14/09/2016 Department of Large Animal Sciences Linear programming - PDF document

14/09/2016 Department of Large Animal Sciences Linear programming head points and case Katarina Nielsen Dominiak Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 1 Department of Large Animal Sciences Characteristics of Linear Programming Absence of random elements • All parameters are assumed to be known constants • Deterministic • No variance on the output • Static • No dynamics in the system as it evolves • Demands linear functions and linear restraints • K.N. Dominiak, AQMHM 2016 Slide 2 Department of Large Animal Sciences Applied linear programming Ration formulation • Least cost versus nutritional requirements of the • animal Benchmarking • Weakness Linear constraints and demand for linear functions Force Direct representation of constraints K.N. Dominiak, AQMHM 2016 Slide 3 1

14/09/2016 Department of Large Animal Sciences Construction of constraints 1. Write down, in words, an accurate description of the relationship that is to be modelled 2. Convert the written description to an arithmetic statement by introducing numbers and activities 3. Re-arrange the arithmetic statement so that all activities are gathered on the left-hand side and all constants are gathered on the right-hand side 4. Tidy up the constraints by gathering similar terms together and ensure that the constraints are specified in linear form K.N. Dominiak, AQMHM 2016 Slide 4 A linear programming model to optimize diets in environmental policy scenarios Moraes, L.E. et al. (2012) Department of Large Animal Sciences Construction of constraints 1. Write down, in words, an accurate description of the relationship that is to be modelled 2. Convert the written description to an arithmetic statement by introducing numbers and activities 3. Re-arrange the arithmetic statement so that all activities are gathered on the left-hand side and all constants are gathered on the right-hand side 4. Tidy up the constraints by gathering similar terms together and ensure that the constraints are specified in linear form K.N. Dominiak, AQMHM 2016 Slide 6 2

14/09/2016 Department of Large Animal Sciences Description in words - background Global food demands increase -> more animal products will be produced Carbohydrate fermentation in dairy cattle produces CH 4 Protein, starch and minerals in feed can lead to excretion of N and minerals if fed at higher levels than animal requirements 72% of total emitted CH 4 in Brazil was from enteric fermentation (1994) K.N. Dominiak, AQMHM 2016 Slide 7 Department of Large Animal Sciences Description in words - background Greenhouse gas (GHG), N, and minerals contaminate the environment Policies and legislations formulated to limit environmental impacts of livestock production - Clean Water Act (US EPA 2003) - Manure applied to crops and pastures at levels the plants can extract (N,P,K) - Kyoto Protocol reduces GHG to level of 1990 - Carbon markets in Europe (and the States) K.N. Dominiak, AQMHM 2016 Slide 8 Department of Large Animal Sciences Description in words – aim Reduce CH 4 emission and excretion of N and minerals Optimal diet cost and feed selection Joint optimization of costs and emisions + excretions Two policies are chosen: - Limit the quantity of CH 4 emitted - Require emission taxes K.N. Dominiak, AQMHM 2016 Slide 9 3

14/09/2016 Department of Large Animal Sciences Construction of constraints K.N. Dominiak, AQMHM 2016 Slide 10 Department of Large Animal Sciences Three scenarios – three models BASEM: Cheapest possible feed – No GHG policies (baseline) Constraints: Decision variables: Nutrient requirements Available feed Defined feed limits and their costs Fibre proportions TAXM: Effect of tax on diet costs and composition, emission and excresion Computes optimal feed mix that balances off tax savings for lower emissions REDM: Feed costs vs emission and excresion if forced to reduce CH 4 emission Decision variables like BASM but with the extra constraint K.N. Dominiak, AQMHM 2016 Slide 11 Department of Large Animal Sciences Simulated herd � � � � � K.N. Dominiak, AQMHM 2016 Slide 12 4

14/09/2016 Department of Large Animal Sciences Real dietary limits and costs kg/kg DM � � � � �� $/kg DM � � � � �� � � � � �� Locally collected K.N. Dominiak, AQMHM 2016 Slide 13 Department of Large Animal Sciences Prediction of emission and manure production How much CH � is emitted? Feed composition – absorbstion � Excrete composition K.N. Dominiak, AQMHM 2016 Slide 14 Department of Large Animal Sciences Construction of constraints K.N. Dominiak, AQMHM 2016 Slide 15 5

14/09/2016 Department of Large Animal Sciences Objective functions BASEM and REDM a = animal group j = feed x = amount of feed (kg of DM) c = cost of feed K.N. Dominiak, AQMHM 2016 Slide 16 Department of Large Animal Sciences Objective functions TAXM is like BASM and REDM but with extra constraint a = animal group j = feed x = amount of feed (kg of DM) c = cost of feed CH � e = expected emission (tonnes) p = tax price per tonne K.N. Dominiak, AQMHM 2016 Slide 17 Department of Large Animal Sciences Constraint equations Nutrient requirements Dietary Feed Limits Fibre and CP max DMI Rolled Barley max Fibre (NDF) min Dietary Forage Proportions Methane Restriction - only REDM K.N. Dominiak, AQMHM 2016 Dias 18 6

14/09/2016 Department of Large Animal Sciences Nutrient requirement constraints Amount Minimum requirement Nutrient content j = feed (19) a = animal category (7) 7*14 = 98 constraint equations i = nutrient (14) K.N. Dominiak, AQMHM 2016 Slide 19 Department of Large Animal Sciences Dietary feed limit constraints �� = � �� � � �� ∙ � � �� ��� j = feed a = cow category (7 kinds) 7*10 = 70 constraint equations l = limit (for 10 kinds of j ) K.N. Dominiak, AQMHM 2016 Slide 20 Department of Large Animal Sciences Methane Restriction constraints (REDM) Maximum emission Total feed (5, 10, 13.5 %) (all cows, all feed) Intercept Methane 1 - CH � reduction (%) BASEM emission emission predictions K.N. Dominiak, AQMHM 2016 Dias 21 7

14/09/2016 Department of Large Animal Sciences Results TAXM Cost of reducing CH � emission through the diet > tax costs Therefore: No differences in emission between BASEM and TAXM CH � K.N. Dominiak, AQMHM 2016 Slide 22 Department of Large Animal Sciences Shadow prices Sensitivity analysis ‘What is the cost of reducing 1 tonne of CH 4 emission?’ Extremely sensitive to feed prices – further analysis required K.N. Dominiak, AQMHM 2016 Slide 23 Department of Large Animal Sciences Results REDM 5% 19,1% 48,5% CH � emission reduced but diet costs increased K.N. Dominiak, AQMHM 2016 Slide 24 8

14/09/2016 Department of Large Animal Sciences Results – trade-offs in diet formulas Because grain/soy = ME (protein, starch, minerals) and forage = NDF (lignin, cellulose, hemicellulose) The REDM results in a trade-off between the two But at what consequences? K.N. Dominiak, AQMHM 2016 Slide 25 Department of Large Animal Sciences Results – trade-offs in diet formulas Consequences of CH 4 reduction Feed ME NDF N+min ex Corn silage Low High Low Soy silage High Low High Grain High Low High Cereal High Low High Soybean meal High Low High Total reduction in DMI K.N. Dominiak, AQMHM 2016 Slide 26 Department of Large Animal Sciences Discussion Animals vs Human in competition for feed ressources ‘N + mineral excresions follows same pattern as intake’ - results in conflicting environmental interests CH 4 vs NO 3 Should mineral diets be based on excresion levels? K.N. Dominiak, AQMHM 2016 Slide 27 9

14/09/2016 Department of Large Animal Sciences And what about animal welfare? Consequences of CH 4 reduction Feed ME NDF N+min ex Corn Low High Low silage Soy silage High Low High Grain High Low High Cereal High Low High Soybean High Low High meal Total reduction in DMI K.N. Dominiak, AQMHM 2016 Slide 28 10