A New Neuro-Fuzzy Adaptive Genetic Algorithm
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1 ec Journal of Electronic Science and Technology of China Vol.1 No.1 A New Neuro-Fuzzy Adaptive Genetic Algorithm ZHU Lili ZHANG Huanchun JING Yazhi (Faculty 302, Nanjing University of Aeronautics and Astronautics, Nanjing China) Abstract Novel neuro-fuzzy techniques are used to dynamically control parameter settings of genetic algorithms (GAs). The benchmark routine is an adaptive genetic algorithm (AGA) that uses a fuzzy knowledge-based system to control GA parameters. The self-learning ability of the cerebellar model ariculation controller(cmac) neural network makes it possible for on-line learning the knowledge on GAs throughout the run. Automatically designing and tuning the fuzzy knowledge-base system, neuro-fuzzy techniques based on CMAC can find the optimized fuzzy system for AGA by the renhanced learning method. The Results from initial experiments show a ynamic Parametric AGA system designed by the proposed automatic method and indicate the general applicability of the neuro-fuzzy AGA to a wide range of combinatorial optimization. Key words genetic algorithm; fuzzy logic control; CMAC neural network; adaptive parameter control It has been acknowledged that mutation probability, crossover probability and population size, etc. have significant impacts on genetic agrithms(ga) s performance [1~3]. Poor settings will cause damage to GA s running effectiveness [2,3]. Empirical and theoretical results about the effect of GA control parameter settings on GA performance and how to set them to improve GA performance have emerged in the literature. Generally, human expertise and knowledge on GA are vague or incomplete [3]. Fuzzy logic based tools are suitable for dealing with this type of knowledge and could implement the control strategies based on experience. However, finding robust control strategies for parameter setting is not a trivial task, since their interaction with GA performance is complex and the optimal ones are problem-dependent [4,5]. Automaticl earning mechanisms for obtaining the rule bases may be used for avoiding the previous problem. Obtaining the fuzzy knowledge-based system used in the fuzzy adaptive genetic algorithm can be accomplished by the automatic fuzzy design techniques, which can incorporate the expert s knowledge [5]. Neuro-fuzzy techniques have been perceived as a promising road to provide fuzzy control systems with the capacity to automatically self-tune their parameters [6]. Cerebellar model articulation controller(cmac) is a type of neural network introduced in Ref.[7] to explain information-processing characteristics of the cerebellum in the brain, which is the part of the brain controlling motor activities of the body. The fast convergence and high accuracy of the CMAC have proven it quite successful when used in on-line learning controlling systems [8~10]. The focus of this paper is to present a new neuro-fuzzy adaptive genetic algorithm that we developed. This work is divided into three parts: introduction of benchmark routine design, discussion of automatic fuzzy system design using CMAC and its optimization for fuzzy adaptive genetic algorithm, evaluation of the proposed neuro-fuzzy AGA. Received
2 64 1 Algorithm Scheme 1.1 Adaptive Genetic Algorithm (AGA) Based on a Fuzzy Rule-Based System In our preliminary studies, we developed a fuzzy AGA, which adopts 6 fuzzy logic controllers (FLC) for adapting control parameters (i.e. selective pressure, crossover probability and mutation probability) of a modified GA. Thus, we use it as the benchmark routine. We select the measure for FLCs inputs as following: one measure of population size (PS), one measure of generation number (GN), two phenotypic measure for both diversity and convergence (PCM1 and PCM2): PS = m/ M (1) GN = t / T (2) PCM1 = f() t fmin () t /[ fmax () t fmin () t ] (3) PCM2 = f( t) f( t i) / [ fmax ( t, t i) fmin ( t, t i) ] (4) where m is current population size, M is maximum population size, t is current generation number, T is the evolution terminating generation number, i is the adaptive control interval of generation. The mean fitness of generation t and t-i is signed as f (t) and f ( t i), respectively. Symbol f ( t min ) is the minimum fitness of generation t and f ( t max ) is the maximum. Moreover, f min ( t, t i) denotes the minimum fitness for both generation t and t-i and f max ( t, t i) denotes the maximum. Since our target of design is combinatorial optimization problems, we adopt a symbolic-coded standard GA as the fundamental algorithm. Major elements of the standard GA can be described as following: 1) symbolic coding representation of chromosomes for conveniently representing combination to be optimized. 2) linear rank-based fitness assignment is applied to non-negative objective function for minimization problems. 3) selective pressure is used to limiting the reproductive range for preventing premature convergence. 4) Stochastic Universal Sampling selection method for minimum spread and zero bias, and a generation gap for modification of Elitist Model. 5) Partly mapping Journal of Electronic Science and Technology of China Vol.1 crossover operation(pmx) for symbolic code series. 6) randomly two-point interchange mutation for transposition representation of combinatorial optimization problems. 7) a maximum generation number as the algorithm s stop criterion. The three FLCs, which inputs are PCM1 and PCM2, dynamically control Pc2, Pm2 and Ps respectively. The other three FLCs, which take PS and GN as their inputs, output Pc1, Pm1 and W1 as their on-line control variants. Here, the on-line crossover and mutation probability (Pc and Pm), and the adaptive selective pressure (SP) may be obtained by: Pc = 0.59 [ W1 Pc1 + (1 W1) Pc2] (5) Pm = [ W1 Pm1 + (1 W1) Pm2] (6) SP = 0.9 Ps (7) where Pc is typically chosen in the interval [0.4, 0.99], Pm is typically chosew in interval [0.001, 0.1], and SP s typical interval is [1.1, 2.0]. Apparently, the range of PS, GN and PCM1 is [0,1], and the range of PCM2 is [ 1,1]. The linguistic label set of these inputs is {Low, Medium, High}. The action interval of all the outputs, including Pc1, Pc2, Pm1, Pm2, Ps and W1 is [0, 1] and its associated linguistic labels shall be {Small, Medium, Big}. For each linguistic term, there is a triangular fuzzy set that defines its meaning. Therefore, these outputs could have up to 81 rules except Ps, which only have 9 rules. The defuzzy method of outputs is centroid. 1.2 Automatic esign of the Fuzzy Knowledge-Based System for ynamic Parametric AGA The 6FLCs-based AGA mentioned previously has shown some strong points on maintaining diversity, sustaining the convergence and controlling convergence speed. However, the definition of the fuzzy knowledge bases is still a challenging feature in the fuzzy control of the GAs. Although we do need to incorporate the expert s knowledge, how to avoid the casualness caused by the designer is another open problem. For these reasons, we decide to turn to a novel class of automatic techniques, neuro-fuzzy techniques, which have been demonstrated to be able to design fuzzy systems for other control applications.
3 No.1 ZHU Lili et al: A New Neuro-Fuzzy Adaptive Genetic Algorithm 65 In this section, we introduce the neuro-fuzzy method for automatically designing fuzzy systems. The dynamic parametric AGA provides a method to unify some of the notions resulting from our preliminary studies. The behaviors of GAs and the interrelations between the genetic operators are very complex, although there are many possible inputs and outputs for the FLCs, robust rule bases are not easily available. Automatic learning mechanisms may be the best choose. In other words, this is a problem of on-line learning qualitative knowledge. Fuzzy knowledge-based approach is well suited to representing qualitative knowledge, while artificial neural networks have been shown to be effective in solving on-line learning problems. Consequently, we do need a neuro-fuzzy system, the hybridization between the architectural and representational aspect of fuzzy knowledge and the learning mechanisms of neural networks. Presenting a striking contrast to other neural networks, cerebellar model neural networks such as the CMAC offer fast training time, guaranteed convergence and a small number of parameters. To develop the CMAC as a fuzzy knowledge on-line learning system, simple modifications and a new training scheme are applied to practical design. We adopt a method of reinforcement learning (RL). Usually, a system is said to be learning when it improves its performance based on a certain performance measure. Suppose that the performance measure is calculated as a function of the parameters of the learning system, which represent its current state. The tasks involving reinforcement learning are faced with a problem concerning the evaluation of their state. The only feedback the system received from its environment is the value of the performance measure corresponding to the current state of the system. The problem remains that the enhanced signal evaluates the performance but does not on its own provide the learning system with the direct information on how it should change itself to improve its performance. As for GAs, the situation is just the same. In addition, we adopt the associative RL, in which the task of the system is to associate different actions to different system states. In this case, the system receives some additional sensory information from the environment regarding the state of the system. 1.3 Automatic Fuzzy System esign Using CMAC Neural Network To incorporate CMAC neural network into fuzzy system design, we must first provide some modifications for fuzzy system parameterization. Next, we must find a suitable CMAC architecture and then determine a local look up table, which is generated by the coarse weights of the CMAC neural network for each layer. Fig.1 shows the framework of our neuro-fuzzy AGA. evahlation sysem inference engine futty rule-base application task expert knowledge and sample-base GA engine adaptive GA parameters GMAC neural network Fig.1 The framework of the proposed neuro-fuzzy AGA The fuzzy system parameterization used for GA control has explicit parameters for membership function shape and rule consequent parts, and implicit parameters for rule number. For our 6FLCs-based AGA, we restricte adjacent membership functions to fully overlap, which is the center of the previous membership function serves as the left base point of the next. We also constrain one membership function to have its center resting at the lower boundaries of the input range, and another one to have its center resting at the upper boundaries of the input range. By using this coding, only n 2 membership function centers
4 66 Journal of Electronic Science and Technology of China Vol.1 need to be specified, where n is the maximum number of partitions for a given dimension. The maximum number of rules in the system is equal to the number of possible combinations of input sets. Each rule has the possibility of generating a single rule for each output variable. Each of the possible rules in the system could have its consequent part and one output fuzzy set per output dimension. The next step is to find an appropriate CMAC architecture. This architecture should express the inference of fuzzy logic based on theoretic analysis of fuzzy control and CMAC with general basis functions. We propose using a multi-layer architecture of vector CMAC, which is a compound of several scalar CMAC architectures. Now suppose the mappings whose domain and N range are bounded subsets I R and Q R, respectively. The elements of the input space I will be denoted by p ; similarly, Q is the output space whose elements are denoted by z. Furthermore, if we define a mapping G( ) : I Q (8) for a scalar CMAC architecture, each p in I is assigned to a unique address v( p), which is the element of the virtual address space V. The value of the output function G ( V )( p) : I R depends on the numbers stored in the virtual array V. If we desire to construct a function that assigns to each p I a vector output z R, we simply construct a scalar CMAC architecture with associated function output d d G ( V )( p) for each output component z d and let the output of the vector CMAC architecture be defined by = ( ) G( V,, V )( p) G ( V )( p),, G ( V )( p) (9) Based on our preliminary studies, we designed the multi-layer vector CMAC as following (Fig.2 shows the architecture): The inputs flow into the output units by a forward feed method. The input of the network is a 4-dimension vector that represents the current state of the GA run, while the output is a 3-dimension vector that means the selected parameters under a stationary control policy. Units in layer 1 represent the centers of the membership functions for each input item. The stimuli functions of these units are their membership functions, respectively, i.e. + x c c x f1( x) = max 0,min, + c c c c (10) where x is the input measure, c is the center of the + membership function, c and c denotes the center of the previous and the next center, respectively. GN PS PCM1 PCM2 Inputs Layer1 Center of the membership function Lower boundary of the input range Upper boundary of the input range Rule Base 1 Rule Base 2 Layer2 Layer3 Pc1 Pm1 W1 Pm2 Pc2 Ps Layer4 Pc Pm SP Outputs Fig.2 The proposed multi-layer architecture of vector CMAC Layer 2 denotes two fuzzy rule bases.efine these two collections of linguistic IF-THEN rules by: R 1 : If GN and PS then Pc1 and Pm1 and W1 R 2 : If PCM1 and PCM2 then Pc2 and Pm2 and P s each rule has a rule code with a weight stamp according to its determination, i.e. 1 = μ( GN) μ( PS) μ( Pc1) μ( Pm1) μ( W1) (11) 2 = μ( PCM1) μ( PCM 2) μ( Pc2) μ( Pm2) μ( Ps) (12) where μ( ) is the value of the membership function. Thus, define the stimuli function of layer 2 by f 2 = ( 1, 2 )' (13) Layer 3 is the defuzzification layer. The fuzzy system used triangular membership functions, the min intersection operator and correlation-product inference procedure. efuzzification of the outputs was
5 No.1 ZHU Lili et al: A New Neuro-Fuzzy Adaptive Genetic Algorithm 67 performed using the fuzzy centroid method, with respect to equation (10), b( f1( x) ) f 3 ( y) = (14) ( f1( x) ) where y is the output item, b is the center of the fuzzy set. Layer 4 is the linear synthesis layer with reference to Eqs.(5)~(7). The utility of CMAC architectures arises from their ability to approximate mappings of the form as Eq. (8). The objective of the automatic fuzzy system design algorithm is to improve the performance of the GA it controls. Therefore, our training set has only one sample, the GA to be controlled. The AGA is considered as a discrete-time dynamical system. Recall that the goal of reinforcement learning is to find a policy for selecting action in a way that the selected sequence of actions will be optimal according to a certain performance measure. Furthermore, in one run, the system state of the AGA is actually a sequence of substates, on which the selection policy for actions is based. The substates transit from one to another by a transition probability, which is only dependent on the current substate and action. Our CMAC training law is then can be divided into two phases. Phase 1: given rule bases, modify the centers of the membership function for all inputs and outputs. Phase 2: based on the results of phase 1, modify rule bases with the fixed centers of the membership functions. 3 Presentation of Evaluation Result As an initial demonstration of the neuro-fuzzy adaptive genetic algorithm, we select a 15-city traveling salesman problem (TSP) as the training task. The TSP is a standard problem for testing new combinatorial optimization algorithm, since it shows almost every aspect of combinatorial optimization. Minimization experiments on the TSP are carried out in order to training the CMAC RL-controller for the 6FLCs-based AGA. To validate our findings, we compare a standard GA with our neuro-fuzzy AGA on the same TSP. We carried out the experiments using the following parameters: the population size is 80 individuals, the maximum population size is 160 individuals, the generation gap is 0.9, the maximum generation number is 500. And for the standard GA, the selective pressure is 2. The number of cities is 15, and we randomly generated the loci of cities on the perimeter of a rectangle, in which 4 loci are set to the 4 vertexes. Thus, we can easily evaluate the performance of both algorithms. Tab.1 shows the training results for rule bases and the centers of membership functions. Fig.4 shows the comparison of performance. The results show an improvement in GA performance, and may indicate that the neuro-fuzzy control method may be universally applicable to other combinatorial applications. Tab.1 The training results of the proposed neuro-fuzzy (a) The trained rule base 1 ITEMS PS GN Pc1 Pm1 W1 CENTERS S S M B M S M B M S RULE S B VB S M BASE 1 M S S M M M M B S S M B VB VS B B S S S B B M M VS M B B B VS B (b) The trained rule base 2 ITEMS PCM1 PCM2 Pc2 Pm2 Ps CENTERS S S S B S S M S B S RULE BASE S B M M B 2 M S S B S M M M M M M B B S B B S M M S B M B S B B B B S B
6 68 Journal of Electronic Science and Technology of China Vol.1 3 Conclusions Evaluations Fig.4 Comparison of performance We have proposed a novel neuro-fuzzy method for controlling GAs. To make dynamic parametric AGA accessible to all, we adopte an automatic fuzzy design technique, which is based on CMAC neural network. This technique is in turn used to design an optimal fuzzy system for GA control. The result is the AGA controlled by the neuro-fuzzy system that exhibited better performance than a standard GA. This indicates that the control knowledge found by the automatic design technique may be universally applicable to control GAs in other combinatorial optimization tasks. In addition, analyzing the optimal centers and the extracted rule bases may lead to better insights on the relationship of GA control parameters to GA performance. Research on performance evaluation and architecture modification for different tasks should be explored and more analysis needs to be done on the resulting systems. References [1] Grefenstette J J. Optimization of control parameters for genetic algorithms [J]. IEEE Transactions on Systems, Man and Cybernetics, 1986, 16(1): [2] Bäck T. The Interaction of Mutation Rate, Selection, and Self-Addaptation Within a Genetic Algorithm [C]. Parallel Problem Solving from Nature 2, Amsterdam:Elsevier Science Publishers, [3] eb K, Agrawal S. Understanding Interactions Among Genetic Algorithm Parameters [C]. Proceedings of the Fifth workshop on Foundations of Genetic Algorithm, Madison, WI, U.S.A., [4] Herrera F, Lozano M. Adaptation of Genetic Algorithm Parameters Based on Fuzzy Logic Controllers [C]. Genetic Algorithms and Soft Computing, Berlin:Physica-Verlag,
7 No.1 ZHU Lili et al: A New Neuro-Fuzzy Adaptive Genetic Algorithm [5] Lee M A, Takagi H. ynamic Control of Genetic Algorithms using Fuzzy Logic Techniques[C]. Proceeding of 5th International Conference on Genetic Algorithms (ICGA 93), Urbana-Champaign, IL, [6] Brown M, Harris C J. Neurofuzzy Adaptive Modeling and Control [M]. Hemel Hempstead Prentice Hall: 1994 [7] Albus J S. A new approach to manipulator control: the cerebellar model articulation controller (CMAC)[J]. Journal of ynamic System, Measurement and Control, Transactions of ASME, 1975, 97(3): [8] Lin C, Chiang C. Learning convergence of CMAC technique [J]. IEEE Transactions on Neural Networks, 1997, 8(6): [9] Hirashima Y, Liguni Y, Adachi N. An adaptive control system design using a memory based learning System [J]. International Journal of Control, 1997, 68(5): [10] Tham C L. Reinforcement learning of multiple tasks using a hierarchical CMAC architecture [J]. Robotics and Autonomous Systems, 1995, 15(4): Brief Introduction to Author(s) ZHU Lili( 朱力立 )was born in He is currently pursuing Ph.. degree in Automation Engineering at the Nanjing University of Aeronautics and Astronautics. His current research interests include: applications involving data fusion, fuzzy control and genetic algorithm. ali_zhu@163.com ZHANG Huanchun( 张焕春 )was born in He is now a Professor of Automation Engineering at the Nanjing University of Aeronautics and Astronautics. His research interests include: computer measurement and control, data fusion, advanced integration technology for measurement and control system. JING Yazhi( 经亚枝 )was born in She is an Associate Professor of Automation Engineering at the Nanjing University of Aeronautics and Astronautics. Her research interests include:computer measurement and control, virtual devices.
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