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x
i
j
Where
denotes the probability that i -th task select the j -th coalition,
.
For solving the Nash equilibrium of mixed strategies, each task t i is allocated to
some coalitions according to its mixed strategy
xx
i
+++=
i
x
i
l
1
and
1
2
, In such a case,
we need to change the utility function of pure strategies, and the expected utility func-
tion is defined as:
xxx x
i
=
(, , , )
i
i
i
l
12
i



x
x
1
i
l
uuu u
=
(, , , )
i
i
i
2
=
ux
i
i
(8)
i
12
l


j
j
j
i
l
x
And we also need to update status of coalition after assigned a task:
busy
=
busy
+
x
i
*
Time
(9)
j
j
j
ji
eex
=−
i
*cos
t
(10)
j
j
j
ij
The fitness function of PSO is defined as follows:
fX
(
)
=
max{max{
uX
(
)
uX
(
*
)}, 0}
(11)
i
i
,
i
i
i
This fitness function is based on the fact: from the point of view of each player, if
it change its strategy, the gain that take pure strategy is less than the gain that take
mixed strategy, and this player will not want to change its strategy. As shown in equa-
tion (10), the value of fitness function of X is zero when X is the best solution X * . The
smaller the value of fitness, the better.
In each time of iteration, the particles update themselves by tracking the two ex-
treme values. One is the optimal solution of each particle, which is called the local
optimal solution, denoted by X i lBest , where N p denotes the number of particles. The
other extreme is the global optimal solution of entire population which is currently
found, denoted by X gBest . During the iteration of PSO, the i -th particle velocity and
position update equation:
i
i
i
i
Vt
( )
+=
wVt c r X
*() **(
+
Xt
( )
k
k
11
lBest
k
(12)
+
cr X
**(
Xt
i
( )
22
gBest
k
Xt
i
( )
+=
Vt
i
( )
++
Xt
i
()
(13)
k
k
k
Where V k i (t) denotes the speed of the i -th particle during the k -th iteration,
X k i (t) denotes the position of the i -th particle during the k -th iteration, X i lBest denotes the
current local optimal solution of i -th particle, X gBest denotes the current global optimal
solution of entire population. r 1 and r 2 denote the random number between 0-1, c 1 and
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