Design and Planning of Waste Collection System

Abstract

The purpose of this chapter is to present the principal factors and objectives to consider when planning a collection system. The prediction and estimation of the amount of waste and the type of waste collection service that is intended to be provided, together with the help of geographic information systems (GIS) to locate containers and design routes, are tools to be used during the adequate design and planning of a waste collection system. Here a specific focus is on waste prediction models, due to its importance on planning, operating, and optimizing waste management system, as well as in the difficulty in predicting, directly, waste generation and its dependence on numerous factors, directly and indirectly, related with the consumption patterns, disposal habits, and urbanization.

Appendices

Appendix A: Forecasting Methods

Let x₁, x₂, …, x_t, … the observed values of the times series, where x_t is the value observed in period t.

1.1 A.1: Naïve Forecast Model

This is the simplest forecasting model. It assumes the value for the next period will equal the one last observed. Let f_{t, 1} be the forecast for period t + 1 after observing x_t, then

$$ {f}_{t,1}={x}_t. $$

1.2 A.2: Moving-Average Models

Among the simpler and commonly used methods are the moving-average methods. These forecast period t as the average of the last N observed values (with N a given parameter). Let f_{t, 1} be the forecast for period t + 1 after observing x_t.

$$ {f}_{t,1}=\frac{1}{N}\sum \limits_{k=t-N}^t{x}_k $$

The choice of N depends on the deviation of the forecast regarding the observed value (the forecast error). For period t, the forecast error e_t is given by

$$ {e}_t={x}_t-{f}_{t-1,1}. $$

There are several ways to model the forecast accuracy (see Appendix B). To find the adequate value for N, one has to choose one of such measures and determine the value that minimizes the accuracy measure.

These methods are adequate for time-series data that fluctuate around a base value b:

$$ {x}_t=b+{e}_t. $$

1.3 A.3: Exponential Smoothing Model

This model is also adequate for time series that may be written as

$$ {x}_t=b+{e}_t. $$

Again consider f_{t, 1} is the forecast for period t + 1 after observing x_t. The simple exponential smoothing method “says” the next forecast (f_{t+1, 1}) is a weighted average between the observed value x_t and the forecast at period t:

$$ {f}_{t+1,1}=\alpha\ {x}_t+\left(1-\alpha \right)\ {f}_{t,1} $$

where α ∈ [0, 1] is the smooth constant.

Let f_{t, k} be the forecast for period t + k at the end of period t, then

$$ {f}_{t,k}={f}_{t,1}. $$

1.4 A.4: Holt’s Model

The Holt’s method divides the time-series data into two components: the level, L_t, and the trend, T_t. These two components can be calculated by the expressions below:

$$ {L}_t=\alpha {x}_t+\left(1-\alpha \right)\left({L}_{t-1}+{T}_{t-1}\right) $$

$$ {T}_t=\beta \left({L}_t-{L}_{t-1}\right)+\left(1-\beta \right){T}_{t-1} $$

where L_t denotes an estimate of the level of the series at period t, T_t denotes an estimate of the trend of the series at period t, and α and β are the smoothing parameters for the level and the trend, respectively, 0 ≤ α, β ≤ 1.

Let f_{t, k} be the forecast for period t + k at the end of period t, then

$$ {f}_{t,k}={L}_t+k\cdotp {T}_t. $$

1.5 A.5: Holt-Winters Method

The Holt-Winters (seasonal) model divides the time-series data into three components: the level, L_t; the trend, T_t; and the seasonal component S_t. These three components are given by:

$$ {L}_t=\alpha \frac{x_t}{S_{t-c}}+\left(1-\alpha \right)\left({L}_{t-1}+{T}_{t-1}\right) $$

$$ {T}_t=\beta \left({L}_t-{L}_{t-1}\right)+\left(1-\beta \right){T}_{t-1} $$

$$ {S}_t=\gamma \frac{x_t}{L_t}+\left(1-\gamma \right){\mathrm{S}}_{t-c} $$

where L_t denotes an estimate of the level of the series at period t, T_t denotes an estimate of the trend of the series at period t, and S_t denotes the seasonal component at period t; c is the frequency/pattern of the seasonality (i.e., for quarterly pattern c = 4; for a yearly pattern c = 12). Lastly, α, β, and γ are the smoothing parameters for the level, the trend, and the seasonality, respectively, 0 ≤ α, β, γ ≤ 1.

Let f_{t, k} be the forecast for period t + k at the end of period t, then

$$ {f}_{t,k}=\left({L}_t+k\cdotp {T}_t\right)\cdotp {S}_{t+k-c}. $$

1.6 A.6: ARIMA Models

Many other forecasting methods are available in the literature. Among the best known are autoregressive integrated moving average (ARIMA) models also named as Box-Jenkins models Hoffman et al. (2013). The general form for this family of models is

$$ \left(1-{\phi}_1\beta -{\phi}_2{B}^2-\dots -{\phi}_p{B}^p\right){\left(1-B\right)}^d{x}_t={\theta}_0+\left(1-{\theta}_1B-{\theta}_2{B}^2-\dots -{\theta}_q{B}^q\right)\cdotp {\varepsilon}_t $$

where ϕ_k is the autoregressive parameter, θ_k is the moving average parameter, B is a backshift operator defined so that B^rx_t = x_t−R, Δ^d = (1 − B)^d is the backward difference operator, and ε_t is an uncorrelated sequence of random errors with mean zero and variance σ².

This generic model can be extended to incorporate seasonal behavior (Box et al. 2015). One chooses a model by specifying the integers p, d, and q, resulting in an ARIMA(p, d, q) model. These integer parameters are determined by examining the sample autocorrelation and partial autocorrelation function. For additional detail see Hyndman and Athanasopoulos (2006).

Appendix B: Measures of Accuracy

In this section, some accuracy measures will be presented. More details can be found in Hyndman (2006).

Let the forecast error e_t be

$$ {e}_t={x}_t-{\overline{x}}_t, $$

where x_t and ${\overline{x}}_t$ are, respectively, the observed and the forecasted values for period t and the percentage error p_t defined by

$$ {p}_t=\frac{e_t}{x_t}\cdotp 100. $$

Let r_t be the relative error defined by

$$ {r}_t=\frac{e_t}{e_t^{\prime }} $$

where ${e}_t^{\prime }$ is the forecast error obtained from the benchmark method.

1.1 B.1: Mean Absolute Error (MAE)

$$ \mathrm{MAE}=\frac{1}{m}\sum \limits_{t=1}^m\left|{e}_t\right| $$

1.2 B.2: Geometric Mean Absolute Error (GMAE)

$$ \mathrm{GMAE}=\sqrt[m]{\mid {e}_1\cdotp {e}_2\cdotp \dots \cdotp {e}_m\mid } $$

1.3 B.3: Mean Square Error (MSE)

$$ \mathrm{MSE}=\frac{1}{m}\sum \limits_{t=1}^m{e}_t^2 $$

1.4 B.4: Mean Absolute Percentage Error (MAPE)

$$ \mathrm{MAPE}=\frac{1}{m}\sum \limits_{t=1}^m\left|{p}_t\right|. $$

1.5 B.5: Geometric Mean Relative Absolute Error (GMRAE)

$$ \mathrm{GMRAE}=\sqrt[m]{\mid {r}_1\cdotp {r}_2\cdotp \dots \cdotp r\mid } $$

Appendix C: Linear Regression Models

1.1 C.1: Simple Linear Regression Model

Let (x_i, y_i), i = 1,…, n, be a set of n observations. The simple linear regression model is given by

$$ {y}_i={\beta}_0+{\beta}_1{x}_i+{\varepsilon}_i $$

where ε_i is the residual value and β₀ and β₁are the least squares estimators computed as.

$$ {\beta}_1=\frac{\sum_{i=1}^n\left({y}_i-\overline{y}\right)\left({x}_i-\overline{x}\right)}{\sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}\ \mathrm{and}\ {\beta}_0=\overline{y}-{\beta}_1\ \overline{x.} $$

with $\overline{x}$ and $\overline{y}$ the averages of x and y, respectively. Residuals ε_i should have mean zero and be uncorrelated with each other and with the independent variable.

Let the forecast variable be $\widehat{y}$ and the observed value y, the coefficient of determination R² is given by:

$$ {R}^2=\frac{\sum_{i=1}^n{\left({\widehat{y}}_i-\overline{y}\right)}^2}{\sum_{i=1}^n{\left({y}_i-\overline{y}\right)}^2}. $$

Standard deviation of the residuals, S_ε:

$$ {S}_{\varepsilon }=\sqrt{\frac{1}{n-2}{\sum}_{i=1}^n{\varepsilon}_i} $$

100 · (1 − α)% prediction interval:

$$ \widehat{y}\pm {z}_{1-\alpha /2}{s}_{\varepsilon}\sqrt{1+\frac{1}{n}+\frac{{\left(x-\overline{x}\right)}^2}{\left(n-1\right){s}_x^2}} $$

where, z_1 − α/2 is (1 − α/2) the critical value of the standard normal distribution, s_ε is the standard deviation of the residuals, x is the value used to calculate $\widehat{y}$, $\overline{x}$ is the mean value of all x observed values, and s_x is standard deviation of all x observed values.

1.2 C.2: Multiple Linear Regression Model

The multiple linear regression models are an extension of the simple linear regression. It assumes that the dependent variable is explained by more than one factor (the independent variables). Given (x_1i, x_2i, …, x_ki, y_i), i = 1,…, n, be a set of n observations. The multiple linear regression model is given by

$$ {y}_i={\beta}_0+{\beta}_1{x}_{1i}+{\beta}_2{x}_{2i}+\dots +{\beta}_k{x}_{ki}+{\varepsilon}_i. $$

The coefficients β₀, β₁, …, β_k measure the effect of each independent variable after taking into account the effect of all other independent variables in the model.

Again, residuals ε_i should have mean zero and be uncorrelated with each other and with each independent variable.

The selection of the independent variables to use in the model is not a straightforward process. Measures of predictive accuracy should be used (e.g., adjusted R², cross-validation, Akaike’s information criterion, Schwarz Bayesian information criterion …). For all technical details about multiple Linear Regression model refer to Jobson (1991).