A formal process of hierarchical functional requirements development for Set-Based Design

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A formal process of hierarchical functional

requirements development for Set-Based

Design

Minghui Sun

Coordinated Systems Lab

Department of Mechanical Engineering

Iowa State University

Ames, IA 50011

Email: minghuis@iastate.edu

Zhaoyang Chen

Coordinated Systems Lab

Department of Mechanical Engineering

Iowa State University

Ames, IA 50011

Email: zchen1@iastate.edu

Georgios Bakirtzis

Autonomous Systems Group

Oden Institute for Computational Engineering & Sciences

The University of Texas at Austin

Austin, TX 78712

Email: bakirtzis@utexas.edu

Hassan Jafarzadeh

Advanced Control Methods

Research and Advanced Engineering

Ford Motor Company

Dearborn, MI 48126

Email: hjafarza@ford.com

Cody Fleming

Coordinated Systems Lab

Department of Mechanical Engineering

Iowa State University

Ames, IA 50011

Email: ﬂemingc@iastate.edu

The design of complex systems is typically uncertain and

ambiguous at early stages. Set-Based Design is a promis-

ing approach to complex systems design as it supports

alternative exploration and gradual uncertainty reduc-

tion. When designing a complex system, functional re-

quirements decomposition is a common and effective ap-

proach to progress the design incrementally. However,

the current literature on Set-Based Design lacks formal

guidance in functional requirements decomposition. To

bridge the gap, we propose a formal process to hierar-

chically decompose the functional requirements for Set-

Based Design. A four-step formal process is proposed to

systematically deﬁne, reason, and narrow the sets, and

eventually decompose the functional requirement into the

sub-requirements. Such a process can be used by the indi-

vidual suppliers working in parallel at multiple levels of

abstraction and guarantee that the resulting system will

eventually satisfy the top-level functional requirements.

An example of designing a cruise control system is ap-

plied to demonstrate the feasibility of the proposed pro-

cess.

1 INTRODUCTION

Modern complex systems are usually developed in

the OEM-supplier mode. The OEM (or a higher-level

supplier) decomposes the top-level functional require-

ments into lower-level ones and assigns them to the lower-

level suppliers that work in parallel. Getting the func-

tional requirements decomposition correct is crucial for

the success of product development.

One of the challenges of functional requirements de-

composition is that the design of complex systems is typ-

arXiv:2210.14434v1 [eess.SY] 26 Oct 2022

ically uncertain and ambiguous at early stages [1]. Set-

Based Design (SBD) [2] is particularly suitable to tackle

this challenge as it provides great support in robust de-

sign alternative development, uncertainty reduction and

resolution [3], and supplier/subsystem autonomy and op-

timality [4]. In SBD, the overall system design problem

is typically decomposed into multiple distinct disciplines,

each utilizing their own sets of possibilities [5]. Teams of

engineers develop a set of design alternatives in parallel at

different levels of abstraction and narrow the prospective

set of alternatives based on additional information until a

ﬁnal solution is converged [6].

Our ultimate goal is to automate functional require-

ments decomposition for SBD because manually decom-

posing functional requirements is time-consuming and

error-prone. This paper, as our ﬁrst step towards this

goal, aims to formalize the functional requirement de-

composition process in the context of SBD. Although

function decomposition may not be nominally related to

SBD, it clearly features set-based reasoning [4].

This paper addresses the following objectives.

Objective 1: Creating methods, leveraging SBD, to

formally decompose the functional requirements.

Objective 2: Provide formal guarantees that the re-

sulting component designs are composable, and the

system as a whole will satisfy the top-level functional

requirements.

Objective 2 is a constraint on Objective 1, and therefore

is addressed ﬁrst in Section 4. We then ensure that this

constraint from Objective 2 is considered when reasoning

about Objective 1 in Section 5. Achieving these objec-

tives would allow design teams to work at different levels

of abstraction in parallel and asynchronously, providing a

path towards more seamless integration for OEMs.

For Objective 1: There is little attention in the SBD

literature to a general formal process of functional re-

quirements decomposition for complex systems. Shall-

cross et al. pointed out that “there is limited SBD research

contributing to requirements development” [7] and SBD

methodologies applied to complex systems are mostly

qualitative [3]. Ghosh and Seering [4] posited that SBD

had not been formally deﬁned, despite many authors hav-

ing studied its process inspired by the example of Toy-

ota. Therefore, in response to Objective 1, we propose a

four-step formal process by applying set-based reasoning

to systematically deﬁne, reason, and narrow the sets, and

eventually decompose the functional requirement into the

sub-requirements. Dullen et al. [8] and Specking et al.

[5] observed that “there has been limited (formal) guid-

ance on how to deﬁne, reason, and narrow sets while im-

proving the level of abstraction of the design”, which is

precisely the proposed process aims to improve.

Furthermore, the proposed process makes two ad-

ditional contributions to the SBD literature. First, most

SBD approaches formulate the functional requirements

as the ranges of the elements in the performance spaces.

Such a formulation applies to many mechanical compo-

nents at lower levels. However, for systems at higher lev-

els, the function is usually deﬁned as a transformation be-

tween the inputs and outputs. Accordingly, the functional

requirements have to be deﬁned as a mapping between

the ranges of the inputs and the outputs. Our process

focuses on the latter formulation as we focus on the re-

quirements development of complex systems. Therefore,

the proposed process solves a different problem than most

current SBD approaches and is a complement to the cur-

rent SBD literature.

Second, according to Eckert et al. [9], there are two

ways to address uncertainties: “buffer” as “the portion of

parameter values that compensates for uncertainties”, and

“excess” as “the value over, and above, any allowances for

uncertainties.” The current SBD literature does not make

explicit distinctions between buffer and excess when ad-

dressing the uncertainties and hence lacks speciﬁcity in

their robustness claims. In our process, buffer and excess

are addressed with a clear distinction in their respective

steps, where buffer is practiced to reduce the controllable

uncertainties, and excess is practiced to accommodate the

possibility of an under-estimated initial characterization

of the uncertainty.

For Objective 2: SBD claims autonomy of teams of

designers is an advantage of SBD [10], but very little SBD

work provides a priori proof of this property. In fact, one

cannot rigorously justify such a claim without an explicit

underlying formalism. Based on the formalism deﬁned in

this paper, we can prove that when the functional require-

ments are decomposed in a speciﬁc way (i.e., composable

and reﬁnement), individual design teams can work inde-

pendently, and the resulting system as a whole will satisfy

the top-level functional requirements.

In summary, we propose in this paper a formal pro-

cess to hierarchically decompose the functional require-

ments in the context of SBD. Individual design teams can

use the proposed process to decompose the functional re-

quirements independently and eventually achieve a sys-

tem that satisﬁes the top-level functional requirements.

We demonstrate the feasibility of the proposed process

with an example of designing a cruise control system

based on existing computational tools.

2 BACKGROUND

2.1 Qualitative SBD approaches

There are qualitative procedural models concerning

requirements development in the literature. Enhanced

function-means modelling (EF-M), a method for function

modeling [11] was used in combined with SBD to manage

platform-based product family design [12]. Functional

decomposition and solutions were generated according to

EF-M. At each level of abstraction, there is a mapping

from FR to DS, and a mapping from DS to the FR to be

assigned to the next level [13]. The two mappings are

conceptually aligned with the approach proposed in this

paper.

A “wayfaring” model was introduced for set-based

requirement generation as a map to discover critical func-

tionalities and create dynamic requirements in [14]. It

was found that prototyping critical functionalities could

guide the design process from the initial concept idea to

arrive at a ﬁnal product with low tooling and production

costs. A novel set-based approach (MBRMA) was devel-

oped to ﬁlter out weak or costly solutions over time and

assess system engineers when adopting trade-off analy-

sis [15]. Although a mathematical formalism contain-

ing the requirements, subsystems, activities, and compo-

nents was deﬁned in this paper, this article did not pro-

vide speciﬁc guidance on how to decompose the require-

ments. Another framework based on a set-based engi-

neering approach allows for building re-usable and adapt-

able engineering methods [16]. The proposed “virtual

methods” can be used to create and validate the early

phase design requirements, and make sure the introduc-

tion of novel technologies to increase the engine subsys-

tem performance can be realized without compromising

the requirements on risk and cost. More frameworks ex-

ist, such as CONGA [17], DMIV [18], MBSS [19], RR-

LeanPD model [20], and a combination of V-model and

SBCE [21].

One weakness of the qualitative approaches is the

ambiguity about the concrete activities needed to ac-

complish the requirements development process, which

makes it challenging for them to be repeated by the gen-

eral industry practitioners. Therefore, a formal process

is needed to provide precise instructions on decompos-

ing the functional requirements and assigning them to the

lower level of abstraction in a transparent and repeatable

way.

2.2 Quantitative SBD approaches

There are many quantitative techniques for design

space exploration in SBD. We focus on the quantitative

approaches that have a discernible feature of hierarchical

decomposition to make a meaningful comparison to the

approach of this paper.

SBD has been applied to the design of a downhole

module to demonstrate whether their method previously

developed in a laboratory setting had the same potential

when practiced in the actual industry setting [22]. The

downhole module was decomposed into a chassis subsys-

tem and a bumper subsystem. The system-level team as-

signed the design “target” to the subsystem based on a

downhole assembly impact model. Mathews et al. ap-

plied a set-based approach for a multilevel design prob-

lem of negative stiffness metamaterials based on Bayesian

Network Classiﬁer [23]. The design process progressed

in a top-down fashion from the macro-level to the meso-

level to eventually the micro-level, which was a typical hi-

erarchical design approach. Both [22,23] have distinctive

features of “hierarchical decomposition”, but they “walk

down” the hierarchy by treating the design space of the

higher-level abstraction as the requirements for the lower-

level design, which is different from our problem formu-

lation.

Jansson et al. [24] combined Set-Based Design with

axiomatic design to manage and evaluate the performance

of multiple design alternatives against the established

functional requirements. This work considered the map-

ping from the design space of the higher level of abstrac-

tion to the functional requirements for the lower level, but

did not show how speciﬁcally the functional requirements

for the lower level of abstraction are derived. A set-based

approach to collaborative design was proposed in [25].

The overall system was decomposed into distributed col-

laborative subsystems, where each design team built a

Bayesian network of his/her local design space and shared

their Bayesian network to identify compatibilities and

conﬂicts to improve the efﬁciency of local design space

search. However, the design problem was formulated to

optimize an objective function at the top level rather than

decomposing the given requirements into the lower level.

A Serious Game was proposed in [26] to illustrate

how the customer requirements of an airplane design

can be decomposed into the design parameters of the

body, tail, wing, and cockpit by applying the principle of

SBD. Although the game was only for educational pur-

poses, it showed a clear process of deriving the ranges

of the design parameters at the lower level from the

ranges of the higher-level design parameters. [27] pre-

sented an Interval-based Constraint Satisfaction Method

for decentralized, collaborative multifunctional design. A

set of interval-based design variables were identiﬁed and

then reduced systematically to satisfy the design require-

ments. SBD was also used by [28,29] to inform sys-

tem requirements and evaluate design options by iden-

tifying the number of potential feasible designs in the

tradespace for each requirement or combination of re-

quirements. The case study was conducted on the design

of a UAV to demonstrate how to assess whether the relax-

ation of the requirements will create better options. The

results demonstrated that SBD provides a comprehensive

tradespace exploration and valuable insights into require-

ment development.

In summary, the functional requirements in these ap-

proaches are ranges of the variables in the performance

space rather than a mapping between ranges of the de-

sign space and the performance space as deﬁned in our

process. Such a difference has a signiﬁcant implication in

the requirements decomposition process. The decomposi-

tion in other approaches is fulﬁlling a mapping from one

set of the ranges (of the performance space) to another

set of ranges (of the design space), while the decomposi-

tion in our process is accomplishing a mapping from one

mapping (between the ranges of the design space and the

performance space of the higher level function) to a set of

mappings (between the ranges of the design space and the

performance space of the sub-functions).

2.3 Design uncertainty

SBD is known for its robustness to design uncertainty

[30,31,32]. In an introduction about the application

of SBD by the U.S. Naval Sea Systems Command, [33]

pointed out that SBD was particularly ﬁt for design prob-

lems where there were many conﬂicting requirements and

a high level of uncertainty in requirements. This obser-

vation was corroborated by [34], “SBD allows design-

ers to develop a set of concepts, so changes in the de-

sign requirements are easier to adjust to.” In an approach

that incorporated Bayesian network classiﬁers for map-

ping design spaces at each level, “design ﬂexibility” was

deﬁned as the size of the subspace that produced satisfac-

tory designs, and “performance ﬂexibility” was deﬁned

as the size of the feasible performance space relative to

the size of the desired performance space [10]. In [35],

a Set-Based Design methodology was proposed to obtain

scalable optimal solutions that can satisfy changing re-

quirements through remanufacturing. The methodology

was demonstrated on a structural aeroengine component

remanufactured by direct energy deposition of a stiffener

to meet higher loading requirements.

There is another line of work that combines SBD

with platform-based design based on Function-Means

modeling techniques to preserve design bandwidth, a sys-

tem’s ﬂexibility that allows its use in different products

[36]. A dynamic platform modeling approach based on

SBD and a function modeling technique were presented

in [37] to represent product production variety streams

inherent in a production operation model. Following the

SBD processes, inferior alternatives were put aside un-

til new information became available and a new set of

alternatives could be reconﬁgured, which eventually re-

duced the risk of late and costly modiﬁcations that propa-

gated from design to production. Modeling platform con-

cepts in early phases and eliminating undesired regions

of the design space was described in [13]. Change was

considered in both the requirements space and the design

space. By applying set-based concurrent engineering, sets

of design solutions were created to cover the bandwidth

of each functional requirement. More work on this topic

can be found in [38,39].

Furthermore, [40] conducted a design experiment on

how delaying decisions using SBD could cause higher

adaptability to requirements changes later in the design

process. As a result, the variable and parameter ranges

were open enough to accommodate the changes in the re-

quirements. A method called Dynamically Constrained

Set-Based-Design efﬁciently provides a dynamic map

for the feasible design space under varying requirements

based on parametric constraint sensitivity analysis and

convex hull techniques [41]. The methodology ultimately

allowed for identifying a robust feasible design space and

a ﬂexible family of solutions. A hybrid agent approach

was applied for set-based conceptual ship design [42]. It

was found that the process was robust to intermediate de-

sign errors. After the errors were corrected, the sets were

still wide enough that the process could move forward and

reach a converged solution without major rework. More

work can be found in [43,44].

According to [45], “excess” is “the quantity of sur-

plus in a system once the necessities of the system are

met”. Later, [9] deﬁnes “excess” as “the value over, and

above, any allowances for uncertainties”, and “buffer” as

“the portion of parameter values that compensates for un-

certainties.” Robustness achieved by addressing these two

concepts has different meanings. Current SBD literature

lacks an explicit distinction between buffer and excess

when addressing the uncertainties, hence lacking speci-

ﬁcity in their robustness claims.

3 PRELIMINARIES

3.1 Notation

We introduce the notation that is used throughout this

paper. For a function fi, we make the following deﬁni-

tions. For function f, the same concepts can be applied

without “(i)”.

a(i)=(a1(i), a2(i), ..., aj(i), ..., an(i))>is a col-

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AformalprocessofhierarchicalfunctionalrequirementsdevelopmentforSet-BasedDesignMinghuiSunCoordinatedSystemsLabDepartmentofMechanicalEngineeringIowaStateUniversityAmes,IA50011Email:minghuis@iastate.eduZhaoyangChenCoordinatedSystemsLabDepartmentofMechanicalEngineeringIowaStateUniversityAmes,IA50011Ema...

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