Solid Waste, Measuring

We usually know where things come from. The food we buy comes from farms, practically everything else we own comes from factories, and the raw materials to make them come from mines and forests.

But were does it all go once we finish using it or it breaks? We put it on the curb and—thanks to trash collectors—it seems to disappear. Yet in reality, nothing disappears: We just change useful things into trash, just as we change natural resources into things we can use. This article examines how mathematics lets us calculate the amounts of trash we must either divert to beneficial uses or dispose in landfills or incinerators.

Types and Fates of Solid Waste

In general, there are two kinds of trash, or solid waste. Industrial solid waste is generated when factories convert raw materials into products. Municipal solid waste is the trash we throw away after we buy and use those products. This article discusses only municipal solid waste, or MSW.

The U.S. Environmental Protection Agency estimates that about 223,230,000 tons of MSW were generated in the United States in 2000. Almost all the waste met one of two fates:

• Disposal at a landfill or incinerator; or
• Recycling or reuse (called diversion).

A small amount of the generated waste was dumped illegally, but these amounts are difficult to quantify, and are not addressed in this article.

Waste Disposal. The most common waste disposal method in the United States is landfilling. A landfill essentially is a big hole in the ground where we bury our garbage. Engineers design a landfill based on the amount and types of solid waste it needs to receive over its projected "lifetime." Once it is full and cannot hold any more trash, a new landfill must be dug. Knowing how much waste is generated by a community helps waste managers and community planners determine how big to make a new landfill, as well as how many trash trucks are needed to collect and transport the waste.

Calculations for a community start with individual households. In 1998 the average person in the United States produced 4.46 pounds of waste per day. From this, the amount of waste produced by a household of four people over a year is calculated as follows.

4.46 lbs / person / day × 365 days / year × 4 people / household = 6,511.6 lbs / household / year

To calculate how much waste must be managed by a community of 160,000 people (that is, 40,000 households of four people each), it is more convenient to convert pounds to tons, starting with the number derived from above.

First,

6,511.6 lbs / household / year 1 ton / 2,000 lbs = 3.3 tons / household / year

So,

3.3 tons / household / year 40,000 households = 132,000 tons / year

Assuming that engineers designed a landfill to be 60 feet from bottom to top, and the community has equipment to compact the waste into the landfill at a rate of 2 cubic yards per ton, how many acres of land will be needed each year to accommodate the community's waste? The next equation shows the calculation.

132,000 tons / year × 2 yd³ / ton × 9 ft³ / yd³ × 1 / 60 ft × 1 acre / 43,560 ft² 0.91 acres / year

This is equivalent to filling a football field with waste each year. And, if the landfill charges the community a disposal fee of $30 for every ton, that generates $3,960,000 every year (132,000 tons / year $30 / ton). This number illustrates how expensive landfill disposal can be, not only from disposal fees, but also in terms of land use constraints and the costs of protecting the environment.

Reuse and Recycling. As the U.S. population continues to increase, Americans keep throwing away more trash each year. To decrease the amount of municipal solid waste (MSW) that is disposed in landfills, consumers, businesses, and industries can divert certain materials from disposal via reuse and recycling. Even better, they can reduce the amount of trash generated in the first place. The more materials that are reduced, reused, and recycled, the less quickly landfills will fill up.

The equations below show how to calculate the percentage of MSW produced (generated) that is diverted from disposal via reuse and recycling. Reduction is difficult to quantify, and is not considered in the following equations.

Generation = Disposal + (Recycling + Reuse)

= Disposal + Diversion

So,

% Diversion = (Diversion / Generation) × 100%

Suppose the community planner contacted the disposal and recycling facilities in her area and found how many tons of MSW were disposed, recycled, and reused in recent years. (Most states require landfills to weigh all waste they receive, and to account for its origin.) Substituting these numbers into the series of equations above, she can derive the percentage of waste generated in 2002 that was being kept from landfills by recycling and reuse. Her results are shown in the table, and the graph is shown below.

As the table shows, by 2002 our sample community was able to keep nearly one-third of its MSW out of its landfill. This number is fairly representative of most communities in the United States that have strong MSW recycling and reuse programs.

"Pay-As-You-Throw." Most U.S. communities have added recycling to their waste management programs to help divert waste from disposal. But community leaders know that not everything can be reused or recycled. A better alternative is to keep things from becoming waste in the first place. And yet there is a catch: consumers like to buy things! Although most people are genuinely concerned about the environment, it will take more than concern to prompt citizens into reducing or rethinking what they buy, and therefore what they will eventually throw away.

One motivation for reducing waste is money: namely, making consumers pay for what they throw away. Some communities have implemented such programs, called "Pay-As-You-Throw," or PAYT. Without PAYT, a household either pays a company every month or year to take their trash, or the household pays for its trash collection in its taxes. In other words, the household pays the same amount no matter whether the family throws away one bag or twenty bags.

But what if the household must pay for every bag or can it sets out for the trash collector? Suppose $300 of its taxes has gone toward trash pickup. Now, what if instead the household pays $1.00 for every bag set out? If the family sets out four bags a week, that would be 4 bags × $1.00 / bag × 52 wks / year = $208 per year.

Although this yields a savings of $92 ($300 - $208), it seems like the cost is greater because the household sees the $4.00 disappear every week and will not see the tax savings until annual tax season.

But what if the household could further reduce its garbage from four bags to three bags a week by being more careful about what is thrown away and by recycling more? In this case, the household would pay only $3.00 per week (3 bags × $1.00 per bag) instead of $4.00. Compared to the $4.00 per week fee, householders would see their dollar savings every time they put out the trash.

This way of looking at monetary benefits is called a market incentive. If appealing to our concern for the environment is not enough to motivate some of us to reduce waste generation, then most of us will do it if we can save money (and hence if it costs us money to not do it).

Applying PAYT in a Sample Community. Essentially, PAYT is as much psychology as it is mathematics, but still there is math involved. The amount charged must be affordable yet enough to cover the cost of disposing the waste generated. How does a community decide a fair charge?

First the community must determine how much waste there will be and how much it will cost to dispose. To find out, start with the amount of waste generated under the old system; that is, before PAYT measures were implemented. Using 2002 as a base year, and using numbers from the previous table, we know this is 320,000 tons. Divide this amount by the number of people in the community to find how much waste is generated by each resident.

For this calculation, the 2002 population must first be estimated. Suppose that census records showed a 1990 population of 154,000 and a 2000 population of 159,000. This is an increase of 5,000 residents over 10 years, or 500 people per year. Because the most recent census year was 2000, and the base year is 2002, there are 2 years' worth of people to add. Hence, 159,000 + (500 × 2) = 160,000.

The amount of waste generated per resident can now be determined.

320,000 tons ÷ 160,000 = 2 tons per resident

It is important to note that this amount includes the trash that business like offices and restaurants generate. This is why the rate of trash generation is so much greater than the previously stated rate of 4.46 pounds, which represents only what people throw away from their homes.

It usually takes a year or two to start a PAYT program, and often the community will grow during that time. How many people will be in the community when the program starts, say in 2004? Use the previous per-year increase in population to make the adjustment for 2 more years into the future. That is, 160,000 + (500 × 2 years) = 161,000. Now, how much waste would those people generate in 2004, the first year of PAYT, if there were no PAYT prior to that year? The answer is 2 tons / person × 161,000 persons = 322,000 tons.

But once the community has a PAYT program in place, the amount of waste generated should start to decrease. Usually a community starting a PAYT program will look to other communities of about the same size that already have such programs, and then assume they will have a similar decrease. Suppose our sample community discovered that waste generation in nearby Payville decreased from 320,000 tons to 176,000 tons after they implemented their PAYT program. The decrease for Payville is 176,000 tons ÷ 320,000 tons × 100 percent = 55 percent.

Now our sample community can apply this rate to the waste it estimates will be generated annually with PAYT, and then determine the monthly total. The annual waste expected with PAYT is 322,000 tons × 0.55 = 177,100 tons. Hence, 177,100 tons / year × 1 year / 12 months = 14,758 tons / month.

Next, estimate how much the PAYT program will cost. Some costs, like building rent, probably will not change much. Others generally will increase over time. For instance, the salary of city employees will probably be greater in 2004 than in 2002. These increases can be estimated the same way population increases were projected: namely, by taking an average of historical increases over a period of time and projecting the annual increase to future years. Conversely, certain costs will decrease as waste decreases. The city may not need as many trucks or as many people to pick up trash as before. Assume that these scenarios have already been considered when estimating costs in the list below.

Now the dollar charge per bag can be determined. First you need to know how much an average bag weighs. A bag full of cat litter will weigh more than a bag of the same size filled with tissue. A simple way to find an average weight per bag is to go to a landfill, select several bags at random, weigh them, add the weight of each, and then divide by the number of bags. Suppose five bags weighed 10, 20, 5, 25, and 40 lbs. Hence, (10 + 20 + 5 + 25 + 40) / 5 = 20 lbs per bag. Converting to tons, 20 lbs / bag ÷ 2,000 lbs / ton = 0.01 ton / bag.

Next, the number of bags that will result from the 14,758 tons that are expected is calculated by 14,758 tons / month × 1 bag / 0.01 ton = 1,475,800 bags / month. Finally, knowing that total costs per month are $872,740, how much is that per bag?

$872,740 ÷ 1,475,800 bags = $0.59 per bag

This price will cover the estimated costs. Now the community leaders must use psychology again and ask whether it is too high, in which case citizens may refuse to accept PAYT. But if the price is too low, the idea of market incentives applies, and citizens won't be enticed to save money by reducing the trash they throw out (that is, because the money potentially saved would be minimal). So the community leaders must adjust the price up or down to yield the best result.

This example gives insight into the mathematics needed to create a waste disposal system that can pay for itself, is good for the environment, and is fair. People who throw away less trash should be paying less than people who throw away more trash.

Richard B. Worth and

Minerva Mercado-Feliciano

Bibliography

U.S. Environmental Protection Agency. Characterization of Municipal Solid Waste in the United States: 1998 Update. Prairie Village, KS: Franklin Associates, a service of McLaren/Hart, 1999.

———. Pay-As-You-Throw: A Fact Sheet for Environmental and Civic Groups. September, 1996.

———. Pay-As-You-Throw—Lessons Learned About Unit Pricing. April, 1994.

———. Pay-As-You-Throw: Throw Away Less and Save Fact Sheet. September, 1996.

———. Pay-As-You-Throw Workbook. September, 1996.

Internet Resources

Your Guide to Waste Reduction, Reuse, and Recycling in Monroe County, Indiana. Monroe County Solid Waste Management District. <http://www.mcswmd.org>.