------------------------------------------------------------
                        α=7.0 —— berlin52 —— 52 nodes       
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——————— Sat, 27 Jan 2024 08:49:06 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
ILP ——— 25 hubs —— 1—35—36—39—40—38—24—5—6—4—12—28—27—26—16—50—20—23—21—31—18—19—45—32—53
total time           237.61               gap                  0.0                  
blossom time         0                    nb blossom           0.0                  
LB <= UB             39079.72<=39079.72   subtour              57.0                 
connectivity cuts    254.0                uc strategy          4                    
uc tolerance         0.01                 tildeV               2:52                 
2-opt strategy       0                                                              
explored_nodes       479                  n lazycuts edges     0.0                  
F                    31.0                 post procedure       true                 
o_i                  0                                                                                                                                                  
Found solution
RING NODES
1—35—36—39—40—38—24—5—6—4—12—28—27—26—16—50—20—23—21—31—18—19—45—32—53

RING
1  --- 35 | cost [854.0]
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 --- 26 | cost [3542.0]
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]
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  --> 21 | cost [1176.0]
3  --> 18 | cost [405.0]
7  --> 21 | cost [1086.0]
8  --> 19 | cost [378.0]
9  --> 19 | cost [924.0]
10 --> 19 | cost [873.0]
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]
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 -->  4 | cost [735.0]
44 --> 35 | cost [288.0]
46 --> 24 | cost [420.0]
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]
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 --- 27 | cost [41.86]
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]
26 --- 50 | cost [44.17]
35 --- 39 | cost [3.71]
36 --- 40 | cost [5.95]
38 --- 39 | cost [5.32]
45 --- 53 | cost [16.8]

BACKUP STAR
2  --> 23 | cost [15.3]
3  --> 19 | cost [6.21]
7  --> 23 | cost [14.73]
8  --> 45 | cost [5.61]
9  --> 45 | cost [10.83]
10 --> 45 | cost [9.870000000000001]
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]
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 -->  6 | cost [7.8]
44 --> 36 | cost [3.33]
46 --> 16 | cost [4.68]
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 = 53.120000000000005
i★ = 23
j★ = 21
k★ = 31

       
——————— Sat, 27 Jan 2024 08:50:41 ————————— seperate y_ij <= y_jj - x_ij on lazy constraints ————————— use blossom = false
BD  —— 25 hubs —— 1-32-45-19-18-31-21-23-20-50-16-26-27-28-12-4-6-5-24-38-40-39-36-35-53-1
total time           62.72                gap                  0.0                  
blossom time         0                    nb blossom           0.0                  
LB <= UB             39079.72<=39079.72   Master/SP costs      37433/1646.72        
Master time          59.62                SP time              3.09                 
subtour              554.0                connectivity cuts    1296.0               
uc strategy          4                    uc tolerance         0.01                 
opt. cuts            175.0                SP method            poly                 
tildeV               2:52                 inst transformation  2                    
2-opt strategy       0                                                              
explored nodes       13108                
F                    31.0                 
                                          o_i                  0                                                              
                                          
Found solution
RING NODES
1—32—45—19—18—31—21—23—20—50—16—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 --- 26 | cost [3542.0]
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]
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  --> 21 | cost [1176.0]
3  --> 18 | cost [405.0]
7  --> 21 | cost [1086.0]
8  --> 19 | cost [378.0]
9  --> 19 | cost [924.0]
10 --> 19 | cost [873.0]
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]
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 -->  4 | cost [735.0]
44 --> 35 | cost [288.0]
46 --> 24 | cost [420.0]
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 --- 27 | cost [41.86]
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]
26 --- 50 | cost [44.17]
35 --- 39 | cost [3.71]
36 --- 40 | cost [5.95]
36 --- 53 | cost [8.75]
38 --- 39 | cost [5.32]

BACKUP STAR
2  --> 23 | cost [15.3]
3  --> 19 | cost [6.21]
7  --> 23 | cost [14.73]
8  --> 45 | cost [5.61]
9  --> 45 | cost [10.83]
10 --> 45 | cost [9.870000000000001]
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]
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 -->  6 | cost [7.8]
44 --> 36 | cost [3.33]
46 --> 16 | cost [4.68]
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 = 53.120000000000005
i★ = 23
j★ = 21
k★ = 31