CYCLE TIME MODELS FOR AISLE CHANGING AS/RS CONSIDERING ACCELERATION/DECELERATION. Jörg Oser, Thomas Drobir TU Graz, Austria.

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CYCLE TIME MODELS FOR AISLE CHANGING AS/RS CONSIDERING ACCELERATION/DECELERATION Jörg Oser, Tomas Drobir TU Graz, Austria Abstrat In multi-aisle AS/RS systems one or more S/R maines sere seeral

An important feature of multi-aisle onfigurations are aisle anging deies wit transfer ars, ure guided aisle switing meanisms or losed loop rail traks wit ured setions for a two-aisle layout.

One or seeral S/R maines ea sering a speified number of aisles an be arranged in arious layouts to meet te required storage apaity and trougput.

1 CYCLE TIME MODELS FOR AISLE CHANGING AS/RS CONSIDERING ACCELERATION/DECELERATION Jörg Oser, Tomas Drobir TU Graz, Austria Abstrat In multi-aisle AS/RS systems one or more S/R maines sere seeral aisles wit storage raks on bot sides of te aisle. An important feature of multi-aisle onfigurations are aisle anging deies wit transfer ars, ure guided aisle switing meanisms or losed loop rail traks wit ured setions for a two-aisle layout. Tese systems offer an eonomi solution, wen te number of storage and retrieal transations is low and storage olume is ig. One or seeral S/R maines ea sering a speified number of aisles an be arranged in arious layouts to meet te required storage apaity and trougput. Introdution and problem statement A long known aisle anging metod onsists of a transfer ar also designated as traerser. It operates in an aisle perpendiular to te atual storage aisles. Figure sows te arrangement of S/R maine riding on te traerser during te aisle anging operation. Te disadantage of tis solution is not only te additional osts of te traerser, but also a substantial loss in floor spae and te extra waiting time wen te S/R moes on and off te traerser. Figure : S/R maine (a), traerser (b) [6]

2 A different option uses ure guided aisle switing meanisms attaed to te S/R maine. Figure from VDI 3658 sows ow te aisle anging meanism works wit a tilting leer attaed to a pair of guidane rollers. In position te roller-pair a,b is mesing wit te rail, so te S/R traels straigt along te front aisle. Roller is out of operation. In position te roller-pair b, meses wit te rail ure setion and dierts te S/R maine into te aisle requested by te PLC. Figure : Aisle anging deie wit tilting leer meanism. [] As an be seen te tilting leer an operate from below or aboe depending on te rail design. Due to te ange of diretion te ure speed as to be substantially lower tan te nominal trael speed in straigt motion. As quite a few trael speed anges will our during one yle wit two ure moements in single ommand (SC) and four su eents in dual ommand (DC) operation wit at least two start and two stop operations in SC, te influene of aelerations and deelerations for aisle anging S/R maines wit ure swit deies for aisle anging will be signifiant. A typial eloity-time-relationsip for a one-leg SC yle is sown in te following diagram. Te grap sows six setions wit different eloities neessary to approa a ertain storage loation. A typial feature is te redued ure speed in setion IV as well as te aeleration and deeleration in setions I, III, V and VII. Figure 3: Veloity-time grap of ure traelling AS/RS

Tus tis resear aims at an improed yle time alulation for multi-aisle AS/RS wit ure trael onsidering aeleration/deeleration operating in SC and DC ommands in a typial layout onfiguration.

eloity witout redued ure speed 8 7 Te results sow a 6-% deiation for E(SC) and -6% for E(DC) operation. Tus, te redued ure speed operation sould not be negleted.

3 Tus tis resear aims at an improed yle time alulation for multi-aisle AS/RS wit ure trael onsidering aeleration/deeleration operating in SC and DC ommands in a typial layout onfiguration. Te failure estimation not onsidering redued ure speeds and aelerations, deelerations is inestigated in figure 4. onst. aeleration/deelaration wit redued ure speed onst. eloity witout redued ure speed 8 7 Te results sow a 6-% deiation for E(SC) and -6% for E(DC) operation. Tus, te redued ure speed operation sould not be negleted. First Come First Sere (FCFS) is onsidered as te referene poliy. E(SC) [s] Figure 4: E(SC) for different operations /3/4 3/3/4 5/3/4 /3/ 3/3/ aisles / eigt / lengt 5/3/ Te rest of te paper is organized as follows. In setion we present a literature reiew, in setion 3 we derie te basi equations for arious speed profiles leading to expressions for single and dual ommands in setion 4. In setion 5 a omparatie publised example is used to erify our results. Finally, a number of ase studies is treated by numerial ealuation in setion 6 and a summary onludes our results in setion 7. Literature reiew Roodbergen e.a. [] present an extensie olletion of resear papers on AS/RS systems wit a systemati approa sowing tat only ery few papers deal wit multi-aisle and operational issues. Early trael time models from Bozer e.a. [] neglet aeleration and onsider aisle aptie systems only. Gudeus [3] inludes aeleration in is model as early as 973 but does not treat multi-aisle systems. His work was followed by Hwang e.a. [4] wit an aeleration yle time model also for aisle aptie systems. Cang e.a. [5] approximates te yle time by a polynomial expression and ealuates arious teories wit a pieewise ariable eloity time funtion desribing real world operating onditions, but is numerial results ould not be erified. Hwang e.a. [6] deelop a trael time model for multi-aisle appliation of transfer ar aisle anging metod. No aeleration effets or ure redution speed are treated due to te transfer ar metod.

4 Benamar e.a. [7] treat te multi-aisle ase using a isual Petri Net Deeloper also wit a transfer ar only and witout modeling a/de effets. Egbelu [8] inestigates te optimal dwell point loation in a two aisle layout and suggests two linear programming optimization funtions. Howeer, Egbelu e.a. [9] reised te results from [8] wit simulation results sowing tat empirial rules ould outperform te optimization formulations. Hale e.a. [] inestigates te optimal dwell point loation in multi-aisle AS/RS witout onsidering a/de and aisle anging effets, wi was done in []. 3 Basi equations In tis apter te speed profile of figure 3 desribing a typial SC is separated into indiidual partitions as follows: - Trael from IO to te beginning of te switing ure - Trael witin an aisle from te beginning of te switing ure - Trael between aisles from ure to ure For a trapezoidal speed profile for trael from IO to te beginning of te switing ure we obtain = atdt+ + (+ )! I aording to figure 5. II III () Figure 5: Trapezoidal trael from IO to ure Figure 6: Triangular trael from IO to ure

5 Figure 5 beomes figure 6 wit a triangular speed profile in ase te minimum trael distane to te ure is "# = $ & and max is not aieed. Te trael distane for figure 6 is sown in ()., $ (-. ) s = atdt+ a(t+ + a t)dt! I, $ (-. ) II () Cure trael wit redued speed results from figure 7. Trael distane s is. s = dt+ a(t t)dt!. I II (3) Figure 7: Cure speed profile for sort trael distanes Te following speed profiles sow te ombination of ure trael wit trapezoidal trael for long distanes in figure 8 and for medium distanes in figure 9 wit triangular trael. max t 3 T Figure 8: Long distane trapezoidal trael Figure 9: Medium distane triangular trael Te trael distane in figure 8 for trapezoidal trael is sown in (4) and for triangular trael aording to figure 9 in (5).

6 3. = dt+! I - 3. / + II + - III + ( ) IV (4) 3. = dt+! I, $ (- ) 3. / + +, II $ (- ) ( ) III (5) Te aisle anging yle is defined by similar speed profiles as desribed in figures 5 and 6 in basi trael assuming a minimum ure speed of. Figure : Aisle anging trapezoidal trael Figure : Aisle anging triangular trael Te referring trael distanes for figure and figure are sow in (6) for trapezoidal trael and in (7) for triangular trael... s = (at+ + )dt+ dt+ a(t t) dt! $. s = (at+ + )dt+ ( + at) dt! I I $ II II. III (6) (7)

4 Single and dual ommands Single and dual ommands require te definition of a basi layout.

Figure a: Basi layout (Layout I) Figure b: AS/RS in ure trael [] Layout I assumes single and dual ommand operation wit te dwell

Te results of furter layouts are desribed as follows but te teory is not presented in tis paper.

Layout II is also used in industry, wereas Layout I is onsidered as a basi referene layout.

7 4 Single and dual ommands Single and dual ommands require te definition of a basi layout. Te simplest layout of te multi-aisle onfiguration is sown in Figure a and b. Figure a: Basi layout (Layout I) Figure b: AS/RS in ure trael [] Layout I assumes single and dual ommand operation wit te dwell point at te I/O loation, bidiretional trael of te S/R maine and redued ure speed. Te results of furter layouts are desribed as follows but te teory is not presented in tis paper. Layout II as I/O stations at ea end of a storage aisle opposite of te ange aisle assuming bidiretional trael and SC/DC operation wit te I/Os as dwell point poliy. A tird layout for iger trougput requirements is a losed loop system wit seeral S/R maines, unidiretional trael and only one I/O. Layout II is also used in industry, wereas Layout I is onsidered as a basi referene layout. Furter basi assumptions are: Continuous retangular storage rak wit randomized storage assignment I/O point in ariable positions depending on te layout Tenial data of S/R maine (speeds, aelerations et) and rak size Simultaneous S/R maine trael in orizontal and ertial diretions Bot triangular and trapezoidal eloity time funtions may our Te full ure eloity is always aieed Aording to tese assumptions, te formulas for SC and DC are deried.

8 Te first partial trael time from IO to ure start follows from equations () and () wit boundary onditions inluding te ure speed. a 4s + < s a a a t l (s) = für j s + - a a < s l j a (8) Te seond partial trael time from te start of te ure to te storage loation in SC follows from s + l < s ψ a l ψ < ψ )a s t (s) = + 4(s l + j für a a (s l ψ < + ) l s l L a a j (9) wit boundary onditions ψ = l + a and ψ = l + a a Calulating te distane funtions from te first partial trael time defined in (8) leads to equation () a 4 t + t 4a s l (t) = j - + t a a für < t ξ ξ < t T lj wit ξ = a a and T lj l + j - = a a () Calulating te distane funtions from te seond partial trael time (9) leads to equation () t a a a a ξ t ξ l l l s j(t) = t + ( )t + + für ξ < t ξ 4 4 4a ξ < t T j t + + l ( ) a a a () wit boundary onditions ξ = + l, a ξ = + l a and l ξ = + a

9 Te probability-density-funtion (pdf) is deried from te following transformation ds ds (z)dz = (z)= df (z) = f s f s dz m ax m ax () Terefore, te transformation result for te orizontal trael time to te j t aisle is a a ξ + t z ξ + t j l f (z) = (z t ) + für ξ + t < z ξ + t Lj ξ + tl < z T j jmax l lj l l l j j j j wit boundary onditions ξ = + l, a ξ = l +, l ξ = + a a and T = t (s = l ) + T (3) jmax l j j j In order to derie te total orizontal trael-time pdf te f j as to be weigted wit te number n j of storage loations witin te region j. J J f (z) = n f (z) mit n = n j j ges j nges j= j= fulfilling te standardizing ondition + j f (z)dz = (4) Expressions for te ertial motion are deried in a similar manner onsidering tree different ases. - Case : IO-loation at zero leel and immediate ertial motion - Case : IO-loation at zero leel and delayed ertial motion - Case 3: IO-loation at random eigt Te total umulatie-density-funtion (df) for te j t aisle follows from te produt of orizontal and ertial distributions. G (z) = F (z) F (z) j j (5) Ealuating an AS/RS wit J aisle leads (5) to G(z) = G (z) = F (z) F (z) J J j j j= j= J J (6) Tese equations finally result in te expeted alue for SC operation from [ ] E(SC) = G(z) dz (7)

10 In a similar proedure te equations of dual ommand yles are deeloped based on figure 3 but due to lengty expressions tey are not exibited ere. SC TB SC PD -, 3-4 SC trael from IO to storage loation (SL) time between SL and retrieal loation (RL) single-ommand trael from RL to IO Pik-up- and deposit-time ure trael in first and seond ure setion 4(5) = 64?, 7(5)+ #=,[(6 :)4 7< =# (5)]?, 6+ (6 :) #=, (8) Equation 8 ontains an approximate solution for E(TB) based on te weigted aerage of te expeted alues of te minimum and maximum trael times for an aisle ange j-k. Figure 3: Speed profile for DC operation 5 Verifiation wit literature results We ae erified our results wit an analytial literature example from HWANG/KO [6] and wit disrete eent simulation using AutoMod and omplete enumeration. HWANG/KO inestigated a ase study aording to figure wit te following basi differenes to our resear - Te time to transfer te S/R maine results from a disontinuous operation of te traerser meanism, wereas we assume ontinuous aisle anging meanisms wit redued ure speed.

11 - Aeleration and deeleration is assumed to be instantaneous wereas we assume limited alues from drie arateristis. AS/RS data are defined by - Horizontal trael =356 ft/min =,88 m/s - Vertial trael = ft/min =,58 m/s - Transfer speed t = 5 ft/min =,54 m/s - Rak lengt L=348 ft = 6 m - Rak eigt H= 88 ft = 7 m Furtermore, we simplified our alulations aording to te assumptions stated aboe to make HWANG/KO s results omparable to our teory wit - Infinite orizontal aeleration - Infinite ertial aeleration - Cure radius R= - Cure speed equals orizontal speed = Numerial ealuation of our teory in apter 4 leads to te following results # Aisles E(SC) HWANG/KO E(SC) DROBIR E(DC) HWANG/KO E(DC) DROBIR,46 min,4 min,6757 min,67 min,4435 min,4394 min,984 min,96 min 3,6849 min,6778 min,737 min,6398 min 4,959 min,949 min 3,84 min 3,438 min 5,85 min,85 min 3,696 min 3,4736 min Table : Comparison of SC and DC, [6] and [] 6 Numerial ealuation 6. SC operation of Layout I In tis setion Layout I wit fie aisles, LxH=x3m and tree lifting ases are onsidered. Figures 4 to 6 sow te df and pdf of one-leg trael time distributions for tree ases.

12 Te ase studies ae been onduted based upon te following alues - Horizontal trael =3, m/s - Vertial trael =,5 m/s - Cure speed =,5 m/s - Rak lengt L= m - Rak eigt H= 3 m - Horizontal aeleration a =,5 m/s² - Vertial aeleration a =,5 m/s² - No. of aisles J=5 Te following one-leg probability- and umulatie-density-funtions ae been alulated based on te teory of apter 4. One-leg means alf of te total trael time of a SC yle. Tree different ertial operation assumptions were ealuated. Case : Figure 4 assumes te IO-point at te lower left orner loation wit E(SC)=x38se.=76se. negleting pik-up- and deposit-times but onsidering additional ure-speed time-losses. Figure 4a: Cdf of one-leg SC trael time Figure 4b: Pdf of one-leg SC trael time Case : Figure 5 assumes te IO-point at te lower left orner loation and a delayed lift operation starting at te end of te ure to minimize struture osillation wit E(SC)=x4se.=8se. negleting pik-up- and deposit-times but onsidering additional ure-speed time-losses.

13 Figure 5a: Cdf of one-leg SC trael time Figure 5b: Pdf of one-leg SC trael time Case 3: Figure 6 assumes te IO-point at alf rak eigt H IO =H/ to minimize total yle time wit E(SC)=x3,5se.=6se. negleting pik-up- and deposit-times but onsidering additional ure-speed time-losses. Figure 6a: Cdf of one-leg SC trael time Figure 6b: Pdf of one-leg SC trael time 6. Comparatie metods of alulating yle times First, a desrete eent siumulation pakage (AutoMod ) was adopted to simulate AS/RS operation in multi-aisle layouts wit simultaneous orizontal and ertial trael onsidering ure dierts and aeleration/deeleration. A seond metod was to use Exel for aeraging disrete yle time alulation wit omplete enumeration of te releant trael equations presented in setion 4 for Layout I. A tird metod was to use te analytial expressions deried in [] for SC and DC operation presented ere for E(SC) at te end of setion 4.

Figure 7 exibits te results of te expeted alue of SC operation E(SC) for Layout I wit LxH=x3m, arying te number of aisles from one to fie and lifting ase.

Exel Analyti AutoMod (+/- CI 95%) 4 4 E(DC) [s] E(SC) [s] AutoMod (+/- CI 95%) 8 6 Exel Analyti 8 6 4 4 /3 /3 3/3 4/3 5/3 /3 no. of aisles / rak eigt [m] /3 3/3 4/3 5/3 no.

14 Figure 7 exibits te results of te expeted alue of SC operation E(SC) for Layout I wit LxH=x3m, arying te number of aisles from one to fie and lifting ase. Figure 8 sows te expeted alues of dual ommand operation E(DC) again for Layout I wit LxH=x3m and lifting ase. Exel Analyti AutoMod (+/- CI 95%) 4 4 E(DC) [s] E(SC) [s] AutoMod (+/- CI 95%) 8 6 Exel Analyti /3 /3 3/3 4/3 5/3 /3 no. of aisles / rak eigt [m] /3 3/3 4/3 5/3 no. of aisles / rak eigt [m] Figure 7: SC operation in Layout I Figure 8: DC operation in Layout I Two AutMod -results for ea ombination no. of aisles / rak eigt reflet te upper and lower bound of te 95% onfidene interal of te simulation results. Furter results are sown in figure 9 for Layout II wit LxH=x3m and in figure wit te expeted alues of trael times E(DC) in dual ommand operation. AutoMod (+/- CI 95%) I/O stations Exel Analyti 4 E(DC) [s] /3 ross-aisle /3 3/3 4/3 5/3 no. of aisles / rak eigt [m] front rak (optional) Figure 9: Layout II wit multiple IO stations Figure : DC operation in Layout II

Layout II exibits sligtly smaller DC yle times as ompared to Layout I. Tus, multiple IO-stations in Layout II offer more trougput and flexibility of operations.

15 Layout II exibits sligtly smaller DC yle times as ompared to Layout I. Tus, multiple IO-stations in Layout II offer more trougput and flexibility of operations. A tird Layout III sows a unidiretional losed loop operation in figure wit LxH=x3m and expeted alues for single ommand trael times E(SC) in figure. 4 AutoMod (+/- CI 95%) Exel Analyti aisle E(SC) [s] /3 /3 3/3 4/3 5/3 no. of aisles / rak eigt [m] ross-aisle I/O-station Figure : Unidiretional losed-loop Layout III Figure : SC operation in Layout III Comparing figures and 7 sows for SC operation a trougput disadantage of a losed loop Layout III ompared to a bidiretional basi Layout I due to longer trael distanes and more ures in losed loop systems. Summary First te basi eloity time relations are deeloped as algebrai expressions onsidering aeleration/deeleration wit triangular and trapezoidal eloity-time funtions wen traeling wit redued ure eloities. Assuming tat a orizontal moe onsists of ronologial superpositions of indiidual time setions furter alulations of SC and DC operation are based on Tebye trael wit te maximum of orizontal and ertial distane (time). Tese expressions are

16 later used to formulate te expeted alues in SC operation. Tey were erified and ompared wit simulation results and omplete enumeration. Applied to te seleted layouts omparatie results are obtained for different rak onfigurations. Numerial examples are presented and ompared to literature results. An extension of tis resear ould inlude arious dwell-point strategies and multiple AS/RS in multiple aisle onfigurations wit one or seeral I/O-loations onsidering aeleration/deeleration and semi-ontinuous diert operation. Referenes [] Roodbergen, Kees Jan and Vis, F.A.Iris., A surey of literature on automated storage and retrieal systems, European Journal of Operational resear 94, (9) [] Bozer, Y.A. and Wite, J.A., Trael time model for AS/RS systems, IIE Transations 6,4, (984) [3] Gudeus,T., Grundlagen der Kommissioniertenik, Girardet Verlag (973) [4] Hwang, Hark and Lee, S.B., Trael time models onsidering te operating arateristis of te storage and retrieal maine, Int.Journal of Prod. Res. 8,, (99) [5] Cang, D.T. and Wen, U.P. and Lin, J.T., Te impat of aeleration/deeleration on trael time models for automated storage/retrieal systems IIE Transations 7, -8 (995) [6] Hwang,Hark and Ko, C.S., A study on multi-aisle system sered by a single storage/retrieal maine Int.Journal of Prod. Res. 6,, (988) [7] Benamar, A. and Sari,Z. and Gouali, N., Performane analysis for multi-aisle automated storage/retrieal systems using isual Petri Net deeloper, Pro. 3 IEEE Int. Symposium on Computational Intelligene in Robotis and Automation, July 6-, 3, Kobe, Japan [8] Egbelu, P., Framework for dynami positioning of storage/retrieal Maines in an automated storage /retrieal system Int.Journal of Prod. Res., 9,, 7-37 (99) [9] Egbelu, P. and Wu, C.T., A omparison of dwell point rules in an Automated storage /retrieal system, Int.Journal of Prod. Res.,3,, ( 993) [] Hale, Treor S. and Hale, Leslie C., Optimal dwell point loations in a multiaisle AS/RS Pro. of te 7 t Ind.Eng.Resear Conferene Banff, Alberta, Canada, May 8- (998) [] Drobir, T. Spielzeitberenung kurengängiger Horegallagersysteme, PD Dissertation 4, TU Graz

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