# Hybrid Algorithms for Inbound and Outbound Truck Scheduling in Cross-Docking Systems
## 3 Problem Statement  
This study takes one condition environment from Yu [4] as the basis. In this environment, in the inbound and outbound areas, there is only single dock for each area, the trucks are with multiple product types, all product types also have different supply/demand values in each truck, and all trucks can enter the docks more than one time. Moreover, this cross-docking system is without a temporary storage area inside, and the assumptions are as follows. 1. All trucks entering the unloading and loading terminals are carried out at the same time and are operated at time zero. 2. All unloading products must be loaded and transported by any outbound truck. Meanwhile, the product types and quantities at both ends will be equal. 3. All trucks entering the loading or unloading terminals are allowed to partially unload or load the products, temporarily exit the terminal, and wait for the next time they enter the terminal for unloading and loading, until the unloading and loading of the single truck is completed. 4. The products are not allowed to be temporarily stored at the cross-dock terminal. 5. Moving one unit of product is assumed to be one unit of time. 6. The time required for sorting and picking within the cross-dock terminal is not considered. 7. The time it takes for each truck to enter or leave the unloading and loading terminals is assumed to be the same. In this kind of condition, we focus on dealing with trucks' scheduling problems to minimize the total operation time of trucks. A truck schedule in a cross-docking system is to decide the arrival time, departure time, and operational time of inbound and outbound trucks. However, a feasible schedule is to arrange the inbound and outbound trucks with the relative products. In other words, the products might have multiple types and different amounts, so the product demand of a single outbound truck may be from more than one inbound truck. Therefore, it is necessary to find out the truck pairs first, which means the match of inbound and outbound trucks within the right product types and the number of products. Figure 1 shows an example with two trucks for the inbound and outbound docks, which have three product types and supply/demand. First, we can let truck A and truck C arrive at the same time, move 5 units of product 1 and 10 units of product 2 from truck A to truck C, and then let truck C leave the outbound dock temporarily. Second, we let truck D arrive, move five units of product 1 from truck A to truck D, and let truck A leave. Third, we let truck B arrive, move 20 units of product 3 from truck B to truck D, and then let truck D leave. Finally, we let truck C come back to the outbound dock and move 15 units of product 2 from truck B to truck C, and then the truck pairs matching procedure will be finished. That is to say, we obtain four truck pairs--[A,C], [A,D], [B,D], and [B,C]--to match the supply and demand. Due to the fact that it takes operation time for each truck pair to load and unload the products, the waiting time of trucks is also included in the operation time. For this reason, we would like to obtain a schedule that can minimize the operation time and waiting time of trucks. In this study, we adopt six algorithms from Yu [4] to figure out the appropriate number of truck pairs as the initial solution. The searching procedure is as follows. 1. Verify all product types from inbound and outbound trucks; if it has one product type both in the single inbound and single outbound trucks, then take these two trucks as a feasible pair. 2. Put all the feasible pairs into six algorithms from Yu [4] and calculate the objective value for each feasible solution. Pick up the solution that has the best objective value to be the best truck pair in this iteration. 3. Update the product types and the number of products. 4. Stop the searching procedure when all product types and the number of products are equal to zero. Otherwise, go back to step 1.