SIMULATION BASED REGRESSION ANALYSIS FOR RACK CONFIGURATION OF AUTONOMOUS VEHICLE STORAGE AND RETRIEVAL SYSTEM. Banu Y. Ekren Sunderesh S.

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1 Prceedings f the 2009 Winter Simulatin Cnference M. D. Rssetti, R. R. Hill, B. Jhanssn, A. Dunkin and R. G. Ingalls, eds. SIMULATION BASED REGRESSION ANALYSIS FOR RACK CONFIGURATION OF AUTONOMOUS VEHICLE STORAGE AND RETRIEVAL SYSTEM Banu Y. Ekren Sunderesh S. Heragu Dept. f Industrial Engineering University f Luisville Luisville, KY 40292, USA ABSTRACT In this study, a simulatin based regressin analysis fr rack cnfiguratin f an autnmus vehicle strage and retrieval system (AVS/RS) is presented. We develp a mathematical functin fr rack cnfiguratin f an AVS/RS that reflects the relatinship between the utput (respnse) and the input variables (factrs) f the system. In the regressin mdel, the utput is the average cycle time fr strage and retrieval and the input variables are the number f tiers, aisles and bays that determine the size f the warehuse. The simulatin mdel f the system is develped using ARENA 12.0, a cmmercial sftware. We use MINITAB statistical sftware t cmplete the statistical analysis and t fit a regressin functin. Tw different appraches are used fr develping the regressin analysis stepwise regressin and the best subsets. We ptimize the regressin functin using the LINGO sftware. We apply this apprach t a cmpany that uses AVS/RS in France. 1 INTRODUCTION Autnmus vehicle strage and retrieval systems (AVS/RSs) represent a relatively new technlgy fr autmated unit-lad strage systems (Malmbrg 2002). In this system, autnmus vehicles (AVs) functin as strage/retrieval (S/R) devices. The mst imprtant difference between an AVS/RS and a traditinal crane-based autmated strage and retrieval system (AS/RS) is the mvement pattern f the S/R device. The AVs in an AVS/RS fllw rectilinear flw patterns fr hrizntal travel. Hwever, in AS/RSs, strage cranes are aisle-captive and capable f simultaneus mvement in the hrizntal and vertical dimensins stre r retrieve unit-lads. The travel pattern in an AS/RS is generally mre efficient within strage racks. Neverthless, an AVS/RS has a significant ptential advantage in the adaptability f system thrughput capacity t transactins demand by changing the number f vehicles perating in a fixed strage cnfiguratin. Fr example, increasing the number f vehicles decreases the transactin cycle times and utilizatin which are als key measures f system perfrmance. It is crucial t design an AVS/RS in such a way that it can efficiently handle the current and future demand requirements while aviding bttlenecks and excess capacity. Due t the relative inflexibility f the physical layut and the equipment, it is imprtant t design it right the first time. T be able t evaluate all pssible design scenaris fr a suitable physical layut f a warehuse with an AVS/RS, we develp a regressin analysis that represents the relatinship between a particular AVS/RS s utput perfrmance average cycle time, and input variables numbers f tiers, aisles and bays. Develpment f a regressin functin makes pssible, a better understanding f the true relatinship between input variables and the utput. In this study, the real data are btained frm a cmpany that uses this system in France. 2 LITERATURE REVIEW An AVS/RS transprts a unit-lad using rail guided autmated vehicles that fllw three-dimensinal rectilinear mvement patterns Figure 1 illustrates the key cmpnents f an AVS/RS. Figure 1a is a three-dimensinal view f an AVS/RS, whereas Figure 1b is a plan view. The majr system cmpnents f the AVS/RS technlgy are lift mechanisms that are munted alng the periphery f the strage rack t prvide vertical mvement. The dminant cst cmpnent f the AVS/RS technlgy is the vehicle. Lifts are used fr vertical mvement f the vehicles. Lifts are typically nt the bttleneck in an AVS/RS but they have a significant influence n thrughput capacity because vertical mvement is generally slwer than hrizntal mvement. They can als cntribute substantially t transactin cycle times /09/$ IEEE 2405

2 In mst unit-lad S/R systems, space-cnserving randm strage plicies are used because f capital cst cnsideratins (Heragu 2008). Under a randm strage plicy, the lcatin f a specific lad in a strage system is a randm variable ver time. This plicy ensures space ccupancy rate maximizatin by allwing different lads t ccupy the same address at different times (Malmbrg 1996). AVS/RS technlgy can prvide cst effective autmatin fr unit-lad S/R systems with varying transactin vlumes because it enables the designer t vary the number f S/R devices (i.e., AVs), depending n the number f S/R transactins in a system. With traditinal crane-based technlgy, aisle-captive S/R devices utilize highly efficient Chebychev mvement patterns within aisles. Hwever, the high cst f crane-based systems (ne crane is required per aisle) raises the threshld fr cst effective autmatin and crane-based AS/RSs can be justified nly fr high thrughput peratins in a stable demand envirnment. Autnmus vehicle Aisle Aisle Aisle Crss aisle P l Lifts.... Secnd tier I/O I/O Lifts P l P l Bays First tier (a) Crss aisle (b) Figure 1: The key cmpnents f an AVS/RS First bays There are varius studies that evaluate the perfrmance f AVS/RSs (Malmbrg 2002, Malmbrg 2003, Ku, Krishnamurthy and Malmbrg, 2007, Ku, Krishnamurthy and Malmbrg, 2008, Fukunari and Malmbrg 2008, Fukunari and Malmbrg 2009, Zhang et al. 2009). Different frm thse studies, we mdel the AVS/RS frm a rack cnfiguratin view. The perfrmance f an AVS/RS als depends n the physical layut because it affects the travel distance f AVs, s the cycle time. The physical layut f an AVS/RS can be defined in terms f three dimensins, number f tiers, aisles, and bays (see Fig. 2). In this study, first, we develp the simulatin mdel f the AVS/RS. Secnd, we cmplete experiments fr the predefined levels f the system s input variables. And last, we develp a regressin analysis using tw different appraches that best reflects the relatin between the utput and inputs. We use ARENA 12.0, a cmmercial simulatin sftware, t develp the simulatin results and MINITAB t develp the regressin functin. 3 PROBLEM DESCRIPTION AND SIMULATION In an AVS/RS, there are tw types f transactins arriving int the system -strage and retrieval transactins. Strage transactins refer t the strage f a unit-lad frm the input/utput (I/O) pint t an available lcatin in the racks. Retrieval transactins refer t the retrieval f a unit-lad frm its current lcatin. All strage transactins are assumed t arrive at the I/O pint and all retrieval transactins end at the I/O pint. The AVS/RS uses lifts t stre r retrieve unit lads frm the bays lcated n tiers ther than the first, and vehicles fr hrizntal mvement within a tier. Vehicles travel in three dimensins at an AVS/RS warehuse; e.g., travel between tiers takes place n the z axis; travel between aisles takes place n the y axis and travel between bays takes place n the x axis (see Fig. 2). Therefre, the sizes f the warehuse, in ther wrds, the number f tiers, aisles and bays are crucial n travel time f AVs. In the AVS/RS warehuse, racks n either side f an aisle cnsist f bays, and each bay can hld three unit-lads. S, the warehuse strage capacity is calculated by multiplying the number f tiers, number f aisles, number f bays, tw (bth sides) and three. The result is then the ttal number f pallet psitins. 2406

3 A Aisle D W T First Tier H Bays x z y First bays Figure 2: Warehuse design f the AVS/RS 3.1 Simulatin Assumptins The actual system has 21 vehicles and 7 lifts. The strage area is divided int 7 znes, depending n the number f lifts. Each zne has 1 lift, and 3 specific vehicles are assigned t each lift. The lifts are lcated at the middle f each individual zne, e.g. the first lift is lcated at the end f the 3rd aisle. The ther assumptins f the AVS/RS are given belw: 1. The dwell pint f a vehicle is the place where the last strage r retrieval transactin is cmpleted. 2. The dwell pint f the lift is where the last vertical mvement is cmpleted. 3. The system uses pure randm strage plicy. 4. The actual distance between tw aisles, the width f a bay and the height f ne tier in an actual AVS/RS installatin are used t calculate the travel times. 5. Each zne has I/O lcatins near the lift. 6. The arrival rates fr strage and retrieval transactins are independent Pissn prcesses and equal. 7. The transactins are served by the vehicles n a first-cme, first-served (FCFS) rule. The vehicles requiring lifts fr vertical mvement are als served by FCFS rder. 8. The vehicle transfer time int the lift is assumed t be zer. 9. The warehuse strage capacity must be greater than 42,000 unit-lads. 3.2 Ntatins The ntatin used in the AVS/RS mdel is given belw: A : number f aisles L : the number f lifts B : number f bays (clumns) per aisle Yx : the distance frm the first bay t the crss aisle D : the distance between tw aisles T T : the lad r unlad transfer time between the lift and the I/O pint. T : number f tiers v v : the velcity f the vehicle W : the width f ne strage bay v L : the velcity f the lift H : the height f ne tier T L/U : the time t lad/unlad t r frm the strage rack V : the number f vehicles : the arrival rate f strage transactins per hur : the arrival rate f retrieval transactins per hur 2407

4 Because f the agreement with the cmpany, we are nt able t give the specific values f the parameters defined abve, except the values summarized belw: V = 21 L = 7 = 225 unit-lads = 225 unit-lads 3.3 Cnstraints The AVS/RS warehuse has a limited area. Therefre, there are cnstraints n the maximum numbers f tiers, aisles and bays. Besides, as mentined in the 9 th simulatin assumptin in Sectin 3.1, there is a cnstraint n the ttal number f unitlad strage psitins. Because ur aim is t minimize the cycle time, we assume that the warehuse capacity shuld be equal t 42,000 unit-lads. This capacity is determined by cnsidering the expected maximum strage demand that may ccur during a perid f time. In ther wrds, the manager expects that in the AVS/RS, the pssible maximum strage psitin that may be needed in a perid is 42,000 unit-lads. S, the plant manager seeks the best rack cnfiguratin that prvides the required number f strage psitins. As mentined previusly, the rack cnfiguratin f an AVS/RS can be defined in terms f three dimensins, numbers f tiers, aisles and bays. If the number f tiers is small, then the warehuse ftprint will be large t accmmdate the required number f strage psitins. If the number f tiers is high, then the ftprint will be small. Under the first type f design (few tiers) the vehicles may becme a bttleneck, because they will need t realize lng hrizntal travels. Hwever, in the secnd cnditin (large number f tiers), the lifts may becme bttleneck because this time lng vertical mvements will be required. Therefre, the perfrmance f the system will als be affected by the number f vehicles and lifts in the system. Thus, the regressin analysis shuld be cmpleted under varius vehicle and lift cmbinatins. Hwever, in this study we cmplete the regressin analysis nly ne cmbinatin f V and L values which are 21 and 7, respectively. This study will be extended as a future study fr develping regressin analysis under varius vehicle and lift cmbinatins f the system. Because the ttal number f strage psitins in the system is calculated by TBA23; when we divide 42,000 unitlads by six, we btain a value f 7,000 fr TBA. This means that the cmbinatin f three input variables, T, B and A, shuld be designed such a way that their prduct shuld prvide fr 7,000 unit-lads. In additin t this cnstraint, because there are limited areas n z, x and y axes, it is assumed that the T, A and B can get the maximum values, 8, 65 and 25, respectively. The warehuse has a rectangular shape. 4 REGRESSION MODELING Because the AVS/RS under study is quite cmplex it makes it difficult fr a manager t identify hw the parameters affect the average cycle time in the system. Regressin analysis enables t understand and evaluate the perfrmance f the system in terms f input variables. Regressin analysis is a statistical tl fr the investigatin f relatinships between utput and input variables. Usually, ne seeks t ascertain the causal effect f ne variable n anther e.g., the effect f a price increase n demand, r the effect f changes in the mney supply n the inflatin rate, etc. T explre such issues, the investigatr assembles data n the underlying variables f interest and emplys regressin t estimate the quantitative effect f the causal variables n the variable that they influence. Regressin analysis with a single explanatry variable is termed simple regressin, with multiple explanatry variables it is termed multiple regressin. We develp a multiple regressin mdel, in this study. The utput f the system is the average cycle time f transactins, which is measured in minutes. Cycle time is the time between when a request riginates until it is cmpleted. Alng with resurce utilizatins, it is a cmprehensive perfrmance measure and emplyed extensively in many industries. AVS/RS cycle times are determined by the strage rack cnfiguratin, vehicle mvement kinematics, strage plicy and vehicle dwell pint plicy. In this study, the vehicle mvement kinematics, strage plicy and vehicle dwell pint plicy are assumed t be fixed (see sectin 3.1). Hwever, the rack cnfiguratin is f interest. The input variables are the numbers f tiers, aisles and bays. While the number f tiers determines the height f the warehuse, the number f aisles and bays determine the ftprint f the warehuse. 4.1 Simulatin Experiments T develp the regressin mdel, first three independent input variables levels are determined. Then, the simulatin mdel is run fr these levels f the input variables. The simulatin mdel is assumed t be a nn-terminating system, allwing us t cnduct a steady state analysis. The warm-up is three mnths. The length f each simulatin run then 1,180,800 minutes (( ) + warm-up perid). The mdel is run fr ten independent replicatins. 2408

5 In the simulatin mdel, the cmmn randm numbers (CRN) variance reductin technique is used. CRN is a ppular and useful variance reductin technique when we cmpare tw r mre alternative cnfiguratins. It requires synchrnizatin f the randm number streams, and uses the same randm numbers t simulate all cnfiguratins. In CRN, a specific randm number used fr a specific purpse in ne cnfiguratin is used fr exactly the same purpse in all ther cnfiguratins. Thus, variance reductin is ensured. Fr the T input variable 4 levels -5, 6, 7 and 8; and fr the A input variable seven levels -35, 40, 45, 50, 55, 60 and 65 are cnsidered. Here, the level f the B input variable varies accrding t the A and T values. Each time it is calculated by dividing 7,000 by TA. Fr example, if T = 7 and A = 40, then B is 25 (7,000/(740)) and if T = 8 and A = 50, B is 18. It shuld be nted that due t the warehuse area cnstraint the maximum values f A and B culd be 65 and 25, respectively. A can get a minimum value f 35 which is calculated by dividing 7,000 by the prduct f maximum values f T and B (7,000/(825) = 35). With the same lgic, the minimum value f B is 13. In designing experiments, where necessary the divisin is runded up t the clsest integer. In additin, we als cmplete experiments at the bundary levels f B. Fr example, at B = 25, A is calculated as 7,000/(25T). In sme cases, fr example at small values f T and A, the B value may be greater than the maximum cnstraint, 25. Under these cnditins, thse experiments are nt cmpleted (e.g., T = 6 and A = 40, B = 29). The levels f the input variables and the cmpleted experiments are summarized in Table 1. Table 1: The input variable levels and cmpleted experiments T A B T A B T A B T A B As seen frm Table 1, a ttal f 21 experiments are cmpleted with ten replicatins each. 4.2 Regressin Functin Analysis First, we trace the factrs t determine whether r nt a curvature exists. At nne f the input levels a curvature is detected. S, we mdel the system by a multiple linear regressin. In a regressin mdel, ANOVA analysis is used t decide which terms t include in the regressin functin. Fr example, if there are three input variables under study, then at mst seven terms can be included in the regressin functin. These pssible terms are, main effects -T, A, B; tw-way interactins -TA, TB, AB; and three-way interactins -TAB. The ANOVA can suggest which terms t include by prviding the statistically significant effect(s). Hwever, in this experimental study, we are nt able t cmplete a full factrial design because f the area cnstraint fr each factr cmbinatin (see Table 1). As we cannt cnduct full factrial design, we cannt apply ANOVA, either (Mntgmery, 1996). Hwever, we may use fur alternative methds fr the regressin analysis. These are: frward selectin, backward eliminatin, stepwise regressin and best subsets. Of these, we use, stepwise regressin and best subsets, t evaluate the pssible regressin functins and determine the best ne (Mntgmery, 1996). Stepwise regressin is the cmbinatin f frward selectin and backward eliminatin methds. The purpse f the stepwise regressin methd is t find a meaningful subset f independent input variables which explains dependent variable efficiently. In this methd, at each iteratin all terms bth in and ut f the mdel are reassessed using their partial F statistics. The term nt in the mdel with the largest partial F statistic, larger than F IN is added t the mdel. The term in the mdel with the smallest partial F statistic, smaller than F OUT is remved frm the mdel. Terms can enter the mdel and be remved frm the mdel mre than nce. The best subset methd finds the pssible n best subsets f i terms (i = 1, 2,, k) f the regressin mdel. Fr each subset, it calculates the cefficient f determinatin -R 2 and adjusted R 2 - values, s that we can chse a subset that has a gd balance f high and small number f terms. R 2 prvides a measure f hw well utputs are likely t be predicted by the regressin mdel. The bigger the value the better fit the mdel is. Hwever, nly cnsidering R 2 is nt adequate t evaluate a regressin functin because the R 2 value always increases with the additin f a new input variable t the functin, even if it is nt significant. If the value is significantly lwer than R 2, it nrmally means that ne r mre explanatry 2409

6 variables are missing. Therefre, usually t be big and clse enugh t the R 2. value is used fr evaluating a regressin functin and it is preferred fr 4.3 Stepwise Regressin Results The statistical analyses are cmpleted in MINITAB at 95% cnfidence level. The stepwise regressin results btained by MINITAB are as in Table 2: Table 2: The stepwise regressin results by MINITAB Step 8 Cnstant T T-value 6.81 P-value TA T-value -2.4 P-value B T-value 6.29 P-value TB T-value P-value S R-Sq R-Sq (adj) Terms having P-values smaller than 0.05 are statistically significant respnse terms. MINITAB shws the significant terms, after cmpleting the stepwise regressin analysis. Accrding t Table 2, the results shw that fur terms are significant n respnse at 95% cnfidence level and shuld be in the regressin functin. These terms are, T, TA, B and TB (P-value < 0.05). The R 2 and values are als seen at the end f the table (90.66% and 90.28%). Because they are big and clse enugh t each ther, we assume that the mdel is acceptable. Hwever, after btaining the significant terms that shuld be in the regressin functin, we als cmplete the usual regressin analysis fr the mdel. Namely, we frm the regressin functin with thse fur terms and evaluate the R 2, and P-values. We als check whether r nt the regressin mdel adequacy is met. The mdel adequacy means that residuals are nrmally distributed, have mean f zer and have a cnstant variance. If ne r mre f these cnditins are nt met, a suitable transfrmatin such as, inverse, lgarithm, natural lgarithm, square rt, inverse, inverse square rt, etc. shuld be applied n the utput t achieve the mdel adequacy. After implementing the regressin analysis, we see that the regressin mdel adequacy is met. In additin, all these fur terms are statistically significant (P-value < 0.05). The regressin functin s R 2 and values are btained as 90.7% and 90.3%, respectively. Therefre, we assume that the regressin mdel is a gd fit. The regressin functin btained by MINITAB is given by: 4.4 Best Subset Regressin Results T TA B TB (1) The secnd apprach, best subset regressin, is als implemented n the utputs. The results btained by MINITAB are shwn in Table 3: 2410

7 Table 3: The subset regressin results by MINITAB Terms R 2 T A B X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X In this table, R 2 and values are calculated fr each alternative design. X in the table shws the terms in the regressin functin. The first clumn shws the ttal number f terms that will be in the regressin functin. Frm Table 3, we can chse the best regressin subset by cnsidering highest R 2 and values. In the table, when the number f terms in the regressin functin decreases, the R 2 and values als decrease. Hwever, this decrease is nt significant in every cnditin. Fr example, when we want t chse the functin with five terms, the R 2 and values are 90.8% and 90.3%, respectively. And, these values becme 90.7% and 90.3% fr the functin with fur terms. Because the R 2 and values d nt vary significantly with the fur term regressin mdel, it is usually better t chse the functin with few terms rather than with mre terms. This is because it is always easier and nt much cnfusing t deal with few terms in a functin. Fr instance, in Table 3 the last subset has seven terms (all the terms) in the regressin functin and has the biggest R 2 and values, 92.3% and 91.7%, respectively. Hwever, usually because all the terms may nt be significant n the respnse, we may want t seek fr an alternative design with fewer terms. When we g thrugh the alternative regressin designs with fewer terms, we btain almst the same R 2 and values as in the previus case. The R 2 and values are usually arund 90% until the fur-term functins. If, we als utilize ur frmer methd s result (stepwise regressin), then we chse the regressin functin with fur terms. This is because the R 2 and values are high enugh and als it includes few terms in the mdel. Hence, the cnsidered terms are: T, B, TA and TB and this functin s R 2 and values are 90.7% and 90.3%, respectively. It shuld be nted that the fitted functin in (1) has high R 2 and values, 90.7% and 90.3%, respectively. It als meets the regressin mdel adequacy. If the R 2 and values were nt as large as we wuld like and/r the mdel adequacy was nt met, then we wuld apply a suitable transfrmatin n the utput and/r try including higher rder term(s) in the regressin mdel. Hwever, the current mdel in (1) prvides all the requirements. Hence, we accept this regressin functin as a gd fit. It shuld als be nted that in functin (1) we can place nly the limited values f A, B and T (see Table 1). Fr example, fr T = 5, A can nly get the values between 55 and 65, and B can get the values between 22 and 25. Fr T = 6, A can nly get the values between 47 and 65, and B can get the values between 18 and 25. Fr T = 7, A can get the values between 40 and 65, and B can get the values between 15 and 25. Fr T = 8, A can get the values between 35 and 65, and B can get the values between 13 and 25. The ther values will nt be represented by the regressin functin. In additin, the chsen values fr A, B and T shuld als prvide ABT = 7,000 unit-lads. Fr example, fr the values f T = 8, A = 58, and B = 15 frm (1) we btain the cycle time as minutes. Frm the functin in (1) we see that whereas TA and TB have negative effects n the utput, T and B have psitive effects. We want t minimize the average cycle time. We ptimize the functin in (1) using the abve cnstraints in LINGO. As a result, the ptimum pints are btained as, A = 65, T = 6 and B = 18 leading t minutes average cycle time. As seen the ptimum pints are btained at the bundary cnditins n the number f aisles. This reasn is prbably because f T A T B A B T A B 2411

8 the acceleratin and deceleratin time delays f the vehicles. Instead f traveling n tw shrt crdinates, x and y, it is better t travel n a lng and a shrt crdinate. 5 CONCLUSION In this study, we presented a regressin analysis fr rack cnfiguratin f an AVS/RS. The case study is applied n a warehuse that uses AVS/RS in France. In the regressin mdel, the system s number f vehicles and lifts are assumed t be fixed. Fr the regressin analysis utput is chsen as the average cycle time f strage and retrieval prcesses and the input variables are the number f tiers, aisles and bays that determine the size f the warehuse. The simulatin mdel f the system is develped using ARENA 12.0, a cmmercial sftware. The statistical analyses are cmpleted in MINITAB statistical sftware. Tw different appraches are used t find the best fit regressin functin; stepwise regressin and the best subsets. Stepwise regressin is the cmbinatin f frward selectin and backward eliminatin methds. The best subset methd finds the pssible n best subsets f i terms (i = 1, 2,, k) f the regressin mdel. In the best subset methd, we chse the best number f terms that reflects the regressin functin by cnsidering the R 2 and values. We als utilize the stepwise regressin results t decide the number f terms in that methd. At last a functin with fur terms is chsen as the system s regressin functin. After deciding n the terms, a regular regressin analysis and adequacy test fr the regressin functin are cmpleted. As a result the functin given in (1) is assumed t be a gd fit fr representing the relatinship between the input variables, number f tiers, aisles, bays, and the utput, average cycle time. Last, we als shw the pssible ptimum pints f this functin. We use the sftware LINGO t minimize the functin. The ptimum pints are btained as A = 65, T = 6 and B = 18 leading t minutes average cycle time. REFERENCES Fukunari, M., and C. J. Malmbrg An efficient cycle time mdel fr autnmus vehicle strage and retrieval systems. Internatinal Jurnal f Prductin Research : Fukunari, M., and C. J. Malmbrg A netwrk queuing apprach fr evaluatin f perfrmance measures in autnmus vehicle strage and retrieval systems. Eurpean Jurnal f Operatinal Research 193: Heragu, S. S Facilities Design. 3rd ed., Clermnt, FL: CRC Press. Ku, P. H., A. Krishnamurthy, and C. J. Malmbrg Design mdels fr unit lad strage and retrieval systems using autnmus vehicle technlgy and resurce cnserving strage and dwell pint plicies. Applied Mathematical Mdelling 31: Ku, P. H., A. Krishnamurthy, and C. J. Malmbrg Perfrmance mdelling f autnmus vehicle strage and retrieval systems using class-based strage plicies. Internatinal Jurnal f Cmputer Applicatins in Technlgy 31: Malmbrg, C. J Strage assignment plicy tradeffs. Internatinal Jurnal f Prductin Research 34: Malmbrg, C. J Cnceptualizing tls fr autnmus vehicle strage and retrieval systems. Internatinal Jurnal f Prductin Research 40: Malmbrg, C. J Interleaving rule dynamics in autnmus vehicle strage and retrieval systems. Internatinal Jurnal f Prductin Research 41: Mntgmery, D. C Design and Analysis f Experiments. 4 th ed. New Yrk: Jhn Wiley & Sns. Zhang, L., A. Krishnamurthy, C. J. Malmbrg, and S. S. Heragu Variance-based apprximatins f transactin waiting times in autnmus vehicle strage and retrieval systems. Eurpean Jurnal f Industrial Engineering 3: AUTHOR BIOGRAPHIES BANU Y. EKREN is a Ph.D. student in the Department f Industrial Engineering at University f Luisville in USA. She received his B.S. Degree in Industrial Engineering frm Dkuz Eylul University, Turkey, and M.Sc. Degree in Industrial Engineering frm Pamukkale University, Turkey, in Her research interests include discrete-event-simulatin, simulatin based ptimizatin, queuing netwrks, lgistics, material handling systems, and facility design prblems. Her current research fcuses n simulatin and analytical mdels fr Autnmus Vehicle Strage and Retrieval Systems. Her e- mail address is <b.ekren@luisville.edu>. 2412

9 SUNDERESH S. HERAGU is Prfessr and the Mary Lee and Gerge F. Duthie Chair in Engineering Lgistics in the Industrial Engineering department at the University f Luisville. He is als Directr f the Lgistics and Distributin Institute (LDI). He cmpleted his B.S. in Mechanical Engineering at Malnad Cllege f Engineering, Hassan, India (affiliated t University f Mysre), MBA at University f Saskatchewan, Saskatn, Canada and Ph.D. in Industrial Engineering at University f Manitba, Winnipeg, Canada. His current research interests are in supply chain management, design f next generatin factry layuts, intelligent agent mdeling f autmated warehuse systems, applicatin f RFID technlgy t imprve intra-plant and inter-plant lgistics, and integratin f design and planning activities in advanced lgistical systems. His address is <s.heragu@luisville.edu>. 2413