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

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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 sucient 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 for-mulated to minimize cost, nitrogen excretion and phosphorus excretion. Compromise pro-gramming 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

Agricultural Systems 67 (2001) 201±215

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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 com-position 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 Uni-ted 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% ecient 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 phos-phorus. 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.

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2. Multiple objective framework and models

To model nutrient excretion, it is necessary to augment the simple LP ration for-mulation model beyond intake levels to include excretion functions. Excretion functions must depend on intake nutrients to maintain the simplicity of the pro-gramming model. This approach also oers 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 fol-lows. 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:

minCX I

iÿ1

iXi 1

subject to :

XI

i1

aijXi5bj8j1;2;. . .;Jÿ1 2a

XI

i1

aiJXi4bj 2b

The objective function speci®ed by Eq. (1) depicts the summation of the prices of theifeed ingredients (denotedi) times their use (denotedXi) in the optimal ration.

Eqs. (2a, b) are typical nutritional lower and upper bound constraints. The technical coecientsaijmeasure the amount of thejth nutrient in theith feed ingredient while

the right hand sides,bj, give the minimum or maximum amount of thejth nutrient

allowable in the ration depending on the indicated sign of the inequality. Notice there are a total ofIfeed ingredients andJnutrients. Note also that j=Jrefers 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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minN0:16FP; UP 3

subject to :

XI

i1

XI

i1

aiJXi4bj 2b

XI

i1

aiJXi 3c

XI

i1

aiJÿ1Xi 3d

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

Table 1

Summary of model notation

De®nition

Indices

i Ingredient

j Nutrient

Parameters

wC Cost weight (no unit)

wN Nitrogen excretion weight (no unit) wP Phosphorous excretion weight (no unit) l Maximum deviation from target values (%) i Price of ingredienti($/kg as fed)

aij Amount of nutrientjin ingredienti(%, Mcal/kg DM, or g/kg DM) bj Required amount of nutrientj(kg, Mcal)

k Phosphorous intake eciency (%) C* _{Target ration cost ($/cow/day)} N* _{Target nitrogen excretion (kg/cow/day)} P* _{Target phosphorous excretion (kg/cow/day)}

Variables

Xi Required level of ingredientiin ration ((Mcal, kg, or g)/cow/day)

Functions

FP(; ) Fecal protein (g/cow/day) UP() Urinary protein (g/cow/day)

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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 relateand

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 pro-gram is:

minPkÿ14:670:6786p0:00196p2ÿ0:317m 4

subject to :

XI

i1

XI

i1

aiJXi4J 2b

XI

i1

aiJÿ2Xip 4e

The parameter k<1 is the rate of eciency with which dairy cows utilize phos-phorous (k=0.5, 0.6, 0.7: NRC, 1989; Wu et al., 2000), while the parameter mis milk production per day, a constant. Phosphorous intake is calculated via the equality relation (4e) and is denoted byp.

Notice that determined this way, the cost, nitrogen excretion, and phosphorous excretion levels are optimal in the sense that there can be no lower values achievable. Once these three programs are solved,C_,_N_{, and} _P*_{are noted and the following}

multi-objective programming model is solved using a MINIMAX formulation:

minl 5

subject to :

XI

i1

aijXi5bj8J1;2;. . .;Jÿ1 2a

XI

i1

aiJXi4bJ 2b

XI

i1

aiJXi 3c

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XI

i1

aiJÿ1Xi 3d

XI

i1

aiJÿ2Xip 4e

XI

i1

iXiC 5f

0:16FP; UP N 5g

kÿ14:670:6786p0:00196p2ÿ0:317m

P 5h

wCCÿC=C4l 5i

wNNÿN=N4l 5j

wPPÿP=P4l 5k

Here,lis a parameter that ensures the solution minimizes the maximum deviation from target values speci®ed for each the three objectives. The new equality con-straints (Eqs. 5f, g, h) are the objective functions from the other three models and merely account for total cost, nitrogen excretion, and phosphorous excretion in the multi-objective programming ration. The constraints speci®ed by Eq. (5i, j, k) in the MINIMAX model measure percentage deviations from the target values when the weights for the goals (thewC,wN, andwP) are equal to one. However, the

model is ¯exible in that the weights can be adjusted to re¯ect a decision maker's subjective importance regarding each objective. In such a case, the weights are used to increase or decrease the relative importance of achieving values close to the target values.

The rations formulated in this study are based on a ``standard'' cow and in general utilize the nutrient requirements speci®ed in the National Research Council's pub-lication the Nutrient Requirements of Dairy Cattle (NRC, 1989). The ``standard'' cow weighs 600 kg, produces 30 kg of milk per day with a fat content of 3.5% and is gaining 300 g per day; in production terms this cow would be considered a mid-lactation cow. This ``standard'' cow was chosen arbitrarily to demonstrate the modeling technique and should not be considered representative of any particular production process.

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Table 2

Nutrient content of ration ingredients and right hand side values for the ration formulation problemsa

Feeds DM

Alfalfa hay 0.91 177.12 1.41 49.59 438.24 12.17 2.08 2.65 23.56 1.08 275.00 26.00 9.00 34.00 2.37 7.58 Mostly legume hay 0.91 148.91 1.23 46.16 490.32 9.40 2.06 2.36 20.52 0.29 169.00 23.00 8.00 37.00 1.63 5.36 Mostly grass hay 0.92 110.72 1.17 37.64 567.30 6.28 1.93 2.10 17.66 0.25 131.00 26.00 8.00 48.00 1.37 6.22 Grass hay 0.92 97.63 1.10 36.12 607.86 4.67 1.87 1.47 16.95 0.77 174.00 35.00 8.00 83.00 1.75 5.07 Straw 0.93 46.50 0.82 27.90 697.50 3.03 0.95 1.30 12.83 1.12 247.00 22.00 6.00 220.00 1.58 2.33 Alfalfa silage 0.40 77.57 1.32 17.84 206.04 2.06 0.49 1.01 10.67 0.21 391.00 27.00 10.00 44.00 0.93 2.38 Mostly legume silage 0.39 68.03 1.23 17.01 211.14 1.74 0.44 0.98 9.93 0.12 268.00 27.00 8.00 41.00 0.86 2.35 Mostly grass silage 0.37 51.94 1.19 14.02 226.31 1.20 0.39 0.85 8.50 0.11 309.00 29.00 8.00 59.00 0.74 2.71 Grass silage 0.36 47.92 1.10 13.90 228.69 0.88 0.38 0.80 8.53 0.29 456.00 34.00 9.00 90.00 0.80 3.05 Corn silage 0.33 26.81 1.61 8.31 162.19 0.27 0.23 0.60 3.34 0.03 179.00 25.00 6.00 33.00 0.30 0.99 Canola meal 0.90 335.38 1.59 77.14 0.00 6.46 9.81 5.06 10.31 0.80 229.00 40.00 7.00 59.00 5.79 9.67 Cottonseed 0.89 220.57 2.40 90.43 0.00 1.28 4.78 3.30 10.63 0.12 70.00 36.00 6.00 17.00 1.96 0.54 Dry distillers grain 0.90 265.80 2.18 143.53 0.00 1.46 6.49 2.88 9.73 2.29 180.00 59.00 8.00 28.00 3.60 1.89 Soybean meal 0.90 473.98 2.07 165.89 0.00 3.21 5.70 2.60 20.97 0.95 227.00 64.00 16.00 44.00 3.49 0.45 Heated soybeans 0.93 400.53 2.07 200.26 0.00 2.48 5.48 2.50 19.80 0.11 163.00 50.00 15.00 31.00 3.42 0.00 Monosodium phosphate 0.99 0.00 0.00 0.00 0.00 0.00 245.03 0.00 0.00 190.08 0.00 0.00 0.00 0.00 0.00 0.00 Dynamate 0.99 0.00 0.00 0.00 0.00 0.00 0.00 114.84 183.15 0.00 0.00 0.00 0.00 0.00 217.80 0.00 Dicalcium phosphate 0.99 0.00 0.00 0.00 0.00 176.42 205.82 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Bakery by-product 0.82 108.11 2.05 21.62 0.00 2.01 2.01 0.98 3.11 5.82 223.00 40.00 7.00 35.00 1.15 8.76 High moisture ear corn 0.67 56.11 1.92 19.64 0.00 0.13 1.20 0.87 3.21 0.05 78.00 22.00 2.00 9.00 0.53 0.47 High moisture shelled corn 0.71 64.97 2.01 22.74 0.00 0.10 1.53 1.29 3.00 0.03 53.00 21.00 1.00 6.00 0.64 0.36 Hominy feed 0.89 92.93 2.14 60.40 0.00 0.31 3.45 3.72 5.13 0.10 95.00 35.00 3.00 11.00 0.89 0.53 Wheat middlings 0.90 172.93 1.92 36.31 0.00 1.04 7.87 1.90 10.30 0.48 149.00 87.00 8.00 120.00 2.24 0.00 Dry shelled corn 0.88 79.29 1.98 41.23 0.00 0.31 2.33 1.06 1.25 0.18 54.00 27.00 3.00 11.00 0.88 0.70

Units of RHS kg g Mcal g g g g g g g g g g g g g RHS 20 3016 32.03 1137 5700 113 36 40 180 36 50 40 10 40 40 50

a _{All measures except DM are on a unit per kg DM basis. CP, crude protein; UIP, undigestible intake protein; NDF, natural detergent ®ber.}

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for variation in dry matter intake. Current NRC requirements do not include neutral detergent ®ber (NDF) requirements for cows. Therefore NDF was derived from Adams et al. (1985) and is a measure of the ®ber requirements of the cow. Net energy of lactation () measures the total energy required by a dairy cow for maintenance and milk production and is speci®ed in mega-calories (Mcal). All mineral requirements are also speci®ed in absolute units, i.e. grams, rather than percentages for the same reason as for the protein fractions.

3. Data

The nutrient and mineral content of the feeds used in this study were derived from the summary analysis of the Northeast DHIA Forage Testing Laboratory for December 1995 (Northeast DHIA, 1995). These data were used as the summary contained measures of means for the nutrients and feeds of interest. A summary of the mean nutrient and mineral contents of each feed used in this study is shown in Table 2. There were 21 feeds used in the study: four types of hay (legume, mixed mostly legume, mixed mostly grass, and grass); straw; ®ve types of silage (legume, mixed mostly legume, mixed mostly grass, grass, and corn); ®ve energy feeds (high moisture ear corn, high moisture shelled corn, dry shelled corn, hominy feed, and wheat middlings); and ®ve protein feeds (canola meal, whole cottonseed, dry dis-tillers grain, soybean meal, and heated soybean meal) and one by-product feed (bakery by-product).

These feeds are those typically available to dairy producers in the northeast Uni-ted States on a continuous basis throughout the production year. Three mineral supplements were also included as potential ration ingredients. These supplements were monosodium phosphate (25% P, 19.2% Na), Dynamate (22% S, 11.6% Mg, 18.5% K), and dicalcium phosphate (21% Ca, 18% P). Prices for all feeds (Table 3) are from the Feed Price List of June 2000 and are representative costs for the feeds delivered on-farm in the Northeast United States (Ishler, 2000).

4. Results and discussion

4.1. Single objective solutions

The simple linear programming rations for minimum cost, minimizing nitrogen excretion, and minimizing phosphorus excretion, based on a phosphorus eciency of 0.5, are presented in the ®rst three columns of Table 4. Beginning with the least cost ration, the minimum cost is $2.84 per day with nitrogen excretion of 276 g/day and phosphorus excretion of 33.63 g/day. The ration consists of two forage sour-ces; alfalfa hay and cereal straw1_{; one protein feed; dry distillers grain; one energy}

1 _{The level of straw included in this ration is relatively high and in most cases this ration formulation}

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feed; wheat middlings; and the bakery by-product. A small amount, 70 g/day, of dicalcium phosphate was also included to provide sucient minerals in the ration. The dry matter intake in the ration is at the upper bound of the allowable range at 20.04 kg.

Formulating the ration to minimize nitrogen excretion reduced the amount of nitrogen excreted by 2.3% but increased the cost of the ration by 28.5% to $3.66 and increased phosphorus excretion by 20.0%. The ration formulated included grass hay and whole cottonseed at the expense of straw, dry distillers grain, wheat mid-dlings and bakery byproduct. Also included was a small amount of monosodium phosphate and Dynamate.

Minimizing phosphorus excretion increases the cost of the ration above the cost minimizing solution to $3.36 with no change in the level of nitrogen excreted. However a 4.6% reduction in phosphorus excretion is noted. The ration mix for the phosphorus minimization solution is dierent again when compared to the two previous solutions. This is due to the dierent valuation placed on nutrients in the objective function.

Table 3

Prices of potential feed ingredients

Feed Price ($/kg as fed)

Alfalfa hay 0.16

Mostly legume hay 0.14

Mostly grass hay 0.13

Grass hay 0.13

Straw 0.10

Alfalfa silage 0.08

Mostly legume silage 0.07

Mostly grass silage 0.06

Grass silage 0.06

Corn silage 0.04

Canola meal 0.17

Cottonseed 0.20

Dry distillers grain 0.15

Soybean meal 0.23

Heated soybeans 0.25

Monosodium phosphate 1.32

Dynamate 0.43

Dicalcium phosphate 0.64

Bakery by-product 0.09

High moisture ear corn 0.07

High moisture shelled corn 0.09

Hominy feed 0.13

Wheat middlings 0.09

Dry shelled corn 0.11

due to the subjective nature of such constraints. If these constraints were included in the model the costs of each ration would increase as relatively higher priced ingredients replace straw in the ration.

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4.2. Multiple objective solutions

Having established the target levels for each of the three objectives, the MINI-MAX model was solved under a variety of weights. Assuming equal weights (i.e

wC=wN=wP=1) for each objective the ration formulated costs $2.90 per day, an

increase over the least-cost ration of $0.06 or 1.8% (Table 4). However, the increase in cost is oset by a marginal reduction in nitrogen excretion of 0.5%, and a modest reduction in phosphorus excretion of 5%, when compared to the cost minimization solution.

Comparing the equal-weight MINIMAX results to the other singular objective function models, it is apparent that better nutrient management can be obtained with a modest increase in the cost of the ration. The ration ingredients of the equal weighted MINIMAX are similar to the least-cost ration except that mixed-mostly grass hay has replaced most of the straw and 40% of the alfalfa hay.

Examination of the remaining MINIMAX models shows that the rations for the unequally weighted MINIMAX formulations are similar to those from the equal-weighted case. In these models, there are marginal reductions in the level of nitrogen excreted, but no change in the level of phosphorus excreted, when compared to the equal-weight case. This, because the level of phosphorous excretion is at the mini-mum generated from the single objective function model. Also, when comparing the equal-weight MINIMAX case with the MINIMAX model where the phosphorus

Table 4

Ration formulations, costs nitrogen excretion, phosphorus excretion and dry matter intake for minimum cost, minimum nitrogen excretion and minimum phosphorus excretion single objective function models and various weightings for the MINIMAX multiple objective formulation based on a phosphorus utili-zation eciency of 50%

Alfalfa hay 6.49 7.46 8.41 3.92 5.17 3.71 3.92

Mixed-mostly-grass hay 7.07 3.85 8.12 7.07

Grass hay 4.85

Straw 4.78 3.60 0.71 2.52 0.71

Whole cottonseed 8.15 1.56

Dry distillers grain 4.48 4.54 4.51 5.25 4.54

Heated soybeans 1.93

Monosodium phosphate 0.13 0.10

Dynamate 0.01 0.01

Dicalcium phosphate 0.07 0.06 0.06 0.06 0.06

Bakery by-product 5.19 1.00 5.56 5.47 4.91 5.56

Wheat middlings 1.46 0.35 0.75 0.35

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objective has a weight of two compared to one for the other objectives, we see that the solutions are identical. The reason for this is that, in the equal-weight case, phosphorus is already at the minimum, therefore, any increase in the weight will not eect the result as it is not possible to reduce phosphorus excretion any further.

On a whole-herd basis, assuming a 100-cow herd, the MINIMAX model reduces nitrogen excretion by 50 kg per year and phosphorus excretion by 56 kg per year. Whilst these results are relatively small, it should be noted that the ration formula-tion is based entirely on the physiological requirements of the cow and does not take into account the palatability of the ration nor the farmer preference for particular ration ingredients; including these constraints into the formulation could lead to higher reductions in nutrients excreted.

It should also be noted the particular set of prices used to generate model results happen to induce rations that (1) do not vary a great deal in terms of nitrogen excretion, and (2) do not achieve the minimum phosphorous intake level. Solving the model using dierent prices substantiated the former observation. The resulting rations diered somewhat in terms of the ingredients selected and suggested a wider range of possible nitrogen excretion levels. The latter observation is apparent through the use of dietary supplements in all the rations presented. It is possible, however, that another set of prices may induce a ration that meets and possibly exceeds the minimum phosphorous requirement without mineral supplementation. A second issue related to (2) is the eciency level assumed and is discussed in the next section.

4.3. Changes in the eciency of phosphorus digestion

An increase in the eciency of phosphorus digestion was also examined in the context of the MINIMAX framework. This was undertaken as there is some uncertainty, in the literature, with regard to the digestive eciency of phosphorus in dairy cattle (Wu et al., 2000). A relative rise in the eciency of phosphorus digestion leads to a reduction in the level of phosphorus required in the diet. In the context of the present model, the value of the parameter k would increase. Tables 4 and 5 report results just as those from Table 4, with the exception that the phosphorus eciency was increased to 60 and 70%.

An increase in phosphorous intake eciency to 60% causes a reduction in the cost of the diets formulated using the single objective function model to minimize phosphorous excretion and all the MINIMAX formulations (Table 5). This occurs simply because phosphorus is a relatively expensive nutrient in the rations and by increasing eciency, intake levels are reduced, which leads to a reduction in the level of costly ingredients in the ration. Note that, for the least-cost and minimum nitrogen excretion rations, there was no change in formulations as phosphorus is not an explicit objective in these two formulations. The ingredients in the rations with higher phosphorus eciencies are similar to those of the lower eciency case. This implies that these ingredients are still relatively less expensive than the excluded ingredients in terms of cost and the nitrogen and phosphorus excretion functions.

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A further increase in eciency of phosphorus digestion to 70% leads to marginal changes in the costs of the rations and the levels of nitrogen excreted when com-pared to the 50% eciency case (Table 6). The ration cost increased by $0.02 and nitrogen excreted fell from 274.76 g to 273.31 g. However, the major reduction is in phosphorus excretion, from 32.09 g in the 50% eciency ration to 13.92 g in the 70% eciency ration. The main change in these rations, when compared to those formulated with lower phosphorus eciencies, is the replacement of the mixed-mostly grass hay with straw and/or corn silage. Consistent with the data in Table 2, this result indicates that corn silage has low phosphorus content with respect to other nutrients required in the diet. Corn silage has a very low phos-phorus content 0.23 g/kg DM and also relatively low values of crude protein, UIP and NDF when compared to other forage sources, such as alfalfa hay and grass hay, hence the inclusion of these forages in the rations formulated at lower e-ciency levels.

One ®nal point to note is that in examining the eect changes in the eciency of phosphorus utilization had on the ration formulation, it was assumed that phos-phorus eciency in all ingredients was equal. However, it would be reasonable to assume this is not the case and that further work measuring eciency of phosphorus utilization in various feed ingredients would enhance the results presented.

Table 5

Alfalfa hay 6.49 7.46 6.74 5.76 6.87 5.03 5.76

Mixed-mostly-grass hay 4.37 1.15 7.10 4.37

Grass hay 4.85 5.36

Straw 4.78 1.74 3.63 1.74

Whole cottonseed 8.15

Soybean meal 2.94

Monosodium phosphate 0.13 0.04

Dynamate 0.01 0.01

Dicalcium phosphate 0.07 0.02 0.02 0.01 0.02

Bakery by-product 5.19 3.36 5.87 6.02 5.17 5.87

Hominy meal 1.11

Wheat middlings 1.46 0.14

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5. Concluding remarks

From the results presented above it is apparent that minimizing nitrogen and phosphorus excretion are competing objectives, because the ration associated with minimizing phosphorus (nitrogen) excretion maximized nitrogen (phosphorus) excretion. A similar situation holds for the cost and nitrogen relationship, minimizing cost (nitrogen excretion) implies maximal nitrogen excretion (cost). Even though cost and phosphorus excretion do not appear to compete as strongly as the nitrogen/ cost relationship they do compete indirectly through the nitrogen excretion objective. The target phosphorus excretion level is obtained with the MINIMAX solution while nitrogen excretion and the ration cost deviate from their target values by approximately 2%. This means that the minimum phosphorus excretion can be obtained with less than a 2% increase in both nitrogen excretion and ration cost.

When comparing the equal-weighted MINIMAX solution to the nitrogen excre-tion minimizaexcre-tion soluexcre-tion the results suggest that for about a 2% increase in nitrogen excretion, ration cost and phosphorus excretion can each be reduced by approximately 21%. Similarly, when comparing the MINIMAX solution to the minimizing phosphorus excretion solution, the target value for phosphorus excre-tion can be obtained for a marginal decrease in nitrogen excreexcre-tion and a nearly 14% decrease in the cost of the ration.

Several conclusions can be drawn from the research presented here. The principal conclusion is that for a minor increase in costs, phosphorus excretion can be reduced

Table 6

Alfalfa hay 6.49 7.46 6.62 7.88 7.84 7.70 7.91

Grass hay 4.85 5.45

Straw 4.78 3.69 3.78 4.04 3.62

Corn silage 2.83 2.01 3.49

Whole cottonseed 8.15 2.53 0.68

Heated soybeans 1.94 0.26 0.11 0.40 0.37

Monosodium phosphate 0.13

Dynamate 0.01 0.01

Dicalcium phosphate 0.07

Bakery by-product 5.19 5.26 5.33 5.51 5.95 5.18 Wheat middlings 1.46

Cost ($/day) 2.84 3.66 3.28 2.92 2.90 2.95 2.93 N excretion (g/day) 276.13 269.93 273.31 275.46 275.33 274.86 275.57 P excretion (g/day) 20.18 24.28 13.57 13.92 14.10 14.07 13.78 Dry matter intake (kg) 20.04 18.70 19.43 19.90 19.87 19.76 19.92

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dramatically with a concurrent marginal reduction in nitrogen excretion. This means that it may be possible for dairy farmers to manage the phosphorus and, to a lesser extent, nitrogen problems of their dairy farm through improved ration formulation. The second conclusion is that it is not possible, even in the multiple-objective programming context, to reduce the individual objectives of cost minimization, nitrogen excretion minimization and phosphorus excretion minimization, to the minimums generated in the single objective programming models due to the com-petition between each objective. Improving the level of one objective always comes at the expense of the level of one of the remaining two objectives. Such tradeos are more consistent with the nature and complexity of real world decisions and the multi-objective formulation oers a more robust set of solutions in this setting.

A ®nal point to consider is that the rations formulated in this research are based entirely on the biological functions presented in the NRCNutrient Requirements of Dairy Cattle and no allowance was made regarding the palatability of the rations formulated. However, many dairy producers have preferences for particular ration ingredients because of habit or availability, hence adding a degree of subjectivity to the ration formulation problem. The addition of these subjective constraints will come at a cost, either economic or nutrient excretion. It was also assumed that the dairy producer purchased all ingredients at the existing market price. The price of ration ingredients is highly variable and regionally speci®c; hence a dierent vector of prices to the one used in this study could lead to an entirely dierent set of rations formulated and a dierent level of nutrients excreted. Therefore, the approach used in this research has the potential to lead to higher reductions in nutrient excretion than those reported in this research when rations are formulated for speci®c farms or regions.

References

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