A subgradient-based branch-and-bound algorithm for the ...

(νj + fj)/sj, where
νj = min
x
i∈I
cijxij :
i∈I
dixij = sj , 0 ≤ xij ≤ 1 ∀ i ∈ I .
The branch that results from requiring the selected plant to be open is then always
investigated first.
We used a FORTRAN code of Ryu and Guignard, but replaced the out-ofkilter
method used in this code for solving the transportation problems by calls
to the network simplex algorithm of CPLEX’s callable library. The code was then
compiled and run on the same machine as BB-SG. Table 3 compares CAPLOC
and BB-SG on the test problem instances from Klose and Görtz (2007) and Table 4
summarizes these results by again averaging over the capacity ratio r. As can be seen
from Table 3 and 4, BB-SG is far superior to CAPLOC. (The number of enumerated
nodes reported for CAPLOC in this table does not include the branches investigated
in the Lagrangean probing.) On the test problems of Table 3, BB-SG was on average
about 55 times faster than CAPLOC. Basically, both algorithms are based on the
same methodology; they apply the same Lagrangean relaxation and use subgradient
optimization for lower bounding. BB-SG however greatly benefits from, firstly, the
best lower bound search and, secondly and mostly, from better branching decisions
based on a reasonable estimate of a primal solution.
Since also the test problem instances from the OR-library are of a size and
difficulty that can be managed by CAPLOC, we also compared both methods on
these test problems. Each of these problem instances is of size |I| × |J| = 1000 ×
100. Table 5 shows the results, which again indicate the superiority of BB-SG. The
ORLIB instances are easier to solve than the instances of Table 3, but still BB-SG
is about 17 times faster than CAPLOC on these instances.
4.3 Comparison with CPLEX and the branch-and-cut method of Avella and Boccia
(2007)
We used CPLEX’s MIP solver in the following way for solving the CFLP (1)–(7).
We started with the weaker aggregate formulation that is obtained by taking for each
j ∈ J the sum of constraints (5) over all i ∈ I. Additional variable upper bounds (5)
were then included “on the fly” whenever violated by the current LP solution. The
model formulation found this way was then passed to the MIP solver, after removing
beforehand those added constraints (5) that showed a positive slack at the final
LP solution. We also passed a feasible solution and upper bound to CPLEX’s MIP
solver. The solution was obtained using a simple rounding procedure, which was
applied each time the current LP relaxation was solved. To this end, the facilities
j ∈ J are sorted according to non-decreasing LP value of the associated variable yj
and then opened as long as yj ≥ 0.9 or the capacity of the open facilities is smaller
than the total demand. CPLEX’s MIP solver was called with default option values
except that (i) CPLEX’s internal primal heuristic was switched off; (ii) the relative
optimality tolerance was set to 10 −6 ; (iii) the parameters CutsFactor, CutPass and
AggCutLim respectively controlling the number of cuts CPLEX may add, the number
of cutting rounds CPLEX may perform, and the number of constraints that can
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