------------------------------------------------------------
                        α=3.0 —— berlin52 —— 52 nodes       
------------------------------------------------------------

——————— Mon, 29 Jan 2024 00:54:32 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
ILP ——— 53 hubs —— 1—22—31—18—3—17—21—42—7—2—30—23—20—50—29—16—46—44—34—35—36—39—40—37—38—48—24—5—15—6—4—25—12—28—27—26—47—13—14—52—11—51—33—43—10—9—8—41—19—45—32—49—53
total time           108.66               gap                  0.0                  
blossom time         0                    nb blossom           0.0                  
LB <= UB             26501.94<=26501.94   subtour              26.0                 
connectivity cuts    80.0                 uc strategy          4                    
uc tolerance         0.01                 tildeV               2:52                 
2-opt strategy       0                                                              
explored_nodes       374                  n lazycuts edges     0.0                  
F                    183.0                post procedure       true                 
o_i                  0                                                                                                                                                  
Found solution
RING NODES
1—22—31—18—3—17—21—42—7—2—30—23—20—50—29—16—46—44—34—35—36—39—40—37—38—48—24—5—15—6—4—25—12—28—27—26—47—13—14—52—11—51—33—43—10—9—8—41—19—45—32—49—53

RING
1  --- 22 | cost [138.0]
2  ---  7 | cost [135.0]
2  --- 30 | cost [1170.0]
3  --- 17 | cost [651.0]
3  --- 18 | cost [405.0]
4  ---  6 | cost [210.0]
4  --- 25 | cost [327.0]
5  --- 15 | cost [75.0]
5  --- 24 | cost [96.0]
6  --- 15 | cost [120.0]
7  --- 42 | cost [228.0]
8  ---  9 | cost [549.0]
8  --- 41 | cost [192.0]
9  --- 10 | cost [249.0]
10 --- 43 | cost [924.0]
11 --- 51 | cost [855.0]
11 --- 52 | cost [1197.0]
12 --- 25 | cost [735.0]
12 --- 28 | cost [546.0]
13 --- 14 | cost [618.0]
13 --- 47 | cost [972.0]
14 --- 52 | cost [957.0]
16 --- 29 | cost [603.0]
16 --- 46 | cost [468.0]
17 --- 21 | cost [759.0]
18 --- 31 | cost [240.0]
19 --- 41 | cost [276.0]
19 --- 45 | cost [225.0]
20 --- 23 | cost [282.0]
20 --- 50 | cost [105.0]
21 --- 42 | cost [870.0]
22 --- 31 | cost [312.0]
23 --- 30 | cost [537.0]
24 --- 48 | cost [48.0]
26 --- 27 | cost [378.0]
26 --- 47 | cost [558.0]
27 --- 28 | cost [330.0]
29 --- 50 | cost [573.0]
32 --- 45 | cost [453.0]
32 --- 49 | cost [150.0]
33 --- 43 | cost [1095.0]
33 --- 51 | cost [1425.0]
34 --- 35 | cost [63.0]
34 --- 44 | cost [240.0]
35 --- 36 | cost [45.0]
36 --- 39 | cost [129.0]
37 --- 38 | cost [129.0]
37 --- 40 | cost [123.0]
38 --- 48 | cost [147.0]
39 --- 40 | cost [129.0]
44 --- 46 | cost [393.0]
49 --- 53 | cost [192.0]

STAR

BACKUP RING
1  --- 31 | cost [4.38]
2  --- 23 | cost [15.3]
2  --- 42 | cost [3.09]
3  --- 21 | cost [8.67]
3  --- 31 | cost [6.2700000000000005]
4  --- 12 | cost [8.82]
4  --- 15 | cost [3.0]
5  ---  6 | cost [1.05]
5  --- 48 | cost [1.41]
6  --- 25 | cost [3.72]
7  --- 21 | cost [10.86]
7  --- 30 | cost [11.58]
8  --- 10 | cost [5.4]
8  --- 19 | cost [3.7800000000000002]
9  --- 41 | cost [7.17]
9  --- 43 | cost [11.700000000000001]
10 --- 33 | cost [15.030000000000001]
11 --- 14 | cost [18.6]
11 --- 33 | cost [21.18]
12 --- 27 | cost [8.49]
13 --- 26 | cost [7.62]
13 --- 52 | cost [8.370000000000001]
14 --- 47 | cost [10.950000000000001]
15 --- 24 | cost [1.68]
16 --- 44 | cost [3.96]
16 --- 50 | cost [3.9]
17 --- 18 | cost [8.16]
17 --- 42 | cost [12.24]
18 --- 22 | cost [3.48]
19 --- 32 | cost [6.6000000000000005]
20 --- 29 | cost [6.3]
20 --- 30 | cost [5.67]
23 --- 50 | cost [3.81]
24 --- 38 | cost [1.35]
25 --- 28 | cost [9.870000000000001]
26 --- 28 | cost [4.7700000000000005]
27 --- 47 | cost [8.76]
29 --- 46 | cost [10.47]
32 --- 53 | cost [2.73]
34 --- 36 | cost [1.02]
34 --- 46 | cost [4.83]
35 --- 39 | cost [1.59]
35 --- 44 | cost [2.88]
36 --- 40 | cost [2.5500000000000003]
37 --- 39 | cost [1.68]
37 --- 48 | cost [1.8]
38 --- 40 | cost [1.05]
41 --- 45 | cost [4.98]
43 --- 51 | cost [15.120000000000001]
45 --- 49 | cost [5.88]
51 --- 52 | cost [18.75]

BACKUP STAR

B = 21.18
i★ = 11
j★ = 51
k★ = 33

       
——————— Mon, 29 Jan 2024 00:56:09 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
BD  —— 53 hubs —— 1-49-32-45-19-41-8-9-10-43-33-51-11-52-14-13-47-26-27-28-12-25-4-6-15-5-24-48-38-37-40-39-36-35-34-44-46-16-29-50-20-23-30-2-7-42-21-17-3-18-31-22-53-1
total time           63.82                gap                  0.0                  
blossom time         0                    nb blossom           0.0                  
LB <= UB             26501.94<=26501.94   Master/SP costs      22626/3875.94        
Master time          57.8                 SP time              6.03                 
subtour              2065.0               connectivity cuts    1764.0               
uc strategy          4                    uc tolerance         0.01                 
opt. cuts            398.0                SP method            poly                 
tildeV               2:52                 inst transformation  2                    
2-opt strategy       0                                                              
explored nodes       6231                 
F                    183.0                
                                          o_i                  0                                                              
                                          
Found solution
RING NODES
1—49—32—45—19—41—8—9—10—43—33—51—11—52—14—13—47—26—27—28—12—25—4—6—15—5—24—48—38—37—40—39—36—35—34—44—46—16—29—50—20—23—30—2—7—42—21—17—3—18—31—22—53

RING
1  --- 49 | cost [192.0]
1  --- 53 | cost [0.0]
2  ---  7 | cost [135.0]
2  --- 30 | cost [1170.0]
3  --- 17 | cost [651.0]
3  --- 18 | cost [405.0]
4  ---  6 | cost [210.0]
4  --- 25 | cost [327.0]
5  --- 15 | cost [75.0]
5  --- 24 | cost [96.0]
6  --- 15 | cost [120.0]
7  --- 42 | cost [228.0]
8  ---  9 | cost [549.0]
8  --- 41 | cost [192.0]
9  --- 10 | cost [249.0]
10 --- 43 | cost [924.0]
11 --- 51 | cost [855.0]
11 --- 52 | cost [1197.0]
12 --- 25 | cost [735.0]
12 --- 28 | cost [546.0]
13 --- 14 | cost [618.0]
13 --- 47 | cost [972.0]
14 --- 52 | cost [957.0]
16 --- 29 | cost [603.0]
16 --- 46 | cost [468.0]
17 --- 21 | cost [759.0]
18 --- 31 | cost [240.0]
19 --- 41 | cost [276.0]
19 --- 45 | cost [225.0]
20 --- 23 | cost [282.0]
20 --- 50 | cost [105.0]
21 --- 42 | cost [870.0]
22 --- 31 | cost [312.0]
22 --- 53 | cost [138.0]
23 --- 30 | cost [537.0]
24 --- 48 | cost [48.0]
26 --- 27 | cost [378.0]
26 --- 47 | cost [558.0]
27 --- 28 | cost [330.0]
29 --- 50 | cost [573.0]
32 --- 45 | cost [453.0]
32 --- 49 | cost [150.0]
33 --- 43 | cost [1095.0]
33 --- 51 | cost [1425.0]
34 --- 35 | cost [63.0]
34 --- 44 | cost [240.0]
35 --- 36 | cost [45.0]
36 --- 39 | cost [129.0]
37 --- 38 | cost [129.0]
37 --- 40 | cost [123.0]
38 --- 48 | cost [147.0]
39 --- 40 | cost [129.0]
44 --- 46 | cost [393.0]

STAR

BACKUP RING
1  --- 32 | cost [2.73]
2  --- 23 | cost [15.3]
2  --- 42 | cost [3.09]
3  --- 21 | cost [8.67]
3  --- 31 | cost [6.2700000000000005]
4  --- 12 | cost [8.82]
4  --- 15 | cost [3.0]
5  ---  6 | cost [1.05]
5  --- 48 | cost [1.41]
6  --- 25 | cost [3.72]
7  --- 21 | cost [10.86]
7  --- 30 | cost [11.58]
8  --- 10 | cost [5.4]
8  --- 19 | cost [3.7800000000000002]
9  --- 41 | cost [7.17]
9  --- 43 | cost [11.700000000000001]
10 --- 33 | cost [15.030000000000001]
11 --- 14 | cost [18.6]
11 --- 33 | cost [21.18]
12 --- 27 | cost [8.49]
13 --- 26 | cost [7.62]
13 --- 52 | cost [8.370000000000001]
14 --- 47 | cost [10.950000000000001]
15 --- 24 | cost [1.68]
16 --- 44 | cost [3.96]
16 --- 50 | cost [3.9]
17 --- 18 | cost [8.16]
17 --- 42 | cost [12.24]
18 --- 22 | cost [3.48]
19 --- 32 | cost [6.6000000000000005]
20 --- 29 | cost [6.3]
20 --- 30 | cost [5.67]
23 --- 50 | cost [3.81]
24 --- 38 | cost [1.35]
25 --- 28 | cost [9.870000000000001]
26 --- 28 | cost [4.7700000000000005]
27 --- 47 | cost [8.76]
29 --- 46 | cost [10.47]
31 --- 53 | cost [4.38]
34 --- 36 | cost [1.02]
34 --- 46 | cost [4.83]
35 --- 39 | cost [1.59]
35 --- 44 | cost [2.88]
36 --- 40 | cost [2.5500000000000003]
37 --- 39 | cost [1.68]
37 --- 48 | cost [1.8]
38 --- 40 | cost [1.05]
41 --- 45 | cost [4.98]
43 --- 51 | cost [15.120000000000001]
45 --- 49 | cost [5.88]
51 --- 52 | cost [18.75]

BACKUP STAR

B = 21.18
i★ = 11
j★ = 51
k★ = 33

       
——————— Fri, 22 Mar 2024 16:00:52 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
BD  —— 53 hubs —— 1-22-31-18-3-17-21-42-7-2-30-23-20-50-29-16-46-44-34-35-36-39-40-37-38-48-24-5-15-6-4-25-12-28-27-26-47-13-14-52-11-51-33-43-10-9-8-41-19-45-32-49-53-1
total time           73.64                gap                  0.0                  
blossom time         0                    nb blossom           0.0                  
LB <= UB             26501.94<=26501.94   Master/SP costs      22626/3875.94        
Master time          68.66                SP time              4.98                 
subtour              1433.0               connectivity cuts    2000.0               
uc strategy          4                    uc tolerance         0.01                 
opt. cuts            286.0                SP method            poly                 
tildeV               2:52                 inst transformation  2                    
2-opt strategy       0                                                              
explored nodes       9400                 
F                    183.0                
                                          o_i                  0                                                              
                                          
Found solution
RING NODES
1—22—31—18—3—17—21—42—7—2—30—23—20—50—29—16—46—44—34—35—36—39—40—37—38—48—24—5—15—6—4—25—12—28—27—26—47—13—14—52—11—51—33—43—10—9—8—41—19—45—32—49—53

RING
1  --- 22 | cost [138.0]
1  --- 53 | cost [0.0]
2  ---  7 | cost [135.0]
2  --- 30 | cost [1170.0]
3  --- 17 | cost [651.0]
3  --- 18 | cost [405.0]
4  ---  6 | cost [210.0]
4  --- 25 | cost [327.0]
5  --- 15 | cost [75.0]
5  --- 24 | cost [96.0]
6  --- 15 | cost [120.0]
7  --- 42 | cost [228.0]
8  ---  9 | cost [549.0]
8  --- 41 | cost [192.0]
9  --- 10 | cost [249.0]
10 --- 43 | cost [924.0]
11 --- 51 | cost [855.0]
11 --- 52 | cost [1197.0]
12 --- 25 | cost [735.0]
12 --- 28 | cost [546.0]
13 --- 14 | cost [618.0]
13 --- 47 | cost [972.0]
14 --- 52 | cost [957.0]
16 --- 29 | cost [603.0]
16 --- 46 | cost [468.0]
17 --- 21 | cost [759.0]
18 --- 31 | cost [240.0]
19 --- 41 | cost [276.0]
19 --- 45 | cost [225.0]
20 --- 23 | cost [282.0]
20 --- 50 | cost [105.0]
21 --- 42 | cost [870.0]
22 --- 31 | cost [312.0]
23 --- 30 | cost [537.0]
24 --- 48 | cost [48.0]
26 --- 27 | cost [378.0]
26 --- 47 | cost [558.0]
27 --- 28 | cost [330.0]
29 --- 50 | cost [573.0]
32 --- 45 | cost [453.0]
32 --- 49 | cost [150.0]
33 --- 43 | cost [1095.0]
33 --- 51 | cost [1425.0]
34 --- 35 | cost [63.0]
34 --- 44 | cost [240.0]
35 --- 36 | cost [45.0]
36 --- 39 | cost [129.0]
37 --- 38 | cost [129.0]
37 --- 40 | cost [123.0]
38 --- 48 | cost [147.0]
39 --- 40 | cost [129.0]
44 --- 46 | cost [393.0]
49 --- 53 | cost [192.0]

STAR

BACKUP RING
1  --- 31 | cost [4.38]
2  --- 23 | cost [15.3]
2  --- 42 | cost [3.09]
3  --- 21 | cost [8.67]
3  --- 31 | cost [6.2700000000000005]
4  --- 12 | cost [8.82]
4  --- 15 | cost [3.0]
5  ---  6 | cost [1.05]
5  --- 48 | cost [1.41]
6  --- 25 | cost [3.72]
7  --- 21 | cost [10.86]
7  --- 30 | cost [11.58]
8  --- 10 | cost [5.4]
8  --- 19 | cost [3.7800000000000002]
9  --- 41 | cost [7.17]
9  --- 43 | cost [11.700000000000001]
10 --- 33 | cost [15.030000000000001]
11 --- 14 | cost [18.6]
11 --- 33 | cost [21.18]
12 --- 27 | cost [8.49]
13 --- 26 | cost [7.62]
13 --- 52 | cost [8.370000000000001]
14 --- 47 | cost [10.950000000000001]
15 --- 24 | cost [1.68]
16 --- 44 | cost [3.96]
16 --- 50 | cost [3.9]
17 --- 18 | cost [8.16]
17 --- 42 | cost [12.24]
18 --- 22 | cost [3.48]
19 --- 32 | cost [6.6000000000000005]
20 --- 29 | cost [6.3]
20 --- 30 | cost [5.67]
23 --- 50 | cost [3.81]
24 --- 38 | cost [1.35]
25 --- 28 | cost [9.870000000000001]
26 --- 28 | cost [4.7700000000000005]
27 --- 47 | cost [8.76]
29 --- 46 | cost [10.47]
32 --- 53 | cost [2.73]
34 --- 36 | cost [1.02]
34 --- 46 | cost [4.83]
35 --- 39 | cost [1.59]
35 --- 44 | cost [2.88]
36 --- 40 | cost [2.5500000000000003]
37 --- 39 | cost [1.68]
37 --- 48 | cost [1.8]
38 --- 40 | cost [1.05]
41 --- 45 | cost [4.98]
43 --- 51 | cost [15.120000000000001]
45 --- 49 | cost [5.88]
51 --- 52 | cost [18.75]

BACKUP STAR

B = 21.18
i★ = 11
j★ = 51
k★ = 33

       
——————— Fri, 22 Mar 2024 16:03:05 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
BD  —— 53 hubs —— 1-22-31-18-3-17-21-42-7-2-30-23-20-50-29-16-46-44-34-35-36-39-40-37-38-48-24-5-15-6-4-25-12-28-27-26-47-13-14-52-11-51-33-43-10-9-8-41-19-45-32-49-53-1
total time           61.51                gap                  0.0                  
blossom time         0                    nb blossom           0.0                  
LB <= UB             26501.94<=26501.94   Master/SP costs      22626/3875.94        
Master time          57.08                SP time              4.43                 
subtour              1602.0               connectivity cuts    1393.0               
uc strategy          4                    uc tolerance         0.01                 
opt. cuts            285.0                SP method            poly                 
tildeV               2:52                 inst transformation  2                    
2-opt strategy       0                                                              
explored nodes       4977                 
F                    183.0                
                                          o_i                  0                                                              
                                          
Found solution
RING NODES
1—22—31—18—3—17—21—42—7—2—30—23—20—50—29—16—46—44—34—35—36—39—40—37—38—48—24—5—15—6—4—25—12—28—27—26—47—13—14—52—11—51—33—43—10—9—8—41—19—45—32—49—53

RING
1  --- 22 | cost [138.0]
1  --- 53 | cost [0.0]
2  ---  7 | cost [135.0]
2  --- 30 | cost [1170.0]
3  --- 17 | cost [651.0]
3  --- 18 | cost [405.0]
4  ---  6 | cost [210.0]
4  --- 25 | cost [327.0]
5  --- 15 | cost [75.0]
5  --- 24 | cost [96.0]
6  --- 15 | cost [120.0]
7  --- 42 | cost [228.0]
8  ---  9 | cost [549.0]
8  --- 41 | cost [192.0]
9  --- 10 | cost [249.0]
10 --- 43 | cost [924.0]
11 --- 51 | cost [855.0]
11 --- 52 | cost [1197.0]
12 --- 25 | cost [735.0]
12 --- 28 | cost [546.0]
13 --- 14 | cost [618.0]
13 --- 47 | cost [972.0]
14 --- 52 | cost [957.0]
16 --- 29 | cost [603.0]
16 --- 46 | cost [468.0]
17 --- 21 | cost [759.0]
18 --- 31 | cost [240.0]
19 --- 41 | cost [276.0]
19 --- 45 | cost [225.0]
20 --- 23 | cost [282.0]
20 --- 50 | cost [105.0]
21 --- 42 | cost [870.0]
22 --- 31 | cost [312.0]
23 --- 30 | cost [537.0]
24 --- 48 | cost [48.0]
26 --- 27 | cost [378.0]
26 --- 47 | cost [558.0]
27 --- 28 | cost [330.0]
29 --- 50 | cost [573.0]
32 --- 45 | cost [453.0]
32 --- 49 | cost [150.0]
33 --- 43 | cost [1095.0]
33 --- 51 | cost [1425.0]
34 --- 35 | cost [63.0]
34 --- 44 | cost [240.0]
35 --- 36 | cost [45.0]
36 --- 39 | cost [129.0]
37 --- 38 | cost [129.0]
37 --- 40 | cost [123.0]
38 --- 48 | cost [147.0]
39 --- 40 | cost [129.0]
44 --- 46 | cost [393.0]
49 --- 53 | cost [192.0]

STAR

BACKUP RING
1  --- 31 | cost [4.38]
2  --- 23 | cost [15.3]
2  --- 42 | cost [3.09]
3  --- 21 | cost [8.67]
3  --- 31 | cost [6.2700000000000005]
4  --- 12 | cost [8.82]
4  --- 15 | cost [3.0]
5  ---  6 | cost [1.05]
5  --- 48 | cost [1.41]
6  --- 25 | cost [3.72]
7  --- 21 | cost [10.86]
7  --- 30 | cost [11.58]
8  --- 10 | cost [5.4]
8  --- 19 | cost [3.7800000000000002]
9  --- 41 | cost [7.17]
9  --- 43 | cost [11.700000000000001]
10 --- 33 | cost [15.030000000000001]
11 --- 14 | cost [18.6]
11 --- 33 | cost [21.18]
12 --- 27 | cost [8.49]
13 --- 26 | cost [7.62]
13 --- 52 | cost [8.370000000000001]
14 --- 47 | cost [10.950000000000001]
15 --- 24 | cost [1.68]
16 --- 44 | cost [3.96]
16 --- 50 | cost [3.9]
17 --- 18 | cost [8.16]
17 --- 42 | cost [12.24]
18 --- 22 | cost [3.48]
19 --- 32 | cost [6.6000000000000005]
20 --- 29 | cost [6.3]
20 --- 30 | cost [5.67]
23 --- 50 | cost [3.81]
24 --- 38 | cost [1.35]
25 --- 28 | cost [9.870000000000001]
26 --- 28 | cost [4.7700000000000005]
27 --- 47 | cost [8.76]
29 --- 46 | cost [10.47]
32 --- 53 | cost [2.73]
34 --- 36 | cost [1.02]
34 --- 46 | cost [4.83]
35 --- 39 | cost [1.59]
35 --- 44 | cost [2.88]
36 --- 40 | cost [2.5500000000000003]
37 --- 39 | cost [1.68]
37 --- 48 | cost [1.8]
38 --- 40 | cost [1.05]
41 --- 45 | cost [4.98]
43 --- 51 | cost [15.120000000000001]
45 --- 49 | cost [5.88]
51 --- 52 | cost [18.75]

BACKUP STAR

B = 21.18
i★ = 11
j★ = 51
k★ = 33

       
——————— Wed, 10 Apr 2024 10:32:33 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
       
——————— Wed, 10 Apr 2024 11:25:26 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
       
——————— Wed, 10 Apr 2024 11:38:17 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
       
——————— Wed, 10 Apr 2024 12:02:15 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
       
——————— Sat, 13 Apr 2024 12:53:57 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
       
——————— Sat, 13 Apr 2024 13:59:48 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false