Stochastic Programming - Index of

140 STOCHASTIC PROGRAMMING
the natural constraint
0 ≤ x t ≤ 1.
So far, this is a deterministic control problem.Itisknown,however,that
predicting the net effects of growth, natural mortality and recruitment is very
difficult. In particular, the recruitment is not well understood. Therefore, it
seems unreasonable to use a deterministic model to describe recruitment, as
we have in fact done above. Let us therefore assume that the growth ratio s
is not known, but rather given by a random vector ˜ξ t in time period t.
To fit into the framework of scenario aggregation, let us assume that we are
able to cut the problem after T periods, giving it a finite horizon. Furthermore,
assume that we have found a reasonable finite discretization of ˜ξ t for all t ≤ T .
It can be hard to do that, but we shall offer some discussion in Section 3.4.
A final issue when making an infinite horizon problem finite is to construct a
function Q(z T +1 ) that, in a reasonable way, approximates the value of ending
up in state z T +1 at time T + 1. Finding Q can be difficult. However, let us
briefly show how one approximation can be found for our problem.
Let us assume that all ˜ξ t are independent and identically distributed with
expected value ξ. Furthermore, let us simply replace all random variables with
their means, and assume that each year we catch exactly the net recruitment,
i.e. we let
(
x t = ξ 1 − z )
t
.
K
But since this leaves z t = z T +1 for all t ≥ T + 1, and therefore all x t for
t ≥ T + 1 equal, we can let
Q(z T +1 )=
∞∑
t=T +1
α t−T −1 x t z t = ξz T +1(1 − z T +1 /K)
.
1 − α
With these assumptions on the horizon, the existence of Q(z T +1 ) and a finite
discretization of the random variables, we arrive at the following optimization
problem, (the objective function amounts to the expected catch, discounted
over the horizon of the problem; of course, it is easy to bring this into monetary
terms):
max ∑ s∈S p(s) [ ∑T
t=0 αt z s t x s t + α T +1 Q(z s T +1 ) ]
s.t. zt+1 s = zs t
0 ≤ x s t ≤ 1,
[
(
1 − x ξ t + ξs t
x s t = ∑ p s′ x s′
s ′ t
∈{s} t p({s} . t)
1 − zs t
K
)]
, with z s 0 = z 0 given,
We can then apply scenario aggregation as outlined before.