Integrated optimization on yard crane scheduling and vehicle positioning at container yards

Highlights

• Simultaneous determination of yard crane schedule and vehicle parking positions.

• Propositions for determining optimal vehicle parking positions.

• A two-stage heuristic algorithm is developed and examined its outperformance.

• Fast convergence of the proposed algorithm contributes to computational competency.

Abstract

A container yard is a storage facility that allows handling resources to improve operational efficiency by facilitating container flows at a container terminal. The container yard system consists of a set of storage blocks with yard cranes performing stacking and unstacking operations for containers to be transported by vehicles. High operational efficiency can be achieved by managing and coordinating the handling operations of yard cranes and vehicles (e.g., the yard crane scheduling, vehicle job dispatching, and coordinating handshakes between yard cranes and vehicles). This study proposes an integrated optimization approach for simultaneously determining the yard crane schedules and the vehicle parking positions under the Chebyshev movement allowing for the simultaneous movement of gantry and trolley of the yard crane. A mixed-integer programming model is formulated to optimize the problem, and the two-stage heuristic algorithm is developed to solve the problem efficiently. Several propositions are also provided to search the optimal boundary of vehicle parking slots for pairs of jobs. Numerical experiments are conducted to show the outperformance of the proposed heuristic algorithm compared to the well-known rule-based heuristics.

Introduction

A container yard system is a key facility temporarily accommodating outbound, inbound, and transshipment containers with a set of container handling equipment. The storage space not only supports the throughput of quayside operations (loading and unloading) but also the handling service for landside operations (receiving and delivery operations). It means that the operational efficiency of a container terminal is conditioned by the yard operation collaboratively achieved by the set of handling equipment. Studies on yard operations are typically categorized into storage space allocation, remarshaling and reshuffling, and equipment scheduling under the handling equipment configurations of Quay Cranes (QCs), vehicles, and Yard Cranes (YCs). Note that this study specifies vehicles as a mode of transportation for containers within a container terminal. The storage space allocation and the remarshaling/reshuffling are also conditioned by the equipment set (Carlo et al., 2014, Gharehgozli et al., 2015, Lehnfeld and Knust, 2014). A container yard consists of many storage blocks for stacking containers, and each block is surrounded by aisles for vehicles to travel and is equipped with YCs to perform stacking and unstacking operations. Hence, the yard operational efficiency is likely dependent on the scheduling operations for YCs and vehicles, including the handshakes between them, which is inevitable when there is a handshake between the two types of equipment.

There are typically two types of block layouts, namely the end-loading and side-loading blocks, as illustrated in Fig. 1. The end-loading block layout requires vehicles to interact with YCs at the ends of the block. This block layout has an advantage for container yards as the traffic controls at seaside and landside are separated so that the seaside blocks are accessed only by vehicles for loading (unstacking) and unloading (stacking) operations whereas the landside is accessed only by road trucks for receiving (stacking) and delivery (unstacking) operations. The representative real-world examples are the Euromax terminal in Rotterdam and Altenwerder terminal in Hamburg. The end-loading block layout requires YCs to perform a round-trip per handling job, possibly resulting in a long waiting time of vehicles. The side-loading block layout is often found in many conventional container terminals such as Pasir Panjang Terminal in Singapore and Modern Terminal in Hong Kong, leading to challenging traffic control requirements for different types of handling equipment. This block layout requires vehicles to temporarily park beside the block for the YCs to perform handing activities typically within a small range of the block in the vicinity of the vehicle to improve the operational efficiency by reducing the YC gantry travels.

A number of leading container terminals (e.g., Tuas Maritime Hubs in Singapore and Yangshan Deepwater Harbor Phase Ⅳ in China) have been applying automation technologies to side-loading blocks. A YC usually consists of three-movement components: (a) the gantry, which moves along the long-side of the block, (b) the trolley, which is attached on the gantry arm and moves perpendicularly to the gantry direction, and (c) the spreader (not shown in the figure), which is attached beneath the trolley and lifts containers from/to the stacking slot. The YC movement traditionally performs in a rectilinear way, as illustrated by solid arrow lines in Fig. 2. The rectilinear movement allows the trolley to move only after completing the gantry movement. Fig. 2 indexes the parking slots where vehicles temporarily stay for job handling as row numbers 0 and 6, and the stacking slots where containers are stacked as row numbers 1–5, together with bay numbers 1–20. The latest technology allows YCs to apply the Chebyshev metric to the two movement components (i.e., gantry and trolley) to reduce travel time. The Chebyshev movements are depicted as dotted arrow lines in Fig. 2. Hoisting and lowering the spreader must be performed only after the movement of gantry or trolley for the safety and stability of container handling.

The Chebyshev movement contributes to the reduction of YC travel time compared to the rectilinear movement (Lee and Kim, 2010). The side-loading block layout provides room for further improvement of the operational efficiency of YCs when looking into the YC movement together with vehicle parking positions. For example, referring to Fig. 2, it is assumed that the unit travel time per slot to be 1 for both gantry and trolley movements and the current trolley position at the slot of (bay 1 & row 5). So the total travel time is shorter when the YC performs a stacking activity to the slot of (bay 11 & row 2) directly from the parking position beside bay 7, (bay 7 & row 0), than having the vehicle and YC first move to beside bay 11, (bay 11 & row 0) before YC operation. Without considering the pick-up and release time, the parking slot of (bay 11 & row 0) requires 12 travel time-units, whereas the YC needs to spend ten travel time-units for the parking slot of (bay 7 & row 0). This shows that the Chebyshev movement can potentially further reduce the YC travel time when the YC operation sequence is coordinated with vehicle parking positions in the side-loading block layout. In the alternative end-loading block layout, there will be little merit to apply the Chebyshev movement due to the long round-trip distance and limited parking positions.

The YC scheduling problem is a representative research theme in the area of equipment scheduling for container terminals as it is the most popular equipment type for handling containers at the yard. A key element for YC scheduling is the reduction of the completion time (i.e., makespan) for incoming jobs, including the waiting time for handshakes. This study proposes a significant element (i.e., vehicle parking positions) contributing to the operational efficiency of a YC at a side-loading block by utilizing the advanced technology (i.e., Chebyshev movement). This study develops an integration scheduling approach for YC scheduling and vehicle parking positions to minimize the makespan. This study contributes to the followings:

• Simultaneous optimization of the job sequencing of YCs and the parking positions of vehicles for a batch of jobs;

• Mixed-integer programming models for optimally integrating the two scheduling elements (i.e., YC sequencing and vehicle positioning) and a two-stage heuristic algorithm for efficiently finding solutions; and

• Myopic decision propositions determining the optimal vehicle parking positions for a pair of job types under the Chebyshev movement.

According to our best knowledge, this is the first study to optimize the two scheduling elements (i.e., YC sequencing and vehicle positioning) simultaneously. The remaining of this study is organized as follows: Section 2 reviews the literature; Section 3 provides mixed-integer programming models to optimize the problem; Section 4 develops a tabu search-based heuristic algorithm to solve the problem efficiently; Section 5 conducts various experiment and examines the efficacy of the proposed algorithm compared to benchmarking algorithms, and conclusions are drawn in Section 6.