.
6 Desirability Function II
Derringer and Suich dened a new class of desirability functions in [2] to gain

exibility for modelling the importance of individual targets.
Their denition of a desirability function for target value problems is given
below:
d
DS
(Y ) :=
8
>
>
>
<
>
>
>
>
:
0; for Y < LSL
(
Y  LSL
T  LSL
)

l
; for LSL  Y  T
(
USL Y
USL T
)

r
; for T < Y  USL
0; for USL < Y
The parameters 
l
and 
r
are weights for deviations to the left respectively
to the right from the target. Values near 0 mean unimportant deviations,
while high values stand for very important targets. In gure 2 characteristic
desirability functions have been plotted, to show the 
exibility of this new
DF class.
For constructing the DI Derringer/Suich also propose the geometric mean.
Mini- or maximisation problems can be handled consistently by using only
one branch of d
DS
. In this case T denes a value for which all lower (higher)
values are accepted as "perfect", thus giving desirability 1. Besides the possi-
bility to model asymmetry herein lays the main advantage of Derringer/Suich
type functions over those of Harrington type. Furthermore it is straightfor-
ward to generalise these functions to more than two segments.
However, they could not get the desired 
exibility for no price: the closed
form of the DFs had to be changed to a piecewise denition. On the plus side
we nd unacceptable target values having desirability 0, thus giving overall
desirability 0 if a single target is unacceptable. The parameters 
l
and 
r
are important for the optimisation step, so they have to be chosen carefully.
Optically there seems to be an easy interpretation of the dierent values of

l;r
. In practice a doubled value of  is told to give a doubled importance
to deviations from the target value. Mathematically it is obvious that this
simplicity is only a "rule of thumb" and has to be questioned.
Due to the superior 
exibility of the Derringer/Suich function class Har-
rington's approach could be discarded. But piecewise denition with non-
dierentiable points at interval borders lead to problems in analytical han-
dling of the new DFs.
10
An implementation of Derringer/Suich desirabilities is found in STAVEX, a
software package for design and analysis of experiments, developed at CIBA-
GEIGY, Basel [5].
Figure 2: Desirability functions of Derringer/Suich type for two dierent
values of 
l
and 
r
, asymmetric case.
specification
de
sir
ab
ilit
y
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
LSL Target USL
βl = βr = 1
βl =
1
3
βr = 3
Desirability functions of Derringer/Suich type
7 Practical optimisation issues
The main achievement using desirability indices has been the conversion of
the original problem into a response surface problem. The founders of the
DFs presented here wanted to use the well known optimisation techniques
for response surfaces, mainly the gradient based ones. For this purpose the
Derringer/Suich DFs had to be rened to be dierentiable everywhere. That
work was done by [1] by simply tting fourth degree polynomials around each
non dierentiable point.
In principle both types of DFs allow the usage of classical techniques for
optimising a real-valued function. In literature those techniques were ap-
plied without any further considerations. See for example [2] or [1]. This
11
behaviour by the scientists may have been supported by the apparently sim-
ple structure of the DFs. Each DF has a single optimum, which seems to
assure unimodality of the respective DI. Unfortunately this turns out to be
an oversimplication.
Figure 3: Desirability function of Derringer/Suich type for Y = X
2
+1; T =
0:5; LSL = 1; USL = 1; 
l
= 1; 
r
= 1 as function of factor X (1-dim)
Desirability as funtion of factor space
Factor Space X
D
es
ira
bi
lity
-2 -1 0 1 2
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
If considered as mappings from factor space into desirability scale DIs are
not unimodal! Figure 3 gives the most simple example for bimodality of a
DI as a function of a single controllable variable. In this special case the DF
equals the DI, because only one target exists. As Figure 3 shows factor space
can even fall apart in disjunct regions of possible desirability. Furthermore
more than one local maximum may exist. It can easily be seen, that not
all local maxima of a DI must be of same height, if more than one target is
considered.
As a consequence classical (gradient based) optimisation techniques may fail,
if their starting point was not chosen very well. After looking at Figure 3 this
is obvious, but it was not considered by many experts and was not mentioned
in literature known to us. Before any gradient based technique can be applied
12
to optimise a DI, unimodality of the response surface has to be assured.
Figure 4: Desirability function of Derringer/Suich type as a function of factor
space (2-dim). Y = X
2
1
+X
2
2
, LSL=-1, T=0, USL=4, 
l
= 
r
=
1
2
.
Desirability index as function of factor space
de
sir
ab
ilit
y
X1
X 2
1
An alternative approach to optimise DIs is used in STAVEX [5]. An exhaus-
tive grid search is performed to nd the optimum settings. This approach
is independent of unimodality of the response, but computing time restricts
this approach to low dimensions of factor space. The cause for using grid
search in STAVEX has been the non dierentiability of the Derringer/Suich
DFs. A modern implementation should use a sophisticated search heuristic
like simulated annealing for DI-optimisation in the case unimodality can not
be assumed.
To give an impression of the beauty and the possible complexity of even
a single DF the example in Figure 4 is given. In there a DF of a single
target is shown as a mapping from a two dimensional factor-space. As can
13
be seen all optimum points lie on a circle in factor space. All these points are
equivalent with respect to desirability. There is no unique optimum. This
shows the need of investigating the structure of possible optima of DIs in
later work. Insights in this eld can lead to specialised search strategies for
DI optimisation, therefore reducing the computing power that is needed in
this process.
8 Aspects of modelling desirability indices
A prerequisite for applying any of the techniques given above is the estimation
of individual models
^
f
i
for the targets Y
i
; i = 1 ; : : : ; Z. These
^
f
i
are inserted
into the individual DFs to calculate the DI for every setting X needed in the
optimisation step.
8.1 Direct modelling
If the DIs can be estimated directly, an important decrease in computational
complexity for solving the MCO can be expected. It is hoped, that not each
of the X
i
which appear in any of the models for dierent targets will be
important in a model for the DI, thus reducing the dimension of the rele-
vant factor space. Nevertheless for both types of DIs mentioned the possible
models come out to be very complicated, even if linear models for the Y
i
are assumed. As an example two targets Y
i
, each a linear function of two
factors X
1;2
, an intercept 
i
and an error 
i
; i = 1 ;2, no interactions are used.
In the Harrington case polynomials of high degree appear in the exponent of
the DI:
D
H
=
q
d
H
1
(Y
1
) 
 d
H
2
(Y
2
)
= (exp(j(
1
+ 
1;1
X
1
+ 
1;2
X
2
+ 
1
)j
n
1
j(
2
+ 
2;1
X
2
+ 
2;2
X
2
+ 
2
)j
n
2
))
0:5
For the sake of simplicity it is assumed here that none of the two targets
needs a further transformation, that is Y
i
= Y
0
i
; i = 1 ;2, which implies
LSL=-1 and USL=1 for the Harrington case. Now it is clear that a model for
 log(D
H
) representing \the truth" has to be of order max
i
(order(d
H
i
(Y
i
)
n
i
))
in the most simple case. Furthermore the error terms will interact with the
eects if n
i
6= 1.
14
Under the same assumptions the Derringer/Suich DI leads to models con-
taining all interactions of the main eects relevant for any of the Y
i
; i =
1; 2; : : : ; Z.
D
DS
=
q
d
DS
1
(Y
1
) 
 d
DS
2
(Y
2
)
= (( 
1
+ 
1;1
X
1
+ 
1;2
X
2
+ 
1
) 

(
2
+ 
2;1
X
1
+ 
2;2
X
2
+ 
2
))
0:5
In this latter case the most simple assumptions were made, too. Only one
branch of the DF is considered for both targets, the exponents were set to
unity. A model for (D
DS
)
2
would have to be of order
Q
i
(order(d
i
)). The
usage of non-integer importance parameters 
l
and 
r
results in models of
non-integer degrees.
As a bottom line direct modelling of DIs does not seem to be an option.
8.2 Sensitivity to modelling errors: a simulation case
study
Another aspect, which has not found the attention it deserves, is the uncer-
tainty in estimating the optimum factor setting X
opt
. To examine the eect
of estimating the underlying functions f
i
; i = 1 ;2; : : : ; Z; with some error a
simulation case study has been performed.
In their paper Derringer and Suich use a set of chemical data to apply their
DFs to. They have four targets Y
1
to Y
4
and three controllable variables X
1
to X
3
. The data were generated using a central-composite design with 20 ex-
periments, to be able to t a second-order model
^
f
i
including all interactions
to each of the four targets. Using their data they got the following models
for the targets Y
i
:
^
f
1
= 139:1 + 16 :5X
1
+ 17 :9X
2
+ 10 :9X
3
 4:0X
2
1
 3:5X
2
2
 1:6X
2
3
+5:1X
1
X
2
+ 7 :1X
1
X
3
+ 7 :9X
2
X
3
; sd = 5 :6;
^
f
2
= 1261:1 + 268:2X
1
+ 246:5X
2
+ 139:5X
3
 83:6X
2
1
 124:8X
2
2
+199:2X
2
3
+ 69 :4X
1
X
2
+ 94 :1X
1
X
3
+ 104:4X
2
X3; sd = 328:7;
^
f
3
= 400:4 99:7X
1
 31:4X
2
 73:9X
3
+ 7 :9X
2
1
+ 17 :3X
2
2
+ 0 :4X
2
3
+8:8X
1
X
2
+ 6 :3X
1
X
3
+ 1 :3X
2
X
3
; sd = 20 :6;
^
f
4
= 68 :9 1:4X
1
+ 4 :3X
2
+ 1 :6X
3
+ 1 :6X
2
1
+ 0 :1X
2
2
 0:3X
2
3
1:6X
1
X
2
+ 0 :1X
1
X
3
 0:3X
2
X
3
; sd = 1 :27:
15
Targets Y
1
and Y
2
are of the type \the more the better". Targets Y
3
and
Y
4
are real target value problems. All DFs in the original paper were lin-
ear. A numerical optimisation results in an optimum factor setting X
DS
opt
=
(0:05; 0:145;0:868)
For a simulation these four estimated functions now worked as \known world".
The experimental design Derringer and Suich performed was 1000 times re-
peated in simulation, using the tted models
^
f
i
as known f
i
; i = 1 ;2; 3; 4;
and adding random errors according to the estimated standard deviations.
Optimally the 1000 simulated optima
^
X
Opt
should scatter around the "true"
X
DS
Opt
, giving overall desirabilities near the optimum desirability. Unfortu-
nately this is not the case for the Derringer/Suich data!
The result of the simulations can be assessed best looking at Figure 5. We
nd the expected scatter around the "true" optimum X
DS
Opt
in each of the
two-dimensional projections, but circa 5% of the
^
X
Opt
show up far away
from X
Opt
. A closer look reveals, that many of these points predicted as
optima give an overall desirability of 0, if evaluated with the known, true
desirabilities! These points are marked with crosses in Figure 5.
The cause for this phenomenon lies in the big standard deviation of target
Y
2
. This results in poorly estimated
^
f
2
in simulations and from time to time
this results in a qualitatively dierent response surface for the DI, giving
nonsense optima.
9 Conclusion
Desirability is a concept for solving MCO based on rescaling and weighting
individual targets. For assigning the weights a strong collaboration between
the statistician and the expert is a must!
Direct modelling of the DI does not seem to be useful for reducing dimension
of relevant factor space.
The unsatisfying results in the simulation study will lead to research con-
cerning the interaction of estimating relations f
i
, setting lower and upper
bounds for poorly tted targets and the sensitivity of the prognoses of X
Opt
.
It is hoped to nd hints how DIs can be turned into self-diagnostic tools,
identifying critical targets or factors and those reducing their sensitivity.
16
References
[1] E. del Castillo, D.C. Montgomery, and D. R. McCarville. Modied desir-
ability functions for multiple response optimization. Journal of Quality
Technology, 28(3):337{344, July 1996.
[2] G. Derringer and R. Suich. Simultaneous optimization of several response
variables. Journal of Quality Technology, 12(4):214{219, 1980.
[3] E.C. Harrington Jr. The desirability function. Industrial Quaily Control,
21(10):494{498, 1965.
[4] M.M.W.B. Hendriks, J. H. de Boer, A.K. Smilde, and D.A. Doornbos.
Multicriteria decision making. Chemometrics and intelligent laboratory
sytems, (16):175{191, 1992.
[5] W. Seewald and C. Weihs. STAVEX User Manual. CIBA-GEIGY Ltd.,
4.100 edition, 1995.
17
Figure 5: Simulated optima for Derringer/Suich data as input model. The
crossing dashed lines give the analytically correct optimum.
0 2 4 6
0
1
2
3
4
5
6
Two-dimensional projections of 1000 estimated optima
Factor 1
F
a
c
to
r 
2 XXX
X X
X
XX
X
X
X
X X
X
X
X X
X
X
X
X
X
X
X
X
X
X
XX
X
X
X
X X
X
X
++
++ ++ ++ +
+
+
+ +
+ +
+ +++ + +++ +++
+
+
+
+
+ +++
+
+ ++
+ +
+ ++ +
+ ++
+ +
+
+ ++
++
++
+++
+
++
+
+
+
+
++
+
+
+
+ +
+
+
+
+
+
++
+
++ +
+
+
+
+
+ ++++++ +
+
++ +
++
+ +
+
+
+ +
+++
+
+
+
++
+
+
+
+
+
+
++
+
+
+
+
+
+
++
+
++ ++ +
+ + +
+
++ ++ + ++
++
+ +
+
++ + + +
++
+ +
++
+
++
+
+ +
+++ +
+
+
+++
+
+
+ +
+
+
+
+
+
+
+
+
+
+
+ +
+
+ +
+++
+
+
+
++
+
++
+
++
+
++ + +
+ +
+
+
+
+
+
+
++
+
+
+
+
+ +
+ +
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+
+
+
+
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+
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+
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+
+
+
+
+
+
+ +
+ + + +
0 2 4 6
-
4
-
3
-
2
-
1
0
1
Factor 1
F
a
c
to
r 
3 +
+
++ +
+ ++
+++
+
+
+
+ ++ +
+ +
+ +
+
+
+
+
+
++
+ +
+
++
++ + ++
+
+
+
+
+
+
+
++
+
+
++ ++
+ +
+
++
+
+
+
+ +
+
+ ++
++
+
+
++
+
+
++
++
+
++
+
+
+
+
+
++
+
+
+
+
++
+
+
+
+
+
+ ++
+
+ +
+
++
+ +
++
+
+
+
+
+
++
+
+
+
+
+
+
+
+
+ +
+
+
+ +
+
+
+
+
+
+
++
+
++ +
+
+
+ +
+
+
+
++ +++
++ ++
+
+
+
+
+
+ +
+ +
+ +
+ +
+
++
++
+
+
+ + +
+
+
+
+
+
+
+ +
++ +
+
+
+
+
+ +
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+
+
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++
+
+
+
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+
+++
+
+
+
+
+
+
+
+
+
+
+
+
+ +
+ +
+
+
+
+
+
+
+
+
+ +
+
+
+
+
+
+
++
+
+ +
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++
++
+
+
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+
X
X
X
X
X
X
XX
X
X
X
X
X
X
X
X X
X
XX
X
X
X
X
X
XX
X
X
X
X
X
X X X
X
X
0 1 2 3 4 5 6
-
4
-
3
-
2
-
1
0
1
Factor 2
F
a
c
to
r 
3
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
XX
X
XX
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X+
+
+ ++
+ + +
+ ++
+
++
++ + +
+
++
++ +
+
+
+
+
+
+
+ +
+
+
+
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+
+
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++
+
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+
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+
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+
+
+++ ++ +
+
++
+
+
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+
++
+
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++
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+
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+
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+ ++ + +
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+
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+ +++
+
+
+
+
+ +
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+
+
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+
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+ +
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+
+
+
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+
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+
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+ ++
+
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+ +
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+
+
+ ++
+ +
+
+
+
+
+
+
+
+
+
+
++
+
+
+
+ +
+
+
++
+
+ +
+
+
+
+ +
18