------------------------------------------------------------
                        α=7.0 —— berlin52 —— 52 nodes       
------------------------------------------------------------

——————— Sat, 06 Jan 2024 20:40:58 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
ILP ——— 27 hubs —— 1—32—45—19—18—31—21—23—20—50—16—46—26—27—28—12—25—4—6—5—24—38—40—39—36—35—53
total time           3606.64 (TL 3600)    gap                  0.089                
blossom time         0                    nb blossom           0.0                  
LB <= UB             42215.01<=46327.09   subtour              261.0                
connectivity cuts    2000.0               uc strategy          4                    
uc tolerance         0.01                 tildeV               2:52                 
2-opt strategy       0                                                              
explored_nodes       4437                 n lazycuts edges     0.0                  
F                    183.0                post procedure       true                 
o_i                  0                                                                                                                                                  
Found solution
RING NODES
1—32—45—19—18—31—21—23—20—50—16—46—26—27—28—12—25—4—6—5—24—38—40—39—36—35—53

RING
1  --- 32 | cost [637.0]
1  --- 53 | cost [0.0]
4  ---  6 | cost [490.0]
4  --- 25 | cost [763.0]
5  ---  6 | cost [245.0]
5  --- 24 | cost [224.0]
12 --- 25 | cost [1715.0]
12 --- 28 | cost [1274.0]
16 --- 46 | cost [1092.0]
16 --- 50 | cost [910.0]
18 --- 19 | cost [1806.0]
18 --- 31 | cost [560.0]
19 --- 45 | cost [525.0]
20 --- 23 | cost [658.0]
20 --- 50 | cost [245.0]
21 --- 23 | cost [1309.0]
21 --- 31 | cost [1050.0]
24 --- 38 | cost [315.0]
26 --- 27 | cost [882.0]
26 --- 46 | cost [3178.0]
27 --- 28 | cost [770.0]
32 --- 45 | cost [1057.0]
35 --- 36 | cost [105.0]
35 --- 53 | cost [854.0]
36 --- 39 | cost [301.0]
38 --- 40 | cost [245.0]
39 --- 40 | cost [301.0]

STAR
2  --> 23 | cost [1530.0]
3  --> 18 | cost [405.0]
7  --> 21 | cost [1086.0]
8  --> 19 | cost [378.0]
9  --> 19 | cost [924.0]
10 --> 45 | cost [987.0]
11 --> 12 | cost [1161.0]
13 --> 27 | cost [555.0]
14 --> 27 | cost [1122.0]
15 -->  5 | cost [75.0]
17 --> 18 | cost [816.0]
22 -->  1 | cost [138.0]
29 --> 50 | cost [573.0]
30 --> 23 | cost [537.0]
33 -->  4 | cost [1551.0]
34 --> 35 | cost [63.0]
37 --> 40 | cost [123.0]
41 --> 19 | cost [276.0]
42 --> 21 | cost [870.0]
43 -->  4 | cost [735.0]
44 --> 35 | cost [288.0]
47 --> 27 | cost [876.0]
48 --> 24 | cost [48.0]
49 --> 32 | cost [150.0]
51 --> 28 | cost [1011.0]
52 --> 28 | cost [1542.0]

BACKUP RING
1  --- 45 | cost [16.8]
4  ---  5 | cost [7.28]
4  --- 12 | cost [20.580000000000002]
5  --- 38 | cost [3.5700000000000003]
6  --- 24 | cost [3.99]
6  --- 25 | cost [8.68]
12 --- 27 | cost [19.81]
16 --- 20 | cost [11.55]
16 --- 26 | cost [35.42]
18 --- 21 | cost [14.35]
18 --- 45 | cost [15.96]
19 --- 31 | cost [23.240000000000002]
19 --- 32 | cost [15.4]
20 --- 21 | cost [19.53]
23 --- 31 | cost [10.64]
23 --- 50 | cost [8.89]
24 --- 40 | cost [5.53]
25 --- 28 | cost [23.03]
26 --- 28 | cost [11.13]
27 --- 46 | cost [36.33]
35 --- 39 | cost [3.71]
36 --- 40 | cost [5.95]
36 --- 53 | cost [8.75]
38 --- 39 | cost [5.32]
46 --- 50 | cost [18.62]

BACKUP STAR
2  --> 21 | cost [11.76]
3  --> 19 | cost [6.21]
7  --> 23 | cost [14.73]
8  --> 45 | cost [5.61]
9  --> 45 | cost [10.83]
10 --> 19 | cost [8.73]
11 --> 27 | cost [12.51]
13 --> 26 | cost [7.62]
14 --> 26 | cost [11.88]
15 -->  6 | cost [1.2]
17 --> 21 | cost [7.59]
29 --> 16 | cost [6.03]
30 --> 20 | cost [5.67]
33 -->  6 | cost [17.04]
34 --> 36 | cost [1.02]
37 --> 38 | cost [1.29]
41 --> 45 | cost [4.98]
42 --> 23 | cost [12.450000000000001]
43 -->  6 | cost [7.8]
44 --> 36 | cost [3.33]
47 --> 26 | cost [5.58]
48 -->  5 | cost [1.41]
49 -->  1 | cost [1.92]
51 --> 12 | cost [5.64]
52 --> 27 | cost [12.780000000000001]

B = 38.230000000000004
i★ = 12
j★ = 28
k★ = 27

       
——————— Sat, 06 Jan 2024 21:41:30 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
BD  —— 26 hubs —— 1-32-45-19-18-31-21-23-20-50-16-46-26-27-28-12-4-6-5-24-38-40-39-36-35-53-1
total time           3603.11 (TL 3600)    gap                  0.162                
blossom time         0                    nb blossom           0.0                  
LB <= UB             39147.64<=46717.15   Master/SP costs      38290/8427.15        
Master time          3561.47              SP time              41.64                
subtour              12549.0              connectivity cuts    2000.0               
uc strategy          4                    uc tolerance         0.01                 
opt. cuts            4709.0               SP method            poly                 
tildeV               2:52                 inst transformation  2                    
2-opt strategy       0                                                              
explored nodes       190247               
F                    183.0                
                                          o_i                  0                                                              
                                          
Found solution
RING NODES
1—32—45—19—18—31—21—23—20—50—16—46—26—27—28—12—4—6—5—24—38—40—39—36—35—53

RING
1  --- 32 | cost [637.0]
4  ---  6 | cost [490.0]
4  --- 12 | cost [2058.0]
5  ---  6 | cost [245.0]
5  --- 24 | cost [224.0]
12 --- 28 | cost [1274.0]
16 --- 46 | cost [1092.0]
16 --- 50 | cost [910.0]
18 --- 19 | cost [1806.0]
18 --- 31 | cost [560.0]
19 --- 45 | cost [525.0]
20 --- 23 | cost [658.0]
20 --- 50 | cost [245.0]
21 --- 23 | cost [1309.0]
21 --- 31 | cost [1050.0]
24 --- 38 | cost [315.0]
26 --- 27 | cost [882.0]
26 --- 46 | cost [3178.0]
27 --- 28 | cost [770.0]
32 --- 45 | cost [1057.0]
35 --- 36 | cost [105.0]
35 --- 53 | cost [854.0]
36 --- 39 | cost [301.0]
38 --- 40 | cost [245.0]
39 --- 40 | cost [301.0]

STAR
2  --> 23 | cost [1530.0]
3  --> 18 | cost [405.0]
7  --> 21 | cost [1086.0]
8  --> 19 | cost [378.0]
9  --> 19 | cost [924.0]
10 --> 45 | cost [987.0]
11 --> 28 | cost [1254.0]
13 --> 27 | cost [555.0]
14 --> 27 | cost [1122.0]
15 -->  5 | cost [75.0]
17 --> 21 | cost [759.0]
22 -->  1 | cost [138.0]
25 -->  4 | cost [327.0]
29 --> 50 | cost [573.0]
30 --> 23 | cost [537.0]
33 -->  4 | cost [1551.0]
34 --> 35 | cost [63.0]
37 --> 40 | cost [123.0]
41 --> 19 | cost [276.0]
42 --> 21 | cost [870.0]
43 -->  6 | cost [780.0]
44 --> 35 | cost [288.0]
47 --> 26 | cost [558.0]
48 --> 24 | cost [48.0]
49 --> 32 | cost [150.0]
51 --> 12 | cost [564.0]
52 --> 27 | cost [1278.0]

BACKUP RING
1  --- 45 | cost [16.8]
4  ---  5 | cost [7.28]
4  --- 28 | cost [29.19]
5  --- 38 | cost [3.5700000000000003]
6  --- 12 | cost [24.43]
6  --- 24 | cost [3.99]
12 --- 27 | cost [19.81]
16 --- 20 | cost [11.55]
16 --- 26 | cost [35.42]
18 --- 21 | cost [14.35]
18 --- 45 | cost [15.96]
19 --- 31 | cost [23.240000000000002]
19 --- 32 | cost [15.4]
20 --- 21 | cost [19.53]
23 --- 31 | cost [10.64]
23 --- 50 | cost [8.89]
24 --- 40 | cost [5.53]
26 --- 28 | cost [11.13]
27 --- 46 | cost [36.33]
35 --- 39 | cost [3.71]
36 --- 40 | cost [5.95]
36 --- 53 | cost [8.75]
38 --- 39 | cost [5.32]
46 --- 50 | cost [18.62]

BACKUP STAR
2  --> 21 | cost [11.76]
3  --> 19 | cost [6.21]
7  --> 23 | cost [14.73]
8  --> 45 | cost [5.61]
9  --> 45 | cost [10.83]
10 --> 19 | cost [8.73]
11 --> 12 | cost [11.61]
13 --> 26 | cost [7.62]
14 --> 26 | cost [11.88]
15 -->  6 | cost [1.2]
17 --> 18 | cost [8.16]
25 -->  6 | cost [3.72]
29 --> 16 | cost [6.03]
30 --> 20 | cost [5.67]
33 -->  6 | cost [17.04]
34 --> 36 | cost [1.02]
37 --> 38 | cost [1.29]
41 --> 45 | cost [4.98]
42 --> 23 | cost [12.450000000000001]
43 -->  4 | cost [7.3500000000000005]
44 --> 36 | cost [3.33]
47 --> 27 | cost [8.76]
48 -->  5 | cost [1.41]
49 -->  1 | cost [1.92]
51 --> 28 | cost [10.11]
52 --> 28 | cost [15.42]

B = 46.050000000000004
i★ = 26
j★ = 27
k★ = 28

       
——————— Sun, 07 Jan 2024 17:41:13 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = true
ILP ——— 27 hubs —— 1—35—36—39—40—38—24—5—6—4—25—12—28—27—26—46—16—50—20—23—21—31—18—19—45—32—53
total time           3609.62 (TL 3600)    gap                  0.086                
blossom time         7.33                 nb blossom           1573.0               
LB <= UB             42357.17<=46327.09   subtour              263.0                
connectivity cuts    2000.0               uc strategy          4                    
uc tolerance         0.01                 tildeV               2:52                 
2-opt strategy       0                                                              
explored_nodes       4307                 n lazycuts edges     0.0                  
F                    183.0                post procedure       true                 
o_i                  0                                                                                                                                                  
Found solution
RING NODES
1—35—36—39—40—38—24—5—6—4—25—12—28—27—26—46—16—50—20—23—21—31—18—19—45—32—53

RING
1  --- 35 | cost [854.0]
1  --- 53 | cost [0.0]
4  ---  6 | cost [490.0]
4  --- 25 | cost [763.0]
5  ---  6 | cost [245.0]
5  --- 24 | cost [224.0]
12 --- 25 | cost [1715.0]
12 --- 28 | cost [1274.0]
16 --- 46 | cost [1092.0]
16 --- 50 | cost [910.0]
18 --- 19 | cost [1806.0]
18 --- 31 | cost [560.0]
19 --- 45 | cost [525.0]
20 --- 23 | cost [658.0]
20 --- 50 | cost [245.0]
21 --- 23 | cost [1309.0]
21 --- 31 | cost [1050.0]
24 --- 38 | cost [315.0]
26 --- 27 | cost [882.0]
26 --- 46 | cost [3178.0]
27 --- 28 | cost [770.0]
32 --- 45 | cost [1057.0]
32 --- 53 | cost [637.0]
35 --- 36 | cost [105.0]
36 --- 39 | cost [301.0]
38 --- 40 | cost [245.0]
39 --- 40 | cost [301.0]

STAR
2  --> 23 | cost [1530.0]
3  --> 18 | cost [405.0]
7  --> 21 | cost [1086.0]
8  --> 19 | cost [378.0]
9  --> 19 | cost [924.0]
10 --> 45 | cost [987.0]
11 --> 12 | cost [1161.0]
13 --> 27 | cost [555.0]
14 --> 27 | cost [1122.0]
15 -->  5 | cost [75.0]
17 --> 18 | cost [816.0]
22 -->  1 | cost [138.0]
29 --> 50 | cost [573.0]
30 --> 23 | cost [537.0]
33 -->  4 | cost [1551.0]
34 --> 35 | cost [63.0]
37 --> 40 | cost [123.0]
41 --> 19 | cost [276.0]
42 --> 21 | cost [870.0]
43 -->  4 | cost [735.0]
44 --> 35 | cost [288.0]
47 --> 27 | cost [876.0]
48 --> 24 | cost [48.0]
49 --> 32 | cost [150.0]
51 --> 28 | cost [1011.0]
52 --> 28 | cost [1542.0]

BACKUP RING
1  --- 36 | cost [8.75]
4  ---  5 | cost [7.28]
4  --- 12 | cost [20.580000000000002]
5  --- 38 | cost [3.5700000000000003]
6  --- 24 | cost [3.99]
6  --- 25 | cost [8.68]
12 --- 27 | cost [19.81]
16 --- 20 | cost [11.55]
16 --- 26 | cost [35.42]
18 --- 21 | cost [14.35]
18 --- 45 | cost [15.96]
19 --- 31 | cost [23.240000000000002]
19 --- 32 | cost [15.4]
20 --- 21 | cost [19.53]
23 --- 31 | cost [10.64]
23 --- 50 | cost [8.89]
24 --- 40 | cost [5.53]
25 --- 28 | cost [23.03]
26 --- 28 | cost [11.13]
27 --- 46 | cost [36.33]
35 --- 39 | cost [3.71]
36 --- 40 | cost [5.95]
38 --- 39 | cost [5.32]
45 --- 53 | cost [16.8]
46 --- 50 | cost [18.62]

BACKUP STAR
2  --> 21 | cost [11.76]
3  --> 19 | cost [6.21]
7  --> 23 | cost [14.73]
8  --> 45 | cost [5.61]
9  --> 45 | cost [10.83]
10 --> 19 | cost [8.73]
11 --> 27 | cost [12.51]
13 --> 26 | cost [7.62]
14 --> 26 | cost [11.88]
15 -->  6 | cost [1.2]
17 --> 21 | cost [7.59]
29 --> 16 | cost [6.03]
30 --> 20 | cost [5.67]
33 -->  6 | cost [17.04]
34 --> 36 | cost [1.02]
37 --> 38 | cost [1.29]
41 --> 45 | cost [4.98]
42 --> 23 | cost [12.450000000000001]
43 -->  6 | cost [7.8]
44 --> 36 | cost [3.33]
47 --> 26 | cost [5.58]
48 -->  5 | cost [1.41]
49 -->  1 | cost [1.92]
51 --> 12 | cost [5.64]
52 --> 27 | cost [12.780000000000001]

B = 38.230000000000004
i★ = 12
j★ = 28
k★ = 27

       
——————— Sun, 07 Jan 2024 18:41:48 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = true
BD  —— 28 hubs —— 1-35-36-39-40-38-24-5-6-4-12-28-27-26-46-16-50-20-30-42-21-31-18-3-41-45-32-53-1
total time           3603.88 (TL 3600)    gap                  0.166                
blossom time         362.35               nb blossom           106439.0             
LB <= UB             39531.41<=47397.15   Master/SP costs      38970/8427.15        
Master time          3579.77              SP time              24.11                
subtour              16884.0              connectivity cuts    2000.0               
uc strategy          4                    uc tolerance         0.01                 
opt. cuts            2781.0               SP method            poly                 
tildeV               2:52                 inst transformation  2                    
2-opt strategy       0                                                              
explored nodes       114713               
F                    183.0                
                                          o_i                  0                                                              
                                          
Found solution
RING NODES
1—35—36—39—40—38—24—5—6—4—12—28—27—26—46—16—50—20—30—42—21—31—18—3—41—45—32—53

RING
1  --- 35 | cost [854.0]
1  --- 53 | cost [0.0]
3  --- 18 | cost [945.0]
3  --- 41 | cost [1729.0]
4  ---  6 | cost [490.0]
4  --- 12 | cost [2058.0]
5  ---  6 | cost [245.0]
5  --- 24 | cost [224.0]
12 --- 28 | cost [1274.0]
16 --- 46 | cost [1092.0]
16 --- 50 | cost [910.0]
18 --- 31 | cost [560.0]
20 --- 30 | cost [1323.0]
20 --- 50 | cost [245.0]
21 --- 31 | cost [1050.0]
21 --- 42 | cost [2030.0]
24 --- 38 | cost [315.0]
26 --- 27 | cost [882.0]
26 --- 46 | cost [3178.0]
27 --- 28 | cost [770.0]
30 --- 42 | cost [2205.0]
32 --- 45 | cost [1057.0]
32 --- 53 | cost [637.0]
35 --- 36 | cost [105.0]
36 --- 39 | cost [301.0]
38 --- 40 | cost [245.0]
39 --- 40 | cost [301.0]
41 --- 45 | cost [1162.0]

STAR
2  --> 42 | cost [309.0]
7  --> 42 | cost [228.0]
8  --> 41 | cost [192.0]
9  --> 41 | cost [717.0]
10 --> 41 | cost [732.0]
11 --> 28 | cost [1254.0]
13 --> 27 | cost [555.0]
14 --> 27 | cost [1122.0]
15 -->  5 | cost [75.0]
17 -->  3 | cost [651.0]
19 --> 45 | cost [225.0]
22 -->  1 | cost [138.0]
23 --> 20 | cost [282.0]
25 -->  4 | cost [327.0]
29 --> 50 | cost [573.0]
33 -->  4 | cost [1551.0]
34 --> 35 | cost [63.0]
37 --> 40 | cost [123.0]
43 -->  6 | cost [780.0]
44 --> 35 | cost [288.0]
47 --> 26 | cost [558.0]
48 --> 24 | cost [48.0]
49 --> 32 | cost [150.0]
51 --> 12 | cost [564.0]
52 --> 27 | cost [1278.0]

BACKUP RING
1  --- 36 | cost [8.75]
3  --- 31 | cost [14.63]
3  --- 45 | cost [15.4]
4  ---  5 | cost [7.28]
4  --- 28 | cost [29.19]
5  --- 38 | cost [3.5700000000000003]
6  --- 12 | cost [24.43]
6  --- 24 | cost [3.99]
12 --- 27 | cost [19.81]
16 --- 20 | cost [11.55]
16 --- 26 | cost [35.42]
18 --- 21 | cost [14.35]
18 --- 41 | cost [23.1]
20 --- 42 | cost [33.39]
21 --- 30 | cost [16.94]
24 --- 40 | cost [5.53]
26 --- 28 | cost [11.13]
27 --- 46 | cost [36.33]
30 --- 50 | cost [15.05]
31 --- 42 | cost [30.73]
32 --- 41 | cost [21.77]
35 --- 39 | cost [3.71]
36 --- 40 | cost [5.95]
38 --- 39 | cost [5.32]
45 --- 53 | cost [16.8]
46 --- 50 | cost [18.62]

BACKUP STAR
2  --> 30 | cost [11.700000000000001]
7  --> 21 | cost [10.86]
8  --> 45 | cost [5.61]
9  --> 45 | cost [10.83]
10 --> 45 | cost [9.870000000000001]
11 --> 12 | cost [11.61]
13 --> 26 | cost [7.62]
14 --> 26 | cost [11.88]
15 -->  6 | cost [1.2]
17 --> 21 | cost [7.59]
19 --> 41 | cost [2.7600000000000002]
23 --> 50 | cost [3.81]
25 -->  6 | cost [3.72]
29 --> 16 | cost [6.03]
33 -->  6 | cost [17.04]
34 --> 36 | cost [1.02]
37 --> 38 | cost [1.29]
43 -->  4 | cost [7.3500000000000005]
44 --> 36 | cost [3.33]
47 --> 27 | cost [8.76]
48 -->  5 | cost [1.41]
49 -->  1 | cost [1.92]
51 --> 28 | cost [10.11]
52 --> 28 | cost [15.42]

B = 46.050000000000004
i★ = 26
j★ = 27
k★ = 28