Directory UMM :Data Elmu:jurnal:A:Agricultural Systems:Vol67.Issue3.2001:

Agricultural Systems 67 (2001) 201±215
www.elsevier.com/locate/agsy

A multi-objective programming approach to feed
ration balancing and nutrient management
P.R. Tozer a,*, J.R. Stokes b
a

Dairy and Animal Science, The Pennsylvania State University, University Park, PA 16802-3503, USA
b
Agricultural Economics, The Pennsylvania State University, University Park, PA 16802-5601, USA
Received 27 July 2000; received in revised form 2 October 2000; accepted 10 October 2000

Abstract
This paper examines the potential to use multiple objective programming to reduce nutrient
excretion from dairy cows through incorporation of nutrient excretion functions into a ration
formulation framework. In a typical ration formulation model, a ration is formulated to
minimize cost while providing sucient nutrients to meet the needs of the animal type being
fed. To reduce the nutrient loading, rations can be formulated to minimize cost, and nitrogen
and phosphorus excretion using multiple objective programming. Rations were initially formulated to minimize cost, nitrogen excretion and phosphorus excretion. Compromise programming was then utilized to examine the impacts on ration formulation of combining the
three individual objectives. The multiple objective ration formulation reduced phosphorus

excretion by 5% and marginally reduced nitrogen excretion with a small increase in ration
cost compared to the single objective minimum cost ration. Multiple objective programming
does have the potential to reduce nutrient excretion. # 2001 Elsevier Science Ltd. All rights
reserved.
Keywords: MINIMAX; Multi-objective programming; Nitrogen; Phosphorous; Rations; Dairy cows

1. Introduction
Linear programming (LP) has formed the basis of livestock ration formulations
since Waugh (1951) de®ned the feeding problem in mathematical form. As Rehman
and Romero (1984) point out, however, LP has many limitations when formulating

* Corresponding author. Tel.: +1-814-863-3917; fax: +1-814-865-7442.
E-mail address: ptozer@das.psu.edu (P.R. Tozer).
0308-521X/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.
PII: S0308-521X(00)00056-1

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P.R. Tozer, J.R. Stokes / Agricultural Systems 67 (2001) 201±215

rations in practice. These problems include the singularity of the objective function
and the rigidity of the constraint set. The singularity of the objective function refers
to the reliance on cost alone as the most important factor in determining the composition of the ration. Lara (1993) also criticizes practical applications of LP due to
the restrictions placed on the decision maker's preferences through a singular
objective function. Lara (1993) explains this problem in the context of available
feeds in that dairy producers often express preferences for feed ration ingredients
that have been used previously over those with which they are unfamiliar.
In reality, producers are likely to have many objectives in mind when formulating
a ration. One additional objective that is important, and likely to become even more
important in the future, is the minimization of nutrient excretion. Overfeeding
nutrients is a problem in many intensive agricultural feeding operations in the United States. Phosphorus overfeeding, for example, often occurs because diets that are
low in phosphorus can lead to reproduction problems in cows. Also, as dairy cows
are not 100% ecient in converting intake nutrients into either tissue or milk, any
excess nutrients are necessarily excreted. For example, dairy cows only utilize
approximately 50% of phosphorus in the ration when fed to expected requirements
(Wu et al., 2000; NRC, 1989). Cost minimizing feed rations can also easily lead to an
increase in fecal and urinary protein, the two sources of excreted nitrogen, through
overfeeding of one or more of the protein fractions of the ration.
While these ideas are supported by research examining cows fed on low quality
pastures, many areas such as the northeastern United States are characterized by

relatively higher quality pastures (Knowlton and Kohn, 1999). In this region, dairy
cows are fed rations comprised of a high quality forage source, such as corn or
alfalfa silage. The feed ingredients typically contain relatively higher levels of phosphorus. Additionally, population concentration in this region and the fact that the
area relies on the Chesapeake Bay watershed implies better nutrient management is
needed.
Overfeeding of nitrogen and phosphorous ultimately leads to an excess of some
nutrients in the soil and this in turn can lead to pollution of watersheds like the
Chesapeake. Leaching and surface run-o€ of nutrients can also cause groundwater
contamination and eutrophication or algal blooms (Sharpley, 2000). Whether induced
by personal environmental concerns or governmental regulation, minimizing excess
nitrogen and phosphorous excretion are legitimate objectives in their own right.
Multiple-objective programming potentially o€ers recourse to reduce the excretion of undesirable nutrients in conjunction with the typical singular objective of
minimizing ration cost. Consistent with Rehman and Romero's (1984) criticism
regarding the rigidity of the constraint set for typical LP feed rations, implementation of a multi-objective formulation necessitates a softening of the constraints. The
purpose of this research then, is to examine the impact on feed rations that accommodate more objectives than just cost minimization. More speci®cally, a nonlinear
multi-objective programming model is developed to balance a dairy cow feed ration
under the guise of cost minimization, phosphorous excretion minimization, and
nitrogen excretion minimization. Alternative assumptions regarding the preferences
of the decision maker are also explored in the context of the formulated rations.

P.R. Tozer, J.R. Stokes / Agricultural Systems 67 (2001) 201±215

203

2. Multiple objective framework and models
To model nutrient excretion, it is necessary to augment the simple LP ration formulation model beyond intake levels to include excretion functions. Excretion
functions must depend on intake nutrients to maintain the simplicity of the programming model. This approach also o€ers the advantage of casting the problem in
a managerial context where producers and/or feed manufactures can assist in the
management of nutrient excretion through one set of control variables, namely,
the ingredients selected for feeding.
While NRC (1989) provides detailed information regarding the excretion of
nitrogen for the dairy cow, such is not the case for phosphorous. As a result, a
nonlinear equation estimated by Morse et al. (1992) is used in the analysis that follows. Morse et al.'s equation relates phosphorous intake and milk production to
phosphorous excretion.
Speci®cation of a multi-objective model necessitates target values for each of the
objectives. One problem that naturally arises is the lack of knowledge regarding
the appropriate levels for these targets. To circumvent this issue, the cost target,
C, the nitrogen excretion target, N, and the phosphorous excretion target, P, are
obtained via separate linear and nonlinear programming models. To clarify, C is
determined by a typical cost minimization LP. The model is:

min C ˆ

I
X

i Xi

…1†

iÿ1

subject to :
I
X

aij Xi 5bj 8 j ˆ 1; 2; . . . ; J ÿ 1

…2a†

iˆ1

I
X
aiJ Xi 4bj

…2b†

iˆ1

The objective function speci®ed by Eq. (1) depicts the summation of the prices of
the i feed ingredients (denoted i) times their use (denoted Xi) in the optimal ration.
Eqs. (2a, b) are typical nutritional lower and upper bound constraints. The technical
coecients aij measure the amount of the jth nutrient in the ith feed ingredient while
the right hand sides, bj, give the minimum or maximum amount of the jth nutrient
allowable in the ration depending on the indicated sign of the inequality. Notice
there are a total of I feed ingredients and J nutrients. Note also that j=J refers to
dry matter as indicated by the constraint Eq. (2b). Table 1 summarizes all model
variable de®nitions and their units.
Similar to the ®nding the minimum cost target, the nitrogen excretion target is
found by minimizing a nitrogen excretion function subject to Eq. (2a) and (2b)
and the two equality relations used to determine total dry matter (DM) and net

energy lactation (NEL) from the ration. The linear program is:

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P.R. Tozer, J.R. Stokes / Agricultural Systems 67 (2001) 201±215

Table 1
Summary of model notation
De®nition
Indices
i
j

Ingredient
Nutrient

Parameters
wC
wN
wP

l
i
aij
bj
k
C*
N*
P*

Cost weight (no unit)
Nitrogen excretion weight (no unit)
Phosphorous excretion weight (no unit)
Maximum deviation from target values (%)
Price of ingredient i ($/kg as fed)
Amount of nutrient j in ingredient i (%, Mcal/kg DM, or g/kg DM)
Required amount of nutrient j (kg, Mcal)
Phosphorous intake eciency (%)
Target ration cost ($/cow/day)
Target nitrogen excretion (kg/cow/day)
Target phosphorous excretion (kg/cow/day)

Variables
Xi

Required level of ingredient i in ration ((Mcal, kg, or g)/cow/day)

Functions
FP(; )
UP()
p(Xi)
(Xi)
(Xi)
C(Xi)
N(Xi)
P(Xi)

Fecal protein (g/cow/day)
Urinary protein (g/cow/day)
Total phosphorous intake (g/cow/day)
Total dry matter intake (g/cow/day)

Total net energy lactation (Mcal/cow/day)
Ration cost in dollars ($/cow/day)
Nitrogen excretion (kg/cow/day)
Phosphorous excretion (kg/cow/day)

min N ˆ 0:16‰FP…; † ‡ UP…†Š
subject to :
I
X
aij Xi 5bj 8 j ˆ 1; 2; . . . ; J ÿ 1

…3†

…2a†

iˆ1
I
X
aiJ Xi 4bj

…2b†

iˆ1
I
X

aiJ Xi ˆ 

…3c†

iˆ1
I
X
aiJÿ1 Xi ˆ 

…3d†

iˆ1

The objective function depicts nitrogen excretion as being functionally related to
fecal (FP) and urinary protein (UP) functions. Eqs. (3c, d) merely accumulate total

P.R. Tozer, J.R. Stokes / Agricultural Systems 67 (2001) 201±215

205

dry matter () and net energy lactation () from the ration. These intakes need to be
calculated given the objective function. The speci®c functions used to relate  and 
to the functions FP(, ) and UP() are linear and are those published by the NRC
(NRC, 1989, pp. 71±77).
Target phosphorous excretion, P, is found by minimizing Morse et al.'s (1992)
nonlinear equation subject to Eqs. (2a, b), and an equality relation that determines
the optimal ration's total phosphorous intake. More speci®cally, the nonlinear program is:
ÿ


…4†
min P ˆ k 14:67 ‡ 0:6786p ‡ 0:00196p2 ÿ 0:317m
subject to :
I
X

aij Xi 5bj 8 j ˆ 1; 2; . . . ; J ÿ 1

…2a†

iˆ1
I
X
aiJ Xi 4J

…2b†

iˆ1
I
X

aiJÿ2 Xi ˆ p

…4e†

iˆ1

The parameter k