Municipal Solid Waste–Flow Allocation Planning with Trapezoidal-Shaped Fuzzy Parameters

Trapezoidal-shaped FCCP

Trapezoidal-shaped fuzzy chance-constrained programming (TFCCP) was first proposed by Liu and Iwamura (1998). In the TFCCP model, the left- and right-hand side coefficients in some specific constraints are presented by the trapezoidal fuzzy numbers, and the reliability of system's ability to meet fuzzy constraints is expressed as a series of predetermined confidence levels. According to Liu and Iwamura (1998), a general TFCCP model could be written as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} Minimize \ f = CX \tag{1{\rm a}} \end{align*}\end{document}

Subject to: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} Pos \left\{{\tilde A} , {\tilde B} \Big| {\tilde A}X \leq {\tilde B} \right\} \geq \alpha \tag{1{\rm b}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} DX \geq E \tag{1{\rm c}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} X \geq 0 \tag{1{\rm d}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} C , A \neq 0 \tag{1{\rm e}} \end{align*}\end{document}

where f is the objective function value; X is a vector of deterministic decision variables; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde A}$$\end{document} is a coefficient matrix in the left-hand side of model constraints; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde B}$$\end{document} is a boundary vector in the right-hand side of model constraints; C, A, D, and E are coefficient matrices or vectors for deterministic auxiliary constraints; Pos{·} denotes possibility of events in {·}; α is a predetermined confidence level. For model structure, Eq. (1b) represents the fuzzy chance constraints with coefficients in both the left- and right-hand sides being described as fuzzy sets. Other equations, such as the objective function Eq. (1a), the general constraints Eq. (1c), and the technical constraints Eqs. (1d) and (1e), are in deterministic forms. The fuzzy parameters in constraint (1b) are expressed as trapezoidal fuzzy numbers, i.e., \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde A} = ( r_{a1} , r_{a2} , r_{a3} , r_{a4} )$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde B} = ( r_{b1} , r_{b2} , r_{b3} , r_{b4} )$$\end{document} , with membership functions being \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu ( {\tilde A} )$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu ( {\tilde B} )$$\end{document} , respectively.

To solve model (1), some specific algorithms for the trapezoidal fuzzy set are defined as: (i) the sum of two trapezoidal fuzzy sets is also a trapezoidal fuzzy set; (ii) the product of a trapezoidal fuzzy set and a scalar number is also a trapezoidal fuzzy set (Liu and Iwamura, 1998). The related operations are represented as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\tilde a} + {\tilde b} = ( r_{a1} + r_{b1} , r_{a2} + r_{b2} , r_{a3} + r_{b3} , r_{a4} + r_{b4} ) \tag{2{\rm a}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \lambda {\tilde a} = \left\{ \begin{matrix} ( \lambda r_{a1} , \lambda r_{a2} , \lambda r_{a3} , \lambda r_{a4} ), \ \ \lambda \geq 0 \\ ( \lambda r_{a4} , \lambda r_{a3} , \lambda r_{a2}, \lambda r_{a1} ), \ \ \lambda \leq 0\end{matrix} \right\} \tag{2{\rm b}} \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde a} = ( r_{a1} , r_{a2} , r_{a3} , r_{a4} )$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde b} = ( r_{b1} , r_{b2} , r_{b3} , r_{b4} )$$\end{document} are the two trapezoidal fuzzy sets, respectively; λ is scalar number. Assume the function K(x,ã) is a general linear function for the deterministic variable x and fuzzy variable ã presented by a trapezoidal fuzzy set, i.e., (r_a1, r_a2, r_a3, r_a4). According to the Eq. (2b), the linear function K(x,ã) can be written as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} K ( x , {\tilde a} ) = L ( x )\cdot{\tilde a} \tag{3} \end{align*}\end{document}

where L(x) is a linear function of the variable x. According to the Liu and Iwamura (1998), L(x) could be converted to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} L^ + ( x ) = \left\{ \begin{matrix}L ( x ), \ \ \ \ L ( x ) \geq 0 \\ 0 , \hfill L ( x ) < 0\end{matrix} \right\} \tag{4{\rm a}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} L^ - ( x ) = \left\{ \begin{matrix}0 , \ \ \ \ \ \ \ L ( x ) \geq 0 \\ - L ( x ) , L ( x ) < 0\end{matrix} \right\} \tag{4{\rm b}} \end{align*}\end{document}

Then, L⁺(x) and L⁺(x) are all nonnegative. Moreover, the linear function K(x,ã) can be written as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} K ( x , {\tilde a} ) = ( L^ + ( x ) - L^ - ( x ) ) \cdot {\tilde a} = ( L^ + ( x ) \cdot {\tilde a} + L^ - ( x ) \cdot {\tilde a}^{\prime} ) \tag{5} \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde a}^{\prime} = ( - r_{a4} , - r_{a3} , - r_{a2} , - r_{a1} )$$\end{document} . Finally, by the addition and multiplications operations of the trapezoidal fuzzy number [i.e., Eqs. (2a) and (2b)], the Eq. (5) can be rewritten as follows (Liu and Iwamura, 1998): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} K ( x , {\tilde a} ) = & \Big ( [ r_{a1} \cdot L^ + ( x ) - r_{a4} \cdot L^ - ( x ) ] , [ r_{a2} \cdot L^ + ( x ) - r_{a3} \cdot L^ - ( x ) ] , \\ & [ r_{a3} \cdot L^ + ( x ) - r_{a2} \cdot L^ - ( x ) ] , [ r_{a4} \cdot L^ + ( x ) - r_{a1} \cdot L^ - ( x ) ] \Big ) \tag{6} \end{align*}\end{document}

For the any confidence level α, the following equations can be obtained: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} K ( x , {\tilde a} ) \leq 0 \leftrightarrow \\ \alpha \cdot [ r_{a1} \cdot L^ + ( x ) - r_{a4} \cdot L^ - ( x ) ] + ( 1 - \alpha ) \cdot [ r_{a2} \cdot L^ + ( x ) - r_{a3} \cdot L^ - ( x ) ] \leq 0 \tag{7} \end{align*}\end{document}

Based on Eq. (7), the fuzzy chance constraints (1b) can be transformed to their respective crisp equivalents at a predetermined confidence level. The transformed deterministic model can be formulated as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} Minimize \ f = CX \tag{8{\rm a}} \end{align*}\end{document}

Subject to: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} ( 1 - \alpha ) ( r_{a1} X - r_{b4} ) + \alpha ( r_{a2} X - r_{b3} ) \leq 0 \tag{8{\rm b}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} DX \geq E \tag{8{\rm c}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} X \geq 0 \tag{8{\rm d}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} C , A \neq 0 \tag{8{\rm e}} \end{align*}\end{document}

Finally, the fixed objective function values and decision variables (i.e., f_opt and x_opt) can be obtained through solving model (8). The related proof of the lemma can be referred to the work of Liu and Iwamura (1998).

A numerical example

To demonstrate the procedures of using the proposed method, a numerical example is presented. Let an FMP model be written as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} Maximize \ f = 55x_1 - 80x_2 \tag{9{\rm a}} \end{align*}\end{document}

Subject to: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} 5 x_1 + x_2 \leq 150 \tag{9{\rm b}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} x_1 + 3.5 x_2 \leq 90\tag{9{\rm c}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} ( 0.7 , 1 , 1.5 , 2 ) \cdot x_1 - ( 4.5 , 6 , 8 , 11 ) \cdot x_2 \leq ( - 3 , - 2 , - 1 , - 0.5 )\tag{9{\rm d}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} ( 5 , 6 , 6.5 , 8 ) \cdot x_1 + ( 5.2 , 7 , 9 , 10.5 ) \cdot x_2 \leq ( 260 , 275 , 285 , 300 )\tag{9{\rm e}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} x_1 , x_2 \geq 0 \tag{9{\rm f}} \end{align*}\end{document}

where (0.7, 1, 1.5, 2), (4.5, 6, 8, 11), (5, 6, 6.5, 8), (5.2, 7, 9, 10.5), (−3, −2, −1, −0.5), and (260, 275, 285, 300) are trapezoidal fuzzy sets, respectively.

The fuzzy constraints (9d) and (9e) could be satisfied at a specific confidence level α, such that the general FMP model can be transformed to a crisp equivalent as follows (Liu and Iwamura, 1998): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} Maximize \ f = 55x_1 - 80 x_2 \tag{10{\rm a}} \end{align*}\end{document}

Subject to: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} 5 x_1 + x_2 \leq 150\tag{10{\rm b}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} x_1 + 3.5x_2 \leq 90\tag{10{\rm c}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} ( 1 - \alpha ) \cdot ( 0.7 x_1 - 11 x_2 + 0.5 ) + \alpha\cdot ( x_1 - 8 x_2 + 1 ) \leq 0\tag{10{\rm d}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} ( 1 - \alpha ) \cdot ( 5 x_1 + 5.2 x_2 - 300 ) + \alpha\cdot ( 6x_1 + 7 x_2 - 285 ) \leq 0 \quad\tag{10{\rm e}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} x_1 , x_2 \geq 0\tag{10{\rm f}} \end{align*}\end{document}

Table 1 shows the obtained solutions at different confidence levels (i.e., α-cut level) for the numerical example. Obviously, as α-cut level increases (from 0.1 to 0.9), x₁ would decrease and x₂ would increase. This will lead to the decrease of the objective function value. This is because, as the confidence level increases, the fuzzy chance constraints would become stricter based on the TFCCP algorithm. It also appears that the high objective-function values can only be obtained in situations when the fuzzy constraints are violated. This also reveals that TFCCP is advantageous in helping evaluate the trade-off between the system costs and reliabilities.

Overview of study case

In this study, an MSW management system in the City of Foshan, China, will be used for demonstrating the applicability of the proposed method. This case has been thoroughly discussed by Su et al. (2008, 2009). The study considers three planning periods with each one containing 5 years. The generated wastes from multiple districts should first be collected and transported to their respective transfer stations and then allocated to various waste treatment/disposal facilities, such as the landfill, the incinerators, and the composting plant (as shown in Fig. 1). Over the past decades, the local managers were facing difficulties of managing waste flows in a cost-efficient manner in light of the impacts of uncertainties, high cost of waste treatment, limited space for landfills, and limited governmental incentives for waste diversion. Therefore, how to build a rational waste management model for tackling these complexities was highly interested by the local managers.

OPEN IN VIEWER

FIG. 1.

Municipal solid waste management system.

Table 2 shows the related parameters based on the paper by Su et al. (2008). Among various treatment facilities, the landfill is used to meet the demand of waste disposal or to receive residues from the incinerators and composting plant; it has a stable cumulative capacity and will be treated as a fixed value. The incinerator and the composting plant have revenue returns but are limited by their daily operating capacities. Because of the significant variations in the capacities and treated amounts for the incinerators and composting plants, the related parameters will be expressed as trapezoidal fuzzy sets. For example, the existing capacities of the Shishan and Xingtan incinerators and the Gaoming composting plant are (850, 1,000, 1,230, 1,400), (820, 980, 1,160, 1,350), and (370, 400, 450, 490) tons (t)/day (d), respectively. As the waste generation rates may keep increasing, the capacity-expansion plans for waste treatment facilities would have to be considered in order to meet the overall waste-disposal demands in the upcoming time periods. The related parameters associated with the expansion options, such as expansion costs and expand capacities, are listed in Table 3. Further, the design safety coefficients for the waste treatment facilities (i.e., the incinerator and composting plant) and the transfer stations need to be incorporated into the management model in order to guarantee the satisfaction of the waste disposal and storage demands. They could be identified through expert consultation and round-table discussions among various stakeholders. As the safety coefficients are subjected to human judgment, they will be described as fuzzy. The design safety coefficients for the incinerators, composting plant, and transfer stations are determined to be (1.35, 1.45, 1.65, 1.8), (1.25, 1.42, 1.5, 1.7), and (1.15, 1.3, 1.4, 1.52), respectively (Su et al., 2008).

To prolong the lifespan of the current landfills and increase the reuse and recycling amounts of wastes, the government has set some specific diversion goals to ensure a certain proportion of the wastes be diverted to incinerators and composting plants. Table 4 shows the three types of diversion goals, representing the three policy scenarios for the future MSW management. The unit transportation costs and the unit operational costs of the stations and facilities, the unit revenues from the incinerators and composting plants, are relatively stable and are treated as fixed values (as shown in Table 5). Other related parameters, such as the distances between the stations and facilities, are referred from the work by Su et al. (2008).

Formulation of a TFCMP model

Based on the general TFCCP and MIP models, a modified TFCMP model for the City of Foshan can be formulated as follows (Su et al., 2008, 2009):

Objective function: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} Min \ f & = \mathop \sum_{j = 1}^{J} \mathop\sum_{l = 1}^{L} \mathop\sum_{k = 1}^{K}LE_{k}XL_{jlk} ( DL_{jl}TR_{k} + TS_{k} + OL_{lk} ) \\ & + \mathop\sum_{j = 1}^{J} \mathop\sum_{i = 1}^{I} \mathop\sum_{k = 1}^{K}LE_{k}XI_{jik} ( DI_{ji}TR_{k} + TS_{k} + OI_{ik} ) \\ & + \mathop\sum_{j = 1}^{J} \mathop\sum_{c = 1}^{C} \mathop\sum_{k = 1}^{K}LE_{k}XC_{jck} ( DC_{jc}TR_{k} + TS_{k} + OC_{ck} ) \\ & + \mathop\sum_{i = 1}^{I} \mathop\sum_{l = 1}^{L} \mathop\sum_{k}^{K}LE_{k}YI_{ilk} ( DRI_{il}TR_{k} + OL_{lk} ) \\ & + \mathop\sum_{c = 1}^{C} \mathop\sum_{l = 1}^{L} \mathop\sum_{k}^{K}LE_{k}YC_{clk} ( DRC_{cl}TR_{k} + OL_{lk} ) \\ & + \mathop\sum_{l = 1}^{L} \mathop\sum_{u = 1}^{U} \mathop\sum_{k = 1}^{K}EL_{luk}BU_{luk} \\ & + \mathop\sum_{i = 1}^{I} \mathop\sum_{v = 1}^V \mathop\sum_{k = 1}^{K}EI_{ivk} BV_{ivk} \\ & + \mathop\sum_{c = 1}^{C} \mathop\sum_{w = 1}^{W} \mathop\sum_{k = 1}^{K}EC_{cwk}BW_{cwk} \\ & - \mathop\sum_{j = 1}^{J} \mathop\sum_{i = 1}^{I} \mathop\sum_{k = 1}^{K}LE_{k}XI_{jik}RI_{ik} \\ & - \mathop\sum_{j = 1}^{J} \mathop\sum_{c = 1}^C \mathop\sum_{k = 1}^{K}LE_{k}XC_{jck}RC_{ck} \tag{11{\rm a}} \end{align*}\end{document}

Subject to:

(1) Constraints of treatment/disposal capacity: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mathop\sum_{k = 1}^K LE_k & \left( \mathop\sum_{j = 1}^J \mathop\sum_{l = 1}^L XL_{jlk} + \mathop\sum_{i = 1}^I \mathop\sum_{l = 1}^L YI_{ilk} + \mathop\sum_{c = 1}^C \mathop\sum_{l = 1}^L YC_{clk} \right) \leq \mathop\sum_{l = 1}^L CAL_L \\ & + \mathop\sum_{l = 1}^L \mathop\sum_{u = 1}^U \mathop\sum_{k = 1}^K \Delta CL_{luk} BU_{luk} , \qquad\qquad\qquad\qquad \tag{11{\rm b}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \tilde{\alpha}_I \mathop\sum_{j = 1}^J & \mathop\sum_{i = 1}^I XI_{jik} \leq \mathop\sum_{i = 1}^I C \tilde{A}I_i \\ & + \mathop\sum_{i = 1}^I \mathop\sum_{v = 1}^V \mathop\sum_{k^{\prime} = 1}^k \Delta CI_{ivk^{\prime}}BV_{ivk^{\prime}} , \forall k , \qquad\qquad\quad\quad\quad\,\; \tag{11{\rm c}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \tilde{\alpha}_C & \mathop\sum_{j = 1}^J \mathop\sum_{c = 1}^C XC_{jck} \leq \mathop\sum_{c = 1}^C C \tilde{A}C_c \\ & + \mathop\sum_{c = 1}^C \mathop\sum_{w = 1}^W \mathop\sum_{k^{\prime} = 1}^k \Delta CC_{cwk^{\prime}}BW_{cwk^{\prime}} , \forall k , \qquad\qquad\qquad\qquad \tag{11{\rm d}} \end{align*}\end{document}

(2) Capacity constraints for transfer stations: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \tilde{\alpha}_J \left( \mathop\sum_{l = 1}^L XL_{jlk} + \mathop\sum_{i = 1}^I XI_{jik} + \mathop\sum_{c = 1}^C XC_{jck} \right) \geq G \tilde{W}_{jk} , \forall j , k ,\tag{11{\rm e}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} & \mathop\sum_j^J \mathop\sum_{l = 1}^L XL_{jlk} + \mathop\sum_j^J \mathop\sum_{i = 1}^I XI_{jik} + \mathop\sum_j^J \mathop\sum_{c = 1}^C XC_{jck} \\ & + \mathop\sum_{i = 1}^I \mathop\sum_{l = 1}^L YI_{ilk} + \mathop\sum_{c = 1}^C \mathop\sum_{l = 1}^L YC_{clk} \leq MTC_k , \forall k , \quad\quad\quad\quad\;\;\; \tag{11{\rm f}} \end{align*}\end{document}

(3) Mass balance equations: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mathop\sum_{l = 1}^L YI_{ilk} = \mathop\sum_{j = 1}^J XI_{jik} FI_k , \forall i , k ,\tag{11{\rm g}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mathop\sum_{l = 1}^L YC_{clk} = \mathop\sum_{j = 1}^J XC_{jck} FC_k , \forall c , k , \tag{11{\rm h}} \end{align*}\end{document}

(4) Regulated diversion rates of waste to various treatment/disposal facilities: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mathop\sum_{j = 1}^J \mathop\sum_{i = 1}^I XI_{ijk} \geq \mathop\sum_{j = 1}^J G \tilde{W}_{jk} GI_k , \forall k ,\tag{11{\rm i}} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mathop\sum_{j = 1}^J \mathop\sum_{c = 1}^C XC_{jck} \geq \mathop\sum_{j = 1}^J G \tilde{W}_{jk} GC_k , \forall k ,\tag{11{\rm j}} \end{align*}\end{document}

(5) Constraints of capacity-expansion options: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} 0 \leq & \mathop\sum_{u = 1}^U BU_{luk} \leq 1 , \quad 0 \leq \mathop\sum_{v = 1}^V BV_{ivk} \leq 1 , \quad 0 \leq \mathop\sum_{w = 1}^W BW_{cwk} \leq 1 , \\ & \forall l , u , i , v , c , w , k ,\tag{11{\rm k}} \end{align*}\end{document}

BU_luk, BV_lvk, and BW_cwk are integers;

(6) Nonnegativity constraints: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} XL_{jlk} \geq 0 , \quad XI_{jik} \geq 0 , \quad XC_{jck} \geq 0 , \quad \forall i , j , c , l , k;\tag{11{\rm l}} \end{align*}\end{document}

where f is net system cost ($); k (k=1, 2,…, K) is index of time periods, where K is number of time periods; k′ is an intermediate index satisfying 1≤k′≤k; LE_k is length of time period k (days); l, i, and c (l=1, 2,…, L; i=1, 2,…, I; c=1, 2,…, C) are indexes of specific landfills, incinerators, and composting plants, respectively; L, I, and C are numbers of landfills, incinerators, and composting plants; j (j=1, 2,…, J) is index of transfer stations, where J is number of transfer stations; u (u=1, 2,…, U) is index of expansion options for landfills, where U is number of options; v (v=1, 2,…, V) is index of expansion options for incinerators, where V is number of options; w (w=1, 2,…, W) is index of expansion options for composting plants, where W is number of options; XL_jlk, XI_jik, and XC_jck are decision variables representing waste flows from transfer station j to landfill l, incinerator i, and composting plant c during period k (t/d), respectively; YI_ilk and YC_clk are decision variables representing residue flows from incinerator i to landfill l and from composting plant c to landfill l during period k (t/d), respectively; BU_luk, BV_ivk, BW_cwk are binary variables (i.e., expressed as 1 or 0, representing yes or no answers) for landfill l with option u, incinerator i with option v, and composting plant c with option w during period k, respectively; CAL_l, CÃI_i, and CÃC_c are existing capacities of landfill l, incinerator i, and composting plant c (t/d), respectively; ΔCL_luk, ΔCI_ivk, and ΔCC_cwk are capacity-expansion amounts for landfill l with option u, incinerator i with option v, and composting plant c with option w during period k (t/d), respectively; DL_jl, DI_ji, DC_jc are transportation distances from transfer station j to landfill l, from transfer station j to incinerator i, and from transfer station j to composting plant c (km), respectively; DRI_il and DRC_cl are transportation distances from incinerator i to landfill l and from composting plant c to landfill l (km), respectively; EL_luk, EI_ivk, and EC_cwk are unit expansion costs for landfill l with option u, incinerator i with option v, and composting plant c with option w during period k ($/t), respectively; FI_k and FC_k are residue rates from incinerator to landfill and from composting plant to landfill during period k (%), respectively; GI_k, and GC_k are maximum allowable diversion rates of waste flows to incinerators and composting plants during period k (%), respectively; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$G{\tilde W}_{jk}$$\end{document} is waste amount generated in the area covered by transfer station j during period k (t/d); MTC_k is maximum transportation capacity during period k (t/d); OL_lk, OI_ik, and OC_ck are unit operational costs for landfill l, incinerator i, and composting plant c during period k ($/t), respectively; RI_ik and RC_ck are unit revenues for incinerator i and composting plant c during period k ($/t), respectively; TR_k is unit transportation cost in period k ($/t·km); TS_k is unit operational cost for transfer stations during period k ($/t); \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde \alpha}_I$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde \alpha}_C$$\end{document} are design safety factors assuring that the waste flows can be completely treated during period k in incinerators and composting plants, respectively; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde \alpha}_J$$\end{document} is the design safety factor for transfer stations ensuring sufficient storage capacities.

The objective function (11a) is to achieve the minimum overall system costs (i.e., the difference between the total costs and revenue) through effectively allocating the waste flows from transfer stations to the treatment/disposal facilities. In detail, the system costs include the transportation and operational costs for the transfer stations, operational costs for the treatment facilities, and revenue for the incinerators and composting plants. The constraints (11b) to (11d) are the treatment capacity constraints for waste treatment/disposal facilities. Different from landfill, the capacity expansions for the incinerator and composting are cumulative processes; an intermediate index k′, which ranges from 1 to k, could be used to reflect such a fact. For example, when k is 1 (i.e., the first stage), k′ should be 1 as well; when k becomes 2 (i.e., the second stage), k′ will range from 1 to 2, showing a cumulative effect of the first and second stages. Moreover, the peak flows resulting from the random arrival and service times of waste-delivery vehicles may also raise a risk of insufficiency in the incinerator and composting facilities. The introduction of the design safety coefficient ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde \alpha}_I$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde \alpha}_C$$\end{document} ) would ensure that the waste flows to incinerators and composting plants can be completely treated. The constraints (11e) and (11f) are the storage and transportation capacity constraints for the transfer stations, respectively, where design safety coefficient \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\tilde \alpha}_J$$\end{document} can ensure the transfer stations have sufficient storage capacities. The constraints (11g) and (11h) show the relationships between the waste amounts and the residues from incinerators and composting plants, respectively. The constraints (11i) and (11j) are the diversion requirements for the waste flows to the incinerators and composting plants, respectively. The local government would require that a proportion of the waste flows must be allotted to the incinerators and composting plants in order to prolong the service lives of landfills and encourage the recycling/reuse of useful organic wastes. The constraints (11k) and (11l) are technical constraints.

Steps of solving a TFCMP model

The proposed TFCMP model is capable of handling uncertainties in fuzzy formats and allowing the violation of constraints in the waste management system within an acceptable limit (i.e., confidence level); the binary integer variables are also incorporated into the models to reflect capacity expansion issues. The obtained solutions (including both continuous and binary variables) at different confidence levels can help decision makers analyze tradeoffs between system economy and reliability. Figure 2 shows the general framework of TFCMP. The detailed procedures of applying a TFCMP model are summarized as follows:

OPEN IN VIEWER

FIG. 2.

Framework of a trapezoidal-shaped fuzzy chance-constrained mixed-integer programming (TFCMP) model.

Step 1: Identify all uncertain variables and acquire the related fuzzy distribution and binary parameter information in the waste management system;

Step 2: Formulate a TFCMP model;

Step 3: Convert the fuzzy chance constraints to their respective crisp equivalents;

Step 4: Generate the final solutions.