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Where, v'
ti
is the volume of traction in ABTS in ∆
t
. We also note that
V' ≥ V
(7)
Thus,
Auction case 1
is the simple case where there is a direct match in volume between the buyer's
order and the seller's offer. In this case, V' = V, and there is no special contribution from ABTS. In
contrast,
Auction cases 2 and 3
, serve to actively increase volume and utility on the parts of buyer and
seller. This is why
V' ≥ V
. The contribution of this chapter is in using ABTS to maximizing
V'
,
i.e.
to
increase the respective transaction volumes, and utilities, of both the buyer and seller agents, by efficient
matching of the requirements and capacities of the buyer and seller agents.
Reverse Auction Functions
In our model of Reverse Auction Functions, there are several different trading situations. One buyer
(
i
) posts his product requirements including the product quantity
p
in t
1
. The product
quantity is variable but the price is fixed in the proposed model. Thus, trading negotiations and match-
ing of trades relates to product quantity rather than product price. Similarly, trading utility is treated as
a function of product volume, and not the product price.
We describe below a number of scenarios. The base conditions of the trading transaction are de-
scribed as follows. Each seller (
j
) will offer its product with the available quantity
s
q
at state P1. At
this point, there are tree possible actions: (1)
Q
, and its price
Q
and
s
q
are not matched and the deal will be over (t
1
and
P2), 2)
Q
and
s
q
are matched and the transaction will be occurred (t
3
and P3), and 3) the seller gathers
other seller's offer to match the buyer's requested quantity (t
4
, t
5
and P4). At P4 the seller
j
collects other
sellers' offer and present the joined offer to the buyer
i
(t
5
). If the joined offer
J
∑
q Q
s
≠
b
,
j
i
j
=
1
and the buyer does not accept the offer, then either the deal is over (t
2
and P2) or the seller tries collect-
ing new seller's group again (P4 and t
5
). If the joined offer
J
∑
q Q
s
=
b
j
i
j
=
1
then the transaction will be occurred ( t
3
and P3).
Similarly, each seller (
j
) will offer its product to a buyer (
i
) with the buyer's needed quantity
Q
at
Q
and
s
q
are not matched and the deal will
state P5. At this point, there are tree possible actions: (1)
Q
and
s
q
are matched and the transaction will take place (t
8
and P7); and (3) the
buyer gathers other buyer's offer to match the seller's requested quantity (t
9
, t
10
and P8). At P8 the seller
(
j
) collects other buyers' offer and presents the joined offer to the seller
j
(t
10
). If the joined offer
be over (t
7
and P6); (2)
I
∑
Q q
b
≠
s
i
j
i
=
1
and the seller does not accept the offer then either the deal is over (t
7
and P6), or the seller tries collect-
ing new seller's group again (P8 and t
10
). If the joined offer
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