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Collaborative Optimization

13

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Xiaoyu GuGraduate Research Assistant,

Student Mem ASMEe-mail: [email protected]

John E. Renaud1

Associate Professor,Senior Mem ASME

e-mail: [email protected]

Leah M. AsheUndergraduate Student

e-mail: [email protected]

Stephen M. BatillAssociate Dean and Professor,

Mem ASMEe-mail: [email protected]

Department of Aerospace and MechanicalEngineering

College of EngineeringUniversity of Notre Dame

Notre Dame, IN 46556

Amrjit S. BudhirajaAssistant Professor

e-mail: [email protected] of Statistics

College of Arts and SciencesUniversity of North Carolina at Chapel Hill,

Chapel Hill, NC

Lee J. KrajewskiWilliam R. and F. Cassie Daley

Professor of Manufacturing Strategye-mail: [email protected] Department ofManagement and Administrative Sciences

Mendoza College of Business,University of Notre Dame,

Notre Dame, IN 46556

Decision-Based CollaborativeOptimizationIn this research a Collaborative Optimization (CO) approach for multidisciplinary sys-tems design is used to develop a decision based design framework for non-deterministicoptimization. To date CO strategies have been developed for use in application to deter-ministic systems design problems. In this research the decision based design (DBD)framework proposed by Hazelrigg [1,2] is modified for use in a collaborative optimizationframework. The Hazelrigg framework as originally proposed provides a single level op-timization strategy that combines engineering decisions with business decisions in asingle level optimization. By transforming this framework for use in collaborative opti-mization one can decompose the business and engineering decision making processes. Inthe new multilevel framework of Decision Based Collaborative Optimization (DBCO) thebusiness decisions are made at the system level. These business decisions result in a set ofengineering performance targets that disciplinary engineering design teams seek to sat-isfy as part of subspace optimizations. The Decision Based Collaborative Optimizationframework more accurately models the existing relationship between business and engi-neering in multidisciplinary systems design. !DOI: 10.1115/1.1432991"

Keywords: Collaborative Optimization (CO), Decision Based Design (DBD),Uncertainty, Optimization

IntroductionIncreasing attention has been paid to the notion that engineering

design is a decision-making process #Chen et al. !3,4", Kim et al.!5", Azarm !6"$. This notion is consistent with the definition ofdecision as a choice from among a set of options and as an irre-vocable allocation of resources. The approach of decision-baseddesign #DBD$ is built upon this notion. Rooted from more thantwo hundred years of research in the field of decision science,economics, operations research and other disciplines, decision-based design #DBD$ provides a rigorous foundation for design,which enables engineers to identify the best trade-off and focus onwhere the payoffs are greatest.Engineering design involves the generation of design alterna-

tives or options and the selection of the best one. Since the num-ber of possible design options is practically infinite for most prod-ucts, human judgement is needed to decide which options toinclude in the consideration of alternative designs and which to

neglect. Moreover an appropriate value measure has to be deter-mined in order to compare and rank order design options. It isimpossible to know exactly how a particular design alternativewill perform before it is built. However the product cannot bebuilt until it is selected. Evidently engineers have to make theirselection a priori, without full knowledge of the consequence ofthis certain selection. Thus design is always a matter of normativedecision making under uncertainty and risk.Optimization techniques have been widely applied to select the

preferred design options from the set of alternatives taken intoconsideration without having to explicitly evaluate all possiblealternatives in the set. There exists a close relationship betweendecision making and optimization. In general, decision makinghas three elements: generation or identification of options, assign-ment of expectations on each option and the application of pref-erences to determine the preferred choice. An optimization prob-lem involves the maximization or minimization of an objectivefunction or functional F(x) in the feasible region of design vari-ables x. A careful comparison between decision making and opti-mization will reveal that options, expectations and preferences areall present in optimization. The option space is equivalent to theset of permissible values of x in the feasible region. The expecta-

1Corresponding author.Contributed by the Design Automation Committee for publication in the Journal

of Computing and Information Science in Engineering. Manuscript received Aug.2000. Associate Editor G. M. Fadel.

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tion of any given x is assigned by F(x), and the preference isstated that more is better #maximization$ or less is better #minimi-zation$. Thus optimization can be used to capture the properties ofdecision-making. This recognition allows the application of rigor-ous decision theory to the case of optimization.

Decision-Based Design „DBD… FrameworkApplication of decision-based design within an optimization

domain, requires practitioners to formulate valid objective func-tions for proper decision-making. Most of the research in the fieldof optimization has been focused on the solution of the optimiza-tion problem such as development, improvement and implemen-tation of search methods to locate the optimum, while little atten-tion has been paid to optimization problem formulation. In fact,the issue of problem formulation is of the same significance, if notmore, as the issue of finding the optimal solution. A solution ob-tained using any search method is no better than the objectivefunction chosen for the optimization. If an irrelevant objective isused, the solution is equally irrelevant #Hazelrigg !1,7"$. Thereforea primary concern in DBD is the development of a mathematicallysound objective function. Recognizing that design is a decision-making process, it is imperative to construct an objective functionwhich is able to capture the preferences of rational decision mak-ers in system design problems involving risk. The decision-baseddesign #DBD$ framework of Hazelrigg !1,2" #Fig. 1 !8"$ providesa basis for exploring this issue.

The Rule of Rational Decisions. The aforementioneddecision-based design framework implements the concept of ra-tional decisions. Rational decisions follow the rule that the pre-ferred decision is the option whose expectation has the highestvalue. In a normative approach, decisions involve options, expec-tations and values. An expectation combines the possible out-comes of an alternative with probabilities of occurrence of eachpossible outcome. Note that in general expectations are notequivalent to outcomes. An outcome refers to what actually hap-pens after a decision is made to select a certain option, while anexpectation refers to what is expected to happen, based on avail-able knowledge, as the result of a decision before the decision ismade. In other words expectations are associated with what willhappen, therefore, expectations relate to the future. In the processof engineering design, it is practically impossible to predict thefuture with precision and certainty. The outcomes of most optionsin the design option space cannot be determined with certainty

prior to the decision to select one option. Hence expectations arealways probabilistic. Engineers are forced to make decisions un-der uncertainty and risk.Unlike problem-solving, decision-making cannot be conducted

in the absence of human values. The purpose of values in decisionmaking is to rank order alternatives according to the preference ofdecision makers. In the case of optimization, engineers seek adesign which maximizes value, and the objective function servesas a numerical value function to automate the process of rankordering. By means of an objective function, a real scalar is as-signed to each design alternative in accordance with the decisionmaker’s preference. In this sense an objective function serves as autility function in the context of economics. Note that utilities aredetermined by preferences and so is the objective function. Anecessary condition for the existence of an objective function indecision-based design is that decision makers or design engineersare rational individuals whose preferences and indifferences be-tween all pairs of outcomes in the design space exists and com-prises a transitive set.

Decision MakingÕOptimization Under Uncertainty. Due tothe nature of engineering design, expectations on design alterna-tives can never be determined with certainty. Particular careshould be taken in the formulation of an objective function whenrisk is present. It is imperative that the objective function #orutility function in the context of economics$ must be valid underconditions of uncertainty and risk. The von Neumann-Morgenstern #vN-M$ utility is such a value measure. Built uponthe notion of von Neumann-Morgenstern lottery and six rigorousaxioms #von Neumann, !9"$, the normative framework for deci-sion making under risk leads to a simple but profound result—theso-called expected utility theorem: ‘‘The utility of a lottery is thesum of the utilities of all possible outcomes of the lotteryweighted by their probabilities of occurrence.’’ It follows that thepreferred choice from among a set of risky options is the optionwith the highest expected utility. In the case of optimization, theaxioms of vN-M utility should be adhered to and the objectivefunction should entail the assignment of a vN-M utility to eachdesign alternative under consideration.

Characteristics of DBD Framework. The normative frame-work of decision-based design implements the concept of rationaldecisions. It facilitates vN-M utility as a measure of value againstwhich design alternatives can be compared and optimal designssought.

Fig. 1 Decision-based design framework in Hazelrigg †8‡

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The DBD framework may view the objective of systems designas one of maximizing profit. Profit, which is also referred to as netrevenue (NR), is revenue generated by the product less all costsgenerated by the product. Revenue is the sum of products soldtimes their prices, in other words, it can be calculated as the prod-uct of the demand q and the price P at which the product is sold.Costs are the sum of things bought multiplied by their prices.Total cost CT consists of: cost of manufacture, and all other lifecycle costs such as costs of research, design, maintenance andrepair and much more. The calculation of net revenue #NR$ can besummarized as Eq. #1$:

NR!P•q"CT (1)

To account for the time value of money since revenues andcosts are spread over periods of time, a better index for profitwould be the present value of net revenue #PNR$ which can beobtained from properly discounting net revenue #NR$ to thepresent and integrating it over time.Recognizing most products are designed to make money, a

valid objective function for optimization or decision making underuncertainty and risk can be established according to the rule ofrational decisions: in the process of decision-based design, theoptimizer should seek to maximize the expected vN-M utility ofthe profit, or net revenue #NR$.The axioms of vN-M utility dictates that utilities are not arbi-

trarily selected. Rather, utilities must provide proper rank orderingof design alternatives in the case of vN-M lottery. In non-deterministic optimization where uncertainty and risk cannot beneglected, the decision maker’s risk preference towards moneymust be taken into consideration. In general, the risk preference ofthe decision maker towards money leads to a utility of moneythat has diminishing marginal value #Bernoulli, !10"$. Thus profitor net revenue #NR$ itself is not a valid utility under suchcircumstances.

Multidisciplinary Enterprise ModelDesign is inherently a multidisciplinary process. Traditionally

multidisciplinary design has focused on disciplines within thefield of engineering analysis, such as aerodynamics, solid mechan-ics, kinematics, control, and many others. In the approach ofdecision-based design, engineers are compelled to look at the de-sign from a broader viewpoint. Indeed, decision-based design en-compasses not only the engineering disciplines but also businessdisciplines including economics, marketing, operation researchand more. In addition to aiming at improving the performance ofa design, engineers must be aware of the substantial impact ofnon-engineering disciplines on the goal of design. Effective com-munications between engineers and experts in the business fieldare vital to produce successful designs.The DBD framework of Hazelrigg !1,2" combines both engi-

neering and business performance simulations in a single levelall-at-once optimization approach. In the current research the Ha-zelrigg framework #Fig. 1$ has been decomposed into the multi-disciplinary enterprise model shown in Fig. 2. The decomposedsystem consists of two major elements: the engineering disciplinesand the business discipline. The work in engineering disciplinesfocuses on predicting the performance of the product for differentdesign configurations, as well as satisfying performance targets setin the business discipline #i.e., management$. The role of the busi-ness discipline centers on providing targets for performance im-provements in order to yield higher profit. These two organiza-tions are coupled through attributes a, total cost CT and demand q.Attributes a refer to the performances of a product. These per-

formances directly influence customer demand. Examples of at-tributes include speed, acceleration, comfort, quality, reliability,safety, etc. Prediction of attributes a can be obtained from theengineering disciplines. Such information will be used as inputinto the business discipline to estimate the demand q for the prod-uct, which will then be used to determine the profit or net revenue.

Fig. 2 Multidisciplinary enterprise model

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In order to compute profit, an estimate of the total cost CT isrequired. CT is again computed in the engineering disciplines.Since total cost is influenced by the number of product manufac-tured and/or sold, estimation of demand q is called for to feedback into the engineering disciplines. In this research it is as-sumed that the demand for the product, the amount of the productmanufactured and the amount of the product sold are equal.The prediction of product attributes a and total cost CT is con-

ducted in the engineering disciplines. The engineering disciplinesmodule is partitioned into a module of product performance and amodule of manufacturing cost and life cycle costs. The set ofvariables which uniquely defines a specific design are referred toas engineering design variables x. Engineers usually have controlover these variables. The estimation of demand q, net revenue NRand, ultimately, the expected utility of net revenue, are managed inthe business discipline. The price of the product not only directlyaffects the amount of profit or net revenue #see Eq. 1$, it is also animportant factor driving the demand q, which is also affected bythe product attributes a. Mathematically speaking, q can be mod-eled as a function of attributes a and the price, P.

q! q̄#a ,P $ (2)Note that price is free to be chosen by the decision maker, there-fore it should be treated as a design variable.In the context of multidisciplinary design, the term ‘‘system

analysis #SA$’’ is often used to describe the process of predictingthe performance of an engineering artifact using numerical simu-lation, which is part of the multidisciplinary enterprise model in-vestigated in this research. In fact, simulation based design toolscan be used to assist in the analyses within the entire multidisci-plinary system. Under these circumstances, the analysis of theenterprise model is no different than a ‘‘system analysis,’’ only ofa larger scale. The research results of multidisciplinary design canbe readily applied to the multidisciplinary enterprise model. Notethat the equations of physical laws involved in the system analysisare sometimes referred to as ‘‘constraints’’ in the field of decisionscience.The performance predictions obtained from a system analysis

#SA$ are referred to as states y in the context of multidisciplinarydesign. If we think of the multidisciplinary enterprise model #Fig.2$ as one big SA, then the attributes a, total cost CT , demand qand net revenue NR are components of the system states. In thispaper states y are used to represent the engineering performancepredictions, calculated in the product performance module of themultidisciplinary enterprise model. Therefore, not all of the statesy are attributes a. Only those states which influence product de-mand q will be included in the set of attributes a. In decision-based collaborative optimization framework #DBCO$, the systemlevel optimization treats attributes a as system level design vari-ables xsys which influence demand q and serve as targets for thediscipline designers to satisfy.Note that variability exists in the engineering design variables x

and all simulation-based design tools employed in each disciplinehave prediction errors associated with them.

System OptimizationThe multidisciplinary enterprise model discussed above belongs

to the type of so-called non-hierarchic systems #or coupled/networked systems$. Such systems, usually of fairly large scale,are characterized by large numbers of design variables x and pa-rameters p, large numbers of requirements or constraints g, and ahigh level of coupling between participating disciplines which areintrinsically linked to one another. Coupling results from the in-formation exchange that are required within the system analysis.This occurs where the output of one discipline is required as inputto another discipline and vice versa. Some of the mathematicalterms, defined by Balling and Sobieszczanski-Sobieski !11", arebriefly reviewed below and will be used hereinafter.

In a coupled system the design variables x can be decomposedinto the set of shared variables xsh and nss sets of local variablesxi , where i ranges from 1 to nss , and nss is the total number ofsubspaces #or disciplines$. The set of shared variables xsh containsdesign variables that are needed by more than one discipline. Theset of local variables xi includes design variables associated withdiscipline i only. The set of xsh and the sets of xi are mutuallyexclusive subsets of the set of design variables x. The term (xsh) iis used to represent the shared design variables that are needed indiscipline i.The set of output from discipline i is denoted by yi . The set of

system states y is composed of all the yi’s in the nss disciplines.The term yi j is introduced to represent the output from disciplinei which is also used as input in another discipline j. Note that yi jare the subset of coupling variables in the set of yi .The vector gi contains the design constraints associated with

discipline i. It is assumed that no constraint will stretch over morethan one discipline. It is also assumed that each inequality con-straint has been formulated such that zero is the permissible value,and the constraint is satisfied when it is greater than or equal tozero. In conventional design optimization, constraints can be for-mulated to guard against system failure or to restrict the designsearch to preferred regions of the design space. This second classof constraint, related to design preference is not used in thedecision-based collaborative optimization framework developedin this research. In this research preferred regions of the designspace are imposed implicitly through the demand function.

Collaborative Optimization „CO…. The Collaborative Opti-mization #CO$ strategy was first proposed by Kroo et al., !12" andBalling and Sobieszczanski-Sobieski !11". The CO strategy hasbeen successfully applied to a number of different design prob-lems including the design of a single-stage-to-orbit launch vehicle#Braun et al., !13"$. Tappeta and Renaud !14" extended this ap-proach and developed three different formulations to provide formultiobjective optimization of non-hierarchic systems.Collaborative Optimization #CO$ is a two level optimization

method specifically created for large-scale distributed-analysis ap-plications. The basic architecture of collaborative optimization#Braun et al., !15"$ is shown in Fig. 3. The CO framework facili-tates concurrent design at the discipline design level. To achievethe concurrency in the subspace level, auxiliary design variables(xaux) i j are introduced as additional design variables to replacethe coupling variables yi j so that the analyses in disciplines i andj can be executed concurrently. Compatibility constraints #or dis-crepancy functions$ d are added to ensure consistency such that(xaux) i j!yi j . Compatibility constraints are usually in the form ofequality constraints.The system level optimizer attempts to minimize a system level

objective function F while satisfying all the compatibility con-

Fig. 3 Basic collaborative optimization architecture „Braunet al., †15‡…

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straints d. System level design variables xsys consist of not onlythe shared variables xsh but also the auxiliary xaux variables.These variables are specified by the system level optimizer andare sent down to subspaces as targets to be matched. Each sub-space, as a local optimizer, operates on its own set of designvariables xssi with the goal of matching target values posed by thesystem level as well as satisfying local constraints gi . The match-ing can be attained by minimizing the discrepancy di betweensome of the local design variables and/or local states and theircorresponding target values, in other words, the objective func-tions at subspace level are identical to the system level #compat-ibility$ constraints. This formulation allows the use of post-optimal sensitivities calculated at the subspace optimum to beused as the gradients of the system level constraints. This impor-tant feature improves the overall efficiency of CO by eliminatingthe need to execute subspace analysis for the sole purpose ofcalculating system constraint gradients by finite differencing.

Decision-Based Collaborative OptimizationDecision-Based CO „DBCO… Framework. In the decision-

based collaborative optimization #DBCO$ framework developedin this research, the multidisciplinary enterprise model #Fig. 2$ isdecomposed along analysis boundaries into several subsystems.The method of collaborative optimization #CO$ is used to deter-mine the optimal design of this complicated model, where anoptimizer is integrated within each subsystem or discipline. Theresulting decision-based collaborative optimization #DBCO$framework rigorously simulates the existing relationship betweenbusiness and engineering in multidisciplinary systems design asshown in Fig. 4. A brief discussion about this framework is givenbelow.

System Level Optimization. In the decision-based collabora-tive optimization framework, the business decisions are made atthe system level. As discussed before, the goal of decision-baseddesign optimization is to maximize the expected utility of netrevenue (E(u(NR))) of the engineering artifact being designed.The system level optimizer in the decision-based collaborativeoptimization framework attempts to increase expected utility ofnet revenue while satisfying compatibility constraints d. Accord-ing to the analyses in the business discipline and the subspace

optimization results, the system level optimizer determines priceP and establishes a set of performance targets for shared designvariables xsh and auxiliary design variables xaux , including de-mand q and total cost CT . Attributes a are treated as auxiliarydesign variables which influence demand. Note that demand q andtotal cost CT are also among the auxiliary variables. Also note thatsince the design variable price P is only needed in the businessdiscipline, it is not a target for any subspace.The business discipline is operated on directly by the system

level optimizer. It is not further decomposed into demand andutility of profit subspaces. The reasons why the business disciplineis dealt with in this manner are as follows: First of all, there is notwo-way coupling between the analysis in the demand subspaceand the utility of profit subspace. The demand q is fed forwardinto the Profit subspace. Secondly, the analyses involved in thisdiscipline are relatively simple and straightforward #Eqs. #1$ and#2$$. Consequently it is relatively easy to obtain sensitivity infor-mation with respect to the profit #or net revenue$ or the utility ofnet revenue. In the cases where an analytical demand model andan analytical utility model are supplied, the sensitivity informationis readily acquired through the application of the chain rule.The system level optimization problem in its standard form is

given in Eq. #3$.

Minimize;w.r.t. xsys

oF!"E#u#NR $$

Subject to: di*!0 i!1,2, . . . ,nss

#xsyso $min%xsys

o %#xsyso $max

xsyso !#xsh

o ,xauxo $

P#0 (3)

Note that maximizing the expected utility E(u(NR)) is equivalentto minimizing its negative value. Note also that net revenue #NR$is determined by the system level design variable xsys and price P.Since system design variables xsys are posed as targets to thesubspaces, the term xsys

0 is used in Eq. #3$ so that they can bedistinguished from subspace design variables in the subproblem

Fig. 4 Decision-based collaborative optimization

Journal of Mechanical Design MARCH 2002, Vol. 124 Õ 5

formulations. The term di* refers to the optimal value of the dis-crepancy function di obtained by the subproblems. The formula-tion of di is discussed in the next section.

Subspace Level Optimization. The subspace optimizer seeksto satisfy the targets sent down by the system level optimizer andreports the discrepancy di* back to the system level. Meanwhilethe subspace optimizers are subject to local design constraints gi .In the field of engineering design, the design constraints normallyguard against failure or restrict the search to preferred region ofthe design space. One example of failure-related constraints is torequire that ‘‘the axial load in a beam not exceed its bucklingload.’’ The statement that ‘‘the mass of the beam should be lessthan 7 kg’’ is an example of constraints based on preference. Theuse of constraints to restrict the search to preferred region of de-sign space is not recommended in decision-based design ap-proach. Firstly, to impose a preferred space, the engineer mustdecide and quantify what level of behavior is unacceptable orundesirable #i.e., not preferable$. This is a matter of decision-making and by imposing constraints of preferences, the designeris removing some degrees of freedom in the design process. Theresulting system optimization may fail to identify the design withthe best trade-off, especially when this constraint is active or nearactive at the optimal solution. In the example of the beam, a beamof mass greater than 7 kg is said to be unacceptable. But it ispossible that a beam of 7.1 kg can support a much higher loadthan a beam of 7 kg. If the goal of the optimization is to find alight beam that can support a large load, the beam of 7.1 kg mightbe a better design. Yet if a constraint is set to ensure the mass ofthe beam to be no greater than 7 kg, the optimizer will not locatethe beam of 7.1 kg, even though it may be more preferred.Upon closer examination, undesirable behaviors #i.e., non-

preferred region$ are often undesirable because such behaviorslead to a decrease in the demand of the product and/or an increasein the cost of the product, which in the end, results in a decreaseof profit. In DBD the market place is used to determine preferredregion of the design space through demand and cost models, andtherefore constraints related to undesirability or preference areeliminated. Therefore the local constraints in the decision-basedcollaborative optimization #DBCO$ framework tend to be thoseconstraints that guard against system failure. Other traditional en-gineering constraints related to consumer preference are elimi-nated from the subspace optimization and are instead incorporatedin the demand model and/or cost model.The subspace optimization problem for discipline 1 of Fig. 4 in

its standard form is given below:

Minimize:w.r.t. xss1

d1!##xsh$1"#xsho $1$

2$&j!2

nss

##xaux$ j1"#xauxo $ j1$

2

$&k!2

nss

#y1k"#xauxo $1k$

2

Subject to: g1'0

#xss1$min%xss1%#xss1$maxxss1!##xsh$1 ,#xaux$ j1 ,x1$ (4)

The subspace must satisfy local constraints gi while attempting tominimize discrepancies in system level targets. Note that attributetargets are imposed in the second and the third terms of the dis-crepancy function.

System Level Constraints Gradient. The gradient of systemlevel constraints plays an important role in forming search direc-tions for the system level optimization. As mentioned earlier, oneimportant feature of CO is that post-optimality sensitivity analysisfrom the converged subspace optimization problem can be used toprovide system level derivatives for compatibility constraints#Kroo et al., !12"$. As a result, both computational expense and

numerical error are reduced. This is possible because the systemlevel design variables are treated as parameters #i.e., targets$ in thesubproblems. Note that for a certain discipline i, depending on thecontributing analysis involved, not all the system level designvariables xsys are necessarily posed as targets to be matched. It ispossible that only a subset of xsys , referred to as (xsys) i , is sentdown as subspace i targets. Generally all the subsets for nss sub-spaces are not mutually exclusive, i.e., their intersections exist.The set of system level design variables xsys is the union of all thesubsets (xsys) i . The gradient of system level constraint di* withrespect to the subset (xsys) i of the system level design variablessent down as targets to discipline i is given below in Eq. #5$. Thegradient of system level constraint di* with respect to those sys-tem level design variables which are not imposed as targets fordiscipline i is apparently zero.

(di*(#xsys

o $ i!"2!#xss* $ i"#xsys

o $ i" (5)

The term (xss* ) i refers to the vector formed by the converged op-timal values of local variables and states in discipline i at the endof subspace optimization. The elements in (xss* ) i are the optimalcounterparts of the system level targets (xsys* ) i .

Test Problem. A preliminary application of the decision-based collaborative optimization framework has been tested on anAircraft Concept Sizing #ACS$ problem. This problem was origi-nally developed by the MDO research group at the University ofNotre Dame #Wujek et al. !16"$. It involves the preliminary sizingof a general aviation aircraft subject to certain performance con-straints. The design variables in this problem are comprised ofvariables relating to the geometry of the aircraft, propulsion andaerodynamic characteristics, and flight regime. Appropriatebounds are placed on all design variables. The problem also in-cludes a number of parameters which are fixed during the designprocess to represent constraints on mission requirements, avail-able technologies, and aircraft class regulations.The original problem has ten design variables and five param-

eters. The design of the system is decomposed into six contribut-ing analyses. This problem has been modified by Tappeta !17" tofit the framework of multiobjective coupled MDO systems. It isfurther modified in this research to be suitable for the case ofdecision-based design #DBD$. For comparison, the rest of thissection gives a brief description of the modified ACS problem byTappeta. It will be referred to as the ACS problem from hereon.The DBD version of the ACS problem will be discussed in thefollowing sections.The Aircraft Concept Sizing #ACS$ problem has three disci-

plines #see Fig. 5$: aerodynamics #CA1$, weight #CA2$ and per-formance #CA3$. It can be observed from the dependency diagramthat the system has two feed-forwards and there are no feed-backsbetween disciplines. Table 1 lists the design variables and theirbounds in the ACS problem. Table 2 lists the usage of designvariables as inputs to each discipline. It can be seen that there arefive shared design variables #x1)x4 and x7$. Table 3 lists theparameters and their values. Table 4 lists the state variables andtheir relations with each discipline. Clearly there are two coupledstates #y2 and y4$. Table 5 contains all the relevant information forthe ACS problem in the standard MDO standard notation intro-duced earlier.The objective in the ACS problem is to determine the least

gross take-off weight within the bounded design space subject totwo performance constraints. The first constraint is that the air-craft range must be no less than a prescribed requirement, and thesecond constraint is that the stall speed must be no greater than aspecified maximum stall speed. In standard form, the optimizationproblem is given below:

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Minimize: F!Weight!y4

Subject to: g1!1"y6

Vstallreq'0

g2!1"Rangereq

y5'0

Vstallreq!70 ft/sec

Rangereq!560 miles (6)

Demand Model and Cost Model. The approach of decision-based design requires engineers to not only focus on the productperformance but also life-cycle costs as well as demand and theprofit obtained over the life-cycle of the product. Thus it is veryimportant to construct a proper demand model and a proper costmodel for the product. The authors are aware that the task to buildsuch models is not an easy one, and engineers are generally nottrained for this task. Since this research is concentrated on theoptimization aspect of decision-based design, it is reasonable toassume that other discipline experts have developed such demandand cost models and made them available to the optimizer.In the case of the Aircraft Concept Sizing #ACS$ problem, nei-

Fig. 5 Aircraft concept sizing problem

Table 1 List of design variables in ACS problem

Table 2 Input design variables of each discipline in ACS

Table 3 List of parameters in ACS problem

Table 4 Lists of states in ACS problem

Journal of Mechanical Design MARCH 2002, Vol. 124 Õ 7

ther a demand model nor a cost model was available from theprevious studies. In order to apply the decision-based collabora-tive optimization framework of Fig. 4, a demand model and a costmodel have been developed. These models are built in a way suchthat they agree with industry trends for this specific class of air-craft and that they lead to reasonable optimization behavior. Al-though they are by no means complete, they serve fairly well asconcept models for the application of DBCO at the current stage.Figure 6 illustrates the demand and cost models of the ACS prob-lem. Only the annual demand, annual cost and annual profit areconsidered in the current research.

Demand Model. The first step in building the demand modelis to identify the attributes that influence the demand q of thisaircraft. The conventional optimization ACS problem #Eq. #6$$tries to minimize gross take-off weight (y4) while satisfying twoperformance constraints, one on stall speed (y6) and the other onaircraft range (y5). Closer examination reveals that the objectivefunction and two constraints, imposed in the original problem, arebased on the designer’s estimate of customer preference forweight, stall speed and range. In the decision-based design ap-proach, it is more appropriate to treat these quantities #take-offweight y4 , aircraft range y5 , and stall speed y6$ as attributes ofdemand. Hence there are no performance constraints in the DBDversion of the ACS problem, and the goal of the optimization is tomaximize profit.It is also assumed that customers are interested in the cruise

speed (x6) of the aircraft as well as how much room they wouldhave on the airplane. A new state variable, fuselage volume (y7),is introduced to reflect the concern for passenger room on theaircraft. In all, there are five attributes of demand in the DBDversion of ACS problem: takeoff weight y4 , aircraft range y5 ,stall speed y6 , fuselage volume y7 , and cruise speed x6 . Notethat demand is also influenced by price P.The demand model developed is a multiplicative model:

q! q̄#a,P $!qB! *i!1

na

ci" cpricewhere a!+y4 ,y5 ,y6 ,y7 ,x6, (7)

na!5

qB!1200

The term qB represents a baseline demand, which is set to 1200.The number of attributes is denoted by na . The affect of eachattribute on the final demand is reflected by the demand coeffi-cient ci . Similarly the term cprice denotes the demand coefficientof price P. The final demand q is the product of all demand coef-ficients and the baseline demand qB .Demand coefficients for each attribute and price are developed

by financial analysts and marketing personnel within the businessdiscipline and vary with time. For the purpose of this study, theyare assumed fixed with respect to time and are given in Fig. 6. Thecurves in Fig. 6 plot the coefficient of demand on the ordinate andthe corresponding attribute #or price$ on the abscissa.1. gross take-off weight (y4)

The lower the take-off weight, the higher the demand; but anaircraft with a very light weight is not desired.2. aircraft range (y5)

The longer the aircraft range, the higher the demand; but after therange reaches more than 600 miles, there is no significant increasein demand when range increases.This formulation is not unlike the original performance constraintg2 #Eq. #6$$, where the coefficient is set to 1 when aircraft rangeequals to 560 miles.3. stall speed (y6)

The lower the stall speed, the higher the demand; but a near-zerostall speed is not necessary.This formulation is not unlike the original performance constraintg1 #Eq. #6$$, where the coefficient is set to 1 when stall speedequals to 70 ft/sec.4. fuselage volume (y7)

The larger the fuselage volume, the higher the demand.5. cruise speed (x6)

The faster the cruise speed, the higher the demand.6. price #P$

Table 5 Design vectors and functions for ACS problem

8 Õ Vol. 124, MARCH 2002 Transactions of the ASME

The lower the price, the higher the demand; if the aircraft is soldfor free (P!0), the demand approaches infinity.

Cost Model. It is assumed that all costs (CT) related to theproduction of the aircraft can be divided into two categories: fixedcost CF and variable cost CV #Krajewski and Ritzman, !18"$. Thefixed cost CF is the part of the total cost CT that remains consis-tent regardless of changes in the amount of product produced, forexample, the annual cost of renting or buying equipment or facili-ties. The variable cost CV is the portion of the total cost CT thatvaries directly with quantity of product produced, such as cost perunit for material and labor. If we assume the quantity of productproduced and sold per year is equal to the demand q for theproduct per year, the total cost CT per year is:

CT!CF$qCV (8)It is assumed that the variable cost CV in the ACS problem is

dependent on five of the seven design variables including wingarea (x2), fuselage length (x3), fuselage diameter (x4), cruisespeed (x6) and fuel weight (x7). A variable cost parameter (pV) iis assigned to each cost-related variable to represent the portion ofvariable cost #per unit$ associated with each variable. The totalvariable cost per unit is the sum of all variable cost parameters:

CV!&i!1

ncost

#pV$ i (9)

where ncost is the number of cost-related variables. The guidelinefor assigning variable cost parameters is: the larger the variable,the higher the cost. The curves in Fig. 6, associated with eachcost-related variable, plot the variable cost parameter #unit: 10,000dollars$ on the ordinate and the corresponding cost-related vari-able on the abscissa. The step jumps in the curves represents theneed to purchase #or rent$ and/or install new equipment #or facili-ties$ when the size of the aircraft exceeds existing productioncapabilities.Substituting Eq. #9$ into #8$, the model of the total cost CT is:

CT!CF$q&i!1

ncost

#pV$ i (10)

and the unit cost Cu can be obtained by dividing both sides of Eq.#10$ by demand q:

Cu!CF

q $&i!1

ncost

#pV$ i (11)

Note that the number of product produced #q$ may have a dis-counting effect on the variable cost parameters pV . For instance,usually the cost per unit for material will decrease when the totalamount of material bought increases. Thus a q-discounting optionhas been included in the determination of variable cost parameterin the cost model.

Note. The demand and cost models developed in this paperare by no means complete. They are conceptual and rather sim-plistic. Future work on the modification of these models will in-clude #and not limit to$ the following issues:1. gross take-off weight (y4)

It has been pointed out to the authors that to a customer highergross weight is actually desirable because it leads to longer air-craft range. Meanwhile higher gross weight leads to higher manu-facture cost. Therefore a modified demand model would includethe gross take-off weight as a slightly favorable feature #i.e., thehigher the take-off weight, the higher the demand$. On the otherhand a modified cost model may include the gross take-off weightas a strongly negative factor #i.e., the higher the take-off weight,the higher the cost$.2. aspect ratio of the wing (x1)

Increasing aspect ratio will increase the wing structural weight,which will in turn lead to an increase in the aircraft gross take-offweight, thus add to the total cost. A modified cost model wouldinclude aspect ratio as another negative factor.3. price #P$

It has been brought to the authors’ attention that in the real world,due to the maintenance requirements such as insurance andhanger, the demand will not approach infinity if the aircraft isgiven out for free. A modified demand model will address thisissue by assigning a definite number to the demand coefficient for

Fig. 6 Concept demand model and concept model

Journal of Mechanical Design MARCH 2002, Vol. 124 Õ 9

price when price is set to zero. This definite number will be asso-ciated with the maximum demand possible for this aircraft.

DBCO Formulation. The decision-based collaborative opti-mization framework has been applied to the DBD version of theAircraft Concept Sizing #ACS$ problem. This application is a pre-liminary study, and focuses on the collaborative optimization fea-ture of the DBCO framework. The issues of propagated uncer-tainty are neglected in this study. The utility of profit is assumedto be the profit itself. Hence the objective of the resulting deter-ministic optimization is to maximize profit #or net revenue$. Dur-ing the optimization, the demand q is treated as a continuousvariable, rather than an integer. At the end of the system optimi-zation, q is rounded to the nearest integer.Two additional disciplines are added in the DBD version of the

ACS problem: cost #CAc$ and business #CAb$. Price P is a newdesign variable and a new state variable #fuselage volume y7$ isintroduced. Table 6 provides the list of input design variables toeach discipline in the DBD version of ACS problem. Clearly de-sign variable x6 #cruise speed$ enters the set of shared variables.Table 7 lists the states y, demand q, total cost CT , and net revenueNR. It also depicts how they are related to each discipline. The setof coupling variables expands to include five additional members:y5 #aircraft range$, y6 #stall speed$, y7 #fuselage volume$, q #de-mand$ and CT #total cost$. Table 8 contains all the design vectorinformation for the DBD version of subspaces 1 #aerodynamics$,2 #weight$, 3 #performance$ and c #cost$ of the ACS problem inMDO standard notation. Table 9 contains the design vector infor-mation of subspace b #business$ since it is operated directly by thesystem level optimizer. The difference between the DBD version

Table 6 Input design variables to each discipline in ACS „DBDversion…

Table 7 Lists of states in ACS problem „DBD version…

Table 8 Design vectors for ACS problem „DBD version…

10 Õ Vol. 124, MARCH 2002 Transactions of the ASME

of ACS problem and the modified ACS problem by Tappeta !17"can be clearly observed by comparing Tables 6, 7, 8 and 9 withTables 2, 4 and 5, respectively.The system level optimization problem, for this application, in

its standard form is detailed in Eq. #12$.

System Level Optimization.

Minimize:w.r.t. xsys

oF!"NR

Subject to: d1*!0

d2*!0

d3*!0

dc*!0

#xsyso $min%xsys

o %#xsyso $max

where xsyso !!x1

o ,x2o ,x3

o ,x4o ,x6

o ,x7o ,y2

o ,y4o ,y5

o ,

y6o ,y7

o ,CTo ,P"

qo! q̄#y4o ,y5

o ,y6o ,y7

o ,x6o ,P $

NR!P•qo"CTo (12)

Note that the system level optimizer calls the business disciplinedirectly to obtain demand q0 and the system level objective NR.There are thirteen system level design variables and four compat-ibility constraints that are evaluated in subspace 1, 2, 3 and c.The subspace optimization problems for each discipline in their

standard forms are given by Eqs. (13) – (16).

Subspace 1 (Aerodynamics) Optimization.

Minimize:w.r.t. xss1

d1!#x1"x1o$2$#x2"x2

o$2$#x3"x3o$2

$#x4"x4o$2$#y2"y2

o$2$#y7"y7o$2

Subject to: #xss1$min%xss1%#xss1$maxwhere xss1!!x1 ,x2 ,x3 ,x4"

!y1 ,y2 ,y7"!CA1!x1 ,x2 ,x3 ,x4" (13)

Subspace 2 (Weight) Optimization.

Minimize:w.r.t. xss2

d2!#x1"x1o$2$#x2"x2

o$2$#x3"x3o$2

$#x4"x4o$2$#x6"x6

o$2$#x7"x7o$2

$#y4"y4o$2

Subject to: #xss2$min%xss2%#xss2$maxwhere xss2![x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7]

!y3 ,y4"!CA2!x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7" (14)

Subspace 3 (Performance) Optimization.Minimize:w.r.t. xss3

d3!#x2"x2o$2$#x7"x7

o$2$#y2"y2o$2

$#y4"y4o$2$#y5"y5

o$2$#y6"y6o$2

Subject to: #xss3$min%xss3%#xss3$maxwhere xss3!!x2 ,x7 ,y2 ,y4"

!y5 ,y6"!CA3!x2 ,x7 ,y2 ,y4" (15)

Subspace c (Cost) Optimization.Minimize:w.r.t. xssc

dc!#x2"x2o$2$#x3"x3

o$2$#x4"x4o$2

$#x6"x6o$2$#x7"x7

o$2$#q"qo$2

$#CT"CTo $2

Subject to: #xssc$min%xssc%#xssc$maxwhere xssc!!x2 ,x3 ,x4 ,x6 ,x7 ,q"

CT!CAc!x2 ,x3 ,x4 ,x6 ,x7 ,q" (16)Note that other than variable bounds, there are no local constraintsfor the subspace optimization problems.

Table 9 Design vectors in the subspace b „business… for ACSproblem „DBD version…

Table 10 Optimal solutions for ACS problem

Journal of Mechanical Design MARCH 2002, Vol. 124 Õ 11

Optimization Results & Discussion. A Sequential QuadraticProgramming #SQP$ method was used for optimization in both thesystem level and the subspace optimization. The SQP solver, fmin-con, was obtained from the Matlab Optimization Toolbox. Theprogram converged to an optimum after thirty-seven system leveliterations. The optimal solution is listed in Table 10. Figure 7shows the system level optimization history of convergence of thesystem level objective function #negative of profit, in subplot 8$,the convergence history of the four compatibility #discrepancy$constraints #d1* , d2* , d3* , and dc* , in subplots 9)12$, and theconvergence history of the seven system level design variables#cruise speed x6

0, aircraft range y50, stall speed y6

0, fuselage vol-ume y7

0, price P, demand q0 and total cost CT0 , in subplots 1)7$.

The abscissa of each subplot is the number of system level itera-tions. Note that the value of profit #not the negative of profit$ wasplotted in subplot 8 for easy reading. For the same reason unit costCu0 was plotted instead of total cost CT

0 in subplot 7.As can be seen from the convergence plots, the system level

optimizer tries to minimize both the negative of profit and theconstraint violations simultaneously. At the beginning of the opti-mization, the system level optimizer sets targets high for price,high for the levels of performance #to ensure high demand$ andlow for cost based on the results of the business analyses. How-ever these targets conflict with one another and lead to a largediscrepancy at the subspace level. Thus the system level opti-mizer, while trying to keep profit as high as possible, was forcedto lower price, downgrade performance and tolerate higher cost sothat the subspace discrepancy could be reduced. Gradually thesystem level optimizer found the best trade-off among the targetsand reached a consistent optimal design. The optimization historyobserved in the ACS problem resembles the existing relationshipbetween business and engineering in multidisciplinary systemsdesign.The demand model and the cost model play an important role in

the decision-based design approach. In order to illustrate the in-fluence of the demand and cost models, a conventional all-at-onceoptimization was performed according to the problem formulationin Eq. #6$. The conventional optimum obtained is also listed inTable 10. Note that fuselage volume (y7) at the conventional op-timum is determined by the optimal fuselage length (x3) and op-timal fuselage diameter (x4). It can be observed that the conven-tional optimum outperforms the DBCO optimal design on lowerweight (y3 ,y4). However it possesses poor characteristics in

many aspects such as smaller aircraft range (y5), higher stallspeed (y6) and smaller fuselage volume (y7). Such an outcome isno surprise since the main concern of the conventional ACS prob-lem is to minimize take-off weight, while the DBD approach takesinto account other performance attributes, because of the DBDobjective of maximizing profit.If we assume that the aircraft configuration at the conventional

optimum design will be sold at the same price as the DBD opti-mum design, the demand, cost and profit of the conventional prod-uct can be obtained according to the demand model #Eq. #7$$ andcost model #Eqs. #10$ and #11$$ developed earlier. These valuesare listed in Table 10 in parentheses because of the assumption.Notice that the unit cost of conventional optimal design is lowerthan the unit cost of the DBD optimal design. However the poorperformance attributes cause the demand for conventional optimaldesign to be much lower than the DBD optimal design. Hence theDBD optimal design leads to higher profit.

ConclusionsIn this research a Decision-Based Collaborative Optimization

#DBCO$ framework which incorporates the concepts of normativedecision-based design #DBD$ and the strategies of CollaborativeOptimization #CO$ has been developed. This bi-level non-deterministic optimization framework more accurately capturesthe existing relationship between business and engineering in mul-tidisciplinary systems design. The business decisions are made atthe system level, which result in a set of engineering performancetargets that disciplinary engineering design teams seek to satisfyas part of subspace optimizations. The objective of the Decision-Based Collaborative Optimization #DBCO$ is to maximize theexpected von Neuman-Morgenstern #vN-M$ utility of the profit ornet revenue #NR$ of a product.A preliminary application of this approach #deterministic case$

has been conducted on a multidisciplinary test problem named theAircraft Concept Sizing #ACS$ test problem. Conceptual demandand cost models have been developed. The corresponding optimi-zation results are discussed and compared with the conventionaloptimization solutions.Future work is being targeted towards a non-deterministic

implementation of the DBCO framework in which the issues ofpropagated uncertainty in such a bi-level optimization frameworkwill be addressed #Gu and Renaud, !19"$. A conceptual utilitymodel for the net revenue #NR$ will be adapted from the literature

Fig. 7 System level convergence plots for ACS problem „DBD version…

12 Õ Vol. 124, MARCH 2002 Transactions of the ASME

in the field of decision-based design #DBD$. The conceptual de-mand and cost models developed in this paper will be modified tobetter reflect the real world.

AcknowledgmentsThis multidisciplinary research effort was supported in part by

the following grants: NSF grant DMI98-12857 and NASA grantNAG1-2240.

NomenclatureCF ! fixed costCT ! total costCu ! unit costCV ! variable cost

E(?) ! expected valueF ! objective function in optimization problem

NR ! net revenueP ! price

PNR ! present value of net revenuea ! vector of attributesci ! demand coefficient for attribute i

cprice ! demand coefficient for priced ! vector of compatibility #or discrepancy$ constraintsdi ! compatibility constraint in discipline ig ! vector of constraintsgi ! vector of constraints in discipline i

ncost ! number of cost-related variablesna ! number of attributesnss ! number of subspaces #disciplines$p ! vector of parameters

(pV) i ! variable cost parameter associated with cost-relatedvariable i

q ! demandqB ! baseline demandu ! utilityx ! vector of design variables

xaux ! vector of auxiliary design variables(xaux) i j ! vector of auxiliary design variables corresponding to

yi jxi ! vector of local design variables for discipline ixsh ! vector of shared design variables

(xsh) i ! vector of shared design variables used in discipline ixssi ! vector of subspace design variables in discipline i

(xss* ) i ! vector of optimal target values in discipline ixsys ! vector of system level design variables

(xsyso ) i ! vector of targets #i.e., system level design variables$

sent down to discipline iy ! vector of state variablesyi ! vector of state variables calculated as output from

discipline i

yi j ! vector of coupling variables calculated as output fromdiscipline i and used as input to discipline j

(?)o ! system level target variable(?)* ! variable at the optimum design

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