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Avoiding Fragmentation in Miniload Automated Storage and Retrieval Systems
Information Control Problems in Manufacturing Proceedigs the IFAC on Proceedigs of of the 15th 15th IFAC Symposium Symposium on Proceedigs of theOttawa, 15th IFAC Symposium on May 11-13, 2015. Canada Information Control Problems in Available online at www.sciencedirect.com Information Control Problems in Manufacturing Manufacturing Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada
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48-3 (2015) 1973–1977 Avoiding FragmentationIFAC-PapersOnLine in Miniload Automated Storage and Retrieval Systems Avoiding Fragmentation Fragmentation in in Miniload Miniload Automated Automated Storage Storage and Retrieval Retrieval Systems Avoiding Avoiding Fragmentation in Miniload Automated Storage and and Retrieval Systems Systems Henri Tokola, Esko Niemi
Henri Tokola, Tokola, Esko Niemi Niemi Henri Esko Henri Tokola, Esko Niemi Department of Engineering Design and Production, Department of Engineering and Production, Aalto University School ofDesign Engineering, Espoo, Department of Engineering Design and Production, Department of Engineering Design and Production, Aalto University School of Engineering, Espoo, Finland (e-mail: [email protected]). Aalto University School of Engineering, Aalto University School of Engineering, Espoo, Espoo, Finland (e-mail: [email protected]). Finland Finland (e-mail: (e-mail: [email protected]). [email protected]). Abstract: This paper studies detailed slotting in miniload automated storage and retrieval systems Abstract: paperThe studies detailed slotting in have miniload automated storageshelves and retrieval retrieval systems (miniload This AS/RSs). systems that are studied a number of identical and theysystems handle Abstract: This paper studies detailed slotting in miniload automated storage and Abstract: This paperThe studies detailed slotting in have miniload automated storage and retrieval (miniload AS/RSs). systems that are are studied number of identical identical shelves and theysystems handle cartons with different widths. The purpose isstudied to find out aaadetailed slotting rule that makes utilisation high (miniload AS/RSs). The systems that have number of shelves and they handle (miniload AS/RSs). The systems that studied have number ofsmall identical shelves and theyFor handle cartons with different widths. The is to find out aaadetailed slotting rule makes utilisation high by reducing the fragmentation ofpurpose the are available storage space into andthat unusable gaps. this cartons with different widths. The purpose is to find out detailed slotting rule that makes utilisation high cartons with widths. The purpose isatodetailed find outslotting a detailed slotting rule that makes utilisation high by reducing the fragmentation of the available storage space into small unusable gaps. For this purpose, the different paper constructs and analyses rule which isand based on a best fit rule, but by reducing the fragmentation of the available storage space into small and unusable gaps. For this by reducing the fragmentation of analyses the available storage into small unusable gaps. For purpose, the paper constructs aa detailed rule which based on aa best fit rule, but which aligns the cartons beingand handled to both ends ofslotting thespace shelf and next is toand a longer carton. The rulethis is purpose, the paper constructs and analyses detailed slotting rule which is based on best fit rule, but purpose, the paper constructs and analyses a detailed slotting rule which is based on aperformance. best fit rule, but which aligns cartons being handled to both ends of the shelf and next to a longer carton. The rule is compared to the first fit rule and to different aligning options in order to validate its The which aligns cartons being handled to both ends of the shelf and next to a longer carton. The rule is which aligns the cartons beingand handled toconstructed bothaligning ends ofoptions thegive shelf and next to10% a longer carton. rule is compared to the fit rule to different in order to its The simulation results show as saving in theThe utilisation compared to the first first fitthat rulethe andrule to thus different aligningcan options in much order as to avalidate validate its performance. performance. The compared to the first fit rule and to different aligning options in order to validate its performance. The simulation results that rule thus can as aa 10% saving in when compared the first aligning. Aligning to a as longer carton instead next to a simulation resultstoshow show thatfitthe therule rulewithout thus constructed constructed can give give next as much much as 10% saving in the theofutilisation utilisation simulation results that rulewithout thus constructed can give next as much a 10% saving in theofutilisation when compared toshow the first fittherule rule aligning. Aligning Aligning to aa as longer carton instead next to to a shortercompared carton gives a 1% benefit. when to the first fit without aligning. next to longer carton instead of next when compared to the firstbenefit. fit rule without aligning. Aligning next to a longer carton instead of next to aa shorter carton gives a 1% shorter carton gives a 1% benefit. Keywords: warehouse automation, automated storage and Hosting retrievalbysystem, avoidance, © 2015,carton IFAC (International Federation of Automatic Control) Elsevier fragmentation Ltd. All rights reserved. shorter gives a 1% benefit. Keywords: automation, automated storage and retrieval system, fragmentation avoidance, simulation warehouse Keywords: warehouse automation, automated storage and retrieval system, fragmentation avoidance, Keywords: warehouse automation, automated storage and retrieval system, fragmentation avoidance, simulation simulation simulation states that most of the papers in this area deal with unit-load 1. INTRODUCTION states that thatwhich most means of the the papers papers ininthis this areaall deal with unit-load unit-load AS/RSs, AS/RSsin which cartons have the states most of area deal with 1. INTRODUCTION 1. thatwhich most of the papers area deal with unit-load AS/RSs, means AS/RSs inthis which all cartons have the same width. Unit-load AS/RSinis not relevant to our paper, In INTRODUCTION modern logistics systems, automated storage and retrieval states 1. INTRODUCTION AS/RSs, which means AS/RSs in which all cartons have the which AS/RSs in which all slot cartons have the same width. width. Unit-load AS/RS isfitnot not relevant tothat our paper, In modern modern logistics systems, automated storage and retrieval retrieval because there themeans cartons alwaysis into the remains systems (AS/RS) aresystems, commonly used. They consist of racks AS/RSs, same Unit-load AS/RS relevant to our paper, In logistics automated storage and width. isfitnot relevant tothat our paper, In modern logistics systems, automated storage and retrieval because thereUnit-load the cartons always into the slot slothave remains systems (AS/RS) are commonly used. They consist of racks racks same after retrieval. Thus, in AS/RS this study, the cartons different and a crane, whichare runs between the racks and automatically because there the cartons always fit into the that remains systems (AS/RS) commonly used. They consist of because there the cartons always fit into the slot that remains systems (AS/RS) are commonly used. They consist of racks after retrieval. Thus, in this study, the cartons have different and a crane, which runs between the racks and automatically widths. This type of in AS/RS with the variable cartons is called inserts and retrieves cartons. Such systems allow improved after retrieval. Thus, this cartons have different and which runs between the racks automatically after retrieval. Thus, this study, study, cartons have is different and aa crane, crane, whichhigh runs between the racks and and automatically widths. This type of in AS/RS with the variable cartons called inserts and retrieves cartons. Such systems allow improved miniload AS/RS. inventory control, floor space utilisation, reduced labour widths. This type of AS/RS with variable cartons is called inserts and retrieves cartons. Such systems allow improved inserts andcontrol, retrieves cartons. Suchutilisation, systems allow improved widths. This type of AS/RS with variable cartons is called miniload AS/RS. inventory high floor space reduced labour costs, flexibility, and security (Johnson and Brandeau, 1996). miniload AS/RS. inventory control, high floor utilisation, reduced labour inventory control, highminiload floor space space utilisation, reduced labour costs, flexibility, and and Brandeau, This studies AS/RS where cartons1996). have miniload AS/RS. costs, paper flexibility, andasecurity security (Johnson (Johnson and Brandeau, 1996). costs, flexibility, andasecurity (Johnson Brandeau,inserted 1996). This paper studies miniload AS/RS where different widths and, cartons are and continuously This paper studies a when miniload AS/RS where cartons cartons have have This paper studies a miniload AS/RS where cartons different widths and, when cartons continuously and retrieved storage, spaceinserted inhave the different widthsfrom and,the when cartonstheare areavailable continuously inserted different widths and,the when cartonsthe areavailable continuously inserted and retrieved from storage, space in the the storage will become fragmented. Therefore, it isspace impossible and retrieved from the storage, the available in anddoretrieved from the storage, the available the storage will become become fragmented. Therefore, it is isspace impossible to complete scheduling, and decisions have toimpossible be inmade storage will fragmented. Therefore, it 1) Initial situation storage will become fragmented. Therefore, it is impossible to do complete scheduling, and decisions have to be made online when a carton arrives into the system.have Fig. to 1 illustrates to do complete scheduling, and decisions be made 1) Initial situation situation 1) to do complete scheduling, and decisions have to be made 1) Initial Initial situation online when carton arrives into the system. 11 illustrates how gaps areaa formed in the storage. The aimFig. of the paper is online when carton arrives into the system. Fig. illustrates onlinegaps when carton into slotting the system. Fig. 1avoid illustrates how in the The of paper is to outa formed how arrives different rules howfind gaps are are formed in the storage. storage. The aim aim of the the paperthis is how gaps are formed in the storage. The aim of the paperthis is to find out how different slotting rules avoid fragmentation. to find out how different slotting rules avoid this to find out how different slotting rules avoid this fragmentation. fragmentation. In computer science that deals with storages and memory, fragmentation. In science that deals with storages and memory, thiscomputer kind of fragmentation is called external fragmentation In computer science that with storages and memory, 2) Three cartons are retrieved In computer science (1969) that deals deals with storages and memory, this kind of fragmentation is called external fragmentation (see e.g. Randell and Wilson et al. (1995)). 2) Three cartons cartons are retrieved retrieved this kind of fragmentation is called external fragmentation 2) this kind ofRandell fragmentation isand called fragmentation 2) Three Three cartons are are retrieved (see (1969) al. Fragmentation is normally by external using et defragmentation (see e.g. e.g. Randell (1969)avoided and Wilson Wilson et al. (1995)). (1995)). (see e.g. Randell (1969)avoided and Wilson et AS/RSs al. (1995)). Fragmentation is by tools. However, unlike computers, the Fragmentation is normally normally avoided by using usingindefragmentation defragmentation Fragmentation is normally avoided by using defragmentation tools. However, However,imposes unlikecosts computers, in AS/RSs AS/RSs delay the defragmentation as it will significantly tools. unlike computers, in the tools. However, unlike computers, in AS/RSs the defragmentation imposes costs ascause it will will significantly delay the current operation and costs it will thesignificantly material handling defragmentation imposes as it delay defragmentation imposes costs as it will significantly delay the current current operationoperation. and it it will will cause the material material handling handling device unnecessary Thus, defragmentation should the operation and cause the 3) Three new cartons are inserted theavoided current operation and it will cause the material handling device unnecessary operation. Thus, defragmentation should be in AS/RSs by proactively reducing fragmentation. device unnecessary operation. Thus, defragmentation should 3) Three new new cartons are are inserted 3) device unnecessary operation. Thus, defragmentation should 3) Three Three new cartons cartons are inserted inserted be avoided in AS/RSs by proactively reducing fragmentation. be avoided in AS/RSs by proactively reducing fragmentation. The literature on AS/RSs is wide as these systems have been be avoided in AS/RSs by proactively reducing fragmentation. The literature literature on AS/RSs is wide wide as as thesethe systems have been studied from on several directions since 1950s. Forbeen an The AS/RSs is these systems have The literature on AS/RSs is wide as thesesystems systems have studied from several directions since the 1950s. Forbeen an overall review of general storage and their studied from several directions since the 1950s. For an studied from several directions since to thelook 1950s. Fortheir an overall review of general storage systems mathematical analysis, it is worthwhile atand Bartholdi overall review of general storage systems and their 4) The storage has small unusable gaps (i.e. storage is overall review of general storage systems and their mathematical is look at Bartholdi and Hackmananalysis, (2014).it Forworthwhile a reviewto of AS/RSs, see mathematical analysis, it is worthwhile to look at Bartholdi 4) The The storage has has small unusable gaps gaps (i.e. storage storage is fragmented) mathematical analysis, it is worthwhile to look at Bartholdi 4) unusable and For aa review see Roodbergen and (2014). Iris (2009). state thatof publications 4) The storage storage has small small unusable gaps (i.e. (i.e. storage is is and Hackman Hackman (2014). ForThey review oftheAS/RSs, AS/RSs, see fragmented) fragmented) and Hackman (2014). For a review of AS/RSs, see Roodbergen and Iris (2009). They state that the publications in the area often focus on the design of AR/RSs or study Fig. 1. Example of how fragmentation fragmented) occurs on a shelf when Roodbergen and Iris (2009). They state that the publications Roodbergen and They state of thatAR/RSs the publications Example of on aa shelf in the area focus on the design or study existing cartons arefragmentation retrieved and occurs new ones AS/RSs as aoften partIris of (2009). general warehousing. Their paper also Fig. Fig. 1. 1. Example of how how fragmentation occurs on inserted shelf when when in the area often focus on the design of AR/RSs or Fig. 1. Example of how fragmentation occurs on a shelf when in the area often focus on the design of AR/RSs or study study existing cartons are retrieved and new ones inserted AS/RSs as a part of general warehousing. Their paper also existing cartons are retrieved and new ones inserted AS/RSs as a part of general warehousing. Their paper also existing cartons are retrieved and new ones inserted AS/RSs as a part of general warehousing. Their paper also2047 Copyright © 2015 IFAC 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2015 responsibility IFAC 2047Control. Peer review© of International Federation of Automatic Copyright 2015 IFAC 2047 Copyright © 2015 IFAC 2047 10.1016/j.ifacol.2015.06.377
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Few of the AS/RS-related papers focus on miniload AS/RS and none, to our knowledge, considers both the fragmentation problem and detailed slotting. As stated by Bartholdi and Hackman (2014), detailed slotting without retrievals can be abstracted as a one-dimensional bin-packing problem: given n numbers, arrange them into as few groups as possible such that each group sums to no more than one. However, in our case there are retrievals too, which complicates the problem. The problem has to be studied in online settings, which rarely appear in the literature. However, there are several papers that study slotting (or assignment) in unit-load AS/RS. Typically, as done e.g. by Kaylan and Medeiros (1988), the papers study zoning in such a way that different kinds of zoning areas are compared. Clearly, the zones near the input/output locations are more valuable than others, but, in our paper, we do not consider zoning or the input/output location. Related to our paper, Hausman et al. (1976) study the closest open location rule, in which the cartons are put into the first empty slot. They study the rule in a unit-load system where fragmentation does not occur. This is similar to the first fit rule, which is used in this paper as a comparison rule.
Therefore a gap is found using either best fit or first fit. The next step is to decide to which side of the gap the carton is aligned. For that, there are three options:
All the combinations of the above fitting and aligning rules are studied and they are the following:
To summarise the above introduction, fragmentation seems to be studied in the computer literature, but not in the case of miniload AS/RSs, as is done here. This paper proposes new slotting rules that avoid fragmentation by cleverly aligning cartons to existing gaps.
The rest of the paper is organised as follows. Next, Section 2 presents the slotting rules that are used in the numerical experiments of Section 3. Section 4 discusses the results of the experiments and, finally, Section 5 draws conclusions.
2. DETAILED SLOTTING RULES IN AS/RS 2.1 Miniload AS/RS system
This paper studies an AS/RS with N identical W-width shelves. Cartons are inserted to and retrieved from the storage. As the studying of slotting into shelves is the aim of the paper, crane movement and input/output locations are not considered in this paper. When cartons are inserted, a decision regarding the slotting position has to be made. Thus, the purpose of the paper is to find out a good detailed slotting rule that makes utilisation high by reducing the fragmentation of the storage into small and unusable gaps. Utilisation is defined here as the ratio of the total width of cartons to the total width of shelves. 2.2 Slotting rules In order to find a good slotting rule, this paper compares seven slotting rules. The rules are based on either best fit (BF) or first fit (FF). Best fit allocates cartons to the smallest free gap into which the carton fits. In first fit, the shelves are gone through in order instead, and the carton is put into the first gap that has enough space.
AL – align left ABL – aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the longer carton if there are cartons on both sides of the shelf. ABS – aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the shorter carton if there are cartons on both sides of the shelf.
BF-AL: best fit that always aligns next to the carton on the left-hand side; BF-ABL: best fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the longer carton if there are cartons on both sides of the shelf. This rule is the focus of the present paper; BF-ABS: best fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the shorter carton if there are cartons on both sides of the shelf; FF-AL: first fit that always aligns cartons next to the carton on the left-hand side; FF-ABL: first fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the longer carton if there are cartons on both sides of the shelf; FF-ABS: first fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the shorter carton if there are cartons on both sides of the shelf.
The different ideas behind the above rules, especially those that are behind the BF-ABL rule, are discussed below. 2.3 Ideas behind slotting rules The ideas behind the slotting rules are discussed here. This is done in three steps. First, the differences between best fit and first fit are discussed. Second, the advantages of aligning on both sides are described. Third, the advantages of aligning next to the longer carton are discussed. The comparison between best fit and first fit is not easy. In general, it is expected that best fit will outperform first fit, as it fills smaller gaps first and saves larger gaps for future use. However, as discovered experimentally by Shore (1975), first fit outperforms best fit with exponentially distributed widths, but not with normally distributed widths. When first fit outperforms best fit, the reason that is hypothesised for this is
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that first fit allocates the shelf from one end and saves the other end for longer cartons. Still, it looks as if best fit should outperform first fit in normal situations, and thus it was selected as a base rule in the BF-ABL rule studied here. By inserting the cartons on both sides of the shelves we avoid the steps shown in Fig. 2, where aligning left and retrieving creates two gaps but aligning with both sides and retrieving creates one larger gap. Aligning with both sides is realised in the rules with the ABL or ABS suffixes.
retrieval percentage (r). This is the probability between 0 and 1 that a random carton is retrieved after the insertion of a carton.
Triangle distribution was selected for the carton width distribution because, according to the data gained from one industrial company, in reality the carton widths seem to closely follow triangle distribution. When the simulation software is run, it follows the following steps:
a) Align left b) Align both sides (AL) (ABL and ABS) Fig. 2. Example of the situation where aligning both sides avoids fragmentation into two gaps. Aligning next to the longer carton is likely to reduce the fragmentation to small gaps in the way that is shown in Fig. 3. The idea is the following. When a carton is aligned next to the longer carton, the retrieval of the shorter carton does not cause a small gap to be created. This rule is applied in the rules with the ABL suffix.
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1.
initialise the storage to be empty at the beginning;
2.
if a random value between 0 and 1 < the retrieval percentage r, then retrieve and remove one random carton from the storage;
3.
generate a new carton and fit the carton into the storage. Generate a carton from triangle distribution. Store the carton in the slot defined by slotting algorithm and aligning type;
4.
repeat steps 2-4 until the next new carton does not fit into the storage;
5.
record the final utilisation of the storage at the end.
This simulation software is used next in an example and then in large-scale experiments. 3.1 Example To demonstrate the differences between the rules, it is shown how BF-ABL and FF-AL differ in a small example. The environment for the example is a single shelf AS/RS with width of 50 units. Table 1 shows the example transaction details, consisting of 5 insertions and 1 retrieval. The shelf contents after the transactions are shown in Fig. 4 for the BFABL rule and in Fig. 5 for the FF-AL rule. For the BF-ABL rule, a final utilisation of 96% was achieved after transaction #6, and, for the FF-AL rule, a final utilisation of 64% was achieved after transaction #5.
Fig 3. Example situations that describe the differences between ABL- and ABS-based rules. Table 1: Transaction details in the example 3. NUMERICAL EXPERIMENTS In this chapter, numerical experiments are used to compare the rules described above in Chapter 2.
Transaction #
Details 1
Insert a carton with a width of 9.6
2
Insert a carton with a width of 5.4
3
Insert a carton with a width of 12.7
4
Retrieve the carton that was inserted in transaction #2
5
Insert a carton with a width of 9.5
6
Insert a carton with a width of 16.0
3.1 Simulation environment Excel spreadsheet/VBA-based simulation software was implemented to demonstrate and compare different slotting rules. The software takes the following parameters:
number of shelves (N)
width of shelves (W)
triangle-distributed carton width o
minimum carton width (𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 )
o
the mode of carton width (𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )
o
maximum carton width (𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 ) 2049
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Fig 6. The effect of the number of shelves on utilisation when the total width of the shelves remains the same. (WxN = 800, r = 0.9, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 10, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 25, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 40)
Fig 4. Shelf content with BF-ABL rule in the example.
Fig 5. Shelf content with FF-AL rule in the example. In this case, the transaction #6 could not be performed because there is not enough continuous space for the carton. 3.2 Experiments
Fig 7. The effect of shelf width on utilisation when there is only one shelf. (N = 1, r = 0.9, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 10, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 25, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 40) 4. DISCUSSION
Next, large-scale experiments were performed to study the differences between rules. In the experiments the default parameters were the following. The retrieval percentage was set to 0.9. The triangle distribution of the carton widths was set to be such that the minimum carton width is 10, the mode of the carton width is 25, and the maximum carton width is 40. The number of shelves and shelf width vary in the experiments. For each combination of parameters, 1000 results were generated and averaged. In the first experiment, the effect of the width of the shelves on the final utilisation was studied with the total width of the shelves, WxN, being fixed to 800 and N varying from 1 to 10. The results are shown in Fig. 6. In the second experiment, the effect of the shelf width on the final utilisation was studied. In the experiment, there was only one shelf and the width of that shelf varied from 40 to 200 in steps of 20. The results are shown in Fig. 7.
This section analyses the numerical results that are shown in Section 3. The purpose is to find out how the BF-ABL rule behaves and in which situations it should be used. The examples in Fig. 4 and Fig. 5 describe how BF-ABL outperforms FF-AL by inserting the cartons on both sides of the shelf. BF-ABL leaves a longer gap in the middle of the shelf, which can be used more easily than two gaps on the different sides of the shelf. Fig. 6 shows how the number of shelves affects the final utilisation in the case of the same total width of the shelves. Generally, the final utilisation will go down as the number of shelves increases. This was expected as the edges of shelves restrict the locations into which cartons can be inserted. Fig. 7 shows how the shelf width affects the final utilisation. If the shelf width is small, there are fewer differences between the best four rules that align the cartons on both sides of the shelf. AL-based rules, which always align cartons
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on the left-hand side of the shelf, get significantly worse when the width of the shelf decreases. When the width of the shelf is 40, all the rules obtain almost the same final utilisation. The reason for this is that in this situation often only one carton will fit into the storage. If different rules are considered, the results in both experiments show that the BF-ABL rule makes the utilisation of the AS/RS higher than the other rules, as it always gets the highest final utilisation in both cases. The difference from the worst rule is about 10% at most. This could be expected, as the BF-ABL rule uses all the ideas described in Section 2.3. BF-ABS is second in both cases and it has a final utilisation around 1% worse than BF-ABL. BF-AL obtains significantly worse solutions than the others when the width of the shelves becomes low. The order of the first fit solutions is the same as with the best fit methods: FF-ABL, FF-ABS, and FF-AL. In the case of low-shelf-width situations, FF-ABL is comparable or better than FF-AL.
Kaylan, A., & Medeiros, D. J. (1988). Analysis of storage policies for miniload AS/RS. Engineering Costs and Production Economics, 13(4), 311-318. Randell, B. (1969). A note on storage fragmentation and program segmentation. Communications of the ACM, 12(7), 365-366. Roodbergen, K. J., & Vis, I. F. (2009). A survey of literature on automated storage and retrieval systems. European Journal of Operational Research, 194(2), 343-362. Sarker, B. R., & Babu, P. S. (1995). Travel time models in automated storage/retrieval systems: a critical review. International Journal of Production Economics, 40(2), 173-184. Shore, J. E. (1975). On the external storage fragmentation produced by first-fit and best-fit allocation strategies. Communications of the ACM, 18(8), 433-440. Wilson, P. R., Johnstone, M. S., Neely, M., & Boles, D. (1995). Dynamic storage allocation: A survey and critical review. Lecture Notes in Computer Science, 986.
5. CONCLUSION The purpose of this paper was to find a slotting rule that avoids fragmentation in miniload AS/RSs. First, the paper introduced how the fragmentation occurs in AS/RS. Then the paper proposed different rules for slotting. After that, the ideas behind different rules were explored. Finally, different rules were compared in numerical experiments. The main results of the paper are the following.
The BF-ABL rule, which is the focus in the paper and which is based on best fit and aligns inserted cartons on both sides of the shelf and next to the longer carton, outperforms the other rules in the experiments. The advantage in terms of utilisation compared to the worst rule, FF-AL, is around 10% at its highest.
Especially in the case of short shelves, the rules with AL, which means always aligning cartons on the left-hand side of the gap, give clearly worse results than the other rules.
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The results described in the paper are based on limited numerical experiments. Thus, in future, more experiments could be performed on the same topic, especially with different width distributions. A mathematical analysis of different aligning strategies could also be interesting.
REFERENCES Bartholdi, J. J., & Hackman, S. T. (2014). Warehouse & Distribution Science: Release 0.96. Supply Chain and Logistics Institute. Hausman, W. H., Schwarz, L. B., & Graves, S. C. (1976). Optimal storage assignment in automatic warehousing systems. Management Science, 22(6), 629-638. Johnson, M. E., & Brandeau, M. L. (1996). Stochastic modeling for automated material handling system design and control. Transportation Science, 30(4), 330-350.
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48-3 (2015) 1973–1977 Avoiding FragmentationIFAC-PapersOnLine in Miniload Automated Storage and Retrieval Systems Avoiding Fragmentation Fragmentation in in Miniload Miniload Automated Automated Storage Storage and Retrieval Retrieval Systems Avoiding Avoiding Fragmentation in Miniload Automated Storage and and Retrieval Systems Systems Henri Tokola, Esko Niemi
Henri Tokola, Tokola, Esko Niemi Niemi Henri Esko Henri Tokola, Esko Niemi Department of Engineering Design and Production, Department of Engineering and Production, Aalto University School ofDesign Engineering, Espoo, Department of Engineering Design and Production, Department of Engineering Design and Production, Aalto University School of Engineering, Espoo, Finland (e-mail: [email protected]). Aalto University School of Engineering, Aalto University School of Engineering, Espoo, Espoo, Finland (e-mail: [email protected]). Finland Finland (e-mail: (e-mail: [email protected]). [email protected]). Abstract: This paper studies detailed slotting in miniload automated storage and retrieval systems Abstract: paperThe studies detailed slotting in have miniload automated storageshelves and retrieval retrieval systems (miniload This AS/RSs). systems that are studied a number of identical and theysystems handle Abstract: This paper studies detailed slotting in miniload automated storage and Abstract: This paperThe studies detailed slotting in have miniload automated storage and retrieval (miniload AS/RSs). systems that are are studied number of identical identical shelves and theysystems handle cartons with different widths. The purpose isstudied to find out aaadetailed slotting rule that makes utilisation high (miniload AS/RSs). The systems that have number of shelves and they handle (miniload AS/RSs). The systems that studied have number ofsmall identical shelves and theyFor handle cartons with different widths. The is to find out aaadetailed slotting rule makes utilisation high by reducing the fragmentation ofpurpose the are available storage space into andthat unusable gaps. this cartons with different widths. The purpose is to find out detailed slotting rule that makes utilisation high cartons with widths. The purpose isatodetailed find outslotting a detailed slotting rule that makes utilisation high by reducing the fragmentation of the available storage space into small unusable gaps. For this purpose, the different paper constructs and analyses rule which isand based on a best fit rule, but by reducing the fragmentation of the available storage space into small and unusable gaps. For this by reducing the fragmentation of analyses the available storage into small unusable gaps. For purpose, the paper constructs aa detailed rule which based on aa best fit rule, but which aligns the cartons beingand handled to both ends ofslotting thespace shelf and next is toand a longer carton. The rulethis is purpose, the paper constructs and analyses detailed slotting rule which is based on best fit rule, but purpose, the paper constructs and analyses a detailed slotting rule which is based on aperformance. best fit rule, but which aligns cartons being handled to both ends of the shelf and next to a longer carton. The rule is compared to the first fit rule and to different aligning options in order to validate its The which aligns cartons being handled to both ends of the shelf and next to a longer carton. The rule is which aligns the cartons beingand handled toconstructed bothaligning ends ofoptions thegive shelf and next to10% a longer carton. rule is compared to the fit rule to different in order to its The simulation results show as saving in theThe utilisation compared to the first first fitthat rulethe andrule to thus different aligningcan options in much order as to avalidate validate its performance. performance. The compared to the first fit rule and to different aligning options in order to validate its performance. The simulation results that rule thus can as aa 10% saving in when compared the first aligning. Aligning to a as longer carton instead next to a simulation resultstoshow show thatfitthe therule rulewithout thus constructed constructed can give give next as much much as 10% saving in the theofutilisation utilisation simulation results that rulewithout thus constructed can give next as much a 10% saving in theofutilisation when compared toshow the first fittherule rule aligning. Aligning Aligning to aa as longer carton instead next to to a shortercompared carton gives a 1% benefit. when to the first fit without aligning. next to longer carton instead of next when compared to the firstbenefit. fit rule without aligning. Aligning next to a longer carton instead of next to aa shorter carton gives a 1% shorter carton gives a 1% benefit. Keywords: warehouse automation, automated storage and Hosting retrievalbysystem, avoidance, © 2015,carton IFAC (International Federation of Automatic Control) Elsevier fragmentation Ltd. All rights reserved. shorter gives a 1% benefit. Keywords: automation, automated storage and retrieval system, fragmentation avoidance, simulation warehouse Keywords: warehouse automation, automated storage and retrieval system, fragmentation avoidance, Keywords: warehouse automation, automated storage and retrieval system, fragmentation avoidance, simulation simulation simulation states that most of the papers in this area deal with unit-load 1. INTRODUCTION states that thatwhich most means of the the papers papers ininthis this areaall deal with unit-load unit-load AS/RSs, AS/RSsin which cartons have the states most of area deal with 1. INTRODUCTION 1. thatwhich most of the papers area deal with unit-load AS/RSs, means AS/RSs inthis which all cartons have the same width. Unit-load AS/RSinis not relevant to our paper, In INTRODUCTION modern logistics systems, automated storage and retrieval states 1. INTRODUCTION AS/RSs, which means AS/RSs in which all cartons have the which AS/RSs in which all slot cartons have the same width. width. Unit-load AS/RS isfitnot not relevant tothat our paper, In modern modern logistics systems, automated storage and retrieval retrieval because there themeans cartons alwaysis into the remains systems (AS/RS) aresystems, commonly used. They consist of racks AS/RSs, same Unit-load AS/RS relevant to our paper, In logistics automated storage and width. isfitnot relevant tothat our paper, In modern logistics systems, automated storage and retrieval because thereUnit-load the cartons always into the slot slothave remains systems (AS/RS) are commonly used. They consist of racks racks same after retrieval. Thus, in AS/RS this study, the cartons different and a crane, whichare runs between the racks and automatically because there the cartons always fit into the that remains systems (AS/RS) commonly used. They consist of because there the cartons always fit into the slot that remains systems (AS/RS) are commonly used. They consist of racks after retrieval. Thus, in this study, the cartons have different and a crane, which runs between the racks and automatically widths. This type of in AS/RS with the variable cartons is called inserts and retrieves cartons. Such systems allow improved after retrieval. Thus, this cartons have different and which runs between the racks automatically after retrieval. Thus, this study, study, cartons have is different and aa crane, crane, whichhigh runs between the racks and and automatically widths. This type of in AS/RS with the variable cartons called inserts and retrieves cartons. Such systems allow improved miniload AS/RS. inventory control, floor space utilisation, reduced labour widths. This type of AS/RS with variable cartons is called inserts and retrieves cartons. Such systems allow improved inserts andcontrol, retrieves cartons. Suchutilisation, systems allow improved widths. This type of AS/RS with variable cartons is called miniload AS/RS. inventory high floor space reduced labour costs, flexibility, and security (Johnson and Brandeau, 1996). miniload AS/RS. inventory control, high floor utilisation, reduced labour inventory control, highminiload floor space space utilisation, reduced labour costs, flexibility, and and Brandeau, This studies AS/RS where cartons1996). have miniload AS/RS. costs, paper flexibility, andasecurity security (Johnson (Johnson and Brandeau, 1996). costs, flexibility, andasecurity (Johnson Brandeau,inserted 1996). This paper studies miniload AS/RS where different widths and, cartons are and continuously This paper studies a when miniload AS/RS where cartons cartons have have This paper studies a miniload AS/RS where cartons different widths and, when cartons continuously and retrieved storage, spaceinserted inhave the different widthsfrom and,the when cartonstheare areavailable continuously inserted different widths and,the when cartonsthe areavailable continuously inserted and retrieved from storage, space in the the storage will become fragmented. Therefore, it isspace impossible and retrieved from the storage, the available in anddoretrieved from the storage, the available the storage will become become fragmented. Therefore, it is isspace impossible to complete scheduling, and decisions have toimpossible be inmade storage will fragmented. Therefore, it 1) Initial situation storage will become fragmented. Therefore, it is impossible to do complete scheduling, and decisions have to be made online when a carton arrives into the system.have Fig. to 1 illustrates to do complete scheduling, and decisions be made 1) Initial situation situation 1) to do complete scheduling, and decisions have to be made 1) Initial Initial situation online when carton arrives into the system. 11 illustrates how gaps areaa formed in the storage. The aimFig. of the paper is online when carton arrives into the system. Fig. illustrates onlinegaps when carton into slotting the system. Fig. 1avoid illustrates how in the The of paper is to outa formed how arrives different rules howfind gaps are are formed in the storage. storage. The aim aim of the the paperthis is how gaps are formed in the storage. The aim of the paperthis is to find out how different slotting rules avoid fragmentation. to find out how different slotting rules avoid this to find out how different slotting rules avoid this fragmentation. fragmentation. In computer science that deals with storages and memory, fragmentation. In science that deals with storages and memory, thiscomputer kind of fragmentation is called external fragmentation In computer science that with storages and memory, 2) Three cartons are retrieved In computer science (1969) that deals deals with storages and memory, this kind of fragmentation is called external fragmentation (see e.g. Randell and Wilson et al. (1995)). 2) Three cartons cartons are retrieved retrieved this kind of fragmentation is called external fragmentation 2) this kind ofRandell fragmentation isand called fragmentation 2) Three Three cartons are are retrieved (see (1969) al. Fragmentation is normally by external using et defragmentation (see e.g. e.g. Randell (1969)avoided and Wilson Wilson et al. (1995)). (1995)). (see e.g. Randell (1969)avoided and Wilson et AS/RSs al. (1995)). Fragmentation is by tools. However, unlike computers, the Fragmentation is normally normally avoided by using usingindefragmentation defragmentation Fragmentation is normally avoided by using defragmentation tools. However, However,imposes unlikecosts computers, in AS/RSs AS/RSs delay the defragmentation as it will significantly tools. unlike computers, in the tools. However, unlike computers, in AS/RSs the defragmentation imposes costs ascause it will will significantly delay the current operation and costs it will thesignificantly material handling defragmentation imposes as it delay defragmentation imposes costs as it will significantly delay the current current operationoperation. and it it will will cause the material material handling handling device unnecessary Thus, defragmentation should the operation and cause the 3) Three new cartons are inserted theavoided current operation and it will cause the material handling device unnecessary operation. Thus, defragmentation should be in AS/RSs by proactively reducing fragmentation. device unnecessary operation. Thus, defragmentation should 3) Three new new cartons are are inserted 3) device unnecessary operation. Thus, defragmentation should 3) Three Three new cartons cartons are inserted inserted be avoided in AS/RSs by proactively reducing fragmentation. be avoided in AS/RSs by proactively reducing fragmentation. The literature on AS/RSs is wide as these systems have been be avoided in AS/RSs by proactively reducing fragmentation. The literature literature on AS/RSs is wide wide as as thesethe systems have been studied from on several directions since 1950s. Forbeen an The AS/RSs is these systems have The literature on AS/RSs is wide as thesesystems systems have studied from several directions since the 1950s. Forbeen an overall review of general storage and their studied from several directions since the 1950s. For an studied from several directions since to thelook 1950s. Fortheir an overall review of general storage systems mathematical analysis, it is worthwhile atand Bartholdi overall review of general storage systems and their 4) The storage has small unusable gaps (i.e. storage is overall review of general storage systems and their mathematical is look at Bartholdi and Hackmananalysis, (2014).it Forworthwhile a reviewto of AS/RSs, see mathematical analysis, it is worthwhile to look at Bartholdi 4) The The storage has has small unusable gaps gaps (i.e. storage storage is fragmented) mathematical analysis, it is worthwhile to look at Bartholdi 4) unusable and For aa review see Roodbergen and (2014). Iris (2009). state thatof publications 4) The storage storage has small small unusable gaps (i.e. (i.e. storage is is and Hackman Hackman (2014). ForThey review oftheAS/RSs, AS/RSs, see fragmented) fragmented) and Hackman (2014). For a review of AS/RSs, see Roodbergen and Iris (2009). They state that the publications in the area often focus on the design of AR/RSs or study Fig. 1. Example of how fragmentation fragmented) occurs on a shelf when Roodbergen and Iris (2009). They state that the publications Roodbergen and They state of thatAR/RSs the publications Example of on aa shelf in the area focus on the design or study existing cartons arefragmentation retrieved and occurs new ones AS/RSs as aoften partIris of (2009). general warehousing. Their paper also Fig. Fig. 1. 1. Example of how how fragmentation occurs on inserted shelf when when in the area often focus on the design of AR/RSs or Fig. 1. Example of how fragmentation occurs on a shelf when in the area often focus on the design of AR/RSs or study study existing cartons are retrieved and new ones inserted AS/RSs as a part of general warehousing. Their paper also existing cartons are retrieved and new ones inserted AS/RSs as a part of general warehousing. Their paper also existing cartons are retrieved and new ones inserted AS/RSs as a part of general warehousing. Their paper also2047 Copyright © 2015 IFAC 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2015 responsibility IFAC 2047Control. Peer review© of International Federation of Automatic Copyright 2015 IFAC 2047 Copyright © 2015 IFAC 2047 10.1016/j.ifacol.2015.06.377
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Few of the AS/RS-related papers focus on miniload AS/RS and none, to our knowledge, considers both the fragmentation problem and detailed slotting. As stated by Bartholdi and Hackman (2014), detailed slotting without retrievals can be abstracted as a one-dimensional bin-packing problem: given n numbers, arrange them into as few groups as possible such that each group sums to no more than one. However, in our case there are retrievals too, which complicates the problem. The problem has to be studied in online settings, which rarely appear in the literature. However, there are several papers that study slotting (or assignment) in unit-load AS/RS. Typically, as done e.g. by Kaylan and Medeiros (1988), the papers study zoning in such a way that different kinds of zoning areas are compared. Clearly, the zones near the input/output locations are more valuable than others, but, in our paper, we do not consider zoning or the input/output location. Related to our paper, Hausman et al. (1976) study the closest open location rule, in which the cartons are put into the first empty slot. They study the rule in a unit-load system where fragmentation does not occur. This is similar to the first fit rule, which is used in this paper as a comparison rule.
Therefore a gap is found using either best fit or first fit. The next step is to decide to which side of the gap the carton is aligned. For that, there are three options:
All the combinations of the above fitting and aligning rules are studied and they are the following:
To summarise the above introduction, fragmentation seems to be studied in the computer literature, but not in the case of miniload AS/RSs, as is done here. This paper proposes new slotting rules that avoid fragmentation by cleverly aligning cartons to existing gaps.
The rest of the paper is organised as follows. Next, Section 2 presents the slotting rules that are used in the numerical experiments of Section 3. Section 4 discusses the results of the experiments and, finally, Section 5 draws conclusions.
2. DETAILED SLOTTING RULES IN AS/RS 2.1 Miniload AS/RS system
This paper studies an AS/RS with N identical W-width shelves. Cartons are inserted to and retrieved from the storage. As the studying of slotting into shelves is the aim of the paper, crane movement and input/output locations are not considered in this paper. When cartons are inserted, a decision regarding the slotting position has to be made. Thus, the purpose of the paper is to find out a good detailed slotting rule that makes utilisation high by reducing the fragmentation of the storage into small and unusable gaps. Utilisation is defined here as the ratio of the total width of cartons to the total width of shelves. 2.2 Slotting rules In order to find a good slotting rule, this paper compares seven slotting rules. The rules are based on either best fit (BF) or first fit (FF). Best fit allocates cartons to the smallest free gap into which the carton fits. In first fit, the shelves are gone through in order instead, and the carton is put into the first gap that has enough space.
AL – align left ABL – aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the longer carton if there are cartons on both sides of the shelf. ABS – aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the shorter carton if there are cartons on both sides of the shelf.
BF-AL: best fit that always aligns next to the carton on the left-hand side; BF-ABL: best fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the longer carton if there are cartons on both sides of the shelf. This rule is the focus of the present paper; BF-ABS: best fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the shorter carton if there are cartons on both sides of the shelf; FF-AL: first fit that always aligns cartons next to the carton on the left-hand side; FF-ABL: first fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the longer carton if there are cartons on both sides of the shelf; FF-ABS: first fit that aligns cartons on both sides of the shelf in such a way that the carton is aligned next to the shorter carton if there are cartons on both sides of the shelf.
The different ideas behind the above rules, especially those that are behind the BF-ABL rule, are discussed below. 2.3 Ideas behind slotting rules The ideas behind the slotting rules are discussed here. This is done in three steps. First, the differences between best fit and first fit are discussed. Second, the advantages of aligning on both sides are described. Third, the advantages of aligning next to the longer carton are discussed. The comparison between best fit and first fit is not easy. In general, it is expected that best fit will outperform first fit, as it fills smaller gaps first and saves larger gaps for future use. However, as discovered experimentally by Shore (1975), first fit outperforms best fit with exponentially distributed widths, but not with normally distributed widths. When first fit outperforms best fit, the reason that is hypothesised for this is
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that first fit allocates the shelf from one end and saves the other end for longer cartons. Still, it looks as if best fit should outperform first fit in normal situations, and thus it was selected as a base rule in the BF-ABL rule studied here. By inserting the cartons on both sides of the shelves we avoid the steps shown in Fig. 2, where aligning left and retrieving creates two gaps but aligning with both sides and retrieving creates one larger gap. Aligning with both sides is realised in the rules with the ABL or ABS suffixes.
retrieval percentage (r). This is the probability between 0 and 1 that a random carton is retrieved after the insertion of a carton.
Triangle distribution was selected for the carton width distribution because, according to the data gained from one industrial company, in reality the carton widths seem to closely follow triangle distribution. When the simulation software is run, it follows the following steps:
a) Align left b) Align both sides (AL) (ABL and ABS) Fig. 2. Example of the situation where aligning both sides avoids fragmentation into two gaps. Aligning next to the longer carton is likely to reduce the fragmentation to small gaps in the way that is shown in Fig. 3. The idea is the following. When a carton is aligned next to the longer carton, the retrieval of the shorter carton does not cause a small gap to be created. This rule is applied in the rules with the ABL suffix.
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1.
initialise the storage to be empty at the beginning;
2.
if a random value between 0 and 1 < the retrieval percentage r, then retrieve and remove one random carton from the storage;
3.
generate a new carton and fit the carton into the storage. Generate a carton from triangle distribution. Store the carton in the slot defined by slotting algorithm and aligning type;
4.
repeat steps 2-4 until the next new carton does not fit into the storage;
5.
record the final utilisation of the storage at the end.
This simulation software is used next in an example and then in large-scale experiments. 3.1 Example To demonstrate the differences between the rules, it is shown how BF-ABL and FF-AL differ in a small example. The environment for the example is a single shelf AS/RS with width of 50 units. Table 1 shows the example transaction details, consisting of 5 insertions and 1 retrieval. The shelf contents after the transactions are shown in Fig. 4 for the BFABL rule and in Fig. 5 for the FF-AL rule. For the BF-ABL rule, a final utilisation of 96% was achieved after transaction #6, and, for the FF-AL rule, a final utilisation of 64% was achieved after transaction #5.
Fig 3. Example situations that describe the differences between ABL- and ABS-based rules. Table 1: Transaction details in the example 3. NUMERICAL EXPERIMENTS In this chapter, numerical experiments are used to compare the rules described above in Chapter 2.
Transaction #
Details 1
Insert a carton with a width of 9.6
2
Insert a carton with a width of 5.4
3
Insert a carton with a width of 12.7
4
Retrieve the carton that was inserted in transaction #2
5
Insert a carton with a width of 9.5
6
Insert a carton with a width of 16.0
3.1 Simulation environment Excel spreadsheet/VBA-based simulation software was implemented to demonstrate and compare different slotting rules. The software takes the following parameters:
number of shelves (N)
width of shelves (W)
triangle-distributed carton width o
minimum carton width (𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 )
o
the mode of carton width (𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )
o
maximum carton width (𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 ) 2049
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Fig 6. The effect of the number of shelves on utilisation when the total width of the shelves remains the same. (WxN = 800, r = 0.9, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 10, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 25, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 40)
Fig 4. Shelf content with BF-ABL rule in the example.
Fig 5. Shelf content with FF-AL rule in the example. In this case, the transaction #6 could not be performed because there is not enough continuous space for the carton. 3.2 Experiments
Fig 7. The effect of shelf width on utilisation when there is only one shelf. (N = 1, r = 0.9, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 10, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 25, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 40) 4. DISCUSSION
Next, large-scale experiments were performed to study the differences between rules. In the experiments the default parameters were the following. The retrieval percentage was set to 0.9. The triangle distribution of the carton widths was set to be such that the minimum carton width is 10, the mode of the carton width is 25, and the maximum carton width is 40. The number of shelves and shelf width vary in the experiments. For each combination of parameters, 1000 results were generated and averaged. In the first experiment, the effect of the width of the shelves on the final utilisation was studied with the total width of the shelves, WxN, being fixed to 800 and N varying from 1 to 10. The results are shown in Fig. 6. In the second experiment, the effect of the shelf width on the final utilisation was studied. In the experiment, there was only one shelf and the width of that shelf varied from 40 to 200 in steps of 20. The results are shown in Fig. 7.
This section analyses the numerical results that are shown in Section 3. The purpose is to find out how the BF-ABL rule behaves and in which situations it should be used. The examples in Fig. 4 and Fig. 5 describe how BF-ABL outperforms FF-AL by inserting the cartons on both sides of the shelf. BF-ABL leaves a longer gap in the middle of the shelf, which can be used more easily than two gaps on the different sides of the shelf. Fig. 6 shows how the number of shelves affects the final utilisation in the case of the same total width of the shelves. Generally, the final utilisation will go down as the number of shelves increases. This was expected as the edges of shelves restrict the locations into which cartons can be inserted. Fig. 7 shows how the shelf width affects the final utilisation. If the shelf width is small, there are fewer differences between the best four rules that align the cartons on both sides of the shelf. AL-based rules, which always align cartons
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on the left-hand side of the shelf, get significantly worse when the width of the shelf decreases. When the width of the shelf is 40, all the rules obtain almost the same final utilisation. The reason for this is that in this situation often only one carton will fit into the storage. If different rules are considered, the results in both experiments show that the BF-ABL rule makes the utilisation of the AS/RS higher than the other rules, as it always gets the highest final utilisation in both cases. The difference from the worst rule is about 10% at most. This could be expected, as the BF-ABL rule uses all the ideas described in Section 2.3. BF-ABS is second in both cases and it has a final utilisation around 1% worse than BF-ABL. BF-AL obtains significantly worse solutions than the others when the width of the shelves becomes low. The order of the first fit solutions is the same as with the best fit methods: FF-ABL, FF-ABS, and FF-AL. In the case of low-shelf-width situations, FF-ABL is comparable or better than FF-AL.
Kaylan, A., & Medeiros, D. J. (1988). Analysis of storage policies for miniload AS/RS. Engineering Costs and Production Economics, 13(4), 311-318. Randell, B. (1969). A note on storage fragmentation and program segmentation. Communications of the ACM, 12(7), 365-366. Roodbergen, K. J., & Vis, I. F. (2009). A survey of literature on automated storage and retrieval systems. European Journal of Operational Research, 194(2), 343-362. Sarker, B. R., & Babu, P. S. (1995). Travel time models in automated storage/retrieval systems: a critical review. International Journal of Production Economics, 40(2), 173-184. Shore, J. E. (1975). On the external storage fragmentation produced by first-fit and best-fit allocation strategies. Communications of the ACM, 18(8), 433-440. Wilson, P. R., Johnstone, M. S., Neely, M., & Boles, D. (1995). Dynamic storage allocation: A survey and critical review. Lecture Notes in Computer Science, 986.
5. CONCLUSION The purpose of this paper was to find a slotting rule that avoids fragmentation in miniload AS/RSs. First, the paper introduced how the fragmentation occurs in AS/RS. Then the paper proposed different rules for slotting. After that, the ideas behind different rules were explored. Finally, different rules were compared in numerical experiments. The main results of the paper are the following.
The BF-ABL rule, which is the focus in the paper and which is based on best fit and aligns inserted cartons on both sides of the shelf and next to the longer carton, outperforms the other rules in the experiments. The advantage in terms of utilisation compared to the worst rule, FF-AL, is around 10% at its highest.
Especially in the case of short shelves, the rules with AL, which means always aligning cartons on the left-hand side of the gap, give clearly worse results than the other rules.
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The results described in the paper are based on limited numerical experiments. Thus, in future, more experiments could be performed on the same topic, especially with different width distributions. A mathematical analysis of different aligning strategies could also be interesting.
REFERENCES Bartholdi, J. J., & Hackman, S. T. (2014). Warehouse & Distribution Science: Release 0.96. Supply Chain and Logistics Institute. Hausman, W. H., Schwarz, L. B., & Graves, S. C. (1976). Optimal storage assignment in automatic warehousing systems. Management Science, 22(6), 629-638. Johnson, M. E., & Brandeau, M. L. (1996). Stochastic modeling for automated material handling system design and control. Transportation Science, 30(4), 330-350.
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