Original paper
Res Eng Design 13 (2001) 42±54
DOI 10.1007/s001630100004
On the valuation of goods and selection of the best design alternative
H.E. Cook, A. Wu
42
Abstract In the planning and early design stages of new
products, the value to the customer for the alternatives
under consideration need to be quanti®ed in the same
units as costs to make rigorous trade-off decisions. According to the S-model, the value of a good is the price at
which demand goes to zero and its demand curve shifts by
a prescribed amount when its value changes. To test these
®ndings, we have investigated the simulated demand for
two lottery tickets. This was necessary because the full
demand curves and the values of commercial products are
not known a priori. The so-called ``endowment effect'' for
the lottery tickets was also observed and explained here as
a direct result of the stochastic nature of the driving forces
for buying and selling a good. The use of the S-model to
examine product value trends over time is explored for
two minivans competing in the real marketplace. The
connections between the S-model and several other engineering design methodologies are discussed.
Keywords Demand á Value á Endowment effect á QFD á
SEU á Marketing research
Introduction
How potential customers value the features of a product is
of great interest to a variety of ®elds including economics,
psychology, marketing, ®nance, and engineering. The
de®nition of value and the method of determining it are
far from uniform across these ®elds, however. Even within
the domain of engineering, which is our interest here, the
de®nitions of value and its surrogates are not consistent.
Value engineers, for example, de®ne value as worth divided by cost (Fowler 1990). In their seminal illumination
of the robust design process, Taguchi and Wu (1980) used
the term ``cost-of-inferior-quality'' to represent the loss of
Received: 8 October 2000 / Revision received: 1 May 2001 /
Accepted: 1 May 2001 / Published online: 14 July 2001
Ó Springer-Verlag 2001
H.E. Cook (&)
Department of General Engineering,
University of Illinois at Urbana-Champaign,
104 S. Matthews Ave., Urbana, IL 61801, USA
E-mail: [email protected]
Tel.: +1-217-244-7992
Fax: +1-217-244-5705
A. Wu
Department of Mechanical and Industrial Engineering,
University of Illinois at Urbana-Champaign, USA
value which occurs when the level of a product attribute is
off its ideal speci®cation. Practitioners of Quality Function
Deployment (QFD) use a zero to ten scale to judge the
value or worth of customer needs (Akio 1990). Utility is a
classical, dimensionless measure of the appeal of a product
(Thurston 1990; Locasio and Thurston 1993).
These design support tools are used to make cost/
bene®t tradeoffs in their respective application domains.
For example, Taguchi's robust design methodology is
widely used in component design (Seventh Symposium on
Taguchi Methods 1989, October 1989, Scottsdale, AZ,
American Supplier Institute, Dearborn, MI). Value engineering methods are used extensively to wring non-value
added costs out of preliminary designs (Tanaka et al.
1993). QFD is widely used to make design trade-offs between alternatives (Tenth Symposium on Quality Function
Deployment 1998, June 1998, Novi MI, QFD Institute, Ann
Arbor, MI). Thurston and co-workers (Thurston 1990;
Thurston 2001; Locasio and Thurston 1993; Carnahan and
Thurston 1998) have pioneered the use of subjective expected utility (SEU) theory in making design trade-offs
when simultaneously considering bene®ts to the customer,
the costs to the manufacturer, and environmental losses.
The S-Model was developed with the objective of unifying Taguchi methods, value engineering, and QFD into
an integrated tool-set having a common formalism for
guiding the planning, design, and development of new
products (Cook and DeVor 1991; Cook 1992; Kolli and
Cook 1994; Cook and Kolli 1994; McConville and Cook
1996; Donndelinger and Cook 1997; Cook 1997; Pozar and
Cook 1997). Aspects of several marketing research tools
(Randall et al. 1974; Green and Ward 1975; Louviere and
Woodworth 1983) were integrated with the S-model to
make a direct assessment of value. As noted by Grif®n and
Hauser (1993) conventional marketing research information is not suf®cient for making detailed cost/bene®t tradeoffs.
The guiding rationale for the S-model was to balance
simplicity and rigor through the use of a phenomenological model of demand written as a Taylor expansion in
terms of the values and prices of the competing products.
There were several reason for taking this approach. The
®rst was that phenomenological models have had great
success in other areas, the theories of diffusion and elasticity, being perhaps the two most notable examples.
Secondly, a Taylor expansion is the simplest and least
presumptive way to formulate the general problem.
Thirdly, the formulation is rigorous in the limit if the
function is analytic in the expansion variables.
H. E. Cook, A. Wu: On the valuation of goods and selection of the best design alternative
The purpose here is three-fold. The ®rst is to illustrate
the use of the model to gain insight into competitive behavior from the construction and analysis of the value
trend curves for competing products. The second is to
examine the S-model predictions that product demand
goes to zero as the price of a product approaches its value
and that demand shifts by a prescribed amount to a
change in value of the good. The third is to examine and
compare the interrelationships between the product development tools listed above and to discuss the general
problem of selecting the best design alternative by incorporating the S-model into the well-known QFD process.
For the convenience of the reader, a review of the S-model
formalism and its key equations are included.
43
Review of S-model
Market segments
The S-model views customers as being within consumer
segments, which are composed of persons who have similar tastes, lifestyle and demographics including, income,
age, and gender. They are assumed to have a single, aggregate value for a good but it may represent an aggregate
quantity taken over a range of multiple, distinct uses of the
good. For example, when water is priced inexpensively, it
is put to many marginal uses of lower value in comparison
to its fundamental value for sustaining life. The use of
market segments, as opposed to an individual perspective,
is a convenient simpli®cation, which is widely used in
planning mass-produced goods. The size of a segment
could, of course, be reduced to the individual level where
value would resemble but not be identical to the concept of
consumer surplus.
Product segments (e.g., minivans, televisions, personal
computers, GPS devices) also exist. A buyer segment is
de®ned as those persons who purchased items from a
particular product segment and will generally be composed of several consumer segments. Marketing research
can be used to determine how each consumer segment
within a buyer segment values the product.
The demand for the good is assumed to increase if price
is reduced or if value is increased. If a person was ready to
buy a speci®c brand today but found that the price had
been increased, he or she might purchase a substitute
brand or simply buy the ®rst brand chosen at a later date
at the higher price.
Fundamental and bottom-line metrics
The coupling between the key elements in the product
realization process is described in Fig. 1. For simplicity in
presentation, the in¯uence of competitors is not shown.
The loop on the right connects the needs of the customer
to the needs of the manufacturer and the loop on the left
connects the needs of society to the needs of the manufacturer. A necessary but not suf®cient condition for the
manufacturer to remain in business is to jointly satisfy the
needs of its customers and society. The challenge to developers of engineering design methodologies is to model
the connections between the elements in the two loops in a
manner that aids product planners and engineers to design
pro®table products in the face of stiff competition. The
Fig. 1. The product realization process couples customer and societal
needs
S-model, because it includes only the linear terms in a
Taylor expansion for demand, represents the simplest
model of how the elements in Fig. 1 are connected when
there are N competitors.
For many products, societal needs are set by governmental regulations covering the manufacture, use, and
disposal of the product. For this class of problems, it is
both convenient and suf®cient to focus on the Customer
and Manufacturer loop provided that the costs for meeting
governmental regulations are included in computing the
total cost of the product.
Demand
Demand and value are the key phenomenological variables
to address in constructing a model of the Customer and
Manufacturer loop because, when they are modeled satisfactorily, all of the other elements in the loop can be
de®ned and readily modeled. Demand, price, and pro®t
(or cash ¯ow) are well-known bottom-line, ®nancial metrics. Value and cost along with the pace of innovation act
as fundamental metrics (Cook 1997) in that they determine
the outcomes for the bottom-line metrics. The demand of
a product i given by Di is taken to be equal to the total
amount of the product sold over a period of time, usually a
year, the assumption being that sales are equal to demand.
In other words, all of the customers who wanted to purchase the product should have been able to do so if they
had the resources. This may not always be the case and it
is important to assure that customers are not being turned
away because of insuf®cient supply. Of course when supply is insuf®cient, price will often rise to maintain a balance with demand. The Taylor expansion is made about a
so-called ``cartel point'' where the N competing products
have the same prices, Pi, values, Vi, and market share, 1/N.
The cartel point was chosen because its high degree of
symmetry reduced the number of independent expansion
coef®cients required by the model to one, noted as K,
which, when divided by N, would be equal to the negative
value of the slope of a cartel member's demand curve with
price.
Res Eng Design 13 (2001)
The basic assumption of the S-model is that demand is
an analytic function of the N values and prices of the
competing products:
Di ˆ fi …V1 ; V2 ; :::; VN ; P1 ; P2 ; . . . PN †
44
…1†
When the prices and values of the products change independently from their levels at the cartel, it follows that the
change in demand for each product i=1, N is given by the
following (Cook 1997):
9
8
<
X
=
1
…2†
dDi ˆ K dVi dPi
dVj dPj
;
:
N j6ˆi
provided that the price and value changes are small. On
writing Eq. 2 as a function of the total variables:
8
9
<
=
X
1
Di ˆ K Vi Pi
Vj P j
…3†
:
;
N j6ˆi
we obtain a useful hyper-plane approximation to the ac- Fig. 2. Fit of a baseline demand curve by a linear approximation and
tual demand surface as a function of the 2N variables of the shift in the linear approximation resulting from a value
value and price. For a monopoly, Eq. 3 becomes as follows improvement of $5
(Cook and DeVor 1991):
…4† historical transaction prices (not list prices). For any given
of the N competing
time period, the average value, V,
It is seen from Eq. 4 that the price where demand goes to products is related to the average price of the products, P,
zero is equal to Vi. The dashed lines in Fig. 2 illustrate the by the expression
use of the linear approximation to a demand curve for a
monopoly. The curve on the left is for a baseline product
1 ‡ E2
VˆP
…6†
and the curve on the right is for an alternative formed
E2
from the baseline by adding a value improvement of $5.
where
The value Vi=$22 given by the intersection of the linear
approximation for the baseline product with the zero de D
mand line represents a marginal value for the product at a E ˆ dD=
…7†
2
demand level of 8 and price level of $15. For a convex
dP=P
downwards demand curve, the marginal value will increase
with price which is in keeping with the notion that mar- de®nes a price elasticity in which the numerator represents
how the fractional change in average demand changes
ginal uses of a product (uses of less value) decrease as
price is increased. As the price of water is increased, for when all of the N products change price by dP, D being the
average
demand.
If
this
elasticity
is
known,
the
negative
example, its marginal uses such as watering the lawn,
slope
of
the
demand
curve,
which
appears
in
Eqs.
3, 4, and
washing the dog, etc. should become less and less preva5,
can
be
computed
from
it
using
the
expression
lent. As the ultimate value of water is priceless, its demand,
as price increases, should approach the horizontal (in
NE2 D
elastic) level needed to sustain life.
Kˆ …8†
P
D i ˆ K ‰ Vi
Pi Š
For automobiles, the price elasticity E2 is approximately
unity (Donndelinger and Cook 1997) and from Eq. 6, we
see that the average value of the automobiles in a segment
is approximately twice their average price. Note that when
E2=1, the average demand for a segment will be reduced by
10% if the average price increases by 10%.
The value given by Eq. 5 represents the buyer segment
value for the good. It is a weighted average of the values
N ‰ Di ‡ DT Š
Vi ˆ
‡ Pi
…5† over all of the consumer segments involved in the purchase
K ‰ N ‡ 1Š
of the good. Of particular interest is how value trends befor product i, i=1, 2, . . . N where DT is the total demand for have when major product changes are made. Competition
between two major brands, A and B, in the minivan market
the N competing products. In using Eq. 5, the demands
over a given historical time period are taken to be equal to is examined in Fig. 3, where the values were determined
from Eq. 5. The prices used in the computations are
the sales over the period and prices are set equal to the
Value trends
If the demands and prices of the products competing
within a segment are known from historical data, the linear set of simultaneous equations represented by Eq. 3 can
be solved for the values of the products. The resulting
expression is given by
H. E. Cook, A. Wu: On the valuation of goods and selection of the best design alternative
asked to make choices between the baseline and the alternative over a series of prices for the latter (McConville
and Cook 1997). The use of four to six price points for
each alternative under consideration strikes a good balance, as a rule of thumb, between statistical accuracy and
time to complete the survey. The price and value of the
baseline product must remain ®xed at P0 and V0, respectively, in keeping with the ®ndings from prospect theory.
The fraction of respondents, f, choosing the alternative is
plotted as a function of price. From the plot, a neutral
price, PN, is determined. This is the price where half of the
respondents choose the alternative and half choose the
baseline. The products for the N±1 competitors are absent
from the choice set and, because the two demands are
equal, it follows from Eq. 2 that
Fig. 3. The values for two production minivans as a function of
model year as computed from Eq. 5
considered proprietary by the manufacturer and are not
included here for that reason. Prices and values were corrected for in¯ation. Also the brand names of the vehicles
are not given here because the results do not apply to the
current products as both have been extensively redesigned
since the period covered in the plot.
Minivan A, noted as mvA, had front wheel drive and
was the recognized market leader in the 1992 model year.
Minivan B, noted as mvB, was a rear wheel drive minivan
for the 1992 and 1993 model years. During the 1994 model
year, the manufacturer introduced a second brand that
had front wheel drive. It also had an improved interior
package, better ride, and fresher styling than the earlier
model, which was more truck-like than car. This new
minivan was in full production by the 1995 model year.
However, both the new and the existing minivans were
sold on the same showroom ¯oor during the 1994, 1995,
and 1996 model years.
The values for mvB shown for those years were computed using a sales-weighted-average for the two brands. It
is seen that by the 1995 model year the value of mvB
exceeded the value of mvA, which had not been upgraded
for several years. In the 1996 model year, mvA received a
major redesign with a signi®cant improvement in interior
room, fresh styling, and the addition of a second rear
sliding door, a feature which was not available on mvB.
The value of mvA increased signi®cantly, becoming
roughly $2,500 more than mvB in the 1996 model year.
Direct value method
Projections of future demand can be made using Eq. 3
provided that the values and prices of future products can
be projected. In order to do this, key elements from choice
theory (Louviere and Woodworth 1983), contingent valuation (Randall et al. 1974) and prospect theory (Kahneman
and Tversky 1979; see also Tversky and Kahnemann 1981)
have been used in conjunction with Eq. 3 to formulate the
direct value (DV) method (Donndelinger and Cook 1997)
of marketing research. In the DV method, one or more
attributes of a baseline product are modi®ed to form an
alternative product having a value V. The baseline and
alternative are described in the survey and respondents are
V
V 0 ˆ PN
P0
…9†
The DV method has been used to determine the value of
a variety of features for automobiles (McConville and
Cook 1996; Donndelinger and Cook 1997; Cook 1997;
Pozar and Cook 1997) construction equipment (Bush
1998; Freeman 2000; Herington 2000) farm equipment
(Silver 1996) and aircraft (LeBlanc 2000). In using the
method, respondents are usually asked only to consider
one attribute change from the baseline at a time to
minimize cognitive stress. A neutral price is found for
each alternative whose value improvement is computed
from Eq. 9. Note, in using Eq. 9 to determine the value
change for the attribute, it is not necessary to know K.
This is important, as the slope of f versus the price of the
alternative need not and likely will not be equal to the
slope of the demand curve for the baseline product in the
marketplace.
Value curves
Once a customer need has been identi®ed, a value curve
for the attribute should be developed. Value curves are
expressed as exponentially weighted parabolas. When
normalized by dividing through by the baseline value V0,
they are of the general form
"
#c
V…g†
‰gC gI Š2 ‰g gI Š2
v…g† ˆ
…10†
V0
‰ gC gI Š 2 ‰ g0 gI Š 2
The curve passes through three points: (1) the critical level
for the attribute, gC, where value goes to zero; (2) the
baseline level for the attribute, g0, where value is V0; and
(3) the ideal level of the attribute, gI, where value is at a
maximum for the attribute. An example is shown in Fig. 4
for the interior noise level in a luxury vehicle cruising at
highway speeds (Pozar and Cook 1997). Ten different
noise levels were evaluated about a baseline noise level of
g0=66 dBA using the DV method. The exponential
weighting coef®cient c=0.59 was determined from the best
®t of Eq. 10 to the points. The values for gC=110 dBA and
gI=40 dBA were pre-determined from human factor
studies, which have demonstrated that 110 dBA is at the
threshold of pain and that noise levels below 40 dBA start
to become too quiet. An advantage of expressing value
curves in the normalized form given by Eq. 10 is that they
45
Res Eng Design 13 (2001)
gram at a major U.S. company. The surveys used are
shown in Appendix A. In using Eq. 4 to analyze the simulation results, demand and supply were set equal to cumulative frequency f, for respondents willing to purchase
or sell the tickets.
The price, P, where a respondent changed from willing
to pay to not willing was assumed to lie half-way between
the respondents maximum stated willingness to pay, P*,
and the next incremental price level. As each of the price
increments was $5, we have P=P*+$2.50. To estimate the
cumulative purchase frequencies for each lottery, we ®rst
arranged the maximum price offered by each respondent
in ascending order against a descending numerical index,
n(P*), given by 78, 77, 76, . . . 1. The cumulative frequencies were then computed using the standard relationship
(DeVor et al. 1992) in the form
46
n …P †
Fig. 4. The normalized value of a luxury vehicle as a function of its f …P ˆ P ‡ 2:50† ˆ m
nT
interior noise level as determined from a DV survey (Pozar and Cook
1997; Society of Automotive Engineers, reprinted by permission)
can be used with a degree of con®dence for similar
products whose baseline value V0 differs somewhat from
that used in developing the original curve. The empirical
weighting coef®cient c is approximately the fraction of
time that the attribute is important when using the
product.
Respondents took the survey sitting in front of a
computer screen using headphones to listen to the noise
levels. They had to sense if the alternative under consideration was louder or quieter than the baseline and to
select the price increase they were willing to pay (WTP) for
a noise reduction or the price reduction they were willing
to accept (WTA) for a noise increase. The noise levels
presented were from recordings of interior noise at highway speeds in an actual luxury vehicle.
0:5
…11†
where nm(P*) was the (minimum) numerical index associated with price P* and nT was the total number of respondents. A similar process was used for sellers except
that the index, n(P*), ran oppositely from 1 to 58, the
number of respondents for the selling survey.
Respondents as buyers
The demand curves were initially constructed separately
for the on-campus and off-campus respondents. The
curves for both appeared equivalent, however, and no
statistically signi®cant differences between the two sample
means were found. Therefore, the responses for both
groups were pooled together for the remaining analyses. In
Fig. 5, the fractional demands for the two tickets are
plotted versus P/EEV, de®ned as their prices divided by
their respective EEVs. A least square ®t of a line through
the quasi-linear P/EEV range from 0.6 to 1.05 intercepts
the axis at 1.09. Points beyond P/EEV=1.0 represent risktaking behavior.
The value of lottery tickets
The products chosen for testing the theoretical relationship between the price intercept and value were two lottery
tickets. Simulated markets for the tickets were used to
develop demand curves. The pay-offs were chosen to be
relatively small so as to avoid signi®cant changes in the
wealth of the respondents had the purchase of the tickets
and pay-offs actually occurred. The reason for this was to
avoid the well-known situation typical of state lotteries in
which potential buyers are offered a remote chance of
winning a large sum if they purchase a ticket priced well in
excess of the expected economic value (EEV) of the payoff.
We do not presume, however, that buyers of such tickets
are irrational. The added value for such a ticket over its
EEV comes from the dream of what the outcome might be,
not from what the outcome will likely be.
Speci®cally, one simulated lottery ticket gave the holder
a 50% chance to win $100 and the other offered an 80%
chance to win $100, their respective EEVs being $50 and
$80. The simulated lotteries were administered as surveys
to 78 students enrolled in a course on product realization.
Of these, 37 were either seniors or graduate students at the
University of Illinois and 41 were graduate engineers
Fig. 5. The demand curves for the simulated purchase of a lottery
taking the course as part of a continuing education pro- ticket
H. E. Cook, A. Wu: On the valuation of goods and selection of the best design alternative
The fact that the values determined for the intercepts
are close to the respective EEVs of the tickets supports the
use of the S-model in representing the aggregate behavior
of buyers in the purchase of a good. As price falls below
value, demand increases with the increase in the driving
force given by V±P. Whether or not these ®ndings apply to
other, more complex goods is speculative because we do
not know the economic values of other goods and are thus
unable to test the prediction. However, the results found
here are strongly supportive of the intercept as a meaningful measure of value.
Use of the intercept as a phenomenological measure of
the value of the product is also intuitive in the absence of
the S-model. The use of V±P as a measure of the driving
force for demand was suggested independently by Anderson and Naurus (1998), for example. The reservation
price for an individual is widely used as a measure of value
(Plott 1990). The intercept here represents the reservation
price for the buyer segment. Demand according to the Smodel is considered to be a stochastic process in which the
probability that an individual in a segment would purchase a good is proportional to V±P (Cook and DeVor
1991). The resulting demand curve is the summation of the
individual purchases. It also follows from the model, that
when V±P is negative the driving force is to sell the good.
Another key property of the S-Model is its prediction of
how a piecewise, linear ®t to the demand curve over a
small range in price shifts when the value of the good
increases by a small amount dV. This property is also
demonstrated in Fig. 5 by the overlap of the two curves in
the vicinity of P/EEV1. The model assumes that the slope
of the demand curve does not change for a small change in
value. The fact that the two curves superimpose in the
quasi-linear region in Fig. 5 means that the two slopes are,
in fact, not the same and suggests that the fractional
change in K is directly related to the fractional change in V
through the relation
dK

K
dV
V
…12†
This empirical relationship does not contradict the
S-model but supports its assumption that if the value
change is small, then a possible change in slope can be
ignored. The modest change in slope in the simulated
experiment is a result of the large difference of $30
between the EEVs of the two tickets.
Respondents as sellers
Students in the same class also participated in a simulated
market in which they were given the lottery tickets and
then surveyed as to their selling price. For each ticket, the
resulting supply curve, Fig. 6, intersected the demand
curve at a price divided by the respective EEV of approximately 0.75. The price PMC  0:75 EEV is known as
the ``market-clearing price'' as it represents the price
where the number of buyers of the ticket would just equal
the number of sellers. This price is seen to lie within the
quasi-linear portion of the curve in Fig. 5.
Although the respondents were similar in many demographic aspects, the behaviors of the demand and
47
Fig. 6. Simulated demand and supply curves for the two lottery
tickets
supply curves in Fig. 6 cannot be explained by assuming
that the respondents had a single value for a given lottery
ticket equal to its EEV. If all the respondents had had the
same value for a ticket, the two curves in Fig. 6 would have
only touched at P/EEV=1 and no sales would have taken
place. We postulate that the value differences arise from
the respondent's differences in risk aversion. At the market-clearing price in Fig. 6, those who were most risk
averse would be sellers and those who were least risk
averse would be buyers. Based upon the curves, the buyers
valued the tickets at VBEEV. They would have purchased
the tickets from sellers who valued the tickets at
Vs0.4 EEV. Thus the S-model is suf®cient to describe the
demand and supply curves in the market-clearing region
in terms of aggregate behavior. It is useful at this juncture
to point out that risk aversion is not the dominant factor
causing buyers and sellers to have different values for
massed produced goods. The manufacturer of an automobile, for example, values this good less than a potential
buyer because the manufacturer has many more automobiles than required for his or her own use. The value of a
mass produced good to the seller can be taken equal to its
variable cost. The net value of a good to the seller is equal
to the cash ¯ow generated.
The average buy and sell prices for the 50% ticket were
$29.17 and $45.43, respectively, and for the 80% ticket,
they were $48.78 and $71.90. These differences and the
shift in the supply curve to the right of the demand curve
generated by the same respondents for the same good are
expressions of the well-known ``endowment effect'' (Thaler
1980; Knetsch and Sinden 1984; Hanemann 1991; Kahneman et al. 1990; Morrison 1998; Kolstad and Guzman
1999), which has been widely observed in controlled experiments for both simulated and actual purchases. [We
use the term ``endowment effect'' here only in a descriptive
manner and do not necessarily imply that it arises solely
from loss aversion (Kahneman et al. 1990)]. Simply stated,
persons generally post a selling price for a good signi®cantly higher than what they were WTP for the good only
Res Eng Design 13 (2001)
48
moments before. This is in contrast to the prediction from
classical economic theory and the explanation remains
unsettled (Morrison 1998).
We wish to emphasize in what follows that the endowment effect is not anomalous behavior for any model
in which the purchase of the good is considered to be a
stochastic process. Consider a market segment of individuals that have a single, true value, VT, for a good. According to the S-model, the probability that a single
individual buys the good is proportional to VT ±P>0 and
the probability that he or she sells it is proportional to
P±VT>0. Thus, if price is near but below VT, then the
probability that any given individual in the segment will
buy the good is small. Likewise, if price is near but above
VT, then the probability that this same individual will sell
the good, if it is in his or her possession, is also small.
Therefore it is statistically likely that there will be a gap
between a person's maximum buy price, PB, for a good and
his or her minimum sell price, PS, for the same good. The
true value of the good to the person can be taken to be
equal to the average of the two limiting prices,
VT=(PB+PS)/2.
Whenever a transaction is freely consummated, the
agreed upon price is such that both the buyer and the
seller perceive receiving a net gain. Thus, when an individual becomes both buyer and seller, in the sense of
purchasing a good and then immediately offering it for
sale, he or she will post a higher price than the price just
paid. The desire is to make a net gain in value from the
sale similar to the net gain made by the purchase. The
stochastic origins of the endowment effect given here are
consistent with this view of human behavior.
The gap between WTP and WTA resulting from the
stochastic nature of demand can be simulated using Monte
Carlo methods. An example is shown in Fig. 7 where the
probability that an individual buys was taken to be
Fig. 7. Monte Carlo simulations of the gap between a single
individual's WTP and WTA
pB=b(VT ±P) and the probability for selling was taken as
pS=b(P±VT). The simulated ®ndings are for b=1/5. The
two lines representing pB and pS are also shown in Fig. 7.
For the P±VT<0 region, the simulated choice is between
buy or not buy and for P±VT>0, the choice is between sell
or not sell. Points on the line equal to unity represent buy
for P±VT<0 and sell for P±VT>0. Points on the line equal to
zero represent not buy for P±VT<0 or not sell for P±VT>0.
The particular simulations shown were made by starting at
jP VT j ˆ 4:5 and moving toward 0 in steps of 0.5. Once a
transition was made from 1 to 0, the simulation was
stopped and the remaining points were set to 0. If, using
the same algorithm, a large number of simulations were
made from 4.5 to 0 and an equal number made from 0 to
4.5, the resulting average demand fractions would follow
the two lines shown. The magnitude of the slope b can be
taken as a measure of the uncertainty that the individuals
in the segment have in the value of the good. Thus, if
individuals were absolutely certain of the value, the slope
would be in®nite and there would be no gap between buy
and sell prices. This represents the condition of classical
economic theory.
Comparisons to other models
Taguchi's model
Taguchi's model for robust design is based upon a quality
loss function which is a sum of two quantities, the cost of
inferior quality, W, and manufacturing cost, which we take
to be equal to variable cost, C. As shown by Cook and
DeVor (1991), the formal relationship connecting the
S-model to Taguchi's model is the expression
X…g† ˆ V…g I †
V…g†
…13†
which equates the cost of inferior quality of an attribute at
an arbitrary level g to the difference between the value for
the attribute at its maximum or ideal level, gI, and the
value for the attribute at level g.
Taguchi suggested estimating W from the costs incurred
to repair the product when the attribute is off target. This
will generally be an underestimate of the true losses because the customer also receives a loss in value when the
product is performing less than expected and when it is
out of service. The S-model expression for W given by
Eq. 13 is a more fundamental way to arrive at this important quantity using the value curve for the attribute of
interest, such as the one shown in Fig. 4 for interior noise.
The target speci®cation, gT, in Taguchi's model is the
attribute level for the minimum in the loss function. The
target speci®cation in the S-model is the level that maximizes cash ¯ow or whatever bottom-line metric is of interest to the manufacturer. The two approaches to
determining the target speci®cation give the same result
when a monopoly is considered. However, for the general
case in which there are several competitors, choosing the
target speci®cation based upon a bottom-line metric is
preferred because it fully accounts for demand, investment, and pricing considerations. For this reason, S-model
value and cash ¯ow considerations have been incorporated
into Taguchi's Design of Experiments formalism (Cook
1997).
H. E. Cook, A. Wu: On the valuation of goods and selection of the best design alternative
Value engineering
Value engineers formally de®ne value as worth divided by
cost, which is more in keeping with a value for the money
measure. In practice they use a de®nition of functional
performance divided by cost because of the dif®culty in
quantifying worth in monetary units. Practitioners focus
on discovering and eliminating non-value-added costs in
preliminary designs (Fowler 1990). The connection between value engineering and the S-model is that worth, as
de®ned by value engineers, can be taken as being equivalent to value as de®ned by the S-model. Thus, use of the
S-model would resolve the value engineer's problem of
quantifying worth in monetary units.
QFD
The ®rst step in the QFD process is to use marketing research to obtain an ordinal rank of customer needs. Design
alternatives are then judged on the basis of their ability to
meet customer needs at low cost. The use of a zero to ten
scale by engineers to rank how well the proposed alternatives meet the customer needs is pragmatic in that it can
be done quickly. It misses a key point, however, which is
that potential customers should be more able to assess
perceived bene®ts than the engineers can. Also, QFD, like
value engineering, uses one set of units for bene®ts and
another for costs, which compromises quantifying the
difference between cost and bene®t in making trade-off
decisions. Locasio and Thurston (1993) have shown how
the problem of having costs and bene®ts in different units
could be overcome in QFD by using SEU. Recently, Cook
(2000) introduced the S-model formalism into the QFD
House of Quality to attack this same shortcoming.
A review of the S-model application to QFD is given
here to demonstrate its use in making trade-off decisions
using cash ¯ow as a metric. In doing this, it is necessary to
relate changes in cost and value to changes in price. For a
monopoly, the change in price needed to maximize cash
¯ow is equal to one-half of the sum of the value and
variable cost changes:
dV ‡ dC
dP ˆ
:
…14†
2
This expression is also approximately correct for an oligopoly based upon Bertrand's classical theory of pricing if
competitors do not change their value and variable cost.
Similarly if competitors do not change value or cost, the
forecast change in demand is given by
dV dC
dD ˆ K
:
…15†
2
The resulting change in annual cash ¯ow, A, for a given
alternative under consideration is given by
dV dC
dM
dA ˆ D0
‡ dD‰P0 C0 Š dF
:
…16†
2
Y
where F is ®xed cost and M is investment, assumed paid in
equal installments over the life of the product Y.
Incorporation of the S-model into the QFD process
using a spreadsheet is illustrated in Table 1. Four alter-
natives (factors), noted by the double index ij, are considered for improving the cash ¯ow from the sale of a
hypothetical automobile. The descriptions of the ij notations for the factors are listed in Table 2. There are N=4
competitors in the segment having a total annual demand
of 800,000 units. The baseline price of the vehicle under
consideration is $20,000 and the average price of four
vehicles is $21,000.
Customer needs are listed under the ``What'' column
and each factor represents a proposed ``How'' for meeting
the customer needs. The results from the ®rst level of QFD
are summarized in topmost section of Table 1. In rank
order beginning with most important, customers want a
more reliable, quieter, better performing, and more fuelef®cient automobile. A plus sign under a factor indicates
that the factor is expected to have a positive in¯uence on
the need, a minus sign is used to signify that a negative
effect is expected, and a zero is used to indicate that no
effect is expected. The second section of Table 1 lists the
system level attribute of the vehicle judged to best meet the
need expressed. In sections two and remaining, the baseline levels are shown in the ®rst column on the left. The
baseline attributes would be determined from measurements on production vehicles. The deviations from baseline would be obtained either from measurements on
prototype vehicles or from computer simulations of performance. Changes in the attributes are linked to changes
in value from baseline in the third section of Table 1. A
reduction of one repair was taken equal to $300 (Donndelinger and Cook 1997). The remaining value computations are described in Appendix B. The ®nal section in
Table 1 represents the computational steps leading to the
forecasts of the change in cash ¯ow for each of the factors.
The 2.2 liter inline-4 DOHC engine with balance shafts is
seen to be preferred to the V-6. The reliability improvement package is seen to make a major improvement over
baseline. Lightweight material A is seen to make a positive
addition to cash ¯ow; whereas, material B makes a negative contribution. At this point, it would be wise to build
several prototype vehicles incorporating factors 11, 21, and
31 to verify the product improvements.
In going through this simulated exercise, it is worth
noting that the fundamental metrics of variable cost and
value were not compared against each other in a traditional cost versus bene®t manner. Instead, they were used
to compute a forecast of cash ¯ow, a bottom-line metric
which formed the basis for making the decisions. Only in
this manner can the in¯uence of the investment level on
the merits of a possible alternative be accounted for
properly. Also the choices were made on how the factors
impacted customer needs in the aggregate. No special
weighting was given to a need based upon its importance
ranking at the top of Table 1 as values determined from
the DV method are complete for quantifying the impact of
the design changes on each customer need.
The ``House of Quality'' construction for QFD has a
``roof '' in which the strength of the interactions between
the alternatives are noted. Such a construction is impossible using an orthogonal spreadsheet. Instead, interactions are displayed using additional columns to the right
of those shown in the topmost section of Table 1. The
49
Res Eng Design 13 (2001)
Table 1. QFD matrix including
fundamental and bottom-line
metrics (reprinted with permission of the QFD Institute, Ann
Arbor)
50
interaction column for factors 11 and 21 would be 1121
and so forth. Moreover, the QFD process shown here can
be replaced with a design of experiments formalism (Cook
1997) and used to make a quantitative assessment of the
interactions between alternatives. In fact, the entire process of alternatives from system to subsystem to component design can be expressed as a waterfall of experiments
(Kolli and Cook 1994). Each level uses the same bottomline metric for assessing the merits of the alternatives
being considered whether they are subsystem or component alternatives.
SEU
A major difference between SEU and the S-model is that
the SEU utilities are assessed from an interview with a
particular individual (a so-called decision maker);
whereas, S-model value represents an aggregate number
determined from a survey of potential customers. The
decision-maker, usually a key executive within the company, is interviewed to assess the SEU utilities for both
costs and bene®ts following a well-de®ned process
(Thurston 1990). Using the approach presented by
Koppleman (1975) [see the discussion on p. 134 in
Ben-Akiva and Lerman (1985)], the decision maker's
utilities can, however, be taken as representing those of
an ``average individual'' thereby converting them into an
aggregate form. (This is but one of several approaches
proposed by Koppleman for arriving at aggregate utilities.)
If the utilities are taken as an aggregate measure, then the
SEU analysis is not compromised by the fact that several
of the axioms by von Neumann and Morgenstern (1947)
(required for an individual to maximize utility) have been
refuted in experimental tests (Kahneman and Tversky
1979; see also Tversky and Kahneman 1981). In this regard, Scott and Antonsson (1999) have made a careful
analysis to show that because of the necessity for aggregation, Arrow's Impossibility Theorem is also not a restriction to making meaningful cost/bene®t trade-offs.
Discussion and summary
Market behavior
For prices near but below the respective EEVs of the lottery tickets, the intercepts and the demand shifts observed
in the simulated markets were in reasonable agreement
with the predictions of the S-model. The S-model yielded a
straightforward explanation of the market-clearing behavior shown in Fig. 6. The most risk averse respondents
would be the suppliers who valued the tickets less than
their respective EEVs. The least risk averse would be the
buyers who valued the tickets at their respective EEVs. As
in any market, those who value the good the least would
sell to those who value the good the most.
The S-model's stochastic view of aggregate demand
predicts an endowment effect as demonstrated here using
Monte Carlo simulations. Explanations offered elsewhere
for the effect have been based upon loss aversion (Kahneman et al. 1990), the nature of movements along indifference curves when there are not comparable substitutions
for a good, which applies mainly to public goods
(Hanemann 1991), and the uncertainty that bidders have
in the value of the good (Kolstad and Guzman 1999).
The explanation here does not rule out the other
mechanisms listed above. The lack of a comparable
substitution mechanism for the effect in Hanemann
(1990), however, should be very weak in the type of market
H. E. Cook, A. Wu: On the valuation of goods and selection of the best design alternative
Table 2. Explanation of attribute indices
studied here. The EEVs of the lottery tickets were well
understood by the respondents, which lessens the contribution to the effect proposed by Kolstad and Guzman
(1999). However, their model is also stochastic and thus a
gap between WTP and WTA should exist simply for this
reason alone. Loss aversion could have contributed to our
®ndings here but we do not think it was necessarily
dominant. The respondents were not told that they had to
give up the tickets for a WTA price. They freely chose to
sell at a price of their own choosing or not to sell. Thus, we
do not see a strong element of seeking compensation for a
perceived loss.
Value trends
The construction of value trends for competing products,
Fig. 3, provides a quantitative assessment of the effectiveness of how design changes over time have impacted
the values of competing products. Based upon conversations with persons who have developed and analyzed value
trends in proprietary applications of Eq. 5, value trend
plots provide important insight into the competitive
landscape, particularly when major product redesigns are
introduced as seen here in Fig. 3. Values computed in this
manner also provide the value for the baseline product, V0.
Formulating the general problem of selecting
the best alternative
The general problem of selecting the best alternative requires an assessment of several types of costs (variable,
®xed, and investments in research, development, tooling
and facilities) as well as a projection of prices and competitive actions. A bottom-line metric such as cash ¯ow,
pro®t, breakeven time, return on investment, or internal
rate of return needs to be used to assess properly the
overall merit of each alternative for the general problem.
The time required to develop an alternative for production
is also a key metric. Taguchi's model, value engineering,
and QFD, did not formally treat demand in their original
formulations and this limited the range of problems that
could be considered. However, both have been tightly
linked to the S-model as described here and elsewhere
(Cook 1997; Cook 2000), which opens these methods to
treating the general problem. SEU can use the logit model
(Ben-Akiva and Lerman 1985) to analyze market share to
expand its range of applicability.
In considering the possible actions of competitors,
value trend plots of the type shown in Fig. 3 can be supplemented by the DV method to gain greater insight. For
example, a DV study (Wu 1998) has shown that the value
of the second rear sliding door is over $1200 per vehicle.
This is almost one-half of the value difference found
between mvA versus mvB in the 1996 model year. When
evaluating a major feature such as the added door, a study
of the impact on the bottom-line of each possible scenario
should be made. With two competitors, competitor A adds
the feature and competitor B either does or does not. Or A
does not add the feature and B either does or does not. If
the value of the feature is higher than its variable cost and
if the investment required is not too large, the outcomes of
such scenario studies will always be to add the feature.
This conclusion, however, assumes that the added price
increment does not make the price go above the maximum
level for the market segment. Thus, under the price constraint, it is necessary to prioritize possible new features
on the basis of their projected pro®tability to determine
which should be incorporated. The cut-off in adding features should be at the point where price approaches its
upper limit for the segment of buyers targeted, assuming
that the resources needed to design, tool, and facilitate the
added features have not been fully consumed before this
point.
The forecast of the bottom-line metric is uncertain and
the range of uncertainty should be evaluated. The most
straightforward means for doing this is to examine the
uncertainty in cash ¯ow using Monte Carlo methods. It is
driven by the uncertainties in the values, prices, and price
elasticities. Finally, we point out that, although almost all
of the S-model applications to date have focused on incremental improvements to existing products, it can be
applied to highly innovative products. A baseline represented by an existing product or service still needs to be
found. The DV method can then be used to develop value
assessments for the innovative product. This will, of
course, be greatly facilitated if respondents can evaluate
prototypes of the innovative product. If the forecast
changes in value, cost, and price are large, a non-linear
model such as logit model may need to be invoked to make
the demand forecasts.
Appendix A
Surveys regarding the simulated purchase of lottery tickets
(Tables 3, 4).
In each of the two surveys, you are asked to state your
willingness to purchase a lottery at a series of different
prices. Assume that your purchase of a ticket entitles you
to be a potential winner in a single drawing (lottery) in
which the chances of winngin $100 are shown at the top of
each column. Each ticket is offered at 20 prices. For each of
the prices shown, the box on the left should be checked if
you would not the buy the ticket and the box on the right if
you would buy the ticket.
In each of the two surveys, assume you have been given
a single ticket with the odds of winning a $100 prize listed
at the top of the column. You have two options: (1) keep
the ticket and participate in the lottery or (2) sell the ticket
at the price offered. For each of the prices shown, please
check tbe box on the right if you would sell your ticket at
the price offered. Otherwise, check the box on the left if
you would not sell your ticket at the price offered.
51
Res Eng Design 13 (2001)
Table 3. Survey used for simulated purchase of lottery tickets
Table 4. Survey used for simulated sale of lottery tickets
52
Table 5. Computation of the
deviation in value from baseline
for interior noise
Table 6. Computation of the
deviation in value from baseline
for acceleration performance
Appendix B
The computations of the value differences from baseline for
interior noise, acceleration performance, and fuel economy
are given in Tables 5, 6, and 7, respectively, for each of the
factors evaluated in the simulated QFD study. Exponentially weighted three point value curves used for the interior
noise and acceleration performance computations were
taken from the results of Pozar and Cook (1997) and
McConville and Cook (1996), respectively. The attribute
variable g for expressing acceleration performance was
Log(1/t) where t is the acceleration tine in seconds. The
reason for this choice is that a driver was assumed to sense
H. E. Cook, A. Wu: On the valuation of goods and selection of the best design alternative
Table 7. Computation of the
deviation in value from baseline
for fuel economy
53
the acceleration force in a psychometric manner similar to
noise. The critical value for acceleration time was taken to
be 40 s, which was deemed so slow as to become a liability
when entering traf®c. The ideal time was taken as 2 s, which
produces an acceleration force of approximately 1 G. Values for fuel economy were computed using a price of fuel of
$1.50 and an annual driving distance of 10,000 miles, which
yields an annual expenditure of $682 for the baseline fuel
economy of 22 mpg. The expenditures were discounted, as
shown in Table 7, using a 5% rate over a period of 7 years.
This discount rate and period of time was based upon the
DV method ®ndings of McConville and Cook (1996).
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