6.4 Framework
6.5.2 Framework Results
Tables 6.1–6.4 show results for all considered instance sets for the RDCST problem. We report obtained average gaps between the best primal and dual bounds, the median runtime to reach these bounds, and the number of instances solved to optimality within the time limit. Dashes in
averagegapin%mediantimeinseconds#optimalsolutions(outof5)BPL3A1A2GA2GLBPL3A1A2GA2GLBPL3A1A2GA2GLSetB\η----303-303----303-303----303-303
R560.00.00.00.00.00.00.00.00.020100101155555555580.00.00.00.00.00.00.00.00.0302111111555555555100.00.00.00.00.00.00.00.00.0302222222555555555120.00.00.00.00.00.00.00.00.0402111111555555555C560.00.00.00.00.00.00.00.00.020100001055555555580.00.00.00.00.00.00.00.00.0302222212555555555100.00.00.00.00.00.00.00.00.071118811656555555555120.00.00.00.00.00.00.00.00.07317151214687555555555E560.00.00.00.00.00.00.00.00.081777633455555555580.00.00.00.00.00.00.00.00.01832118182881012555555555100.00.00.00.00.00.00.00.00.05064692196153212729555555555120.00.00.00.00.00.00.00.00.015722917506751566455355555555555
R10100.00.00.00.00.00.00.00.00.0302111111555555555150.00.00.00.00.00.00.00.00.0403433433555555555200.00.00.00.00.00.00.00.00.0407445444555555555250.00.00.00.00.00.00.00.00.0417676677555555555C10100.00.00.00.00.00.00.00.00.0402222222555555555150.00.00.00.00.00.00.00.00.0425565565555555555200.00.00.00.00.00.00.00.00.061014172317101512555555555250.00.00.00.00.00.00.00.00.0272236434332292420555555555E10100.00.00.00.00.00.00.00.00.0717556445555555555150.00.00.00.00.00.00.00.00.0431059118112148222527555555555200.00.00.00.30.00.40.00.00.040583392619824329196180140254555453555250.00.00.02.81.62.80.00.00.314253171361---6517511419555120554
Table6.1:ResultsforinstancesbyGouveiaetal.[68](B:delay-bound,η:parameterforupperLPandintegerbounds,gap:gapbetweenbestprimalanddualbound,BP:stabilizedbranch-and-price,L3:layeredgraphapproachwithconnectioncutsonG 0L,A1:basicALFvariant,A2G(L):improvedALFvariantwithconnectioncutsonG 0(G 0L),bestresultsareprintedbold).
averagegapin%mediantimeinseconds#optimalsolutions(outof5) BPL3A1A2GA2GLBPL3A1A2GA2GLBPL3A1A2GA2GL SetB\η----303-303----303-303----303-303 R1001000.00.00.00.00.00.00.00.00.0459465464555555555 1500.00.00.00.00.00.00.00.00.04105658566555555555 2000.00.00.00.00.00.00.00.00.061965107696555555555 2500.00.00.00.00.00.00.00.00.06628695695555555555 C1001000.00.00.00.00.00.00.00.00.0822835344627292524555555555 1500.00.90.00.00.00.00.00.00.0106276916223171798072545555555 2000.02.10.00.00.00.00.00.00.013774767544169201146188121535555555 2500.03.90.00.00.00.00.00.00.014-175596257585553505555555 E1001000.00.00.00.00.00.00.00.00.016593370991962661146159159555555555 1500.02.10.01.91.72.00.00.00.0409162672-6249-419650415535232555 2000.010.40.35.05.35.60.40.30.0245-4051---544848433471504000445 2500.011.70.78.17.35.52.61.41.5425-----650858896399502000343 R100010000.00.00.00.00.00.00.00.00.0122451268107109555555555 15000.02.40.00.00.00.00.00.00.018958637677355336434535555555 20000.00.90.00.00.00.00.00.00.02035489212010211710545555555 25000.020.00.00.00.00.00.00.00.025609910787676545555555 C100010000.02.70.00.00.00.00.00.00.01760181910121191010535555555 15000.06.80.00.00.00.00.00.00.029-10812112280747559505555555 20000.013.20.00.00.10.00.00.00.039-32518201549833552710309505545555 25000.064.50.00.00.00.00.00.00.033-215424635337280228200505555555 E100010000.08.90.00.00.00.00.00.00.026-14520101105171089147123515555555 15000.016.10.04.73.42.60.10.00.050-625--33779391256636504123455 20000.033.40.24.46.94.20.80.00.188-3153---761422132376504101354 25000.0-0.98.85.64.45.36.63.6218-5952---3929--503000312 Table6.2:ResultsforinstancesbyGouveiaetal.[68](B:delay-bound,η:parameterforupperLPandintegerbounds,gap:gap betweenbestprimalanddualbound,BP:stabilizedbranch-and-price,L3:layeredgraphapproachwithconnectioncutsonG0 L,A1: basicALFvariant,A2G(L):improvedALFvariantwithconnectioncutsonG0 (G0 L),bestresultsareprintedbold).
averagegapin%mediantimeinseconds#optimalsolutions(outof30)BPL3A1A2GA2GLBPL3A1A2GA2GLBPL3A1A2GA2GLSetB\η----303-303----303-303----303-303
T10160.00.00.00.00.00.00.00.00.0001000000303030303030303030300.00.00.00.00.00.00.00.00.02111899898303030303030303030500.00.00.00.00.00.00.00.00.037414044433844403030303030303030301000.00.00.00.70.70.80.00.00.0576136145177151117172142303030282928303030T30160.00.00.00.00.00.00.00.00.0201111111303030303030303030300.00.00.00.00.00.00.00.00.09219101199109303030303030303030500.00.00.00.00.00.00.00.00.020341736068616578603030303030303030301000.00.00.21.80.80.52.80.30.047362580329453283283356297303029252727272930T50160.00.00.00.00.00.00.00.00.0502111111303030303030303030300.00.00.00.00.00.00.10.00.119437161513131414303030303030293029500.00.00.00.00.00.00.00.00.044332826374525766453030303030303030301000.00.01.41.20.71.12.83.90.52027682271542446407587582356303022252525262627T70160.00.00.00.00.00.00.00.00.0802111111303030303030303030300.00.00.00.00.00.00.00.00.031446172016161514303030303030303030500.00.00.00.00.00.00.00.00.079502275873547871493030303030303030301000.10.91.30.71.91.03.32.20.32418732356398424370464715350282925262525262726T99160.00.00.00.00.00.00.00.00.01502122122303030303030303030300.00.00.00.00.00.00.00.00.066442141813131614303030303030303030500.00.00.00.00.00.00.10.00.0150442386364576763603030303030302930301000.30.60.51.41.10.50.90.90.15267191622357453333502574323282826252627282829
Table6.3:ResultsforrandominstancesfromSection3.13.1(B:delay-bound,η:parameterforupperLPandintegerbounds,gap:gapbetweenbestprimalanddualbound,BP:stabilizedbranch-and-price,L3:layeredgraphapproachwithconnectioncutsonG 0L,A1:basicALFvariant,A2G(L):improvedALFvariantwithconnectioncutsonG 0(G 0L),bestresultsareprintedbold).
averagegapin%mediantimeinseconds#opt.solutions(outof18/10/5/5/5) BPL3A1A2GA2GLBPL3A1A2GA2GLBPL3A1A2GA2GL SetB\η----303-303----303-303----303-303 B-Ran3140.00.00.00.00.00.00.00.00.0000000000181818181818181818 4270.00.00.00.00.00.00.00.00.0000000000181818181818181818 B-Cor400.00.00.00.00.00.00.00.00.0000000000181818181818181818 540.00.00.00.00.00.00.00.00.0000000000181818181818181818 C-Ran3970.00.00.00.00.00.00.00.00.01825333333101010101010101010 5410.00.00.00.00.00.10.71.50.1481061091111199910109999999 C-Cor500.00.00.00.00.00.00.11.00.023275456511101010109109910 680.00.00.00.00.90.00.60.90.14821933444510101010910999 D-Ran5540.00.00.00.00.00.00.00.00.07639434455555555555 7550.00.10.00.01.00.00.11.90.114517251987892910545545433 D-Cor660.00.00.00.00.00.00.00.00.011328435435555555555 900.00.00.00.00.00.00.00.00.0199326444444555555555 Berlin52-Ran190.00.00.00.00.00.00.00.00.0000000000555555555 260.00.00.00.00.00.00.00.00.0000000000555555555 Berlin52-Cor1650.00.00.00.00.00.00.00.00.0171011011555555555 2250.03.30.00.00.00.00.00.00.0451553332334545555555 Brazil58-Ran200.00.00.00.00.00.00.00.00.0000000000555555555 270.00.00.00.00.00.00.00.00.0102111111555555555 Brazil58-Cor39790.039.40.00.00.00.00.20.00.044-155610568505555454 54250.0-0.00.00.00.00.71.50.067-324253165203711127434102505555344 Table6.4:ResultsforinstancesbyLeggierietal.[111](B:averagedelay-bound,η:parameterforupperLPandintegerbounds,gap: gapbetweenbestprimalanddualbound,BP:stabilizedbranch-and-price,L3:layeredgraphapproachwithconnectioncutsonG0 L,A1: basicALFvariant,A2G(L):improvedALFvariantwithconnectioncutsonG0 (G0 L),bestresultsareprintedbold).
gap and time columns represent 100% and 10 000 seconds, respectively. Parameterδcontrolling some of the stopping criteria for the layered graph extension phase in A2G and A2GL is set to 0.01 which turned out to work well in preliminary tests, see Section 6.4 and Algorithm 6.2 for details. Furthermore, parameterηcontrolling the calculation of upper LP and integer bounds is varied in{−3,0,3}. Some tests were performed to initialize the first layered graph in a more sophisticated way, e.g. based on the initial heuristic solutions, but following the proposed trivial way mostly yielded the best results.
In general, lower absolute values ofηresult in more frequent upper bound computations and thus higher runtime overhead in the first phase. In principle, we only need upper LP bounds to measure the quality of our lower LP bounds. However, in many cases a synergy effect could be observed: The layered graph extensions arising from LP solutions when redirecting arcs to higher layers often result in a faster convergence of the lower LP bound. Currently, we are not sure about the reasons for this effect but this will be analyzed in more detail in future work.
Additionally, it would be enough to compute an upper integer bound immediately before entering the final MIP phase since we know that this bound has to be the best one over all reduced layered graphs. But then we would have to solve a complete MIP model on a possibly already large layered graph without a previously known primal bound used for pruning the branch-and-bound tree. Indeed we can use the initial heuristic primal branch-and-bound which, however, may be quite weak. On the other hand, if we obtain upper bounds from time to time on smaller layered graphs then first we have more possibilities to improve them by heuristics and second we can utilize the best bound in further MIP computations. In many cases, this approach could significantly accelerate the overall primal bound calculations, although we have to repeatedly solve growing MIPs. Finally, in case of limited runtime obtaining upper integer bounds usually yields tight gaps already in the first layered graph extension phase.
To summarize, when comparing different values ofηfor A2G and A2GL, in most cases it is beneficial to compute upper primal bounds (η = 3) but there are still situations where these bounds are not needed in the final MIP phase and thus the calculations of them only produce runtime overhead. Similarly, due to the synergy effect discussed above it is also preferable to calculate upper LP bounds (η =−3) instead of completely ignoring them (η = 0). Finally, we could not observe any significant improvements by further increasing|η|.
One would expect that the basic ALF variant A1 produces too much overhead by iteratively solving MIPs quickly increasing in size. This can be clearly observed e.g. in the results for the random instances in Table 6.3. However, this is mostly not the case for the Gouveia instances in Tables 6.1 and 6.2 where A1 can obviously compete with A2G and A2GL. Here, the same argumentation as above holds: A1 benefits from the steadily tightening series of lower and upper bounds obtained by the MIPs. Especially computing lower integer bounds clearly provides even stronger bounds on the optimal value than just lower LP bounds.
Compared to L3 on the full layered graph ALF can dramatically improve the performance on instances with a large set of achievable delay values and huge bounds, as shown in Tables 6.2 and 6.4, in many cases even by orders of magnitude. Additionally, ALF consumes substantially less memory since the graphs it works on are significantly smaller than the full layered graphs, see Table 6.5. However, BP is still superior on the instances in Table 6.2. As results in Tables 6.1 and 6.3 indicate, for instances with low delay-bounds the framework causes too much
computa-tional overhead and does not pay off in the end. Finally, the most robust and best performance of ALF is observed for the instances by Leggieri et al. [111] in Table 6.4. Here, all ALF approaches outperform BP and L3 in all cases except for the Brazil58-Cor instances withB = 5425where also the reduced layered graphs become quite large due to the huge delay-bounds.
Table 6.5 provides statistics of the ALF experiments on some selected instance sets: average numbers of solved LPs and MIPs, average sizes of preprocessed original graphsG0and full lay-ered graphsGLused in approach L3, and average sizes of reduced layered graphsGiLin the last iteration of ALF relative toGLin percent. Since A2G and A2GL only differ by the sets of valid inequalities added in the final MIP phase the presented values are the same for both. According to Lemma 6.3.2, a solution to the LP relaxation on a reduced layered graph when redirecting arcs to lower layers is optimal if it is feasible for the original problem; this explains the zeroes in the columns of solved MIPs for A2G(L). In general, the number of ALF iterations stays quite low, both for the basic A1 and the improved A2G(L). Even if A1 solved in average up to 32 MIPs for Brazil58 instances the according runtimes are quite moderate and especially much better than solving one MIP on the full layered graph, see Table 6.4. The reason for this immediately becomes obvious when looking at the reduced layered graph sizes of ALF approaches. Particu-larly, for instances with large delay-bounds, e.g. R1000, E1000, and Brazil58-Cor, the reduced layered graphs are tiny compared to their full counterparts. This indicates that actually only a small part of the complete layered graph is relevant to prove optimality. However, instance sets R5 and E5 with low delay-bounds represent the opposite behavior: Here, ALF sometimes needs more than 90% ofGLand thus it makes more sense to just solve one MIP onGLwhich is clearly visible in Table 6.1.