![](bgb.png)
Table 6: Computation time on fictive instances (ms).
instance
enhanced
model
FCNDA
enhanced
model
EFCNDA
dynamic
programming
Fi 10 205 166 322
Fi 11 327 77 652
Fi 12 993 332 1409
Fi 13 1130 120 3800
Fi 14 1369 347 12 403
Fi 15 1450 98 39 654
Fi 16 2691 280 164 243
gramming was more efficient. As for FCNDA, the
MIP formulation proved to be efficient, with an av-
erage computation time of 900 ms. Although the in-
stances with a higher degree are harder to solve, these
instances stay tractable in practice. One should fa-
vor a MIP based approach, regardless of the problem,
when dealing with high degree nodes. As the fictive
instances have less arcs, the MIP approaches seem
more sensible to the overall number of arcs than to
the maximum degree of the instances.
6.3 Operational Considerations
We compared the optimal solutions of both problems.
Results are displayed in Table 7, the column labeled
”arcs with rule broken” denotes the number of arcs
where the maintenance rule (illustrated in figure 5) is
violated when FCNDA is solved.
Table 7: Optimal solution costs and characteristics.
instance
Solution
EFCNDA
Solution
FCNDA
arcs with
rule broken
Ar 1 6156.6 6087.3 6
Ar 2 10 357.3 9870.0 8
Ar 3 6546.2 6125.8 14
Ar 4 6720.8 6461.9 14
Ar 5 5081.8 5081.8 0
Ar 6 6546.5 6544.2 1
Ar 7 9348.0 8638.6 18
Ar 8 12 328.3 12 248.4 4
Ar 9 25 619.1 24 422.8 15
An optimal EFCNDA solution is on average 3.7 %
more expensive than a FCNDA optimal solution (see
Table 7). This can be seen as an acceptable capital
expenditure over-cost if it is compensated by future
easier maintenance activities.
The maintenance rule is violated in almost every
real-life instance we tried (8 out of 9). On average,
it is not respected in 6.2 % of the arcs, which is sig-
nificant. This suggests that optimal FCNDA solutions
will be much harder to repair in case of failure on one
duct.
7 CONCLUSION
We introduced two combinatorial problems related to
FTTH network design, one unconstrained by main-
tenance consideration and the other one constrained.
Regarding the unconstrained problem, one integer
programming based solving algorithm was proposed.
Adding valid inequalities leads to a more tractable
problem. We proposed two solution methods for the
constrained problem. These methods are complemen-
tary, as they prove efficient in different contexts: the
dynamic programming approach is generally faster
in graphs where nodes have a small degree, whereas
the mixed integer programming, embedding efficient
valid inequalities, is generally faster otherwise.
On a complexity level, the unconstrained problem
seems harder to solve than the constrained problem.
Our numerical experiments confirmed this tendency
on real-life instances. From the operational point of
view, the maintenance rule can be considered as a
reasonable compromise between capital expenditure
over-costs for the network deployment and mainte-
nance savings.
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