multicriteria optimization of aircraft panels: determining viable genetic algorithm configurations

M u It i criteria 0 p t i m izat i o n of Ai rc raft Panels: Determining Viable Genetic Algorithm Configurations Robert Flynn Polytechnic University, Dept. of Computer Science, Six Metro Tech Center, Brooklyn, NY 11201

Porter D. Sherman Sikorsky Aircraft Corporation, 6900 Main Street, Stratford, CT 06497-9129

This article compares different Genetic Algorithm (GA) configurations for multicriteria optimization using the design of helicopter flat panels as a test bed. The results show that an efficient set of GA configurations can be determined through a two-step process and that the techniques used produced results as optimal as provided by expert rule- based techniques but found them in l / l O O the time. Phase I of the experiment established and tested a set of GA baseline configurations; Phase 2 expanded the most optimal baseline results into a set of preferred configurations. Over 700 GA configurations were tested and the effects of the different configurations are compared to previous studies based o n single criterion optimization. Guidelines for identifying and testing GA configurations are summarized. 0 1995 John Wiley & Sons, Inc.

I. INTRODUCTION

The purpose of this experiment was to develop an approach for identifying efficient genetic algorithm (GA) configurations. GAS have proven to be an effective optimization tool for many diverse optimization problems’ but a pre- cise method for configuring a GA has yet to be established because a GA configuration requires multiple control parameters each with a wide range of settings. Examples of these control parameters are: randomization of the initial population, scaling window size, generation gap values, trial size, population size, crossover rate, and mutation rate. The problem domain is optimizing the design of aircraft panels and has multiple objective functions. The design of aircraft panels is a challenging optimization problem in that the multiple objectives need to be simultaneously minimized. The first phase of the experiment establishes a set of baseline GA configurations using ‘rule-of-thumb’ values that were compiled from GA-related publications. The baseline configurations were

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 10, 987-999 (1995) 0 1995 John Wiley & Sons, Inc. CCC 0884-8173/95/011987-13

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tested by applying them to the optimization of aircraft panel designs. The second phase identified the baseline configurations that produced the lowest results; expanded the parameter ranges of these configurations and tested them against the problem domain. The phase 2 configurations that produced the lowest results were categorized as the “preferred” wt. The “preferred” configurations found the lowcst aircraft panel design paramcters out of all the configurations tried. When the GA optimization process was compared to the results of rule-based aircraft panel optimization, the GA’s results were as good but the GA produced results in less time (by approximately 100: I ) , required less design expert interaction, and was morc versatile. Also the GA designs were lower in weight (by more than I .5 Ibs.) than another panel design process that used a pseudo-optimization method. The experiments in this study provide a method for deriving “preferred” GA configurations.

11. GENETIC ALGORITHMS

Holland’ established a theoretical foundation for the design of artificial systems based on reproductivc plans that mimic natural selection by producing increasingly fit populations. This paradigm has been given the name genetic algorithm (GA) and it uses fitness functions to guide stochastic searches transcending over large populations. The basic flow of the G A can be summarized in Figure I .

A. Creating a Population

The initial population used to start a GA optimization is usually created by a random generator. The members of the population are called genc~s and they contain the values of the input variables that define ii problem domain. It is only

Figure 1. High level flow of GA.

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during step one (the initialization phase) that a randomized population is generated; all succeeding populations will be composed of recombinations of the top ranked, fittest genes.

B. Evaluation

Each member of a newly generated population is a group of values that must be evaluated (tested) for optimality. Evaluation is performed by applying the values contained in each member of the current population to IZ fitness (objective) functions. F o r j members of a population, the results zL of n objective functions are stored in j resultant vectors 4:

ql = [ z , , z2, . . . , Z n l

42 = [ z , , z2, . . . , z,,l

Therefore for each member of the population there is a resultant vector 4 of dimension n.

C. Ranking

The fitness results are used as the criteria in which to order the genes. The GA ranks the members of a population by ordering the members according to a single value associated with each member. In single criteria optimization this value is usually the direct result of the fitness function. In multicriteria optimization the multiple results of the objective functions have to be preprocessed and mapped into a single value. This preprocessing of individual results to derive the rank can also improve optimization The preprocessing technique used in these experiments is Pareto Optimality, introduced in 1896 by the Italian economist P a r e t ~ . ~ The Pareto technique has been used successfully with GAS in nonlinear, single objective optimization.s Pareto optimality (to maximize)' is defined as

x , y E s x dominates y if {fi(x) 2 fi(y) for all i a& fj(x) > f j ( y ) for somej} (2)

For these experiments, Pareto Optimality was used to determine the dominance of each objective result zL [from n different objectives ( I ) ] , across the whole population where there are j members (genes) subject to recombination. For example gene a may be optimal (nondominated) for objective 1 ( I 5 a I j ) ; gene b may be optimal (nondominated) for objective 2 ( I 5 b 5 j ) ; and so forth for each of the n objectives. To capture this multidimensional optimality (i.e., identifying dominated and nondominated objectives across the whole gene population) Pareto Optimality (P) creates a ranking vector p composed of [rl, r2, . . . , r,] . These r ' s represent the optimality (dominance or nondominance) of

990 FLY” AND SHERMAN

an objective in terms of a rank. For each objective, Pareto Optimality finds the gene which is optimal. Doing this allows one to rank the genes, objective by objective. We can summarize the information for the scores of any gene’s rank in terms of the ranking vector p and then use this information to determine a composite score for the gene. Formally to determine r, P is applied to the contents of resultant vectors qj (1):

where P is taken across each z k for all population membersj, to produce the Pareto ranking rk for each of the n fitness function results. This will produce j vectors of ranks: each vector is n dimensional and is associated to one of j genes. For example P will rank z I for dominance across all the zl’s in the q, vectors and store each resulting rank in rI of each related p j vector. At the beginning of the ranking process each r is initialized to zero and during the ranking process P will increment rm by 1 every time zIn is dominated. When (3) is finished, for any vector pk of ranks associated with gene k , if r,, = 0 then z,,, (the result of the objective function associated with gene k ) was the Pareto optimal. But if z,, > 0 then it is not Pareto optimal because it has been dominated by at least one other result. A single rank Rk is determined for each gene k by summing the members in each ranking vector p k :

for all ri in P I . After (4) is completed, the vector Rj is passed to the GA’s selection module.

D. Selection

The G A selection process looks at the population ranking vector Ri and decides how many of the top-ranked members (genes with the lowest sc’ores) should be selected for recombination. Once the fittest members are selected they are used to create what is referred to as a “gene pool” which is the focal point for the process of recombination.

E. Recombination

The process of recombination is performed on the members of the gene pool to insure that the G A searches a larger portion of the problem domain. The key recombination operations are crossover and mutation:&*

I . C r o s s o v e r Recombinntion

Crossover entails randomly selecting two genes from the gene pool, fixing a dividing point in the genes, and swapping all the bits to the right of the

AIRCRAFT PANELS 99 1

dividing point between the two genes. The crossover process continues at a predefined rate until a new population is generated.

2. Mutation Recombination

Mutation arbitrarily changes gene values at a predefined rate. This insures that the GA will continually search in new areas of the problem domains thus avoiding “local minimum” traps.

111. PROBLEM DOMAIN

A. Flat Panel Design for Aircraft

An aircraft panel is comprised of an outer skin, stiffeners fastened horizon- tally to the skin, and frames fastened vertically to the skin (see Fig. 2). The stiffeners and frames are used to keep the panel and the bays (formed by the stiffeners and frames) from buckling under shear and compression loads experi- enced during flight. Buckling of the panels and/or bays have a detrimental affect on the operation of an aircraft and loading limits are specified relative to the mission that the aircraft is designed to serve. Optimizing the design of aircraft panels is a complex task that includes minimizing the weight and manufacturing cost while meeting the load requirements on the panel. Key design parameters that can be optimized are skin thickness, stiffener area and moment of inertia, frame area and moment of inertia, and number of stiffeners and frames required. The materials used in these panels can be aluminum, titanium, and composites. The relationships between the materials, the geometry, the weight, and the loading on these panels determine the multiple objectives of the panel design. The multiple fitness requirement for the aircraft panels in this research is governed by four object functions which encompass panel geometry, panel loading, manufacturing constraints, and construction-time versus weight:

(1) PBUCKLE (Buckling of N Panel)-This deals with maximum load combination (compression and shear) that a panel can take before it buckles. The panel load must be kept within a specified range.

4

+ +

Figure 2. Flat Aircraft Panel Geometry.

992 FLY" AND SHERMAN

(2) BBUCKLE ( B i d l i n g (J ' r r Buy)-This deals with maximum load combination (compression and shear) that a panel bay can take before it buckles. The bay is the portion of the panel between two adjacent frames and stiffeners. The bay load must be kept within a specified tange. WGT (I'rrricl Weight)-Aircraft weight is restricted therefore panel weight must be minimized a s much as possible. S & F ( Nurnhrr of' Frnrrres r i n d .St(fliJnrJr.v)-The installation of a stiffener or frame has fixed labor cost associated with it. It is desired to minimize the number of stiffeners and/or frames because of these labor costs.

( 3 )

(4)

These four key factors are tightly coupled with over 20 "free" variables (e.g., stiffener spacing, thickness, materials. material constraints, etc.) but for this initial investigation, the GA optimization limited 7 continuous input variables. The design material used in these experiments was aluminum though 13 different material configurations (? metal (it steel is included) and 10 composite) are possible. We only used one stiffener and frame shape though there are seven basic shapes of stiffeners and frames, each shape requiring different cross- sectional moments of inertia and area equations.

IV. THE DESIGN OF THE EXPERIMENT

A. Code

A generic GA application package written in 'C' was used: GENESIS Copyright 0 1986, 1990 by John Grefenstette,'."' and ported over to the RS6000 AIX platform. Objective functions were coded in 'C' and added to the GA as the evaluation module.

B. Static Control Parameters (1 ) The elitist strategy was always specified.',6 This strategy always passes the

current "best" structure to the ensuing generation to insure that it will not inadvertently be lost in recombination.

(2) Gray coding was used throughout these experiments because it has been proven superior to binary coding in avoiding "Hamming cliffs""'." (i.e., in Gray coding contiguous integers differ by a single bit where in binary representation Hamming code distances can be as large the total number of bits in the representation).

C. Dynamic Control Parameters

DeJong'? introduced two performance metrics to measure single fitness function GA performance: on-line performance and off-line performance. On- line represents the average performance of all tested structures over the whole GA run and shows how efficient the CA is running (i.e.. how quickly the GA locates the best regions to search). Off-line represents the current best result and does not care about efficiency so much as finding a good solution (i.e., the GA can search any region, without penalty, to find a solution). The experi-

AIRCRAFT PANELS 993

ments in this article focus on off-line performance because in aircraft panel design the most efficient design is more important than the efficiency of the search.

DeJong’2,’3 developed “rules-of-thumb’’ that applied to both on-line and off-line performance:

population sizes which range between 50-100 members are the best for near optimal solutions especially when they are propagated through 10-20 generations

0 crossover rate (tc 0.6 is a good start 0 mutation @ 0.001 is a good start

Further research Grefenstte’ embellished DeJongs’s initial work by adding more GA parameter guidelines:

keep mutation rates under 0.01 (because > 0.5 is harmful > 0. I create a purely

crossover rate of 0.45 was good for off-line performance increase the number of trials if the GA does not converge

0 decrease the number of trials if the GA is over converging.

random search)

Schaffer et al. l 4 presented an exhaustive study on the interaction among population, crossover, and mutation in single fitness function GA optimization for on-line performance:

use population sizes which range between 20-30 0 use crossover rate between 0.75 and 0.95

use mutation rate between 0.005 and 0.001

Rules-of-thumb derived from these single criterion results were used to configure the baseline multicriteriu GA configurations in these experiments in an effort to measure how well they would generalize:

Cross-over Rate: This value controls the frequency at which crossover is applied during recombination. For example, a low cross-over rate value only allows a small amount of crossover to occur. The range of cross-over rate values that were used in these experiments were: 0.2 through 0.7. Mutation Rate Vulue: This value controls the frequency at which mutation is applied during recombination. For example a very large mutation rate creates a random search by randomly altering all the genes. Two mutation rate values were used in these experiments: 0.05 and 0.001. Trial Size: This parameter regulates how many GA trials will be allowed to occur during one optimization run. The G A will stop after the specified number of trials unless it has already converged (i.e., it has found near optimal solutions early in the search). Three trial sizes were used in these experiments: l K , 3K, and 6K. Generafion Gap V a h : This determines what fraction of the population that will be replaced during recombination. A gap of 0.5 means only half the population was replaced; a gap of 1 means the whole population was replaced. We used generation gaps of 0.5 and 1 . Population Size: This parameter regulates how many elements will comprise a

FLYNN AND SHERMAN 994

( 6 )

(7)

population. If a population of 50 were selected then for each trial a new population of SO genes will be produced from the recombination operation. Three population sizes were used in these experiments: 50, 150, and 200. Initial Random Seed: This is used to randomly select the initial population of the GA. To avoid the possibility of the GA starting in a local optima, three different random seeds were used in each test group. Scaling Window Value: As each generation finds better solutions, the performance measure must be updated continually to reflect the progress and keep forcing the search to find the next level of solutions. The scaling window value represents the number of generations that will be allowed to occur before the internal performance measure is re-set to a “better” performance value. Two scaling windows were used in these experiments: 5 and 2.

D. Framework for Testing

An experimental matrix was setup which used three GA test groups for the Y axis (see Fig. 3). Each Y test group was repeated for three different random seeds. Initially the X axis consisted of three blocks of trials ( IK, 3K, and 6K). Each trial block contained three blocks of populations (50, 150, and 200). Each population block contained four variations of crossover and mutation.

E. Definitions

Average o f t h e top ofthe populations means that the top 10 results of each population were averaged. Ten results were chosen as a measure of robustness: a low average combined with the elitist strategy implies that the search was localizing on a “good” solution.

Filtered Hits are all the population elements with weight <2.1 lbs. (for a 30 in. X 30 in. panel) and loads within an accepted range (i.e., filtered results focused on one objective while the three others met minimum restrictions). Any weight under 2.1 lbs. is considered a good result because the original nonoptimized panel weight was about 3.6 Ibs. The filter was peripheral to the GA and did not affect the outcome of the optimization.

F. PHASE#l

A range of rule-of-thumb configurations were tested: crossover fluctuated between 0.3 and 0.5, mutation fluctuated between 0.05 and 0.001, population fluctuated between 50, 150, and 200, and the size of a trial fluctuated between IK , 3K, and 6K. The first four configurations looked like this:

c-rate 0.5 0.3 0.S 0.3 m-rate .OOl ,001 .05 .OS

A total of 324 unique configurations were tested in PHASE # I : 36 configurations x 3 random seeds x 3 scaling windows.

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k

GGlSW2 stands for generation gap of 1 and scaling window of 2 GGlSWS stands for generation gap of 1 and scaling window of 5 GG.SSW5 stands for genenuon gap of 0.5 and scaling window of 5

G. PHASE#2

Fifteen configurations that produced optimal results were identified from PHASE # I . These configurations were incorporated into PHASE #2 and expanded into a total of 42 configurations. Crossover was emphasized in these new configurations because it is the focal point in GA recombination. The following cross-over rates were used: 0.2, 0.3 , 0.4, 0.5, 0.6, and 0.7. The mutation rates were restricted to: 0.05 and 0.001. A total of 378 unique configurations were tested in PHASE #2: 42 configurations x 3 random seeds x 3 scaling windows.

I. Results

Phase 1: 0 Smaller generation gap of 0.5 did not perform well-the 0.5 test group's results

Only one configuration with a population size of 50 was competitive (it had c-

The 3K block did better than the IK block and some of the best results occurred

The 6K appeared to be the most stable block maintaining the best minimum

were worse than the other groups in 99% of the configurations.

rate of 0.5 and a m-rate of 0.05).

in a population size of 200 with a c-rate of 0.5 and m-rate of 0.001.

averages.

W.*M h b.

2.b

2.2

f i ~ g 1 . 4 6 b I 21 n 13 24 16 % 21 19 II I 37 Y

Tliab 3K 3K 3K 3K 0U (K (K 8K BK BK (K OK (K 0K 0K (K

c-,a. 0.4 0.5 0.0 0.7 0.0 0.2 0.3 0.4 0.5 0.8 0.4 0.5 0.0 0.4 0.5 0.0 Popllaion 50 50 50 50 50 50 5 0 5 0 50 50 1 5 0 1 5 0 1 5 0 2 0 0 2 0 0 2 0 0

Y.,o. .m .a .05 .m .mi .M .m .m .m .os .mi .mi .mi .mr ,001 .mi

Graph 1.

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Phase 2:

I) G A test groups with a generation gap of I consistently produced lower results

0 The IK configuration only did well in one of the random seeds. 0 Only four 3K configurations did well (with population size of SO, a m-rate of

0 6 K configurations consistently did well with all three population sizes. 0 Sixteen configurations were identified as “preferred” because they outper-

formed the other 26. Graph 1 shows the “preferred” configurations and compares the best top weight average by generation gap (as defined in Fig. 3). Graph 2 compares the lowest filtered average weights obtained in PHASE # 2 . The lowest filtered weight found overall was 1.9014 Ibs. (in experiment 36). 73% of the experiments that found weights <2.0 Ibs. were “preferred” configurations.

higher in all the trial blocks.

0.05, and c-rates that fluctuated between 0.4 and 0.7).

2, Ohseruatioris

Parameter Selection

0 J. Grefenstette’ obtained better results when the c-rate was decreased as population increases. In this study. this did not always hold true because good results were achieved with many different ranges of c-rate on the higher populations. Some of the worst results were at a c-rate of0 .2 for large populations of 150 and 200 with m-rates of 0.001.

0 It has been suggested that a high c-rate and a low m-rate ratio work best with small population sizes.’ This study found that when population sizes of SO were used the best results were obtained from a mutation rate of 0.0s taken over a range of c-rates: 0.2 through 0.7. The higher populations of IS0 and 200 performed well with a m-rate of 0.001 when taken over a range of c-rates 0.2 through 0.6.

0 DeJong’? was able to improved performance by lowering the generation gap. This experiment found that a lower generation gap hurt performance.’

I . - i V I

4 1 21 22 2b 29 34 31

€.P.,bn.”l

Graph 2.

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Compared to rule-based o p t i m i ~ a t i o n : ~ ~ ’ ~ ~

0 The rule-based optimization required continual involvement of an aircraft design expert from conception of the rulebase to debugging the system when the in- ferencing stalled. The GA only required the design expert to initially create the objective functions.

0 The rule-based system was severely hampered by the presence of multiple objectives. It was configured for three free variables and of these variables two had to always be fixed. The GA efficiently handled seven continuous free variables without having to fix any of them. The rule-based optimization process had to loop continually, feeding back information and updating the input variables. This extensive looping required an average of approximately 2 hr of computer time on a high end S U N SPARC station for each configuration. The ‘C’ coded GA in this research averaged about 45 sec per configuration using a low end RS6000 workstation. The rule-based system was severely hampered by the presence of multiple objectives: at times the inference engine stalled because the search space exceeded the ranges of the rules. In this study the Pareto ranking technique efficiently ranked the multiple objective results. The G A always produced results that were uniformly minimized across the four objectives.

Compared to constrained, partially optimized designs: l 6

The GA consistently reduced the weight (by over 1.5 Ibs. per panel) and lowered manufacturing costs of the aircraft panels when compared t o panels designed by a simplified, pseudo-optimizing method that required certain variables to be fixed. These results are significant because the GA optimization was above to allocate compression and shear loading optimally across the panel skin, panel stiffeners, and panel frames without causing failure of any individual member. Combined compression and shear loading could not be optimized in the simplified method: all shear was assigned to the skin and all compression was assigned to the stiffeners.

V. METHODOLOGY

This is a summary of guidelines suggested to establish a baseline and a set of “preferred” configurations for multicriteria GA optimization research:

0 0

0

Initially use a n elitist strategy, gray code, and a generation gap > 0.5. Trial sizes of 3K and 6K both produced good results (in this study 82% of the unfiltered best average results occurred in the 6K range). Population size and Scaling windows For small pop. (50) and large trial size (6K) try a smaller scaling window (approx. 2). For large pop. (150-200) and large trial size (6K) try a larger scaling window (approx. 5) .

For a small pop. (50) a m-rate of 0.05 worked best. For a large pop. (150-200) a m-rate of 0.001 worked best. The c-rate should be experimented with over a range within 0.2 to 0.7 (Note that 0.2 and 0.7 are at the extremes tested in these experiments and the best results appear to be generated by 0.4 through 0.06).

0 Population size and Mutation rate

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A minimum of three different random seeds should be used to generate initial populations for each test case. (In this study it took 33 min to process 16 configurations over 3 random seeds). Different results were obtained when different seeds were used with the same configuration.

0 Multiple objective functions may require that a software filter be written into the objective function module of the GA to capture results that may not be optimal across all the objectives but meet the design constraints. In general there are usually several key objectives that will drive a multicriteria optimization. Taking the average results over the generations gives a good indication of how the GA is performing across all the objective functions. But a filter can be set to look at the results given by the top half of every generation so that it can capture the extreme (but not totally optimal) values at the GA searches the problem space. Depending on the requirements of the problem these filtered values may fit the solution better than the optimal across all the objective functions.

VI. CONCLUSIONS The genetic algorithm proved to be a fast, robust optimization method, the GA produced comparable results to rule-based optimization in a time factor of approximately 100 : 1, and the GA produced better results than a simplified pseudo- optimization method. Previous conclusions about GA parameter selection that were derived from single fitness function experiments can be applied to multiple fitness problems. The experimental data also identifies certain relationships between parameters that can exist and that should be exploited early in GA configurations.

GA configuration methodology is still evolving and needs to be tested in many different applications and experiments. Multicriteria ranking methods other than Pareto have also been successfully tried3 but the field of multicriteria optimization using GAS is still relatively young. The authors are presently investigating other ranking methods for use in multiobjective GA-based optimization and plan to further embellish the control parameter methodology presented here.

References

1. D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison-Wesley, Reading, MA. 1989.

2. J.H. Holland, Adaptation in Natural andArt@ciul Systems, University of Michigan Press, Ann Arbor, MI., 1975

3. J.D. Schaffer, Some Experiments in Machine Learning using Vector Evaluuted Genetic Algorithms, Ph.D. Dissertation, Vanderbilt University, Nashville, TN, 1984.

4. V. Pareto, Cours d’eronomie politique, Volumes 1 & 2, F. Rouge, Lausanne, Switzerland, 1896.

5. G.E. Liepins, M.R. Hilliard, “Genetic algorithms: Foundations and applications,” Annals of Operation Research, 21 31-58 (1989).

6. K.A. DeJong, “Adaptive systems design: A genetic approach,” IEEE Trans. Sys . , Man, Cyhern. SMC-10 9 566-574 (1980).

7. J.J. Grefenstette, “Optimization of control parameters for genetic algorithms,” IEEE Trans. Sys . , Man, Cybern SMC-16 I 122-128 (1986).

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8. W.M. Jenkins, “The genetic algorithm-or can we improve design by breeding?”, IEE Colloquium on Aritijical Intelligence in Civil Engineering, IEE Digest #011, London, UK, 1992, 1-4.

9. L.D. Davis, Ed., Handbook of Genetic Algorithms, Van Nostrand Reinhold Com- pany, New York, 1991.

10. J.J. Grefenstette, and L.D. Davis, D. Cerys, GENESIS and OOGA: Two Genetic Algorithm Systems, TSP Publications, Melrose, MA, 1991.

11. J.D. Schaffer, R.A. Caruana, Representation and hidden bias: Gray vs. binary coding for genetic algorithms, Proceedings of the 5th International Conference on Machine Learning, Morgan Kaufman, Los Altos, CA, 6/12-6114, 1988, 153-161.

12. K.A. DeJong, Analysis of the Behavior of a Class of Genetic Algorithms, Ph.D. Dissertation, University of Michigan, Ann Arbor, 1975.

13. K.A. DeJong, “Learning with genetic algorithms: An overview,” Mach. Learn., #3, 121-138 (October 1988).

14. J.D. Schaffer, R.A. Caruana, L.J. Eshelman, and R. Das, “A study of control parameters affecting online performance of genetic algorithms for funciton optimization,” Proceedings of the 3rd Internotional Conference on Genetic Algorithms, Morgan Kaufman, San Mateo, CA, 614-617, 1989.

15. C. Kassapoglou, A. DiNicola, J. Chou, “Structural evaluation of composite fuse- lage structure fabricated using a therm-X process,” Proceedings of the 46th AHS Forum, Washington, DC, 1990.

16. Personal conversations with C. Kassapoglou, Structures Engineer, Structures Re- search Section, Sikorsky Aircraft, Stratford, CT, 1994.

multicriteria optimization of aircraft panels: determining viable genetic algorithm configurations

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