GENETIC ALGORlIHMS WITH AN APPLICATION TO NONLINEAR TRANSPORTATION PROBLEMS

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1 101 GENETIC ALGORlIHMS WITH AN APPLICATION TO NONLINEAR TRANSPORTATION PROBLEMS MATTHEW HOBBS ISOR, VICTORIA UNIVERSITY WELLINGTON Abstract In this paper I will describe the theory and implementation of Genetic Algorithms. This is followed by a description of an attempt to solve the nonlinear transportation problem using a genetic algorithm approach, (Michalewicz, Vignaux, & Hobbs [1990]). This is a continuation of earlier research in which a cv«tem was built for the standard linear transportation problem (Vignaux & Michalewicz [1990]). Results of this technique are promising. 1 G enetic Program m ing Much of problem solving in O perations Research involves optim ization using one of a variety of techniques. The m ost extensive algorithm in practice is that of Linear Programming. Like LP, m ost algorithm s apply only to relatively sm all problem domains. Som e algorithm s guarantee an optim al solution while others are heuristics and aim to find an approxim ate or good solution. T his still leaves a wide range of problems for which there are no useful algorithm s. In general, algorithm s take advantage of particular structural properties of the problem in order to solve it. This approach fails for m ost com plex problem s. In may cases the information m ay be little more than the payoff of any particular solution (p.? cost) and the expression of the solution (e.g.? p"'h ct:cn schedule). In this case the procedure for payoff optim ization is unclear. If we have a given solution, w ith a determ ined payoff, how do we know if it is optim al? How do we determine what part of the solution expression should be changed if it is not? W ith only the one solution we can usually answer none of these questions. The key is to incorporate comparisons with other solutions into the algorithm - of tbe type described above are usually probabilistic search tech niques. In the sim plest case (with the least number of assum ptions m ade) com pletely random search operates by randomly selecting another solution (independent to the history of the search) and remembering only the best solution so far. The obvious

2 102 disadvantage is the tim e costs involved (when the feasible set is of any size). A more sensible approach is to som ehow bias the random search to home in on the best solutions, quickly. An area where random search can be improved is in usiii6 of the search to select new (trial) solutions. The questions becom es one of asking how do we m ost efficiently use m e history to produce the trial solutions? Sim ulated Annealing is an algorithm whereby the history at any stage in the algorithm is the current solution. A trial solution is chosen by m odifying the current solution in som e way (m utation; and the new current solution is chosen probabilistically between the trial and current solutions, according to their relative payoff. In Sim ulated A nnealing the search history used at any step is only the current solution, previous solutions are discarded. G enetic Algorithm s use the idea that previous solutions should not be discarded im m ediately and that useful in fo ;:.. can be obtained by m aintaining groups of solutions as the search history. The prim ary problem is how to com bine the pool of solutions to create new (better) solutions. T he concept of m aintaining a pool of solutions in order to generate new trial solutions is analogous to that of natural evolution with genetic code being the expression of an individual solution. New solutions are created by com bining code from existing solutions (crossover). The optim izational tendency occurs when current solutions are discarded from the pool (e.g. die) according to lack of payoff. A lgorithm s that include ideas such as these are called G enetic A lgorithm s (Holland [1975], D avis [1987], G oldberg [1989]). A typical G enetic A lgorithm starts w ith a population of random ly generated solutions (the initial population) and repeatedly applies genetic operators m odeled on natural genetic processes (e.g. crossover, m utation) to the population. Consider this extract from D avis [1987]:... the m etaphor underlying genetic algorithm s is that of natural evolution. In evolution, the problem each species faces is one of searching for beneficial adaptions to a com plicated and changing environm ent. T he knowledge that each species has gained is em bodied in the m akeup of the chrom osom es of its m em bers. The operations that alter this chrom osom al m akeup are applied when parents reproduce; am ong them are random m utation, inversion of chrom osom al m aterial, and crossover - exchange of chrom osom al m aterial oetween two parents chrom osom es." Theory Any run of a genetic algorithm is essentially a sim ulation of the evolution of a set of solutions exposed to constraints and performance (the environm ent). Those individuals (solutions) better suited to the environm ent, in general, will have a greater chance of survival and therefore have a greater chance of producing offspring. A lthough this m ay seem initially as a hill-clim bing process, the algorithm (at the sam e tim e) m aintains population diversity within the search space. The ability to m aintain diversity enables the algorithm to escape local optim a and ie a p tall buildings w ithout too much ado. Conversely, near a global optim um, the algorithm would have difficulty hom ing in to the precise solutions, due to the blindness of the random m utations.

3 103 The expression of a solution ( chromosome) for various problems will vary considerably but can be considered as a collection of elem ents (genes). Ideally we would like to be able to treat each gene separately, for optim ization purposes. T his is not usually feasible due to the often highly nonlinear or constrained nature of the problem. The only information we have m ay be the payoff of the chrom osom e jt = whole. A feature of the genetic algorithm is in that although it appears to involve only com petition between chrom osom es, due to the nature of the genetic operators (e.g. crossover) - that m anipulate genes directly - in fact com petition occurs at the gene level. This makes sense when we consider that new solutions are created with only a subset of genes differing from their parent(s). A useful term to use is that of a schema - see Holland [1975] or G oldberg [1989]. A schem a (in this context) can be considered as a sim ilarity tem plate for chrom o somes defining a set of fixed genes. For exam ple fixing gene number 3 at yes' and gene number 7 at no (and leaving all other genes free) defines a schem a for the chrom osom es. To be m ore accurate than the previous paragraph, com petition is on the schem a level. A lthough a genetic algorithm processes only n structures per generation, it also processes of th e order of n3 schem ata. Holland has nam ed tlys im portant result implicit parallelism. It is generally accepted that for any problem to be attem pted by a genetic algorithm there must be five com ponents, as follows. 1. A genetic representation of solutions to the problem (chrom osom e), 2. A way to crcatc random solutions (for the initialization). 3. An evaluation function (the environm ent ), rating solutions by fitness. 4. G enetic operators to alter child com position during reproduction, and 5. Values for the param eters that the genetic algorithm uses (population size, probabilities of applying genetic operators, etc.) For a variety of reasons m ost applications of the genetic algorithm have used binary (bit string) representations for the solutions (chromosomes). But the -im plicit parallelism result is not restricted to bit strings (see A ntonisse [1989]). The application in this paper uses a m atrix of real numbers. Although it is true that any structure that can be put on com puter can be represented as a binary vector (string), the difference is that the schem a for the richer structure are much m ore relevant, and there are much less of them. Im plem entation A genetic algorithm will typically be of the following form. 1. create an initial set of random solutions (the initial population); 2. evaluate the solutions ^see now good they are); 3. put the better solutions into a set:

4 try to combine these solutions to produce child solutions; 5. discard poorer solutions to m aintain population size; 6. repeat the above three steps until we get a good solution or resources run out. There are two main genetic operators, mutation and crossover. The m utation operator arbitrarily alters one or m ore com ponents of a selected structure this increases the variability of the population (introducing new - or lost - schem a). In general, each gene of each chrom osom e in the population undergoes a random change with a probability equal to the m utation rate. The crossover operator com bines the features of two parent structures to form two sim ilar offspring. It operates by sw apping corresponding segm ents of a chrom osom e representing the parent solut ions (w ith the ob jective of com bining good schem ata to produce better schem ata - with more fixed genes). W hen problem s are found to be too specialized or com plex for the standard algorithm s it is necessary to take a heuristic approach to the problem. G enetic algorithm s can be a powerful tool in such cases due to the fact that it can com bine the inform ation from various solutions (and at the sam e tim e, m aintain the diversity required to try to span the search space). Currently there is no program ming language specific to this problem dom ain so system s have had to be set up in other languages as available. It should be noted that when the param eters to the problem are changed (e.g. a different problem of the sam e type) the programs set up will require very little change. For exam ple if the objective function (goal) of the problem changes it is only necessary to change the evaluation procedure and no other. Such system s are m ore flexible to changes than m ost. For exam ple a problem beine solved using Linear Program m ing is fine until nonlinearity is introduced, in which case another algorithm is required. G enetic algorithm s can com pete very effectively against standard nonlinear algorithm s on nonlinear problems. T hey m ay be best used in com bination with nonlinear algorithm s for such problem s. For exam ple using a nonlinear algorithm to home-in on the local optim a. There are a number of applications for which genetic algorithm s have been applied including: physical design of circuit boards; travelling salesm an (nodecovering) problem; com binatorial problem s in general; gam e strategy; nonlinear problem s; optical design; keyboard configuration; m achine learning; sim ulation of evolution: and gas pipeline optim ization. In fact any problem that m eets the five properties listed previously can be attem pted by a genetic algorithm. 2 A G enetic Algorithm for the Nonlinear Transportation Problem In testin g the use of the genetic algorithm on the linear transportation problem (see V ignaux Iz M ichalewicz [1989],[1990]) it is possible to compare its solution with the known optim um found using the standard algorithm and therefore to determ ine how efficient or otherw ise the genetic algorithm is in absolute term s. Once we m ove to nonlinear objective functions, the optim um m ay not be known. Testing is reduced to com paring the results with those of other nonlinear solution m ethods that may them selves have converged to a local optim um.

5 105 The genetic algorithm will be com pared with the GAM S nonlinear system (w ith MINOS optim izer) as a typical exam ple of an industry-standard efficient m ethod of solution. T his system, being essentially a gradient-driven m ethod, found som e of the problem s set up difficult or im possible to solve. In these cases m odifications to the objective functions were m ade so that the m ethod could at least find an approximate solution. T he genetic system itself was w ritten by Z. M ichalewicz in the CT programming language. The param eters required include (as well as the problem description): number of iterations, population size, m utation and crossover rates, and random number starting seed. The behaviour of nonlinear optim ization algorithm s depend m arkedly on the form of the objective function. It is clear that different solution techniques m ay behave differently. In the experim ents the overall ob jective function is the sum of the arc objective m ncuons, thus there were no cross term s. Six different arc objective functions were used including: step function (w ith 5 equal steps): discretely changing slope (3 zones of a particular gradient); square; square-root: a function w ith a peak; and a linearly increasing sine function. T hey were used in conjunction w ith five problem structures in which the supply and dem and vectors and the corresponding parameter m atrix were randomly generated. T he solutions to these problem s are unknown. The objective function for the transportation problem was thus of the form / < * «) where f(x) is one of the six arc objective functions. Experiments and results The population size was fixed at 40. T he m utation rate was 20% w ith the proportion of mutation-1 being 50%, and the crossover rate was 5%. Problem s were run for 10,000 generations. The system s were tested on the six functions for five random ly generated cost matrices. See the second graph page for a graphical display of the results. The genetic system was run on SPARC sta tio n /S U N term inals while GAM S was run on an O livetti 386 with m ath co-processor. A lthough tim e com parisons between the two m achines are difficult to m ake it should be noted that in general GAM S finished each run well before the genetic system. An exception is case A (in which GAMS evaluates numerous arc-tangent functions) when the genetic algorithm took no more than 15 m inutes to com plete while GAM S averaged at about tw ice that. For cases A,B, and D. where the GAM S param eter m eant that m ultiple runs had to be performed to find the best G A M 6! solution, the genetic system overall was much faster. Conclusions The transportation problem was chosen as it provided a relatively sim ple convex feasible set. This means that it is easier to ensure feasibility in the solutions. The

6 106 procedure was then to look at the effects that the objective function alone lias on the solving of the problem. For the class of practical problem s the genetic system is, on average, better than G A M :.j_, ij% in case A and by 11.6% in case B. For the reasonable functions the results were different. In case C (the square function), the genetic system performed worse by 7.6% while in case D (the square-root function), the genetic system was worse by only... F r the other of functions the genetic system is dom inant. The genetic system resulted in im provem ents of 32.9% and 55.1% over GAM S, averaging over the five problems. T his dem onstrates the superiority of the genetic m ethod over other system s which are very often lim ited to certain classes of problem functions. G AM S did well on the sm ooth /m on otonic (reasonable) functions, it is these cases where the gradient m easurem ent techniques axe m ost apt. In case C GAMS bettered the genetic system with much less cost in solving tim e. For the practical problem s, the gradient techniques have difficulty seeing around the corner to new zones of better costs. The genetic algorithm, taking a more structural approach, is able to jum p betw een zones readily, resulting in much better solutions. For the other problems, although they are both =mooth, have significant structural features. Like the practical problem s, but even m ore so, the genetic system did much better the GAM S. REFER ENCES A ntonisse, J., A New Interpretation of Schem a Notation that Overturns the Binary Encoding Constraint, Proceedings of the Third International Conference on G enetic A lgorithm s (J. David Schaffer, editor), George Mason University, June pp D avis, L., (editor), G enetic A lgorithm s and Sim ulated A nnealing, Pitm an, L ondon Goldberg, D.E., G enetic Algorithm s in Search. O ptim ization and Machine Learning, Addison W esley, H olland, J., "A daptation in Natural <uiu A u ih cim S ystem s, Ann Arbor: University of M ichigan Press, M ichalew icz, Z., V ignaux, G.A., Hobbs, M. A G enetic A lgorithm for the Nonlinear Transportation Problem, subm itted to the O perations Research Society of A m erica (O R SA ) Journal on C om puting. Vignaux, G.A., M ichalewicz, Z., G enetic Algorithm s for the Transportation Problem, Proceedings of the 4th International Sym posium on M ethodologies for Intelligent System s, Charlotte, October 12-14, Vignaux, G.A., M ichalewicz, Z., A G enetic Algorithm for the Linear Transportation Problem, subm itted to IEEE Transactions on M an, System s, and Cybernetics.