Among 7 types of considered event, except job addition and job cancellation, five remain types can be defined their impacts on the initial schedule via the above proposed method. In detail, each type will be given an example case based on the numerical example in Chapter 4.
5.3.3.1 Machine breakdown
In the event of machine breakdown, the machine becomes unavailable for a duration of time. All operations assigned on that machine, whose start time is after the breakdown time, are not able to be processed by the existing schedule until machine repair is finished.
In the case that machine k is breakdown at the time t. This research assumes that the time tr to repair the breakdown machine can be predefined. Set Oij is the operation affected directly by the event, p′ij is the remain processing time of Oij∗ that have not been processed yet and RSij is the time when the remain of Oij∗ starts to be processed.
In the numerical example, the case considered is that machine M1 (k = 1) is breakdown at t = 4 and the repair time is tr = 2. The directly affected operation is O31 with the remain processing time is p′11 = 3. Two values of latest completion time of O31 are LCC31 = LCT31 = 7 (defined in part 5.3.1). Applying RSR, O31 will continue to be processed on machine 1 after its repair process is finished (s′13 = t + tr = 4 + 2 = 6), so the real completion time of O31 is RC31 = RS31+ p′11= 6 + 3 = 9.
Based on Equation 5.6 and 5.7, ∆C= ∆T= 9 − 7 = 2. That means, the makespan value is changed from 11 to 13, simultaneously the maximum tardiness is also change from 2 to 4. This can be proven by Figure 5.5.
Figure 5.5 Machine breakdown example
Hence, if α and β are equal or less than 2, in other words, α and β are equal or less than
∆C and ∆T then the machine breakdown is assigned in Group 2. Otherwise, it is assigned in Group 1.
In some cases, machine breakdown may not affect on both Cmax and Tmax, if the directly affected operation still be finished before its latest completion time. It will be also assigned in Group 3.
5.3.3.2 Delay in job release time or Delay in machine available time
With the event of delay in job arrival, the release time of job is changed, later than the old one. Maybe this is because of material unavailability or change of production approval. Therefore, the start time and completion time of operation are also changed.
With the event of delay in machine arrival, the available time of a machine is later.
Hence, the start time and completion time of operation are also changed.
Based on the predictive scheduling method, the start time of the first operation on a machine is belong to the release time of the operation and the available time of machine.
These two events (delay in arrival and delay in machine available time) have the same effect on the schedule, so they are considered together in this part. Notation ri and r′i = ri + (delay time) are the initial release time and the new release time of job i when it is delay in arrival; RSij is the real start time of Oij∗.
In the numerical example, considering the case that job 2 arrival is delay from ri = 0 to r′i = 4 (or that can be available of machine M3 is changed from 2 to 4). As a result, the start time of O21 is delay to RS21 = 4 and the completion time of it is also delay to RC21= 4 + 1 = 5. Two values of latest completion time of O21 defined in part 5.3.1 are LCC21 = LCT21 = 4.
Based on Equation 5.6 and 5.7, ∆C= ∆T= 5 − 4 = 1. That means, the makespan value is changed from 11 to 12, simultaneously the maximum tardiness is also change from 2 to 3. This can be proven by Figure 5.6. Hence, if α and β are equal or less than 1, then the event is assigned in Group 2. Otherwise, it is assigned in Group 1.
Figure 5.6 Delay in arrival example
In some cases, the delay may not affect on both Cmax and Tmax, if delay time is equal or less than the slack time of directly affected operation, so it still be finished before its latest completion time. It will be assigned in Group 3.
5.3.3.3 Processing time variation
The event of processing time variation is easy to happen and often happens on more than one operation. Because of orocessing time of operations are predicted, there may be deviation between predicted and actual values. This makes change of the operation’s completion time.
It is difficult and waste of time to consider the individual effect of each variation. The proposed method of event evaluation plays a vital role to deal with this problem. The scheduler or the manager does not need to defined exactly each variation. It only needs to control the complete time of each operation, which is culmulated from many previous variations. Once there is any operation whose completion time later than its LC, it is shown that the schedule is influenced.
In the numerical example, considering the case that processing time values of some operations are changed as in Table 5.5.
Table 5.5 Processing time variations Operation Deviation Completion time
Old New
O21 +1 3 4
O12 +1 5 7
O31 +1 7 8
From these variations, the corresponding ∆C and ∆T of each job are defined.
∆C21= ∆T21= 4 − 4 = 0
∆C31= ∆T31= 8 − 7 = 1
∆C12= 7 − 7 = 0;∆T21= 7 − 9 = −2
The impact of all above events on the initial schedule is defined by the largest value of
∆C and the largest value of ∆T between operations whose processing time is altered.
∆C= max(∆C21, ∆C31, ∆C12) = max(0, 1, 0) = 1
∆T= max(∆T21, ∆T31, ∆T12) = max(0, 1, −2) = 1
That means the makespan value is changed from 11 to 12, simultaneously the maximum tardiness is also change from 2 to 3. This can be proven by Figure 5.7.
Figure 5.7 Processing time variation example
Hence, if α and β are equal or less than 1, in other words, α and β are equal or less than
∆C and ∆T then the event of processing time variation is assigned in Group 2. Otherwise, it is assigned in Group 1.
In some cases, the processing time variations may not affect on both Cmax and Tmax, so it still be finished before its latest completion time. It will be assigned in Group 3.
5.3.3.4 Tightenning the due time
The event of tightening the due time means that the due time of job is shortenned, sooner than the old one. It is often based on requirement of customer or change of jobs’
priorities. It only affects on the maximum tardiness value.
Set di′ is the new due time and Ti′ = ci− di′ is the new tardiness of job i that is tightenned.
Now, the deviation of maximum tardiness value is defined by:
∆T= Ti′− Tmax (5.8)
In the numerical example, considering the case that due time of job 3 is tightenned from d3 = 11 to d3′ = 8. The new tardiness of job 3 is T3′ = c3− d3′ = 11 − 8 = 3. Based on Equation 5.8, ∆T= T3′− Tmax = 3 − 2 = 1. That means the maximum tardiness increases by 1. If β are equal to or less than 1, in other words, β are equal to or less than
∆T then the event is assigned in Group 2. Otherwise, it is assigned in Group 1.
Another case is considered, in that, due time of job 3 is tightenned from d3 = 11 to d3′ = 10. The new tardiness of job 3 is T3′ = c3− d3′ = 11 − 10 = 1. Based on Equation 5.8,
∆T= T3′− Tmax = 1 − 2 = −1. The negative value of ∆T means that the event does not affect on the initial schedule, so it is assigned in Group 3.
5.3.3.5 Job cancellation
A job comes unpredictably in the schedule duration and it may be urgent or not. An urgent job needs immediately attention to be scheduled as soon as possible. This study focuses on objective of minimizing simultaneously the makespan and the maximum tardiness. The event of job cancellation can not affect on both objectives, so it does not alter the performance of initial schedule, but it may cause some idle durations on the
shop floor as well as result waste of capacity. Because of this reason, all events regarding to cancelling a scheduled job will be assigned in Group 2.
5.3.3.6 Job addition
The event of job addition happens when a job comes unpredictably in the schedule duration. It surely makes effects on the initial schedule, so it is only classified into Group 1 or Group 2. The major issue is that whether it is urgent and rescheduling action should be conducted as soon as possible or should be delay to next periodic rescheduling point.
However, in real-world production, there may be more than one job arrive during a scheduling period, total impact evaluation of these event cannot be defined through two parameters ∆C and ∆T as other events. The classification will be done as follows:
If the release time of the new job i∗ is over the next periodic rescheduling time point t, the job will be considered to be insert at the next rescheduling period (Group 2). If not, the tightness parameter ki∗ of the job if it must wait for next period is considered.
ki∗ = di∗ − t
∑nj=1i∗ pi∗j (5.9)
where di∗ is the due date of job i∗, pi∗j is the average processing time of operation Oi∗j among its set of doable machines. If ki∗ is larger than the initial schedule average tightness, job i∗ will be classified into Group 2. Otherwise, it will be classified into Group 1 and need to be reschedule immediately.
For example, in the numerical example, at the time t = 3, there are a new job (job 4) is approval. It will be release at r4 = 5 and its due time is d4 = 13. The doable machines and processing time of each operation of this job are shown in Table 5.6.
Table 5.6 Information of the new arrival job Job Release
Time Due
Time Oper- ration
Processing time
M1 M2 M3 Average
4 5 14
41 - 3 1 2
42 1 - 3 2
43 2 4 - 3
The tightness of job 1, 2, 3 is respectively 1.0, 1.0 and 1.2, so the average tightness value of current schedule is approximately 1.07. Three case of period time is considered:
Case 1: Assume that the periodic rescheduling time point is t = 4. Job 4 is release after this point, so it is classified into Group 2 and will be added in schedule at next period.
Case 2: Assume that the rescheduling period time point is t = 6. Job 4 is release before this point and the tightness if job 4 is considered at next period via Equation 5.9 is:
k4∗ = (14 − 6)/(2 + 2 + 3) = 1.14 > 1.07
Hence, job 4 is classified into Group 2 and will be added in schedule at next period.
Case 3: Assume that the rescheduling period time point is t = 7. Job 4 is release before this point and the tightness if job 4 is considered at next period via Equation 5.9 is:
k4∗ = (14 − 7)/(2 + 2 + 3) = 1 < 1.07
Hence, job 4 is classified into Group 1 and the rescheduling is implemented immediately at time t = 3.