This paper develops a hybrid demon algorithm for a two-dimensional orthogonal strip packing problem. This algorithm combines a placement procedure based on an improved heuristic, local search, and demon algorithm involved in setting one parameter. The hybrid algorithm is tested on a wide set of benchmark instances taken from the literature and compared with other well-known algorithms. The computation results validate the quality of the solutions and the effectiveness of the proposed algorithm.
Cutting and packing are a very active field of research within operational research, computer science, mathematics, and management science. The two-dimensional cutting and packing problem is widely applied in optimally cutting raw materials such as glass, textile, steel, and paper and transportation and logistics fields. For example, in textile or glass industries, rectangular components have to be cut from large sheets of material. In warehousing, goods have to be placed on shelves. In newspapers paging, articles and advertisements have to be arranged in pages [
The two-dimensional orthogonal strip packing problems (2SP) addressed in this paper consist of packing rectangular pieces into a large rectangular sheet of fixed width and unlimited height in order to minimize the used height, where the rectangular pieces are placed orthogonally without overlap and no rotations are allowed. This problem is of significance both from a theoretical and a practical point of view because it arises in various production processes and has many applications in the glass, steel, paper, and textile industries, and they also have indirect applications in other fields [
2SP is known to be NP-hard, some exact algorithms are proposed by Martello et al. [
In comparison to the literature on construction heuristic algorithms to packing problems, metaheuristic algorithms are paid more and more attention recently. Bortfeldt [
The remainder of the paper is organized as follows. Section
It has been reported that construction heuristic algorithm is one of the best heuristics while combining with a simulated annealing algorithm [
New scoring function score (
Conditions |
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| |
---|---|---|---|---|
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Case (1): |
4 | 4 | −2 or −1 |
Case (2): |
3 | 3 | 0 | |
Case (3): |
2 |
|
0 | |
Case (4): |
1 |
|
0 | |
Case (5): |
0 | 0 | +1 |
Placement by the different scoring rules.
Case (3) has higher score than case (4)
Case (4) has higher score than case (3)
In addition, one piece
The least waste strategy.
The simulated annealing algorithm (SA) was invented to allow computer simulation of equilibria in statistical physics. It is a powerful randomized search algorithm, and the computational results have shown that ISA [
Demon algorithm is a simulated annealing based algorithm that uses computationally simpler acceptance function. This paper applies demon algorithm for 2SP. In order to solve it, the hybrid demon algorithm can be stated as in Algorithm
HDA() (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
LS() (1) sort all unpacked pieces by non-increasing ordering of length size to obtain (2) (3) (4) (5) swap the order of pieces (6) (7) (8) (9)
HDA first searches a better solution according to a local search algorithm
HDA only involves in setting one parameter
Effect of
Effect of
Effect of
Figures
Effect of the least waste strategy on the data set
Effect of the least waste strategy on the data set Nice1~6 and Path1~6.
Effect of the least waste strategy on the data set gcut.
In this section, we present the results obtained in a set of experiments we conducted in order to evaluate the performance of the hybrid demon algorithm (HDA) proposed in this paper. This paper uses the same data sets C, N, NT and CX, 2sp, BWMV, Nice, and Path as Leung et al. [
The algorithm was implemented in Visual C++6.0 and the experimental tests were run on a computer with an Intel core 2 CPU 2.13 GHz and 0.99 GB RAM. GRASP [
Results obtained by GRASP, SVC, ISA, and HDA on C.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
C11 | 16 | 20 | 20 | 20 | 20 | 20.0 | 20 | 0.0 | 0.0 | 0.0 | 0.0 |
C12 | 17 | 20 | 20 | 20 | 21 | 20.0 | 20 | 0.0 | 5.0 | 0.0 | 0.0 |
C13 | 16 | 20 | 20 | 20 | 20 | 20.0 | 20 | 0.0 | 0.0 | 0.0 | 0.0 |
|
|||||||||||
C21 | 25 | 40 | 15 | 15 | 15 | 15.0 | 15 | 0.0 | 0.0 | 0.0 | 0.0 |
C22 | 25 | 40 | 15 | 15 | 15 | 15.0 | 15 | 0.0 | 0.0 | 0.0 | 0.0 |
C23 | 25 | 40 | 15 | 15 | 15 | 15.0 | 15 | 0.0 | 0.0 | 0.0 | 0.0 |
|
|||||||||||
C31 | 28 | 60 | 30 | 30 | 30 | 30.0 | 30 | 0.0 | 0.0 | 0.0 | 0.0 |
C32 | 29 | 60 | 30 | 31 | 31 | 31.0 | 30.9 | 3.3 | 3.3 | 3.3 | 3.0 |
C33 | 28 | 60 | 30 | 30 | 30 | 30.0 | 30 | 0.0 | 0.0 | 0.0 | 0.0 |
|
|||||||||||
C41 | 49 | 60 | 60 | 61 | 61 | 61.0 | 61 | 1.7 | 1.7 | 1.7 | 1.7 |
C42 | 49 | 60 | 60 | 61 | 61 | 61.0 | 61 | 1.7 | 1.7 | 1.7 | 1.7 |
C43 | 49 | 60 | 60 | 61 | 61 | 60.9 | 61 | 1.7 | 1.7 | 1.5 | 1.7 |
|
|||||||||||
C51 | 73 | 60 | 90 | 91 | 91 | 91.0 | 91 | 1.1 | 1.1 | 1.1 | 1.1 |
C52 | 73 | 60 | 90 | 91 | 91 | 90.8 | 91 | 1.1 | 1.1 | 0.9 | 1.1 |
C53 | 73 | 60 | 90 | 91 | 91 | 91.0 | 91 | 1.1 | 1.1 | 1.1 | 1.1 |
|
|||||||||||
C61 | 97 | 80 | 120 | 122 | 121 | 121.0 | 121 | 1.7 | 0.8 | 0.8 | 0.8 |
C62 | 97 | 80 | 120 | 121 | 121 | 121.0 | 121 | 0.8 | 0.8 | 0.8 | 0.8 |
C63 | 97 | 80 | 120 | 122 | 121 | 121.0 | 121 | 1.7 | 0.8 | 0.8 | 0.8 |
|
|||||||||||
C71 | 196 | 160 | 240 | 244 | 242 | 242.0 | 241 | 1.7 | 0.8 | 0.8 | 0.4 |
C72 | 197 | 160 | 240 | 243 | 242 | 241.0 | 241 | 1.3 | 0.8 | 0.4 | 0.4 |
C73 | 196 | 160 | 240 | 243 | 242 | 242.0 | 241 | 1.3 | 0.8 | 0.8 | 0.4 |
|
|||||||||||
Average | 83.19 | 82.95 | 82.84 |
|
0.95 | 1.03 | 0.76 |
|
Tables
Table
Table
Results obtained by GRASP, SVC, ISA, and HDA on N.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
N1 | 10 | 40 | 40 | 40 | 40 | 40.0 | 40 | 0.0 | 0.0 | 0.0 | 0.0 |
N2 | 20 | 30 | 50 | 50 | 50 | 50.0 | 50 | 0.0 | 0.0 | 0.0 | 0.0 |
N3 | 30 | 30 | 50 | 51 | 50 | 50.1 | 50 | 2.0 | 0.0 | 0.2 | 0.0 |
N4 | 40 | 80 | 80 | 81 | 81 | 80.0 | 80 | 1.3 | 1.3 | 0.0 | 0.0 |
N5 | 50 | 100 | 100 | 102 | 101 | 101.0 | 100 | 2.0 | 1.0 | 1.0 | 0.0 |
N6 | 60 | 50 | 100 | 101 | 101 | 100.9 | 100.7 | 1.0 | 1.0 | 0.9 | 0.7 |
N7 | 70 | 80 | 100 | 101 | 101 | 100.0 | 100 | 1.0 | 1.0 | 0.0 | 0.0 |
N8 | 80 | 100 | 80 | 81 | 81 | 81.0 | 81 | 1.3 | 1.3 | 1.3 | 1.3 |
N9 | 100 | 50 | 150 | 151 | 151 | 150.9 | 151 | 0.7 | 0.7 | 0.6 | 0.7 |
N10 | 200 | 70 | 150 | 151 | 151 | 150.8 | 151 | 0.7 | 0.7 | 0.5 | 0.7 |
N11 | 300 | 70 | 150 | 151 | 151 | 150.7 | 150.8 | 0.7 | 0.7 | 0.5 | 0.5 |
N12 | 500 | 100 | 300 | 304 | 301 | 301.0 | 301 | 1.3 | 0.3 | 0.3 | 0.3 |
N13 | 3152 | 640 | 960 | 965 | 963 | 960.0 | 960 | 0.5 | 0.3 | 0.0 | 0.0 |
|
|||||||||||
Average | 179.15 | 178.62 | 178.18 |
|
0.95 | 0.63 | 0.41 |
|
Table
Results obtained by GRASP, SVC, ISA, and HDA on NT.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
n1a | 17 | 200 | 200 | 200 | 202 | 200.0 | 200 | 0.0 | 1.0 | 0.0 | 0.0 |
n1b | 17 | 200 | 200 | 209 | 200 | 211.2 | 200 | 4.5 | 0.0 | 5.6 | 0.0 |
n1c | 17 | 200 | 200 | 200 | 200 | 200.0 | 200 | 0.0 | 0.0 | 0.0 | 0.0 |
n1d | 17 | 200 | 200 | 200 | 200 | 200.0 | 200 | 0.0 | 0.0 | 0.0 | 0.0 |
n1e | 17 | 200 | 200 | 200 | 200 | 200.0 | 200 | 0.0 | 0.0 | 0.0 | 0.0 |
|
|||||||||||
n2a | 25 | 200 | 200 | 206 | 205 | 204.0 | 201.3 | 3.0 | 2.5 | 2.0 | 0.7 |
n2b | 25 | 200 | 200 | 206 | 209 | 209.4 | 210 | 3.0 | 4.5 | 4.7 | 5.0 |
n2c | 25 | 200 | 200 | 208 | 209 | 208.5 | 206.3 | 4.0 | 4.5 | 4.3 | 3.2 |
n2d | 25 | 200 | 200 | 209 | 207 | 207.8 | 205.9 | 4.5 | 3.5 | 3.9 | 3.0 |
n2e | 25 | 200 | 200 | 206 | 205 | 206.7 | 206.1 | 3.0 | 2.5 | 3.3 | 3.1 |
|
|||||||||||
n3a | 29 | 200 | 200 | 209 | 208 | 206.1 | 206.3 | 4.5 | 4.0 | 3.1 | 3.2 |
n3b | 29 | 200 | 200 | 208 | 207 | 209.0 | 208.9 | 4.0 | 3.5 | 4.5 | 4.5 |
n3c | 29 | 200 | 200 | 205 | 207 | 206.1 | 205 | 2.5 | 3.5 | 3.1 | 2.5 |
n3d | 29 | 200 | 200 | 207 | 208 | 204.3 | 204.5 | 3.5 | 4.0 | 2.2 | 2.3 |
n3e | 29 | 200 | 200 | 207 | 207 | 208.0 | 208.1 | 3.5 | 3.5 | 4.0 | 4.1 |
|
|||||||||||
n4a | 49 | 200 | 200 | 206 | 205 | 206.0 | 205.9 | 3.0 | 2.5 | 3.0 | 3.0 |
n4b | 49 | 200 | 200 | 207 | 205 | 205.0 | 204.7 | 3.5 | 2.5 | 2.5 | 2.3 |
n4c | 49 | 200 | 200 | 205 | 205 | 206.0 | 205.3 | 2.5 | 2.5 | 3.0 | 2.7 |
n4d | 49 | 200 | 200 | 206 | 205 | 204.8 | 204.9 | 3.0 | 2.5 | 2.4 | 2.5 |
n4e | 49 | 200 | 200 | 205 | 205 | 206.0 | 206.1 | 2.5 | 2.5 | 3.0 | 3.1 |
|
|||||||||||
n5a | 73 | 200 | 200 | 205 | 204 | 205.1 | 205.5 | 2.5 | 2.0 | 2.6 | 2.8 |
n5b | 73 | 200 | 200 | 204 | 204 | 203.6 | 203.1 | 2.0 | 2.0 | 1.8 | 1.6 |
n5c | 73 | 200 | 200 | 206 | 204 | 204.4 | 204.5 | 3.0 | 2.0 | 2.2 | 2.3 |
n5d | 73 | 200 | 200 | 204 | 205 | 205.0 | 204.7 | 2.0 | 2.5 | 2.5 | 2.3 |
n5e | 73 | 200 | 200 | 206 | 205 | 204.7 | 204.9 | 3.0 | 2.5 | 2.3 | 2.5 |
|
|||||||||||
n6a | 97 | 200 | 200 | 204 | 203 | 202.8 | 202.7 | 2.0 | 1.5 | 1.4 | 1.3 |
n6b | 97 | 200 | 200 | 204 | 204 | 203.0 | 202.9 | 2.0 | 2.0 | 1.5 | 1.5 |
n6c | 97 | 200 | 200 | 204 | 204 | 203.6 | 203.2 | 2.0 | 2.0 | 1.8 | 1.6 |
n6d | 97 | 200 | 200 | 204.1 | 202 | 203.8 | 203 | 2.1 | 1.0 | 1.9 | 1.5 |
n6e | 97 | 200 | 200 | 204 | 203 | 203.5 | 203 | 2.0 | 1.5 | 1.8 | 1.5 |
|
|||||||||||
n7a | 199 | 200 | 200 | 202 | 202 | 201.0 | 201 | 1.0 | 1.0 | 0.5 | 0.5 |
n7b | 199 | 200 | 200 | 203 | 202 | 202.0 | 201 | 1.5 | 1.0 | 1.0 | 0.5 |
n7c | 199 | 200 | 200 | 203 | 202 | 201.9 | 201 | 1.5 | 1.0 | 1.0 | 0.5 |
n7d | 199 | 200 | 200 | 203 | 202 | 201.9 | 201 | 1.5 | 1.0 | 1.0 | 0.5 |
n7e | 199 | 200 | 200 | 203 | 202 | 201.9 | 201 | 1.5 | 1.0 | 1.0 | 0.5 |
|
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t1a | 17 | 200 | 200 | 200 | 200 | 200.0 | 200 | 0.0 | 0.0 | 0.0 | 0.0 |
t1b | 17 | 200 | 200 | 200 | 211 | 200.0 | 200 | 0.0 | 5.5 | 0.0 | 0.0 |
t1c | 17 | 200 | 200 | 200 | 210 | 200.0 | 200 | 0.0 | 5.0 | 0.0 | 0.0 |
t1d | 17 | 200 | 200 | 200 | 200 | 211.8 | 200 | 0.0 | 0.0 | 5.9 | 0.0 |
t1e | 17 | 200 | 200 | 200 | 209 | 200.0 | 200 | 0.0 | 4.5 | 0.0 | 0.0 |
|
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t2a | 25 | 200 | 200 | 204 | 207 | 207.0 | 206.3 | 2.0 | 3.5 | 3.5 | 3.2 |
t2b | 25 | 200 | 200 | 208 | 205 | 207.0 | 205.8 | 4.0 | 2.5 | 3.5 | 2.9 |
t2c | 25 | 200 | 200 | 208 | 206 | 206.0 | 207.4 | 4.0 | 3.0 | 3.0 | 3.7 |
t2d | 25 | 200 | 200 | 206 | 207 | 209.3 | 204.4 | 3.0 | 3.5 | 4.7 | 2.2 |
t2e | 25 | 200 | 200 | 206 | 207 | 207.4 | 205.9 | 3.0 | 3.5 | 3.7 | 3.0 |
|
|||||||||||
t3a | 29 | 200 | 200 | 207 | 208 | 209.0 | 209 | 3.5 | 4.0 | 4.5 | 4.5 |
t3b | 29 | 200 | 200 | 209 | 207 | 208.1 | 207.9 | 4.5 | 3.5 | 4.1 | 4.0 |
t3c | 29 | 200 | 200 | 206 | 207 | 206.6 | 206.3 | 3.0 | 3.5 | 3.3 | 3.2 |
t3d | 29 | 200 | 200 | 207 | 208 | 206.4 | 206.4 | 3.5 | 4.0 | 3.2 | 3.2 |
t3e | 29 | 200 | 200 | 208 | 206 | 205.0 | 205 | 4.0 | 3.0 | 2.5 | 2.5 |
|
|||||||||||
t4a | 49 | 200 | 200 | 205 | 205 | 205.0 | 204.7 | 2.5 | 2.5 | 2.5 | 2.3 |
t4b | 49 | 200 | 200 | 205 | 205 | 206.1 | 205.9 | 2.5 | 2.5 | 3.1 | 3.0 |
t4c | 49 | 200 | 200 | 206 | 205 | 204.9 | 204.7 | 3.0 | 2.5 | 2.5 | 2.3 |
t4d | 49 | 200 | 200 | 206 | 205 | 205.7 | 205.4 | 3.0 | 2.5 | 2.8 | 2.7 |
t4e | 49 | 200 | 200 | 207 | 205 | 205.2 | 205.5 | 3.5 | 2.5 | 2.6 | 2.8 |
|
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t5a | 73 | 200 | 200 | 206 | 204 | 204.4 | 204.5 | 3.0 | 2.0 | 2.2 | 2.3 |
t5b | 73 | 200 | 200 | 204 | 204 | 204.0 | 204.4 | 2.0 | 2.0 | 2.0 | 2.2 |
t5c | 73 | 200 | 200 | 205 | 204 | 205.0 | 205.5 | 2.5 | 2.0 | 2.5 | 2.8 |
t5d | 73 | 200 | 200 | 204 | 205 | 204.9 | 204.7 | 2.0 | 2.5 | 2.5 | 2.3 |
t5e | 73 | 200 | 200 | 204 | 204 | 204.0 | 204.7 | 2.0 | 2.0 | 2.0 | 2.3 |
|
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t6a | 97 | 200 | 200 | 204 | 204 | 203.2 | 203.7 | 2.0 | 2.0 | 1.6 | 1.8 |
t6b | 97 | 200 | 200 | 204 | 202 | 203.4 | 203.2 | 2.0 | 1.0 | 1.7 | 1.6 |
t6c | 97 | 200 | 200 | 204 | 204 | 203.0 | 202.7 | 2.0 | 2.0 | 1.5 | 1.3 |
t6d | 97 | 200 | 200 | 204 | 204 | 203.5 | 203.7 | 2.0 | 2.0 | 1.8 | 1.8 |
t6e | 97 | 200 | 200 | 205 | 204 | 203.5 | 203.5 | 2.5 | 2.0 | 1.8 | 1.8 |
|
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t7a | 199 | 200 | 200 | 203 | 201 | 201.2 | 201 | 1.5 | 0.5 | 0.6 | 0.5 |
t7b | 199 | 200 | 200 | 203 | 202 | 201.0 | 201 | 1.5 | 1.0 | 0.5 | 0.5 |
t7c | 199 | 200 | 200 | 204 | 202 | 201.0 | 201 | 2.0 | 1.0 | 0.5 | 0.5 |
t7d | 199 | 200 | 200 | 202 | 202 | 202.0 | 201 | 1.0 | 1.0 | 1.0 | 0.5 |
t7e | 199 | 200 | 200 | 203 | 202 | 201.7 | 201 | 1.5 | 1.0 | 0.8 | 0.5 |
|
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Average | 204.64 | 204.54 | 204.48 |
|
2.32 | 2.27 | 2.24 |
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Table
Results obtained by GRASP, SVC, ISA, and HDA on CX.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
50cx | 50 | 400 | 600 | 613 | 603 | 620.2 | 607.3 | 2.2 | 0.5 | 3.4 | 1.2 |
100cx | 100 | 400 | 600 | 617 | 616 | 615.8 | 617.3 | 2.8 | 2.7 | 2.6 | 2.9 |
500cx | 500 | 400 | 600 | 605 | 604 | 601.0 | 601 | 0.8 | 0.7 | 0.2 | 0.2 |
1000cx | 1000 | 400 | 600 | 602 | 601 | 600.0 | 600 | 0.3 | 0.2 | 0.0 | 0.0 |
5000cx | 5000 | 400 | 600 | 600 | 600 | 600.0 | 600 | 0.0 | 0.0 | 0.0 | 0.0 |
10000cx | 10000 | 400 | 600 | 600 | 600 | 600.0 | 600 | 0.0 | 0.0 | 0.0 | 0.0 |
15000cx | 15000 | 400 | 600 | 600 | 600 | 600.0 | 600 | 0.0 | 0.0 | 0.0 | 0.0 |
|
|||||||||||
Average | 605.29 |
|
605.29 | 603.66 | 0.88 |
|
0.88 | 0.61 |
Section
Results obtained by GRASP, SVC, ISA, and HDA on 2sp.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
cgcut1 | 16 | 10 | 23 | 23 | 23 | 23.0 | 24 | 0.0 | 0.0 | 0.0 | 4.3 |
cgcut2 | 23 | 70 | 63 | 65 | 65 | 65.0 | 65 | 3.2 | 3.2 | 3.2 | 3.2 |
cgcut3 | 62 | 70 | 636 | 661 | 661 | 660.2 | 662.2 | 3.9 | 3.9 | 3.8 | 4.1 |
|
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gcut1 | 10 | 250 | 1016 | 1016 | 1016 | 1016.0 | 1016 | 0.0 | 0.0 | 0.0 | 0.0 |
gcut2 | 20 | 250 | 1133 | 1191 | 1187 | 1187.0 | 1187 | 5.1 | 4.8 | 4.8 | 4.8 |
gcut3 | 30 | 250 | 1803 | 1803 | 1803 | 1803.0 | 1803 | 0.0 | 0.0 | 0.0 | 0.0 |
gcut4 | 50 | 250 | 2934 | 3002 | 3017 | 3010.5 | 3002 | 2.3 | 2.8 | 2.6 | 2.3 |
gcut5 | 10 | 500 | 1172 | 1273 | 1273 | 1273.0 | 1273 | 8.6 | 8.6 | 8.6 | 8.6 |
gcut6 | 20 | 500 | 2514 | 2627 | 2632 | 2632.0 | 2629.5 | 4.5 | 4.7 | 4.7 | 4.6 |
gcut7 | 30 | 500 | 4641 | 4693 | 4693 | 4693.0 | 4693.8 | 1.1 | 1.1 | 1.1 | 1.1 |
gcut8 | 50 | 500 | 5703 | 5912 | 5876 | 5890.4 | 5884.2 | 3.7 | 3.0 | 3.3 | 3.2 |
gcut9 | 10 | 1000 | 2022 | 2317 | 2317 | 2317.0 | 2317 | 14.6 | 14.6 | 14.6 | 14.6 |
gcut10 | 20 | 1000 | 5356 | 5964 | 5973 | 5964.8 | 5964.9 | 11.4 | 11.5 | 11.4 | 11.4 |
gcut11 | 30 | 1000 | 6537 | 6899 | 6891 | 6884.4 | 6883.6 | 5.5 | 5.4 | 5.3 | 5.3 |
gcut12 | 50 | 1000 | 12522 | 14690 | 14690 | 14690.0 | 14690 | 17.3 | 17.3 | 17.3 | 17.3 |
gcut13 | 32 | 3000 | 4772 | 4994 | 4977 | 4965.9 | 4963.2 | 4.7 | 4.3 | 4.1 | 4.0 |
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ngcut1 | 10 | 10 | 23 | 23 | 23 | 23.0 | 23 | 0.0 | 0.0 | 0.0 | 0.0 |
ngcut2 | 17 | 10 | 30 | 30 | 30 | 31.0 | 31 | 0.0 | 0.0 | 3.3 | 3.3 |
ngcut3 | 21 | 10 | 28 | 28 | 28 | 28.0 | 28 | 0.0 | 0.0 | 0.0 | 0.0 |
ngcut4 | 7 | 10 | 20 | 20 | 20 | 20.0 | 20 | 0.0 | 0.0 | 0.0 | 0.0 |
ngcut5 | 14 | 10 | 36 | 36 | 36 | 36.0 | 36 | 0.0 | 0.0 | 0.0 | 0.0 |
ngcut6 | 15 | 10 | 29 | 31 | 31 | 31.0 | 31 | 6.9 | 6.9 | 6.9 | 6.9 |
ngcut7 | 8 | 20 | 20 | 20 | 20 | 20.0 | 20 | 0.0 | 0.0 | 0.0 | 0.0 |
ngcut8 | 13 | 20 | 32 | 33 | 34 | 34.0 | 34 | 3.1 | 6.3 | 6.3 | 6.3 |
ngcut9 | 18 | 20 | 49 | 50 | 51 | 52.0 | 51 | 2.0 | 4.1 | 6.1 | 4.1 |
ngcut10 | 13 | 30 | 80 | 80 | 80 | 80.0 | 80 | 0.0 | 0.0 | 0.0 | 0.0 |
ngcut11 | 15 | 30 | 50 | 52 | 52 | 52.0 | 52 | 4.0 | 4.0 | 4.0 | 4.0 |
ngcut12 | 22 | 30 | 87 | 87 | 87 | 87.0 | 87 | 0.0 | 0.0 | 0.0 | 0.0 |
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beng1 | 20 | 25 | 30 | 30 | 30 | 31.0 | 30.7 | 0.0 | 0.0 | 3.3 | 2.3 |
beng2 | 40 | 25 | 57 | 57 | 57 | 57.0 | 57 | 0.0 | 0.0 | 0.0 | 0.0 |
beng3 | 60 | 25 | 84 | 84 | 84 | 84.0 | 84 | 0.0 | 0.0 | 0.0 | 0.0 |
beng4 | 80 | 25 | 107 | 107 | 107 | 107.0 | 107 | 0.0 | 0.0 | 0.0 | 0.0 |
beng5 | 100 | 25 | 134 | 134 | 134 | 134.0 | 134 | 0.0 | 0.0 | 0.0 | 0.0 |
beng6 | 40 | 40 | 36 | 36 | 36 | 36.0 | 36 | 0.0 | 0.0 | 0.0 | 0.0 |
beng7 | 80 | 40 | 67 | 67 | 67 | 67.0 | 67 | 0.0 | 0.0 | 0.0 | 0.0 |
beng8 | 120 | 40 | 101 | 101 | 101 | 101.0 | 101 | 0.0 | 0.0 | 0.0 | 0.0 |
beng9 | 160 | 40 | 126 | 126 | 126 | 126.0 | 126 | 0.0 | 0.0 | 0.0 | 0.0 |
beng10 | 200 | 40 | 156 | 156 | 156 | 156.0 | 156 | 0.0 | 0.0 | 0.0 | 0.0 |
|
|||||||||||
Average | 1539.95 | 1539.05 | 1538.64 |
|
|
2.8 | 3.02 | 3.04 |
Table
Table
Results obtained by GRASP, SVC, ISA, and HDA on BWMV.
Instances | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
C01 | 20 | 10 | 60.3 | 61.4 | 61.4 | 61.3 | 61.3 | 1.8 | 1.8 | 1.7 | 1.7 |
40 | 10 | 121.6 | 121.9 | 122 | 121.8 | 121.8 | 0.2 | 0.3 | 0.2 | 0.2 | |
60 | 10 | 187.4 | 188.6 | 188.6 | 188.6 | 188.6 | 0.6 | 0.6 | 0.6 | 0.6 | |
80 | 10 | 262.2 | 262.6 | 262.6 | 262.6 | 262.6 | 0.2 | 0.2 | 0.2 | 0.2 | |
100 | 10 | 304.4 | 305 | 304.9 | 304.9 | 304.9 | 0.2 | 0.2 | 0.2 | 0.2 | |
|
|||||||||||
C02 | 20 | 30 | 19.7 | 19.8 | 19.8 | 19.9 | 19.8 | 0.5 | 0.5 | 1.0 | 0.5 |
40 | 30 | 39.1 | 39.1 | 39.1 | 39.1 | 39.1 | 0.0 | 0.0 | 0.0 | 0.0 | |
60 | 30 | 60.1 | 60.3 | 60.1 | 60.1 | 60.1 | 0.3 | 0.0 | 0.0 | 0.0 | |
80 | 30 | 83.2 | 83.3 | 83.2 | 83.2 | 83.2 | 0.1 | 0.0 | 0.0 | 0.0 | |
100 | 30 | 100.5 | 100.6 | 100.5 | 100.5 | 100.5 | 0.1 | 0.0 | 0.0 | 0.0 | |
|
|||||||||||
C03 | 20 | 40 | 157.4 | 163.5 | 164.6 | 164.0 | 163.7 | 3.9 | 4.6 | 4.2 | 4.0 |
40 | 40 | 328.8 | 334.2 | 333.9 | 333.8 | 333.8 | 1.6 | 1.6 | 1.5 | 1.5 | |
60 | 40 | 500 | 506.6 | 506.9 | 505.8 | 505.9 | 1.3 | 1.4 | 1.2 | 1.2 | |
80 | 40 | 701.7 | 709.7 | 710.1 | 709.2 | 709.5 | 1.1 | 1.2 | 1.1 | 1.1 | |
100 | 40 | 832.7 | 840.2 | 839.9 | 837.8 | 838.4 | 0.9 | 0.9 | 0.6 | 0.7 | |
|
|||||||||||
C04 | 20 | 100 | 61.4 | 63.3 | 63.8 | 63.9 | 63.4 | 3.1 | 3.9 | 4.1 | 3.3 |
40 | 100 | 123.9 | 126.2 | 126.2 | 126.1 | 125.8 | 1.9 | 1.9 | 1.8 | 1.5 | |
60 | 100 | 193 | 196.6 | 195.6 | 195.5 | 195.5 | 1.9 | 1.3 | 1.3 | 1.3 | |
80 | 100 | 267.2 | 272 | 270.5 | 269.8 | 270.1 | 1.8 | 1.2 | 1.0 | 1.1 | |
100 | 100 | 322 | 327.3 | 325.3 | 324.6 | 324.6 | 1.6 | 1.0 | 0.8 | 0.8 | |
|
|||||||||||
C05 | 20 | 100 | 512.2 | 533.9 | 537.9 | 534.6 | 534.1 | 4.2 | 5.0 | 4.4 | 4.3 |
40 | 100 | 1053.8 | 1074.4 | 1076.4 | 1073.6 | 1073.5 | 2.0 | 2.1 | 1.9 | 1.9 | |
60 | 100 | 1614 | 1645.5 | 1647.6 | 1643.4 | 1644.0 | 2.0 | 2.1 | 1.8 | 1.9 | |
80 | 100 | 2268.4 | 2290.5 | 2288.9 | 2289.0 | 2289.3 | 1.0 | 0.9 | 0.9 | 0.9 | |
100 | 100 | 2617.4 | 2651.1 | 2653.5 | 2644.4 | 2646.4 | 1.3 | 1.4 | 1.0 | 1.1 | |
|
|||||||||||
C06 | 20 | 10 | 159.9 | 167.2 | 169.6 | 169.6 | 168.7 | 4.6 | 6.1 | 6.1 | 5.5 |
40 | 10 | 323.5 | 333.4 | 332.6 | 334.0 | 333.2 | 3.1 | 2.8 | 3.2 | 3.0 | |
60 | 10 | 505.1 | 519.9 | 517.2 | 519.0 | 518.5 | 2.9 | 2.4 | 2.8 | 2.6 | |
80 | 10 | 699.7 | 718.4 | 714.7 | 715.5 | 715.4 | 2.7 | 2.1 | 2.3 | 2.2 | |
100 | 10 | 843.8 | 865.1 | 860.6 | 861.1 | 861.2 | 2.5 | 2.0 | 2.1 | 2.1 | |
|
|||||||||||
C07 | 20 | 30 | 490.4 | 501.9 | 501.9 | 501.9 | 501.9 | 2.3 | 2.3 | 2.3 | 2.3 |
40 | 30 | 1049.7 | 1059 | 1059.9 | 1059.0 | 1059.4 | 0.9 | 1.0 | 0.9 | 0.9 | |
60 | 30 | 1515.9 | 1529.6 | 1530 | 1529.6 | 1529.6 | 0.9 | 0.9 | 0.9 | 0.9 | |
80 | 30 | 2206.1 | 2222.2 | 2222.1 | 2222.1 | 2222.1 | 0.7 | 0.7 | 0.7 | 0.7 | |
100 | 30 | 2627 | 2644 | 2644 | 2645.4 | 2644.1 | 0.6 | 0.6 | 0.7 | 0.7 | |
|
|||||||||||
C08 | 20 | 40 | 434.6 | 458.3 | 461.2 | 458.6 | 458.0 | 5.5 | 6.1 | 5.5 | 5.4 |
40 | 40 | 922 | 954.3 | 956.5 | 951.9 | 951.8 | 3.5 | 3.7 | 3.2 | 3.2 | |
60 | 40 | 1360.9 | 1405 | 1403.5 | 1399.4 | 1403.1 | 3.2 | 3.1 | 2.8 | 3.1 | |
80 | 40 | 1909.3 | 1971.5 | 1965 | 1954.7 | 1960.9 | 3.3 | 2.9 | 2.4 | 2.7 | |
100 | 40 | 2362.8 | 2436.8 | 2425 | 2410.8 | 2418.3 | 3.1 | 2.6 | 2.0 | 2.4 | |
|
|||||||||||
C09 | 20 | 100 | 1106.8 | 1106.8 | 1106.8 | 1106.8 | 1106.8 | 0.0 | 0.0 | 0.0 | 0.0 |
40 | 100 | 2189.2 | 2190.6 | 2190.6 | 2190.6 | 2191.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
60 | 100 | 3410.4 | 3410.4 | 3410.4 | 3410.4 | 3410.4 | 0.0 | 0.0 | 0.0 | 0.0 | |
80 | 100 | 4578.6 | 4588.1 | 4588.1 | 4588.1 | 4588.1 | 0.2 | 0.2 | 0.2 | 0.2 | |
100 | 100 | 5430.5 | 5434.9 | 5434.9 | 5434.9 | 5434.9 | 0.1 | 0.1 | 0.1 | 0.1 | |
|
|||||||||||
C10 | 20 | 100 | 337.8 | 350.5 | 351.5 | 350.4 | 350.1 | 3.8 | 4.1 | 3.7 | 3.6 |
40 | 100 | 642.8 | 664.4 | 667 | 664.0 | 663.7 | 3.4 | 3.8 | 3.3 | 3.3 | |
60 | 100 | 911.1 | 934.7 | 936.6 | 933.1 | 933.2 | 2.6 | 2.8 | 2.4 | 2.4 | |
80 | 100 | 1177.6 | 1209.9 | 1212.4 | 1204.1 | 1205.2 | 2.7 | 3.0 | 2.3 | 2.3 | |
100 | 100 | 1476.5 | 1512.3 | 1514 | 1504.2 | 1506.5 | 2.4 | 2.5 | 1.9 | 2.0 | |
|
|||||||||||
Average | 1043.34 | 1043.19 |
|
1041.92 | 1.77 | 1.80 | 1.66 |
|
Table
Results obtained by GRASP, SVC, ISA, and HDA on Nice and Path.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
Nice1 | 25 | 1000 | 1000 | 1034 | 1037 | 1040.7 | 1034.8 | 3.4 | 3.7 | 4.1 | 3.5 |
Nice2 | 50 | 1000 | 1001 | 1047 | 1038 | 1047.2 | 1037.7 | 4.6 | 3.7 | 4.6 | 3.7 |
Nice3 | 100 | 1000 | 1001 | 1041 | 1035 | 1036.5 | 1030.9 | 4.0 | 3.4 | 3.5 | 3.0 |
Nice4 | 200 | 1000 | 1001 | 1037 | 1026 | 1030.9 | 1023 | 3.6 | 2.5 | 2.9 | 2.2 |
Nice5 | 500 | 1000 | 1000 | 1024 | 1017 | 1015.0 | 1008 | 2.4 | 1.7 | 1.5 | 0.8 |
Nice6 | 1000 | 1000 | 999 | 1020 | 1014 | 1011.0 | 1004 | 2.1 | 1.5 | 1.2 | 0.5 |
|
|||||||||||
Nice1t | 1000 | 1000 | 1001 | 1026 | 1015 | 1011.0 | 1006 | 2.5 | 1.4 | 1.0 | 0.5 |
1000 | 1000 | 1001 | 1022 | 1016 | 1010.0 | 1005 | 2.1 | 1.5 | 0.9 | 0.4 | |
1000 | 1000 | 1000 | 1020 | 1013 | 1011.0 | 1005 | 2.0 | 1.3 | 1.1 | 0.5 | |
1000 | 1000 | 1000 | 1019 | 1013 | 1010.0 | 1005 | 1.9 | 1.3 | 1.0 | 0.5 | |
1000 | 1000 | 1000 | 1022 | 1014 | 1010.0 | 1005 | 2.2 | 1.4 | 1.0 | 0.5 | |
1000 | 1000 | 1001 | 1020 | 1014 | 1010.0 | 1005 | 1.9 | 1.3 | 0.9 | 0.4 | |
1000 | 1000 | 1000 | 1022 | 1014 | 1010.0 | 1006 | 2.2 | 1.4 | 1.0 | 0.6 | |
1000 | 1000 | 1001 | 1021 | 1016 | 1012.0 | 1007 | 2.0 | 1.5 | 1.1 | 0.6 | |
1000 | 1000 | 1000 | 1022 | 1017 | 1012.0 | 1005 | 2.2 | 1.7 | 1.2 | 0.5 | |
1000 | 1000 | 1001 | 1027 | 1016 | 1012.0 | 1007 | 2.6 | 1.5 | 1.1 | 0.6 | |
|
|||||||||||
Nice2t | 2000 | 1000 | 1001 | 1016 | 1008 | 1006.0 | 1004 | 1.5 | 0.7 | 0.5 | 0.3 |
2000 | 1000 | 1001 | 1015 | 1011 | 1005.0 | 1005 | 1.4 | 1.0 | 0.4 | 0.4 | |
2000 | 1000 | 1000 | 1016 | 1008 | 1007.0 | 1005 | 1.6 | 0.8 | 0.7 | 0.5 | |
2000 | 1000 | 1000 | 1014 | 1007 | 1006.0 | 1003 | 1.4 | 0.7 | 0.6 | 0.3 | |
2000 | 1000 | 1000 | 1015 | 1008 | 1006.0 | 1003 | 1.5 | 0.8 | 0.6 | 0.3 | |
2000 | 1000 | 1000 | 1016 | 1002 | 1005.0 | 1004 | 1.6 | 0.2 | 0.5 | 0.4 | |
2000 | 1000 | 1001 | 1016 | 1007 | 1007.0 | 1004 | 1.5 | 0.6 | 0.6 | 0.3 | |
2000 | 1000 | 1001 | 1014 | 1006 | 1006.0 | 1003 | 1.3 | 0.5 | 0.5 | 0.2 | |
2000 | 1000 | 1001 | 1016 | 1008 | 1007.0 | 1006 | 1.5 | 0.7 | 0.6 | 0.5 | |
2000 | 1000 | 1001 | 1016 | 1009 | 1007.0 | 1005 | 1.5 | 0.8 | 0.6 | 0.4 | |
|
|||||||||||
Nice5t | 5000 | 1000 | 1000 | 1010 | 1003 | 1003.0 | 1003 | 1.0 | 0.3 | 0.3 | 0.3 |
5000 | 1000 | 1001 | 1011 | 1005 | 1003.0 | 1003 | 1.0 | 0.4 | 0.2 | 0.2 | |
5000 | 1000 | 1001 | 1010 | 1002 | 1003.0 | 1003 | 0.9 | 0.1 | 0.2 | 0.2 | |
5000 | 1000 | 1000 | 1009 | 1005 | 1002.0 | 1003 | 0.9 | 0.5 | 0.2 | 0.3 | |
5000 | 1000 | 1001 | 1011 | 1006 | 1003.0 | 1004 | 1.0 | 0.5 | 0.2 | 0.3 | |
5000 | 1000 | 1000 | 1009 | 1001 | 1002.0 | 1001 | 0.9 | 0.1 | 0.2 | 0.1 | |
5000 | 1000 | 1001 | 1011 | 1004 | 1003.0 | 1003 | 1.0 | 0.3 | 0.2 | 0.2 | |
5000 | 1000 | 1000 | 1011 | 1004 | 1002.0 | 1003 | 1.1 | 0.4 | 0.2 | 0.3 | |
5000 | 1000 | 1001 | 1010 | 1004 | 1003.0 | 1003 | 0.9 | 0.3 | 0.2 | 0.2 | |
5000 | 1000 | 1000 | 1010 | 1005 | 1003.0 | 1003 | 1.0 | 0.5 | 0.3 | 0.3 | |
|
|||||||||||
Path1 | 25 | 1000 | 1001 | 1042 | 1042 | 1042.0 | 1041.8 | 4.1 | 4.1 | 4.1 | 4.1 |
Path2 | 50 | 1000 | 1000 | 1019 | 1014 | 1014.7 | 1011.9 | 1.9 | 1.4 | 1.5 | 1.2 |
Path3 | 100 | 1000 | 1000 | 1027 | 1022 | 1022.6 | 1025.1 | 2.7 | 2.2 | 2.3 | 2.5 |
Path4 | 200 | 1000 | 1002 | 1023 | 1018 | 1017.7 | 1013 | 2.1 | 1.6 | 1.6 | 1.1 |
Path5 | 500 | 1000 | 1000 | 1034 | 1022 | 1020.0 | 1016 | 3.4 | 2.2 | 2.0 | 1.6 |
Path6 | 1000 | 1000 | 1002 | 1026 | 1018 | 1011.0 | 1010 | 2.4 | 1.6 | 0.9 | 0.8 |
|
|||||||||||
Path1t | 1000 | 1000 | 999 | 1019 | 1011 | 1007.0 | 1003 | 2.0 | 1.2 | 0.8 | 0.4 |
1000 | 1000 | 1001 | 1018 | 1010 | 1006.0 | 1005 | 1.7 | 0.9 | 0.5 | 0.4 | |
1000 | 1000 | 1001 | 1018 | 1013 | 1008.0 | 1006 | 1.7 | 1.2 | 0.7 | 0.5 | |
1000 | 1000 | 1000 | 1016 | 1009 | 1006.0 | 1003 | 1.6 | 0.9 | 0.6 | 0.3 | |
1000 | 1000 | 1003 | 1024 | 1017 | 1010.0 | 1010 | 2.1 | 1.4 | 0.7 | 0.7 | |
1000 | 1000 | 1002 | 1018 | 1013 | 1010.0 | 1005 | 1.6 | 1.1 | 0.8 | 0.3 | |
1000 | 1000 | 999 | 1019 | 1012 | 1008.0 | 1004 | 2.0 | 1.3 | 0.9 | 0.5 | |
1000 | 1000 | 1000 | 1020 | 1012 | 1008.0 | 1006 | 2.0 | 1.2 | 0.8 | 0.6 | |
1000 | 1000 | 999 | 1019 | 1012 | 1006.0 | 1003 | 2.0 | 1.3 | 0.7 | 0.4 | |
1000 | 1000 | 1002 | 1018 | 1011 | 1008.0 | 1006 | 1.6 | 0.9 | 0.6 | 0.4 | |
|
|||||||||||
Path2t | 2000 | 1000 | 1000 | 1015 | 1009 | 1006.0 | 1002 | 1.5 | 0.9 | 0.6 | 0.2 |
2000 | 1000 | 1002 | 1016 | 1010 | 1007.0 | 1004 | 1.4 | 0.8 | 0.5 | 0.2 | |
2000 | 1000 | 1000 | 1015 | 1011 | 1006.0 | 1003 | 1.5 | 1.1 | 0.6 | 0.3 | |
2000 | 1000 | 999 | 1014 | 1007 | 1003.0 | 1000 | 1.5 | 0.8 | 0.4 | 0.1 | |
2000 | 1000 | 1002 | 1018 | 1012 | 1008.0 | 1006 | 1.6 | 1.0 | 0.6 | 0.4 | |
2000 | 1000 | 1002 | 1016 | 1011 | 1007.0 | 1003 | 1.4 | 0.9 | 0.5 | 0.1 | |
2000 | 1000 | 998 | 1011 | 1007 | 1004.0 | 1001 | 1.3 | 0.9 | 0.6 | 0.3 | |
2000 | 1000 | 998 | 1014 | 1010 | 1004.0 | 1003 | 1.6 | 1.2 | 0.6 | 0.5 | |
2000 | 1000 | 1001 | 1017 | 1010 | 1008.0 | 1003 | 1.6 | 0.9 | 0.7 | 0.2 | |
2000 | 1000 | 1003 | 1018 | 1009 | 1009.0 | 1005 | 1.5 | 0.6 | 0.6 | 0.2 | |
|
|||||||||||
Path5t | 5000 | 1000 | 1000 | 1010 | 1002 | 1003.0 | 1001 | 1.0 | 0.2 | 0.3 | 0.1 |
5000 | 1000 | 998 | 1009 | 1003 | 1001.0 | 1001 | 1.1 | 0.5 | 0.3 | 0.3 | |
5000 | 1000 | 1000 | 1011 | 1002 | 1003.0 | 1002 | 1.1 | 0.2 | 0.3 | 0.2 | |
5000 | 1000 | 995 | 1006 | 998 | 997.0 | 997 | 1.1 | 0.3 | 0.2 | 0.2 | |
5000 | 1000 | 1004 | 1016 | 1005 | 1006.0 | 1006 | 1.2 | 0.1 | 0.2 | 0.2 | |
5000 | 1000 | 1000 | 1009 | 1003 | 1002.0 | 1001 | 0.9 | 0.3 | 0.2 | 0.1 | |
5000 | 1000 | 998 | 1009 | 1001 | 1001.0 | 1000 | 1.1 | 0.3 | 0.3 | 0.2 | |
5000 | 1000 | 996 | 1007 | 998 | 999.0 | 998 | 1.1 | 0.2 | 0.3 | 0.2 | |
5000 | 1000 | 997 | 1007 | 999 | 999.0 | 998 | 1.0 | 0.2 | 0.2 | 0.1 | |
5000 | 1000 | 1002 | 1013 | 1004 | 1004.0 | 1005 | 1.1 | 0.2 | 0.2 | 0.3 | |
|
|||||||||||
Average | 1016.63 | 1009.75 | 1007.36 |
|
1.65 | 0.96 | 0.72 |
|
Results obtained by GRASP, SVC, ISA, and HDA on ZDF.
Instance | meanh | Gap | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
|
LB | GRASP | SVC | ISA | HDA | GRASP | SVC | ISA | HDA | |
zdf1 | 580 | 100 | 330 | 333 | 331 | 330.0 | 330 | 0.9 | 0.3 | 0.0 | 0 |
zdf2 | 660 | 100 | 357 | 360 | 358 | 357.0 | 357 | 0.8 | 0.3 | 0.0 | 0 |
zdf3 | 740 | 100 | 384 | 387 | 385 | 384.0 | 384 | 0.8 | 0.3 | 0.0 | 0 |
zdf4 | 820 | 100 | 407 | 410 | 408 | 407.0 | 407 | 0.7 | 0.2 | 0.0 | 0 |
zdf5 | 900 | 100 | 434 | 437 | 434 | 434.0 | 434 | 0.7 | 0.0 | 0.0 | 0 |
zdf6 | 1532 | 3000 | 4872 | 5251 | 5085 | 5081.8 | 5066 | 7.8 | 4.4 | 4.3 | 4 |
zdf7 | 2432 | 3000 | 4852 | 5163 | 5083 | 5084.7 | 5017 | 6.4 | 4.8 | 4.8 | 3.4 |
zdf8 | 2532 | 3000 | 5172 | 5544 | 5386 | 5549.0 | 5397 | 7.2 | 4.1 | 7.3 | 4.4 |
zdf9 | 5032 | 3000 | 5172 | 5476 | 5468 | 5404.0 | 5408 | 5.9 | 5.7 | 4.5 | 4.6 |
zdf10 | 5064 | 6000 | 5172 | 5570 | 5462 | 5419.0 | 5433 | 7.7 | 5.6 | 4.8 | 5 |
zdf11 | 7564 | 6000 | 5172 | 5562 | 5516 | 5419.0 | 5439 | 7.5 | 6.7 | 4.8 | 5.2 |
zdf12 | 10064 | 6000 | 5172 | — | 5651 | 5454.0 | 5403 | — | 9.3 | 5.5 | 4.5 |
zdf13 | 15096 | 9000 | 5172 | — | 5600 | 5415.0 | 5415 | — | 8.3 | 4.7 | 4.7 |
zdf14 | 25032 | 3000 | 5172 | — | 5468 | 5286.0 | 5353 | — | 5.7 | 2.2 | 3.5 |
zdf15 | 50032 | 3000 | 5172 | — | 5960 | 5172.0 | 5273 | — | 15.2 | 0.0 | 2 |
zdf16 | 75032 | 3000 | 5172 | — | 5931 | 5172.0 | 5203 | — | 14.7 | 0.0 | 0.6 |
|
|||||||||||
Average | 3907.88 | 3773.03 |
|
5.34 | 2.67 |
|
Table
A hybrid demon algorithm for 2SP is presented in this paper. This algorithm improves the scoring rule presented by Leung et al. [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Nature Science Foundation of China (Grant no. 71272085).