MULTIOBJECTIVE OPTIMIZATION AND QUANTITATIVE TRADE-OFF ANALYSIS IN XEROGRAPHIC SYSTEM DESIGN

1 Copyright © 1997 by ASME
Proceedings of DETC’97:
1997 ASME Design Engineering Technical Conferences
September 14-17,1997,Sacramento,California
DETC97/DTM-3885
MULTIOBJECTIVE OPTIMIZATION AND QUANTITATIVE TRADE-OFF ANALYSIS IN
XEROGRAPHIC SYSTEM DESIGN
Sudhendu Rai
Member of Research & Technology Staff
Wilson Center for Research & Technology
Xerox Corporation, Webster NY14580
ABSTRACT
A complex engineering system such as a xerographic marking
engine is an aggregate of interacting subsystems that are coupled through a
large number of constraints and design variables. The traditional way of
designing these systems is to decouple the overall design into smaller
subsystems and assign teams to work on these subsystems. This approach is
critical to making the project manageable and enabling concurrent
development. However, if the goal is to design systems that can deliver best
possible performance, i.e. if the performance limits are being pushed to the
extreme, characterizing the interactions becomes critical.
Multiobjective optimization is a design methodology that addresses
the issue of designing large systems where the goal is to simultaneously
optimize a finite number of performance criteria that come from one or more
disciplines and are coupled through a set of design variables and constraints.
This approach to design makes explicit and quantitative the inherent trade-offs
that need to be made in doing coupled system design. It also enables the
determination of the attainable limits of performance from a given system.
This paper will discuss the multiobjective optimization
methodology and optimal methods of performing quantitative trade-off
analysis. These design methods will be applied to problems from the
xerographic design domain and results will be presented.
INTRODUCTION
The traditional way of designing large engineering
systems has been to break it down into smaller subsystems;
work on them in somewhat of an isolated manner and put them
together to produce the whole system. The subdivision into
smaller subsystems was essential in the past to make the project
manageable and enable concurrent development and worked
well since the performance requirement was not high and the
market pressures on cost and product development time were
low. However, if the goal is to design products whose
performance is being pushed to extreme, it is no longer
reasonable to take the traditional compartmentalized view to
product design. It becomes necessary to focus on the
interactions between the various subsystems and characterize
the trade-offs between various objectives in a quantitative and
precise manner.
The area of integrated design has been receiving the
attention of several researchers in the recent past. For example,
integrated structure/control system design efforts have focussed
on trying to look at the interactions between controller design
and mechanical design and focus on simultaneous optimization
of the performance characteristics coming from the two
disciplines. (For examples see, Khot & Venkayya et.al [4],
Miller & Shim [7], Hale & Lisowski [3], Bodden & Junkins
[1], Meirovitch [6], Rai & Asada [12]). Other researchers have
focussed on the design of structural optimization of mechanical
systems that can improve static and dynamic characteristics of
the structures. (For examples see, Eschenauer et. al. [2], Koski
[5], Rai & Asada [11]). In a separate report prepared for ARPA,
the authors Whitney et. al. [19] highlight the need for a systems
perspective to electromechanical product design.
This paper will focus on some of the issues involved
in the design of large xerographic marking engines. A design
methodology will be presented and corroborated with practical
examples that illustrate how some of these issues can be
resolved. The first section of the paper will briefly describe the
xerographic marking process. In the second section some of
the issues involved in designing systems that can deliver
extremely high performance will be examined. The design
methodology of multiobjective optimization will then be
described in the context of xerographic system design and
illustrated with relevant examples.
1.0 The Xerographic Marking Engine
The section provides an overview of the xerographic
marking process. For a more in-depth description, the reader is
referred to Pai & Springett [9]. The xerographic subsystems

are usually described as an aggregate of the following
subsystems.
Photoreceptor: A photoreceptor is a transducer that consists of
a layer of dielectric over a photosensitive substrate. The
photoreceptor can be uniformly charged to a given potential
and then discharged pixel by pixel by shining light on it, to
produce a latent charged image on it.
Charging: This subsystem is responsible for laying out charge
uniformly on a photoreceptor. It usually consists of a corona
emitting element that generates corona when a high voltage is
applied to it and a biased shield and grid for guiding the corona
to the photoreceptor.
Image Processing: This subsystem is responsible for converting
an analog image obtained from a scanner (or an electronically
created image) into a binary on/off bitmap in a way that the
image produced using this bitmap best resembles the original
image visually.
Exposure: This subsystem is responsible for discharging a
uniformly charged photoreceptor to produce a latent image on
it. It usually consists of a laser beam that is selectively turned
on and off and guided on a moving charged photoreceptor to
produce the latent image.
Development: This subsystem is responsible for depositing
toner particles on the latent image to convert it to a real image
on the photoreceptor. There are many variants of the
development system but essentially it consists of a set of biased
rolls that supply charged toner particles that are transported in
the presence of electric fields created by the latent image and
the biased rolls so that the latent image on the photoreceptor
develops into a real toned image.
Transfer: If the image is developed on the photoreceptor, it is
necessary to transfer it to a substrate such as paper. The transfer
is accomplished by applying a charge of opposite polarity to the
paper (or any other substrate) that attracts the toner off the
photoreceptor to the paper.
Fusing: The fusing subsystem fuses the toner on the paper.
There are many different ways of accomplishing it but a
commonly used approach is to pass the paper through a set of
heated pressure rolls that melts the toner and fuses it to the
paper to produce the final image.
The traditional way of designing xerographic marking
engines has been to break it down into the above subsystems
and working on them in somewhat of an isolation. This
approach has worked well in the past primarily because the
performance expected out of these subsystems was not very
high. However, if the goal is to design extremely high
performance systems, it is necessary to take a more holistic
view of system design and focus on the interactions between
the individual subsystems. This is shown schematically in
Figure 1.
Photoreceptor
Image
Processing
Charge Expose Develop Transfer Fuse
Paper
Photoreceptor
Image
Processing
Charge Expose Develop Transfer Fuse
Paper
Image
Print
Traditional Isolated Subsystem View
“Systems” view where the subsystems are coupled
Figure 1: The isolated subsystem view vs. the “systems” view
of the xerographic marking engine.
The objective of this paper is to address the problem of system-
level design optimization of xerographic marking engine so that
the interactions can be quantitatively and precisely understood
and the attainable performance limits of the marking
technology can be explored.
2.0 Issues In The Design Of Large Xerographic
Systems
2.1 Coupled Subsystems
The xerographic marking engine is a highly coupled
system. The final image output is the result of an interaction
with several subsystems such as image processing, charge,
exposure, development, transfer, fuser and photoreceptor
subsystems. These subsystems are all coupled with each other
through the photoreceptor subsystem and the image they
interact with. The study of these individual subsystems has
been pursued by the xerographic community for several
decades and a significant understanding in the underlying
physics of each subsystem exists. However, when the goal is to
look at the overall system that is an aggregate of these
subsystems, there is relatively less understanding about how
these systems are coupled with each other and how they
interact.
From the systems viewpoint, the goal is to design
machines that can deliver high image quality output at low
costs. What is the optimal way of prioritizing the relative
importance of the individual subsystems that can deliver the
best possible performance is not known apriori? How the
system level performance goals on cost and performance

translate to requirements and allocations on the individual
subsystems in an optimal manner is a significantly difficult
problem that can be determined only through a quantitative
consideration of the interactions and trade-offs between
different subsystems?
2.2 Objective Conflict
Image quality is a performance that has several
dimensions such as line quality, contrast, variations in print
colors from the desired colors etc. The design goal is
simultaneously optimize on all dimensions of quality to achieve
customer satisfaction. However, in a typical xerographic
machine these objectives are conflicting in nature. It is not
possible to simultaneously optimize all performance measures.
The problem therefore is to make judicious trade-offs between
the conflicting measures and determine the corresponding
design and operating setpoints that will deliver the desired
performance.
2.3 Preference Articulation
The existence of conflict in design necessitates that
design decisions be made that best resolve the conflicting
objectives. Conflict resolution is best achieved if preferences
are articulated explicitly and precisely. If design decisions are
made with implicit assumptions that have not been articulated
precisely, then there is no guarantee that the design achieved is
optimal.
Image quality is a performance that has both objective
and subjective components. In the presence of subjective and
objective components of performance, it is often hard to
explicitly articulate the preferences of the design decision-
maker. Methods are needed that can enable quantification of
design preferences in the presence of uncertainties.
2.4 Family of Designs
Traditional optimization methods seek to determine
the best possible design solution. However, in real systems, the
search for “the best” design is an utopian ideal. There are
usually several designs that meet the requirements of image
quality and cost and it is essential to select from “some
optimal” set of designs. In other words, there is no unique
design from the “systems viewpoint” but instead there is a
family of designs from which the design-decision-maker has to
choose his design.
3.0 Multiobjective Design Optimization of Xerographic
Systems
The system design problem is characterized by the
presence of several metrics of performance that have to be
simultaneously optimized, a set of design variables that
influences them and the existence of equality and inequality
constraints.
Multiobjective optimization is a design methodology
that addresses the issue of designing large systems where the
goal is to simultaneously optimize a finite number of
performance criteria that come from one or more disciplines
and are coupled through a set of design variables and
constraints. This section will provide a brief description of the
multiobjective optimization design methodology fundamentals.
3.1 Pareto-Optimality
Problem Statement
Let us assume that the design variables of interest are
represented by the vector:
x=[x1,x2,...,xn]T
(1)
The performance criteria are represented by the vector:
f(x)=[f1(x),f2(x),...,fm(x)]T
(2)
The equality constraints are given as:
g(x)=[g1(x),g2(x),...,gp(x)]T
= 0 (3)
and the inequality constraints are denoted by:
h(x)=[h1(x),h2(x),...,hq(x)]T
<= 0 (4)
The design goal is to find a vector of design variables
x* that simultaneously minimizes all the components of the
objective function vector f(x) without violating the constraints
specified by equations (3) and (4). The multi-objective
optimization is different from the single objective problem in
the sense that instead of minimizing only one objective function
in single objective optimization, the design goal is to
simultaneously minimize a finite number of performance
criteria that are represented above as components of the vector
f. In other words, the multi-objective optimization problem is a
vector minimization problem when contrasted with the
traditional scalar function optimization problem.
It is the typical characteristic of multi-objective
optimization problems stated above that not all of the
performance criteria (described above as the vector f(x)) can be
simultaneously minimized. In other words, the multiple criteria
often conflict with each other so that none of the feasible
solutions can simultaneously minimize all of the criteria. To
deal with such a situation, it is necessary to introduce the
concept of Pareto-optimality (Pareto [10]).
A vector x* is Pareto-optimal if and only if there is no
other vector x with the characteristics:
fj(x) <= fj(x*) for all j     m}
and
fj(x) < fj(x*) for at least one j  {1, ... , m}(5)
A Pareto-optimal solution (for the multi-objective
minimization problem) is such that it is not possibly to move

feasibly from that solution to any other point in the design
space without increasing at least one of the performance
criterion.
Illustration
The concept of Pareto-optimality is best illustrated for
the case when two objective functions are being simultaneously
optimized. Figure 2 shows the mapping of the functions from
the design variable space to the space of performance criteria.
The boundary of Pareto-optimal solutions is shown (in bold) in
the space of performance metrics. It is obvious that within the
space of feasible solutions shown in the figure, only the
solutions lying on the Pareto-optimal curve have the
characteristic property that moving on the curve, f1 and f2
cannot be simultaneously decreased - one objective can be
decreased only at the expense of another. Also, each point on
the Pareto-optimal curve corresponds to one set of design
variables x which is termed as a Pareto-optimal design.
For problems involving three metrics, the Pareto-
optimal solutions lie on a surface that can be plotted in three
dimensions. However, for problems with higher dimensions, it
is no longer possible to visually show all the solutions and one
has to look at the numerical values of the objective functions to
make the appropriate trade-offs




Design Space Design Objective Space
Pareto-optimal
curve or “trade-off”
curve
Figure2: Mapping from the design variable space to the design
objectives space showing the Pareto-optimal curve
3.2 Direct Methods of Obtaining Pareto-optimal
solutions
The multi-objective problem of generating Pareto-
optimal solutions for the performance criteria vector f(x) is
accomplished by formulating a scalar substitute problem,
min p[f(x)] (6)
where the function p is called a preference function
whose arguments are the components of the vector f(x). There
are numerous methods of formulating the preference function
such that minimizing the preference function yield a Pareto-
optimal solution.
This section will describe some of these methods.
a. Method of Objective Weighting
The method of objective weighting is one of the most
intuitive ways of forming the preference function. In this
approach, a linear combination of the individual objective
functions is used as the preference function. Thus:
p[f(x)] = [wjfj(x)] = wT
f(x) (7)
The weights are usually chosen to lie between 0 and
1.0 and are normalized so that their sum is equal to 1 ie.
0.0 <= wi =< 1.0, wj = 1.0 (8)
This approach will find the Pareto optimal solutions if
the feasible objective space is convex. It cannot find those
solutions where the objective space is non-convex. This is
illustrated in Figure 3.
Figure 3: Solution region with both concave and convex
Pareto-optimal solutions
In Figure 3, the solutions represented by the concave
portion of the Pareto-optimal solution curve cannot be
determined the weighted objective approach. However, the
solutions that lie on the convex region of the Pareto-optimal
curve can be determined by the weighted objective approach.
Another limitation of this approach to generating the Pareto-
optimal solutions is that it is often not possible to generate
uniformly spaced points on the trade-off curve (or hypersurface
in higher dimensions) by uniformly varying the weights.
b. Method of Distance Functions (or Goal
Programming)
Vector norms or measures of distance in the space of
objective functions are often used to scalarize the vector
minimization problem into a scalar minimization problem. The
substitute problem is then written as:
p[f(x)] = [ |fj(x) - yj|n
]1/n
1 <= n =<  (9)
Usually, the value of n is chosen to be 1 or 2. The
value yj are often called the demand level or aspiration level.
This approach is not guaranteed to yield a Pareto-optimal
solution. That is obvious from Figure 4. Solutions for aspiration
level y1 and y2 will lead to Pareto-optimal solutions (shown as
black dots) on the Pareto optimal solution curve (or surface for
Concave feasible Pareto-
optimal solution boundary
solution boundary
Convex feasible Pareto-
Convex feasible Pareto-
f1
f2

higher order vectors) whereas the solution for aspiration level
y3 will not be Pareto-optimal but will lie inside the feasible
region and coincide with y3.
Figure 4: Solutions generated using the method of distance
functions
3. Method of Min-Max Formulation with Objective
Weighting
The weighted min-max formulation tries to minimize
the maximum deviation of the functions from the individual
minimum weighted by some weighting. If fj
*
denotes the
minimum value of fj(x) then the weighted deviation of fj(x) is
measured by :
zj = wj|fj(x) - fj
*
| / |fj
*
| j = 1, ... , m (10)
The min-max formulation attempts to find the
minimum of the maximum zj. In the first step, the individual
minima for each function fj are computed. Then a slack
variable  is introduced and the following constrained
optimization problem is solved.
min [  ]
subject to.
zj <  for each j = 1, ... , m (11)
This approach of generating the Pareto-optimal
solutions has the advantage that both convex and concave
regions of Pareto-optimal solution curve of surface can be
determined ( Steur [18]). In other words the entire set of Pareto-
optimal solutions can be generated by choosing appropriate
values of the weights.
For a more detailed exposition the reader is referred to
[2],[13],[17] and [18]. Another interesting account of
multiobjective optimization activity in the former USSR can be
found in [8].
3.3Example of Pareto-optimal Xerographic Design
Setpoint Determination
This section will describe the application of
multiobjective optimization design methodology to determine
the design setpoint solutions that simultaneously optimize two
conflicting image quality attributes.
Problem Statement
The creation of a latent image on a photoreceptor and
its subsequent development is one of the fundamental functions
of a marking engine.(For a description of various development
methods, the reader is referred to Schaffert [14], Schein [16]).
The latent image is created on the photoreceptor as a result of
exposure by a light source of a uniformly charged
photoreceptor. As the photoreceptor carrying the latent image
moves through the development region, the fields created as a
result of the potential difference between the image on the
photoreceptor and the development rolls causes charged toner
particles to migrate from the donor rolls to the latent image on
the photoreceptor and develop it, as shown schematically in
Figure 5. (This developed image is subsequently transferred
and fused to a substrate such as paper or transparency).
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Cloud of Charged
Toner Particles
+ + + + + + + + + + + + + + + +
_ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _
Moving
Photoreceptor
Latent Image
Figure 5: Schematic view of the image development in a
xerographic marking engine
A critical problem during the image development is
that it is often not possible to develop all types of images with
extremely high fidelity and resolution. In other words, if one
tries to determine an operating condition setpoint that develops
fine lines extremely well, then thick lines or solid areas areas
do not develop that well and vice-versa. (For an elaborate
discussion, see Scharfe [15]).
In one approach to xerographic development, the
region between the photoreceptor and the development rolls is
filled with a cloud of charged toner particles and it is this cloud
that is responsible for the development of the latent image. If
the cloud is brought too close excessive background can occur.
There are several xerographic system design variables that
f1
y3
y1
y2
f2

affect the image quality such as the potential of the
photoreceptor surface, the charge on the toner particles that
make up the cloud, the potential of the various components of
the developer housing, the speed of the photoreceptor and the
geometry of the development region including gaps and
diameters of the rolls.
In this example, two xerographic design metrics will
be considered. The first will be a measure of background and
the other will be a measure of difference of the developed line
width from the desired latent image line width. The former
quantitative measure is derived from measurements taken
experimentally from the cloud in the development region and
the latter is determined by measuring the developed line width
and taking its difference from the line width of the latent image.
The effect of 11 design variables on both these metrics
was studied experimentally. These design variables denoted by
x1 to x11 in this paper consist of variables such as voltages,
gaps, toner charge and photoreceptor speed. The performance
indices consisting of a scavenging metric and line width change
are denoted by p1 and p2 respectively.
Based on experimental observations, a regression
relationship between the metrics and the design variables was
derived. Multiobjective optimization of the two metrics was
done based on the experimentally derived models of the
xerographic phenomenon and the Pareto-optimal solutions were
obtained.
The results of one such exercise is presented in Figure
6 and Figure 7. Figure 6 shows the results derived using the
min-max approach described in a previous section. It was
observed that the Pareto-optimal curve was generated
uniformly by a variation of the weights. The weighted sum
approach and the goal programming approach to multiobjective
optimization was also studied and the results are shown in
Figure 7.
The study was performed in the Microsoft Excel
environment. The results of the experiments were captured in
the Excel spreadsheet and the Excel regression tool was used to
generate a regression model of the image development metrics.
The same Excel optimizer was used for multiobjective
optimization studies to generate the Pareto-optimal or “trade-
off” curve.
The trade-off curve obtained as a result of this exercise
shows that these two metrics are conflicting in nature and that it
is not possible to simultaneously minimize both measures of
performance. The setpoints corresponding to the points on the
trade-off curve are shown in the spreadsheet of the Figures 6, 7
and 8.
The setpoints were chosen from the Pareto-optimal set.
Traditionally, when design decisions were made without
treating the performance metrics simultaneously, it was often
the case that the solution would lie inside the region bounded
by the Pareto-optimal boundary where there would be
possibility of simultaneously improving both the performance
metrics.
The other advantage of this approach is that it provides
insights into the optimal trade-offs between the multiple
performance criteria that are inherent in the system physics. By
exploring the Pareto-optimal solution frontier the designer can
then refine his(/her) utility function with respect to the solution
set and converge on a satisfactory solution.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 SlackVariable P1 P2
27.58621 22 -500 0 0.852515 9 147.9403 -100 1.382488 650 1.496726 8.1E-05 75.62211722 -0.086831095
27.58621 22 -500 0 0.852515 9 140.9945 -100 1.382488 650 1.496726 0.000161 74.32493264 -0.086751428
27.58621 22 -500 0 0.852515 9 89.19343 -100 1.382488 650 1.496726 0.000755 64.65058557 -0.086157279
27.58621 22 -500 0 0.852515 12.46542 75 -100 1.382488 650 1.496726 0.001462 59.89387123 -0.085450063
27.58621 22 -500 0 0.852515 25 75 -100 1.382488 650 1.81076 0.006754 48.78076009 -0.080157428
27.58621 22 -500 0 0.852515 25 75 -100 1.382488 661.3399 2.245089 0.012875 42.43986539 -0.074036264
27.58621 22 -500 0 2.131287 25 75 -100 2.192438 850 2.245089 0.048523 10.74114496 -0.03836457
27.58621 22 -480.299 0 2.131287 25 75 -100 4.147465 850 2.245089 0.078326 -7.979488273 -0.008507845
27.58621 22 -335.681 0 2.131287 25 75 -100 4.147465 850 2.245089 0.115611 -28.49983591 0.028930557
27.58621 22 -250.974 0 2.131287 25 75 -100 4.147465 850 2.245089 0.137358 -40.51925119 0.05085941
27.58621 21.63205 -200 0 2.131287 25 75 -100 4.147465 850 2.245089 0.151586 -48.4089234 0.065282239
27.58621 18.52364 -200 0 2.131287 25 74.99999 -100 4.147465 850 2.245089 0.161744 -53.95657163 0.075644382
27.58621 14.40915 -200 0 2.131287 25 75 -100 4.147465 850 2.245089 0.175039 -61.29979778 0.089360396
27.58621 11.80854 -200 0 2.131287 25 75 -100 4.147465 850 2.245089 0.183277 -65.94115812 0.098029739
27.58621 10.83684 -200 0 2.131287 25 75 -100 4.147465 850 2.245089 0.186299 -67.67538035 0.101268997
27.58621 10 -200 0 2.131287 25 75 -138.575 4.147465 850 2.245089 0.210519 -75.77935341 0.12790329
27.58621 10 -200 0 2.131287 25 75 -157.286 4.147465 850 2.245089 0.219589 -78.98565849 0.139468783
27.58621 10 -200 0 2.131287 25 75 -197.799 4.147465 850 2.245089 0.150854 -85.92817116 0.164511187
27.58621 10 -200 0 2.131287 25 75 -187.141 4.147465 850 2.245089 0.220352 -84.10178838 0.157923224
27.58621 10 -200 0 2.131287 25 75 -194.183 4.147465 850 2.245089 0.199351 -85.30855315 0.162276157
27.58621 10 -200 0 2.131287 25 75 -196.587 4.147465 850 2.245089 0.175472 -85.7204005 0.163761735
27.58621 10 -200 0 2.131287 25 75 -197.799 4.147465 850 2.245089 0.150854 -85.92817116 0.164511187
27.58621 10 -200 0 2.131287 25 75 -198.53 4.147465 850 2.245089 0.125938 -86.05343097 0.164963013
27.58621 10 -200 0 2.131287 25 75 -199.019 4.147465 850 2.245089 0.100871 -86.13718791 0.165265133
27.58621 10 -200 0 2.131287 25 75 -199.369 4.147465 850 2.245089 0.075718 -86.19713743 0.165481378
27.58621 10 -200 0 2.131287 25 75 -199.632 4.147465 850 2.245089 0.050511 -86.24216708 0.165643805
27.58621 10 -200 0 2.131287 25 75 -199.836 4.147465 850 2.245089 0.025268 -86.27723023 0.165770281
Trade-offs between P1 and P2
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-100 -50 0 50 100
P1
P2
Series1
Figure 6: Trade-off curve and Pareto-optimal solutions using
the weighted min-max approach
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Series1
Figure 7: Trade-off curve using the weighted sum approach

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P1
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Series1
Figure 8: Trade-off curve using the goal programming
approach.
4.0 Conclusions
This paper demonstrates the use of multiobjective
design methodology for quantitatively studying the optimal
trade-offs that confront designers who have to make design
decisions in the face of coupled and conflicting performance
requirements in the design of a complex engineering systems
such as a xerographic marking engine. It was shown that
instead of a unique solution to the system design problem,
there exists a family of Pareto-optimal designs from which the
design choices have to be made. It was also shown that the
methodology can be used on experimental data. This is often
the case when the engineering processes being studied are
either not amenable to precise modeling and even if such first
principle models exist, they are computationally expensive for
performing rigorous optimization studies.
Future work includes extending the methodology to
more performance indices when it is not possible to visualize
the Pareto-optimal hypersurface of design solutions and the
designer has to make decisions by interacting with the decision
support system.
Acknowledgements:
The author is grateful to Howard Mizes for providing
experimental data, and several other members of the Wilson
Center for providing valuable input leading to a better
understanding of the problem.
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