3.1. Impact of PM on Operational Completion Time
We assume that machines (i.e.,
) have PM activities executed after operations
. The feasible schedule with preventive maintenance is shown in
Figure 2. A feasible schedule that has executed PM activities. The PM activities are indicated with dark blocks. From the perspective of operation
, the other operations can be divided into two sets. The set consisting of
is called the associated set of
. The set consisting of
is called the non-associated set of
. For the operations in the associated set, if PM activity occurs before or after one operation, it may cause the delay of
. However, the completion time of the operation
can only have its delay prevented when a PM activity is executed after the operation
. As shown in
Figure 2, there are four precedence constrain chains to
. In addition, the influence can be delivered to
through any chain.
In chain 1, i.e., , the idle time between operations can absorb the impact of PM. The idle time between and can be obtained by . The completion time of operation affected by chain 1, is denoted as .
In chain 2, i.e., , the completion time of the operation , affected by chain 2, is denoted as .
In chain 3, i.e., , the completion time of the operation , affected by chain 3, is denoted as .
In chain 4, i.e., , the completion time of the operation , affected by chain 4, is denoted as .
where , , are all equal to 1, and the rest of the binary variables are 0.
The completion time of the operation caused by PM activities is denoted as
The sequence of PM is determined when the preventive maintenance plan is developed. In addition, the time of PM is often considered a constant. The calculation of PM’s influence on operation completion time is a deterministic problem.
The PM activities have two influences on the initial schedule : ① PM can reduce the possibility of machine failure and increases the stability of schedule; ② PM can reduce the idle time between operations and may increase the makespan of schedule.
3.2. Influence of Machine Failure on Operational Completion Time
The machine breakdowns will lead to the interruption of the processing of the jobs. Since this paper assumes that the jobs continue to be processed after the machines are repaired, it can be considered that the machine breakdowns substantially increase the time the job stays on the machine; at the operational level, it can be regarded as the increase of operation processing time. Assume that these operations (i.e.,
) will suffer from breakdowns. The actual schedule is shown in
Figure 3, and the breakdowns are indicated with red blocks. Through the same above principle, there are four precedence constrain chains to
. The effect can be delivered to
through any chain.
The completion time of , affected by chain 1, is denoted as .
The completion time of , affected by chain 2, is denoted as .
The completion time of , affected by chain 3, is denoted as .
The completion time of , affected by chain 4, is denoted as where , and , are all equal to one and the rest of the binary variables are zero.
The completion time of operation caused by machine breakdowns is denoted as . Compared to the PM activities, the CM activities are not only affected by subsequent operations of machine dimension, but are also affected by subsequent operations of job dimension, which will lead to more serious disorder.
Due to stochastic failures, the number and time of machine failures in actual scheduling cannot be obtained before the completion of production. Even though the realized number of breakdowns may be infinite, the complexity of computing increases exponentially with the number of associated operations. For example, the number of operations in the associated operation set of is six. Meanwhile, machine breakdown may occur during the operation . Thus, the number of total scenarios is .