a tabu search technique applied to scheduling container transfers

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A Tabu search technique applied to schedulingcontainer transfersP. Preston a & E. Kozan aa School of Mathematical Sciences , Queensland University of Technology , Brisbane, Qld.4001, AustraliaPublished online: 21 Mar 2007.

To cite this article: P. Preston & E. Kozan (2001) A Tabu search technique applied to scheduling container transfers,Transportation Planning and Technology, 24:2, 135-153, DOI: 10.1080/03081060108717664

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A TABU SEARCH TECHNIQUEAPPLIED TO SCHEDULING

CONTAINER TRANSFERS

P. PRESTON and E. KOZAN*

School of Mathematical Sciences, Queensland University of Technology,Brisbane Qld. 4001 Australia

(Received 19 August 1999; In final form 20 May 2000)

A scheduling model is designed and applied to seaport container terminals taking intoaccount factors such as container handling equipment, labour resources, storagecapacities and terminal layout. Major factors influencing container transfer efficiency areanalysed to optimise resource usage resulting in lower operating costs while achieving adesired level of customer service. A Tabu Search (TS) heuristic is used to compare thebenchmark of the Port of Brisbane with a Genetic Algorithm (GA) heuristic solution.

Keywords: Scheduling; Heuristics; Containers; Seaports; Tabu Search; GeneticAlgorithm

INTRODUCTION

Berthing time of a container carrying ship accounts for a considerableproportion of its journey. Decreasing the turnaround time at portwould result in a reduction in travelling time thus a reduction in thecost of transporting containers. This paper considers reducing the timeat port by reducing the time to transfer the containers from the storagearea to the berth. Specifically this transfer is modelled mathematicallyand the handling schedule determined using Tabu Search. The resultsare compared with a Genetic Algorithm and the current practice at thePort of Brisbane.

*Corresponding author.

135

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136 P. PRESTON AND E. KOZAN

Containers first appeared as a method for transporting cargo inAmerica around the late 1930's. They were introduced as demountablewagons that could be removed from their chassis and placed onanother rail or road chassis. This was primarily used to eliminate thereloading which occurred when the cargo was required to changetrains or from truck to train. This led to increased security, a reductionin transportation costs, and enabled door-to-door delivery of goods.The shipping industry was quick to recognise the economic appeal ofcontainerisation and employed containers as a means of shippingcargo. The introduction of containerisation caused some dramaticchanges to the layout at seaport terminals. These changes includealterations to the storage area and the introduction of specialisedcontainer handling equipment. However that main change has been inthe storage area. Storage methods have undergone significantmodifications to take full advantage of the containers stacking ability.This means more cargo can be stored at the port requiring a smallerarea of land.

The general container transfer port consists of berth spaces ofvarying lengths, container storage areas, rail and road links, yardmachines (e.g., forklifts), and shore cranes. A simplified version of theport layout is provided in Figure 1. [See UNCTAD (1990), (1991) and(1993) for more details on port layout and systems including handlingequipment and storage facilities.]

When a container ship arrives at the port, management will allocatea number of yard machines to service it (i.e., transfer the importcontainers to storage or road/rail links and the export containers fromthe storage area to the berth). The problem is to determine theschedule in which to transfer the containers.

The method of assigning containers to machines for loading onships for export most widely employed is the use of "gut instinct" orheuristics. These approaches may seem to be effective but may actuallyincrease the berthing time of the ships. A better assigning techniquemay involve an analytical model. Many papers have consideredanalytical models to replace "gut instinct" methods of loading andunloading containers. For example, Daganzo (1989) devised a mixedinteger model assigning shore cranes to ship holds. He discussed bothstatic and dynamic cases and expected a feasible solution in 'real time5

although it is untested for large traffic flows. Daganzo (1990)

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SCHEDULING CONTAINER TRANSFERS 137

Berthing and Marshaling Area 1 Berthing and Marshaling Area 2

12 3 4 5 6 7J

1!

If

t

-

i

•

\

1

1

-

tt

(

t1

>ra

^—

A

1 I

1!T1u1

s

To RailTerminal Links

Road Terminal

i

Ij|

01

[-P

« -

a

1

sA

i

1

,cr

1

iita:. ,

J—

1

Rail Terminal

FIGURE i The port layout.

constructed a model that minimised delays by assuming turnaroundtime is a function of berth space and crane availability only. Itcombines the 'first come first served' (FCFS) heuristic with a simplifiedversion of the previous model. This paper involves some assumptionsthat may prove to be invalid (i.e., berth length infinite and theincoming ship requires the same length of berth space as the previousship).

Gendreau et al. (1995) discussed using the travelling salesmanproblem with deadlines, for a single machine. This model was modifiedto have the objective of minimising the total loading time. It groups

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the containers and assigns a crane to service each group to be loadedon the same pre-determined ship.

Other works take the approach of re-arranging the storageoperation to minimise berthing time. Taleb-Ibrahimi et al. (1993)discussed strategies to minimise loading time of export containers byusing a pre-determined storage arrangement. They propose to have a"roughpile" temporary storage area where early containers are storeduntil a permanent storage area is assigned.

THE MODEL

The notations used in defining our model are as follows:

lock This is defined as the time required by the yard machines to"lock on" to a container before picking it up. It is assumedthat the time to "unlock" a container after moving it is thesame.

move When a container is stored below one or more others and isrequired for loading the upper containers are moved to atemporary storage location to remove the desired container.Move is the time required moving containers to the adjoiningtemporary position.

cw, rw The width of a column and row, respectively, in the storagearea.

ti Time container i is scheduled for handling (movement).xh yt The row and column respectively of the storage area partition

container i is stored.zi31 The vertical storage position of container i is stored at time tt.

This is measured as the number of containers stored oncontainer i which delays access by handling equipment.

ztj The vertical storage position of container i is stored at timetf.

mac,- The yard machine container i is scheduled to be transferredby.

ship/ The ship container i is to depart on.depart* Departure time of ship s.arrive* Arrival time of ship s.

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v™ac Velocity of the yard machine, mac.travel,- The time required to transport container i between the

storage area, marshalling area, track area and/or intermodalterminal.

Since travelling time is independent of the vertical storage positionand primarily a function of the row and column storage position it isconstant (since vmac is assumed constant for this model, althoughmachine type and container type can greatly affect velocity).

, , Xj * rw + y,- * cw , ,travel,- = lock -\ : \- lock. Where m = mac;.

ytn

This equation defines the travelling time for all containers. Thetravelling time includes the time to lock and drop off the container atthe start and end of the journey.

setu = I ° If ZiJ = °\ %i,t * (4 * lock + move) + 2 * lock + move Otherwise

This is the time required to move containers stored above the nextscheduled container. If the desired container is on top there is no set-up time. If the desired container is not on top of the other containers,the set-up time incorporates the time required to move thesecontainers to an adjoining position. The set-up time has twocomponents: one of these is "lock-unlock" time and the other ismoving time. In this case, the yard machine must first move the topcontainers to a temporary storage position, then move the desiredcontainer from the storage area, and finally return the top containersfrom the temporary storage position back to the storage are. Figure 2shows a case of the set-up for a container on the second level for athree high stack i.e., where zut — 1.

The difficulty in the optimising container scheduling problems canbe explained with the following example. If in the situation in Figure 2the bottom container is loaded before the two containers stored aboveit, then the set-up time for this container is 2.5 minutes. If, on the otherhand, it is loaded after both the containers stored on top of it the set-up time is 0 minutes. While 2.5 minutes may not seem a long timewhen 500 containers are being loaded to a ship and 33% (or 167) of

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move

travelling setup

FIGURE 2 Set-up arrangement for container on the second level for a three highstack.

these are likely to be stored below 2 other containers, this could resultin more than 400 extra minutes needed to load a ship.

A distinguishing feature of this research is the use of identicalparallel machines where each job (container) is processed (transferredfrom storage to berth) by exactly one of m identical machines (yardmachine). The meaning of 'identical machines' is that each machinedoes the same job - however, some machines (reach stackers) cantransfer containers quicker than others (forklifts). Using the param-eter vmac for the different types of yard machine reflects this non-identical behaviour. Also because of the use of parallel machineswe do not just want to minimise the total 'working' time of all themachines but to reduce the time of the machine doing the most 'work'.The difference between these is illustrated in Figure 3.

In (a) the 'working' times for machines 1, 2 and 3 are 15, 10, 20minutes respectively. This gives a total of 45 minutes. However, theproject cannot be finished until the last machine finishes, which in thiscase is machine 3 after 20 minutes. This means the completion timeof the project is 20 minutes. On the other hand, in (b) all machinesare in use for 15 minutes giving a total work time of 45 minutes.While this is the same time as in (a), project completion for (b) isjust 15 minutes.

We have chosen an objective function to reflect the need to minimisethe completion time rather than simply the total service time. This isdone by minimising

z = max [machine 1, machine 2 , . . . , machine mac]

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Machine 1 Machine 2 Machine 3

-45.

Machine 1

Machine 2

Machine 3

-20-

(a)

Machine 1 Machine 2 Machine 3

-45-

Machine 1

Machine 2

Machine 3

(b)

FIGURE 3 Illustration of the objective.

In the example shown in Figure 3, this will minimise z = max[machine1, machine 2, machine 3].

linimise z = <

f 20 (a)iimmise z — <

max[15,10,20] (a)

max[15,15,15] (b)

15 (b)

=» z =

To further relate this to the problem at hand, the objective of thismodel is to minimise the time ships spend at the berth. We will

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minimise time spent transferring containers from a storage area to shipor ship to the storage area. This transfer time is the sum of the set-upand travelling time for each container. Since the idea is to minimise thecompletion of the transfers, we want to find the minimum project timewhich is the time for the yard machine that is in use longest by addingthe transfer times of the containers allocated to it.

Minimise max Y"̂ (travel,-+ setup,-) (1)mac *—'

{i|mac,= mac}

This equation is designed to find the maximum time any yard machineis in service and minimise this value. This will minimise the time theship spends at the port and also minimise the total working time of allyard machines. The total working time of all yard machines can beexpressed as ^,-(travel,-+setup,-). It is seen that by minimising theobjective in Eq. (1) above it is in effect simultaneously minimising(travelf+setup;) for each machine and therefore also minimising^,-(travel,-+setup,-), the total working time of all machines.

The model is subject to the following physical and modellingconstraints. The physical constraints are those required to obeyphysical limitations of the port system while the modelling constraintsare those used for this particular model.

If Xi = x? and yt = y? then zit, •£ z?t, Vf ^ i'. (2)

The location constraints in Eq. (2) are used to satisfy the physicalcondition that only one container can be stored in a given storageposition. The solution program will either move one of the containersto another position or store them on top of the other. The initialstorage locations are also checked for feasibility of the heightparameter - ensuring if then z^, > 0 there are in fact containersoccupying those positions.

If mac,- = mac? then /,- ̂ t/. V7 ̂ / (3)

Equation (3) defines the machine constraints which are used to satisfythe physical condition that each yard machine can only be scheduledto handle only one container at a time.

If xi = Xf and yi = ye and ziyl < z?,, and f,- < /?then ztjf = z?,, - 1 Vf? > f,-, i ^ i. (4)

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This Eq. (4) is a modelling constraint used to modify the parameter Zfit

when container i is stored above i' and is scheduled to be loaded beforecontainer /'.

arrive1 + N (travelling time,- + setup time,) < depart* Vs (5){/|ship,=i}

The ship constraint Eq. (5) ensures that each ship has a time windowwithin which loading and unloading service should begin and end.This interval (arrive'', depart*) represents the time the ship is at portand available for loading. Time windows can be fixed, ships will arriveat a certain date and must depart at a certain date, or flexible, willarrive at a certain date but the departure date can be negotiateddepending on number of imports and exports amongst other factors.Generally the time window is fixed because this allows the shippingcompany to know exactly what port costs are. With flexibletime windows the shipping company may decide to pay extra toallow their ship to be serviced (imports unloaded and exportsloaded) quicker or to allow them to stay in port longer. [See Kozanet a!., 1996 and 1999 for more information on flexible or soft timewindows.]

While the model may seem similar to many job shop machinescheduling models, it is very different in the way the set-up time isdetermined. The set-up time in general job shop machine schedulingproblems is dependent only on the job immediately preceding the jobin question. In this model the set-up time is dependent on the order ofscheduling of the containers (if any) initially stored on top of thecontainer in question. For this reason the solution is dependent on theorder of the whole sequence not just the immediate predecessors ofcertain jobs.

The model is formulated as a mixed integer non-linear programmingproblem and can be solved by branch-and-bound techniques for smallsize problems. However, the problem is known to be NP-hard, thus itscomputation complexity increases exponentially with the number ofcontainers in the schedule. This makes it difficult to solve in areasonable time with the current exact solution techniques such asbranch-and-bound or tree searches. This implies that for large sizereal life problems, heuristic techniques have to be used. A Genetic

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Algorithm (GA) has been applied previously to this problem (Kozanand Preston, 1999) with promising results but the solution times werefound to be quite large in some cases.

The Tabu Search (TS) used in this paper uses a neighbourhoodwhere the container's identification number was part of a list. This listwas then divided according to machines by machine 1 taking 1st, 11thetc., containers, Machine 2 taking the 2nd, 22nd etc., containers, andso on. This way containers must all be transferred and can only bedone once. The move between neighbourhoods consisted of choosing arandom container and assessing changing its position in the list to thebeginning, middle or end. When a container is moved it joins the Tabulist and cannot be moved again for 100 iterations.

This procedure is explained with the following example. Forsimplification we assume there are only 50 containers to be loadedby 10 yard machines and the initial solution vector is simply thenumbers 1-50 in order. We also assume that the initialisationprocedure is performed correctly so the location constraint (2) issatisfied. The machine constraint (3) is satisfied by definition of theneighbourhood. Hence, the initial allocation gives machine 1 totransfer containers 1, 11, 21, 31 and 41, machine 2 to transfer 2, 12,22, 32 and 42 and so on with machine 10 to transfer 10, 20, 30, 40and 50.

Selecting at random a container (say 15) to change positionscreates the neighbourhood. We now evaluate the objective functionwhen 15 is inserted at the beginning, middle and at the end. Ifthe beginning is found to be best the new solution vector is[15,1,2,..., 14,16,17,...,48,49,50]. Similarly if the middle isbest it is [1,2,3,..., 14,16,17,.. .,24,25,15,26,27,.. .,48,49,50]or [1,2,3,..., 14, 16,17,..., 48,49,50,15] if the end is best. Container15 is then put into the Tabu list and will not be moved again for 100iterations. This is repeated until either the stopping criterion, itera-tion limit of 500 or 5 successive iterations without improvement,are satisfied. Figure 4 illustrates the procedure. This type of neigh-bourhood is used because of knowledge of the system: intuitivelyit makes sense for the containers on top to be loaded first andthose on the bottom last, and this algorithm should achieve thatobjective.

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Initialisation

Randomly Generate

Initial Solution

Fix Solution to Ensure

Constraints Satisfied

Evaluate Objective

Function

Select Container to

Move for New

Neighbourhood

Evaluate New

Neighbourhoods

Select best

Neighbourhood and

Transform Solution

FIGURE 4 Flow chart of the Tabu Search procedure.

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The solution times were found to be in the order of one minutewhich is much better than the GA where at least 10 minutes wasrequired. The next section details the current situation at the Port ofBrisbane and details the benchmark set-up considered. This isfollowed by the analysis of the results obtained.

APPLICATION TO THE PORT OF BRISBANE

The Port of Brisbane is the fastest growing city port in Australia. Aconsiderable amount of this growth is in containerised commodities.Since 1976 total trade through the Port of Brisbane doubled whilecontainer trade increased 400% (Port of Brisbane, 1992). This increaseis partly due to Brisbane's proximity to booming Asian markets, sinceBrisbane is less than 12 sailing days to Singapore, Japan, and HongKong—which is four-five days less than Australian southern ports.Thus, cargo unloaded in Brisbane can be delivered to destinations inthe south-east before cargo from the same vessel is unloaded inSydney.

To cope with this rapid growth the Port of Brisbane has plannedexpansions in berth length, container storage areas, container carryingmachinery and improved road and rail links. Currently there are fourcontainer terminals with five gantry cranes, 30 forklifts, 10 reachstackers, and storage area for about 9000 TEU's. In addition there arepower outlets for 1500 reefer containers.

Using data supplied by the Port of Brisbane it is estimated that anaverage of 486 containers are exported with each ship. It is found thateach of these 'average' ships have a transfer time of 673 minutes. Withthis in mind, and using current resources and storage practices of theport, the benchmark set-up (for 500 containers) was established inTable I and the model simulated a number of times.

TABLE I The benchmark set up for comparison

Containers for export: 500

Containers in storage (% occupied) 500(22%)Yard machines used 10Storage levels 3Storage policy Fixed

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Since little previous research has been done in this area, it is difficultto determine how good the solutions obtained are, so the benchmarkset-up, as outlined in Table I above, is compared with current practices.Different terminal set-ups, as outlined below, are also examined.

A random storage policy (container stored in a random manner butstill conforming to location constraint given in Eq. (2)) is compared toa fixed storage policy (containers stored as close to the berth aspossible also ensure Eq. (2) remains feasible) for the different layouts.Changing the maximum allowed container storage height, in the range1 — 5, gave a comparison of different storage layouts. This determinedan optimal storage height. The number of containers already instorage is also taken into account and varied. This was done to give anidea of what would happen in a busy port where the storage area isoften highly utilised. The final alteration to the benchmark was tochange the number of yard machines. This could be used to determineif the time saved with extra yard machines would cover the cost ofpurchasing additional machines.

Figure 5 shows the simulation results using the benchmark set-upoutlined in Table I.

Since the Port currently does not use the fixed storage policy, thisset-up has also been simulated using random storage locations. Theresults are shown in Figure 6.

The simulation runs shown in Figures 5 and 6 have an average of400.559 minutes (a = 2.205) for the fixed storage policy and 635.092(cr = 21.331) for the random storage policy. Thus, it can be concludedthat fixed storage policy is better than random storage policy. This,however, is not of great surprise since one would expect the policywhere containers are stored closest to the berth to be better than arandom storage allocation. The solutions for the random storageallocation also compare favourably with the observed times at the Portof Brisbane. The loading sequence for the random storage policydetermined using Tabu Search is found to be 28 minutes quicker thancurrent practice. With this in mind the random storage strategy couldbe seen as being approximately equivalent to the current practice soimprovements to this benchmark would be expected to transfer toimprovements in current practice.

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& 405--:;;%>

0 10 20 30 40 50 60 70 80 90 100

Simulation Run

FIGURE 5 The simulation runs for the benchmark problem.

700

—«—TimeAverage

Run Avg-• Current

40 60

Simulation Run

FIGURE 6 Simulations of the benchmark for the random storage policy.

Another issue is the effect the number of containers already in thestorage area has on the loading time. The model has been simulatedfor varying storage area utilisation in the range 0% to 70% forrandom and scheduled storage policies. The results are shown

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graphically in Figure 7 with the average of 100 simulation runs plottedand error bars denoting one standard deviation.

This confirms the above belief that the fixed policy is significantlybetter than random storage allocation (as expected) for all levels ofstorage utilisation. It also shows there is not a significant change intransfer time for different storage area fullness with the randomstorage policy. The fixed policy shows exponential growth risingsharply once 40% of the storage area becomes full. This would bemainly due to containers being further from the berth (since a greaterproportion is already occupied) and also having containers storedbelow others that are not loaded to the current ship.

The next set of simulations varied the number of yard machines(6-16) available for use. These results are shown in Figure 8.

It is observed from this figure that the transfer times increaseexponentially as the number of yard machines is decreased. This trendis the same for both random and fixed storage strategies.

Further analysis was performed to determine the effect of themaximum number of levels allowed in the storage area on the transfertimes. Simulations were run with no other containers in the storagearea and also with 230 containers (10% when 3 levels are used) for amaximum number of levels between 1-5. Figures 9 and 10 show that

0% 10% 20% 30% 40% 50% 60% 70%

Storage Area Utilisation

FIGURE 7 Average transfer times for 500 containers with varied storage utilisation.

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!«00

I

-Random-Fixed

!0 12

Number of Machines

FIGURE 8 Transfer times with varied number of yard machines.

600- -

jS 550 ; -

- Random

1 2 3 4

Maximum Storage Level

FIGURE 9 Transfer times with varied maximum storage level-storage area empty.

for the random storage policy there is little change when extra levelsare allowed and that single level storage gives the best result. The fixedpolicy with the storage area empty found 3 levels to be the best thoughthere is not a significant difference between 2-5 levels. With 230containers in the storage area the single level storage is almostcompletely full (only 38 free spaces) so the random and fixed policy's

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700

° Random

- Fixed

4501 2 3 4 5

Maximum Storage Level

FIGURE 10 Transfer with varied maximum storage level-230 containers in storage.

are almost identical. The fixed policy in this case found four levels tobe optimal but there is no significant change between three to fivelevels. These figures also show that the increase in storage area usagehad no effect on the random storage policy but around a 30-40minute increase for the fixed policy (except for single level storagewhere an increase of 80 minutes was observed).

CONCLUSION

The results presented in this paper show that a fixed storage policy -where containers are stored in the closest rows to the berth - is betterthan a random storage policy. The storage area fullness does not havea significant effect on random storage policy but when it reaches 50%capacity in the fixed policy it starts to increase the transfer timesdramatically. When using the fixed storage policy, decreasing themaximum height at which containers can be stored shows marginalincreases in the transfer times until a single level where a greaterincrease is noticed. The random storage policy shows little differencebetween the different levels. Increasing the number of yard machines

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t- Rand_TS

- * - Fixed_TS

i— Rand_GA

—e— Fixed GA

0% 10% 20% 30% 40% 50% 60% 70%

Storage Area Utilisation

FIGURE 11 Comparison of average transfer times varied storage utilisation using TSand GA.

decreases the transfer times exponentially. In most cases the TS resultswere better than the GA results. Figure 1! provides an example of this.

In this case the TS found lower solutions for all levels of utilisationfor the random storage policy and for less than 50% utilisation for thefixed policy. The other advantage in using TS is the CPU time whichwas 10 times faster (1 minute as opposed to 10 minutes for GA).

The model was verified by making comparison with a simulationanalysis whose results were used to reflect actual conditions pertainingin the Port of Brisbane.

The ship time costs and the impact of negotiating a ship schedule (ina time window) on overall cost, investment and operations should bebalanced with the ideal operating strategies for a seaport. The modeloutput can be used to evaluate the cost of the change of the operatingstrategies.

References

Alvarez-Vales, R. and Tamarit, J. M. (1989) "Heuristic Algorithms for Resource-constrained Project Scheduling: A Review and an Empirical Analysis". Advances inProject Scheduling, Elsevier Science Publishers, B. V., Amsterdam.

Castilho, D. B. and Daganzo, C. F. (1993) "Handling Strategies for Import Containersat Marine Terminals". Transportation Research, 27B, 151-166.

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a tabu search technique applied to scheduling container transfers

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