DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

Manuscript for IEEE Transactions on Automatic Control 19
∑
i∈b
≡
≡ [
b (T
i
∑
i∈a
∑
i∈a
i
i
b
b (T
) − CCT(T
b
b (T
) +
b
∑
i∉a
And, for group a,
∑
i∈a
b (T
i
b
) >
b
) − CCT(T
i
b
b (T
) − CCT(T
b
b
) ≤
∑
i∈b
b (T
) − CCT(T
i
)] − [
∑
i∈a
∑
i∈a
b (T
So, the inequality in (20) should hold.
∑
i∉a
a
∑
i∉a
b
i
a
) − CCT(T
b
i
) >
b (T
a
a
∑
i∈a
i
a
)
b (T
a
) +
) − CCT(T
) − CCT(T
a
a
∑
i∉a
)] >
b (T
i
∑
i∉a
a
i
) − CCT(T
b (T
a
) −
∑
i∉a
a
b (T
i
)
b
. (18)
)
) . (19)
b i (T ) < bi
(T )
(20)
But, this inequality is not possible because b i (T) is an increasing function with T.
Through this property the two-tier auctioning model can be transformed into a multi-tier
model, in which there are multiple brokers arbitrating components and the seller. A broker bids
to its superior broker or seller for T≥T s(m) as in (21), in which T s(m) denotes the maximal of
optimal completion time solutions of a group s(m) and s(m) subordinate components and brokers
of broker m. In this way, the search space becomes reduced as the bidding process goes to the
superior. In this multi-tier auctioning model communications and computations are more
distributed through the brokers overcoming the problems of the two-tier model.
Multi-tier auctioning model
Broker’s bid
b
m
(T ) = −∞
=
∑
a
a∈s(
m )
b (T )
if T < max{arg max
else
T
∑
a
a∈s(
m )
b (T ) − CCT(T )}
(21)