1 Utilizing Lamarckian Evolution and the Baldwin Effect in Hybrid Genetic Algorithms

Utilizing Lamarckian Evolution and the Baldwin Effect inHybrid Genetic AlgorithmsChristopher R. Houck - chouck@Jeffery A. Joines - jjoine@Michael G. Kay1 - kay@Box 7906Dept. of Industrial EngineeringNorth Carolina State UniversityRaleigh N.C. 27695-7906(919) 515-2008(919) 515-1543 FAXAbstract — Genetic algorithms(GA) are very efficient at exploring the entire search space; how-ever, they are relatively poor at finding the precise local optimal solution in the region at which the algorithm converges. Hybrid genetic algorithms are the combination of improvement proce-dures, usually working as evaluation functions, and genetic algorithms. There are two basic strat-egies in using hybrid GAs, Lamarckian and Baldwinian learning. Traditional schema theory does not support Lamarckian learning, i.e., forcing the genetic representation to match the solution found by the improvement procedure. However, Lamarckian learning does alleviate the problem of multiple genotypes mapping to the same phenotype. Baldwinian learning uses improvement procedures to change the fitness landscape, but the solution that is found is not encoded back into the genetic string. This paper empirically examines the issues of using Lamarckian and Baldwin-ian learning in hybrid GAs. In the empirical investigation conducted, a general trend was observed where increasing use of Lamarckian learning led to the quicker convergence of the genetic algorithm to the best known solution for a series of test problems.Keywords - Lamarckian Evolution, Baldwin Effect, Local Improvement Procedures, Hybrid GAs1 - Corresponding authorI.I NTRODUCTIONGenetic algorithms (GAs) are a powerful set of global search techniques that have been shown to produce very good results on a wide class of problems. Genetic algorithms search the solution space of a function through the use of simulated “Darwinian” evolution, i.e., the survival of the fittest strategy. In general, the fittest individuals of any population tend to reproduce and survive to the next generation, thus improving successive generations. However, inferior individuals can, by chance, survive and also reproduce. Genetic algorithms have been shown to solve linear and nonlinear problems by exploring all regions of the state space and exponentially exploiting prom-ising areas through mutation, crossover, and selection operations applied to individuals in the population. Whereas traditional search techniques use characteristics of the problem (objective function) to determine the next sampling point (e.g., gradients, Hessians, linearity and continuity), GAs make no such assumptions. Instead, the next sampled points are determined based on sto-chastic sampling/decision rules, rather than a set of deterministic decision rules. Therefore, evalu-ation functions of many forms can be used, subject to the minimal requirement that the function can map the population into a partially ordered set. A more complete discussion of genetic algo-rithms, including extensions and related topics, can be found in the books by Davis[6], Goldberg[7], and Michalewicz[18].This paper examines the use of genetic algorithms utilizing local improvement procedures (LIPs) to solve a wide class of problems. Section II describes how GAs can be hybridized by incorporat-ing local improvement procedures into the algorithm. Incorporating local improvement proce-dures gives rise to the concepts of Larmarckian evolution and the Baldwin Effect which are explained in Section II. Also, the concept of a one-to-one genotype to phenotype mapping is introduced, where genotype refers to the space the GA searches while the phenotype refers to the space of the actual problem. The concepts of Section II are tested in Section III on three different classes of problems: multi-modal continuous function optimization, the location-allocation prob-lem, and the manufacturing cell formation problem. The hybrid-GA is shown to be a very effi-cient and effective optimization method, especially when a one-to-one mapping is enforced (i.e., Larmarckian evolution is employed).II.H YBRIDIZING GA S WITH L OCAL I MPROVEMENT P ROCEDURESMany researchers [1,3,6,12,14,18] have shown that GAs perform well for global searchingbecause they are capable of quickly finding and exploiting promising regions of the search space, but they take a relatively long time to converge to a local optimum. Local improvement proce-dures, e.g., two-opt switching for combinatorial problems and gradient descent for unconstrained nonlinear problems, quickly find the local optimum of a small region of the search space, but are typically poor global searchers. Local improvement procedures have been incorporated into GAs in order to improve the algorithm’s performance through what could be termed “learning.” Such hybrid GAs have been used successfully to solve a wide variety of problems [1,6,12,14].There are two basic models of evolution that can be used to incorporate learning into a GA: the Baldwin Effect and Lamarckian evolution. The Baldwin Effect allows an individual’s fitness (phenotype) to be determined based on learning, i.e., the application of local improvement. Like natural evolution, the result of the improvement does not change the genetic structure (genotype) of the individual. Lamarckian evolution, in addition to using learning to determine an individuals fitness, changes the genetic structure of an individual to reflect the result of the learning. Both “Baldwinian learning” and “Lamarckian learning” have been investigated in conjunction with hybrid GAs.A. Baldwin EffectBaldwinian learning utilizes a LIP to determine the fitness of an individual; however, the genetic structure of the individual is not changed to reflect the application of the LIP. The Baldwin Effect, as utilized in genetic algorithms, was first investigated by Hinton and Nolan [9] using a flat land-scape with a single well representing the optimal solution. Individuals were allowed to improve by random search, which in effect transformed the landscape to include a funnel around the well. Hinton and Nolan showed that without learning the genetic algorithm fails; however, with the ran-dom search the GA is capable of finding the optimal.Whitley et al. [22] demonstrate that “exploiting the Baldwin Effect need not require a needle in a haystack and improvements need not be probabilistic.” They show how using a LIP can in effect change the landscape of the fitness function into flat landscapes around the local basins. This transformation increases the likelihood of allocating more individuals to certain basins or local optima. In a comparison of Baldwinian and Lamarckian learning, Whitley et al.[22] show that utilizing either form of learning is more effective than the standard GA approach without the local improvement procedure (a bitwise steepest ascent algorithm performed on a binary representa-tion). Whitley et al. [22] argue that, while Lamarckian learning is faster, it may be suspectable to premature convergence to a local optimum as compared to Baldwinian learning. Three numerical optimization problems were used to test this conjecture; however, the results were inconclusive.They believe that these test functions were too easy and that harder problems may in fact show that a Baldwinian search strategy is preferred.B. Lamarckian EvolutionLamarckian learning forces the genotype to reflect the result of some form of improvement. This results in the inheritance of acquired or learned characteristics that are well adapted to the envi-ronment. The improved individual is placed back into the population and allowed to compete for reproductive opportunities. However, Lamarckian learning inhibits the schema processing capa-bilities of genetic algorithms [8,21,22]. Changing the genetic information in the chromosomes results in a loss of inherited schema, altering the statistical information about hyperplane parti-tions implicitly contained in the population.While Lamarckian learning may disrupt the schema processing of a genetic algorithm, Baldwin-ian learning certainly aggravates the problem of multiple genotype to phenotype mappings. A genetic algorithm works on both genotypes and phenotypes. A genotype refers to the composition of the values in the chromosome or individual in the population. Whereas a phenotype refers to the solution that is constructed from a chromosome. In a direct mapping, there is no distinction between genotypes and phenotypes. For example, to optimize the function , a typical representation for the chromosome would be a vector of real numbers (x 1, x 2), which pro-vides a direct mapping to the phenotype. However, in some problem instances, a direct mapping is not possible or desired [19]. The most common example of this is the TSP problem with an ordinal representation. Here, the genotype is represented by an ordered list of n cities to visit, e.g.,(c [1]c [2]...c [n ]). However, the phenotype of the TSP is a tour, and any rotation of the chromosome yields the same tour; thus, any rotation of a genotype results in the same phenotype. For example,the two tours (1,2,3,4) and (3,4,1,2) have different genotypes since their gene strings are different,but both strings represent the same tour and thus have the same phenotype.It has been noted that having multiple genotypes map to the same phenotype may confound the GA [10,19]. This problem also occurs when a LIP is used in conjunction with a GA. Consider the example of optimizing . Suppose a simple gradient based LIP is used to determine the fitness5x cos 12sin x 2–x sinof a chromosome. Then any genotype between [-π/2, 3π/2] will have the same phenotype value of 1 atπ/2.III.E XPERIMENTATIONTo investigate the trade-off of disrupted schema processing in Lamarckian learning and of multi-ple genotype mapping to the same phenotype in Baldwinian learning, a series of experiments using LIPs as evaluation functions were run on several different test problems. For each test prob-lem the GA was run with a varying levels of chromosome “forcing” after the evaluation. The GA was run with no forcing (i.e., a pure Baldwinian search strategy), denoted 0% update, as well as 100% forcing (i.e., a pure Lamarckian search strategy). Combinations of both strategies were used to determine if there was a trend between the two extremes, or if combinations of Baldwin-ian learning and Lamarckian learning were beneficial. In these experiments, individuals were “forced” with a probability of 5%, 15%, 25%, and 50% to match the resulting solution. Each run of the GA was replicated 10 times, with common random seeds. The genetic algorithm used in these experiments (described in [12]) runs in the Matlab computing environment, and was run using default parameters.A. Test ProblemsSeven different problems were used to ensure that (1) any effect observed held across different classes of problems as well as different instances of the problem and (2) was not due to some unknown structure of the specific class of problem selected, specific instance chosen, or the par-ticular LIP used.Corona Problems: The first problem set consists of three instances from a family of continuous, nonlinear, multi-modal functions developed by Corona et al. [5] to test the efficiency and effec-tiveness of simulated annealing as compared to stochastic hill-climbing and Nelder-Mead optimi-zation methods. This parametrized family of test functions is easy to compute and contains a large number of local minima. The function can be best described as an n-dimensional parabola with rectangular pockets removed with the global optima always occurring at the origin. These func-tions are parameterized with the number of dimensions (n), the number of local optima, and the width and height of the rectangular pockets. The same three problems (Cor2, Cor4, and Cor10) used in Corona et al. [5] and Houck et al. [12] were chosen for these experiments. A sequentialquadratic programming (SQP) optimization method was used as the LIP which is available in the Matlab environment.Location-Allocation Problem:The continuous location-allocation problem, a nonlinear, integer program as described in [11], will be used as the second problem set. The location-allocation problem (LA) is a type of multifacility location problem in which both the location of n new facil-ities and the allocation of the flow requirements of m exiting facilities to the new facilities are determined so that the total transportation costs are minimized. The location-allocation problem is a difficult optimization problem because its objective function is neither convex nor concave resulting in multiple local minima. Optimal solution techniques are limited to small problem instances (less than 25 existing facilities for general l p distances [16] and up to 35 for rectilinear distances [17]).The alternate location-allocation method developed by Cooper [4] will be used as the LIP for this problem. This method quickly finds a local minimum solution given a set of starting locations for the new facilities. This procedure works by starting with a set of new facility locations, it then determines an optimal set of allocations based on those locations. The optimal new facility loca-tions for these new allocations are then determined by solving n single facility location problems. This method continues until no further allocation changes are made. The individuals in the GA are a vector of real values representing the starting locations for the new facilities. T wo represen-tative problems (a 200 EFS by 20 new NFS and a 250 EFs by 25 new NFs) taken from [11] will be used in this experiment.Cell Formation Problem: The final problem set consists of two manufacturing cell-formation problems, non-linear integer programming problems described by [13]. The cell formation prob-lem is concerned with assigning a group of machines to a set of cells and a series of parts needing processing by these machines into families. The goal is to create a set of autonomous manufactur-ing units that eliminate inter-cell movements (i.e., a part needing processing by a machine outside its assigned cell). Consider an m machine and n part cell formation problem with k maximum cells, the problem is to assign each machine and part to one and only one cell or family, respec-tively. Joines et al. [13] developed an integer programming model with the following set variabledeclarations that eliminates the classical binary assignment variables and constraints.This representation has been shown to outperform other cell formation methods utilizing the same objective [13]. A greedy one-opt switching method as described in 14 will be used as the LIP Given an initial assignment of machines and parts to cells and families, respectively, the heuristic uses a set of improvement rules developed by Ng [20] to determine the improvement in the objec-tive function if machine i was switched from its current cell to any of the remaining k-1cells. The procedure repeats this step for each of the m machines as well for the parts and families. Two instances of the problem (King [15] and Chandrasekharan [2]) will be used as representative problems. The King data set is a 16x43 problem while the Chandrasekharan (Chan.) data set is a 100x40 problem.IV .R ESULTSThe results of these experiments were examined for effects in solution quality as well as computational efficiency. The results, in terms of solution quality, are provided in Table I. For comparison on the importance of using LIP with GAs, the table also shows the results when no local improvement (NLI) is used. For each problem instance, the first row provides the mean functional value found by the GA, the second row the mean of the number of generations it took the GA to find the best solution (not necessarily the optimal), and the third row is how many of the replications found the best known solution.Table I: Solution Qualityx i l ,=machine i is assigned to cell l y j l ,=part j is assigned to family lTable I: Solution QualityAs Table I shows, the use of a local improvement procedure improves the quality of the final solution found by the genetic algorithm. Also, the use of the Baldwin learning, 0%, does not perform as well as using some amount of Lamarckian learning. When some form of Lamarckian learning is employed, the genetic algorithm converges to the best known solution, except for 1 trial of both the 50% and 100% location-allocation problems.The results of these experiments with regard to computational efficiency are shown in Fig-ures 1-3. Each figure shows a box and whiskers plot of the generation the GA found the best solu-tion (not necessarily the optimal) for each problem instance. The boxes have lines at the lower quartile, median, and upper quartile values, and the whiskers show the range of the rest of the data. As can be seen, the genetic algorithm converges to the optimal solution much quicker when the genotypes are forced to match the phenotype through chromosome updating. In Figure 3(b), the logarithm of the number of generations until convergence is shown rather than the raw num-ber of generations until convergence. Without any forcing, this test problem takes a disproportion-ately larger number of generations to converge to the optimal. Each of these figures shows aFig. 1. Convergence on the Corana FunctionFig. 2. Convergence on the Location-Allocation ProblemFig. 3. Convergence for Cell Formation Problemscommon trend, with increasing levels of Lamarckian learning, the genetic algorithm convergesquicker, and as seen in Table I, the GA is converging to the best known solution.V.C ONCLUSIONSLocal improvement procedures were shown to enhance the performance of the genetic algorithm. Even though Lamarckian learning disrupts the schema processing of the genetic algorithm, it reduces the problem of a one-to-one genotype to phenotype mapping. Whereas, Baldwinian learn-ing does not affect the schema processing capabilities of the genetic algorithm, but results in a large number of genotypes mapping to the same phenotype.In the empirical investigation conducted, a general trend was observed where increasing use of Lamarckian learning led to the quicker convergence of the genetic algorithm to the best known solution for a series of test problems. By forcing the genotype to reflect the phenotype, the GA converges more quickly and to better solutions than by leaving the chromosome unchanged after evaluating it. This may seem counterintuitive since forcing the genotype to be equal to the pheno-type might have forced the GA to converge prematurely to one of these local optima. However, for this class of problems and procedures, Lamarckian learning outperformed Baldwinian learn-ing.R EFERENCES[1]H.Bersini and B.Renders. Hybridizing genetic algorithms with hill-climbing meth-ods for global optimization: Two possible ways. In1994 IEEE International Sympo-sium Evolutionary Computation, pages 312–317, Orlando, Fl, 1994.[2]M.Chandrasekharan and R.Rajagopalan. Zodiac-an algorithm for concurrent for-mation of part families and mac hine cells.International Journal of Production Re-search, 25(6):835–850, 1987.[3]P.Chu and J.Beasley. A genetic algorithm for the generalised assignment problem.Technical report, The Management School Imperial College London, 1995.[4]L.Cooper. The transportation-location problems.Operations Research, 20:94–108,1972.[5] A.Corana, M.Marchesi, C.Martini, and S.Ridella. Minimizing multimodal func-tions of continuous variables with the "simulated annealing" algorithm.ACM Trans-actions on Mathematical Software, 13(3):262–280, 1987.[6]L.Davis.Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York,1991.[7] D.Goldberg.Genetic Algorithms in Search, Optimization & Machine Learning. Ad-11dison Wesley, Reading, MA, 1989.[8] F.Gruau and D.Whitley. Adding learning to the cellular development of neural net-works.Evolutionary Computation, 1:213–233, 1993.[9]G.Hinton and S.Nolan. How learning can guide plex Systems,1:495–502, 1987.[10] A.Homaifar, S.Guan, and G.E. Liepins. A new approach on the traveling salesmanproblem by genetic algorithms. In Proceedings of the 5th ICGA, pages 460–466,Champaign, IL, 1993.[11] C.R. Houck, J.A. Joines, and M.G. Kay. Comparison of genetic algorithms, ran-dom restart, and two-opt switching for solving large location-allocation problems.Computers and Operations Research, To Appear, 1996.[12] C.R. Houck, J.A. Joines, and M.G. Kay. A genetic algorithm for function optimi-zation: A Matlab implementation.ACM Transactions on Mathmatical Software,Submitted, 1996.[13]J.Joines, C.Culbreth, and R.King. Manufacturing cell design: An integer program-ming model employing genetic.IIE Transactions, 28(1), 1996.[14]J.Joines, R.King, and C.Culbreth. The use of a local improvement heuristic with agenetic algorithm for manufacturing cell design. Technical Report NCSU-IE Tech-nical Report 95-06, North Carolina State University, 1995.[15]J.King. Machine-component grouping formation in group technology.InternationalJournal of Management Science, 8(2):193–199, 1980.[16]R.Love and H.Juel. Properties and solution methods for large location-allocationproblems with rectangular distances.Journal Operations Research Society, 33:443–452, 1982.[17]R.Love and J.Morris. A computational procdure for the exact solution of location-allocation problems with rectangular distances.Naval Research Logistics Quartly,22:441–453, 1975.[18]Z.Michalewicz.Genetic Algorithms + Data Structures = Evolutionary Programs.Springer-Verlang, New York, 2nd edition, 1994.[19]R.Nakano. Conventional genetic algorithm for job shop problems. In Proc. of the 4thICGA, pages 474–479, San Mateo, CA, 1991.[20]S.Ng. Worst-case analysis of an algorithm for cellular manufacturing.EuropeanJournal of Operational Research, 69(3):384–398, 1993.[21] D.Whitley. A genetic algorithm tutorial.Statics and Computing, 4:65–85, 1994.[22] D.Whitley, S.Gordon, and K.Mathias. Larmarckian evolution, the Baldwin effectand function opimization. In Y.Davidor, H.Schwefel, and R.Manner, editors,Par-allel Problem Solving from Nature-PPSN III, pages 6–15. Springer-Verlag, 1994.。