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

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Linear programming head points and case

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 1

Katarina Nielsen Dominiak 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

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 2

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

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 3

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Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 4

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

A linear programming model to optimize diets in environmental policy scenarios Moraes, L.E. et al.(2012)

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 6

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

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Description in words - background

Global food demands increase -> more animal products will be produced Carbohydrate fermentation in dairy cattle produces CH4 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 CH4 in Brazil was from enteric fermentation (1994)

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences Slide 7

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 Department of Large Animal Sciences Slide 8

Description in words – aim

Reduce CH4 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 CH4 emitted
• Require emission taxes

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences Slide 9

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Construction of constraints

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 10

Three scenarios – three models

BASEM: Cheapest possible feed – No GHG policies (baseline) 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 CH4 emission Decision variables like BASM but with the extra constraint

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences

Constraints: Nutrient requirements Defined feed limits Fibre proportions Decision variables: Available feed and their costs

Slide 11

Simulated herd

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences Slide 12

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Real dietary limits and costs

Locally collected

kg/kg DM $/kg DM

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences Slide 13

Prediction of emission and manure production

How much is emitted?

Feed composition – absorbstion Excrete composition

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences

CH

Slide 14

Construction of constraints

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 15

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Objective functions

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016

BASEM and REDM

a = animal group j = feed x = amount of feed (kg of DM) c = cost of feed

Slide 16

Objective functions

TAXM is like BASM and REDM but with extra constraint

e = expected emission (tonnes) p = tax price per tonne

CH

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences

a = animal group j = feed x = amount of feed (kg of DM) c = cost of feed

Slide 17

Constraint equations

Dias 18

Nutrient requirements Dietary Feed Limits Methane Restriction - only REDM DMI Fibre and CP max Rolled Barley max Fibre (NDF) min Dietary Forage Proportions

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences

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Nutrient requirement constraints

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016

Amount Nutrient content Minimum requirement

j = feed (19) a = animal category (7) i = nutrient (14)

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7*14 = 98 constraint equations

Dietary feed limit constraints

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016

• ∙
• =

j = feed a = cow category (7 kinds) l = limit (for 10 kinds of j)

Slide 20

7*10 = 70 constraint equations

Methane Restriction constraints (REDM)

Dias 21

Total feed (all cows, all feed) Methane emission predictions reduction (%) BASEM emission Maximum emission (5, 10, 13.5 %) 1 - CH

K.N. Dominiak, AQMHM 2016 Department of Large Animal Sciences

Intercept

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Results TAXM

Cost of reducing emission through the diet > tax costs Therefore: No differences in emission between BASEM and TAXM

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016

CH CH

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Shadow prices

Sensitivity analysis ‘What is the cost of reducing 1 tonne of CH4 emission?’ Extremely sensitive to feed prices – further analysis required

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 23

Results REDM

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016

CH emission reduced but diet costs increased

5% 19,1% 48,5%

Slide 24

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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?

Results – trade-offs in diet formulas

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 25

Results – trade-offs in diet formulas

Consequences of CH4 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

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 26

Discussion Animals vs Human in competition for feed ressources ‘N + mineral excresions follows same pattern as intake’ CH4 vs NO3

• results in conflicting environmental interests

Should mineral diets be based on excresion levels?

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 27

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Consequences of CH4 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

And what about animal welfare?

Department of Large Animal Sciences K.N. Dominiak, AQMHM 2016 Slide 28