The cross-docking is a very important subject in logistics and supply chain managements. According to the definition, cross-docking is a process dealing with transhipping inventory, in which goods and products are unloaded from an inbound truck and process through a flow-center to be directly loaded onto an outbound truck. Cross-docking is favored due to its advantages in reducing the material handing cost, the needs to store the product in warehouse, as well decreasing the labor cost by eliminating packaging, storing, pick-location and order picking. In cross-docking, products can be consolidated and transported as a full load, reducing overall distribution costs. In this paper, we focus on a truck scheduling at the multi-door, multi-crossdocking network with inventory constraints and process capability constraints. In this model, a truck can visit severals docks for loading or unloading many types products. This situation is very common in reality. This study also developed an exact mathematical model using mixed-integer linear programming (MILP) with the objective of minimizing the makespan to obtaint the benchmark in small scale problems. Large scale problems are solved through Simulated Annealing (SA) algorithm and Tabu Search (TS) algorithm. Performance of these algorithms will be compared to benchmarks obtained from solver as well as to each other.

As the global markets on supply chain has seen an influx of competitors during the past few years, it is pertinent that manufactures, retailers and distributors strive to optimize costs to increase their competitiveness. Driven by such demand, the idea of cross docking was hailed. It was defined by [

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In this paper, the study will focus on the cross-docking process which involve separate, multiple docks that have the capability to handle different types of products. All cross docks allow temporary storages, but at the end of the day, the inventory in all cross docks has to be zero. In addition, the layout is symmetrical, meaning there are an equal number of inbound and outbound doors for each dock. We also assume that the inbound doors and outbound doors are separate, meaning each set has single purpose. At all times, each door can only process one truck and preemption is not allowed. Furthermore, the number of loaded products has to be equal or larger than the demand.

In our model, the super scripts R and S represent for variables relating to process of receiving and shipping, respectively.

α Time for handling an unit item

γ Transition time of the truck between the docks

Subject to

The

The

This imposed the time window constraints on receiving trucks. The receiving truck cannot enter the dock before its allowed soonest enter time in

The

The

The

The

The

The

We also apply the same physical constraints for the shipping trucks, which creates

The total unload at a dock is assumed to be affected right after the receiving truck

The same idea is applied for constructing the constraints of shipping trucks. However; the total load to a shipping truck up to time

By forcing the inventory level is always greater or equal to zero and smaller than capacity through

To solve the small-scale problems, CPLEX Optimizer engine which is developed by IBM company was used to create the benchmark. However, due to the NP-hard property of the original problems, when the size increases metaheuristic algorithms must be adopted. In this study, TS and SA are also implemented and results obtained from CPLEX are used as benchmarks. During TA and SA, this study applies two common following algorithms for creating initial solutions and assignment process. The

TS and SA are chosen is due to its simplicity in the process of creating new solution in the process of exploring and exploiting, and the foundation of such process is the neighborhood search method. In this paper, the neighborhood search is implemented through two swapping methods in

The pseudo code of TS is described in

For simulated annealing algorithm, we verify its performance with two versions. The first one employs the sigmoid function which is presented in

To conduct result analysis, ten data sets with different scales are considered. The data set information and the results obtained from CPLEX are given in

Problem | Receiving truck | Shipping truck | Product | Total quantity | Cmax | Runtime (second) |
---|---|---|---|---|---|---|

1 | 3 | 4 | 4 | 140 | 112 | 20 |

2 | 3 | 4 | 5 | 64 | 36 | 37 |

3 | 3 | 4 | 8 | 180 | 110 | 57 |

4 | 3 | 4 | 9 | 188 | 110 | 81 |

5 | 3 | 4 | 10 | 444 | 277 | 217 |

6 | 5 | 4 | 6 | 1030 | 476 | 9478 |

7 | 6 | 4 | 8 | 491 | 221 | 9193 |

8 | 5 | 7 | 10 | 430 | 208 | 10812 |

9 | 6 | 7 | 10 | 2020 | Unsolved | |

10 | 6 | 11 | 15 | 1252 | Unsolved |

For small-scaled problems, CPLEX works quite well in terms of run time, which only takes less than 2 minutes to solve. When there is increase in the number of trucks and product quantity, the run time grows exponentially as can be seen from the data set 6 to 10.

The comparison between results for both TA and SA and CPLEX are shown in the

Problem | TA | SA (Sigmoid) | SA (Metropolis) |
---|---|---|---|

1 | 0.00% | 0.00% | 0.00% |

2 | 0.00% | 0.00% | 0.00% |

3 | 0.00% | 0.00% | 0.00% |

4 | 0.00% | 0.00% | 0.00% |

5 | 7.94% | 7.94% | 7.94% |

6 | 9.03% | 10.08% | 9.03% |

7 | 9.50% | 13.57% | 9.50% |

8 | 0.48% | 1.92% | 0.48% |

In the

On an overall viewpoint, SA Metropolis algorithm yields most promising results when comparing with 2 other methods in gap.

In conclusion, to solve the problem of truck scheduling in crossdocking network, 3 approaches are taken. The first is using MILP in conjunction with CPLEX to solve for the exact solution. However, because of its restriction to small-sized problems, TS and SA are implemented to search for the makespan of large-sized problems. The two metaheuristics exhibit the tradeoff between producing a consistent and good result and having short run time. In general, the results from the approaches proved to be not only optimal and feasible to the constraints of the system, but also managed to adhere and comply to several practical conditions. The result also proves the credibility and feasibility of the model as well as the algorithm. Regarding the all-encompassing and real-life adherent nature of the proposed model, not only does it make a solid contribution to the topic’s literature but also serve as a foundation for further development of the program into software. Further study on this topic can be expanded to include the interior operations of the crossdocking network. Another direction is to expand the problem downstream by combining the truck scheduling problem with the vehicle routing problem to the customers. Although the algorithm obtained reliable results, this study still encountered some challenges in handling the most difficult constraint in the crossdocking problem, the concurrency of load and unload. This concurrency creates challenges in ensuring the feasibility of system state as well as the solutions deriving from the neighborhood. The feasibility is only assured through very carefully checked and revised mechanism. This process sometime takes long time for specific cases.