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\title{Operations Research}
\author{Homework 7}
\date{Due via Moodle or Mailbox Thursday, November 12, 2020}
\maketitle


\begin{enumerate}

\item \emph{(HL, Exercise 10.3-3.)}
A college student has $7$ days remaining before final examinations
begin in her four courses, and she wants to allocate this study time
as effectively as possible.  She needs at least $1$ day on each
course, and she likes to concentrate on just one course each day, so
she wants to allocate $1, 2, 3$, or $4$ days to each course.  Having
recently taken an OR course, she decides to use dynamic programming to
make these allocations to maximize the total grade points to be
obtained from the four courses.  She estimates that the alternative
allocations for each course would yield the number of grade points
shown in the following table.
\begin{center}
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{\textbf{Estimated Grade
  Points}$\vphantom{1^{1^1}}$} \\
\textbf{Study Days}$\vphantom{1^{1^1}}$ & 
  \textbf{Course 1} & \textbf{Course 2} & 
  \textbf{Course 3} & \textbf{Course 4} \\
\hline
\textbf 1$\vphantom{1}^{\vphantom{\tfrac11}}$ & 1 & 5 & 4 & 4 \\
\textbf 2 & 3 & 6 & 6 & 4 \\
\textbf 3 & 6 & 8 & 7 & 5 \\
\textbf 4 & 8 & 8 & 9 & 8 \\
\hline
\end{tabular}
\end{center}
Solve this problem by dynamic programming.

\item \emph{(HL, Exercise 10.3-10.)} Consider the following
\emph{integer} nonlinear programming problem.
\begin{gather*}
  \text{Maximize } z= 32 \, x_1 - 2 \, x_1^2 + 30 \, x_2 + 20 \, x_3
\intertext{subject to}
  3 \, x_1 + 7 \, x_2 + 5 \, x_3 \leq 20
\intertext{and}
  x_1, x_2, x_3 \text{ are nonnegative integers}.
\end{gather*}
Use dynamic programming to solve this problem.

\item This problem is ``extra credit'' up to a total of 10 points for
this homework set.

\emph{(From HL, Exercise 10.3-12.)} Re-solve the Local Job Shop
employment scheduling problem from class when the cost of changing the
level of employment from one season to the next is changed to $100$
times the square of the difference in employment levels.

You may use Sympy and modify the example file on the course web site.
If you use the file provided, please be sure to adapt the annotation
text.  If the comments are left unchanged and diverge from the
computation results, your solution will be downgraded even if the
final result is correct.

\end{enumerate}

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