int. j. prod. r es., 2000, vol. 38, no. 6, 1339 ± 1356
Simulation study of an automated storage/retrieval system JEROEN P. VAN DEN BERG{ and A. J. R. M. (NOUD) GADEMANN ‡* In this paper we present a simulation study of an automated storage/retrieval system and examine a wide variety of control policies. We compare several storage location assignment policies. For the class-based storage policy, we apply a recent algorithm that enables us to evaluate the trade-o between storage space requirements and travel times. We also study a new storage location policy which combines low storage space requirements with short mean travel times. Furthermore, we study the sequencing of storage and retrieval requests whereby we focus on the trade-o between e cient travel of the S/R machines and response time performance.
1.
Introduction Automated storage/retrieval systems (AS/RSs) are widely used in industry. An AS/RS is a product-to-picker storage system that consists of one or multiple parallel aisles with two high bay pallet racks alongside each aisle. A storage/retrieval (S/R) machine or autom ated stacker crane travels within the aisle and performs storages and retrievals. The S/R machine moves along rails that are mounted to the ¯ oor and the ceiling. In a typical con® guration, the S/R machine carries at most one pallet. Pallets for storage arrive at the input station and wait at an accumulator conveyor until the S/R machine transports these to their storage locations. This makes it obligatory to always select the ® rst pallet from the input queue, i.e. ® rst come, ® rst served (FCFS). The S/R machine deposits retrieved loads at the output station. The S/R machine has three independent drives for horizontal, vertical and fork movement. Hence the travel time of the S/R machine is measured by the maximum of the isolated horizontal and vertical travel times. In many applications the S/R machine is restricted to one aisle. Due to its unit-load capacity, the operational characteristics of the S/R machine are limited to single command cycles and dual command cycles. In a single command cycle either a storage or a retrieval is performed between two consecutive visits of the input and output station. In a dual command cycle the S/R machine consecutively performs a storage, travels empty to a retrieval location and performs a retrieval. The empty travel between the storage and retrieval location is referred to as interleaving travel . Revision received August 1999. { Berenschot Management Consultants, Distribution Logistics Group, PO Box 8039, 3503 RA Utrecht, The Netherlands. ‡ Centre for Production, Logistics and Operations Management, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands, and ORTEC Consultants BV, PO Box 490, 2800 AL Gouda, The Netherlands. * To whom correspondence should be addressed. e-mail:
[email protected] International Journal of Production Research ISSN 0020± 7543 print/ISSN 1366± 588X online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals/tf/00207543.html
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Multi-load S/R machines enable multiple storage and retrieval requests to be performed per cycle, thereby reducing the mean transaction time. A miniload AS/RS is an AS/R S that is designed for the storage and order-picking of small items. The items are stored in modular storage drawers or in bins. These containers may be subdivided into multiple compartments, each containing a speci® c product. The order-picker resides at the end of the aisle at a pick station. The literature on the planning and control of AS/RSs has focused on two topics: Request sequencing and storage location assignment. Hausman et al. (1976) are one of the ® rst to consider request sequencing. They study sequencing with single command cycles only. Graves et al. (1977) study the e ects of performing dual command cycles. They observe travel time reductions up to approximately 30%. Han et al. (1987) show that the AS/RS throughput performance may be improved by cleverly sequencing the retrieval requests, so that the interleaving travel time between storage and retrieval locations in a dual command cycle is reduced. The authors suggest two approaches for sequencing storage and retrieval requests. (1) Select a subset (wave) of storage and retrieval requests; sequence the requests in the wave. Release the next wave when all storage and retrieval requests in the current wave have been completed. (2) Re-sequence the requests every time a new request comes in and employ due dates or priorities to ensure that a retrieval at the far end of the aisle is not excessively delayed. We will refer to these approaches as wave sequencing and dynam ic sequencing, respectively. In this paper we will focus on dynamic sequencing. Several simulation studies have been presented of an AS/RS under the dynamic sequencing approach. Schwarz et al. (1978) substantiate the equivalence of the closest open location (COL) rule and randomized storage under realistic conditions. Moreover, they compare the closest open location rule and the class-based storage policy with two or three classes. Finally they consider imperfect information on the turnover rates of products. Linn and Wysk (1987) systematically evaluate a number of storage and retrieval selection rules when the product demand shows seasonal trend. Seidmann (1988) presents a dynamic control approach that adapts its policies based on the number of requests waiting and changes in turnover rate. Linn and Wysk (1990) present an expert system that modi® es its controls depending on the utilization rate of the AS/RS. Linn and Xie (1993) consider urgency rules for an AS/ RS in an assembly environment with given due dates. The second topic that has received considerable attention in the literature is the assignment of incoming stock to storage locations. Hausman et al. (1976) present three storage location assignment policies: class-based storage, randomized storage and dedicated storage. The class-based storage policy distributes the products, based on their demand rates, among a number of classes and reserves a region within the storage area for each class. Accordingly, an incoming load is stored at an arbitrary open location within its class. Randomized and dedicated storage are in fact extreme cases of the class-based storage policy: random ized storage considers a single class and dedicated storage considers one class for each product. The class-based storage policy and the dedicated storage policy attempt to reduce the mean travel times for storage/retrieval by storing products with high demand at locations that are easily
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accessible. The demand for a product may be estimated by the cube-per-order index (COI) (see Heskett 1963) . Van den Berg (1996) presents a polynomial time dynamic programming algorithm that distributes products and locations among classes such that the mean single command travel time is minimized. The algorithm outperf orms previous algorithms (e.g. Graves et al. 1977, Hausman et al. 1976, Rosenblatt and Eynan 1989, and Eynan and Rosenblatt 1994) . The algorithm holds for any demand curve, any travel time metric, any warehouse layout and any positions of the input station and output station. Moreover, it allows that the inventory level varies and determines the storage space requirements per class by imposing a risk-level on stock over¯ ow. In this paper we use simulation to evaluate the performance of various control policies for the AS/RS. Simulation is mandatory to adequately model all operational features of the AS/RS, since existing analytical models only apply to special instances. Some examples of state of the art analytical models are: Bozer and White (1984) for randomized storage and FCFS sequencing of the retrievals; Han et al. (1987) for randomized storage and the nearest-neighbour rule for selecting storage locations and sequencing retrieval requests; Seidmann (1988) for the same problem but with the relaxed assumption that retrievals are not necessarily selected according to the ® rst in ® rst out (FIFO) rule when a requested product is stored at multiple locations; Eynan and Rosenblatt (1993) for class-based storage together with the nearest-neighbour rule for selecting storage locations and retrievals within the same class; Pan and Wang (1996) for class-based storage in a squarein-time rack. We study several aspects of the control of warehousing systems. One aspect is the location assignment for incoming products. Here we consider the class-based storage policy (among other policies) and we use the dynamic programming algorithm developed in Van den Berg (1996) to determine an optimal class-partition. We also introduce a new policy (continuous storage) . A second aspect of AS/RS control is the sequencing of storage and retrieval requests. For this problem we develop policies based on the heuristics presented in Van den Berg and Gademann (1999) . A third aspect is the response time or due date performance. In the literature most publications on sequencing and scheduling procedures have focused on the system throughput. In this paper we also evaluate the due date performance. This paper is organized as follows. In } 2 we make assumptions. Subsequently, in } 3 we sketch the control policies and rules that will be examined in the simulations. Next, in } 4 we present and discuss the simulation results. We end with some conclusions. 2.
Assumptions In this paper we use simulation to study the dynamic sequencing approach, i.e. requests arrive randomly and need to be ful® lled within short response times. We base our simulation on the product set, turnover rates and travel speeds of the AS/ RS in the spare parts distribution centre of the Yamaha Motor Co., situated in The Netherlands. For each product the turnover over the last year is known. The turnover distribution is such that 20% of the products represents 78% of the total turnover. We are interested in the performance of an AS/RS that consists of one aisle, while the AS/ RS originally consists of six aisles. Therefore, we uniformly selected 1526 products
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which generate an appropriate workload for one crane. These products are stored in one aisle with two storage racks each having 50 £ 16 locations, which gives a total of 1600 locations. All locations have identical height and width: 1 m £ 1 m, so that the rack dimensions become: 50 m £ 16 m. The S/R machine has independent drives for horizontal and vertical travel. The maximum crane speeds are 80 m min– 1 in horizontal direction and 24 m min– 1 in vertical direction. The time to pick up or deposit a unit-load is 15 s. This pick up/deposit time incorporates the time lost by acceleration/deceleration. As a result, we may assume that acceleration/deceleration is instantaneous. Hence, the travel times of the S/R machine may be measured by the Chebyshev or maximum metric, i.e. the travel time is equal to the maximum of the isolated horizontal and vertical travel times. We assume that the input and output stations are at the lower-end corner of the aisle. Van den Berg and Gademann (1999) have shown that the relative travel time savings due to sequencing rules are considerably larger in this situation than for remote input and output stations. The warehouse manager of Yamaha estimates that the products are present approximately 95% of the time for fast moving products down to 85% of the time for slow moving products. We assume that the turnover rates of all products, the product mix and the ordering policies of the products do not change. We determine the storage space requirements with the method described in Van den Berg (1996) . In this method the number of available unit-loads at an arbitrary epoch is approximated by a Normal distribution. Subsequently, a su ciently large storage space is determined, so that the given set of products ® ts during a speci® ed fraction a of the time. 3.
Control policies In this section, we present the policies and rules that will be evaluated in the simulation. We distinguish the following policies and rules: (1) (2) (3) (4)
storage location assignment policies; request selection rules; open location selection rules; urgency rules.
A storage location assignm ent policy imposes constraints on the selection of open locations for incoming unit-loads. In case the policy does not make a unique selection, an open location selection rule is applied. A request selection rule determines the sequence in which the storage and retrieval requests are executed, by successively selecting the next request. We attempt to prevent excessive delay of retrievals, by employing urgency rules. In }} 3.1± 3.4 we elaborate on the various policies and rules that are examined in the simulation. 3.1.
Storage location assignm ent policies In the simulation study we consider the following storage location assignment policies: (1) (2) (3) (4)
randomized storage; class-based storage; dedicated storage; continuous storage.
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The ® rst three policies are well known in the literature and in practice. The classbased storage policy partitions the products among a number of classes and reserves a region within the rack for each class. The randomized storage policy allows a product to be stored anywhere in the rack. The dedicated storage policy assigns speci® c locations to each product, which may only be occupied by that product. The continuous storage policy is newly presented in this study. This policy combines low storage space requirements with short mean travel times. For the class-based storage policy (CL-K, where K denotes the number of classes), we determine the sizes of the classes and the distribution of the products among the classes with the dynamic programming algorithm presented in Van den Berg (1996) . Accordingly, each class-region is large enough to hold the products assigned to that class during a speci® ed fraction a of the time. The randomized storage policy (RAN) is the special case CL-1. For the dedicated storage policy (DED) we reserve one storage location for each product, since at most one unitload of each product is available at the same time. For the continuous storage policy (CON) we estimate the required storage space out as if it were randomized storage. Let t in denote the travel time from location j and t j j to the input and output station, respectively. Now, the continuous storage policy out ranks all rack locations on non-decreasing …tin j ‡ tj † and all products on nonincreasing demand per reserved location. Then, it calculates for each product i the smallest integer value li such that P…Q i– 1 ø li – 1† ù a , where Q i– 1 is a random variable denoting the number of present unit-loads of products 1, . . . , i – 1 at an arbitrary epoch. Recall from } 2 that we approximate the number of available unitloads at an arbitrary epoch by a Normal distribution. Now, when a unit-load of product i comes in, it is assigned to an open location with index j ù li . In other words, the location of product i is chosen such that su cient locations closer to the input and output stations are saved for products with a higher demand per location. In fact the CON policy becomes equivalent to the DED policy when a ˆ 1. Next we consider the exception that all open locations lie outside the designated region. This issue does not apply to the randomized storage policy. Preliminary simulations indicate that the best results are obtained in this situation when open locations are selected closest to the designated region, but preferably not closer to the input and output stations. In other words, one should prevent to assign unit-loads to locations that are designated to faster moving products. This may ® ll up the space for fast moving products more rapidly, which may cause a signi® cant increase in travel time. In the simulation studies we apply this exception rule. 3.2.
Request selection rules A clever sequence of the storage and retrieval requests may result in less travel time than a FCFS sequence. In the simulation study we will examine the following three rules which select the next request after the current request has been completed: (1) longest waiting retrieval (LWR); (2) nearest retrieval (NR); (3) nearest storage/retrieval (NSR). LWR and NR are well known rules. Both the LWR rule and the NR rule alternate storage and retrieval requests, as long as both are available. Clearly, the next storage request must be selected according to FCFS. However, the rules have
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freedom in the selection of the next retrieval. The LWR rule selects a retrieval request according to the FCFS sequence. The NR rule selects the retrieval request which is positioned closest to the S/R machine, thus the nearest neighbour among the retrievals. The NSR rule considers all retrieval positions as well as the input station and selects the position which is closest to the S/R machine. The next request will be a storage if the input station is the closest position and it will be a retrieval otherwise. The NSR rule will always select a storage after a retrieval, since we consider coinciding input and output stations. However, a storage may either be succeeded by another storage or by a retrieval. The NSR rule is derived from the Nearest Neighbour heuristic in Van den Berg and Gademann (1999) for the wave sequencing approach. Note that the NSR rule may perform single command cycles, even if the input and output stations are at the same position and both storage and retrieval requests are available. At ® rst sight it may seem to be more e ective to perform only dual command cycles in this situation. However, this is not necessarily true. We may for example decide to perform a storage request in a single command cycle even though a retrieval is available, because a subsequent storage request is stored closer to the retrieval position, as may be seen in the following example. Example 1. Consider the situation in ® gure 1. In this situation there are three storage requests waiting, for classes A, C and B, respectively, and there is one retrieval in class C. If we decide to perform the ® rst storage in a dual command cycle, then we ® nd one dual command cycle addressing classes A and C and two single command cycles addressing classes B and C, respectively. However, if we decide to perform the ® rst storage in a single command cycle, then we ® nd a dual command cycle addressing class C and two single command cycles addressing classes A and B. The latter will require less travel time. In the above example we yielded travel time savings by performing time-e cient single command storage cycles. However, if the number of retrievals is at least as large as the number of storages, then all storage requests can be included in dual command cycles and no single command storage cycles will be necessary. Hence, for
Figure 1.
Rack partitioned into three classes. Three storage requests are available for classes A, C and B, respectively. One retrieval requst is available in class C.
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coinciding input and output stations, the NSR-rule only allows single command storage cycles when the number of storage requests exceeds the number of retrieval requests. Finally we consider the situation that no requests are available. In this situation the S/R machine will travel to its dwell point position. In the simulation study we will consider a stable system, i.e. over a long time period the numbers of storage and retrieval requests are equivalent. Park (1992) shows that the optimal dwell point position is at the input station if the probability of the ® rst operation after an idle period being a storage is at least 0.5. If a new request comes in while the S/R machine is traveling towards the dwell point position, then the S/R machine immediately interrupts its travel and starts performing the request. Hereby, we assume that the S/R machine travels in a straight line to the dwell point position. 3.3.
Open location selection rules If the request selection rule decides that the next operation is a storage, then we have to select an open location where the unit-load is to be stored. Let s1 , . . . , sm and r1 , . . . , rn …m ù 1, n ù 0† denote the storage and retrieval requests that are currently available, respectively. If there is at least one open location in the AS/R S, then the storage location assignment policies in }} 3.1 identify one or more open locations from which a location for s1 should be selected. Let L denote the set of open locations identi® ed by the storage location assignment policy. Hence, if L contains more than one open location, then we have to make a selection. For this, we will examine the following open location selection rules in the simulation study: (1) (2) (3) (4)
random open location; closest open location; nearest neighbour; shortest leg;
The random open location rule (ROL) randomly selects an open location i 2 L for s1 . The closest open location rule (COL) selects an open location i 2 L for s1 such out that …t in i ‡ t i † is minimal, breaking ties arbitrarily (see Hausman et al. 1976) . The nearest-neighbour rule (NN) and the shortest leg rule (SL) have been suggested by Han et al.(1987) . These rules look ahead one more retrieval. For each open location i 2 L for s1 , the rules ® rst determine which of the currently available retrievals would be performed after s1 , if s1 were stored in location i. Now the NN rule selects the open location i* such that ti*j* is minimal, breaking ties arbitrarily, where j* represents the position of the retrieval that is considered to be performed after storing s1 in location i*. Likewise, the SL rule selects an open location i* and a position j* such that t in i* ‡ ti *j * is minimal. For both rules it holds that if no new retrievals come in while processing storage s1 , then the retrieval at position j* will in fact be performed immediately after s1 . Otherwise, one of the newly arrived requests might be selected instead. Han et al. (1987) report that the SL rule was outperf ormed by the NN rule in the long run, because it appeared to drive the open locations to the positions furthest from the input and output station. Han et al. (1987) identify the no cost zone, which in contains the rack locations i for which t in i ‡ tij ˆ t j . All locations within this zone give rise to optimal selections for the SL rule. Thus, for one retrieval j there may exist multiple open locations i 2 L which can be visited without extra travel time. Han et al. (1987) do not mention how to break ties in this situation. In the simulation
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study, we will break ties by selecting the location i 2 L furthest from the input and output station in an attempt to preserve open locations close to the input and output stations. For both rules, if the subsequent request is considered to be a storage as well, so that storage s1 is performed in a single command cycle, then we consider the out open location i 2 L for s1 with the smallest …t in i ‡ t i † as it would have been selected by the COL rule. We will consider the DED and CON policies only in combination with the COL rule. The COL rule will establish that incoming unit-loads will be stored as close as possible to the designated region, yet not closer to the input and output stations. 3.4.
Urgency rules We may yield travel time savings by relaxing the FCFS sequence for retrieval requests. However, this may result in extensive waiting times for some retrieval requests. Han et al. (1987) suggest that due dates or priorities should be employed, to ensure that a retrieval at the far end of the aisle is not excessively delayed. We attempt to tackle the trade-o between e cient travel and limited response times, by imposing urgency rules. In our simulation we assign a due date to each storage or retrieval request which is ten minutes after its moment of arrival. From the practical point of view, ten minutes will be regarded as a high performance. Besides, with a lead time of, for example, one hour only few orders will become urgent and basically all requests will become urgent with a lead time of one or two minutes. In most distribution warehouses a lead time of several hours is acceptable. Even in situations with tighter time constraints such as production warehouses and reserve storage areas (used for replenishing the pick face), lead times of over ten minutes are generally acceptable. In some situations, we may obtain a better due date performance by performing urgent storage (retrieval) requests in single command cycles although dual command cycles are possible. However, this will give rise to less e cient travel of the S/R machine, which might cause that the number of available storage and retrieval requests grows rapidly. For this reason we assume that, when possible, dual command cycles will be performed even if only storage requests or only retrieval requests are urgent. We de® ne the following urgency rule for retrieval requests: . Retrieval requests that are within two minutes from their due date are urgent.
If there are urgent retrieval requests, then these must be performed before any of the retrieval requests that are not urgent. However, if there are multiple urgent retrieval requests, then there still is freedom to sequence these requests. We will consider two interpretations of the above rule. (a) Apply the same sequencing rule as for non-urgent requests when selecting the next urgent retrieval request. (b) Apply the LWR rule for selecting the next urgent retrieval request. Note that in the former interpretation, some retrieval requests may still be excessively delayed when the system is heavily occupied, which will not occur with the latter interpretation. However, due to its more e cient travel in comparison with the LWR rule, it may be that the system overload is resolved more rapidly with the former interpretation.
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4.
Simulation study In this section we discuss how the simulation study has been implemented. The simulation has been programmed in C‡‡. As we mentioned in } 2, of each product there is a single unit-load. Consequently, storage and retrieval requests for a certain product come in alternately. We introduce the following notation: qi ¶i T UNR i T UNS i
the long-term fraction of the time that product i is present in the AS/RS turnover rate of product i, i.e. the number of retrieval requests per unit of time random variable that represents the time between the arrival of a storage request for product i and the arrival of the subsequent retrieval request for product i random variable that represents the time between the arrival of a retrieval request for product i and the arrival of the subsequent storage request for product i.
In the simulation T UNR i has a negative exponential distribution with parameter / q ¶i i and T UNS i is constant and equal to …1 – qi † / ¶i . Accordingly, the expected time between two consecutive retrieval requests is E…T UNS i † ‡ E…T UNR i † ˆ …1 – qi † / ¶i ‡ qi / ¶i ˆ 1 / ¶i , which is in accordance with the turnover rate ¶i . We have chosen a constant T UNS i , since T UNS i generally is short in comparison with T UNR i . A short T UNS i may cause that a new storage request is released, while the preceding retrieval has not been performed yet. We attempt to minimize the number of times that this will occur, by choosing T UNS i to be deterministic (so, in this case, T UNS i is a random variable with a constant distribution) . Each simulation run concerns 1 000 000 requests of which approximately half are storages and half are retrievals. This resembles a period of approximately 10 years of warehouse operation. To obtain a representative distribution of the products over the storage locations we perform an initial run of 100 000 requests prior to each simulation run. We use a subrun method to determine the con® dence intervals for each of the parameters. We have one set with arrivals of the storage and retrieval requests, that is used in all simulations. Hence, the arrival process is independent of the behaviour of the AS/ RS operation.
5.
Simulation results Firstly, in } 5.1 we select the appropriate space requirements for each policy. Subsequently, in } 5.2 we consider the storage location assignment policies. Next, in } 5.3 we examine how we may further improve the performance of the AS/R S by applying clever open location selection rules. Then in } 5.4 we investigate the behaviour of the various request selection rules. Finally, in } 5.5 we increase the occupancy rate of the system and evaluate the due date performance of the policies. 5.1.
Space requirements In this section we examine the space requirements for each policy. We will examine three policies: the continuous storage policy (CON) which represents the policies with one `class’ , and the class-based storage policy with three classes (CL-3)
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and six classes (CL-6). For the dedicated storage policy (DED) we already know that the number of required locations is equal to the number of products, which is 1526. For the three policies, we use the longest waiting retrieval (LWR) rule for sequencing the requests. This means that the storage and retrieval requests are performed alternately as long as both are available and that the retrieval requests are performed in a FCFS sequence. For CON we use the closest open location (COL) rule and for the CL-K policies we use the random open location (ROL) rule to select an open location within the class region. Throughout this paper we will denote a policy in the following format: Storage location assignment policy / Open location selection rule / Request selection rule Furthermore, we introduce the following notation for the performance indicators: TT RS RR OR
mean travel time per request mean response time for storage requests mean response time for retrieval requests occupancy rate of the S/R machine.
We compute T T by taking the total travel time of the S/R machine, other than the travel time to the dwell point position when the S/R machine is idle, and dividing it by the number of storage and retrieval requests that have been completed. In tables 1, 2 and 3 we vary the con® dence level a , which represents the fraction of the time during which each class region is su ciently large to hold its products for
a
No of locations
No of space shortages
1400 1410 1414 1422 1432 1526
2624 119 17 0 0 0
0.99 0.999 0.999 8 0.999 99 0.999 999 6 1
TT 18.11 § 0. 09 17.45 § 0. 04 17.34 § 0. 03 17.25 § 0. 02 17.24 § 0. 02 17.20 § 0. 02
Table 1. Number of rack locations, number of space shortages and mean travel time T T (s) in the simulation for the CON/COL/LWR policy for various a -values and the DED/ COL/LWR ( a ˆ 1). a 0.99 0.999 0.999 8 0.999 99 0.999 999 6
No of locations
No of space shortages
No of storages in other class
1412 1422 1430 1440 1452
45 0 0 0 0
1754 73 13 3 2
TT 19.73 § 0. 03 19.76 § 0. 03 19.81 § 0. 02 19.86 § 0. 02 19.92 § 0. 03
Table 2. Number of rack locations, number of space shortages, number of storages outside the designated class-region and mean travel time T T (s) in the simulation for the CL-3/ROL/LWR policy.
Simulation of autom ated storage/retrieval system a 0.99 0.999 0.999 8 0.999 99 0.999 999 6
No of locations
No of space shortages
No of storages in other class
1424 1436 1446 1454 1472
0 0 0 0 0
1207 125 91 91 48
1349 TT 17.75 § 0.02 17.84 § 0.02 17.86 § 0.02 17.88 § 0.02 17.89 § 0.02
Table 3. Number of rack locations, number of space shortages, number of storages outside the designated class-region and mean travel time T T (s) in the simulation for the CL-6/ROL/LWR policy.
three di erent policies. Consequently, a determines the space requirements. Table 1 depicts for the CON/COL/LWR policy, the space requirements and the number of times a storage could not be performed in the simulation run because the rack was full (the total number of storages was 500 000) and the mean travel time T T per request for various values of a . Note that the CON/COL/LWR policy becomes equivalent to the DED/COL/LWR policy if a ˆ 1. Tables 2 and 3 show the same results for the CL-3/ROL/LWR and the CL-6/ROL/LWR policies, respectively, as well as the number of times that a unit-load had to be stored outside its class region. For the situation with one `class’ (table 1) it seems that a ˆ 0. 999 99 is the most appropriate selection, which gives 1422 locations. This is su ciently large to hold the products throughout the simulation and it gives a competitive result for the mean travel time. It is a surprising result, that the mean travel time decreases, when the total storage space increases. In all further simulations with the RAN and CON policies, we will use a ˆ 0.999 99. For class-based storage with three classes we select a ˆ 0.999, which gives 1422 locations. For this a -value no space shortages occur and only a limited number of unit-loads have to be stored outside their class region (73 out of 500 000) . Finally for class-based storage with six classes we select a ˆ 0.99, which gives 1424 locations. Accordingly, the space requirements are comparable for the three policies. For a ˆ 0.99, we found that 1207 out of 500 004 unit-loads could not be stored within their class region, i.e. 0.2%, which seems an acceptable amount. It is also surprising that T T appears to increase for increasing a , while the opposite was true for CON. For all policies we use less than the 1600 locations that are available in the two storage racks. Accordingly, we select the required number of locations i with the out smallest …tin i ‡ ti †. The remaining locations will not be used in the simulation. In a practical situation, these locations might be used for reserve storage. 5.2.
Storage location assignm ent In this section we evaluate the results for the storage location assignment policies. The policies that we consider are: RAN, CL-3, CL-6, DED and CON. As a basic situation, we ® rst consider the situation where the sequence of the requests is determined by the LWR rule. The overall arrival rate of the storage and retrieval requests in the simulations is 0.8 requests per minute. Table 4 shows that the mean travel time per request T T is the smallest for the DED and CON policies. The CL-6 policy performs slightly worse, while the CL-3 requires on average 2.5 s longer per request. The RAN policy performs considerably worse (more than double T T ). Table 4 also shows that the S/R machine has a high occupancy rate of 87% under the
1350 Policy RAN/ROL/LWR CL-3/ROL/LWR CL-6/ROL/LWR DED/COL/LWR CON/COL/LWR
J. P. van den Berg and A. J. R. M. Gademann TT 35.56 § 0.05 19.76 § 0.03 17.75 § 0.02 17.20 § 0.02 17.25 § 0.02
RS 4: 1: 1: 1: 1:
41.9 § 4. 2 37.0 § 0. 5 27.7 § 0. 4 25.4 § 0. 4 25.6 § 0. 4
RR 4: 1: 1: 1: 1:
44.9 § 4.3 46.0 § 0.6 35.9 § 0.4 33.5 § 0.4 33.8 § 0.4
OR 0.872 0.662 0.635 0.628 0.628
Table 4. Mean travel time T T (s), mean response time for storages and retrievals with 95% con® dence intervals RS and RR (min : s) and, occupancy rate OR of the S/R machine.
RAN policy. The other policies give rise to considerably lower occupancy rates of 62± 68%. Since the S/R machine ful® ls incoming requests more rapidly under the latter four policies, this results in massive reductions of the average response times. 5.3.
Open location selection rules Next, we investigate how we can further improve the performance of the AS/RS by applying alternative open location selection rules. We examine these rules in combination with the RAN, CL-3 and CL-6 policies. We do not consider the DED and CON policies here, since these policies are only used in combination with the COL rule. Table 5 shows the mean travel time per request and the mean response times for storage and retrieval requests for several policies. In all experiments we use the LWR rule for sequencing the requests. We gather from table 5 that the ROL rule is outperf ormed by the three other open location selection rules. If we look at the RAN policy, then we see that the mean travel time per request T T may be reduced by up to 3 s when using the NN rule, a reduction of more than 8%. Although these time-savings may seem small, the mean response times for storages and retrievals are reduced by more than 1 min, a reduction of more than 27%. Such a large response time reduction is typical for systems with a high occupancy rate. Policy
TT
RS
RR
OR
RAN/ROL/LWR RAN/COL/LWR RAN/NN/LWR RAN/SL/LWR
35.56 § 0.05 34.94 § 0.05 32.62 § 0.05 33.67 § 0.05
4: 4: 3: 3:
41.9 § 4.2 16.0 § 3.6 14.1 § 1.7 34.2 § 2.3
4: 4: 3: 3:
44.9 § 4.3 14.9 § 3.6 27.7 § 1.9 37.7 § 2.3
0.872 0.864 0.833 0.847
CL-3/ROL/LWR CL-3/COL/LWR CL-3/NN/LWR CL-3/SL/LWR
19.76 § 0.03 19.51 § 0.03 19.17 § 0.03 19.22 § 0.03
1: 1: 1: 1:
37.0 § 0.5 35.6 § 0.5 33.5 § 0.5 33.7 § 0.5
1: 1: 1: 1:
46.0 § 0.6 44.3 § 0.5 42.6 § 0.5 42.5 § 0.5
0.662 0.658 0.654 0.655
CL-6/ROL/LWR CL-6?COL/LWR CL-6/NN/LWR CL-6/SL/LWR
17.75 § 0.02 17.65 § 0.02 17.45 § 0.02 17.47 § 0.02
1: 1: 1: 1:
27.7 § 0.4 27.1 § 0.4 26.0 § 0.4 26.2 § 0.4
1: 1: 1: 1:
35.9 § 0.4 35.4 § 0.4 34.3 § 0.4 34.3 § 0.4
0.635 0.634 0.631 0.631
Table 5. Mean travel time T T (s), mean response time for storages and retrievals with 95% con® dence intervals RS and RR (min : s), and the occupancy rate OR of the S/R machine for four open location selection rules in combination with randomized storage and classbased storage.
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For the class-based storage policies, the open location selection rules show similar results, except that the magnitude of the time-savings is smaller. This follows immediately from the fact that the selection of open locations is more restricted for class-based storage than for randomized storage. Han et al. (1987) observe that the NN rule performs better than the SL rule for randomized storage. They conjecture that the inferiority of the SL rule follows from the fact that it seems to drive the open locations away from the I/O station. We anticipate this, by selecting the open location most distant from the I/O station when breaking ties. Note that there will be often multiple open locations with the same shortest leg travel time, namely all open location in the no cost zone. We see that even with this modi® cation, the NN rule performs better than the SL rule in combination with randomized storage. For class-based storage the performance of the NN rule and the SL rule is comparable. 5.4.
Request selection rules Subsequently, we investigate whether we may improve the performance of the AS/RS by relaxing the FCFS-sequence of the retrievals. In the previous section we have seen that the NN rule performs better than the other open location selection rules. Accordingly, we will use this rule in combination with the RAN and CL-K policies and focus on the request selection rules. For the DED and CON policies we will use the COL rule. Table 6 shows how the request selection rules a ect the travel time and the response times for various policies. We see that the NR rule, which selects the retrieval request closest to the S/R machine, reduces the mean travel time per request and also the mean response times for storages and retrievals in comparison with the LWR rule. The travel time and response time reductions are remarkably small. This Policy
TT
RS
RR
RAN/NN/LWR RAN/NN/NR RAN/NN/NSR
32.62 § 0.05 32.22 § 0.06 32.09 § 0.06
3 : 14.1 § 1. 7 3 : 10.7 § 1. 6 3 : 03.7 § 1. 5
3 : 27.7 § 1.9 3 : 20.0 § 1.7 3 : 24.9 § 1.7
CL-3/NN/LWR CL-3/NN/NR CL-3/NN/NSR
19.17 § 0.03 18.97 § 0.03 18.91 § 0.03
1 : 33.5 § 0. 5 1 : 32.0 § 0. 4 1 : 26.3 § 0. 4
1 : 42.6 § 0.5 1 : 37.4 § 0.4 1 : 45.8 § 0.4
CL-6/NN/LWR CL-6/NN/NR CL-6/NN/NSR
17.45 § 0.02 17.27 § 0.02 17.20 § 0.02
1 : 26.0 § 0. 4 1 : 24.8 § 0. 4 1 : 19.6 § 0. 3
1 : 34.3 § 0.4 1 : 30.1 § 0.3 1 : 38.6 § 0.4
DED/COL/LWR DED/COL/NR DED/COL/NSR
17.20 § 0.02 17.04 § 0.02 17.01 § 0.02
1 : 25.4 § 0. 4 1 : 24.2 § 0. 4 1 : 18.3 § 0. 3
1 : 33.5 § 0.4 1 : 29.1 § 0.3 1 : 39.6 § 0.4
CON/COL/LWR CONCOL/NR CON/COL/NSR
17.25 § 0.02 17.09 § 0.03 17.06 § 0.02
1 : 25.6 § 0. 4 1 : 24.4 § 0. 4 1 : 18.4 § 0. 3
1 : 33.8 § 0.4 1 : 29.3 § 0.3 1 : 39.8 § 0.4
Table 6. Mean travel time TT (s), and mean response time for storages and retrievals with 95% con® dence intervals RS and RR (min : s) for ® ve policies in combination with three request selection rules.
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J. P. van den Berg and A. J. R. M. Gademann
is likely due to the low occupancy of the system. Typically in the considered situations there will only be a limited number of retrieval requests waiting, which signi® cantly reduces the possible travel time savings. Furthermore, we gather from table 6 that the NSR rule further reduces the mean travel time per request in comparison with the NR rule. This validates our conjecture that performing e cient single command storage cycles, even though retrieval requests are available, reduces the mean travel time per request. The reduction of T T results in a further reduction of the mean response time of the storage requests. However, the mean response time of the retrieval requests increases. This is because the NSR rule may save retrievals for e cient dual commands later, and thus may successively perform a number of storage requests even when retrieval requests are available. The NR rule and the LWR rule do not allow this. The reduction of T T when using the NSR rule in comparison with the LWR rule is the largest for the RAN policy. In fact, only for the RAN policy the mean retrieval response time reduces when using the NSR rule instead of the LWR rule. For the other policies the increased e ciency does not reduce the waiting time for retrievals. 5.5.
Urgency rules In the previous section, we considered the mean response times for storage and retrieval requests. In this section we will focus on the maximum response times. In many practical situations it is not critical when a request has to wait for a few minutes, but a more extensive response time may cause delay in production or delay of a truck departure. In this section we arbitrarily assume that 10 min is an acceptable response time for both storage and retrieval requests. Accordingly, we assign a due date to each incoming request which is 10 min after its arrival time. Subsequently, we attempt to ful® l as many requests before their due date as possible, by applying urgency rules. We introduce the following notation: RS max RRmax LS LR
maximum response time of any storage request in the simulation maximum response time of any retrieval request in the simulation number of storage requests that are completed after their due date in the simulation number of retrieval requests that are completed after their due date in the simulation.
Table 7 shows the number of storage and retrieval requests that are ful® lled after their due date (there are approximately 500 000 storages and 500 000 retrievals in the simulation) and the maximum response times. As may be expected, the number of late retrievals increases when using NR and NSR in comparison with LWR. Accordingly, in the considered situation the LWR rule may be preferred over the other two rules, especially since the reduction of T T is minor in the simulations. However, it may be expected that the travel time savings obtained by NR and NSR may increase when the number of available requests increases. Accordingly, it may be interesting to examine a situation with a higher occupancy rate of the S/R machine. We increase the occupancy rate by increasing the combined arrival rate of the storage and retrieval requests from 0.8 request/min to 1 request/min. We present the results for this situation in tables 8 and 9. We do not consider the RAN policy, since this policy will give rise to an instable system and we do not consider the DED policy since its behaviour is similar to that of the CON policy.
1353
Simulation of autom ated storage/retrieval system Policy
LS
RS max
LR
RRmax
RAN/NN/LWR RAN/NN/NR RAN/NN/NSR
15 293 14 460 12 354
32 : 23 31 : 56 30 : 30
16 128 21 113 22 329
35 : 53 2 : 42 : 45 2 : 20 : 28
CL-3/NN/LWR CL-3/NN/NR CL-3/NN/NSR
139 177 53
14 : 26 13 : 36 12 : 23
85 1 278 2 585
14 : 00 34 : 19 43 : 49
CL-6/NN/LWR CL-6/NN/NR CL-6/NN/NSR
109 68 8
13 : 11 13 : 49 11 : 04
69 783 1 970
13 : 45 30 : 53 44 : 23
DED/COL/LWR DED/COL/NR DED/COL/NSR
301 80 5
16 : 19 14 : 45 11 : 08
236 852 2 651
14 : 45 36 : 45 42 : 53
CON/COL/LWR CON/COL/NR CON/COL/NSR
145 84 5
15 : 08 14 : 53 11 : 04
90 883 2 719
14 : 08 37 : 26 42 : 08
Table 7. Number of storage and retrieval request with response times exceeding 10 min and the maximum response times observed in the simulation (h : min : s) for ® ve policies.
Policy CL-3/NN/LWR CL-3/NN/NR CL-3/NR/NR/NR CL-3/NN/NR/LWR CL-3/NN/NSR CL-3/NN/NSR/NSR CL-3/NN/NSR/LWR CL-6/NN/LWR CL-6/NN/NR CL-6/NN/NR/NR CL-6/NN/NR/LWR CL-6/NN/NSR CL-6/NN/NSR/NSR CL-6/NN/NSR/LWR CON/COL/LWR CON/COL/NR CON/COL/NR/NR CON/COL/NR/LWR CON/COL/NSR CON/COL/NSR/NSR CON/COL/NSR/LWR
TT
RS
RR
OR
18.73 § 0.03 18.28 § 0.03 18.32 § 0.03 18.34 § 0.03 18.16 § 0.03 18.21 § 0.03 18.23 § 0.03
2 : 31 .8 § 1.8 2 : 21 .1 § 1.3 2 : 23 .2 § 1.4 2 : 23 .8 § 1.5 2 : 06 .3 § 1.0 2 : 08 .3 § 1.1 2 : 08 .8 § 1.1
2 : 39.2 § 1.7 2 : 19.2 § 1.0 2 : 22.8 § 1.2 2 : 23.6 § 1.6 2 : 32.1 § 1.1 2 : 37.9 § 1.4 2 : 39.3 § 1.6
0.810
17.24 § 0.02 16.85 § 0.02 16.87 § 0.02 16.88 § 0.02 16.70 § 0.03 16.73 § 0.02 16.75 § 0.02
2 : 13 .9 § 1.3 2 : 05 .8 § 1.0 2 : 06 .8 § 1.1 2 : 07 .2 § 1.1 1 : 52 .2 § 0.8 1 : 53 .4 § 0.8 1 : 53 .6 § 0.8
2 : 20.4 § 1.2 2 : 04.8 § 0.7 2 : 06.9 § 0.9 2 : 07.3 § 1.0 2 : 17.6 § 0.9 2 : 20.7 § 1.0 2 : 21.6 § 1.1
0.785
17.27 § 0.02 16.88 § 0.03 16.91 § 0.02 16.91 § 0.02 16.78 § 0.03 16.82 § 0.02 16.85 § 0.02
2 : 16 .7 § 1.5 2 : 07 .5 § 1.2 2 : 08 .8 § 1.0 2 : 09 .2 § 1.1 1 : 51 .4 § 0.8 1 : 52 .9 § 0.9 1 : 53 .4 § 0.9
2 : 23.4 § 1.5 2 : 05.7 § 0.9 2 : 08.3 § 1.0 2 : 08.8 § 1.1 2 : 22.8 § 0.9 2 : 27.0 § 1.2 2 : 28.4 § 1.3
0.785
0.803 0.803 0.804 0.801 0.801 0.802
0.779 0.779 0.779 0.776 0.777 0.777
0.779 0.779 0.780 0.778 0.778 0.779
Table 8. Mean travel time T T (s), mean response time for storages and retrievals with 95% con® dence intervals RS and RR (min : s), and the occupancy rate OR of the S/R machine for the CL-3, CL-6 or COL policies in combination with two urgency rules under a high occupancy rate.
1354
J. P. van den Berg and A. J. R. M. Gademann
When we compare table 8 with table 6, we see that the policies that combine the NN rule with the NR rule or the NSR rule show a larger travel time reduction with the increased occupancy rate than other policies. This may be explained by the fact that these policies consider all open locations in the designated area and all retrieval positions. Consequently, these policies bene® t more from the increased number of retrieval requests that are waiting in the system. Furthermore, we gather from table 8 that the CL-6/NN/LWR policy outperf orms the CON/COL/LWR policy for the increased occupancy rate. This is a remarkable results, since for the lower occupancy rate in table 6 the CON/COL/LWR policy performed better. This follows from the fact that the performance of the NN rule used in the CL-6/NN/LWR policy improves when the occupancy rate of the system increases. For each of the policies we employed urgency rules to prevent excessive waiting times for some retrievals. We considered two urgency rules. One urgency rule uses the same request selection rule for sequencing the urgent retrieval requests as for non-urgent requests. The other urgency rule uses the LWR rule for sequencing the urgent requests. We gather from table 8 that the urgency rules cause a small increase of T T . Accordingly, we look at table 9 to see whether the urgency rules improve the due date performance. We gather that the policies which apply the LWR rule give the smallest number of late retrievals. None of the urgency rules is able to achieve the performance of the Policy
LS
RS max
LR
RRmax
CL-3/NN/LWR
7648
27 : 32
7 619
28 : 41
CL-3/NN/NR CL-3/NN/NR/NR CL-3/NN/NR/LWR
4443 5205 5535
26 : 19 27 : 51 28 : 52
9 173 8 376 10 041
1 : 30 : 27 54 : 52 29 : 33
CL-3/NN/NSR CL-3/NN/NSR/NSR CL-3/NN/NSR/LWR
2230 2497 2743
26 : 26 26 : 12 23 : 39
12 377 12 588 15 207
1 : 35 : 44 1 : 12 : 36 29 : 02
CL-6/NN/LWR
4332
27 : 17
4 169
23 : 57
CL-6/NN/NR CL-6/NN/NR/NR CL-6/NN/NR/LWR
2498 2662 2863
24 : 46 25 : 58 26 : 02
6 254 4 851 5 707
1 : 25 : 06 56 : 02 23 : 46
CL-6/NN/NSR CL-6/NN/NSR/NSR CL-6/NN/NSR/LWR
968 1049 1050
24 : 31 23 : 15 22 : 55
9 023 7 844 9 241
1 : 23 : 29 56 : 27 22 : 49
CON/COL/LWR
5412
25 : 14
5 200
24 : 21
CON/COL/LWR CON/COL/NR/NR CON/COL/NR/LWR
2920 3319 3601
22 : 42 22 : 44 22 : 54
6 932 5 809 7 014
2 : 06 : 40 1 : 17 : 09 24 : 55
CON/COL/NSR CON/COL/NSR/NSR CON/COL/NSR/LWR
913 954 1092
23 : 30 23 : 32 18 : 51
11 331 10 447 12 679
2 : 03 : 14 56 : 57 25 : 01
Table 9. Number of storage and retrieval requests with response times exceeding 10 min and the maximum response times observed in the simulation (h : min : s) for two policies.
Simulation of autom ated storage/retrieval system
1355
LWR rule. However, if we compute the total number of late requests L S ‡ L R from table 9, then it appears that some of the policies signi® cantly improve the policies with the LWR rule, e.g. the */NN/NR and */NN/NR/NR policies (* is a wild-card for CL-3, CL-6 or CON). Unfortunately, these policies cause excessive lateness for some of the retrievals as follows from the large values for RRmax . All in all, the LWR rule seems to be the most dependable request selection rule. It results in mean response times that are a little longer than with some of the other request selection rules. However, it prevents excessive waiting times for some retrievals and it con® nes the number of late retrievals. The proposed urgency rules cannot achieve a comparable performance. 6.
Conclusions In this paper we have presented a simulation study of an automated storage/ retrieval system (AS/RS). We examined various elements of AS/RS control: storage location assignment policies, request selection rules, open location selection rules and urgency rules. The extensive simulation study both shows the isolated e ects of various policies as well as compares several combinations of policies and rules. This extensive comparison of policies and rules provides a solid base for selecting the most suitable policy in a speci® c case. We considered the following storage location assignment policies: randomized storage, class-based storage with three and six classes, dedicated storage and the newly presented continuous storage policy. The continuous storage policy combines low storage space requirements with short expected travel times. Due to its nature this policy seems applicable very generically. We gathered that the continuous storage policy and the class-based storage policy with six classes in combination with the nearest-neighbour rule for selecting open locations for storages seem to perform well in comparison with the other policies. In the simulations, the dedicated storage policy outperf ormed these policies, but this may be contributed to the fact that the AS/RS contains only one unit-load per product. In systems with multiple unit-loads per products, the space requirements for dedicated storage will increase relative to the other policies, and thereby the travel times. This disadvantage does not hold for the continuous storage policy. Since the e ectiveness of the class-based storage policy in general depends on the number of classes that is used, the continuous storage policy seems to be the most robust and e ective in general. We gathered that the nearest-neighbour rule (see Han et al. 1987) gives the best results for selecting an open location within the storage area for randomized storage or within a class-region for class-based storage. We also determined that when an incoming unit-load cannot be stored within its designated region, it is better to assign it to a location further away from the input and output station, than to a location that is nearer than its designated region. The latter is likely to ® ll up the storage space for fast moving products, which may result in increased mean travel times. For sequencing the requests, we have considered two rules that do not perform the retrieval requests in a ® rst come, ® rst served (FCFS) sequence. It appeared that these rules may reduce the mean travel time per request and the mean response times. However, the rules may cause considerable waiting times for some retrievals. This holds in particular for the NSR rule, which, in order to increase throughput, may skip a retrieval temporarily to perform a single command storage. When response times are the most important criterion, the longest waiting retrieval (LWR) rule will most probably show a good performance. We proposed two urgency rules that
1356
Simulation of autom ated storage/retrieval system
attempt to anticipate this by giving priority to retrievals with long waiting times. In this case, precisely what a good due date performance is depends on the situation; one could look at the mean response time, the maximum response time (or lateness) or the number of late requests. In the simulations we considered all three criteria and found that these were satis® ed better when using a FCFS sequence for the retrievals than by applying urgency rules. For further research one might investigate more sophisticated urgency rules that provide a better due date performance. Although on average a FCFS sequence for the retrievals gives good results, in some situations it may be better to allow small changes in the sequence of the requests which reduce the travel time while satisfying the due dates for the respective requests. This is particularly true when the system is heavily occupied. Then the sequencing problem could be observed as a static problem similar to the wave sequencing approach for which we may determine a sequence with an e ective due date performance.
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