Minimizing Environmental Impacts of Livestock Production Using Diet Optimization Models - Sustainable Animal Agriculture - page 75

i and j denote the criteria or goal (Dubeau et al .,
2011). Efficient solutions can also be described
through the idea of Pareto efficiency (Tamiz
et al ., 1999), which characterizes a solution as
efficient if and only if there is no other solution
that improves the objective value of a deter-
mined goal without negatively affecting other
goals. Results from the three criteria model
(Eqn 6.8), when applied to the growing pig data
from Quebec (Dubeau et al ., 2011), show that
when a nitrogen excretion reduction of 25% and
a phosphorus excretion reduction of 16% were
achieved, the diet cost increased by 10%.
A classical application of multi-criteria pro-
gramming is goal programming. The formula-
tion of a goal-programming model involves a
'pay-off ' matrix because optimization goals can
conflict. In the goal-programming framework,
each objective is optimized individually, and an
objective function is created to minimize devia-
tions from the result of the individual optimiza-
tions. The individual objective function of each
goal is then turned into an equality constraint,
which will restrict the system along with
resource constraints (Romero and Rehman,
1989). The goal-programming model can be
described mathematically as:

be specified. However, the derivation of the
weights is usually a complex task and can often
be particular to each system to be optimized
(Hannan, 1985). Moreover, the weight attribu-
tion for each deviation goal can be subjective
and lead to distinct efficient solutions (Gass,
1987). Therefore, the use of multi-criteria
programming is often limited to systems for
which weights are specified with confidence or
the most efficient solution vector can be
identified graphically or numerically (Dubeau
et al ., 2011).

Incorporation of environmental policies
in the diet optimization model

The minimization of environmental impacts
from livestock production units is traditionally
enforced by the implementation of a regulatory
policy. Therefore, the model structure should
incorporate the effects of the policy in the opti-
mization. The regulatory policy can be incorpo-
rated into the model through different strategies.
Pomar et al . (2007) and Jean dit Bailleul et al .
(2001) included an environmental term in the
objective function to represent a tax on phos-
phorus and nitrogen excretion in pigs. In a
recent publication, Moraes et al . (2012) modified
a base traditional least-cost diet formulation
model by using two different regulatory policy
strategies. Methane emissions were taxed or
capped and diets were formulated to minimize
costs and methane emissions simultaneously.
The marginal costs of methane emissions miti-
gation were derived using the shadow prices
from a methane emissions constraint and a cap
and trade policy scheme was simulated using the
carbon credit market. The effects of these two
policy strategies on feed selection for diet
formulation, system profitability and sensitivity
between environmental impacts were exten-
sively examined. In the next section, the model
by Moraes et al . (2012) is described in detail, and
the results generated by the model structure are
examined. The use of this modelling framework
provides a comprehensive example of the utiliza-
tion of optimization models in minimizing the
environmental impacts of livestock in an opti-
mal manner. The modelling framework also pro-
vides information for determining whether

n

= å 1

min wn ( + p )

j j j

j

Subject to
z

(6.9)

() x +-= n pg

j j j j

x S

np

Î

, ³ 0

j j

where w j is the weight of goal j, n j is the negative
deviation from goal j, p j is the positive deviation
from goal j , x is the vector of decision variables,
g j is the objective target for the goal j, z j ( x ) is
the objective function for the goal j and S is the
feasible region as defined in Eqn 6.2.
Andre et al . (2009) developed goal-
programming models capable of incorporating
environmental and economic goals in policy
development. A model application in Spain was
presented, in which macroeconomic and envi-
ronmental goals (gas emissions) were applied in
an equilibrium model. The formulation of goal-
programming models requires that a ranking be
established between criteria or that weights,
which assign a level of importance to each goal,