Column 6 represents the number of instances
(among the 100 instances in a series) for which a per
mutation π
t
with the largest dimension and relative
volume of the stability box S B (π
t
,T) provides an op
timal solution for the instance 1p
∗

∑
w
i
C
i
with the
actual processing times p
∗
= (p
∗
1
,... , p
∗
n
) ∈ T. From
the experiments, it follows that, if the maximal pos
sible error of the processing times is no greater than
0.4%, then the dominance digraph (J ,A ) is a com
plete circuitfree digraph. Therefore, the permutation
π
t
∈ S
max
provides an optimal solution for the instance
1p
∗

∑
w
i
C
i
.
The average (maximum) relative error ∆ of the ob
jective function value γ
t
p
∗
calculated for the permu
tation π
t
∈ S
max
constructed by the algorithm MAX
STABOX with respect to the optimal objective func
tion value γ
p
∗
deﬁned for the actual job processing
times is givenin column 7 (in column 8, respectively).
For all series presented in Tables 2 and 3, the aver
age (maximum) error ∆ of the value γ
t
p
∗
of the objec
tive function γ =
∑
n
i=1
w
i
C
i
obtained for the permu
tation π
t
∈ S
max
with the largest dimension and rel
ative volume of a stability box was not greater than
0.012069 (not greater than 0.013812).
The CPUtime for an instance of a series is pre
sented in column 5. This time includes the time for
the realization of the O(n
2
) algorithm for construct
ing the dominance digraph (J ,A ) using condition (3)
of Theorem 1 and the time for the realization of the
O(nlogn) algorithm MAXSTABOX for construct
ing the permutation π
t
∈ S
max
and the stability box
S B (π
t
,T). This CPUtime grows rather slowly with
n, and it was not greater than 79.06 s for each instance.
5 CONCLUSIONS
In (Sotskov and Lai, 2011), an O(n
2
) algorithm has
been developed for calculating a permutation π
t
∈ S
with the largest dimension and volume of a stabil
ity box S B (π
t
,T). In Section 3, we proved Proper
ties 1 – 6 of a stability box allowing us to derive an
O(nlogn) algorithm for calculating such a permuta
tion π
t
∈ S
max
. The dimension and volume of a sta
bility box are efﬁcient invariants of uncertain data T,
as it is shown in simulation experiments on a PC re
ported in Section 4.
ACKNOWLEDGEMENTS
The ﬁrst and third authors were supported in this re
search by National Science Council of Taiwan.
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