4.1 Data description
Data used in cases study presented in section 4.1 are detailed.
Table 1 shows the processing time of each order in each machine. Empty cell means this order cannot
be processed in this machine.
Table 1 processing time of each order in each machine which can be processed
Order 3
Order 4
Extruder-01
Order 1
225
421.88
292.5
292.5
Extruder-02
257.14
482.14
334.29
334.29
Extruder-03
225
421.88
292.5
292.5
Extruder-04
257.14
482.14
334.29
334.29
300
375
Sewer-01
Sewer-02
Order 2
Order 5
Order 6
Order 7
Order 8
Order 9
281.25
428.57
514.29
257.14
428.57
514.29
257.14
500
400
375
292.5
334.29
642.86
334.29
334.29
642.86
334.29
281.25
225
Order 10
292.5
500
240
300
Sewer-03
300
375
500
400
375
Sewer-04
300
375
500
400
375
500
500
Trellis-01
600
600
600
600
Trellis-02
600
600
600
600
Packaging-01
50
25
25
40
25
20
Table 2 shows the set-up time between two orders in each machine. Empty cell means that at least one
order of two both cannot be processed in the machine.
Table 2 Set-up time between two orders in each machine
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 1
Extruder-01
70
60
70
Extruder-02
70
60
70
1
Extruder-03
70
60
70
Extruder-04
70
60
70
10
70
1
Sewer-01
36
21
26
11
26
Sewer-03
36
21
26
11
26
Sewer-04
36
21
26
11
26
5
5
5
5
5
10
70
1
60
70
70
1
60
60
70
70
60
Sewer-02
Trellis-01
Trellis-02
Packaging-01
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 2
Extruder-01
70
70
60
Extruder-02
70
70
60
Extruder-03
70
70
60
Extruder-04
70
70
60
Sewer-01
70
70
10
70
70
70
10
70
10
70
70
70
10
70
36
21
35
30
Sewer-03
36
21
35
30
10
Sewer-04
36
21
35
30
10
5
0
0
0
0
Sewer-02
35
10
10
Trellis-01
Trellis-02
Packaging-01
70
70
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 3
Extruder-01
60
70
10
Extruder-02
60
70
10
Extruder-03
60
70
10
Extruder-04
60
70
10
60
70
70
70
70
60
0
10
70
10
70
60
60
0
0
0
Sewer-01
Sewer-02
Sewer-03
Sewer-04
Trellis-01
5
5
0
Trellis-02
5
5
0
Packaging-01
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 4
Extruder-01
70
60
10
Extruder-02
70
60
10
Extruder-03
70
60
10
Extruder-04
70
60
10
70
70
70
70
10
10
70
10
70
70
70
70
70
10
10
10
Sewer-01
Sewer-02
Sewer-03
Sewer-04
Trellis-01
5
5
5
Trellis-02
5
5
5
Packaging-01
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 5
Extruder-01
Extruder-02
10
70
70
70
70
10
60
60
70
Extruder-04
10
70
70
70
70
10
60
60
70
Sewer-01
21
21
25
30
5
Sewer-03
21
21
25
30
5
Sewer-04
21
21
25
30
5
5
0
0
0
0
Extruder-03
Sewer-02
Trellis-01
Trellis-02
Packaging-01
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 6
Extruder-01
Extruder-02
70
10
70
70
70
70
70
10
70
Extruder-04
70
10
70
70
70
70
70
10
70
Sewer-01
26
35
25
35
Extruder-03
Sewer-02
20
35
20
Sewer-03
26
35
25
35
20
Sewer-04
26
35
25
35
20
Trellis-01
Trellis-02
Packaging-01
5
Order 1
0
Order 2
0
Order 3
Order 4
Order 5
0
Order 6
Order 7
0
Order 8
Order 9
Order 10
Order 7
Extruder-01
1
70
60
70
Extruder-02
1
70
60
70
Extruder-03
1
70
60
70
Extruder-04
1
70
60
70
Sewer-01
11
Sewer-03
Sewer-04
60
10
70
70
70
10
70
70
70
30
30
35
35
11
30
30
35
35
11
30
30
35
35
60
60
60
Sewer-02
Trellis-01
Trellis-02
Packaging-01
5
Order 1
0
Order 2
0
Order 3
Order 4
Order 5
0
Order 6
0
Order 7
Order 8
Order 9
Order 10
Order 8
Extruder-01
Extruder-02
70
70
10
10
60
70
70
60
10
70
70
10
10
60
70
70
60
10
Trellis-01
5
5
5
Trellis-02
5
5
5
Extruder-03
Extruder-04
Sewer-01
Sewer-02
Sewer-03
Sewer-04
Packaging-01
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7
Order 8
Order 9
Order 10
Order 9
Extruder-01
Extruder-02
70
10
70
70
60
10
70
60
70
Extruder-04
70
10
70
70
60
10
70
60
70
Sewer-01
26
10
5
20
35
Extruder-03
Sewer-02
10
20
Sewer-03
26
10
5
20
35
Sewer-04
26
10
5
20
35
Trellis-01
Trellis-02
Packaging-01
5
Order 1
0
Order 2
0
Order 3
Order 4
Order 5
0
Order 6
0
Order 7
Order 8
Order 9
Order 10
Extruder-01
60
70
0
10
Extruder-02
60
70
0
10
60
Extruder-03
60
70
0
10
Extruder-04
60
70
0
10
Trellis-01
0
5
5
Trellis-02
0
5
5
70
70
60
10
70
10
70
60
70
70
60
Sewer-01
Sewer-02
Sewer-03
Sewer-04
Packaging-01
Order 10
4.2 Results
Results data are shown taking into account two possible objective functions, makespan minimization
and delay minimization.
Every instance has been implemented on the modeler AIMMS 3.13 and solved with CPLEX v12.4 in a
PC Intel Core I3, OS Windows 7 and NVIDIA graphic card of 4 GB.
4.2.1 Makespan cases study
Four instances are presented in makespan case study. Table 3 shows the features of each instance.
Instance 1 is composed by 10 order and every machines where objective function is minimizing
makespan. Instance 2 is similar to instance 1 with the exception that extruder 1 is unable (i.e., because of
maintenance activities). Instance 3 is similar to instance 1 with the exception that order 9 and 10 are not
considered. Instance 4 is similar to instance 3 with the exception that objective function is minimizing
every finish time in each stage.
Table 3, Instance features in minimizing makespan
Instance
Orders
Machines
Objective function
Instance 1
10
4 Extruders, 4 sewers, 2
trellis, 1 packaging
Minimize makespan
Instance 2
10
3 Extruders, 4 sewers, 2
trellis, 1 packaging
Minimize makespan
Instance 3
first 8 orders
4 Extruders, 4 sewers, 2
trellis, 1 packaging
Minimize makespan
Instance 4
first 8 orders
4 Extruders, 4 sewers, 2
trellis, 1 packaging
Minimize every finish time
In table 4 solving time, iteration, constraints, variables and optimal solution are shown. A relevant
solving time difference between instance 1 and 2 is shown because of the machine removal. A relevant
constraint and variable amount between instances 1 and 2 in respect of 3 and 4 are shown because of the
order removal. Solving time between instance 1 and 3 is decreased considerably. Solving time between
instance 4 and 3 is increased because of the fact that objective function has more variables to minimize.
Instance 4 has not a good objective function since minimum makespan is not ensured.
There are not relevant differences between instance 1 makespan and instance 3 makespan since trellis
stages is a bottleneck.
Table 4, Computational data
Instance
Solving time in
seconds
Iterations
Constraints
Variables
(integers)
Optimal solution in
minutes, makespan
Instance 1
65
461.782
485
316 (262)
1497.5
Instance 2
5490
38.130.055
455
295 (241)
1698.14
Instance 3
1.68
9.747
323
221 (117)
1496.5
Instance 4
355
2.573.473
323
232 (177)
1518.18
Figure 1 to figure 4 shows orders configuration for each instance. In figures are shown each order
processed in the corresponding machine. Void between orders is at least the set-up time, that is, set-up
time and idle time is shown between orders.
Fig. 1 Instance 1, configuration orders
Fig. 2 Instance 2, configuration order
Fig. 3 Instance 3, configuration order
Fig. 4 Instance 4, configuration order
4.2.2 Due date cases study
Two instances are presented for due date case study. First one shows the performance of advance de
due date to an earliest data. Second one shows the performance of a higher penalty in a relevance order.
Instance 5 shows who model performances with the first eight orders. Every order has a due date of
1500 minutes. Every order accomplish as is shown in the figure 5.
Fig. 5 Instance 5, configuration order with the same due date for every order
Instance 5 is composed by 310 constraints, 228 variables (which 177 integers), solving in 125
iterations and spending a solving time of 0.06 seconds.
Now, in the same instance, due date for order 5, 7 and order 8 were changed to 1350, so that, with the
previous configuration there will be a delay in mentioned orders. Before executing, configuration changes
as is shown in figure 6. Orders 5, 7 and 8 are manufactured before its due date.
Fig. 6 Instance 5, configuration order changing due date for order 4 and 6
Once due date in orders 5, 7 and 8 are changed, model is solved in 137 iterations spending a solving
time of 0.08 seconds.
In instance 6, there exist an earliness due date for every order, that is, every order must finish before
than 1200. As is shown in figure 7, order 8 has a delay.
Fig. 7 Instance 6, configuration order with the same due date for every order
The same tardiness penalty was considered for every order. Now we assume order 8 cannot be delayed
because of the relevance, so a higher penalty is added to order 8 in order not to have a delay, certainly a
penalty of 50 units (rest of penalty orders are 1 unit). Figure 8 shows a new configuration where order 8 is
not delayed, in opposition, order 4 is delayed.
Instance 6 is composed with the same amount of constraints and variables as the previous instance and
it is solved in 4474 iterations spending a solving time of 0.94 seconds.
Fig. 8 Instance 6, configuration order adding a high penalty to order 8
Once a penalty of order 8 is changed, model is solved in 4609 iterations spending a solving time of
0.70 seconds.
Distinctiveness is worth highlighting. Solving time in makespan instances are greater than solving time
in due date instances, which is more efficient working (from the standpoint of scheduling) with due dates.