Agriculture Reference
In-Depth Information
the expected value of the requirement of nutri-
ent
i
,
k
a
i
is the value of the standard normal vari-
able when the probability of the constraint to be
satisfied is
a
i
(and can be found in Z tables),
s
bi
is
the standard deviation of the requirement of the
i
th nutrient,
x
is the vector of solutions and
S
is
the feasible region as described in Eqn 6.2. The
chance constraint, along with other nutrient
constraints, defines the feasible region
S
.
The development of a diet formulation
model with probabilistic constraints determin-
ing uncertainty in the feed composition matrix
follows the same concept of Eqn 6.6. The differ-
ence from the random nutrient requirements
model is that now
a
ij
are the random variables.
For an application of stochastic programming in
diet formulation, in which single and joint
chance constraints determined the inclusion of
uncertainty in the feed composition matrix,
refer to St-Pierre and Harvey (1986a, b). A deter-
ministic equivalent chance constraint can usu-
ally be derived and used to identify optimal
solutions in the stochastic programming frame-
work (Charnes and Cooper, 1963; Symonds,
1968). However, stochastic programming mod-
els and their deterministic equivalents might
exhibit non-linear formulation, increasing the
complexity in the model optimization (Kall and
Wallace, 1994). Linear approximations of
chance constraints have been proposed in the
literature (Olson and Swenest, 1987) and were
investigated by St-Pierre and Harvey (1986a), in
which non-linear chance constraints were speci-
fied to introduce uncertainty in feed composition.
In the non-linear programming framework, spe-
cialized optimization algorithms are required in
model optimization. The interpretation of solu-
tions and sensitivity analysis are not straightfor-
ward, as in the linear programming, for example
the existence of local versus global optimum.
A valuable element in the examination of opti-
mum scenarios is the evaluation of dual values,
especially shadow prices. In the non-linear
programming framework, the calculation and
interpretation of shadow prices are more com-
plex than in linear programming models. From
an environmental perspective, shadow prices
have valuable information because they can be
interpreted as the marginal costs of mitigation
strategies and can be used, in an economic
framework, to derive abatement cost curves.
Therefore, the inclusion of uncertainty into the
linear programming model might increase
the complexity in the interpretation of results
and, from and an environmental perspective,
reduce the usefulness of sensitivity analysis.
Nevertheless, in cases where solutions and
shadow prices are derived with confidence, and
can be directly interpreted, the inclusion of
uncertainty can represent the variation in bio-
logical variables better. Moreover, probabilistic
assessments can be incorporated into decision
making. For instance, the trade-off between
increasing the chance of meeting animal nutri-
ent requirements and increasing environmental
impacts can be examined.
Multi-criteria programming
The simplex algorithm can be modified to opti-
mize a linear program when there is more
than one objective function to be maximized or
minimized. Multiple criteria programming usu-
ally involves the minimization of deviations or
variations from specific goals, which are usually
set by individual optimizations. From an envi-
ronmental perspective, multi-criteria linear pro-
gramming can be used to minimize livestock
environmental impacts and diet costs. Recently,
a multi-criteria programming model was devel-
oped to minimize diet costs and nitrogen and
phosphorus excretion from growing pigs
(Dubeau
et al
., 2011). The model can be repre-
sented, in matrix notation, as:
ì
T
min
min
min
Z
Z
Z
=
=
=
cx
px
px
1
ï
T
í
2
r
ï
T
(6.8)
3
h
Subject to
Î
x
S
where
c
denotes the vector of feed prices,
x
is the
vector of feeds,
p
r
is the vector of the nitrogen
content of feeds,
p
h
is the vector of the phospho-
rus content of feeds and
S
denotes the feasible
region, as described in Eqn 6.2.
In order to find the solution to this problem,
efficient solutions that comprise the efficient set
must be identified. A solution is efficient if there
exists no other feasible solution
x
¢ such that
z
i
(
x
¢) £ z
i
(
x
) for all
i
, and z
j
(
x
¢) < z
j
(
x
) for at least
one
j
, where z denotes the objective function,
