I samples of any particular feed ingredient.

Transcription

1 01999 Applied Poultry science, Inc THE USE OF AN ELECTRONIC SPREADSHEET TO SOLVE LINEAR AND NON-LINEAR "STOCHASTIC" FEED FORMULATION PROBLEMS' GENE M. PEST12 Depament of Poultry Science, The University of Georgia, Athens, GA Phone: (706) FM: (706) ANDREW E SEILA Terry College of Business, The University of Georgia, Athens, GA Primarv Audience: Nutritionists, Small Producers, InstructorsjStudents DESCRIPTION OF PROBLEM Ingredient composition tables used in feed formulation usually include only average nutrient composition values. Since raw mate- tion, table values do not exactly represent all I samples of any particular feed ingredient. Minimum nutrient restrictions can be expected to be met on average using least-cost linear programming techniques. However, 1 Supported b State and Hatch funds allocated to the Georgia Agricultural Stations of The University o r Georgia. 2 To whom correspondence should be addressed

2 Research Report PEST1 and SEILA 111 mulated with average values can be expected to contain less than minimum restrictions 50% of the time. Researchers recognized the problem of ingredient variability in the 1970s and a nonlinear or "stochastic" programming technique was presented as one possible solution to dealing with ingredient variability in feed formulation [l]. The benefits of stochastic formulation for producers who need to guarantee levels of nutrients in feeds has been detailed [2]. The addition of optimization routines to several electronic spreadsheets - Microsoft Excel [3], Lotus Version 4.0 [4], and Quattro Pro [q- makes them viable alternatives to commercial feed formulation programs for teaching and research purposes, particularly for solving non-linear models. Add-ins such as What's Best [6] and What-If Solver [I are available at a modest price to provide optimization capabilities to spreadsheets that lack them or to extend standard capabilities to very large models. The purpose of this paper is to show how to construct linear and non-linear feed formulation example problems using Microsoft Excel [3]. MATERIALS AND METHODS The linear feed formulation problem chosen has been detailed [8]. It was developed as a simple example of a corn and soybean mealbased broiler grower diet. The Microsoft Excel [3] spreadsheet in Figure 1 contains the ingredient composition matrix for the ingredients normally in the feed (corn, soybean meal, poultry by-product meal, poultry fat, etc.). Figure 1 is the resulting spreadsheet after optimization. The entries in cells B5 through K16 as well as B18 through K18 and M6 through M16 are coefficients. The levels of the nutrients in Column L are calculated from the quantities of each ingredient in the solution (Row 19) and the nutrient composition matrix (Cells B5 to K16). The limit for each nutrient appears in Column M, and the total cost of the feed ($/ton) is presented in Cell (22. The footnotes to Figure 1 show the formulas used to build the spreadsheet. The feed mix was optimized using the built-in solver in Microsoft Excel 97 (also known as Excel 8.0). To perform the optimization, the "Solver...'I option was chosen from the "Tools" menu, producing the dialog box appearing in Figure 2. The cell containing the objective value, the objective value, and the constraints are entered in this dialog box along with the selection of a maximization or minimization for the objective. Next, the researcher should select the "Options" button, producing the dialog box shown in Figure 3. Here, researchers enter the parameters for the optimization process. Since all relationships in this model are linear, it is important to check the "Assume Linear Model" check box so the computer will use the simplex method, rather than a more computationally difficult nonlinear algorithm. Once these options and values have been entered, clicking the "OK" button to close the Solver Options dialog box and clicking the "Solve" button in the Solver Parameters dialog box will begin the optimization and produce the dialog box appearing in Figure 4. The user has the option here to keep or discard the optimized values and to produce any combination of three auxiliary reports for sensitivity analysis. In solving the problem, the total cost in Cell C23 was minimized by adjusting the quantities in Cells B19 to K19 subject to the constraints in Figures 2, 3, and 5. Figures 5 and 6 are tables produced as a by-product of the optimization. Figure 5 summarizes the final solution and the status of each constraint and shows how each constraint was entered into the problem. Figure 6 gives the shadow prices for all constraints. The non-linear feed formulation problem chosen has been detailed by Rahman and Bender [l]. It was developed from a poultry feed formulation problem which utilizes the variances for the protein content of each ingredient. Therefore, the management objective of this problem was to determine the least-cost formula that guarantees (or provides) an 80% chance of having the specified protein level in any given batch of feed. The Microsoft Excel [3] spreadsheet in Figure 7 contains the ingredient composition matrix as in the linear problem (Figure l), an additional column (H) that displays the average content of the nutrients, and an additional row (11) that contains the variances for protein in each ingredient. The footnotes to Figure 7 show the formulas used to produce the spreadsheet. The value in Cell D20 is the standard normal deviate, calculated from the probability input in Cell C20, the desired probability of success in meeting the specified protein limit. Ingredients not included in the

3 JMR 112 STOCHASTIC FEED FORMULATION CELLS CALCULATED BY FORMULA: L6 = SUMPRODUCT ($B$19:$K$19,B6:K6) B20=B19*B5 L7 = SUMPRODUCT ($B$19$K$19,B7K7) L8 = SUMPRODUCT ($B$19$K$19,B8:KS) c20 = C19'CS D20=D19'D5 L9 = SUMPRODUCT ($B$19$K$19,B9K9) L10 = SUMPRODUCT ($B$19$K$19,BlOKlO) E20 = E19'ES F20=F19*F5 L11= SUMPRODUCT ($B$19:$K$19,Bll:Kll) Gu)=G19*GS L12 = SUMPRODUCT ($B$19$K$19,B12K12) H20 = H19*HS L13 = SUMPRODUCT ($B$19$K$19,B13K13) I20 = 119'15 L14 = SUMPRODUCT ($B$19:$K$19,B14K14) 520 = J19*J5 L15 = SUMPRODUCT ($B$19SK$19,B15:KlS) K20 =K19'K.5 L16 = SUMPRODUCT ($B$l9$K$19,B16:K16) FIGURE 1. The basic linear feed formulation problem as programmed in Microsofl Excel 97 [3] for a broiler grower feed. The quantities in Cells 619 through K19 were first input as each.

4 Research Report PEST1 and SEILA 113 FIGURE 2. The "Solver Parameters' dialog box in Microsoft Excel 97 [3] for the linear feed mix problem solution (except one, soybean meal or SBM) were excluded to simpld) the appearance of the problem. All the calculations in the nonlinear problem (Figure 7) are consistent with those in the linear problem (Figure l), except Cell F6 and the cells it is derived from. Cell F6 contains the non-linear relationship that dictates the non-linear solution to the problem. The solver in Excel 97 was used to optimize this non-linear problem. After selecting "Solver..." from the 'Tools" menu, the Solver Parameters dialog box in Figure 8 was entered and the options in Figure 9 were also used. In this case, the "Assume Linear Model" check box was not checked since this model is not linear. The model was optimized by closing the Solver Options dialog box and selecting "Solve" from the Solver Parameters dialog box. In solving the problem, the total cost in Cell C18 was minimized by adjusting the quantities in Cells C14 to E14 subject to the constraints in Figures 8 through 11. When the Microsoft Excel [3] solution did not agree with the published solution [l], the GAMSl program [9] validated the solution. The shadow price for soybean meal is shown to be ($) in Figure 11. RESULTS AND DISCUSSION The final value for the objective function in the Microsoft Excel [3] solution for the linear feed mix problem (Figure 5) was identical to the solution of the UFFDA Program [8] to the nearest O.OOO1 $/ton and the ingredient amounts in the solution were identical to 0.01%. The sensitivity analysis showed that if the price of corn increases $32.90/ton to $ or decreases $63.73 to $64.27/ton, it will stay at the same level of 67.44% of the diet. If the protein level of the diet were to change from 19.8 to 20.8%, the cost of the diet would increase by $2.80 to $166.08/ton (Figure 3). With a Zenith Data Systems ZNote notebook computer having an microprocessor running at 25 Mhz and 8 MB of RAM [lo], Microsoft Excel [3] took about 6 sec to solve the problem; more modern computers complete the problem much more quickly. The final value for the objective function in the Microsoft Excel solution for the nonlinear feed mix problem (Figure 7) was slightly dieferent from the solution given by Rahman and Bender [l], but was essentially identical to the solution of the GAMSl program [9, 111. In order to be 80% sure of having 24% protein in the feed, the average protein content would have to be %. With a Zenith Data Systems Z-Note notebook computer having an microprocessor running at 25 Mhz and 8 MB of RAM [lo], Microsoft Excel [3] took about 4 sec to solve the problem. It is clear from these results that Microsoft Excel can provide a fast and accurate way to solve both linear and non-linear feed formula-

5 114 JAPR STOCHASTIC FEED FORMULATION FIGURE 3: The Solver Options dialog box in Microsoft Excel 97 [3] for the linear feed mix problem FIGURE 4. The Solver Results dialog box in Microsoft Excel 97 [3] for the linear feed mix problem tion problems. The short time that Microsoft Excel took to solve these problems is practically insignificant compared to the time it takes to load the program from a fmed disk drive. The time necessary to load the programs and solve problems will vary greatly from machine to machine, but should all be measured in seconds, not minutes. Similar results can be expected from the other popular spreadsheet programs available that include optimization routines (Lotus Version 4.0 [4], and Quattro Pro [5]), either built in or with available add-ins. During the winter of 1997, one company was offering a commercial feed formulation program capable of solving stochastic models for $18,000. This figure compares to $50 for Microsoft Excel at our university and $300 from many computer stores in the United States. Although the specialized commercial product has many extra features important to large feed manufacturers when simultaneously reformulating several

6 I I ~ _II - Research Report PEST1 and SEILA 115 MicrosoA Excel 8.0 Answer Report Worksheet: [Broiler297 formal.xls]broill.dat Report Created: 4/13/97 9:15:30 PM Cell (Min) Cell Name Original Value Final Value 5C$22 Least Cost = SEM-48 $ $ Adjustable Cells Cell Name Original Value Final Value $B$19 QuantltyCorn - - _ IcIlAEnXSBM o 2481 o 2481 $D$19 QuantJQ-M$al c300 $E$19 QuaGLfat I 0-oi 78 o gj 78 - $FU9 Quan!ily stone $G$19 Quantllyphos $HU9 Quantity Salt I $1$19 Quantity Premixes $J$19 Quantity-Met ! $KS19 Quantity HCI 'IGURE 5. The Microsoft Excel 97 [3] "Answer Report" for the linear solution of a broiler grower feed problem

7 0 ~ -..I.I_._._. I 116 JAPR STOCHASTIC FEED FORMULATION Microsoft Excel 8.0 Sensitivity Report Worksheet: [Broiler297 forrnat.xls]broill.dat Report Created: 4/13/97 9:15:32 PM Adjustable Cells Final Reduced Obiective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease...I $B$19 Quantity. Corn a $C$19 Quan!?!!xE!M-48-0?_4_fll- IJ.mo0.I_ KG19 Quan?!!LEa! E $E QuantiQ... Fat _ $F$19 Quantity stone $G$19 Quantity Phos no x~ _- $H$19 Quantity Salt E BF5 _._-- $1$19 Quantity Premixes E $J$19 Quantity Met E $K$19 Quantity HCI Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease. $L$6. Welght Values too $L$7 M E (KcaVg) Values o.oao651e9~ - $L$8 Protein Values i9.aooo O.rn a u-59 Calcium Values a ,.... $L$10 Avail. Phos. Values o o $LIS12 Arginine Values moo l l E $L$13 Lysine. Values $L$15 Threonine Values ~ E+30 $L$ 6. TrYP!oqha!cvG!!u!!22~-e~~. 0 0._ E+30 $L$14 Met + Cys Values IGURE 6. The Microsoft Excel 97 [3] Sensitivity Report for the linear solution of a broiler grower feed feeds, such programs and spreadsheets with Solver should give identical answers for the same problem. The construction and solving of linear feed formulation problems in electronic spreadsheets is not as easy as with other programs designed specifically for teaching feed formulation such as UFFDA [8]. But the process is much easier than with dedicated optimization programs currently available to students such as GAMS1 [9]. Most animal agriculture students and producers are capable of learning to use spreadsheets in a much shorter period of time than they can learn the specialized optimization programs designed for econometricians and operations researchers. Commercial software for feed formulation is now available and capable of running on microcomputers. The commercial programs provide many valuable features for producers. However, the features are not necessary for research or instructional purposes and add to the complexity of the programs and their cost. Student versions of optimization programs capable of solving linear and non-linear problems are difficult to use without special training. Inexpensive linear programs designed for instructing feed formulation are available, but are not capable of solving nonlinear models. The spreadsheets have some advantages in that many students, faculty, and scientists already own and use them for other purposes and are familiar with them. Built-in solvers in the most popular spreadsheets are capable of solving linear, non-linear, and integer programming problems with up to 200 variables and up to 100 constraints. Add-in packages that extend these limits are available from several vendors for prices ranging from $500 to $5,000. The addition of custom formatting

8 Research Report PEST1 and SEILA 117 Rotein Protein SD Fat Fiber weight Sigm$2TjA O.Oo u.oooa S S Quantities Cost per ingredient Robability CELLS CALCULATED BY FORMULA: F6 = SUMPRODUCT ($Cc$13$E$13,c6:E6) Ell = (E7*E13)"2 + D18* SQRT(SUM( C1l:Ell) C15 = C4*C13 F7= H7= SUMPRODUCT ($C$13:$E$l3,C?E7) D15 =D4*D13 F8 = H8 = SUMPRODUCT ($C$13$ES13,C8:E8) E15 = E4*313 F9= H9 = SUMPRODUCT ($C$13$E$13,C9:E9) F15 = F4*F12 F10 = H10 = SUMPRODUCT ($C$13$E$13,ClO:ElO) C17 = SUM(Cl5:Fl5) C11= (C7*C13) 2 Dl8 = -NORMSINV(($C$18)) D11= (D7*D13)"2-1GURE7. The basic non-linear stochastic feed formulation problem of Rahman and Bender [ 11 as programmed n Microsoft Excel 97 [3]. The quantities in Cells C13 through E13 were first input as 0.3 each.

9 118 JAPR STOCHASTIC FEED FORMULATION FIG mix problem

10 l-l_-l_--- ll" l_l 1-1 _ ~ Research Report PEST1 and SEILA 119 Microsoft Excel 8.0 Answer Report Works h e et : [RAND B. x Is] N o n I i n e a r S o I uti o n Report Created: 3/4/98 12:29:50 PM Target Cell (Min) Cell Name Original Value Final Value $C$17 Total Cost M&BW Adjustable Cells Cell Name Original Value Final Value $C$13 Qu-antities M&BW $D$13 Quantities SBG' $E$13 Quantities Corn Con st raint s Cell Name Cell Value Formula Status Slack $F$6 Protein Value $ -_I F $6>=$G$6 Binding $F$8 Fat Value -l -lll_ll -. l l.._...l._l $F$8>=$G$8 Not Binding " $F$9 Fiber Value $F$9<=$G$9 -_1 Not Binding -- ll_l_-_l $F$IO Weight Value - 1 $F$IO=$G$IO Bind3.-I-" 0 $C$13 Quantities M&BM $C$13>=0 --- Not Bindm I $D$13 Quantities SBM $D$13>=0 ~----_l_l_l _I_-_ Bindin!EO0 -IGURE 10. The Microsoft Excel 97 [3] "Answer Report" for the non-linear solution of the problem of Rahman and Bender [l] and macro language subroutines to simplify using spreadsheets for feed formulation could make them much easier to use by the more inexperienced at computerized feed formulation, inside or outside of formal schools. With the use of macros, spreadsheets could be used to create turnkey applications. Spreadsheets are very easy to customize, so that features can be added by operators with minimal training. For instance, equations could be added to the spreadsheet in Figure 1 to calculate the total savings that could be expected from using alternative ingredients. Clearly, small producers could benefit from using the spreadsheet to formulate feed. Most are capable of customizing spreadsheets to fit their particular applications. Electronic spreadsheets on microcomputers are capable of giving small and large producers the right answer if they are capable of inputting the right data and asking the right questions. Animal nutritionists' main interests are in biology, not econometrics. As a result, the economic interpretation of feeding practices has been practically ignored in animal nutrition texts and in our teaching programs. Most animal nutrition texts have only a few paragraphs or pages on feed formulation, despite the fact that feed formulation is the primary objective of animal nutrition and the major cost in animal production. The development and implementation of easy, customized software models should facilitate the teaching of (and increase interest in) the economic ramifications of choices in animal nutrition and feeding.

11 120 JAPR STOCHASTIC FEED FORMULATION Microsoft Excel 8.0 Sensitivity Report Worksheet: [RANDB.xls]Nonlinear Solution Report Created: 3/ :29:51 PM Adjustable Cells Final Reduced Cell Name Value Gradient $C$13 Quantities M&EM $D$13 Quantities SBM $E$13 Quantities Corn Constraints Final Lagrange Cell Name Value Multiplier $F$6 Protein Value $F$8 Fat Value $F$9 Fiber Value $F$10 Weight Value IGURE 11. The Microsoft Excel 97 [3] 'Sensitivity Report" for the non-linear solution of the problem oi Rahman and Bender [l] CONCLUSIONS AND APPLICATIONS 1. Electronic spreadsheet programs that run on desktop microcomputers are now capable of solving both hear and non-linear (stochastic) feed formulation problems. 2. Spreadsheet programs are widely available (to students and large and small producers) as well as much less expensive than commercial software. In contrast, besides being expensive, sophisticated feed formulation software may be difficult to learn to use for students and small producers. 3. Many students and producers are already familiar with spreadsheets and will find simple formulation models easy to set up and usen. For larger producers needing more complex models with several feed formulations solved simultaneously, specialized software is still necessary. 4. The spreadsheet models described here may be obtained free of charge from anywhere on the World Wide Web (

12 Research Report PEST1 and SEILA Rahman, SA and F.E Bender, Linear programming approximation of least-cost feed mixes with probability restrictions. Amer. J. Ag. Eon D Alfonso, T.H., W.B. Roush, and JA Ventura, kast-cost ultry rations with nutrient variability: A comparison oenear programming with a margin of safety and stochastic programming models. Poultry Sci. 71: Microsoft Corporation, One Microsoft Way, Redmond, WA Lotus Development Corporation, 55 Cambridge Parkway, Cambridge, MA Borland International, Inc., 1800 Green Hills Road, P.O. Box 66OOO1, Scotts Valley, CA REFERENCES AND NOTES 6. Lindo S terns, Inc., 1415 North Dayton Street, Chicago, IL 6G Frontline Systems, Inc., P.O. Box 4288, Incline Village, NY Pest& G.M. and B.R Miller, Animal feed formulation: Economic and computer applications. Van Nostrand Reinhold, New York, NY. 9. RosenlhaL R Review of the GAMSNINOS modellin language and optimization program: OR/MS Today 3% Zenith Data Systems Corporation, Hilltop Road, St. Joseph, MI Bender, F.E, C. Kahan, and W.C. Mylander, Optimization for profit: A decision maker s guide to Iinear programming. Haworth Press, Inc., Binghamton, NY.