ABSTRACTFor designs involving computer intensive systems
analyses, approximation techniques are commonlyused to create a simplified approach to evaluating thesystem behavior. These techniques help in reducingthe product development time and in finding theoptimal solutions. Two important types ofapproximation techniques are the Design ofExperiments (DOE) and the Artificial NeuralNetworks (ANN). While these techniques have theirown unique features, they have certain importantadvantages as well as disadvantages over each other.In this paper, an integration strategy is presented inwhich both methods complement one another inachieving affordable current systems design. Theproposed strategy is verified by comparing the DOEand ANN approaches to the approximations of typicalnonlinear behaviors in design. The high speed civiltransport (HSCT) aircraft design is used as anexample in this study.
1. INTRODUCTIONTo make decisions in the early stages of a design
process, a great deal of analysis is required. This isespecially significant for designs that require computerintensive concurrent analyses. Approximationtechniques are constantly used as they create asimplified approach to evaluating the behavior ofcomplex systems and help in identifying the optimalsolutions. A good approximation method not onlygenerates results that are accurate enough to produce areliable design, but also decreases the productdevelopment time by reducing the intensivecomputations.
Among the existing approximation techniques,there are various forms, ranging from theapproximation of derivatives (Barthelemy andSobieszczanski-Sobieski, 1983, Bloebum, et al., 1992,Renaud and Gabriels, 1994) to the approximation ofdesign space (Chen, 1996a, Engelund, et al., 1993,Malone and Mason, 1991). While the former approachwould reduce the computational cost for optimization,the latter approach is beneficial for optimization aswell as concept explorations (rapid designevaluations). The latter is the focus of this study.
In the area of approximating the design space, thestatistical Design of Experiments (DOE) techniques(Box, et al., 1978, Montgomery, 1991) and theArtificial Neural Networks (ANN) methods (Smith,1993) are the two most widely used techniques.Through previous studies (Chen, 1995, Chen, et al.,1996a, b), some insight has been gained on using theDOE techniques, specifically the Response SurfaceMethodology (RSM) (Box and Draper, 1987, Khuriand Cornell, 1987), for achieving potential timesavings in a design process. It is found that the DOEtechniques provide an effective way to formalize thedesign knowledge in a design problem as well as toreduce the size of a complex problem. In complexsystems design, the DOE techniques have been appliedto select the design parameters that have the mostsignificant impact on the system performance(Engelund, et al., 1993), approximate finite elementanalysis (Schoffs, 1987), and serve as fast analysismodules in concept exploration (Chen, et al., 1996a).
Effective approximations of the systemperformance over the design space using the ANNtechniques, have been achieved in the field ofstructural analysis (Batill and Swift, 1993, Hajela andBerke, 1992). Researchers have also beeninvestigating the application of ANN in modelingcomplex systems (Tsoukalas and Ikonomopoulos,1991), analyzing nonlinearities in functional mappings(Hajela. P and Lee. H, 1996) and in the optimization ofmixed (discrete/continuous) design variable systems(Sellar, et al, 1994). Studies show that ANN possessesthe ability of modeling different shapes of functions.
Although DOE and ANN techniques have theirown unique features, they have certain importantadvantages and disadvantages over each other. In thispaper, an integration strategy is presented to utilizethese two techniques complementary for achievingaffordable current systems design. The proposedstrategy is verified by comparing the DOE and ANNapproaches to the approximations of typical nonlinearbehaviors in design. In the following sections, anintroduction to the DOE and the ANN techniques isprovided. Advantages and disadvantages of these twomethods are discussed. The integration strategy isthen presented, followed by the verification studiesusing the example problem.
2.1 Design of Experiments (DOE)The DOE techniques are formal techniques which
support the design and analysis of experiments(Montgomery, 1991). Among the various DOEtechniques, the Response Surface Methodology (RSM)(Box and Draper, 1987, Khuri and Cornell, 1987) is acollection of statistical techniques which support thedesign of experiments and fitting response surfacemodels. These models are mathematical functionsrepresenting relationships between independent systemvariables X (factors) and dependent performancevariables Y (responses). A typical second-orderresponse surface model is shown in Figure 1. Theresponse surface model is built based on the results of"experiments". In problems using computersimulation tools, performing 'experiments' isequivalent to performing a number of simulations withdifferent input settings for X (design factors). Standardexperiments include full factorial design, fractionalfactorial design, central composite design, Plackett-Burman design, and orthogonal arrays, etc.(Montgomery, 1991, Phadke, 1989). Theseexperiments are designed to use the minimum numberof combinations to obtain information about the factoreffects at each of the factor level. The coefficients inthe response surface models, such as (3j, pH and PJJ inFigure 1, are usually determined by the least squaredmethod (Heiberger, 1989).
y = Po + ?PiXj + SPiiXj
Figure 1 - 2nd Order Response Surface Model
As introduced in Section 1, DOE techniques havebeen applied by a number of researchers in designingcomplex engineering systems. The advantages ofusing DOE techniques, specifically the responsesurface models, are summarized as follows:
• Using DOE techniques, it is possible to study thesignificance of different design factors throughstatistical analysis. In a problem with a largenumber of design variables, it is important toapply this technique to narrow the designvariables to the most critical ones.
• The response surface models can be used toapproximate the performance at any point in thedesign space, and to quickly answer what-ifquestions. It can be used to replace computerintensive programs, and to serve as a fastanalysis module in a design program.
• In optimization, the smoothing capability of theexperimental designs allows the models to workwell on noisy, erratic or staircase functions. Thechance of finding an optimal solution is thusincreased.
In spite of the advantages listed above, thecapability of the DOE techniques to model highlynonlinear behaviors is limited. Though datatransformation techniques for achieving a goodaccuracy are available, they are complicated to apply.
2.2 Artificial Neural Networks (ANN)The ANN technique is inspired by the cognitive
and data processing capabilities that are characteristicof biological neural networks. A basic networkcomprising three layers: the input, the hidden and theoutput layers, is shown in Figure 2. Each node in theinput layer brings into the network the value of oneindependent variable (Xj). The nodes in the input layerdo not perform any processing on the input, thusserving only as a fan-out. The hidden nodes calculatesynaptic connection by multiplying the input signalsby the connection weights (w^ ). It is common toprocess the weighted sum of the inputs by anactivation function, to obtain an output signal Y. Thebasic activation function is a sigmoid function. Thehyperbolic tangent function, expressed in Eqn. 1., is asigmoid function where, for any value of u, y rangesfrom -I to +1. Each output node performs a similarcalculation and generates the resulting value as anestimate of the independent variable it represents.
Input Layer Hidden Layer Output LayerFigure 2 - ANN Architecture
e - ey = (i)
Backpropagation is a commonly used trainingprocess that propagates the error information backward
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from the output nodes to the hidden nodes. The erroris calculated by comparing the output from the outerlayer and the actual target value from the training data.The steepest decent method is commonly used to findthe values of connection weights that would minimizethe network's error to a user-specified value. Once theconnection weights are determined, the network istrained and could be used as a model for the system.
Previous applications of ANN in engineeringdesign have demonstrated the following potentialadvantages of using neural networks:
• Neural networks' mapping function is veryflexible. It is supported by mathematical proofsthat with the sigmoid activation function, themodel computed by the neural network canapproximate any shapes of mapping function.
• Neural networks possess the ability ofgeneralizing from a scarce set of data points andgiving good results at new data points.
• Neural networks possess the ability of selforganizing to adapt to changes in theenvironment as well as the ability of faulttolerance.
The major disadvantage of ANN techniques lies inthe fact that, generally, ANN requires a large amountof training data to achieve a good accuracy. Anotherproblem with the ANN applications is related to itsindeterminacy due to the use of random numbers thatpredict the initial weights of neurons. Besides, theselection of the number of layers and the number ofnodes is often arbitrary and is based on a test-and-errorapproach.
3. INTEGRATION STRATEGYAmong the existing applications, either DOE or
ANN technique is used for approximations in design.Due to the reasons discussed in Section 2, drawbackscannot be avoided when one of the methods is used.To take advantages of both methods and overcometheir limitations, an integration strategy is proposed.As shown in Figure 3, this strategy first employs theDOE techniques to generate a number of designexperiments and create response surface models torepresent the performance behaviors under study. Ifthe response surface models are not accurate enough,the ANN technique is implemented. Here, a neuralnetwork is trained on a set of data points whichcorrespond to the design experiments generatedpreviously based on DOE techniques. Additionalpoints would be added to achieve the desired accuracy.If the results are still not accurate enough, the networkis trained again by changing its architecture. When theresults from either the DOE or the ANN techniques areaccurate enough, optimization programs areintroduced to determine the optimal values for theinput factors based on the fast analysis modules.
With the proposed integration strategy, it isanticipated that the limitation of the ANN i.e., the
necessity for a large set of training data, could beovercome by using the experiments designed by theDOE techniques. With these experiments, neuralnetworks could probe the entire parameter space ofdesign settings in its training process. As the neuralnetworks are efficient in modeling highly nonlinearbehaviors, they overcome the limitations of the DOEtechnique. This combination is essentially free as thesame training sets can serve as the basis for bothmethods.
Design experiments based on DOE methods
Simulate the Analysis Programs |
Generate Output Data (responses'
Create Response Surface Models
noIs the model accurate
enough ?____4 Train the Artificial
Neural Networksyes
Replace the Simulation Programby the fast analysis modules of
optimization
Figure 3 - The Integration Strategy
To further verify our approach, DOE and ANNapproaches to the approximations of typical nonlinearbehaviors are compared in this study. Theapplicability of these two methods is investigated fromthe following aspects:
1. When a large set of training data is available, howgood is the model fit by ANN compared to the onefit by DOE ?
2. When a limited training data is available, how goodis the model fit by ANN compared to the one fit byDOE?
3. When applying the ANN, what is the differencebetween the model fit by the random data in thedesign space or those points designed based onDOE techniques?
4. When used for optimization, how good is theperformance of using a fast analysis module builtby DOE or ANN compared to using the realsimulation program?
The high speed civil transport (HSCT) aircraftsystem design problem is used as an example in thefollowing section for investigations.
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4.1 HSCT Design ModelThe HSCT aircraft system design was used in our
previous studies of applying the statistical DOEmethods to the design of complex systems (Chen et. alI996a, b, 1995). The same example problem is usedhere to compare the approximations based on DOEand ANN techniques.
In the early stages of designing a HSCT aircraft,the problem statement could be given as follows:
Given the mission requirements, the engine cycleconcept and the generic HSCT baseline model, itis necessary to identify the appropriate number ofpassengers and flight range, and developconcurrently the airframe configuration andpropulsion system top-level specifications whichmeet HSCT overall design requirements includingperformance requirements, economiccompetitiveness, and environmentalconsiderations as well as downstream designconsiderations.
Based on the problem statement, the optimizationmodel formulated by the compromise DecisionSupport Problem (DSP) is provided in the Appendix.The compromise DSP is a multiobjectivemathematical construct which is a hybrid formulationbased on mathematical programming and goalprogramming (Mistree, et al., 1993). The compromiseDSP is concerned with finding the values of the systemvariables to satisfy the design requirements includingboth the system constraints which are the "demands"and the system goals which are the "wishes". Theobjective is to minimize the deviation function Z,which is a function of the deviations of different goalsAj(x) ( from target values Gj using lexicographicminimization (Ignizio, 1985). In summary, the HSCTdesign model in the Appendix has nine factors, whichinclude seven control factors (system variables) andtwo noise factors. The system has a total of sevenresponses (performance parameters), which aremodeled as thirteen nonlinear constraints and fivenonlinear goals. For evaluations of systemperformance, the FLight Optimization System(FLOPS) (McCuIlers, 1993), a concurrent systemsanalysis program in which the engine cycle analysis,the overall vehicle synthesis and the mission analysisare integrated, is used. Our intention here is not toadvocate replacing FLOPS with approximationfunctions for aircraft design, rather FLOPS is usedhere as an example of a computer intensive analysisroutine. Detailed discussions of the technicalbackground behind the HSCT system is made in Chen,1995.
To compare the results of using the DOE and ANNtechniques, responses with typical nonlinear behaviorsare chosen for the investigations in this study. Basedon our previous studies, three out of the sevenresponses are picked. They are (1) the lift-over-dragratio (LOD), which has a lower-order nonlinearbehavior, (2) the take-off field length (FAROFF),which has a higher-order nonlinear behavior withresponse varying over a wide range, and (3) theNitrous-Oxide emissions (NOX), that has a higher-order nonlinear behavior with the responses varyingwithin a small range.
4.2 Approximations Using DOE TechniquesTwo types of design of experiments are used to
study the accuracy of DOE techniques forapproximating nonlinear behaviors. The first type,central composite design (CCD) which uses 531experiments for 9 factors, represents the situation inwhich a large number of simulations are affordable.The second type, the nonstandard 85 experiments, arecreated based on the heuristic rules when only a smallnumber of experiments is affordable. Both are usedfor fitting second-order response surfaces. The centralcomposite designs are first order fractional factorialdesigns augmented by additional "stars" and centerswhich allows the estimation of a second order surface.These experiments are designed within the bounds ofeach factor as specified in the design model (seeAppendix). The coefficients of the quadratic equationbased on the CCD directly indicate the significance ofthe first-order effects (linear terms), interaction effects(interaction terms), and second-order effects (quadraticterms). The nonstandard 85 experiments are designedbased on the results from the linear order screeningexperiments and the information on the interestingregion of responses. Details of the sequentialexperiment strategy and the heuristic rules for creatingnonstandard experiments are provided in (Chen et. al1996).
NORMAN® is the program used in this study toautomate the sequencing of the experiments and togenerate the response surface models based on thesimulation results from FLOPS. The accuracy of theresponse surface models is checked by studying theANOVA (ANalysis Of VAriance). Note the accuracychecking here is only based on those data points usedfor fitting the response surface model, not forrepresenting the accuracy across the whole designspace. The results of the regression coefficient R forall the three responses, based on both 531 and 85experiments, are shown in Table 1. The regressioncoefficient, R, is used for measuring the accuracy ofcurve fitting. The closer R is to 1, the more significantis the regression and the more accurate the regressionmodel is. R**2, also called as the coefficient ofdetermination, measures the proportion of the variation
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around the mean explained by the regression (S.O.S1
regression/S.O.S. total). R**2_adj is the ratio ofMSE2 regression/MSB total. It is noted from Table 2that the accuracy of the models for all the threeresponses is satisfactory, with the accuracy ofFAROFF and NOX comparatively better than that ofLOD. As concluded from our previous studies, theaccuracy of the results based on the nonstandardexperiments is very close to those based on thestandard experiments.
Table 1 - Regression of the DOE ResponsesRegressionCoefficients
RA2RA2_ADJRA2_PRESS
FAROFF531 85
0.9960 0.99760.9955 0.99350.9824 0.9799
ron531 85
0.9874 0.98740.9647 0.96470.8920 0.8920
NOX531 85
0.9837 0.99490.98190.98590.9783 0.9706
4.3 Approximations using ANN TechniquesTo study the accuracy of ANN techniques for
approximating typical nonlinear behaviors, threedifferent sets of data are used to train the network,
(a) Model I - standard CCD (531 experiments)(b) Model II - nonstandard 85 experiments(c) Model III - 85 random experiments.The experiments used for Models I and II are
exactly the same sets as those used to create responsesurface models in Section 4.2. Set (c) is added tostudy the difference between the models created by theexperiments designed using DOE techniques and thoserandomly generated.
The network training is implemented by Matlab®.The sigmoid function used in the network is thehyperbolic tangent function (Eqn. 1) and hence all thedata are normalized between -1 and +1. It is observed
Table 2 - Comparison of the ANN architectureResponse
FAROFF
LOD
NOX
Model
IIIIII
IIIIII
IIIIII
hiddenlayers
111
111
111
hiddennodes
1203030
2003030
603030
epochs
53792359238
13734177<52!
11905152f21S
Sum of Squares (S.O.S.) is a measure of the deviation of the experimentaldata from the mean value of the data. The total S.O.S. is contributed fromboth the S.O.S. of regression and the S.O.S. of residuals.
The mean square of error (MSE), square root of S.O.S./d.o.f., is a parameterwhich captures the influence of d.o.f. A lower MSE of residuals means amore significant regression and greater accuracy of the model.
that training the ANN is a long process. The mainreason is the lack of any rules to choose the number ofhidden layers and the number of hidden nodes. TheANN architecture used for the various models aresummarized in Table 2. As expected, it is observedthat as the size of the training data increases, thenumber of hidden nodes have to be increased in orderto achieve proper training. This is clear from Table 2,where the number of hidden nodes used to train ModelI is more than those needed for Models II and III. Allthe networks are trained on a try-and-error basis. Theresults shown in Table 2 are the best from the tests. Allthe networks use only one hidden layer. It is noted thatwhen two hidden layers are used, overfitting occurs inall networks.
4001
Experiment Number
Figure 4. Error of FAROFF for Model IIThe target error for training the network is set at
0.05. The actual error would be represented by theleast squared error (LSE) which is the sum of thesquare of the errors. Figure 4 is a graphical illustrationof the error for each training point, when the ANNmodel for the response FAROFF is trained usingModel II. In Table 3, the least squared errors of thethree training models for each response, are presented.As expected, the LSE for Model I is the least for allthe responses. The reason for this result is due to thecomprehensive training data used.
Table 3 - LSEs of the ANN Responses
Response
FAROFF (ft)LODNOX (Ib)
Model I
47.31530.00496
13.839
Model II
120.920.008094
72.802
Model III
222.3390.0528698.802
4.4 A Comparison between DOE andANN Approximations
The accuracy of the approximations based on DOEand ANN techniques for typical nonlinear behaviors isfurther compared by confirmation tests at additionaldata points across the design space. 121 grid pointsover two critical control factors, i.e., aspect ratio (AR)and thrust-weight ratio (THRUST), are selected acrossthe design range while the remaining seven factors arefixed at their middle levels. The ranges for AR andTHRUST are [1.5-3.1] and [0.28-0.48], respectively.
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A comparison of the least squared errors (LSEs) forthe 121 grid points based on various approximationmodels is made. In Table 4, results are provided forapproximation models obtained from DOE (85experiments) and the ANN (models I, II and III). It isnoted that the LSEs of the DOE models for LOD andNOX are all smaller than that of ANN models, whileFAROFF is the opposite. This indicates that DOE isefficient for modeling typical engineering behaviors,but falls short in modeling behaviors that are highlynonlinear with values varying within a large range.Among the ANN models, it is observed that the ANNmodels obtained based on the 531 CCD designs(Model I) are better than the models obtained by thenonstandard 85 experiments set (Model II), while bothare better than Model III which uses the 85 randomexperiments set. This indicates that the ANN can betrained effectively by a data set that is acomprehensive representation of the design space.
18000130008000
1.5
>0.48
HRUST
38000g 33000it 28000•= 23000"• 18000
130008000
This data set could be designed based on the DOEtechniques as introduced in Section 4.2.
Table 4 - Comparisons of Grid Points LSEs
Response
FAROFF (ft)LODNOX (Ib)
DOE85 nonstd
1900.630.01989
58.706
ANN ModelI
1529.140.02128
87.882
II
1784.270.0363
182.296
III
1811.860.0463336.23
The graphical representation of this comparison isprovided in Figure 5. On the left is the accurateresponse behavior obtained by actual FLOPSsimulations at the grid points (11x11). In the middle isthe grid plot based on the response surface modelcreated by the 85 nonstandard experiments. On theright is the one based on the ANN trained by the 85nonstandard experiments (model II).
38000£ 33000~ 28000£ 23000g 18000
130008000
2.14 278^^0-28AR 3.1
(a) FAROFF - from FLOPS
1'52.14 ~~—'0.28AR 2'78 3.1
(b) FAROFF - from DOE (85 nonstandard)
2.142^8AR
(c) FAROFF - from ANN (Model II)
oo
(d) LOD - from FLOPS (e) LOD - from DOE (85 nonstandard) (f) LOD - from ANN (Model II)
F10000
^5"=, 9000X
0.48 = 8°°°' 7000THRUST 6000
0̂.28 1 -5'AR"
(g) NOX - from FLOPS (h) NOX -from DOE (85 nonstandard) (j) NOX - from ANN (Model II)
Figure 5 - Grid Plots of Results based on FLOPS, RSM and ANN
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It is observed that, for low-order nonlinear behavior(LOD) and high-order nonlinear behavior with theresponses varying within a small range (NOX), theapproximations using both DOE and ANN are all veryclose to the true behavior (FLOPS). However, it isseen that for response FAROFF, which has a high-order nonlinear behavior with the responses varyingwithin a large range, the values at the upper corner arenot predicted accurately. This is because of tworeasons: (1) the modeling is based on experimentswhere all the nine factors are varied within theirbounds whereas the grid design rotates only twofactors and keeps the remaining seven factors at theirnominal levels, and (2) the response FAROFF variesover a wide range [8000 - 38000] feet while the othertwo responses have relatively smaller ranges.
4.5 Verifications of DOE and ANN Models inOptimization
The compromise DSP in Appendix A is solvedfrom different starting points using different analysismodules, including the original simulation program(FLOPS), the response surface models, and the neuralnetwork models. Note that ANN models are only usedfor the three selected responses, i.e., FAROFF, LODand NOX. Three different starting points, the lowerbounds, the middle values, and the upper bounds, ofthe seven design variables are used for thiscomparison.
From Figure 6 it can be noted that the constraintviolation quickly converges to 0 when using theresponse surface models. However, when usingFLOPS, the convergence depends heavily on theinitial starting points. It is seen from the figure that intwo of the three runs using FLOPS, the constraintviolation does not converge to 0.
It is also seen that the constraint violationconverges to 0 in all the three test runs when using theneural network models trained by the CCD design(model I) and the nonstandard 85 experiments (ModelII). However, this is not the case when the neuralnetwork models trained by the 85 random experiments(Model HI) are used. In this case, the constraintviolation does not converge to 0 when the upperbounds of the design variables are used as the initialstarting points. This matches well with our observationfrom the grid plots that the responses based on theANN model III are non-smooth, whereas the responsesbased on the ANN models I and II are smoother andmore accurate. This also indicates that using the set of85 nonstandard experiments selected across theinteresting design region to train the ANN is a betterapproach than using a set of random experiments.
Since the constraint violations and deviationfunction values, plotted in Figure 6, are based on theapproximations using DOE and ANN models,
confirmation tests are conducted by the FLOPS tocalculate the true values of constraint violation anddeviation function at the optimal design points. It isfound that the design solution achieved using responsesurface models is even better than using the FLOPSprogram. The deviation function value evaluated byFLOPS at the best feasible design generated with theDOE models is 0.056, which is better than 0.064, thesmallest deviation function value obtained usingFLOPS. When using the ANN models, the best resultis 0.1432 from model I, 0.1859 from model II, and0.2957 from model III, respectively. In general , theresults based on DOE models are better than thosebased on ANN models, while those ANN modelsbased on the "designed" training data are better thanthe ones based on the data picked randomly.
Using FLOPS
S 1.50E-01 • L
1 i.oofc-oi • V -i_o—D_n_a_• 1 -S^S^j^^^f
> « nnc_A^ ^^ ̂ ^
1 2 3 4 5Iteration No.
Using DOE(85 nonstandard)
\\) 03 E+03 J-H——I——I——I——h
3 1 s fcteration No.
2 3 4 5 6Iteration No.
H—I—H7 8 9
Figure 6 - Convergence of the ConstraintViolation and the Deviation Function
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5. CLOSUREIn this paper, the applications of the two
approximation techniques, Design of Experiments(DOE) and Artificial Neural Networks (ANN), areinvestigated. Advantages and disadvantages of thesetwo methods are analyzed. An integration strategy,which combines the advantages of these twotechniques, is proposed. The comparisons of thesetwo approaches to the approximations of typicalnonlinear behaviors are discussed in detail using thehigh speed civil transport (HSCT) aircraft systemdesign as an example. The major observations are asfollows :
• The DOE technique is efficient for modelingtypical engineering behaviors, but falls short inmodeling behaviors that are highly nonlinear withvalues vary within a large range. Thisobservation is made from the ANOVA analysisfor the regression of the three responses understudy (Section 4.2).
• The ANN in general needs a large set of sampledata to be trained properly. This is the maindisadvantage of the ANN technique. This isproved from the results shown in Table 4, wherethe least squared error of the ANN model I (531experiments) is smaller than the other two models(85 experiments) (Section 4.4). The sameobservation is made when the compromise DSP issolved by different ANN models, Section 4.5.The tradeoff is, of course, the time required fortraining increases as the size of the trainingsample increases. Meanwhile, more time isneeded for testing the proper number of layers andnodes.
• The ANN model trained by the experimentsdesigned using the DOE techniques is better thanthe model trained by random experiments. This is,again, from the results shown in Table 3, where,the least squared error based on the ANN Model IIis less than that from Model III. Model II is basedon the training by the 85 nonstandardexperiments, which cover the interesting range ofresponses. On the other hand, the ANN Model IIIis trained by the 85 random set of experiments.
• When using either the DOE or the ANN models inthe optimization, the solutions converge muchquicker compared to using the computer intensiveanalysis program. In Section 4.5, it isdemonstrated that DOE and ANN techniques offersimplified design models which smoothens thesystem behavior and increases the chance offinding the optimal solutions.
From this study, it can be concluded that the ANNtechnique is very good in efficient highly nonlinearbehaviors, provided that a large and a comprehensiveset of data is used to train the networks. The DOEtechnique, which is not very accurate in approximatinghighly nonlinear behaviors, can be used to
approximate low-order linear behaviors and select therequired set of data to train the network. For thesereasons, an integration strategy is proposed in thispaper. This strategy employs DOE techniques in theinitial stage to create smooth low-order nonlinearfunctions and utilize ANN techniques in the later stageto model highly nonlinear behaviors based on theexperiments selected by DOE technique. Although,the compromise DSP technique is used as an examplefor the optimization of the design system, the approachin this paper can be extended to the cases in whichother types of optimization programs are used.
ACKNOWLEDGMENTThe support from NSF grant DMI 9624363 is
gratefully acknowledged. We thank Rudi Cartuyvelsfrom Numerical Technologies, Belgium for the use ofNORMAN® in creating response surface models.
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APPENDIX: THE COMPROMISE DECISIONSUPPORT PROBLEM FOR HSCT DESIGN
Given:
Find:
Satisfy:
Approximation Models of GW(X), FUEMAX(X), PI(X),FAROFF(X), SFC(X), LOD(X), and TNOX(X)# passengers NPT = 300 and Range ORANGE = 5000 nm(identified through the screening experimentation).Mission profile (altitude, range, reserve fuel, etc.)Engine type= Mixflow turbo fanGeneric HSCT baseline configurationOverall Design Requirements (Table 1)Noise Factors turbine entry temperature and burner efficiency areat their mid levels, Xg=Xg=0 (normalized)
The system variables (top-level design specifications), X^ i=I, 7AR \i THRUST X2 SW X3
OPRDES X4 FPRDES X5
BPRDES X6 TTRDES X7
The values of the deviation variables associated with goals G(X):the gross weight, GW: d i", d)+
the fuel weight, FUEMAX: d2",d2'f
the NOx emission, TNOX: d3~, d3+
the SFC at cruise speed, SFC: d4", d4+
the lift-over-drag, LOD: d$~, d$+
The system constraints, C(X), as determined by FLOPS nrresponse surface models
+=1.0The bounds on the system variables (normalized):-1 < A R < 1 -1 < THRUSTS 1-I <SW<1 -1 S OPRDES <1-1 < FPRDES < 1 -1SBPRDES < 1-1 < TTRDES S 1 (actual bounds see Table 1)
Minimize:The sum of the deviation variables:Z = {fjWI+), f2(d2
+), f3(d3+), f4(d4
+), f5(d5-)J
1324American Institute of Aeronautics and Astronautics
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