Our exact method BMR for the VRPTW was tested on Solomon instances Solomon [1987], which are divided into six classes (classes C1, RC1 and R1 with tight time windows and strict vehicle capacity, and classes C2, RC2 and R2 with wide time windows and loose vehicle capacity). We considered all 100-customer instances and instances with 50 customers of classes C2, RC2 and R2.
The travel costs dij are computed as dij = b10eijc/10, where eij is the Euclidean
distance between vertices i and j; the travel times tij are integer values computed as
Because instances of classes C1, RC1 and R1 have tight time windows, we did not find it worth running procedure H2. Thus, on such instances H2 was skipped. In addition, we ignored SCs and WSR3s for all classes of instances.
BMR uses the best upper bounds reported inRopke[2005] andDanna and Pape[2005]. Such upper bounds are obtained by running the heuristic of Pisinger and Ropke[2007] with different parameter settings (Ropke [2010]). Whenever BMR uses the upper bound, its computing time is added to the total computing time of BMR. For instance R211 with 100 customers, the upper bound used was found byDesaulniers et al.[2008].
BMR uses the following parameter setting • in H1: M axit1 = 100 and M axit2 = 50;
• in H2: ∆(N
i) = 8, M axit1 = 100, and M axit2 = 50;
• in H3: ∆(N
i) = 10, ∆b = 300, ∆(F ) = ∆(B) = 5 × 107, ∆(R) = 1.5 × 106,
M axit1 = 100, and M axit2 = 50;
• in H4, ∆a = 1 × 104, ∆b = 300, ∆(F ) = ∆(B) = 5 × 107, ∆(R) = 1.5 × 106,
∆(C ) = 20.
We compare BMR with the methods ofJepsen et al.[2008] andDesaulniers et al.[2008], hereafter called JPSP and DHL, respectively. Desaulniers et al.[2008] presented three versions of their algorithm. We consider the version labeled “ESPPRC SRC” as it could solve more instances than the others. According to SPEC (http://www.spec. org/benchmarks.html), our machine is three times faster than the Intel Pentium 4 3.0 GHz PC of JPSP and twice as fast as the Linux PC Dual Core AMD Opteron at 2.6 GHz of DHL.
In Tables4.1,4.2, and4.3, we report on detailed computational results on the Solomon instance considered. The columns of three tables report the instance name (Inst), the optimal value (z∗ - in bold if solved for the first time by BMR), the upper bound used (zU B) and the time to compute it (T ) in columns under heading Upper Bound. For each
bounding procedure Hk, k = 1, . . . , 4 (if run), we report the lower bound (LBk) and
the cumulative computing time spent up to Hk. Columns |F3|, |B3|, and |R3| report
the cardinalities (in thousands) of the setsF3,B3 andR3; if we could not completely generate a set, an empty circle is displayed. The number of SR3s inequalities (SR3) added in H4 is shown. The number of routes (| eR|) in the final reduced problem Pe and the time taken by Cplex to solve it (Tcpx) are shown. Finally, the total computing
time in seconds (Ttot) of the methods compared are reported in the last three columns
of the table. Ttot under BMR is equal to the sum of the time to compute the upper
bound, the time spent up to H4 and Tcpx.
Table 4.3 shows that BMR was able to solve four instances open so far. The only open Solomon instance is R.208.100, where BMR ran out of memory. Notice that the
lower bounds achieved using the ng-routes in algorithm CCG are close to the bounds achieved using elementary routes and the time taken to perform H2 is limited. Table4.4 compares BMR, JPSP, and DHL. For each class, Table4.4 reports the class name (Class), the number of customers (n), the number of instances (N P ), the number of instances solved by each of the three methods (Solved) and the average computing time in seconds (T ) (n.a. means data are not available). In the last three rows, the average computing time of the methods over all instances, the instances solved by JPSP, and the instances solved by DHL are shown.
Table 4.4 shows that BMR outperforms JPSP and DHL: all instances solved by the other methods were solved by BMR and the average time is significantly lower. Table4.5reports the optimal solutions BMR found for the four 100-customer instances (RC204, RC208, R204, and R211) open before this paper. We give the optimal solution value, the number of vehicles used, and, for each route, the cost and the sequence of visited customers.
We also report a computational analysis about some components of the algorithm proposed (i.e., parameter ∆(Ni) and dominance and fathoming rules). Table 4.6 re-
ports the results obtained on a selected set of instances by varying parameter ∆(Ni)
in procedure H2. In particular, the table shows the value of lower bound LB2 using
∆(Ni) = 5, 8, 10, 12, and the corresponding computing time, compared with LB1, LB3,
and LB4. The last line of the table reports the average percentage deviations of the
different lower bounds. The table shows that using ∆(Ni) = 8 gives a good trade-off
between quality of the lower bound and computing time. Indeed, the lower bound is on average about 5 percent greater than LB1 and about 0.5 percent lower than LB3.
Tables4.7and4.8report statistics on the number of states fathomed by the dominance and fathoming rules applied in procedures GenP4 (called by procedure H4) and GenPF (called to generate the final route sets eR). The tables show the following columns: total number of forward (backwards) states in millions generated in computing F (B) (States), percentage of states eliminated by Dominance 1 (%Dom1), percentage of states fathomed by Fathoming rule x (%F ath x, x = 1, 2, 4, 5), final cardinality (in millions) of the set F (B) generated (|F |, |B|). The dominance and fathoming rules are reported in the tables using the same order of application in BMR.
Regarding procedure GenP4, table 4.7shows that the dominance and fathoming rules are very effective in reducing the number of states. Indeed, on average about 90% of the states generated (for both sets F and B) are eliminated. In particular, the new Fathoming 1 and Fathoming 2 rules eliminates on average 80% of the states not dominated by Dominance 1.
Concerning procedure GenPF, table 4.8shows that both Fathoming 5 and 4 eliminate on average about 90% of the states generated.