DISPATCHING VEHICLES CONSIDERING UNCERTAIN HANDLING TIMES AT PORT CONTAINER TERMINALS

DISPATCHING VEHICLES CONSIDERING UNCERTAIN HANDLING TIMES AT PORT CONTAINER TERMINALS Vu Duc Nguyen Department of Industral Engneerng, Pusan Natonal Unversty, Busan 609-735, South Korea duc.se@pusan.ac.r Kap Hwan Km The correspondng author Department of Industral Engneerng, Pusan Natonal Unversty, Busan 609-735, South Korea apm@pusan.ac.r Abstract Ths paper consders the problem of vehcle dspatchng at port contaner termnals n a dynamc envronment. The problem deals wth the assgnment of delvery orders of contaners to vehcles whle tang nto consderaton the uncertanty n the travel tmes of the vehcles. Thus, a real-tme vehcle dspatchng algorthm s proposed for adaptaton to the dynamc changes n the states of the contaner termnals. To evaluate the performance of the proposed algorthm, a smulaton study was conducted by consderng varous values of decson parameters under the uncertanty n travel tmes. Further, the performance of the proposed algorthm was compared wth those of heurstc algorthms from prevous studes. 1 Introducton Because of globalzaton, many cargoes today are transported from one area of the world to another. Over the last decade, cargo transportaton by contanershps has rapdly grown n popularty because of ts cost effcency. In contaner termnals, contaners are transferred between contanershps and the storage yard va dschargng and loadng operatons. Durng dschargng operatons, contaners n a contanershp are unloaded from the contanershp and staced n the storage yard, and vce versa durng loadng operatons. In ths paper, we consder a port contaner termnal n whch three man types of handlng equpment, quay cranes (QCs), vehcles, and yard cranes (YCs), are used for shp operatons. Fgure 1 llustrates the layout of a seaport contaner termnal that conssts

of areas for the QCs, vehcle drvng, and YCs. Contaner termnals have complcated handlng systems, and thus, there are many sources of uncertantes durng ther operaton. In partcular, the travel tmes of vehcles may not be consdered as beng constant any more. Ths paper attempts to schedule the delvery operatons of vehcles whle tang nto account the uncertanty n the travel tmes of the vehcles. Fgure 2 shows the dschargng and loadng processes n port contaner termnals. Fgure 1: Layout of Seaport Contaner Termnal.

Fgure 2: Dschargng and Loadng Processes. Vehcle dspatchng problems have been addressed by many researchers. Egbelu and Tanchoco [1] presented a dspatchng method for sngle-load automated guded vehcles (AGVs) that ncorporated a varety of prorty rules. Egbelu [3] suggested a demanddrven rule n whch AGVs are frst assgned delvery tass that are allocated to machnes wth the smallest number of tass already present n ther nput buffers. Blge and Ulusoy [5] presented a method for smultaneously schedulng the operaton of machnes and the transfer of materals by AGVs. Km et al. [6] suggested an AGV dspatchng method n whch the prmary crteron for selectng the next delvery tas s to balance the worload across dfferent worstatons. Van der Meer [9] undertoo a smulaton study to evaluate varous dspatchng rules, ncludng rules usng pre-arrval nformaton, for automated lftng vehcles (ALVs) n contaner termnals. Lm et al. [10] ntroduced an AGV dspatchng method usng a bddng concept n whch the dspatchng decsons are made through communcaton between related vehcles and machnes. Km and Bae [12] presented a mxed-nteger programmng model for assgnng optmal delvery tass to AGVs and suggested a heurstc algorthm for solvng the mathematcal model. Usng a smulaton study, the heurstc algorthm was performed and compared wth other dspatchng rules. Bsh et al. [15] proposed a vehcle dspatchng technque to mnmze the total tme taen to serve a shp. They developed easly mplementable heurstc algorthms and dentfed both the absolute and the asymptotc worst-case performance ratos of these heurstcs. Brsorn et al. [16] presented an alternatve formulaton of the AGV assgnment problem that does not nclude due tmes and that s based on a rough analogy to nventory management; they proposed an exact algorthm for solvng the formulaton. Grunow et al. [17] descrbed a smulaton study of AGV dspatchng strateges n a seaport contaner termnal, where AGVs can be used n ether sngle- or

dual-carrer mode. They compared a typcal, on-lne dspatchng strategy adopted from flexble manufacturng systems wth a pattern-based, off-lne heurstc algorthm. Nguyen and Km [19] developed a mathematcal formulaton of the dspatchng problem for ALVs. They suggested a heurstc algorthm and compared ts solutons to optmal solutons. Angelouds and Bell [20] presented a flexble AGV dspatchng algorthm capable of operatng under uncertan condtons wthn a detaled contaner termnal model. Several performance ndcators were descrbed, focusng on generc features of vehcle operatons as well as the assessment of uncertanty levels nsde the termnal. From the results of the smulatons, t was found that ther technque outperforms wellnown heurstcs and alternatve algorthms. However, n contaner termnals there are many uncertan factors. Smulatons have been used as a powerful tool for analyzng the performance of port contaner termnals n complex dynamc envronments. Varous levels of detal and the uncertantes n contaner termnals can be expressed n smulaton models. Much research on the development of smulaton models of contaner termnals has been publshed (Cho [2]; Yun and Cho [7]; Tahar and Hussan [8]). Hartmann [11] ntroduced an approach for generatng scenaros that can be used as nput data for smulaton models of port contaner termnals. Through a smulaton study, Vs and Hara [13] and Yang et al. [14] compared the performance of two types of automated vehcle, namely AGVs and ALVs. Lee et al. [18] undertoo a smulaton study comparng varous handlng systems consstng of dfferent types of transport vehcle (prme movers and shuttle carrers) and dfferent storage-yard layouts (wth and wthout a chasss lane besde blocs). Ths paper s organzed as follows. Secton 2 ntroduces the shp operaton and the method of operatonal control for vehcles n contaner termnals. A heurstc algorthm for solvng the vehcle dspatchng problem bearng n mnd the uncertantes s proposed n Secton 3. Secton 4 presents the results of a smulaton experment for comparng the proposed heurstc algorthm wth other algorthms and analyzng the performance of the proposed heurstc algorthm. Fnally, some conclusons and ssues for further research are set out n Secton 5. 2 Shp Operaton and Operaton Control Method for Vehcles Before a shp arrves at a port contaner termnal, all nformaton regardng the nbound and outbound contaners s sent to the termnal by a shppng agent. Based on ths nformaton, a lst of the sequence of dschargng and loadng operatons for ndvdual contaners s then made. When the shp actually arrves, shp operatons are usually performed on the bass of the dschargng and loadng sequence lst. For a dschargng operaton, after recevng a contaner from a QC, the vehcle delvers t to the desgnated storage yard. When the vehcle arrves at the yard, t wats at the transfer pont (TP) n the yard for the contaner to be pced up by YC. A YC pcs up the contaner and stacs t n an empty slot n a bay. Loadng operatons are performed n exactly reverse order. Durng the dschargng operaton, a vehcle and a QC must

converge when the QC releases an nbound contaner onto the vehcle, and the vehcle and a YC must converge when the YC pcs up the nbound contaner from the vehcle. Smlar convergences occur durng the loadng operaton. Ths necessty for synchronzaton frequently causes delays to transport operatons n contaner termnals. In contaner termnals, a vehcle can be consdered a resource that has to be effcently dspatched. To adapt to a changng envronment, a dspatchng decson must be made whenever an mportant event occurs. Shp operaton planners develop the sequence of dschargng and loadng operatons for each QC. The sequence s ntally put nto LIST A n Fgure 3. Among the tass (dschargng and loadng operatons) n LIST A, a pre-specfed number of the most mmedate tass for each QC are moved to LIST B. The tass n LIST B are canddates for assgnaton to vehcles. Whenever a vehcle commences travel to pc up a contaner for a tas, the tas s removed from LIST B, and the next urgent tas from the correspondng LIST A s moved to LIST B. Note that for each QC, the same number of mmedate tass must be mantaned n LIST B, unless LIST A becomes manly empty. The dspatchng algorthm s trggered when a vehcle becomes dle. When a vehcle completes a delvery tas, the vehcle reports the completon of the tas to the control system (CS). The CS wll then trgger the dspatchng algorthm for assgnng delvery tass to vehcles. Followng the outcome of dspatchng, f the vehcle s assgned a delvery tas, t wll commence travelng to the pcup poston of the assgned tas. Otherwse, the vehcle wll move to the parng area to awat the next assgnment. When a QC completes a delvery tas, completon of the tas s reported to the CS, and the next tas s added to the QC s tas lst. The CS wll then trgger the dspatchng algorthm for reassgnng delvery tass to vehcles. When an dle vehcle s assgned a tas, the vehcle wll commence travel. When the vehcle arrves at the desgnated QC, ts arrval wll be reported to the CS. At the quay, the vehcle checs the status of the QC. For loadng tass, f the QC s not avalable, the vehcle has to wat untl t becomes so. If t s avalable, the QC pcs up the outbound contaner from the vehcle. Smlarly, for dschargng tass, the vehcle has to wat untl the QC becomes avalable, and the QC releases the nbound contaner onto the vehcle. The change n status of the vehcle s then reported to the CS. When the vehcle departs from the QC wth an unloaded contaner, t wll go to a desgnated bloc to delver t. When the vehcle completes the delvery of a loadng contaner, the dspatchng procedure s trggered for assgnng another tas to the vehcle. When no tas s assgned, the vehcle becomes dle and moves to the parng area. When a tas s assgned, t moves to the pcup poston of the next assgned tas. All changes n the system status are reported to the CS.

(LIST A) A sequence lst of remanng dschargng and loadng tass for QC1 A sequence lst of remanng dschargng and loadng tass for QC2 A sequence lst of remanng dschargng and loadng tass for QC3 (LIST B) A pooled lst of mmedate tass for dspatchng Fgure 3: Varous Lsts of Tass for Dspatchng. 3 Heurstc Algorthm Consderng Uncertantes The vehcle dspatchng problem was formulated as a mxed-nteger programmng (MIP) model, and a detaled descrpton of ths formulaton can be found n Km and Bae [12]. Ther suggested algorthm assumed determnstc handlng and travel tmes for the equpment. The present paper extends ther dspatchng heurstc algorthm by relaxng ths assumpton. The followng frst ntroduces a formulaton of the dspatchng problem and the heurstc algorthm by Km and Bae [12]. A loadng operaton cycle by a QC begns wth the pcup of a contaner from a vehcle, whle a dschargng operaton cycle ends wth the release of a contaner onto a vehcle. For a QC operaton to be performed wthout delay, a vehcle must be ready at a specfed locaton beneath the assocated QC before the transfer of a contaner commences. Let e be an event representng the moment at whch a vehcle transfers the th contaner of QC (the th operaton of QC ). When the th operaton of QC s a loadng operaton, event e corresponds to the begnnng of the pcup of a contaner from a vehcle. When the th operaton of QC s a dschargng operaton, t corresponds to the begnnng of the release of a contaner onto a vehcle. The tme of event e s denoted Y. A delay to an operaton occurs when the correspondng vehcle does not arrve at the requested moment, whch s the tme of the event wth no delays to QC operaton and s represented by s,.e., the earlest possble event tme. Three types of events are undergone by vehcles durng a shp operaton: The ntal event, whch represents the current state of each vehcle; the event when a vehcle begns to receve a contaner from a QC or when a vehcle begns to transfer a contaner to a QC; and the stoppng event, when a vehcle completes all of ts assgned tass. The notatons related to shp operatons are summarzed as follows:

V = The set of vehcles. K = The set of QCs. O e = The ntal event of vehcle v, v v V. F e = The stoppng event of a vehcle v, v v V. Note that, although the number of stoppng events of vehcles s the same as the number of vehcles, stoppng events wth dfferent subscrpts do not need to be dstngushed from each other. e = The event that corresponds to the begnnng of a pcup (or release) of a contaner from (onto) a vehcle for the tas related to the th operaton of QC. Assume that there exst m tass for QC. = The set of e for = 1,2,,m and K. = The locaton at whch event O e occurs. le ( ) represents the ntal locaton of T l ( e ) v vehcle v. Here, l ( e ) represents the poston at whch the th contaner of QC wll be transferred. The locaton at whch a vehcle completes ts fnal F delvery tas s denoted le ( v ). lj t = The pure travel tme from l ( e ) to l ( e l j ). lj C = The tme requred for a vehcle to be ready for l e after undergong j e, whch s a random varable. For example, f both l e and e are related to loadng j operatons, then the startng moment (event) for evaluatng c lj s the pcup of the th lj contaner of QC by QC. Included n C are the travel tme from the apron to the locaton of the next contaner (the j th contaner of QC l) n the marshallng yard, the release tme of the contaner by a YC, and the travel tme of the vehcle to QC l. Let S and D be the sets of O F e v and e v, v V, respectvely. A feasble dspatchng decson s then a one-to-one assgnment between all the events n S T and those n D T. Let K = { O} K, K = { F} K, and x lj be a decson varable that becomes 1 f e s assgned to e l j, for K and l K. For l, K, the assgnment of e to mples that the vehcle that has just delvered the th contaner of QC s scheduled to delver the j th contaner of QC l. Let α be the travel cost per unt tme of a vehcle, and β be the penalty cost per unt tme for a delay n the completon tme. It s assumed that α << β. Further, let m O and m F equal V. The dspatchng problem can then be formulated as follows: l e j Mnmze Subject to m ml lj lj t x + β E Ym s m K = 1 l K j= 1 K + α [( ) ] (1)

ml l K j= 1 m K = 1 l lj lj Yj Y C M x lj x = 1, for K and = 1,, m (2) lj x = 1, for l K and j = 1,, m (3) l ( + ) ( 1), for K, l K, = 1,, m, and j = 1,, ml (4) O Y v = 0, for v = 1,, V (5) Y+ 1 Y s+ 1 s, for K and = 1,, m 1 (6) y s, for K and = 1,, m (7) lj x = 0 or 1, for K, l K, = 1,, m, and j = 1,, ml (8) Because α << β, the sum of the delays to QC operatons wll be mnmzed frst. For the same value of the sum of the delays, the total travel dstance of the vehcles wll be mnmzed. Constrants (2) and (3) force the one-to-one assgnment between all the events n S T and those n D T. Constrant (4) mples that two events that are served consecutvely by the same vehcle must be set apart by at least the tme requred for the vehcle to travel and transfer a load between the two events. That s, x lj can be 1 only f l lj Fj Yj Y C. Note that x, K, s not restrcted by constrant (4). Constrant (6) mples that two events that are served by the same QC must be set apart by at least the tme requred for the QC to perform all the movements between the two events. Constrant (7) sgnfes that the actual event tme s always more than or equal to the lj earlest possble event tme. A feasble soluton of ( x ) s a one-to-one assgnment from a node n S T to a node n D T. Let us express the above formulaton n a more general way as follows: Mnmze f(x, Y) subject to g(x, Y, C) = 0 (9) lj lj Km and Bae [12] solved the problem by settng C = c and ncreasng the values of Y by the smallest possble ncrements so that delay cost could be mnmzed. Once the lj lj values of Y ( Y = y ) and C ( C = c ) are gven, (9) becomes an assgnment problem, wth some assgnments forbdden because of constrant (4). That s, for a gven set of y l and y j, f the nequalty l ( lj lj yj y + c) 0 holds, then x cannot equal 1. The remanng problem s how to ncrease the values of Y. Km and Bae [12] fxed them as follows: Suppose that Y (whch are equal to s n the ntal stage), = 1,..., K, and = 1,, m are gven and the events are sequenced n ncreasng order of Y. We denote event (j) as the j th event n the sequence, y as the tme of event (), c j as the tme requred for a

vehcle to be ready for event (j) after t goes through event () (correspondng to the lj notaton of c ), t j as the pure travel tme from the locaton of event () to the locaton of event (j), and x j as the decson varable for the assgnment of event (j) from event (). Let T be a subset of T, whch ncludes only the frst ξ ξ events n the sequence. The constrant subset ξ of (2) (4) can then be wrtten as follows: (Constrant subset ξ ) j S T ξ j j D T ξ x = 1, for j D T ξ (10) x = 1, for S T ξ (11) y j ( y + cj ) M ( xj 1), for S T ξ and j Tξ (12) x = 0 or 1, for j S T ξ and j D Tξ (13) In the algorthm, for gven values of y, the feasblty of each s checed one at a tme from the constrant subset 1 to the constrant subset T. In the process, f an nfeasble constrant subset s found, the nfeasblty s resolved by ncreasng an event tme so that one or more x j can be allowed to be 1 by constrant (12). Durng teratve procedures of the algorthm, attempts to mnmze the delay to QC operaton are made by ncreasng y by the least possble amount. However, after a feasble soluton to constrant subset T, whch s equvalent to constrants (2) and (3), s found, the total travel tme of vehcles, whch s the frst term of objectve functon (1), wll be mnmzed by applyng the assgnment problem technque. A smlar procedure wll be followed n the algorthm descrbed n ths paper. In Km and Bae [12], because c j s constant, for a gven set of values of Y, t s clear whether the constrant yj ( y + cj) 0 s satsfed or not. However, n constrant (4), because C j s a random varable, t s not certan whether or not yj ( y + Cj) 0 holds. Thus, we defne a probablty functon P j = P{ yj ( y + Cj) 0}, whch can be easly evaluated once P(C j ) s gven, and modfy the constrant subset ξ as follows: (Revsed constrant subset ξ) xj 1 Pj θ, for S T ξ and j D T (14) ξ { ( [ ]/ )} ( 1) yj y + E Cj Pj M xj, for S T ξ and j T ξ (15) and (10), (11), and (13).

Constrant (14) mples that e can be connected to e j only f P j θ. A hgher penalty s gven to the assgnment wth a lower probablty of tmely delvery. That s, c j n (12) s replaced wth E[ C ]/ P. j j A detaled heurstc algorthm can then be descrbed as follows: O Step 0. Intalzng. Set y = s and y = 0, for all vehcles. Set ξ = 0. Step 1. Next Tas. ξ = ξ + 1. If ξ > m (m = the total number of tass n sequence T), then go to Step 4. Otherwse, sequence the events n ncreasng order of y and go to Step 2. Step 2. Feasblty Checng. Chec the exstence of a feasble soluton to revsed constrant subset ξ. If there s a feasble soluton, then go to Step 1. Otherwse, go to Step 3. Step 3. Delayng Event Tme. π * = mn max ( EC [ ξ] P ξ) ( yξ y),0 (, ξ ): P ξ θ. Let ξ { }, where { } S T ξ 1 Denote y γ γ γ as the event tme of event (ξ). Then update * λ yj = yj + π, for j λ. ξ Go to Step 2. Step 4. Tas Assgnment. After settng the assgnment cost of event to j so as to be equal to (t j /P j ), solve the assgnment problem wth the objectve of mnmzng the total assgnment cost. Stop. Feasblty Chec: In ths step, for gven values of y, the feasblty can be checed by solvng a maxmum cardnalty matchng problem n a bpartte graph (Evans and Mnea [4]). When the maxmum cardnalty s the same as S T, revsed constrant subset ξ has a feasble soluton. When solvng the matchng problem n the bpartte graph, arcs from node to j are lned n the graph only f Pj θ. Delayng Event Tme: To satsfy the revsed constrant subset, one or more addtonal x j must be allowed to become 1 by relaxng constrant (15). In other words, the tme for event ξ s delayed so that at least one x ξ, for < ξ, becomes 1, denoted π *. The process ξ s repeated untl the current constrant subset becomes feasble. 4 Smulaton Experments A smulaton model was developed usng Plant-Smulaton software. Detaled operaton of the hypothetcal contaner termnal (Fgure 1) can be descrbed as follows. When a shp arrves, t s assgned a berth f there s one avalable for the shp to enter. Otherwse, the shp must wat untl one becomes avalable. When the shp enters a berth, a prespecfed number of QCs are assgned to the shp. A dschargng and loadng sequence

for contaners s then generated for each QC. Based on the specfed sequence, QCs start to dscharge and load contaners. The wharf of the model termnal n Fgure 1 has one berth and three QCs. The yard conssts of sx storage blocs, and two YCs of the same sze are deployed at each bloc. The total number of vehcles s sx. The vehcles are shared between all the QCs, that s, a poolng strategy s used for dspatchng vehcles. From LIST A for each QC, the eght most mmedate tass are moved to LIST B for dspatchng. That s, the number of loong-ahead tass s 24. The total number of contaners transferred by QCs durng one smulaton run s about 1000. The detaled movements of QCs and YCs (gantry, trolley, and hostng movements) are modeled n the smulaton. The travel tme of vehcles s assumed to follow a unform dstrbuton: U(E[C j ] ± Δ E[C j ]). E[C j ] s calculated usng the travel dstance from the poston of event to that of event j dvded by the speed of vehcles, and Δ s a constant referred to here as the uncertanty factor. The uncertanty factor s set to be 0.2 n the experments. The threshold of the connectng probablty, θ, has a value of 0.5 to 1.0. The performance measures compared n the smulaton experments are the total delay tme of QCs, the total travel tme of vehcles, the total travel tme of empty vehcles, and the vehcle throughput, whch s the number of delvery tass performed per hour. The performance of the proposed heurstc algorthm supportng the uncertantes of travel tmes (LADP-un) was compared wth that of the Greedy algorthm and that of the heurstc algorthm suggested by Km and Bae [12] for the determnstc case (LADP-de). For the Greedy algorthm, whenever a vehcle becomes dle, t s assgned the delvery tas that ncurs the mnmum assgnment cost (t j /P j ) of all the tass n LIST B that can be performed by vehcles wthout volatng constrants. Table 1 lsts the total delay tme of QCs, the total travel tme of vehcles, the total travel tme of empty vehcles, the vehcle throughput, and the computatonal tme for each algorthm. As can be seen n Table 1, LADP-un showed the best performance, and both LADP-un and LADP-de sgnfcantly outperformed the Greedy algorthm n terms of the total delay tme of QCs and the vehcle throughput. However, the Greedy algorthm was the best n terms of the total travel tme of vehcles and the total travel tme of empty vehcles, because both the LADP algorthms frst attempted to mnmze the total delay tme of QCs before then attemptng to mnmze the total travel tme of vehcles as a secondary objectve. In addton, the Greedy algorthm spent the least computatonal tme solvng nstance problems, and LADP-un too relatvely longer than LADP-de n terms of computatonal tme per nstance. Fgure 4 shows the mprovement n the performance of LADP-un over that of other algorthms (Greedy and LADP-de). Its performance was a 55.11% mprovement on that of the Greedy algorthm and 18.45% on that of LADP-de n terms of the total delay tme of QCs. The vehcle throughput of LADP-un was 25.83% larger than that of the Greedy algorthm and 4.15% greater than that of LADP-de.

Algorthm Table 1: Comparson of LADP-un, Greedy, and LADP-de Algorthms. Total delay tme of QCs (s) Total travel tme of vehcles (s) Total travel tme of empty vehcles (s) Vehcle throughput (moves/hour) Computatonal tme per nstance (ms) Greedy 306574 87557 41378 728 17 LADP-de 168748 89443 43078 879 1186 LADP-un 137616 89319 42452 916 1525 Fgure 4: Improvement n Performance of LADP-un Over That of Greedy and LADP-de Algorthms. Fgures 5 8 show the changes n the total delay tme of QCs and computatonal tme, the total travel tme of vehcles, the total empty travel tme of vehcles, and the vehcle throughput, respectvely, for varous thresholds of the connectng probablty. Fgure 5 shows that the total delay tme of QCs decreases rapdly as the threshold of the connectng probablty decreases. As the threshold of the connectng probablty decreases, the number of arcs connectng nodes n the bpartte graph ncreases. As a result, the average computatonal tme per nstance ncreases. However, because the feasble soluton of the problem becomes larger, the soluton qualty mproves, as can be observed n Fgure 5. When the threshold of the connectng probablty falls below a certan value (0.6, as ndcated n Fgure 5), the change n the reducton n the total delay tme becomes smaller.

Fgure 5: Effects of Threshold of Connectng Probablty on Total Delay Tme of QCs and Average Computatonal Tme per Instance. Fgure 6: Effects of Threshold of Connectng Probablty on Total Travel Tmes of Vehcles.

Fgure 7: Effects of Threshold of Connectng Probablty on Total Empty Travel Tmes of Vehcles. Fgure 8: Effects of Threshold of Connectng Probablty on Vehcle Throughput. Smlarly, the changes n the total travel tme of vehcles and total empty travel tme of vehcles are shown n Fgures 6 and 7. The total travel tme of vehcles and the total empty travel of vehcles ncreases qucly as the threshold of the connectng probablty ncreases and reaches 0.9. However, the vehcle throughput decreases as the threshold of

the connectng probablty ncreases, as shown n Fgure 8. Note that the results n Fgures 5 8 compare LADP-de and LADP-un because the case n whch the threshold equals 1 corresponds to LADP-de. The results show that LADP-un outperforms LAPDde n ts objectve values at the expense of greater computatonal tme. For example, Fgure 5 shows that the percentage dfference between the two algorthms n the total delay tme of QCs ncreased from 7.14% to 18.45% as the threshold of the connectng probablty decreased from 0.9 to 0.5. 5 Conclusons Ths paper has dscussed the vehcle dspatchng problem n port contaner termnals whle tang nto account the uncertanty n the travel tmes of vehcles. A heurstc algorthm (LADP-un) was proposed for solvng the problem. Smulaton models were developed to evaluate the performance of the proposed heurstc algorthm under varous condtons. The performance of LADP-un was compared wth a greedy heurstc rule (Greedy) and a heurstc algorthm for the case wth determnstc travel tmes (LADP-de). From the expermental results, t was found that LADP-un outperformed the other algorthms n terms of the total delay tme of QCs and the vehcle throughput. It was also found that the total delay tme of QCs, the total travel tme of vehcles, and the empty travel tme of vehcles decreased rapdly when the threshold of the connectng probablty decreased. Moreover, the vehcle throughput ncreased as the threshold was reduced. Ths study manly ntroduced the schedulng problem for vehcles. As part of future studes, the combned schedulng problem for YCs and QCs as well as for vehcles may be addressed. Acnowledgements Ths study was supported by the Korean Mnstry of Educaton & Human Resources Development through the Research Center of Logstcs Informaton Technologes (LIT). References [1] Egbelu, P. J. and Tanchoco, J. M. A., Characterzaton of Automatc Guded Vehcle Dspatchng Rules, Internatonal Journal of Producton Research, 22, 359 374 (1984). [2] Cho, D. W., A Computer Smulaton Model for Contaner Termnal Systems, Journal of the Korean Insttute of Industral Engneers, 11, 173 187 (1985). [3] Egbelu, P. J., Pull Versus Push Strategy for Automated Guded Vehcle Load Movement n a Batch Manufacturng System, Journal of Manufacturng Systems, 6, 209 221 (1987).

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