Research paperInternal resource requirements: The better performance metric for truck scheduling?☆
Introduction
Cross-docking is a warehouse strategy that has helped to improve the efficiency in many of today’s supply chains. The basic idea of cross-docking is to transfer incoming cargo directly to outgoing trailers. Through this synchronization of in- and outbound flows, the costly storing and order picking functions of traditional warehouses can be minimized (e.g., [51, p. 12] and [54, p. 1708]). Since the cargo usually spends less than 24h in the cross-docking terminal, reduced inventory holding costs, required storage space, and handling costs, as well as a faster inventory turnover can be realized (e.g., [1, p. 292] and [17, p. 55]). Companies from various industries such as retailing (e.g., Wal-Mart [49]), express and small parcel delivery (e.g., DHL [9] and UPS [24]), automotive industry (e.g., Renault [45] and Toyota [55]), and less-than-truckload logistics service industry [27] operate cross-docking terminals in their distributions networks.
Besides its increasing practical relevance, cross-docking also received a lot of academic attention, especially in the last two decades. Scholars have studied manifold strategic (e.g., location and design of a cross-dock), tactical (e.g., material flow through cross-docking distribution networks), and operational (e.g., assigning trucks to dock-doors, determining times at which trucks are processed) decision problems related to cross-docking terminals. A vast number of publications have dealt with the so-called truck scheduling problem. It aims to compute feasible and appropriate truck schedules by determining where, i.e., at which dock-door, and when trucks should be processed.
When comparing truck scheduling in academia and industry, it can be seen that ongoing research is detached from industry practice. Ladier and Alpan [32] recently observed that academic research on truck scheduling often incorporates unrealistic assumptions. For instance, the majority of reviewed truck scheduling models assume that an infinite number of internal resources, such as workers and material handling equipment, is available. This assumption clearly does not reflect reality, where both workforce and material handling equipment are usually limited. By neglecting resource scarcity, these models are not suitable to address two major concerns of cross-docking practitioners: (i) determining the number of resources needed, and (ii) scheduling the resources in an efficient way. In addition, a discrepancy in terms of applied performance measures can be observed between truck scheduling in practice and theory. Practitioners usually strive for efficient truck schedules, i.e., plans which deploy a minimum number of resources and finish the workload “in time”. Therefore, practitioners usually use performance measures that are directly related to the resource requirements and utilization to steer cross-docking operations. These metrics, however, are rarely used by academic researchers. Instead, the makespan, defined as the time span that elapses from the start of the first operation until the completion of the last operation, followed by the travel distance, defined as the total distance traveled by the cargo inside the facility, are the two most frequently used performance metrics in truck scheduling models [32, p. 158]. It is often argued that minimizing the travel distance results in a minimum workload and ultimately minimizes the working time, since it requires a shorter time for a worker to complete the task. The following simplified example, however, shows that neither of the two metrics necessarily lead to a truck schedule that deploys a minimum amount of resources. For the sake of simplicity, the total processing time is used as a surrogate objective function for the total travel distance.
Consider a situation with five inbound trucks. The trucks’ arrival and departure times are given in Table 1 and must not be violated. Loading operations of outbound trucks start at 10:00. In order to ensure a smooth loading process, all inbound trucks must be processed by 10:00. A total of three dock-doors can be used for unloading operations. Travel distance differences are reflected by door-dependent processing times. Unloading an inbound truck requires exactly one worker equipped with a forklift (denoted as an operator in the following).
Given the truck information for the example, Fig. 1 shows truck schedules for various performance measures.
When aiming to minimize the makespan, unloading operations start relatively late and finish early. This leads to a compact truck schedule and expedites parallel processing on all available dock-doors. It requires three operators to execute the plan. When applying the total travel distance (or total processing time) as the objective (or surrogate objective) in the example, each truck is assigned to the dock-door with the shortest processing time. Different from the makespan minimization, the whole 2 h horizon is used for unloading operations. Due to the 5 min overlap of trucks 2, 3, and 4 from 09:05 to 09:10, a total of three operators must be deployed to execute the plan. Lastly, Fig. 1c shows a feasible truck schedule that can be executed with a minimum number of operators. The workload is distributed evenly over the planning horizon in order to avoid large peak workloads. It requires at most two operators at a time to handle the workload, which is 33% less than when applying the makespan or travel distance as the key performance measure. Since the “manpower is very often the first cost center of a logistic platform where the operations are done manually” [32, p. 147], this decrease helps to reduce the operational costs and increase the efficiency significantly.
The example suggests that when using the makespan or travel distance as the performance metrics in a truck scheduling model, the generated plan is not efficient per se. The most frequently used performance measures, hence, fail to support cross-docking practitioners in efficiently scheduling the internal resources. This paper addresses this practical need by utilizing the internal resources as the key performance metrics in a truck scheduling model. The problem considered in this paper is how to schedule a set of inbound trucks with time windows at a multidoor cross-docking platform, where the departure times of outbound trucks follow a given schedule. The goal is to identify a feasible truck schedule that can be executed with a minimum number of internal resources.
The main contributions of this paper are as follows. A novel operational cross-docking problem of simultaneously scheduling inbound trucks and internal resources (e.g., workers or material handling equipment) is modeled as a machine scheduling problem with unrelated parallel machines and resource constraints. The model, in the following referred to as resource-constrained truck scheduling problem with fixed outbound departures (TSFD-RC), allows for the scheduling of internal resources in a more efficient way, and hence addresses a major managerial issue in cross-docking facilities. We present a discrete-time model formulation for this problem. Moreover, a column generation-based solution procedure is developed. This heuristic solution procedure is shown to perform very well on realistic large-sized instances and clearly outperforms standard MIP solvers. We also derive managerial insights through a large numerical experiment. In this context, the effects of exogenous factors (e.g., number of dock-doors, width of time windows, etc.) and the service level on the number of deployed resources are analyzed. Furthermore, we analyze the solution robustness of the novel model and how it affects the throughput times of trucks.
The remainder of the paper is structured as follows: Section 2 provides an overview of the literature that is relevant to our work. In Section 3, we describe our model assumptions and propose a mixed-integer programming formulation. Section 4 investigates the model complexity, provides a lower bound on the number of deployed resources, and proposes a column generation-based solution procedure for the problem. Section 5 contains a computational study. The study benchmarks the heuristic solution procedure and derives managerial insights. Section 6 concludes the paper with an outlook on future research opportunities.
Section snippets
Literature review
Van Belle et al. [52] and Buijs et al. [14] provide a comprehensive review on strategic, tactical, and operational decision problems in cross-docking facilities. Ladier and Alpan [32] review the literature on the most common operational decision problems in cross-docking facilities, such as the dock-door assignment problem [11], [26], [30], [38], [39] and truck scheduling problem (Ladier and Alpan [32] use a different terminology and further distinguish between sequencing and scheduling
Problem description
In the resource-constrained truck scheduling problem with fixed outbound departures (TSFD-RC), we consider a cross-docking terminal with multiple inbound and outbound dock-doors and an exclusive service mode, that is in-/outbound trucks can only be processed at in-/outbound doors, respectively. We assume that the set of outbound trucks has already been scheduled, i.e., the truck-to-door assignment and the start times of outbound trucks are known. Hence, the problem reduces to a scheduling
Complexity, bounds, and solution procedure
This section sets out to propose a solution procedure for the TSFD-RC. Specifically, a column generation-based solution procedure is developed in Section 4.3. Prior to that, we discuss the model complexity in Section 4.1 and propose a lower bound for the TSFD-RC in Section 4.2.
Computational performance
This section contains the numerical experiments for evaluating the performance of both the TSFD-RC and the proposed column generation-based solution procedure. The performance of the column generation-based solution procedure is analyzed in Section 5.2. The numerical study in Section 5.3 sets out to investigate the solution quality of the TSFD-RC and to derive managerial insights by benchmarking the TSFD-RC model with respect to various key performance indicators (KPIs) against frequently used
Conclusion
This study investigates the resource-constrained truck scheduling problem with fixed outbound departures (TSFD-RC), which aims to identify a feasible inbound truck schedule that can be executed with a minimum number of internal resources. We propose a discrete-time mixed-integer programming model for this novel problem, prove the complexity status of the problem, and develop a column generation-based solution procedure.
From a managerial perspective, cross-docking operations can be organized
Funding sources
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
CRediT authorship contribution statement
Pascal Wolff: Conceptualization, Methodology, Investigation, Writing - review & editing. Simon Emde: Conceptualization, Methodology, Investigation, Writing - review & editing. Hans-Christian Pfohl: Conceptualization, Methodology, Investigation, Writing - review & editing.
Declaration of competing interest
Authors declare that they have no conflict of interest.
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Area: Production Management, Scheduling and Logistics. This manuscript was processed by Associate Editor Kress. Research Paper: Production Management, Scheduling and Logistics.