\section{The Juice Logistics Model }\label{S:Welches}

\newpar	
Transportation problems constitute one of the most common applications of linear programming. In
this project, we investigate a modified version of one such problem faced by the Welch's grape processing company in the early 1990's. \footnote{Based upon Schuster and Allen, \cite{Schuster_Welches}, (1998)}
At that time, Welch's sought to minimize costs associated with seasonal fluctuations in grape
production, the transport of raw materials between plants, and storage costs from one growing season to the next. New policies, based upon linear programming, resulted
in a net savings for Welch's of more than \$130,000 during the first year of adoption.

\newpar
Assume the company collects
``raw,'' unprocessed grape juice, hereafter referred to simply as ``juice,'' from grape
growers at two different plants, labeled plant 1 ($k=1$) and plant 2 ($k=2$). The former is located
in an urban setting, the
latter in a rural region.
The collection process takes place
over a 12-month period, with month $i=1$ corresponding to September, when the grape harvest is largest.
The majority of grapes are collected from growers who belong to a regional
Grape Cooperative Association (GCA). However, during the winter months, when local harvesting is low, the company frequently seeks grape sources elsewhere, including overseas.

\newpar
Simultaneous to the collection of juice at the plants is the processing of two different products,
jam ($j=1$) and juice
concentrate ($j=2$). Company policy dictates that a minimum percentage of juice must be processed for each product during a given
month at each particular plant. To ensure that enough juice is on hand, the company is free to transfer
juice from one plant to another during any given month. Because of transportation costs, the company might be motivated to maintain a large juice inventory at each plant. The drawback of such a practice, however, is the carryover cost associated with the storage of unused juice into the next season.

\newpar
The objective of the company is to minimize costs that stem from three different sources: cost associated with shipping finished products to stores, cost stemming from moving juice between the two plants, and cost due to storing juice at each plant.

\newpar
We define the decision variables for this model using the following notation. Assume all amounts are measured in tons.
	\bi
	\item $TS_{i,j,k}$, where $1 \leq i \leq 12$ and $1 \leq j,k \leq 2$: Amount of product $j$ shipped to stores from plant $k$ at month $i$.
	\item $TI_{i,k,m}$, where $1 \leq i \leq 12$ and $1 \leq k,m \leq 2$: Amount of grape juice transferred into plant $k$ from plant $m$ during month $i$. A stipulation, of course, is
that $TI_{i,k,k}=0$ for $1 \leq k \leq 2$ and $1 \leq i \leq 12$.
\item $TO_{i,k,m}$, where $1 \leq i \leq 12$ and $1 \leq k,m \leq 2$: Amount of grape juice transferred out of plant $k$ and into plant $m$ during month $i$. Again, we assume that $TO_{i,k,k}=0$ for $1 \leq k \leq 2$ and $1 \leq i \leq 12$.
    This family of decision variables may not appear necessary since we can interpret negative values of $TI_{i,k,m}$ as the amount of juice transferred out of plant $k$ and into plant $m$, but adding these variables, along with appropriate constraints, permits us to construct a model in which all decision variables are nonnegative. For example, a negative value of $TI_{i,k,m}-TO_{i,k,m}$ means that
    plant $k$ suffers a net loss of juice to plant $m$, even though each individual quantity
    in this difference is nonnegative.
	\item $EI_{i,k}$,  where $1 \leq i \leq 12$ and $1 \leq k \leq 2$: The amount of grape juice inventory at plant $k$ at end of month $i$.
	\ei

\newpar
Associated with the first three of these four families of variables are various costs, which are as follows:
	\bi
	\item %CT
    The cost of transporting juice from plant $k$ to the other plant during month $i$ is \$65 per ton.
	\item Manufacturing Costs: For each plant, jam costs \$150 per ton to produce and juice concentrate
    \$175 per ton.
	\item The cost of storing juice in each plant from one season to the next:  Assume
this amount is \$500 per ton for either plant.
	\ei

\be
\item Formulate the objective function for this LP, which consists of the total cost of manufacturing the two products at the plants, transferring the juice from one plant to the other, and storing juice from one season to the next. Your function will involve triple summations.

\newpar
There are several important quantities that determine the LP's constraints.
	\bi
	\item At the start of every year, each plant is required to have 5 tons of juice on hand.
	\item The amount of juice delivered to each plant at the start of month $i$ is approximately
\begin{align*}
P_1(i) &= 15 \left(1+\sin\left(\frac{\pi}{6}i\right) \right) \\
\intertext{for plant 1, and}
P_2(i) &= 50 \left(1+\sin\left(\frac{\pi}{6}i\right) \right)
\end{align*}
for plant 2.
Note that the delivery amounts can be created as lists in Maple using the following syntax: \newline

\noi
\mf{$>$ P1:=[seq(15*(1+sin(Pi*i/6)),i=1..12)]:} \newline
\mf{$>$ P2:=[seq(50*(1+sin(Pi*i/6)),i=1..12)]:}
\newpar

    \item At plant 1, 3 tons of juice are required to produce 1 ton of jam. At plant 2, 4 tons of juice are needed. At either plant, 2 tons of juice are required to create one ton of juice concentrate.
    \item Monthly product demands require that at least 10 tons of jam and 15 tons of juice
    concentrate be produced.
    \item At most 50 tons can be shipped from one plant to another during any given month.
    \item Each plant can produce at most 150 tons of the combined products each year.
	\ei

\newpar
\item Use the the decision variables and model parameters, to formulate each of
the following families of constraints.
	\be
	\item At the end of each year, the juice inventory
    at each plant must equal 70 tons.
    \item During each month, the amount transferred into one plant equals the amount
    transferred out of the other.
    \item Production of the products at the plants must meet monthly demand.
    \item At most 50 tons can be shipped from one plant to the other during any given month.
    \item Each plant is limited in how much jam and juice concentrate it can produce annually.
    \item Each month, the amount of juice transferred into one plant equals the amount transferred out of the other.
    \item A balance of juice exists from one month to the next.
        \be
        \item At the end of month 1, the ending inventory at each plant is the initial inventory
        of 70, plus the juice transferred in, less the juice transferred out, less juice lost to production of each product, plus
        juice obtained via normal delivery.
        \item At the end of month $i$, where $2 \leq i \leq 12$, the ending inventory is
        that from the previous month, plus the juice transferred in, less the juice transferred out, less juice lost to production of each product, plus
        juice obtained via normal delivery.
        \ee
    \ee

\newpar
\item Now use Maple to solve the LP that minimizes the objective from (1), subject to the
constraints from (2).

\ee 