ScienceDirect. Testing Optimization Methods on Discrete Event Simulation Models and Testing Functions

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1 Available online at ScienceDirect Procedia Engineering 69 ( 2014 ) th DAAAM International Symposium on Intelligent Manuacturing and Automation, 2013 Testing Optimization Methods on Discrete Event Simulation Models and Testing unctions Pavel Raska, Zdenek Ulrych Department o Industrial Engineering - aculty o Mechanical Engineering, University o West Bohemia, Univerzitni 22, Pilsen , Czech Republic Abstract The paper deals with testing o selected heuristic optimization methods and their evaluation. We have proposed dierent techniques which express the success o the optimization method in dierent ways (the method success, the dierence between optimum and local extreme, the distances o quartiles, the number o simulation experiments until the optimum was ound). These evaluation techniques use box plot characteristics calculated rom the repeated optimization experiments The Authors. Published by Elsevier Ltd The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility o DAAAM International Vienna. Selection and peer-review under responsibility o DAAAM International Vienna Keywords: Simulation Optimization; Evaluation; Heuristic Optimization Methods; Discrete Event Simulation Models; Testing unction 1. Introduction Many industrial companies are solving the problem o how to design their system (production system, logistic, etc.) as eectively as possible. We have to say that this problem is aected by many internal or external company actors. There exist many possible scenarios how to solve these problems, but which o these scenarios is the best (optimal)? Is it possible to imagine how a change in the subsystem aects the entire system? We have to say that many o these NP-hard problems are impossible to solve using only the human actor or by static calculation. One o the possible answers to the previous question is the use o discrete event simulation in connection with optimization. Many present simulation sotware packages use their own integrated simulation optimizers which are black-boxes. We can list some problems o these integrated simulation optimizers: e.g. the user cannot set or tune the parameters Corresponding author. Tel.: ; ax: address: praska@kpv.zcu.cz The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility o DAAAM International Vienna doi: /.proeng

2 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) o the optimization method even in the case where the user has appropriate inormation about the obective unction type. The simulation optimizer switches optimization methods (e.g. neural network) to ind an eective method and the user does not know which methods were selected and the success o the method; the user cannot implement or modiy a possible eective optimization method; no possibility to save experimental data rom the simulation to a database (knowledge database generation), etc. We have developed our own simulation optimizer (automatically varies the simulation models input parameters values to achieve the best coniguration o the simulation models) considering the needs o the department to solve practical simulation proects [1], [2] and the module or testing the implemented optimization methods and the optimization methods parameters settings (this module does the same as the irst module but it is mainly ocused on evaluation o optimization method behavior). We have proposed and implemented some evaluation techniques to better understand the behavior o the optimization methods. 2. Developed Application Implemented Optimization Methods Many integrated simulation optimizers use similar optimization methods e. g. Simulated Annealing, Tabu Search, etc. Ater the literature review we selected commonly used optimization methods to compare their eiciency in searching or the global optimum in the Search space. These methods are: Random Search (a new candidate solution is generated in the search space with uniorm distribution - Monte Carlo method) Stochastic Hill Climbing (candidate solutions individuals in the population - are generated in the neighborhood o the best candidate solution rom the previous population. Generating new possible solutions is perormed by mutation) [3] Stochastic Tabu Search (i a new candidate solution is generated, it becomes an element o the Tabu List. This solution cannot be visited again i the aspiration criterion is not satisied. 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) [4], [5] Stochastic Local Search (a candidate solution is generated in the neighborhood o the best candidate solution.) [3] Stochastic Simulated Annealing (a candidate solution is generated in the neighborhood o the candidate solution known 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) [4], [5] We implemented the basic principle o evolutionary algorithms into some o these optimization methods (generating a whole population instead o one possible solution in order to avoid getting stuck on a local optimum). Previous testing o optimization methods conirmed that generating one solution leads to premature convergence (depending on obective unction type). We united dierent variants o selected optimization methods. The user can choose a dierent variant o the optimization method by clicking on the checkbox. We have also tested other optimization methods used in optimization o continuous simulation: Downhill Simplex (this heuristic 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) [5], [6] Dierential Evolution (the selection is carried out between the parent and its ospring (the ospring is created through a crossover between the parent and the new 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) [6], [7], [8], [9]

3 770 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) Evolution Strategy (the optimization method uses Steady State Evolution population consists o children and parents with good itness. The individual - child - is generated in the neighborhood o the other individual parent. The method uses the Rechenberg 1/5th-rule. The population is sorted according to the obective values - Rank-Based itness Assignment. The optimization method uses Tournament selection) [6], [10], [11] 3. Discrete event simulation models and testing unctions We have tested the optimization methods abilities to search or the global optimum or three discrete event Arena simulation models. These models relect real production systems in Czech industrial companies. Each model has a speciic obective unction considering the simulated system and the simulation goal. The entire search space o each simulation model was mapped to ind the global optimum o the obective unction. The Manuacturing System and Logistics model - 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, 28 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 Penalty model - 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 (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 Assembly Line model - 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 input simulation model parameters (controls) are the number 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. We tested the implemented optimization methods on our standard testing unctions. All testing unctions were minimized. De Jong s unction the unction is a continuous, convex and unimodal testing unction. The unction deinition: n 2 X x (1) 1 where X denotes the obective unction; denotes index o control; n denotes the dimension o the search space; x denotes the value o control. Rosenbrock s unction - Rosenbrock s (Rosenbrock's valley, Rosenbrock's banana) unction is a continuous, unimodal and non convex testing unction. The unction deinition:

4 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) n1 X (2) ( x 2 2 x 1) 1 x 1 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). The unction deinition: [12] n X 1 x sin( x ) sin 2 2m (3) 1 : n, 0 x (4) We selected m 5 in our simulation model. Ackley s unctions - Ackley s unction is a multimodal test unction. This unction is a widely used testing unction or premature convergence. The unction deinition: 1 n n 1 n 2 X 20exp 0.02 x exp cos2 x 20 exp 1 1 n 1 (5) 1: n, 30 x 30 (6) ig. 1. Obective unction - The Manuacturing System and Logistics Discrete Event Simulation Model - Number o orklits or Large Parts = 14; Obective unction The Penalty Discrete Event Simulation Model. ig. 2. Obective unction - The Assembly Line (discrete event simulation model); Obective unction - Rosenbrock s unction.

5 772 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) ig. 3. Obective unction - Michalewicz s unction; Obective unction - Ackley s unction. 4. Evaluation o optimization experiments The behavior o optimization algorithms is random, so we had to perorm many optimization experiments to identiy the pure nature o the optimization algorithms. Considering the number o simulation experiments we can divide the number o simulation experiments as ollows: 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. We speciied the same conditions which had to be satisied or each optimization method, e.g. the same termination criteria, the same search space where the optimization method can search or the global optimum. 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). The second module is ocused on testing the behavior o 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 optimization method parameters, number o replications, and export the obective unction chart to image. The results o optimization series are exported to MS Excel workbook. Excel was selected because o its wide usage, speciying ormulas, visualizing the data to charts, etc. Ater inishing the series boxplot characteristics are calculated (the smallest observation sample minimum Q 1, lower quartile Q 2, median Q 3, upper quartile Q 4, and largest observation - sample maximum Q 5 ) and three boxplot 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. We have to propose evaluation techniques which express the ailure o the optimization method in dierent ways due to the large volume o data (over 4 billion simulation experiments). Each criterion value is between [0, 1]. I the ailure is 100[%] the criterion equals 1 thereore we try to minimize all speciied criteria. The user can set up the weight o each criterion. Other parameters necessary or evaluation are automatically loaded rom simulation optimization results Optimization Method Success The irst criterion is the value o not inding the known VTR (value to reach). This value is expressed by:

6 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) s n 1 s succ (7) 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. Average Method Success o inding Optimum can be ormulated as ollows: avg 1 s 100 % s 1i i1 (8) where i denotes the index o one series, 1 denotes the value o the irst criterion, s denotes the number o i perormed series. The average optimization method success o inding the optimum o testing unctions is shown in ig. 4. ig. 4. Average Optimization Method Success Simulation Optimization Results o Testing unctions; Average Method Success Simulation Optimization Results o Discrete Event Simulation Models. 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 in a small search space (we have to evaluate each possible solution in all the search spaces to obtain the optimum hence the search space cannot be too huge). 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 optimization methods were not successul in inding the optimum, because the area around the optimum is straight and the method could not obtain inormation about raising or decreasing the obective unction terrain. 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 (the irst criterion equals zero, so the optimum was ound in each optimization experiment). The percentage o absolutely successul series compared to all perormed series is shown in ig. 5. The Evolution Strategy has problems with the multimodal Ackley s unction (the method was aected by the number o individuals randomly chosen rom the population or the tournament). This setting aects the exploration (the procedure which allows search operations to ind new and maybe better solution structures). The opposite o exploration is the exploitation o the search space (the process o improving and

7 774 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) combining the traits o the currently known solutions, as done by the crossover operator in evolutionary algorithms, or instance). The behavior o Stochastic Hill Climbing, Stochastic Local Search and Stochastic Tabu Search is similar (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 (the copying o identical individuals suppresses the diversity o new promising individuals exploration vs. exploitation. Random Search looks successul, but there were only two possible settings generating the same individual possibility. ig. 5. Percentage o Absolutely Successul Series Compared To All Perormed Series - Testing unctions; Percentage o Absolutely Successul Series Considering All Perormed Series - Discrete Event Simulation Models 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 equals zero in this case). 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 second criterion. The second criterion is calculated using the ormula: 2 X X Best X X Worst (9) where X denotes the obective unction value o the global optimum o the search space; X Best denotes the obective unction value o the best solution ound in a concrete series; X Worst denotes obective unction value o the worst solution o the search space.

8 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) The dierence between the optimum and the local extreme is shown in ig. 6. The charts contain only series where the irst criterion equals zero (no optimum was ound in the series). The average o the second criterion is shown or each optimization method these values express the ailure o the optimization method. ig. 6. Average o the Second Criterion (Dierence between Optimum and Local Extreme) - Testing unctions; Average o the Second Criterion - Discrete Event Simulation Models The Distances o Quartiles The 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: 3 Q1 where X X w4 Q1 Q2 w 3 3 Q2 Q3 w2 Q 3 3 Q4 w1 Q 3 4 Q5 (10) 3 X X Worst denotes the obective unction value o the global optimum o the search space; w 43 denotes the weight (penalty) o obective unction values between sample minimum Q 1 and lower quartile Q 2 ; w 33 denotes the weight o obective unction values between lower quartile Q 2 and median Q 3 ; w 23 denotes the weight o obective unction values between median Q 3 and upper quartile Q 4 ; w 13 denotes the weight o 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 ig. 7. ig. 7. Average o the Third Criterion (Distances o Quartiles) - Testing unctions; Average o the Third Criterion - Discrete Event Simulation Models.

9 776 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) The eort is to minimize the value o the third criterion. I the irst criterion equals zero then the third criterion equals zero. The Downhill Simplex optimization method provided the worst optimization results o all tested optimization methods due to rounding o the coordinates. Pseudo gradient optimization methods ound solutions o similar quality. Simulated Annealing provides a worse solution than the Evolution Strategy The Number o Simulation Experiments Until the Optimum Was ound The ourth criterion evaluates the speed o inding the optimum the number o perormed simulation experiments until the optimum/best solution was ound in each series see ig. 8. ig. 8. Average o the ourth Criterion (Number o Simulation Experiments until the Optimum Was ound) - Testing unctions; Average o The ourth Criterion - Discrete Event Simulation Models. The ourth criterion when the obective unction is minimized can be ormulated as ollows: Q1 1 w4 Q1 Q2 w3 Q2 Q3 w2 Q3 Q4 w1 Q4 Q m ~ X (11) where w denotes the weight o number o simulation experiments until the optimum was ound between sample 4 4 minimum Q 1 and lower quartile Q 2 ; w denotes the weight o number o simulation experiments until the optimum 3 4 was ound between Q 2 and median Q 3 ; w denotes the weight o number o simulation experiments until the 2 4 optimum was ound between Q 3 and upper quartile Q 4 ; w denotes the weight o number o simulation experiments 1 4 until the optimum was ound between Q 4 and largest observation Q 5 ; m ~ denotes the number o easible solutions in X the search space. 5. Conclusion The goal o the research is to compare selected optimization methods - Random Search, Stochastic Hill Climbing, Stochastic Tabu Search, Stochastic Local Search, Downhill Simplex, Simulated Annealing, Dierential Evolution and Evolution Strategy. The success o heuristic optimization methods depends on the obective unction landscape. Evolution Strategy is a suitable optimization method or all the tested obective unctions (little propensity to bad methods or tuning parameters). This optimization method achieves good values or the second criterion (distance between ound local optimum and global optimum or VTR). The alternative to Evolution Strategy 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. The strategy o Random Search is simple

10 Pavel Raska and Zdenek Ulrych / Procedia Engineering 69 ( 2014 ) and eective with a small search space, but i the search space is huge (NP-hard) we can say it is lucky to ind the optimum. Pseudo-gradient optimization methods (Stochastic Hill-Climbing, Stochastic Local Search, Stochastic Tabu Search) achieve almost the same results or the simple obective unction landscape. Dierential Evolution uses the elitism strategy (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 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 and with the subsidy o the Motivation System (POSTDOC) o University o West Bohemia. The paper uses the results o the proect CZ.1.07/2.3.00/ Reerences [1] V. Votava, Z. Ulrych, M. Edl, V. Trkovsky, M. Korecky, Analysis and Optimization o Complex Small-lot Production in new Manuacturing acilities Based on Discrete Simulation, in: Proceedings o 20th European Modeling & Simulation Symposium EMSS 2008, Campora San Giovanni, 2008, pp [2] P. Kopeček, Heuristic Approach to Job Shop Scheduling, in: DAAAM International Scientiic Book 2012, Published by DAAAM International, Vienna, Austria, 2012, pp [3] P. Maer, Modern Methods o Production Scheduling (Czech language: Moderní metody rozvrzhování výroby), PhD. Thesis, University o Technology, aculty o Inormation Technology, Brno, [4] A. J. Monticelli, R. Romero, E. N. Asada, undamentals o Simulated Annealing, in: Modern Heuristic Optimization Techniques, IEE Press ed., M. E. El-Hawary, Ed., New Jersey, John Wiley & Sons, 2008, pp [5] T. Weise, E-Book Global Optimization Algorithms - Theory and Application 2nd Edition, Available: Accessed: [6] J. Tvrdík, Evolutionary algorithms - Textbooks (Czech language Evoluční algoritmy - učební texty), University o Ostrava, Available: Accessed: [7] K. P. Wong a Z. Y. Dong, Dierential Evolution, in: Modern Heuristic Optimization Techniques, M. E. El-Hawary, Editor, New Jersey, John Wiley & Sons, 2008, pp [8] X. Zeng, W.-K. Wong a S. Y.-S. Leung, An operator allocation optimization model or balancing control o the hybrid assembly lines using Pareto utility discrete dierential evolution algorithm, Computers & Operations Research, Vol. 39, Issue. 5, May 2012, pp [9] L. Xiangtao a Y. Minghao, An opposition-based dierential evolution algorithm or permutation low shop scheduling based on diversity measure, Advances In Engineering Sotware, Vol. 55, January 2013, pp [10] V. Miranda, undamentals o Evolution Strategies and Evolutionary Programming, in: Modern heuristic optimization techniques, M. E. El- Hawary, Editor, New Jersey, John Wiley & Sons, 2008, pp [11]. Koblasa,. Manlig, J. Vavruška, Evolution Algorithm or Job Shop Scheduling Problem Constrained bythe Optimization Timespan, Applied Mechanics and Materials, 2013, pp [12] H. Pohlheim, GEATbx: Example unctions, Available: Accessed