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Cite this article: Tensa, M., Edmonds, K., Ferrero, V., Mikes, A., Soria Zurita, N., Stone, R., DuPont, B. (2019)
‘Toward Automated Functional Modeling: An Association Rules Approach for Mining the Relationship between Product
Components and Function’, in Proceedings of the 22nd International Conference on Engineering Design (ICED19),
Delft, The Netherlands, 5-8 August 2019. DOI:10.1017/dsi.2019.177
ICED19
INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN, ICED19
5-8 AUGUST 2019, DELFT, THE NETHERLANDS
ICED19 1
TOWARD AUTOMATED FUNCTIONAL MODELING: AN
ASSOCIATION RULES APPROACH FOR MINING THE
RELATIONSHIP BETWEEN PRODUCT COMPONENTS AND
FUNCTION
Tensa, Melissa; Edmonds, Katherine; Ferrero, Vincenzo; Mikes, Alex; Soria Zurita, Nicolas;
Stone, Rob; DuPont, Bryony
Oregon State University
ABSTRACT
The objective of this research is to support DfX considerations in the early phases of design. In order to
do conduct DfX, designers need access to pertinent downstream knowledge that is keyed to early stage
design activities and problem knowledge. Product functionality is one such “key” connection between
early understanding of the design problem and component choices which dictate product performance
and impact, and repositories of design knowledge are one way to archive such design knowledge.
However, curation of design knowledge is often a time-consuming activity requiring expertise in product
modeling. In this paper, we explore a method to automate the populating of design repositories to support
the overall goal of having up-to-date repositories of product design knowledge. To do this, we mine
information from an existing repository to better understand the relationships between the components,
functions, and flows of products. The resulting knowledge can be applied to automate functional
decompositions once a product’s components have been entered and thus reliably provide that “key”
between early design activities and the later, component dependent characteristics.
Keywords: Functional modelling, Design for X (DfX), Machine learning, Data-driven design, Product
respository
Contact:
DuPont, Bryony
Oregon State University
United States of America
bryony.dupont@oregonstate.edu
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1 INTRODUCTION
The modern global theater of product design has become increasingly complex. Designers today
are challenged with creating and innovating designs that require more than simple problem-solving.
Traditionally, product design has focused on the principle of designing for an intentional end-function
of a product. Today’s designers have additional design considerations to contend with such as how
to improve the sustainability of a product, increase efficiency in use, reduce assembly time, reduce
the product physical profile. These areas of design are termed Dfx or Design for X. The increasing
complexity of product design promotes a need to improve the availability of design knowledge beyond
the cognitive memory of modern designers. Design repositories are one way to increase auxiliary design
knowledge, this data can be used to reduce design time, increase design uniformity, and increase design
fidelity. A foundation of engineering design is functional decomposition. We hypothesize that providing
designers with a better understanding of product functionality from data mining existing products, we
can enable designers to identify component and function correlations early in the design process. We
suggest that by bridging the gap between component and function in an automated way, we can offer
external knowledge to aid in design creativity, fidelity, and increase solution breadth. This relationship
between component and function will provide the foundation that will help designers identify how
component and function influence all Dfx objectives.
Figure 1. Black box electric tea kettle
Understanding functionality is an essential part of engineering design. Most product design efforts
start with the need to design for an overarching function that solves a design problem. This overarching
function can be considered a Black Box that is responsible for converting input flows into output flows,
as shown for a traditional electric tea kettle in Figure 1. The designer is often able to intuit the input and
output flows of a given black box design. For example, if tasked with creating an electric tea kettle, a
designer will know the black box function is to heat water, and that this design will require inputs of
electrical energy and water and outputs of thermal energy and heated water. The next step in the design
process is to identify discrete system functions that help execute the main function. Identification of
system functions is often completed through a functional decomposition. This decomposition is used to
visualize the flows, internal secondary functions, and the intra-relations of discrete product functions,
an example of which is shown in Figure 2. In addition to product functionality, the designer needs to
determine the components and sub-assemblies that will make up the product. Component solutions are
closely interrelated to function.
In this work, we explore how to increase and refine design knowledge by using association data
mining on an existing Design Repository. This approach will enable us to understand the associations
between product components and the probable functions they perform. To do this, we apply association
rule learning using an Apriori algorithm to search repository data and determine the probabilities of
relationships between components, functions, and flows within the Design Repository1, focusing on a
subset of 12 Black and Decker products. These associations, once identified, can be used to inspire
design of new products by allowing designer to access design a multitude of design solutions both
typical and novel. This research also provides the basis for expanding design knowledge, as tied to our
hypothesis, by first limiting the time required to add new products to the Design Repository. Second
by enhancing auxiliary design knowledge, and providing component-based design suggestions based
1The Design Repository is a database of design information compiled from over 130 products and
housed at Oregon State University. A rudimentary web interface to access the information is available at:
ftest.mime.oregonstate.edu/repo/browse
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on function. Furthermore, this work creates a prototype model for future work in automating functional
modeling of user added products to the Design Repository.
Figure 2. Simple functional model for an electric tea kettle
2 LITERATURE REVIEW
Design creativity is measured by the novelty, fluency and the number of ideas in a given
design (Torrance 1962). Commonly used metrics to measure group idea generation are the quantity
of non-redundant ideas and a quality rating (Mullen, Johnson, and Salas 1991). Shah et al. devel-
oped a set of metrics specifically for the evaluation of engineering idea generation techniques
including quantity, quality, novelty, and variety of ideas (Shah, Smith, and Vargas-Hernandez 2003;
Shah, Kulkarni, and Vargas-Hernandez 2000). They noted the fact that engineering design must meet
a particular need and function thus requiring an expanded set of measures. Systematic design method-
ologies (Kevin N. Otto and Wood 2006;Pahl 2007;Ullman 2018;Cagan, Nussbaum, and Vogel 2002)
provide a path for linking early design choices or decisions (e.g., customer needs gathering and interpre-
tation, setting of engineering specifications) to an appropriately abstracted product function. The same
methodologies and other stand-alone techniques offer a path to generate a set of component solutions
from the derived functionality to define potential product variants.
2.1 Internal & external design search methods
There are many different concept generation methods available to design engineers. These meth-
ods fall into two categories, internal and external searching methods. Well-known internal searching
methods include brainstorming techniques, such as the 6-3-5 method and the function based morpho-
logical matrix introduced by Zwicky (K. Otto 2003;Zwicky 1969). A shortcoming of these and many
other internal searching methods is the designer is limited to their own knowledge and there is a lack of
external stimulation which may prompt the development of new avenues for creative thought.
An external search approach, referred to as catalog design, is based on a catalog of physical ele-
ments (components, assemblies, etc.) that can be browsed for solutions that match required performance
specifications. A significant benefit of catalog design is the ability to utilize design knowledge that falls
outside human memory (Roth 2002;Ward and Seering 1993). Designers of source inspiration from pre-
vious solutions in the concept generation phase (Eckert and Stacey 2000). One such example of a design
catalog is a Design Repository.
Design repositories (S. Szykman et al., 2000) are currently still in their infancy, but their impact on
data-driven design processes can be substantial. Currently the Design Repository at OSU is the largest
known online Design Repository publicly available. Repositories store large datasets of products, pro-
viding a comprehensive collection of DfX information which provides the fodder for predictive models
to estimate downstream impacts earlier in the design process (Wisthoff et al. 2016). Previous researchers
have used emerging technologies to enable a transition to the use of repositories in the design of prod-
ucts, through means such as driving concept generation through morphology (Wisthoff et al. 2016;
Matt R. Bohm, Vucovich, and Robert B Stone 2008). Researchers have also explored the differences
between design databases and repositories finding that repositories contain more comprehensive infor-
mation and are dynamic. Repositories also provide the best means of gathering, recording, and storing
artifact data (Matt R. Bohm, Robert B. Stone, and Simon Szykman 2005;Simon Szykman et al. 1999).
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2.2 External design search methods: Oregon State University Design Repository
In this study, the Design Repository hosted by the Design Engineering Lab at Oregon State Univer-
sity will be used, which contains information about 130 consumer based electro-mechanical products at
various levels of abstraction (Matt R. Bohm, Haapala, et al. 2010; Matt R. Bohm, Vucovich, and Robert
B Stone 2008; Matt R Bohm et al. 2008;Matt R. Bohm, Robert B. Stone, and Simon Szykman 2005).
Product information is captured and separated into categories that allow innovative ways to approach
design, with the goal of enabling designers to use data-driven methods to develop insight during the
initial phase of concept generation. The underlying data schema captures a breadth of information, such
as the color, system, function flow, manufacturing process, parameters, and other valuable information.
The Design Repository stores product information as a PostgreSQL database.
Currently, the Design Repository contains over 5000 individual artifacts. Various iterations of the
repository have refined the classification of the components and their functions to follow a standardized
taxonomy (Matt R. Bohm, Haapala, et al. 2010;Matt R. Bohm, Vucovich, and Robert B Stone 2008;
Matt R. Bohm, Robert B. Stone, and Simon Szykman 2005). In this work, we used the functional
model with the functional basis and the component basis as the standardized design language
(Matt R. Bohm and Robert B. Stone 2009;Robert B. Stone and Wood 2000). Artifacts are given both
a common name and a specific component basis taxonomy name (Kurtoglu et al. 2009). For example,
a common name of a component might be pin 1 and pin 2, but the component basis taxonomy would
be pin for standardization of the dataset. The data from the repository can be extracted through queries
that are written to return the information of interest. We queried the repository for an initial set of data
which include the standardized component basis, the function it performs, and the flows in and out of
the component for a set of Black and Decker power tools. This work is a preliminary analysis towards
the verification of algorithms aimed at automatically generating a functional model based on an input
of product components.
2.3 Association rules & Apriori algorithm
Association rule mining is a form of data mining that searches information and determines
the probabilities of relationships between variables (Borgelt 2012). In our analysis, we will uti-
lize association rule mining, which finds finds non-obvious relationships between large datasets
(Matt R. Bohm and Robert B. Stone 2009;Agrawal et al. 1993); most significantly, how one choice
makes another choice more or less likely. In a supermarket example, an association rule may be that
a purchaser buying hamburger buns is likely to also buy sliced cheese (Brin et al. 1997).
In product development, association rule mining (ARM) has been used to determine user
preference of product functionality and product portfolio identification (Jiao and Zhang 2005;
Agard and Kusiak* 2004). Association rule mining, in tandem with decision trees, can be used when
developing convergent products that combine multiple user desired functions into a singular product
(Lee, Song, and Park 2012). In terms of useful direct designer knowledge, ARM has been com-
bined with user surveying and multi-objective genetic algorithms to generate soft rules for designs
in specific product families. Rules such as: expected product shape, user preferences button shape
(Fung et al. 2012). As shown, association rule mining can be used to identify important product func-
tions, determine customer needs, and develop general design rules for new product development. These
studies have provided a great understanding of the product design space, but are subject to drawbacks.
First, some of these studies are subject to imperfect or limited data, often relying on surveying tech-
niques and/or online text mining (such as customer reviews). Though this type of data can capture the
sentiment of the market, it may not prove useful information for developing generalized external design
knowledge. The work presented in this paper, moves away from user need identification and uses vetted
product system information from a Design Repository to disseminate useful external designer knowl-
edge. Second, the literature presented focuses mainly on market identification. There is research gap
between understanding the market, and the development of useful design knowledge. The aim of this
work is to use ARM to offer novice designers the tools and information needed to design successfully in
all product families. The case study presented in this paper is a stepping stone in providing: automated
product functional modeling, suggest functions based on designer-supplied components, and provide
suggestions on meeting designer-defined Dfx goals.
In our example, we will use association rule mining to identify the correlations between com-
ponents and particular product functions; it is anticipated that each function will be associated with
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multiple components. The goal is to discover patterns in previous product designs, then use the pat-
terns make rules determining the relationship between functions and flows for components. We use the
Apriori algorithm, a common ARM algorithm, to find correlations within the Design Repository and
establish statistically prevalent relationships for component-function-flow associations. Association rule
mining requires large datasets to ensure validity, and we anticipate a set of 250 elements and a func-
tion set of 100 functions, resulting in a suitably large transaction space. The result of this task is a set
of rules indicating the likelihood of components contributing to specific product functions. There are
three main measurements that determine the associations, and these are derived through a probabilistic
analysis of items and itemsets. In our research, an item is a component or a function, while an itemset
is the combination of the component-function-flow that it performs. The first measure of association is
called Support, which is the prevalence of an item within all of the itemsets. The second measure of
association is called Confidence, which is the probability of two items appearing in the same item set.
In the context of our work, Confidence is determined by the prevalence of combinations of components
with functions and flows. The third measure of association is called Lift, which accounts for the pop-
ularity of the function and flow in the Confidence measurement of the combination with components.
If a function and flow is popular, it could have a high Confidence, so Lift accounts for this so as not to
conflate associations by simple prevalence.
3 METHODOLOGY
The focus of this work is to mine the Design Repository to better inform component-function-flow
relationships, increase designer knowledge, and ultimately inform decisions made early in the design
process. This goal is achieved by implementing association rule learning to gain insight into the current
product data in the repository. Ultimately this work will allow for easier incorporation of new products
into the Design Repository through automation. The methodology used in this paper is presented in
Figure 3. We first query the repository database for the selected data; the components, functions, and
flows from constituent products (for this particular study, twelve Black and Decker consumer prod-
ucts are selected). Then we apply association rules using an Apriori algorithm to the extracted set of
data to determine the probabilities of associations between component basis terms (henceforth referred
to as components), functions, and flows. The outputs of the algorithm are a list of components, the
functions, and flows associated with that component, and the three measures of association: Support,
Confidence, and Lift (previously defined in the literature review section). Finally, we apply these results
to probabilistically determine the functions and flows for a given set of components for a new product.
Figure 3. Methodology
The Apriori algorithm was selected as it is an efficient way to find association rules and allows
verification of its effectiveness with a computationally simple example. The Apriori algorithm uses the
measures of association to limit the number of itemsets that are explored based on a minimum thresh-
old for Support and Confidence, thus decreasing computational complexity. The algorithm executes in
Python using the library called PyFIM (Borgelt 2012). The Python Apriori algorithm requires an input
of data as well as a declaration of minimum thresholds for the three measures of association: Support,
Confidence, and Lift. The three measures of association: Support,Confidence, and Lift. Initially, these
thresholds are set low in order to maximize the data returned to begin learning the relationships between
components and functions.
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3.1 Case study
We apply the above methodology to a set of Black and Decker products extracted from the
Design Repository. The Black and Decker subset of data was selected for analysis because the prod-
ucts have similar components and similar functionality, so we can find meaningful connections between
component-function-flow associations through repetition. Additionally, we expect potential problems, as
well as meaningful correlations, can be easily identified with a smaller set of data before applying these
techniques to larger and more complex data. The Black and Decker dataset has twelve different tool sys-
tems and 50 unique components with 824 total combinations of components, functions, and flows,152
of which are unique. The products include: sander attachment, can opener, circular saw attachment,
drill attachment, dustbuster, jigsaw attachment, mini router attachment, power pack, screwdriver, slice
right, palm sander, jigsaw, and rice cooker. We test automatic generation of component-function-flow
relationships by applying the results of the Apriori algorithm to a list of components from a product
in the Design Repository that was not analyzed with the Black and Decker data set. The component-
function-flow information already exists and can easily be verified against. The verification case study
product was a Delta jigsaw because it is in a similar design family as the Black and Decker products
from the data set. Each unique component from the Delta Jigsaw was compared to the components in
the results from the Black and Decker set to determine the most likely the function and flow. The result
are compared to the component, function, and flows of the Delta Jigsaw to determine the accuracy of
the algorithm. In summary, the steps to get these results are:
•Step One: Query the repository for component, function and flow combinations within a group of
products.
•Step Two: Find most likely functions and flows for a component from association rules between
the items (Apriori algorithm).
•Step Three: Compare results of the mined Black and Decker product component associations to
the known function-component relationships and flows from the verification case study (the Delta
jigsaw).
4 RESULTS AND DISCUSSION
4.1 Black and Decker dataset
The Black and Decker dataset was chosen for the similarity between product functions. There
were a total of 866 combinations of components, functions, and flows with 283 unique combinations.
The product similarity is indicated by the large number of total combinations, but smaller number of
unique combinations. The small dataset provides more consistency in the results for this initial explo-
ration in better understanding the steps toward automation. The Apriori algorithm found 67 associations
between component, function, and flow to have 100% Confidence, 49 associations between the values
of 100% and 50%, and 450 associations below the value of 50%. The 11 associations with 100% Con-
fidence have only one function and flow linked to them. For example, it was found that the component,
bracket, will always be associated with the function secure and the flow solid. These high Confidence
values create opportunities to easily develop rules that could be applied for automating functional mod-
eling. Low Confidence values are the result of either a low Support value for a particular combination
or the inclusion of many combinations in the dataset. Because most of the combinations are not unique,
the Confidence measurement is not always the most indicative of a positive correlation between com-
ponents, functions, and flows. When dividing each occurrence by the total number of combinations,
Confidence for a few occurrences will become very small with a large dataset. This confirms the impor-
tance of using the Lift measurement, which accounts for the frequency of functions and flows when
determining how meaningful a relationship is.
The two highest Lift values are 86,600, for the combinations of cushion - decrement rotational and
diode - guide electrical. These both have 100% Confidence and very low support values, which means
that these combinations only occurred once in our dataset. Extremely high values for Lift can also be
an indication of a small dataset. Table 1 shows an example of our results for the component reservoir,
all of the combinations have the same Confidence value, but the high Lift values are for function flow
combinations that are more likely to occur in the dataset, such as store liquid and import liquid. The
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Confidence value is low for reservoir because it appears in the dataset in multiple combinations but with
low prevalence for each.
Table 1. Reservoir values for lift and confidence
Component Function Confidence Lift
reservoir store liquid 0.1 82.4
reservoir export mixture 0.1 82.4
reservoir import liquid 0.1 82.4
reservoir mix liquid 0.1 82.4
reservoir mix solid 0.1 82.4
reservoir store solid 0.1 41.2
reservoir export thermal 0.1 20.6
reservoir import solid 0.1 3.582
4.2 Verification case study: Delta Jigsaw
For this analysis, we featured five unique components that existed in the training dataset (Black
and Decker) and in the verification case (the Delta jigsaw). Because the actual functions and flows of
the jigsaw are known, the function and flows that the apriori algorithm outputs can be verified by direct
comparison of results. Table 2 compares a sample of the results from the Black and Decker products
and the Delta Jigsaw. The goal is to see how accurate the data is when compared to a product that was
not included in the original learning dataset.
Table 2. Sample of verification dataset
Head Body Confidence (%) Support (%) Lift (%) Jigsaw
Blade export mechanical 16.67 0.35 1443.33
separate solid 16.67 0.35 1202.78 ×
export solid 16.67 0.35 759.65 ×
import solid 16.67 0.35 601.39 ×
transfer mechanical 16.67 0.35 313.77 ×
guide mechanical 5.56 0.12 481.11
secure solid 5.56 0.12 54.67
guide solid 5.56 0.12 45.82
Cam change mechanical 66.67 0.23 2510.14 ×
transfer mechanical 33.33 0.12 627.54 ×
Electric Wire transfer electrical 66.13 4.73 938.82 ×
secure solid 17.74 1.27 174.60
position solid 12.90 0.92 87.99
guide solid 3.23 0.23 26.61
Screw couple solid 93.85 7.04 913.15 ×
position solid 6.15 0.46 41.96
Spring position solid 47.37 1.04 323.00
guide solid 21.05 0.46 173.63 ×
store mechanical 10.53 0.23 4557.89
supply mechanical 10.53 0.23 4557.89
secure solid 5.26 0.12 51.79
couple solid 5.26 0.12 51.21
On the left side of Table 2, the Black and Decker dataset is shown with the Apriori results. The
Confidence value for all combinations of components, functions, and flows for the learning dataset are
scaled relative to each other by the same factor as Support. However, the Lift values scale the Confidence
by the frequency of a function and flow combination. The right side of Table 2, indicated whether or not
the jigsaw has the function and flow combination.
The table is organized by component from high to low Confidence. When there is an identical
Confidence, it is then organized from high to low Lift. This reflects the importance we put on the different
parameters. As the jigsaw results indicate, the associations with higher confidence or lift are indicating
the more likely associations. We chose a sample of components from the larger jigsaw dataset. The
results show the diversity of combinations between the learning set and the verification. For example,
cam, as well as cap and washer, which was not shown, all had identical function and flows in the
learning and verification set. The components blade and screw returned approximately twice as many
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associations for the Black and Decker set than then verification set. The component screw has a very
high Confidence for couple solid (93.85) meaning it is significantly more likely to happen then position
solid due to the higher Lifts and Supports. Lastly, the component spring only matched one function flow
from the jigsaw, guide solid. The electric wire was the only component to have a function and flow
that was not in the learning set. It has couple solid instead of secure, position, or guide solid. These
are all very similar terminology, so there could be issues with standardized language. While there has
been significant previous work to attempt to standardize language for functional basis and component
basis taxonomy, no functional model is going to be created or replicated with 100% accuracy, so we see
differences in the functional models. The Design Repository is built by different researchers, and their
errors and subjectivity contribute to the variation in data in the repository, which is extracted into the
results of automated analysis. The learning set contains products with different functionality than the
jigsaw, this difference in functionality accounts for excess function and flow results in the learning set.
However, these distinct results are important because they can provide the designer with novel solutions
that they may not otherwise consider. The verification process using the Delta jigsaw has provided the
first step towards automating functional modelling. However more data and analysis will be needed to
be able to identify how many and which function and flows to choose from each component.
5 CONCLUSIONS AND FUTURE WORK
In this research, we used the Black and Decker dataset from the Design Repository to determine
associations between component, function, and flow. The Apriori algorithm was implemented in Python
to learn the association rules, which were compared to the Delta jigsaw, another product from the Design
Repository. We found some similarities between the associations between the results from the Apriori
algorithm and the Delta jigsaw, but our results were restricted by the limited size and scope of the data
set used for this initial exploration. In future research, we will continue to query the Design Repository
to expand our analysis to the entirety of the product information available, which will increase the accu-
racy of the association rule learning and assist in the eventual implementation of automatic functional
modeling. As seen in our data, the higher the Confidence and lift values the more likely the combinations
will appear in our verification data. Increasing our learning data set will help improve the validity of the
association rules values, which will improve our confidence to assign function and flows to a particular
component. This methodology could benefit other designers as it begins to bring to light to different
potential concepts.
With improved component-function-flow association values, we can begin to develop grammar
rules for applying the results of the associations that appear in our data. For example, in our Black and
Decker dataset we found the component solder was always associated with the function-flow couple
solid. These singular component, function and flow associations can be turned into grammar rules and
coupled with additional data mining techniques to determine the order of function and flow in a func-
tional model. A fairly straight forward example would be that a flow must always be imported before it
is exported. While we applied this methodology specifically to a Design Repository, this methodology
could be used on any product dataset to determine associations. Confidence and Lift values could be
used to determine the most likely component and function and flow associations in the dataset used.
We plan to apply this methodology to the the entirety of the repository data in order to develop
grammar rules and apply data mining to create a process of automated functional modeling. The results
of association rule learning are the functional representation of a component. Given a component, we
found the most likely functions and flows. Once these functional representations of each component are
determined, they can be combined using grammar rules into the functional model. Automated functional
modeling will allow for increased standardization in functional modeling and improved consistency
between user created functional models. Furthermore, automation will increase the accuracy of gener-
ated function models, increase data fidelity, decrease decomposition time, and increase function data
used to precipitate future research. This will also allow the repository adapt as trends in components
change over time. One such avenue would be to create an algorithm that can automatically capture
the functional model of an existing (or proposed) product then directly provide component and sub-
assembly suggestions that relate to downstream impact of DfX performance measures. This proposed
research will increase the available auxiliary design knowledge which will improve innovation and cre-
ativity of designers, students, and other users of the repository. In this case, the algorithm will expand
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functional design solutions beyond the cognitive capabilities of the designer. Within the engineering
education community, this approach could help with teaching the abstraction process that is critical
to concept generation in systematic design approaches. Furthermore, we intend to verify automated
supplied design knowledge through user feedback.
In the current state of the Design Repository, no information regarding the final user, the user
interactions with the product, failure modes caused by the user, and humans error exists. Future work
will include the development of new categories and relationships that include the user, user interactions,
human error, and their relationships with the systems in the database expanding the work presented in
the Function-Human Error Design Method (FHEDM) (Zurita et al. 2018). Such approach would enable
safety, comfort, and performance objectives of the human-system interactions to become part of the
up-front selection of design decision based on the human factors engineering assessment of user-system
interactions.
ACKNOWLEDGMENTS
This material is based upon work supported by the National Science Foundation under Grant
No. CMMI-1826469. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of the National Science
Foundation.
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