An Approximation Algorithm for Quay Crane Scheduling with Non-
interference Constraints in Port Container Terminals
a,Der-Horng Lee ∗, Hui Qiu Wang a and Lixin Miao b
a Department of Civil Engineering, National University of Singapore, Singapore 117576
b Research Center for Modern Logistics, Graduate School at Shenzhen, Tsinghua
University, P. R. China 518055

1. Introduction
Nowadays competition between port container terminals, especially between
geographically close ones, is rapidly increasing. How to improve the competitiveness of
port container terminals is, therefore, an immediate challenge, with which port operators
are confronted. In terms of port competitiveness, the makespan of a container vessel, which
is the latest completion time among all handling tasks of the container vessel, is a critical
success factor (Steenken et al., 2004). In reality, quay crane scheduling significantly affects
the makespan of a container vessel since quay cranes are the interface between land side
and water side in any port container terminals. Thus this paper aims to study quay crane
scheduling problem to enhance the competitiveness of port container terminals.

As illustrated in Figure 1, container vessels are typically divided longitudinally into holds
that open to the deck through a hatch. Holds are about eight containers deep, and containers
can also be stacked (about six high) on deck (Daganzo, 1989). The interference between
quay cranes is that quay cranes cannot cross over each other because they are on the same
track. In practice, only one quay crane can work on a hold at any time. Generally, a quay
crane can move to another hold until it completes the current one. The average processing
time of a hold is about two hours and the travel time of a quay crane between two holds is
about one minute. The quay crane scheduling problem in port container terminals is to
determine a handling sequence of holds for quay cranes assigned to a container vessel in
fulfilling pre-specified objectives and satisfying various constraints. For instance, there are
ten holds in a container vessel, and two quay cranes are allocated to handle the container
vessel. Table 1 illustrates a feasible quay crane schedule for this instance. It shows the
handling sequence of holds for every quay crane, the processing time of each hold and the
time schedule for handling every hold.


∗Corresponding author
Tel: +65 6516 2131
Fax: +65 6779 1635
E-mail: cveleedh@nus.edu.sg
1

Fig. 1. The illustration of the QCSNIP.

Table 1 An illustration of a quay crane schedule
Quay Crane 1 Quay Crane 2
Operation
Sequence
Hold
Number
Processing
Time of a
Hold (min)
Completion
Time of the
Quay Crane
(min)
Operation
Sequence
Hold
Number
Processing
Time of a
Hold (min)
Completion
Time of the
Quay Crane
(min)
1 1 98 98 1 2 81 81
2 3 119 217 2 5 178 259
3 4 52 269 3 10 171 430
4 9 101 370 4 8 162 592
5 7 114 484
6 6 81 565

Daganzo (1989) studied the static and dynamic quay crane scheduling problems for
multiple container vessels. Daganzo (1989) assumed that container vessels were to divide
into holds, and only one quay crane could work on a hold at a time. Quay cranes could be
moved freely and quickly from hold to hold, and container vessels could not depart until all
their holds had been handled. The objective was to serve all these container vessels, while
minimizing their aggregate cost of delay. Exact and approximate solution methods for quay
crane scheduling were presented in Daganzo (1989). Furthermore, Peterkofsky and
Daganzo (1990) developed a branch and bound solution method for the static quay crane
scheduling problem. Nevertheless, both papers did not consider the interference between
quay cranes, which means the quay cranes could unrealistically cross over each other.

Lim et al. (2004) augmented the static quay crane scheduling problem for multiple
container vessels by taking into account non-interference constraints. They assumed that
containers from a given area on a container vessel were a job, and there was a profit value
when a job was assigned to a quay crane. The objective was to find a crane-to-job matching
Sea side
The front of the
container vessel
The tail of the
container vessel
Quay
crane ……
1 2 …… H
Container vesselHold
H-1
1 K
Land side
K: The number of quay cranes
H: The number of holds
2