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,