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Proceedings of the 2002 IEEE International Conference on Robotics 8 Automation Washington, DC * May 2002

Crane and Shuttle Optimization in Warehousing Systems

Francesco Amatol, F'rancesco Bade2, Pasquale Chiacchio3
Abstract We consider a modern warehousing system composed of a number of aisles, each one served by a crane and sided by a pair of racks, and a shuttle charging one unit load at the time which moves along a mono-dimensional path placed orthogonally with respect to the aisles axis. Two control algorithms to optimize the operations of the cranes (moving within the aisles of the warehouse) and the operations of the shuttle (moving on a straight line placed between the aisles and the picking /refilling area) respectively are proposed. To evaluate the performance of the proposed control algorithms we define three different cost indices; the first two indices look to the crane/aisle and shuttle areas as they were independent systems, while the third index measures the performance of the whole warehousing system. Finally extensive simulations are performed on the model developed in [2] to show the effectiveness of the proposed control algorithms. kind of operation is called, in the warehousing system context, a Dual Command (DC) machine cycle [3], [4],[5],

P I .

Concerning the shuttle, it moves along a mono-dimensional path placed orthogonally with respect to the aisles axis and performs picking actions (from the main input buffer, the aisles output locations and the picking/refilling output locations) and deposit actions (into the main output buffer, the aisles input locations, the picking/refilling input locations). We assume that the shuttle can charge one unit load at the time. This paper is devoted to investigate the simultaneous optimization of the DC cycles operations of the cranes and the picking/deposit actions performed by the shuttle. and [5] consider the DC optimization probThe papers [4] lem, under the assumption that the rack is continuous, that is it is composed of an infinite number of pointwise locations each one addressed by a pair of real numbers.
Buller IN

1 Introduction
In the last fifteen years, a big effort has been spent to find optimal strategies for planning and control of warehousing systems (see [l]and the bibliography therein). This issue has become more and more challenging with the growth and diffusion of modern computer technology which allows to implement various complex operations in a completely automated way. This work focuses on the control issues; i.e. we assume to face with an existing warehouse for which the basic rule which assigns a given product to a rack storage location has already been established in a previous phase. More precisely the current paper is devoted to the control problem, while in [2] model building has been considered. Our goal, in this paper, is that of finding an optimal and, at the same time, simple rule to sequence the storage and retrieval requests. We refer to a warehouse consisting of a number of aisles, each one served by a crane, of a shuttle and some picking/refilling positions and Figure 1);on both sides of each aisle there is a storage rack. As said, each aisle is served by a crane, moving both vertically and horizontally at the same time, which performs the following operations: i) picking of the load to be stored at the buffer input of the aisle, referred as the input/output (I/O) point of the aisle; ii) storage of the load into the assigned location S of the rack; iii) movement to location R where a retrieval has been requested; iv) retrieval of the load stored in R v) coming back to the 1/0 point. This
'Francesco Amato is with the Dipartimento di Informatica, Matematica, Elettronica e Tkasporti, UniversitiL degli Studi Mediterranea di Reggio Calabria, Via Graziella, Localitlr Fe0 D e Vito, 89100 Reggio Calabria, ITALY, Tel. +39/081/7683513, Fax +39/081/7683186, e-mail: 2fiancesco B a d e is with Dipartimento di Ingegneria Elettrica e dell'hformazione, Universith degli Studi di Salerno, Via Ponte Don Melillo, 1, 84084, Fisciano (SA), ITALY. Tel. +39/089/96-4400, Fax +39/06/233227957, e-mail: 3Pasquale Chiacchio is with the Dipartimento di Informatica e Sistemistica, Universith degli Studi di Napoli Federico 11, Via Claudio 21, 80125, Napoli, ITALY.

p Jrq


BayPR 2

BayPR 3

BagPR 4

.Figure 1: Warehouse layout

By following the guidelines of [4], [5],[6], in this paper we develop an algorithm to optimally sequence the retrieval orders. However we start from the more restrictive assump tion (which is based on the experience of many real world situations) that the load to be stored arrives at the 1/0 point with the address of the destination location already assigned. This means that it is not possible to choice, as assumed in the previous papers, the destination among a number of empty locations. Moreover we propose a procedure, based on parametric identification techniques, to compute precisely the throughput of the crane/aisle subsystem which comes out from the application of our technique. The continuous approximation allows us to express the throughput improvement as a function of a small number of meaningful parameters. Concerning the shuttle, we propose an algorithm to optimize the sequence of picking and deposit operations, so to minimize the time required by each cycle. Assuming again a continuous structure for the locations placed along the shuttle path, we evaluate the theoretical improvement of the shuttle sub-system throughput, with respect to the

0-7803-7272-7/02/$17.00 0 2002 IEEE


case in which no optimization is considered, obtained by the application of the proposed algorithm. A key point in the development of our technique is that, as we shall show, the cranes optimization is independent (in a sense) from the shuttle optimization and therefore they can be performed separately. The goodness of the proposed control algorithms, derived under the simplifying continuity assumption, will be validated via numerical simulations performed on a real warehouse model.

2 Modern Automated Warehousing Systems Control: Preliminary Results

The main goal of the paper is to propose an optimal strategy for improving the throughput of the warehouse; in the sequel we shall focus first on the cranes/aisles sub-system and then on the shuttle sub-system.

2.1 The Crane/Aisle sub-system We abstractly assume to deal with a single aisle, one-sided rack warehouse under the assumption that a DC cycle is adopted. For a single aisle one-sided rack warehouse, the throughput is defined as the inverse of the mean time required t o perform the cycle; obviously, a simple way to improve the throughput could be that of reducing the dimension of the aisle, and therefore of the corresponding rack. However, in a real warehouse (consisting of two or more twesided aisles), this approach would lead t o build more aisles if one desires to remain unchanged the storage capacity of the warehouse. This way of proceeding requires further replication of expensive subsystems and therefore should be avoided; in particular, in an automatic system, each aisle is equipped with its own crane, which generally mounts a rather expensive technology on-board. The above discussion leads t o the following conclusion: it is of mandatory importance to improve the throughput without reducing the rack dimension. In the sequel we shall make the following assumptions: AI We deal with a unit load system; a unit load is regarded as an indivisible part (at least from the system point of view) of a certain product. In the sequel wewill identify the unit load of a given product with the pallet which holds it (in other words we shall use the word load and pallet with the same meaning). The crane can carry one unit load at the time (one in the storage phase and one in the retrieval phase). Each location of the rack can contain exactly one unit load of the given product; A2 The crane travels simultaneously in both rack direction, with constant speed (accelerations and decelerations are not considered); A3 A random storage assignment is used. This means that no specific rule is followed when assigning a storage location to an incoming load; the problem of throughput optimization under more sophisticated storage strategies (for example class based [7]) will be discussed in future papers; A4 The rack is continuous; this means that it is composed of an infinite number of virtual locations. Each location is addressed by a pair of real numbers.

sh horizontal speed of the S/R machine [mlmin]; vertical speed of the S/R machine [mlmin]; L rack length [m]; rack I height [m]; t h = L/sh time to reach the horizontal end of the rack [min]; tv = H / s , time to reach the vertical end of the rack [min]; T = ma{th,tv} [min]; b = min {th/T, t,/T} rack shape factor (dimensional); E[SC]expected time for the crane Single Command (SC) cycle, i.e. time to travel from the 1 / 0 point (the retrieve location) to / 0 point) [min]; the storage location (the 1 E[TB]expected time for navel-Between (TB), i.e. time to travel from the storage location to the retrieve location [min]; rc time the crane needs to store or retrieve a load [min]; TW average time the crane has to wait at the aisle input buffer for an incoming load [min]; E[DCc]expected time for the crane DC cycle [min]. Without loss of generality we ;assumethat t h < t,; therefore the rack is the rectangular region depicted in Figure 2 (a). If we locate the 1/0 point a t the left-lower corner of the rack and f a a pair of coordinate axis with origin at the 1/0 point, the rack region can be identified with the set of points belonging to the interval [0, b] x [0,11. According t o Figure 2 (b), it is simple t o recognize that E[DCc]can be computed according to the following expression

E[DCc]= E[SC] E[TB] E[Sq

= 2E[SC:] E[TB]


In [4] it is shown that

E [ T B ] =( i + i b 2 - k b 3 ) T .
Therefore from (1) we have


(:- -t ; - 30 ) T .



If we consider a set of n DC cycles the crane throughput

is defined as
^IC =

+ 4n~c +nW nE[DCc:]

E[DCc] 4%



+ rw


Now let us assume of obtaining a reduction of the TravelBetween (TB) time equal to (Y * 100%; the new throughput can be written as

Obviously, 7~ > ^IC, therefore we let TC = (1 &)^IC, where , &* 100% represents the percentage throughput improvement. We have

Remark 1 Assumption Ad is standard in the study of warehousing systems (see for example [3], f d ] ) . Indeed it is not possible to develop a general theory if we try to take into account the discrete nature of the rack structure.
The following notation will be used:

Pc = ^IC

7c - ^IC


E[DCc]- aE[TB] 4TC

+ 4TC + rw

+w - 1 .



Figure 2: (a>Rack shape; (b) Computation of E[DC].

b, t h and rc which, f o r a given warehouse, are fixed, and o n the optimization parameters CY and niy; CY is increased when the T B time is optimized, whale is reduced when the shuttle operations are optimized.

Remark 2 Note that pc depends o n the design parameters

Remark 3 It is important t o recall that, whale the values of E[DCc], E [ T B ]are exactly evaluated in (3) and (4), the value r, is known and a will be evaluated in the next section, the parameter % can only be evaluated via simulation
and therefore only when a whole model of the warehouse is available.

In Figure 3 the behaviour of the function &(CY) parameterized in is depicted for b = 0.316, t h = 0.360 and TC = 0.04 (these are the parameter values of the real warehousing system we shall consider later in the paper). Note that, as expected, / 3 c is an increasing function of CY and a decreasing function of We note that for .rW = 0.5 a TB time reduction of 60% (that is CY = 0.6) corresponds t o a throughput improvement of 12%, which, according to [ 5 ] , leads, into a large warehouse, to the elimination of an aisle without reducing the overall warehouse performance. Also note that in correspondence of a smaller niy there is a strong improvement = 0.2 we have that of the throughput; for example for pc > 0.14.

A7 Along the mono-dimensional path of the shuttle there is a continuous distribution of picking/deposit location; we place the origin of the path abscissa in correspondence of the main input buffer and the end in correspondence of the main output buffer. The following notation will be used: ss speed of the shuttle [m/min]; Ls length of the shuttle path [m]; ts = L s / s s time to go from the origin of the path to the end of the path [min]; E [ P ]expected time to go from a given point to the next picking location [min]; E[D]expected time to go from the picking location to the next = E[D]when no optimization aldeposit location [min] ( E [ P ] gorithm is used); TS time the shuttle needs to pick up o r deposit a load [min]; E[DCs] expected time to go from a given point to the deposit location in the shuttle DC cycle [min] (E[DCs] = 2E[P]= 2E[D] when no optimization algorithm is used). At the moment, we assume to have a sequence of n pair of pick-up and deposit orders LISToyd = {(Pi, Di), i = 1 , . . . ,n), where P i and D, denote the location of the it h pick-up and deposit respectively, and to follow a FCFS policy; therefore each pair (Pi, Di) is executed according to the ordered list LIST,,d. In this case we have
E[DCs]= E[P] E[D]= 2 E [ P ] .



In the sequel we shall show that 1 E[P]= -t, ,

therefore, from (7) we have


2 (9) 3 Therefore, when no optimization is performed, the shuttle throughput is

E[DCs]= -t, .

0 (8 0 16
0 I4





To prove formula (8), without loss of generality, we consider a path of unit length; the final result will be then multiplied by t,. Let us denote by x E [0,1/2] the position of the picking location. Denote by 2, the stochastic variable whose realization represents the distance of an arbitrary chosen point of the interval [0,1]from the point x. The pdf G(x,z ) of 2, is given b Y z<o G ( x ,2) := f'rob(Zz 5 z )

O W t

Therefore the cdf g ( x , z ) has the following expression








Figure 3: Behaviour of pc(o) parameterized in Tw 2.2 The Shuttle Sub-system

We assume that the shuttle cycle is composed of a picking operation followed by a deposit operation. Therefore, also in the shuttle case, we talk of DC cycle. We make the following further .assumptions: A6 The shuttle travels with constant speed;

g ( x , z ) := -

0 k<O 2 o<z<x 1 x<l-s 0 z>l-x


The expected value, E ( x ) , of the distance of an arbitrary chosen point in the interval [0,1] from x E [O,1/21 is therefore given by

1 E ( x ) = Jdl-' zg(s, z)dz = x2 - x + -




By simmetry arguments, it is simple to recognize that E ( x ) , for z E [1/2,1], has the same expression as (13). Therefore, by integrating over x, we obtain E[P]as follows

E[P]= 2t,


E(z)d;c = -t,

1 3

2.3 A Performance Index for the Whole Warehouse So far we have introduced two partial performance indices:
the crane/aisle sub-system throughput

and the shuttle/picking refilling sub-system throghput


Due to Assumption AS, the only degree of freedom, in the optimization procedure, is the clever sequencing of the retrieval orders. The approach followed in this paper can be roughly described as follows: consider a list of NC storage and retrieval orders; optimize the sequencing of the NC retrievals; when the N c DCs have been performed consider a new list of N c orders and repeat the same procedure. This approach belongs t o the category of the so-called static approaches. An alternative way to proceed, which is not considered here, is known in the literature as the dynamic approach [SI, [9]. It consists of re-sequencing the list of orders whenever a new pair storage/retrieval is available; in this case it is necessary to assign due dates to the retrievals, to avoid that retrievals from locations at the end of the rack are never executed. To describe the algorithm we propose, let us denote by R the set composed of the retrieval location addresses, by S the ordered list of storage location addresses and by NC the cardinality of R and S.

Algorithm 1 Fkom the above formulae it is evident that an improvement Step 1 Take the first element s E S. Select the element of of the crane throughput 7% can be obtained, as observed R with the minimum TB distance f s measured by in Section 2.1, via the reduction of E[DCc]; in the same the Chebichev metric. Recall that, given two points way an improvement of the shuttle throughput ys can be PI = (21, y1) and P 2 = ( 2 2 , yz), the Chebichev distance obtained by reducind E(DCs]; note that the improvement dc(P1,P2) between PI and Pz is defined as of 7s also brings a benefit to the index ~c through the dc(P1,Pz) := m:ur{lzl -mI,Iy1 - yzl} ; reduction of In any case, from the point of view of the indices defined W e need to use Chebichev metric because the crane moves simultaneously in both directtons. above, the crane and shuttle optimizations can be performed independently each other. Step 2 Perform the DC cycle unth storage in s and retrieval from r; Now we shall introduce a further index which looks at the warehouse as a whole. This index is a measure of the Step 3 Update R = R - { r } , S = S - {s} ; f R = 0 then stop; otherwise goto Step 1. warehousing system throughput, since it considers com- Step 4 I plete missions, starting from the main input buffer and The performance of an optimization procedure similar to ending at the main output buffer or at some of the rack Algorithm 1 has been evaluated in [5]. However in that locations. This index is defined as follows study it was assumed that the destination address of the incoming load could be chosen among a number of empty number of complete operations rw = (17) locations; moreover the theoretical computation of the extime pected performance of that procedure was obtained in an approximate way. Fkom the point of view of index (17) there is interaction The aim of this section is that of computing exactly the between the crane and shuttle optimizations. For this reaexpected performance of the procedure described in Algoson the separate optimization of the two subsystems leads rithm 1 under the Assumptions Al-A4, AS. This result to a suboptimization of index ( 1 7 ) l . On the other hand will be then validated in Section 5 on a real warehouse by finding an optimization policy which looks at the system the analysis of the case study. as a whole is a very complex (and may be not achievable Note that, at Step 1 of Algorithm 1, one choices, between by simple strategies) objective, and is beyond the scope of the elements of R, the one which guarantees the minimum this paper. distance between the storage address and the retrieval adAccording to the above considerations, in the sequel we dress. Therefore the first question to be answered is the shall develop two strategies: i) reduction of the T B time in order to minimize E[DCc], i. e. to maximize the parameter a introduced in Section 2.1; ii) reduction of by optimization of the shuttle operations. The impact of the sen points in the same region? optimization policies on the index (17) will be a posteriori evaluated in Section 5 for the real life warehouse modeled in Section 3. 3.1 A procedure for the computation of &(b) Experience (see formulae (2a)-(2b)) suggestes that aver3 An Optimization Algorithm for the Crane/Aisle age distances are always polynomial functions of the shape Sub-system factor b. Therefore we assume the following structure for In this section we shall describe the proposed optimization the function dk(b) (an a posteriori analysis shows that it policy, under the following further assumption. suffices to stop at the fifth term) A8 Each load arrives at the 1/0 point with the storage location already assigned. The next procedure makes use of a parametric identiSimulation experiments show that an almost perfect decoupling fication based method to evaluate coefficient vector between crane and shuttle optimizations is reached for very small values of z. ak = (aok alk a2k a3k a4k a5k) .




o '

1 1 ,







Figure 4: Behaviour of a N ( b ) parameterised in b Procedure 1 Step 1 Grid the interval [ O , l ] by building the set B = {O,l/n, 2/n,... , I } . Step2 F o r b t E B , 1 = 1 , ...,n + l , i ) Grid the rectangle [0,bt] x [0,1]by building the set P = { ( x i ,U<),i = 1,. . ., U}; ii) For each point ( x i , y i ) E P compute numerically the average value of the smallest distance between k points arbitray cho_sen in the rack [0,bt] x [0,1] and the point ( x i , % ) ,say d k (bt, (x<,Yi));
iii) Compute the number J k ( b l ) =

performed. Therefore we have computed to the formula

(YN (b)


We note that, for b = 0.3,


2 0.6 for N c >_ 12.

Step 3 Build the matrices

4 An Optimization Algorithm for the Shuttle Sub-System The goal of this sub-section is the development of an optimal strategy for improving the throughput of the subsystem composed by the shuttle and the picking/refilling locations. The proposed algorithm makes a clever sequencing of the scheduled picking/deposit operations to improve the throughput of the shuttle system.

Algorithm 2
Step 0 (Initialization) Choice N s such that n / N s =: m a t c h E N. Let
S T A R T = The initial location of the shuttle COUNT = 0 .

Step 1 Consider the first N s elements of LISTord, (Pi,Di), i = 1,. . .,N s . Let Step 4 Compute a k = Htv, where H t denotes the pseudoinverse of H . By applying Procedure 1we found the coefficients for &(b), k = 1,... ,20, reported in Table 1. Now let us denote by E [ T B N ( ~the ) ] mean TB time necessary to perform the NC DC cycles by using Algorithm 1. When we couple the first storage location with the first retrieval location, we have to find the minimum among NC distances; concerning the second coupling we have to find the minimum among NC - 1distances and so on. Therefore
Nleft = NS L I S T = {(Pi,Di), i = 1, . . ,Nleft) LIsTord = LIsTord - L I S T .

Step 2 Compute:
D I S Z = dist(START,Pi) dist(Pi,0,) i = 1,. . . ,N l e f t . k = arg min DIST,
i d , ...,


pcuvr = p k ,

Dcurr = Dk

Step 3 Execute (Pcurp,Dcurr). Let

In Figure 4 the behaviour of a N ( b ) is depicted. Remember that Q * 100 has been defined in Section 2 as the percentage reduction of the TB time after the application of an optimal policy with respect t o the case in which no optimization is

Step 4 if Nleft # 0 goto Step 2; elseif C O U N T < nbotch Step 1; else end of the algorithm.



R e m a r k 4 Note that P i ' s represents the coordinates of the buffer locations where the pallets have t o be picked and D,' s the coordinates of the buffer locations where the pallets have to be stored; therefore, in order to implement the control algorithm described above, it i s necessary the knowledge of the tokens located in the fusion places of the shuttle net subsystem, in other words the knowledge of (part of) the state of the plant. Therefore, as in the classical control context, the control algorithm we finally use is state dependent.

Table 2: Behaviour of

0s versus NS

To evaluate analytically the expected time to execute a DC

under the application of Algorithm 2, let us assume t o deal with a unit length path and denote by 22 the stochastic variable whose realization represents the distance of an arbitrary chosen picking location from the starting point z E [0,1/2] plus the distance of an arbitrary chosen deposit location from the picking location. The cdf Go(x,z) of the stochastic variable Z," has the following expression
GO(X, Z)

Now if we have a block of IUS picking/deposit pairs, the first time we select the minimum between N S distances, the second time the minimum between NS - 1 distances, etc. Therefore the expected time to perform a set of N S DC cycles, in presence of optimization, is given by

:= Prob(Z,O 5



1 0


Now the shuttle throughput in presence of optimization is given by 2 therefore the shuttle throughput improvement ps is given b Y

+ $xz - L x 2 + Sz A 2 X 2 + 5 Ix - 1 -9.2 + f- f x + 1) z + ( - i X Z


The corresponding pdf is

o_<z<x x<z<l-x l-z<z<l+x 1 x 5 z < 2 -x 2> -2 - x (21)

d o(x,z)= go(x,z) := ;~FG

=Il o

1 0

$ z + $x


O<z<x x52<1-x l - x <z<l+x l + x 5 z <2-2 222-2


where E$[DCs] is given by (24). In Table 2 the values of ps for NS = 2,3, ...,10, r s = 0.067 and t , = 0.181 (these are the parameter values of the real warehousing system we shall consider later in the paper) are reported.

5 Case Study: Control of a Medium-Size

Now, by ZEk we denote the stochastic variable whose realization represents the minimum between k distances (corresponding to k picking/deposit pairs) starting by point x. It is simple to recognize that t.he cdf of z g k is given by
Gf(z, z ) = 1 - (1 - Go(z,z ) ~ )


From (23) it is possible to derive the pdf of Zzk

& ( z , z ) = k (1 - G 0 ( x , z ) ) d -G0(z,z) dz = k ( 1 - G o ( z ,z ) ) ~ - ' g o ( x , z ) .

Therefore the expected travel time, starting from z, corresponding to the minimum travel time between k possible picking/deposit pairs is

E f [ D C s ] ( z= ) t,


z k (1- G O ( Z , ~ ) ) ~ g-o' ( x , z)dz

Finally, by integration over x, we obtain Eo[DCs],the expected time to perform a shuttle DC cycfe by choosing among k picking/deposit pairs the one with minimum travel time from an arbitrary starting point in [0,1]

E f [ D C s ]= 2



Warehouse The main characteristics of the warehouse model used in the simulations and detailed in Section 3 are the following: sh = s, = 50[m/min]; rack location size= l[m] x l[m] x l[m]; number of x location= 18, thus L = 18[m]; number of y location= 57, thus H = 57[m]; speed of shuttle machine sg = 100[m/min]; length of the shuttle path Ls = 18.10 [m] time to move the pallet to the next buffer position TB = 0.083[min]; time to load/unload the pallet on the crane machine rc = 0.04[min]; time the shuttle needs to load/unload a pallet rs = 0.067[min]. Note that, according to the notation of Section 4, we have: t h = 0.36 [mzn]; t, = 1.14 [min]; T = t, = 1.14 [min]; b = 0.316; ts = 0.181 [min]. A simulation experiment has been conducted under the following assumptions. A number of 40 complete DC missions sets is considered for each crane machine. A complete DC mission consists of a storage mission starting from the first location of the main input buffer or from a picking/refilling area output location and terminating at an aisle rack location and of a retrieval mission starting from an aisle rack location and terminating at the last location of the main output buffer or at a picking/refilling area input location.


0 A partial DC mission consists of the part of a complete mission which starts at the input buffer of the aisle and terminates at the output buffer of the same aisle (this is the classical DC cycle considered in Section 2.1). 0 For each set of complete DC missions the experiment is repeated a number of times by permuting the list of storage orders S. For each set of the 40 complete DC missions, two simulation experiments, according to Algorithm 1 with N c = 1 and N c = 20, have been executed. In both simulations we not performthe shuttle optimization. We denote and DCN,=ZOthe average time t o perform a by DCN,==~ partial DC mission in the two cases. We obtained:

where YW and ?W are the throughputs before and after the optimization respectively. Therefore, concerning the throughput improvement of the whole warehouse, we have

pw * 100% = kZ-?? * 100%= 12.8%.

6 Conclusions We have proposed two control algorithms t o manage the cranes and the shuttle movements; the expected performance of such control algorithms has been derived under the simplifying assumption that the distribution of the locations within the warehouse is continuous; such assumption is standard in the warehousing system context and is necessary in order to obtain general results holding for any rack shape and depending on a few meaningful parameters. The evaluation of the control system performance on the real world warehouse, whose model has been developed in the first part of the paper, both validates the theoretical analysis previously obtained and shows the effectiveness of the proposed control algorithms. Open problems which are not discussed in this paper and will be investigated in future works are: i) Use of the cranes optimization algorithm in a dynamic way, but assigning due dates to the retrieval orders. ii) Development of control algorithms which take into account the different working conditions of the warehouse. Indeed it may be happen that a set of only retrieval orders or only storage orders can be assigned from the high-level scheduler; in this case by using an ad hoc algorithm better results could be achieved.


Now let us compare the theoretical and simulation results. According to the theory developed in Section 2.1, in the case of one-sided aisle and b = 0.3 we have that cy~~(0.3) = 0.69 (see Figure 4). From Figure 3 we have that the theoretical improvement pc of the throughput should = 0.9 and cy = 0.69). be about 11% (for Concerning the simulation results, taking into account that the computed average time to perform a simulated partial DC cycle also incorporates the term 47- q, we have (see formulae (4)-(6))


n L


- 0.714

2 2 -= 0.791. ?c = D C N c = 2 O 2.53
Therefore the real improvement is

J. P. Van den Berg, A literature survey on planning and control of warehousing systems, IEE Transactions, vol. 31, pp.

which is in accordance with the theoretical result. For the same sets of DC missions considered above, we have executed another simulation experiment in which also the shuttle optimization has been considered; indeed Algorithm 2 with N S = 5 has been applied. We denote by DCNc=?O,NsfS the average time to perform a partial DC mission in this case. We obtained:

Modeling of as/rs via coloured petri nets, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Como, Italy, pp. 1029-1034, July 2001. [3] S. C. Graves, W. H. Hausman, and L. B. Schwarz, Storageretrieval interleaving in automatic warehousing systems, Management Science, vol. 23, no. 9, pp. 935-945, 1977. Y . A. Bozer and J. A. White, Travel-time models for [4] automated storage/retrieval systems, IEE k n s a c t i o n s , vol. 16, no. 4, pp. 329-338, 1984. [5] M. H. Han, L.F. McGinnis, J. S. Shieh, and J. A. White, On sequencing retrievals in an automated storage/retrieval system, IEE Transactions, vol. 19, no. 3, pp. 56-66, 1987. H. F. Lee and S. K. Schaefer, Retrieval sequencing for [6) unit-load automated storage and retrieval systems with multiple o enin s Int. J. of Production Research, vol. 34, no. 10, pp.

1-13, 1999. [2] F.Basile, C.Carbone, and P.Chiacchio,

Thus we have:

yc = - 2

DCN~=ZO,N~ 2.42 =~

-2 - - 0.826.

In this case the real improvement is

which is in accordance with the theoretical result, obtained by looking at Figure 3 with cy = 0.69 and = 0.5. Concerning the throughput improvement index (17) associated to the global optimization of the warehouse, we compare the case in which no optimization is performed N c = 1, N s = 1) with the the full optimization case = 20, N s = 5); we obtain
YW = 0.465,

timal storage assignment in automatic warehousing swstems, Management Science, vol. 22, no. 6, pp. 629-638, 1976. (81 A. Seidmann, Intelligentcontrol schemes for automated storage and retrieval systems, Int. J. of Production Research, vol. 26, no. 5, pp. 931-952, 1988. [9] R. J. Linn and X. Xie, A simulation analysis of sequencing rules in a pull-based assembly facility, Int. J. of Production Research, vol. 31, no. 10, pp. 2355-2367, 1993.

2843-26d2, 1996. [7] W. H. Hausman, L. B. Schwarz, and S. C. Graves, Op-

y w

= 0.525,