Optimal routing in an automated storage/retrieval ... - Springer Link

IIE Transactions (1999) 31, 407±415

Optimal routing in an automated storage/retrieval system with dedicated storage JEROEN P. van den BERG* and A.J.R.M. (NOUD) GADEMANN Faculty of Mechanical Engineering, Production and Operations Management Group, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands E-mail: [email protected] Received October 1995

We address the sequencing of requests in an automated storage/retrieval system with dedicated storage. We consider the block sequencing approach, where a set of storage and retrieval requests is given beforehand and no new requests come in during operation. The objective for this static problem is to ®nd a route of minimal total travel time in which all storage and retrieval requests may be performed. The problem of sequencing a list of retrievals is equivalent to the Traveling Salesman Problem (TSP), and thus NP-hard in general. We show that the special case of sequencing under the dedicated storage policy can be solved in polynomial time. The results apply to systems with arbitrary positions of the input and output stations. This generalizes the models in the literature, where only combined input/output stations are considered. Furthermore we identify a single command area in the rack. At the end we evaluate the model against heuristic procedures.

1. Introduction Automated Storage/Retrieval Systems (AS/RS's) are widely used in warehouses and distribution centers around the world. Some advantages of AS/RS's over traditional warehousing systems are high space utilization, reduced labor costs and improved inventory control. An AS/RS consists of one or multiple parallel aisles with storage racks alongside. Usually, in every aisle a Storage/ Retrieval (S/R) machine travels and performs the storage and retrieval of goods. But sometimes an S/R machine can travel between aisles, thus serving more than one aisle. The control of AS/RS's has received considerable attention in the literature. Hausman et al. [1] discuss three storage policies for the AS/RS: randomized storage, class-based storage and dedicated storage. The latter two policies attempt to reduce the mean travel time by allocating incoming loads to open locations based on product demand. The demand may be measured by the turnover rate per storage volume or the Cube per Order Index (COI), as is discussed by Heskett [2]. Whereas Hausman et al. [1] consider only single command cycles, Graves

*Current address: Berenschot, P.O. Box 8039, 3503 RA Utrecht, The Netherlands Tel.:+31 30 291 6822, Fax: +31 30 291 6826, Email: [email protected] 0740-817X

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1999 ``IIE''

et al. [3] study the e€ects of performing dual command cycles. They observe travel time reductions of up to approximately 30%. Hwang and Lim [4] present an ecient method for ®nding the optimal dwell point position for the S/R machine when idle. When sequencing requests on an AS/RS, we have to make a trade-o€ between eciency and urgency. If we only consider eciency, then some requests may come late. Likewise, if we only look at urgency and sequence requests according to FCFS, then this will deteriorate eciency. Han et al. [5] suggest two approaches for sequencing storage and retrieval requests in a dynamic environment: (1) Select a block of the most urgent storage and retrieval requests; sequence the requests in the block and when all storage and retrieval requests in the block have been completed, select the next block, et cetera. (2) Re-sequence the requests every time a new request is received and employ due dates or priorities. We will refer to these approaches as block sequencing and dynamic sequencing, respectively. Han et al. [5] investigate the block sequencing approach and remark that the problem of sequencing a list of retrievals is equivalent to the Traveling Salesman Problem (TSP), and thus NP-hard in general, see Garey and Johnson [6]. They suggest the well-known nearest neighbor heuristic for selecting retrievals for dual com-

408 mand cycles and present an analytical study. They also discuss the no cost zone for the Chebyshev metric, i.e., the area in the rack with open locations (i.e., empty locations) that may be visited for a storage without extra travel time according to the Chebyshev metric, while traveling from the input station to the retrieval location. Based on this, they present the shortest leg heuristic. However, this heuristic was outperformed by the nearest neighbor heuristic due to the fact that it appeared to ®ll up the area close to the Input/Output (I/O) station. Lee and Schaefer [7] use a Linear Assignment Problem to solve the block sequencing problem when there is an equal number of storage and retrieval requests. The solution of the Linear Assignment Problem may correspond to an infeasible sequence since a location may be used for storage while the product has not been retrieved yet. The authors use the ranking algorithm of Murty [8] which repetitively ®nds the next best solution of the Linear Assignment Problem. Since this might require excessive computation times until a feasible solution is found, they impose a limit on the number of iterations and apply a heuristic that constructs a feasible solution at each iteration. Several simulation studies have been presented that investigate the dynamic sequencing approach in an AS/ RS, these include the work of Schwarz et al. [9], Linn and Wysk [10,11], Seidmann [12] and Linn and Xie [13]. In this paper we address the block sequencing approach in an AS/RS that operates under the dedicated storage policy. Van den Berg [14] identi®es situations where the dedicated storage policy provides competitive results with respect to average travel times and space requirements. We consider the problem of ®nding a sequence of all storage and retrieval requests with the least travel time for the S/R machine. The sequencing problem is known to be NP-hard in general. We show that the special case of block sequencing with dedicated storage can be solved to optimality in polynomial time. We do so by modeling the problem as a Transportation Problem. The results apply to systems with arbitrary positions of the input and output stations, either separate or combined. In some operations it is preferred to position the input and output stations of the AS/RS at separate positions around the rack. This especially holds for point-of-use storage systems that supply material to a close-by production unit. In such a system, incoming products enter the system from one end of the aisle and retrieved products leave from the other end to reduce the ¯ow of material through the production unit. The authors are not aware of any work that addresses the sequencing problem in an AS/RS with the input and output stations at separate positions. Besides the importance of the results for static problems, the formulation of the transportation problem is insightful in that it may serve as a stepping stone or building block for designing ecient heuristics for dynamic routing problems. In particular we identify a single command area for which locations in the warehouse it is never advantageous

van den Berg and Gademann to be visited in a dual command cycle. At the end we evaluate the model against two heuristic procedures for some randomly generated instances. In Section 2 we make assumptions and introduce notation. Subsequently in Section 3, we examine the problem and formulate it as a Transportation Problem. In Section 4 we develop two new heuristics and evaluate their performance against the optimal solutions found with the Transportation Problem.

2. Assumptions We consider an AS/RS that consists of one aisle with two storage racks alongside. If the AS/RS consists of multiple parallel aisles with one Storage/Retrieval (S/R) machine in every aisle, then we may address each aisle separately. A single-load S/R machine travels within the aisle to perform the storage and retrieval requests. Goods are stored on storage modules such as pallets or totes. We will refer to a storage module containing products as a unit-load. Unit-loads for storage arrive at the input station and wait at an accumulator conveyor until the S/R machine transports these to a rack location. Consequently, storages are performed according to a First Come First Served (FCFS) routine. We assume that the block sequencing approach is applied. Accordingly, lists of storage and retrieval requests are given that need to be ful®lled and no new requests are added to these lists during the processing of the requests. We assume that the AS/RS operates under the dedicated storage policy. Hence, we know the storage location of an incoming unitload. The retrievals may be executed in any sequence. If multiple unit-loads of a requested product are available in the rack, then the unit-loads for the retrieval requests on the list are selected according to the First In First Out (FIFO) rule, i.e., the unit-loads that are longest in storage will be retrieved ®rst. This means that the FIFO rule applies to subsequent blocks of storages and retrievals, but not necessarily to the retrievals within a block. The S/R machine deposits retrieved unit-loads at the output station. We refer to the rack location where a storage (retrieval) is to be performed, as the storage position (retrieval position). We allow arbitrary positions of the input and output stations. We assume that the time to pick up or deposit a unit-load is independent of the sequence of the requests. Accordingly, we minimize the makespan for performing all storage and retrieval requests by minimizing the travel time. Consequently, we choose to minimize the travel time as our objective. We use the following notation: m = number of storages; n = number of retrievals; Si = storage position corresponding to storage i ˆ 1; . . . ; m. The storages Si are indexed according to the sequence of arrival;

Optimal routing in an automated storage/retrieval system Rj = retrieval position corresponding to retrieval j=1; . . . ; n; I = the position of the input station; O = the position of the output station; SP = the starting position of the S/R machine; t…A; B† = travel time between position A and position B. We make the following assumptions: (1) The starting position of the S=R machine is given. (2) The S/R machine is initially empty. (3) The S/R machine ends at the position of the last operation. (4) There is at least one retrieval request …n  1†. (5) All storages are initially present at the input station. (6) All unit loads at positions R1 ; R2 ; . . . ; Rn are initially available in the rack. (7) All storage positions S1 ; . . . ; Sm ; are initially empty. Furthermore, we assume that the travel time of the S/R machine satis®es the triangle inequality, which states that the travel time between arbitrary positions A, B and C is such that t…A; B†  t…A; C† ‡ t…C; B†. If a problem instance only has storage requests, contrary to assumption 4, then it becomes trivial because the sequence of the storage requests is ®xed. Assumption 2 may be relaxed since if the S/R machine is initially loaded, then it ®rst travels to the destination of the unit-load and we de®ne that position as the starting position of the S/R machine. Also assumption 3 may be relaxed. A minor modi®cation (cf. Remark 1 at the end of Section 3) allows the method presented in this paper to ®nd an optimal route when the ending position of the S/R machine is speci®ed.

3. The transportation problem A route may be seen as a concatenation of loaded travels, with empty travels in-between. An empty travel may have length 0, if the ending position of a loaded travel coincides with the starting position of the subsequent loaded travel. We may associate a unique route with each sequence of the loaded travels, since it follows from the triangle inequality that direct (i.e., straight) loaded and empty travels constitute a shortest route for any speci®ed sequence of the loaded travels. In the context of routing an S/R machine in an AS/RS with a combined I/O station, the notion of single and dual command cycles is well-known. Since we deal with arbitrary positions of the input and output stations, the notion of a command cycle needs to be generalized. We de®ne a command cycle as any trip of the S/R machine between two successive visits of the input station and/or the output station. In a single command cycle a single rack location is visited, so either one storage or one retrieval is performed. In a dual command cycle two rack locations are visited, so both a storage and a retrieval are performed. Note that in case of separate input and output

409

stations a command cycle not necessarily starts and ends at the same position. In case of a combined I/O station this de®nition coincides with the usual de®nition of a command cycle. In this section we show how the problem of ®nding a sequence of given storage and retrieval requests which minimizes the total travel time may be modeled as a Transportation Problem. For each storage i; a loaded travel …I ! Si † must be performed, i ˆ 1; . . . ; m: Likewise, for each retrieval j; a loaded travel …Rj ! O† must be performed, j ˆ 1; . . . ; n. We refer to these travels as si and  rj ; respectively: The travel times of the loaded travels are ®xed, the travel times of the empty travels follow from the sequence of the loaded travels. Accordingly, we want to ®nd a sequence of the loaded travels which minimizes the total empty travel time. Figure 1 depicts the route: SP ! s1 ! s2 ! r2 !  r1 ! r4 ! s3 ! r3 : Recall that the sequence of the storage requests is ®xed (FCFS), while the sequence of the retrieval requests may be chosen freely. Since the retrieval travels  r1 and r4 are performed in single command cycles, these cycles may be inserted after any visit of O without altering the length of the route. So for instance the route r3 !  r4 ! r1 also is a feasible SP ! s1 ! s2 ! r2 ! s3 !  concatenation of the travels in Fig. 1. Therefore, without loss of generality, we restrict ourselves to the following set S of sequences. De®nition 3.1. For given loaded travels s1 ; . . . ; sm ; and  r1 ; . . . ;  rn , let S be the set of all sequences of these loaded travels with the following two properties: (1) Storage travel si precedes storage travel si‡1 in the sequence, for i ˆ 1; . . . ; m ÿ 1 (FCFS). (2) All  rj ; j 2 f1; . . . ; ng, which are preceded by an empty travel …O ! Rj †; i.e., the single command retrieval cycles, are sequenced according to increasing index immediately after the ®rst retrieval. r1 ; . . . ; rn ; which Note that any sequence of s1 ; . . . ; sm ; and  satis®es property 1, may be converted into a sequence in S

Fig. 1. A route with three storages and four retrievals. The continuous arcs represent the loaded travels and the dashed arcs represent the empty travels.

410 of the same length by altering the positions of the single command retrieval cycles. Hence, S does contain an optimal sequence of the loaded travels. According to assumption 3 the route ends after the last request. Hence, the ending position of the route is either Sm ; if the last request is a storage, or O, if the last request is a retrieval. Accordingly we introduce a virtual ending position EP ; which may be reached by an empty travel of zero length from either Sm or O after all requests have been ful®lled. Since there is one empty travel before the ®rst loaded travel, one between each two consecutive loaded travels, and one after the last loaded travel, a route consists of …m ‡ n ‡ 1† empty travels and …m ‡ n† loaded travels. Departure positions for empty travels are: SP ; S1 ; . . . ; Sm and O. Arrival positions for empty travels are: I; R1 ; . . . ; Rn and EP : One empty travel departs from each rj ends in O, so that n empty travels SP ; S1 ; . . . ; Sm . Each  depart from O. One empty travel arrives in each R1 ; . . . ; Rn ; EP : Finally, there are m empty travels arriving in I, before each s1 ; . . . ; sm . We formulate a Transportation Problem (TP) as follows. Let G ˆ …V1 ; V2 ; E† be a complete bipartite graph with V1 a set of …m ‡ 3† vertices representing the departure positions of the empty travels and V2 a set of …n ‡ 2† vertices representing the arrival positions of the empty travels within the route. The set V1 consists of: one vertex representing SP ; m vertices representing S1 ; . . . ; Sm ; and two vertices O and On , both representing O: Hereby, the vertex O represents the ®rst …n ÿ 1† visits of O and the vertex On represents the n-th visit of O: We make this distinction, because after the ®rst …n ÿ 1† visits of O; an empty travel to one of the R1 ; . . . ; Rm or to I may follow, while after the n-th visit of O; an empty travel to I or to EP may follow. In the TP, the number of departures from vertex O is …n ÿ 1† and the number of departures from all other vertices in V1 is one. The set V2 consists of: n vertices representing R1 ; . . . ; Rn ; one vertex representing I and one vertex representing EP . The number of arrivals in vertex I is m and the number of arrivals in all other vertices in V2 is one. The number of transports between any two vertices in a solution of the TP represents the number of empty travels between the corresponding positions. Figure 2 shows graph G for a problem with three storages and four retrievals. The number of departures or arrivals is denoted beside the vertices. The costs associated with the edges in set E are equal to the travel times of the corresponding empty travels, as de®ned in Table 1. Some empty travels are forbidden. The corresponding edges have been assigned costs 1 in Table 1 and have been omitted from Fig. 2. The following theorem shows that any solution of the TP corresponds to a route without disjoint subtours. This result is not trivial. For instance, it is a well-known result

van den Berg and Gademann

Fig. 2. Transportation problem with three storages and four retrievals.

Table 1. Costs related to edges in the transportation problem From

To

Cost

SP SP SP Si Si Sm Si Sm O O O On On On

Rj I EP Rj I I EP EP Rj I EP Rj I EP

t…SP ; Rj † j ˆ 1; . . . ; n t…SP ; I† 1 t…Si ; Rj † i ˆ 1; . . . ; m, j ˆ 1; . . . ; n t…Si ; I† i ˆ 1; . . . ; m ÿ 1 1 1 i ˆ 1; . . . ; m ÿ 1 0 t…O; Rj † j ˆ 1; . . . ; n t…O; I† 1 1 j ˆ 1; . . . ; n t…O; I† 0

that when the Traveling Salesman Problem is solved with a similar approach, a solution with multiple disjoint subtours may be found. Theorem 3.1. Each feasible solution of the TP corresponds to a sequence in S, and vice versa. Proof. In the Appendix. Corollary 3.1. An optimal solution of the TP corresponds to an optimal sequence of the storage and retrieval requests.

Optimal routing in an automated storage/retrieval system Proof. Theorem 3.1 states that each solution of the TP corresponds to a sequence in S and vice versa. The costs associated with a solution of the TP represent the empty travel time. Since the loaded travel times are ®xed, the optimum solution of the TP corresponds to a sequence that is optimal in S. Since we already concluded that S contains an optimal solution, the sequence is optimal. Remark 3.1. We may model the situation where the S/R machine has a speci®ed ending position EP (contrary to assumption 3) by rede®ning the costs in Table 1 of the transport from Sm to EP to be: t…Sm ; EP † and the costs of the transport from On to EP to be: t(O,EP). As we mentioned in the introduction, the block sequencing problem is equivalent to the Traveling Salesman Problem, and thus NP-hard in general. We have shown in this section that the block sequencing problem with dedicated storage is a special case, which is polynomially solvable. Special cases of the TSP have been shown to be polynomially solvable, as by Burkard et al. [15]. Only few polynomially solvable special cases have been described in the literature that concern the ®eld of warehousing. Examples include the problem of ®nding an optimal route for manual order-pickers in a parallel-aisle warehouse, discussed by Ratli€ and Rosenthal [16], and the problem of ®nding an optimal sequence for order-picking in a carousel, discussed by Bartholdi and Platzman [17].

411

associated single command cycles, if t…A; B† ‡ t…O; I† < t…A; I† ‡ t…O; B†: If the dual command cycle addresses position Sm , then it is not necessary to return to the input station since it is the last storage. In this situation a dual command is obligatory if retrieval requests still have to be performed. The nearest neighbor heuristic and the cheapest insertion heuristic are well-known heuristics for ®nding feasible solutions of the Traveling Salesman Problem as is discussed by Golden and Stewart [18]. In this section we develop their equivalents for the problem of sequencing storage and retrieval requests in an AS/RS with arbitrary positions of the input and output stations. In contrast to the procedures presented in the literature, the heuristics do not only consider empty travels between storage and retrieval positions, but also consider the input and output stations as possible ending and starting positions for empty travels. According to Theorem 3.1 each feasible solution of the TP de®nes a feasible route. Hence, we may de®ne a heuristic that successively selects empty travels from A 2 fSP ; S1 ; . . . ; Sm ; O; On g to B 2 fR1 ; . . . ; Rn ; I; EP g until it has found a feasible solution. Hereby, dual command cycles which require more travel time than the two corresponding single command cycles are forbidden. Hence, the heuristic may only select an empty travel …A ! B†; with A 2 fSP ; S1 ; . . . ; Sm g and B 2 fR1 ; . . . ; Rn g, if: t…A; B† ‡ t…O; I† < t…A; I† ‡ t…O; B† or A ˆ Sm :

4. Computational results In this section we present two heuristics for the problem of sequencing requests in an AS/RS and compare their performance with the optimal solutions found with the TP. The results with the exact algorithm give additional insights in the computation times and some properties of optimal solutions. The heuristics that we present are based on the insights from the TP-model. Although there is no need for heuristics for the block sequencing problem, the heuristics appear to be very powerful and may therefore be valuable tools for sequencing requests in a dynamic situation. If the input and the output stations coincide, then it follows from the triangle inequality that a dual command cycle never requires more travel time than the corresponding single command storage cycle and single command retrieval cycle. However, if the input and output stations are at separate positions, then a dual command cycle is not always advantageous. This is due to the fact that, after a dual command cycle is performed, eventually the S/R machine has to return to the input station to pick up the next storage, i.e., the empty travel …O ! I† must be performed. In other words, a dual command cycle that addresses positions A 2 fSP ; S1 ; . . .; Smÿ1 g and B 2 fR1 ; . . . ; Rn g saves time in comparison with the two

…1†

We de®ne two variations of this heuristic. The Nearest Neighbor Heuristic one by one selects empty travels commencing in SP , S1 ; . . . ; Smÿ1 , Sm =On , O, successively, with the least resulting travel time, until a feasible solution to the TP is found. The selection of the travels commencing in Sm and On is performed at the same time, since the selection of one uniquely speci®es the other. The Cheapest Insertion Heuristic successively selects empty travels with the least travel time which may commence and end at any position, until a feasible solution to the TP is found. Again the selection of the empty travels commencing in Sm and On takes place at the same time. In both heuristics an empty travel …A ! B†; with A 2 fSP ; S1 ; . . . ; Sm g and B 2 fR1 ; . . . ; Rn g is only allowed if it satis®es condition (1). We evaluate the heuristics using randomly generated problem instances. We consider a rack with dimensions 1  b (in time units), where 0 < b  1 is the shape factor as discussed by Bozer and White [19]. We choose SP ˆ …0:5; 0:5† and I ˆ …0; 0†: In the problem instances in Table 2 we choose O ˆ …0; 0†, i.e., I and O coincide, and in the problem instances in Table 3 we choose O ˆ …1; 0†, i.e., I and O are at separate positions. The …x; y†-coordinates of the storage and retrieval positions are randomly selected from uniform distributions U ‰0; 1Š and U ‰0; bŠ, respectively. The travel times of the

412

van den Berg and Gademann

S/R machine are measured with the Chebyshev or maximum metric. Tables 2 and 3 show the mean travel times for processing a block of m storages and n retrievals. The mean travel times are found by taking the average over 1000 problem instances. `NN' represents the Nearest Neighbor Heuristic, `CI' represents the Cheapest Insertion Heuristic and `TP' represents the Transportation Problem. Moreover, we compare the mean travel times for the TP and the two heuristics to the expected travel time for a situation in which storages and retrievals are performed according to a FCFS sequence. We compute the associated travel times using the following expressions for the expected travel times [19]

1 1 …2† E…t…SP ; R†† ˆ ‡ b2 ; 4 12 1 1 E…t…S; I†† ˆ ‡ b2 ; …3† 2 6 1 1 …4† E…t…O; R†† ˆ ‡ b2 ; 2 6 1 1 1 …5† E…t…S; R†† ˆ ‡ b2 ÿ b3 ; 3 6 30 where S 2 fS1 ; . . . ; Sm g and R 2 fR1 ; . . . ; Rn g: For each con®guration we select a sequence of storages and retrievals that minimizes …2†; . . . ; …5†. From the results in Table 2 we may compute that for the con®guration with the I/O station at a corner of the

Table 2. Mean travel times in an AS/RS with length 1 and height b and the I/O station at (0,0) m

n

5 5 10 15 20 5 5 10 15 20 5 5 10 15 20

b

5 15 10 5 20 5 15 10 5 20 5 15 10 5 20

Loaded travel time

1.0 1.0 1.0 1.0 1.0 0.75 0.75 0.75 0.75 0.75 0.5 0.5 0.5 0.5 0.5

Empty travel time

6.7 13.3 13.3 13.3 26.7 5.9 11.9 11.9 11.9 23.8 5.4 10.8 10.8 10.8 21.6

FCFS

NN

CI

TP

2.2 8.7 4.5 8.8 9.2 1.9 7.7 4.0 7.9 8.1 1.8 7.0 3.6 7.2 7.3

2.0 7.5 3.0 8.0 4.8 1.7 6.6 2.7 7.1 4.2 1.6 6.0 2.4 6.3 3.7

1.6 7.4 2.5 7.2 4.0 1.4 6.6 2.2 6.5 3.5 1.2 5.9 2.0 5.9 3.1

1.4 6.6 2.2 6.9 3.5 1.3 5.8 2.0 6.2 3.0 1.1 5.2 1.7 5.6 2.6

Table 3. Mean travel times in an AS/RS with length 1 and height b, the input station at (0,0) and the output station at (1,0) m

5 5 10 15 20 5 5 10 15 20 5 5 10 15 20

n

5 15 10 5 20 5 15 10 5 20 5 15 10 5 20

b

1.0 1.0 1.0 1.0 1.0 0.75 0.75 0.75 0.75 0.75 0.50 0.50 0.50 0.50 0.50

Loaded travel time 6.7 13.3 13.3 13.3 26.7 5.9 11.8 11.8 11.8 23.8 5.4 10.8 10.8 10.8 21.6

Empty travel time FCFS

NN

CI

TP

6.3 13.0 13.0 13.0 26.3 5.7 11.6 11.6 11.6 23.5 5.2 10.6 10.6 10.6 21.5

5.6 11.7 11.2 12.2 22.3 5.3 10.9 10.7 11.1 21.4 5.0 10.2 10.1 10.3 20.5

5.5 11.7 11.1 11.8 22.0 5.2 10.8 10.6 10.9 21.2 4.9 10.2 10.1 10.2 20.4

5.4 11.5 10.9 11.6 21.6 5.1 10.7 10.4 10.8 20.9 4.9 10.1 10.0 10.1 20.3

Number of dual commands 2.5 3.5 5.7 3.4 12.4 1.9 3.1 4.5 2.9 10.4 1.0 2.0 2.8 1.8 6.7

Optimal routing in an automated storage/retrieval system rack, FCFS requires on average 79% more empty travel time, NN requires on average 29% more empty travel time and CI requires on average 12% more travel time than the TP. From the results in Table 3 we may compute for a con®guration with I and O at separate corners of the rack that FCFS requires on average 11% more empty travel time, NN requires on average 3% more empty travel time and CI requires on average 1% more travel time than the TP. For the con®guration with the I/O station at one corner of the rack, the computational results suggest that the TP performs considerably better than the two heuristics. For the con®guration with I and O at separate corners of the rack the computational results suggest that the two proposed heuristics perform almost as well as the TP. This seems to be inherent to the problem as shown in Table 3 by the fact that the FCFS sequence is relatively close to the optimum when I and O are at separate corners of the rack. This may be explained by the fact that the empty travel time that is saved by combining a storage and a retrieval in a dual command cycle may be lost again some time later due to the extra travel from O to I: If the I/O station is positioned at a corner of the rack, then an optimal solution will contain as many dual command cycles as possible. It is interesting to see how many dual command cycles will occur in an optimal solution if I and O are at distant positions. For the TP solutions, we represented in Table 3 the average number of dual command cycles per block of m storages and n retrievals. Obviously, only dual command cycles visiting positions A and B will occur for which t…A; B† ‡ t…O; I†  t…A; I† ‡ t…O; B† or A ˆ Sm ; since otherwise the two single command cycles would require less travel time and the solution would not be optimal. Command cycles for which t…A; B† ‡ t…O; I† ˆ t…A; I† ‡ t…O; B† are supposed to be performed in two single command cycles. Table 3 shows that a considerable number of dual command cycles occurs in the optimal solutions even though I and O are at distant positions. Furthermore, we see that the number of dual command cycles increases more than proportionally as the number of storages and retrievals increases. Clearly, a larger number of storages and retrievals improves the possibilities for ecient dual command cycles. Table 3 shows that the time-savings of the TP compared with the other heuristics and FCFS grow for increasing b: This may be explained by the following lemma, which distinguishes the area in the rack where no time is gained by performing dual command cycles. Lemma 4.1. Let S 2 fSP ; S1 ; . . . ; Smÿ1 g and R 2 fR1 ; . . . ; Rn g be given. If t…I; S† ‡ t…S; O† ˆ t…I; O† or t…I; R† ‡ t…R; O† ˆ t…I; O†, then a dual command cycle with S and R is never shorter than the individual single command cycles.

413

Proof. Let t…O; S† ‡ t…S; I† ˆ t…O; I†. Accordingly, we derive the stated result with the triangle inequality by comparing the empty travel times: t…O; I† ‡ t…S; R† ˆ t…O; S† ‡ t…S; I† ‡ t…S; R†  t…S; I† ‡ t…O; R†: Due to the symmetry of the problem the lemma also holds when t…I; R† ‡ t…R; O† ˆ t…I; O†: j We will refer to the area of locations A for which t…I; A† ‡ t…A; O† ˆ t…I; O† as the single command area. Figure 3 shows the single command area for the Chebychev metric in a rack with the input station at (0,0) and the output station at (1,0). It is immediately seen from Fig. 3 that the fraction of the rack area in which no time is saved by performing dual command cycles increases as b decreases and fewer e€ective dual command cycles will be possible. Also note that the single command area vanishes if I and O coincide. In the numerical examples the cheapest insertion heuristic consistently outperformed the nearest neighbor heuristic. This can be explained by the fact that the cheapest insertion heuristic considers all locations that must be visited, while the nearest neighbor heuristic considers only one. However, the nearest neighbor heuristic does not perform signi®cantly worse than the cheapest insertion heuristic. Furthermore the nearest neighbor heuristic has a straightforward interpretation in a dynamic situation, because it addresses the storage requests on a ®rst come ®rst served basis. Therefore the nearest neighbor heuristic may be of a particular interest in dynamic situations.

5. Conclusions We considered block sequencing for an AS/RS that operates under a dedicated storage policy. We addressed the

Fig. 3. Single command area in a rack with coordinates in time. The diagonal lines indicate the path of the S/R machine when it travels with full horizontal and vertical speed.

414 situation where storage and retrieval requests are performed in blocks and all requests in a block must be executed before commencing with the next block. We studied the problem of ®nding a sequence with the least travel time for a given block of storage and retrieval requests. Due to the dedicated storage policy, the loaded travel times are ®xed, so that we need to minimize the empty travel times. We identi®ed the empty travels that may occur in an optimal route and we incorporated these empty travels in a Transportation Problem (TP). The TP enables us to ®nd an optimal solution in polynomial time. This is a notable result, since the problem is a special case of the Traveling Salesman Problem, which is NP-hard in general, [6]. This is the ®rst research that allows arbitrary positions of the input and output stations. Based on the TP-model, we designed two heuristics that seem to give near-optimal results on average. The heuristics may be valuable tools for other, in particular dynamic, situations. Furthermore, we identi®ed a single command area in the rack. If a storage or retrieval position lies within this area, then a dual command cycle is never shorter than the two corresponding single command cycles. From computational results we observed that the performance of the TPalgorithm improves relative to the heuristics as the fraction of the rack occupied by the single command area decreases.

References [1] Hausman, W.H., Schwarz, L.B. and Graves, S.C. (1976) Optimal storage assignment in automatic warehousing systems. Management Science, 22(6), 629±638. [2] Heskett, J.L. (1963) Cube-per-order index ± a key to warehouse stock location. Transportation and Distribution Management, 3, 27±31. [3] Graves, S.C., Hausman, W.H. and Schwarz, L.B. (1977) Storageretrieval interleaving in automatic warehousing systems. Management Science, 23(9), 935±945. [4] Hwang, H. and Lim, J.M. (1993) Deriving an optimal dwell point of the storage/retrieval machine in an automated storage/retrieval system. International Journal of Production Research, 31(11), 2591±2602. [5] Han, M.-H., McGinnis, L.F., Shieh, J.S. and White, J.A. (1987) On sequencing retrievals in an automated storage/retrieval system. IIE Transactions, 19(1), 56±66. [6] Garey, M.R. and Johnson, D.S. (1979) Computers and Intractability: A Guide to the Theory of NP-completeness, W.H. Freeman, New York. [7] Lee, H.F. and Schaefer, S.K. (1996) Retrieval sequencing for unitload automated storage and retrieval systems with multiple openings. International Journal of Production Research, 34(10), 2943±2962. [8] Murty, K.G. (1968) An algorithm for ranking all the assignments in order of increasing costs. Operations Research, 16(3), 682±687. [9] Schwarz, L.B., Graves, S.C. and Hausman, W.H. (1978) Scheduling policies for automatic warehousing systems: simulation results. AIIE Transactions, 10(3), 260±270. [10] Linn, R.J. and Wysk, R.A. (1987) An analysis of control strategies for an automated storage/retrieval system. INFOR, 25(1), 66±83.

van den Berg and Gademann [11] Linn, R.J. and Wysk, R.A. (1990) An expert system framework for automated storage and retrieval system control. Computers and Industrial Engineering, 18(1), 37±48. [12] Seidmann, A. (1988) Intelligent control schemes for automated storage and retrieval systems. International Journal of Production Research, 26(5), 931±952. [13] Linn, R.J. and Xie, X (1993) A simulation analysis of sequencing rules in a pull-based assembly facility. International Journal of Production Research, 31(10), 2355±2367. [14] van den Berg, J.P. (1996) Planning and control of warehousing systems. Ph.D. thesis, University of Twente, Fac. of Mech. Eng., Enschede, The Netherlands, Ch. 7. [15] Burkard, R.E., Deineko, V.G., van Dal, R., van der Veen, J.A.A. and Woeginger, G.J. (1996) Well-solvable special cases of the TSP: a survey. Memorandum, Eindhoven University of Technology, Dep. of Math. and Comp. Science, Eindhoven, The Netherlands. [16] Ratli€, H.D. and Rosenthal, A.S. (1983) Order-picking in a rectangular warehouse: a solvable case of the traveling salesman problem. Operations Research, 31(3), 507±521. [17] Bartholdi, III, J.J. and Platzman, L.K. (1986) Retrieval strategies for a carousel conveyor. IIE Transactions, 18(2), 166±173. [18] Golden, B.L. and Stewart, W.R. (1985) Empirical analysis of heuristics, in Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G. and Shmoys, D.B. (ed), The Traveling Salesman Problem, John Wiley &Sons, Chichester, Ch. 7. [19] Bozer, Y.A., and White, J.A., (1984) Travel-time models for automated storage/retrieval systems. IIE Transactions, 16(4), 329± 338.

Appendix Proof of Theorem 3.1 The proof is constructive. Let L be the set of loaded travels and let E be the set of empty travels associated with a solution to the TP without any transports with cost 1. Observe that such a solution exists, since any sequence of the requests corresponds to a solution of the TP without transports with cost 1. We de®ne a directed multigraph GT ˆ …fSP ; S1 ; . . . ; Sm ; O; R1 ; . . . ; Rn ; I; EP g; L [ E† containing all loaded and empty travels (see Fig. A1). We may construct a route in GT by starting in SP and successively selecting the next (empty or loaded) travel until we arrive in EP . The selection of the next travel may be done according to the following rules:  At vertices SP ; S1 ; . . . ; Sm ; R1 ; . . . ; Rn ; the selection of the next travel is trivial, since there is only one outgoing travel.  At vertex I; we select the loaded travel si with the smallest index which has not been performed yet, in order to satisfy property 1 of De®nition 1.  At vertex O; we select the empty travel …O ! Rj † with the smallest index j which has not been performed yet, in order to satisfy property 2 of De®nition 1; if no such empty travel exists, then we select an empty travel …O ! I† which has not been performed yet.

Optimal routing in an automated storage/retrieval system

415

i ˆ 1; . . . ; m ÿ 1; j ˆ 1; . . . ; n: Accordingly, vertex O is visited at least once, so that also the retrievals rj for which there exists an empty travel …O ! Rj † have been performed. (2) The set E contains an empty travel …On ! EP †. In this situation the constructed route from SP to EP contains n visits to O. This means that all retrievals  rn have been performed, as well as any stor1 ; . . . ;  rages si for which there exists an empty travel …Si ! Rj †, with i 2 f1; . . . ; mg and j 2 f1; . . . ; ng. If m ˆ 0; then we have found a route with all travels. Otherwise, since there cannot be an empty travel …Sm ! EP † there must be an empty travel …Sm ! Rj †, with j 2 f1; . . . ; ng; so that we ®nd that in particular sm has been performed. This implies that also the storages s1 ; . . . ; smÿ1 have been performed, since the storages are selected according to increasing index.

Fig. A1. Graph GT corresponding to the route in Fig. 1. The dashed arcs depict the empty travels, the continuous arcs depict the loaded travels.

Only at the n-th visit of O, we perform the empty travel that corresponds to the transport which commences in node On of the TP: The TP is constructed in such a way that all vertices in GT have an equal number of incoming and outgoing travels, except SP and EP : Accordingly, the above selection rules construct a route from SP to EP : Next, we show that this route contains all travels. We distinguish the following two situations: (1) The set E contains the empty travel …Sm ! EP †. Consequently, the constructed route from SP to EP visits Sm . This means that all storages s1 ; . . . ; sm have been performed, since the storages are selected according to increasing index, as well as any retrievals  rj for which there exists an empty travel …SP ! Rj † or …Si ! Rj †, with i 2 f1; . . . ; m ÿ 1g and j 2 f1; . . . ; ng. There are at most …n ÿ 1† empty travels …O ! Rj †; j ˆ 1; . . . ; n; so that there must be at least one empty travel …SP ! Rj † or …Si ! Rj †;

In both situations the constructed route from SP to EP contains all loaded travels. Thus, for any solution of the TP we may construct a route which contains all loaded travels and satis®es the properties in De®nition 3.1 and thus corresponds to a sequence in S. The reverse implication holds, since for an arbitrary route in S, we may identify the empty travels in the route and assign one transport between the two associated vertices in graph G of the TP. Graph G is constructed in such a way that this results in a feasible solution of the TP:

Biographies Jeroen P. van den Berg received an M.Sc. in Applied Mathematics and a Ph.D. in Mechanical Engineering at the University of Twente in The Netherlands. The title of his Ph.D. thesis was Planning and control of Warehousing Systems. Since 1997 he has been a Management Consultant for Berenschot in Utrecht, The Netherlands. His interests include logistics, warehousing, supply chain management, IT and warehouse management systems. Noud Gademann graduated in 1987 and received his Ph.D. in 1993, both in Applied Mathematics at the University of Twente in The Netherlands. The title of his Ph.D. thesis was Linear Optimization in Random Polynomial Time. Since 1993 he has been an Assistant Professor in the Production and Operations Management group of the Faculty of Mechanical Engineering at the University of Twente. His interests include combinatorial optimization, probabilistic algorithms, warehousing systems and production management.