Agriculture Reference
In-Depth Information
of
c
and W
c
is the variance-covariance matrix
of
c
. It can be easily shown that the expected value
of
c
T
x
is equal to m
c
T
x
and that the variance of
c
T
x
is equal to
x
T
The probabilistic constraint, along with other
nutrient constraints, defines the feasible region
S
.
In the case where the nutrient requirement
(
b
i
) is the random variable, a certain degree of
certainty can be specified to the random nutri-
ent requirement through the specification of a
probability distribution function. The probabil-
istic constraint can then be formulated to incor-
porate the uncertainty into a parametric form.
Using probability theory (Roussas, 2003), the
chance-constrained equivalent to Eqn 6.5 can
be represented as:
W
c
x
, where superscript
T
represents
the transpose of the vector. The stochastic pro-
gramming model, in which
c
is a random vector,
can then be represented, in matrix notation, as:
T
Ω
Subject to
m
min
xx
c
(6.4)
T
x
≤
e
c
x
∈
n
∑
min
cx
where
x
is the vector of decision variables,
x
T
W
c
x
is the variance of
c
T
x,
m
c
T
x
is the expected value
of
c
T
x
,
e
is the reference value chosen for the
maximum expected diet cost and
S
is the feasible
region, as in Eqn 6.2.
Various objective functions can be formu-
lated to accommodate distinct optimization
objectives (Shapiro
et al
., 2009). For example, the
risk someone is willing to take can be included in
the objective function, and a stochastic program-
ming model can be specified based on the deter-
mined risk (Hazell and Norton, 1986). Stochastic
programming models have been traditionally
applied in diet formulation to incorporate uncer-
tainty in the feed composition matrix, through
the use of chance constraints (Chen, 1973;
St-Pierre and Harvey, 1986a, b). The idea behind
chance-constrained programming is the formu-
lation of a model in which a determined con-
straint is met at a certain probability level. An
application to the diet formulation problem
would be the formulation of a constraint in
which the requirement of a determined nutrient
is met with certain probability. In the context of a
least-cost diet, a model with probabilistic con-
straints can be described algebraically as:
jj
j
=
1
Subject to
(6.6)
n
∑
ax
≥
F
−
1
()
a
ij
j
b
i
i
j
=
1
x
∈
S
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
,
F
b
i
is
the distribution function of the random variable
b
i
,
a
i
is the probability of meeting the
i
th nutrient
requirement,
x
is the vector of solutions and
S
is
the feasible region. The chance constraint, along
with other nutrient constraints, defines the
feasible region
S
.
The probability distribution function assig-
ned to the random variable in the chance con-
straint is determined by the random variable
distribution. For instance, assume that the require-
ment of the
i
th nutrient in a population of ani-
mals follows a normal distribution, with expected
value
bi
).
Then, our knowledge about the nutrient require-
ment can specified through the probability distri-
bution of a normal distribution, and our random
variable
b
i
can be standardized into a standard
normal random variable. In this framework, the
chance-constrained programming model can be
represented mathematically as:
m
bi
and variance
s
bi
, i.e.
b
i
~N(
m
bi
,
m
2
2
n
∑
min
cx
jj
j
=
1
Subject to
n
(6.5)
∑
min
cx
⎛
⎜
⎞
⎟
≥
n
jj
∑
Pr
ax
≥
b
a
j
=
1
ij
j
i
i
Subject to
j
=
1
(6.7)
x
∈
S
n
∑
ax
≥+
m
k
s
ij
j
bi
a
b
i
i
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 requirement of nutrient
i, a
i
is the pro-
bability of meeting the
i
th nutrient requirement,
x
is the vector of feeds and
S
is the feasible region.
j
=
1
x
∈
S
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, m
bi
is
