USING ARTIFICIAL NEURAL NETWORKS TO APPROXIMATE A DISCRETE EVENT STOCHASTIC SIMULATION MODEL

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1 USING ARTIFICIAL NEURAL NETWORKS TO APPROXIMATE A DISCRETE EVENT STOCHASTIC SIMULATION MODEL Robert A. Kilmer Department of Sytem Engineering Unite State Military Acaemy Wet Point, NY 1996 Alice E. Smith Department of Inutrial Engineering Univerity of Pittburgh Pittburgh, PA ABSTRACT: A computer imulation moel may be regare a a tochatic function that map a et of input to a et of output; in many cae computer imulation moel are quite computationally expenive. It woul be beneficial to have fat, accurate approximation of computer imulation moel to perform uch tak a quick turnaroun tuie, enitivity analye, moel aggregation/reuction, an imulation optimization. Thi paper examine the ue of two metho, artificial neural network (ANN) an multiple linear regreion, for approximating a lot ize - reorer point inventory ytem imulation. INTRODUCTION Computer imulation ha been ientifie in urvey a the mot wiely ue tool of inutrial engineer an management cientit (Pritker, 1992); however, computer imulation can become computationally expenive in term of proceing time an/or memory requirement. For thee reaon it woul be beneficial to have a quick proceing metho that provie accurate approximation of computer imulation. Thi paper provie ANN approximation of a computer imulation an compare the reult with firt an econ orer linear regreion moel for interpolation an extrapolation. There i a nee to reuce the computational buren of running computer imulation particularly for avance tak uch a imulation optimization. The work in thi paper i bae upon two aumption: (1) there i little or no obervational ata of the real ytem uner conieration an (2) an accurate, but low, mechanitic computer imulation of the ytem exit. The coure of action that we are invetigating for thi ituation i to obtain ata from the mechanitic computer imulation in orer to formulate a fat operating empirical moel of the ytem.

2 There are three major ifference between uing ANN to approximate tochatic imulation an uing ANN to perform function approximation. Firt, a given et of input yiel ifferent output, compouning training. Secon, training an teting ata i computationally expenive to generate, an therefore mut be leverage. Thir, training an teting ata i uually eigne, i.e. i not choen uniformly ranomly from the problem omain. (Hurrion, 1992) howe that it i poible to fit a neural network to moel the generalize repone of parameter change in a viual interactive imulation. Even though hi reearch wa not a ytematic examination of the ANN approach to approximating computer imulation, it howe how thi approach coul ait imulation uer. The firt iue examine in thi paper involve a comparion of ANN approximation with the type of multiple linear regreion typically ue in repone urface metho. The econ iue coniere i ata preentation metho, e.g., preenting iniviual replication or mean from the computer imulation to the ANN. Since ANN uually o better with more training ata, it eem reaonable to try to evelop ANN moel uing the iniviual replication of imulation output. On the other han, ince the operation of the ANN an regreion moel both provie eterminitic reult (i.e., for any given input there i only one output repone) both try to approximate the expecte value of the output of the imulation. THE INVENTORY SYSTEM SIMULATION The ytem examine i the inventory ytem ecribe in the experimental eign an optimization chapter of (Law an Kelton, 1991). Thi ytem i a lot ize-reorer point ytem (i.e., an (S,) inventory ytem where S = orer quantity an = reorer point). Law an Kelton hel all input parameter fixe except for = reorer point, an = S - = reorer quantity. It houl be note that i jut a convenient reparameterization of the reorer quantity, S. The only output meaure wa the average annual total cot, tc, of operating the inventory ytem. For comparion purpoe, the work preente in thi paper wa limite to the ame input parameter an output meaure. The mechanitic moel of the inventory ytem wa built with the SIMAN imulation language a a tochatic, icrete-event, terminating computer imulation. The ata ue to evelop the empirical approximation moel conite of two ifferent et of input parameter. The firt ata et conite of 4 combination of an, where = {2, 6} an = {1, 5}. The econ et conite of 36 combination of an, where = {, 2, 4, 6, 8, 1} an = {5, 2, 4, 6, 8, 1}. Our tet et of 42 point wa contructe with all poible combination of 21 equally pace value of between an 1 (i.e.,, 5, 1,..., 1) an 2 equally pace value of between 5 an 1 (i.e., 5, 1, 15,..., 1). Ten replication of the computer imulation were mae for each combination of an for the tet et an both training et; the mean tc of each et of ten provie the correct, or target, anwer, a wa one in Law an Kelton. Moel A an B tet the ability of approximation technique to extrapolate beyon the four ata point ue to evelop the approximation tool. Moel C an D examine the ability of the approximation technique to interpolate between the 36 ata point. Two ifferent metho of preenting the target ata to the

3 approximation tool were ue: one ue the mean tc value over the ten replication (moel A an C), an the other ue the iniviual tc value from each replication (moel B an D). Thee are ummarize in Table 1. TABLE 1: DATA PRESENTATION METHODS USED Moel Label Data Set Target Data Preentation Metho A 4 point mean B 4 point iniviual replication C 36 point mean D 36 point iniviual replication THE REGRESSION MODELS Moel A an B ue full, firt-orer multiple linear regreion moel. Moel, C an D ue full, econ-orer multiple linear regreion moel. Thee regreion moel were electe to permit comparion with Law an Kelton reult, an becaue they are the type typically ue in repone urface metho (Donohue, 1988). The linear regreion moel obtaine in each cae are hown in Table 2. The F tet for regreion wa ignificant for α =.1 for regreion moel B, C an D. In aition, the regreion moel obtaine by Law an Kelton for moel B* an D* are alo provie in Table 2 for valiation/verification purpoe. There i cloe agreement in the coefficient of the regreion equation for moel B an B* an for moel D an D*. It houl be note that the lat three value in the equation for D* are milabele in Law an Kelton. Thi can be een by olving for the minimum of the function. The minimum value for the Law an Kelton function i foun at = 148 an = -42. By making the change given in Table 2, row D*, the minimum value i at = 29 an = 68. Thee reult are conitent with the contour iagram of the function foun in Law an Kelton, an in Figure 1c in thi paper. Moel Label Regreion Equation TABLE 2: REGRESSION MODELS P Value of F Statitic (egree of freeom) R 2 A Cot = (2,1).8 B Cot = *E -12 (2,37).76 B* Cot = Not Given NA C Cot = *E -9 (5,3).78 D Cot = *E -11 (5,354).77 D* Cot = Not Given NA * Reult from Law an Kelton Note that regreion moel A an B are the ame moel, an that C an D are the ame moel. Thi i jut a manifetation of the regreion attempting to etimate the expecte value of the output for any particular input value. The benefit of incluing all of the ata in regreion (moel B an D) i that lack of fit tet can be performe, although thi wa not one here.

4 THE ANN MODELS The ANN moel ue were feeforwar, fully connecte, traine with the backpropagation algorithm with a moothing factor uring training. All of the ANN moel ha two input noe, one correponing to an the other correponing to, an one output noe correponing to tc. The pecific architecture i given in Table 3. All of the ANN' were traine up to a maximum of 5, epoch with a convergence criterion of.5 for the tolerance value. The reult of training the ANN moel for the variou ata et in each experiment are provie in Table 3. A can be een from the table, the ANN that wa traine on the mean ata require fewer hien noe an fewer preentation of the ata to train the network, which wa expecte ince the ata ha alreay been conveniently preprocee. It wa not expecte that the ANN learn all the ata point from the iniviual replication to.5, ince for each combination of an, there were 1 ifferent tc target. TABLE 3: ANN MODELS Moel Label Architecture # Data Point Not Learne To.5 / Total # Of Training Iteration A 1 hien layer, / 4 2,849 1 noe B 2 hien layer, 1 / 4 8,32 1 noe each C 1 hien layer, 1 noe / 36 1,111 D 2 hien layer, 1 noe each 48 / 36 24,29 RESULTS The contour plot (line connect point of equal tc) given in Figure 1 permit a viual comparion of the regreion an ANN moel with the original computer imulation. Figure 1a plot the mean of 1 replication of the computer imulation at the 42 tet point. Figure 1b plot the mean of 1 replication of the 36 point (i.e., the training et for moel C). Figure 1c an 1 are the plot of the regreion an ANN moel, repectively, that were contructe uing the mean of 1 replication of the 36 ata point (moel C). The bet one coul hope for from the ANN or the regreion woul be a contour plot ientical to 1b. Thee plot how that the ANN (1) i cloer than the econ orer regreion (1c) to approximating the computer imulation. Each regreion moel an each ANN were tete with the 42 value from the tet et. The experimental reult for both the regreion moel an the ANN moel are provie in Table 4. Moel A an B were ue to examine the extrapolation ability of the approximation metho uing only four combination of an for training. The relatively high MAE for both the ANN an the regreion inicate the anger in trying to preict repone in region where training ata ha not been provie. Moel C an D were ue to examine the interpolation ability of the approximation metho. The reult in Table 4 inicate that the ANN, regarle of training on mean ata or on iniviual

5 replication, outperforme the correponing regreion moel. Thee reult were ignificant at α =.1 for a matche t tet. Thi inicate that ANN moel can provie better reult for interpolating from a computer imulation than the multiple firt an econ orer regreion moel commonly ue in repone urface metho. Although not performe here, higher orer regreion moel might yiel better reult than thee lower orer moel Fig. 4a. Direct imulation 42 point. Fig. 4b. Direct imulation 36 point Fig. 4c. Regreion on 36 point. Fig. 4. ANN on 36 point. TABLE 4: COMPARISON OF TEST SET RESULTS Moel Label # of an Point # of Training Vector Regreion MAE ANN MAE A B C D

6 Table 4 alo permit a comparion of the metho of preenting target ata for eveloping each approximation moel. For the regreion approximation technique, both metho of preenting output ata (mean an replication) yiel the ame accuracy in term MAE. While the mean metho (moel A an C) ha lightly better R 2 value than the correponing iniviual replication metho (moel B an D), they ha le ignificant level for the F tatitic (ee Table 2). The ANN traine on the iniviual replication ha a very lightly larger MAE. However, it houl be note that thi i not a fair comparion ince the ANN traine on iniviual replication ue more hien noe an require many more training iteration in orer to obtain the achieve preciion. Our current work fully invetigate the two preentation metho for ANN approximation of imulation moel. CONCLUSIONS The reult of thi imple tuy emontrate that for mall number of input parameter an output meaure the ANN approach can outperform the firt an econ orer linear regreion moel typically ue in imulation repone urface metho. Training the ANN on iniviual replication may prouce comparable reult, but at the expene of increae training time, an poibly larger network ize. Our current work expan the two input imulation to four input, an fully explore the alternative preentation metho of mean an iniviual replication. We alo evelop an ANN to preict variance a well a expecte value for tc. Thi network i ue to evelop ANN preiction interval for interpolate point. Finally we have ue our methoology on a full cale, real worl imulation of a hopital emergency room. Future work will take the ANN moel of the imulation an ue them to perform imulation optimization. REFERENCES Donohue, M., (1988). The Ue of Correlate Simulation Experiment in Repone Surface Optimization. Ph.D. Diertation, Virginia Polytechnic Intitute an State Univerity. Hurrion, R. D., (1992). Uing a neural network to enhance the eciion making quality of a viual interactive imulation moel, Journal of Operation Reearch Society, 43 (4), Law, A. an Kelton, D., (1991). Simulation Moeling an Analyi. McGraw- Hill, New York. Pritker, A. A., (1992). Moelling for imulation analyi, in Hanbook of Inutrial Engineering (G. Salveny, E.). John Wiley & Son, New York.