A cutting stock problem with
intermediate rolls and usable leftovers in stainless steel coil slitting
Jinwoo
Naa, Byung-In Kima,*
aDepartment of Industrial
and Management Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Gyeongbuk,
37673, Republic of Korea
*Corresponding author. E-mail address: bkim@postech.ac.kr
(B.-I. Kim)
Benchmark
Instances (Click)
Last
updated: 2026/09/16
1. Introduction
We
introduce a variant of the 1.5D-CSP
with intermediate rolls and usable leftovers in stainless steel coil slitting.
The main features of this problem are as follows:
(1) 1.5D-CSP with intermediate rolls.
(2) Usable leftovers generated from intermediate rolls.
(3) Strong heterogeneity in sizes of available stocks.
In stainless steel coil slitting, a raw coil may
either be processed as a whole or first split into two intermediate rolls, each
serving different groups of orders with distinct characteristics. This flexibility
fundamentally changes the structure of the optimization problem because it
eliminates the separability typically assumed in classical CSP problems,
which allows optimization to be decomposed by material characteristics. Furthermore, the widths of intermediate
rolls depend on cutting patterns determined by order widths, slit losses, and
leftover management, and usable leftovers may arise from both intermediate
rolls. As a result, integrating these decisions significantly increases both
the complexity and the practical relevance of the problem.
2. Benchmark instances
Benchmark instances were
constructed
from a real-world dataset
provided by
our industrial collaborator. The dataset contains
historical order and raw material information, including order widths, demands,
grades, and widths and weights of inventory and candidate coils. To ensure a comprehensive evaluation
of the proposed model under different production scales, the instances were categorized
into three size groups: small (S), medium (M), and large (L).
The
classification was determined based on the number of orders, number of order
groups, total demand, and number and total weight of eligible inventory and
candidate coils (both hot rolled and cold rolled).
For each size category, multiple instances were independently constructed by
randomly sampling orders from the industrial dataset. The corresponding order
groups and eligible inventory and candidate coils were then identified according
to the actual classification and compatibility rules. Table 1 represents the
real-world industrial settings and Table 2 summarizes the characteristics of
industrial and benchmark instances.
Table 1. Real-world industrial setting
|
Category |
Description |
Value |
|
Width |
Minimum rolling width (mm) |
580 |
|
|
Maximum rolling width (mm) |
1200 |
|
|
Maximum narrow rolling width (mm) |
700 |
|
|
Minimum cold-rolled split part
width (mm) |
200 |
|
|
Maximum cold-rolled split part
width (mm) |
1600 |
|
|
Minimum hoop width (mm) |
50 |
|
|
Maximum hoop width (mm) |
300 |
|
Capacity |
Total rolling capacity (ton) |
6150 |
|
|
Narrow width rolling capacity
(ton) |
1050 |
Table 2. Characteristics of instances.
|
|
Real |
Small |
|||||||||
|
Instance set |
|
S1 |
S2 |
S3 |
S4 |
S5 |
S6 |
S7 |
S8 |
S9 |
S10 |
|
# order |
140 |
3 |
3 |
3 |
4 |
4 |
4 |
5 |
5 |
5 |
5 |
|
# groups |
65 |
3 |
3 |
3 |
3 |
4 |
4 |
4 |
5 |
5 |
5 |
|
total
demand (t) |
4899 |
51 |
43 |
98 |
138 |
114 |
128 |
115 |
69 |
121 |
145 |
|
# inventory
hot-rolled coil |
558 |
24 |
24 |
22 |
17 |
27 |
36 |
24 |
40 |
27 |
37 |
|
inventory
hot-rolled coil weight (t) |
9101 |
410 |
410 |
379 |
279 |
454 |
613 |
410 |
673 |
454 |
629 |
|
# inventory
cold-rolled coil |
95 |
1 |
1 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
|
inventory
cold-rolled weight (t) |
1699 |
15 |
15 |
15 |
0 |
15 |
0 |
10 |
15 |
10 |
15 |
|
# candidate
hot-rolled coil |
21 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
candidate
hot-rolled coil weight (t) |
369 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
# candidate
cold-rolled coil |
9 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
candidate
cold-rolled coil weight (t) |
158 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
Medium |
Large |
||||||||
|
Instance set |
|
M1 |
M2 |
M3 |
M4 |
M5 |
L1 |
L2 |
L3 |
L4 |
L5 |
|
# order |
|
17 |
15 |
15 |
16 |
25 |
115 |
120 |
120 |
119 |
120 |
|
# groups |
|
16 |
11 |
11 |
14 |
19 |
60 |
56 |
59 |
62 |
58 |
|
total
demand (t) |
|
506 |
646 |
735 |
508 |
1130 |
3318 |
3922 |
4088 |
4170 |
4333 |
|
# inventory
hot-rolled coil |
|
262 |
174 |
174 |
262 |
274 |
558 |
558 |
558 |
558 |
558 |
|
inventory
hot-rolled coil weight (t) |
|
4273 |
2773 |
2773 |
4273 |
4465 |
9101 |
9101 |
9101 |
9101 |
9101 |
|
# inventory
cold-rolled coil |
|
89 |
0 |
0 |
89 |
89 |
95 |
95 |
95 |
95 |
95 |
|
inventory
cold-rolled weight (t) |
|
1599 |
0 |
0 |
1599 |
1599 |
1699 |
1699 |
1699 |
1699 |
1699 |
|
# candidate
hot-rolled coil |
|
10 |
5 |
5 |
10 |
10 |
21 |
21 |
21 |
21 |
21 |
|
candidate
hot-rolled coil weight (t) |
|
179 |
90 |
90 |
179 |
179 |
369 |
369 |
369 |
369 |
369 |
|
# candidate
cold-rolled coil |
|
4 |
0 |
1 |
4 |
5 |
9 |
9 |
9 |
9 |
9 |
|
candidate
cold-rolled coil weight (t) |
|
68 |
0 |
18 |
68 |
86 |
158 |
158 |
158 |
158 |
158 |
|
*Inventory and candidate coils include
only those eligible for at least one order in the corresponding instance. |
|||||||||||
3. Computational results
The
proposed algorithm was evaluated through computational experiments by comparing
it with five methods: (1) the MILP model without a warm start, (2) the MILP
model with a warm start solution initialized by the constructive heuristic, (3)
the constructive heuristic, (4) SA using relocation or swap moves involving one
or two raw materials, and (5) standard ALNS using only heuristic repair
operators. Both MILP approaches and the constructive heuristic were executed
once per instance. SA, ALNS, and the matheuristic were each executed
independently five times per instance using distinct random seeds of
(1000+replication index). The same set of seeds was used for all algorithms.
All algorithms were implemented in C++, with IBM ILOG CPLEX 22.1 as the MILP solver.
Default CPLEX settings were used throughout the experiments, except for the
time limits. The thread setting remained default, with up to 20 parallel
threads reported by CPLEX. The experiments were conducted on a computer
equipped with an Intel Core i5-14600KF 3.5 GHz processor and 64 GB of RAM. The
parameter settings for the MILP model and all algorithms are summarized in
Table 3.
Table 3. Parameter settings for MILP and all algorithms.
|
Category |
Description |
Value |
|
|
Objective weight factor |
Unusable leftover ( |
5 |
|
|
|
Usable leftover ( |
1 |
|
|
|
Candidate coil usage ( |
1 |
|
|
Acceptance |
Initial temperature ( |
|
|
|
|
Cooling factor ( |
0.99 |
|
|
Destroy size |
Destroy ratio for heuristic
repair |
1–10% |
|
|
|
Destroy size for sub-MILP repair |
40–50 materials |
|
|
Operator weight adaptation |
Reaction factor |
0.5 |
|
|
|
Score for finding best solution |
10 |
|
|
|
Score for improving current solution |
3 |
|
|
|
Score for accepted new solution |
1 |
|
|
|
weight update period |
100 |
|
|
Time limit |
|
small |
other |
|
|
MILP time limit |
5 s |
600 s |
|
|
Sub-MILP time limit |
1 s |
30 s |
|
|
Algorithm time limit |
5 s |
600 s |
|
Iteration |
Maximum number of non-improving ALNS iterations ( |
400 |
|
Table
4 compares the results obtained using the MILP without a warm start, the MILP
with a warm start initialized by the constructive heuristic, and the
matheuristic. For
each method, the lower bound (LB), CPU time (in seconds), and gap are
provided. The LB denotes the larger of the two lower
bounds obtained from the MILP with and without a warm start. The definition of the gap is calculated
as follows: 
, where OFV denotes the objective function value. For
the matheuristic, Avg.CPU and Avg.GAP denote the averages over multiple
independent runs, while Std.GAP denotes the standard deviation of the gaps
across these runs.
Table 4. Comparison of results with MILP and matheuristic.
|
|
MILP |
MILP with warm start |
Matheuristic |
|||||||
|
Instances |
LB |
CPU (s) |
GAP (%) |
CPU (s) |
GAP (%) |
Avg.CPU (s) |
Avg.GAP (%) |
Std.GAP (%) |
||
|
S1 |
16536.9 |
0.2 |
0.00 |
0.2 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S2 |
16578.5 |
0.2 |
0.00 |
0.1 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S3 |
29674.8 |
0.3 |
0.00 |
0.2 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S4 |
63160.0 |
0.2 |
0.00 |
0.5 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S5 |
32325.2 |
0.2 |
0.00 |
0.3 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S6 |
52837.2 |
0.2 |
0.00 |
0.2 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S7 |
31368.4 |
0.5 |
0.00 |
0.4 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S8 |
29457.5 |
0.4 |
0.00 |
0.2 |
0.00 |
0.3 |
0.00 |
0.00 |
||
|
S9 |
37241.7 |
0.2 |
0.00 |
0.2 |
0.00 |
0.1 |
0.00 |
0.00 |
||
|
S10 |
35651.4 |
0.6 |
0.00 |
0.6 |
0.00 |
0.6 |
0.00 |
0.00 |
||
|
M1 |
125150.0 |
600.0 |
3.08 |
600.0 |
3.11 |
600.0 |
3.11 |
0.00 |
||
|
M2 |
149913.8 |
600.0 |
7.12 |
600.0 |
7.13 |
600.0 |
7.17 |
0.07 |
||
|
M3 |
460222.7 |
313.1 |
0.01 |
445.8 |
0.01 |
600.0 |
0.01 |
0.00 |
||
|
M4 |
158642.4 |
158.6 |
0.01 |
70.9 |
0.01 |
600.0 |
0.47 |
0.16 |
||
|
M5 |
569590.8 |
600.0 |
0.32 |
600.0 |
0.33 |
600.0 |
0.40 |
0.07 |
||
|
L1 |
1085061.0 |
600.0 |
10.71 |
600.0 |
25.04 |
600.0 |
13.75 |
1.06 |
||
|
L2 |
928350.7 |
600.0 |
14.75 |
600.0 |
14.45 |
600.0 |
18.10 |
0.76 |
||
|
L3 |
1212602.5 |
600.0 |
14.23 |
600.0 |
29.18 |
600.0 |
19.93 |
1.23 |
||
|
L4 |
1099049.5 |
- |
- |
600.0 |
32.49 |
600.0 |
27.57 |
1.67 |
||
|
L5 |
1087076.1 |
- |
- |
600.0 |
44.21 |
600.0 |
36.65 |
1.41 |
||
|
REAL |
1276935.2 |
- |
- |
600.0 |
39.14 |
600.0 |
28.99 |
0.98 |
||
Table
5 compares the SA, ALNS, and the
matheuristic.
Table 4. Comparison of results with the constructor, SA, ALNS, and matheuristic.
|
Constructor |
SA |
ALNS |
Matheuristic |
|||||||||||
|
Instance |
CPU (s) |
GAP (%) |
Avg.CPU (s) |
Avg.GAP (%) |
Std.GAP (%) |
Avg.CPU (s) |
Avg.GAP (%) |
Std.GAP (%) |
Avg.CPU (s) |
Avg.GAP (%) |
Std.GAP (%) |
|||
|
S1 |
0.1 |
34.01 |
0.1 |
0.00 |
0.00 |
0.1 |
0.00 |
0.00 |
0.3 |
0.00 |
0.00 |
|||
|
S2 |
0.1 |
37.10 |
0.1 |
0.00 |
0.00 |
0.1 |
0.00 |
0.00 |
0.3 |
0.00 |
0.00 |
|||
|
S3 |
0.1 |
154.53 |
3.8 |
0.03 |
0.06 |
5.0 |
0.09 |
0.14 |
0.3 |
0.00 |
0.00 |
|||
|
S4 |
0.1 |
28.80 |
5.0 |
19.42 |
0.00 |
4.1 |
15.54 |
8.69 |
0.3 |
0.00 |
0.00 |
|||
|
S5 |
0.1 |
34.28 |
0.2 |
0.00 |
0.00 |
1.0 |
0.00 |
0.00 |
0.3 |
0.00 |
0.00 |
|||
|
S6 |
0.1 |
99.93 |
0.3 |
0.00 |
0.00 |
0.6 |
0.00 |
0.00 |
0.3 |
0.00 |
0.00 |
|||
|
S7 |
0.1 |
44.51 |
0.1 |
0.00 |
0.00 |
0.1 |
0.00 |
0.00 |
0.3 |
0.00 |
0.00 |
|||
|
S8 |
0.2 |
22.96 |
0.1 |
0.00 |
0.00 |
0.1 |
0.00 |
0.00 |
0.3 |
0.00 |
0.00 |
|||
|
S9 |
0.2 |
5.96 |
0.1 |
0.00 |
0.00 |
1.1 |
0.05 |
0.12 |
0.1 |
0.00 |
0.00 |
|||
|
S10 |
0.1 |
87.73 |
4.1 |
0.93 |
0.52 |
5.0 |
1.18 |
0.02 |
0.6 |
0.00 |
0.00 |
|||
|
M1 |
0.1 |
76.69 |
600.0 |
12.20 |
0.90 |
600.0 |
12.46 |
1.97 |
600.0 |
3.11 |
0.00 |
|||
|
M2 |
0.1 |
71.62 |
600.0 |
20.68 |
7.03 |
600.0 |
15.16 |
3.76 |
600.0 |
7.17 |
0.07 |
|||
|
M3 |
0.1 |
22.91 |
600.0 |
1.14 |
0.82 |
600.0 |
1.80 |
0.40 |
600.0 |
0.01 |
0.00 |
|||
|
M4 |
0.2 |
71.87 |
600.0 |
3.92 |
3.37 |
600.0 |
3.52 |
1.95 |
600.0 |
0.47 |
0.16 |
|||
|
M5 |
0.2 |
18.65 |
600.0 |
1.25 |
0.25 |
600.0 |
1.48 |
0.38 |
600.0 |
0.40 |
0.07 |
|||
|
L1 |
0.2 |
41.67 |
600.0 |
17.80 |
0.90 |
600.0 |
20.86 |
1.07 |
600.0 |
13.75 |
1.06 |
|||
|
L2 |
0.3 |
64.84 |
600.0 |
28.05 |
1.34 |
600.0 |
33.66 |
1.53 |
600.0 |
18.10 |
0.76 |
|||
|
L3 |
0.2 |
55.32 |
600.0 |
25.48 |
0.99 |
600.0 |
27.11 |
1.21 |
600.0 |
19.93 |
1.23 |
|||
|
L4 |
0.3 |
66.75 |
600.0 |
32.55 |
2.23 |
600.0 |
34.06 |
2.04 |
600.0 |
27.57 |
1.67 |
|||
|
L5 |
0.3 |
69.58 |
600.0 |
38.82 |
0.88 |
600.0 |
39.45 |
0.70 |
600.0 |
36.65 |
1.41 |
|||
|
REAL |
0.3 |
89.46 |
600.0 |
39.20 |
3.75 |
600.0 |
39.38 |
0.88 |
600.0 |
28.99 |
0.98 |
|||
Appendix 1. Detailed
model sizes.
|
Instance |
Variable |
Binary |
Integer |
Constraint |
Nonzero |
|
S1 |
2,158 |
962 |
94 |
3,651 |
10,240 |
|
S2 |
2,158 |
962 |
94 |
3,651 |
10,244 |
|
S3 |
2,368 |
1,046 |
136 |
3,945 |
11,390 |
|
S4 |
2,119 |
946 |
86 |
3,596 |
10,021 |
|
S5 |
2,385 |
1,072 |
100 |
3,988 |
11,267 |
|
S6 |
2,375 |
1,068 |
98 |
3,974 |
11,177 |
|
S7 |
2,796 |
1,236 |
182 |
4,563 |
13,482 |
|
S8 |
2,712 |
1,222 |
126 |
4,465 |
12,746 |
|
S9 |
2,802 |
1,258 |
144 |
4,591 |
13,252 |
|
S10 |
2,812 |
1,262 |
146 |
4,605 |
13,382 |
|
M1 |
87,009 |
40,130 |
4,928 |
134,785 |
408,565 |
|
M2 |
66,227 |
30,358 |
3,692 |
104,231 |
310,331 |
|
M3 |
63,897 |
29,426 |
3,226 |
100,969 |
296,705 |
|
M4 |
79,228 |
36,434 |
4,540 |
123,308 |
372,066 |
|
M5 |
113,017 |
51,406 |
8,376 |
172,069 |
535,519 |
|
L1 |
704,541 |
314,818 |
71,372 |
1,033,403 |
3,415,465 |
|
L2 |
712,808 |
315,998 |
77,274 |
1,042,850 |
3,482,138 |
|
L3 |
706,134 |
314,922 |
72,752 |
1,035,100 |
3,437,962 |
|
L4 |
732,139 |
326,918 |
74,766 |
1,073,101 |
3,558,513 |
|
L5 |
719,052 |
319,558 |
76,398 |
1,052,654 |
3,503,422 |
|
REAL |
812,886 |
360,802 |
87,724 |
1,187,732 |
3,968,710 |
Appendix 2.
Detailed computational results.
|
Instance |
Method |
Replication |
Avg.CPU (s) |
Best |
Average |
Worst |
Std |
|
S1 |
MILP |
1 |
0.2 |
16536.9 |
16536.9 |
16536.9 |
0.0 |
|
MILP (warm start) |
1 |
0.2 |
16536.9 |
16536.9 |
16536.9 |
0.0 |
|
|
Constructor |
1 |
0.1 |
22160.6 |
22160.6 |
22160.6 |
0.0 |
|
|
SA |
5 |
0.1 |
16536.9 |
16536.9 |
16536.9 |
0.0 |
|
|
ALNS |
5 |
0.1 |
16536.9 |
16536.9 |
16536.9 |
0.0 |
|
|
Matheuristic |
5 |
0.3 |
16536.9 |
16536.9 |
16536.9 |
0.0 |
|
|
S2 |
MILP |
1 |
0.2 |
16578.5 |
16578.5 |
16578.5 |
0.0 |
|
MILP (warm start) |
1 |
0.1 |
16578.5 |
16578.5 |
16578.5 |
0.0 |
|
|
Constructor |
1 |
0.1 |
22728.4 |
22728.4 |
22728.4 |
0.0 |
|
|
SA |
5 |
0.1 |
16578.5 |
16578.5 |
16578.5 |
0.0 |
|
|
ALNS |
5 |
0.1 |
16578.5 |
16578.5 |
16578.5 |
0.0 |
|
|
Matheuristic |
5 |
0.3 |
16578.5 |
16578.5 |
16578.5 |
0.0 |
|
|
S3 |
MILP |
1 |
0.3 |
29674.8 |
29674.8 |
29674.8 |
0.0 |
|
MILP (warm start) |
1 |
0.2 |
29674.8 |
29674.8 |
29674.8 |
0.0 |
|
|
Constructor |
1 |
0.1 |
75532.1 |
75532.1 |
75532.1 |
0.0 |
|
|
SA |
5 |
3.8 |
29674.8 |
29684.7 |
29716.7 |
18.0 |
|
|
ALNS |
5 |
5.0 |
29681.1 |
29701.4 |
29775.3 |
41.4 |
|
|
Matheuristic |
5 |
0.3 |
29674.8 |
29674.8 |
29674.8 |
0.0 |
|
|
S4 |
MILP |
1 |
0.2 |
63160.0 |
63160.0 |
63160.0 |
0.0 |
|
MILP (warm start) |
1 |
0.5 |
63160.0 |
63160.0 |
63160.0 |
0.0 |
|
|
Constructor |
1 |
0.1 |
81347.3 |
81347.3 |
81347.3 |
0.0 |
|
|
SA |
5 |
5.0 |
75426.6 |
75426.6 |
75426.6 |
0.0 |
|
|
ALNS |
5 |
4.1 |
63160.0 |
72973.3 |
75426.6 |
5485.8 |
|
|
Matheuristic |
5 |
0.3 |
63160.0 |
63160.0 |
63160.0 |
0.0 |
|
|
S5 |
MILP |
1 |
0.2 |
32325.2 |
32325.2 |
32325.2 |
0.0 |
|
MILP (warm start) |
1 |
0.3 |
32325.2 |
32325.2 |
32325.2 |
0.0 |
|
|
Constructor |
1 |
0.1 |
43406.2 |
43406.2 |
43406.2 |
0.0 |
|
|
SA |
5 |
0.2 |
32325.2 |
32325.2 |
32325.2 |
0.0 |
|
|
ALNS |
5 |
1.0 |
32325.2 |
32325.2 |
32325.2 |
0.0 |
|
|
Matheuristic |
5 |
0.3 |
32325.2 |
32325.2 |
32325.2 |
0.0 |
|
|
S6 |
MILP |
1 |
0.2 |
52837.2 |
52837.2 |
52837.2 |
0.0 |
|
MILP (warm start) |
1 |
0.2 |
52837.2 |
52837.2 |
52837.2 |
0.0 |
|
|
Constructor |
1 |
0.1 |
105636.8 |
105636.8 |
105636.8 |
0.0 |
|
|
SA |
5 |
0.3 |
52837.2 |
52837.2 |
52837.2 |
0.0 |
|
|
ALNS |
5 |
0.6 |
52837.2 |
52837.2 |
52837.2 |
0.0 |
|
|
Matheuristic |
5 |
0.3 |
52837.2 |
52837.2 |
52837.2 |
0.0 |
|
|
S7 |
MILP |
1 |
0.5 |
31368.4 |
31368.4 |
31368.4 |
0.0 |
|
MILP (warm start) |
1 |
0.4 |
31368.4 |
31368.4 |
31368.4 |
0.0 |
|
|
Constructor |
1 |
0.1 |
45330.8 |
45330.8 |
45330.8 |
0.0 |
|
|
SA |
5 |
0.1 |
31368.4 |
31368.4 |
31368.4 |
0.0 |
|
|
ALNS |
5 |
0.1 |
31368.4 |
31368.4 |
31368.4 |
0.0 |
|
|
Matheuristic |
5 |
0.3 |
31368.4 |
31368.4 |
31368.4 |
0.0 |
|
|
S8 |
MILP |
1 |
0.4 |
29457.5 |
29457.5 |
29457.5 |
0.0 |
|
MILP (warm start) |
1 |
0.2 |
29457.5 |
29457.5 |
29457.5 |
0.0 |
|
|
Constructor |
1 |
0.2 |
36219.8 |
36219.8 |
36219.8 |
0.0 |
|
|
SA |
5 |
0.1 |
29457.5 |
29457.5 |
29457.5 |
0.0 |
|
|
ALNS |
5 |
0.1 |
29457.5 |
29457.5 |
29457.5 |
0.0 |
|
|
Matheuristic |
5 |
0.3 |
29457.5 |
29457.5 |
29457.5 |
0.0 |
|
|
S9 |
MILP |
1 |
0.2 |
37241.7 |
37241.7 |
37241.7 |
0.0 |
|
MILP (warm start) |
1 |
0.2 |
37241.7 |
37241.7 |
37241.7 |
0.0 |
|
|
Constructor |
1 |
0.2 |
39461.7 |
39461.7 |
39461.7 |
0.0 |
|
|
SA |
5 |
0.1 |
37241.7 |
37241.7 |
37241.7 |
0.0 |
|
|
ALNS |
5 |
1.1 |
37241.7 |
37261.9 |
37342.5 |
45.0 |
|
|
Matheuristic |
5 |
0.1 |
37241.7 |
37241.7 |
37241.7 |
0.0 |
|
|
S10 |
MILP |
1 |
0.6 |
35652.5 |
35652.5 |
35652.5 |
0.0 |
|
MILP (warm start) |
1 |
0.6 |
35652.5 |
35652.5 |
35652.5 |
0.0 |
|
|
Constructor |
1 |
0.1 |
66927.5 |
66927.5 |
66927.5 |
0.0 |
|
|
SA |
5 |
4.1 |
35652.5 |
35984.4 |
36067.4 |
185.6 |
|
|
ALNS |
5 |
5.0 |
36067.4 |
36070.8 |
36084.3 |
7.6 |
|
|
Matheuristic |
5 |
0.6 |
35652.5 |
35652.5 |
35652.5 |
0.0 |
|
|
M1 |
MILP |
1 |
600.0 |
129007.4 |
129007.4 |
129007.4 |
0.0 |
|
MILP (warm start) |
1 |
600.0 |
129035.9 |
129035.9 |
129035.9 |
0.0 |
|
|
Constructor |
1 |
0.1 |
221122.6 |
221122.6 |
221122.6 |
0.0 |
|
|
SA |
5 |
600.0 |
138774.5 |
140423.5 |
141567.9 |
1120.9 |
|
|
ALNS |
5 |
600.0 |
138943.9 |
140746.9 |
144958.4 |
2471.0 |
|
|
Matheuristic |
5 |
600.0 |
129035.9 |
129035.9 |
129035.9 |
0.0 |
|
|
M2 |
MILP |
1 |
600.0 |
160584.9 |
160584.9 |
160584.9 |
0.0 |
|
MILP (warm start) |
1 |
600.0 |
160606.5 |
160606.5 |
160606.5 |
0.0 |
|
|
Constructor |
1 |
0.1 |
257287.3 |
257287.3 |
257287.3 |
0.0 |
|
|
SA |
5 |
600.0 |
166324.2 |
180913.9 |
194550.7 |
10531.8 |
|
|
ALNS |
5 |
600.0 |
166568.4 |
172638.3 |
179206.1 |
5635.2 |
|
|
Matheuristic |
5 |
600.0 |
160606.9 |
160655.9 |
160833.9 |
99.6 |
|
|
M3 |
MILP |
1 |
313.1 |
460268.7 |
460268.7 |
460268.7 |
0.0 |
|
MILP (warm start) |
1 |
445.8 |
460268.7 |
460268.7 |
460268.7 |
0.0 |
|
|
Constructor |
1 |
0.1 |
565664.7 |
565664.7 |
565664.7 |
0.0 |
|
|
SA |
5 |
600.0 |
462173.5 |
465489.0 |
471340.1 |
3764.7 |
|
|
ALNS |
5 |
600.0 |
466635.1 |
468511.5 |
471093.8 |
1847.5 |
|
|
Matheuristic |
5 |
600.0 |
460270.7 |
460280.5 |
460291.5 |
10.2 |
|
|
M4 |
MILP |
1 |
158.6 |
158656.7 |
158656.7 |
158656.7 |
0.0 |
|
MILP (warm start) |
1 |
70.9 |
158656.7 |
158656.7 |
158656.7 |
0.0 |
|
|
Constructor |
1 |
0.2 |
272658.8 |
272658.8 |
272658.8 |
0.0 |
|
|
SA |
5 |
600.0 |
160493.2 |
164866.2 |
174009.4 |
5340.1 |
|
|
ALNS |
5 |
600.0 |
161985.6 |
164231.4 |
169423.2 |
3089.8 |
|
|
Matheuristic |
5 |
600.0 |
158936.9 |
159395.3 |
159524.8 |
256.4 |
|
|
M5 |
MILP |
1 |
600.0 |
571386.3 |
571386.3 |
571386.3 |
0.0 |
|
MILP (warm start) |
1 |
600.0 |
571444.4 |
571444.4 |
571444.4 |
0.0 |
|
|
Constructor |
1 |
0.2 |
675845.3 |
675845.3 |
675845.3 |
0.0 |
|
|
SA |
5 |
600.0 |
574814.1 |
576734.1 |
578446.8 |
1443.1 |
|
|
ALNS |
5 |
600.0 |
575275.4 |
578020.4 |
581210.0 |
2180.6 |
|
|
Matheuristic |
5 |
600.0 |
571369.0 |
571851.2 |
572400.3 |
395.0 |
|
|
L1 |
MILP |
1 |
600.0 |
1201272.9 |
1201272.9 |
1201272.9 |
0.0 |
|
MILP (warm start) |
1 |
600.0 |
1356769.3 |
1356769.3 |
1356769.3 |
0.0 |
|
|
Constructor |
1 |
0.2 |
1537224.8 |
1537224.8 |
1537224.8 |
0.0 |
|
|
SA |
5 |
600.0 |
1265343.1 |
1278215.8 |
1287563.2 |
9760.3 |
|
|
ALNS |
5 |
600.0 |
1297540.3 |
1311391.4 |
1328155.0 |
11652.3 |
|
|
Matheuristic |
5 |
600.0 |
1218600.8 |
1234271.3 |
1244331.4 |
11555.2 |
|
|
L2 |
MILP |
1 |
600.0 |
1065313.0 |
1065313.0 |
1065313.0 |
0.0 |
|
MILP (warm start) |
1 |
600.0 |
1062519.8 |
1062519.8 |
1062519.8 |
0.0 |
|
|
Constructor |
1 |
0.3 |
1530293.9 |
1530293.9 |
1530293.9 |
0.0 |
|
|
SA |
5 |
600.0 |
1167508.2 |
1188750.3 |
1200481.6 |
12472.5 |
|
|
ALNS |
5 |
600.0 |
1218751.8 |
1240798.0 |
1256544.3 |
14159.1 |
|
|
Matheuristic |
5 |
600.0 |
1085959.6 |
1096397.5 |
1103102.3 |
7063.3 |
|
|
L3 |
MILP |
1 |
600.0 |
1385121.5 |
1385121.5 |
1385121.5 |
0.0 |
|
MILP (warm start) |
1 |
600.0 |
1566425.3 |
1566425.3 |
1566425.3 |
0.0 |
|
|
Constructor |
1 |
0.2 |
1883458.3 |
1883458.3 |
1883458.3 |
0.0 |
|
|
SA |
5 |
600.0 |
1501785.9 |
1521537.7 |
1531105.4 |
12018.1 |
|
|
ALNS |
5 |
600.0 |
1527313.8 |
1541355.4 |
1562738.4 |
14703.3 |
|
|
Matheuristic |
5 |
600.0 |
1432699.8 |
1454245.6 |
1473615.6 |
14876.9 |
|
|
L4 |
MILP |
0 |
- |
- |
- |
- |
- |
|
MILP (warm start) |
1 |
600.0 |
1456128.0 |
1456128.0 |
1456128.0 |
0.0 |
|
|
Constructor |
1 |
0.3 |
1832667.5 |
1832667.5 |
1832667.5 |
0.0 |
|
|
SA |
5 |
600.0 |
1433493.6 |
1456753.8 |
1495645.1 |
24509.5 |
|
|
ALNS |
5 |
600.0 |
1440726.8 |
1473393.1 |
1489470.0 |
22424.5 |
|
|
Matheuristic |
5 |
600.0 |
1384956.9 |
1402039.2 |
1432380.0 |
18404.3 |
|
|
L5 |
MILP |
0 |
- |
- |
- |
- |
- |
|
MILP (warm start) |
1 |
600.0 |
1567629.6 |
1567629.6 |
1567629.6 |
0.0 |
|
|
Constructor |
1 |
0.3 |
1843503.1 |
1843503.1 |
1843503.1 |
0.0 |
|
|
SA |
5 |
600.0 |
1498788.7 |
1509043.1 |
1524336.1 |
9516.7 |
|
|
ALNS |
5 |
600.0 |
1502837.6 |
1515902.2 |
1522057.5 |
7626.5 |
|
|
Matheuristic |
5 |
600.0 |
1469124.3 |
1485460.5 |
1504454.2 |
15277.8 |
|
|
REAL |
MILP |
0 |
- |
- |
- |
- |
- |
|
MILP (warm start) |
1 |
600.0 |
1776769.0 |
1776769.0 |
1776769.0 |
0.0 |
|
|
Constructor |
1 |
0.3 |
2419225.0 |
2419225.0 |
2419225.0 |
0.0 |
|
|
SA |
5 |
600.0 |
1698107.9 |
1777555.3 |
1818781.5 |
47926.3 |
|
|
ALNS |
5 |
600.0 |
1761767.9 |
1779815.0 |
1792496.5 |
11249.3 |
|
|
Matheuristic |
5 |
600.0 |
1631798.7 |
1647166.0 |
1661452.4 |
12517.5 |