THE PROBLEM

FEED FORMULATION TO CONTROL RISK

W.B. Roush, Ph.D. Gardnerville, Nevada wbroush@yahoo.com

Uncertainty and risk are inherent in biological variability. Early in use of linear programs for feed formulation it was recognized that nutrient variability posed a problem for meeting the nutrient requirements of animals. A linear program solution based on average nutrient values has a 50% probability of meeting or not meeting the nutrient requirements. In order to compensate for the risk, some nutritionists incorporate a margin of safety in the ingredient database or matrix. Nott and Combs ( 1967) suggested an adjustment of the nutrient means by subtracting ( or adding) a fraction (they suggested 0.5) of the standard of deviation from (or to) the nutrient mean which would provide a probability of 69% or greater in meeting the nutrient requirement. The adding or subtracting depends on whether the constraint is a maximum or minimum value (Roush, 2006).

With linear programming there is an assumption that all the variables are known with certainty (Hanna, et al., 2009). When variability is included in the problem, the assumption of certainty is violated. A consequence of this violation is an over formulation of the requested probability and dietary requirements. With the over formulation there is an increased dietary cost.

THE SOLUTION

The solution to accurately meeting nutrient requirements where variability is involved is to apply a nonlinear algorithm. Stochastic programming uses such an algorithm. Several papers have addressed stochastic programming as a method of constraining dietary risk (Van de Panne and Popp, 1963; St. Pierre and Harvey, 1986; St. Pierre, 1991). The philosophy and assumptions about linear and stochastic programming are discussed elsewhere (D' Alfonso et al., 1992; Cravener et al., 1994; Roush, et al., 1994; 1996). An Excel Spreadsheet program was developed using Visual Basic for Applications (VBA) to illustrate the value of formulating diets with stochastic programming. The nonlinear algorithm, Solver which is available within Excel spreadsheets, was used to illustrate the value of using stochastic programming. Figures 1 through 5 represent examples of the Excel (VBA) program for feed formulation.

THE VALUE

The value of stochastic programming is the accuracy and precision of formulating what is requested. When nutrient insurance was incorporated in the formulation problem, such as insuring a 69% probability of meeting the nutrient requirement, there is a difference in the answers obtained by linear and stochastic programming. As shown in Figure 5, a linear program

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will produce a cost of $168.36 while a stochastic program would produce a lower cost diet, $167.09 based on the same requested diet formulation parameters. Examination of the requested and calculated probabilities of the two methods for protein reveals that the linear program with a margin of safety over formulates the requested 69% probability to a 78% level. The stochastic program more accurately and precisely produces the requested protein probability of 69%. If the formulator wants to have the diet formulated at the 78% probability for protein, then the stochastic program would more accurately provide that probability.

TAKE HOME MESSAGE

Stochastic programming represents an accurate and precise method of formulation for animal diets.

LITERATURE CITED

Cravener, T.L., W.B. Roush, and T.H. D'Alfonso. 1994. Laying hen production responses to least cost rations formulated with stochastic programming or Linear programming with a margin of safety. Po ult. Sci. 73: 1290-129 5.

D 'Alfonso, T .H., W. B. Roush, and J. A. Ventura. 1992. Least cost poultry rations with nutrient variability: A comparison of linear programming with a margin of safety and stochastic programming models. Poult. Sci. 71 :255-262.

Nott, H. and G.F. Combs. 1967. Data processing feed ingredient composition data. Feedstuffs 39:21-22.

Hanna, M.E., R.M. Stair, B. Render 2009. Quantitative Analysis for Management. Massachusetts 10th Edition. Prentice Hall, Upper Saddle River, New Jersey. 768 pp.

Roush, W.B. 2006. Advancements in empirical models for prediction and prescription. In: Mechanistic Modelling in Pig and Poultry Production. R. Gous, T. Morris and C. Fisher, eds. CAB International. pp. 97-113.

St. Pierre, N.R., 1991. Ration balancing: From Costs to profits. In: Maryland Nutr. Conf. Feed Manuf., College Park, MD., Univ. of Maryland. pp. 73-80.

St. Pierre, N.R. and W.R. Harvey, 1986. Incorporation of uncertainty in composition of feed into least cost ration models. 1. Single chance constrained programming, and 2. Joint chance constrained programming. J. Dairy Sci. 69:3051-3073.

Van de Panne, C. and W. Popp. 1963. Minimum cost cattle feed under probabilistic protein constraints. Management Sci. 9:405-430.

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Figure 1. Menu for Stochastic Formulation Program developed in Excel with Visual Basic for Applications (VBA)

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25 26 27 28 ·29· 30 31 32 33

.. 34· 35 36 37 38 39" 40

Oats Wheat Middlings Mollasses Anim·a1 Tallow Vegetable Oil *Poultry Fat Blerided Fat

Cottonseed meat: solvent Peanut meal, de hulled *Poultry by Product Meal Meat and Bone Meal Fish Meal , 65% Brewers Dried Yeast Corn Distillers Dried Soluables

&%. Whey wJ@ Alfalfa Meal

41 b....A. .......... w;...~liW....._.m. ...... ..a"'"""".ilb.-'1&...di....-.A~~....afl:m ... .....@~di,-.;;~~.,~hl.......a--w .. ,_...,,,,,w,,ili_..AJ

Figure 2. Price Sheet for Feed Formulation program developed in Excel with Visual Basic for Applications (VBA)

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Choose .Ingredients.

In redient

Wheat Barley Milo Oats Wheat Middlings Mollasses Animal Tallow Ve etable Oil

Blended Fat

Soybean Meal, 44 Corn Gluten Feed Corn Gluten Meal Cottonseed meal, Peanut meal deht

Meat and Bone Me Fish Meal, 65% Brewers Dried Yee

Cost Ton

CDm D,stdlers Dne 221,00 Whey Alfalfa Meal

Amount Relation

3 3 3 3 3 1 1 1

3 1 1 3 3

*Limestone $34,00 3 * Dk:alciurn Phosph $220,00 3

·Return to Menu

Level

0.0300

*Salt, Iodized $55.00 2 0,0040 *V&M Mix $2.136.00 2 0.0010

~1::i_"lethioriine $2 200 0_0 _________ 3 __________ _ · *L-1,,-sine $2.400.00 3

L-Threonine L-Tryptophan L-Valine Your Choice Your Choice

◄

Figure 3. Ingredient Selection Sheet for Feed formulation developed in Excel with Visual Basic for Applications

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Arginine Lysine Met+Cys

" Threonine Tryptophan

19.8000 0.6900 0.8500 0.5000 0.4250 0.5000 0.1835 0.5000 1 :2395 0.5000 1.0336 0.6900 0.7999 0.6900 --·

_ __ 0~_6_45_5+--___::__;_0.5000

"-·---- ·~·· 0.1980 --·· 0.5000 ____ _

19.8000 0.8500

----- 0.4250 · ------------------ rf fo§s --

1.2395 0.5000 1.0336 0.6900 -

- --___ --'-_~:=~!=-=E 1;;[········ 1

Figure 4. Nutrient designation with probability selection. Requested probabilities for ME, protein, lysine, and methionine +cystine at a 69% are shown.

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*Corn -----+-------"-'-0.6~--'--53-e-) ____ 0.64 ~P.011.ltryf~t . .. ...... .......... 0.02!J l 0.02 *So bean Meal, 50% 0.2479: ·· 0:2427 *PotJ.ltrY .. ~~ Product Meal 0.0300: 0.0306 Corn Distillers Dried Soluables 0.0300) 0.0300

0.0115 ) 0.0115 0.0112 [ 0.0113 0.0040 ! 0.0040 0.0010: 0.0010 0.00171 0.0016 o ooo~; o.oom

···········································---~

Figure 5. Comparison of formulated rations using linear and stochastic programming. The more accurate stochastic formulation of requested probabilities for ME, protein, lysine and methionine and cystine is shown.

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