An interval-parameter mean-CVaR two-stage stochastic programming approach for waste management under uncertainty

Abstract

In this research, approaches of interval mathematical programming, two-stage stochastic programming and conditional value-at-risk (CVaR) are incorporated within a general modeling framework, leading to an interval-parameter mean-CVaR two-stage stochastic programming (IMTSP). The developed method has several advantages: (i) it can be used to deal with uncertainties presented as interval numbers and probability distributions, (ii) its objective function simultaneously takes expected cost and system risk into consideration, thus, it is useful for helping decision makers analyze the trade-offs between cost and risk, and (iii) it can be used for supporting quantitatively evaluating the right tail of distributions of waste generation rate, which can better quantify the system risk. The IMTSP model is applied to the long-term planning of municipal solid waste management system in the City of Regina, Canada. The results indicate that IMTSP performs better in its capability of generating a series of waste management patterns under different risk-aversion levels, and also providing supports for decision makers in identifying desired waste flow strategies, considering balance between system economy and environmental quality.

1 Introduction

Municipal solid waste (MSW) management issues have increasing bearing on aesthetics, socio-economy, human health, and amenity of many communities, states and nations around the world (Kollikkathara et al. 2009). Hence, an effective planning for MSW management is important for facilitating sustainable socioeconomic development. However, various complexities exist in such an MSW management system, including the collection techniques to be used, the levels of service to be offered, and the facilities to be adopted (Dai et al. 2011). Many system parameters such as waste generation rate, facility capacity, diversion goal, and waste treatment cost may appear uncertain and cannot be expressed as deterministic values. Moreover, such uncertainties may be further multiplied by the complex features of the system components, as well as by their associations with economic penalties if the promised targets are violated (Li et al. 2009). Furthermore, there is an inherent necessary to reflect and deal with system risks associated with such uncertainties (e.g., variations in waste generation rates). Therefore, incorporation of various uncertainties, economic penalties and system risks within a general mathematical programming framework is desired for supporting MSW management and planning under such complexities.

Recently, a number of inexact optimization techniques were developed for dealing with uncertainties in planning problems, including fuzzy, stochastic, and interval mathematical programming (Ahmed 2004; Kall and Mayer 2005; Schultz and Tiedemann 2006; Qin et al. 2007; Cheng et al. 2009; Xu et al. 2009; Tan et al. 2010a, b; Shao et al. 2011; Zhu and Huang 2011; Cai et al. 2011; Noyan 2012). Among them, two-stage stochastic programming is a typical stochastic mathematical programming method. It is effective for tackling optimization problems where an analysis of policy scenarios is desired and the model’s coefficients are random with known probability distributions (Chen et al. 2012). In two-stage stochastic programming, a decision is first undertaken before values of random variables are known and, then, after the random events have happened and their values are known, a second decision is made in order to minimize ‘‘penalties’’ that may appear due to any infeasibility. The fundamental idea behind two-stage stochastic programming is the concept of recourse, which is the ability to take corrective actions after a random event has taken place (Huang and Loucks 2000). Over the past decades, two-stage stochastic programming with the expected recourse function was widely explored (Huang and Loucks 2000; Seifi and Hipel 2001; Maqsood and Huang 2003; Ahmed et al. 2004; Guo and Huang 2009a). For example, Maqsood and Huang (2003) developed an inexact two-stage stochastic programming model for planning solid waste management systems, where this method could tackle uncertainties in waste-generation rates that can be presented as probability density functions. However, the traditional two-stage stochastic programming model is risk neutral in the sense that it considers optimization as an expected criterion (Ahmed 2004). Concretely, the risk-neutral two-stage stochastic programming model takes the expected system cost as the objective function without considering risk-averting issues which may lead to excessive waste when the waste generation rate is in an extra-high condition. Therefore, a risk-averting method quantifying the effects of the variability of random waste generation rate would generate more robust solutions compared to the existing risk-neutral approaches. Conditional value-at-risk (CVaR), having appealing features such as sub-additivity and convexity, can effectively quantify risks based on known probability distributions of random variables (Rockafellar and Uryasev 2000). This approach is enhanced from a value-at-risk model, which has become an essential tool for quantifying portfolio market risks (Hsu et al. 2011). The proposed CVaR model can not only handle the expected loss of an event under extreme conditions, but also calculate the associated risks through an optimization model.

More recently, many studies incorporating CVaR into constraints of two-stage stochastic programming were conducted. Most of them focused on the introduction of CVaR into objective functions of two-stage stochastic programming models [i.e., mean-CVaR two-stage stochastic programming (MTSP)] (Noyan 2012). For example, Ahmed (2006) explored the computational suitability of various mean-risk objective functions in addressing risk in stochastic programming models, where the absolute semi-deviation risk and the quantile deviation risk measures (i.e. CVaR) were considered. Schultz and Tiedemann (2006) introduced CVaR and integer programming into a general two-stage stochastic programming framework, and transformed it into an explicit mixed-integer linear programming model when the probability distribution is discrete and finite. Fabian (2008) proposed a decomposition framework for handling a two-stage stochastic model that contains CVaR objectives and constraints. Schultz (2011) formulated a mean-CVaR stochastic integer programming model, and explored the structural properties of these optimization problems and the corresponding solution algorithms. Noyan (2012) considered a MTSP model and employed it for supporting disaster management, where uncertain information in the MTSP model was addressed through a discrete scenario method. The proposed MTSP model can not only effectively analyze predefined policy scenarios and deal with uncertainties that can be expressed as probability density functions, but also reflect expected losses associated with many extreme events and analyze trade-offs between system costs and risks. However, no application of MTSP to waste management was reported. Besides, MTSP model required probabilistic distributions for uncertain parameters, whereas in many real-world cases the quality of available information was mostly not satisfactory enough to be presented as probabilities; when uncertainties can only be obtained as interval parameters rather than probabilistic distributions, the MTSP approach may become inapplicable. In comparison, the interval mathematical programming methods were effective for handling uncertain parameters that could not be quantified as distribution functions in the model’s left- and/or right-hand sides as well as in the objective function (Tan et al. 2010a). In interval mathematical programming method, interval numbers are acceptable as its uncertain inputs. Although interval mathematical programming is effective in dealing with interval numbers in objective function coefficients and constraint parameters, it may become infeasible when the right-hand sides are highly uncertain. Moreover, interval mathematical programming cannot reflect the trade-off between system cost and risk generated from the serious waste generation condition, which is common in many real-world waste management problems.

Therefore, one potential approach for better accounting for the uncertainties, economic penalties, and system risk at extreme probability levels is to incorporate the interval mathematical programming and MTSP within a general optimization framework. This leads to an interval-parameter mean-CVaR two-stage stochastic programming (IMTSP) method. The objective of this study is to develop such an IMTSP method and apply it to MSW management. The IMTSP method will be able to handle uncertainties presented as intervals and probability distributions; moreover, it can also help control the system risk and avoid problems of excessive waste. The model will then be used for planning a real MSW management system in the city of Regina, Canada. The proposed method can generate optimal solution under variable risks, which can be used for facilitating the reflection of trade-offs between system costs and risks.

2 Methodology

A general form of the two-stage stochastic programming model can be formulated as follows (Birge and Louveaux 1988):

$$ \mathop {\hbox{min} }\limits_{{\mathbf{x}}} \, {\mathbb{E}}\left[ {f\left( {{\mathbf{x}},\omega } \right)} \right] = \mathop {\hbox{min} }\limits_{{\mathbf{x}}} \, \left\{ {{\mathbf{c}}^{T} {\mathbf{x}} + {\mathbb{E}}_{\omega \in \Upomega } \left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right]: \, {\mathbf{x}} \in X} \right\}, $$

(1a)

where $f\left( {{\mathbf{x}},\omega } \right) = {\mathbf{c}}^{T} {\mathbf{x}} + Q\left( {{\mathbf{x}},\omega } \right)$ is the total cost function of the first-stage problem [i.e. model (1a)], and

$$ Q\left( {{\mathbf{x}},\omega } \right) = \mathop {\hbox{min} }\limits_{{\mathbf{y}}} \, \left\{ {{\mathbf{q}}\left( \omega \right)^{T} {\mathbf{y}}\left( \omega \right): \, {\mathbf{D}}\left( \omega \right){\mathbf{y}}\left( \omega \right) + {\mathbf{T}}\left( \omega \right){\mathbf{x}} \ge {\mathbf{h}}\left( \omega \right), \, {\mathbf{y}} \in Y} \right\} $$

(1b)

is the second-stage problem. Here ${\mathbf{c}} \in {\mathbb{R}}^{{n_{1} }} ,\,X \subseteq {\mathbb{R}}^{n1} ,\,Y \subseteq {\mathbb{R}}^{n2} ;\,\omega$ is a random variable from probability space $\left( {\Upomega ,\mathcal{F},P} \right)$ with $\Upomega \to {\mathbb{R}}^{k} ,\,{\mathbf{q}}:\Upomega \to {\mathbb{R}}^{{n_{2} }} ,\,{\mathbf{h}}:\Upomega \to {\mathbb{R}}^{{m_{2} }} ,\,{\mathbf{D}}:\Upomega \to {\mathbb{R}}^{{m_{2} \times n_{2} }}$ and ${\mathbf{T}}:\Upomega \to {\mathbb{R}}^{{m_{2} \times n_{1} }} ;\,\,{\mathbb{E}}$ denotes the expectation operates; ${\mathbf{x}}$ and ${\mathbf{y}}$ are the vectors of first-stage and second-stage decision variables, respectively.

Model (1) is a risk neutral two-stage stochastic programming model in the sense that it is concerned with the optimization of an expectation objective. A common approach to address risks is to consider a weighted mean-risk criterion, where a dispersion statistic is used as a proxy for risk (Birbil et al. 2008). Markowitz (1952) developed a classical mean–variance portfolio optimization model, where variance was used as the risk measure. The disadvantage of this mean–variance formulation is that it considers the under-and-over-performances equally. When a typical dispersion statistics such as variance is used as risk measures, the mean-risk approach may lead to inferior solutions (Noyan 2012). CVaR as a downside risk measure can be proposed to remedy this drawback (Ahmed 2006). Therefore, we have the following mean-CVaR model:

$$ \hbox{min} \, \left\{ {{\mathbb{E}}\left[ {f\left( {{\mathbf{x}},\omega } \right)} \right] + \lambda {\text{CVaR}}_{\beta } \left[ {f\left( {{\mathbf{x}},\omega } \right)} \right]:{\mathbf{x}} \in X} \right\}, $$

(2)

where ${\text{CVaR}}_{\beta }$ denotes the conditional value-at-risk at confidence level $\beta ;\,\lambda$ is a non-negative weight to trade-off expected cost with risk, which is specified by decision makers corresponding to their risk preferences. Birbil et al. (2008) indicated that ${\text{CVaR}}_{\beta } \left( {Z + a} \right) = {\text{CVaR}}_{\beta } \left( Z \right) + a$ for $a \in {\mathbb{R}}$ and $Z \in \mathcal{Z},$ which $\mathcal{Z}$ is a linear space of F-measurable functions on a probability space $\left( {\Upomega ,\mathcal{F},P } \right).$ Then, we have ${\text{CVaR}}_{\beta } \left[ {f\left( {{\mathbf{x}},\omega } \right)} \right] = {\mathbf{c}}^{T} {\mathbf{x}} + {\text{CVaR}}_{\beta } \left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right],$ and

$$ \begin{gathered} {\mathbb{E}}\left[ {f\left( {{\mathbf{x}},\omega } \right)} \right] + \lambda {\text{CVaR}}_{\beta } \left[ {f\left( {{\mathbf{x}},\omega } \right)} \right] \hfill \\ = {\mathbf{c}}^{T} {\mathbf{x}} + {\mathbb{E}}\left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right] + \lambda \left\{ {{\mathbf{c}}^{T} {\mathbf{x}} + {\text{CVaR}}_{\beta } \left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right]} \right\} \hfill \\ = \left( {1 + \lambda } \right){\mathbf{c}}^{T} {\mathbf{x}} + {\mathbb{E}}\left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right] + \lambda {\text{CVaR}}_{\beta } \left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right]. \hfill \\ \end{gathered} $$

(3)

Obviously, the essential issue for obtaining the numerical solutions of model (2) is to calculate the minimization of ${\text{CVaR}}_{\beta } \left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right]$ firstly. Let $Q\left( {{\mathbf{x}},\omega } \right)$ denote a loss function associated with decision vector ${\mathbf{x}} \in X \subset {\mathbb{R}}^{n}$ and random vector $\omega \in {\mathbb{R}}^{m} .$ For convenience, the underling probability of $\omega$ will be assumed to have a density function $p( \cdot )$ (Rockafellar and Uryasev 2002). Suppose that $Q\left( {{\mathbf{x}},\omega } \right)$ is continuous in ${\mathbf{x}}$ and measurable in $\omega ,$ and that ${\mathbb{E}}\left\{ {\left| {Q\left( {{\mathbf{x}},\omega } \right)} \right|} \right\} < \infty$ for each ${\mathbf{x}} \in X$. For a confidence level $\beta$ and a fixed decision variable ${\mathbf{x}},$ the value-at-risk, denoted by ${\text{VaR}}_{{{\upbeta}}} \left( {\mathbf{x}} \right)$, can be defined as follows:

$$ {\text{VaR}}_{\beta } \left( {\text{x}} \right) = \hbox{min} \, \left\{ {\alpha \in {\mathbb{R}}: \, \int\limits_{{Q({\mathbf{x}},\omega ) \le \alpha }} {p(\omega )d\omega \ge \beta } } \right\} $$

(4)

where $\int_{{Q\left( {{\mathbf{x}},\omega } \right) \le \alpha }} {p(\omega )d\omega }$ denotes the probability of $Q\left( {{\mathbf{x}},\omega } \right)$ not exceeding a threshold $\alpha$. ${\text{CVaR}}_{\beta } \left( {\mathbf{x}} \right)$ is defined as the expected value of the loss that exceeds ${\text{VaR}}_{\beta } \left( {\mathbf{x}} \right),$ which can be presented as follows:

$$ {\text{CVaR}}_{\beta } \left( x \right) = \frac{1}{1 - \beta }\int\limits_{{Q\left( {{\mathbf{x}},\omega } \right) \ge {\text{VaR}}_{\beta } \left( {\text{x}} \right)}} {Q\left( {{\mathbf{x}},\omega } \right)p(\omega )d\omega } . $$

(5)

The CVaR is a coherent risk measure (Ahmed 2006). The problem involved ${\text{CVaR}}_{\beta } \left( {\text{x}} \right)$ is different to proceed because of its convoluted and implicit version. Rockafellar and Uryasev (2000) developed the following function:

$$ F_{\beta } \left( {{\mathbf{x}},\alpha } \right) = \alpha + \frac{1}{1 - \beta }\int\limits_{{\omega \in {\mathbb{R}}^{m} }} {\left\{ {\hbox{max} \left\{ {0, \, \left[ {Q\left( {{\mathbf{x}},\omega } \right) - \alpha } \right]} \right\}} \right\}p(\omega )d\omega } , $$

(6)

where $F_{\beta } \left( {{\mathbf{x}},\alpha } \right)$ is shown to be convex and continuously differentiable with respect to $\alpha ,$ and $\mathop {\hbox{min} }\nolimits_{\text{x}} \, \left\{ {{\text{CVaR}}_{\beta } \left( {\text{x}} \right)} \right\} = \mathop {\hbox{min} }\nolimits_{\alpha } \, \left\{ {F_{\beta } \left( {{\mathbf{x}},\alpha } \right)} \right\}$. Through introducing an auxiliary variable $z,$ the minimization of ${\text{CVaR}}_{\beta }$ is equivalent to the following model (Tong et al. 2010):

$$ \mathop {\hbox{min} }\limits_{x} \, \left\{ {\alpha + \frac{1}{1 - \beta }{\mathbb{E}}_{{\omega \in {\mathbb{R}}^{m} }} \left[ {z(\omega )} \right]} \right\}, $$

(7a)

subject to

$$ {\mathbf{x}} \in X, $$

(7b)

$$ z(\omega ) \ge Q\left( {{\mathbf{x}},\omega } \right) - \alpha , $$

(7c)

$$ z(\omega ) \ge 0. $$

(7d)

By coupling model (1) and model (7) within the mean-CVaR model [i.e., model (2)], a mean-CVaR two-stage stochastic programming (MTSP) model can be formulated as follows:

$$ \hbox{min} \, f = \left( {1 + \lambda } \right){\mathbf{c}}^{T} {\mathbf{x}} + {\mathbb{E}}\left[ {Q\left( {{\mathbf{x}},\omega } \right)} \right] + \lambda \left\{ {\alpha + \frac{1}{1 - \beta }{\mathbb{E}}_{{\omega \in {\mathbb{R}}^{m} }} \left[ {z(\omega )} \right]} \right\} $$

(8a)

subject to

$$ z(\omega ) \ge Q\left( {{\text{x}},\omega } \right) - \alpha $$

(8b)

$$ z(\omega ) \ge 0 $$

(8c)

$$ \alpha \ge 0 $$

(8d)

$$ {\mathbf{x}} \in X $$

(8e)

with

$$ Q\left( {{\mathbf{x}},\omega } \right) = \hbox{min} \, {\mathbf{q}}\left( \omega \right)^{T} {\mathbf{y}}\left( \omega \right) $$

(8f)

subject to

$$ {\mathbf{D}}\left( \omega \right){\mathbf{y}}\left( \omega \right) + {\mathbf{T}}\left( \omega \right){\mathbf{x}} \ge {\mathbf{h}}\left( \omega \right) $$

(8g)

$$ {\mathbf{y}} \in Y $$

(8h)

By letting random variables (i.e. $\omega$) take discrete values $\omega_{s}$ at probability levels $p_{s}$ ($s = 1,2, \ldots v$ and $\sum {p_{s} } = 1),$ the above MTSP model can be equivalently formulated as the following linear programming model (Noyan 2012):

$$ \hbox{min} \, f = \left( {1 + \lambda } \right){\mathbf{c}}^{T} {\mathbf{x}} + \sum\limits_{s = 1}^{v} {p_{s} \left( {{\mathbf{q}}_{s} } \right)^{T} {\mathbf{y}}_{s} } + \lambda \left( {\alpha + \frac{1}{1 - \beta }\sum\limits_{s = 1}^{v} {p_{s} z_{s} } } \right) $$

(9a)

subject to

$$ {\mathbf{T}}_{s} {\mathbf{x}} + {\mathbf{D}}_{s} {\mathbf{y}}_{s} \ge {\mathbf{h}}_{s} ,\quad \, s = 1, \ldots ,v $$

(9b)

$$ {\mathbf{x}} \in X $$

(9c)

$$ {\mathbf{y}}_{s} \ge 0,\quad \, s = 1, \ldots ,v $$

(9d)

$$ z_{s} \ge \left( {{\mathbf{q}}_{s} } \right)^{T} {\mathbf{y}}_{s} - \alpha ,\quad \, s = 1, \ldots ,v $$

(9e)

$$ \alpha \in {\mathbb{R}}, \, z_{s} \ge 0,\quad \, s = 1, \ldots ,v $$

(9f)

Obviously, model (9) can deal with uncertainties in the right-hand sides presented as probabilistic distributions when coefficients in the left-hand sides and the objective function are deterministic. However, in real-world optimization problems, the quality of information that can be obtained is mostly not satisfactory enough to be presented as probabilities (Li et al. 2009). Such complexities cannot be solved through model (9). Interval mathematical programming is effective in tackling uncertainties expressed as interval values with known lower and upper bounds but unknown distribution functions (Huang et al. 1992). Therefore, through incorporating interval mathematical programming and MTSP within a general optimization framework, an interval-parameter MTSP (or IMTSP) method can be formulated as follows:

$$ \hbox{min} \, f^{ \pm } = \left( {1 + \lambda } \right)\left( {{\mathbf{c}}^{ \pm } } \right)^{T} {\mathbf{x}}^{ \pm } + \sum\limits_{s = 1}^{v} {p_{s} \left( {{\mathbf{q}}_{s}^{ \pm } } \right)^{T} {\mathbf{y}}_{s}^{ \pm } } + \lambda \left( {\alpha^{ \pm } + \frac{1}{1 - \beta }\sum\limits_{s = 1}^{v} {p_{s} z_{s} } } \right) $$

(10a)

subject to

$$ {\mathbf{T}}_{s}^{ \pm } {\mathbf{x}}^{ \pm } + {\mathbf{D}}_{s}^{ \pm } {\mathbf{y}}_{s}^{ \pm } \ge {\mathbf{h}}_{s}^{ \pm } ,\quad \, s = 1, \ldots ,v $$

(10b)

$$ z_{s} \ge \left( {{\mathbf{q}}_{s}^{ \pm } } \right)^{T} {\mathbf{y}}_{s}^{ \pm } - \alpha^{ \pm } ,\quad \, s = 1, \ldots ,v $$

(10c)

$$ \alpha^{ \pm } \in \left\{ {{\mathbb{R}}^{ \pm } } \right\}, \, z_{s} \ge 0,\quad \, s = 1, \ldots ,v $$

(10d)

$$ {\mathbf{x}} \in X $$

(10e)

$$ {\mathbf{y}}_{s} \ge 0,\quad \, s = 1, \ldots ,v $$

(10f)

where superscript “±” means interval-valued feature; the “−” and “+” superscripts represent lower and upper bounds of an interval parameter/variable, respectively. The objective function value of model (10) includes the expected cost and risk cost of excess waste amount. Model (10) can not only effectively reflect random uncertainty but also guarantee solutions to be more stable and reliable.

Model (10) can be transformed into two deterministic submodels that correspond to the lower and upper bounds of the desired objective function value. This transformation process is based on an interactive algorithm, which is different from the best/worst case analysis (Huang et al. 1992). Interval solutions associated with varying levels of constraint–violation risks can then be obtained by solving the two submodels sequentially. The submodel corresponding to the lower-bound objective function value $\left( {f^{ - } } \right)$ can be firstly formulated as follows:

$$ \begin{gathered} \hbox{min} \, f^{ - } = \left( {1 + \lambda } \right)\left( {\sum\limits_{j = 1}^{{k_{1} }} {c_{j}^{ - } x_{j}^{ - } } + \sum\limits_{{j = k_{1} + 1}}^{{n_{1} }} {c_{j}^{ - } x_{j}^{ + } } } \right) + \sum\limits_{s = 1}^{v} {\left( {\sum\limits_{j = 1}^{{k_{2} }} {p_{s} q_{js}^{ - } y_{js}^{ - } } + \sum\limits_{{j = k_{2} + 1}}^{{n_{2} }} {p_{s} q_{js}^{ - } y_{js}^{ + } } } \right)} \\ + \lambda \left( {\eta^{ - } + \frac{1}{1 - \alpha }\sum\limits_{s = 1}^{v} {p_{s} z_{s} } } \right) \\ \end{gathered} $$

(11a)

subject to

$$ \begin{gathered} \sum\limits_{j = 1}^{{k_{1} }} {\left| {t_{js} } \right|^{ + } {\text{Sign}}\left( {t_{js}^{ + } } \right)x_{j}^{ - } } + \sum\limits_{{j = k_{1} + 1}}^{{n_{1} }} {\left| {t_{js} } \right|^{ - } {\text{Sign}}\left( {t_{js}^{ - } } \right)x_{j}^{ + } } \hfill \\ \, + \sum\limits_{j = 1}^{{k_{2} }} {\left| {d_{js} } \right|^{ + } {\text{Sign}}\left( {d_{js}^{ + } } \right)y_{js}^{ - } } + \sum\limits_{{j = k_{2} + 1}}^{{n_{2} }} {\left| {d_{js} } \right|^{ - } {\text{Sign}}\left( {d_{js}^{ - } } \right)y_{js}^{ + } } \ge h_{s}^{ - } \quad \forall s \hfill \\ \end{gathered} $$

(11b)

$$ \sum\limits_{j = 1}^{k2} {\left| {q_{js} } \right|^{ - } {\text{Sign}}\left( {q_{js}^{ - } } \right)y_{js}^{ - } } + \sum\limits_{{j = k_{2} + 1}}^{n2} {\left| {q_{js} } \right|^{ - } {\text{Sign}}\left( {q_{js}^{ - } } \right)y_{js}^{ + } } \le z_{s} + \eta^{ - } , \quad \forall s $$

(11c)

$$ z_{s} \ge 0, \quad \forall s $$

(11d)

$$ \eta^{ - } \ge 0 $$

(11e)

$$ x_{j}^{ - } \ge 0,\quad \, j = 1,2, \ldots ,k_{1} $$

(11f)

$$ x_{j}^{ + } \ge 0,\quad \, j = k_{1} + 1,k_{1} + 2, \ldots ,n_{1} $$

(11g)

$$ y_{js}^{ - } \ge 0, \quad \forall s,\quad \, j = 1,2, \ldots ,k_{2} $$

(11h)

$$ y_{js}^{ + } \ge 0, \quad \forall s,\quad \, j = k_{2} + 1,\;k_{2} + 2, \ldots ,n_{2} $$

(11i)

where $x_{j}^{ \pm } ,\; \, j = 1,2, \ldots ,k_{1}$, are interval variables with positive coefficients in the objective function; $x_{j}^{ \pm } ,\; \, j = k_{1} + 1,k_{1} + 2, \ldots ,n_{1}$, are interval variables with negative coefficients; $y_{js}^{ \pm } , \, j = 1,2, \ldots ,k_{2}$ and $s = 1,2, \ldots ,v$, are interval random variables with positive coefficients in the objective function; $y_{js}^{ \pm } \ge 0, \, j = k_{2} + 1,k_{2} + 2, \ldots ,n_{2}$ and $s = 1,2, \ldots ,v$, are interval random variables with negative coefficients. Solutions of $\eta_{\text{opt}}^{ - }$, $x_{{j,{\text{ opt}}}}^{ - }$ ($j = 1,2, \ldots ,k_{1}$), $x_{{j,{\text{ opt}}}}^{ + }$ ($j = k_{1} + 1,k_{1} + 2, \ldots ,n_{1}$), $y_{{js,{\text{ opt}}}}^{ - }$ ($j = 1,2, \ldots ,k_{2}$), and $y_{{js,{\text{ opt}}}}^{ + }$ ($j = k_{2} + 1,\;k_{2} + 2, \ldots ,n_{2}$) can be obtained through submodel (11). Based on the above solutions, the second submodel for $f^{ + }$ can be formulated as follows:

$$ \begin{gathered} \hbox{min} \, f^{ + } = \left( {1 + \lambda } \right)\left( {\sum\limits_{j = 1}^{{k_{1} }} {c_{j}^{ + } x_{j}^{ + } } + \sum\limits_{{j = k_{1} + 1}}^{{n_{1} }} {c_{j}^{ + } x_{j}^{ - } } } \right) + \sum\limits_{s = 1}^{v} {\left( {\sum\limits_{j = 1}^{{k_{2} }} {p_{s} q_{js}^{ + } y_{js}^{ + } } + \sum\limits_{{j = k_{2} + 1}}^{{n_{2} }} {p_{s} q_{js}^{ + } y_{js}^{ - } } } \right)} \\ + \lambda \left( {\eta^{ + } + \frac{1}{1 - \alpha }\sum\limits_{s = 1}^{v} {p_{s} z_{s} } } \right) \\ \end{gathered} $$

(12a)

subject to

$$ \begin{gathered} \sum\limits_{j = 1}^{{k_{1} }} {\left| {t_{js} } \right|^{ - } {\text{Sign}}\left( {t_{js}^{ - } } \right)x_{j}^{ + } } + \sum\limits_{{j = k_{1} + 1}}^{{n_{1} }} {\left| {t_{js} } \right|^{ + } {\text{Sign}}\left( {t_{js}^{ + } } \right)x_{j}^{ - } } \hfill \\ \, + \sum\limits_{j = 1}^{{k_{2} }} {\left| {d_{js} } \right|^{ - } {\text{Sign}}\left( {d_{js}^{ - } } \right)y_{js}^{ + } } + \sum\limits_{{j = k_{2} + 1}}^{{n_{2} }} {\left| {d_{js} } \right|^{ + } {\text{Sign}}\left( {d_{js}^{ + } } \right)y_{js}^{ - } } \ge h_{s}^{ + } , \quad \forall s \hfill \\ \end{gathered} $$

(12b)

$$ \sum\limits_{j = 1}^{{k_{2} }} {\left| {q_{js} } \right|^{ + } {\text{Sign}}\left( {q_{js}^{ + } } \right)y_{js}^{ + } } + \sum\limits_{{j = k_{2} + 1}}^{{n_{2} }} {\left| {q_{js} } \right|^{ + } {\text{Sign}}\left( {q_{js}^{ + } } \right)y_{js}^{ - } } \le z_{s} + \eta^{ + } , \quad \forall s $$

(12c)

$$ z_{s} \ge 0, \quad \forall s $$

(12d)

$$ \eta^{ + } \ge \eta_{\text{opt}}^{ - } $$

(12e)

$$ x_{j}^{ + } \ge x_{{j,{\text{ opt}}}}^{ - } ,\quad \, j = 1,2, \ldots ,k_{1} $$

(12f)

$$ 0 \le x_{j}^{ - } \le x_{{j,{\text{ opt}}}}^{ + } ,\quad \, j = k_{1} + 1,k_{1} + 2, \ldots ,n_{1} $$

(12g)

$$ y_{js}^{ + } \ge y_{{js,{\text{ opt}}}}^{ - } , \quad \forall s,\quad \, j = 1,2, \ldots ,k_{2} $$

(12h)

$$ 0 \le y_{js}^{ - } \le y_{{js,{\text{ opt}}}}^{ + } , \quad \forall s,\quad \, j = k_{2} + 1,k_{2} + 2, \ldots ,n_{2} $$

(12i)

Solutions of $\eta_{\text{opt}}^{ + } ,$ $x_{{j,{\text{ opt}}}}^{ + }$ ($j = 1,2, \ldots ,k_{1} ),$ $x_{{j,{\text{ opt}}}}^{ - }$ ($j = k_{1} + 1,\;k_{1} + 2, \ldots ,n_{1} ),$ $y_{{js,{\text{ opt}}}}^{ + }$ ($j = 1,2, \ldots ,k_{2} ),$ and $y_{{js,{\text{ opt}}}}^{ - }$ ($j = k_{2} + 1,\;k_{2} + 2, \ldots ,n_{2}$) can be obtained through submodel (12). Through integrating solutions of submodels (11) and (12), interval solution for model (10) under a set of $p_{s}$ ($s = 1,2, \ldots ,v$) levels can be expressed as follows: $x_{{j,{\text{ opt}}}}^{ \pm } = \left[ {x_{{j,{\text{ opt}}}}^{ - } , \, x_{{j,{\text{ opt}}}}^{ + } } \right], \quad \forall j;$ $y_{{js,{\text{ opt}}}}^{ \pm } = \left[ {y_{{js,{\text{ opt}}}}^{ - } , \, y_{{js,{\text{ opt}}}}^{ + } } \right], \quad \forall j,s;$ $f_{\text{ opt}}^{ \pm } = \left[ {f_{\text{ opt}}^{ - } , \, f_{\text{ opt}}^{ + } } \right]$. Figure 1 shows the flow chart for the formulation of the IMTSP method and its method of solution.

Fig. 1

General framework of the IMTSP method

3 Case study

The City of Regina is located in southeast Saskatchewan and is the provincial capital with a population of approximately 179,000 within the city limits. The amount of waste generated by the residential sector is considerably smaller than the amount of waste generated by the industrial, commercial and institutional (IC&I) and construction and demolition (C&D) sectors, which contributes about 70 % of the waste compared with 30 % for the residential sector (Dai et al. 2012). In 2008, the residential sector generated a total of approximately 68,000 tonnes of solid waste and of this diverted about 11,000 tonnes (16.5 %) of material through recycling or returned for deposit beverage container program (City of Regina 2011). For the long-term planning exercise, a general waste generation rate of [1.00, 1.17] kg/capita/day is used for the residential sector of the City (Li and Huang 2009). Residential waste collection service is provided to 52,000 households (76 % of the total) in Regina, as part of the general city services that are funded by property taxes. Collection of solid waste from residences is the most costly and visible part in MSW management system. The average frequency of collection is around once per week. The average garbage collection/transportation cost was [56, 71] $/tonne for manual collection or [29, 43] $/ton for the automatic one. Costs for waste collection and transportation are estimated based on the existing conditions in the collection areas; the average container size, collection frequency, collection mode (automatic and manual), and collection time (per load) were considered. In this study, a projected interval of [38, 47] $/ton is used for approximating the cost of waste collection and transportation (Guo and Huang 2009b).

The main approaches for waste treatment include recycling, composting and landfill. Recycling is the high profile activity that the public most identifies in terms of sustainable development and waste minimization. Therefore, the city operates many recycling programs to encourage residents to reduce the amounts of waste that end up at the landfill (Li and Huang 2009). For example, a big blue bin project was started in 1990. It provides residents with an option to recycle paper and cardboard. The amount of paper and cardboard recycled was enough to make 88,524,588 cartons in 2008. Over the last three years, the revenue from the sale of paper has declined from $40/tonne to $35/tonne in 2007 and 2008, while the operation cost of the program has increased (City of Regina 2011). A white metal goods recycling program was operated for collecting and selling the scrap metal. Between 2006 and 2008, this program has diverted approximately 2,700 tonnes of material and generated $246,000 in gross revenue. Potentially recyclable material constituted approximately 41.79 % of the waste stream, while the observed recycling rate of residential waste was only 8.72 % in Regina, indicating a high potential for further improvements (Dai et al. 2012). Composting is the conversion of organic waste into a soil product for use elsewhere (Li and Huang 2009). The city is operating a backyard composting program to naturally process grass clippings (by leaving them on the lawns). The waste stream generated in the city contained a high percentage of compostable materials, including 20,000 tonnes of yard wastes (25.64 %) and 20,900 tonnes of organic wastes (26.79 %); the potentially compostable material constituted approximately 52.43 % of the waste stream (City of Regina 2011). Composting on-site removes this valuable material from the waste stream, saves space in the landfill and reduces the use of fossil fuels for waste transport. Composting returns nutrients and moisture to the soil, improves soil consistency, and impedes weed growth, all without the use of herbicides, fertilizers, and excessive watering. Revenue can be obtained from compost produced by composting technology. The residue ([8, 10] % of original waste) from composting facilities will be finally disposed by landfills. The landfill is located in the northeast quadrant of the city at the corner of Fleet Street and McDonald Street. It is the only solid waste landfill in the study region and which occupies 97 ha, with an actual landfilling area of 60 ha. In 2008, Regina’s citizens generate about 1,000 kg per capita of MSW, more than 80 % of which is disposed in the landfill with less than 20 % diverted (City of Regina 2011). Landfill deliveries have increased from 2005 to 2008 with 478,474 tonnes in 2005 rising to 645,456 tonnes in 2008, where the IC&I sector and C&D sector produced 61 % of the total waste flow, and the residential sector generated approximately 39 % (Dai et al. 2012). The total number of vehicle transactions has increased over the last 4 years from 137,128 in 2006 to 178,097 vehicles in 2008, with an average load of [3.48, 3.62] tonnes, respectively. In Canada, establishment of waste diversion targets and relevant regulations were currently a growing trend. In March 2010, that city’s Council adopts, in principle, the Waste Plan Regina’s Residential Option, and set a waste diversion target of 34 % by 2020, and 48 % by 2025 (City of Regina 2011).

The study time is 15 years (from 2012 to 2027), which is further divided into three 5-year periods. Table 1 presents the waste-generation rates and the associated probabilities of occurrence in the three planning periods, indicating that the waste-generation amounts are highly uncertain, presented by intervals with the associated probabilities. Tables 2 and 3 contain regular costs for allowable waste flows, operating costs for waste management facilities, and penalty costs for surplus waste flows, as well as revenues from waste management facilities over the three planning periods (Li and Huang 2009; City of Regina 2011; Dai et al. 2012). Table 4 shows the capacity-expansion options and the relevant capital costs for composting and recycling facilities (City of Regina 2011). It is also indicated that the expansion costs decrease along with time. Since the planning problem under consideration is dynamic with multiple stages, discount factors are necessary for each period to obtain a total present value for the object function. In this project, all the cash flows are counted in year 2012 dollars.

Table 1 Waste generation rates under different probability levels

Table 2 Costs and revenues for allowable waste flows

Table 3 Costs and revenues for excess waste flows

Table 4 Capacity expansion options and the relevant capacity costs

Because uncertainties and risks exist in a variety of system components and a link to the pre-regulated policy as formulated by local authorities is desired, the IMTSP method developed in Sect. 2 is considered to be suitable for tackling this type of management problem. The model includes continuous and binary decision variables. The binary variables represent the development or expansion options for waste-management-facilities in different periods (i.e. ${\text{BC}}_{ck}^{ \pm }$ and ${\text{BR}}_{rk}^{ \pm }$); their solutions can be used for answering the questions related to timing, sizing and siting for waste-management-facility development and/or expansion under uncertainty. The continuous variables represent the optimized waste flows from the city to the waste-management-facilities. Furthermore, the continuous variables include two subsets: those (the first-stage ones, ${\text{X}}_{ik}^{ \pm }$) that must be determined before the random variables (i.e. waste-generation rates) are disclosed, and those (the second-stage ones, $Y_{iks}^{ \pm }$) that will be determined after the random variables are disclosed. The modeling formulation based on the IMTSP method for the city’s MSW management can be presented as follows:

$$ \begin{gathered} \hbox{min} \, f^{ \pm } = \left( {1 + \lambda } \right)\left[ {\sum\limits_{i = 1}^{3} {\sum\limits_{k = 1}^{3} {{\text{L}}_{k} {\text{X}}_{ik}^{ \pm } \left( {{\text{TR}}_{ik}^{ \pm } + {\text{OP}}_{ik}^{ \pm } } \right)} } + \sum\limits_{i = 2}^{3} {\sum\limits_{k = 1}^{3} {{\text{L}}_{k} {\text{X}}_{ik}^{ \pm } \left( {{\text{FE}}_{i}^{ \pm } {\text{FT}}_{ik}^{ \pm } - {\text{RE}}_{ik}^{ \pm } } \right)} } } \right] \\ + \sum\limits_{i = 1}^{3} {\sum\limits_{k = 1}^{3} {\sum\limits_{s = 1}^{9} {p_{s} {\text{L}}_{k} {\text{Y}}_{iks}^{ \pm } \left( {{\text{DR}}_{ik}^{ \pm } + {\text{DP}}_{ik}^{ \pm } } \right)} } } + \sum\limits_{i = 2}^{3} {\sum\limits_{k = 1}^{3} {\sum\limits_{s = 1}^{9} {p_{s} {\text{L}}_{k} {\text{Y}}_{iks}^{ \pm } \left( {{\text{FE}}_{i}^{ \pm } {\text{DT}}_{ik}^{ \pm } - {\text{RM}}_{ik}^{ \pm } } \right)} } } \\ + \left( {1 + \lambda } \right)\left[ {\sum\limits_{c = 1}^{3} {\sum\limits_{k = 1}^{3} {{\text{CCE}}_{ck}^{ \pm } {\text{BC}}_{ck}^{ \pm } } } + \sum\limits_{r = 1}^{3} {\sum\limits_{k = 1}^{3} {{\text{CRE}}_{rk}^{ \pm } {\text{BR}}_{rk}^{ \pm } } } } \right] + \lambda \left( {\alpha^{ \pm } + \frac{1}{1 - \beta }\sum\limits_{s = 1}^{9} {p_{s} z_{s} } } \right) \\ \end{gathered} $$

(13a)

subject to

$$ z_{s} + \alpha^{ \pm } \ge \left[ {\sum\limits_{i = 1}^{3} {\sum\limits_{k = 1}^{3} {{\text{L}}_{k} {\text{Y}}_{iks}^{ \pm } \left( {{\text{DR}}_{ik}^{ \pm } + {\text{DP}}_{ik}^{ \pm } } \right)} } + \sum\limits_{i = 2}^{3} {\sum\limits_{k = 1}^{3} {{\text{L}}_{k} {\text{Y}}_{iks}^{ \pm } \left( {{\text{FE}}_{i}^{ \pm } {\text{DT}}_{ik}^{ \pm } - {\text{RM}}_{ik}^{ \pm } } \right)} } } \right], \quad \forall s $$

(13b)

(Auxiliary variable of CVaR constraints)

$$ \sum\limits_{k = 1}^{3} {{\text{L}}_{k} \left[ {\left( {{\text{X}}_{1k}^{ \pm } + {\text{Y}}_{1ks}^{ \pm } } \right) + \sum\limits_{i = 2}^{3} {{\text{FE}}_{i}^{ \pm } \left( {{\text{X}}_{ik}^{ \pm } + {\text{Y}}_{iks}^{ \pm } } \right)} } \right]} \le {\text{LC}}^{ \pm } , \quad \forall s $$

(13c)

(Total landfill capacity constraints)

$$ {\text{X}}_{2k}^{ \pm } + {\text{Y}}_{2ks}^{ \pm } \le {\text{CC}}^{ \pm } + \sum\limits_{c = 1}^{3} {\sum\limits_{k = 1}^{{k^{'} }} {\Updelta {\text{CC}}_{ck}^{ \pm } {\text{BC}}_{ck}^{ \pm } } } , \quad \forall s,\;k,\;k^{'} = 1,2,3 $$

(13d)

(Composting capacity constraints)

$$ {\text{X}}_{3k}^{ \pm } + {\text{Y}}_{3ks}^{ \pm } \le {\text{RC}}^{ \pm } + \sum\limits_{r = 1}^{3} {\sum\limits_{k = 1}^{{k^{'} }} {\Updelta {\text{RC}}_{rk}^{ \pm } {\text{BR}}_{rk}^{ \pm } } } , \quad \forall s,k,k^{'} = 1,2,3 $$

(13e)

(Recycling capacity constraints)

$$ {\text{X}}_{2k}^{ \pm } + {\text{Y}}_{2ks}^{ \pm } \ge {\text{DC}}_{k}^{ \pm } {\text{W}}_{ks}^{ \pm } , \quad \forall k,s $$

(13f)

$$ {\text{X}}_{3k}^{ \pm } + {\text{Y}}_{3ks}^{ \pm } \ge {\text{DR}}_{k}^{ \pm } {\text{W}}_{ks}^{ \pm } , \quad \forall k,s $$

(13g)

(Constraints of waste flow to the composting and recycling)

$$ \sum\limits_{i = 1}^{3} {\left( {{\text{X}}_{ik}^{ \pm } + {\text{Y}}_{iks}^{ \pm } } \right)} \ge {\text{W}}_{ks}^{ \pm } , \quad \forall k,s $$

(13h)

(Waste disposal demand constraints)

$$ \alpha \ge 0 $$

(13i)

$$ \, z_{s} \ge 0, \quad \forall s $$

(13j)

$$ 0 \le {\text{Y}}_{iks}^{ \pm } \le {\text{X}}_{ik}^{ \pm } \le {\text{X}}_{ik,\hbox{max} } , \quad \forall i,k,s $$

(13k)

(Non-negativity and technical constraints)

$$ {\text{BC}}_{ck}^{ \pm } \left\{ \begin{gathered} = 1,\quad {\text{if capacity expansion for composting facility is undertaken}} \hfill \\ = 0,\quad {\text{if otherwise}} \hfill \\ \end{gathered} \right., \quad \forall c,\;k $$

(13l)

$$ {\text{BR}}_{rk}^{ \pm } \left\{ \begin{gathered} = 1,\quad {\text{if capacity expansion for recycling facility is undertaken}} \hfill \\ = 0,\quad {\text{if otherwise}} \hfill \\ \end{gathered} \right.,\, \quad \forall r,\;k $$

(13m)

(Binary constraints)

$$ \sum\limits_{c = 1}^{3} {{\text{BC}}_{ck}^{ \pm } } \le 1, \quad \forall k $$

(13n)

$$ \sum\limits_{r = 1}^{3} {{\text{BR}}_{rk}^{ \pm } } \le 1, \quad \forall k $$

(13o)

(Expansions for composting and recycling facilities may occur in any given time period)

where $f^{ \pm }$ is mean-risk function value ($); λ and β are risk parameters; ${\text{L}}_{k}$ is length of time period k (week); k is time period, k = 1, 2, 3; c is the name of expansion option for composting facility, c = 1, 2, 3; r is the name of expansion option for recycling facility, r = 1, 2, 3; s is the level of waste generation, s = 1, 2,…, 9; i is type of waste management facility, i = 1, 2, 3, where i = 1 for the landfill, 2 for composting facility, and 3 for recycling facility; ${\text{TR}}_{ik}^{ \pm }$ is cost of collection and transportation for allowable-waste flow to facility i during period k ($/tonne), i = 1, 2, 3; ${\text{DR}}_{ik}^{ \pm }$ is cost of collecting and transporting excess waste flow from the city to facility i during period k ($/tonne) (the second-stage cost parameter), where ${\text{DR}}_{ik}^{ \pm } \ge {\text{TR}}_{ik}^{ \pm }$, i = 1, 2, 3; ${\text{OP}}_{ik}^{ \pm }$ is regular operating cost of waste management facility i during period k ($/tonne), i = 1, 2, 3; ${\text{DP}}_{ik}^{ \pm }$ is operating cost of facility i for excess waste flow during period k ($/tonne) (the second-stage cost parameter), where ${\text{DP}}_{ik}^{ \pm } \ge {\text{OP}}_{ik}^{ \pm }$, i = 1, 2, 3; ${\text{FE}}_{i}^{ \pm }$ is residue flow from facility i to the landfill (% of incoming mass to facility i), i = 1, 2, 3; ${\text{FT}}_{ik}^{ \pm }$ is disposal cost for allowable residues generated by facility i during period k ($/tonne), i = 2, 3; ${\text{DT}}_{ik}^{ \pm }$ is disposal cost for excess waste residues generated by facility i during period k ($/tonne) (the second-stage cost parameter), where ${\text{DT}}_{ik}^{ \pm } \ge {\text{FT}}_{ik}^{ \pm }$, i = 2, 3; ${\text{RE}}_{ik}^{ \pm }$ is revenue generated by processing allowable waste flows in facility i during period k ($/tonne), i = 2, 3; ${\text{RM}}_{ik}^{ \pm }$ is revenue generated by processing excess waste flows in facility i during period k ($/tonne), (the second-stage revenue parameter) i = 2, 3; ${\text{W}}_{sk}^{ \pm }$ is residential waste-generation rate with probability p _s in period k (tonne/week); p _s is probability of waste generation rate (${\text{W}}_{ks}^{ \pm }$) with level s (%); ${\text{X}}_{ik}^{ \pm }$ is allowable-waste flow to facility i during period k (tonne/week) (the first-stage decision variable), i = 1, 2, 3; ${\text{X}}_{ik,\hbox{max} }$ is maximum allowable-waste flow from the city to facility i during period k (tonne/week); ${\text{Y}}_{isk}^{ \pm }$ is amount by which the allowable-waste level is exceeded when the waste-generation rate is ${\text{W}}_{sk}^{ \pm }$ with probability p _s (tonne/week) (the second-stage decision variable); ${\text{LC}}^{ \pm }$ is existing landfill capacity (tonne); ${\text{CC}}^{ \pm }$ is existing composting facility capacity (tonne/week); ${\text{RC}}^{ \pm }$ is existing recycling facility capacity (tonne/week); ${\text{DC}}_{k}^{ \pm }$ is waste flow transferred from composting facility in period k; ${\text{DR}}_{k}^{ \pm }$ is waste flow transferred from recycling facility in period k; $\Updelta {\text{CC}}_{ck}^{ \pm }$ is available expanded/developed capacity for composting facility with expansion option c in period k (tonne/week); $\Updelta {\text{RC}}_{rk}^{ \pm }$ is available expanded/developed capacity for recycling facility with expansion option r in period k (tonne/week); ${\text{BC}}_{ck}^{ \pm }$ is binary decision variable for expanding composting facility with option c at the start of period k; ${\text{BR}}_{rk}^{ \pm }$ is binary decision variable for expanding recycling facility with option r at the start of period k; ${\text{CCE}}_{ck}^{ \pm }$ is capital cost of expanding composting facility by option c in period k ($/tonne); ${\text{CRE}}_{rk}^{ \pm }$ is capital cost of expanding recycling facility by option c in period k ($/tonne).

4 Result analysis

In this study, there are two risk-related parameters: λ and β, where three β levels (0.8, 0.9 and 0.99) were examined under each λ level. Table 5 shows the solutions of waste-flow allocation over periods 1–3 under β = 0.99. An excessive waste flow would be generated if the allowable-waste-flow level is exceeded (i.e. excessive flow = generated flow − assigned quota). The waste-flow patterns (including allowable and excess flows) would vary dynamically due to temporal and spatial variations in waste-generation/management conditions. When the level of waste-generation rate is medium to high in period 1, the optimized allowable flows to the landfill would be 1,188 tonnes/week under λ = 0.5, 1397 tonnes/week under λ = 1, and [1557, 1698] tonnes/week under λ = 5. The optimal excessive flows would be [141, 283] tonnes/week under λ = 0.5, [0, 142] tonnes/week under λ = 1 and 0 tonnes/week under λ = 5, respectively. Thus, the total waste flows to the landfill would be [1329, 1471], [1397, 1539], and [1557, 1698] tonnes/week under λ = 0.5, 1 and 5, respectively. Figure 2 presents the optimal waste-allocation patterns from the city to the landfill, composting and recycling facilities under different risk parameters. The results indicate that a plan for waste-flow allocation would be related to both waste-generation rate and capacity-expansion scheme under varied risk parameters. Figure 3 shows the varying trend of the excessive waste flows allocated to the landfill, composting and recycling facilities under different λ and β levels. In the case of excess waste, the allotment to the landfill should be assigned initially, and then to the incinerating and composting facilities. Since the incinerating and composting facilities have higher regular and penalty costs for treating wastes, allotment of wastes to the landfill would be more economical. Moreover, the related outputs are affected significantly by the risk parameters. For example, the excessive waste flows to any waste management facilities would increase when λ is fixed and β changes from 0.8 to 0.99 or β is fixed and λ changes from 0.5 to 5. Therefore, decreasing the value of λ and/or β means the more excess waste flow, which may lead to a higher level of system risk; conversely, increasing the value of λ and/or β means the less excess waste flow, which may lead to a lower level of system risk.

Table 5 Solution of the IMTSP method for continuous variables for β = 0.99

Fig. 2

Optimal waste-flow allocation pattern for a landfill, b composting facility and c recycling facility

Fig. 3

Excess waste flows to a landfill, b composting facility and c recycling facility for different risk parameters

Table 6 presents the solutions of facility expansion schemes under different risk parameters. When β = 0.8 or 0.9, the expansion schemes for composting and recycling facilities would be constant even though the weight λ vary between 0.5 and 5. In comparison, if β = 0.99, they would vary under different λ levels. For example, when λ = 0.5, a centralized composting facility would be expanded by an increment of 483 tonnes/week at the starts of periods 1 and 2 respectively, with 70 % of the capacity (i.e. 338 tonnes/week) being dedicated to the residential waste since the IC&I (industrial, commercial and institutional) and rural sectors would not be included within the scope of this study. This facility should be developed with a capacity of 189 tonnes/week at the start of period 3. When λ = 1 and 5, a centralized composting facility would be developed with a capacity of 483 tonnes/week at the starts of periods 1 and 2 respectively; then this facility should be expanded by an increment of 385 tonnes/week at the start of periods 3. Consequently, the total expanded capacities for the composting facility would be 1155, 1351 and 1351 tonnes/week under λ = 0.5, 1 and 5, respectively. Comparatively, the recycling facility would be expanded once with an increment of 350 tonnes/week at the start of period 1 when λ = 0.5, and expanded with the same capacity at the start of period 2 when λ = 1 and 5, respectively.

Table 6 Solutions of IMTSP method for binary variables under different risk parameters

Figure 4 shows the mean-risk function value of $f^{ \pm }$ for different risk parameters. The results indicate that the optimal mean-risk function would increase when the risk parameters λ or β increase. For example, the mean-risk function value would increase from $[96, 122] × 10⁶ to $[747, 941] × 10⁶ when β is fixed at 0.8 and λ increases from λ = 0.5 to λ = 10. Besides, the mean-risk function value would increase from $[96, 122] × 10⁶ to $[107, 133] × 10⁶ when λ is fixed at 0.5 and β increases from β = 0.8 to β = 0.99, respectively. Figure 5 presents the value of CVaR for different λ and β levels. Generally, CVaR would increase as β increases. When λ is fixed at 5 and β changes from β = 0.8–0.99, the value of CVaR would increase from $[69.7, 88.4] × 10⁶ to $[90.8, 110.4] × 10⁶. In addition, CVaR decreases as λ increases. For example, when β is fixed at 0.99 and λ changes from 0.5 to 10, the value of CVaR would decrease from $[90.8, 110.4] × 10⁶ to $[80.9, 100.6] × 10⁶. This is because of the changing trade-off between the expectation total cost and the CVaR criterion.

Fig. 4

Mean-risk function value of $f^{ \pm }$ for different risk parameters

Fig. 5

Value of CVaR _β for different risk parameters

Figure 6 illustrates the total expected cost under different λ and β levels. It is indicated that the total expected cost would increase in responding to the lager risk parameters λ and/or β. This implies that a higher system cost would guarantee a lower system risk. Conversely, if the decision maker aims towards an economic plan, a higher expected loss could be confronted. In the real-world application, decision makers may need to choose between a more risky solution with a lower system cost and a more conservative solution with a higher system cost. Figure 7 presents the recourse cost under different λ and β levels. According to these results, increasing λ leads to a more risk-averse policy with higher total expected costs and lower recourse costs in general. Therefore, a more conservative (i.e., averting risks) policy would lead to a lower excessive waste flow and a lower recourse cost.

Fig. 6

Total expected cost for different risk parameters

Fig. 7

Recourse cost for different risk parameters

5 Discussion

In order to better reflect the effect of risk parameters λ and β on the optimal results, sensitivity analysis is conducted in this research. Figure 8 shows the total expected cost under different λ and β levels, where the upper and lower parts correspond to the upper and lower bounds of total expected cost, respectively. It appears that different combination of λ and β levels would notably influence the value of total expected cost. For example, the total expected cost would be $[62.1, 77.5] × 10⁶ when β = 0.5 and λ = 0.1, $[66.9, 82.8] × 10⁶ when β = 0.9 and λ = 3, and $[77.1, 95.0] × 10⁶ when β = 0.99 and λ = 10. A higher confidence level (β) and a higher acceptable risk level (λ) would give rise to a higher total expected cost. Conversely, a lower confidence level and a lower acceptable risk level would correspond to a lower total expected cost. Figure 9 presents the distribution of CVaR under different λ and β levels. The specified β level of CVaR _β quantifies the mean value of the worst (1 − β) % of the total costs. When β increases, CVaR _β accounts for the risk of larger realizations. Thus, larger β values would lead to more conservative policies, which give higher weights to worse scenarios. The specified λ level represents the risk preference of decision makers. When λ = 0, decision makers would not consider the variability of the uncertain recourse costs based on a risk-neutral attitude. When λ = 100, decision makers would seriously consider the variability based on a risk-aversive attitude. Moreover, increasing the value of λ would increase the relative importance of the risk related terms and thus would lead to risk-averting policies.

Fig. 8

Distribution of the lower and upper bounds of total expected cost under different λ and β levels

Fig. 9

Distribution of the lower and upper bounds of the CVaR under different λ and β levels

Sensitivity analysis results of Figs. 8 and 9 proclaim the trade-offs between costs and risk levels. A plan with low levels of the parameter λ and/or the parameter β would correspond to lower system cost, implying that the decision makers have an optimistic attitude; however, it might be associated with a higher risk levels (i.e., excess of waste generation rate). Conversely, a plan with high levels of the parameter λ and/or the parameter β would better resist from waste excess, and it would result in a lower risk of system failure. Therefore, the IMTSP encourages decision makers to assign the β and λ to adjust risk control levels based on their preferences, and ensure management policies be made with reasonable consideration of both system stability and cost. Also, Figs. 8 and 9 indicate another fact that β and λ have different sensitivity levels. When λ is less than 0.5 and/or β is less than 0.7, the variation degree of total expected cost would increase gradually and the variation degree of CVaR decrease gradually, with an increase of λ value. This implies that the risk constraint in effect is poor. When λ changes from 0.5 to 3 and β is larger than 0.7, the variation degree of total expected cost and CVaR are both sharp. This implies that the risk constraint in effect is robust. When λ is larger than 5, the values of total expected cost and CVaR under different β would be nearly constant, which indicates that the risk constraint is of significant effect. Thus, the λ and β values in the CVaR constraint should be properly chosen to avoid the invalidation of risk control. The decision makers should incorporate their implicit knowledge (such as socio-economic conditions) and preferences about the risk and costs into the WSM management problems for generating more practical decisions.

Uncertainty sources are important for MSW management systems. The random characteristics of various processes and conditions, the errors in acquiring the modeling parameters, and the imprecision of the related system constraints are all possible sources of the uncertainties (Li et al. 2009). Waste generation rate arises as a direct consequence of human activities and consumption level of individual residents. However, the total population and consumption level may be of the stochastic characteristic in the planning horizon. Accordingly, waste generation rate varies from <1,000 tonne/week (i.e., the extra low scenario) to >2,000 tonne/week (i.e., the extra high scenario), which could be expressed as a random variable. Thus, waste generation rate could be regarded as the stochastic sources for two stage stochastic model. Moreover, the object of this research was to develop an inexact optimization model for allocating the waste flow generated by city residents. Thus, waste generation rate was one of the most important parameters to be considered for the developed IMTSP model. On the other hand, the variability of random waste generation rate may result in the risk of economic loss from extreme events. If the extra high scenario of waste generation rate happens, waste disposal facilities might not have capacity to deal with the extra solid waste. The excess waste would become dangerous for city environment and human health. Thus, the decision makers have to take expensive measures, such as transporting it to neighboring city, to deal with the excess waste. CVaR as a kind of risk measure method is incorporated within the objective function of IMTSP method to describe the expected losses on extreme conditions. Therefore, the IMTSP can help decision makers analyze the trade-off between system cost and risk.

To better reflect advantages of the proposed IMTSP method, ITSP is applied to the study case for comparison. The total expected cost of ITSP model would be $[61.34, 77.42] × 106, which would be lower than the one of IMTSP. Solutions of binary variables are the same as for ITSP and IMTSP methods (λ and β are fixed at 0.5 and 0.8, respectively). Comparatively, solutions of continuous variables are different for ITSP and IMTSP. Table 7 presents solutions of continuous variables for ITSP. Different from IMTSP, ITSP searches for the optimal solutions to obtain the minimum cost without any consideration of risks. Consequently, it is incapable of analyzing trade-offs between system costs and risks (Maqsood et al. 2005). A lower level of allowable waste flow (i.e. ${\text{X}}_{ik,opt}^{ \pm }$) would be generated by ITSP to obtain minimum costs, leading to a higher level of excessive waste flow (i.e. ${\text{Y}}_{isk,opt}^{ \pm }$). Moreover, even though a lower cost would be obtained by ITSP compared with IMTSP, the obtained solutions would be limited in a number of aspects. First, for a basic municipal facility, such as a landfill, a fixed amount of waste flow normally should be guaranteed to ensure its good running; an overly high excess of waste flow, as suggested by the ITSP model, would cause serious consequences to such a facility. Second, the solutions of ITSP may lead to unbalanced allocation patterns among different MSW management facilities; this could provoke serious contradictions and harm implementation of waste management policies. Third, although the total expected cost is minimum, the large deficit amount at a high waste flow condition (i.e., ${\text{Y}}_{391,opt}^{ \pm }$) would occur at a probability level of p _s  = 0.004. If this really happens, a high second-stage penalty would have to be paid and high system infeasibility may need to be faced. On the other hand, solutions of ITSP seem to be exactly the same as those of IMTSP when the value of λ is zero, which indicates that ITSP is a special or simplified case of IMTSP.

Table 7 Solution of the ITSP model for continuous variables

Without consideration of interval mathematical programming, the MTSP method can be used to solve the study case. Its deterministic parameters can be derived by averaging the lower and upper bounds of intervals, such as the capacity of MSW management facilities. Accordingly, the solutions are fixed values rather than discrete intervals although their varying trends are similar to the solutions of IMTSP. Moreover, the solutions of MTSP are special cases in the solutions obtained from IMTSP. In such a case, the decision alternative would be restricted to a single solution. Meanwhile, the effectiveness and flexibility of the alternative would be reduced since impact of uncertainties is not considered. Robust risk analysis method (RRAM) was introduced by Chen et al. (2012), which incorporated interval mathematical programming, two-stage stochastic programming and variance within a general modeling framework. Thus, the main difference between IMTSP and RRAM is the quantitative risk measure for the second-stage random variable, where the former uses the CVaR and the latter uses the variance. Moreover, compared with IMTSP, the disadvantage of RRAM is that it considers the under-and-over-performances of the probability distribution equally. For example, the high waste generation rate may result in system risks, while the low waste generation rate is safe to the management system. Therefore, quantitatively evaluating the right tail of distributions of waste generation rate is necessary; however consideration the left tail of distributions must not only waste the expected cost but also be of no use for the system stability. One of the difficulties in planning a MSW management system is that those interval and stochastic uncertainties may be further multiplied by their associations with economic penalties if the promised targets are violated. Fortunately, two-stage stochastic programming and multi-stage stochastic programming can both effectively deal with such a problem. In this research, a time interval of 15 years is considered, which is divided into three 5-year periods. This implies that multistage stochastic programming is a suitable method, because it extends the two-stage stochastic programming by permitting modified decisions in each time stage based on the information of sequentially realized uncertain events (Li et al. 2009). Moreover, scenario-tree based multistage stochastic programming can not only reflect the correlations between the waste generation rates of the different time periods, but also model the decision process and define all possible scenarios. However, the main objective of this study is to control risks associated with the recourse actions and analyze trade-offs between expected costs and system risks. This objective can be achieved by the proposed two-stage stochastic programming associated with CVaR. Moreover, more scenarios are necessary for CVaR to quantify the risks from the extreme high waste generation rate. If a multi-stage stochastic programming is taken place the two-stage stochastic programming, the complicated scenario-tree would result in a concern upon the huge computational efforts. In other hand, to merge above advantage of multi-stage stochastic programming, future studies would be undertaken to investigate the performances of a hybrid model by integrating the multi-stage stochastic programming, CVaR and interval mathematical programming within a general framework.

6 Conclusions

In this study, an interval-parameter mean-CVaR two-stage stochastic programming (IMTSP) method was developed for MSW management under uncertainty. This method incorporated interval mathematical programming, two-stage stochastic programming and CVaR within a general framework. In the developed IMTSP, uncertainties could be presented as probabilistic distribution functions and discrete intervals. Also, CVaR as a kind of risk measure method was successfully incorporated within the objective function of IMTSP to describe the expected losses under extreme conditions. In general, the proposed IMTSP method has several advantages: (i) its objective function simultaneously takes expected cost and system risk into consideration. Both discrete random variables and intervals can be used to reflect uncertainties. Therefore, the solution results would possess characteristics of CVaR, interval mathematical programming, and two-stage stochastic programming. The MTSP and ITSP model do not have such an advantage; (ii) it can be used for supporting quantitatively evaluating the right tail of distributions of waste generation rate, which can better quantify the system risk compared to RRAM method; (iii) it is useful for helping decision makers analyze the trade-offs between cost and risk and identify desired waste management strategies under complex uncertainties; and (iv) it is effective to penalize the second-stage costs that are above the expected values, as well as to capture the notion of risk in stochastic programming.

The developed IMTSP was applied to the long-term planning of MSW management system in the City of Regina, Canada. The initial allocation target value could be used to reflect decision maker’s opinion on waste allocation, and the risk acceptance level (i.e. λ) and the percentile level (i.e., β) could reflect decision maker’s preference upon system cost and risk aversion. Moreover, in the case of excess waste, the allotment to the landfill should be assigned initially, and then to the incinerating and composting facilities. The study results demonstrated that IMTSP method could generate a series of waste management patterns under different risk-aversion levels, gain in-depth insights into the effects of uncertainties, and maintain a balance between economic cost and environmental protection. Thus, the IMTSP method was valuable for supporting: (i) the adjustment or justification of the existing waste flow allocation patterns, (ii) the long-term capacity planning for waste-management facilities, and (iii) the strong bases for selecting desired waste management policies. In choosing appropriate solutions for practical applications, the decision makers should incorporate their implicit knowledge (such as socio-economic conditions) and preferences about the risk and costs into the WSM management problems for generating more practical decisions.

Although this study is the first attempt for the planning of waste management systems through the developed IMTSP method, the results suggested that it would be an effective tool for decision makers in conducting long-term MSW planning. However, the IMTSP also has much space for improvement. Multistage stochastic programming is a more suitable method for the multi-period planning, and it can reflect the correlations between the waste generation rates of the different time periods. Thus, it is an interesting work to integrate the CVaR, interval mathematical programming and multi-stage stochastic programming instead of two-stage stochastic programming. Moreover, discrete distribution values of stochastic variables (i.e., waste generation rates) are different to estimate. The multi-uncertainty (i.e., probability intervals) can used to define the parameter of waste generation rates. Thus, many other uncertainty methods which can used to solve the multi-uncertainty have potentials to be further integrated into an IMTSP framework for reflecting more complex conditions.