Optimal storage locations in a carousel storage and retrieval system

Location Science,Vol. 4, No. 4, pp. 237-245, 1996 8 1997 ElsevierScience Ltd

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OPTIMAL STORAGE LOCATIONS IN A CAROUSEL STORAGE AND RETRIEVAL SYSTEM RAYMOND

G. VICKSON*

Department of Management Sciences, Faculty of Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

and ALLEN

FUJIMOTO

A.T. Kearney Ltd, 20 Queen Street West, Toronto, Ontario MSH 3R3, Canada (Revised version received 30 October 1996)

Abstract-This paper examines the problem of optimal product location in a single bi-directional carousel storage and retrieval system. For items with independent demands, it establishes the long-run average optimality of a simple demand rate ranking and partitioning scheme which has previously been suggested in the literature through heuristic reasoning and simulation studies. 0 1997 Elsevier Science Ltd. Key words: Carousel, optimal packing, ranking, partitioning.

1. INTRODUCTION Carousel storage and retrieval systems are sometimes used in warehouses for the storing and order picking of small, light, and highly demanded items. They consist of a series of bins which are linked together in a closed loop and mounted on an oval track. Each bin has several shelves (usually equal in number and equally spaced) on which the various stocked items are stored. Carousels operate under computer control by rotating the track in order to bring a requested item to a fixed location (the picking station), where it is then retrieved by a manual or automated picker. Most carousels are bi-directional, meaning that they can rotate in both a clockwise and counter-clockwise direction. In bi-directional carousels, the rotational direction is chosen by the controller to minimize the distance traveled by the next item to be picked. Typical carousels contain 30-50 bins, having about lo-12 shelves each. They usually require between one and two minutes to complete one revolution. Figure 1 illustrates a typical carousel configuration. An important practical problem in carousel management concerns the locations at which to stock the demanded items. Some facilities simply use a random storage strategy, wherein replenished items are put into any available bin. However, a more effective strategy is to use dedicated storage, with each separate item being assigned permanently to its own bin. In this case, there arises the issue of how to assign individual items to bins so as to minimize the *Author for correspondence. Supported by NSERC grant A9304 from the Natural Sciences and Engineering Research Council of Canada and by a MRCO grant from the Manufacturing Research Corporation of Ontario. 237

R. G. VICKSON and A. FUJIMOTO

238 Pick

shelf rotate desired item to pick station Fig. 1. A carousel storage and retrieval system.

long-run average travel distance (or time) between successive picks. This paper treats the problem of optimal product location in a single bi-directional carousel. We develop a very simple and intuitive stocking policy to minimize the long-run average travel time for a sequence of single independent and identically distributed retrievals. This policy was suggested in Lim et al. (1985) but has not previously been proven to be optimal. The arguments in Lim et al. are loose; the storage policies recommended are based largely on undocumented simulation results and the authors’ intuition. The policy has also been discussed at length in Fujimoto (1991), in which an extensive series of tabu search (Glover, 1989, 1990) runs was used to amass evidence of its effectiveness. Our work here completes these investigations by proving optimal@ of the policy. Problems resembling the carousel stocking problem have appeared in the computer science literature. Grossman and Silverman (1973) and Yue and Wong (1973) discussed optimal strategies for organization of an archival store, disk space allocation, and pagination. Yue and Wong, in particular, developed a ranking and partitioning scheme for access probabilities that is the same as ours. However, the methods and results in Grossman and Silverman and Yue and Wong cannot be applied to our problem because these references assumed monotonicity of the distance function d(i,j) between requests at locations i and j; that is, they assumed that i
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We do not discuss the closed-loop mode in this paper, but merely note that the same product location strategy applies to it as well; see Vickson and Lu (1996). This means the problem of optimal product location may be separated from that of carousel operation. The structure of this paper is as follows. Section 2 introduces the problem and states a simple algorithm for its optimal solution. In Sections 3 and 4 we prove results necessary in establishing the validity of the algorithm. Finally, Section 5 summarizes our results and presents suggestions for future research.

2. STATEMENT OF THE PROBLEM Following Lim et al. (1985) Bartholdi and Platzman (1986) and Stern (1986) we assume in the sequel that each product consists of a number of identical units, and thus serves to cover a cycle of several demand requests. For example, a product may be a case of cartons, with demands occurring at the individual carton level. The model thus assumes that replenishment of products on the carousel is done off-line, perhaps during evenings or weekends. We consider a carousel having n bins of m shelves each. There are N = nm stored products, each of which occupies a single shelf. We assume that the items i = 1,2, .... N are requested independently and at random. This assumption of independent demands is common in warehousing studies; see Lim et al. (1985) and Stern (1986) in the carousel warehousing context and Bozer and White (1984) Egbelu (1991) Graves et al. (1977) Hackman and Rosenblatt (1990), Hausman et al. (1976) and Jarvis and McDowell (1991) in the context of rectangular warehouses*. Problems with demands that are correlated between different products are much more difficult. Grossman and Silverman (1973) illustrate the difficulties that occur even in very small-scale problems with correlated requests from an archival store in a computer system. It is our belief that exact optimal solutions are virtually impossible to obtain for problems involving correlated demands on realistically sized carousels. However, the policies obtained below are reasonable heuristics if a few of the largest demands are positively correlated, because they strive to keep the items with high demands in close proximity. We denote the product demand rates as Ai and normalize them so that ;I, represents the probability that the next request is for product i. Here ;i, > 0 for all i, and Cz I 1, = 1. It should be noted that this normalization is not needed in applying the packing procedure; it is used only in the interpretation of the objective R below as an average distance (in bins). There are two levels to the problem. These are: ??the ??the

grouping problem: how to group the products for packing into different bins location problem: how to assign the groups to the bins

For a grouping G = {Gi, i = l,..., n}, each group Gi contains m products. The total request probability of group G; is Pr(G,) = P; = ,z. i,. Each group becomes, in effect, a single product, and the problem then becomes the one of assigning the total request probabilities Pi to bins. Any such assignment

simpler

*Unlike the case of computer file access, where retrievals may be many thousands of times more frequent than in warehouses, it is unusual for warehouse management to have reliable estimates of demand correlations. Often, crude average demand rates are all that are available.

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corresponds to a permutation 0 of (1, 2, .... n). We let pi denote the total request probability assigned to bin i under permutation r~and grouping G; that is, pi is the Pk value put into bin i. (Here we suppress the dependence ofpi on (G, a) for notational simplicity.) Under open-loop operation, the expected rotational distance between successive picks for given G and cr is R(GJJ) = ii, ,,p, pip, d(i,j),

(1)

where

d(i, j) =

(j-i\,

if (j-ii

n- Ij-ii,


if Ij-i(

>n/2

(2) is the shortest distance between bins i and j. The optimal expected rotational distance is R*= rn$ { rn)

R(G,a)].

(3)

The solution of the lower-level problem R(G)= min R(G,a)

(4)

for given G follows from a result of Bergmans (1972). The solution of the higher-level problem R*= rn$ R(G)

(5)

is our main contribution in this paper. It should be noted that the algorithm below depends only on the fact that d(i,j) is a function of the difference Ii-j1 and possesses certain symmetry and monotonicity properties. The algorithm thus applies equally well to the more general problem of minimizing the longrun average travel time t&j), where t(ij) has the form t(i,j) = $(d(i,j)), with $(d)aO being any strictly increasing function such that 4(O) = 0. This permits the algorithm to be used even when there are acceleration effects that may cause the travel time t(i,j) to differ significantly from the travel distance d(i,j). The problem in equation (3) is solved completely by the following simple ranking and partitioning scheme. Algorithm Step 1. Sort the products in order of descending demand: I, a&>...

>&.

Step 2. Partition the products into groups of m each, starting from the top of the list: group G, = items 1,2,...,m group GZ= items m+ 1,m+2,...,2m Step 3. Assign group G1 to an arbitrary bin, then assign G2 next to G,, G3 next to G, on the opposite side, Gq next to G2, Gs next to G3, and so forth. We follow Lim et al. (1985) and refer to the resulting probability profile

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...PSPIP. PZPh... or

in Step 3 as an organ pipe arrangement (OPA). Bergmans (1972) established the optimal@ of OPA for certain types of location problems involving linear or circular patterns of points. His approach used a necessary condition for optimality which, while correct, was incorrectly proved. Since we need this result for other purposes in the sequel, we will include a corrected proof of it in the next section. Following this, we use this result to prove optimality of the ranking and partitioning scheme given in Steps 1 and 2 of the algorithm. Considered in the context of general partitioning problems, Steps 1 and 2 of the algorithm produce a partition referred to as an Extremal Partition by Anily and Federgruen (1991). However, the form of our objective (1) is different from any of those treated by Anily and Federgruen, so their results cannot be applied to our problem. 3. OPTIMALITY OF THE OPA We define a symmetry line of the carousel to be a line through its center that divides it symmetrically into halves. If n is even, both ends of a symmetry line either pass through bins or bisect the arc between two bins (recall that we think of the bins as points). If n is odd, a symmetry line passes through a bin at one end and bisects the arc between two bins at the other. A symmetry line L divides the bins into three sets, A, B and C, where A and B are the bins on opposite sides of L and the possibly empty set C is the set of bins lying on L itself. (For notational simplicity, we suppress the L-dependence of A, B and C.) For any j EA we denote byj’ its mirror image with respect to L in the other set B. Figure 2 shows a symmetry line whose mirror image pairs are (2,3), (1,4), (n,5), .... (id’). Following Bergmans (1972), we say that an assignment G has the pairwise majorizing property (PMP) with respect to a symmetry line L if Pi2:Pi'

(6)

holds for all i on the same side of L. Bergmans’ fundamental, but incorrectly proved, necessary condition for optimal@ is the following. L

Fig. 2. A symmetry line and mirror image points.

R. G. VICKSON

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and A. FUJIMOTO

Theorem 1. An assignment is optimal only if the PMP holds for all symmetry lines. Proof. Suppose that (T is an assignment in which the PMP is violated for some symmetry line L. Then the bins in A are split into two non-empty, disjoint subsets A+ = { &A:pi>p;,) and A- = (jEA:pj 2pip holds at least one i E A +. Using the symmetry properties d(i,j) = d(j,i) = d(i’,j’) and d(i,j’) = d(j,i’), the expected rotational distance R becomes R = RI + Rt2 + RX+ 2 ,Z_ jz_

[(pipj+ pi,p,,,)d(i,j) + (pip,/ + Pi~Pj)d(ivj’)l,

(7)

where R, and R2 denote similar sums but with both summation indices i and j in A+ or A-, and R3 denotes a similar sum with at least one of i or j in C if C is not empty.’ The factor 2 arises in (7) because each (i,j) pair contributes pipjd(i,j) +pipjd(j,i) = 2pipjd(i,j) in (1). Now consider the new assignment 0’ obtained by interchanging all pairs pj,pj, for j E A-. Its expected rotational distance R’ is given by equation (7), with R,, R2 and R3 unchanged and with pj,pf swapped in the fourth term. This implies R-R’ ~2 i,%+ js%_ (P, -p;,)(pi,-Pj)[d(i,j’)-d(i,j)l.

(8)

Since d(i,j’) > d(i,j), each term in (8) is nonnegative, and at least one term is positive. Thus R’-CR, and so the original assignment is not optimal. ?? Remark. Bergmans applied this argument for a single pi, pi’ interchange with j E A-. His argument fails if the set A- contains more than one element, because when only one member j of A- is exchanged with its mirror image, the remaining members of A- interact with j to form a contribution having the wrong sign. It is possible to construct numerical examples of this type in which every single pairwise pi, pi, interchange for j E A- actually increases the value of R. However, these ‘wrong sign’ effects cancel out when we interchange all pairs simultaneously. Optimal@ of the OPA follows from Theorem 1, and for the sake of completeness we indicate very briefly the type of argument involved in showing this. If P2 is put in a bin 2
pj’

4. OPTIMALITY

OF THE RANKING

AND PARTITIONING

SCHEME

The ranking and partitioning scheme presented above determines unique bin probabilities Pf,P* z,...,Pz with the property that PT is as large as possible, then Pz is as large as possible after removing the first group, and so forth. Optimality of this approach follows from the next result, which constitutes the main contribution of this paper. Theorem 2. An optimal grouping must have PI = P:, P2 = Pi, . .. . P, = Pz. ProoJ Consider a grouping G with group probabilities PI > Pz> ... > P,. Suppose that P,
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and R2

=

j 2

;

k2

i Pdkd(.i,k).

w

Since P, 1 such that & <&,. Let pr =qt +A, andpi = qi + &, and note that q1 > qi. Consider now a new packing with products a and b interchanged. This has probabilities p,’ = q1 + A,, and pi’ = qi + & in bins 1 and i, respectively. The new packing might not be an OPA - and if it is not, it can be improved later by a further rearrangement - but the probability p,’ in bin 1 is still the largest. The new expected rotational distance is R’ = RI ’+ RZ, where R,‘=2p,’

j#Z:1.i Pjd(lJ)+2Pi’ js,i Pjd(i,j>+2pl’pi’d(l,i).

(11)

We have

j;,

R-R’=%k-AJ 1

i pj[d(i,j)-d(l,j)]+(q, .

-q;)d(l,i)

Let L be the symmetry line that bisects the arc from bin 1 to bin i, and let A be the set of bins j # 1 on the same side of L as bin 1. Using the symmetry properties d&j) = d(l,j’) and d(l,j) = d(i,j’) forj &A, we have ,.J, i p&W) -41 A = .,fk Qj-pj*)[d(i,j)

-41 J>l.

(13)

The PMP implies pj apjp for all j E A, because the bins in A are on the same side of L as bin 1 (whose probability is the largest) > . Furthermore, d(i,j) > d(lj) for all j EA. It follows that all terms in the summation on the right of equation (12) are nonnegative, and the last term (q, -qi)d(l,i) is positive. This implies R’ r, such that &,<&,. The change in the expected distance due to interchanging products a and b can be analyzed by the expressions above, but with bin 1 replaced by the P, bin and with L taken as the symmetry line that bisects the arc from the P, bin to the Pi bin. As before, we obtain R’ CR, so the original grouping is not optimal. Optimal@ thus requires Pi = Pi* for i = r also, and hence for all i = 1, .... n. W

5. CONCLUSIONS, AND TOPICS FOR FUTURE RESEARCH This paper establishes the optimality of a simple ranking and partitioning scheme for locating products in a bi-directional carousel. The scheme is easily implemented by an inexperienced user, as it involves no more than sorting the products according to their annual demand rates and then packing them one-by-one on the carousel, starting from the top of the sorted list. It is also important to pack the bins in an OPA, but again, this is quite straightforward. The sorting phase is the most arduous part of the procedure for manual implementation, but this offers no barrier to the use of the method, even with hundreds of products, if the product data are available in something as simple as an electronic spreadsheet. In reality, carousel storage and retrieval systems sometimes operate by batching together a number of orders and then picking them sequentially. Exact and heuristic procedures are

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given in Bartholdi and Platzman (1986) and in Stern (1986) for determining the best routing of the carousel through the pick list in such situations. Our optimal packing scheme should still be effective because it strives to keep the most frequently demanded items in close proximity. In Lim et al. (1985) it is argued that the approach remains valid in principle because it optimizes the travel distance to the first item picked in a list. However, this is not quite true, because the first accessed bin would be something akin to a minimum of several random variables, whose probability distribution would be different from that of the individual bins. The ranking and partitioning scheme continues to have strong intuitive appeal, but more research is needed into the question of optimal packing for batched retrievals. Another extension of the work here is to the case of multiple carousels that feed a common picking station. Several related questions that arise are: (1) should the most highly demanded items be concentrated on one carousel, or should they be dispersed among many? (2) should the ranking and partitioning scheme be applied on a carousel-by-carousel basis, or should products be alternated among the different carousels? (3) is, in fact, any version of the ranking and partitioning scheme still valid in the multiple carousel case? We are currently studying these issues by simulation and will offer experimentally derived insights in a future paper. Acknowledgements-The authors are grateful to anonymous referees for their comments and suggestions. The research in this paper was supported by the Natural Sciences and Engineering Research Council of Canada and the Manufacturing Research Corporation of Ontario. It was carried out in association with the Waterloo Management of Integrated Manufacturing Systems (WATMIMS) research-group.

REFERENCES Anily, S. & Federgruen, A. (1991) Structured partitioning problems. Operafions Research, 39, 130-149. Eartholdi, J. J. & Platzman, L. K. (1986) Retrieval strategies for a carousel conveyor. IIE Transactions, 18, 166-173. Bergmans, P. P. (1972) Minimizing expected travel time on geometrical patterns by optimal probability rearrangements. Information and Control, 20, 331-350. Bozer, Y. A. & White, J. A. (1984) TravelAtime models for automated storage/retrieval systems. IIE Transactions, 16, 329-338.

Egbelu, P. J. (1991) Framework for dynamic positioning of storage/retrieval machines in an automated storage/ retrieval system. International Journal of Production Research, 29, 12-37. Fujimoto, A. (1991) Models for determining storage strategies for carousel storage and retrieval systems. M.A.Sc. Thesis, Department of Management Sciences, University of Waterloo. Glover, F. (1989) Tabu search - part I. ORSA Journal on Computing, 1, 190-206. Glover, F. (1990) Tabu search - part II. ORSA Journal on Computing, 2,4-32. Graves, S. C., Hausman, W. H. & Schwartz, L. B. (1977) Storage-retrieval interleaving in automatic warehousing systems. Management Science, 23,935-945. Grossman, D. D. & Silverman, H. F. (1973) Placement of records on a secondary storage device to minimize access time. Journal of the Association for Computing Machinery, 20, 429-438. Hackman, S. T. & Rosenblatt, M. J. (1990) Allocating items to an automated storage and retrieval system. IIE Transactions. 22. 7-14.

Hausman, W. H., Schwartz, L. B. & Graves, S. C. (1976) Optimal storage assignment in automatic warehousing systems. Management Science, 22, 629-638. Jarvis, J. M. & McDowell, E. D. (1991) Optimal product layout in an order picking warehouse. IfE Transactions, 23,93-102.

Lim, W. K., Bartholdi, J. J. and Platzman, L. K. (1985) Storage schemes for carousel corweyors under real time control. Material Handling Research Center Tech. Report MHRC-TR-85-10, Georgia Institute of Technology. Stern, H. L. (1986) Parts location and optimal picking rules for a carousel conveyor automatic storage and retrieval system. Proceedings of the 7th International Conference on Automation in Warehousing, San Francisco, California, pp. 185-192.

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Vickson, R. G. and Lu, X. (1996) Optimal product and server locations in one-dimensional storage racks, European J. of Operational Research, forthcoming. Yue, P. C. & Wong, C. K. (1973) On the optimality of the probability ranking scheme in storage applications. Journal of the Association for Computing Machinery,

20, 624-633.