Chapter 10 USING INDIFFERENCE POINTS IN ENGINEERING DECISIONS

Abstract

Multi-criteria decision support methods are common in engineering design. These methods typically rely on the specification of importance weights to accomplish trade-offs among competing objectives. Such methods can have difficulties, however: they may not be able to select all possible Pareto optima, and the direct specification of importance weights can be arbitrary and ad hoc. The inability to reach all Pareto optima is shown to be surmountable by the consideration of trade-off strategy as an additional parameter of a decision. The use of indiffer- ence points to select a best solution, as an alternative to direct specification of importance weights, is presented, and a simple truss design example is used to illustrate the concepts.

Full text

Chapter 10
USING INDIFFERENCE POINTS
IN ENGINEERING DECISIONS
Michael J. Scott and Erik K. Antonsson
Proceedings of the
11th International Conference on Design Theory and Methodology
ASME, Paper Number DETC2000/DTM-14559, (September, 2000)
Abstract Multi-criteria decision support methods are common in engineering design. These
methods typically rely on the specification of importance weights to accomplish
trade-offs among competing objectives. Such methods can have difficulties,
however: they may not be able to select all possible Pareto optima, and the direct
specification of importance weights can be arbitrary and ad hoc. The inability
to reach all Pareto optima is shown to be surmountable by the consideration of
trade-off strategy as an additional parameter of a decision. The use of indiffer-
ence points to select a best solution, as an alternative to direct specification of
importance weights, is presented, and a simple truss design example is used to
illustrate the concepts.
Keywords: multicriteria analysis, engineering design, design decision-making,
aggregation functions, trade-offs, strategies
1. Introduction
Multi-criteria decision making is an important part of design. There are
many methods, both informal and formal, that support such design decision
making, such as Pugh charts [5], QFD [3], and the Analytic Hierarchy Process,
or AHP [6]. These design decision methods share several key features. All rely
on the aggregation of preferences to choose among designs, and most methods
allow for the assignment of importance to individual attributes through the
use of weights. These importance weights are meant to allow for meaningful
comparison of many options when two or more attributes must be traded-off
against each other. Among decision methods, weighted-sum aggregation of
preferences is common, as is direct specification of importance weights.
225

226 IMPRECISION IN ENGINEERING DESIGN
Multi-criteria decision methods are related to multi-criteria optimization and
the calculation of the Pareto frontier. Decision methods can be used to avoid
unnecessary computation by optimizing directly to the most desirable config-
uration without calculating other Pareto points. It is implicit in the use of any
decision method that the selection of its parameters, usually weights, enables
the selection of the most desirable points.
Decision methods are important for decision support, and are crucial for
semi-automated design, yet their underlying decision representations have rarely
been examined or justified. Even if standard decision methods worked all the
time, a formal investigation of the underlying mathematics of decision would
still be warranted. It shall be demonstrated below that weighted-sum methods
have serious drawbacks; in fact, any method that relies exclusively on impor-
tance weights to define a decision runs the risk of missing “optimal” options.
A complete model of a decision requires an additional parameter to specify
the level of compensation between criteria [7]. Also, the direct specification of
importance weights is an ad hoc process, and the answers it produces may not
always be reliable. Several relevant results are presented here:
In addition to importance weights, the level of compensation between
attributes is a parameter that defines a decision.
For decision-making that conforms to the axioms of rational design [4],
a parameterized family of functions (with compensation parameter s)
was shown to span a complete range of degrees of compensation [8].
The compensation parameter sincreases with the level of compensation,
which is demonstrated formally in two different ways. The compensa-
tion parameter, together with weights, defines a decision.
By the use of these functions, a weight/strategy pair to select any Pareto
optimal point can always be found.
The ability to choose any Pareto point is not present when degree of
compensation is pre-selected (as with the use of a weighted sum).
The concept of indifference points as a structured alternative to ad hoc speci-
fication of parameters is presented below, and its application illustrated by an
example. The notion of level of compensation is a less intuitive concept than
importance weighting, and a structured method is even more essential when
both compensation and importance weights are considered. The relation of
these results to multi-criteria optimization will be discussed below.
2. Example: simple truss structure
Consider the structure shown in Figure 10.1. This is a pin-jointed two-
member bracket to support a load of one kilogram (1000 g) at a distance of

Using Indifference Points in Engineering Decisions 227
w
h
t
y=0.5
x
L
=
1
Figure 10.1 Example: a simple truss structure.
1 meter from a wall (L=1). The positions of the wall mounts are fixed, with
the lower support one half meter below the upper support (y=0.5). Both
members are made of aluminum (6061-T6). The designer controls four design
variables:
x∈[0.1m,0.9m] distance from wall to pin
t∈[5 mm,20 mm] thickness of bending member
h∈[5 mm,20 mm] height of bending member
w∈[5 mm,20 mm] width of (square) compression member
For this example, the performance measures to consider are the mass (M)
of the structure, and the safety factor (S). The example is simple enough that
both can be expressed analytically, but let us start by treating the performance
calculation as a black box. The details of the calculations are presented in the
Appendix. For purposes of this paper, the design problem is to minimize the
mass of the structure while maximizing the factor of safety.
Further suppose that no additional advantage is gained fromfactors of safety
above ten. Also, designs with safety factors below one should not be consid-
ered. Using optimization or other means, it can be determined that the min-
imum mass achievable with a factor of safety of one is 123 grams, while the

228 IMPRECISION IN ENGINEERING DESIGN
100 200 300
0.0
0.2
0.4
0.6
0.8
1.0
µmass
mass [g] 0246810
0.0
0.2
0.4
0.6
0.8
1.0
safety factor
µsafety
Figure 10.2 Preferences for mass and safety factor
minimum mass achievable with a factor of safety of ten is 302 grams. The best
designs will be trade-offs between the safety factor and the mass.
Assuming that both 123 g (the lowest possible mass for the acceptable range
of safety factors) and 302 g (the mass corresponding to the highest safety fac-
tor) are acceptable, it is common to normalize the performance measures. Here
we follow the approach of the Method of Imprecision, or MoI [9, 7], and spec-
ify preferences on the performance measures, which incidentally normalizes
the performances to the interval [0,1]. The results presented apply to any nor-
malization scheme, or to no normalization at all. Let the preferences for mass
and safety be as follows:
µmass(M) = 302 −M
179
µsafety(S) = S−1
9
so that µmass(123) = 1,µmass (302) = 0,µsafety(1) = 0,andµ
safety(10) = 1,
as shown in Figure 10.2. Note that these simple linear preferences are cho-
sen for convenience; all the results presented here hold for more complicated
preferences as well.
Weighted sum
As was discussed above, a common way to select a best design is to assign
importance weights to the two criteria, and then use a weighted sum to aggre-
gate preferences; the best designs will have the highest overall preference. Let
the importance weights assigned to mass (ω1) and safety (ω2) be normalized

Using Indifference Points in Engineering Decisions 229
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
µmass
µsafety
ω1>0.658
0.492 <ω
1<0.658
ω1<0.492
Figure 10.3 Three “best” points found using weighted sum exploration

230 IMPRECISION IN ENGINEERING DESIGN
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
µmass
µsafety
Figure 10.4 Pareto frontier with three “best” points.
so that their sum is one. Employing this approach, there are only three possible
“best” points, summarized in the table below:
ω1mass safety x t h w
[g] factor
ω1>0.658 123 1 0.71 5 5 5
0.492 <ω
1<0.658 129 1.6 0.82 5 5 5
ω1<0.492 302 10 0.9 5 9.34 8.16
According to this weighted sum aggregation, all other possible points are worse
when both mass and safety factor are considered. These points are shown on
the graph in Figure 10.3.
The three “best” points shown in Figure 10.3 do not represent the entire
range of reasonable trade-offs between the two performance measures mass

Using Indifference Points in Engineering Decisions 231
and safety. To make the notion of a “reasonable” decision more precise, we
use the idea of Pareto optimality:
Definition 1 The alternative Adominates the alternative Bif Aperforms no
worse than Bon all attributes, and better than Bon at least one attribute. In
this case, regardless of the weights or the strategy, it is always better to choose
Aover B. A feasible solution is undominated (or Pareto optimal) if there is no
other feasible solution which dominates it.
Figure 10.4 shows a plot of all the feasible Pareto optimal points, normalized
with respect to preference, with the three points from Figure 10.3 retained.
(The calculation of the Pareto frontier is detailed in the Appendix). Examining
the entire Pareto frontier, we see that it is made up of two concave sections.
The weighted sum approach fails to identify any Pareto points on the concave
sections of the frontier, though it is quite possible that one of those points
represents the most desirable compromise. Some difficulties of the weighted
sum have been discussed previously in an optimization context by [2] and [1],
among others.
Clearly, aweighted sum approach to multi-criteria decision making is prob-
lematic if it cannot identify all possible best solutions. In the example pre-
sented here, the performance calculations are all analytic expressions which
are easily evaluated, and the Pareto frontier is thus easily discovered. In such
simple cases, a designer can choose to informally explore regions of the per-
formance space that the formal decision model does not identify. When the
design is more complicated, perhaps because evaluation is more costly (say,
each point is a finite element calculation rather than an analytic expression),
or because there are more than two competing objectives, informal exploration
becomes much more difficult, and designers may rely more on automated tech-
niques such as optimization. In these more complicated design situations, it is
particularly important that design decision methods provide reliable guidance.
If the preference aggregation is valid, it is not necessary to compute the entire
Pareto frontier.
In the example presented above, it is clear that the choice of a point on
the Pareto frontier depends on the trade-off between safety and mass. As is
suggested by the problems exhibited by the weighted sum, a formal model of a
design decision is more complex than a simple matter of choosing importance
weights.
3. Compensation strategies: how to consider all
designs
In the preceding section it was seen that a weighted sum cannot always
identify all Pareto points for a design. This is one instance of a more general
result about the aggregation of preference. All existing support methods for

232 IMPRECISION IN ENGINEERING DESIGN
multi-criteria decision making ultimately rely on the aggregation of disparate
preferences with aggregation functions. [4] presented axioms that an aggrega-
tion function must obey in order to be appropriate for rational design decision
making. [8] showed that the operators that satisfy these axioms are a restricted
set of weighted means, and that, in particular, there is a family of aggregation
operators Psthat spans an entire range of possible operators between min and
max, given by:
Ps(α1,α
2;ω
1,ω
2)=ω
1
α
1
s+ω
2
α
2
s
ω
1+ω
21
s
Here, the values α1,α2are individual preference values to be aggregated.
The values αiare the result of applying preferences µito performances xi:
αi=µi(xi), or more generally, α=µ(~x). The parameter scan be interpreted
as a measure of the level of compensation,ortrade-off, and is sometimes re-
ferred to as the trade-off strategy. Higher values of sindicate a greater willing-
ness to allow high preference for one criterion to compensate for lower values
of another. The parameters ω1and ω2are importance weights, both assumed to
be positive without loss of generality, and as they may be normalized, the ratio
ω=ω2
ω1is sufficient to characterize the relative importance of two attributes.
The definition is for two attributes, but can be extended to more than two. It is
readily shown [8] that
P−∞ =lims→−∞ Ps=min
P0=lims→0Ps= geometric mean (αω1
1αω2
2)
1
ω1+ω2
P1=lims→1Ps= arithmetic mean ω1α1+ω2α2
ω1+ω2
P∞=lims→+∞Ps=max
Thus the common weighted sum is one instance of this family of design-
appropriate aggregation functions, with the compensation parameter sequal
to 1.
Several results about this family of aggregation functions can be proven [7],
including:
For any Pareto optimal point in a given set, there is always a choice of
a weight ratio ωand a trade-off strategy sthat selects that point as the
most preferred.
For any fixed strategy s, there are Pareto sets in which some Pareto points
can never be selected by any choice of weights ω. Thisiswhatoccurs
in the truss example above, where s=1cannot select all Pareto points.
This is related to the well-known result that non-convex portions of a
Pareto surface are unreachable by weighted-sum minimization.

Using Indifference Points in Engineering Decisions 233
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
µmass
µsafety
Figure 10.5 Normalized Pareto frontier
The second result does not say that for every strategy s, every Pareto set has
some Pareto points that are unreachable. In the truss example, for instance,
s=−1can select every Pareto point if the correct weights are chosen. This
can be seen in Figure 10.5. Here, the three “optimal” points found earlier by
the weighted sum method are circled. Setting s=−1and varying the weights
allows for a more varied range of “best” designs:

234 IMPRECISION IN ENGINEERING DESIGN
ω1mass [g] safety xh w
0.1 262 7.5 0.9 8.09 7.59
0.2 248 6.7 0.9 7.65 7.38
0.3 236 6.1 0.9 7.30 7.21
0.4 226 5.6 0.9 6.99 7.06
0.5 217 5.15 0.9 6.70 6.91
0.6 207 4.7 0.9 6.41 6.76
0.7 196 4.2 0.9 6.05 6.57
0.8 183 3.65 0.9 5,64 6.34
0.9 164 2.95 0.9 5.07 6.01
These points are shown as squares on the graph in Figure 10.5.
By allowing the weight assigned to one attribute to be much larger than the
weight assigned to the other, points much closer to the extremes of the Pareto
frontier can be reached with s=−1:
ω1mass [g] safety xhw
0.01 288 9.05 0.9 8.89 7.96
0.05 272 8.1 0.9 8.41 7.74
0.95 155 2.55 0.89 5 5.77
0.99 132 1.7 0.83 5 5.10
These points are shown as triangles on the graph in Figure 10.5.
4. Using indifference points to determine
strategies and weights
As was mentioned above, the direct specification of importance weights is
an ad hoc process. If a designer says that safety is twice as important as mass,
how are we to know that aggregation with any strategy will choose the best al-
ternatives? The difficulty is only compounded by the consideration of trade-off
strategies. In this section a technique is presented for determining the correct
parameter pair of compensation strategy sand weight ratio ω2
ω1for a particular
decision. Rather than direct specification, the technique relies on the use of
indifference points to establish the appropriate parameters.
Two points are considered indifferent if they have the same preference; it
is not necessary that the numerical preferences be known. When a single in-
dividual has complete decision-making authority, strategies and weights can
be considered simultaneously, and their values can be calculated from indiffer-
ence points. The procedure is as follows:
1. Determine preferences α1=µ1(xi)and α2=µ2(xi)such that
Ps(α1,1; ω1,ω
2)=P
s
(1,α
2;ω
1,ω
2)=0.5

Using Indifference Points in Engineering Decisions 235
(The values for s,ω1,andω
2are to be determined.) In other words, at
which value α1is there indifference between a design with a preference
equal to α1for the first performance attribute and a preference values
of 1on the second attribute, and a design that achieves preferences of
0.5on both attributes (and thus, by idempotency [4], has a combined
preference of 0.5)? A similar question is asked for α2. Sometimes it is
easier to ask for values of xiand calculate αi; sometimes it is easier to
seek αidirectly. Either approach to determining the indifference points
is acceptable.
To see how the selection of indifference points might work on the truss
example, start with the preference values from Section 2.:
µmass [g] safety
0302 1
0.5214 5.5
1126 10
The reference design is thus the one that yields a mass of 214 g and a
safety factor of 5.5. It is important to note that the reference design does
not need to be physically realizable. In the truss example, there is no fea-
sible design with preferences (0.5,0.5). To determine the value of α1,
ask “If we start from the reference design and increase the safety factor to
10, how much can the mass increase so that the new design has the same
overall preference as the reference design?” The answer could be, say,
that the mass can increase to 260 g; since µ1(260) = 0.29,α1=0.29,
indicating that there is indifference between the two points with per-
formances (x1,x
2) = (260,10) (preferences (α1,α
2)=(0.29,1))and
(214,5.5) (preferences (0.5,0.5)).
The corresponding question is asked for α2: “Comparing the reference
design to one where the mass is 126 g, what is the safety factor that
achieves indifference withthe reference design (214,5.5)?” If the answer
is that a safety factor of 3 (µ2(3) = 0.22) together with a mass of 214 g
is indifferent to a safety factor of 5.5 together with a mass of 126 g, then
α2=0.22.
If the person providing the indifference points is comfortable thinking
in terms of preferences between 0 and 1, then it is not necessary to re-
fer to performance values. Instead the question can be asked directly:
“If we start with a reference design where preference for both mass and
safety is 0.5, and we increase the preference for mass to 1, how low a
preference for safety achieves indifference with the reference design?”
This latter form of questioning is particularly useful when several at-
tributes are combined hierarchically and thus groups of attributes must

236 IMPRECISION IN ENGINEERING DESIGN
be compared. It is always possible, even in hierarchical aggregation, to
specify particular values of all performance attributes in order to specify
indifference points.
2. Let ω=ω2
ω1.
3. If α1=α2,thenω=1:
(a) If α1=0.5,thens=−∞.
(b) If α1=0.25,thens=0.
(c) If α1>0.25,thens∈(−∞,0).Solveα
1
s+1=2(0.5)snumer-
ically.
(d) If α1<0.25,thens∈(0,∞).Solveα
1
s+1=2(0.5)snumeri-
cally.
4. If α16=α2,thenω6=1. Note that if s=0:
α
1
m=0.5=α
2
1−m⇒α
2
1−logα10.5=0.5
Thus:
(a) If α21−logα10.5=0.5,thens=0,andb=1−logα10.5
logα10.5
(b) If α21−logα10.5>0.5,thens<0.
If α21−logα10.5<0.5,thens>0.
Solve numerically for sfrom
1+ωα2s
1+b1
s
=α
1
s+ω
1+ω1
s
=0.5
which reduces to
(α1
s−0.5s)(α
2
s−0.5
s)=(1−0.5
s
)
2
Once this is solved numerically for s,thenωcan also be deter-
mined.
Applying steps 2–4 to the indifference values α1=0.29 and α2=0.22 de-
termined above, the parameters that determine the aggregation are found to
be:
s∗=−0.5ω∗=1.23,or(ω
1
,ω
2)=(0.45,0.55)
The best point on the Pareto frontier for that trade-off is:
mass [g] safety α1α2xt h w
223 5.45 0.44 0.49 0.9 5 6.90 7.01

Using Indifference Points in Engineering Decisions 237
Note that this best point depends only on the preferences for safety and mass,
and on the answers given above to the two indifference questions. Also note
that this point, which is shown as a star on the graph in Figure 10.5, is in-
tuitively appealing as an overall optimum, while the points provided by the
weighted sum method are not.
It should be noted that if either α1or α2is close to 0, then the (s, ω)pair
is quite sensitive to small differences in α1and α2. In these cases, it might be
preferable to elicit other indifference points to determine sand ω. In the pro-
cedure described above, points that are equivalent to (0.5,0.5) are chosen; the
procedure can easily be modified to consider indifference to some other refer-
ence point. Indeed, if the procedure is applied more than once with different
reference points, the redundant information serves as a check on the accuracy
of the specification.
5. Conclusion
Weighted sum aggregation with importance weights is common to many
methods for engineering design decision making. Two important difficul-
ties with these methods are the inability of weighted-sum methods to select
all Pareto points, and the arbitrary nature of direct assignment of importance
weights.
It is shown here that a complete model of an engineering decision depends
not only on the importance weights, but also on the level of compensation, or
trade-off strategy. A weighted sum, or any other pre-determined aggregation
procedure, is overly and inappropriately constraining. The appropriate strategy
(or degree of compensation among attributes) is situation-dependent, and a
“one-size-fits-all” decision method that dictates an aggregation method can
lead to incorrect results. An easily computed family of preference aggregation
functions is completely determined by two parameters that represent the trade-
off strategy (degree of compensation) and the importance weighting.
The choice of strategy and importance weights, or even the choice of im-
portance weights alone, can be ad hoc and arbitrary if accomplished by direct
specification. A simple procedure is presented, along with straightforward cal-
culations, to establish the proper importance weights and degree of compensa-
tion to reach rational engineering decisions.
The results in this paper regarding decision methods are related to previ-
ously known results about multi-criteria optimization [2, 1]. The preference
aggregation operators presented here could be used to explore a non-convex
Pareto frontier. From an optimization point of view, once an aggregated pref-
erence function (or utility function) is determined, locating the optimum is
straightforward. If some point lies on a non-convex region of a Pareto frontier,
and a utility function is constructed using a weighted sum, then that point is an

238 IMPRECISION IN ENGINEERING DESIGN
inferior point. From a design decision point of view, however, it is appropri-
ate to question the choice of a weighted sum to aggregate the preferences. The
method presented in this paper, unlike a weighted sum, has the great advantage
that it does not aprioriexclude any Pareto points from consideration. Thus,
if the aggregated preference is optimized, the selected “optimum” is actually
what the designer desires, and is not artificially constrained by the geometry of
the design and performance spaces.
Acknowledgements
This material is based upon work supported, in part, by the National Science
Foundation under NSF Grant Number DMI-9813121. Any opinions, findings,
conclusions, or recommendations expressed in this publication are those of the
authors and do not necessarily reflect the views of the sponsors.

Using Indifference Points in Engineering Decisions 239
References
[1] CHEN,W.,WIECEK,M.,AND ZHANG, J. Quality utility - a compromise
programming approach to robust design. ASME Journal of Mechanical
Design 121, 2 (1999), 179–187.
[2] DAS,I.,AND DENNIS, J. A closer look at drawbacks of minimizing
weighted sums of objectives for pareto set generation in multicriteria opti-
mization problems. Structural Optimization 14, 1 (1997), 63–69.
[3] HAUSER,J.,AND CLAUSING, D. The House of Quality. Harvard Busi-
ness Review 66, 3 (May/June 1988), 63–73.
[4] OTTO,K.N.,AND ANTONSSON, E. K. Trade-Off Strategies in Engi-
neering Design. Research in Engineering Design 3, 2 (1991), 87–104.
[5] PUGH,S.Total Design. Addison-Wesley, New York, NY, 1990.
[6] SAATY,T.L.The Analytic Hierarchy Process: Planning, Priority Setting,
Resource Allocation. McGraw Hill, New York, NY, 1980.
[7] SCOTT,M.J.Formalizing Negotiation in Engineering Design. PhD thesis,
California Institute of Technology, Pasadena, CA, June 1999.
[8] SCOTT,M.J.,AND ANTONSSON, E. K. Aggregation Functions for Engi-
neering Design Trade-offs. Fuzzy Sets and Systems 99, 3 (1998), 253–264.
[9] WOOD,K.L.,AND ANTONSSON, E. K. Computations with Imprecise
Parameters in Engineering Design: Background and Theory. ASME Jour-
nal of Mechanisms, Transmissions, and Automation in Design 111, 4 (Dec.
1989), 616–625.

240 IMPRECISION IN ENGINEERING DESIGN
A Appendix: Details of the Truss Example
Here are the details for the example of the bracket shown in Figure 10.1.
Material properties and performance measures
The material is aluminum (6061-T6), which has the following relevant ma-
terial properties:
Young’s modulus (E)69 ·109Pa
Density (ρ)2660 kg/m3
Yield stress (σ)275 ·106Pa
There are four design variables:
x∈[0.1m,0.9m] distance from wall to pin
t∈[5 mm,20 mm] thickness of bending member
h∈[5 mm,20 mm] height of bending member
w∈[5 mm,20 mm] width of (square) compression member
The first performance measure is total mass (in kilograms, here, so the load
Pbelow is equal to 1):
M=ρhtL +w2qx2+y2= 2660 ht +w2px2+0.25
The safety factor Shas two components, the safety factor for the bending mem-
ber Sb, and the safety factor for the compression member Sc. Since the yield
stress in the bending member is σ, and the maximum stress in the bending
member is 12P(L−x)
th2, the factor of safety in bending is the ratio:
Sb=σth2
12P(L−x)=σth2
120(1 −x)
Similarly, using the Euler buckling load, the safety factor in the compression
member is:
Sc=π2Exyw4
12PL(x
2+y2)
1.5=π2Exyw4
120(x2+0.25)1.5
The safety factor for the entire design is defined to be the minimum of the two:
S=min σth2
120(1 −x),π2Exyw4
120(x2+0.25)1.5!

Using Indifference Points in Engineering Decisions 241
100 150 200 250 300 350
1
2
3
4
5
6
7
8
9
10
infeasible
region
feasible
region
mass [g]
safety factor
Figure A.1 Pareto frontier of best performances

242 IMPRECISION IN ENGINEERING DESIGN
Calculation of the Pareto frontier
The design problem is to minimize the mass while maximizing the factor
of safety; both are analytic expressions. First, note that mass is linear in both
tand h, while the factor of safety in bending is linear in tbut quadratic in
h. Thus, as long as no other design variables reach the maximum acceptable
dimensions, it will always be preferable to increase hrather than t. Setting
t=tmin =5mmreduces the problem to the three design variables x,h,and
w.Bothhand wcan be expressed as functions of xand a safety factor, and thus
finding the minimum possible mass for a given safety factor requires solving a
rational equation in x. These solutions yield a Pareto frontier of designs, which
isshowninFigureA.1.
The values of x,h,andwwhich generate these optimal designs are included
in Figure A.2. It can be seen from Figure A.2 that at each Pareto point, at least
one domain constraint is active: in particular, for low mass, htakes its mini-
mum acceptable value of 5 mm, while for higher mass, xtakes its maximum
acceptable value of 0.9 m. Nevertheless, along most of the Pareto frontier two
design variables are changing as the frontier is traversed.

Using Indifference Points in Engineering Decisions 243
100 150 200 250 300
1
2
3
4
5
6
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mass [g]
S
x
h
w
S,x,h,w
Figure A.2 Pareto frontier with design variable values