Hierarchical Control of Equipment in Automated Container Terminals - Computational Logistics - page 7

The resulting schedule minimizes the makespan. The schedule provides the
time at which each task must be finished. The time allowed for carrying out
a task, as well as the meeting time is then sent by the supervisory controller
to the relevant controller in the lower level.
- The lower level
In the lower level, the system is driven by the continuous-time dynamics of
each piece of equipment. A controller of a piece of equipment can do partic-
ular tasks. When the supervisory controller asks a lower-level controller how
much time is required to do a task, it uses minimal-time control to determine.
After receiving the scheduled time for meeting and completing the task from
the supervisory controller, the controller of each piece of equipment will try
to finish its task within this time interval. Based on the continuous-time
dynamics and a certain cost function defined over a time horizon, optimal
control is applied for the continuous-time control of the equipment.

The details of the controller in the higher-level and lower-level are given next.

3.1 The Higher-level Controller

In the higher level, the goal is to minimize the makespan. The minimal makespan
is determined by the operation time s i,j of task τ i,j described in (1)-(6). In the
higher level, the scheduling problem can be described as follows:

N

min
S

J i ( x i,j ,s i,j ) (10)

i =1

subject to the discrete-event dynamics (1)-(6) and the time constraints due to
the time required by the lower level to carry out tasks, where N is the number
of containers, J i is defined as the transport time associated with container i and
S =[ s 1 , 1 , ..., s 1 , 6 ,s 2 , 1 , ..., s 2 , 6 ,s N, 1 , ..., s N, 6 ] T describes the vector with variables
representing the time required for doing task τ i,j associated with container i .
For the sake of simplicity, we assume here that there is no reshuing of con-
tainers in the vessel. Considering N containers that are transported successively,

i =1 J i = N− 1

i =1 J i + J N .Solvingmin i =1 J i is based on the result of solving

min N− 1

i =1 J i . Therefore, the optimal control problem of N containers can be
solved recursively from container 1 to container N . The completion time of the
handling of container N is given by x N, 4 , which gives the time at which container
N is unloaded in the storage yard. The scheduling problem of transporting N
containers in the higher level can therefore be written as follows:

min
S

x N, 4 (11)