QUAY CRANE SCHEDULING
PROBLEM IN PORT
CONTAINER TERMINAL
Wenjuan Zhao, Xiaolei Ma
What is QC Scheduling Problem

Determine the sequence of discharging and
loading operations in a ship by each Quay Crane
and the time schedule for the operation.
Major Input: Ship Stowage Plan
Ship bay
Deck
Hatch
Hold
Problem characteristics

Similar to m-parallel machine problem

Different from it with unique characteristics
 Precedence
 Tasks
relationships among tasks
on deck and in hold from the same bay
 Certain
tasks cannot be performed simultaneously
 Cranes
could not cross with each other
Problem inputs

Ship stowage plan (with all constraints)

Time required to carry each task

Crane travel time between different tasks

Crane ready time
Problem Notations

Indices
i, j Tasks to be performed
k QCs where k  1,..., K .

Problem Data
pi
rk
li
l k0
lkT
t ij
The time required to perform task i
The earliest available time of QC
The location of task (expressed by the ship bay number)
The starting position of QC
The final position of QC k
The travel time of a QC from location ( li ) of task i to
location ( l j ) of task j
Problem Notations

Sets of indices
 The set of all tasks
 The set of pairs of tasks not be performed simultaneously

 The set of ordered pairs of with precedence relationship
Decision variables
X ijk
Yk
Di
Z ij
W
1 if crane k performs task j right after task i; 0 otherwise
The completion time of QC k
The completion time of task i
1 if task j starts later than the completion time of task i; 0
Time at which all tasks are completed
Problem Formulation
K
Minimize 1W   2  Yk
k 1
Subject to:
Yk  W k  1,..., K ,
(1) Define makespan
X
X
j
j
k
0j
 1 k  1,...K ,
(2) Start from task 0
k
iT
 1 k  1,...K ,
(3) End at task T
 X
k
i
k
ij
 1 j ,
(4) Each task assigned to one QC
k
k
X

X
 ij  ji  0 i  , k  1,...K , (5) Flow balance
j
j
Di  tij  p j  D j  M (1  X ijk ) i, j , k  1,..., K , (6) Time constraint
Di  p j  D j (i, j )  ,
(7) Precedence constraint
Problem Formulation
Di  Dj  p j  M (1  Zij ) i, j ,
(8) Define Zij
Z ij  Z ji  1 (i, j )  ,
(9) Non-simultaneous constraint
k
 X
v 1 u
k
v
uj
  X uiv  M(Zij +Z ji ) i, j  , li  l j , k  1,...K , (10) Non-interference
v 1 u
Dj  t kjT  Yk  M (1  X kjT ) j , k  1,..., K , (11) QC completion time
rk  Dj  t0k j  p j  M (1  X 0k j ) j , k  1,..., K , (12) QC starting time
X ijk , Zij  0 or 1 i, j , k  1,..., K ,
(13) Binary variables
Yk , Di  0 i ,k  1,..., K ,
(14) Non-negative
Reference:
(1) Kim, K.H., Park, Y.M., 2004. A crane scheduling method
for port container terminals. European Journal of
Operation Research 156, 752–768.
(2) Lee, D.H., Wang H.Q., Miao L.X., 2008. Quay crane
scheduling with non-interference constraints in port container
terminals. Transportation Research Part E 44, 124–135.