Agriculture Reference
In-Depth Information
Briefly, the diet formulation linear programme
can be represented algebraically as:
proposed by Jean dit Bailleul
et al
. (2001), and it can
be described mathematically as (Pomar
et al
., 2007):
n
n
(
)
∑
=
∑
min
c
+
b
r
x
mn
i
cx
jj
j
j
j
1
Subject to
x
∈
j
j
=
1
(6.3)
Subject to
(6.2)
n
S
∑
ax
≥≤
,
b
,
for
i
=
12
,
, ...,
m
ij
j
i
j
=
1
where
c
j
is the cost of feed
j, r
j
is the amount of
phosphorus resulting from the
j
th feed,
x
j
is the
amount of feed
j, b
is the unit cost associated
with the excess and unavailable phosphorus
fraction,
x
is the vector of solutions and
S
is the
feasible region, as described in Eqn 6.2. Values
of
b
can be viewed as taxes for phosphorus
excretion.
Phosphorus intake decreased with increas-
ing
b
's, as expected, and diet cost increased with
higher
b
values. The marginal reduction of
phosphorus was calculated from the ratio
between the change in total phosphorus and
ingredient cost at critical values of
b
, i.e. values
of
b
where the solution vector changed basis.
When the methodology was applied to Canadian
data (Pomar
et al
., 2007) during June 2002 in
Quebec, the first change in solution basis was
achieved at a
b
equal to US$0.00696 kg
−1
.
Phosphorus dietary content decreased from
5.287 to 5.181 g kg
−1
and dietary costs increased
from US$196.1 to US$196.8 t
−1
. For this critical
value of
b
, the marginal reduction of P excretion
was 143 g US$
−1
(Pomar
et al
., 2007).
x
≥≥=
012
,
j
,
,...,
n
j
where
c
j
is the cost of feed
j, x
j
is the amount of
feed
j, a
ij
is the content of nutrient
i
in feed
j, b
i
is
the animal's requirement for nutrient
i, n
repre-
sents the number of feeds and
m
represents the
number of constraints. The constraints define a
feasible region
S
that must contain the solution.
Other constraints may be included to define the
feasible region
S
.
The control in formulating the objective
function and constraints allows great flexibility in
specifying the objective of optimization and the
feasible region
S
. In the linear programming
framework, results are assessed with the vector of
solutions and the level of scarce resources usage
(left side of constraint equations). Sensitivity
analyses are used to examine the marginal costs
of restricting resource usage and ranges for which
model solutions are invariant. Shadow prices rep-
resent the change in the objective function opti-
mal value when a constraint equation is relaxed
or strengthened. Therefore, they represent the
marginal cost of strengthening or relaxing a con-
straint by one unit. The classical application of
the linear programming model to diet optimiza-
tion is the formulation of minimum cost and
maximum profit diets. Recently, extensions of the
least-cost diet formulation model were developed
to minimize the environmental impacts of live-
stock (Jean dit Bailleul
et al
., 2001; Pomar
et al
.,
2007; Moraes
et al
., 2012). For instance, by
including an environmental term in the tradi-
tional linear programming algorithm, Pomar
et al
. (2007) formulated diets with reduced levels
of phosphorus, consequently reducing phospho-
rus excretion by pigs. Similarly, Jean dit Bailleul
et al
. (2001) modified the traditional least-cost
formulation algorithm to decrease nitrogen excre-
tion by pigs. In both studies, the objective function
included a term representing the cost associated
with the calculated nutrient excess and unavaila-
ble fractions. The model proposed by Pomar
et al
.
(2007) applies the same concept as the model
Stochastic programming
The linear programming diet formulation model
can be extended through the inclusion of uncer-
tainty, to the composition of feeds, to the animal
nutrient requirements or to the feed costs. For
example, if we assume that feed costs are not
known with certainty over time, which is a rea-
sonable assumption, they become stochastic ele-
ments in our model. By assuming that they vary
according to some probability distribution func-
tion, we specify some degree of certainty about
the values of the vector {
c
j
}. We can, therefore,
construct a stochastic programming model to
determine diets that have minimum variance in
feed costs under a reference maximum expected
cost. Let
c
represent the {
c
j
} vector, and assume
that
c
~N(
m
c
,
W
c
), where
m
c
is the expected value
