Agriculture Reference
In-Depth Information
where
V
i
{
x
i
} is the value derived by having daily
gain
u
i
given that the body weight is
x
i
, a
i
{
x
i
, u
i
} is
the dietary cost for live weight
x
i
and daily gain
u
i
at the
i
th stage.
Extensions of this simple dynamic problem
are presented by Kennedy (1986) and Glen
(1980), including maximizing returns per unit
of time and the use of an animal replacement
model in conjunction with this feeding rate
model. The objective function in the dynamic
program is constructed based on the objective
of the optimization. The flexibility in specifying
the model is tremendous, enabling the develop-
ment of a large variety of modelling structures
applied to the minimization of diet costs and live-
stock environmental impacts. The minimization
or maximization of various types of objective
functions can be performed; the restriction in
the technique use relies on the ability of the deci-
sion maker in describing the problem mathe-
matically. For instance, the determination of
optimal policies in the dynamic programming
model can be hampered by the difficulty in the
definition of the model final conditions (Soetaert
and Herman, 2009). Moreover, realistic repre-
sentations of the model state space may have
very large dimensions, exposing the modeller to
a complex problem, often referred to as the curse
of dimensionality (Kristensen
et al
., 2008). As a
result, if the behaviour of state variables is
confidently known over time, other modelling
techniques may represent simpler strategies, for
example linear programming.
Dynamic programming
Planning and resource allocation are tradi-
tional applications of dynamic programming in
forestry and fisheries management (Kennedy,
1986). A classical application in livestock sci-
ence is the animal replacement problem. Several
dynamic programming and Markov decision
models (Kristensen, 1991; Kristensen and
Søllested, 2004) have been used to find optimal
replacement policies in animals of different spe-
cies. Dynamic programming applications in
feed formulation are not as widely used as linear
programming applications; however, the
strength of dynamic programming is that deci-
sions can be made over time. Optimum combi-
nations of feeds can be identified and resources
that optimize some utility function, as adopted
policies evolve over time, can be characterized.
An application of dynamic programming in diet
optimization is the identification of a sequence
of optimum feeding strategies in relation to dis-
tinct production rates. The example from
Kennedy (1986) illustrates the optimal fatten-
ing of a steer over time. The problem is based on
the choice between sequences of rations that
will produce different daily gains at different
costs when fed to animals of various body
weights. In this example from Kennedy (1986), it
is assumed that the six daily gains range from
0.25 to 1.5 kg at intervals of 0.25 kg. The length
of each stage, or the time between decisions, is
assumed to be 28 days and the initial steer body
weight is 300 kg. It is a four-stage scenario, in
which at the last stage steers are worth US$3 kg
-1
live weight. For specific diet costs and rates of
gain, refer to Kennedy (1986). The maximum
body weight specified was 440 kg, and the prob-
lem then becomes the identification of the
sequence of diets that maximize the net revenue.
The model can be solved by backward induction
techniques and be mathematically represented
by the recursive equation:
Linear programming
Linear programming is the most widely used
optimization technique in diet formulation.
Since the development of the simplex algo-
rithm by Dantzig (1963), several extensions of
this method have been adapted to the diet for-
mulation problem. In a brief mathematical
description, linear programming can be char-
acterized in terms of an objective function,
which describes the contributions of each
decision variable to the optimal value, and
constraints, which limit the use of scarce
resources. For linear programming assump-
tions and an extensive mathematical and eco-
nomical description, refer to Winston (1987).
{}
=
Vx
max[
ax u
Vx
−
{
,
}
i
i
i
i
+
{
28
321
+
u
}],
i
+
1
i
i
for
i
=
,,
with
{}
Vx
=
3
x
,
for
i
=
4
(6.1)
44
4
Subject to
∈
(
)
u
x
x
025 050
.
,
.
,...,
150
.
i
=
+
300
1
28
u
i
≤
440
i
