CROSS-DOCKING: SCHEDULING OF INCOMING AND OUTGOING SEMI TRAILERS

CROSS-DOCKING: SCHEDULING OF INCOMING AND OUTGOING SEMI TRAILERS 1 th International Conference on Production Research P.Baptiste, M.Y.Maknoon Département de mathématiques et génie industriel, Ecole polytechnique Montréal Montréal, Québec, Canada. Abstract This paper presents the scheduling of incoming and outgoing semi trailers in a transshipment platform. A set of incoming semi trailers, containing products for different destinations, arrives to the cross-docking. The incoming and outgoing doors are limited; consequently, the semi trailers have to be sequenced. This paper studies the simultaneous scheduling of incoming and outgoing semi trailers for a single inbound and outbound door. The objective is to maximize the direct flow from incoming semi trailers to outgoing semi trailers. The problem is decomposed into three parts. Tabu search is integrated with dynamic programming and a heuristic method is proposed to solve the three cases of the problem. In addition, few examples are performed and the results are shown. Keywords: Cross-docking, Scheduling incoming and outgoing semi trailers, scheduling, Tabu search, heuristic methods, dynamic programming 1 INTRODUCTION Transshipment is a logistic activity between producer and consumer in supply chain process that aims to reduce the costs by reducing inventory level. It breaks down incoming items, process and consolidates them for reshipment. Transshipment aims to reduce the cost and the lead time. In this paper, the authors focus on transshipment platforms. There are two sets of semi trailers beside transshipment platform: the incoming and the outgoing. The two sets have to be sequenced in a manner so that the platform respects just-in-time requirements. There are two ways to transfer products through the platform: moving directly from inbound to outbound door or using a temporary inventory area in the platform. Transshipment efficiency can be measured by the amount of products that passes directly through the platform. The second approach could not be acceptable. In addition to the inventory cost, there are additional movements of the products from inbound door to the storage area and from storage area to the outbound door. Therefore, there are several extra movements which occupy the transshipment facilities but do not enhance transshipment efficiency. This paper studies the problem of simultaneous sequencing inbound and outbound semi-trailers in transshipment in three parts. The three cases differ by the knowledge of incoming or outgoing sequence. In general three approaches are used to solve the problems: dynamic programming, Tabu search and Heuristic. The approach is being tested for some examples and the results are analyzed. 1.1 Literature review Generally in research papers, two aspects of transshipment are studied: strategically and operational. The research at the strategically level concerns mainly the platform location or the assignment of transportation jobs. Operational aspects deal with the efficiency of operational activities. One of the operational activities of transshipment is semi trailers transportation. Ping Chen et al. consider delivery and pickup time windows, warehouse capacities and inventory-handling to minimize the total transportation and inventory cost in a transshipment network [1]. Young Hae Lee et.al. [2] proposed an integration model of transshipment and semi trailers scheduling to obtain more robust program. Lim et.al develop polynomial algorithm for transshipment problem considering just in time objective in the transshipment model [3]. Amano et al. [4] presents modal-shift transportation planning in cross docking network. It contains sets of facilities, orders and carriers with the objective to find a feasible schedule for carriers with minimum total cost which respect to deadlines. Steepest decent algorithm is used to solve the problem. Changing the physical layout can enhance transshipment performance. According to Bartholdi et al. [5] cross docking is a labor intensive area and workers performance depends on how well semi trailers are assigned to doors; moreover, a good layout reduces travel distances without creating congestion. They conclude that changing layout reduce reduces shipping and handling cost within transshipment platform. Time span in scheduling increases the transshipment efficiency. Li et al. [6] propose a problem in which each container should be filled in exact time. Machine scheduling problem is used as a model. In this model the transshipment platform is divided into loading and unloading areas. The arriving dates for incoming semi trailer are variable. The received Items are then either shipped away directly or sent to the exportation area in order to be loaded for reshipment. In this problem the time to start unloading semi trailer is scheduled in order that each loading semi trailer is completed at its due date. Douglas L. Mc Williamsa et al. [7] have studied the problem of parcel industry. The research focuses on the transfer operation platforms. Parcels are unloaded and shipped to outbound semitrailers. The objective is to minimize time interval from the first unloaded parcel till the last loaded parcel. An integrated simulation model that integrates a genetic algorithm is proposed to find the solution. Wooyen Yu et al. consider the scheduling of inbound and outbound semi-trailers in order to minimize completion time while the storage is located at shipping dock. Two approaches are proposed to obtain the results: mathematical model and heuristic algorithms. Mathematical models are used to solve small size

problems while they are not practical for the large problems. In contrast, the heuristic methods are used to solve the large problems [8]. Scheduling the sequence of loading and unloading semi trailers also increases the efficiency of the transshipment platform. In fact, synchronize loading and unloading sequences decreases inventory level and increases direct product flow from inbound to outbound door. As a result it increases transshipment efficiency. In this paper scheduling of loading and unloading semi trailers in cross docking are studied. 2 PROBLEDESCRIPTION: In practice, transshipment has various layouts. In this research the layout is being restricted to one inbound and one outbound door. This restriction is not realistic (in a real transshipment platform) but can be used as a baseline for other layouts. In this model an incoming semi trailer arrives at inbound door and unloads products for various destinations. If the outgoing semi trailer is going to the fine destination, the products are moved directly to outbound semi trailer (direct transit of products), in the other hand, the products are moved to a temporary storage (products in temporary storage). In studied model, the following assumptions are considered: Each trailer leaves the inbound door when it is fully unloaded. On the other side, each trailer leaves the outbound door when it is fully loaded. The internal operations of cross docking such as sorting and merging are not considered. The storage capacity is assumed unlimited. Each outbound semi trailers leaves only for one destination. All incoming and outgoing semi trailers are available at time zero. The total numbers of arriving and departing products are equal. The products differ by their destination. Loading, unloading and transfer time are constant and are not considered. Inside incoming semi trailers there are products for different destinations while for outgoing semi trailers there are products for just one destination. In addition, the information about type of products and the quantity of arriving products for each destination are known. For this problem, the following decision variables are considered: 1. Incoming sequence 2. Outgoing sequence 3. Unloading sequence of semi trailers 4. Unloading policy The first two variables are obvious, but the third variable corresponds to the fact that unloading order of an incoming semi trailer contains items to be shipped in different destinations, which can influence the efficiency. Obviously, items that can be shipped in the active destination (current outgoing semi trailer) have to be unloaded first. This variable can be free or fixed (due to technical constraints for the unloading operations). For the last case, the optimal decision is usually evident. The fourth variable (unloading policy) corresponds to the following situation: an outgoing semi trailer is positioned at the output door, and items are already waiting on the ground for the same destination. The manager can choose to ship those items or wait till an incoming semi trailer arrives with items that can be shipped directly to this destination. Obviously, the loading and unloading sequences (variables 1 and 2) and products movement policies (variables 3 and 4) are two important factors which affect transshipment performance. For the fourth variable, there are two different policies. At the first policy, products already on the ground are systematically used to complete a semi trailer (fewer inventories). At the second policy, items already on the ground remain for the last semi trailer for their destination. The optimal policy is a combination of the two extreme policies. In general some times, it is better to use inventory, compare to waiting for direct transshipment. Three cases are proposed. The definition of each case is as follows: Case 1: the sequences of incoming and outgoing semi trailers are known a priori and only variables 3 and 4 are examined. Case 2: The sequences of incoming semi trailers are known a priori and variables 2, 3 and 4 are examined. Case 3: no sequences are known a priori and variables 1, 2, 3 and 4 are examined. With the above assumptions the resolution approaches are discussed in the next section. 3 RESOLUTIONS 3.1 First case resolution approach Objective in the first case is to obtain the optimal policy (maximizing direct transiting products) when the sequence of loading and unloading semi trailers are known. Complete enumeration technique is used to obtain optimal policy. A graph is used to present all the possibilities of assignments. The graph nodes are used to present assignment state and the arcs are used to present forthcoming possibilities. The following variables are saved in each node: - Vector of variables indicates possible direct transiting product for each destination. (PDT) - Vector of variables indicates the number of products which are now in temporary storage, for each destination. (PTS) - cost (the total number of direct transiting product from beginning to the current node) (C) - Order number of loaded semi trailer. (ON) For each arc, unloading semi trailer order number and cost (the number of direct transferred product for forthcoming assignment) are saved. For the Initial node, the PDT variables are equal to the first semi trailer products content for each destination and the rest of the variables default numbers are zero. This algorithm starts with the initial node, afterwards, in each iteration, it increments the order number of loading semi trailer and reads all the generated nodes for current loading semi trailer. For each node, it generates all the possible arcs for remaining loading semi trailers order;

1 th International Conference on Production Research therefore, all the future possible nodes are obtained. This process repeats as long as all the loading semi trailers are chosen. At the last iteration the node which has the highest cost is chosen as the final node. The path from the initial node up to the final node is loading and unloading optimal policy In practice, two domination rules are used to decrease computational time. The first rule proposes that if for all destinations the summation of direct transiting and storage products for two or more nodes are the same, the node with the highest cost dominates the others. However, the second rule is applied when two or more nodes have the same cost. The node with higher summation of direct transiting products for all destinations dominates the others. Dynamic programming is used to solve this algorithm. The steps are as follows: Optimal policy algorithm Step 1: create an initial node (discussed before) Step 2: do as long as all outbound semi trailers are assigned. Step 3: obtain all forthcoming assignments. Step 4: check domination rule. Step 5: go to step 2. Step 6: find the node with highest cost in the last iteration Step 7: find the path for optimal policy. Example: In this example there are 6 incoming semi trailers containing the products for 3 destinations. The first semi trailer contains 3 products for destination A, 2 and 0 for destinations B and C, respectively. Table 1 presents the incoming semi trailer orders. The sequence of outgoing semi trailers is C-B-A-A-B-A and the trailer capacity is 5 units. 0 i i 0 5 1 2 Node(i) with cost 0 Optimal Path Dominated Node 8 3 5 4 5 6 7 7 Figure 1 : Optimal policy algorithm for the example The optimal path is 1-2-4-7-11-14-16 with the cost of units. The nodes, and 16 are dominated nodes. 3.2 Second case resolution approach In this case, two methods are proposed. Using Tabu search integrated with optimal policy algorithm and a heuristic method. The first method proposes the following algorithm: Loading semi trailer sequence algorithm: Step 1: Run optimal policy algorithm for the initial value Step2: Select two loading semi trailers order number Step3: Swap the order number and save it in Tabu list (if they are not for same destination and are not in Tabu list). Step 4: Run optimal policy algorithm. Step 5: Save the cost and optimal path if it is improved. Step 6: Go to step 2 or stop if the cost is not modified after 20 iterations. 8 7 11 13 14 15 16 Table 1 : Example Order Destination A B C 1 3 2 0 2 0 4 1 3 4 0 1 4 4 1 0 5 2 0 3 6 2 3 0 The algorithm flow chart is presented in figure 2. START RUN OPTIMAL POLICY ALGORITHM RANDOMLY SWAP 2 LOADING SEMI TRAILER DESTINATION ORDER The graph is presented in figure 1: LIMIT=0 RUN OPTIMAL POLICY ALGORITHM LIMIT=LIMIT+1 YES COST IMPROVE LIMIT=20 STOP Figure 2: loading semi trailer sequence algorithm

In loading semi trailer algorithm, the optimal policy is not the combination of sub optimal policies. In other words, sometimes selecting the arc with lower cost would lead us to the node with highest cost. On the other hand in proposed heuristic method it is supposed that sub optimal assignments lead the process to the optimal result. In heuristic algorithm, the preceding graph is used to illustrate all the assigned possibilities. Furthermore, deep first search method is used searching for the best assignment. The algorithm is described as follows and the flow chart is presented in figure 3: Loading semi trailer sequence heuristic algorithm Step 1: Create an initial state (the value of direct transiting products for each destination equals to the first unloaded semi trailer products, the rest is zero) Step 2: For all loading semi trailers Step 3: For all destinations Step 4: Do as long as direct transiting product is more than semi trailer capacity Step 5: For current state, calculate the cost if the summation of direct transiting products and in storage products for selected destination equals to or greater than semi trailer capacity. Persevere the results, if the cost is improved. Step 6: Unload the next semi trailer and update values. Step 6: Go to step 4 Step 7: Save the best assignment Step 8: Set best assignment as current state and go to step 2 Step 8: The final list is the optimal assignment Example: In the previous example, for the given incoming sequence, the solution obtained with the first algorithm is 23 with B-A- A-C-B-A sequence. For the second algorithm the obtained sequence is B-A-A-C-A-B with the value of 23. 3.3 Third case resolution approach The previous two methods are developed to obtain the good sequence of loading and unloading semi trailers. Tabu search integrates with loading semi trailer algorithm or loading semi trailer heuristic algorithm to obtain good unloading sequence. The algorithm is presented in the following steps: Loading and unloading sequence algorithm Step 1: Run loading semi trailer sequence algorithm/ loading semi trailer sequence heuristic algorithm for initial value Step2: Select two unloading semi trailers order numbers Step3: If they are not in Tabu list, Swap the order number and save it in Tabu list. Step 4: Run loading semi trailer algorithm/ loading semi trailer heuristic algorithm Step 5: Save the cost and optimal path if it increase. Step 6: Go to step 2 or Stop if the cost is not improved after 20 iterations. The figure 4 presents the algorithm: START START FOR ALL LOADING ORDER RUN LOADING SEMI TRAILER SEQUENCE ALGORITHM OR RUN LOADING SEMI TRAILER SEQUENCE HEURISTIC ALGORITHM FOR INITIAL ANSWER FOR ALL DESTINATIONS RANDOMLY SWAP 2 UN LOADING SEMI TRAILER ORDER CALCULATE COST KEEP IT IF IT IS IMPROVED DIRECT TRANSITING PRODUCT > SEMI TRAILER CAPACITY UNLOAD NEXT SEMI TRAILER AND UPDATE VALUES LIMIT=0 RUN LOADING SEMI TRAILER SEQUENCE ALGORITHM OR RUN LOADING SEMI TRAILER SEQUENCE HEURISTIC ALGORITHM YES COST IMPROVE LIMIT=LIMIT+1 SET BEST CHOSEN DE AS CURRENT STATE OBTAIN LIST IS OPTIMAL POLICY LIMIT=20 END STOP Figure 3: Loading semi trailer sequence heuristic algorithm Figure 4: loading and unloading sequence algorithm

1 th International Conference on Production Research Example: In the previous example, for the given data, the solution obtained with the first algorithm is 27 with A-B-C-A-B-A and 3-6-4-5-1-2 sequences. For the second algorithm the obtained sequence is A-B-A-A-C-B and 4-1-6-2-5-3 with the cost value of 26. 4 EXPERIMENT 0 0 80 70 60 50 40 30 20 0 3-3-3-3 6-3-2-1 4-4-2-2 4-4-3-1 1 2-1 2-2 3-1 3-2 Medium size problems (contains loaded and unloaded semi trailers) are considered as the test problems. Four different destination combinations (3-3-3-3, 6-3-2-1, 4-2-2-2 and 4-4-3-1) are selected to cover the problem diversity. For each combination 4 sets of data are generated. The loading and unloading trailers capacities are considered as units. Each algorithm is run for all generated data and combinations. The results are shown in table 2 and are summarized in figure 5 and table 3. The results depend on the combination of destinations (Figure 5). The improvement for case 3-3-3-3 is 13.75% and for the case 4-4-3-1 is 34.75% (Table 3). The results indicate that the combinations of destinations are important to implement cross docking semi trailer scheduling. Table 2 : Experimental results for different sequence combination Destination Data Set 3-3-3-3 6-3-2-1 4-4-2-2 4-4-3-1 1 Results 2-1 2-2 3-1 3-2 1 61 68 62 78 76 2 5 66 61 73 6 3 67 6 71 7 7 4 62 71 74 74 75 1 66 80 78 86 87 2 65 82 78 85 84 3 63 83 86 85 86 4 62 7 82 88 8 1 47 75 74 74 80 2 4 73 73 80 80 3 47 74 74 80 80 4 4 75 72 81 83 1 47 76 78 7 81 2 47 74 75 7 7 3 46 78 77 7 82 4 4 78 73 86 86 For case 2 of the problem, integrated Tabu search with dynamic programming show better performance compare to the heuristic method with improvement between 6.25% to 2.25% and 4.75% to 28.5% respectively. In contrast, for case 3 heuristic method show better performance. Figure 5 : Experimental results for different sequence combination Figure 6 and 7 present the Tabu search results for selected problem (4-4-3-1, data test 1) for case 3-1 and 3-2. It seems that integrated Tabu search with heuristic method reach to good value with less iteration. Table 3 : Summary of improvements approach implementation Combination 3-3-3-3 6-3-2-1 4-4-2-2 4-4-3-1 average 1 62.25 64 48 47.25 55.375 2-1 68.5 81 74.25 76.5 75.0625 percentage 6.25% 17.00% 26.25% 2.25% 1.6% 2-2 67 81 73.25 75.75 74.25 percentage 4.75% 17.00% 25.25% 28.50% 18.88% 3-1 76 86 78.75 80.75 80.375 percentage 13.75% 22.00% 30.75% 33.50% 25.00% 3-2 74.75 86.5 80.75 82 81 percentage.50% 22.50% 32.75% 34.75% 25.63% Claculated Value 0 80 70 60 50 40 Tabu search iteration results 1 3 5 7 11 13 15 17 1 Iteration Figure 6 : Tabu search results for selected problem (4-4-3-1, data test 1) Case 3-1 In addition, for test data, on average there is almost 20 percent improvement when the sequence of outgoing semi trailers is planned compare to 25 percent improvement,in average when both sequences are planned.

Claculated Value Tabu search iteration results (Heuristic Method) 0 80 70 60 1 2 3 4 5 6 7 8 11 13 14 Iteration Figure 7 : Tabu search results for selected problem (4-4-3-1, data set 1) Case 3-2 5 CONCLUSION Transshipment platform is a place where the products from incoming semi trailers are unloaded and then loaded for reshipment. Efficiency of such platform is related to the ratio of direct moves (only one manipulation). Scheduling the incoming and outgoing semi trailers can increases transshipment efficiency. This research explores the particular case of a platform with a single incoming door and a single outgoing door. Three cases of this problem are studied and dynamic programming and heuristic methods are proposed as a two major function to solve the problems. A medium size problem is defined as test data for numerical results. Engineering, 2006, article in press. [3] Lim A., Miao Z., Rodrigues B., Xu A., Transshipment through Cross docks with Inventory and Time Windows, Wiley Inter science, 2005. 724-733. [4] Amano M., Yoshizumi T., Okano H., 2003, The modal-shift transportation planning problem and its fast steepest descent algorithm, Proc. of the Winter Simulation Conference. 1720-1728. [5] Bartholdi J.J., Gue K.R., Reducing labor costs in an LTL cross docking terminal. Operation Research, 2002, 48,823-832. [6] Li Y., Rodrigues B., Cross docking JIT scheduling with time windows, Journal of the Operational Research Society,2004 55, 1342 1351 [7] Mc Williams D.L., Stanfield P.M., Geiger C.D., The parcel hub scheduling problem: A simulation-based solution approach,computers and Industrial Engineering 2005,4, 33 4 [8] Yu W., Egbelu P.J., Scheduling of inbound and outbound trucks in cross docking system with temporary storage, European journal of operation research,2006, Article in press. When both schedules are known, an optimal algorithm based on dynamic programming finds the optimal use of the temporary inventory. When one or both sequence is unknown, two different approaches have been proposed. The first uses a stochastic algorithm for the schedule and an optimal evaluation function. The second is a heuristic. The heuristic is much faster and as efficient that the stochastic algorithm. The heuristic algorithm reaches the good results in a very few iterations, moreover, because of the assumptions for heuristic algorithm, it run in shorter time rather than other algorithm. For generated case it is shown that by scheduling both incoming and outgoing semi-trailers, there is 25% improvement in transshipment performance. Most of this improvement can be obtained with scheduling only outgoing semi trailers (20%). To conclude, it is shown that scheduling loading and unloading semi trailers increases transshipment efficiency. Those results have to be extended to general platform, with more than one incoming door and more than one outgoing door. 6 REFERENCES [1] Chen P., Guo Y., Lim A., Rodrigues B.,Multiple cross docks with inventory and time windows, Computers and Operations Research 2006, 33, 43 63. [2] Lee Y.H., Jung J.W., Lee K.M., Vehicle routing scheduling for cross-docking in the supply chain, Computers and Industrial