\be
\item For the coal model without transshipment, suppose that demand at City 3 decreases from 500,000 to 300,000 tons, in which case the problem is no longer balanced. Determine the least costly manner in which to deliver
    coal from the mines to cities and still meet demand. Where will the excess coal remain under this scenario? (Hint: Define an ``imaginary'' City 5, whose demand is the excess of total supply over total demand, i.e., 200,000 tons. Set to zero the transportation cost from each mine to City 4.)
\item Prove the assertion immediately prior to Theorem \ref{Th:Transportation_Transshipment_IntegralSolutions}. Namely, if $A$ is a square, $n$ by $n$ having determinant equal to $\pm 1$ and if both $A$ and $b \in \R^n$ have integer-valued entries, then the solution vector of $\vecx$ of $A \vecx=\vecb$ also has
    integer-valued entries. (Hint: Use Cramer's Rule. See Appendix \ref{C:Linear_Algebra_Appendix}.)

\item  \emph{Import substitution}\index{import substitution} refers to a nation's practice of producing a certain commodity domestically so as to reduce the amount of the commodity imported from abroad.
    \footnote{Based upon Blandford, Boisvert, and Charles \cite{Blandford_Import_Substitution}, (1982).}
    Suppose a country desires to reduce its dependence upon overseas sources for corn and soy. The country has three available plantations for producing a mixed crop of these items, which are then transported to four different markets. The plantations are labeled  1, 2, and 3, and have 9, 10, and 10 acres available, respectively, for growing either crop. The demand for corn and soy at each of the four markets is summarized in Table
    \ref{Ta:Exercises_Transportation_Transshipment_Import_Demand}.

    \begin{table}[h]
\caption{Market demand for corn and soy, measured in tons}\label{Ta:Exercises_Transportation_Transshipment_Import_Demand}
\begin{tabular}{|c|c|c|}
\hline
Market/Commodity & Corn & Soy \\ \hline
1 & 2 & 5  \\ \hline
2 & 5 & 8  \\ \hline
3 & 10 & 13  \\ \hline
4 & 17 & 20  \\
\hline
\end{tabular}
\end{table}

\noi Each plantation produces 2 tons of corn per acre and 4 tons of soy per acre. The costs of growing corn and soy are \$200 per acre and \$300 per acre, respectively, regardless of the plantation. The cost of transporting either good from plantation $i$ to market $j$ is $20i+30j$ dollars per ton.

\begin{enumerate}
\item Suppose crop 1 represents corn and crop 2 represents soy. Let $x_{ijk}$, where $1 \leq i \leq 3$, $1 \leq j \leq 4$, and $1 \leq k \leq 2$, represent the amount of crop $k$, measured in tons, that is transported from plantation $i$ to market $j$. Let $y_{ik}$, where $1 \leq i \leq 3$, and $1 \leq k \leq 2$, represent
    the acreage at plantation $i$ that is devoted to producing crop $k$. Use this notation to determine an expression that represents the total cost due to growing the two crops on the three plantations and transporting them to the different markets.
\item Determine a sequence of inequalities that stipulates the acreage supply at each plantation is not exceeded.
\item Determine a second sequence of inequalities that guarantees demand at each market is met.
\item Determine a third sequence of inequalities that guarantees the amount of each crop, in tons, transported from a given plantation to a given market does not exceed the available amount of the crop that can be produced at that plantation.
    \item Determine the values of the quantities $x_{ijk}$ and $y_{jk}$ that minimize total cost due to import substitution of corn and soy.
\end{enumerate}

\ee


