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\section*{Design of robotic dairy barns}

The most important recent development in the dairy industry is robotic milking.
Dairy barns with milking robots are becoming more and more interesting
from an economical point of view. Today already
over 200 robots have been installed on commercial Dutch farms.  As
robotic milking barns (RMB) are expensive, it is important to develop
models which make it possible to discuss the optimal layout of an RMB
and the optimal capacities of the various facilities in the barn
depending on the herd size before actually constructing it.

In an experimental farm in Duiven in the Netherlands the agricultural
research center IMAG-DLO is investigating the behaviour of the cows in
an RMB. Based on extensive measurements and observations, it was
concluded that it is necessary to incorporate the
stochastic behaviour of the cows in the design of an RMB.  Another
aspect, which makes this design complex, is the interaction between the
facilities in the barn:  increasing the capacity of bottleneck
facilities will shift queues and alter the location of bottlenecks,
possibly forcing the designer to increase the capacity elsewhere.


Below we describe the RMB
and discuss some of the performance aspects.
The basic layout of the RMB we are dealing with is shown in figure \ref{layout}.

\begin{figure}[hbt]
\begin{center}
\includegraphics[width=1.0\linewidth]{layout}
\caption{Layout of the experimental barn in Duiven.
         Dimensions in mm. ({\scriptsize source: IMAG-DLO})}
\label{layout}
\end{center}
\end{figure}

In the barn we distinguish five facilities.

\begin{itemize}
\item
{\bf The Milking robot.}
The milking robot is shown in figure \ref{robot}.  Milking robots are
different from the ordinary milking machines in one crucial aspect: the
robot uses sensors to find the teats of the cow and then connects the
cups to the teats with a robot arm.

\begin{figure}[hbt]
\begin{center}
\includegraphics[width=1.0\linewidth]{robot}
\caption{A milking robot ({\scriptsize source: IMAG-DLO})}
\label{robot}
\end{center}
\end{figure}

There are at least two good reasons for using robots. First, it saves a
serious amount of labour and second, it makes it possible to go from
milking twice a day to three or even more times a day. When cows are
milked three times a day their production is increased by about 15
percent.

\item
{\bf The Concentrate feeder.} Each cow is allowed to
receive only a limited amount of `concentrate.'  So
the Concentrate feeder must have the equipment to be able to identify
the cows and to decide how much concentrate to give to the cow.  Cows
are very fond of the concentrate. Therefore the Concentrate feeder can
be and is used to get the cows to pass through the Milking robot. In
the present design in Duiven the cows can only reach the Concentrate
feeder via the robot.

\end{itemize}
\noindent
The three more conventional facilities are:

\begin{itemize}
\item
{\bf The Forage lane.} Forage lanes are cheap. There are no limitations on
foraging. The only condition is that there must be enough eating
positions at the forage feeder to prevent the cows from becoming
aggressive.

\item
{\bf The Water troughs.} A `high-yielding' cow may drink
upto 180 liters a day. Water troughs are cheap, but of
significant physiological importance.

\item
{\bf The Cubicles.} In the cubicles the cows can lay down, rest, and avoid
confrontations.  They spend roughly 50 percent of their time in the
cubicles.  Cubicles only require space, some fencing and bedding
material (wood shavings, sand, rubber mattresses).

\end{itemize}

\noindent
Further we introduce one more, artificial, facility
that we will call:

\begin{itemize}
\item
{\bf Walking.}
The space in between the facilities is used for walking, idling or
grouping.  This takes nearly 25 percent of their time, so 5 to 6 hours
a day. In that time they cover at most a few kilometers, so a better
word for the facility might be `Standing.' Anyway, the walking area
should be large enough to accommodate somewhat more than 25 percent of
the herd.
\end{itemize}

From extensive measurements in the experimental barn in Duiven
information has been obtained on the behavior of the cows. There is
data available on the times spent in the
\href{http://www.win.tue.nl/~iadan/ors2/Robot.txt}{robot},
\href{http://www.win.tue.nl/~iadan/ors2/Concentrate.txt}{concentrate feeder},
\href{http://www.win.tue.nl/~iadan/ors2/Forage.txt}{forage lane},
\href{http://www.win.tue.nl/~iadan/ors2/Water.txt}{water troughs} and the
\href{http://www.win.tue.nl/~iadan/ors2/Cubical.txt}{cubicles}, and on
the '\href{http://www.win.tue.nl/~iadan/ors2/transition.txt}{transition}' behavior of
the cows between these facilities. From the measurements in Duiven we
also know that Walking takes (approximately) 23.8 percent of the time.
This implies that the mean time to `walk' from one facility to the next
is 5.36 minutes.

Waiting is something cows do not like. In this respect they do not
differ from humans.  When a cow is waiting for a facility and another
cow arrives, aggressive behaviour might occur. Particularly at the
scarce facilities Milking robot and Concentrate waiting has to be
limited.  Some waiting
for Cubicles is not really a problem because Walking seems to be an
alternative for the Cubicles.
So in the discussion about the design we focus on mean
waiting times and the queue lengths.  Given the transition behavior,
service times, capacities and herd size we should be able to evaluate
waiting times, queue lengths and utilizations for the various
facilities. From these we can discuss and judge the design under
consideration.\\

Now you are asked to:
\begin{itemize}
\item
Develop a model to evaluate the performance of the RMB (clearly discuss assumptions and limitations);
\item
Develop an (approximate) MVA approach to evaluate the performance of the RMB;
\item
Use this model to design an RMB for a herd of 50 cows; the design
criterion that may be used is an upper limit of (say) 2 minutes for
the mean waiting time in each facility. Describe your procedure used to
determine the `optimal' design.
\item
Cows are milked 3 or 4 times a day. The model will probably predict
that the number of visits to the robots is (much) greater (check!).
The reason is that cows do not always want to be milked, but they just
want to get some concentrate food.
In that case they unnecessarily occupy the robots. This loss
of capacity can be reduced by using a `selective gate' in front of
the robots: cows are only allowed to pass if they have to be milked.
Use your model to evaluate the effect of using a selective gate.
\item
Typically cows in a herd are socially ranked. The low-ranking cows tend to
wait longer in front of facilities than the dominant ones (they may be
pushed away). Incorporate social rank into the model and demonstrate
its effect on the performance of the RMB.
\item
Write a clear report on your findings.
\end{itemize}

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