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Real-time trailer scheduling for crossdock operations.

Abstract

Crossdocks play an important role in supply chain operations. Due to the need to decrease transportation lead time to coordinate with other supply chain activities such as just-in-time, make-to-order, or merge-in-transit strategies, shortening the total transfer time at crossdocks is increasingly important. In this research, we use real-time information about freight transferring within a crossdock to schedule waiting inbound trailers in order to reduce the time freight spends in a crossdock. We use dynamic simulation models to compare the performance of several strategies. These are first-come, first-served, look-ahead, minimum processing time, and minimum total time policies. We examine these under different trailer arrival headways, crossdock layouts, and destination distributions. Our simulation results show that our time-based algorithms save more time than the first-come, first-served and look-ahead policies. In addition, these algorithms appear to result in higher service reliability and productivity.

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Crossdocking is a type of hub-and-spoke network operation used to distribute goods from suppliers and manufacturers to vendors and retailers. Through consolidation processes, shipments from various suppliers can be rearranged to create full truckload shipments destined to different retailers in order to attain transportation economies. In addition, retailers' average inventory levels and order cycle times are reduced.

The crossdock is the hub of its distribution network. Any delay in its freight handling can hinder the performance of the whole network. Hence, minimizing the time and/or costs incurred when transporting freight from inbound trailers to outbound trailers in the hub is the main challenge of crossdock operations.

In the past, because of the lack of real-time information about incoming and outgoing shipments, a crossdock supervisor could only draw on his or her past experience to assign waiting trailers to receiving doors. Therefore, the efficiency of typical crossdock operations was, by definition, sub-optimal. For example, case studies such as Gue (1999), Bartholdi and Gue (2000), and Brown (2003) report improvements from 7 percent to 23 percent when applying more efficient scheduling methods. Due to the development of technologies, real-time information about the contents, locations, and destinations of shipments in a crossdock is readily available. For example, with advance shipping notices, information about the contents of incoming trailers is known before they arrive; with radio-frequency identification (RFID) tags attached on freight and RFID readers installed at receiving and shipping doors, information about the movement of freight within the crossdock is available any time. In collaborating with a warehouse management system, this real-time information should allow for the development of more efficient operations.

This research uses real-time information to schedule trailer unloading in order to decrease total freight transfer time. Trailer scheduling policies directly impact freight wait time and travel time and thus affect the performance of crossdock operations. Even though trailer scheduling is important, few studies about this topic have been published. To date those studies focus mainly on minimizing travel distance for the worker. From a practical viewpoint, the travel time from a receiving door to a shipping door might take less than five minutes, but the wait time for one unit of freight in an incoming trailer to be unloaded and for the outgoing trailer to be fully loaded and ready to leave might exceed an hour. Therefore, instead of only measuring travel time/distance to assign waiting trailers, taking into account the wait time that a waiting trailer will impose on itself and other freight should have the potential to increase the efficiency of crossdocking.

In the second section of this article we introduce types of crossdocks and related papers about door assignment and trailer scheduling. In the third section, we propose two time-based algorithms that aim at considering processing time or transfer time to assign waiting trailers. To evaluate these algorithms we built detailed simulation models to imitate the transfer of freight under our algorithms and two other scheduling policies: the first-come, first-served (FCFS) and look-ahead policies. We describe the models and data sets in the fourth section. Simulation results are compared in section five. We conclude by discussing the time-saving effects our algorithms can achieve for crossdock and supply chain operations.

LITERATURE REVIEW

Types of Crossdocks

The crossdocking system, like the just-in-time (JIT) system, is designed to reduce inventory and processing costs, and also aims to attain full-truckload transportation economies within shorter order cycle times. Crossdocking attains these benefits by transferring shipments directly from inbound trailers to outbound trailers with no storage in between. Usually, shipments are shipped within twenty-four hours, but sometimes it could be less than one hour (Apte 2000; Gue 2001).

Crossdocks can be classified by the types of freight handled, the timing of information flows, or the types of staging involved. Here we discuss only the staging types. In Bartholdi, Gue, and Kang's (2001) article, crossdocks are categorized into single-stage, two-stage, and free-stage types. In a single-stage crossdock, pallets are put into queues corresponding to their receiving or shipping doors (see Figure la). So the ratio of receiving doors to shipping doors is close to one. Sandal (2005) and Bartholdi, Gue, and Kang (2001) both use the door ratio of one in their staging queuing studies. In a two-stage crossdock, workers put pallets on the first staging lanes corresponding to the receiving doors. Another set of workers sorts them to the second staging lanes corresponding to the shipping doors (see Figure 1b). Accordingly, more time and labor are required than for the single-stage type. Free staging areas are usually used in the less-than-truckload (LTL) trucking industry. The receiving and shipping doors in an LTL crossdock could be on both sides (see Figure 1c). LTL terminals range in size from six to eight doors to more than 200, even more than 500 doors (Gue 1999). The ratio of receiving doors to shipping doors is close to one half (Gue 1999; Taylor and Noble 2004).

Door Assignment and Trailer Scheduling

Because of the quick freight-transferring characteristic of crossdocks, sizable and dense freight handling is ordinary and necessary. With many trailers arriving at a crossdock during the course of operations, dispatchers need to determine how to handle them efficiently to make the best use of the crossdock's capacity. Assigning trailers to unloading doors, arranging receiving/shipping door locations, optimizing the number of workers and the number of facilities, using staging strategies, and coordinating inbound/outbound schedules are all possible ways to improve these operations. In this article, our focus is on how to assign incoming trailers to receiving doors to help reduce total transfer time for freight.

Allocating shipping doors' locations according to their demands is a good way to keep workers' travel distances short. Tsui and Chang (1990) were the first ones to formulate this door assignment problem. In their paper, they minimize the weighted distances between receiving and shipping doors for LTL crossdocks. Solution algorithms have also been proposed by Tsui and Chang (1992) and Bermudez and Cole (2001). However, this door assignment method works only when the number of incoming and outgoing trailers equals the number of doors. To extend this problem, Lim, Ma, and Miao (2006a, 2006b) segregate trailers into groups by specifying their arrival times and departure times to accommodate the number of trailers greater than the capacity of a crossdock. Because of the specification of trailer departure times, in their study penalty costs are incorporated to account for the costs of unfulfilled shipments.

[FIGURE 1 OMITTED]

Arguing that congestion in front of receiving and shipping doors could increase labor costs and delays if solely considering shortest distance, Bartholdi and Gue (2000) identify three types of congestion that can occur in an LTL crossdock (these are interference among forklifts, dragline congestion, and floor space congestion) and minimize the total labor cost, which includes travel and congestion costs. In their model, they use aggregated demand for each destination (shipping door) and average arriving quantities for each incoming trailer to allocate receiving and shipping doors applying their cost function. Their result represents a good crossdock layout in which trailers have similar loads.

The above door assignment algorithms are suitable for tactical planning that is usually updated once a month or every couple of months. At the operational level, when only one receiving door is available, and several trailers are waiting for unloading, we need to find a way to choose a single trailer from a waiting line. This is the trailer scheduling problem explored in this study. Conventionally, the FCFS rule is the way to assign the next trailer to unload. This rule is fair with respect to the wait times of the trailers, but may not be beneficial to the overall operation of crossdocks.

A scheduling idea similar to the minimizing weighted distance criterion is proposed by Gue (1999). His look-ahead scheduling algorithm turns static criteria into rules that are applicable in a dynamic environment: Each incoming trailer is assigned ranks for each shipping door according to the weighted distances of its contents before or when it is in the trailer waiting line. When a receiving door is available, the look-ahead scheduling algorithm will search for the trailer in the trailer waiting line with its first choice for that receiving door. If none exists, it finds the trailer that would have the second lowest weighted distance when assigned to that receiving door. This process continues until an assignment is made. For example, waiting trailers one, two, and three have their first three priorities as (A, D, E), (B, A, C) and (A, C, B), respectively. When receiving door A is available, waiting trailer one will be chosen because receiving door A can give waiting trailer one the lowest weighted travel distance and it arrives at the trailer waiting line prior to waiting trailer three. On the other hand, if receiving door C is available, since no waiting trailers have receiving door C as first priority, the waiting trailer with its second priority for receiving door C (waiting trailer three) will be chosen. This algorithm is convenient to implement as long as the information about the destinations of pallets in incoming trailers is available. Gue claims a 15 to 20 percent saving in labor cost due to travel, compared to the FCFS rule. Brown (2003) also applies the look-ahead rule in her study.

TIME-BASED TRAILER SCHEDULING ALGORITHMS

From the point of view of crossdock operations, minimizing workers' travel costs is important, and hence this is the main objective of the look-ahead algorithm. However, from the perspective of a whole supply chain, minimizing transfer time could be the most critical issue for a crossdock, especially for time-oriented logistics strategies and high-value and perishable items (Cook 2007). Time-oriented logistics strategies like just-in-time, make-to-order, and merge-in-transit all require short lead time to achieve their feature of flexibility. High-value products have high corresponding holding costs, and transferring perishable products faster can maintain their freshness and quality. In addition, crossdocks work as hubs of a supply chain and hence the less time freight stays at a crossdock, the more efficient the supply chain will be.

Our time-based approaches are concerned with the impact of a new unloading trailer on the total processing time or the transfer time needed for existing pallets in the crossdock and the pallets from the new unloading trailer. The trailer scheduling algorithms require dynamic information. Whenever a receiving door is available and there is more than one new trailer waiting, the algorithms calculate and compare the total processing time or the whole transfer time needed for each alternative waiting trailer. The waiting trailer with the lowest processing or transfer time will be chosen. Two time-based algorithms are introduced below. The first one considers the total processing time for all pallets in the crossdock, while the second one considers the whole transfer time.

A Criterion Based on the Processing Time during the Unloading of a Waiting Trailer

Pallets' travel time between receiving doors and shipping doors, wait time at receiving doors, and wait time at shipping doors are considered here, and the total of these three times is called the "processing time." Since the destinations of pallets in an unloading trailer are given, the total travel time of the trailer depends on which receiving door is assigned, and this also affects the wait time at receiving doors. The wait time at shipping doors is another situation. A pallet arriving at an outgoing trailer will stay there until the outgoing trailer is fully loaded and leaves the crossdock. If more pallets are sent to an outgoing trailer and the trailer becomes full, then the wait time of the pallets already in the trailer is reduced. Therefore, the mix of pallet destinations in each new waiting trailer impacts the wait time at shipping doors.

We would like to assign a waiting trailer that can minimize the processing time, taking into account the new waiting trailer and all pallets at receiving doors and shipping doors at the time of the assignment. Figure 2 shows a simplified example of how to make the assignment using this time-based algorithm.

In Figure 2, trailers one, two, and three are waiting in the trailer waiting line. When receiving door C becomes available, receiving doors A and B still have two and one pallets, respectively, waiting to be transported. We call these pallets "existing" pallets. The first part of the symbol under an existing pallet represents its destination (shipping door) and the second part displays the cumulative time, starting when a newly assigned trailer begins its unloading and ending when the pallet arrives at its destined shipping door. For instance, the first pallet in receiving door A, will go to shipping door D and the time arriving at shipping door D will be [T.sub.A1]. Notice that the value of [T.sub.A1] changes over time, so real-time information about its value is needed for each new trailer assignment. The symbol under a pallet of a waiting trailer has the same meaning as an existing pallet's and its cumulative time changes if the waiting trailer is assigned to a different receiving door because of different travel distances between receiving and shipping doors. For each waiting trailer assignment, if there are multiple waiting trailers, we compare the processing time that occurs due to transporting the pallets in each waiting trailer and the existing pallets as well as the wait time for all pallets at shipping doors during the time span of unloading the waiting trailer. Of course, the wait time will be shorter if some pallets can leave early from shipping doors before completion of unloading the waiting trailer. This becomes a time-saving advantage to a waiting trailer when applying this time-based approach.

[FIGURE 2 OMITTED]

Let us return to the calculation of processing time for all waiting trailers in Figure 2. The time span for unloading waiting trailer one is [T.sub.16], and six pallets at shipping door D can leave early during the course of unloading. If we assume the second pallet of waiting trailer one will be the sixth pallet at shipping door D, we know the first six pallets take only [T.sub.12] time units in the crossdock and thus they save the total wait time at shipping door D of 6* ([T.sub.16]-[T.sub.12]) time units. Besides the six pallets, the other nine pallets will stay at the crossdock while waiting or being transported and thus consume a total of 9*[T.sub.16] time units. Thus the processing time for waiting trailer one is 15* [T.sub.16]-6*([T.sub.16]-[T.sub.12]). On the other hand, the processing time for waiting trailers two and three are 15*[T.sub.26] and 15*[T.sub.36] respectively, since no pallets can leave early during their unloading. These three processing times are compared and the trailer with the lowest value is chosen.

The procedures for assigning a new waiting trailer to an available receiving door are illustrated in Figure 3.

There are two other important features about this algorithm:

First, in order to determine how much time can be saved by each alternative trailer assignment, we need to know which pallets will be able to leave early. In our example, the second pallet at receiving door A and the second and third pallets in waiting trailer one all have the possibility of being the one that results in the outgoing trailer at shipping door D leaving early. To find out the right leaving time and leaving pallets, sorting the arrival times of all pallets going to shipping door D is necessary.

Second, when there are no time-saving effects for any waiting trailers during an assignment, the cumulative times to transport the pallets in each trailer are compared--similar to the criterion used in the look-ahead algorithm. This implies that the performance of our minimum processing time-based method is at least equal to or even better than the look-ahead algorithm.

A Criterion Based on the Total Transfer Time during the Unloading of a Waiting Trailer

The second criterion considers not only the processing time in the above subsection, but also a trailer's wait time in the trailer waiting line. The procedures of obtaining the total transfer time (i.e., the processing time plus the wait time in the trailer waiting line) are similar except that we calculate the total transfer time instead of the processing time when comparing all waiting trailers at the last step. Considering the wait time in the waiting trailer line can avoid delays for those waiting trailers with high processing values and ensure that trailers are assigned in the most time-saving way.

SIMULATION MODEL AND DATA

Crossdock Layouts

Three different crossdock layouts--four receiving doors and four shipping doors (four to-four doors), eight receiving doors and eight shipping doors (eight-to-eight doors), and four receiving doors and eight shipping doors (four-to-eight doors) to represent staging crossdocks and a free-stage crossdock--are considered to test four trailer scheduling policies. These four policies are the FCFS, look-ahead, minimum processing time, and minimum total time policies. The crossdock dimensions are based on Sandal's work (2005): all crossdocks are seventy-five feet wide; each door is fifteen feet wide and has eight feet of space from its neighbor doors. Staging crossdocks (four-to-four doors and eight-to-eight doors) have their receiving doors and shipping doors on different sides, while in the free-stage setting (four-to-eight doors), these two kinds of doors are distributed on both sides and receiving doors are located near the middle of the crossdock to reduce travel distance.

Transferring Processes

The transferring processes in our simulation model are as follows: When an incoming trailer arrives at the crossdock, it will be immediately assigned to a receiving door if one is available and no other trailers are waiting. If none are available, the incoming trailer will be put into a trailer waiting line. Whenever a new incoming trailer arrives or a receiving door becomes empty, the simulation model checks if it needs to apply a trailer scheduling policy to assign a waiting trailer to an available receiving door, as illustrated in Figure 4. The waiting trailer then is moved to the receiving door. Each receiving door is allocated one worker and one forklift. The worker unloads a pallet from the incoming trailer, moves it to its destined shipping door, and uploads it to the outgoing trailer waiting at the shipping door. In the operations of crossdocking, each shipping door stands for a specific destination for parking outgoing trailers and that destination typically does not change for several months. After uploading a pallet, the worker goes back to his original receiving door to start his or her next movement. A pallet placed in an outgoing trailer has to wait until the outgoing trailer is fully loaded with pallets. At that time, the pallet leaves the simulation model and we stop counting its time in the crossdock.

[FIGURE 3 OMITTED]

Model Basics and Assumptions

The four simulation models are built separately using the Arena simulation package (version 11). Incoming and outgoing trailers are assumed to be forty-eight feet long, each carrying twenty-eight pallets. We simulate incoming trailer arrivals as an exponential distribution with parameters ranging from five to thirty-five minutes for each data set. Once an outgoing trailer is fully loaded, it leaves the simulation model and another new and empty outgoing trailer replaces it immediately. Ten replications are run for each scenario and the average performances are calculated across the ten replications. Every replication starts with all doors empty and terminates at the 1000th minute.

Except for the stochastic characteristic of the trailer arrivals, other process times are deterministic, including travel times between doors, unloading, and uploading times.

[FIGURE 4 OMITTED]

Data Sets

Four data sets are generated and used in this study. The distributions of freight mixture and destination are set before sampling and are shown in Table 1. The steps of data generation are as follows: First, we draw the number of destinations for each incoming trailer. For example, we get two destinations. Then we draw the actual destinations, such as destination two and four. Finally, the destination of each pallet is assigned according to the relative percentages in column five of Table 1. For example, if we get destinations two and four for a trailer in data set one, there will be sixteen pallets bound for destination two and twelve pallets bound for destination four. The reasons we create these types of data sets are as follows:

* Not all incoming trailers have their pallets going to four or eight destinations every time. Hence, we assume that the number of destinations found on each trailer is determined according to a probability.

* After a period of crossdocking operation, the distribution of pallets' destinations can be found and we assume that the relative percentages among destinations should be stable even if all four or eight destinations are not used by a particular trailer.

* Pallets in a trailer going to the same destination are assumed to be grouped rather than scattered.

In our four data sets, data sets one and two are for the four-to-four door setting and data sets three and four are both for the four-to-eight and eight-to-eight door setting. Data sets one and three have more skewed destination distributions and hence more unbalanced demands (as shown in Table 1), while data sets two and four are relatively balanced. (1) These two types of data sets can help us check the sensitivity of the four policies relative to different demand distributions.

Performance Measures

Since our algorithms are time-oriented, we mainly measure the average "total time" needed for a pallet starting from arriving at the trailer waiting line to leaving a crossdock at the shipping side. In addition, "travel time" and "throughput" are also measured. The travel time is the average time needed for moving a pallet from a receiving door to a shipping door, which is the same measure as travel distance, the decision criterion of the look-ahead algorithm. The throughput represents the average number of pallets leaving a crossdock in a l000-minute operation period, not including pallets still waiting in outgoing trailers. The above three measurements are all obtained after a ten-replication run.

Because we have four or eight shipping doors in our simulations and each door has different demand density, we get four or eight different values from these shipping doors for the travel time and the total time measurements. Therefore, we calculate weighted average values for comparison. The average travel time of a shipping door is multiplied by the number of pallets that have left the crossdock via the shipping door, and the sum of the above weighted values for all the shipping doors are then divided by their throughput to get a weighted average travel time. The same procedure applies to the calculation of the weighted average for the total time.

SIMULATION RESULTS

Four-to-Four Door Scenarios

Tables 2 and 3 show the average travel times, average total times, and average throughputs for the four trailer scheduling policies under different trailer arrival headways and under data sets one and two. Because of the property of the look-ahead algorithm, when there are more choices from the trailer waiting line, it can find a better trailer to shorten the total travel distance of the assignment. Therefore, in most cases the look-ahead algorithm has advantages over the other policies with respect to travel time. However, when the trailer arrivals are sparse (headways exceeding thirty minutes), the four policies do not have obvious differences.

On the other hand, our two time-based algorithms perform well on the average "total time" as expected, especially our minimum total time policy. When the headways are less than thirty minutes, the performances of the time-based methods are better than the other two on both data sets. As for the average "throughput," the look-ahead policy is the best in most scenarios and the two time-based methods are better than the FCFS policy.

Four-to-Eight Door Scenarios

A little different from the results in the previous subsection, the minimum processing time algorithm incurs the shortest average travel time in this four-to-eight door layout and thus produces the highest average throughput among these four policies, in most cases under data sets three and four, as shown in Table 4 and 5. Under the four-to-eight door layout, the minimum processing time algorithm can increase the throughput up to 30 percent compared to the FCFS policy. As for the average "total time," the minimum total time algorithm still has its advantage. Again, these four policies have similar performances when trailer arrivals are sparse.

Eight-to-Eight Door Scenarios

Because of the additional receiving doors in this crossdock layout, the average queuing lengths under the seven headway scenarios are smaller than those under the four-to-four and four-to-eight door scenarios. When the headways are twenty minutes, the average queuing length is about 0.26 trailers, which makes all of the scheduling policies perform similarly (see Table 6 and 7). Except for the cases in which the headways are equal to or greater than twenty minutes, we find that the two time-based algorithms still consistently perform better on total time, and the throughputs of the minimum processing time algorithm exceed that of the look-ahead algorithm in most cases. The increase of throughput attains up to 15 percent for the minimum processing time algorithm compared to the FCFS policy.

CONCLUSION

Two time-based trailer scheduling algorithms are proposed and compared with the look-ahead and FCFS policies under different trailer arrival headways, crossdock layouts, and pallet destination distributions. The overall time improvements shown in Table 8 provide evidence that our time-based algorithms can save time on transferring pallets relative to the FCFS and look-ahead policies when the average number of waiting trailers is greater than 0.65 (the headways shorter than thirty minutes in the four-to-four and four-to-eight door settings and shorter than twenty minutes in the eight-to-eight door setting). Our simulation results also show that it is not necessary to use scheduling techniques when the average number of waiting trailers is below 0.65 because there are not many choices from the trailer waiting line. The time-saving effect from the two time-based trailer scheduling algorithms can be as high as 63 percent, 56 percent, and 30 percent in the four-to-four, four-to-eight, and eight-to-eight door scenarios, respectively, compared to the FCFS policy. All these improvements are attainable without expanding facilities or manpower. These methods can result in noticeable influences on a supply chain:

* Reliable on-time delivery is an important criterion for rating a crossdock operation. With shorter total transfer time, the time-based algorithms appear to perform with higher reliability.

* In our simulations for the four-to-eight and eight-to-eight doors, the minimum processing time algorithm generates the highest throughputs in most cases. This means higher productivity and less transferring times for those crossdocks.

* The best travel time saving from the look-ahead algorithm compared to the minimum total time algorithm is about 0.05 minutes under the four-to-four door crossdock. If we compare the total times of these two algorithms under the five-minute scenario of data set one, the minimum total time method saves about 124 minutes. If the average inventory holding cost of pallets for the 124 minutes is higher than the labor cost of the 0.1-minute round-trip saving for a worker, it will be justified to adopt the minimum total time method. Under the situations in which the travel time using the look-ahead algorithm is higher than or equal to the time-based algorithms, adopting one of the time-based methods will be a better choice.

The main purpose of this study is to evaluate the performance of the four trailer scheduling policies on transferring freight. To avoid distraction, we do not consider staging processes in the four-to-four and eight-to-eight layouts. In fact, if allocating staging lines in crossdocks, the real travel distances should be shorter than the travel distances used in our simulation models, and accordingly this should strengthen the advantage of our time-based methods.

In future research, real-world cost and data should be considered and methods to increase throughput of larger crossdocks should be examined. Other issues involve setting arrival and departure schedules for outgoing trailers to take into account delivery reliability.

REFERENCES

Apte, U. M. and S. Viswanathan (2000), "Effective Cross Docking for Improving Distribution Efficiencies," International Journal of Logistics: Research and Applications, 3(3), pp. 291-302.

Bartholdi, J. J. and K. Gue (2000), "Reducing Labor Costs in an LTL Crossdocking Terminal," Operation Research, 48(6), pp.823-832.

Bartholdi, J. J., K. R. Gue, and K. Kang (2001), "Staging Freight in a Crossdock," Proceedings of the International Conference on Industrial Engineering and Production Management.

Bermudez, R. and M.H. Cole (2001), A Genetic Algorithm Approach to Door Assignments in Breakbulk Terminal, Technical Report, Mack-Blackwell Transportation Center, University of Arkansas.

Brown, A. M. (2003), Improving the Efficiency of Hub Operations in a Less-than-Truckload Distribution Network, Masters Thesis, The Virginia Polytechnic Institute and State University.

Cook, S. (2007), "Cross-docking Increase as Supply Chain Shifts to Demand Chain," Global Logistics & Supply Chain Strategies, 11 (11), pp.46-47.

Gue, K. R. (1999), "The Effects of Trailer Scheduling on the Layout of Freight Terminals," Transportation Science, 33, pp. 419-428.

Gue, K. R. (2001), "Crossdocking: Just-in-Time for Distribution," Lecture Note, Naval Postgraduate School.

Lim, A., H. Ma, and Z. Miao (2006a), "Truck Dock Assignment Problem with Time Windows and Capacity Constraint in Transshipment Network through Crossdocks," Proceedings of the Computational Science and Its Applications--ICCSA 2006, pp. 688-697.

Lim, A., H. Ma, and Z. Miao (2006b), "Truck Dock Assignment Problem with Operational Time Constraint within Crossdocks," Proceedings of the 19th International Conference on Industrial Engineering and Other Applications of Applied Intelligent Systems, IEA/AIE 2006, pp. 262-27l.

Sandal, S. (2005), Staging Approaches to Reduce Overall Cost in a Crossdock Environment, Master Thesis, University of Missouri-Columbia.

Taylor, G. D. and J. S. Noble (2004), "Determination of Staging Needs in a Crossdock Environment," Proceeding of 2004 Industrial Engineering Research Conference.

Tsui, L. Y. and C. Chang (1990), "A Microcomputer Based Decision Support Tool for Assigning Dock Doors in Freight Yards," Proceedings of the 12th Annual Conference on Computers & Industrial Engineering, pp. 309-312.

Tsui, L.Y. and C. Chang (1992), "Optimal Solution to a Dock Door Assignment Problem," Computers & Industrial Engineering, Vol. 23, pp. 283-286.

ENDNOTE

(1) The destination distributions of data set two and four are set to be uniform. However, due to the randomness of drawing, the destinations of these two data sets are not uniformly distributed.

Mr. Wang is a Ph.D. candidate, Institute of Transportation Studies, University of California, Irvine, Irvine, California 92697; e-mail jianafuw@uci.edu. Ms. Regan is associate professor, Department of Computer Science and Institute of Transportation Studies, University of California, Irvine; e-mail aregan@uci.edu. The authors are grateful to three anonymous reviewers for many useful comments. This work was supported by both the University of California Transportation Center and the Ministry of Education of Republic of China (Taiwan). We gratefully acknowledge this generous support.
Table 1. Sampling Probabilities and Results of Four Data Sets

 Freight Mixture
 Distribution

 Number of
 Destinations Probability

Data Set 1 One 0.25
(Used in Two 0.45
4-to-4 Door Three 0.2
Case) Four 0.1
Total 1.00
Data Set 2 One 0.25
(Used in Two 0.45
4-to-4 Door Three 0.2
Case) Four 0.1
Total 1.00
Data Set 3 One 0.25
(Used in Two 0.35
4-to-8 Door Three 0.2
Case and Four 0.1
8-to-8 Door Five 0.04
Case) Six 0.03
 Seven 0.02
 Eight 0.01
Total 1.00
Data Set 4 One 0.25
(Used in Two 0.35
4-to-8 Door Three 0.2
Case and Four 0.1
8-to-8 Door Five 0.04
Case) Six 0.03
 Seven 0.02
 Eight 0.01
Total 1.00

 Destination Distribution

 Destinations Probability

Data Set 1 Shipping Door 1 0.33
(Used in Shipping Door 2 0.15
4-to-4 Door Shipping Door 3 0.4
Case) Shipping Door 4 0.12
Total 1.00
Data Set 2 Shipping Door 1 0.25
(Used in Shipping Door 2 0.25
4-to-4 Door Shipping Door 3 0.25
Case) Shipping Door 4 0.25
Total 1.00
Data Set 3 Shipping Door 1 0.15
(Used in Shipping Door 2 0.09
4-to-8 Door Shipping Door 3 0.08
Case and Shipping Door 4 0.13
8-to-8 Door Shipping Door 5 0.15
Case) Shipping Door 6 0.2
 Shipping Door 7 0.08
 Shipping Door 8 0.12
Total 1.00
Data Set 4 Shipping Door 1 0.125
(Used in Shipping Door 2 0.125
4-to-8 Door Shipping Door 3 0.125
Case and Shipping Door 4 0.125
8-to-8 Door Shipping Door 5 0.125
Case) Shipping Door 6 0.125
 Shipping Door 7 0.125
 Shipping Door 8 0.125
Total 1.00

 Sampling Results

 Amount
 Destinations (Pallets)

Data Set 1 Shipping Door 1 703
(Used in Shipping Door 2 341
4-to-4 Door Shipping Door 3 890
Case) Shipping Door 4 306
Total 2240
Data Set 2 Shipping Door 1 676
(Used in Shipping Door 2 620
4-to-4 Door Shipping Door 3 415
Case) Shipping Door 4 529
Total 2240
Data Set 3 Shipping Door 1 430
(Used in Shipping Door 2 172
4-to-8 Door Shipping Door 3 206
Case and Shipping Door 4 319
8-to-8 Door Shipping Door 5 285
Case) Shipping Door 6 452
 Shipping Door 7 136
 Shipping Door 8 240
Total 2240
Data Set 4 Shipping Door 1 294
(Used in Shipping Door 2 271
4-to-8 Door Shipping Door 3 322
Case and Shipping Door 4 228
8-to-8 Door Shipping Door 5 294
Case) Shipping Door 6 208
 Shipping Door 7 233
 Shipping Door 8 390
Total 2240

Table 2. Average Travel Time, Total Time, and Throughput for the
Four-to-Four Crossdock Layout Using Data Set 1 under Different
Scheduling Methods and Trailer Arrival Headways

Travel Time (mi Total Min Proc. Look-
Headway Min Time (#) Time (#) ahead FCFS

 5 1.49 1.47 1.44 1.53
 10 1.50 1.48 1.45 1.53
 15 1.51 1.49 1.47 1.53
 20 1.51 1.50 1.49 1.53
 25 1.51 1.51 1.50 1.51
 30 1.51 1.50 1.50 1.51
 35 1.51 1.51 1.51 1.51

Total Time (minutes)
 Total Min Proc. Look-
Headway Min Time Time ahead FCFS

 5 260.06 339.83 384.23 432.02
 10 122.18 222.81 264.86 329.64
 15 118.74 179.32 200.20 234.76
 20 127.24 146.32 153.78 165.78
 25 122.02 127.47 130.80 133.94
 30 123.64 125.59 125.53 127.43
 35 125.72 125.84 126.32 127.14

Throughput (pallets/1000 minutes)

 Total Min Proc. Look-
Headway Min Time Time ahead FCFS

 5 1120 1143 1148 1114
 10 1104 1131 1139 1104
 15 1075 1083 1092 1064
 20 1004 1002 1003 985
 25 865 868 856 850
 30 720 722 719 714
 35 621 618 618 618

(#) "Min total time" stands for the minimum total time algorithm, and
min proc. time stands for the minimum processing time algorithm.

Table 3. Average Travel Time, Total Time, and Throughput for the
Four-to-Four Crossdock Layout Using Data Set 2 under Different
Scheduling Methods and Trailer Arrival Headwavs

Travel Time (minutes)

Headway Min Total Min Proc. Look-
 Time Time ahead FCFS

 5 1.50 1.48 1.45 1.53
 10 1.50 1.48 1.46 1.53
 15 1.51 1.49 1.47 1.53
 20 1.51 1.51 1.49 1.52
 25 1.52 1.51 1.50 1.52
 30 1.53 1.52 1.52 1.53
 35 1.53 1.53 1.53 1.53

Total Time (minutes)

Headway Min Total Min Proc. Look-
 Time Time ahead FCFS

 5 260.31 349.20 393.00 421.61
 10 121.92 227.08 288.56 319.90
 15 118.31 181.29 208.47 227.59
 20 128.75 150.51 158.54 167.16
 25 126.62 129.92 138.14 141.51
 30 127.98 129.80 130.80 132.10
 35 133.08 133.17 134.68 134.39

Throughput (pallets/1000 minutes)

Headway Min Total Min Proc. Look-
 Time Time ahead FCFS

 5 1117 1136 1159 1095
 10 1104 1129 1134 1078
 15 1075 1096 1101 1041
 20 996 999 997 977
 25 850 864 860 843
 30 711 708 712 712
 35 618 624 624 619

Table 4. Average Travel Time, Total Time, and Throughput for the
Four-to-Eight Crossdock Layout Using Data Set 3 under Different
Scheduling Methods and Trailer Arrival Headways

Travel Time (minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 1.04 0.99 1.07 1.32
 10 1.06 1.02 1.09 1.31
 15 1.16 1.13 1.16 1.29
 20 1.22 1.20 1.21 1.30
 25 1.25 1.24 1.24 1.28
 30 1.28 1.28 1.29 1.30
 35 1.28 1.28 1.29 1.31

Total Time (minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 270.68 303.71 380.13 419.03
 10 137.41 187.97 256.62 308.71
 15 141.65 158.65 186.20 214.70
 20 141.32 146.13 151.21 163.93
 25 145.38 146.33 147.52 151.45
 30 150.23 150.15 151.07 152.62
 35 159.47 159.47 159.38 160.55

Throughput (pallets/ 1000 minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 1499 1549 1445 1199
 10 1422 1493 1413 1201
 15 1274 1309 1288 1163
 20 1044 1072 1055 1010
 25 854 868 862 838
 30 687 690 690 680
 35 601 601 607 598

Table 5. Average Travel Time, Total Time, and Throughput for the
Four-to-Eight Crossdock Layout Using Data Set 4 under Different
Scheduling Methods and Trailer Arrival Headways

Travel Time (minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 1.00 0.97 1.04 1.32
 10 1.06 1.00 1.05 1.30
 15 1.14 1.12 1.12 1.30
 20 1.23 1.22 1.21 1.30
 25 1.27 1.26 1.25 1.30
 30 1.28 1.27 1.27 1.29
 35 1.27 1.27 1.26 1.27

Total Time (minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 263.77 287.92 361.56 419.89
 10 134.29 175.54 250.25 307.14
 15 136.59 157.58 184.74 218.46
 20 142.86 149.86 156.04 168.89
 25 147.18 149.06 149.81 155.36
 30 148.45 148.92 149.61 150.66
 35 155.13 155.17 154.28 154.59

Throughput (pallets/1000 minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 1520 1569 1428 1234
 10 1441 1522 1420 1231
 15 1314 1331 1307 1164
 20 1064 1082 1104 1028
 25 856 864 868 837
 30 683 692 686 683
 35 602 602 602 596

Table 6. Average Travel Time, Total Time, and Throughput for the
Eight-to-Eight Crossdock Layout Using Data Set 3 under Different
Scheduling Methods and Trailer Arrival Headways

Travel Time (minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 1.61 1.60 1.58 1.85
 10 1.66 1.64 1.63 1.83
 15 1.76 1.76 1.76 1.85
 20 1.81 1.80 1.80 1.82
 25 1.84 1.84 1.84 1.84
 30 1.84 1.84 1.84 1.84
 35 1.85 1.85 1.85 1.85

Total Time (minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 310.54 316.17 333.64 359.64
 10 148.48 162.02 182.14 212.00
 15 131.98 132.87 134.61 143.13
 20 137.07 136.90 136.54 138.41
 25 148.52 148.49 148.32 148.30
 30 161.38 161.34 161.38 161.37
 35 174.13 174.13 174.13 174.13

Throughput (pallets/1000 minutes)

 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 2044 2071 2046 1803
 10 1921 1952 1940 1758
 15 1547 1557 1557 1490
 20 1134 1134 1136 1137
 25 889 889 889 889
 30 698 698 698 698
 35 601 601 601 601

Table 7. Average Travel Time, Total Time, and Throughput for the
Eight-to-Eight Crossdock Layout Using Data Set 4 under Different
Scheduling Methods and Trailer Arrival Headways

Travel Time (minutes)
 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 1.64 1.63 1.62 1.85
 10 1.69 1.66 1.65 1.84
 15 1.78 1.77 1.77 1.85
 20 1.82 1.82 1.82 1.83
 25 1.83 1.83 1.83 1.83
 30 1.81 1.81 1.81 1.81
 35 1.82 1.82 1.82 1.82

Total Time (minutes)
 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 300.17 307.82 330.66 359.56
 10 150.09 164.67 181.52 213.71
 15 133.43 133.37 136.54 145.86
 20 139.70 139.41 139.94 140.67
 25 150.84 150.80 150.81 150.84
 30 159.95 159.95 159.95 159.95
 35 168.82 168.82 168.82 168.82

Throughput (pallets/1000 minutes)
 Min Total Min Proc. Look-
Headway Time Time ahead FCFS

 5 2008 2031 2019 1809
 10 1902 1934 1932 1757
 15 1532 1535 1546 1498
 20 1147 1148 1145 1144
 25 876 876 876 876
 30 686 686 686 686
 35 593 593 593 593

Table 8. Improvement on Total Time for Each Scenario Compared
to the FCFS Policy

Four-to-Four Scenarios with
Data Set 1

 Min Total Min Proc. Look-
Headway Time Time ahead

 5 39.80% 21.34% 11.06%
 10 62.94% 32.41% 19.65%
 15 49.42% 23.62% 14.72%
 20 23.25% 11.74% 7.24%
 25 8.89% 4.82% 2.34%
 30 2.97% 1.44% 1.49%
 35 1.12% 1.03% 0.65%

Four-to-Eight Scenarios with
Data Set 3

 Min Total Min Proc. Look-
Headway Time Time ahead

 5 35.40% 27.52% 9.28%
 10 55.49% 39.11% 16.87%
 15 34.02% 26.11% 13.27%
 20 13.79% 10.86% 7.76%
 25 4.01% 3.38% 2.59%
 30 1.56% 1.61% 1.02%
 35 0.67% 0.67% 0.72%

Eight-to-Eight Scenarios with
Data Set 3

 Min Total Min Proc. Look-
Headway Time Time ahead

 5 13.65% 12.09% 7.23%
 10 29.96% 23.58% 14.09%
 15 7.79% 7.17% 5.96%
 20 0.97% 1.09% 1.35%
 25 -0.15% -0.12% -0.01%
 30 -0.01% 0.02% -0.01%

Four-to-Four Scenarios with
Data Set 2

 Min Total Min Proc. Look-
Headway Time Time ahead

 5 38.26% 17.18% 6.79%
 10 61.89% 29.01% 9.80%
 15 48.02% 20.34% 8.40%
 20 22.98% 9.96% 5.16%
 25 10.52% 8.19% 2.38%
 30 3.12% 1.74% 0.99%
 35 0.97% 0.91% -0.22%

Four-to-Eight Scenarios with
Data Set 4

 Min Total Min Proc. Look-
Headway Time Time ahead

 5 37.18% 31.43% 13.89%
 10 56.28% 42.85% 18.52%
 15 37.47% 27.87% 15.43%
 20 15.41% 11.27% 7.61%
 25 5.27% 4.06% 3.57%
 30 1.47% 1.16% 0.70%
 35 -0.35% -0.37% 0.20%

Eight-to-Eight Scenarios with
Data Set 4

 Min Total Min Proc. Look-
Headway Time Time ahead

 5 16.52% 14.39% 8.04%
 10 29.77% 22.95% 15.06%
 15 8.53% 8.57% 6.39%
 20 0.69% 0.90% 0.53%
 25 0.00% 0.02% 0.02%
 30 0.00% 0.00% 0.00%
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Title Annotation:Crossdocks
Author:Wang, Jiana-Fu; Regan, Amelia
Publication:Transportation Journal
Geographic Code:1USA
Date:Mar 22, 2008
Words:7530
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