Formulation of the Dairy Replacement Problem by using a Markovian Decision Process Approach and Linear Programming.

The "replacement problem" has been frequently studied in literature utilising different methodologies. The method traditionally used to solve this problem is to formulate it as a decisional Markovian Process and to optimise it using Dynamic Programming (DP). There are, however, problems associated with the use of DP. The decision to cull is based solely upon the expected performance of the animal in question and that of it's replacement. Therefore, information concerning the cows performance in relation to the remaining herd has been ignored. This is especially relevant were the producer is trying to increase yields by genetic selection.
A developed methodology (Yates and Rehman, 1998) suggests the use of Linear Programming (LP) to solve the "replacement problem", but the model presented is simplified.
The purpose of this project was to include more initial states to simulate a greater variation in genetic merit and other representative aspects.
Important parameters evaluated are:
- the age at first calving which was shown to be a significant factor in the occurrence of mastitis;
- the estimation of conception rates, calf survival rates, calf mortality rate, culling rate, first service pregnancy rates, failure to conceive rates;
- the prediction of annual milk yields, milk yield for all the different genetic states and for each parity and year;
- breeding costs, milk and meat prices.
The model was developed over a 15 years decision horizon. Whilst the genetic merit, in terms of milk production, of an individual animal is different from that of any other animal, it was possible to group animals into similar bands based on their production. In this model were taken into account two different production levels (high and low milk yield) and cows with 7 different parities (age in terms of lactations).
The total herd increase in milk yield also depends upon the conception and mortality rates, known collectively as the "replacement rate". This genetic turn-over is therefore a gradual process dependent upon the past and current state of the herd and the one desired in the future.
Before specifying the Markovian decision process involved in this model, it was first necessary to define all the possible genetic states over the 15 year period which are 584, the model so specified was constructed using XPRESS, a software package to solve Linear Programming problems.
Other factors were included like land, capital and labour utilisation, nutritional requirements.
The proposed solution can be seen as an innovation to introduce in dairy farms to help the farmers in taking decisions to optimise the management from an economic point of view and taking into account various problems that can raise in a long term planning.