**Cplex**. Optimal solution \(z=60\) found in about 8 seconds. Using 4 threads on an old laptop. Log is below.**CBC**. Hit timelimit of 2 hours. Objective = 62 (non-optimal). Also using 4 threads on the same machine. The strange thing is that with CBC the best possible bound is not changing at all. Not by a millimeter. See the highlighted numbers in the log below. CBC is a very good solver, but sometimes I see things like this.**Gurobi**and**Xpress**solve this problem very fast. Ran on NEOS via an MPS file.**SCIP**has also problems with this model. Ran on NEOS via an MPS file.

All runs were with default settings. It looks like the commercial solvers do a much better job than the open source and academic codes. Sometimes you get what you pay for.

#### Cplex Log

Tried aggregator 2 times. MIP Presolve eliminated 4 rows and 5 columns. MIP Presolve modified 3 coefficients. Aggregator did 500 substitutions. Reduced MIP has 999 rows, 1499 columns, and 2497 nonzeros. Reduced MIP has 500 binaries, 0 generals, 0 SOSs, and 0 indicators. Presolve time = 0.02 sec. (7.38 ticks) Found incumbent of value 500.000000 after 0.03 sec. (8.85 ticks) Probing time = 0.00 sec. (0.20 ticks) Tried aggregator 1 time. Reduced MIP has 999 rows, 1499 columns, and 2497 nonzeros. Reduced MIP has 500 binaries, 0 generals, 0 SOSs, and 0 indicators. Presolve time = 0.00 sec. (5.26 ticks) Probing time = 0.00 sec. (0.19 ticks) MIP emphasis: balance optimality and feasibility. MIP search method: dynamic search. Parallel mode: deterministic, using up to 4 threads. Parallel mode: deterministic, using up to 3 threads for concurrent optimization. Tried aggregator 1 time. LP Presolve eliminated 999 rows and 1499 columns. All rows and columns eliminated. Presolve time = 0.02 sec. (0.71 ticks) Initializing dual steep norms . . . Root relaxation solution time = 0.02 sec. (1.15 ticks) Nodes Cuts/ Node Left Objective IInf Best Integer Best Bound ItCnt Gap * 0+ 0 500.0000 0.0000 100.00% Found incumbent of value 500.000000 after 0.06 sec. (16.78 ticks) 0 0 55.2664 490 500.0000 55.2664 0 88.95% * 0+ 0 75.0000 55.2664 26.31% Found incumbent of value 75.000000 after 0.08 sec. (21.35 ticks) 0 0 55.2825 353 75.0000 Cuts: 349 222 26.29% 0 0 55.4160 311 75.0000 Cuts: 349 455 26.11% 0 0 55.4160 275 75.0000 Cuts: 349 775 26.11% * 0+ 0 72.0000 55.4160 23.03% Found incumbent of value 72.000000 after 0.59 sec. (116.10 ticks) 0 0 55.4160 182 72.0000 Cuts: 349 1032 23.03% 0 0 55.4160 141 72.0000 Cuts: 349 1208 23.03% 0 0 55.4160 121 72.0000 Cuts: 192 1336 23.03% 0 0 55.4160 105 72.0000 Cuts: 128 1410 23.03% 0 0 55.4160 98 72.0000 Cuts: 85 1472 23.03% 0 0 55.4160 61 72.0000 Cuts: 74 1494 23.03% 0 0 55.4160 67 72.0000 Cuts: 160 1571 23.03% 0 2 55.4160 56 72.0000 55.4160 1571 23.03% Elapsed time = 1.33 sec. (273.54 ticks, tree = 0.01 MB, solutions = 3) 1239 854 55.4160 59 72.0000 55.4160 3845 23.03% Cuts: 38 * 1471+ 1230 71.0000 55.4160 21.95% Cuts: 16 Found incumbent of value 71.000000 after 3.39 sec. (827.52 ticks) * 1474+ 1230 70.0000 55.4160 20.83% Found incumbent of value 70.000000 after 3.39 sec. (828.63 ticks) * 1481+ 1230 69.0000 55.4160 19.69% Found incumbent of value 69.000000 after 3.41 sec. (831.79 ticks) * 1490+ 1230 68.0000 55.4160 18.51% Found incumbent of value 68.000000 after 3.44 sec. (834.87 ticks) * 1731+ 1033 67.0000 58.7491 12.31% Found incumbent of value 67.000000 after 6.63 sec. (1824.17 ticks) * 1731+ 688 65.0000 59.0933 9.09% Found incumbent of value 65.000000 after 7.25 sec. (2079.31 ticks) * 1731+ 458 62.0000 60.0000 3.23% Found incumbent of value 62.000000 after 7.47 sec. (2162.68 ticks) * 1731+ 305 61.0000 60.0000 1.64% Found incumbent of value 61.000000 after 8.20 sec. (2495.45 ticks) * 1731+ 0 60.0000 60.0000 0.00% Found incumbent of value 60.000000 after 8.58 sec. (2603.40 ticks) * 1731 0 integral 0 60.0000 60.0000 10141 0.00% Found incumbent of value 60.000000 after 8.61 sec. (2604.50 ticks) Cover cuts applied: 150 Implied bound cuts applied: 14 Flow cuts applied: 37 Mixed integer rounding cuts applied: 249 Gomory fractional cuts applied: 26 Root node processing (before b&c): Real time = 1.31 sec. (273.18 ticks) Parallel b&c, 4 threads: Real time = 7.30 sec. (2331.56 ticks) Sync time (average) = 0.25 sec. Wait time (average) = 0.01 sec. ------------ Total (root+branch&cut) = 8.61 sec. (2604.75 ticks) MIP status(101): integer optimal solution

#### CBC Log

Calling CBC main solution routine... Integer solution of 74 found by feasibility pump after 0 iterations and 0 nodes (2.78 seconds) Integer solution of 72 found by RINS after 0 iterations and 0 nodes (2.96 seconds) 128 added rows had average density of 31.601563 At root node, 128 cuts changed objective from 55.26645 to 55.416026 in 10 passes Cut generator 0 (Probing) - 367 row cuts average 2.1 elements, 0 column cuts (12 active) in 0.022 seconds - new frequency is 1 Cut generator 1 (Gomory) - 598 row cuts average 26.1 elements, 0 column cuts (0 active) in 0.090 seconds - new frequency is 1 Cut generator 2 (Knapsack) - 14 row cuts average 11.9 elements, 0 column cuts (0 active) in 0.037 seconds - new frequency is -100 Cut generator 3 (Clique) - 0 row cuts average 0.0 elements, 0 column cuts (0 active) in 0.003 seconds - new frequency is -100 Cut generator 4 (MixedIntegerRounding2) - 368 row cuts average 11.2 elements, 0 column cuts (0 active) in 0.023 seconds - new frequency is 1 Cut generator 5 (FlowCover) - 394 row cuts average 2.8 elements, 0 column cuts (0 active) in 0.041 seconds - new frequency is 1 Cut generator 6 (TwoMirCuts) - 598 row cuts average 35.5 elements, 0 column cuts (0 active) in 0.072 seconds - new frequency is -100 After 0 nodes, 1 on tree, 72 best solution, best possible 55.416026 (3.94 seconds) Integer solution of 70 found by heuristic after 4138 iterations and 57 nodes (14.24 seconds) Integer solution of 69 found by heuristic after 12712 iterations and 287 nodes (23.32 seconds) Integer solution of 66 found by heuristic after 18149 iterations and 487 nodes (25.94 seconds) After 1005 nodes, 555 on tree, 66 best solution, best possible 55.416026 (31.73 seconds) Integer solution of 65 found by heuristic after 32591 iterations and 1015 nodes (33.93 seconds) Integer solution of 64 found by heuristic after 43252 iterations and 1405 nodes (39.79 seconds) Integer solution of 63 found by heuristic after 52409 iterations and 1805 nodes (44.62 seconds) After 2017 nodes, 1104 on tree, 63 best solution, best possible 55.416026 (45.77 seconds) After 3045 nodes, 1653 on tree, 63 best solution, best possible 55.416026 (52.29 seconds) After 4099 nodes, 2212 on tree, 63 best solution, best possible 55.416026 (55.02 seconds) After 5163 nodes, 2769 on tree, 63 best solution, best possible 55.416026 (57.08 seconds) . . . After 131221 nodes, 37229 on tree, 63 best solution, best possible 55.416026 (610.94 seconds) After 132282 nodes, 37230 on tree, 63 best solution, best possible 55.416026 (613.18 seconds) After 133298 nodes, 37231 on tree, 63 best solution, best possible 55.416026 (615.50 seconds) After 134320 nodes, 37318 on tree, 63 best solution, best possible 55.416026 (621.40 seconds) Integer solution of 62 found by heuristic after 2839018 iterations and 134502 nodes (621.99 seconds) After 135334 nodes, 37422 on tree, 62 best solution, best possible 55.416026 (627.07 seconds) After 136352 nodes, 37407 on tree, 62 best solution, best possible 55.416026 (631.83 seconds) After 137391 nodes, 37395 on tree, 62 best solution, best possible 55.416026 (637.74 seconds) After 138400 nodes, 37385 on tree, 62 best solution, best possible 55.416026 (642.25 seconds) . . . After 1566320 nodes, 37408 on tree, 62 best solution, best possible 55.416026 (7177.35 seconds) After 1567325 nodes, 37412 on tree, 62 best solution, best possible 55.416026 (7182.33 seconds) After 1568336 nodes, 37421 on tree, 62 best solution, best possible 55.416026 (7187.07 seconds) After 1569358 nodes, 37402 on tree, 62 best solution, best possible 55.416026 (7192.93 seconds) After 1570410 nodes, 37400 on tree, 62 best solution, best possible 55.416026 (7199.07 seconds) Thread 0 used 30010 times, waiting to start 2696, 0 locks, 0 locked, 0 waiting for locks Thread 1 used 30010 times, waiting to start 2480, 0 locks, 0 locked, 0 waiting for locks Thread 2 used 30010 times, waiting to start 2083, 0 locks, 0 locked, 0 waiting for locks Thread 3 used 30010 times, waiting to start 1770, 0 locks, 0 locked, 0 waiting for locks Main thread 6737 waiting for threads, 0 locks, 0 locked, 0 waiting for locks Exiting on maximum time Partial search - best objective 62 (best possible 55.416026), took 19832056 iterations and 1570713 nodes (7202.49 seconds) Strong branching done 3540964 times (6897 iterations), fathomed 137838 nodes and fixed 779663 variables Maximum depth 198, 2057372 variables fixed on reduced cost Time limit reached. Have feasible solution. MIP solution: 6.200000e+01 (1570713 nodes, 7202.62 CPU seconds, 7202.62 wall clock seconds)