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Response surface approximation of Pareto optimal frontin multi-objective optimization

Tushar Goel a,*, Rajkumar Vaidyanathan a,1, Raphael T. Haftka a, Wei Shyy a,2,Nestor V. Queipo b, Kevin Tucker c

a Department of Mechanical and Aerospace Engineering, University of Florida, P.O. Box 116250, Gainesville, FL 32611-6250, United Statesb Applied Computing Institute, Faculty of Engineering, University of Zulia, Venezuela

c NASA Marshall Space Center, MS/TD64, MSFC, AL 35812, United States

Received 31 March 2005; received in revised form 6 February 2006; accepted 6 July 2006

Abstract

A systematic approach is presented to approximate the Pareto optimal front (POF) by a response surface approximation. The data forthe POF is obtained by multi-objective evolutionary algorithm. Improvements to address drift in the POF are also presented. Theapproximated POF can help visualize and quantify trade-offs among objectives to select compromise designs. The bounds of this approx-imate POF are obtained using multiple convex-hulls. The proposed approach is applied to study trade-offs among objectives of a rocketinjector design problem where performance and life objectives compete. The POF is approximated using a quintic polynomial. The com-promise region quantifies trade-offs among objectives. 2006 Elsevier B.V. All rights reserved.

Keywords: Pareto optimal front; Response surface approximation; Multi-objective evolutionary algorithms; Rocket injector design; Pareto drift

1. Introduction

Practical engineering design often involves multiple dis-ciplines and lacks the benefit of closed form analytical solu-tions. The design scope is frequently defined by multiple andsometimes conflicting design objectives, along with a sub-stantial number of design variables. Multi-objective optimi-zation problems usually have many optimal solutions,known as Pareto optimal solutions[1]. Each Pareto optimal

solution represents a different compromise among designobjectives. Hence, the designer is interested in finding manyPareto optimal solutions in order to select a design compro-mise that suits his preference structure. There are a numberof different methods available for solving multi-objective

optimization problems. One popular approach is condens-ing multiple objectives into a single, composite objectivefunction by methods like weighted sum, geometric mean,perturbation, Tchybeshev, minmax, goal programming,and physical programming[13]. Another approach is tooptimize one objective while treating other objectives asconstraints[4]. These approaches give one Pareto optimalsolution in each simulation.

There are numerous multi-objective evolutionary algo-

rithms (MOEAs)[5] that can be made to find multiple Par-eto optimal solutions in a single simulation run[5]. Some ofthe latest ones include strength Pareto evolutionary algo-rithm (Zitzler and Thiele)[6], Pareto archived evolutionarystrategies (PAES by Knowles and Corne) [7], elitist non-dominated sorting genetic algorithm (NSGA-II by Debet al.) [8], and controlled elitist non-dominated sortinggenetic algorithm (Deb and Goel) [9].

Most of the popular evolutionary algorithms are basedon the concept of Pareto dominance and involve a finitesize of population at each generation [5]. Due to the finite

0045-7825/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.07.010

* Corresponding author. Tel.: +1 352 392 6780; fax: +1 352 392 7303.E-mail address:[email protected](T. Goel).

1 Currently with General Motors, Bangalore, India.2 Present address: Department of Aerospace Engineering, The Univer-

sity of Michigan, Ann Arbor, MI 48109, United Sates.

www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 196 (2007) 879893
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size of the population, some of the good solutions makeway for the other solutions. If the lost solutions are thePareto optimal solutions, the loss may not be repairedand sub-Pareto optimal solutions are obtained as the finalsolution. This problem is named here Pareto drift. A rem-edy for this problem is suggested in the form of maintain-

ing an archive of the Pareto optimal solutions.To solve multi-objective optimization problems one cancouple EAs with exact function evaluations or with surro-gate models of computationally expensive function evalua-tors. With reference to the former scenario, Sasaki et al.[10,11] used EAs coupled with CFD analysis to designthe supersonic wings for multiple objectives. Makinenet al.[12]used EAs coupled with CFD analysis to solve air-craft wing design problems. Deb and Goel [1315] usedevolutionary algorithms and a posterior local search toobtain Pareto optimal solution set of simple mechanicalcomponent shapes. Ishibuchi et al.[16]used a combinationof evolutionary algorithms and local search for finding Par-

eto optimal solutions in flowshop scheduling test problems.Obayashi et al.[17]used EAs coupled with CFD to designsupersonic wings. In all these works, evolutionary algo-rithms are directly coupled with exact evaluation ofdesigns. For computationally expensive problems, directcoupling of function evaluators with EAs will be impracti-cal because multi-objective evolutionary algorithms requiremany analyses. Therefore, surrogates such as responsesurface approximations are often adopted[1835].

Once such surrogate models are available, the computa-tional burden of performing optimization and generatinga multitude of trade-off solutions is substantially reduced.

For example, Madsen et al. [18], Vaidyanathan et al.[19,20], Shyy et al.[21]and Papila et al.[22,23]used poly-nomial- or neural network-based surrogate models asdesign evaluators for the optimization of propulsion com-ponents such as turbulent flow diffuser, supersonic turbineand swirl coaxial injector element. Dornberger et al. [24]used neural networks and polynomial response surfacesto approximate the design objectives and a modificationof genetic algorithm to find the Pareto optimal solutionsfor the design of turbine blades. Bramanti et al. [25]usedneural network models to approximate the design objec-tives and then coupled these models with evolutionaryalgorithm to find multiple trade-off solutions to electro-magnetic problems. Wilson et al. [26] and Cappelleriet al. [27] used surrogate modeling (response surfaceapproximations and kriging) for approximating the objec-tives while designing piezomorph actuators. Farina et al.[28,29]used evolutionary strategies along with multiquad-rics interpolation-based response surface approximationsto optimize the shape of electromagnetic components likeC-core and magnetizer for single and multiple objectives.Emmerich et al. [30,31] proposed using local metamodel(Kriging approximation based on a few nearest neighbors)to evaluate objectives required for optimization of analyti-cal test functions and an airfoil shape design problem using

evolutionary algorithms. Ong et al. [32] used radial basis

functions to approximate the objective function and con-straints and then used a combination of evolutionary algo-rithm and sequential quadratic programming to findoptimal solutions of an aircraft wing design problem withsingle objective. Recently, Nain and Deb [33] combinedartificial neural networks and evolutionary algorithms to

reduce the computational cost while finding trade-off solu-tions for standard multi-objective optimization test prob-lems. In a latest effort, Knowles and Hughes [34], andKnowles [35] combined a Gaussian process based globaloptimizer EGO with evolutionary multi-objective optimi-zation algorithms to significantly reduce the computationalcost in optimization of analytical test problems.

As previously stated, a major advantage of using surro-gate models is that function evaluation becomes inexpen-sive, making it feasible to evaluate a large number ofPareto optimal designs (e.g.[26]). Once many Pareto opti-mal solutions are obtained, the Pareto optimal front (POF)can be represented by its own response surface and the

domain of application can be identified. Though thedesigner has many Pareto optimal solutions, he still facesthe problem of selecting the compromise solution. Theresponse surface approximation of the Pareto optimalfront would allow the designer to visualize and assesstrade-offs among the objectives, to explore compromisesolutions, and to take decisions based on realistic goals.

This paper presents a methodology to construct aresponse surface approximation of the Pareto optimalfront based on surrogate models. The methodology is dem-onstrated using a four-objective single element gasgasinjector design problem motivated by liquid-rocket injector

applications. Major interests here are to probe the interac-tions and trade-offs among objectives, assessing correla-tions and conflicts among them. The objective functionevaluations require the solution of a fluid dynamics prob-lem involving turbulence and combustion. The solution isobtained by numerically solving the NavierStokes equa-tions, aided by the turbulence closure and chemical kineticschemes. A pressure-based, finite difference, NavierStokessolver, FDNS500-CVS[3638], is used in this study. Steadystate is considered for all problems.

The objectives of this paper can be summarized asfollows:

1. demonstrate the construction of the Pareto optimalfront response surface approximation (Pareto optimalfront RSA), addressing the problem of its region ofapplicability in function space;

2. discuss the Pareto-drift problem and suggest a remedy inthe context of dominance based evolutionary algorithmsand

3. illustrate the objective function trade-offs for liquid-rocket injector design problem.

The paper is organized as follows: Section 2 gives thegeneral description of terms used in the paper. Section 3

briefly describes the NSGA-II algorithm. Section 4

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discusses the problem of Pareto drift and proposes one pos-sible remedy. A method for constructing a response surfaceapproximation for the Pareto optimal front is outlined inSection5and illustrated with the help of a single elementliquid-rocket injector design problem in Section6. Finally,Section7 summarizes the main conclusions of the paper.

2. Basic terminology

This section defines some of the terms used in this paper.More precise definitions of the terms can be found in Refs.[1,5,39,40].

2.1. Multi-objective optimization and Pareto optimality

2.1.1. Search space

Search space or design space is the set of all possiblecombinations of the design variables. If all design variablesare real, the design space is given as x 2 RN (Nis the num-

ber of design variables). The feasible domain Sis the regionin design space where all constraints are satisfied.

2.1.2. Multi-objective optimization problem formulation

Multi-objective optimization problem is formulated as

Minimize Fx; where F fj: 8j 1;M; x xi: 8i 1;N

Subject to:

Cx6 0; where C cp: 8p 1; P;

Hx 0; where H hk: 8k 1; K:

1

2.1.3. Domination criteriaA feasible design x(1) dominates another feasible designx(2) (denoted as x(1)

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all simulations a tournament selection operator with tour-nament size oftwowas used. An extensive parametric studywas conducted to select the parameters used in NSGA-IIalgorithm[42]. Based on the above mentioned studies, fol-lowing parameters were set for the simulations:

Population size (npop) 100Generations 250Crossover probability (Pcross) 1.00Distribution parameter (for crossover) 20Mutation probability (Pmut) 0.25Distribution parameter (for mutation) 200

4. Pareto drift in MOEAs

Most of the popular multi-objective evolutionary algo-rithms (MOEAs) are based on dominance criteria. Thesealgorithms investigate the POF using a finite population

size and implement diversity preserving mechanisms suchas niching, clustering, crowding distances etc. [5, Chapters56]to find the complete POF. The solutions obtained ateach generation are characterized into dominated andnon-dominated solutions. These non-dominated solutionsare non-dominated with respect to the current set of solu-tions and may include the Pareto optimal solutions as wellas sub-optimal solutions. In non-elitist MOEAs, the geneticoperators may destroy some of these solutions to explorethe design space. Introducing elitism in MOEAs alleviatesthis problem to some extent, but when the number ofnon-dominated solutions in the combined population

exceeds the population size, as happens commonly in elitistMOEAs, some of the non-dominated solutions have to bedropped. If the solution thus lost is Pareto optimal, it maynot be recovered during the course of the optimization anda suboptimal solution can appear to be a non-dominatedsolution. This problem of losing Pareto optimal solutionsis defined as Pareto drift.

Fig. 1 shows one instance of this behavior for a two-objective optimization problem while using NSGA-II [8].All the solutions evaluated so far are shown by the dotsand the non-dominated solutions at the final generationof a NSGA-II simulation are shown by asterisks. Most ofthe solutions obtained after the final generation of aNSGA-II simulation are apparently non-dominated withrespect to all the evaluated solutions, but a few solutionsin the population at the final generation were dominated.This is clear in the zoomed view in Fig. 2, where most ofthe final generation solutions were dominated by the solu-tions evaluated during the search. This problem is commonto all dominance based methods.

One remedy to this problem is to maintain andcontinuously update an (unbounded-sized) archive of allthe non-dominated solutions obtained so far, similar tosome evolutionary strategies (e.g., PAES[7]) which main-tain a bounded-sized archive. The NSGA-II algorithm is

augmented with the archiving strategy and referred as

archiving NSGA-II (NSGA-IIa). The implementation canbe summarized as follows:

1. Initialize an archive with all the non-dominated solu-tions after Step 3 in the NSGA-II algorithm.

2. After Step 7 in the NSGA-II algorithm, update thearchive as follows: Compare archive solutions with rank-1 solutions in

the combined population. Remove all dominated solutions from the archive. Add all rank-1 solutions in the current population

which are non-dominated with respect to the archive.

The final Pareto optimal set is the archive solutions set.

This archiving of the Pareto optimal solutions retains the

Fig. 1. Demonstration of Pareto-drift problem: solutions at last genera-tion of NSGA-II simulation (reported Pareto optimal front) vs. all

evaluated solutions. (For zoom in region, see Fig. 2.)

Fig. 2. Pareto-drift problem: loss of Pareto optimal solutions duringNSGA-II simulations (zoomed in view). Full Pareto optimal front isshown inFig. 1.


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information of the good solutions obtained so far and

improves the convergence to the Pareto optimal front.On the negative side, the time and memory requirementsincrease substantially due to continuous increase in thearchive size and the need to compare all the archive solu-tions with the current generation non-dominated solutions.

5. Generating the Pareto optimal front response surface

approximation

Fig. 3 illustrates the methodology used for generatingthe Pareto optimal front RSA. The starting point is theidentification of design variables and their allowable

ranges, performance criteria and constraints. Once theproblem is defined, the designs are evaluated (objectivesand constraints) through an experiment or numerical sim-ulation as required by the problem. Because the cost ofdesign evaluation is often very high, computationally inex-pensive surrogate models are developed for both objectivesand constraints. Polynomial response surface approxima-tions were used in the context of this work. On the otherhand, for multi-objective problem, identifying correlatedobjectives may help reduce the complexity of the designproblem. Highly correlated objectives may be dropped3

or a representative objective can be used for all the corre-lated objectives (principal component analysis) hencereducing the dimensionality of the problem [42]. Afterdefining the multi-objective optimization problem, the Par-eto optimal solutions are generated. In this paper, aMOEA based hybrid method (NSGA-IIa + e-constraintstrategy [4], Appendix B) [42] is suggested as the multi-objective optimizer to generate Pareto optimal solutions.

The Pareto optimal front is then approximated by aresponse surface fit to the available Pareto optimal solu-tions. Since the quality of the response surface approxima-

tion depends on the number of data points, it is imperative

to generate a large number of the Pareto optimal solutionsto ascertain good quality. The Pareto optimal front RSArepresents one objective as a function of the other objec-tives, so it is important to choose this objective judiciously.Also note that the equation developed for the Pareto opti-mal front RSA is valid only in a limited region of functionspace. To identify the region where this response surfacerepresents the Pareto optimal front, convex hull(s) is (are)fitted to the approximating data [44]. When the responsesurface approximation to Pareto optimal solutions resultsin non-convex Pareto optimal front, identification of theboundary of the Pareto optimal front RSA using a single

convex hull, may not be appropriate. Then it is more aptto fit several convex hulls to the subsets of Pareto optimalsolutions such that the non-convex boundary of the Paretooptimal front RSA is properly approximated. The multiplesubsets (clusters) of the Pareto optimal set can be effectivelyidentified using clustering. Details of the particular cluster-ing method used here are given inAppendix C. It is impor-tant to note that a sufficient number of subsets (clusters)should be selected to adequately capture the non-convexboundary.

This Pareto optimal front RSA shows the interactions ofthe different objectives for the global optimal values andhelps understand the physics of the problem. Once theclose form solution is available, the designer can visualizeand assess trade-offs among different objectives. This infor-mation can be used to refine the utility functions (impor-tance associated with different objectives) to come upwith more useful and practical designs.

6. An application: liquid-rocket single element

injector design

The proposed methodology was applied to design a sin-gle element injector of liquid-rocket engine. A schematicdiagram of the hybrid Boeing element injector under con-

sideration is shown inFig. 4.

Perform

correlation check

Evaluate F, H, C

fm =f(x1, x2,, xn)

cp = c(x1, x2,, xn)

hk= h(x1, x2,, xn)

Optimization problem

Min F={f1, f2,,fm}

fi =f(X)

s. t. C(X) 0

H(X) = 0

Global Pareto optimalresponse surface

F* = F*(f1, f2, , fm)

x1x2

xn

F

H

C

Generate Pareto

solutions

f1f

3

f2

Identify bounds

on F* using

convex hulls Clustering

f1f

3

f2

Perform

correlation check

Evaluate F, H, C

fm =f(x1, x2,, xn)

cp = c(x1, x2,, xn)

hk= h(x1, x2,, xn)

Optimization problem

Min F={f1, f2,,fm}

fi =f(X)

s. t. C(X) 0

H(X) = 0

Global Pareto optimalresponse surface

F* = F*(f1, f2, , fm)

x1x2

xn

F

H

C

x1x2

xn

F

H

C

Generate Pareto

solutions

f1f

3

f2

Generate Pareto

solutions

f1f

3

f2

Identify bounds

on F* using

convex hulls Clustering

f1f

3

f2

Clustering

f1f

3

f2

Response surface

method

Fig. 3. Flowchart of proposed method of constructing Pareto optimal front response surface approximation.

3 Only the objectives which are strongly correlated to another should bedropped to reduce the dimensionality. If the objectives are weakly

correlated, dropping objectives will cause loss of information.

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6.1. Problem modeling

The injector design has two primary objectives:improvement of performance and life. As discussed byVaidyanathan et al. [20], the performance of the injectoris indicated by the axial length of the thrust chamber, while

the survivability of the injector is associated with the ther-mal field inside the thrust chamber. A visual representationof the objectives is shown in Fig. 5. High temperaturesinduce high thermal stresses on the injector and the thrustchamber and thus reduce the life of the components butimprove the performance of the injector. Consequently,the objectives under consideration are:

1. Combustion length (Xcc) is the distance from the inlet,where 99% of the combustion is complete. It is desirableto keep the combustion length as small as possible asthis directly affects the size and efficiency of the

combustor.2. Face temperature (TFmax) is the maximum temperature

of the injector face. It is desirable to reduce temperatureto increase the life of the injector.

3. Wall temperature (TW4) is the wall temperature at 3 in.(fourth probe) from the injector face. Higher values ofthe wall temperature reduce the life of the injector, sothis objective is minimized.

4. Tip temperature (TTmax) is the maximum temperatureon the post tip of the injector. It is desirable to keep thistemperature low to maximize life.

It can be seen that the dual goal of maximizing the per-formance and the life is now cast as a four-objective designproblem. As discussed by Vaidyanathan et al. [20] theseobjectives pose different and some times conflicting require-ments on the design scenarios, hence there cannot be a sin-gle optimal solution for this problem.

There are four design variables for the injector designproblem shown in Fig. 4. These design variables, theirbaseline values, and ranges are given as follows.

1. Hydrogen flow angle (a) the maximum angle variesbetween 0 to 20. The baseline hydrogen flow angle is10.

2. Hydrogen area(DHA) the increment with respect to thebaseline cross-section area (0.0186 in2) of the tube carry-ing hydrogen. The increment varies from 0% to 25% ofthe baseline hydrogen area.

3. Oxygen area (DOA) the decrement with respect to the

baseline cross-section area (0.0423 in2

) of the tube carry-ing oxygen. The area varies between 0% and (40)% ofthe baseline area.

4. Oxidizer post tip thickness (OPTT) varies between X00

to 2X00. The baseline value of tip thickness X00 is0.01 in.

All the variables were linearly normalized between 0 and1. More information about the design variables can befound in the work by Vaidyanathan et al. [20].

The boundary conditions applied in all CFD simula-tions are as follows. Both fuel and oxidizer flow in, throughthe inlet (west) boundary where the mass flow rate is fixedfor both streams. The nozzle exit, at the east boundary, ismodeled by an outlet boundary condition. The southboundary is modeled with the symmetry condition. Allwalls (both sides of the oxygen post, the outside of the fuelannulus, the outside chamber wall, and the faceplate) aremodeled with the no slip adiabatic wall boundary condi-tion. This is a computationally expensive simulation-basedproblem so for optimization purposes it is advisable todevelop surrogate models for the objective functions. Itwas shown by Vaidyanathan et al. [19,20], Shyy et al.[21], and Papila et al. [22,23] that accurate response sur-faces for complex problems like single element injectors

can be developed.

Fig. 4. Schematic of hybrid Boeing element injector (US patent 6,253,539) and design variables.

Fig. 5. Performance measures in single element liquid-rocket injector.


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6.2. Response surface approximation of objective functions

For constructing the surrogate models, Vaidyanathanet al. [20] conducted a design of experiments (orthogonalarrays [46]) to pick the design data points. CFD simula-tions were carried out for 54 data points. Simulations for

two designs failed to produce valid results. Response sur-faces for all the objectives but TTmax were approximatedin normalized variable space using 38 design data pointsand tested using the remaining 14 designs. For the objectiveTTmaxall 52 designs were used to fit the response surface.The response surfaces were fit using standard least-squaresregression with a quadratic polynomial using JMP [47]. Areduced cubic response surface was fitted to the objectiveTTmax and this was cross-validated using PRESS [39].Vaidyanathan et al. [20] obtained the following relationsbetween design variables and objective functions

TFmax 0:692 0:477a 0:687DHA 0:080DOA

0:0650OPTT 0:167a2 0:0129DHAa

0:0796DHA2

0:0634DOAa

0:0257DOADHA 0:0877DOA2

0:0521OPTTa 0:00156OPTTDHA

0:00198OPTTDOA 0:0184OPTT2;

4

Xcc 0:153 0:322a 0:396DHA 0:424DOA

0:0226OPTT 0:175a2 0:0185DHAa

0:0701DHA2

0:251DOAa

0:179DOADHA 0:0150DOA2


0:0752OPTTDOA 0:0192OPTT2; 5

TW4 0:758 0:358a 0:807DHA 0:0925DOA

0:0468OPTT 0:172a2 0:0106DHAa

0:0697DHA2 0:146DOAa



0:0151OPTTDOA 0:0173OPTT2; 6

TTmax 0:370 0:205a

0:0307D

HA 0:108D

OA 1:019OPTT 0:135a

2 0:0141DHAa

0:0998DHA2 0:208DOAa


0:353OPTTa 0:0497OPTTDOA

0:423OPTT2

0:202DHAa2

0:281DOAa2 0:342DHA2a

0:245DHA2DOA 0:281DOA

2DHA

0:184OPTT2a 0:281DHAaDOA:

7

The quality of the response surface approximations isgiven in Table 1 [20]. The response surfaces for all theobjectives had very high adjusted coefficient of multipledetermination which indicate good prediction capabilities.The rms error on the training and the test data points forresponse surfaces fitted to objectives TFmaxand TW4werevery low while the errors for the objectives Xccand TTmax(PRESS error instead of rms error) were relatively higher

but the values were reasonable.For the chosen application, side constraints do not

require any modeling; however, in a general case whenclosed form solution of objectives and constraints is notavailable, one can develop surrogate models for bothobjectives and constraints and use these surrogates in opti-mization as was done by Mack et al. [48] for design ofradial turbine used in liquid-rocket engine.

6.3. Define multi-objective optimization problem

Using correlation analysis, Goel et al. [42] found that

the objectives TFmaxand TW4 were strongly correlated inthe design space (Appendix D). Hence, TW4 was droppedfrom the objectives list4 and a multi-objective optimizationproblem was formulated with the remaining three objec-tives TFmax, Xcc, and TTmax(Eqs.(4), (5) and (7), respec-tively). The constraints on this problem were simplebounds on the variables (between 0 and 1).

6.4. Generate Pareto optimal solutions

The three-objective optimization problem was solvedusing NSGA-IIa and e-constraint strategy. At the end ofa NSGA-IIa simulation, the total number of optimal solu-tions in the archive was 5724 and the population at the finalgeneration had 100 non-dominated solutions. For thisproblem, the archive corresponds to less than 350 KB ofmemory (using double precision type for all variables andfunctions) which is modest for current computational capa-bilities. This indicates that the size of archive may not be acritical issue in terms ofmemoryrequirements for complexproblems with more objectives, variables and constraints.Future improvements in computer hardware will further

Table 1Accuracy of response surface approximation of objectives (refer toAppendix Afor definition of accuracy measures)

TFmax Xcc TW4 TTmax

# of observations 38 38 38 52R2adj 1.000 0.995 1.000 0.989ra 0.00566 0.0205 0.00803 0.0303

Mean 0.495 0.497 0.514 0.591r(14 pts) 0.00460 0.0178 0.00669 PRESS 0.0388

4 Choice of dropping TW4 instead of TFmax was arbitrary since thecorrelation was very strong. Principal component analysis to identify three

orthogonal vectors would have been more appropriate.

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render this issue ofmemory size trivial. The computationalexpense of comparing and replacing non-dominating solu-tions in the archive can be reduced by implementing effi-cient search algorithms.

A comparison of the solutions in archive with the solu-tions at the final iteration is shown inFig. 6. The majority

(64 out of 100) of the non-dominated solutions at final iter-ation were dominated by the optimal solutions in thearchive. To improve robustness of the results, the NSGA-IIa simulations were repeated with ten random initial seeds.Results from the NSGA-IIa and the NSGA-II werecompared for each simulation. It was observed that onan average 63 out of 100 (range = [53,68], standard devia-tion = 5) non-dominated solutions in the NSGA-II finaliteration were dominated by the corresponding archivesolutions. This demonstrates the effectiveness of the archiv-ing strategy in preventing the loss of potential Pareto opti-mal solutions.

To further improve the convergence to the Pareto opti-

mal front, the archives from the ten NSGA-IIa simulationswere combined. The dominated and duplicate designs wereremoved from the combined archive to get 29,650candidate optimal designs. To assess the improvementsby using multiple simulations, each archive from theNSGA-IIa simulation was compared with the combinedarchive. It was observed that on an average 2346 solutions

(range = [2169,2620], standard deviation = 137) from theindividual archives were dominated by the combinedarchive of the solutions.

Goel et al. [42] demonstrated that local search using e-constraint strategy can further improve the solutionsobtained using MOEA. Following the procedure outlined

in Appendix B, 118,600 solutions were obtained in acombined pool of solutions from local search and theNSGA-IIa simulations. There were 88,553 non-dominatedsolutions. After removing the duplicates, 87,149 Paretooptimal solutions were found. This final solution set dom-inated 29,483 solutions out of 29,650 solutions from thecombined pool of the solutions obtained after multipleNSGA-IIa simulations. The results manifest the improve-ments using the hybrid approach of using MOEA and localsearch[42].

The improvements in the solutions quality (given as thedistance between the dominated solution and the solutionwhich dominates this solution) from each step (archiving,

multiple simulations and hybridization) are quantified bycomputing average and maximum improvements and aretabulated in Table 2. The results of comparison betweenthe NSGA-II and the NSGA-IIa (archiving), and betweenindividual archive and combined archive (multiple simula-tions) are averaged over 10 simulations. For all steps, theaverage improvements were much smaller than the maxi-mum improvements. The effect of subsequent steps on con-vergence to the Pareto optimal front (the average andmaximum improvements in the solution quality) reducedwhich demonstrated the convergence to the Pareto optimalfront. Relatively smaller reduction in maximum improve-

ments demonstrates that different steps are effective in con-verging the far-from-optimal solutions to Pareto optimalfront.

To elucidate the impact of this three step method toachieve converged Pareto optimal solution, a simpleexample is presented as follows. Suppose a designer decidesto give equal importance to all objectives (typicalpreference structure), the objective function f to min-imize is TFmax+Xcc+ TTmax. If she uses only NSGA-IIsimulations results, the best solution obtained is f=0.8106 (TFmax= 0.3790,Xcc= 0.3431, TTmax= 0.0885).After implementing the archiving strategy, her bestresult is f= 0.7692 (TFmax= 0.3811,Xcc= 0.3518,TTmax= 0.0363). With multiple simulations her resultimproves to f= 0.7670 (TFmax= 0.3819,Xcc= 0.3503,

Table 2Average and maximum improvements in solution quality (distance between dominated solution and the solution which dominates this solution) by (a)archiving, (b) multiple simulations and (c) hybridization (MOEA + local search)

Average improvement Maximum improvement

Mean* Std dev.* Min Max Mean* Std dev.* Min Max

Archiving 0.156 0.039 0.090 0.249 0.560 0.073 0.436 0.690Multiple simulations 0.050 0.015 0.032 0.083 0.466 0.041 0.415 0.527Hybridization 0.008 0.324

*Based on 10 simulations.

Fig. 6. Comparison of the solutions in the archive with the solutions atfinal iteration (it demonstrates that the archive alleviates the Pareto drift).This result is shown for one representative NSGA-II/a simulation.

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TTmax= 0.0348) and after hybridization, her final solutionis f= 0.7667 (TFmax= 0.3817,Xcc= 0.3504, TTmax=0.0346). Solving this single objective optimization problemusing sequential quadratic programming gives the sameresult as the final solution.

6.5. Representative designs

The procedure discussed previously resulted in a Paretooptimal solution set with 87,149 solutions. In order to illus-trate alternative representative design concepts, a hierarchi-cal clustering algorithm (given inAppendix C) was used inthe function space. Nine compromise solutions wereselected. These solutions vis-a-vis Pareto optimal solutionset are shown inFig. 7.Fig. 7shows that the representativesolutions were uniformly selected from the Pareto optimalfront. It was observed that small values of the face temper-ature TFmax in general were accompanied by longer com-bustion length Xcc and higher tip temperature TTmax.

However, a small compromise in the face temperaturecan substantially reduce the combustion length and tiptemperature. Also it was observed that the combustionlength can be substantially reduced by allowing a high ther-mal environment in the combustion chamber. Selecteddesigns represent these distinct regions.

Objective function values and design parameter valuesfor the nine representative designs are given in Table 3.Some interesting results about the physics of the problemwere observed fromTable 3. In general, the smaller valuesof the flow angle yield low face temperature, but highertemperature on the injector tip and longer combustion

length. Increase in the flow angle of the injector causesan increase in the temperature of the face of injector anda reduction in the length of the combustor. It is also seenthat if the cross-section area of the oxygen tube is reduced,for low values of flow angle and high value of cross-sectionarea of hydrogen tube, the temperature at the face increasesand the combustion length reduces. There is a strong effectof cross-interactions among different variables, which

impact the optimal design. This does not allow drawingconclusion about the effect of individual parameters. Amore detailed analysis of the results is given by Vaidyana-than et al.[49].

6.6. Pareto optimal front response surface approximation

Visualization of the Pareto optimal front in three-dimensions is cumbersome and only qualitative inferencesabout different design domains and function trade-offscan be made. This problem is alleviated by approximatingthe Pareto optimal front with a polynomial-based responsesurface. All 87,149 Pareto optimal solutions were used to fitthe response surface. It was unknown which of the objec-tives can be represented as a function of the remainingobjectives more accurately. Hence, each objective functionwas represented in terms of the other two objective func-

tions considering different order polynomials. Since thecost of fitting a polynomial response surface model is verysmall compared to the cost of identifying Pareto optimalsolutions, it is recommended to try all (Nobj) combinationsbefore selecting the final form of relationship betweenobjectives.

The quality indicators of different response surfaceapproximations are presented inTable 4. Several observa-tions can be made fromTable 4. First, choosing the properobjective to be represented as a function of the remainingtwo objectives influences the quality of the approximationsubstantially. In this case, TFmax= TFmax(Xcc, TTmax)gave the best quality of the response surface and TTmax=TTmax(TFmax,Xcc) gave the worst results. Second, asexpected, increasing the order of the polynomial gives abetter fit. With increasing order of the polynomial, R2adjincreased and rms error and number of points with higherrors (>0.1) reduced. The maximum error for Xcc=Xcc(TFmax, TTmax) reduced with increase in the order ofpolynomial but the effect was unclear for the remainingtwo approximations. Finally, a quintic response surfaceapproximation of TFmax= TFmax(Xcc, TTmax) was selectedas Pareto optimal front RSA. The R2adj was very good. ThePareto optimal front RSA fits the data very well. The coef-ficients of the Pareto optimal front RSA and the corre-

sponding t-statistics are given in Table 5. High values ofFig. 7. Nine representative trade-off solutions obtained using clustering.

Table 3Objective functions and design variables for nine representative Paretooptimal designs (refer toFigs. 4 and 5)

a DHA DOA OPTT TFmax Xcc TTmax

0.000 1.000 0.629 0.803 0.015 0.957 0.9370.000 1.000 0.202 0.669 0.033 0.655 0.9550.000 1.000 0.308 0.281 0.044 0.689 0.703

0.017 1.000 0.897 0.000 0.067 1.023 0.3930.246 1.000 0.163 0.000 0.171 0.503 0.3650.624 0.665 0.000 0.000 0.498 0.260 0.2180.300 0.104 0.000 0.260 0.730 0.122 0.5620.472 0.090 0.000 0.000 0.818 0.076 0.2501.000 0.018 0.562 0.000 0.936 0.117 0.013

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t-statistics for all the terms showed statistical significanceof the coefficients.

The contour plot of the Pareto optimal front RSATFmax= TFmax(Xcc, TTmax) along with the Pareto optimalsolutions illustrates its application domain in Fig. 8. It isimportant to note thatTable 5represents the Pareto opti-mal front for limited ranges of Xcc and TTmax. Only theregions bounded by the Pareto optimal solutions shown

as dots in Fig. 8 represent the Pareto optimal front. Thismeans that the zones with extreme values of the objectiveTFmax(>1.00 or 0.40. The insightgained in the present investigation has further elucidatedthe interactions of the objectives in specific ranges, besidesthose observed by Vaidyanathan et al. [20].

It is also evident fromFig. 8that for a good compromiseamong all the objectives, the values of normalized Xcc

should be between 0.350.60 and TTmaxshould be between

Table 4Accuracy of response surface approximations of the 87,149 Pareto optimal solutions (refer to Appendix Afor definition of accuracy measures)

TFmax=f(Xcc, TTmax) Xcc=f(TFmax, TTmax) TTmax=f(TFmax,Xcc)

Mean response 0.475 0.357 0.369Quadratic R2adj 0.977 0.883 0.754

RMS error 0.0560 0.0980 0.161Max error 0.236 0.400 0.452

# of points with error > 0.1 5512 25,840 55,157

Cubic R2adj 0.984 0.926 0.899RMS error 0.0462 0.0783 0.103Max error 0.210 0.356 0.298# of points with error > 0.1 4001 18,762 27,287

Quartic R2adj 0.991 0.969 0.937RMS error 0.0349 0.0506 0.0815Max error 0.245 0.300 0.555# of points with error > 0.1 1974 4964 17,712

Quintic R2adj 0.993 0.983 0.956RMS error 0.0316 0.0371 0.0679Max error 0.262 0.280 0.273# of points with error > 0.1 1304 1806 11,967

Table 5Coefficients of response surface representing Pareto optimal frontTFmax= TFmax(Xcc, TTmax)

Term Coefficient t-ratio

Intercept 1.03 1632.7

Xcc 0.27 23.5TTmax 1.09 127.1X2cc 13.34 166.6XccTTmax 0.62 68.15

TT2max 2.53 10.07X3cc 33.41 130.4

X2ccTTmax 2.99 12.2

XccTT

2

max 4.47 10.84TT3max 0.95 23.22

X4cc 30.74 90.15

X3ccTTmax 2.38 5.22

X2ccTT2max 2.68 7.09

XccTT3max 8.46 23.77

TT4max 2.17 16.6X5cc 10.84 79.23

X4ccTTmax 5.57 13.08

X3ccTT2max 14.32 16.53

X2ccTT3max 12.16 14.29

XccTT4max 1.76 3.75

TT5max

0.58 5.33

Fig. 8. Contour plot of the Pareto optimal front response surfaceapproximation TFmax= TFmax(Xcc, TTmax) with Pareto optimal solutions.

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0.00 and 0.30 of its normalized value. For this range ofvalues, TFmaxvaries between 0.10 and 0.40 of its normal-ized value. This information is very useful for a designer,in selecting the trade-offs among all the objectives orimproving the already existent preference structure. Forexample, the optimization with equal importance to all

the objectives had led to a design with TFmax= 0.3817,Xcc= 0.3504, TTmax= 0.0346. But following the informa-tion about the Pareto optimal front RSA, he may allowan increase in objective TTmax to reduce the other twoobjectives. Once the close form relation is available to thedesigner relating different objectives, a sensitivity analysiscan be done to find a more exact quantification of thetrade-offs for the selected design. At the optimal designwhen all objectives have equal importance, the derivativesusing the response surface are given as oTFmax/oXcc= 1.26;oTFmax/oTTmax= 1.04. This indicates thatincreasingXccby a unit amount keeping TTmaxconstant ismore (21%) effective in reducing TFmax than reducing

TTmax while keeping Xcc constant. However, since objec-tives Xccand TTmaxare not independent, one must ensurethat changes in the design objectives do not violate theboundary of the Pareto optimal front.

The boundary of the application regions of this non-convex response surface was identified using multiple con-vex hulls of the Pareto optimal solutions. One convex hullwas identified in each cluster of points. The convex hullidentifies the set of points on the boundary such that allthe points in the cluster are within the bounded region.For this problem, convex hulls using 9 clusters were notsufficient to represent the boundary of the Pareto optimal

response surface. A significant non-Pareto optimal regionwas bounded by the convex hulls. Hence convex hulls werefitted to the data from 30 clusters. These 30 clusters, cent-roids of the clusters and the corresponding bounding con-

vex hulls are shown on the contour plot of the responsesurface in Fig. 9. The points defining the boundary ofone representative convex hull are given in Table 6. Theboundary of the response surface is identified adequatelywith the convex hulls in most of the regions, but there were

some small regions where the convex hulls bounded non-Pareto optimal front. The non-Pareto optimal front regioncan be further reduced by using a large number of clusters;however there is a trade-off between the number of clusters(and convex hulls) and the non-Pareto optimal front regionfor non-convex problems. In order to select a combinationofXcc and TTmax, where the Pareto optimal front RSA isvalid, the selected point must be bounded by at least oneconvex hull.

6.7. Comparison of 3-objective vs. 4-objective optimization

problem results

To verify the effect of reducing the objectives, a singleNSGA-IIa simulation based hybrid strategy was used tosolve optimization problems with all 4-objectives and 3-objectives, which resulted in 18,812 and 16,467 uniquenon-dominated solutions. These two non-dominated solu-tion sets were compared. Most of the solutions obtainedwere non-dominated with respect to one another. Foursolutions from the 4-objective NSGA-IIa simulation resultswere dominated by solutions obtained for the 3-objectiveoptimization, and eight solutions obtained from the 3-objective simulation were dominated by the 4-objectivesimulation results. The number of dominated solutionsfor both populations was very small when the total numberof solutions is considered. The diversity of the solutions inobjective function space was also comparable for the twocases. This verifies that there were no adverse effects intro-duced by reducing the number of objectives.

7. Concluding remarks

The Pareto optimal front in multi-objective optimiza-tion problems is useful to visualize and assess trade-offsamong different design objectives. In addition to identifycompromise solutions, this also helps the designer set real-

istic design goals. In this paper, a systematic approach for

Fig. 9. Convex-hulls to bound the domain where the Pareto optimal frontRSA TFmax= TFmax(Xcc, TTmax) is valid. Dots show the centers of theclusters for each convex hull and thick lines show the boundary of each

cluster.

Table 6Data points in function space defining the boundary of one representativeconvex hull

Xcc TTmax TFmax

0.093 0.937 0.3870.108 0.937 0.3800.098 1.035 0.384

0.069 1.094 0.3880.066 1.094 0.3890.055 0.951 0.4050.055 0.942 0.4060.055 0.940 0.4060.073 0.938 0.3970.093 0.937 0.387


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providing an approximate closed form solution to the Par-eto optimal front and associated issues are presented. Thisapproach comprise of the following steps: (i) evaluation ofdesign by direct simulation or surrogate models, (ii) deter-mination of the Pareto optimal solutions by multi-objectiveoptimizer, (iii) approximation of Pareto optimal front

using response surface and (iv) approximation of theboundary using convex-hull(s) of the Pareto optimal solu-tions data. The selection of the objective to be representedas a function of the other objectives has a great bearing onthe accuracy of fit of the Pareto optimal front RSA andhence should be carefully selected by trying the differentpossibilities. The Pareto optimal front RSA does not repre-sent the Pareto front for all combinations of the objectivesso the relevant domain should be properly bounded. Con-vex hulls fit to the Pareto optimal solutions data can beused to locate the boundary of the domain where theresponse surface represents the Pareto optimal front. Fora non-convex Pareto optimal front, the bounds should be

identified using convex hulls fitted to the multiple clustersof the Pareto optimal solutions. The centers of the clusterscharacterize alternative optimal design concepts which canbe selected for further assessment.

The problem of Pareto drift (losing Pareto optimal solu-tions during course of optimization) in the dominancebased MOEAs is identified. Implementation of an archiv-ing strategy to preserve all good solutions is suggested asa remedy. Specifically, NSGA-II algorithm is modified toimplement the archiving strategy and the improvementsin the quality of the solutions and convergence to Paretooptimal front were demonstrated. On the negative side,

the computational cost of finding the Pareto optimal solu-tions has increased due to continuous update of thearchive.

The proposed approach is exemplified using a four-objective liquid-rocket single-element injector design prob-lem. The response surface method was employed in twoaspects. First, response surfaces based on the CFD solu-tions were used to evaluate the design objectives [20]. Thisresponse surface related the design variables and the objec-tives. Second, the Pareto optimal front was approximatedby response surface. This response surface representedone objective in terms of others objectives. The complexityof the multi-objective optimization problem was reducedby removing one (TW4) of the two correlated objectives(TFmaxand TW4). The resulting multi-objective optimiza-tion problem was solved using a combination of NSGA-IIa and e-constraint strategy. The solution of the multi-objective optimization problem was the Pareto optimalsolutions data which represented a non-convex Paretofront. The Pareto optimal front was approximated by aquintic response surface TFmax= TFmax(Xcc, TTmax). Thedomain of application of this Pareto optimal front RSAwas bounded by convex hulls fitted to multiple clusters ofthe Pareto optimal solutions data.

The analysis of Pareto optimal solutions revealed impor-

tant information about the effect of design variables on the

objectives. For example, smaller values of the flow angleyields low face temperature, but higher temperature oninjector tip and longer combustor length. Interactionsamong different objectives as identified by the Pareto opti-mal response surface manifested the importance of eachdesign objective in different regions of the function space.

For example, the face temperature can be reduced moreeffectively by allowing small increase in combustion lengthfor small combustors, whereas allowing small rise in tiptemperature is more effective for moderate length combus-tors. The Pareto optimal response surface also helped inthe visualization of trade-offs among designs. It wasobserved that a good compromise region with equal impor-tance to all objectives, will have 0.35 < Xcc< 0.60, 0.00

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