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Sequencing methods for automated storage and retrieval systems with dedicated storage
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Pergamon
Computers ind. Engng Vol. 32, No. 2, pp. 351-362, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: S0360-8352(96)00298-7 0360-8352/97 $17.00 + 0.00
SEQUENCING METHODS FOR AUTOMATED STORAGE AND RETRIEVAL SYSTEMS WITH DEDICATED STORAGE HEUNGSOON FELIX LEE t* and SAMANTHA K. SCHAEFER: tDepartment of Mechanical and Industrial Engineering, Southern Illinois University, Edwardsville, IL 62026-1805, U.S.A. -'Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130-4899, U.S.A. (Received 28 August 1996) Abstraet--ln this paper, we study the effect of sequencing storage and retrieval requests on the performance of automated storage and retrieval systems (AS/RS) where a storage request is assigned a predetermined storage location. By exploiting this unique operating characteristic, we present several optimum and heuristic sequencing methods under static and dynamic approaches. Applications of such sequencing methods include unit-load AS/RS with dedicated storage, miniload AS/RS, and potentially unit-load AS/RS with randomized storage. We find that the sequencing methods can significantly reduce travel time by a storage and retrieval machine, thereby, increasing throughput, and that the dynamic heuristic method is simple and fast, yet considerably outperforms the others. © 1997 Elsevier Science Ltd. All rights reserved
I. INTRODUCTION
Since automated storage and retrieval systems (AS/RS) were introduced in the 1950s, the technology has advanced far beyond its original function, which was to eliminate the walking that accounted for 70% of manual retrieval time. AS/RS have been adopted not only as alternatives to traditional warehouses but also as a part of advanced manufacturing systems, and the number is expected to grow rapidly. This is because AS/RS have many benefits including savings in labor costs, improved material flow and inventory control, improved throughput level, high flow-space utilization, and increased safety and stock rotation [1-3]. Typically, AS/RS consist of a series of storage aisles each of which is served by a storage and retrieval (S/R) machine or crane. Each aisle is supported by a pickup and delivery (P/D) station customarily located at the end of the aisle and accessed by the S/R machine and the external handling system. A S/R machine usually operates in two modes: single cycle (SC) and dual cycle (DC). For each of the modes, a S/R machine starts at the P/D station, stores and/or retrieves a load, and returns to the P/D station to complete a cycle. In a SC, a S/R machine either stores or retrieves, while in a DC, it both stores and retrieves in one cycle. In a DC, a S/R machine picks up a load from a P/D station, travels to a storage location to store it, travels to another location to retrieve a load, and then returns to the P/D station to deliver it. The effectiveness of AS/RS depends on the methods of control that govern the scheduling of storages and retrievals. A common practice in sequencing storage and retrieval requests is that both requests are processed in first-come-first-served (FCFS) manner. The FCFS assumption is reasonable for storages, since most AS/RS are interfaced with a conveyor loop for input and output. In this case, it is difficult to change the sequence of loads presented for storage. However, the FCFS assumption is less compelling for retrievals since retrieval requests are just electronic messages and can be easily resequenced. In a DC, storage and retrieval requests can be paired to decrease the time spent travelling between the storage and retrieval locations. By minimizing the travel time, we can increase system *To whom all correspondence should be addressed. 351
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throughput (i.e., the number of storages or retrievals performed per period) and reduce AS/RS operating costs such as wear of mechanical parts and electric power cost. According to Han et al. [4], a 50% or more decrease in the travel-between time component of a dual cycle leads to an increase in throughput of 10-15%. Such an increase in throughput could help to handle peak demand in the operation phase, and eliminate an aisle in a multi-aisle system in the design phase, which leads to considerable savings. In this paper, we present several sequencing methods for AS/RS with dedicated storage, i.e., AS/RS for which storage requests are assigned predetermined storage locations. The proposed sequencing methods can be applied to two common types of AS/RS: unit-load AS/RS and miniload AS/RS. In unit-load AS/RS with dedicated storage, each load is assigned to a specific location or set of locations in the storage rack. These locations may be determined by activity and inventory levels or by stock number. Such assignments are normally made to aid in the retrieval process and may increase throughput [5]. Miniload AS/RS are an important subset of AS/RS and are used to handle small loads that are contained in bins or drawers within the storage system. The S/R machine is designed to retrieve the bin and deliver it to a P/D station at the end of the aisle so that the individual items can be withdrawn from the bins by a picker. The bin or drawer is then stored in its original location in the rack. Thus, miniload AS/RS follow dedicated storage. In addition to the above two types of AS/RS, the proposed methods can be useful for unit-load AS/RS with randomized storage, where a load can be stored in any open location, commonly in the open location closest to the P/D station. Such a randomized storage system is used to increase the utilization of the rack [5]. Han et al. [4] show that finding the optimum retrieval sequence for this type of unit-load AS/RS is not computationally tractable. This is because it involves not only finding the retrieval sequence, but also finding storage locations at the same time. One way to overcome this complexity is a 2-step procedure in which the first step is to determine locations for storage requests and the second step is to determine sequencing with the storage locations specified from the previous step. The proposed methods in this paper can be used for the second step and we introduce an efficient optimum sequencing method. In the quest to reduce the length of time required to serve a load retrieval or storage request, dwell point positioning policies have been cultivated. The dwell point is the position the S/R machine takes when idle. Egbelu [6] has developed two dwell point rules based on the probability of future requests. When Egbelu and Wu [7] compared these new rules with four simple established techniques, they were able to successfully decrease the travel time of the S/R machine. Linn and Wysk [8, 9] have developed expert systems that can strategically alter scheduling rules to achieve specific goals for production control in computer integrated manufacturing. They have discovered that as demand levels increase, policy alterations in retrieval sequencing and queue selection were at least twice as likely to be required as adjustments for other selection functions. This finding establishes the need for the development of proper retrieval sequencing techniques as part of overall strategic and tactical control rule application mechanisms to improve system performance. Han et al. [4] studied retrieval sequencing heuristic methods for unit-load AS/RS where storages can be stored in any open location, knowing that finding the optimum sequence is not computationally tractable. They showed evidence that the heuristics can improve throughput by 5-8%. The remainder of the paper is organized as follows. Section 2 presents various solution methods to sequence storage and retrieval requests for the AS/RS with dedicated storage under both static and dynamic approaches. Section 3 discusses performance measures to evaluate improvements by the sequencing methods. Section 4 describes the experimental results with the proposed sequencing methods. Finally, Section 5 concludes with a brief summary and future research issues. 2. SOLUTION METHODS
The list of storage and retrieval requests changes as old requests are processed and new requests arrive. Storage requests are processed in the order received. We take two approaches to the sequencing methods for retrieval requests: static and dynamic approaches. The static approach
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selects a block of retrievals and sequences them, and when the block of retrievals are completed, selects another block and so forth. Alternatively, the dynamic approach resequences the list each time a S/R machine completes a cycle by including new requests that arrive during the cycle. We first present sequencing methods for the static approach and then for the dynamic approach. We assume that each S/R trip for a cycle originates and terminates at the P/D station. This assumption is commonly used for both unit-load AS/RS and miniload AS/RS [4, 10, 11]. 2.1. S t a t i c approach
Suppose that n is the block size for retrievals and there are m storages waiting to be processed. If there are more than n retrievals in the current list, we consider the first n retrievals for resequencing. If m > n, we need consider only the first n storages since subsequent storages cannot be processed in this block due to the FCFS discipline for storages. Thus, we consider only the case where m ~< n below. Each storage or retrieval request is associated with a bin location, which is translated into a pair of travel times in horizontal and vertical directions from the P/D station. Let (h~i, vsi) and (hrj, vrj) be the pair for the ith arrived storage (i = 1. . . . . m) and thejth arrived retrieval (j = 1. . . . . n), respectively. Thus, with simultaneous travel in both the horizontal and vertical directions, it takes 2max(h~g, v~) to process the ith storage request with a single cycle. The objective is to minimize the total travel time required to process m storages and n retrievals in the current block. For each retrieval in the block, the following decisions must be made. Should the retrieval be performed in either SC or DC, and if it is performed in DC, with which storage request should it be paired? We can show that the optimum solution has m retrievals performed in DC and (n - m) retrievals in SC. This is because DC time is no greater than the sum of two individual SC times, one SC storage time and one SC retrieval time. That is, the following equation holds: max(h~i, v~i) + max(lh~i - hrj[, Ivs,-vrjl) + max(hrj, vrj) ~< 2max(hsi, v~) + 2max(h~j, vrj).
(1)
The proof of equation (1) can be shown by considering each of the following four cases: (h~ > or < vs~) and (h,j > or < v~j). With this result, we can formulate this problem as the assignment problem [12]. Since the assignment problem requires balancing (i.e., the number of storages equals the number of retrievals to be paired), we need to create (n - m) fictitious storages. We number these storage m + 1. . . . . n). When a retrieval is paired with a fictitious storage, a S/R machine performs SC for the retrieval. We first define necessary notation. Z = total travel time, Xo = assignment variable which is set to 1 if storage i is paired with retrieval j, set to 0 otherwise for i, j = 1. . . . . n, Cij = travel time to process a pair of storage i and retrieval j. Minimize
Z= ~ ~ CoX~
(2)
i=lj=l
Subject to: ~ X u = 1,
for i = 1,2 . . . . n.
(3)
for j = 1 , 2 . . . . n.
(4)
for i , j = 1 . . . . .
(5)
i= I
~X,j=I, j=l
A'u = 0, I
n
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The values of C,~ depends on whether i is associated with an actual storage or a fictitious one. For the former case (i.e., i ~< m), it is assigned DC time, while for the latter case, it is assigned SC retrieval time for retrieval j. Thus, we write C~j as C,j=cj+cij+Cj =2Cj
forl
fori/>m,
(6)
C, = max(hs,, v~)
(7)
C,j = max(lh~,- h,jl, Iv~,- v,jl)
(8)
Cj = max(hrj, vrj)
(9)
where
We can optimally solve the above assignment problem using the Hungarian method [12]. We also present a heuristic method to solve this assignment problem, which requires less computation but can be very effective for both static and dynamic approaches. Using equations (6)-(9), we can rewrite the objective equation (2) as follows:
Z=
~ ~ CoXiJ+ i=jj=l
=
~, ~ CuXo
(10)
i=m+lj=l
~,~,(c~+cij+cj)Xij+ ~ ~2c,X~ i=.jj= I
(11)
i=m+ Ij= I
Which when simplified becomes: (12) ]=1
i=j
i=lj=l
When m = n, equation (12) can be further simplified to: n
n
z = r c,+ f cj+ i=j
j=l
(13) i=lj=l
Thus, when m = n, the optimum sequence minimizing the total travel time is equivalent to the optimum sequence minimizing the total travel-between time. This makes sense since the sum of the times to travel from the origin to the storage point and from the retrieval point to the origin remains constant within a block. From equation (12), we develop the following heuristic algorithm, letting D,j = c~j for m = n and D~ = c o - c~ for m < n. Algorithm 1. Static heuristic algorithm.
Step Step Step Step
1. 2. 3. 4.
Let set J = {1, 2 . . . . . n} be the block of retrievals and i = 1. Find k that gives the smallest D~ for all j in J. Perform a DC for storage i and retrieval k and J = J - {k}. If i = m, then perform a SC retrieval for each j in J and terminate. Otherwise, increase i by 1 and go to Step 2.
This heuristic algorithm takes much less computation than the above optimum method. It requires at most only n ( n - 1)/2 comparisons. In contrast, the Hungarian method to solve the above assignment problem requires O(n 3) comparisons and arithmetic operations. Although we are not able to achieve tight theoretical error bounds for this heuristic, empirical results shows that it usually provides near-optimal solutions as we will discuss in Section 5.
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2.2. Dynamicapproach In the dynamic approach, we resequence the retrieval order reflecting arrivals of new requests as the system operates over time. We can naturally extend the solution methods for the static approach to the dynamic approach. Suppose that there are m storages and n retrievals (m, n > 0) in the storage and retrieval lists at time 0.
Algorithm 2. Dynamic assignmentalgorithm. Step 1. Step 2. Step 3. Step 4.
Solve the assignment problem with the current lists of storages and retrievals. Perform a DC with the first storage and a retrieval that is paired with the first storage in the solution from Step 1. Remove the storage and/or retrieval performed in the cycle from the respective list, and add to each list the new storages and retrievals that arrive during the cycle. If one of the two lists is empty, then perform a SC and go to Step 3. If both lists are empty, then wait for the first request to arrive and then repeat this step. Otherwise, go to Step 1.
Recall that when the number of storages in the list, m, is greater than the number of retrievals, n, in Step 1, we ignore storages n + 1. . . . . m and solve the smaller-size assignment problem. Similarly, we develop the dynamic heuristic algorithm simply by replacing the assignment problem in Step 1 of the above algorithm with the static heuristic algorithm, which is algorithm 1. In order to find the next DC pair, the dynamic assignment algorithm has a complexity of O(n 3) since it solves the assignment problem each time. On the other hand, the dynamic heuristic algorithm has a complexity of O(n). This heuristic requires at most 2n arithmetic operations and 2n comparisons to compute D~s and find the smallest D u. It is difficult to show the optimality of a dynamic sequencing method because of the dynamic nature of predicting future requests. It seems to be reasonable to make relative comparison among the dynamic methods under consideration. We present experimental results in Section 4 for such comparison. 3. MEASURESTO IMPROVEMENTSBY SEQUENCING In order to measure improvements achieved by the proposed sequencing methods, we use commonly adopted AS/RS with the following characteristics: (1) any opening in the methods aisle is equally likely to be selected for a storage or retrieval request; (2) a S/R machine simultaneously travels in both directions with constant speeds; and (3) the P/D station is located at the base and at the end of the aisle. We let L be the length and H be the height of the AS/RS aisle, and Vh and Vv the horizontal and vertical travel speeds of the S/R machine. Then, the times required to travel the full length and height, respectively, are given by th = L/Vh and tv = H/Vv. Using these travel times, we define two parameters, T = max(th, tv) and b = min(th/T, iv~T), which are referred to as the maximum travel time and shape factor, respectively. With a continuous approximation to the discrete rack face, Bozer and White [10] derived the expected travel times of SC and DC when no resequencing method is applied, i.e., when retrievals are processed FCFS as follows: E(SC) = Expected Single Cycle Travel Time.
E(SC) = [1
+lb2]T
(14)
E(TB) = Expected Travel-Between Time for the Dual Cycle. E(TB) = [1 + ~b 12
_-~ob3]T
(15)
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Heungsoon Felix Lee and Samantha K. Schaefer
E(DC) = Expected Dual Cycle Travel Time. 1 2-1b3]T E(DC) = I4 + ~b
= E(SC) + E(TB)
(16)
By setting T to 1, we normalize the rack. The size (shape) of the normalized rack becomes 1.0 x b, where b ~< 1. We can decrease travel time only by decreasing the travel-between time, equation (15), of the dual cycle. Thus, equation (14) serves as a lower bound on expected DC travel time for any resequencing method for both static and dynamic approaches. Denoting E(DC)s as expected dual cycle travel time achieved by a certain sequencing method, we can express ~, the reduction rate of E(TB), as follows: =
E(DC) - E(DC)s E(TB)
(17)
W e relate reduction of E(TB) to possible increase in throughput as follows:
E(DC) + 4r - 1 = E(DC) - ~E(TB) + 4r
(18)
where fl is the increase in throughput that can be realized by ~t reduction of E(TB), and r is the P/D time, i.e., the time taken to pick or drop a load. When r is negligible and b varies from 0.2 to 1, equation (18) shows that the maximum increase in throughput ranges from 33.5% for b = 0.2 to 35% for b = 1. If as Han et al. [4] suggest, a 12% increase in throughput is considered significant, the expected travel-between time must decrease by at least 40%. If r is 0.2, which is a typical P/D time for an AS/RS, the maximum throughput improvement possible decreases to 22%. If throughput improvement of 12% or more is considered significant, the expected travel-between time must be decreased by at least 60%. 4. EXPERIMENTS We conducted experiments to determine how much the proposed optimum and heuristic methods can reduce the cycle time of a S/R machine, and consequently, improve system throughput under both the static and dynamic approaches. We used three different values for shape factor, b: 1, 0.6, and 0.2. We normalized the rack, i.e., set T to 1. We tried numerous values for block size n, ranging from 2 to 100. We limited the experiments to the case where the number of storages equals the number of retrievals in the block (i.e., m = n). For each value of b, we generated 1000 requests for both storages and retrievals. Each request is associated with a pair of travel times in horizontal and vertical directions that were randomly generated from uniform distributions (0, 1) and (0, b), respectively. Thus, when n = 10, we applied the static methods 100 times, once for each block of 10 storages and retrievals, and recorded the average of 1000 DC times performed. For the dynamic methods, we assumed that one storage and one retrieval arrive during a DC. Therefore, when n = 10, we started with the block of the first 10 storages and the first 10 retrievals and determined the first DC to perform. After performing the DC, we updated the block by removing the processed storage and retrieval and adding the 1lth generated storage and retrieval, and determined the next DC to perform. We continued this process until the lists of 1000 requests are exhausted. Block size n = 1 means that the S/R machine performs retrievals FCFS and its expected DC travel time, equation (16), was used as a baseline for any improvements made. We presented two dynamic sequencing methods in Section 2.2 in which storages are processed FCFS as is commonly practiced. In our experiments, we also investigated the situation in which storages can be resequenced in order to study its impact on reduction of the travel-between time.
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Table 1. Sequencing methods ASSTA: HRSTA: ASFCDYN: HRFCDYN: ASDYN: HRDYN:
STATIC ASSIGNMENT ALGORITHM STATIC HEURISTIC ALGORITHM DYNAMIC ASSIGNMENT ALGORITHM in which storages are performed FCFS, while retrievals are performed in any order DYNAMIC HEURISTIC ALGORITHM in which storages are performed FCFS, while retrievals are performed in any order DYNAMIC ASSIGNMENT ALGORITHM in which storages and retrievals may be performed in any order DYNAMIC HEURISTIC ALGORITHM in which storages and retrievals may be performed in any order
For this type of AS/RS, a P/D station needs to be equipped with more sophisticated sensors and material handling devices in order to access any load waiting for storage We modified the dynamic methods to allow storage resequencing. For the dynamic assignment algorithm (algorithm 2), we modified Step 2 to pick a DC pair that gives the smallest Do among the DC pairs in the assignment solution instead of picking a DC pair with the first storage. For the dynamic heuristic algorithm we modified Step 2 of algorithm 1 to pick a DC pair that gives the smallest D,~ for all i, j instead of fixed i. It is not beneficial to apply storage resequencing under the static methods. This is because storage resequencing does not affect total travel time required to process the entire block of storages and retrievals. We coded all the proposed methods in Microsoft Fortran and ran on a 486-DX PC. Table 1 summarizes the various resequencing methods that were experimented with in this section. Figures 1-6 graphically present the average DC time for each method over various values of block size n and shape factor b. We discuss the outcomes of each method below.
4.1. Static approach The sequencing methods significantly affect the performance of the AS/RS. Under the static assignment algorithm (ASSTA), the average travel-between time decreased by over 45% when block size n was 5, compared with the baseline E(TB) (i.e., FCFS for both storages and retrievals). This amount of reduction is translated into an increase in throughput of at least 13% when P/D 0 " ~
0
0
kLI 1.6
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~'~
• Heuristic
~ - ~
v E (SC) o E (DC)
o
< 1.4
1.2 0
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20
40
60
80
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Block size Fig. 1. Static sequencing methods for b = 1.
358
Heungsoon Felix Lee and Samantha K. Schaefer 1,6
- -
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1.4
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< 1.2
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B l o c k size Fig. 2. Static sequencing methods for b = 0.6. 1.4
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1.2
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Fig. 3. Static sequencing methods for b = 0.2.
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S e q u e n c i n g m e t h o d s for A S / R S 1.8
0--0---0
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• ASDYN • ASFCDYN
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A HRDYN 0 HRFCDYN I
v E (SC) o E (DC)
< 1.4
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Block size Fig. 4. D y n a m i c s e q u e n c i n g m e t h o d s for b = I.
1.6
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1.4 • ASDYN • ASFCDYN A HRDYN ~ HRFCDYN V E (SC) i~
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< 1.2
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Block size Fig. 5. D y n a m i c s e q u e n c i n g m e t h o d s for b = 0.6.
360
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~
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• ASDYN • ASFCDYN A HRDYN 1.2
O HRFCDYN V E (SC) O E (DC)
<
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time r = 0 and of 9% when r = 0.2 [see equation (18)]. Figures 1-3 graphically illustrate this improvement for b = 1, b = 0.6, and b = 0.2. When block size n was steadily increased, average travel-between time exponentially decreased, converging to its lower bound, E(SC). Improvement of approx. 88% was observed when block size was increased to 100. The improvement became more evident for smaller shape factor b, since the rack size is smaller and requests generated are located more closely. When b = 0.2, average travel-between time was reduced by more than 50% even for n=5. Overall, the results of the static heuristic algorithm (HRSTA) are similar to those of the assignment algorithm, except that it has a 3-9% larger travel-between time than the other optimum algorithm. This difference alters throughput by less than 4% when r = 0 and by less than 2% when r = 0.2. The heuristic algorithm took much less computation, taking only seconds for each run regardless of block size n. On the other hand, ASSTA took several minutes for large n, requiring progressively longer runs for larger n.
4.2. Dynamic approach The impact of the sequencing methods is even more significant under the dynamic approach. The dynamic approach consistently outperformed the static approach for every block size n. This result intuitively makes sense since the dynamic method uses more information to make a decision on sequencing by considering newly arriving requests. The dynamic assignment algorithm (ASFCDYN) reduced travel-between time 4-14% more than its static counterpart. The difference was larger at smaller n and gradually decreased as n increased. This is because the impact of a newly added request on sequencing is smaller for larger n, i.e., it is seldom selected for the next DC pair when n is large. Thus, the sequence from the static method deviates less from that of the dynamic method for large n. The dynamic heuristic counterpart (HRFCDYN) performed better than ASFCDYN most of the time, achieving up to 7% additional reduction of E(TB). The former outperformed the latter for
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all 11 n when b = 0.2, for all but n = 2 when b = 0.6, and for all but n -- 2, 3, 5 when b = 1. For even these exceptions of n, the former was outperformed by the latter by no more than 1% of E(TB). This is a very useful finding since HRFCDYN is far simpler than ASFCDYN and takes just a fraction of the computation time. The former took a few minutes on a PC to find 1000 pairs of DCs, but the latter took several hours for large n. HRFCDYN reduced E(TB) at least 44% even when n is as small as 3, which corresponds to an increase in throughput of at least 13% when r = 0 and of at least '9% when r = 0.2. This result is practically important since the number of loads or bins waiting to be stored, m, is no larger than 5 in most AS/RS [11, 13, 14] due to a limited length of conveyor interface to unit-load AS/RS or a limited number of pick positions for miniload
AS/RS. ASDYN and HRDYN, which allow both storages and retrievals to be performed in any order, resulted in even shorter average travel times. They reduced E(TB) about 4-17% more than their dynamic counterparts which process storages FCFS. This reduction corresponds to additional increase in throughput of about 2-7% when r = 0 and of about 1-4% when r = 0.2. These performance improvements are greater at smaller n and gradually decreased as n increased. This makes sense since the effect of additional sequencing freedom is greater when there are the smaller number of requests to be sequenced. It is also interesting to find that HRDYN, which is much simpler and faster, consistently outperforms ASDYN for every n and b, with an increased reduction in E(TB) ranging from 0 to 6%. For n = 5, HRDYN led to a travel-between time reduction of at least 73% and an increase in throughput of at least 23% for r = 0 and at least 14% for r = 0.2. With these results, one can easily conduct investment assessment to justify the purchase of a more expensive material handling device that picks a storage load in any order. 5. CONCLUSIONS
Recently more AS/RS have been adopted to replace conventional material handling in warehouses and production systems. AS/RS require large capital investments, therefore, the users are interested in making the most of them. In this paper, we studied the effect of sequencing storage and/or retrieval requests on the reduction of travel time by a S/R machine, and consequently throughput, for AS/'RS where storage locations are predetermined. Applications include miniload AS/RS, unit-load AS/RS with dedicated storage, and potentially unit-load AS/RS with randomized storage. We found that different types of AS/RS require different sequencing to be employed and that the proposed sequencing methods can significantly affect performance. We presented variious methods for both static and dynamic approaches. For the static approach, we formulated the ,;equencing problem as the assignment problem for which efficient algorithms exist to find an optimum solution. We also derived a heuristic method from the assignment problem formulation. For the dynamic approach, we studied two cases: (1) storages are processed in FCFS and (2) in any order. For each case, we developed the dynamic sequencing methods by building upon the static methods and applying them repeatedly over time. We found that the static assignment method reduced E(TB) by up to 45% even when block size is 5 or less, consequently, increasing throughput by up to 9% with consideration of typical pick or drop time of a load. The static heuristic solution performed well and the maximum deviation from the static optimum did not exceed 9% of E(TB). We found that the dynamic methods clearly outperformed the static methods, achieving about 10-20% additional reduction of E(TB). With FCFS storages, most of the time the dynamic heuristic method outperformed the dynamic assignment method with up to a 7% additional reduction of E(TB). When storages are sequenced with the use of a more sophisticated material handling interface at the P/D station, an additional 4-17% reduction of E(TB) was achieved over FCFS storages. In this case, the dynamic heuristic method always outperformed the dynamic assignment method by up to 7 % of E(TB). This heuristic method achieved reduction of at least 73% of E(TB) and increase in throughput of at least 14%, even when block size is just 5. The dynamic heuristic method is also a very attractive choice in light of computation time. It is far simpler and faster, requiring only a fraction of computation time of its counter part relying
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on the assignment algorithm. In order to find each dual cycle pair, the former takes O(n) while the latter takes O(n3). In future research, it is worthwhile to study sequencing methods with multi-criterion objectives or other constraints. For instance, for AS/RS which feed parts to Just-in-Time Production systems, each retrieval request may have a target date to meet. A desirable goal of the sequencing methods for such AS/RS is to increase throughput and at the same time minimize tardiness or earliness. In another instance, the proposed dynamic heuristic algorithm can significantly reduce travel time, but certain retrieval requests can experience long delays before being processed. One simple remedy is to set a limit on the maximum delay that a request can tolerate. Within this limit, the sequencing method attempts to minimize travel time. Acknowledgement--This research is supported in part by research grants from Southern Illinois University at Edwardsville. REFERENCES 1. Allen, S. L. A selection guide to AS/R systems. Industrial Engineering, March 1992, 28. 2. DeWeerdt, L. Productivity gains through AS/RS retrofit. Production & Inventory Management, September 1991, 29. 3. Forger, G. How Ford cuts orderpicking cycles 60% with centralized storage. Modern Material Handling, May 1991, 52. 4. Han, M. H., McGinnis, L. F., Shieh, J. S. and White, J. A. On sequencing in an automated storage/retrieval system. liE Transactions, 1987, 19, 56. 5. Tompkins, J. A. and White, J. A. Facilities Planning. Wiley, New York, 1984. 6. Egbelu, P. J. Framework for dynamic positioning of storage/retrieval machines in an automated storage/retrieval system. International Journal of Production Research, 1991, 29, 17. 7. Egbelu, P. J. and Wu, C.-T. A comparison of dwell point rules in an automated storage/retrieval system. International Journal of Production Research, 1991, 31, 2515. 8. Linn, R. J. and Wysk, R. A. An expert system based controller for an automated storage/retrieval system. International Journal of Production Research, 1990, 28, 735. 9. Linn, R. J. and Wysk, R. A. An expert system framework for automated storage and retrieval system control. Computers and Industrial Engineering, 1990, 18, 37. 10. Bozer, Y. A. and White, J. A. Travel time models for automated storage/retrieval systems, liE Transactions, 1984, 16, 329. 11. Bozer, Y. A. and White, J. A. Design and performance models for end-of-aisle order picking systems. Management Science, 1990, 36, 852. 12. Kuhn, H. W. The Hungarian method for solving the assignment problem. Naval Research Logistics Quarterly, 1955, 2, 83. 13. Groover, M. P. Automation, Production Systems and Computer-Integrated Manufacturing. Prentice-Hall, Englewood Cliffs, NJ, 1987. 14. Randhawa, S., McDowell, E. and Wang, W. Evaluation of scheduling rules for single- and dual-dock automated storage/retrieval system. Computers and Industrial Engineering, 1991, 20, 401.
Pergamon
Computers ind. Engng Vol. 32, No. 2, pp. 351-362, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: S0360-8352(96)00298-7 0360-8352/97 $17.00 + 0.00
SEQUENCING METHODS FOR AUTOMATED STORAGE AND RETRIEVAL SYSTEMS WITH DEDICATED STORAGE HEUNGSOON FELIX LEE t* and SAMANTHA K. SCHAEFER: tDepartment of Mechanical and Industrial Engineering, Southern Illinois University, Edwardsville, IL 62026-1805, U.S.A. -'Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130-4899, U.S.A. (Received 28 August 1996) Abstraet--ln this paper, we study the effect of sequencing storage and retrieval requests on the performance of automated storage and retrieval systems (AS/RS) where a storage request is assigned a predetermined storage location. By exploiting this unique operating characteristic, we present several optimum and heuristic sequencing methods under static and dynamic approaches. Applications of such sequencing methods include unit-load AS/RS with dedicated storage, miniload AS/RS, and potentially unit-load AS/RS with randomized storage. We find that the sequencing methods can significantly reduce travel time by a storage and retrieval machine, thereby, increasing throughput, and that the dynamic heuristic method is simple and fast, yet considerably outperforms the others. © 1997 Elsevier Science Ltd. All rights reserved
I. INTRODUCTION
Since automated storage and retrieval systems (AS/RS) were introduced in the 1950s, the technology has advanced far beyond its original function, which was to eliminate the walking that accounted for 70% of manual retrieval time. AS/RS have been adopted not only as alternatives to traditional warehouses but also as a part of advanced manufacturing systems, and the number is expected to grow rapidly. This is because AS/RS have many benefits including savings in labor costs, improved material flow and inventory control, improved throughput level, high flow-space utilization, and increased safety and stock rotation [1-3]. Typically, AS/RS consist of a series of storage aisles each of which is served by a storage and retrieval (S/R) machine or crane. Each aisle is supported by a pickup and delivery (P/D) station customarily located at the end of the aisle and accessed by the S/R machine and the external handling system. A S/R machine usually operates in two modes: single cycle (SC) and dual cycle (DC). For each of the modes, a S/R machine starts at the P/D station, stores and/or retrieves a load, and returns to the P/D station to complete a cycle. In a SC, a S/R machine either stores or retrieves, while in a DC, it both stores and retrieves in one cycle. In a DC, a S/R machine picks up a load from a P/D station, travels to a storage location to store it, travels to another location to retrieve a load, and then returns to the P/D station to deliver it. The effectiveness of AS/RS depends on the methods of control that govern the scheduling of storages and retrievals. A common practice in sequencing storage and retrieval requests is that both requests are processed in first-come-first-served (FCFS) manner. The FCFS assumption is reasonable for storages, since most AS/RS are interfaced with a conveyor loop for input and output. In this case, it is difficult to change the sequence of loads presented for storage. However, the FCFS assumption is less compelling for retrievals since retrieval requests are just electronic messages and can be easily resequenced. In a DC, storage and retrieval requests can be paired to decrease the time spent travelling between the storage and retrieval locations. By minimizing the travel time, we can increase system *To whom all correspondence should be addressed. 351
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throughput (i.e., the number of storages or retrievals performed per period) and reduce AS/RS operating costs such as wear of mechanical parts and electric power cost. According to Han et al. [4], a 50% or more decrease in the travel-between time component of a dual cycle leads to an increase in throughput of 10-15%. Such an increase in throughput could help to handle peak demand in the operation phase, and eliminate an aisle in a multi-aisle system in the design phase, which leads to considerable savings. In this paper, we present several sequencing methods for AS/RS with dedicated storage, i.e., AS/RS for which storage requests are assigned predetermined storage locations. The proposed sequencing methods can be applied to two common types of AS/RS: unit-load AS/RS and miniload AS/RS. In unit-load AS/RS with dedicated storage, each load is assigned to a specific location or set of locations in the storage rack. These locations may be determined by activity and inventory levels or by stock number. Such assignments are normally made to aid in the retrieval process and may increase throughput [5]. Miniload AS/RS are an important subset of AS/RS and are used to handle small loads that are contained in bins or drawers within the storage system. The S/R machine is designed to retrieve the bin and deliver it to a P/D station at the end of the aisle so that the individual items can be withdrawn from the bins by a picker. The bin or drawer is then stored in its original location in the rack. Thus, miniload AS/RS follow dedicated storage. In addition to the above two types of AS/RS, the proposed methods can be useful for unit-load AS/RS with randomized storage, where a load can be stored in any open location, commonly in the open location closest to the P/D station. Such a randomized storage system is used to increase the utilization of the rack [5]. Han et al. [4] show that finding the optimum retrieval sequence for this type of unit-load AS/RS is not computationally tractable. This is because it involves not only finding the retrieval sequence, but also finding storage locations at the same time. One way to overcome this complexity is a 2-step procedure in which the first step is to determine locations for storage requests and the second step is to determine sequencing with the storage locations specified from the previous step. The proposed methods in this paper can be used for the second step and we introduce an efficient optimum sequencing method. In the quest to reduce the length of time required to serve a load retrieval or storage request, dwell point positioning policies have been cultivated. The dwell point is the position the S/R machine takes when idle. Egbelu [6] has developed two dwell point rules based on the probability of future requests. When Egbelu and Wu [7] compared these new rules with four simple established techniques, they were able to successfully decrease the travel time of the S/R machine. Linn and Wysk [8, 9] have developed expert systems that can strategically alter scheduling rules to achieve specific goals for production control in computer integrated manufacturing. They have discovered that as demand levels increase, policy alterations in retrieval sequencing and queue selection were at least twice as likely to be required as adjustments for other selection functions. This finding establishes the need for the development of proper retrieval sequencing techniques as part of overall strategic and tactical control rule application mechanisms to improve system performance. Han et al. [4] studied retrieval sequencing heuristic methods for unit-load AS/RS where storages can be stored in any open location, knowing that finding the optimum sequence is not computationally tractable. They showed evidence that the heuristics can improve throughput by 5-8%. The remainder of the paper is organized as follows. Section 2 presents various solution methods to sequence storage and retrieval requests for the AS/RS with dedicated storage under both static and dynamic approaches. Section 3 discusses performance measures to evaluate improvements by the sequencing methods. Section 4 describes the experimental results with the proposed sequencing methods. Finally, Section 5 concludes with a brief summary and future research issues. 2. SOLUTION METHODS
The list of storage and retrieval requests changes as old requests are processed and new requests arrive. Storage requests are processed in the order received. We take two approaches to the sequencing methods for retrieval requests: static and dynamic approaches. The static approach
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selects a block of retrievals and sequences them, and when the block of retrievals are completed, selects another block and so forth. Alternatively, the dynamic approach resequences the list each time a S/R machine completes a cycle by including new requests that arrive during the cycle. We first present sequencing methods for the static approach and then for the dynamic approach. We assume that each S/R trip for a cycle originates and terminates at the P/D station. This assumption is commonly used for both unit-load AS/RS and miniload AS/RS [4, 10, 11]. 2.1. S t a t i c approach
Suppose that n is the block size for retrievals and there are m storages waiting to be processed. If there are more than n retrievals in the current list, we consider the first n retrievals for resequencing. If m > n, we need consider only the first n storages since subsequent storages cannot be processed in this block due to the FCFS discipline for storages. Thus, we consider only the case where m ~< n below. Each storage or retrieval request is associated with a bin location, which is translated into a pair of travel times in horizontal and vertical directions from the P/D station. Let (h~i, vsi) and (hrj, vrj) be the pair for the ith arrived storage (i = 1. . . . . m) and thejth arrived retrieval (j = 1. . . . . n), respectively. Thus, with simultaneous travel in both the horizontal and vertical directions, it takes 2max(h~g, v~) to process the ith storage request with a single cycle. The objective is to minimize the total travel time required to process m storages and n retrievals in the current block. For each retrieval in the block, the following decisions must be made. Should the retrieval be performed in either SC or DC, and if it is performed in DC, with which storage request should it be paired? We can show that the optimum solution has m retrievals performed in DC and (n - m) retrievals in SC. This is because DC time is no greater than the sum of two individual SC times, one SC storage time and one SC retrieval time. That is, the following equation holds: max(h~i, v~i) + max(lh~i - hrj[, Ivs,-vrjl) + max(hrj, vrj) ~< 2max(hsi, v~) + 2max(h~j, vrj).
(1)
The proof of equation (1) can be shown by considering each of the following four cases: (h~ > or < vs~) and (h,j > or < v~j). With this result, we can formulate this problem as the assignment problem [12]. Since the assignment problem requires balancing (i.e., the number of storages equals the number of retrievals to be paired), we need to create (n - m) fictitious storages. We number these storage m + 1. . . . . n). When a retrieval is paired with a fictitious storage, a S/R machine performs SC for the retrieval. We first define necessary notation. Z = total travel time, Xo = assignment variable which is set to 1 if storage i is paired with retrieval j, set to 0 otherwise for i, j = 1. . . . . n, Cij = travel time to process a pair of storage i and retrieval j. Minimize
Z= ~ ~ CoX~
(2)
i=lj=l
Subject to: ~ X u = 1,
for i = 1,2 . . . . n.
(3)
for j = 1 , 2 . . . . n.
(4)
for i , j = 1 . . . . .
(5)
i= I
~X,j=I, j=l
A'u = 0, I
n
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The values of C,~ depends on whether i is associated with an actual storage or a fictitious one. For the former case (i.e., i ~< m), it is assigned DC time, while for the latter case, it is assigned SC retrieval time for retrieval j. Thus, we write C~j as C,j=cj+cij+Cj =2Cj
forl
fori/>m,
(6)
C, = max(hs,, v~)
(7)
C,j = max(lh~,- h,jl, Iv~,- v,jl)
(8)
Cj = max(hrj, vrj)
(9)
where
We can optimally solve the above assignment problem using the Hungarian method [12]. We also present a heuristic method to solve this assignment problem, which requires less computation but can be very effective for both static and dynamic approaches. Using equations (6)-(9), we can rewrite the objective equation (2) as follows:
Z=
~ ~ CoXiJ+ i=jj=l
=
~, ~ CuXo
(10)
i=m+lj=l
~,~,(c~+cij+cj)Xij+ ~ ~2c,X~ i=.jj= I
(11)
i=m+ Ij= I
Which when simplified becomes: (12) ]=1
i=j
i=lj=l
When m = n, equation (12) can be further simplified to: n
n
z = r c,+ f cj+ i=j
j=l
(13) i=lj=l
Thus, when m = n, the optimum sequence minimizing the total travel time is equivalent to the optimum sequence minimizing the total travel-between time. This makes sense since the sum of the times to travel from the origin to the storage point and from the retrieval point to the origin remains constant within a block. From equation (12), we develop the following heuristic algorithm, letting D,j = c~j for m = n and D~ = c o - c~ for m < n. Algorithm 1. Static heuristic algorithm.
Step Step Step Step
1. 2. 3. 4.
Let set J = {1, 2 . . . . . n} be the block of retrievals and i = 1. Find k that gives the smallest D~ for all j in J. Perform a DC for storage i and retrieval k and J = J - {k}. If i = m, then perform a SC retrieval for each j in J and terminate. Otherwise, increase i by 1 and go to Step 2.
This heuristic algorithm takes much less computation than the above optimum method. It requires at most only n ( n - 1)/2 comparisons. In contrast, the Hungarian method to solve the above assignment problem requires O(n 3) comparisons and arithmetic operations. Although we are not able to achieve tight theoretical error bounds for this heuristic, empirical results shows that it usually provides near-optimal solutions as we will discuss in Section 5.
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2.2. Dynamicapproach In the dynamic approach, we resequence the retrieval order reflecting arrivals of new requests as the system operates over time. We can naturally extend the solution methods for the static approach to the dynamic approach. Suppose that there are m storages and n retrievals (m, n > 0) in the storage and retrieval lists at time 0.
Algorithm 2. Dynamic assignmentalgorithm. Step 1. Step 2. Step 3. Step 4.
Solve the assignment problem with the current lists of storages and retrievals. Perform a DC with the first storage and a retrieval that is paired with the first storage in the solution from Step 1. Remove the storage and/or retrieval performed in the cycle from the respective list, and add to each list the new storages and retrievals that arrive during the cycle. If one of the two lists is empty, then perform a SC and go to Step 3. If both lists are empty, then wait for the first request to arrive and then repeat this step. Otherwise, go to Step 1.
Recall that when the number of storages in the list, m, is greater than the number of retrievals, n, in Step 1, we ignore storages n + 1. . . . . m and solve the smaller-size assignment problem. Similarly, we develop the dynamic heuristic algorithm simply by replacing the assignment problem in Step 1 of the above algorithm with the static heuristic algorithm, which is algorithm 1. In order to find the next DC pair, the dynamic assignment algorithm has a complexity of O(n 3) since it solves the assignment problem each time. On the other hand, the dynamic heuristic algorithm has a complexity of O(n). This heuristic requires at most 2n arithmetic operations and 2n comparisons to compute D~s and find the smallest D u. It is difficult to show the optimality of a dynamic sequencing method because of the dynamic nature of predicting future requests. It seems to be reasonable to make relative comparison among the dynamic methods under consideration. We present experimental results in Section 4 for such comparison. 3. MEASURESTO IMPROVEMENTSBY SEQUENCING In order to measure improvements achieved by the proposed sequencing methods, we use commonly adopted AS/RS with the following characteristics: (1) any opening in the methods aisle is equally likely to be selected for a storage or retrieval request; (2) a S/R machine simultaneously travels in both directions with constant speeds; and (3) the P/D station is located at the base and at the end of the aisle. We let L be the length and H be the height of the AS/RS aisle, and Vh and Vv the horizontal and vertical travel speeds of the S/R machine. Then, the times required to travel the full length and height, respectively, are given by th = L/Vh and tv = H/Vv. Using these travel times, we define two parameters, T = max(th, tv) and b = min(th/T, iv~T), which are referred to as the maximum travel time and shape factor, respectively. With a continuous approximation to the discrete rack face, Bozer and White [10] derived the expected travel times of SC and DC when no resequencing method is applied, i.e., when retrievals are processed FCFS as follows: E(SC) = Expected Single Cycle Travel Time.
E(SC) = [1
+lb2]T
(14)
E(TB) = Expected Travel-Between Time for the Dual Cycle. E(TB) = [1 + ~b 12
_-~ob3]T
(15)
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E(DC) = Expected Dual Cycle Travel Time. 1 2-1b3]T E(DC) = I4 + ~b
= E(SC) + E(TB)
(16)
By setting T to 1, we normalize the rack. The size (shape) of the normalized rack becomes 1.0 x b, where b ~< 1. We can decrease travel time only by decreasing the travel-between time, equation (15), of the dual cycle. Thus, equation (14) serves as a lower bound on expected DC travel time for any resequencing method for both static and dynamic approaches. Denoting E(DC)s as expected dual cycle travel time achieved by a certain sequencing method, we can express ~, the reduction rate of E(TB), as follows: =
E(DC) - E(DC)s E(TB)
(17)
W e relate reduction of E(TB) to possible increase in throughput as follows:
E(DC) + 4r - 1 = E(DC) - ~E(TB) + 4r
(18)
where fl is the increase in throughput that can be realized by ~t reduction of E(TB), and r is the P/D time, i.e., the time taken to pick or drop a load. When r is negligible and b varies from 0.2 to 1, equation (18) shows that the maximum increase in throughput ranges from 33.5% for b = 0.2 to 35% for b = 1. If as Han et al. [4] suggest, a 12% increase in throughput is considered significant, the expected travel-between time must decrease by at least 40%. If r is 0.2, which is a typical P/D time for an AS/RS, the maximum throughput improvement possible decreases to 22%. If throughput improvement of 12% or more is considered significant, the expected travel-between time must be decreased by at least 60%. 4. EXPERIMENTS We conducted experiments to determine how much the proposed optimum and heuristic methods can reduce the cycle time of a S/R machine, and consequently, improve system throughput under both the static and dynamic approaches. We used three different values for shape factor, b: 1, 0.6, and 0.2. We normalized the rack, i.e., set T to 1. We tried numerous values for block size n, ranging from 2 to 100. We limited the experiments to the case where the number of storages equals the number of retrievals in the block (i.e., m = n). For each value of b, we generated 1000 requests for both storages and retrievals. Each request is associated with a pair of travel times in horizontal and vertical directions that were randomly generated from uniform distributions (0, 1) and (0, b), respectively. Thus, when n = 10, we applied the static methods 100 times, once for each block of 10 storages and retrievals, and recorded the average of 1000 DC times performed. For the dynamic methods, we assumed that one storage and one retrieval arrive during a DC. Therefore, when n = 10, we started with the block of the first 10 storages and the first 10 retrievals and determined the first DC to perform. After performing the DC, we updated the block by removing the processed storage and retrieval and adding the 1lth generated storage and retrieval, and determined the next DC to perform. We continued this process until the lists of 1000 requests are exhausted. Block size n = 1 means that the S/R machine performs retrievals FCFS and its expected DC travel time, equation (16), was used as a baseline for any improvements made. We presented two dynamic sequencing methods in Section 2.2 in which storages are processed FCFS as is commonly practiced. In our experiments, we also investigated the situation in which storages can be resequenced in order to study its impact on reduction of the travel-between time.
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Table 1. Sequencing methods ASSTA: HRSTA: ASFCDYN: HRFCDYN: ASDYN: HRDYN:
STATIC ASSIGNMENT ALGORITHM STATIC HEURISTIC ALGORITHM DYNAMIC ASSIGNMENT ALGORITHM in which storages are performed FCFS, while retrievals are performed in any order DYNAMIC HEURISTIC ALGORITHM in which storages are performed FCFS, while retrievals are performed in any order DYNAMIC ASSIGNMENT ALGORITHM in which storages and retrievals may be performed in any order DYNAMIC HEURISTIC ALGORITHM in which storages and retrievals may be performed in any order
For this type of AS/RS, a P/D station needs to be equipped with more sophisticated sensors and material handling devices in order to access any load waiting for storage We modified the dynamic methods to allow storage resequencing. For the dynamic assignment algorithm (algorithm 2), we modified Step 2 to pick a DC pair that gives the smallest Do among the DC pairs in the assignment solution instead of picking a DC pair with the first storage. For the dynamic heuristic algorithm we modified Step 2 of algorithm 1 to pick a DC pair that gives the smallest D,~ for all i, j instead of fixed i. It is not beneficial to apply storage resequencing under the static methods. This is because storage resequencing does not affect total travel time required to process the entire block of storages and retrievals. We coded all the proposed methods in Microsoft Fortran and ran on a 486-DX PC. Table 1 summarizes the various resequencing methods that were experimented with in this section. Figures 1-6 graphically present the average DC time for each method over various values of block size n and shape factor b. We discuss the outcomes of each method below.
4.1. Static approach The sequencing methods significantly affect the performance of the AS/RS. Under the static assignment algorithm (ASSTA), the average travel-between time decreased by over 45% when block size n was 5, compared with the baseline E(TB) (i.e., FCFS for both storages and retrievals). This amount of reduction is translated into an increase in throughput of at least 13% when P/D 0 " ~
0
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358
Heungsoon Felix Lee and Samantha K. Schaefer 1,6
- -
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time r = 0 and of 9% when r = 0.2 [see equation (18)]. Figures 1-3 graphically illustrate this improvement for b = 1, b = 0.6, and b = 0.2. When block size n was steadily increased, average travel-between time exponentially decreased, converging to its lower bound, E(SC). Improvement of approx. 88% was observed when block size was increased to 100. The improvement became more evident for smaller shape factor b, since the rack size is smaller and requests generated are located more closely. When b = 0.2, average travel-between time was reduced by more than 50% even for n=5. Overall, the results of the static heuristic algorithm (HRSTA) are similar to those of the assignment algorithm, except that it has a 3-9% larger travel-between time than the other optimum algorithm. This difference alters throughput by less than 4% when r = 0 and by less than 2% when r = 0.2. The heuristic algorithm took much less computation, taking only seconds for each run regardless of block size n. On the other hand, ASSTA took several minutes for large n, requiring progressively longer runs for larger n.
4.2. Dynamic approach The impact of the sequencing methods is even more significant under the dynamic approach. The dynamic approach consistently outperformed the static approach for every block size n. This result intuitively makes sense since the dynamic method uses more information to make a decision on sequencing by considering newly arriving requests. The dynamic assignment algorithm (ASFCDYN) reduced travel-between time 4-14% more than its static counterpart. The difference was larger at smaller n and gradually decreased as n increased. This is because the impact of a newly added request on sequencing is smaller for larger n, i.e., it is seldom selected for the next DC pair when n is large. Thus, the sequence from the static method deviates less from that of the dynamic method for large n. The dynamic heuristic counterpart (HRFCDYN) performed better than ASFCDYN most of the time, achieving up to 7% additional reduction of E(TB). The former outperformed the latter for
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all 11 n when b = 0.2, for all but n = 2 when b = 0.6, and for all but n -- 2, 3, 5 when b = 1. For even these exceptions of n, the former was outperformed by the latter by no more than 1% of E(TB). This is a very useful finding since HRFCDYN is far simpler than ASFCDYN and takes just a fraction of the computation time. The former took a few minutes on a PC to find 1000 pairs of DCs, but the latter took several hours for large n. HRFCDYN reduced E(TB) at least 44% even when n is as small as 3, which corresponds to an increase in throughput of at least 13% when r = 0 and of at least '9% when r = 0.2. This result is practically important since the number of loads or bins waiting to be stored, m, is no larger than 5 in most AS/RS [11, 13, 14] due to a limited length of conveyor interface to unit-load AS/RS or a limited number of pick positions for miniload
AS/RS. ASDYN and HRDYN, which allow both storages and retrievals to be performed in any order, resulted in even shorter average travel times. They reduced E(TB) about 4-17% more than their dynamic counterparts which process storages FCFS. This reduction corresponds to additional increase in throughput of about 2-7% when r = 0 and of about 1-4% when r = 0.2. These performance improvements are greater at smaller n and gradually decreased as n increased. This makes sense since the effect of additional sequencing freedom is greater when there are the smaller number of requests to be sequenced. It is also interesting to find that HRDYN, which is much simpler and faster, consistently outperforms ASDYN for every n and b, with an increased reduction in E(TB) ranging from 0 to 6%. For n = 5, HRDYN led to a travel-between time reduction of at least 73% and an increase in throughput of at least 23% for r = 0 and at least 14% for r = 0.2. With these results, one can easily conduct investment assessment to justify the purchase of a more expensive material handling device that picks a storage load in any order. 5. CONCLUSIONS
Recently more AS/RS have been adopted to replace conventional material handling in warehouses and production systems. AS/RS require large capital investments, therefore, the users are interested in making the most of them. In this paper, we studied the effect of sequencing storage and/or retrieval requests on the reduction of travel time by a S/R machine, and consequently throughput, for AS/'RS where storage locations are predetermined. Applications include miniload AS/RS, unit-load AS/RS with dedicated storage, and potentially unit-load AS/RS with randomized storage. We found that different types of AS/RS require different sequencing to be employed and that the proposed sequencing methods can significantly affect performance. We presented variious methods for both static and dynamic approaches. For the static approach, we formulated the ,;equencing problem as the assignment problem for which efficient algorithms exist to find an optimum solution. We also derived a heuristic method from the assignment problem formulation. For the dynamic approach, we studied two cases: (1) storages are processed in FCFS and (2) in any order. For each case, we developed the dynamic sequencing methods by building upon the static methods and applying them repeatedly over time. We found that the static assignment method reduced E(TB) by up to 45% even when block size is 5 or less, consequently, increasing throughput by up to 9% with consideration of typical pick or drop time of a load. The static heuristic solution performed well and the maximum deviation from the static optimum did not exceed 9% of E(TB). We found that the dynamic methods clearly outperformed the static methods, achieving about 10-20% additional reduction of E(TB). With FCFS storages, most of the time the dynamic heuristic method outperformed the dynamic assignment method with up to a 7% additional reduction of E(TB). When storages are sequenced with the use of a more sophisticated material handling interface at the P/D station, an additional 4-17% reduction of E(TB) was achieved over FCFS storages. In this case, the dynamic heuristic method always outperformed the dynamic assignment method by up to 7 % of E(TB). This heuristic method achieved reduction of at least 73% of E(TB) and increase in throughput of at least 14%, even when block size is just 5. The dynamic heuristic method is also a very attractive choice in light of computation time. It is far simpler and faster, requiring only a fraction of computation time of its counter part relying
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on the assignment algorithm. In order to find each dual cycle pair, the former takes O(n) while the latter takes O(n3). In future research, it is worthwhile to study sequencing methods with multi-criterion objectives or other constraints. For instance, for AS/RS which feed parts to Just-in-Time Production systems, each retrieval request may have a target date to meet. A desirable goal of the sequencing methods for such AS/RS is to increase throughput and at the same time minimize tardiness or earliness. In another instance, the proposed dynamic heuristic algorithm can significantly reduce travel time, but certain retrieval requests can experience long delays before being processed. One simple remedy is to set a limit on the maximum delay that a request can tolerate. Within this limit, the sequencing method attempts to minimize travel time. Acknowledgement--This research is supported in part by research grants from Southern Illinois University at Edwardsville. REFERENCES 1. Allen, S. L. A selection guide to AS/R systems. Industrial Engineering, March 1992, 28. 2. DeWeerdt, L. Productivity gains through AS/RS retrofit. Production & Inventory Management, September 1991, 29. 3. Forger, G. How Ford cuts orderpicking cycles 60% with centralized storage. Modern Material Handling, May 1991, 52. 4. Han, M. H., McGinnis, L. F., Shieh, J. S. and White, J. A. On sequencing in an automated storage/retrieval system. liE Transactions, 1987, 19, 56. 5. Tompkins, J. A. and White, J. A. Facilities Planning. Wiley, New York, 1984. 6. Egbelu, P. J. Framework for dynamic positioning of storage/retrieval machines in an automated storage/retrieval system. International Journal of Production Research, 1991, 29, 17. 7. Egbelu, P. J. and Wu, C.-T. A comparison of dwell point rules in an automated storage/retrieval system. International Journal of Production Research, 1991, 31, 2515. 8. Linn, R. J. and Wysk, R. A. An expert system based controller for an automated storage/retrieval system. International Journal of Production Research, 1990, 28, 735. 9. Linn, R. J. and Wysk, R. A. An expert system framework for automated storage and retrieval system control. Computers and Industrial Engineering, 1990, 18, 37. 10. Bozer, Y. A. and White, J. A. Travel time models for automated storage/retrieval systems, liE Transactions, 1984, 16, 329. 11. Bozer, Y. A. and White, J. A. Design and performance models for end-of-aisle order picking systems. Management Science, 1990, 36, 852. 12. Kuhn, H. W. The Hungarian method for solving the assignment problem. Naval Research Logistics Quarterly, 1955, 2, 83. 13. Groover, M. P. Automation, Production Systems and Computer-Integrated Manufacturing. Prentice-Hall, Englewood Cliffs, NJ, 1987. 14. Randhawa, S., McDowell, E. and Wang, W. Evaluation of scheduling rules for single- and dual-dock automated storage/retrieval system. Computers and Industrial Engineering, 1991, 20, 401.