global optimization algorithms

Global OptimizationGeneral issues in global optimizationClassification of algorithmsThe DIRECT algorithmRelationship to EGOLipschitzian optimizationExploration and ExploitationApplication to High Speed Civil Transport

Global optimization issuesOptimization problem is NP-hardNo-free-lunch theorem (Wolpert and Macready)No single algorithm can do well on all problemsIf an algorithm is improved for one problem, it will suffer for others.Great opportunity for engineers to use problem knowledge to tailor algorithms.Big headache for journals because they get many worthless new algorithms.

Global Optimization

Global optimization algorithms by Thomas Weise

Global trendIf there is global trend, the only issue is not to be trapped in a local minimumIf there are many important basins there is no choice but to explore thoroughly.

global optimization algorithms

EGO and DIRECTEGO is an algorithm that relies on a surrogate to connect all the data in design space.DIRECT is an algorithm that does not try to exploit any correlation between the function value at one point and its value at nearby points.EGO will do well for functions with global trend and not too many deep local optima.DIRECT will do well for cheap functions with many local optima.We combine them by using DIRECT to calculate new sampling points for EGO.

DIRECT AlgorithmJones, D.R., Perttunen, C.D. and Stuckman, B.E. (1993), Lipschitzan optimization without the Lipschitz constant, Journal of Optimization Theory and Application 79, 157181.S. E. COX, R. T. HAFTKA, C. A. BAKER, B. GROSSMAN, W. H. MASON and L. T. WATSON A Comparison of Global Optimization Methods for the Design of a High-speed Civil Transport, Journal of Global Optimization 21: 415433, 2001.Matlab: http://www4.ncsu.edu/~ctk/Finkel_Direct/

Lipschitzian OptimizationOptimizer divides space into boxes and samples the vertices of eachOne box is further divided based on a predicted maximum rate of change of the function, K

DIRECTThe function value at the middle of each box and its largest dimension are used to determine potentially optimal boxesEach potentially optimal box is dividedLipschitzian optimization for all possible Lipschitz constants

DIRECT Box Division.

Exploration vs. ExploitationDIRECT uses convex hull of box sizes to balance exploitation vs. exploration.With enough function evaluations every region in design space will be explored.This is clearly not feasible for high dimensional spaces.Coxs paper compares DIRECT to repeated local optimization with random starts.

Optimization of a High Speed Civil Transport26 design variables defining the shape of the wing and fuselage and tail and the flight trajectoryLong range (NY Tokyo type flight) with 250 passengers for minimum takeoff gross weightConstraints on takeoff and landing maneuvers, engine out conditions.CFD analysis plus structural optimization needed for each design.

Results

Results

Results

Results

Deterministic

1.1 A Classification of Optimization Algorithms23

State SpaceSearch

Branch andBound

AlgebraicGeometry

Probabilistic

(Stochastic) Hill Climbing

RandomOptimization

Monte CarloAlgorithms

Soft Computing

EvolutionaryComputation (EC)

ArtificialIntelligence (AI)

ComputationalIntelligence (CI)

MemeticAlgorithms

SimulatedAnnealing (SA)

Tabu Search(TS)

ParallelTempering

StochasticTunneling

Direct MonteCarlo Sampling

EvolutionaryAlgorithms (EA)

HarmonicSearch (HS)

SwarmIntelligence (SI)

Ant ColonyOptimization (ACO)

Particle SwarmOptimization (PSO)

DifferentialEvolution (DE)

Standard GeneticProgramming

Linear GeneticPrograaming

Grammar GuidedGenetic Prog.

Figure 1.1: The taxonomy of global optimization algorithms.

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DOT Optimum ValuesOptimization #Weight

DIRECT

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26DV CompLFOPCV3DOTDIRECT0.0005FunctionCallsFunctionCallsFunctionCalls10.8035918360.81783119710130.813163144572481.83788428920.83107612420.813595140210680.810726258569516.59815035230.84316322680.81417943858720.812379174569925.60698278440.80834429700.81703277949580.812369232571922.94560908350.82847416200.81737139578730.812822145572159.97697039860.87859416470.8164087290.812707146571485.670.83378128620.81645731878690.812762144571520.12311200180.8346738640.8149955029870.812634144570496.85141599690.80431412960.81802084889300.81279145572614.594137557100.79566413500.81174764968430.812801145568223.354749402110.83846723220.81494785048670.812823145570463.495272211120.81210318360.818774610910180.812823145573142.227658673130.8442118090.82290449048730.812823145576033.143255196140.7916458320.81397215037260.812823145569780.505203333150.80948720790.82217036286420.812823145575519.253928897160.82630318090.8182432497870572770.274814807170.80551814850.817028395756571919.876527657180.86124811880.8190701404127741519573349.098308244190.81263816740.8165036156811571552.530926817200.80446635100.8244685914789577128.013998339210.82330417820.8196429649785573750.075416459220.825223313770.8169124481811571838.713701412230.86618413230.8176006471900572320.452958468240.82816916200.8132674049961569287.183396587250.85431823220.8149532765991570467.293561644260.80510128620.8169937823962571895.647590877270.83823615120.8138260303899569678.221235689280.81021922680.8158884803874571121.936197627290.81446614850.8137067128989569594.698957392300.81620912150.8159564575731571169.520251436310.80182131050.8191566449668573409.651460367320.8093219440.8150458483843570532.093799774330.85654214850.8153873574899570771.1502133340.80498611880.8179174084898572542.185889117350.81802835640.8179976721839572598.370462364360.81821235640.8143924527785570074.716896187370.84725517550.81290168451018569031.179156821380.81241440230.8154367116840570805.698142317390.81595613500.8127175597840568902.291808992400.81205723220.818871292573209.7410.81360537260.8215100689927575057.048226779420.80669425920.8156229101871570936.03709067430.8026631860.8164490334813571514.323389837440.84441416200.8196284043957573739.883022055450.83376815930.8172322813783572062.596890843460.84232611340.81741391171535572189.738168497470.80385228620.8116875111902568181.257773652480.81490838880.8197135779731573799.504548622490.79001835100.8110230124990567716.108677323500.83385717280.81832159081307572825.113594304510.86044632400.816380242959571466.169428669520.80030614850.8150302287896570521.160065352530.8237518370.8148386199926570387.033959184540.80987424300.8140379041928569826.53289462550.87679219980.81063824621268567446.772333597560.86855712690.8174569427723572219.859861847570.82904115930.8141665214840569916.564983016580.8020698640.815686672813570980.670377018590.85427519170.81673610841014571715.275848286600.8222631050.81292150431041569045.053012555610.79963825920.8141851642962569929.614963335620.88051538880.8182959923867572807.194601383630.80816313770.8211014045784574770.983170515640.8349097290.8138343591867569684.051396025650.80923234020.8135642871955569495.000996508660.81468916470.8166150606814571630.542401249670.8077515390.8152375611983570666.292750665680.83081422950.8161747959814571322.357159569690.82205939960.8184548153783572918.370736529700.82160621060.81625312541068571377.187800226710.82410212150.8173236291102572126.540276607720.84305315390.81603048311101571221.33818633730.820075314850.81551615491020570861.308415778740.84247719170.81714681661097572002.771605812750.8054137800.8217908415904575253.589058296760.81827819980.8193869994870573570.899593994770.80946618360.8134999069841569449.934838936780.80158241580.8148874027812570421.181887516790.83911215660.8207814603786574547.022176235800.81773210800.818032183900572622.528131354810.83620627270.8131113951075569177.97651146820.82570124840.822562713785575793.899084821830.83348617280.8147607271132570332.508928349840.85116913500.81026981791048567188.872530209850.802528260.8136224685929569535.727939185860.80579622410.8166284729757571639.931041187870.82370215660.8164574929957571520.245039813880.8192914040.8184974811839572948.23676424890.79806924030.81406617861364182569846.325013876900.83346314580.8151999765756570639.983522204910.87693923760.8170789973929571955.298131466920.86819215930.8190631053810573344.173692373930.80432728080.8167803461071571746.242167929940.8180297560.817087245755571961.071477898950.80203911070.8143144733988570020.131338204960.81418717820.81423119921045569961.83946402970.88445813770.8137847316784569649.312138846980.848909118880.8148818156809570417.270917645990.80206712420.8141686991814569918.0893886471000.86577114310.81250740771011568755.185418922TotalTotalTotalFunc.2188349080844103Min0.7900180.81026981790.810726Average0.8254420860.81627977750.8126178667WeightMin553012.6567188.872530209567508.2Average577809.4602571395.844248077568832.506666667

26DV Comp0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

DOT Optimum Values

PLOTDOT PlanformDIRECT Planform000011.542.38211.542.36623.083.67523.083.63534.624.53134.624.46346.165.06746.164.98557.7015.36557.7015.33765.935.49465.935.49269.2415.52169.2415.52680.7815.50880.7815.52392.3215.38492.3215.412103.865.406103.865.48115.45.507115.45.57126.945.517126.945.548138.485.451138.485.458150.025.51150.025.541175.0255.52175.0255.51200.035.038200.034.914207.935.119207.934.986225.0325.524225.0325.48250.0365.438250.0365.499275.0394.03275.0394.1173000300066.191671.685678.4199.41282.2329.41290.64812.82492.77812.824102.87616.235103.32416.236115.10419.647113.87119.648127.33323.059124.41723.06139.56126.471134.96326.472151.7929.882145.5129.884164.01833.294156.05633.296176.24636.706166.60336.707180.84140.118177.14940.119183.67843.529183.12443.531186.51546.941185.64946.943189.35150.353188.17550.355192.18853.765190.753.767195.02557.176193.22557.179197.86260.588195.7560.591200.69964198.27564.003213.21264209.07264.003212.98460.588209.66360.591212.75757.176210.25457.179212.52953.765210.84453.767212.30150.353211.43550.355212.07446.941212.02546.943211.84643.529212.61643.531211.62640.118213.20740.119212.87636.706213.79736.707214.77133.294214.38833.296216.66529.882214.97929.884218.55926.471215.56926.472220.45423.059216.16423.06222.34819.647218.38719.648224.24316.235221.36416.236226.13712.824224.34112.824228.0319.412227.3189.412229.9266230.2956197.75131.803196.74327.429201.38231.892200.37427.518205.01331.982204.00527.608208.53432.023207.52627.649212.05432.065211.04627.691215.3332.058214.32227.684218.60632.051217.59827.677221.77231.997220.76427.623224.93731.942223.92927.568224.93727.406223.92923.032221.77227.351220.76422.977218.60627.297217.59822.923215.3327.29214.32222.916212.05427.283211.04622.909208.53427.325207.52622.951205.01327.366204.00522.992201.38227.456200.37423.082197.75127.545196.74323.171197.75131.803196.74327.429193.15940.711193.58545.668196.7940.8197.21645.757200.42140.89200.84745.847203.94240.931204.36845.888207.46240.973207.88845.93210.73840.966211.16445.923214.01440.959214.4445.916217.1840.905217.60645.862220.34540.85220.77145.807220.34536.314220.77141.271217.1836.259217.60641.216214.01436.205214.4441.162210.73836.198211.16441.155207.46236.191207.88841.148203.94236.233204.36841.19200.42136.274200.84741.231196.7936.364197.21641.321193.15936.453193.58541.41193.15940.711193.58545.668000011.542.38411.542.36823.083.67823.083.63834.624.53434.624.46746.165.07146.164.98957.7015.3757.7015.34165.935.49865.935.49769.2415.52669.2415.5380.7815.51280.7815.52792.3215.38892.3215.417103.865.411103.865.484115.45.512115.45.574126.945.522126.945.553138.485.455138.485.462150.025.515150.025.545175.0255.524175.0255.515200.035.042200.034.918207.935.123207.934.99225.0325.529225.0325.484250.0365.443250.0365.503275.0394.034275.0394.12130003000000011.54-2.38211.54-2.36623.08-3.67523.08-3.63534.62-4.53134.62-4.46346.16-5.06746.16-4.98557.701-5.36557.701-5.33765.93-5.49465.93-5.49269.241-5.52169.241-5.52680.781-5.50880.781-5.52392.321-5.38492.321-5.412103.86-5.406103.86-5.48115.4-5.507115.4-5.57126.94-5.517126.94-5.548138.48-5.451138.48-5.458150.02-5.51150.02-5.541175.025-5.52175.025-5.51200.03-5.038200.03-4.914207.93-5.119207.93-4.986225.032-5.524225.032-5.48250.036-5.438250.036-5.499275.039-4.03275.039-4.1173000300066.191-671.685-678.419-9.41282.232-9.41290.648-12.82492.778-12.824102.876-16.235103.324-16.236115.104-19.647113.871-19.648127.333-23.059124.417-23.06139.561-26.471134.963-26.472151.79-29.882145.51-29.884164.018-33.294156.056-33.296176.246-36.706166.603-36.707180.841-40.118177.149-40.119183.678-43.529183.124-43.531186.515-46.941185.649-46.943189.351-50.353188.175-50.355192.188-53.765190.7-53.767195.025-57.176193.225-57.179197.862-60.588195.75-60.591200.699-64198.275-64.003213.212-64209.072-64.003212.984-60.588209.663-60.591212.757-57.176210.254-57.179212.529-53.765210.844-53.767212.301-50.353211.435-50.355212.074-46.941212.025-46.943211.846-43.529212.616-43.531211.626-40.118213.207-40.119212.876-36.706213.797-36.707214.771-33.294214.388-33.296216.665-29.882214.979-29.884218.559-26.471215.569-26.472220.454-23.059216.164-23.06222.348-19.647218.387-19.648224.243-16.235221.364-16.236226.137-12.824224.341-12.824228.031-9.412227.318-9.412229.926-6230.295-6197.751-31.803196.743-27.429201.382-31.892200.374-27.518205.013-31.982204.005-27.608208.534-32.023207.526-27.649212.054-32.065211.046-27.691215.33-32.058214.322-27.684218.606-32.051217.598-27.677221.772-31.997220.764-27.623224.937-31.942223.929-27.568224.937-27.406223.929-23.032221.772-27.351220.764-22.977218.606-27.297217.598-22.923215.33-27.29214.322-22.916212.054-27.283211.046-22.909208.534-27.325207.526-22.951205.013-27.366204.005-22.992201.382-27.456200.374-23.082197.751-27.545196.743-23.171197.751-31.803196.743-27.429193.159-40.711193.585-45.668196.79-40.8197.216-45.757200.421-40.89200.847-45.847203.942-40.931204.368-45.888207.462-40.973207.888-45.93210.738-40.966211.164-45.923214.014-40.959214.44-45.916217.18-40.905217.606-45.862220.345-40.85220.771-45.807220.345-36.314220.771-41.271217.18-36.259217.606-41.216214.014-36.205214.44-41.162210.738-36.198211.164-41.155207.462-36.191207.888-41.148203.942-36.233204.368-41.19200.421-36.274200.847-41.231196.79-36.364197.216-41.321193.159-36.453193.585-41.41193.159-40.711193.585-45.668000011.54-2.38411.54-2.36823.08-3.67823.08-3.63834.62-4.53434.62-4.46746.16-5.07146.16-4.98957.701-5.3757.701-5.34165.93-5.49865.93-5.49769.241-5.52669.241-5.5380.781-5.51280.781-5.52792.321-5.38892.321-5.417103.86-5.411103.86-5.484115.4-5.512115.4-5.574126.94-5.522126.94-5.553138.48-5.455138.48-5.462150.02-5.515150.02-5.545175.025-5.524175.025-5.515200.03-5.042200.03-4.918207.93-5.123207.93-4.99225.032-5.529225.032-5.484250.036-5.443250.036-5.503275.039-4.034275.039-4.12130003000

PLOT0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

DOT DesignDIRECT Design

Chart2002.3822.3663.6753.6354.5314.4635.0674.9855.3655.3375.4945.4925.5215.5265.5085.5235.3845.4125.4065.485.5075.575.5175.5485.4515.4585.515.5415.525.515.0384.9145.1194.9865.5245.485.4385.4994.034.11700669.4129.41212.82412.82416.23516.23619.64719.64823.05923.0626.47126.47229.88229.88433.29433.29636.70636.70740.11840.11943.52943.53146.94146.94350.35350.35553.76553.76757.17657.17960.58860.5916464.0036464.00360.58860.59157.17657.17953.76553.76750.35350.35546.94146.94343.52943.53140.11840.11936.70636.70733.29433.29629.88229.88426.47126.47223.05923.0619.64719.64816.23516.23612.82412.8249.4129.4126631.80327.42931.89227.51831.98227.60832.02327.64932.06527.69132.05827.68432.05127.67731.99727.62331.94227.56827.40623.03227.35122.97727.29722.92327.2922.91627.28322.90927.32522.95127.36622.99227.45623.08227.54523.17131.80327.42940.71145.66840.845.75740.8945.84740.93145.88840.97345.9340.96645.92340.95945.91640.90545.86240.8545.80736.31441.27136.25941.21636.20541.16236.19841.15536.19141.14836.23341.1936.27441.23136.36441.32136.45341.4140.71145.668002.3842.3683.6783.6384.5344.4675.0714.9895.375.3415.4985.4975.5265.535.5125.5275.3885.4175.4115.4845.5125.5745.5225.5535.4555.4625.5155.5455.5245.5155.0424.9185.1234.995.5295.4845.4435.5034.0344.1210000-2.382-2.366-3.675-3.635-4.531-4.463-5.067-4.985-5.365-5.337-5.494-5.492-5.521-5.526-5.508-5.523-5.384-5.412-5.406-5.48-5.507-5.57-5.517-5.548-5.451-5.458-5.51-5.541-5.52-5.51-5.038-4.914-5.119-4.986-5.524-5.48-5.438-5.499-4.03-4.11700-6-6-9.412-9.412-12.824-12.824-16.235-16.236-19.647-19.648-23.059-23.06-26.471-26.472-29.882-29.884-33.294-33.296-36.706-36.707-40.118-40.119-43.529-43.531-46.941-46.943-50.353-50.355-53.765-53.767-57.176-57.179-60.588-60.591-64-64.003-64-64.003-60.588-60.591-57.176-57.179-53.765-53.767-50.353-50.355-46.941-46.943-43.529-43.531-40.118-40.119-36.706-36.707-33.294-33.296-29.882-29.884-26.471-26.472-23.059-23.06-19.647-19.648-16.235-16.236-12.824-12.824-9.412-9.412-6-6-31.803-27.429-31.892-27.518-31.982-27.608-32.023-27.649-32.065-27.691-32.058-27.684-32.051-27.677-31.997-27.623-31.942-27.568-27.406-23.032-27.351-22.977-27.297-22.923-27.29-22.916-27.283-22.909-27.325-22.951-27.366-22.992-27.456-23.082-27.545-23.171-31.803-27.429-40.711-45.668-40.8-45.757-40.89-45.847-40.931-45.888-40.973-45.93-40.966-45.923-40.959-45.916-40.905-45.862-40.85-45.807-36.314-41.271-36.259-41.216-36.205-41.162-36.198-41.155-36.191-41.148-36.233-41.19-36.274-41.231-36.364-41.321-36.453-41.41-40.711-45.66800-2.384-2.368-3.678-3.638-4.534-4.467-5.071-4.989-5.37-5.341-5.498-5.497-5.526-5.53-5.512-5.527-5.388-5.417-5.411-5.484-5.512-5.574-5.522-5.553-5.455-5.462-5.515-5.545-5.524-5.515-5.042-4.918-5.123-4.99-5.529-5.484-5.443-5.503-4.034-4.12100

DOTDIRECT26 DV case

DIRECT

DOTweight =0.817831197count =1013weight =0.8135951402count =1068weight =0.8141794385count =872weight =0.8170327794count =958weight =0.8173713957count =873weight =0.8164573187count =869weight =0.814995502count =987weight =0.8180208488count =930weight =0.8117476496count =843weight =0.8149478504count =867weight =0.8187746109count =1018weight =0.8229044904count =873weight =0.8139721503count =726weight =0.8221703628count =642weight =0.8182432497count =870weight =0.817028395count =756weight =0.8190701404count =1277weight =0.8165036156count =811weight =0.8244685914count =789weight =0.8196429649count =785weight =0.8169124481count =811weight =0.8176006471count =900weight =0.8132674049count =961weight =0.8149532765count =991weight =0.8169937823count =962weight =0.8138260303count =899weight =0.8158884803count =874weight =0.8137067128count =989weight =0.8159564575count =731weight =0.8191566449count =668weight =0.8150458483count =843weight =0.8153873574count =899weight =0.8179174084count =898weight =0.8179976721count =839weight =0.8143924527count =785weight =0.8129016845count =1018weight =0.8154367116count =840weight =0.8127175597count =840weight =0.8215100689count =927weight =0.8156229101count =871weight =0.8164490334count =813weight =0.8196284043count =957weight =0.8172322813count =783weight =0.8174139117count =1535weight =0.8116875111count =902weight =0.8197135779count =731weight =0.8110230124count =990weight =0.8183215908count =1307weight =0.816380242count =959weight =0.8150302287count =896weight =0.8148386199count =926weight =0.8140379041count =928weight =0.8106382462count =1268weight =0.8174569427count =723weight =0.8141665214count =840weight =0.815686672count =813weight =0.8167361084count =1014weight =0.8129215043count =1041weight =0.8141851642count =962weight =0.8182959923count =867weight =0.8211014045count =784weight =0.8138343591count =867weight =0.8135642871count =955weight =0.8166150606count =814weight =0.8152375611count =983weight =0.8161747959count =814weight =0.8184548153count =783weight =0.8162531254count =1068weight =0.817323629count =1102weight =0.8160304831count =1101weight =0.8155161549count =1020weight =0.8171468166count =1097weight =0.8217908415count =904weight =0.8193869994count =870weight =0.8134999069count =841weight =0.8148874027count =812weight =0.8207814603count =786weight =0.818032183count =900weight =0.813111395count =1075weight =0.822562713count =785weight =0.814760727count =1132weight =0.8102698179count =1048weight =0.8136224685count =929weight =0.8166284729count =757weight =0.8164574929count =957weight =0.8184974811count =839weight =0.8140661786count =1364weight =0.8151999765count =756weight =0.8170789973count =929weight =0.8190631053count =810weight =0.816780346count =1071weight =0.817087245count =755weight =0.8143144733count =988weight =0.8142311992count =1045weight =0.8137847316count =784weight =0.8148818156count =809weight =0.8141686991count =814weight =0.8125074077count =1011

LFOPCV3weight =0.844617687# of calls=1161weight =0.8400561159# of calls=1323weight =0.839836476# of calls=1539weight =0.8391380551# of calls=999weight =0.8362063067# of calls=2727weight =0.8323292374# of calls=1647weight =0.8314723928# of calls=1620weight =0.8265232746# of calls=4131weight =0.8254193123# of calls=2619weight =0.8257010675# of calls=2484weight =0.8372579407# of calls=1485weight =0.8359205481# of calls=3132weight =0.8347034675# of calls=1755weight =0.8347646022# of calls=1755weight =0.8334855054# of calls=1728weight =0.8572385644# of calls=1674weight =0.855177069# of calls=1620weight =0.8526663458# of calls=2835weight =0.8516985801# of calls=1539weight =0.8511685386# of calls=1350weight =0.8737779309# of calls=1701weight =0.8028082497# of calls=3078weight =0.8039760277# of calls=2349weight =0.803565308# of calls=1998weight =0.8024996392# of calls=2862weight =0.806761462# of calls=1188weight =0.8066805704# of calls=1485weight =0.8072440136# of calls=1242weight =0.8062236332# of calls=1971weight =0.8057959377# of calls=2241weight =0.8275852575# of calls=2376weight =0.8264774414# of calls=3024weight =0.8258304857# of calls=1890weight =0.8263141308# of calls=972weight =0.8237015169# of calls=1566weight =0.824815197# of calls=2241weight =0.8251413018# of calls=1539weight =0.8229757192# of calls=2295weight =0.8233113877# of calls=1917weight =0.819290293# of calls=1404weight =0.7978609161# of calls=945weight =0.7994746904# of calls=945weight =0.7991958392# of calls=1620weight =0.799443787# of calls=1755weight =0.7980685299# of calls=2403weight =0.8566813684# of calls=1809weight =0.8493270335# of calls=1485weight =0.8370076561# of calls=1674weight =0.8361493778# of calls=1620weight =0.8334632717# of calls=1458weight =0.8750816127# of calls=918weight =0.8754537759# of calls=1944weight =0.8759684183# of calls=1890weight =0.8758954608# of calls=1809weight =0.8769392199# of calls=2376weight =0.8699480946# of calls=2538weight =0.8696694228# of calls=1593weight =0.8737638001# of calls=1458weight =0.8704400406# of calls=1674weight =0.8681919378# of calls=1593weight =0.8043102822# of calls=2187weight =0.8048156486# of calls=1431weight =0.8048424745# of calls=702weight =0.8041501101# of calls=3348weight =0.8043269046# of calls=2808weight =0.8156954608# of calls=1971weight =0.8155525767# of calls=1755weight =0.8150607857# of calls=1566weight =0.8146274944# of calls=1674weight =0.8180288601# of calls=756weight =0.8022113075# of calls=2430weight =0.8077144091# of calls=2160weight =0.801914744# of calls=2619weight =0.8043364814# of calls=1728weight =0.8020393244# of calls=1107weight =0.8255482598# of calls=2349weight =0.8253166798# of calls=1188weight =0.8132173036# of calls=1134weight =0.8161805702# of calls=2754weight =0.8141865355# of calls=1782weight =0.9002351938# of calls=1944weight =0.894000466# of calls=1620weight =0.8843036789# of calls=2187weight =0.882942689# of calls=4050weight =0.8844579242# of calls=1377weight =0.8740188931# of calls=702weight =0.8734968825# of calls=1377weight =0.8717385621# of calls=1566weight =0.8519995603# of calls=1998weight =0.8489089873# of calls=1188weight =0.8038038941# of calls=1674weight =0.8021568604# of calls=1107weight =0.8019628547# of calls=1512weight =0.802043928# of calls=1242weight =0.8020673895# of calls=1242weight =0.8769983433# of calls=1350weight =0.8667641085# of calls=2079weight =0.8671075564# of calls=2214weight =0.8676705835# of calls=1701weight =0.8657713568# of calls=1431

Sheet1DOT PlanformDIRECT 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DOT DesignDIRECT Design

26DV CompLFOPCV3DOTDIRECTFunctionCallsFunctionCallsFunctionCalls10.8035918360.81783119710130.81316314420.83107612420.813595140210680.81072625830.84316322680.81417943858720.81237917440.80834429700.81703277949580.81236923250.82847416200.81737139578730.81282214560.87859416470.8164087290.81270714670.83378128620.81645731878690.81276214480.8346738640.8149955029870.81263414490.80431412960.81802084889300.81279145100.79566413500.81174764968430.812801145110.83846723220.81494785048670.812823145120.81210318360.818774610910180.812823145130.8442118090.82290449048730.812823145140.7916458320.81397215037260.812823145150.80948720790.82217036286420.812823145160.82630318090.8182432497870170.80551814850.817028395756180.86124811880.8190701404127741519190.81263816740.8165036156811200.80446635100.8244685914789210.82330417820.8196429649785220.825223313770.8169124481811230.86618413230.8176006471900240.82816916200.8132674049961250.85431823220.8149532765991260.80510128620.8169937823962270.83823615120.8138260303899280.81021922680.8158884803874290.81446614850.8137067128989300.81620912150.8159564575731310.80182131050.8191566449668320.8093219440.8150458483843330.85654214850.8153873574899340.80498611880.8179174084898350.81802835640.8179976721839360.81821235640.8143924527785370.84725517550.81290168451018380.81241440230.8154367116840390.81595613500.8127175597840400.81205723220.818871292410.81360537260.8215100689927420.80669425920.8156229101871430.8026631860.8164490334813440.84441416200.8196284043957450.83376815930.8172322813783460.84232611340.81741391171535470.80385228620.8116875111902480.81490838880.8197135779731490.79001835100.8110230124990500.83385717280.81832159081307510.86044632400.816380242959520.80030614850.8150302287896530.8237518370.8148386199926540.80987424300.8140379041928550.87679219980.81063824621268560.86855712690.8174569427723570.82904115930.8141665214840580.8020698640.815686672813590.85427519170.81673610841014600.8222631050.81292150431041610.79963825920.8141851642962620.88051538880.8182959923867630.80816313770.8211014045784640.8349097290.8138343591867650.80923234020.8135642871955660.81468916470.8166150606814670.8077515390.8152375611983680.83081422950.8161747959814690.82205939960.8184548153783700.82160621060.81625312541068710.82410212150.8173236291102720.84305315390.81603048311101730.820075314850.81551615491020740.84247719170.81714681661097750.8054137800.8217908415904760.81827819980.8193869994870770.80946618360.8134999069841780.80158241580.8148874027812790.83911215660.8207814603786800.81773210800.818032183900810.83620627270.8131113951075820.82570124840.822562713785830.83348617280.8147607271132840.85116913500.81026981791048850.802528260.8136224685929860.80579622410.8166284729757870.82370215660.8164574929957880.8192914040.8184974811839890.79806924030.81406617861364900.83346314580.8151999765756910.87693923760.8170789973929920.86819215930.8190631053810930.80432728080.8167803461071940.8180297560.817087245755950.80203911070.8143144733988960.81418717820.81423119921045970.88445813770.8137847316784980.848909118880.8148818156809990.80206712420.81416869918141000.86577114310.81250740771011TotalTotalTotalFunc.2188349080843921Min0.7900180.81026981790.810726Average0.8254420860.81627977750.8126178667WeightMin553012.6567188.872530209567508.2Average577809.4602571395.844248077568832.506666667