Konrad-Zuse-Zentrum fur ¨ Informationstechnik Berlin
T OBIAS ACHTERBERG1 T IMO B ERTHOLD? G REGOR H ENDEL
Rounding and Propagation Heuristics for Mixed Integer Programming
1 ?
IBM Deutschland, Germany,
[email protected] Supported by the DFG Research Center M ATHEON Mathematics for key technologies in Berlin.
ZIB-Report 11-29 (July 2011)
Takustraße 7 D-14195 Berlin-Dahlem Germany
Rounding and Propagation Heuristics for Mixed Integer Programming Tobias Achterberg∗, Timo Berthold†, Gregor Hendel‡ July 11, 2011
Abstract Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB 2010 benchmark instances.
1
Introduction: primal heuristics for MIP
Mixed integer programming (MIP) is to solve the optimization problem x ˜opt = argmin{cT x | Ax ≤ b, l ≤ x ≤ u, xj ∈ Z for all j ∈ J},
(1)
ˆ n (with R ˆ := R ∪ {±∞}), and with A ∈ Rm×n , b ∈ Rm , c ∈ Rn , l, u ∈ R J ⊆ N = {1, . . . , n}. In state-of-the-art MIP solvers, primal heuristics play a major role in finding and improving integer feasible solutions at an early stage of the solution process. Knowing good solutions early during optimization helps to prune the search tree and to simplify the problem via dual reductions. Further, it proves the feasibility of a problem and a practitioner might be satisfied with a solution that is proven to be within a certain gap to optimality. This article gives an overview about rounding and propagation heuristics for MIP that are integrated into SCIP [1], which is a state-of-the-art noncommercial solver and framework for mixed integer programming. The ZI Round heuristic has been introduced by Wallace [5], more details on the other primal heuristics can be found in [2, 1, 3].
∗ IBM
Deutschland, Germany,
[email protected] Institute Berlin, Takustr. 7, 14195 Berlin, Germany,
[email protected], supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin. ‡ Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany,
[email protected] † Zuse
1
2
Rounding heuristics
The goal of rounding heuristics is to convert a fractional solution x ¯ of the system Ax ≤ b, l ≤ x ≤ u into an integral solution, i.e., xj ∈ Z ∀j ∈ J. All rounding heuristics described in this section use the notion of up- and down-locks. For a MIP (1), we call the number of positive coefficients ξj+ := |{i : aij > 0}| the up-locks of the variable xj ; the number of negative coefficients is called the down-locks ξj− of xj . Simple Rounding is a very cheap heuristic that iterates over the set of fractional variables of some LP-feasible point. It only performs roundings, which guarantee to keep all linear constraints satisfied. Consider an integer variable xj , j ∈ J, with fractional LP solution x ¯j . If ξj− = 0, we can safely set x ˜j := b¯ xj c without violating any linear constraint. Analogously, if ξj+ = 0, we can set x ˜j := d¯ xj e. If all integer variables with fractional x ¯j can be rounded that way, then x ˜ will be a feasible MIP solution. In contrast to Simple Rounding, Rounding also performs roundings which potentially lead to a violation of some linear constraints, trying to recover from this infeasibility by further roundings later on. The solutions that can be found by Rounding are a superset of the ones that can be found by Simple Rounding. Like Simple Rounding, the Rounding heuristic takes up- and down-locks of an integer variable with fractional LP value x ¯j into account. As long as no linear constraint is violated, the algorithm iterates over the fractional variables and applies a rounding into the direction of fewer locks, updating the activities Ai x ˜ of the LP rows after each step, Ai being the i-th row of A. If there is a violated linear constraint, hence Ai x ˜ > bi for some i, the heuristic will try to find a fractional variable that can be rounded in a direction such that the violation of the constraint is decreased, using the number of up- and down-locks as a tie breaker. If no rounding can decrease the violation of the constraint, the procedure is aborted. ZI Round[5] reduces the integer infeasibility of an LP solution step-by-step by shifting fractional values towards integrality, but not necessarily rounding them. For each integer variable xj with fractional solution value x ¯j , the heuristic calculates bounds for both possible rounding directions of x ¯j such that the obtained solution stays LP-feasible. x ¯j is shifted by the corresponding bound into the direction which reduces the fractionality min{¯ xj −b¯ xj c, d¯ xj e− x ¯j } most. The set of fractional variables might be processed several times by ZI Round. It either terminates with a MIP solution x ˜ or aborts if the integer infeasibility could not be decreased anymore or if a predefined iteration limit has been reached.
3
Propagation heuristics
The goal of propagation heuristics is to construct a feasible MIP solution x ˜ from scratch or from a start solution, while applying domain propagation to exclude variable values that would lead to an infeasibility or to a solution that is inferior to the incumbent. In contrast to diving or LNS heuristics (see, e.g., [2]), propagation heuristics do not solve any LP during search. The Shifting heuristic is similar to Rounding, but it tries to continue in the case that no rounding can decrease the violation of a linear constraint. In this case, the value of a continuous variable or an integer variable with integral
2
value will be shifted in order to decrease the violation of the constraint. To avoid cycling, the procedure terminates after a certain number of non-improving shifts. A shift is called non-improving, if it neither reduces the number of fractional variables nor the number of violated rows. Shift-and-Propagate tries to find a MIP solution by alternately fixing variables and propagating these fixings. Starting with an initial, typically infeasible, solution in which each variable is assumed to be at one of its bounds, it iterates over all integer variables xj , j ∈ J in nondecreasing order of their impact on the activity of the linear constraints. In each step, the heuristic fixes a variable xj to a value x ˜j such that the overall infeasibility gets maximally reduced. The fixing xj = x ˜j is then propagated to reduce further variable domains. In case that the domain propagation detects the infeasibility of the current partial solution, the domain of xj is reset to its previous state and the variable is postponed. The procedure is aborted when a predefined iteration limit is exceeded.
4
Improvement heuristics
Improvement heuristics consider the incumbent solution as a starting point and try to construct an improved solution with better objective value. Oneopt is a straightforward improvement heuristic: given a feasible MIP solution x ˜, the value of an integer variable xj , j ∈ J, can be decreased for cj > 0 or increased for cj < 0 if the resulting solution is still feasible. If more than one variable can be shifted, they are sorted by non-decreasing impact |cj δj | on the objective and sequentially shifted until no more improvements can be obtained. Here, δj ∈ Z denotes how far the variable can be shifted into the desired direction without losing feasibility. Oneopt often succeeds in improving solutions which were found by the rounding heuristics described in Section 2, since their defensive approach to round into the direction of fewer locks tends to over-fulfill linear constraints, sacrificing solution quality. The Twoopt heuristic attempts to improve a feasible MIP solution x ˜ by altering the solution values of pairs of variables. Only variables which share a pre-defined ratio of LP rows are considered as pairs. Each step of the heuristic consists of improving the objective value by shifting one variable, and then compensating the resulting infeasibilities by shifting a second variable, without completely losing the objective improvement. Similarly to Oneopt, pairs are processed in non-decreasing order of their impact on the objective.
5
Computational experiments
In MIP solving, the running time to optimality and the number of branch-andbound nodes are typical measures for comparison. For primal heuristics, the time needed to find a first feasible solution, an optimal solution, or a solution within a certain gap to optimality are favorable measures that concentrate on the primal part of the solution process. Nevertheless, the trade-off between speed and solution quality is not well covered by any of them. We suggest a new performance measure that takes into account the overall solution process. The goal is to measure the progress of convergence towards the optimal solution over the entire solving time. Let x ˜ be a solution for a MIP,
3
and x ˜opt be an optimal (or best known) solution for that MIP. We define the primal gap γ ∈ [0, 1] of x ˜ as: if |cT x ˜opt | = |cT x ˜| = 0, 0, T T if c x ˜opt · c x ˜ < 0, (2) γ(˜ x) := 1, T T |c Tx˜opt −c x˜T| , else. max{|c x ˜opt |, |c x ˜|} Let tmax ∈ R≥0 be a limit on the solution time of a MIP solver. Considering a log file of a MIP solver for a certain problem instance within a fixed computational environment, we define its primal gap function p : [0, tmax ] 7→ [0, 1]: ( 1, if no incumbent found until point t, (3) p(t) := γ(˜ x(t)), with x ˜(t) being the incumbent solution at point t. p(t) is a step function that changes whenever a new incumbent is found. It is zero from the point on at which the optimal solution is found. We define the primal integral P (T ) of a run until a point in time T ∈ [0, tmax ] as: ZT P (T ) :=
p(t) dt = t=0
I X
p(ti−1 ) · (ti − ti−1 ),
(4)
i=1
where ti ∈ [0, T ] for i ∈ 1, . . . , I − 1 are the points in time when a new incumbent solution is found, t0 = 0, tI = T . We suggest to use P (tmax ) for measuring the quality of primal heuristics. It favors finding good solutions early. The fraction P (tmax )/tmax can be seen as the average solution quality during the search process. Spoken differently, the smaller P (tmax ) is, the better is the expected quality of the incumbent solution if we stop the solver at an arbitrary point in time. We used the benchmark set of the MIPLIB 2010 [4] as test set for our experiments. Since we are interested in the primal part of the solution process, we excluded the four infeasible instances triptim1, enlight14, ns1766074, and ash608gpia-3col; further, mspp16 was excluded since it terminated for memory reasons during presolving for all our tests. Thus, 82 test instances remained. We performed four different runs: SCIP without any primal heuristics (Heur OFF), SCIP using only the rounding and propagation heuristics which are described in this paper (R&P), SCIP using all default heuristics except the ones described in this paper (NoR&P), and SCIP with default settings (DEF). All experiments were conducted with a time limit of one hour, a memory limit of 4 GB on a 3.00 GHz Intel® Core™2 Extreme CPU X9650 with 6144 KB Cache and 8 GB RAM. Figure 1 exemplarily shows the primal gap function p(t) for the four settings applied to the instance n3seq24. A square shows when a new primal solution is found and its quality. It can be seen that the two settings that use rounding and propagation heuristics find solutions earlier and hence have a smaller primal integral P (tmax ). For all four settings, SCIP found the optimal solution within an hour, but timed out without proving optimality. Interestingly, with disabled heuristics (Heur OFF), the optimal solution was found in the smallest amount of time. Nevertheless, the behavior of the default settings (DEF) seems favorable since primal solutions of reasonable quality are found much earlier. 4
NoR&P, P (tmax ) = 107282.2 p(t) in %
p(t) in %
DEF, P (tmax ) = 42166.8 75 50 25 0 0
1,000
2,000
75 50 25 0
3,000
0
1,000
t in sec.
75 50 25 0 1,000
2,000
3,000
Heur OFF, P (tmax ) = 105009.6 p(t) in %
p(t) in %
R&P, P (tmax ) = 79724.7
0
2,000 t in sec.
3,000
75 50 25 0 0
t in sec.
1,000
2,000
3,000
t in sec.
Figure 1: The solving process depicted for the instance n3seq24 DEF NoR&P R&P Heur OFF φ(P ) 0.44 0.47 0.61 1.0 φ(t1 ) 8.54 17.48 10.88 57.66 φ(topt ) 215.63 218.18 236.92 263.46 φ(tsolved ) 712.90 676.10 746.48 838.60 Table 1: Computed mean values for all four settings Table 1 shows aggregated results of our experiments. The first row shows the evaluation of the normalized primal integral over the entire testset. For each instance, the corresponding integral P (tmax ) is divided by the integral obtained with the setting Heur OFF. This reference value is then used to compute the geometric mean φ(P ) for each setting. The remainder of the table shows the geometric means of the time until the first (t1 ) and the best (topt ) primal solution were found and the mean solving time (tsolved ). The advantage of rounding and propagation heuristics can be best seen in the time to first solution t1 : they are valuable for finding start solutions. The impact on the time to the best solution and the overall solving time is much smaller, the latter even showing a degradation. The primal integral P (tmax ) implies the following ranking: To get good primal solutions, using cheap and expensive heuristics together (DEF) is slightly better than using only expensive heuristics (noR&P), which is considerably better than using only cheap ones (R&P), which is much better than using no heuristics (Heur OFF). This behavior is only partially expressed by the other measures. Detailed computational results including charts for the development of the primal gap function p(t) can be found in the Appendix.
5
References [1] T. Achterberg. Constraint Integer Programming. PhD thesis, TU Berlin, 2007. [2] T. Berthold. Primal Heuristics for Mixed Integer Programs. Master’s thesis, TU Berlin, 2006. [3] G. Hendel. New rounding and propagation heuristics for mixed integer programming. Bachelor’s thesis, TU Berlin, 2011. [4] T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter. MIPLIB 2010. Mathematical Programming C, 3(2):103–163, 2011. [5] C. Wallace. ZI round, a MIP rounding heuristic. J Heuristics, 16:715–722, 2009.
6
Appendix In Table 2, the results of the experiment are explicitly shown for each of the 82 instances in our test set. Different performance measures are evaluated for the four settings Heur OFF, R&P, NoR&P, and DEF, see above. For each instance, the first row shows the time until the first (t1 ), and the best (topt ) solution were found, the overall solving time (tsolved ) and the normalized integral value (P (tmax )). The second row shows the development of p(t) over time. Note that according to the definition of the normalized integral, P (tmax ) will always be 1.00 for the setting without primal heuristics. For the figures, a square at 3600 seconds indicates that the solver hit the time limit, because the incumbent solution was not proven to be optimal. Note that the light shaded areas, which indicate the period without an incumbent solution, are cut off at the top. DEF
533.64 3582.95 3600.0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.03
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
226.27 226.27 226.27
1.26
Gap
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
536.83 536.83 536.84
2.98
0.8
0.8
0.6
0.6
0.6
0.4
0.2
0.2
0
0
0
100
200 300 Time(sec.)
400
500
0.33
60
60 Gap
80
20
0
0 500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
45.63
48.99
1.24
200 300 Time(sec.)
400
500
0.45
40
20 0
100
14.12 592.35 3516.76
80
40
0
41.34
100
200 300 Time(sec.)
400
2.10
80
60
60
40
56.39
58.69
1.27
35.13
0
38.71
42.33
1.06
33.05
6
6
6
0
10
20 30 Time(sec.)
40
0
50
1.34
Gap
6
Gap
8
4 2
0
10
20 30 Time(sec.)
137.01 3485.08 3485.09
40
0
50
0.60
10
20 30 Time(sec.)
3600.0 3600.0 3600.0
40
1.34
30
30
30
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
500.97 688.34 3600.0
0.72
Gap
30
Gap
40
0
20 10 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
571.07 1905.29 3600.0
0.82
758.96
794.2 3600.0
1.11
6
6
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.07 917.53 3600.0
0.05
Gap
6 Gap
6 Gap
8
2
4 2
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.03 2690.61 3600.0
0.07
0.07
0.07 3600.0
0.21
15
15
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
Gap
15 Gap
15 Gap
20
5
10 5
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
7
0
0
40
50
1.00
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
1.00
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
3600.0 3600.0 3600.0
20
10
20 30 Time(sec.)
4
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20
5
10
2 0
20
10
0
613.21 1980.27 3600.0
8
4
1.00
20
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
8
2
36.93
10 0
8
4
1.00
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
2685.16 2685.16 2685.17
40
10
500
4
0
50
40
20
400
2 0
40
20
0
36.82
8
2
200 300 Time(sec.)
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
8
4
100
20 0
8
4
0
32.64 630.78 3600.0
80
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0.4
0
500
20 0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.2 0
0.76 986.43 1886.23
Gap
0
Gap
0.4
Gap
0.8
0.6
Gap
0.8 Gap
1
0.2
0
180.29 180.29 180.29
1
0.4
1.00
20
0
1
Gap
Gap
0
1
0
Gap
Gap
Gap
40
20
0
P (tmax )
2680.8 3594.77 3600.0
40
20
0
3600.0 3600.0 3600.0
Gap
1.14
topt tsolved
40
20
10
beasleyC3
3600.0 3600.0 3600.0
t1
40
0
bab5
Heur OFF
topt tsolved P (tmax )
60
2
app1-2
0.67
t1
60
41.0
Gap
air04
R&P
topt tsolved P (tmax )
60
0.77 353.51 1274.95
Gap
aflow40b
0.45
t1
60
185.25 185.25 185.25
Gap
acc-tight5
NoR&P
topt tsolved P (tmax )
256.63 3287.83 3600.0
Gap
30n20b8
t1
Gap
Problem Name
1.00
10 5
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
DEF
4.52 386.06 419.45
0
0
0
200
400 Time(sec.)
95.03
600
800
169.6
0.46
2.25
400 Time(sec.)
600
0
800
30.69 223.84
0.44
Gap 0
200
400 Time(sec.)
600
40
0
800
0.97 120.42 125.46
0.42
4.05
20
20
10
10
0
0
0
40
60 80 Time(sec.)
100
120
0.49
1.32
0
20
40
60 80 Time(sec.)
100
105.5 131.83
120
0.44
10 5
0
20
40
60 80 Time(sec.)
100
0
120
0.77
20
40
60 80 Time(sec.)
100
50
100 150 Time(sec.)
0.79
0.26
0
120
0.82
3.15
Gap 50
100 150 Time(sec.)
20
40
60 80 Time(sec.)
100
709.23 709.23 709.24
0.47
1.01
0
50
100 150 Time(sec.)
1.38
0.8
0.6
0.6
8.43 219.18 219.19
0
2,000
0.10
7.21
500
145.7
1,000 1,500 Time(sec.)
145.7
0.08
200
400 Time(sec.)
0.01 234.81 3600.0
600
0.21
500
1,000 1,500 Time(sec.)
8.34 316.47 316.48
0.24
0.09
200
400 Time(sec.)
6.57 3600.0
0
600
0.10
0
0.0
200
400 Time(sec.)
7.36 3600.0
600
0.21
60
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
31.48 449.62 3189.21
0.60
Gap
60 Gap
60 Gap
60
0
40 20
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
10.14 374.06 2818.98
0.50
40.03 301.69 3544.72
1.17
20
20
10
10
10
0
0
0
20.74 3600.0
0.61
4.95 144.23 3600.0
20 Gap
10
0
0.49
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.96
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
8
200
1.00
0
500
1,000 1,500 Time(sec.)
2,000
1.00
0
200
400 Time(sec.)
600
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
20
10
0 0
100 150 Time(sec.)
7.27 186.98 3600.0
20
10
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
10.43 102.04 3596.58
20
1.00
10
0
Gap
8.58
Gap
20
Gap
20
Gap
30
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
50
13.75 636.55 3600.0
30
0
120
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
30
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
100
20 0
30
0
60 80 Time(sec.)
7.28 3600.0 80
20
0
6.91
80
40
40
20
0
80
20
20
40
80
40
0
95.84 778.98 778.99
20
0
1.00
0.4
0
2,000
Gap
Gap
Gap
0 0
120
0.2 0
40
20
20 0
0
2,000
40
40
0.4 0.2
0
100
Gap
1,000 1,500 Time(sec.)
Gap
0.8
0.6
Gap
0.8
0.6
Gap
0.8
500
60 80 Time(sec.)
1495.95 1495.95 1495.96 1
0
40
10
0
200
1
0
20
20
2067.97 2067.97 2067.97
0.2
0
243.42 244.21 244.22
1
0.4
1.00
10
1
0.4
800
98.17 171.45
0
120
10
200
600
5 0
245.34 245.34 245.34
0 0
400 Time(sec.)
15
20
10
0
200
100
10
0
120
184.61 212.53 212.55
Gap 0
60 80 Time(sec.)
5 0
20
10
40
15 Gap
10
20
2.55 136.23 179.03
15
200
10
0
Gap
20
Gap
0
Gap
20
Gap
20 10
0
80.44 149.69 30
Gap
Gap Gap Gap
200
30
393.34 393.34 393.34
Gap
0
1.00
20
20
30
181.68 206.85 206.85
Gap
40
30
1.36 102.26 129.01
Gap
Gap
Gap
60
20
P (tmax )
635.66 745.69 944.05
60
40
20
0.2
danoint
0.69
60
40
0
csched010
838.8 897.29
topt tsolved
60
20
cov1075
2.5
t1
80
0
core2536-691
topt tsolved P (tmax )
80
5
bnatt350
0.02
Heur OFF
t1
80
15
bley xl1
R&P
topt tsolved P (tmax )
Gap
binkar10 1
0.07
t1
80
1.1
Gap
bienst2
NoR&P
topt tsolved P (tmax )
2.51 619.04 701.01
Gap
biella1
t1
Gap
Problem Name
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
10
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
DEF
19.44 104.69
0.39
0.01
0
5
10
82.59
15 20 Time(sec.)
25
94.24
0
30
0.99
19.53
Gap
Gap 5
10
15 20 Time(sec.)
57.52
25
80.3
0
30
1.02
13.48
40 20
0
5
10
15 20 Time(sec.)
37.05
25
76.56
0
30
0.87
18.71 80
60
60
40 20
20
40 Time(sec.)
60
241.0
0
80
0.70
20 0
20
40 Time(sec.)
60
23.55 128.95 158.15
0
80
1.15
0 0
50
100 150 Time(sec.)
200
10.62 1938.46
0.05
Gap
Gap
20
0
20
40 Time(sec.)
60
0.71
50
100 150 Time(sec.)
315.02 315.02 452.68
1.62
66.63
0 50
100 150 Time(sec.)
66.63 3600.0
200
0.34
0.6
0.6
0.6
0.6
0.4
0.2
0.2
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
131.96 131.96 131.96
1.01
Gap
1 0.8
Gap
1 0.8
0.4
130.36 130.36 130.36
0.99
0.8
0.8
0.6
0.6
0.6
40
60 80 Time(sec.)
100
0.58 1679.56 3600.0
0
120
0.95
0.2 0
20
40
60 80 Time(sec.)
100
0.43 2429.37 3600.0
0
120
0.96
Gap
Gap
20
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.0 2004.04 3600.0
0.14
40
60 80 Time(sec.)
100
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.36 2908.79 3193.88
0.18
0.87
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.15
60
0.0
0 500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
28.62 1988.89
0.19
0.12
Gap
60
Gap
60
Gap
60
0
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
27.81 1863.8
0.42
0.0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
27.13 1545.83
0.22
23.28
40
40
40
40
0
5
10
15 20 Time(sec.)
0.01 127.69 3600.0
25
0.33
20
0
0
0.3
0
5
10
15 20 Time(sec.)
65.23 3600.0
25
0.39
0.0
Gap
60
Gap
60
20
5
10
15 20 Time(sec.)
94.46 3600.0
0
25
0.40
27.08
60
60
60
60
20
0
0 0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
Gap
80
Gap
80
40
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
9
0
40
60 80 Time(sec.)
100
120
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0
5
10
15 20 Time(sec.)
25
1.00
40 20
20 0
20
43.43 3600.0
80
20
0
20 0
80
40
1.00
26.77 1478.99
60
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40
0 0
60
20
0
20
20 0
1.00
3.22 1832.67 3276.45 80
0
200
20 0
0.0 650.65 3600.0
20
100 150 Time(sec.)
40
80
40
50
60
80
20
0
0.94 2866.82 2964.95
80
40
1.00
0.4
0
120
40
20 0
20
60
40
20
80
0.2 0
0.38 2392.02 2489.41
60
40
0.4
Gap
20
Gap
0.8
0.6
Gap
0.8 Gap
1
0
60
131.23 131.23 131.23
1
0.2
40 Time(sec.)
0.4
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1
0.4
20
0.2 0
1
0.4
0
193.77 193.77 2564.5
1 0.8
0
1.00
20 0
1
0.2
81.94
30
40
0.8
0.4
25
60
40
0
200
15 20 Time(sec.)
15.26 201.53 248.37
20 0
10
40
0
80
60
40
20
5
20 0
2.58 205.23 235.44
60
40
40
Gap
0
Gap
80
60
Gap
80
60 40
0
50.11
80
Gap
Gap Gap Gap
0
100
131.17 131.17 131.17
Gap
20
100
10.62
Gap
40
100
2.47 220.59
Gap
1.00
100
Gap
Gap
13.51
Gap
31.43 125.07
60
40 20
60
iis-bupa-cov
P (tmax )
1.14
60
0
iis-100-0-cov
0.55
60
40
0.2
gmu-35-40
98.04
60
60
glass4
18.47
topt tsolved
80
0
ex9
0.02
t1
80
20
enlight13
0.45
Heur OFF
topt tsolved P (tmax )
80
0
eilB101
24.89 108.87
t1
80
20
eil33-2
R&P
topt tsolved P (tmax )
Gap
0.02
t1
Gap
dfn-gwin-UUM
NoR&P
topt tsolved P (tmax )
Gap
t1
Gap
Problem Name
0 0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
DEF
0.35 224.63 657.33
20
0
0 0
100
79.06
200 Time(sec.)
300
400
79.07
0.31
14.97
300
0
400
83.63
0.32
15.55
Gap
40
0 0
100
93.52
200 Time(sec.)
300
400
93.53
0.36
40
40
20
20
20
0
0
0
0
50
100 150 Time(sec.)
200
18.4 3600.0
1.00
0
0.03
50
100 150 Time(sec.)
200
12.3 3600.0
1.00
Gap
60
Gap
60
Gap
60
40
0.01
50
100 150 Time(sec.)
40.73 3600.0
1.01
80
80
60
60
60
40
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
205.4 3600.0
0.22
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
10.32 1588.24 3600.0
0.14
40 20 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.22
2.08
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
54.51 424.76 500.01
0.46
0
4.33 354.17 376.22
0.33
80
80
60
60
60
100
200 300 Time(sec.)
0
400
0.10
Gap
80
60
Gap
80 Gap
100
0
40 20
0
100
200 300 Time(sec.)
0
400
52.88 372.95 472.83
0.23
4.56
100
200 300 Time(sec.)
316.8 405.42
0.16
80
80
60
60
60
40 20
0
0
0
50
100
150 200 250 Time(sec.)
300
0.11 157.72 159.86
350
0.21
0.38
50
100
150 200 250 Time(sec.)
300
35.74 184.27
350
0.13
0.12
40
20
20 0
0
0.0
20
40
60 80 100 120 140 Time(sec.)
0.13 131.97
0.01
0
0.01
20
40
50
100
150 200 250 Time(sec.)
300
37.37 148.26
0.11 130.49
0.01
0.21
6.5
0.0
20
40
60 80 100 120 140 Time(sec.)
5.16 122.18
0.01
60
60
60
0.01
0
100
44.99
200 300 Time(sec.)
0
400
46.64
0.69
36.8
Gap
60
Gap
80
Gap
100
80
0
40 20
0
100
50.29
200 300 Time(sec.)
0
400
53.43
0.95
0.01
100
45.64
200 300 Time(sec.)
46.78
0.69
39.03
60
60
60
40
20
20
0
0
0
10
20 30 Time(sec.)
40
50
0
10
20 30 Time(sec.)
10
40
50
Gap
60
Gap
80
40
0
50
100
0
20
40
0
100
50.58
80
0
1.00
150 200 250 Time(sec.)
300
350
1.00
60 80 100 120 140 Time(sec.)
1.00
40
0
400
80
20
400
20 0
80
40
200 300 Time(sec.)
1.01 432.24 3193.47
100
80
20
100
20 0
0
100
40
0
40
80
20
1.00
29.95 178.24
100
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40
0
350
20 0
60 80 100 120 140 Time(sec.)
0
20 0
40 Gap
Gap
40
0
0
Gap
0
Gap
80
60
Gap
80 Gap
100
20
1.00
289.64 369.28 403.33
100
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40
0
400
100
20
0
20 0
100
40
1.00
196.31 265.93 318.38
100
20
200
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
100
40
100 150 Time(sec.)
20 0
100
40
50
60
20 0
1.00
35.91 3049.12 3600.0
60 Gap
40
239.2
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.0 3403.14 3600.0
60
400
20 0
Gap
0
Gap
80
60
Gap
80
20
300
2.88 3600.0 100
20
0
0.68
100
40
200 Time(sec.)
40
0
200
100
20
100
20 0
100
40
0
101.17 239.19
60
Gap
Gap Gap Gap
83.62
200 Time(sec.)
80
4.62 316.41 419.39
Gap
100
80
4.4 246.77 317.26
Gap
0
1.00
20
20
80
0.01
Gap
40
80
0
mine-166-5
Gap
60
Gap
60
40
20
P (tmax )
42.08 406.79 666.64
60
40
20
mik-250-1-100-1
0.19
topt tsolved
60
0
mcsched
0.02 177.67 583.23
t1
80
20
map20
Heur OFF
topt tsolved P (tmax )
80
60
map18
0.37
t1
Gap
macrophage
R&P
topt tsolved P (tmax )
80
0.0
Gap
m100n500k4r1
0.23
t1
80
14.8
Gap
lectsched-4-obj
NoR&P
topt tsolved P (tmax )
0.01 245.66 641.41
Gap
iis-pima-cov
t1
Gap
Problem Name
200 300 Time(sec.)
400
53.46
1.00
40 20 0
0
10
20 30 Time(sec.)
40
50
0
10
20 30 Time(sec.)
40
50
DEF
0.84
t1
24.65 438.38 603.12
60
0
0 0
200 400 600 800 1,000 1,200 1,400 Time(sec.)
394.86 3481.54 3600.0
0.19
Gap
0
0.93
0.05
0
0.99
111.95 80
60
60
40 20
0
0
0
0
50
100 150 Time(sec.)
200
0.41
0
50
100 150 Time(sec.)
200
110.48 2962.52 3600.0
1.05
Gap
80
60
Gap
80
60
Gap
80
20
50
100 150 Time(sec.)
2.42 818.07 3600.0
0.31
60
60
60
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.40
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
994.89 1697.2 3600.0
1.02
Gap
Gap
20
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.12
0.76
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.09 109.28 1032.29
0.08
0.06
0 500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
34.61 523.35
0.09
77.12
60
40 20
0
0
7.91
0
20
40
60 80 100 Time(sec.)
120
15.99 235.22
140
1.06
1.62
0
20
40
60 80 100 Time(sec.)
120
15.11 163.08
140
0.42
Gap
60
Gap
60
Gap
60
20
7.86
40
60 80 100 Time(sec.)
120
11.17 279.51
140
1.07
7.74
20
20
6 8 10 Time(sec.)
12
26.84 2372.34 3600.0
14
0
16
1.02
25.18
Gap
20
Gap
20
Gap
30
4
10
0
2
4
6 8 10 Time(sec.)
12
39.38 3600.0
14
0
16
0.96
2
4
6 8 10 Time(sec.)
12
24.28 181.57 3600.0
14
0
16
0.93
26.23 1 0.8
0.6
0.6
0.6
0.6
0
31.86
0.2 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
47.19 150.31
1.17
Gap
1 0.8
Gap
1 0.8
0.4
0.4 0.2
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
28.59 166.45 392.23
0.98
36.45
92.39 260.96
1.60
60
60
60
0
0
50
100 150 200 Time(sec.)
250
0
Gap
60 Gap
80
20
40 20
0
50
100 150 200 Time(sec.)
11
250
0
120
140
1.00
0
2
4
6 8 10 Time(sec.)
12
0
14
16
1.00
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
24.99 282.79 447.85
80
40
60 80 100 Time(sec.)
0.4
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
80
20
40
0.2 0
80
40
20
29.55 3600.0
1
0.2
0
10
0
0.8
0.4
1.00
10.52 304.19
30
2
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40
0 20
30
0
0
20 0
30
10
1.00
89.77 799.61 80
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20 0
80
0
0
40
80
20
1.00
1030.03 1278.57 3600.0
80
40
200
60
40
20 0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
60
40
20
100 150 Time(sec.)
40
0 0
31.25 3590.62 3600.0
60
40
50
20
20
Gap
0 0
Gap
60
Gap
80
40
0
106.9 2202.45 3600.0
80
20
1.00
40
0
200
80
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20 0
80
40
0
205.9 277.44 100
40
1.00
10 0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
234.6 285.85
20
200 400 600 800 1,000 1,200 1,400 Time(sec.)
3600.0 3600.0 3600.0
100
40
0
20
10
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
103.0 203.92 220.85
Gap
Gap Gap Gap
1.00
100
0.07 148.95 631.79
Gap
3600.0 3600.0 3600.0
Gap
Gap Gap
0.96
200 400 600 800 1,000 1,200 1,400 Time(sec.)
20
0
40
0 0
100
31.46 1297.75 3600.0
Gap
0.24
10 0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
2.41 818.48 3600.0
Gap
0
200 400 600 800 1,000 1,200 1,400 Time(sec.)
840.17 1064.75 3600.0
Gap 0
0.04 141.49 288.18
0
neos-1396125
0
1.00
20
20
20
10
40
Gap
20
10
neos-1337307
40
20
Gap
60
Gap
60
40
P (tmax )
28.06 1525.87 1647.92
60
60
neos-1109824
1.02
topt tsolved
80
0
n4-3
0.01 1265.59 1514.87
t1
80
20
n3seq24
Heur OFF
topt tsolved P (tmax )
80
0
n3div36
0.71
t1
80
20
mzzv11
R&P
topt tsolved P (tmax )
Gap
msc98-ip
NoR&P
topt tsolved P (tmax )
0.01 1657.66 1911.35
Gap
mine-90-10
t1
Gap
Problem Name
1.00
40 20
0
50
100 150 200 Time(sec.)
250
0
0
50
100 150 200 Time(sec.)
250
DEF
0.42
t1
2.55 202.86 238.96
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
3.38
237.45 2946.48 3600.0
Gap
1419.96 2345.89 3600.0
2.74
80
60
60
40 20
37.64 149.83
0.92
0.37
20 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
13.56
87.4
0.31
0.16
60 40
20
20
0
0
0
10
20 Time(sec.)
30
0.51
10
20 Time(sec.)
223.77 403.65 428.65
0.70
160.06
100
200 300 Time(sec.)
56.85
65.37
400
1.11
1
0
500
0
18.08
100
200 300 Time(sec.)
80.7
400
93.09
1.43
9.95
10
20 Time(sec.)
0
30
0.51
0
100
200 300 Time(sec.)
45.51
400
55.88
0.75
14.7
0
20
40 Time(sec.)
60
80
0.81
0
20
40 Time(sec.)
1396.76 1396.76 1396.76
60
80
1.66
Gap
40
Gap
40
Gap
40
0
20
40 Time(sec.)
801.06 801.06 801.07
60
0
80
0.95
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.2
0.2
0
0
0
200
400
600 800 1,000 1,200 Time(sec.)
43.97 132.83 563.83
0.19
200
400
600 800 1,000 1,200 Time(sec.)
33.92 115.29 528.25
0.15
Gap
Gap
20
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.11 823.18 823.19
0.01
200
400
0.70
13.86 631.21 631.21
0.02
0.11 389.54 389.54
0.05
60
60
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
158.93 252.71 3600.0
0.57
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
2.04
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
Gap
Gap 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
2.15
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
12
80
1.00
200
400
600 800 1,000 1,200 Time(sec.)
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
20
10 0
0
298.99 354.52 3045.95
20
10
60
40
0 0
504.37 1331.99 3004.87
20
10 0
0
40 Time(sec.)
20
20
127.13 3241.0 3600.0
20
40
Gap
0
0
Gap
60
Gap
60
Gap
80
20
20
3600.0 3600.0 3600.0
80
40
0
20
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
80
20
1.00
10 0
80
40
500
30
20
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
400
66.26
127.81 2527.55 3600.0
10 0
200 300 Time(sec.)
0.4
0
600 800 1,000 1,200 Time(sec.)
30
10 0
1.00
0.2 0
92.69 2179.78 3600.0
30
20
0
0
Gap
0
Gap
0.8
Gap
1
Gap
1
0.4
100
842.15 842.15 842.16
1
0.2
30
20
0
1
0.4
0
54.83
40
20
20 Time(sec.)
1
0
500
60
0
10
2
60
20
0
319.77 414.48 456.84
60
0
1.00
40
60
20
101.2
20 0
1
11.12
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
60
511.8 556.23
0
500
0
8.86
2 Gap
Gap 0
0.98
40
0
30
2
1
32.1 124.88
20 0
1.00
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
60 Gap
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20 0
Gap
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40
Gap
0
Gap
80
60
Gap
80
40
0
797.54 797.54 797.55
60
Gap
Gap Gap Gap
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
80
684.53 684.53 684.53
Gap
2.73
1.00
20 0
100
11.88
Gap
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
100
162.55 403.84 421.71
Gap
0
100
0.16
Gap
20
100
10
net12
Gap
Gap
40
0
P (tmax )
22.12 1538.69 3600.0
40
20
30
neos-934278
0.53
40
20
0
neos-916792
98.44 124.33
topt tsolved
40
2
neos-849702
2.1
t1
60
60
neos-686190
topt tsolved P (tmax )
60
0
neos-476283
0.22
Heur OFF
t1
60
20
neos18
R&P
topt tsolved P (tmax )
60
246.59 3517.45 3517.46
Gap
neos-1601936
NoR&P
topt tsolved P (tmax )
2.17 193.92 206.58
Gap
neos13
t1
Gap
Problem Name
10 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
DEF
NoR&P
360.34 360.34 431.38
1.51
t1
422.65 422.65 554.26
100
200 300 Time(sec.)
1.76
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
15.41 381.75
0.02
0
0.09
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
2.14 302.55
0.83
0.01
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
75.28 632.39
0.00
40
40
1.5
0
2
7.76
Gap
40
Gap
40
Gap
60
1 Time(sec.)
20
0
0.5
1 Time(sec.)
1191.82 1191.82 1903.62
1.5
0
2
2.57
0.5
1 Time(sec.)
695.84 695.84 695.85
1.5
0
2
1.50
463.8 1 0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.55
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
81.71 629.33 629.34
0.53
0
1.00
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.16
0
3600.0 3600.0 3600.0
1.00
0
3600.0 3600.0 3600.0
1.00
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.2
0.2
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
65.03 183.69 682.52
0.85
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
22.07 416.86 746.31
0.75
71.14
20
0
100
200 300 Time(sec.)
0.03 436.38 442.66
0.19
20
0
400
0
100
200 300 Time(sec.)
470.8 874.02
1.14
7.29 740.62 743.26
0.31
0
100
200 300 Time(sec.)
0.03 380.38 382.35
0.20
60
40 20
200
400
0
600 800 1,000 1,200 1,400 Time(sec.)
0.01 1313.26 1417.1
0.10
Gap
80
60
Gap
80
60
0
40 20
0
200
400
0
600 800 1,000 1,200 1,400 Time(sec.)
3.6 1123.98 1272.91
0.21
200
400
0.0 1824.41 1909.51
0.26
80
80
80
60
60
60
60
0
0
500
1,000 Time(sec.)
1,500
0
Gap
80 Gap
100
20
40 20
0
500
1,000 Time(sec.)
13
1,500
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1.00
0
100
200 300 Time(sec.)
400
1.00
0
200
400
600 800 1,000 1,200 1,400 Time(sec.)
17.46 1250.58 1473.57
100
40
1.00
40
0
600 800 1,000 1,200 1,400 Time(sec.)
100
20
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20 0
100
40
0
148.18 1502.84 1504.68
80
0
1.00
20
0
400
60
20
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
46.61 433.98 784.68
80
40
0
40
20
0
400
1.00
0.4
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40 Gap
Gap
40
2
0.2 0
Gap
0
Gap
0.8
Gap
1
Gap
1
0.4
1.5
3600.0 3600.0 3600.0
1
0.2
1 Time(sec.)
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
1
0.4
0.5
20
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
30.58 2548.98 3600.0
20 Gap
Gap
10
1.00
0.4
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
334.67 1552.74 1552.82
20
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.2 0
Gap
0
Gap
1 0.8 Gap
1 0.8
0
1.00
463.8 585.43
1
0.2
400
20
0
0.8
0.4
200 300 Time(sec.)
0.84 219.48
60
0.5
0
0.12
60
0
100
10
0
60
20
0
6.88 714.09 3600.0
Gap
Gap
Gap
0
Gap
Gap Gap Gap
1.68
20
10
0
3600.0 3600.0 3600.0
Gap
400
20
10
155.61 541.74 541.75
Gap
200 300 Time(sec.)
20
0
3600.0 3600.0 3600.0
Gap
100
1.56 1437.89 3600.0
10
0
pg5 34
10 0
0
20
40
opm2-z7-s2
0.65
10 0
400
1.00
20
30
0
ns1830653
200 300 Time(sec.)
P (tmax )
237.76 293.44 328.04
30
20
ns1758913
100
topt tsolved
30
0
ns1688347
0
3.57 1099.34 3459.53
20
ns1208400
1.20
20
10 0
400
338.6 401.51
t1
30
0.01
Gap
noswot
0
Heur OFF
topt tsolved P (tmax )
285.64
Gap
10
1.78 3598.52 3600.0
Gap
newdano
1.77
t1
20 Gap
Gap
20
0
R&P
topt tsolved P (tmax )
Gap
topt tsolved P (tmax )
Gap
netdiversion
t1
Gap
Problem Name
1.00
40 20
0
500
1,000 Time(sec.)
1,500
0
0
500
1,000 Time(sec.)
1,500
DEF t1
pigeon-10
0.0 152.42 3600.0
NoR&P
topt tsolved P (tmax ) 4.84
t1
60
60
60
40
20 0
1.22
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
26.38 108.41 3600.0
1.08
28.73
20
10
10
0
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
6.09
61.76
0.84
0.04
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
11.06
81.57
1.04
0.06
Gap
20
Gap
20
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
10.52
54.15
0.87
80
80
60
60
60
40 20
0
0
0
0
2
4
6 Time(sec.)
8
10
207.1 303.18
0.05
0
2
4
6 Time(sec.)
8
10
12.31 260.18 361.65
0.05
Gap
80
60
Gap
80 Gap
100
20
7.38
2
4
6 Time(sec.)
8
611.5 679.32
0.23
60
60
100
200
300 400 Time(sec.)
500
37.42 213.09
600
0.11
0.0
0
100
200
300 400 Time(sec.)
500
37.12 251.08
0
600
0.08
0.02
10
5
0
0
0
20
40 60 Time(sec.)
80
0.43
11.19
200
300 400 Time(sec.)
500
600
0.13
0
20
40 60 Time(sec.)
80
65.31 283.58
0.74
0.01
4.63
20
40 60 Time(sec.)
0
80
71.61 209.67
0.48
16.27
60
1.2
0.08
10
20 Time(sec.)
30
76.76 115.69
40
0.09
0
10
20 Time(sec.)
30
0.78
8.05 101.26 113.47
0.13
20
40 60 Time(sec.)
0.01 1893.13 2808.4
80
0.07
20 0
100
0
20
40 60 Time(sec.)
5.87 1182.11 1925.61
80
0.13
Gap
40
36.61
10
20 Time(sec.)
30
0
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
14
500
600
1.00
20
40 60 Time(sec.)
80
1.00
0
20
40
60 80 100 120 140 160 Time(sec.)
67.8
1.00
20
92.94
40
0.19
0
10
20 Time(sec.)
30
87.99 102.99 113.03
40
1.00
40
0
20
40 60 Time(sec.)
80
0.08
20 0
100
0
20
40 60 Time(sec.)
33.42 2738.63 3600.0
80
100
1.00
60
40 20
0
41.35
0 0
60
40 20
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0.26
20
20 0
56.33
0.0 1942.44 3600.0
60
60
300 400 Time(sec.)
40
50.96
0
100
200
40
0
60 80 100 120 140 160 Time(sec.)
40 Gap
Gap 0
40
20
1.06
40
20
20
38.9
0
40
100
20 0
40
20 0
0
40
0
66.71
Gap
Gap 20
1.06
35.66
0
60 80 100 120 140 160 Time(sec.)
40
40
0
40
Gap
0.08
20
Gap
59.2
0
Gap
24.98
0
60 80 100 120 140 160 Time(sec.)
40 20
Gap
0.77
40
Gap
60
Gap
60
Gap
60
20
0
35.81 206.25 80
0
1.00
5 0
80
20
10
10
80
40
8
92.13 223.25
80
40
6 Time(sec.)
15
10
5 0
100
31.31 210.66
5
4
40
0 0
15 Gap
15
10
2
20
20
Gap
0 0
Gap
60
Gap
60
Gap
80
40
0
278.95 380.74 453.91
80
20
1.00
40
0
10
80
40
63.14
20 0
80
40
1.00
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
7.9
100
40
0
7.13
100
20
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
10
0
100
40
0
52.25 3600.0
20
Gap
Gap
1.20
40 20
0
30
0.0 160.41 245.61
Gap
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
30
0.01
Gap
0
30
7.43
Gap
20
26.69 138.89 3600.0
Gap
Gap Gap
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
40
30
0.06
Gap
40
Gap
60
Gap
80
0
rmine6
1.00
100
20
rmatr100-p5
31.44 3600.0
80
15
rmatr100-p10
P (tmax )
31.44
100
0
reblock67
3.60
topt tsolved
80
20
ran16x16
0.0 113.37 3600.0
t1
100
23.65 171.43 3600.0
Gap
rail507
Heur OFF
topt tsolved P (tmax )
80
0
qiu
6.94
t1
100
20
pw-myciel4
R&P
topt tsolved P (tmax )
218.41 218.41 3600.0
Gap
Gap
Problem Name
40 20 0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
DEF
NoR&P
22.77 117.16 139.87
0.55
t1 28.92
20
0
0
0
50
100 150 Time(sec.)
200
485.0 1626.56
250
0.24
200
0
250
20
0.20
0.1
0 0
50
100 150 Time(sec.)
200
313.1 1415.92
250
0.34
100
200
300 400 Time(sec.)
500
600
0.48
0
100
200
300 400 Time(sec.)
500
3.61 245.47 3600.0
0
600
0.74
0.44
100
200
300 400 Time(sec.)
500
600
2.34
Gap
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
285.96 625.36 1244.68
0.66
277.48 1086.96 1660.38
0.88
60
60
60
60
40
20
20
0
0
0
200
400 600 Time(sec.)
800
1,000
0.46
0
200
400 600 Time(sec.)
800
10.69 2626.06 3600.0
1,000
0.29
Gap
80
Gap
100
80
Gap
100
80
40
200
400 600 Time(sec.)
800
2.02 2871.37 3600.0
0.56
60
60
40
20
20
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
7.2 231.91 286.09
0.52
7.18
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
60.19 163.48
0.47
0
0.16
50
100 150 Time(sec.)
0
200
1.28 3600.0
0.44
0
30.85 108.43
0.61
16.79
50
100 150 Time(sec.)
0
200
276.61 702.24 1596.85
0.08
50
100 150 Time(sec.)
0
200
0.16 3600.0 3600.0
0.44
60
60
0
0.01
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20.01
48.35
0.79
Gap
60 Gap
60 Gap
80
20
40
3.24
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
3.24
3.77
0.59
0.02
0
13.02
43.76
0.79
5.42
60
60
60
60
20
0
0 0
5
10 Time(sec.)
15
1.16 212.69 591.87
20
0.44
Gap
80
Gap
80
40
40
5
10 Time(sec.)
15
1.26 102.86 529.93
0
20
0.37
5
10 Time(sec.)
15
11.09 194.58 515.85
20
1.01
14.09
30
30
30
30
20
10
10
10
0
0
0
50
100 150 Time(sec.)
200
0
50
100 150 Time(sec.)
15
200
Gap
40
Gap
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
140.4
1.00
0
50
100 150 Time(sec.)
200
1.00
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
12.16
1.00
0
5
10 Time(sec.)
15
26.16 439.05
40
20
1.00
40
0 0
40
20
1,000
20
20 0
0
11.74
80
20
800
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
80
40
400 600 Time(sec.)
20
20 0
200
3600.0 3600.0 3600.0
80
40
1.00
10
0
80
20
0
40.38
80
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20
10
0
0
40
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
Gap 10
1.00
20 0
20
Gap
20
Gap 0
10.09
600
Gap
0
Gap
60 Gap
60
Gap
80
40
500
209.69 2312.39 3600.0
80
0
300 400 Time(sec.)
40
0
1,000
80
20
200
20 0
80
40
100
267.51 950.24 2374.85
100
0
0
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
80
20
1.00
5 0
100
40
981.9
250
15
5 0
200
12.64 484.77 3600.0
15
10
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
100 150 Time(sec.)
40
0 0
16.59 572.78 3600.0
5 0
50
20
20
15
10
40
Gap
0 0
Gap
Gap
40
0
13.84 384.62
60
Gap
Gap Gap Gap
100 150 Time(sec.)
60
2.04 1153.66 3600.0
Gap
50
20
303.35 1417.32 1419.37
Gap
0
40
60
40
3.63 479.02 3600.0
Gap
40
60
10
timtab1
1.00
60
80
20
tanglegram2
255.2 259.82
80
0
tanglegram1
P (tmax )
30.15
80
5
sp98ir
0.67
topt tsolved
20
3.37 646.23 1704.96
15
sp98ic
29.43 132.75 140.52
t1
80
0
satellites1-25
Heur OFF
topt tsolved P (tmax )
60
40
20
20
roll3000
0.57
Gap
Gap
40
0.09
Gap
rococoC10-001000
96.07 108.68
t1
60
Gap
Gap
60
R&P
topt tsolved P (tmax )
Gap
topt tsolved P (tmax )
Gap
rocII-4-11
t1
Gap
Problem Name
20
1.00
20 10
0
50
100 150 Time(sec.)
200
0
0
50
100 150 Time(sec.)
200
DEF
NoR&P
topt tsolved P (tmax )
172.12 825.49 1695.51
0.72
t1
166.6 805.37 1051.57
0.96
topt tsolved
242.79 1159.38 1438.29
3
3
3
2
2
2
1 0
200
400 600 Time(sec.)
0
800
0.84
Gap
3
Gap
4
1 0
200
400 600 Time(sec.)
609.93 2635.11 3600.0
0
800
0.84
200
400 600 Time(sec.)
3600.0 3600.0 3600.0
1.00
60
60
60
0.34
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
49.47 3600.0
0.42
0.1
Gap
20
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
43.81 3600.0
0.24
0.33
40
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
0
0.45
Table 2: The results for every instance
16
800
1.00
15.05
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
54.43 3600.0
1.00
40
20
0
400 600 Time(sec.)
40
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
62.66 3600.0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
200
20 0
40
20
0
40 20
20
Gap
0
Gap
60
Gap
80
40
0
3600.0 3600.0 3600.0
80
20
1.00
2
0
800
80
40
P (tmax )
1 0
80 Gap
Gap
231.51 935.35 1333.29
t1
4
698.86 2842.75 3600.0
Gap
Heur OFF
topt tsolved P (tmax )
4
0
zib54-UUE
0.69
t1
4
1
vpphard
R&P
topt tsolved P (tmax )
Gap
Gap
unitcal 7
t1
Gap
Problem Name
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
20
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 Time(sec.)
