1996 Multi-Shuttle Automated Storage Retrieval Systems

Multi-shuttle automated storage/retrieval systems

RUSSELL D. MELLER1 and ANAN MUNGWATTANA2

1Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36849-5346, USA2Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0118,USA

Received September 1995 and accepted August 1996

Multi-shuttle automated storage/retrieval systems have been developed for use in factories and distribution centers because theyare more e�cient than single-shuttle systems (owing to less empty travel). This improved e�ciency results in more agile support(¯exible response, less waiting time, etc.) for the production system the storage/retrieval system serves. In this paper we developanalytical models to estimate the throughput in multi-shuttle systems. Throughput improvements greater than 100% are illustratedwhen triple-shuttle systems are compared with single-shuttle systems.

1. Introduction

Automated storage and retrieval systems (AS/RSs) aremajor material handling support systems that have beenwidely used in automated factories and distribution cen-ters. The basic components of the type of AS/RS weconsider are storage racks, storage/retrieval (S/R) ma-chines, input/output (I/O) stations, and interface con-veyors [1]. AS/RSs are operated and controlled by acomputer that can collect important information andprovide a high degree of inventory visibility, which canthen be used by manufacturing, distribution, accounting,sales, etc. Consequently, owing to the computerizedhandling, fewer workers are required, which results inlabor cost savings, and owing to computerized account-ing, fewer errors and improved material ¯ow and in-ventory control result. However, even with all thesepractical advantages, owing to their high initial capitalinvestment and inadequate throughput performance,these systems have been used only to a limited extent inpractice.

Let us consider the material control system require-ments in today's agile manufacturing climate. The in-creased emphasis placed on the rapid con®guration ofproduction systems, as well as the continuing need for¯exibility in production scheduling, necessitates a mate-rial control system that provides high material visibilityand accuracy of information. Also, consider the impactthat electronic data interchange (EDI) information and`just-in-time' (JIT) delivery schedules are having on dis-tribution centers. No longer are shipments packed weeksahead and delivered to a customer's warehouse in fulltruckloads. Now it is much more common to pack same-day shipments, ship directly to the point-of-need in the

customer's company, and to do so in less-than-truckloadshipments. Thus, the practical advantages of an AS/RSlisted above seem to be required even more today than inthe past.

Because throughput capacity is a concern with AS/RSs,multi-shuttle AS/RSs have been developed to increase thethroughput capacity of a system. See Fig. 1 for an ex-ample of a twin-shuttle S/R machine. The throughputcapacity of an AS/RS increases, in general, as the numberof shuttles increases because, correspondingly, theamount of empty travel decreases. Of course there is adiminishing return on capacity for each additional shut-tle, and the capital investment of the system also increasesas the number of shuttles increase. Consequently, al-though there are twin- and triple-shuttle systems cur-rently in use (e.g., the Russell Corporation, a majormanufacturer of activewear in the United States, uses atriple-shuttle system in its Alexander City distributioncenter and a twin-shuttle system in its Marianna dis-tribution center), it is believed that there are no systems inpractice with more than three shuttles (K.B. Brewster,Eskay Corporation, personal communication; R. Ward,Director, Material Handling Industry of America, perso-nal communication).

Thus, to meet the requirements of agile manufacturingand EDI-based distribution, S/R systems must increasetheir throughput and responsiveness. In this paper weillustrate how multi-shuttle systems may be used toachieve this objective. In the next section we review theresearch literature on AS/RSs. In Section 3 we developour analytical models for various multi-shuttle AS/RScommand cycles under heuristic operating policies andillustrate the accuracy of our models by comparing theirresults with Monte Carlo simulation results. Finally, in

0740-817X Ó 1997 ``IIE''

IIE Transactions (1997) 29, 925±938

Section 4 we present our concluding remarks and anoutline for future research. Note that we concentrate oure�orts on triple-shuttle systems in this paper. The inter-ested reader is directed to [2] for the details on twin-shuttle models (that paper also presents an application ofthe triple-shuttle model developed in this paper).

2. Literature review

Single-shuttle systems can perform up to one storage andone retrieval in a cycle, which is called a dual-command(DC) cycle. In a twin-shuttle system the S/R machine canperform up to two storages and two retrievals in a cycle,which is called a quadruple-command (QC) cycle. Like-wise, triple-shuttle systems can perform up to three stora-ges and three retrievals in a cycle, which is called a sextuple-command (STC) cycle. Fig. 2 illustrates an STC cycle tour.

It is typical for the S/R machine to visit all of the storagelocations before visiting the retrieval locations [3].

Bozer and White [1] extended [3] and [4] by developinganalytical models for expected single- and dual-commandcycles in racks of general shapes. They also consideredalternative I/O locations and various dwell-point strate-gies for the S/R machine.

The above models are based on the assumption thatstorage and retrieval requests are processed with a ®rst-come-®rst-served (FCFS) policy. Han et al. [5] showedthat throughput can be improved by relaxing the FCFSassumption for retrieval sequencing and instead use anearest neighbor (NN) policy in an attempt to reduce thetravel-between times (storage requests are still assumed tobe processed FCFS because most AS/RSs are interfacedwith an FCFS input conveyor). Han et al. [5] developedan analytical model for the NN policy to estimate itsperformance, and used Monte Carlo simulation formodel validation. Their results indicate that the NNpolicy with a set of 15±20 retrievals and one open locationreduces travel-between times on average by 60% relativeto FCFS. A 60% reduction in travel-between times yieldsa 16% decrease in travel time (or a 12% increase inthroughput once P/D times are accounted for) [5]. Eynanand Rosenblatt [6] extended [5] by applying the NNpolicy to the class-based warehouse. Recently, Keserlaand Peters [7] presented a Monte Carlo simulation studyof a twin-shuttle AS/RS operating under various oper-ating policies. Their study indicates that there is the po-tential for a 45% increase in throughput if a twin-shuttlesystem is used in place of a single-shuttle system. In thispaper we develop analytical models for multi-shuttle AS/RSs (concentrating on triple-shuttle systems). We presentour models in the next section.

3. Multi-shuttle analytical models

In developing our multi-shuttle analytical models, weintegrate our results with the previous results of twosingle-shuttle models: the FCFS model of Bozer andWhite [1] and the NN model of Han et al. [5]. The as-sumptions and notation from these two models are usedthroughout this research and include the following:

1. The rack is considered to be a continuous rectan-gular pick face with length L and height H . The I/O sta-tion is located at the lower left-hand corner.

2. Randomized storage is used and the rack is ap-proximately 100% utilized. Therefore, any point withinthe pick face is equally likely to be selected for a retrievalor be available for a storage.

3. The S/R machine travels simultaneously in the hor-izontal and vertical directions. In calculating the traveltime, constant velocities sh and sv, are used for horizontaland vertical travel, respectively.

Fig. 1. A twin-shuttle S/R machine.

Fig. 2. A sextuple-command cycle tour.

926 Meller and Mungwattana

4. Pick-up and deposit (P/D) times associated with loadhandling are ignored because the P/D time is generallyindependent of the rack shape and the travel velocity ofthe S/R machine.

5. `Creeping' time (the time to move the S/R machineover one opening) is ignored because it is small comparedwith travel-between time.

6. An FCFS operating policy is implemented.

Bozer and White [1] show how Assumptions 1 and 2can be relaxed. The travel metric of Assumption 3 holdsfor any system and acceleration/deceleration e�ects canbe accounted for by approximations (see [8,9]; [10] for anapplication). Assumptions 4 and 5 can be addressed byadding the appropriate factors to the expected travel time(see [2] for an application of our model to the RussellCorporation's Alexander City Distribution Center AS/RS). Finally, Assumption 6 will be addressed later in thissection.

Using these system parameters the expected time usingFCFS for one-way travel (SA), single-command (SC)travel, travel-between (TB) travel and dual-command(DC) travel is determined as follows:

1. Compute the dimensions of the rack in time: i.e.,th � L=sh and tv � H=sv.

2. Let T and b be the scaling factor and the shape factorfor the rack, respectively, where T � max�th; tv� andb � min�th=T ; tv=T �.

3. Compute the normalized travel times for the b� 1rack as follows:

E�SA� � �12� 1

6b2�; �1�

E�SC� � �1� 13

b2�; �2�E�TB� � �1

3� 1

6b2 ÿ 1

30b3�; �3�

E�DC� � �43� 12 b2 ÿ 1

30 b3�: �4�Note that the associated time-scaled versions of (1)±(4)are obtained by multiplying through by the scalingfactor T .

3.1. Multi-shuttle FCFS results

If the storages and retrievals are sequenced sequentially(i.e., all storages before the retrievals) and strictly FCFS,the expected quadruple-command (QC) cycle time isequivalent to the expected time for a single-commandcycle and three travel-between times. That is,

E�QCFCFS� � E�SC� � 3E�TB�: �5�Likewise, for a sextuple-command (STC) cycle in a triple-shuttle system,

E�STCFCFS� � E�SC� � 5E�TB�: �6�To compare the throughput of systems equitably with

di�erent numbers of shuttles we need to compare cycletimes carefully. Assume that the single-shuttle system

performs DC cycles only, the twin-shuttle system per-forms QC cycles only, and the triple-shuttle system per-forms STC cycles only. Therefore, the average travel timeof performing a storage and a retrieval in a single-shuttlesystem is equal to E(DC), the average travel time ofperforming a storage and a retrieval in a twin-shuttlesystem is equal to 1

2E(QC), and the average travel time of

performing a storage and a retrieval in a triple-shuttlesystem is equal to 1

3E(STC). The following observation

(its proof is presented in Appendix A) indicates that in-creasing the number of shuttles increases the e�ciency ofthe system.

Observation 1. Using FCFS results, 13

E(STC)< 12

E(QC)< E(DC), for all values of b.

In Table 1 we present analytical and Monte Carlo si-mulation results (denoted by a hat over the cycle time ofinterest) for the expected time to store and retrieve oneunit load under the three types of shuttle system. Thesimulation results obviously correspond very closely tothe analytical results. Note that the Monte Carlo resultshere (and in all future references to Monte Carlo resultsin this paper) are the average over 32 000 randomlygenerated cycles in a continuously represented b� 1normalized rack. Averaging over the four values of b, wesee that expected cycle time is reduced by 25% or 33% byusing a twin-shuttle (QC) or triple-shuttle system (STC)over a single-shuttle system (DC), respectively. However,further improvements can be achieved by using a mod-i®ed strategy of storing and retrieving the unit loads, aspresented in the next section.

3.2. Improving multi-shuttle system performance

To increase the e�ciency of twin- and triple-shuttle sys-tems, we use an improved strategy for storing and re-trieving the unit loads. For example, a STC cycle can bemodi®ed and signi®cantly improved by performing thesecond storage at the same location as the ®rst retrievaland the third storage at the same location as the secondretrieval) as shown in Fig. 3 (this concept for twin-shut-tles was originally presented to the authors by Y. A.Bozer (personal communication) and later seen in [7]),

Table 1. Analytical and Monte Carlo simulation travel times

Method b � 0.25 b � 0.50 b � 0.75 b � 1.0

E(DC) 1.3641 1.4542 1.6005 1.8000EÃ (DC) 1.3694 1.4596 1.6054 1.804312E(QC) 1.0253 1.0979 1.2133 1.3667

12

E(QC) 1.0268 1.0998 1.2153 1.368913E(STC) 0.9123 0.9792 1.0842 1.2222

13EÃ (STC) 0.9148 0.9807 1.0867 1.2252

Multi-shuttle automated storage/retrieval systems 927

where the broken lines indicate the STC cycle withoutsuch a modi®cation. In a twin-shuttle system we refer tothe improved cycle as a modi®ed quadruple-command(MQC) cycle, which under the assumption of FCFS-se-quencing of storage and retrieval locations has an ex-pected value equivalent to a single-command cycle andtwo travel-between times, thus eliminating a travel-be-tween time. In a triple-shuttle system the correspondingcycle is called a modi®ed sextuple-command (MSC) cycle,which under the assumption of FCFS-sequencing ofstorage and retrieval locations has an expected valueequivalent to a single-command cycle and three travel-between times, thus eliminating two travel-between times.Using the modi®ed quadruple-command or sextuple-command expected travel times, the average time to storeand retrieve a unit load decreases by 34% and 49%, re-spectively, compared with dual-command cycles (refer to[2] for details).

Because we have multiple retrieval and storage requestsin a multi-shuttle system, we may have the opportunity toincrease throughput by sequencing the operations withina cycle (as opposed to the FCFS sequencing considered sofar). Note that with MQC or MSC cycles, because onlyone open location is required for each command cycle,the optimum ordering of storage and retrievals is a verysmall Traveling-Salesman Problem (TSP) [11]. However,the problem we are considering is more di�cult than astandard TSP because we also have to choose the openlocation from m open locations to minimize the traveltour (essentially m separate TSPs). Obviously, when thenumber of open locations increases, the complexity of theproblem increases as well. Also note that for QC and STCcycles, because there are multiple storage locations tochoose from m open locations, determining the optimumtour is more di�cult still.

Later in this section we discuss operating policies toimprove multi-shuttle systems based on heuristically se-quencing the storages and retrievals. Note that for brevityand for implementation issues we do not present anythingother than very simple heuristic policies (see [7] and [12]for more complex heuristics). We now derive the neces-sary relationships for our analytical models.

3.3. Analytical models for expected travel time estimation

In this section we develop analytical models and deriveexpressions for the expected AS/RS machine travel timebetween several locations. We then use the derived ex-pressions to estimate the expected travel time in multi-shuttle systems that are implemented with the heuristicoperating policies proposed in the next section.

The additional notation that will be used throughoutthis model follows:

km�z��Km�z�� p.d.f. (c.d.f.) of the smallest one-waytravel time from the I/O station to oneof the m open locations;

E(SWm� � the expected smallest one-way traveltime from the I/O station to one of them open locations;

lm�z��Lm�z�� p.d.f. (c.d.f.) of the largest one-waytravel time from the I/O station to oneof the m open locations;

E(SLm� � the expected largest one-way travel timefrom the I/O station to one of the mopen locations;

E(TBLm� � the expected largest travel-between timebetween one point and m locations.

First, we derive expressions for E(SWm� and E(SLm�. Inour analysis we vary the number of open locations, m,between 1 and 10 (given the results presented in [5] andour observations at the Russell Corporation, this seemssu�cient). To do so, we use order statistics from [13] andthe p.d.f. (g�z�) and c.d.f. (G�z�) of travel time from theI/O station to any point in the rack from [1]. We thenderive an expression for E(TBLm�. To do so we use thep.d.f.(f �z�) and c.d.f.(F �z�) of travel-between time from [1].

The one-way travel times between the I/O station andm open locations are z1, z2, z3, . . ., zm, each distributedwith p.d.f. g�z�. Let

z�1� � the smallest one-way travel time of (z1, z2, z3,. . ., zm);

z�2� � the second smallest one-way travel time of(z1, z2, z3, . . ., zm);

. . . � . . . ;z�m� � the largest one-way travel time of (z1, z2, z3,. . ., zm).

The quantities z�1�; z�2�; . . . ; z�m� are random variables andare called order statistics of the sample. When using thenearest neighbor policy, we choose the location with thesmallest one-way travel time, SWm, which we de®ne as

SWm � min�z1; z2; z3; . . . ; zm� � z�1�: �7�We want to ®nd the expression for Km�z�; where

Km�z� � Pr�z�1� � z� � 1ÿ Pr�z�1� > z�: �8�Since

Pr�z�1� > z� � Pr�z1; z2; z3; . . . ; zm are all > z�; �9�

Fig. 3. A modi®ed sextuple-command cycle tour.


then,

Km�z� � 1ÿ �1ÿ G�z��m 0 < z � 1: �10�The probability density function, km�z� � K 0m�z�, is givenby

km�z� � m�1ÿ G�z��mÿ1g�z� 0 < z � 1: �11�Therefore the expected smallest one-way travel time,E(SWm�, is expressed as

E�SWm� �Z 1

0

zm�1ÿ G�z��mÿ1g�z�dz: �12�

For example, letting m = 2 in (12) yields

E�SW2� � 1

3b2 ÿ 2

15b3 � 1

3

� �: �13�

Note that it would require a great deal of e�ort toobtain a closed-form expression for (12) whenm � 1; 2; . . . ; 10. Therefore, we evaluate (12) numericallywhen m � 1; 2; . . . ; 10 with a variety of b values as shownin Table 2. (If numerical methods, such as MATLAB [14],are not available, such values can be used with linearinterpolation to estimate the value for a particular b va-lue.) Note that E(SW1� is equivalent to the expected one-way travel time, E(SA).

We also derive an expression for the expected largestone-way travel time for use as an upper bound later. Asde®ned, z�m� is the largest travel time of the sample,z1; z2; z3; . . . ; zm, each distributed with p.d.f. g�z�. Thec.d.f. of z�m� is given by

Pr�z�m� < z� � Pr�z1; z2; z3; . . . ; zm are all � z� � �G�z��m:�14�

Thus

Lm�z� � �G�z��m: �15�The p.d.f. of the largest one-way travel time, lm�z�, is

lm�z� � mGmÿ1�z�g�z�: �16�Therefore the expected largest one-way travel time,E(SLm�, is

E�SLm� �Z 1

0

zm�G�z��mÿ1g�z�dz: �17�

For instance, letting m = 2 in (17) yields

E�SL2� � 2

15b3 � 1

6

� �: �18�

We evaluate (17) numerically with di�erent values of mand b as shown in Table 3.

Finally, we also develop a model for E(TBLm�. From(17), we substitute F �z� and f �z� in place of G�z� and g�z�,respectively. Therefore

E�TBLm� �Z 1

0

zm�F �z��mÿ1f �z�dz: �19�

We evaluate (19) numerically with di�erent values of mand b as shown in Table 4. Note that the correspondingexpected smallest travel-between time between a pointand m locations, E(TBm�, was previously derived by Hanet al. [5] as

E�TBm� �Z 1

0

zm�1ÿ F �z��mÿ1f �z�dz; �20�

which is evaluated numerically for various values of mand b in [5].

Table 2. The expected smallest one-way travel time, E(SWm)

m b � 0.25 b � 0.50 b � 0.75 b � 1.0

1 0.5104 0.5417 0.5938 0.66672 0.3521 0.4000 0.4646 0.53333 0.2754 0.3317 0.3964 0.45714 0.2308 0.2907 0.3520 0.40635 0.2017 0.2626 0.3199 0.36946 0.1813 0.2417 0.2953 0.34107 0.1661 0.2253 0.2756 0.31838 0.1544 0.2119 0.2594 0.29959 0.1450 0.2007 0.2458 0.2838

10 0.1372 0.1991 0.2341 0.2703

Table 4. The expected largest travel-between time, E(TBLm)

m b � 0.25 b � 0.50 b � 0.75 b � 1.0

1 0.3432 0.3708 0.4130 0.46672 0.4698 0.4883 0.5296 0.59373 0.5439 0.5554 0.5916 0.65904 0.5940 0.6011 0.6321 0.70055 0.6307 0.6352 0.6615 0.72976 0.6590 0.6619 0.6843 0.75187 0.6818 0.6836 0.7057 0.76938 0.7005 0.7016 0.7180 0.78359 0.7162 0.7170 0.7310 0.7953

10 0.7297 0.7303 0.7423 0.8005

Table 3. The expected largest one-way travel time, E(SLm)

m b � 0.25 b � 0.50 b � 0.75 b � 1.0

1 0.5104 0.5417 0.5938 0.66672 0.6687 0.6833 0.7229 0.80003 0.7504 0.7567 0.7839 0.85714 0.8001 0.8028 0.8211 0.88895 0.8334 0.8345 0.8468 0.90916 0.8571 0.8577 0.8659 0.92317 0.8750 0.8752 0.8808 0.93338 0.8889 0.8890 0.8928 0.94219 0.9000 0.9000 0.9027 0.9474

10 0.9091 0.9091 0.9109 0.9524


In the next section we use the values from Table 2 toestimate the total travel time under our operating heur-istics. In addition, the values from Tables 3 and 4 are inthe re®nement of our analytical models.

3.4. Expected travel time estimation in a triple-shuttlesystem

In this section we present a model of the expected traveltime in a triple-shuttle system operating under variousstrategies. The results for twin-shuttle systems operatingunder the same strategies are presented in [2].

3.4.1. Estimation of travel time under STC and theNN policy

Suppose that the NN policy is implemented with the STCcycle (STCNN). We can estimate the expected travel timeof such a cycle as follows:

Step 1. With three open locations (e.g., s1; s2 and s3), theS/R machine moves from the I/O station to theclosest location, say, s1. The expected travel timefrom the I/O station to s1 is equal to the expectedsmallest one-way travel time with 3 open loca-tions; that is, using m � 3 in (12),

E�SWs1I=O� � E�SW�m�3��

� 1

4� 1

2b2 ÿ 2

5b3 � 3

28b4

� �:

Step 2. After storing the ®rst unit load, the S/R machinemoves to the second storage location, which, say,is s2 because it is closer to s1 than s3. The ex-pected travel time between s1 and s2 is estimatedas the expected smallest travel-between time from[5] with m � 2; that is using m � 2 in (20),

E�TBs2s1� � E�TB�mÿ1�2��

� 1

3b2 ÿ 31

105b3 � 5

42b4 ÿ 11

630b5 � 1

5

� �:

Step 3. The S/R machine then moves to the last storagelocation, s3. The expected travel time between s2and s3 is estimated as a single-shuttle expectedtravel-between time [1]; that is, using (3),

E�TBs3s2� � E�TB�mÿ2�1�� 1

3� 1

6b2 ÿ 1

30b3

� �:

Step 4. From s3, the S/R machine moves to the closest ofthe three retrieval points (e.g., r1; r2 and r3), say,r1. The expected travel time between s3 and r1 canbe expressed by evaluating (20) with m � 3; that is,

E�TBr1s3� � E�TB3� � 3

Z 1

0

z�1ÿ F �z��2f �z�dz:

This expression is then evaluated with numerical methods(or approximated with linear interpolation [5]).

Step 5. From the ®rst retrieval location, r1, the S/Rmachine moves to the closest of the last two re-trieval locations, say, r2. The expected travel timebetween r1 and r2 is estimated as the expectedtravel time between s1 and s2, which is theminimum of two expected travel-between times;that is,

E�TBr2r1� � E�TB2�

� 1

3b2 ÿ 31

105b3 � 5

42b4 ÿ 11

630b5 � 1

5

� �:

Step 6. The S/R machine then moves to the last retrievallocation, r3. The expected travel time between r2and r3 is estimated to equal the expected traveltime between s2 and s3; that is,

E�TBr3r2� � E�TB� � 1

3� 1

6b2 ÿ 1

30b3

� �:

Step 7. Finally, the S/R machine returns to the I/O sta-tion. The expected travel time from r3 to the I/Ostation is estimated to equal the expected one-way travel time; that is,

E�SAI=Or3 � � E�SA� � 1

2� 1

6b2

� �:

Therefore the expected total travel time of STC with theNN policy is estimated as:

E�STCNN� � E�SW�m�3�� E�TB�mÿ1�2�� E�TB�mÿ2�1�� E�TB3� � E�TB2�� E�TB� � E�SA�: �21�

The above analytical model is for three open locations;however, it can be generalized to m open locations by re-placing the current value of m � 3 in Steps 1, 2, and 3 withthe appropriate value of m. We evaluate (21) numericallywith di�erent values of m and b as shown in Table 5.

3.4.2. Estimation of travel time under MSC and theNN policy

Suppose that the NN policy is implemented with theMSC (MSCNN). We can estimate the expected travel timeof such a cycle as follows:

Table 5. The expected travel time of STCNNm

m b � 0.25 b � 0.50 b � 0.75 b � 1.0

3 2.0698 2.3245 2.6511 3.01474 1.8462 2.1156 2.4356 2.77535 1.7360 2.0079 2.3158 2.63906 1.6690 1.9383 2.2354 2.54687 1.6233 1.8881 2.1762 2.47898 1.5899 1.8494 2.1302 2.42599 1.5641 1.8184 2.0931 2.3831

10 1.5433 1.8008 2.0622 2.3475


Step 1. With three open locations (e.g., s1; s2 and s3), theS/R machine moves from the I/O station to theclosest location, say, s1. The expected travel timeis equal to the expected smallest one-way traveltime with m � 3; that is, using m � 3 in (12),

E�SWsI=O� � E�SW�m�3��

� 1

4� 1

2b2 ÿ 2

5b3 � 3

28b4

� �:

Step 2. After storing the ®rst unit load, the S/R machinemoves to the retrievel location that is closest tos1. Among the three retrieval locations (e.g.,r1; r2; and r3), assume that r1 is closest to s1. Thetravel time between s1 and r1 is estimated as theexpected smallest travel-between time from [5]with m � 3; that is, using m � 3 in (20),

E�TBr1s � � E�TB3� � 3

Z 1

0

z�1ÿ F �z��2f �z�dz:

This expression is then evaluated numerically (or ap-proximated with linear interpolation [5]).

Step 3. The S/R machine then moves to the closest of thetwo remaining retrieval locations, say, r2. Theexpected travel time between r1 and r2 is esti-mated as the expected smallest travel-betweentime from [5] with m � 2; that is, using m � 2 in(20),

E�TBr2r1� � E�TB2�

� 1

3b2 ÿ 31

105b3 � 5

42b4 ÿ 11

630b5 � 1

5

� �:

Step 4. The S/R machine then moves to the last retrievallocation, r3. The expected travel time between r2and r3 is estimated to equal the single-shuttleexpected travel-between time from [1]; that is,from (3),

E�TBr3r2� � E�TB� � 1

3� 1

6b2 ÿ 1

30b3

� �:

Step 5. Finally, the S/R returns to the I/O station. Theexpected travel time is estimated as the expectedone-way travel time; that is,

E�SAI=Or3 � � E�SA� � 1

2� 1

6b2

� �:

Thus, the expected total travel time of MSC with the NNpolicy is estimated as:

E�MSCNN3 � �E�SW�m�3�� E�TB3�

� E�TB2� � E�TB� � E�SA�: �22�The above model can be generalized to the case with mopen locations by replacing m � 3 with the appropriatevalue of m in Step 1 above. We evaluate (22) numericallywith di�erent values of m and b in Table 6.

We now compare the E�MSCNN3 � values from Table 6

to the E�STCNN3 � values from Table 5. We see that the

improvement gained is about 27% using MSCNN3 instead

of STCNN3 . We present this comparison graphically in

Fig. 4. In addition, in Fig. 4 we also present the valuesobtained from a Monte Carlo simulation. Note that ourmodel for STCNN

3 seems very accurate whereas the dif-ference between our model of MSCNN

3 and the simulationexhibit some signi®cant di�erences, including a bias tounderestimate the travel times.

In order to analyze the di�erence in travel times moreaccurately for E�MSCNN

m �, we consider each of the travel

times legs comprising E�MSCNNm �: E�SWs

I=O�, E�TBr1s �,

E�TBr2r1�, E�TBr3

r2�, and E�SAI=Or3 �. Table 7 shows each

average travel time leg from the analytical model andsimulation model, and the percentage error (negativevalues indicate that the analytical model underestimatesthe travel time).

An important factor for the travel time under MSCNNm

is the number of open locations, m. When m is changed, ita�ects every travel time leg (as exhibited in the simulatedtravel legs). However, our model only recognizes theimpact of m in one of the travel leg estimations

Table 6. The expected travel time of a MSCNNm tour

m b � 0.25 b � 0.50 b � 0.75 b � 1.0

1 1.7449 1.9104 2.1390 2.41792 1.5866 1.7687 2.0098 2.28453 1.5099 1.7004 1.9416 2.20834 1.4653 1.6594 1.8972 2.15755 1.4362 1.6313 1.8651 2.12066 1.4158 1.6104 1.8405 2.09227 1.4006 1.5940 1.8208 2.06958 1.3889 1.5806 1.8046 2.05079 1.3795 1.5694 1.7910 2.0350

10 1.3717 1.5678 1.7793 2.0215

Fig. 4. Improvement of MSCNN3 over STCNN

3 comparisons withsimulation.


Table 7. E(MSCmNM) travel time analysis (b � 1:0)

m � 1 m � 2

Travel time Model Sim % Error Travel time Model Sim % Error

E�SWsI=O� 0.6667 0.6657 0.1502 E�SWs

I=O� 0.5333 0.5334 )0.0187

E�TBr1s � 0.2781 0.2879 )3.4040 E�TBr1

s � 0.2781 0.2784 )0.1078

E�TBr2r1� 0.3397 0.3632 )6.4703 E�TBr2

r1� 0.3397 0.3618 )6.1083

E�TBr3r2� 0.4667 0.4587 1.7441 E�TBr3

r2� 0.4667 0.4628 0.8427

E�SAI=Or3 � 0.6667 0.6883 )3.1382 E�SA

I=Or3 � 0.6667 0.7342 )9.1937

Total time 2.4179 2.4638 )1.8630 Total time 2.2845 2.3706 )3.6320

m � 3 m � 4

Travel time Model Sim. % Error Travel time Model Sim. % Error

E�SWsI=O� 0.4571 0.4567 0.0876 E�SWs

I=O� 0.4063 0.4046 0.4202

E�TBr1s � 0.2781 0.2788 )0.2511 E�TBr1

s � 0.2781 0.2845 )2.2496

E�TBr2r1� 0.3397 0.3626 )6.3155 E�TBr2

r1� 0.3397 0.3622 )6.2120

E�TBr3r2� 0.4667 0.4633 0.7339 E�TBr3

r2� 0.4667 0.4605 1.3464

E�SAI=Or3 � 0.6667 0.7609 )12.3801 E�SA

I=Or3 � 0.6667 0.7718 )14.7224


m � 5 m � 6


E�SWsI=O� 0.3694 0.3680 0.3804 E�SWs

I=O� 0.3410 0.3399 )0.3236E�TBr1

s � 0.2781 0.2895 )3.9378 E�TBr1s � 0.2781 0.2938 )5.3438

E�TBr2r1� 0.3397 0.3631 )6.4445 E�TBr2

r1� 0.3397 0.3621 )6.1861E�TBr3

r2� 0.4667 0.4610 1.2364 E�TBr3

r2� 0.4667 0.4580 1.8996

E� SAI=Or3 � 0.6667 0.7927 )15.8950 E�SA

I=Or3 � 0.6667 0.8003 )16.6937


m � 7 m � 8


E�SWsI=O� 0.3183 0.3179 0.1258 E�SWs

I=O� 0.2995 0.2988 0.2343

E�TBr1s � 0.2781 0.2989 )6.9588 E�TBr1

s � 0.2781 0.3044 )8.6399

E�TBr2r1� 0.3397 0.3632 )6.4703 E�TBr2

r1� 0.3397 0.3632 )6.4703

E�TBr3r2� 0.4667 0.4550 2.5714 E�TBr3

r2� 0.4667 0.4574 2.0332

E�SAI=Or3 � 0.6667 0.8079 )17.4774 E�SA

I=Or3 � 0.6667 0.8133 )18.0253


m � 9 m � 10


E�SWsI=O� 0.2838 0.2834 0.1411 E�SWs

I=O� 0.2703 0.2690 0.4833

E�TBr1s � 0.2781 0.3073 )9.5021 E�TBr1

s � 0.2781 0.3112 )10.6362

E�TBr2r1� 0.3397 0.3648 )6.8805 E�TBr2

r1� 0.3397 0.3647 )6.1885

E�TBr3r2� 0.4667 0.4545 2.6843 E�TBr3

r2� 0.4667 0.4541 2.7747

E�SAI=Or3 � 0.6667 0.8156 )18.2565 E�SA

I=Or3 � 0.6667 0.8195 )18.6455



(E�SWsI=O�). From Table 7, we can conclude the following

when comparing the simulated to the estimated travellegs:

1. The values of E�SWsI=O� are very close for all values

of m because the S/R machine starts from the I/Ostation, which is a ®xed location.

2. The di�erence between E�TBr1s � increases as the

number of open locations increases. This is ex-pected, because as m increases, the ®rst storage lo-cation is closer to the I/O station. Therefore thedistance from the ®rst storage location (s) to the ®rstretrieval location (r1) is less closely related to theminimum of three random travel-between times.

3. The di�erence between the E�TBr2r1� and E�TBr3

r2�values are fairly constant regardless of the value ofm, with E�TBr2

r1� di�erences always being higher andunderestimated whereas E�TBr3

r2� di�erences arealways overestimated.

4. The di�erence between the E�SAI=Or3 � values is higher

as m increases because of the nature of the NNpolicy, which always selects the location that isclosest to the current location. Thus it seems that thedistance from the last retrieval location to the I/Ostation may be closer to the maximum of three SAdistances. That is, the simulated value for the legapproximated by E�SA� seems to approach the ex-pected largest one-way travel time (E�SL3�), whichis 0.8571 for b � 1:0 (from Table 3).

Let us consider the expected travel time between thelast retrieval location and the I/O station, E�SA

I=Or3 �. The

lower bound of this travel time is equal to E�SA� and theupper bound is E�SL3�. The value of E�SA

I=Or3 � can be

characterized as follows: it begins a small amount abovethe lower bound and increases as m increases, and it ®-nally converges on the upper bound when m is sub-stantially high. Therefore E�SA

I=Or3 � can be approximated

with a simple exponential function as

�E�SAI=Or3 � � minb��maxbÿminb� � �1ÿ eÿmKb=rb�; �23�

where minb is the lower bound (E�SA�), maxb is the upperbound (E�SL3�), m is the number of open locations, rb isthe quantity �maxbÿminb�, and Kb is a constant. Notethat for m � 1, E�SA

I=Or3 � is approximated as minb. We

obtained the value for Kb by trial-and-error for b � 1:0 as0.0404. Moreover, because Kb is a function of b, we de-termine Kb with the following equation:

Kb � 0:0404� �rb ÿr1:0�: �24�Figure 5 shows the comparison between �E�SA

I=Or3 � from

our adjusted analytical model and simulation. Obviouslythe errors ®rst reported in Table 7 with respect to thistravel leg are greatly reduced with our adjusted model.

Now we combine �E�SAI=Or3 � with the other travel time

legs to obtain the expected cycle time values, �E�MSCNNm �,

for b � 1:0. We also extend the adjusted analytical modelof the NN policy for the other b values, i.e.,b � 0:75; 0:50; and 0.25. Table 8 displays the accuracy ofthe adjusted analytical model compared with the simu-lation model.

Notice that all approximations slightly underestimatethe simulated value because we do not adjust the traveltime between the storage location and the ®rst retrievallocation, E�TBr1

s �. However, because the error of�E�MSCNN

m � is relatively small (< 2%) over all b and mvalues, we concluded that any further adjustments wereof marginal value.

3.4.3. Estimation of travel time under MSCand the RNN policy

The Reverse Nearest Neighbor (RNN) policy is, as thename implies, a reversal of the NN policy. The RNNpolicy determines the rack location sequence (in reverse)by choosing the last retrieval location such that it is theclosest of the three retrieval locations to the I/O point.Then it chooses the next to last retrieval location by usingthe nearest neighbor criterion, while the ®rst retrievallocation is then the only retrieval location remaining.Finally, the storage location is chosen as the open loca-tion nearest the I/O point. The tour is formed by con-necting the I/O-to-storage sub-tour and the ending sub-tour that starts at the ®rst retrieval location and proceedsthrough the other two locations while ending at the I/Opoint (the retrieval-I/O sub-tour). The RNN policy is aheuristic policy, but the following observation (the proofis provided in Appendix A) indicates that RNN wouldseem to be a good heuristic because it connects together aminimum length I/O-to-storage sub-tour and a retrieval-I/O sub-tour that has a common characteristic as theminimum length retrieval-I/O sub-tour.

Fig. 5. Adjusted and simulated values of �E�SAI=or3 � for the

�E�MSCNNm � model (b � 1:0�.


Observation 2. The minimum length retrieval-I/O sub-tourwill visit the retrieval location that is closest to the I/O last(as in the RNN policy).

We now illustrate how to estimate the expected traveltime of MSC, E�MSCRNN

3 �, under such a policy.

Step 1. With three retrieval locations (e.g., r1; r2; andr3), assume that r3 is the closest of the three tothe I/O station. The expected time to travel fromthe I/O station to r3 is equal to E�SW3�; that is,(12) with m � 3.

Step 2. The expected travel time between r3 and theclosest of the two remaining retrieval points, say,r2, is approximated with E�TB2�; that is, (20)with m � 2.

Step 3. The expected travel time between r2 and the re-maining retrieval location, r1, is approximatelyequal to E�TB�; refer to (3).

Step 4. With three open locations (e.g., s1; s2 and s3), theS/R machine chooses the open location closest tothe I/O station. Thus, the expected travel timefrom the I/O station to the closest open locationis equal to E�SW�m�3��; that is, (12) with m � 3.

Step 5. The expected travel time between the open lo-cation and the ®rst storage location is approxi-mately equal to E�TB�; refer to (3).

Therefore, the total expected travel time of an MSC tourunder RNN is estimated as

E�MSCRNN3 � � E�SW3� � E�TB2� � E�TB�

� E�SW�m�3�� E�TB�: �25�The above model can be generalized to the case with mopen locations by replacing m � 3 with the appropriatevalue of m in Step 4 above.

We must again analyze the di�erences between ouranalytical model and the simulation results because it isagain obvious that the analytical model underestimatesthe travel time (see [2] for details). In this model we be-lieve that the errors occur because the travel time betweenthe storage location and the ®rst retrieval location is es-timated as E�TB� regardless of the number of open lo-cations, m. However, as m increases, this is less likely tobe valid. That is, when m increases, the storage locationwill be closer to the I/O station, and thus the E�TB� as-sumption of two uniformly distributed points will nothold.

We use the same technique as we used earlier with theE�MSTNN

m � model to adjust our approximation of

E�TBr1s � with the RNN policy. The value of E�TBr1

s � can

be characterized in a similar fashion to E�SAI=Or3 � with the

NN policy; that is, when m increases, E�TBr1s � also in-

creases, eventually approaching its upper bound. The

Table 8. �E�MSCNNm � from the adjusted analytical model and simulation

b � 1:0 b � 0:75

m Model Sim. %Error m Model Sim. % Error

1 2.4179 2.4638 )1.8630 1 2.1390 2.1793 )1.88412 2.3503 2.3706 )0.8563 2 2.0752 2.1024 )1.30923 2.2980 2.3223 )1.0464 3 2.0307 2.0654 )1.70684 2.2664 2.2936 )1.1859 4 2.0055 2.0363 )1.53375 2.2451 2.2743 )1.2839 5 1.9890 2.0220 )1.65976 2.2339 2.2541 )0.9006 6 1.9770 2.0033 )1.33137 2.2168 2.2429 )1.1637 7 1.9675 1.9940 )1.34808 2.2062 2.2371 )1.3813 8 1.9595 1.9883 )1.46799 2.1972 2.2256 )1.2761 9 1.9526 1.9775 )1.2740

10 2.1891 2.2185 )1.3252 10 1.9463 1.9699 )1.2105

b � 0:50 b � 0:25

m Model Sim % Error m Model Sim. % Error

1 1.9104 1.9421 )1.6583 1 1.7449 1.7640 )1.09462 1.8663 1.8831 )0.9027 2 1.7132 1.7094 0.22373 1.8286 1.8572 )1.5643 3 1.6720 1.6883 )0.97594 1.8102 1.8345 )1.3400 4 1.6517 1.6676 )0.95975 1.7989 1.8241 )1.4019 5 1.6394 1.6595 )1.22646 1.7904 1.8074 )0.9521 6 1.6305 1.6452 )0.90137 1.7831 1.7990 )0.8919 7 1.6232 1.6403 )1.05268 1.7765 1.7931 )0.9370 8 1.6170 1.6340 )1.05449 1.7703 1.7835 )0.7485 9 1.6113 1.6272 )0.9876

10 1.7723 1.7745 )0.1218 10 1.6061 1.6187 )0.7873


lower bound of E�TBr1s � is equal to E�TB� and an upper

bound is equal to E�TBL10�, which was developed inSection 3.3. Because both s and r1 are non-®xed locations,our function to adjust E�TBr1

s � must be di�erent thanour function to adjust E�SA

I=Or3 � in the NN model

(E�TBr1s � approaches its upper bound more slowly). We

approximate E�TBr1s � with an exponential function of the

form

�E�TBr1s � � minb��maxbÿminb�

� �1ÿ �eÿfm=�maxb �minb�2g�K�; �26�where minb is the lower bound (E�TB�), maxb is the upperbound (E�TBL10�), m is the number of open locations,and K is a constant. Note that for m � 1, E�TBr1

s � isapproximated as minb. Again, the value of K is obtainedby trial-and-error, and for b � 1:0, min1:0 � 0:4667,max1:0 � 0:8055, the appropriate value of K is equal to0.0100. (For this model it was determined that the valueof K could remain a constant for each value of b.) Thecomparison between �E�TBr1

s � from the adjusted analyticalmodel and simulation is presented in [2], which indicatesthe approximation to be good one.

Finally, we add the other travel time legs of the RNNpolicy to �E�TBr1

s �. We also extend this adjusted model tothe di�erent values of b (0.75, 0.50, and 0.25). We notethat for b � 0:75, 0.50, and 0.25, the travel time from the

adjusted analytical model increases when m is between 6and 10. This obviously cannot occur in the real systembut does occur in our model because �E�TBr1

s � from theadjusted analytical model increases faster than the re-duction of E�SWs

I=O�. Therefore we modi®ed our modelso that whenever we predict an increase in the expectedtravel time as m increases to m� 1, we assign�E�MSCRNN

m�1 � equal to �E�MSCRNNm �. This ®nal modi®ca-

tion is presented in Table 9. It should be clear that this®nal modi®cation results in a very accurate approxima-tion.

3.5. Expected travel time estimation in a twin-shuttlesystem

The twin-shuttle models are very similar to their triple-shuttle counterparts and, owing to lack of space, we referthe interested reader to [2] for the presentation of thetwin-shuttle model. We note that all ®nal (adjusted)models compare well with the simulation results [2]. InFig. 6 we compare the analytical model results for NNand RNN with the optimal times based on total enu-meration of all tours (OPT). We also present the Keserlaand Peters [7] results (K&P). Note that we expect theKeserla and Peters heuristic to produce lower tour timesthan NN or RNN because it is based on a more complexheuristic [7], but we are unable to determine how their

Table 9. The ®nal approximation of �E�MSCRNNm � from the adjusted analytical model and simulation.

b � 1:0 b � 0:75


1 2.3969 2.4337 )1.5121 1 2.1127 2.1383 )1.19722 2.3082 2.3253 )0.7350 2 2.0466 2.0482 )0.07713 2.2521 2.2790 )1.1803 3 2.0053 2.0134 )0.40294 2.2200 2.2497 )1.3193 4 1.9850 1.9888 )0.18895 2.2006 2.2344 )1.5146 5 1.9747 1.9769 )0.11336 2.1884 2.2204 )1.4409 6 1.9696 1.9656 0.20277 2.1808 2.2084 )1.2477 7 1.9674 1.9551 0.63118 2.1761 2.1976 )0.9761 8 1.9670 1.9455 1.10629 2.1736 2.1962 )1.0295 9 1.9670 1.9427 1.2508

10 2.1723 2.1887 )0.7478 10 1.9670 1.9393 1.4284

b � 0:50 b � 0:25


1 1.8683 1.8733 )0.2669 1 1.6889 1.6781 0.64362 1.8124 1.8028 0.5334 2 1.6361 1.6115 1.52863 1.7790 1.7844 )0.3023 3 1.6008 1.6012 )0.02234 1.7684 1.7715 )0.1724 4 1.5916 1.5943 )0.17255 1.7669 1.7654 0.0854 5 1.5916 1.5952 )0.22576 1.7669 1.7582 0.4948 6 1.5916 1.5910 0.03777 1.7669 1.7506 0.9311 7 1.5916 1.5887 0.18258 1.7669 1.7433 1.3538 8 1.5916 1.5862 0.34049 1.7669 1.7422 1.4177 9 1.5916 1.5844 0.4544

10 1.7669 1.7403 1.5285 10 1.5916 1.5836 0.5052


heuristic tours outperform the optimal tours for m � 2.Also, we are unable to determine why the average simu-lated expected cycle time would increase as m increases.We conjecture that it is due to a small sample size (1000)in their simulation experiments. The results in Fig. 6indicate that our heuristic tour times are within 5% ofthe optimal tour times.

4. Conclusions

Twin- and triple-shuttle S/R machines have been devel-oped over the past few years to improve the throughputof AS/RSs. We showed that a twin- or triple-shuttlesystem operating under quadruple-command or sextuple-command policies, respectively, has a higher throughputcapacity than a single-shuttle system operating underdual-command. Furthermore, the performance of thetwin- and triple-shuttle systems can be signi®cantly en-hanced by using an improved strategy of storing and re-trieving at the same location when possible, which weterm modi®ed quadruple-command (MQC) and modi®edsextuple-command (MSC).

Instead of using modi®ed command cycles with FCFSretrieval within a cycle, we propose implementing oper-ating policies called nearest neighbor (NN) and reversenearest neighbor (RNN). With such policies the traveltime is reduced by reducing the travel time between sto-rage and retrieval locations. The NN and RNN policiesexhibit good performance and are also practical to im-plement. In Fig. 7 we illustrate the average travel time toperform one storage and one retrieval over the variouspolicies with one, two, or three shuttles. Implementing atriple-shuttle AS/RS properly can double the expectedthroughput of a comparable single-shuttle system.

The triple-shuttle models were used to evaluate a cur-rent system in operation and then to persuade the RussellCorporation to implement the modi®ed sextuple-com-mand policy with the nearest-neighbor heuristic. Not

only did throughput increase by over 20%, the impact onthe production system was marked. Waiting time forgoods stored in the AS/RS decreased by over 67% be-cause the crane utilization, originally at 94%, was reducedto 77% with this modi®cation. Such decreases in waitingtime resulted in signi®cant labor savings [2].

Acknowledgements

This material is based on work supported by the NationalScience Foundation under Grant No. DMII-9412646 andthe Russell Corporation under Auburn University Admin-istrative Digest No. 1222±94CG. This paper is a summaryand extension of Mr Mungwattana's Master's Thesis, whichwas completed in the Industrial Engineering Department ofAuburn University. Mr Mungwattana's graduate work wassupported by the Royal Thai Government.

References

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Fig. 6. Twin-shuttle travel time comparisons �b � 1�: Fig. 7. 13

E�MSC�R�NNm � and 1

2E�MQC�R�NN

m � versus E�DCNNn�3;m�

�b � 1:0�:


[8] Gudehus, T. (1973) Grundlagen der Kommisoniertechnik: Dynamikder Waren-Verteil und Lagersysteme [Principles of Order Picking:Operations in Distribution and Warehouse Systems], W. Girardet,Essen, W. Germany.

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Appendix A

Proof of Observation 1

Let

DDQ � travel time saved using QC instead of DC;

DQS � travel time saved using STC instead of QC;

DDS � travel time saved using STC instead of DC:

Therefore

DDQ � E�DC� ÿ 1

2E�QC�;

DQS � 1

2E�QC� ÿ 1

3E�STC�;

DDS � E�DC� ÿ 1

3E�STC�:

Note that if DDQ;DQS;DDS > 0 8 b (where 0 � b � 1),then the observation holds.Thus, ®rst:

DDQ � E�DC� ÿ 1

2E�QC�

� �E�SC� � E�TB�� ÿ 1

2E�SC� � 3

2E�TB�

� �� 4

3� 1

2b2 ÿ 1

30b3

� �ÿ 1� 5

12b2 ÿ 1

20b3

� �� 1

3� 1

12b2 � 1

60b3:

) DDQ > 0 8 b:

Second:

DQS � 1

2E�QC� ÿ 1

3E�STC�

� 1

2E�SC� � 3

2E�TB�

� �ÿ 1

3E�SC� � 5

3E�TB�

� �

� 1� 5

2b2 ÿ 1

20b3

� �ÿ 8

9� 7

18b2 ÿ 1

18b3

� �� 1

9� 1

36b2 � 1

180b3

) DQS > 0 8 b:

Third, by de®nition, DDS � DDQ � DQS; thus,

DDS � 4

9� 1

9b2 � 1

45b3 ) DDS > 0 8 b:

Therefore, Observation 1 must hold. j

Proof of Observation 2

Let the travel time from the I/O point to the three retrievallocations, i; j; k, be represented by ti; tj and tk, respectively.Without loss of generality, assume ti < tj � tk (traveltimes are de®ned relative to S/R machine Tchebychevtravel time and ties are broken based on which point isclosest in Euclidean travel time). Note that once it hasbeen decided which retrieval location is scheduled second,the retrieval-I/O sub-tour is determined by examining thetravel times from the two remaining retrieval locations tothe I/O point. The trip that is represented by the shortestof these two travel times is included in the minimum re-trieval-I/O sub-tour, thus completing the retrieval-I/Osub-tour itself. This leads to three cases, depending onwhich retrieval location is scheduled second.

Case 1 ( j scheduled second): There are two sub-tours toconsider with j scheduled second:

k ! j! i! I/O, which has a travel time equal to�1�tkj � tji � ti; and

i! j! k ! I/O, which has a travel time equal to�2�tij � tjk � tk:

Because travel times are symmetric and, by de®nitionti < tk, the minimum retrieval-I/O sub-tour with j sched-uled second must be k ! j! i! I/O.

Case 2 (k scheduled second): Case 2 can be proved in thesame manner as Case 1. The minimum retrieval-I/O sub-tour with k scheduled second must be j! k ! i! I/O.

Case 3 (i scheduled second): We prove by contradictionthat Case 3 cannot hold in an optimal retrieval-I/O sub-tour. There are two sub-tours to consider with i scheduledsecond. Consider ®rst the retrieval-I/O sub-tourk ! i! j! I/O, which has a travel time equal to

t� � tki � tij � tj:

Assume that this is the optimal sub-tour with i scheduledsecond. Compare this sub-tour with the sub-tourk ! j! i! I/O, which does not schedule i second andhas a travel time equal to


t0 � tkj � tji � ti:

Because travel times are symmetric, t� ÿ t0 � �tki ÿ tkj��tj ÿ ti�. Note that because tj > ti by de®nition,t� ÿ t0 > �tki ÿ tkj�. Clearly, the quantity t� ÿ t0 can onlybe less than zero if tki < tkj. However, for this to be true, imust lie between j and k (in terms of travel time), whichcontradicts the original assumption that ti < tj. Thereforethe ®rst retrieval-I/O sub-tour with i scheduled secondcan never be optimal.

Finally, consider the retrieval sub-tour with the retrievallocations reversed, namely, j! i! k ! I/O. Becausetk � tj, if tk 6� tj, then k ! I/O cannot be included in theminimum sub-tour. If tk � tj, then it can be shown withthe same logic as was used above that this sub-tourcannot be optimal. Therefore neither retrieval-I/O sub-tour with i scheduled second can ever be optimal. Since iis visited last in the minimum retrieval-I/O sub-tour with jor k scheduled second, and i can never be scheduledsecond in an optimal retrieval-I/O sub-tour, the hypoth-esis must be true. j

Biographies

Russell D. Meller received his B.S.E., M.S.E., and Ph.D. in Industrialand Operations Engineering from The University of Michigan. Hisdissertation on facility layout was awarded the 1994 Institute of In-dustrial Engineers Outstanding Dissertation Award and First Prize inthe 1993 College on Location Analysis Dissertation Prize Competitionfrom The Institute of Management Sciences. In 1996 he received aCAREER Development Grant from the National Science Foundation.Dr Meller's professional experience includes consulting with the Sys-teCon Division of Coopers and Lybrand, General Electric, and theRussell Corporation. Currently he is an Assistant Professor of In-dustrial and Systems Engineering at Auburn University. His researchinterests include facilities layout, automated material handling systems,and operations research applications in forestry. He is a member ofIIE, INFORMS, and Alpha Pi Mu.

Anan Mungwattana is a Ph.D. student in the Industrial and SystemsEngineering department at Virginia Polytechnic Institute and StateUniversity. He holds an M.S. degree in Industrial Engineering fromAuburn University and received his B.E. degree from Kasetsart Uni-versity in Thailand. He had worked with National Thai Company inThailand before coming to the United States. Currently he is workingat the National Institute of Standards and Technology as a guest re-searcher in the Manufacturing Systems Integration Division. He is amember of IIE.


1996 Multi-Shuttle Automated Storage Retrieval Systems

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