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\documentclass{simiode}
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\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Employee Attrition}




\markboth{Employee Attrition}{Employee Attrition}

\author{Lyle Smith III,
Bryan College\\
Dayton TN  37321 USA\\
 lylesmith333@gmail.com 
}

\makeStitlePDFLaTex

\begin{abstract}
This scenario models the loss of employees and the employer's attempt to retain them through stock options. It most naturally is solved with a first-order linear decay model with two populations. This is intended for introductory students and is paired well with exponential growth/decay material. We also introduce the 5\% rule to sort possible factors into assumptions and modeling factors. Finally, iterative methods can be used to find a balance between growth and decay.
\end{abstract}


\section*{SCENARIO DESCRIPTION}
A large online retailer has a problem.  It hires 100,000 ``last mile'' delivery workers to deliver its packages.  However, these part-time employees leave the company at a rate of 9\% per month.  We will construct a model that predicts the number of employees in this job during a given month.

\section*{Part 1: Losing Employees}
\begin{enumerate}[A.]\setlength\hangindent{0.6cm}
	\item Make a list of every factor you can think of that will affect the employee population change.  We will assume no new hires for now.
	\item Using the 5\% rule, determine which factors should be included in the model and which can be assumed to be insignificant. Label each factor from part A. as either A for assumed to be small or M for included in the model.
	
	\begin{center}\begin{tabular}{| p{15cm}|}\hline
		\textbf{The 5\% Rule}: If you believe the factor will effect the model by 5\% or more, it must be included in the mathematical formulation of the model.  If you believe the effect will be less than 5\%, list the factor as an assumption, i.e. assume the effect is minimal.\\\hline
	\end{tabular}\end{center}
	\item Formulate a mathematical model (an equation) for the population of employees.  Use $A(t)$ to represent the population.  Be sure to include all of the model factors from part B.
	\item What initial condition will you use?
	\item Predict the behavior of your model.  What will happen in the second month?  What will the behavior be for large times?
	\item Use the \texttt{DSolve} command in Mathematica to find the solution function for your model's Initial Value Problem (IVP).  We would like to store the resulting function to use later.  The syntax for this is\\
\texttt{A1[t\_] = A[t] /. First@DSolve[\{A'[t]==????,A[0]==????\},A[t],t]}
	\item Plot the resultant function to check your answer.
	\item Does your result match your prediction from part E?
	\item Find the population at time $t=120$ months.  What is wrong with this prediction?  What assumption did we violate?
\end{enumerate}

\section*{Part 2: Stock Options}
The workforce will only support the hiring of 5,000 employees per month, so a new strategy is employed to retain workers.  In addition to hiring 5,000 per month, 10\% of the employees are offered stock options because it is believed that this will reduce the attrition rate to 3\% (for these employees only).

\begin{enumerate}[A.]\setlength\hangindent{0.6cm}
	\item What new factors might now affect the total population?
	\item Again, divide these new factors into assumptions and model components.
	\item Formulate a mathematical model (an equation) for this second population of employees, call it $B(t)$.  A new model for $A(t)$ may also be necessary.
	\item What initial condition will you use, $B(0)$?
	\item Predict the behavior of your model.
	\item Find the solution functions for your model for $A(t)$ and $B(t)$.
	\item Plot the resultant functions to check your answer.
	\item Does your result match your prediction?
	\item What is the end behavior for the total employee population, i.e. $A(t)+B(t)$ for $t\to\infty$? 
\end{enumerate}

\section*{Part 3: Achieving Balance}
The plan to offer stock options to 10\% of the employees has succeeded, but too well.  We would like to reach exactly 100,000 employees in the steady state.
\begin{enumerate}[A.]\setlength\hangindent{0.6cm}
	\item What percentage of the employee population should be offered stock options in order to maintain this level?
	\item What are the resultant functions at this stock option level?
	\item Plot both population levels and the total population on the same graph.
	\item Offer a summary recommendation to the company based on your work.
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}
We have placed solutions and our comments in a Mathematica notebook, 1-109-T-Mma-EmployeeAttrition-TeacherVersion.nb along with a pdf, 1-109-T-Mma-EmployeeAttrition-TeacherVersion.pdf, for those who do not read Mathematica notebooks.
}
 
 



\end{document}
