UC Berkeley Research Reports

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1 UC Berkeley Researc Reports Title Crane Double Cycling in Container Ports: Affect on Sip Dwell Time Permalink ttps://escolarsip.org/uc/item/9qp7p7jq Autors Goodcild, Anne V. Daganzo, Carlos F. Publication Date escolarsip.org Powered by te California Digital Library University of California

2 Institute of Transportation Studies University of California at Berkeley Crane Double Cycling in Container Ports: Effect on Sip Dwell Time Anne V. Goodcild and Carlos F. Daganzo RESEARCH REPORT UCB-ITS-RR July 2005 ISSN

3 Crane Double Cycling in Container Ports: Effect on Sip Dwell Time Anne V. Goodcild, Carlos F. Daganzo Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, Loading sips as tey are unloaded (double-cycling) can improve te efficiency of a quay crane and tus container port. Tis paper describes te double-cycling problem, and presents two solution algoritms and simple formulae to estimate reductions in te number of operations, and operating time. Te problem is formulated as a sceduling problem. Small problems can be solved to optimality wit a standard numerical solver, but problems of typical size are computationally burdensome and terminated after 10 ours wit optimality gaps larger tan 50%. A formula for an improved lower bound to te optimal solution is developed and sows te optimality gaps are actually below 2.5% in all cases. Te paper presents a greedy algoritm tat can obtain solutions in seconds. A formula for an upper bound to te greedy algoritm s performance can be used to accurately predict crane performance. Te problem is extended to include an analysis of double-cycling wen sips ave deck atces. Results are presented for many simulated vessels, and compared to empirical data from a real-world trial. Te paper demonstrates tat analytical metods can be used in addition to numerical metods to provide greater insigt. More importantly, te paper demonstrates tat double-cycling can create significant efficiency gains. Key words: logistics, freigt transportation, terminal operations, container port terminal

4 Introduction Te volume of goods moved by container troug te US transportation system as grown dramatically over te last 15 years, but our infrastructure as not. Tis as led to significant freigt congestion at transportation terminals, and greater interest in freigt from governmental planning agencies. Strategies are required tat aid te movement of freigt troug te system, and specifically troug terminals. Quay cranes are te most expensive single unit of andling equipment in port container terminals, and because of tis, one of te key operational bottlenecks at ports is quay crane availability (Crainic and Kim 2004). By improving quay crane utilization, ports can reduce sip turn-around time, improve port productivity and improve trougput in te freigt transportation system. Tis work addresses te key bottleneck to port productivity; quay crane utilization. Double-cycling is a tecnique tat can be used to improve te utilization of quay cranes by converting empty crane moves into productive ones. Instead of using te current metod, were all relevant containers are unloaded before any are loaded (single-cycling), containers are loaded and unloaded simultaneously (see Fig. 1). Tis allows te crane to carry a container wile moving from te apron to te sip (one move) as well as from te sip to te apron; doubling te number of containers transported in a cycle (or two moves). Tis crane efficiency improvement can be used to reduce sip turn-around time and terefore improve port trougput, and address te capacity problem. In teir efforts to increase productivity, ports ave undertaken various projects suc as renovating and adding terminals, constructing and expanding intermodal facilities, and implementing new IT infrastructure (Mangelluzzo 2005). Because crane productivity is so important, ports ave also invested in various crane utilization improvement strategies. For example, dual oist cranes ave been developed tat separate te crane s cycle into two subcycles tat can be operated independently. Tese are currently in use at many ports, including Rotterdam and Norfolk. Operational canges are of interest because tey provide benefits at less cost. Te concept of double-cycling is recognized by some in te industry (Ward 2005), (Yau 2003), and its potential to improve efficiency is intuitively understood. Altoug small scale trials ave been undertaken at San Pedro and Tacoma, as well as at ports in Asia and Europe, a broad implementation of double cycling as not occurred. Tere are several reasons for tis. First, small scale trials ave understated te benefits of double cycling wic, as we will sow, increase wit te number of containers. Second, te lack of implementation tools tat detail te necessary operational canges leave some operators doubting tat te benefits will overcome te operational costs. Tird, te absence of a rigorous analysis of te benefits of double cycling leaves te question of its impact on crane operations open. Tis paper aims to address tis final point by providing a quantitative analysis of te impact of double cycling on crane efficiency. We empasize tis point troug comparison wit empirical data. To address point two, we will assume te sip s loading plans are given. Creating vessel loading plans is a complicated process, wit many competing objectives. Sipping lines use software tools to create loading plans tat accommodate vessel stability requirements, placement constraints on azardous materials, refrigerated containers, above, and below deck storage, and may include strategies to minimize te number of cycles necessary to unload containers at subsequently visited ports. From tese tools, a sequence of operations is generated for te crane operator and foreman (wo directs landside operations). Wen

5 considering te benefits of double cycling, tis paper assumes tat existing planning tools ave been used to create a loading plan, as is current practice, and tat tis loading plan as made no accommodations for double cycling. We terefore consider canges only to te crane s sequence of operations. In practice, tis would be determined at a planning stage, and te crane operator would be given a sequence of operations to carry out, in te same way tey are wen performing single-cycle operations. Tis way we may demonstrate tat doublecycling is feasible and beneficial witout disrupting current operations. Altoug no academic researc as addressed te problem of double cycling, a significant amount of researc as addressed oter problems of port design and operation. Port researc typically focuses on strategic design planning issues suc as te number of berts (Sconfeld and Sarafeldien 1985), te size of storage space (Kim and Kim 2002), and te number of various pieces of equipment to install (Vis et al 2001). Operational planning and control problems include bert sceduling (Park and Kim 2003), quay-crane sceduling (Daganzo 1989), stowage planning and sequencing (Cristiansen et al 2004), storage space planning (Castilo and Daganzo 1993), and dispatcing of yard cranes and prime movers (Kim and Bae 1999). To date, most of tis work utilizes queueing teory and stocastic models (Daganzo 1990), simulation (Lai and Leung 2000), and classical operations researc tecniques suc as routing, network, and sceduling problems (Kim and Kim 2002). In te next section, a framework is set for analysis of te problem. Te section also includes an example tat is used to illustrate te problem s basic properties. In Section 2 a sceduling formulation, and te results from its implementation wit a numerical solver are presented. A lower bound to te problem is developed in Section 3. Section 4 presents a new algoritm and bounds its results. Te problem is extended to accommodate sips wit deck atces in Section 5. Section 6 presents te results of simulated vessel loadings and unloadings, tools to convert benefits from cycles to time, and compares time-savings to empirical results. Te paper concludes wit a discussion. 1. Modeling Framework Te layout of containers on a sip can be modeled as a 3-dimensional matrix. Containers are stacked on top of one anoter, and arranged in rows. One row stretces across te widt of te sip. Large container vessels today typically old 20 stacks of containers across te widt of te sip, and up to 20 stacks along te lengt of te sip (40-foot equivalent units). Of course, we expect tese figures to increase wit te penetration of Malacca-max carriers. Figure 2 gives a top and side view of a typical vessel (altoug te number of container stacks is not representative). Te complete operating cycle of te crane can be broken down into te following components; a) locking to or unlocking from a container, b) lengtwise motion along te sip, c) orizontal motion across te sip, d) vertical motion. It is important to point out tat te number of locking and unlocking operations is not affected by double cycling. Te same number of containers will need to be picked up and put down

6 wit single or double cycling. In tis researc we will assume tat dockside containers are ready for loading wen required, and containers being unloaded can be quickly removed from te immediate area. In te initial analysis it will be assumed tat sips lack atc-coverings, or doors on te deck tat separate above-deck and below-deck storage. Tis assumption will be relaxed in Section 5. Consider te case were a sip arrives in port wit a set of containers on board to be unloaded and a loading plan for containers to be loaded. Te loading plan indicates te placement of containers on te sip. Given are u c and l c, te number of containers to be unloaded and loaded, respectively, in eac stack labeled c. Figure 3 is an example problem tat will be used for illustrative purposes. A reandle is a container tan must be moved to access containers below it, but will ten be stowed again before te sip departs. Note tat if any reandles are necessary, tese are included in te total number of loads and unloads. Note tat tis will over-estimate te amount of work necessary to unload and load a set of containers because te container is considered moved from te vessel to te sore and back to a location on te vessel. In Fig. 3, u A = 3 and l A = 2. Te set of stack labels, c, is called S. An ordering of tese stacks can be described by a permutation function, Π. A permutation is a one-to-one correspondence between te set of n {1,, N} and c S suc tat Π(n) = c, or n = Π -1 (c). For example, in Fig. 3, te set of stack labels S = {A, B, C, D} a permutation of tese is {B, A, C, D} given by te function Π e were Π e (1) = B, Π e (2) = A, Π e (3) = C, and Π e (4) = D. If we consider te time it takes to unload and load a sip to be a measure of crane productivity, ten te goal of double cycling is to reduce te total turn-around time. A proxy for tis is te number of cycles. We will discuss time-savings in section 7. Te number of cycles necessary to complete loading and unloading will be represented by te variable w. We will consider double cycling witin one row of te sip. We will complete unloading and loading of one row before moving te crane lengtwise along te sip to te next row. Tus we will not require te crane to move laterally along te sip witin one cycle. We will restrict our attention to special cases of te generic double-cycling metod described below: Coose an unloading permutation, Π. Unload all containers in te first stack of te permutation, ten all containers in te second stack of te permutation, proceed in tis fasion until all stacks ave been unloaded. Coose a loading permutation, Π, and load te stacks in tat order. Load all containers in te first stack, ten in te second, etc.. Loading can start in any stack as soon as it is empty or it contains just containers tat sould not be unloaded at tis port. Once loading as begun in a stack, continue loading until tat stack is complete. Figure 4(a) is a queueing diagram for a single-cycling operation were te stacks of Fig. 3 are andled in te order {A,B,C,D} bot for loading and unloading. Time is expressed in cycles. Note tat loading operations must wait until cycle w = 10, wen unloading is finised. Te process requires w = 20 cycles. If we double-cycle, we can still plot te unloading curve on te same diagram. Now, using te same sequence for unloading and loading, Π = Π = {A,B,C,D}, we can sift te

7 loading curve to te left as far as possible witout overlapping te unloading curve. Figure 4(b) sows te maximum sift. Loading can start as early as w = 4 and te process would require only 14 cycles. Te same number of cycles is obviously obtained if we start loading eac stack as early as possible, as in Fig. 4(c). Tis introduces some delay as te loading operations must wait one cycle for te unloading operations to be completed in stack B, but does not cange te completion time. Wit single-cycling one cycle is required for every container. Wit double-cycling, owever, te number of cycles will depend on te sequence. Fig. 4(d) sows tat if te loading and unloading sequence is {B,A,C,D}, ten te completion time is w = 13. In te next section we formulate te problem of determining Π and Π as a mixed integer program and discuss te results of its implementation using a numerical solver. 2. Sceduling Approac Te problem is formulated as a job sop sceduling problem were jobs are equated wit te loading or unloading of stacks and te crane operates as two macines. Since all work on loading and unloading is assumed to be continuous, we just look for te job completion times. Start times are identified from te known job durations. We use te following notation: u c - number of containers to unload in stack c S l c - number of containers to load in stack c S X c - completion time, i.e. cycles, of unloading c S Y c - completion time of loading c S w - total completion time X kj - binary variable to track ordering of unloading jobs (1 if j S is unloaded after k S and 0 oterwise) Y kj - binary variable to track ordering of loading jobs (1 if j S is loaded after k S and 0 oterwise) M - a large number Te sceduling problem (SP) is to minimize te completion time of all jobs subject to constraints tat uniquely identify te permutations Π and Π, and a feasible set of job start and end times. It is assumed tat te process starts at time zero. Te formulation is: minimize w subject to w Y c c S, (1a) Y c X c l c c S, (1b) X k X j + MX kj u k j,k S, (1c) X j X k + M(1-X kj ) u j j,k S, (1d) Y k Y j + MY kj l k j,k S, (1e) Y j Y k + M(1-Y kj ) l j j,k S, (1f) X c u c c S, (1g) X kj, Y kj = 1, 0 j,k S. (1)

8 Tese constraints completely define te double-cycling problem. Constraints (1a) ensure tat te process ends wen te last stack is loaded or later. Constraints (1b) ensure tat stacks are only loaded wen empty. Constraints (1c), (1d) and (1) ensure tat every stack is unloaded after te previous one in Π as been unloaded. Tis is acieved by specifying for every pair of stacks (j,k) tat eiter stack k is unloaded before stack j (if X kj = 1) or te reverse (if X kj = 0), and tat te time difference between te two events is large enoug to unload te second of te two stacks. Constraints (1e), (1f) and (1) are equivalent to (1c), (1d) and (1) but for loading jobs. Constraints (1g) ensure tat eac unloading completion time allows for enoug time to at least unload tat stack. Te SP is in te family of job sop sceduling problems. Tese problems are notoriously difficult to solve. Heuristic solution algoritms ave been developed for some versions. For a summary of current solution tecniques see Pinedo (2002), or Applegate and Cook (1991). Tese metods are not readily applicable to te SP. In our problem, tere are two macines; an unloading crane and a loading crane (in reality tis is one device). Eac stack must visit te unloading crane before te loading crane, and containers are only loaded into stacks from wic all relevant containers ave been unloaded. Sip data were generated to evaluate te algoritm. Te numbers of containers to load and unload in eac stack were assumed to be independent and uniformly distributed between 0 and 10. A standard numerical solver (AMPL wit CPLEX 8.0) was ten used to determine te optimal permutation. Table 1 sows te number of stacks, solution time, number of runs, and optimality gap. Runs were carried out on a 1.4 MHz Dell Poweredge 1650 server, running Linux wit 2 GB of real memory and 1 GB of swap space. For small problems (10 stacks and fewer), te problem could be solved optimally. However, average solution times were found to grow quickly wit problem size (number of stacks). Problems wit more tan 10 stacks (about alf te size of typical problems) could not be solved optimally in less tan 10 ours. Te optimality gaps of Table 1 are te percent difference between te solution at ten ours and te igest lower bound known. Te igest lower bound known is te solution to an infeasible relaxation to te problem. For example, if te solution was 100 cycles, and te igest lower bound known 75 cycles, ten te optimality gap would be 25%. Tese gaps increased wit problem size and were rater large, even for N = 15, so te quality of te 10-our solutions were uncertain. Tis sows tat, for small problems, te numerical solver provides a good solution. For typical problems owever, furter analysis is desirable. 3. Lower Bound We will now determine a lower bound for te number of cycles, wic can be used to better judge te quality of te numerical solver s solutions. Define: Y = N n= 1 u ( n) uc, Λ = lπ n) = Π = c S N n= 1 l ( c. c S Recall from Fig. 4(a) tat using single-cycling, te number of cycles necessary to complete a row is:

9 Υ + Λ. Tis is intuitive; one cycle for eac container. We ave assumed te crane must start and finis on te dock. For double-cycling, wit a specific loading permutation Π and unloading permutation Π, te number of cycles must be at least Λ + u Π'(1) wic satisfies: Λ + u Π Λ + min ( u ). (2) '(1) c c Te rigt-and side of (2) is a general lower bound tat applies to all permutations. It is also a lower bound if we force te same permutation for loads and unloads ( Π = Π). More specifically, if w 1 * is te optimal number of cycles wit te same permutation for loading and unloading and w 2 * is te optimal number of cycles wit different permutations, we can write: * * w w Λ + min ( u ). (3) 1 2 c c 1. Te new lower bound reveals tat te quality of te solution determined by te numerical solver (in 10 ours) is quite ig. Revised optimality gaps, using (3) for te lower bound, are given in Table 2. Currently, rows do not accommodate more tan 20 stacks, but it is useful to understand te beavior of te algoritm across a range of problem sizes. Vessels will continue to grow in size, and, in addition, tecnologies may cange to allow for more efficient lateral movement of te crane. Tis is wy our results were extended to 250 stacks. For current vessels, te generic algoritm is not fast enoug to be used for real-time applications. Terefore, we will discuss an alternative algoritm tat can be used for real-time applications, tat is simpler, and provides greater flexibility and insigt. Te algoritm yields an upper bound formula for te required number of cycles. Te formula can be evaluated witout running te algoritm, and tus is useful for planning purposes. 4. Greedy Strategy and an Upper Bound We propose to unload and load eac stack as soon as possible, assuming tat te loading and unloading sequences are given by te same greedy permutation, Π = Π = G. Tis greedy permutation is obtained by ordering te stacks in descending order of te variable d c were: d c = l c u c ( if Λ Υ), (4a) d c = u c l c ( if Λ < Υ). (4b) Te rationale for (4a) is tat we want te unloading operations to run aead of te loading operations as muc as possible. We will now introduce some terminology wic will allow us to prove te existence of an upper bound on te number of cycles required using te greedy strategy. Associated wit tis strategy, we define functions U(w) and L(w) for te number of stacks completed as illustrated by te solid curves on te top and bottom parts of Fig. 5. We

10 also define L -1 (c) and U -1 (c) for integer c as te inverse functions, and U c = u G(c) and L c = l G(c) as te orizontal steps of te curves. See Fig. 5 and te notations along te bottom of te abscissa s axis. Note tat U(Y) = L(Λ) = N. Define w G as te number of cycles wit G. * * THEOREM 1. If Λ Y, ten Λ + min ( u ) w w w Λ + max ( u ). c c 2 1 G c c * PROOF. In view of (3), it suffices to prove w w G Λ + max ( u ). Since 1 c c * w G w 1 it suffices to sow tat Λ + max c ( uc ) wg. Consider a orizontal sift, L, to curve L, tat will ensure curve L is to te rigt of curve U (as in te dotted curve in te top of Fig. 5), i.e. tat L -1 (c-1) + L U -1 (c) for c = 1,... N. Obviously te least possible delay, L 0, is max i (U -1 (c) L -1 (c-1)) = max c (U c + (U -1 (c-1)-l -1 (c-1)). Since Λ Y, te greedy strategy satisfies (U -1 (c-1) L -1 (c-1)) 0 c. Hence L 0 max c (U c ). Terefore max c (U c ) + Λ = Λ + max c ( uc ) is an upper bound to te number of cycles. * * COROLLARY 1. If Λ Y, ten Υ + min ( l ) w w w Υ + max ( l ). c c 2 1 G c c PROOF. G in tis case is defined by (4b). Tat Corollary 1 is true sould be obvious by symmetry. If one were to record te process of unloading and loading te row, and ten play tis recording in reverse, te reversed movie would display a sequence of operations wit te same total time as for a problem in wic te role of loads and unloads is switced. Tus, for every problem instance wit loads greater tan unloads, tere is a dual instance were unloads are greater tan loads. Figure 6(b) sows te reverse movie of Fig. 6(a). Note tat te greedy strategy continues to be greedy in reverse, and tat everyting said up to tis point, including te bounds, continues to old wit time running backward. Note tat in Fig. 6(b) te greedy strategy wit time running backward implies a reverse ordering of {d c }, as specified in (4b), tus Corollary 1 olds. * * Define u = min c ( uc ), l = min c ( lc ), u = max c ( uc ), and l = max c ( lc ). Ten given Teorem 1 and Corollary 1 we can write: * * * * max( Λ + u, Υ + l ) w w wg max( Λ + u, Υ + ). (5) 2 1 l For rows of large sips were (Λ,Y) >> (u *, l * ), te greedy strategy is very close to bot te upper and lower bounds since all te members of (5) are relatively close to max(λ,y). More specifically, note tat if u c and l c are bounded by a constant (stack size) ten te percent optimality gap vanises as te number of stacks (problem size) tends to infinity. Tus te greedy strategy is asymptotically optimal. Figure 7 compares te performance of te sceduling and greedy algoritms. Te percentage difference between eac strategy s solution and te minimum is sown. In general, bot algoritms perform well. As expected, te greedy algoritm s performance improves wit row size. Wit te sceduling problem we could not provide an upper bound formula for te savings tat double-cycling would provide over single-cycling. Using te

11 greedy strategy, we ave been able to quantify te improvements double-cycling provides in all cases. Results for te greedy strategy are comparable to tose of te sceduling algoritm (sligtly better for te largest problems), but can be obtained more quickly. In te next section, we consider deck atces as an extension to te problem considered so far. 5. Deck Hatces Most container sips currently ave atc coverings, as sown in Figure 8. Tese are large steel plates tat separate above-deck and below-deck storage. In Fig. 8(a) we ave sown a vessel two atces wide, wit tree stacks of containers above and below eac atc. In Fig. 8(b) we ave sown a vessel tree atces wide, wit five stacks above and below eac atc. Typical sips are tree atces wide, eac atc containing 5 or 6 stacks of containers. Hatces cange te nature of te problem already addressed because te stacks are no longer independent. To access te containers below a atc all containers must be unloaded from above te atc, and before loading containers atop a atc all containers below te atc must be loaded. Te following algoritm (te greedy atc strategy) will be considered. As in te atcless case, assume tat eac stack on te sip is given an initial label. To carry out te strategy it is necessary to: 1. Order te atces using a greedy strategy using te same metod as for te atcless case. Treat te atces as stacks, considering only te containers atop te atces. 2. Order te stacks witin eac atc using a greedy strategy, considering only te containers below deck. Te algoritm is ten carried out as follows: 1. Apply te greedy strategy to te containers above deck, treating atces as stacks, pausing eac time all containers above atc ave been removed. 2. During te t pause, unload and load te containers below te t atc using double-cycling. As wit te atcless case, tis metod may not provide te least number of cycles to complete loading and unloading operations on te row. Te value of te metod is its simplicity; parts of te unloading and loading process are equivalent to te atcless sip problem already addressed. Eac piece below a atc is equivalent to a atcless sip, and te containers above a atc to a stack of a atcless sip. Tese relationsips will allow us to use te analysis of te atcless sip to develop bounds for te atced case. First it is necessary to define some notation. atc label S te set of stacks for atc N te number of stacks of atc H te set of atces F te number of atces

12 u c te number of containers to unload below atc in stack c S u te number of containers to unload above atc in stack c S c l c te number of containers to load below atc in stack c S l te number of containers to load above atc in stack c S c u = containers to unload below atc u c c S u = containers to unload above atc u c c S l = containers to load below atc l c c S l = containers to load above atc l c c S Y = u c containers for unloading below deck H c S Y = u containers for unloading above deck Λ = Λ = H c S l c c H c S c H c S containers for loading below deck l containers for loading above deck w A = te number of cycles above deck w B = te number of cycles below deck w B, = te number of cycles below atc Also define te greedy permutation for atces; P(f) = ; P -1 () = f for f = 1... F; H and te greedy permutation for containers below atc, H: Π(n ) = c S ; Π -1 (c ) = n for c S, {n = 1...N } THEOREM 2. An upper bound on te optimum number of cycles for te atced case is max{ u, l } + max{ Λ, Υ} + max { u, l } + max c S { u c, l c}. H PROOF. We know from (5) tat w A = number of cycles for a row of F stacks wit data given by {u,l } max{λ,y} + max {u,l }. Likewise w B = w B, = (number of cycles for a row of N stacks wit data given by H H { u c, l c} ) max{ u, l } + max c S { u c, l c}. Obviously ten, te total number of H cycles wit te algoritm satisfies w A + w B max{ u, l } + max{ Λ, Υ} + max { u, l } + max { u c, l c}. H c S

13 THEOREM 3. A lower bound on te optimum number of cycles for te atced case is max{ u, l } + max{ Λ, Υ} + min { u, l } + min c S { u c, l c}. H PROOF. We know from (5), but treating eac atc as a stack, tat te lower bound on te number of cycles above deck is max{ Λ, Υ} + max { u, l }. We also know from (5), but treating eac atc as a vessel, tat te lower bound on te number of cycles below deck is max{ u, l } + max { u c, l c}. H H c S Clearly, for rows were max{ u, l } + max{ Λ, Υ} >> max { u, l } + max { u c, l c}, bot te upper and lower bounds are close to te solution provided by te greedy strategy and max{ u, l } + max{ Λ, Υ} provides a reasonable estimate of te total unloading and H loading time for one row. As wit te atcless case, te gap between te upper and lower bound is quite small, and decreases wit te size of te row. Tus, te simple greedy algoritm is reasonably efficient. If one only double-cycles below deck, as is current practice, te benefits of double-cycling will be reduced by rougly te ratio of containers above deck to containers below deck. 6. Evaluation Tis section addresses te magnitude of double cycling benefits. We present simulation results tat estimate benefits in cycles for rows of current vessels. We also present tools to convert benefits from number of cycles to an amount of time, and compare tese tools to data collected in a real-world trial of double cycling. c S 6.1 Simulation Results To understand te benefits on a larger scale, a simulation program was developed to generate sip data for many vessels and calculate te number of cycles necessary to complete operations in eac row using various double cycling strategies. For all results sown ere, double-cycling only takes place below deck. Tis is current practice, and matces te realworld trial at te Port of Tacoma. Loading plans were generated from sets of parameters intended to approximate te universe of actual and future sips. For more details see Goodcild (2005). Results are sown in Figure 9. As expected, te number of cycles using double cycling is always less tan single cycling. Of course, if tere are no containers to load, or unload, ten double cycling provides no benefit. Similarly, tere exist vessel loadings were double cycling will cut te number of cycles almost by alf (if we were to ave no containers above deck). But, on average, tere is a 21% reduction in te number of cycles wen using te greedy strategy. Of course, tis percentage would be greater if we could also double-cycle above deck. Eac time we replace two single cycles wit one double cycle, we reduce te distance traveled by te crane, and save some time.

14 6.2 Time-Savings We will use te following notation. Please refer to figure 10. W time saved for eac replacement of two single cycles by one double cycle S r - number of cycles required to turn-around row r {1 R} using single-cycling D r - number of cycles required to turn-around row r {1 R} using double-cycling d r - number of cycles moving two containers wile operating on row r {1 R} V - oist speed of te trolley wen not moving a container V l speed of te crane wen moving lengtwise along te vessel V t orizontal travel speed of te trolley wen not moving a container d V vertical distance from te apron to te maximum eigt container reaces d L lateral distance between two rows of te vessel b orizontal distance from landside veicle to te edge of te vessel P widt of te vessel T r time required to position landside veicle after departure of previous veicle For eac double-cycle we save some empty-crane travel relative to te two corresponding single cycles, but we also experience a sligt landside-repositioning penalty. Te time penalty, T r, is incurred because after dropping a container for unloading onto a landside veicle, te crane must wait for a container for loading to be positioned below te crane. Wit single cycling, tis can be done simultaneously wit oter crane operations. Any minor vertical motion necessary between dropping a container on te landside veicle and picking up te next container (to remove te trolley from te work area) will also be accounted for in tis penalty. Te total distance traveled by te crane is reduced by one complete empty cycle between te apron and te position above eiter te container to load or te container to unload. Of course, some orizontal and vertical motions will take place simultaneously. An upper bound on te time saved per cycle, W, is provided by summing te orizontal and vertical travel times, te lower bound by taking te maximum of te two: d b ( 1 ) P ( 1 ) 3 d P 3 V V Tr > W > 2 max + r V Vt Vt, V V (6) t Vt For eac double cycle te vertical distance is reduced by 2d V. Te orizontal distance is reduced by twice te distance between te apron and te closer of te two containers. Tis is 2( b + (1/ 3) P), assuming a uniform distribution of locations. If double cycling only below deck, te maximum eigt reaced wile double cycling will be te clearance eigt of te vessel. Te number of cycles saved by double-cycling in eac row is d r = S r D r. After completing loading and unloading operations on a row te crane moves along te vessel to te next row. If tere are R rows on a vessel, te time consumed wit lateral motion is 2*(R-1)*d L /V l wit single cycling, and (R-1)*d L /V, exactly alf, wit double cycling. b T

15 6.3 Validation In June 2003, te Center for te Commercial Deployment of Transportation Tecnologies, Transystems, te Port of Tacoma, and Wasington United Terminals, worked togeter on a full-scale demonstration of te Efficient Marine Terminal concept. Double cycling of container cranes is a key element of tis concept and on June 28, one bay of a Hanjin vessel was loaded and unloaded simultaneously using double cycling (TranSystems Corp. 2003). Double cycling occurred below deck only. During tis trial, te adjusted average time for a single cycle was 1 minute and 45 seconds, and for a double cycle, 2 minutes and 50 seconds. We now compare te difference in tese empirical cycle times to te differences obtained using te expressions developed above. Parameter values based on te trial at Tacoma are given in Table 3 (Garcia, 2005). Te lower bound is 25.4 seconds. Te upper bound, 39.8 seconds. Te upper bound is very close to te empirical difference of 40 seconds. We expect tese empirical results to better matc te upper bound as we assume a constant (maximum) speed of te crane. Clearly te specific results depend on te parameters of eac crane, vessel, and container arrangement, but we ave demonstrated tat our tools provide results tat matc empirical data. A 21% reduction in te number of cycles reduces operating time by approximately 8%. A 35% reduction in cycles would reduce operating time by 13%. We can estimate te financial benefits of double cycling by comparing productive sip time (moving containers) to unproductive time. Intercontinental freigt rates are estimated at $.045 per TEU-km (Notteboom 2004). At tis rate, eac our of idle vessel time costs te sip operator $10,000. Te 40 second benefit of eac double cycle is wort approximately $100. During typical operations, double cycling could provide approximately $350 per our, saving about $15,000 on a large container vessel. Wit tree cranes per vessel, tis is comparable to te total crane operator s wages for loading and unloading te vessel. 7. Discussion Troug te researc presented in tis paper we ave developed an understanding of te impact of double-cycling on loading and unloading operations bot wit respect to te number of cycles, and amount of time. For very small problems an exact, optimal solution is obtained. For larger problems, a sub-optimal strategy is developed tat can be solved quickly. Results for tis strategy are tigtly bounded by simple formulae, and comparable to results obtained after 10 ours using a numerical solver. Savings in cycles and time can be quickly calculated allowing for a flexible and transparent process. Te results obtained are applicable to a wide range of scenarios including large and small vessels, tose wit many or few port visits, and a varying number of cranes operating per sip. An alternative approac tat focused on providing more accurate information for specific cases would require detailed information on eac container. Our approac relies on only a small number of parameters, and sensitivities to tese parameters are easily measured. Favorable comparisons to empirical results support our approac.

16 Double-cycling wile unloading and loading te vessel creates an opportunity to doublecycle landside equipment. Cassis used to deliver containers to te apron can ten carry an unloaded container to local storage. Typically, tese cassis return to local storage empty. Future work will consider double-cycling wit landside equipment in more detail. Our ongoing researc focuses on addressing planning level decisions suc as te impact of doublecycling on te need for landside transport veicles, apron space, and port productivity. We are also working on designing loading plans to create more opportunities to double cycle. Finally, we would like to empasize a key result; tat any amount of double cycling will reduce te time required to turn around te vessel, at $100 per cycle te savings can be very significant. Wile te benefits of double cycling will not solve current port congestion tey can be implemented quickly and, in conjunction wit oter measures, can ease congestion before more long-term infrastructure projects come on line. Acknowledgements Tis work as been funded in part by te National Science Foundation, Grant CMS References Applegate, D., W. Cook A Computational Study of te Job-Sop Sceduling Problem. ORSA Journal on Computing Castilo, B. D., C. F. Daganzo Handling Strategies for Import Containers at Marine Terminals. Transportation Researc B: Metodology Cristiansen, M. et al. Fortcoming. Maritime Transportation, in Handbooks in Operations Researc and Management Science: Transportation, C. Barnart and G. Laporte (eds) Nort- Holland, Amsterdam, Te Neterlands. Crainic T.G., K.H. Kim. Fortcoming. Intermodal Transportation, in Handbooks in Operations Researc and Management Science: Transportation, C. Barnart and G. Laporte (eds) Nort-Holland, Amsterdam, Te Neterlands. Daganzo, C.F Te Crane Sceduling Problem, Transportation Researc B: Metodology Daganzo, C.F Te Productivity of Multipurpose Seaport Terminals. Transportation Science Garcia, B Personal Conversations. TranSystems Corp. Goodcild, A.V An Analysis of Double Cycling and its Impact on Port Operations. PD Dissertation. University of California at Berkeley. Kim, K.H., J.W. Bae A Dispatcing Metod for Automated Guided Veicles to Minimize Delay of Containersip Operations. International Journal of Management Science

17 Kim, K.H., H.B. Kim Te Optimal Sizing of te Storage Space and Handling Facilities for Import Containers. Transportation Researc B: Metodology Lai, K.K., J. Leung Analysis of Gate House Operations in a Container Terminal, International Journal of Modeling and Simulation Mongelluzzo, B Ports Rus to Keep Up, Journal of Commerce 6(9) Notteboom, T Container Sipping and Ports: An Overview. Review of Network Economics 3(2) Park, Y.M., K.H. Kim A Sceduling Metod for Bert and Quay Cranes, OR Spectrum Pinedo, M Sceduling: teory, algoritms, and systems, Prentice Hall, Upper Saddle, N.J.. Sconfeld, P., O. Sarafeldien. (1985) Optimal Bert and Crane Combinations, Journal of Waterway, Port, Coastal and Ocean Engineering TranSystems Corp Efficient Marine Terminal: Full Scale Demonstration. (available at Vis, I.F.A., R. de Koster, K.J. Roodbergen Determination of te Number of Automated Guided Veicles Required at a Semi-automated Container Terminal, Journal of te Operational Researc Society Ward, T Personal Conversations, JWD Group. Yau, V Personal Conversations, American Presidential Lines.

18 Tables and Figures Single-Cycling Double-Cycling Unload container (a) Return witout container Unload container (b) Load container Figure 1 (a) Unloading using single-cycling (b) Unloading and loading wit double-cycling. stack label Plan View A B C D E F I H G stack container Side View Figure 2 Plan and side views of a simplified sip (number of containers sown not representative of typical sip size).

19 Containers to Unload: u A Reandles Containers to Load: l D A B C D A B C D Container to be reandled Key: Container to unload Container to stay on vessel Container for load Figure 3 Detailed plan for containers to be unloaded and loaded.

20 (a) (b) Stacks completed D C B A Y Λ Stacks completed D C B A Number of cycles (c) Number of cycles (d) Stacks completed D C B A Stacks completed D C A B u B Λ Number of cycles Number of cycles Unloads Loads Figure 4 Turn-around time wit different metods: (a) single-cycling wit ordering {A,B,C,D} unloading starts at w = 10, 20 cycles (b) double-cycling wit ordering {A,B,C,D} unloading starts at w = 4, 14 cycles. (c) doublecycling wit ordering {A,B,C,D} unloading starts at w = 3, 14 cycles. (d) double-cycling ordering {B,A,C,D} unloading starts at w = 3, 13 cycles.

21 Number of Stacks Average CPU Solution Time in minutes (limited to 10 ours (600 minutes)) Number of runs represented Optimality Gap ** ** ** ** ** ** ** ** ** 15 * % 20 * % 25 * % 30 * % 35 * % 40 * % 45 * % 50 * % 100 * % 150 * % 200 * % 250 * % Table 1 Solution time and optimality gap for sips of increasing size. ** indicates te optimal solution was obtained so optimality gap is zero. * indicates run was terminated at 600 minutes. Number of Stacks Solution obtained after 10 ours Difference between solution and lower bound Lower Bound % % % % % % % % % % % % Table 2 Te difference between te 10-our solution, and te lower bound

22 5 4 U(Y) = N C Number of Stacks Completed L U 2 U(w) U 3 U 4 L -1 (1) + L A D B Specific Stacks Completed Unloads Loads 0 U -1 (0) U U -1 (1) U -1 (2) U -1 (3) U -1 (4) Number of Cycles, w 5 4 L(Λ ) = N C Number of Stacks Completed L(w) L 2 L 4 = 0 L 3 A D B Specific Columns Completed Unloads 0 L -1 (0) L L -1 (1) L -1 (2) L -1 (3) Number of cycles, w L -1 (4) Figure 5 Curves L and U of te greedy strategy

23 (a) 5 Number of Stacks Completed U 2 U 3 U 4 L 4 = 0 L 3 L 2 C A D B Specific Stacks Completed Unloads Loads U 1 L Number of cycles, w (b) 4 2 of Number cycles, w Number of Stacks Completed 5 D C A B Unloads Loads Specific Stacks Completed Figure 6(a) Definition of te greedy strategy (b) Rotated version of part (a).

24 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Numerical Solver Greedy Algoritm Figure 7 Performance comparison of te sceduling and greedy algoritms. (a) Plan View Side View (b) Figure 8 (a) Plan view and side view of a sip wit atc coverings (b) Rear view of a simple atced sip.

25 275 save 0% 225 Number of moves save 21% save 50% greedy strategy single cycling greedy strategy single cycling Number of moves using single cycling Figure 9 Comparison in te number of moves required to turn-around a row using single cycling and te greedy strategy. Red bars sow te distance between te greedy strategy te general lower bound. Savings of 0%, 21%, and 50% are indicated. orizontal distance vertical distance d V b P Figure 10 Horizontal and vertical motion of te crane. Parameter V V t Value 300 feet per minute 500 feet per minute

26 P 130 feet d V 75 feet b 60 feet T r 15 seconds (Ward 2005) Table 3 Parameter values for evaluation, based on Port of Tacoma trial.