SIMULATION OPTIMIZER AND OPTIMIZATION METHODS TESTING ON DISCRETE EVENT SIMULATIONS MODELS AND TESTING FUNCTIONS

SIMULATION OPTIMIZER AND OPTIMIZATION METHODS TESTING ON DISCRETE EVENT SIMULATIONS MODELS AND TESTING UNCTIONS Pavel Raska (a), Zdenek Ulrych (b), Petr Horesi (c) (a) Department o Industrial Engineering - aculty o Mechanical Engineering, University o West Bohemia, Univerzitni, 06 14 Pilsen (b) Department o Industrial Engineering - aculty o Mechanical Engineering, University o West Bohemia, Univerzitni, 06 14 Pilsen (c) Department o Industrial Engineering - aculty o Mechanical Engineering, University o West Bohemia, Univerzitni, 06 14 Pilsen (a) praska@kpv.zcu.cz, (b) ulrychz@kpv.zcu.cz, (c) tucnak@kpv.zcu.cz ABSTRACT The paper deals with testing o selected optimization methods used or optimization o speciied obective unctions o three discrete event simulation models and our selected testing unctions. The developed simulation optimizer uses modiied optimization methods which automatically adapt input parameters o discrete event simulation models. Random Search, Hill Climbing, Tabu Search, Local Search, Downhill Simplex, Simulated Annealing, Dierential Evolution and Evolution Strategy were modiied in such a way that they are applicable or discrete event simulation optimization purposes. The other part o the application is ocused on testing the implemented optimization methods. We have proposed some evaluation techniques which express the success o the optimization method in dierent ways. These techniques use calculated box plot characteristics rom the series o optimization experiments. Keywords: simulation optimization, heuristic optimization methods, discrete event simulation models, testing unction 1. INTRODUCTION Many o today s industrial companies try to design their own production system as eectively as possible. The problem is that this intention is aected by many actors. We can say the problem is NP-a hard problem in most cases. A possible answer to the problem is using discrete event simulation and simulation optimization. The use o discrete event simulation ocuses on the invisible problems in the production system many times and also avoids bad decisions made by the human actor. The next question is eectively inding a suitable solution to the modelled problem. We can use a simulation optimizer to ind an optimal/suboptimal easible solution respecting the deined model constraints. The basic problem o global optimization can be ormulated as ollows: X ~ ~ X arg min ~ X X : X X X X (1) where X X X denotes the global minimum o the obective X unction; denotes the obective unction value o the candidate solution the range includes real X ~ numbers; denotes the Search space. This optimal solution is represented by the best coniguration (input parameters values) o the simulation model. Current simulation sotware (Arena, Witness, PlantSimulation etc.) uses its own simulation optimizers. These integrated optimization modules are black-boxes but many o them use similar optimization methods. We have tested the ollowing optimization methods: Random Search, Hill Climbing, Tabu Search, Local Search, Downhill Simplex, Simulated Annealing, Dierential Evolution and Evolution Strategy. These methods were modiied in such a way that they are applicable or discrete event simulation optimization purposes. The goal o our research is to compare some o these widely used optimization methods. Hence we have designed our own simulation optimizer. We have to say that it is not possible to implement exactly the same optimization methods which are used in these simulation optimizers. Another reason or testing the optimization methods and designing our own simulation optimizer was that our department ocuses on modelling and optimizing production and non-production processes in industrial companies (Kopecek, 01; Votava, Ulrych, Edl, Korecky and Trkovsky, 008). Some proects have to be solved with diiculty without the use o special simulation optimization tools because o the large complexity o the discrete event simulation model. The problem is that some integrated simulation optimizers cannot aect all the parameter types o the designed simulation model. Proceedings o the European Modeling and Simulation Symposium, 01 50

. SELECTED OPTIMIZATION METHODS We have transormed some o the selected optimization methods to use the principle o evolutionary algorithms. These optimization methods generate a whole population (instead o one possible solution) in order to avoid getting stuck on a local optimum. Previous testing o optimization methods conirms that generating one solution leads to premature convergence in most cases (depending on obective unction type). Dierent variants o selected optimization methods obtained rom a literature review were united into the algorithm. The user can combine dierent variants o optimization methods by setting the optimization method parameters..1. Random Search A new candidate solution is generated in the search space with uniorm distribution (Monte Carlo method). This method is suitable or cases where the user has no inormation about the obective unction type. The user is able to perorm a number o simulation experiments... Downhill Simplex This method uses a set o n + 1 linearly independent candidate solutions (n denotes search space dimension) - Simplex. The method uses our basic phases Relection, Expansion, Contraction and Reduction. (Tvrdík 004; Weise 009).. Stochastic Hill Climbing Candidate solutions are generated (populated) in the neighbourhood o the best candidate solution rom the previous population. Generating new possible solutions is perormed by mutation. This method belongs to the amily o local search methods..4. Stochastic Tabu Search The newly generated candidate solution is an element o the Tabu List during the optimization process. This candidate solution cannot be visited again i the aspiration criterion is not satisied (this eature prevents the method rom becoming stuck at a local optimum). The method uses the IO method o removing the candidate solution rom the Tabu List. The user can set whether the new candidate solution is generated using mutation o the best candidate solution rom the previous population or the new solution is generated using mutation o the best ound candidate solution. (Monticelli, Romero and Asada 008; Weise 009).5. Stochastic Simulated Annealing A candidate solution is generated in the neighbourhood o the candidate solution rom the previous iteration. This generating could be perormed through the mutation o a randomly selected gene or through the mutation o all genes. Acceptance o the worse candidate solution depends on the temperature. Temperature is reduced i the random number is smaller than the acceptance probability or the temperature is reduced i and only i a worse candidate solution is generated. I the temperature alls below the speciied minimum temperature, temperature is set to the initial temperature. (Monticelli, Romero and Asada 008; Weise 009).6. Stochastic Local Search A candidate solution is generated in the neighbourhood o the best candidate solution..7. Evolution Strategy This optimization method uses Steady State Evolution population consists o children and parents with good itness. A candidate solution (child) is generated in the neighbourhood o the candidate solution (parent) and it is based on the Rechenberg 1/5th-rule. The population is sorted according to the obective values (Rank-Based itness Assignment). The optimization method uses Tournament selection. (Koblasa, Manlig and Vavruska 01; Miranda 008; Tvrdík 004).8. Dierential Evolution Selection is carried out between the parent and its ospring. The ospring is created through a crossover between the parent and the new candidate solution (individual) which was created through the mutation o our selected individuals and the best one selected rom the population BEST method. The optimization method uses General Evolution and the Ali and Törn adaptive rule. The user can deine the probability o a crossover between the new candidate solution and the parent. (Tvrdík 004; Wong, Dong, 008). DEVELOPED APPLICATION We have developed our own simulation optimization application which addresses the problems listed in the irst chapter. The application contains seven dierent global optimization methods. This application contains two modules. The irst module is a simulation optimizer which enables optimization o developed simulation models in ARENA or PlantSimulation simulation sotware. The obective unction o the models is speciied within the discrete simulation models. The user can also test a speciied obective unction without the need o creating the simulation model igure 1. Best ound solution obective unction value Speciied constraints Optimization process Evolution Strategy Speciied obective unction igure 1: Graphical user interace o simulation optimizer - irst module Proceedings o the European Modeling and Simulation Symposium, 01 51

The application was created in Visual Basic 010. This programming language was used or connection to ARENA simulation sotware and Microsot Access database. The data rom simulation experiments results and settings are stored in this database. The ile contains inormation about: 1. Controls identiication, names, low and high boundaries, type (discrete vs. continuous), initial values o controls and comments.. Constraints speciication o the constraint unction through using the mathematical operator buttons and the list o controls. User can validate the built expression.. Obective unction - speciication o the obective unction without the need o a simulation sotware tool. Obective unction is composed o mathematical operators and selected controls rom the list o all controls. User can validate the built expression. 4. Optimization experiment setting minimization vs. maximization o obective unction, Termination criterion (Value to reach, number o simulation experiments, speciied time, sub-optimum improvement ratio etc.), parameters settings o selected optimization method, low and high boundaries o selected optimization method parameters, number o replications, creation o a knowledge base o a simulation model, etc. The second module is designed or testing the behaviour o the implemented optimization method in terms o setting the parameters or the optimization method. The user can speciy the range o optimization method parameters. Ater inishing the number o optimization experiments replications (series o concrete optimization method setting) the data are exported to MS Excel workbook. We have also developed an application which enables D visualization o simulation experiments when there are two controls and one obective unction. Simulation experiments are represented by the points in the D chart o the obective unction. The obective unction surace is generated rom the data obtained rom simulation experiments. Another possibility is to generate a whole D chart rom the data obtained rom the simulation runs o all possible settings o the simulation model input parameters complete search space. 4. DISCRETE EVENT SIMULATION MODELS AND OBJECTIVE UNCTIONS The testing o optimization methods which search or global optima was applied to three discrete event simulation models. These models relect real production systems o industrial companies. Discrete event simulation models were built in Arena simulation sotware. We speciied dierent obective unctions considering the simulated system. All possible solutions and their obective unction values were mapped to ind the global optimum in the search space. 4.1. The Manuacturing System and Logistics This discrete event simulation model represents the production o dierent types o car lights in a whole production system. The complex simulation model describes many processes; or example, logistics in three warehouses, production lines, 8 assembly lines, painting, etc. The obective unction is aected by the sum o the average utilization o all assembly lines and average transport utilization. The obective unction is maximized. Controls are the number o orklits responsible or: transport o small parts rom the warehouse to the production lines and assembly lines, transport o large parts rom the warehouse to the assembly lines and the transport o the inal product rom the assembly lines to the warehouse. The obective unction landscape o this model when the number o orklits or transport o large parts = 14 is shown in igure. Obective unction value Number o orklits - inal parts Number o orklits - inal parts igure : Obective unction - The Manuacturing System And Logistics Discrete Event Simulation Model - Number o orklits or Large Parts = 14 4.. The Penalty This simulation model represents a production line which consists o eight workstations. Each workstation contains a dierent number o machines. Each product has a speciic sequence o manuacturing processes and machining times. The product is penalized i the product exceeds the speciied production time. A penalty also occurs i the production time value is smaller than the speciied constant. The penalty unction is shown in igure where T denotes production time; T min denotes required minimum production time; T max denotes required maximum production time; T crit denotes critical production time; k denotes the penalty or early production (slope o the line - constant); k denotes the penalty or exceeding 1 the speciied production time (slope o the line - constant); P denotes the penalty or exceeding the 1 speciied production time (constant); P denotes the penalty or exceeding the speciied critical production time (constant); P denotes the penalty o the product. Proceedings o the European Modeling and Simulation Symposium, 01 5

igure : Penalty unction This rule is deined because premature production leads to increasing storage costs the JIT product. The obective unction is aected by the total time spent by the product in the manuacturing system. The obective unction is minimized. Controls o the production line simulation model are the arrival times o each product in the system. The obective unction is shown in igure 4. igure 5: Obective unction - The Assembly Line discrete event simulation model 5. TESTING UNCTIONS We also tested implemented optimization methods on our standard testing unctions. All testing unctions are minimized. 5.1. De Jong s unction It is a continuous, convex and unimodal testing unction. The unction deinition: n x 1 X () X where denotes the obective unction; denotes index o control; n denotes the dimension o the search space; x denotes the value o control. The obective unction is shown in igure 6. igure 4: Obective unction The Penalty Discrete Event Simulation Model 4.. The Assembly Line This model represents an assembly line. Products are conveyed by conveyor belt. The assembly line consists o eleven assembly workplaces. Six o these workplaces have their own machine operator. The rest o the workplaces are automated. A speciic scrap rate is deined or each workplace. At the end o the production line is a sorting process or deective products. The obective unction relects the penalty which is aected by the number o deective products and the palettes in the system. The obective unction is maximized. The obective unction is shown in igure 5. The input simulation model parameters (controls) are the numbers o ixtures in the system and the number o ixtures when the operator has to move rom the irst workplace to the eleventh workplace to assemble waiting parts on the conveyor belt. igure 6: Obective unction - De Jong s unction 5.. Rosenbrock s unction Rosenbrock s (Rosenbrock's valley, Rosenbrock's banana) unction is a continuous, unimodal and non-convex testing unction. The unction deinition: n1 X () 100 ( x x 1) 1 x 1 The obective unction is shown in igure 7. Proceedings o the European Modeling and Simulation Symposium, 01 5

igure 7: Obective unction - Rosenbrock s unction 5.. Michalewicz s unction Michalewicz s unction is a multimodal test unction (n! local optima). The parameter m deines the "steepness" o the valleys or edges. Larger m leads to a more diicult search. or very large m the unction behaves like a needle in a haystack (the unction values or points in the space outside the narrow peaks give very little inormation on the location o the global optimum). (Pohlheim 006) m n x (4) X sin( x ) sin 1 1 : n, 0 x (5) We selected m 5 in our simulation model. The obective unction is shown in igure 8. igure 8: Obective unction - Michalewicz s unction 5.4. Ackley s unctions Ackley s unction is a multimodal test unction. This unction is a widely used testing unction or premature convergence. (Tvrdík 004) 1 n n 1 n X 0exp 0.0 x exp cos x 0 exp 1 1 n 1 (6) 1: n, 0 x 0 (7) The obective unction is shown in igure 9. igure 9: Obective unction - Ackley s unction 6. EVALUATION METHOD Simulation experiments results are saved to a database ile during simulation experiments i the user uses a simulation optimizer. Simulation experiments results are visualized in the obective unction chart and stored in the table placed in the application. The graphical user interace o the irst module is shown in igure 1. I the second module is used the simulation experiments data are exported to MS Excel workbook ater inishing the series (series - replications o optimization experiments with concrete optimization method setting). Excel was selected because o its wide usage Considering the number o simulation experiments we can divide the number o simulation experiments igure 10: 1. Simulation experiment simulation run o simulation model.. Optimization experiment perormed with concrete optimization method setting to ind optimum o obective unction.. Series replication o optimization experiments with concrete optimization method setting. The second module ocuses on testing the behaviour o the implemented optimization method in terms o setting the parameters or the optimization method. The user can set up the parameters o a selected optimization method, low and high boundaries o the selected optimization method parameters, number o replications, and export the obective unction chart to image igure 11. The same conditions had to be satisied or each optimization method, e.g. the same termination criteria, the same search space. I the optimization method has the same parameters as another optimization method, we set up both parameters with the same boundaries (same step, low and high boundaries). Proceedings o the European Modeling and Simulation Symposium, 01 54

igure 10: The Number o Simulation Experiments X 0 denotes the ound optimum (local in this case). These characteristics are visualized in the box plot chart igure 1. Three box plot charts are generated - Best obective unction value, Range o provided unction obective values during the simulation experiments, and Number o experiments required to ind global (local) optimum. Visualization can help the user to ind a suitable setting o optimization method more quickly. Due to the large volume o data (over 4 billion simulation experiments) we have to propose evaluation techniques (criteria) which express the ailure o the optimization method in dierent ways. Each criterion value is between [0, 1]. I the ailure is 100[%] the criterion equals 1 thereore we try to minimize all speciied criteria. We implemented the graphical user interace to MS Excel workbook which enables the user to set up the weights o each criterion and other parameters o the evaluation. These parameters are automatically loaded rom the simulation experiments results. We used the VBA or MS Excel. 6.1. Optimization Method Success The irst criterion 1 is the value o not inding the known VTR (value to reach). This value is expressed by: s n 1 s succ (8) igure 11: GUI o the Second Module Box plot characteristics (the smallest observation sample minimum Q 1, lower quartile Q, median Q, upper quartile Q 4, and largest observation - sample maximum Q 5) are calculated or each perormed setting o the optimization method parameters igure 1. where s denotes the number o perormed series, n succ denotes the series where the VTR was ound. Simulation runs o all possible settings o simulation model input parameters were perormed. This means that we have evaluated all possible solutions o the search space hence we can determine the global optimum (VTR) in the search space. Average Method Success o inding Optimum can be ormulated as ollows: avg 1 s i1 s 1i 100 % (9) igure 1: Example o Results rom Simulation Optimization Experiments Provided by Evolution Strategy Displayed in Box Plot Chart The Assembly Line Simulation Model where i denotes the index o one series, 1 denotes the i value o the irst criterion (Optimization method success the best value is zero), s denotes the number o perormed series. The average optimization method success o inding the optimum o testing unctions is shown in igure 1. Proceedings o the European Modeling and Simulation Symposium, 01 55

We can see that the Evolution Strategy and Simulated Annealing are successul optimization methods. Random Search also achieves good results. It was aected by doing many simulation experiments by this method. The probability o inding the optimum increases with a high number o simulation experiments. This strategy is simply random and i the search space is huge (NP-hard) we can say it is lucky to ind the optimum. This method is usable when the user has no inormation about the obective unction type. We have to evaluate each possible solution in the search space to obtain the optimum hence the search space cannot be too huge. igure 1: Average Optimization Method Success Simulation Optimization Results o Testing unctions Average optimization method success o inding the optimum o discrete event simulation models is shown in igure 14. We can say that Simulated Annealing and Evolution Strategy are quite successul optimization methods again. Random Search was not successul in the case o the Penalty model because o the larger search space. The Penalty discrete event simulation model has a complicated obective unction landscape. The area around the optimum is straight and the method could not obtain inormation about rising or decreasing the obective unction terrain. igure 14: Average Method Success Simulation Optimization Results o Discrete Event Simulation Models Previous charts express the average success o optimization methods o all optimization methods settings. These charts also contain bad settings thereore we separated the bad series rom the good series. The next chart contains the iltered series with the best ound irst criterion value only (in this case 1 = 0 so the optimum was ound in each optimization experiment). The percentage o absolutely successul series compared to all perormed series is shown in igure 15. It is obvious that the avourite, Evolution Strategy, has problems with the multimodal Ackley unction. The success o this method was aected by the number o individuals randomly chosen rom the population or the tournament exploration vs. exploitation o the search space. The irst approach is to generate other new solutions which have not been investigated beore - exploration. Since computers have only limited memory, the already evaluated solution candidates usually have to be discarded in order to accommodate new ones. Exploration is a metaphor or the procedure which allows search operations to ind new and maybe better solution structures. Exploitation, on the other hand, is the process o improving and combining the traits o the currently known solutions, as done by the crossover operator in evolutionary algorithms, or instance. Exploitation operations oten incorporate small changes into already tested individuals leading to new, very similar solution candidates or try to merge building blocks o dierent, promising individuals. They usually have the disadvantage that other, possibly better, solutions located in distant areas o the problem space will not be discovered. (Michalewicz 004) The behaviour o Hill Climbing, Local Search and Tabu Search is similar considering the similar pseudo gradient principle. Substandard results were achieved with the Downhill Simplex method. This optimization method works by calculating the points o the centroid (center o gravity o the simplex). We have to modiy this optimization method in such a way that it is applicable or discrete event simulation optimization purposes where the step in the search space is deined. We use the rounding o coordinates o the vector (new calculated point) to the nearest easible coordinates in the search space and this leads to deviation rom the original direction. We perormed other simulation experiments with smaller steps and the success o inding the optimum was higher than beore. This problem can be solved by using a calculation with the original points and the obective unction value will be calculated by the approximations o the obective value o the nearest easible points in the search. Dierential Evolution uses the elitism strategy in our case. This leads to copying o identical individuals which suppresses the diversity o new promising individuals. Random Search looks successul, but there were only two possible settings generating the same individual possibility. This evaluation can be modiied by using the coeicient which recalculates the value o success depending on the number o perormed series. The termination criterion was the number o possible Proceedings o the European Modeling and Simulation Symposium, 01 56

solutions in the search space when there is little search space. This led to increasing the probability o success o this optimization method. the obective unction value o the best solution ound in concrete series; denotes obective unction value o the worst solution (element) o the search space. X Worst The dierence between the optimum and the local extreme is shown in igure 17 (testing unctions) and igure 18 (discrete event simulation models). The charts contain only series where the 1 = 0 (no optimum was ound in the series). The average o second criterion is shown or each optimization method these values express the ailure o the optimization method. Output. o unction can take value 0,1 igure 15: Percentage o Absolutely Successul Series Compared To All Perormed Series - Testing unctions igure 17: Average o the Second Criterion - Dierence between Optimum and Local Extreme - Testing unctions igure 16: Percentage o Absolutely Successul Series Considering All Perormed Series - Discrete Event Simulation Models 6.. The Dierence between Optimum and Local Extreme The second criterion is useul when there is no series which contains any optimum or the solution whose obective unction value is within the tolerance o optimum obective unction value. The irst criterion 1 equals zero in this case. The unction where the output o the unction can take value 0,1. This unction evaluates the dierence between the obective unction value o the best solution ound in the series and the optimum obective unction value. The eort is to minimize. The list o ound optimums considering obective unction value using the comparator unction is sorted in ascending order. Ater that the value o the second criterion is calculated using the ormula: X Best X X Worst X (10) where X denotes the obective unction value o the global optimum o the search space; X Best denotes igure 18: Average o the Second Criterion - Dierence between Optimum and Local Extreme - Discrete Event Simulation Models 6.. The Distances o Quartiles Third criterion expresses the distance between quartiles o a concrete series. Weights are used or evaluation purposes. These weights penalize the solutions) placed in quartiles. Values o the weights were deined based on the results o the simulation experiments. The user can deine the weight value. The sum o weights equals one. The third criterion when the obective unction is minimized can be ormulated as ollows: Proceedings o the European Modeling and Simulation Symposium, 01 57

Q1 X w4 Q1 Q w Q Q w Q Q4 w1 Q 4 Q (10) 5 X X Worst where denotes the obective unction value o the global optimum o the search space; X w 4 denotes the weight (penalty) o obective unction values between sample minimum Q 1 and lower quartile Q ; w denotes the weight o obective unction values between lower quartile Q and median Q ; w denotes the weight o obective unction values between median Q and upper quartile Q 4; denotes the weight o w1 obective unction values between upper quartile Q 4 and largest observation - sample maximum Q 5; X Worst denotes obective unction value o the worst solution (element) o the search space. The evaluation o optimization experiments using the third criterion is shown in igure 19 and in igure 0. igure 19: Average o the Third Criterion - Distances o Quartiles - Testing unctions worst optimization results o all tested optimization methods due to rounding the coordinates. Pseudo gradient optimization methods ound solutions o similar quality. Simulated Annealing provides a worse solution than the Evolution Strategy. 6.4. The Number o Simulation Experiments Until the Optimum Was ound The ourth criterion 4 evaluates the speed o inding the optimum the number o perormed simulation experiments until the optimum/best solution was ound in each series. The eort is to minimize 4 ( ). The ourth criterion when the obective unction is minimized can be ormulated as ollows: ~ X 4 0,1 Q1 1 w4 Q1 Q w Q Q w Q Q4 w1 Q4 Q 4 4 4 4 5 4 m where w 4 4 (11) denotes the weight (penalty) o number o simulation experiments until the optimum was ound between sample minimum Q 1 and lower quartile Q ; w 4 denotes the weight o number o simulation experiments until the optimum was ound between lower quartile Q and median Q ; w denotes the weight o number o simulation experiments until the optimum was ound between median Q and upper quartile Q 4; denotes the weight o number o w 1 4 simulation experiments until the optimum was ound between upper quartile Q 4 and largest observation - sample maximum Q 5; denotes the number o m ~ X easible solutions in the search space. The evaluation o optimization experiments using the third criterion is shown in igure 1 and in igure. 4 igure 0: Average o the Third Criterion - Distances o Quartiles - Discrete Event Simulation Models 0,1 ). I the irst criterion equals zero 1 then the third criterion equals zero 0 (absolutely successul series). The Downhill Simplex optimization method provided the The eort is to minimize ( igure 1: Average o the ourth Criterion 4 - Number o Simulation Experiments until the Optimum Was ound - Testing unctions Proceedings o the European Modeling and Simulation Symposium, 01 58

and with the subsidy o the Motivation System (POSTDOC) o University o West Bohemia. The paper uses the results o the proect CZ.1.07/..00/09.016 carried out with the support o Ministry o Education, Youth and Sports. igure : Average o The ourth Criterion 4 - Discrete Event Simulation Models 7. CONCLUSION The goal o our research is to compare selected modiied optimization methods (Random Search, Hill Climbing, Tabu Search, Local Search, Downhill Simplex, Simulated Annealing, Dierential Evolution and Evolution Strategy) used in the developed simulation optimizer and used in the second module which is ocused on testing the implemented optimization methods. Optimization methods generate whole populations instead o one possible solution which prevents premature convergence. The success o optimization methods depends on the obective unction landscape. Evolution Strategy is a suitable optimization method or all the tested obective unctions (a little propensity to bad tuning o the method parameters). This optimization method achieves good values or speciied criteria. The alternative to Evolution Strategy optimization methods is Simulated Annealing. Simulated Annealing has the ability to escape rom the local extreme thanks to the implemented approach o setting the temperature to the initial temperature. We can expect to ind good results using Random Search i there is a small search space. I the dimension o the search space is bigger, there is little probability o success. Optimization methods based on pseudogradient searching such as Hill-Climbing, Local Search, Tabu Search achieve almost the same results or the simple obective unction landscape due to their similar nature. Dierential Evolution avoids repressing the diversity o solutions (elitism - an advantage o this approach is the aster inding o a easible solution but not the inding o the global optimum). The range o provided simulation optimization results using this optimization method is better than the optimization methods based on pseudo-gradient searching. ACKNOWLEDGEMENTS This paper was created with the subsidy o the proect SGS-01-06 Integrated production system design as a meta product with use o a multidisciplinary approach and virtual reality carried out with the support o the Internal Grant Agency o University o West Bohemia REERENCES Koblasa,., Manlig,., Vavruska, J., 01. Evolution Algorithm or Job Shop Scheduling Problem Constrained bythe Optimization Timespan. Applied Mechanics and Materials, 09, 50-57. Kopecek, P, 01. Heuristic Approach to Job Shop Scheduling. DAAAM International Scientiic Book 01, pp. 57-584. October 4-7, Zadar (Croatia). Michalewicz, Z., ogel, D. B., 004. How to Solve It: Modern Heuristics, Berlin: Springer, Miranda, V., 008. undamentals o Evolution Strategies and Evolutionary Programming. In: El-Hawary, M.E., ed. Modern heuristic optimization techniques. New Jersey: John Wiley & Sons, 4 60. Monticelli, A.J., Romero, R., Asada, E., 008. undamentals o Tabu Search. In: El-Hawary, M.E., ed. Modern heuristic optimization techniques. New Jersey: John Wiley & Sons, 101 10. Monticelli, A.J., Romero, R., Asada, E., 008. undamentals o Simulated Annealing. In: El-Hawary, M.E., ed. Modern heuristic optimization techniques. New Jersey: John Wiley & Sons, 1 144. Pohlheim, H., 006. Genetic and Evolutionary Algorithm Toolbox or use with MATLAB. GEATbx. Available rom: http://www.geatbx.com/docu/cnindex-01.html [accessed 0 November 011] Tvrdik, J., 004. Evolutionary Algorithms - textbook. Virtual inormation center or Ph.D. students, Ostrava University. Available rom: http://pr.osu.cz/doktorske_studium/dokumenty/ev olutionary_algorithms.pd [accessed 6 ebruary, 011] Votava, V., Ulrych, Z., Edl, M., Korecky, M., Trkovsky, V., 008. Analysis and Optimization o Complex Small-lot Production in new Manuacturing acilities Based on Discrete Simulation. Proceedings o 0th European Modeling & Simulation Symposium EMSS 008, pp. 198-0. September 17-19, Campora San Giovanni (Amantea, Italy). Weise, T., 009. Global Optimization Algorithms - Theory and Application nd Edition. Thomas Weise - Proects. Available rom: http://www.itweise.de/proects/book.pd [accessed ebruary, 01] Wong, K.P., Dong, Z.Y., 008. Dierential Evolution. In: El-Hawary, M.E., ed. Modern heuristic optimization techniques. New Jersey: John Wiley & Sons, 171 186. Proceedings o the European Modeling and Simulation Symposium, 01 59