Empirical Examination of the Functional Basis and
Design Repository
Benjamin W Caldwell, Chiradeep Sen, Gregory M Mocko, Joshua D
Summers and Georges M Fadel
Clemson University, USA
This paper investigates the use of functional basis within the design repository
through the observation of eleven functional models. It also examines the amount
of information contained in the functional model of a hair dryer at various
hierarchical levels. Two experiments show that the secondary level of the
functional basis hierarchy is used most often because the secondary level provides
significantly more information than the primary level, and the tertiary level does
not provide enough additional information to be useful.
Introduction
Functional Analysis in Design
Functional modeling is a well-accepted activity in conceptual design [1-3].
According to Pahl and Beitz, a function is “the intended input/output
relation of a system whose purpose is to perform a task”. A function is a
solution-neutral representation of the process of converting a set of inputs
to a set of outputs. Input and output flows are broadly classified in early
research as that of materials, energies and signals. The overall function of a
product is described by linking together multiple functions via their flows,
creating a function structure [1].
The use of functional analysis as a conceptual design tool is discussed in
literature, particularly by European researchers [1, 4], as a means for
broadening the search for solutions. Significant advances in function-based
design have been made by Collins et al. [5], Hundal [6], Kirschman and
Fadel [7], Szykman et al. [8], Little et al. [9], Otto and Wood [2], and
J.S. Gero and A.K. Goel (eds.), Design Computing and Cognition ’08,
© Springer Science + Business Media B.V. 2008
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Stone and colleagues at Missouri University of Science and Technology
[10-12].
Functional Basis
Recent research efforts have identified the need for a finite vocabulary of
terms to increase consistency in functional models [7, 8, 11]. The
functional basis is one such vocabulary, which contains 54 functions and
45 flows, arranged in a three-level hierarchy that can be used to describe
the function of products in a consistent manner. The functional basis seeks
to support archival of design knowledge and comparison of products
functionally as well as provide a standard stopping point for functional
decomposition [12].
The hierarchy of functions and flows within the functional basis, shown
in Table 1 and Table 2, respectively, was originally created to allow
designers to describe function at various levels of detail. Hirtz and
colleagues [12] explain that original design problems may use higher-level
terms since the details of the product are not known. Adaptive and variant
designs, however, may use more specific, lower-level terms since the
details about a functional model are already known. In addition, the
authors of the functional basis state that the secondary level provides the
most specific function detail that is practical for engineering design [12].
However, these claims about the hierarchy are not supported by
experimental evidence.
The Function-Based Design Repository
The functional basis and related research has led to the development of a
function-based design repository at Missouri University of Science and
Technology, hereafter referred to as the repository or the design repository
(available at http://function.basiceng.umr.edu/repository/). This web-based
design repository, populated through reverse engineering and disassembly
of consumer products, contains a functional description of 115 products
[10]. Functional models for approximately half of these products are
present in graphical form in the repository, and all of the products in the
repository contain customer requirements, function-component
relationships, and component-assembly relationships in matrix form.
Empirical Examination of the Functional Basis and Design Repository 263
Table 1 Functional Basis Function
Hierarchy
Branch
Channel
Separate
Distribute
Import
Export
Transfer
Guide
Connect
Couple
Control
Magnitude
Mix
Actuate
Regulate
Change
Stop
Convert
Provide
Convert
Store
Signal
Supply
Sense
Indicate
Support
Table 2 Functional Basis Flow
Hierarchy
Divide
Extract
Remove
Transport
Transmit
Translate
Rotate
Allow DOF
Join
Link
Increase
Decrease
Increment
Decrement
Shape
Condition
Prevent
Inhibit
Contain
Collect
Supply
Detect
Measure
Track
Display
Process
Stabilize
Secure
Material
Human
Gas
Liquid
Solid
Plasma
Mixture
Signal
Status
Control
Energy
Human
Acoustic
Biological
Chemical
Electrical
Electromagnetic
Hydraulic
Magnetic
Mechanical
Object
Particulate
Composite
Gas-Gas
Liquid-Liquid
Solid-Solid
Solid-Liquid-Gas
Colloidal
Auditory
Olfactory
Tactile
Taste
Visual
Analog
Discrete
Optical
Solar
Rotational
Translational
Pneumatic
Radioactive/
Nuclear
Thermal
Motivation and Research Questions
The functional basis and design repository has been continually developed
for over a decade [8, 10-12], Several tools have been developed that
operate on information stored in the repository, including automated
concept generation [13], and functional similarity [14]. More recently,
researchers have critically evaluated the role of function in product
development and suggest it is not sufficient for completely describing
products [15, 16].
Previous research has focused on exploring the theoretical foundations
of the functional basis and the development of tools employing the
264
B.W. Caldwell et al.
functional basis. However, there exists a gap in evaluating the practical
usage of the functional basis, specifically the use of the hierarchical
organization of functional terms. To address this gap, the approach taken
in this research is largely empirical and is supported by experiments on
functional models within the design repository. The research questions
(RQ) that this paper seeks to answer are:
RQ1: To what extent has the functional basis’ controlled vocabulary
been used to describe the products in the design repository?
RQ2: What information value is obtained by using the hierarchy of
terms defined in the functional basis?
Answering these questions will provide a deeper exploration of the
claim made by Hirtz et al. [12] that the secondary level is the most
frequently used level for engineering design. This is accomplished by
evaluating the frequency of terms (RQ1) and further explained by
computing the information content across levels of the hierarchy (RQ2).
Experimental Approach
Two experiments, which focus on the use of the functional basis within the
design repository, are discussed in this paper. In the first experiment, the
frequency of usage of each term of the vocabulary is evaluated for eleven
functional models selected from the design repository. This experiment
addresses the first research question. In the second experiment information
content of a functional model is computed at different levels of the
functional basis hierarchy, addressing the second research question.
The experiments completed in this research are demonstrated using a
consumer hair dryer (see Figure 1).
Fig. 1. Hair Dryer Functional Model
The hair dryer was selected as an example because it is an
electromechanical consumer product, similar to many of the products in
Empirical Examination of the Functional Basis and Design Repository 265
the design repository. It demonstrates the use of many of the functional
basis’ commonly-used functions, and the hair dryer example has been used
in previous research papers related to functional modeling [17].
Experiment 1: The Use of the Hierarchy within the Design
Repository
Objective
The intent of this experiment is to understand how the functional basis’
hierarchy is currently used by analyzing functional models in the design
repository. The design repository is the largest implementation of the
functional basis, so it is fitting to analyze models stored in the design
repository. By exploring how the hierarchy is used, this study will give
insight into the hierarchy’s usefulness and each level’s ability to represent
the functionality of a particular product.
Experimental Protocol
In this study, ten randomly selected functional descriptions in addition to
the hair dryer are analyzed. The products were limited to those with a
downloadable functional model. These eleven functional models were
analyzed by counting the frequency of use of each unique term in the
collection of models. The terms were categorized as functions verbs,
function nouns, or flows according to the guidelines in this section. A
sample function block with input and output flows, taken from the hair
dryer functional model, is shown in Figure 2 and used to explain the
experimental procedure. Articles, prepositions, and conjunctions are
ignored in this experiment, which is indicated in Figure 2 (strikethrough
text).
Fig. 2. Sample Function Block from Hair Dryer
Function verbs are inside a function block and are part of the functional
basis’ function set (see Table 1). Function verbs were easy to identify
266
B.W. Caldwell et al.
because they were always the first word inside a function block. In the
sample, the only function verb is “Convert” (bold text).
Function nouns are inside a function block, may include an adjective
describing the noun, and are usually part of the functional basis’ flow set
(Table 2). A function can contain more than one function noun. In the
sample, the two function nouns are “Electrical Energy” and “Thermal
Energy” (single underline text).
Flows are arrows that enter or exit function blocks and were usually part
of the functional basis’ flow set (Table 2). Any label associated with an
arrow was considered a flow. In some cases, an arrow had more than one
label, separated by a comma, probably for the purpose of reducing the
number of arrows cluttering the functional model. In these cases each
label was considered a flow. In some cases, flows were not labeled in the
functional model; unlabeled flows were counted the same as their most
recently labeled flow. Flows were counted each time they entered a
function block; flows that exited a block but did not enter another block
(outputs of the entire system) were also counted. For example, in Figure 1,
the flow “Human Energy” would be counted five times because it enters
four function blocks (“Import”, “Guide”, “Export”, and “Convert”) and it
is an output of the system (via “Export”). The two flows in Figure 2 are
“Electrical Energy” and “Thermal Energy” (double-underline text).
The eleven functional models include: Black and Decker Jigsaw
Attachment, Brother Sewing Machine, Cassette Player, Delta Circular
Saw, Delta Nail Gun, Dryer, Digger Dog, Garage Door Opener, Oral B
Toothbrush, Shopvac, and Supermax Hair Dryer.
Results
The results from this experiment are shown in Tables 3, 4, and 5. The first
column in each table lists the exact term used in the eleven functional
models. The second column gives the total number of times that the term
was used in the eleven functional models, and the third column gives this
value as a percent. The shaded rows indicate terms that are not defined in
the functional basis.
Observations
The following observations are made (see Table 6):
Empirical Examination of the Functional Basis and Design Repository 267
• Approximately half of the terms available in the functional basis were
used in the eleven functional models. The first row in Table 6 shows
how many terms were available and how many were used.
• The functional models followed the functional basis well for function
verbs, but not as well for function nouns and flows. Every function
verb used in these eleven models was a functional basis term; however,
15 unique function nouns terms and 32 unique flow terms were used
that are not part of the functional basis. Of the 353 instances of function
nouns, 90.7% were functional basis terms; of the 513 flow instances,
only 75% of the flow terms were from the functional basis vocabulary.
• There was a high frequency of use of a few terms. The five most
frequent function verbs, shown in Table 6, account for 67.7% of all
function verbs. The five most frequent function nouns account for
62.6% of all function nouns (69.1% of functional basis nouns used). The
five most frequent flows account for 51.9% of all flows (69.1% of
functional basis flows used). Thus, excluding non-functional basis
terms, about two thirds of function verbs, function nouns, and flows can
be accounted for by five functional basis terms in their respective
categories.
• The secondary level of the hierarchy is used most often. The use of the
hierarchy within the eleven functional models can be seen in the last
row of Table 6. Secondary terms are used 95%, 79%, and 66% of the
time for function verbs, function nouns, and flows, respectively. If nonfunctional basis terms are ignored, then these percentages increase to
95%, 87%, and 88%, respectively. Therefore, when a functional basis
term is selected, about 92% of the time it is secondary.
• Most function verbs are a type of channel, and most function nouns
and flows are a type of energy. Table 7 and Table 8 further demonstrate
how the hierarchy is used by these eleven functional models. The
percentages in these tables represent the number of functions nouns,
verbs, or flows that are labeled with either the corresponding term or a
term hierarchically below that term. For example, in Table 8, the
function noun status signal (secondary) is comprised of 0.6% auditory
status signal (tertiary), 0.6% visual status signal (tertiary), and 2.8%
status signal (secondary), for a total of 4.0%. Similarly, the 11.0%
signal (primary) is the total of 4.0% status signal (secondary), 5.9%
control signal (secondary), and 1.1% signal (primary). It can be seen
from these tables that 57.3% of all function verbs used are hierarchically
under channel, 57.2% of all function nouns used are hierarchically under
energy, and 46.2% of all flows used are hierarchically under energy.
B.W. Caldwell et al.
268
Table 3 Verb Results Table 4 Noun Results
Term
export
import
transfer
convert
guide
actuate
change
transmit
distribute
regulate
store
couple
supply
secure
separate
stop
process
position
indicate
translate
rotate
support
mix
track
Total
#
48
45
45
44
21
15
14
9
8
8
8
5
5
4
3
3
3
3
2
2
2
1
1
1
300
%
16
15
15
14.7
7
5
4.7
3
2.7
2.7
2.7
1.7
1.7
1.3
1
1
1
1
0.7
0.7
0.7
0.3
0.3
0.3
100
Term
electrical energy
mechanical energy
solid
control signal
human energy
rotational energy
human material
status signal
gas
electromagnetic
human force
acoustic energy
mixture
translational energy
signal
magnetic energy
torque
pneumatic energy
thermal energy
auditory status signal
solid-gas mixture
visual signal
energy
solid-solid
hand
garage door
reaction
(time)
button signal
garage
on/off
safety signal
(clothes)
(lint)
blade
button
limit signal
motion
weight
Total
#
81
58
40
21
21
15
12
10
9
6
6
5
5
5
4
4
4
3
3
2
2
2
1
1
8
5
3
2
2
2
2
2
1
1
1
1
1
1
1
353
Table 5 Flow Results
%
22.9
16.4
11.3
5.9
5.9
4.2
3.4
2.8
2.5
1.7
1.7
1.4
1.4
1.4
1.1
1.1
1.1
0.8
0.8
0.6
0.6
0.6
0.3
0.3
2.3
1.4
0.8
0.6
0.6
0.6
0.6
0.6
0.3
0.3
0.3
0.3
0.3
0.3
0.3
100
Term
electrical energy
mechanical energy
human material
control signal
human energy
solid
human force
status signal
pneumatic energy
solid-solid
acoustic energy
rotational energy
electromagnetic
thermal energy
gas
magnetic energy
material
translational
auditory signal
visual signal
torque
hand
on/off
air
blade
ff/rew
play/stop
weight
thread
saw blade
debris
power pack
wrench
brads
debris and air
toothpaste
toothpaste/debris
wood
dirty teeth
heat
hot air
sewed material
teeth
alignment
analog video
feed speed
forward/reverse
intensity
noise
sewing/bobbin
speed
stereo audio
stitch width
Total
#
89
75
40
34
28
21
20
13
10
9
8
8
6
6
5
4
3
3
1
1
1
19
14
8
8
7
7
7
6
5
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
513
%
17.3
14.6
7.8
6.6
5.5
4.1
3.9
2.5
1.9
1.8
1.6
1.6
1.2
1.2
1
0.8
0.6
0.6
0.2
0.2
0.2
3.7
2.7
1.6
1.6
1.4
1.4
1.4
1.2
1
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
100
Empirical Examination of the Functional Basis and Design Repository 269
Table 6 Summary of Experiment 1 Observations
Vocabulary
terms used
Unique nonfunctional basis
terms
Percent of
terms from the
functional basis
High frequency
terms
Use of the
hierarchy
Function Verb
24 out of the 54
available
Function Noun
22 out of the 45
available
Flow
19 out of the 45
available
0
15
32
100%
90.7%
75%
Elec Energy: 22.9%
Mech Energy: 16.4%
Solid:
11.3%
Human Energy: 5.9%
Control Signal: 5.9%
Primary:
1.4%
Secondary:
78.8%
Tertiary:
7.6%
Power Conj.:
2.8%
Other:
9.3%
Elec Energy: 17.3%
Mech Energy: 14.6%
Human Mat’l: 7.8%
Control Signal: 6.6%
Human Energy: 5.5%
Primary:
0.6%
Secondary:
66.1%
Tertiary:
4.3%
Power Conj.:
4.1%
Other:
25.0%
Export:
Import:
Transfer:
Convert:
Guide:
Primary:
Secondary:
Tertiary:
16.0%
15.0%
15.0%
14.7%
7.0%
0.3%
95.0%
4.7%
Table 7 Hierarchy Distribution
for Function Verb
Primary Verb Secondary
Separate
Branch
3.7%
Distribute
Import
Export
Channel 57.3%
Transfer
Guide
Couple
Connect 2.0%
Mix
Actuate
Control
Regulate
13.3%
Magn
Change
Stop
Convert 14.7% Convert
Store
Provide 4.3%
Supply
Sense
Signal
2.0% Indicate
Process
Stabilize
Support 2.7% Secure
Position
Verb
1.0%
2.7%
15.0%
16.0%
18.0%
8.3%
1.7%
0.3%
5.0%
2.7%
4.7%
1.0%
14.7%
2.7%
1.7%
1.0%
1.0%
1.3%
1.0%
Table 8 Hierarchy Distribution for Function
Noun and Flows
Primary Noun
Flow Secondary
Human
Gas
Liquid
Material 19.5% 19.3%
Solid
Plasma
Mixture
Status
Signal 11.0% 9.6%
Control
Human
Acoustic
Biological
Chemical
Electrical
ElectroEnergy 57.2% 46.2% magnetic
Hydraulic
Magnetic
Mechanical
Pneumatic
Radioactive
Thermal
Noun Flow
3.4% 7.8%
2.5% 1.0%
11.3% 4.1%
2.3%
4.0%
5.9%
5.9%
1.4%
1.8%
2.9%
6.6%
5.5%
1.6%
22.9% 17.3%
1.7% 1.2%
1.1% 0.8%
22.1% 16.8%
0.8% 1.9%
0.8% 1.2%
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B.W. Caldwell et al.
Experiment 1 Analysis
The lack of use of all functional basis terms does not indicate that the
unused terms are not needed. The products in the design repository are
limited to a specific domain of products, consumer electromechanical
products, so some terms may not be used. However, if a more extensive
study of products from additional domains produces similar results, then
this may indicate that not all functional basis terms are needed.
The high use of additional terms in the eleven functional models
demonstrates the ability of functional models to contain specific details
about a product since, in most of the instances, a more-specific term was
used. For example, “on/off”, which was used fourteen times in seven of
the functional models, contains more detail than the functional basis
tertiary term discrete control signal. In many cases, it appears that if
functional basis terms had been used, it would be nearly impossible to
understand the meaning of the functional model. For example, in the hair
dryer functional model (Figure 1), if both “on/off” and “intensity” were
labeled as control signal or discrete control signal, an observer would have
difficulty understanding that one flow referred to the on/off switch and the
other to the temperature control.
The high frequency of function noun and flow terms – electrical energy,
mechanical energy, solid, and human material – make sense as all of the
products analyzed are consumer electromechanical products. The high
frequency use of convert also seems reasonable because functions
represent a transformation of energy, material, or information through a
system. However, the extensive use of channel and its hierarchical
subordinates (import, export, transfer, and guide) was not expected. Over
half (57.3%) of all function verbs pertain to channel, which is defined as,
“To cause a flow (material, energy, signal) to move from one location to
another location.” [12]. This statistic seems to indicate that the primary
means of accomplishing the overall product function is to move material,
energy, or signals from one place to another. In addition, 31.0% of the
function verbs are either import or export, showing the importance of
moving material, energy, or signals specifically into or out of the product.
The use of primary, secondary, and tertiary terms, approximately 1%,
92%, and 6% of the time, respectively, if only functional basis terms are
considered, supports the claim that the secondary level is the most practical
level for engineering design. However, an explicit set of rules does not
exist that tells designers which level to use when creating functional
models. This suggests that designers intuitively recognize the value of the
secondary level and predominantly use it for functional modeling.
Empirical Examination of the Functional Basis and Design Repository 271
Experiment 2: Information Content in Functional Models
Objective
The objective of this experiment is to further analyze the key findings from
Experiment 1, which demonstrates the high use of secondary terms in the
design repository and suggests that the secondary level has an inherent
value that is greater than the primary or tertiary levels. To complete this
experiment, first, two information content metrics are developed for
quantifying the information in functional models. Second, information
content is computed for a functional model described at various levels in
the hierarchy. The findings of this experiment are then used to provide a
plausible justification of the high frequency of secondary terms in the
functional models.
Relevant basics of information theory
The information metric of functional models relies upon the basics of
information theory [18]. A discrete source of information produces a
message as a sequence of discrete events from a finite set of possible
events, called a vocabulary. The occurrence of each event in the message is
controlled by a set of probabilities. Information obtained from the
occurrence of an event ‘i’ is defined in terms of the probability of that
event, pi, as [18, 19]:
I ( p ) = log(1 pi ) , where b is a positive number
The following properties of information (IPs) have been discussed in
literature [20, 21]. These properties are summarized by Carter [22] as:
IP-1: Information is always a non-negative quantity.
I ( pi ) ≥ 0
IP-2: If an event has probability of 1, no information is obtained from
its occurrence, meaning, since the event is fully predictable, its
occurrence does not add any information.
I (1) = 0
IP-3: If two independent events occur (whose joint probability is the
product of their independent probabilities), the total information
obtained should be the sum of their individual information.
B.W. Caldwell et al.
272
I ( p1 ⋅ p2 ) = I ( p1 ) + I ( p2 )
IP-4: Information is a monotonic continuous function of the
probabilities, which means a slight increase in the probabilities
should always result into a slight increase in information.
Conceptually, when b equals 2, the magnitude of information represents
the number of ‘binary’ questions (answered in yes/no format) that must be
asked in order to determine the event exactly. Starting with ‘x’ choices in
the vocabulary, the questions will essentially form a binary search tree
through the vocabulary, each time eliminating half of the search domain
(x/2, x/4, x/8… etc.) and asking if the occurred event belonged to the left
branch or the right branch.
‘I’ is accepted as a measure of information since, conceptually, the
knowledge of an event of probability pi saves the observer asking, hence
stands for answers to, ‘I’ number of questions. With the base of logarithm,
b as 2, the unit of information is bits.
Functional models as discrete sources of information
For simplicity, consider the functional model shown in Figure 3. This
model is obtained by (1) removing arrows and nouns from Figure 2,
retaining only the verbs and (2) replacing non-standard terms, if any, with
primary level terms of the functional basis.
Fig. 3. Hair Dryer Functional Model (Primary Verbs only)
The model in Figure 3 can be treated as a discrete source, thereby
allowing the application of information theory, based on the following
observations:
1. Each verb behaves as a discrete event, since it either appears fully in
the model or not at all; there is no intermediate (continuous) state.
2. Each verb is chosen from a finite vocabulary (the functional basis).
3. The occurrence of each verb is controlled by a probability
distribution. The distribution is assumed to be uniform in this paper.
Empirical Examination of the Functional Basis and Design Repository 273
4. The functional model as a whole behaves like a message, since a
reader encounters one verb at a time when traversing through the
model.
Thus, each verb carries some information with it, expressed as a
function of the verb’s probability. The information from the functional
model can be then computed as the arithmetic sum of information obtained
from individual verbs. Since the functional basis does not provide a
formalism for modeling and analyzing the “connectedness” of the
functional models. The connectivity between the verbs does influence the
information theory-based analysis, but is not addressed in this study.
Thus, flow arrows are omitted in Figure 3. However, the nouns behave in a
similar manner as the verbs and add to the functional model’s information.
Information metric for functional models
Based on the analogy in Section 4.3, the information in a functional model
is found by adopting the definition of information from Section 4.2. For
the primary functional model (Figure 3), the following is computed:
Size of available vocabulary from Table 1,
x =8
Therefore, probability (assumed uniform) of each verb is given by
pi = 1 x = 1 8
Therefore, information in each verb is given by
I i = log 2 ( 1 pi ) = log 2 (8) = 3 bits
Total number of verbs in the functional model, as counted from Figure 3
y = 18
Therefore, total information in the model is given by
I = y log 2 ( 1 pi ) = 18 × 3 = 54 bits
The results are summarized in Table 9. Conceptually, the functional
model in Figure 3 represents answers to 54 binary questions, hence 54 bits
of information can be said to be encoded in the functional model. Since the
probability distribution of all verbs in the vocabulary is assumed uniform,
1 = x . Therefore, the metric of information, expressed in terms of the
pi
size of the vocabulary,
B.W. Caldwell et al.
274
I = y log 2 ( x)
is used as the expression for information of a functional model in the
remainder of this paper, where y is the number of verbs and x is the size of
the vocabulary in the level.
Table 9 Calculation of information in hair dryer functional model (primary)
Primary information
Size of vocabulary
8
Probability
1/8
Unit information
3
Instances
18
Total information
54.000
Verification against requirements of information
The metric presented in Section 4.4 is verified against IR-1 through IR-4
discussed in Section 4.2. From Table 9:
1. Information in a functional model is always positive.
2. A functional model carries no information if the probability of one
verb is 1. In that case, only one verb gets repeatedly used in each
block (and only one noun in each arrow, if nouns are used). Such a
model can be summed up as just one verb (and two nouns),
irrespective of number of verbs (or nouns) in the functional model.
Therefore, an additional block (same verb) does not change the
functional model. Thus, each additional block carries information of
zero. Therefore, the entire model carries total information of zero.
3. Information (I1) from two independent verbs of probabilities pi and pj
is equal to the information obtained based on their joint probability
(I2).
I1 = log 2 ( 1 pi ) + log 2 ( 1 p j ) = log 2 ( 1 pi ×
1
pj
)
4. Any increase in the vocabulary size (x) results in an increase in
information, since a larger vocabulary allows for a finer resolution in
choosing verbs.
Therefore, the metric satisfies all conditions mentioned in Section 4.2.
Change in information with hierarchy
Secondary (Figure 4) and tertiary (Figure 5) functional models are
obtained by replacing the primary level verbs of Figure 3 with verbs from
Empirical Examination of the Functional Basis and Design Repository 275
respective levels. In order to ensure that each higher level verb is
represented in the lower levels, secondary verbs that have not been
categorized in the tertiary level (Table 1) are propagated, as is, to the
tertiary level. For example, in Table 1, the secondary verbs ‘distribute’,
‘import’ and ‘export’ are all propagated in the empty cells in column 3.
Fig. 4. Hair Dryer Functional Model (Secondary Verbs only)
Fig. 5. Hair Dryer Functional Model (Tertiary Verbs only)
Information based on available vocabulary
As the functional model is traversed from the primary to the secondary and
tertiary levels, the size of the available vocabulary (x) increases from 8 to
21 and 35, respectively. Information based on these sizes is computed in
Table 10 and Table 11 for the secondary and tertiary level, respectively.
4.6.2. Information based on hierarchically reduced vocabulary
Once a higher level verb is used in the functional model, the choice in the
lower levels is limited to those verbs that are children of the higher level
verb. This is defined in this paper as the hierarchically reduced
vocabulary. In the case of the hair dryer, four of eight primary verbs are
used in the primary functional model (Figure 3). These four used verbs
expand into only eleven verbs in the secondary level (Table 1). Of these
eleven secondary verbs, eight are used in the functional model (Figure 4).
These eight used verbs expand into twelve verbs in the tertiary level.
276
Table 10 Calculation of information in
hair dryer functional model (secondary
level)
Secondary information
Size of vocabulary
21
Probability
1/21
Unit information
4.392317
Instances
18
Total information
79.062
B.W. Caldwell et al.
Table 11 Calculation of information in
hair dryer functional model (tertiary
level)
Tertiary information
Size of vocabulary
35
Probability
1/35
Unit information
5.129283
Instances
18
Total information
92.327
The information content for the hierarchically reduced vocabulary at
secondary and tertiary levels is recomputed in Table 12 and Table 13,
respectively.
Table 12 Calculation of information in
hair dryer functional model (secondary
level, hierarchically reduced vocabulary)
Secondary information
Size of vocabulary
11
Probability
1/11
Unit information
3.46
Instances
18
Total information
62.27
Table 13 Calculation of information in
hair dryer functional model (tertiary
level, hierarchically reduced vocabulary)
Tertiary information
Size of vocabulary
12
Probability
1/12
Unit information
3.58
Instances
18
Total information
64.53
Results
Figure 6 and Figure 7 show the trend in information based on the size of
available and hierarchically reduced vocabularies, respectively.
Observations
The following observations are made on the results:
• The amount of information increases from primary to secondary to
tertiary functional models. This trend is consistent between both
approaches of measurement, available and hierarchically reduced, as
seen in Figure 6 and Figure 7.
• The increase in information from primary to secondary level ( ∆I1,2 ) is
significantly higher than the increase from secondary to tertiary level
( ∆I 2,3 ). Table 14 summarizes this observation.
Empirical Examination of the Functional Basis and Design Repository 277
Fig. 6. Change in information with
available vocabulary
Fig. 7. Change in information with
hierarchically reduced vocabulary
Table 14 Increase in information across levels of hierarchy
Increase in
information
Available vocabulary
Hierarchically reduced
vocabulary
∆I1,2
79.062 – 54 = 25.062
62.270 – 54 = 8.270
∆I 2,3
92.327 – 79.062 = 13.265
64.529 – 62.270 = 2.259
• The information produced per unit size of vocabulary in any level of
the functional basis reduces with increasing hierarchy levels. For a
given level, this quantity is given by
log 2 ( x )
1
= y⋅
x
x
This quantity reduces with increasing size of the vocabulary (x). The
values for this quantity for the primary, secondary and tertiary level are
6.75 bits/verb, 3.76 bits/verb and 2.64 bits/verb, respectively.
• Information in a functional model can be increased in two ways:
a. Increase the number of verbs (y)
b. Increase the size of the vocabulary (x)
The increase in information is faster in the first case (linear with y) than
the second case (logarithmically with x).
Experiment 2 Analysis
The information metric is an indicator of the degree of specificity captured
in a functional model. The higher size of vocabulary at higher levels
278
B.W. Caldwell et al.
results from increasing specificity in describing the functional terms (see
Table 1 and Table 2), allowing the functional model to capture more
details about the product than the lower levels. In case of information, this
trend is manifested as a higher value of ‘x’, resulting into a higher value of
information, ‘I’.
The additional gain in information by introducing a new level in the
functional basis, in general, gradually diminishes. Since information is a
logarithmic function of the size of the vocabulary (x), it grows slower than
the size (x) itself. Therefore, a larger increment in information can be
obtained by adding a new level only when the ratio of sizes between the
new level to the currently highest level is higher the ratio of sizes between
the currently highest level and its immediate lower level.
The trend of increase of information with levels of the hierarchy
(observation 1) counters the decreasing trend of information per unit size
of the vocabulary (observation 3). If information per unit size of the
vocabulary is accepted as the incentive of using a level, observation 3
suggests that the incentive of adding a higher level to the functional basis
(or using an existing high level) is always lower than the incentive of using
a lower level, even though the magnitude of information monotonically
increases with levels. This analysis predicts the presence of an optimum
level in the hierarchy, which produces a reasonable balance between
information and incentive.
Any operation on the functional model that results into more blocks and
arrows is a better means of increasing information than adding words or
levels to the vocabulary. Additional words or levels not only cause an
increasingly slower growth of information, they reduce the incentive of the
level too. Functional decomposition, for example, is a means of breaking
down a function into multiple, interacting sub-functions, the net effect of
which is same as the original function. Thus, functional decomposition is
identified as promising means of increasing information in a functional
model.
Conclusions
The benefit of using the primary and tertiary levels in the hierarchy is
minimal. This claim has been demonstrated in Experiment 1 by showing
that approximately 92% of the terms used in the functional models are
from the secondary level. This is further analytically supported in
Experiment 2, where the presence of an optimum level in the hierarchy has
been predicted. The secondary level constitutes an optimum level in the
Empirical Examination of the Functional Basis and Design Repository 279
hierarchy, because it offers significantly higher information than the
primary level, yet, provides significantly higher incentive than the tertiary
level. Therefore, a flat vocabulary, which combines all secondary terms
with select primary and tertiary terms, is suggested.
Limitations and Future Work
The conclusions in this paper are applicable only to consumer-based
electromechanical products. To address this limitation, the experiments
will be conducted on a broader range of products (e.g., heavy machinery,
aerospace, automobile) to determine if the conclusions hold generally for
mechanical design.
Secondly, all models stored in the design repository are prepared
through reverse engineering of existing products, making functional
decomposition unnecessary for their creation. Functional decomposition is
expected to produce more information about the product than committing
to a particular level of the functional basis hierarchy.
Finally, the information content, presented in Experiment 2, is limited to
function verbs. However, the information captured within a functional
model is dependent on the verbs, nouns, and connectedness of the model.
Current research is focused on addressing the last two information sources.
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