Neural Networks and Genetic Algorithms Multiobjective acceleration
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Neural Networks and Genetic Algorithms Multiobjective acceleration Neural Networks and Genetic Algorithms Multiobjective acceleration Presentation Transcript

  • 1 A Hybrid Multi-Objective Evolutionary Algorithm Using an Inverse Neural Network A. Gaspar-Cunha(1), A. Vieira(2), C.M. Fonseca(3) (1) IPC- Institute for Polymers and Composites, Dept. of Polymer Engineering, University of Minho, Guimarães, Portugal (2) ISEP and Computational Physics Centre, Coimbra, Portugal (3) CSI- Centre for Intelligent Systems, Faculty of Science and Technology, University of Algarve, Faro, Portugal HYBRID METAHEURISTICS (HM 2004) ECAI 2004, Valencia, Spain August, 2004 Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 2 INTRODUCTION Most real optimization problems are multiobjective Example: Simultaneous minimization of the cost and maximization of the performance of a specific system Dominated solution Cost Single optimum (maximal performance) Performance Single optimum (minimal cost) Instituto Superior de Engenharia do Porto Multiple optima (both objectives optimized) PARETO FRONTIER (set of non-dominated solutions) Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 3 INTRODUCTION Computation time required to evaluate the solutions Start Engineering problems: Initialise Population i=0 Black Box Numerical modelling routines • Finite elements • Finite differences • Finite volumes • etc Evaluation Assign Fitness Fi Convergence criterion satisfied? i=i+1 no Selection HIGH COMPUTATION TIMES yes Stop Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Recombination Dept. Polymer Engineering University of Minho
  • 4 INTRODUCTION OBJECTIVES: • Develop an efficient multi-objective optimization algorithm • Reduce the number of evaluations of objective functions necessary • Compare performance with existing algorithms Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 5 CONTENTS • Multi-Objective Evolutionary Algorithm (MOEA) • Artificial Neural Networks (ANN) • Hybrid Multi-Objective Algorithm (MOEA-IANN) • Results and Discussion • Conclusions Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA6 How to deal with multiple criteria (or objectives)? Single objective (for example, weighted sum)  0 ≤ wj ≤ 1   ∑ wj = 1   0 ≤ Fj ≤ 1 0 ≤ FOi ≤ 1  q FOi = ∑ w j F j j =1 Decision made before the search Pareto Frontier Multiobjective optimization Decision made after the search Objective 2 200 1 190 180 2 170 5 6 3 4 160 500 Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve 1000 Objective 1 1500 Dept. Polymer Engineering University of Minho 2000
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA7 Basic functions of a MOEA: Maintaining a diverse nondominated set (Density estimation) Density C2 Archiving Fitness C1 Instituto Superior de Engenharia do Porto Preventing nondominated solutions from being lost (Elitist population - archiving) Guiding the population towards the Pareto set (Fitness assignment) Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA8 Reduced Pareto Set G.A. with Elitism (RPSGAe) Start RPSGAe sorts the population individuals in a number of pre-defined ranks using a clustering technique, in order to reduce the number of solutions on the efficient frontier. Initialise Population i=0 a) Rank the individuals using a clustering Evaluation algorithm; b) Calculate Assign Fitness Fi i=i+1 the fitness using a ranking function; c) Copy the best individuals to the external population; Convergence criterion satisfied? no Selection yes Stop Instituto Superior de Engenharia do Porto Recombination d) If the external population becomes full: - Apply the clustering algorithm to the external population; - Copy the best individuals to the internal population; Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA9 Clustering algorithm example NR = r N=15; Nranks=3 N ranks r=2; NR=10 r=1; NR=5 C2 N C2 1 1 2 1 12 1 2 1 2 1 1 2 1 C1 1 C1 Gaspar-Cunha, A., Covas, J.A. - RPSGAe - A Multiobjective Genetic Algorithm with Elitism: Application to Polymer Extrusion, in Metaheuristics for Multiobjective Optimisation, Lecture Notes in Economics and Mathematical Systems, Gandibleux, X.; Sevaux, M.; Sörensen, K.; T'kindt, V. (Eds.), Springer, 2004. Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA10 Clustering algorithm example r=3; NR=15 Fitness - Linear ranking : C2 1 2( SP − 1) ( N + 1 − i ) FOi = 2 − SP + N 23 12 3 2 1 3 2 FO(1) = 2.00 FO(2) = 1.87 31 23 FO(3) = 1.73 1 C1 RPSGAe • Number of Ranks - Nranks Parameters: • Limits of indifference of the clustering algorithm - limit • N. of individuals copied to the external population - Next Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA11 Reduced Pareto Set G.A. with Elitism (RPSGAe) Internal population Next Internal population (Generation n) External population External population (Generation n) Generation 1 Generation 2 Generation 3 Generation 4 Next Generation 5 Generation n Order of the RPSGAe: O(Nranks q N2) Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA12 How the basic functions are accomplished in the RPSGAe : 1. Guiding the population towards the Pareto set Fitness assignment: ranking function based reduction of the Pareto Set on the 2. Maintaining a diverse nondominated set Density estimation: ranking function based on the reduction of the Pareto Set 3. Preventing nondominated solutions from being lost Elitist population: periodic copy of the best solutions (to the main population), selected with the method of Pareto set reduction Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 13 ARTIFICIAL NEURAL NETWORKS – ANN Artificial Neural Networks • ANN implemented by a Multilayer Preceptron is a flexible scheme capable of approximating an arbitrary complex function; • The ANN builds a map between a set of inputs and the respective outputs; • A feed-forward neural network consists of an array of input nodes connected to an array of output nodes through successive intermediate layers; • • Each connection between nodes has a weight, initially random, which is adjusted during a training process; The output of each node of a specific layer is a function of the sum on the weighted signals coming from the previous layer; Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Input Layer Hidden Layer Output Layer P1 C1 P2 C2 ... ... Pi Cj Dept. Polymer Engineering University of Minho
  • 14 HYBRID MULTI-OBJECTIVE ALGORITHM Two possible approachs to reduce the computation time 1. During evaluation – Some solutions can be evaluated using an approximate function, such as Fitness Inheritance, Artificial Neural Networks, etc (this reduce the number of exact evaluations necessary). 2. During recombination – Some individuals can be generated using more efficient methods (this produce a fast approximation to the optimal Pareto frontier, thus the number of generations is reduced). Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 15 HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN Use of ANN to “Evaluate” some Solutions Start Artificial Neural Network Initialise Population i=0 Parameters to optimise P1 Convergence criterion satisfied? i=i+1 no P2 C2 ... Pi Assign Fitness Fi C1 ... Evaluation Criteria Cj Selection yes Stop Instituto Superior de Engenharia do Porto Recombination Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 16 HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN Use of ANN to “Evaluate” some Solutions – Method A Proposed by K. Deb et. al Neural Network learning using some solutions of the p generations Neural Network learning using some solutions of the p generations p generations r generations p generations r generations RPSGA with RPSGA with Neural exact Network function evaluation evaluation RPSGA with RPSGA with Neural exact Network function evaluation evaluation Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve ... ... p generations RPSGA with exact function evaluation Dept. Polymer Engineering University of Minho
  • 17 HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN Use of ANN to “Evaluate” some Solutions – Method B Neural Network learning using some solutions of the p generations Neural Network learning using some solutions of the p generations eNN > allowed error p generations r generations RPSGA with RPSGA with: exact • All solutions function (N) evaluated evaluation by Neural Network • M evaluated by exact function Instituto Superior de Engenharia do Porto M e NN = S j =1 (C NN i, j i =1 ∑ ∑ − Ci , j ) 2 S M eNN > allowed error p generations r generations RPSGA with RPSGA with: exact • All solutions function (N) evaluated evaluation by Neural Network • M evaluated by exact function Faculty of Science and Technology University of Algarve ... ... p generations RPSGA with exact function evaluation Dept. Polymer Engineering University of Minho
  • HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN 18 Use of an Inverse ANN as “Recombination” operator Start Recombination operators: Initialise Population • Crossover i=0 • Mutation • Inverse ANN (IANN) Evaluation Criteria Variables C1 V1 C2 V2 Selection ... ... Recombination Cq VM Assign Fitness Fi Convergence criterion satisfied? i=i+1 no yes Stop Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 19 HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN Set of Solutions Generated with the IANN Selection of n+q solutions from the • 3.q extreme solutions • n interior solutions For j = 1, ..., q (where, q is the number of criteria) : ∆C2 c Criterion 2 present population to generate: b e1 C j = C 'j + ∆C j Points 1, 2, …, n: a 1 2 3 a 4 e2 b c Criterion 1 ∆C1 Point ej to a: C j = C j + ∆C j ' Point ej to b: C j ( j =i ) = C 'j ∧ C j ( j ≠i ) = C 'j + ∆C j Point ej to c: C j ( j =i ) = C j − ∆C j ' Instituto Superior de Engenharia do Porto ∧ C j ( j ≠i ) = C 'j + ∆C j Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN Set of Solutions Generated with the IANN Use of IANN to generate new solutions c e1 a 1 2 3 a 4 e2 Parameter 2 Criterion 2 ∆C2 b b c Criterion 1 ∆C1 Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve 2 1 b a c 4 e1 a 3 e2 b c Parameter 1 Dept. Polymer Engineering University of Minho 20
  • HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN 21 MOEA-IANN Algorithm Parameters  Number of Ranks - Nranks  N. of individuals copied to the external population - Next  Limits of indifference of the clustering algorithm – limit  Criteria variation at beginning - ∆Cinit  Criteria variation at end - ∆Cf  N. of generations which individuals are used to train the IANN – Ngen  Rate of individuals generated with the IANN – IR Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 22 RESULTS AND DISCUSSION – Test problems K. Deb et. al - Test Problem Generator Minimize f1 ( x1 ) , Minimize f 2 ( x2 ) ,   Minimize f q −1 ( xq −1 ), f q ( x ) = g ( xq ) h( f1 ( x1 ) , f 2 ( x2 ) ,  , f q −1 ( xq −1 ), g ( xq ) ), Minimize Subject to x xi ∈ ℜ i , for i = 1, 2,  , q. 2 Criteria 2C-ZDT1 (Convex): M = 30; xi ∈ [0, 1] 1.00 f1 ( x1 ) = x1  = g × 1 −   where, g ( x 2 , , x M Instituto Superior de Engenharia do Porto 0.60 f1   g  ) = 1+ 9 ∑ M i =2 f2 f 2 ( x 2 , , x M ) 0.80 0.40 0.20 xi M −1 Faculty of Science and Technology University of Algarve 0.00 0 0.2 0.4 0.6 0.8 f1 Dept. Polymer Engineering University of Minho 1
  • 23 RESULTS AND DISCUSSION – Test problems 2 Criteria 2C-ZDT2 (Non-convex): M = 30; xi ∈ [0, 1] 1.00 f1 ( x1 ) = x1   f1  2  = g × 1 −    g      where, g ( x 2 , , x M ) = 1 + 9 ∑ M i=2 0.60 f2 f 2 ( x 2 , , x M ) 0.80 0.40 0.20 xi 0.00 M −1 0 0.4 0.6 1 1.00 f1 ( x1 ) = x1 0.60 f1  −  f 1  sin (10 π f1 )   g  g    ) = 1+ 9 ∑ M 0.20 f2  f 2 ( x 2 ,  , x M ) = g × 1 −   -0.200 xi 0.4 0.6 -0.60 M −1 0.2 -1.00 i=2 f1 Instituto Superior de Engenharia do Porto 0.8 f1 2C-ZDT3 (Discrete): M = 30; xi ∈ [0, 1] where, g ( x 2 , , x M 0.2 Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho 0.8 1
  • 24 RESULTS AND DISCUSSION – Test problems 2 Criteria 2C-ZDT4 (Multimodal): M = 10; x1 ∈ [0, 1]; xi ∈ [-5, 5] 1.40 1.20 f1 ( x1 ) = x1  = g × 1 −   f1   g  f2 f 2 ( x 2 , , x M ) 1.00 0.80 0.60 ( where, g ( x 2 , , x M ) = 1 + 10 ( M − 1) + ∑i = 2 xi2 − 10 cos( 4 π xi ) M 0.40 ) 0.20 0.00 0 0.2 0.4 0.6 0.8 1 0.6 0.8 1 f1 2C-ZDT6 (Non-uniform): M = 10; xi ∈ [0, 1] 1.00 f1 ( x1 ) = 1 − exp(−4 x1 ) sin 6 (6 π x1 )   f1  2  = g × 1 −    g      where, g ( x 2 , , x M ) Instituto Superior de Engenharia do Porto  ∑M xi = 1 + 9 i = 2  M −1  0.60 f2 f 2 ( x 2 , , x M ) 0.80 0.40     0.25 Faculty of Science and Technology University of Algarve 0.20 0.00 0 0.2 0.4 f1 Dept. Polymer Engineering University of Minho
  • 25 RESULTS AND DISCUSSION – Test problems 3 Criteria 3C-ZDT1 (Convex): M = 30; xi ∈ [0, 1] f1 ( x1 ) = x1 1.0 f 3 ( x 3 , , x M )  = g × 1 −   where, g ( x3 , , x M f1 f 2 g ) = 1+ 9 ∑     M i =3 f3 f 2 ( x2 ) = x2 0.5 0.0 0.2 0.4 xi 0.6 f2 M −1 0.4 0.6 0.8 0.8 1.0 0.2 0.0 0.0 f1 1.0 3C-ZDT2 (Non-convex): M = 30; xi ∈ [0, 1] f1 ( x1 ) = x1 1.0 f 3 ( x3 ,  , x M )   f f 2  = g × 1 −  1 2     g       where, g ( x3 ,  , xM Instituto Superior de Engenharia do Porto ) =1+ 9 ∑ M i =3 xi M −1 Faculty of Science and Technology University of Algarve f3 f 2 ( x2 ) = x 2 0.5 0.0 0.2 0.4 0.6 f2 0.4 0.6 0.8 0.8 1.0 1.0 Dept. Polymer Engineering University of Minho f1 0.2 0.0 0.0
  • 26 RESULTS AND DISCUSSION – Test problems 3 Criteria 3C-ZDT3 (Discrete): M = 30; xi ∈ [0, 1] f1 ( x1 ) = x1 1.00 f 2 ( x2 ) = x2 0.75 0.50  f1 f 2  f1 f 2    sin (10 π f1 f 2 )  −   g  g    0.00 0.0 0.2 0.4 0.6 ∑i = 3 x i M f2 where, g ( x3 , , x M ) = 1 + 9 0.25 f3  f 3 ( x 3 ,  , x M ) = g × 1 −  M −1 -0.25 0.8 1.0 1.00 1.0 0.8 0.6 0.4 0.2 f1 1.0 0.75 1.000 0.8 0.8125 0.50 0.6250 0.25 0.4375 f3 0.6 0.2500 f2 0.00 0.06250 0.4 -0.1250 -0.25 -0.50 0.0 0.2 0.4 f1 0.6 0.6 0.4 0.8 0.2 1.0 Instituto Superior de Engenharia do Porto f2 0.8 1.0 -0.3125 0.2 0.0 0.0 -0.5000 0.2 0.4 0.6 0.8 1.0 f1 0.0 Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho 0.0 -0.50
  • 27 RESULTS AND DISCUSSION – Test problems 3 Criteria 3C-ZDT4 (Multimodal): M = 10; x1,2 ∈ [0, 1]; xi ∈ [-5, 5] 18 f1 ( x1 ) = x1 16 14 f 2 ( x2 ) = x2 f 3 ( x 3 , , x M ) 12  = g × 1 −   f1 f 2 g f3 10     8 6 4 ( where, g ( x3 , , x M ) = 1 + 10 ( M − 1) + ∑i =3 xi2 − 10 cos( 4 π xi ) M ) 0.0 0.2 0.4 0.6 f2 3C-ZDT6 (Non-uniform): M = 10; xi ∈ [0, 1] 0.4 0.6 0.8 0.8 1.0 0.2 f1 1.0 f 1 ( x1 ) = 1 − exp(−4 x1 ) sin 6 (6 π x1 ) 1.0 f 2 ( x 2 ) = 1 − exp(−4 x 2 ) sin 6 (6 π x 2 )   f f 2  = g × 1 −  1 2     g       where, g ( x3 , , x M ) Instituto Superior de Engenharia do Porto 0.8  ∑ xi = 1 + 9 i =3  M −1  M 0.6 f3 f 3 ( x 3 , , x M ) 2 0.0 0 0.4     0.25 Faculty of Science and Technology University of Algarve 0.2 0.0 0.2 0.4 0.6 f2 0.4 0.6 0.8 0.8 1.0 1.0 Dept. Polymer Engineering University of Minho f1 0.2 0.0 0.0
  • 28 RESULTS AND DISCUSSION – Metrics Hypervolume Metric (Zitzler and Thiele - 1998) This metric calculates the dominated space volume, enclosed by the nondominated points and the origin. S metric: Volume of the space dominated by the set of objective vectors C2 Hypervolume C1 Criteria C1 and C2 to maximize Instituto Superior de Engenharia do Porto However, is not possible to say that one set is better than other Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 29 RESULTS AND DISCUSSION – Algorithm Parameters Influence of algorithm parameters on performance Parameter Tested values(*) Best results Influence Selected limit 0.01; 0.05; 0.1; 0.2 [0.01; 0.2] Small 0.01 ∆ Cinit 0.3; 0.4; 0.5; 0.6 [0.3; 0.5] Small 0.5 ∆ Cf 0.0; 0.1; 0.2; 0.3 [0.0; 0.3] Small 0.2 Ngen 5; 10; 15; 20 [5; 10] Small 5 IR 0.35; 0.50; 0.65; 0.80 [0.35; 0.8] Small 0.8 (*) 5 runs for each tested parameter value • The influence of the algorithm parameters on its performance is very small. • Each optimisation run was carried out 21 times using the algorithm parameters selected and different seed values. Algorithm Parameters: - N = 100 - Ne = 100 - Nranks = 30 - Next = 3N/Nranks = 10 - cR = 0.8 - mR = 0.05 Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • 30 RESULTS AND DISCUSSION – Method B Use of ANN to “Evaluate” some Solutions – Method B S metric, 22000 evaluations Number of evaluations Test problem Method B RPSGAe Decrease (%) Method B RPSGAe Decrease (%) ZDT1 0.851 0.849 0.24 10000 19000 47.4 ZDT2 0.786 0.773 1.68 15300 22000 30.5 ZDT3 2.736 2.554 7.13 18000 22000 18.2 ZDT4 0.1116 0.0807 38.29 5000 22000 77.3 ZDT6 0.599 0.571 4.90 12500 22000 43.2 • The S metric after 22000 evaluations decrease when Method B is used • The number of evaluations necessary to attain identical level of the S metric decreases considerably when Method B is used Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho
  • RESULTS AND DISCUSSION – 2 Criteria Test Problems MOEA - Inverse ANN 2C-ZDT1 1 S metric 0.8 0.6 0.4 IANN RPSGAe 0.2 0 0 50 100 150 Generations 200 250 300 • The Inverse ANN approach has the largest improvement during the first generations, i.e., when the solution is far from the optimum; Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho 31
  • 32 RESULTS AND DISCUSSION – 2 Criteria Test Problems MOEA - Inverse ANN 2C-ZDT2 2C-ZDT3 2.5 S metric 3 0.8 S metric 1 0.6 0.4 IANN RPSGAe 0.2 2 1.5 1 0 0 0 100 Generations 200 300 0 2C-ZDT4 0.15 100 Generations 200 300 2C-ZDT6 0.8 0.6 0.1 S metric S metric IANN RPSGAe 0.5 0.05 IANN RPSGAe 0 0.4 0.2 IANN RPSGAe 0 0 Instituto Superior de Engenharia do Porto 100Generations 200 300 Faculty of Science and Technology University of Algarve 0 100 Generations 200 Dept. Polymer Engineering University of Minho 300
  • RESULTS AND DISCUSSION – 3 Criteria Test Problems MOEA - Inverse ANN 3C-ZDT1 0.8 S metric 0.6 0.4 IANN 0.2 RPSGAe 0 0 50 100 150 200 250 300 Generations Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho 33
  • 34 RESULTS AND DISCUSSION – 3 Criteria Test Problems MOEA - Inverse ANN 3C-ZDT2 0.8 1.5 S metric S metric 0.6 0.4 0.2 IANN RPSGAe 1.2 0.9 0.6 IANN RPSGAe 0.3 0 0 0 100 Generations 200 300 0 3C-ZDT4 0.06 100 Generations 200 300 3C-ZDT6 0.4 0.3 0.04 S metric S metric 3C-ZDT3 1.8 0.02 0.2 0.1 IANN RPSGAe 0 IANN RPSGAe 0 0 Instituto Superior de Engenharia do Porto 100Generations 200 300 Faculty of Science and Technology University of Algarve 0 100 Generations 200 Dept. Polymer Engineering University of Minho 300
  • 35 CONCLUSIONS • Algorithm parameters have a limited influence on its performance • Good performance of the proposed algorithm • The number of generations needed to reach identical level of performance is reduced thus, the computation time is reduced by more than 50%. • Most improvements of the IANN approach accomplished during the first generations Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho are
  • 36 ANY QUESTION!? Instituto Superior de Engenharia do Porto Faculty of Science and Technology University of Algarve Dept. Polymer Engineering University of Minho