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Anders Ringgaard Kristensen, I PH KVL Anders Ringgaard Kristensen, I PH Outline Short survey of animal replacement models A 4-level model for optimal feeding level, fattening policy and slaughtering of organic steers (BKN): Model

  1. Anders Ringgaard Kristensen, I PH KVL Anders Ringgaard Kristensen, I PH Outline � Short survey of animal replacement models � A 4-level model for optimal feeding level, fattening policy and slaughtering of organic steers (BKN): � Model structure Animal replacement � Decisions being optimized models � A 3-level model for optimal replacement of sows � Model structure Anders Ringgaard Kristensen � Integration with Dynamic Linear Model � A dairy cow model with one calving disease � Model structure � Decisions being optimized Advanced Herd Management 2006 1 Advanced Herd Management 2006 2 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Short survey of animal replacement models Short survey of animal replacement models Anim al replacem ent m odels: Breeding anim als Anim al replacem ent m odels: Grow ing anim als � � Dairy cows: Dairy heifers � Traits: � Slaughter pigs � Age (lactation number and stage) � Steers � Milk yield � Traits considered: � Pregnancy status � Age � Diseases � Weight � Decisions � Pregnancy status (heifers … ) � Replace � � Inseminate Decisions considered: � Treat (for diseases) � Feeding level � Sows � Insemination (heifers) � More or less like cows (except milk yield … ) � Slaughter (pigs, steers) Advanced Herd Management 2006 3 Advanced Herd Management 2006 4 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows Sow m odel: Litter size and Markov property Circum venting the Markov violation � Define the state as the combined value of present and � Litter size at next parity depends on all previous litter previous litter size. size observations. � If each may be “Low”, “Average”, “High” (9 � In other words, the Markov property is not satisfied. combinations): � Low, Low � Low, Average � Low, High � Average, Low � Average, Average � Average, High � High, Low � High, Average � High, High � Only a computational problem – the “curse of dimensionality” once again. Advanced Herd Management 2006 5 Advanced Herd Management 2006 6 1

  2. Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows A com prehensive exam ple State space � � A sow replacement model Litter size � Developed for practical use at prototype level � Age (parity) � 3 levels: Decisions at 2 levels � Rematings � Estimation at herd level � Bayesian updating (Dynamic linear model) � Decision making Advanced Herd Management 2006 7 Advanced Herd Management 2006 8 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows Litter size, age dependence The Markov property � Let i n be the state at stage n � The Markov property is satisfied if and only if � P( i n+ 1 | i n , i n- 1 , … , i 1 ) = P ( i n+ 1 | i n ) � In words: The distribution of the state at next stage depends only on the present state – previous states are not relevant. � This property is crucial in Markov decision processes. Advanced Herd Management 2006 9 Advanced Herd Management 2006 10 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows The Markov property in sow s Lacking Markov property: � The lacking Markov property must be considered when � Does ”Age” satisfy the Markov property � the state is defined Yes! � Does ”Rematings” satisfy the Markov property? � Straight forward solution: � Yes, almost. The probabilty of conception hardly depends on � Define the state as i n = ( y 1 , y 2 , … , y n ) previous results at all. � For a sow in parity 8 this means e.g. 15 8 = 2.5 x 10 9 state � Does ”Litter size” satisfy the Markov property? combinations. � No, certainly not. Several high (or low) will further increase (or decrease) our expectations to future litter size. Example: � Prohibitive � Sow 1: 12 – 14 – 16 – 15 – 16 piglets � Sow 2: 6 – 5 – 7 – 6 – 16 piglets � We prefer sow 1 Advanced Herd Management 2006 11 Advanced Herd Management 2006 12 2

  3. Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows From registration to inform ation From registrations to inform ation � Interpretation � Registration: Litter size λ i = y i Data: Λ = { y 1 , y 2 , … , y n } � Processing: Ψ () - Kalman filtering, DLM � � Information: I = Ψ ( Λ ) = ( i 1 , i 2 ) Decision: Θ � Decision strategy: I → Θ � � Ψ (): As little loss of information as possible (preferably none). Advanced Herd Management 2006 13 Advanced Herd Management 2006 14 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows Litter size m odel ( Toft & Jørgensen) : Observation equation = µ + + ε Y M n ( ) n n n ⎡ ⎤ ( ) M n = + ε = − µ [ ] I M n ( ) Y = θ = n n n n 1 1 ⎢ ⎥ I Z ε µ = − θ − − θ + θ − θ ⎣ ⎦ n n 2 exp( ( n 1 ) ) n n 1 2 3 4 n = − + M n ( ) aM n ( 1 ) e M → σ 2 ( ) ( , 0 ) M n N ε → τ 2 N ( , 0 ) n Advanced Herd Management 2006 15 Advanced Herd Management 2006 16 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows System equation DLM: Kalm an filtering � Each time an observation y n is made, the current estim ate of M ( n ) is updated according to the − = ⎡ ⎤ ⎡ ⎤ ⎥ + ⎡ ⎤ a 0 M n ( 1 ) e Kalman filtering method. θ = θ + M F e ⎢ ⎥ ⎢ ⎢ ⎥ ε ε � I = Ψ ( y 1 , y 2 ,… − , y n ) = ( M ( n )) 1 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ n n 0 0 − n 1 n � E( y n+ 1 | y 1 , y 2 ,… , y n )= E( y n+ 1 | M ( n )) = ZF ( M ( n ),0)’ − σ ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ 2 2 0 , ( 1 a ) 0 � V( y n+ 1 | y 1 , y 2 ,… , y n ) is known (Eq. (12)) → ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ e N ⎝ τ ⎠ � The processing of data into information reduces ⎣ ⎦ ⎣ 2 ⎦ 0 0 the dimension from n to 1 without loss of information (given the litter size model) Advanced Herd Management 2006 17 Advanced Herd Management 2006 18 3

  4. Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows Biological param eters Model structure � � Founder level Litter size parameters: � � Stage: Life span of a sow Herd level estimation � State: Dummy � Censoring � � Decision: Dummy Other assumptions � � Conception rates (parity, remating) Child level 1 � Piglet mortality (parity) � Stage: Production cycle from weaning � Sow weight (parity) � State: M ( n ) � Involuntary culling (parity) � Decision: Mating method � Feed intake Advanced Herd Management 2006 19 Advanced Herd Management 2006 20 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows A 3-level model for optimal replacement of sows Model structure Prototype � � Child level 2 A plug-in for the MLHMP software � Stages: ”Mating”, ”Gestation”, ”Suckling” � Herd interface � State: � Herd specific parameters and prices � Mating: ”Healthy”, ”Diseased” � Reads sow data � Gestation: ”Pregnant”, ”Infertile”, ”Diseased” � � Presents results Suckling: Present litter size � Decision: � Built for use in practice � Mating: Allow m matings, m ∈ { 1,… ,5} � Demonstration � Gestation: Dum my � Suckling: ”Keep”, ”Replace” Advanced Herd Management 2006 21 Advanced Herd Management 2006 22 Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH Anders Ringgaard Kristensen, I PH A 3-level model for optimal replacement of sows DLM and MDP Estim ating Disease Costs using � First processing: Monitoring & filtering � Second processing: Decision making MLHMP � MDP with DLM integrates both parts The follow ing slides are m ade by Doron Bar, Cornell University, NY Advanced Herd Management 2006 23 Advanced Herd Management 2006 24 4

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