\section*{A Mixed Integer Linear Programming
Model\\ of Municipal Waste Management}\label{S:Recycling}

\newpar	
Faced with limited landfill space, a municipality decides it must improve its recycling program over the next few years. \footnote{Based upon a model of Baetz and Neebe, presented in A Planning Model for the Development of Waste Material Recycling Programmes \emph{The Journal of the Operational Research Society}, \textbf{45} (1994), 1374-1384. }
The current waste disposal process used by the city is summarized in Figure
\ref{Fi:waste_management}.

\begin{figure}[h]
\centering
\psfrag{city}[Bc][Bc]{City}
\psfrag{collection}[Bc][Bc]{Collection Site}
\psfrag{recycled}[Bc][Bc]{Recycled Materials}
\psfrag{WTE}[Bc][Bc]{WTE Plant}
\psfrag{landfilla}[Bc][Bc]{Landfill}
\psfrag{landfillb}[Bc][Bc]{Backup Landfill}
\includegraphics[width=.9\linewidth]{images/p06x01.eps}
\caption{Waste management system.}\label{Fi:waste_management}
\end{figure}

\newpar
Garbage is collected by the city and brought to a central collection site. There, it is sorted, with
a portion of the material removed for recycling, a portion sent directly to the municipal landfill, and the
remainder incinerated at the city's waste-to-energy (WTE) plant. Leftover residue from this incineration is transported to a city landfill. In addition, the city maintains a second,
backup landfill should the first become full at a future date.

\newpar
Costs to the city stem from a variety of sources. In terms of recycling there is a startup cost associated with recycling each new item, e.g. cardboard, plastic, etc., as well as an annual cost dependent upon the amount of material processed. These facts suggest the city not increase its current level of recycling. However,
both waste transportation and landfill usage costs are high. Although the WTE plant produces some energy to offset these costs, the corresponding revenue is minimal.

\newpar
As a result of this dilemma, the city decides to embark on a three-year trial recycling and waste management plan. For each year of the plan, a minimum specified percent of the total collected waste will be processed for recycling. The exact manner in which this is to accomplished
will be determined using a mixed integer linear programming model whose goal is to
minimize total costs to the city over the three-year period. Because the waste collected by the city can
be subdivided into 10 waste-component categories, the model will stipulate how much of each component is
designated for recycling, incineration, and direct landfill deposit during each of the three time periods.

\newpar
Data for the 10 waste by-products categories is provided in Table \ref{Ta:Waste_Product_Data}.

\newpar
We will use the following notation for the decision variables in this model. In what follows,
we assume ``tons'' means thousands of tons, ``cubic meters'' represents thousands of cubic
meters, $1\leq i \leq 10$ denotes waste product $i$, and $1\leq t\leq 3$ records the time period, which
is in years.
	\bi
	\item $X_{it}$: Number of tons of waste component $i$ recycled during time period $t$.
	\item $Y_{it}$: Number of tons of waste component $i$ sent directly to landfill during time period $t$.
	\item $Ys_{it}$: Number of tons of waste component $i$ sent directly to secondary landfill during time
	period $t$.
	\item $Z_{it}$: Number of tons of waste component $i$ sent to WTE and then residue sent to landfill
	during time period $t$.
	\item $Zs_{it}$: Number of tons of waste component $i$ sent to WTE and then residue sent to
	secondary handful
	during time period $t$.
	\item $LV_{t}$: Total volume of all waste components plus residue sent to landfill during period $t$.
	\item $LVs_{t}$: Total volume of all waste components plus residue sent to
	secondary landfill during period $t$.
	\item $R_{it}$: Binary decision variable, equal to 1 if recycling of waste component $i$ is initiated
	at start of time period $t$ and equal to 0 otherwise.
	\item $P_{t}$: Binary decision variable, equal to 1 if secondary landfill is opened
	at start of time period $t$ and equal to 0 otherwise.
	\ei

\newpar
Here is information regarding costs associated with these variables.
	\bi
	\item Unit recycling costs: The collection and processing cost is estimated to be \$75 per ton for
	all products. Recycling revenues vary by material and are given by the data in Table \ref{Ta:Waste_Product_Data}.
	\item Unit landfill costs: This includes only collection and transportation to the landfill. The
	collection cost is approximately \$40 per ton,
	and the round-trip driving cost for one 10-ton-capacity truck to
	deliver
	to either landfill is \$50 per round trip.
	\item Unit incineration cost: This is the sum of the collection/transportation cost to the WTE,
	plus the WTE processing cost,
	less energy revenue from incineration, plus transportation cost of the residue to the landfill.
	The collection cost is the
	same as used to compute unit landfilling cost, and the round trip driving cost for one truck to deliver from the collection
	facility to the WTE is \$10 per round trip. The WTE processing cost is estimated at \$35 per ton, and
	the energy revenue
	depends upon the particular waste product. To compute this value, use the energy content data plus the
	facts that one ton equals 2200 lb. and energy is worth \$.00000215 per BTU. The residue factor
	is the ratio of weight of residue to weight prior to combustion and is a constant depending
	only
	upon the particular waste product. Finally, the round trip driving cost for one truck to deliver from the
	WTE to either landfill is \$20 per round trip.
	\item Landfill disposal costs are \$9 per cubic meter for either landfill.
	\item The current recycling start-up cost associated with waste product $i$ is given in Table \ref{Ta:Waste_Product_Data}.
	\item The current cost associated with opening the secondary landfill is \$40 million.
	\ei
With the exception of unit recycling costs, all costs are expected to increase at a rate of 5\% per year.
Unit recycling costs are expected to increase only at an annual rate of 3\%.

\newpar
Finally, there are several parameters associated with the model.
	\bi
	\item The population of the city is currently 500,000 and expected to increase by 2\% annually. Each person
	produces an average of .75 tons of waste per year.
	\item The percentages of each waste product are found in Table \ref{Ta:Waste_Product_Data}.
	\item The city desires to send at least 25\% of its waste to the WTE plant. The plant itself
	processes approximately 350 tons per day and typically operates throughout the year with about
	1\% downtime.
	\item The landfill and secondary landfills have capacities of 500,000 and 600,000 cubic meters,
	respectively.
	(For purposes of the model, these values should be converted to thousands of cubic meters.)
	\item The raw density of waste product $i$ is measured prior to incineration, has units of tons per
	cubic meters, and is provided by the data.
	\item The incinerated density for waste product $i$ is measured post-incineration, has units of
	tons per cubic meters, and is 1.67 tons per cubic meters for all products.
	\ei
	

\newpar
The city can now use all of this information to express a three-year waste management
plan as a vector in $\R^3$. Each entry of this vector is
between 0 and 1 inclusive and represents the minimum percent of total waste to be recycled during
that respective time period. For example, a waste management plan of $[.1,.2, .4]$
indicates that the city plans to recycle at least 10\% of its total waste the first year, 20\%
the second year, and
40\% the third year.

\newpar
Set up and solve a mixed integer linear programming model whose goal is to minimize the costs
to the city over a three-year period, in units of thousands of dollars,
if it uses a waste management plan
$[.1,.2, .4]$. This model should incorporate the following constraints:
	\be
	\item Logical conditions on the binary decision variables.
	\item Constraints involving landfill capacity.
	\item Constraints involving system capacity: collected waste is diverted to a limited number of sources;
	there are recycling goals as well as upper and lower bounds to the waste processed at the WTE plant.
	\ee
Summarize all findings by product and by time period and determine whether the backup landfill is utilized.

\newpar
Suppose the city is concerned that residents may be resistant
to increased recycling requirements during the upcoming year. At the same time,
the city recognizes its limited amount of landfill space.
Determine a waste management plan in which no recycling takes
place during the first year and in which the following conditions are met:
	\be
	\item During the second year of new plan, the city must recycle at least 10\% of its waste.
	\item During the third year of the new plan, the city must recycle between 10 and 30 \% more
	than it did during the second year.
	\item No more than 50\% of the landfill amount used in the original plan is utilized
	in the new model!
	\ee
\newpage

\begin{table} \centering
\caption{\large{{\textbf{Waste product data}}} }
\label{Ta:Waste_Product_Data}
\scalebox{.7}{
\rotatebox{90}{
\begin{tabular}{||l||c|c|c|c|c|c|c|c||}
\hline \hline
Material &
\% of waste stream &
\begin{tabular}{l} \% of material \\ currently recycled \end{tabular} &
Residue factor &
\begin{tabular}{l} Raw density \\ (tons/cubic meter) \end{tabular} &
\begin{tabular}{l} Incinerated density \\ (tons/cubic meter) \end{tabular} &
\begin{tabular}{l} Recyclable material  \\ revenues \\ (\$ /ton) \end{tabular} &
\begin{tabular}{l} Unit energy \\ content values  \\ (BTU/lb.) \end{tabular} &
\begin{tabular}{l} Recycling program \\ development cost  \\ (1000's of \$) \end{tabular} \\ \hline
Newsprint & 25 & 50 & .08 & .41 & 1.67                                & 20 & 7200 & 500 \\ \hline
Fine paper & 15 & 0 & .08 & .41 & 1.67                                & 30 & 7200 & 1000 \\ \hline
Cardboard & 4 & 0 & .06 & .25 & 1.67                                  & 20 & 7000 & 1000 \\ \hline
Glass & 8 & 25 & .99 & .97 & 1.67                                     & 45 & 60 & 500 \\ \hline
Non-ferrous metals & 1 & 75 & .98 & .80 & 1.67                          & 1000 & 0 & 500 \\ \hline
Ferrous metals & 2 & 50 & .99 & 1.60 & 1.67                            & 30 & 300 & 500 \\ \hline
Food wastes & 15 & 0 & .08 & 1.44 & 1.67                                & 0 & 2000 & 2500 \\ \hline
Yard wastes & 12 & 0 & .06 & .52 & 1.67                                 & 0 & 2800 & 1000 \\ \hline
Plastics & 4 & 0 & .20 & .32 & 1.67                                   & 200 & 7000 & 5000 \\ \hline
Remainder & 14 & 0 & .90 & 1.67 &                                     1.67 & 30 & 5000 & 1000 \\ \hline
\end{tabular}
}
}
\end{table} 