Robust berth scheduling at marine container terminals via hierarchical optimization

Abstract

In this paper, we present a mathematical model and a solution approach for the discrete berth scheduling problem, where vessel arrival and handling times are not known with certainty. The proposed model provides a robust berth schedule by minimizing the average and the range of the total service times required for serving all vessels at a marine container terminal. Particularly, a bi-objective optimization problem is formulated such that each of the two objective functions contains another optimization problem in its definition. A heuristic algorithm is proposed to solve the resulting robust berth scheduling problem. Simulation is utilized to evaluate the proposed berth scheduling policy as well as to compare it to three vessel service policies usually adopted in practice for scheduling under uncertainty.

Introduction

Marine container terminals are essential components of global supply chains and terminal operators face many challenges on day to day operations to remain efficient and competitive. Most of these challenges are due to the interactions between different operations and facilities within the terminals. For instance, berth scheduling with respect to vessel characteristics considering the storage, loading, and unloading operations at the yard side is a daily challenge for terminal operators. Another challenge comes from assigning vessels to the available berths depending on different service policies followed by the terminal operator [1], [2]. Terminal operators should overcome these challenges to improve their performance measures and stay competitive. Specifically, there is a growing industry interest to expand the metrics in reliability and understand the gap in container delivery and liner schedule reliability.¹ It is, therefore, crucial to evaluate the causes and the effects of shipment delays.

To stay competitive, terminal operators should develop accurate and reliable berth schedules in order to avoid its customers' shipment delays. Berth scheduling refers to the allocation of a specific vessel to a particular physical location within the port for loading/unloading processes. This paper concentrates in the development of a berth schedule that explicitly accounts for the uncertainties in vessel arrivals and handling times. Even though port operators may have an estimated vessel arrival time window, it is difficult to know the exact arrival time in advance (e.g., delay due to weather, delay at port of origin). In addition, due to a number of operational factors (e.g., quay crane breakdowns, yard congestion, changes to stowage plan, etc.) terminal operators can usually only estimate an upper and a lower bound on the vessels' handling times.

A significant amount of research has been conducted to analyze and improve berth scheduling policies (we refer to [2] for an excellent and recent literature review and classification on seaside operations at marine container terminals). The research studies on berth scheduling could be classified considering four main assumptions on the inherent characteristics of the problem [1], [2]. The first assumption is on the definition of the berth space as discrete or continuous space. In case of discrete space, the wharf is divided into a specific number of berths. On the other hand, continuous space defines the entire wharf as the berth space. We note that a third case exist (hybrid) where the wharf is discretized in berths but multiple vessels can be served at each berth simultaneously [2]. The second assumption is on the dynamic or static nature of vessel arrivals. Particularly, while static vessel arrivals refer to the case where all vessels are at the port when the schedule is developed, the dynamic vessel arrivals formulation assumes that arrival times are known within a time window. The third assumption refers to deterministic versus stochastic nature of handling times. While handling times are assumed to be known in deterministic handling time formulations, handling times are assumed to be random variables in stochastic handling time formulations. The fourth assumption refers to performance measurements or to uncertainty of the problem parameters; i.e., handling and arrival times. Most of the studies in the current literature focus on and vary in their assumptions on the first two characteristics and limited research has been published on analyses of the latter two characteristics.

In the setting of this paper, discrete berth space and uncertain vessel arrivals and handling times are assumed. We propose a mathematical formulation (extending the work by Konur and Golias [3]) that simultaneously minimizes the average and range of the total service time required to serve all the vessels at the terminals. The problem is initially formulated as a bi-objective optimization problem that contains two optimization problems in the definitions of each of the two objective functions. To overcome this complexity, the problem is decomposed into a bi-objective bi-level optimization problem. The revised formulation simplifies the problem and provides means to address both objectives in isolation. To solve the resulting problem, we propose a heuristic approach that combines exact and heuristic solution methods. The proposed scheduling strategy is compared to three commonly used scheduling strategies under arrival and handling time uncertainty: First Come First Serve Early Start (FCFS-S), First Come First Serve Early Finish (FCFS-F), and Expected Arrival and Handling Time Scheduling (EAHTS).

To the best knowledge of the authors, this study is the first in analyzing robustness in case of uncertain vessel arrival and handling times for the berth scheduling. The rest of the paper is structured as follows. Section 2 reviews the berth scheduling literature. Section 3 explains the proposed robust scheduling approach and provides the mathematical formulations. The proposed solution algorithm is explained in Section 4. Section 5 present results from a number of numerical examples and compares the proposed approach to three commonly used berth scheduling policies under arrival and handling time uncertainty. Summary of contributions, results and findings, and possible future research directions are noted in Section 6.