Introduction

Cross-dock is a center that transships freight between trucks with minimal use of storage in between. Cross-docking reclaims transportation efficiency by bundling arriving freight into full truckloads. Lowering the entire inventory level is another economical advantage. However, cross-docking is beneficial as long as savings in inventory and transportation expenses do not overwhelm the cost of material handling at the platform (Bartholdi et Gue (2000) ;Gue (1999)).

The problem of material handling at cross-dock deals with decisions on how to transfer freight inside the terminal. It is a process that has a direct interaction with platform key resources ( i.e., operators, internal transporters). Therefore, the cost of material handling is defined based on the effective use of platform resources.In transhipment process, double hand- ling refers to an additional retrieval and displacement for freight in temporary storage area.

It is an inefficient phenomenon that influences important platform performance indicators (e.g., handling rate, output rate, operator requirement, processing time, usage of temporary storage) (Schwind (1995)). This paper aims to present a scheduling model that reduces the cost of double handling in the transhipment process.

Cross‐dock  scheduling problems Dock‐door assignment • Platform Chrtcs:  # doors = # dest. • Double‐handling: N/A • Transhipment method: Automatic, Manual Truck sequencing

• Platform Chrtcs:  # doors ¹ # dest.

• Double‐handling: Negligible impact • Transhipment method: Automatic Truck sequencing  + Internal transshipment • Platform Chrtcs:  # doors ¹ # dest. • Double‐Handling: Considerable impact • Transhipment method: Manual

Figure 3.1 Cross-dock scheduling problems

According to the literature, most studies on cross-dock scheduling are quite recent (see Van Belle et al. (2012) ; Boysen et Fliedner (2010) for an extensive review). This research can be classified in three categories according to the impact of double-handling on the operational cost. Figure 3.1 illustrates this classification.

In the first group of studies, cross-dock has a sufficient amount of doors to process all trucks. With this platform characteristics, all of the freight directly displaced within doors.The scheduling problem focuses on optimizing total transfer and congestion. Gue (1999) and Tsui et Chang (1992) have studied platform layout design for receiving and shipping doors. For a known door layout with the same goal, much research has been performed in operational scheduling to assign trucks to platform doors (see Cohen et Keren (2009) ; Oh et al. (2006) ; Bartholdi et Gue (2000) ; Chmielewski et al. (2009) ; Lim et al. (2006) ; Miao et al. (2009)).

In the second and the third groups of studies, the platform has a limited amount of docks. Therefore, the aim of the scheduling model is to determine a processing order for the trucks. The second group of studies focuses on truck loading and unloading schedule. The schedu- ling problem is to determine a sequence of vehicles at the dock to minimize total operational time (The time between unloading the first truck to loading the last one). This problem is

an operational issue for distribution centers in courier industries in which highly automated conveyors and sortation systems are used to process a large volume of parcels every day. In this transhipment method, double-handling has negligible impact on total operational cost. Thus, scheduling problem is expressed as time indices models (see Yu et Egbelu (2008) ; Boy- sen et al. (2010) ; Vahdani et Zandieh (2010) ; Soltani et Sadjadi (2010) ; McWilliams et al. (2005) ; McWilliams (2009) ; Konur et Golias (2013)).

The third group of research develops scheduling models for cross-dock facilities in Less- than-TruckLoad (LTL) industries. In these terminals, labor intensive transporters (e.g., fork- lifts, pallet jacks) are employed. With this handling approach, product transhipment is costly and double-handling has a significant impact on the total operational expense (Bartholdi et Gue (2000)).

Unlike studies in truck sequencing for courier industries, the modeling approach based on reducing attendance time of the vehicle at the platform does not provide a precise measure of operational cost due to the following reasons :

First, the actual processing cost is directly related to the cost of each transfer operation. This means that direct transhipment or transferring via storage have different transferring costs. Even for fixed sequences of inbound and outbound trucks, various material handling decisions result in having different transhipment costs. Thus, a detailed transhipment plan is required.

Second, reducing the truck processing time at the platform dock may increase double- handling. In other words, decreasing vehicle presence at the inbound door may force the operator to transfer the remaining freight to a storage area. Also, at the outbound door, it may force the operators to use the products in storage to fill the truck. However, these products could be directly transferred if we synchronize the process of loading and unloading trucks with material handling decisions.

In literature, Boysen (2010), has investigated special LTL cross-dock in food industries with cooling requirements, that forbid internal storage. Therefore, all products are directly transferred within doors. The study considers a platform setting with single receiving and shipping dock. A dynamic programming model has been represented to minimize operational cost.

The problem of double-handling was first introduced by Maknoon et Baptiste (2009), the authors have proposed a heuristic as a resolution approach for the platform with a single receiving and shipping door. Further, for the same platform setting, the problem is proven to be NP-Hard in the strong sense. A polynomial algorithm was introduced to schedule product transhipment when the loading and unloading order of trucks is known (Sadykov (2012)). This paper is in line with previous studies, as we focus on the product displacement route. The

presented scheduling simultaneously considers internal transhipment and the trucks’ loading and unloading order.

In Section 3.2, we suggest a mathematical formulation that minimizes double-handling in internal transhipment. In Section 3.3, we introduce a series of valid inequalities that are added to the model. In Section 3.4, we present a path branching algorithm. Computational results are reported in Section 3.5, followed by the conclusion in Section 3.6.

Eliminated state Path representing operational plan 1 :   States eliminated by property 1 3:    States eliminated by property 3 2:    Path fixed by property 2 1 2 3 4 5 6 1 7 2 3 4 5 6 7 Outgoing Sequence Incoming  Sequence Truck

Figure 3.2 Model Schema