A large Markovian linear program to optimize replacement policies and dairy herd net income for diets and nitrogen excretion.

The purpose of the study was 2-fold: 1) to propose a novel modeling framework using Markovian linear programming to optimize dairy farmer-defined goals under different decision schemes and 2) to illustrate the model with a practical application testing diets for entire lactations. A dairy herd population was represented by cow state variables defined by parity (1 to 15), month in lactation (1 to 24), and pregnancy status (0 nonpregnant and 1 to 9 mo of pregnancy). A database of 326,000 lactations of Holsteins from AgSource Dairy Herd Improvement service (http://agsource.crinet.com/page249/DHI) was used to parameterize reproduction, mortality, and involuntary culling. The problem was set up as a Markovian linear program model containing 5,580 decision variables and 8,731 constraints. The model optimized the net revenue of the steady state dairy herd population having 2 options in each state: keeping or replacing an animal. Five diets were studied to assess economic, environmental, and herd structural outcomes. Diets varied in proportions of alfalfa silage (38 to 98% of dry matter), high-moisture ear corn (0 to 42% of dry matter), and soybean meal (0 to 18% of dry matter) within and between lactations, which determined dry matter intake, milk production, and N excretion. Diet ingredient compositions ranged from one of high concentrates to alfalfa silage only. Hence, the model identified the maximum net revenue that included the value of nutrient excretion and the cost of manure disposal associated with the optimal policy. Outcomes related to optimal solutions included the herd population structure, the replacement policy, and the amount of N excreted under each diet experiment. The problem was solved using the Excel Risk Solver Platform with the Standard LP/Quadratic Engine. Consistent replacement policies were to (1) keep pregnant cows, (2) keep primiparous cows longer than multiparous cows, and (3) decrease replacement rates when milk and feed prices are favorable. The optimal policy called for the replacement of open cows between 7 and 12 mo in lactation depending on parity, diet, and market conditions. Under favorable market conditions, net revenue was greatest with the greatest concentrate diet, which was $15.24 and $52.32/mo per cow greater than the optimal net revenue realized with the intermediate and the no-concentrate (all-forage) diets, respectively. A suboptimal solution to limit the N excretion to 12 kg/mo per cow when market conditions were favorable resulted in a diet with the second-greatest amount of concentrates being the one with the greatest net revenue. Under unfavorable market conditions, the diet with the greatest concentrate content had the least net revenue compared with all the others. A suboptimal solution for a maximum N excretion of 12 kg/mo per cow with unfavorable market conditions resulted in the least-concentrate diet having the greatest net revenue ($22/mo per cow), followed by the second-greatest concentrate diet ($20/mo per cow) and the all-forage diet ($18/mo per cow). The implementation of a Markovian linear program for dairy decision making provides both robustness and versatility in operations research. The model could become a valuable tool for economic decision making for dairy farms.