Dynamic programming

The value of the goal function is

VT(t+ 1 )(w)=

r w

∑tn+=^10 kn

+

s t

∑

n= 0

kn

s

k



w w

∑tn+=^10 kn



=

=

p

w(q^1

∑tn+=^10 kn

+

s

t

∑

n= 0

kn

s

k∑

t+ 1

n= 0 kn^1

∑tn+=^10 kn

)=

=

p

w(q^1

∑tn+=^10 kn

+

s

t

∑

n= 0

kn

s

k∑

t+ 1

n= 1 kn

∑tn+=^10 kn

)=

=

p

w(q^1

∑tn+=^10 kn

+

s

t+ 1

∑

n= 1

kn

s

∑tn+=^11 kn

∑tn+=^10 kn

)=

p

wq^1

∑tn+=^10 kn

( 1 +

t+ 1

∑

n= 1

kn)=

p

w

s

t+ 1

∑

n= 0

kn.