Mixed Integer Linear Programming in Process Scheduling: Modeling ...

136 FLOUDAS AND LIN
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(a) Discrete-time representation
(b) Continuous-time representation
Figure 3. Discrete and continuous representations of time.
with these time intervals, as illustrated in part (a) of figure 3. Therefore, activities that
affect the schedule or changes of the state of the manufacturing system, such as the start
of a task or change of inventory, can only take place at a specific instance of each time
interval, for example, at the beginning of each time interval. This is essentially an approximation
of time. In many cases, a short length of time has to be used for the duration of
the time intervals in order to achieve a reasonable degree of accuracy. For example, when
all the processing times are fixed, their greatest common factor (GCF) can be selected as
the duration of the time intervals, otherwise only suboptimal solutions can be obtained.
When very small time interval length needs to be used and/or the time horizon under
consideration is long, the number of the discretized time intervals can be very large. This
leads to very large combinatorial problems, which are computationally very expensive
to solve or even intractable. These two main drawbacks, namely the approximation nature
and the computational difficulty in the solution of the resulting large combinatorial
models, substantially restrict the application of the discrete-time approach, especially to
real-world problems.
To address the aforementioned drawbacks of the discrete-time approach, research
efforts have emerged to develop more effective continuous-time formulations for process
scheduling. In contrast to the discrete-time approach, the continuous-time models introduce
the key concept of event or variable time interval. The exact definition of event varies
from one formulation to another, but essentially, they all correspond to a time instance in
the continuous domain of the horizon. By associating the events to continuous variables
which can take potentially any value in the time horizon, the activities or changes of system,
for example, the start and the end of a task, are allowed to take place at any time in
the horizon, which renders the capability of modeling the time most accurately. More importantly,
by eliminating a major fraction of the inactive event-time interval assignments
that is characteristic in the solution of a discrete-time model, a usually much smaller
number of events or variable time intervals need to be introduced to model the production
process compared to the number of fixed time intervals required by the discrete-time
approach. Consequently, the continuous-time approach leads to mathematical programming
problems of much smaller size, which, in many cases, requires less computational
effort for their solution. On the other hand, however, because of the variable nature of the
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