AIAA_2010_Malak_Galvan Manuscript

8/7/2019 AIAA_2010_Malak_Galvan Manuscript

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American Institute of Aeronautics and Astronautics

1

Using Predictive Modeling Techniques

to Solve Multilevel Systems Design Problems

Richard J. Malak, Jr.1 and Edgar Galvan.2

Texas A&M University, College Station, TX, 77843

A challenging aspect of systems design is the search for a combination of concepts for

system components that together yield desirable system-level characteristics. These system-

level characteristics depend not only on the concepts designers choose, but also the details of

how they implement them. This yields a challenging search problem through a

heterogeneous and discontinuous space of system alternatives. Although designers can use

design optimization methods for this task, they can be slow because they entail solving a

different optimization problem for every valid combination of component-level concepts. In

this paper, we present a new approach to systems design based on the use of abstract

predictive models. Under this approach, designers abstract multiple physically

heterogeneous component-level concepts into a unified model that captures the salient

characteristics of the possible implementations of each concept. This enables them to search

the space of system alternatives quickly and effectively. We demonstrate the new approach

on a utility cart system design problem and compare it to optimization-based approaches

common in the literature. The new approach yields a system design as desirable as the one

we obtain from the best-performing optimization-based approach, but in less than one-tenth

the computational time.

1 Assistant Professor, Department of Mechanical Engineering, 3123 TAMU, College Station, TX 7743-3123.2 Graduate Research Assistant, Department of Mechanical Engineering, 3123 TAMU, College Station, TX 7743-3123.


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Nomenclature

b = damping coefficient

d = helical spring wire diameter

D = helical spring coil diameter

l = leaf spring length

a = leaf spring width

h = leaf spring thickness

Di = diameter of the gear

w = gear width

k = spring constant

Ng1 = first gear ratio

Ng2 = second gear ratio

Ng3 = third gear ratio

Rsusp = suspension reliability

Rtrans = transmission reliability

Csusp = suspension cost

Ctrans = transmission cost

dtow = distance towed

R = overall reliability

C = overall cost

tr = rise time

ts = settling time

ymax = maximum vehicle displacement

m = total vehicle mass

= max torque

= engine speed at max torque

= number of active helical spring coils

= number of leaves


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I. IntroductionURRENT trends are toward engineered systems with increased functionality, larger numbers of interacting

components, and elevated expectations about cost, reliability, sustainability, and performance. To achieve these

high expectations, system designers must explore a broad space of alternatives to locate the few exceptional

solutions despite strict limitations on the time and design resources available. System designers often use

optimization methods to assist in this task. These methods are particularly useful for searching a well-defined

parameter space for the best set of parameter values. However, the problem ofdesigning an engineered systemas

opposed to the problem ofrefining a systemis not so well defined. To be successful, designers require methods

that enable them to search the heterogeneous space of alternatives that occur in systems design problems.

The focus of this paper is on the point in a design process at which designers have determined the system

organizationi.e., how components interact to produce system-level functionality and behaviorbut have not yet

determined how to implement the components physically. For example, designers may have settled on an aircraft

with two wings and two wing-mounted engines, but have not yet determined the specific type of engine, how to

actuate control surfaces, etc. Although designers have determined a system decomposition and hierarchy at this

point in a design process, considerable design freedom remains and it is not straightforward for them to apply

optimization methods to this problem. The challenge lies in the heterogeneous nature of the search space. Each

arrangement of component-level concepts represents a distinct search space, with no regularity between each search

space. Moreover, because each concept can involve different technologies and operate on different principles, there

exists no unified parametric description of the design alternatives. Consequently, a straightforward application of

optimization methods entails an optimization run for each combination of component-level concepts. For a complex

system, the number of valid combinations of component concepts can be unmanageably large. Introducing

additional concepts results in a combinatorial explosion in the number of valid system configurations. Technological

incompatibilities may mean the number of valid combinations is lower than the worst-case number. However, even

a dozen valid combinations can be intractable if it takes a large amount of design resources to evaluate each one.

System designers would benefit from an approach that allows for extensive search of the space of system

alternatives despite the challenge presented by heterogeneous search space.

In this paper, we demonstrate the use of predictive modeling in combination with optimization methods to

support systems design and compare this approach to those based strictly on optimization methods. Predictive

C


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models capture associations between variables but do not have explanatory power or indicate causality 1. This makes

them promising as a mechanism for dealing with the mixed discrete-continuous optimization problem. As we

demonstrate, designers can abstract the key properties of many implementations of a component into a single

predictive model that they can use to perform system-level design exploration via optimization methods. The

system-level optimization problem yields the identity of the best concept and target specifications for designing it in

detail. Designers can follow this system-level problem by optimizing the components to these targets. Unlike classic

multilevel optimization procedures, the process does not require iteration between levels. This is because the

knowledge designers formalize into the predictive models ensures that the component target specifications are

technically feasible. Prior investigation into elements of this approach have proved promising 2-4, but the potential

benefits of abstracting physically heterogeneous concepts into a single model has not been studied. As we

demonstrate in this paper, this capability leads to gains in computational efficiency without sacrifices in solution

quality.

In the next section, we formulate the mixed discrete-continuous systems design problem that is the focus of the

paper. In section III, we describe how designers would solve this problem using two existing optimization-based

approaches: an all-at-once optimization formulation and a formulation based on analytical target cascading 5, 6.

Section IV is a description of the approach based on the use of predictive models to describe the capabilities of

several implementation concepts for a particular component. In section V we present a system design example on

which we compare the three approaches. Section VI contains an analysis and discussion of the comparison results.

II. Design Problem FormulationSystems design is the process of transforming a problem statement into a detailed description of a system

capable of solving the stated problem 7, 8. The full systems design process involves many steps, both qualitative

(planning, problem clarification, alternative generation, etc.) and quantitative (engineering analysis, optimization,

etc.). In this paper we focus on the problem of designing the components of a fixed system hierarchy. Although the

fixed hierarchy imposes constraints on each component, system designers retain considerable design freedom at this

point in a project.

Once designers have determined the system hierarchy, they must (1) determine the best concept for each

component in the hierarchy and (2) design each of the chosen component concepts in detail. These sub-problems are


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dependent because the best implementation designers can

achieve for a given concept will influence which

arrangement of concepts is best overall. Figure 1 is a

mathematical formulation of the overall problem. It

consists of models at three levels of abstraction: a utility

model, system-level analysis models, and component-

level analysis models.

Utility Model. To be compatible with optimization

algorithms, designers must formalize their decision-

making preferences in a computer-interpretable form. A

utility function, denoted , is a scalar function that

relates system-level attributes to a utility value that

designers seek to maximize. System attributes are

properties or measures of effectiveness that designers

consider when determining relative preference among

different system implementations. For an automobile,

system-level attributes might include fuel economy, top speed, and production cost. Designers define a utility

function such that attribute vectors that are more preferred lead to larger utilities. There are multiple approaches by

which designers can do this. One approach is to focus on profit maximization as an objective, in which case a

designers utility function computes profit and may include an adjustment to capture the designers risk attitude 5, 9.

In this case, designers may consider also to be a function of enterprise-level design variables, such as

production quantities, product sales price, etc. Under multi-attribute utility theory (MAUT), one elicits a utility

function by answering a series of questions involving hypothetical choices involving lotteries 10. We use MAUT in

the example of Section V.

System-Level Analysis Models. Designers compute system attributes as a function of component attributes

using system-level analysis models, denoted for where is the number of attributes designers

use to describe the system in question. Each system-level analysis model takes one or more component-level

attributes as inputs. Each model may involve different formalisms as is appropriate for computing the attribute of

Maximize: With respect to:

Component design concepts, for Component design variables, for

Subject to:

Design variable bounds,o for o for

Engineering constraints,o for o for

Where:

is the number of design variables for the concept of the component

is the number of concepts for the component is a vector of system attributes for is a system attribute,

computed from component attributes

is a system-level analysis model is a vector of component

attributes

for is a vector ofattributes for the component

is a component-level analysis model is a vector of design variables for the

concept of the component

Figure 1. System design formulation for a fixed

system hierarchy.


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A. All-at-Once OptimizationUnder an AAO optimization approach, designers

formulate the design problem as a single optimization

problem that parallels the formulation of Figure 1. We

depict this in Figure 2. The main practical consideration is

that designers must repeat the AAO method for every

combination of component concepts. Thus, this is a multi-

stage problem: (1) iterate through all combinations of

component concepts to find the best implementation of

each concept relative to the system-level utility function

and (2) select the combination with the highest utility overall. Thus, the first stage involves classical nonlinear

optimization methods using the AAO framework and the second stage is a simple selection from a list of

alternatives.

B. Multilevel Optimization using Analytical Target CascadingIn an effort to manage complexity in a

systems design process, designers

sometimes decompose an optimization

problem in a way that reflects the

organization of engineering expertise on a

design project. Designers can decompose a

problem among various collaborators

along disciplinary boundaries 11-13, or

according to system structure 6, 14. We

focus in this paper on an approach called

Analytical Target Cascading (ATC)

because it follows a hierarchical system

structure 6, 14.

Analytical Target Cascading Approach

Utility Maximization

Problem

x1,c1

ATC System-Level Problem

ztarget

System-Level

Analysis

ATC Component-

Level Problem

ATC Component-

Level Problem

ATC Component-

Level Problem

y1 x2,c2 y2 xm,cm ym

y1,target y1,actual ym,target ym,actualy2,target y2,actual

y

z

Component 1,

Concept c1Analysis

Component 2,

Concept c2Analysis

Component m,

Concept cmAnalysis

Figure 3. Schematic of Analytical Target Cascading (ATC)

based approach for a specified selection of component concepts.

All-At-Once Approach

System Analysis

z

Component 1,

Concept c1Analysis

x1,c1

y1

x2,c2

y2

xm,cm

ym

Component 2,

Concept c2Analysis

Component m,

Concept cmAnalysis


Problem

Figure 2. Schematic of All-at-Once (AAO)

approach for a specific selection of component

concepts.


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One challenge in applying ATC to the design problem defined in Figure 1 is that ATC is based on the notion of

target achievement rather than utility maximization. Researchers have demonstrated the use of ATC together with

utility functions in the context of enterprise-driven optimization using on customer demand models 5, 9. We are

unaware of prior application of ATC in concert with MAUT.

Like the AAO approach, ATC requires a two-phase strategy in which one first optimizes the system for every

combination of component concepts and then chooses the best overall combination. The distinction is that under

ATC, designers decompose the optimization of a particular combination of component concepts according to the

system hierarchy. Figure 3 is a schematic of the ATC-based approach for one selection of component concepts.

Designers must re-execute this for each combination of concepts. The utility maximization problem yields a target

system-level attribute vector, , that serves as an input to the ATC sub-problem. The final solution, the output

of the ATC procedure, is the one that minimizes deviation from this target. This procedure for coordinating the

utility-level and ATC problems is similar to one demonstrated previously in the literature5, 9, but is simplified due to

the fact that there are no enterprise-level variables in the current problem.

C. LimitationsThese optimization-based approaches share one significant limitation: they search the heterogeneous space of

systems by enumerating all combinations of component concepts. This limitation is not severe if there are few

combinations or if each combination is very fast to evaluate. However, complex systems can involve dozens or

hundreds of components. Even with just a couple concepts for each component, the number of combinations can be

unreasonable. Add to this the likelihood of complex engineering analyses, such as computational fluid dynamics,

finite element modeling, and system dynamics simulations, and one can conclude that for most systems the

computational burden per combination is heavy. Parallelization of the optimization runs can help reduce the time

impact of the problem (each combination is an independent optimization problem), but this comes at an expense in

terms of computing resources and still may be insufficient to make a full search practical.

It is straightforward to compute a worst-case bound on the number of combinations designers need to evaluate. If

there are components in a system, denoted , and each component has associated with it physically

heterogeneous implementation concepts, then the upper bound on the number of optimization runs is


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. This upper bound can grow fairly quickly as the numbers of components and component concepts grows.

Introducing just one new component concept can increase the computational burden by a factor of.

In practice, this upper bound is likely to be fairly conservative given that technical incompatibilities may

preclude certain combinations of component concepts. For example, one cannot power a DC motor directly with

hydraulic fluid and therefore designers would not try to optimize a system involving a DC motor concept for an

actuator component and a fluid-power concept for power transmission. Nonetheless, technical incompatibilities

alone may not eliminate so many combinations that computation becomes trivial. System designers would benefit

from a better approach to this problem.

IV. Combining Optimization with Abstract Predictive TechniquesPredictive modeling techniques can be useful for improving upon existing optimization methods in systems

design. Predictive models allow a user to predict unknown values of one variable as a function of the others. They

imply nothing about causality and have no explanatory power 1, 15. Construed broadly, predictive modeling includes

response surface models based on computer experiments, a practice sometimes called meta-modeling. Several

authors report computational gains when applying this predictive approach to optimization problems 16-19. However,

although this approach can reduce the computational burden of a single optimization run, it does not solve the more

fundamental problem of potentially having an excessive number of combinations to optimize.

We consider a different approach for utilizing predictive modeling in which system designers abstract many

component concepts into a single predictive model. This enables designers to recast the heterogeneous systems

design problem into a homogeneous search space. This type of abstraction is not possible in every case (see the

discussion in Section VI), but it can yield considerable reductions in the numbers of independent optimization runs

that designers require. If designers can abstract as few as two concepts into a single model they can reduce the

required number of optimization runs for a system with components by a factor of.

The Abstract Predictive (AP) approach involves a three-step process of (1) abstraction and predictive modeling,

(2) predictive system-level optimization, and (3) component optimization. Figure is a schematic of this approach.

Unlike the illustrations of the AAO and ATC approaches (Figures 2 and 3, respectively), the procedure ofFigure is

sufficient to explore all combinations of concepts without iteration.


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A. Abstraction and Predictive Model GenerationThe modeling approach is based on prior work involving predictive modeling in conceptual design 2, 3. The prior

work involved only one concept per model and relied on data about prior implementations of a particular concept

(e.g., catalog and spec sheet data) rather than drawing samples from an engineering model. Here, we extend the

approach to the current problem.

Figure 4 is a summary of the abstraction and

predictive modeling procedure. In Step 1,

designers identify the principal abstraction of the

component by identifying attributes of the

componenti.e., measurable properties of the

component that are inputs to higher-level models

or for which decision makers have preferences

directly. In addition, designers must classify each

attribute as a dominator or a parameter attribute. This classification relates to how changes in the attribute value

affect system-level preferences. An attribute is a dominator if it is something that designers generally prefer to

1. Identify component attributes and each as a dominator or aparameter

2. For each concepti. Generate samples of design variables

ii. Run component-level analysis model on samples togenerate attribute data

iii. Filter attribute data using parameterized Pareto dominanceiv. Fit a model to the non-dominated attribute data

3. Abstraction procedurei.

Generate samples of each concept model at the samelocationsii. Perform parameterized Pareto dominance on the data

iii. Fit a model to the non-dominated dataFigure 4. Procedure for creating an abstract predictive

model for a particular component

Abstract Predictive Approach


Problem

x1,c1

System-Level

Analysis

Component-Level

Optimization

y1 x2,c2 y2 xn,cn yn

y1,target yn,targety2,target

y

zy1AP Component Model 1

AP Component Model 2

AP Component Model m

y2

ym

Component m,

Best Concept

Analysis

Component-Level

Optimization

Component-Level

Optimization

Component 2,

Best Concept

Analysis

Component 1,

Best Concept

Analysis

Concept

Analysis

Concept

AnalysisConceptAnalysis

Component 1

Concepts

Concept

AnalysisConcept

AnalysisConcept

Analysis

Component 2

Concepts

Concept

AnalysisConcept

AnalysisConcept

Analysis

Component m

Concepts

A&PM

Procedure

A&PM

Procedure

A&PM

Procedure

(1) Abstraction & Predictive Modeling Phase

(2) Predictive System-Level Optimization Phase

(3) Component Optimization Phase

Figure 5. Schematic of Abstract Predictive (AP) approach.


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maximize or minimize, assuming all other factors are equal.

Cost, reliability, lifetime, environmental impact, and fuel

efficiency are examples of dominator attributes. All other

attributes are classified as parameterattributes. These are

attributes with no problem-independent preference

association. For example, maximizing the bore of a

hydraulic cylinder is desirable in some applications (those for which power output is paramount), but the opposite is

preferred in other situations (when speed is paramount).

In Step 2, designers generate individual predictive models for each concept. The third sub-step data filtering

using parameterized Pareto dominanceis particularly important when designers use samples from analysis models

as their data source. Dominance-based filtering eliminates data corresponding to provably bad solutionsones that a

designer never would implement. The result is a more accurate predictive relationship among the component-level

attributes. For example, suppose designers seek to predict production cost as a function of efficiency and maximum

power output for an electric motor and they have the data points ($75, 0.9, 750 W) and ($300, 0.9, 750 W). To

include the second point during fitting would yield overly pessimistic predictions about what production costs are

achievable at particular levels of efficiency and power output.

Parameterized Pareto dominance is an extension of classical Pareto dominance to the situation in which some

attributes are not dominators2. This is necessary to support dominance analysis in a multilevel systems context.

Figure 6 contains the definition of parameterized Pareto dominance. It is mathematically sound in that it will not

eliminate any solution that could be part of the optimal system4.

Step 3 is the extension to support abstraction across component concepts. The procedure involves a second

application of parameterized Pareto dominance because different concepts can dominate in different regions of the

attribute space. This procedure yields the overall non-dominated set that can be the basis for an accurate predictive

model.

Definition (Parameterized Pareto Dominance): An

alternative having attributesis parametrically Paretodominated by one with attributes if , and , where isthe set of parameter attribute indexes and is the (non-empty) set of dominator attribute indexes.

Figure 6. Definition of parameterized Pareto

dominance.


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acceleration is found at each time step using a model of vehicle dynamics and the engine power curve given the

current wheel speed and gear ratio. When the engine reaches a predetermined speed the model shifts to a higher gear

if it is available. If no higher gear is available the utility cart stops accelerating. The simulation ends after sixty

seconds and outputs the final distance traveled.

Ride Quality. We evaluate ride quality using the time response of the UC system, modeled using a quarter-car

model (Figure 8), to a typical speed bump at 15 mph (approx. 24 kph). We consider ride quality to be an aggregate

of three system behavioral attributes: settling time, vertical displacement, and rise time.

Objectives are to minimize settling time and vertical displacement, and to maximize rise

time. Tradeoffs among these attributes are determined using a utility function (described

below). The quarter-car model enables us to relate suspension component attributes and

system-level attributes. The vehicle is modeled as spring mass damper system that

ignores the effects of the tires mass and springiness. The resulting transfer function is

where is the damping coefficient is the spring constant, and is the vehicle mass. The input is a half-

cycle sinusoid that represents the speed bump. The rise time, settling time and vertical displacement are determined

from the model output. The displacement information is also used to determine the spring reliability.

2. Utility ModelWe use techniques from the MAUT literature to determine a utility function for the UC system. The overall

utility function is:

where is the utility function for the settling time attribute at , is for the rise time

attribute, is for the vertical displacement attribute, is for thetow distance attribute, is for the reliability attribute, is for the cost attribute, and is the

system level attribute vector. The design problem is to find the component-level attribute values that maximize

system utility, .

b

u(t)

Figure 8. Quarter

Car Model.

x

k

m


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3. Component ConceptsIn the interest of scope, we limit the example to the design of the transmission and suspension components. We

consider three physically heterogeneous suspension concepts and two for the transmission (forward speeds only),

yielding six unique combinations of concepts.

Transmission. We consider two three-speed transmission concepts:

3-Speed 8-Gear (T3-8): Figure 9(a). Basic three speed transmissionsystem with 8 gears total. The engine shaft turns the four layshaft

gears. The three remaining gears on the layshaft turn the gears

corresponding to each speed. Design variables consist of the widths

and diameters of the eight gears.

3-Speed 6-Gear (T3-6): Figure 9(b). Simplified three speedtransmission system with six gears total. The engine shaft has three

gears which turn the gears corresponding to each speed. In this

concept there is no layshaft, reducing cost at the expense of reducing the compactness of the transmission.

Design variables consist of the widths and diameters of the six gears.

Both transmission concepts are subject to geometric constraints to ensure proper meshing of the gears. Given the

design variable values and operating conditions, we compute five component-level attributes:

Cost: The sum of material costs and parts. Classified as a dominator attribute (less is better). Reliability: The probability that the transmission will function without failure under anticipated operating

conditions. Classified as dominator attribute (more is better).

Three Gear Ratios: The three available ratios of transformation from transmission input shaft to output shaft.Classified as a parameter attribute.

Suspension. We consider three concepts for the suspension design. We assume identical suspension designs are

used on all four wheels. All suspension concepts have the same shock absorber configuration.

Figure 9. Layout of both

transmission concepts: (a) 3-

speed 8-gear transmission; (b)

3-speed 6-gear transmission

(a)

(b)

TO

DIFFERENTIALFROM

ENGINE

LAY SHAFT

TO

DIFFERENTIALFROM

ENGINE


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B. Design Approaches for Comparison1. All-at-Once Formulation

In the AAO optimization approach, an optimizer at the system level searches the component-level design

variable space directly as illustrated in Figure 2. Each combination of component-level concepts has a different set

of design variables and requires different component-level models. Consequently, the entire optimization process is

repeated for each of the six combinations of design concepts. The execution time for each optimization run is

recorded along with the optimization results.

2. Analytical Target Cascading FormulationThe ATC-based optimization approach entails multiple coordinated optimizers as illustrated in Figure 3. The

utility-level problem is formulated as

subject to:

where is the vector of system-level attributes, is the system-level utility function, and and are

system-level constraint functions. This problem is solved once for all concepts and yields a target for the ATC

procedure, .

The ATC problem is partitioned into system-level and component-level problems based on an object partitioning

of the UC system into a system level and two component-level problems (the transmission and the suspension). At

the ATC system level, an optimizer minimizes deviation between the target identified at the utility level, ,

and what is feasible, . This search is conducted over the space of component-level attributes. Component

attribute targets are cascaded down to the component optimizers, which search the space of component design

variables for a particular design concept. Since component-level variables and models are different across the

concepts, designers must repeat the entire process for each combination of components.

3. Abstract Predictive FormulationThe AP approach allows a designer to search the design space through high-level properties of the system

common to all component concepts. We formulate the AP approach according to the steps defined in Figure 4. We


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With the abstract predictive models in place, designers can use them in place of the component level analysis.

Following Figure 7, the design problem becomes

subject to:

where

Constraints and are imposed on the attributes by the physical constraints of the

components. For example, it is not possible to have a spring constant less than zero. They also serve to ensure the

validity of inputs to the predictive models (e.g., to ensure that inputs are not from parts of the attribute space for

which no data was collected and for which the model may be a poor fit). The output of this optimization

problem, , are the target attribute values for the components during detailed design optimization (c.f., process flow

in Figure ). However, before designing the components in detail, one must determine which concept is best for each

component.

Designers can identify the best concept through a straightforward procedure of comparing predictions to the

component-specific predictive models to the target attribute vector. The best concept for the component is the

one that minimizes

where

is the element of that corresponds to the attribute

predicted by the component attribute prediction models. For example, for the suspension, one would evaluate

for

.

Once designers have identified the best concept for a component, they can use standard engineering optimization

techniques to identify design variable settings (c.f., Figure ). One can formulate this as a target-achievement type

problem, where the design objective is to minimize deviation for the target attribute vector for that component. For


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the component, designers can formulate this as

, where is

the analysis model and is the vector of component-level design variables, and is the best concept.

C. Optimization ResultsTable 3 contains the results from the UC design

problem. The data includes the final system-level

and component-level attributes, computation time

and utility value of the most preferred design for

each of the three solution approaches. In each case,

the system using the 3-speed 6-gear transmission

and leaf spring suspension is the most preferred.

We measure computational time as the elapsed

time for all computational activities related to each

approach. These times are measured on a mid-

range PC workstation with a 2.93 GHz clock speed.

Although run times will vary depending on

hardware and software configurations, one can

expect the relative results to hold across platforms.

For the AAO approach, the computational time we report is the time required to optimize each of the six

combinations of concepts. For the ATC approach, this is the time to identify a target attribute vector at the utility

level plus the time to apply ATC to each of the six

concept combinations (the same target applies to all

concepts, so this portion of the time is counted only

once). Table 4 is a breakdown of the computational time

for each operation in the AAO and ATC approaches.

Optimization times for individual concepts ranged from

785 s (about 13 min) to 2969 s (about 50 min).

The AP approach involves a greater number of steps,

AAO ATC AP

Utility 0.841 0.664 0.808

System-Level Attributes

Cost ($US) 901 365 1080Reliability 0.996 0.949 0.997

Distance Towed (m) 256 56 337Settling Time (s) 1.86 1.03 2.03

Rise Time (s) 0.55 0.58 0.54

Displacement (m) 0.072 0.069 0.060

Suspension (LS)

Spring Constant, (N/m) 6752 1194 4113Damping, (Ns/m) 2249 1670 2045

Reliability, 0..996 0.998 0.997Cost, ($US) 178 56 119

Transmission (T3-6)

Gear Ratio 1, 1.66 1.66 2.02Gear Ratio 2, 1.32 1.03 1.89Gear Ratio 3, 0.99 1.05 0.82Reliability, 0.999 0.995 0.999Cost, ($US) 723 310 961

Total Computation Time (s) 13659 5789 1331

Speedup Factor (rel. to AAO) 1 2.35 10.3Pct Utility Difference w/ AA0 0 21% 4%

Table 3. Results for utility cart design problem using three

approaches. The most preferred design in each case is the

T3-6 transmission combined with the leaf spring

suspension concept.

Computational Time (s)

AAO ATC

Utility Level Optimization n/a < 1

Concept Combinations:Transmission Suspension

T3-8 HS 2485 1196T3-8 LS 1754 1006T3-8 HLS 1590 785T3-6 HS 2969 885T3-6 LS 2235 810T3-6 HLS 2626 1107

Total 13659 5789

Table 4. Computational time by step for AAO

and ATC approaches.


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combinations to twelve. Computational time for the AAO and ATC approaches grows invariably as more concepts

and components are added. One can think of this as there being an average computational cost associated with

evaluating any concept combination for a given problem and the overall computational cost is this average time

multiplied by the number of combinations. Thus, as the number of combinations grows, so does the overall

computation time. In contrast with this growth pattern, the AP approach as a whole takes only a small amount of

time more than it does to complete the ATC optimization for one concept combination. Adding concepts while

keeping the number of components fixed may affect the times associated with steps 1-5 in the AP procedure (c.f.,

Table 5), but it would have little or no effect on the two steps that dominate the computation time (steps 7 and 9).

Adding components with multiple concepts likely would impact all steps of the AP procedure, but there is no reason

for one to believe the degree of impact would scale directly with the number of combinations as it does for the other

approaches. Most likely, the AP approach is most sensitive to the number of samples one takes in step 1 (which in

turn can affect the dominance and model fitting times as well as model accuracy and final solution quality) and the

number of attributes required in the predictive model. The precise computational characteristics of the AP approach

are a subject for future study.

The practical advantages of the AP approach can be limited somewhat by engineering details that make certain

combinations of component-level concepts incompatible with each other. Moreover, just because designers have

four concepts for one component and three for another does not mean they have twelve valid system combinations

in total. This does not detract from the preceding results, but does mean the actual advantages of the AP approach

will be less than one would obtain from a worst-case combinatorial analysis.

Another limitation on the practical advantages of the AP approach is that designers may be unable to abstract all

concepts for a given component into a unified model. For example, we use the three gear ratios in the predictive

transmission model in our example, which means we are unable to represent a four-speed transmission. We still

would be able to abstract concepts for four-speed transmissions into a single model, but would have to deal with

three-speed and four-speed transmissions independently during optimization. Thus, the AP approach would be

beneficial, but to a lesser extent than if we could abstract all transmission concepts into a single model.


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VII. SummaryIn this paper, we have presented and demonstrated a new approach for solving system design problems that

involve concept selection at the component level. This approach involves the use of abstract predictive models that

capture the capabilities of multiple component-level concepts into a single model. This has the effect of converting a

heterogeneous and combinatorial problem into a single search over a homogeneous space. In comparison with other

approaches common in the literature, the new approach offers significant computational advantages (a more than

ten-fold speedup) without major sacrifices in solution quality. Although our comparison is limited to one system

design problem, there is reason for one to believe the relative results will generalize to other problems. Future work

will involve additional assessment of the computational characteristics of the approach and demonstration on

problems of greater complexity.

Acknowledgments

This work is supported by the Department of Mechanical Engineering at Texas A&M University.

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