IFTCIP: An Integrated Optimization Model for Environmental Management under Uncertainty

Abstract

In this study, an interval-fuzzy two-stage chance-constrained integer programming (IFTCIP) method is developed for supporting environmental management under uncertainty. The IFTCIP improves upon the existing interval, fuzzy, and two-stage programming approaches by allowing uncertainties expressed as probability distributions, fuzzy sets, and discrete intervals to be directly incorporated within a general mixed integer linear programming framework. It has advantages in uncertainty reflection, policy investigation, risk assessment, and capacity-expansion analysis in comparison to the other optimization methods. Moreover, it can help examine the risk of violating system constraints and the associated consequences. The developed method is applied to the planning for facility expansion and waste-flow allocation within a municipal solid waste management system. Violations of capacity constraints are allowed under a range of significance levels, which reflects tradeoffs between the system cost and the constraint-violation risk. The results indicate that reasonable solutions for both binary and continuous variables have been generated under different risk levels. They are useful for generating desired decision alternatives with minimized system cost and constraint-violation risk under various environmental, economic, and system-reliability conditions. Generally, willingness to take a higher risk of constraint violation will guarantee a lower system cost; a strong desire to acquire a lower risk will run into a higher system cost.

Appendices

Appendix A

1.1 Solution Method

A two-step solution method is proposed for solving the developed IFTCIP model. Then, model 5 can be transformed into two deterministic submodels, which correspond to the lower and upper bounds of the objective function value, respectively. Interval solutions, with varying levels of constraint-violation risk, can then be obtained by solving the two submodels sequentially. When the objective is to be minimized, the submodel with λ ⁺ (corresponding to f ⁻) can be firstly formulated and solved (assume that B ^± > 0, and f ^± > 0) as:

$${\text{Max}}\quad \lambda ^ + $$

(8a)

subject to:

$${\sum\limits_{j = 1}^{k_{1} } {c^{ - }_{j} } }x^{ - }_{j} + {\sum\limits_{j = k_{1} + 1}^{n_{1} } {c^{ - }_{j} x^{ + }_{j} } } + {\sum\limits_{h = 1}^v {p_{h} } }{\left( {{\sum\limits_{j = 1}^{k_{2} } {d^{ - }_{j} } }y^{ - }_{{jh}} + {\sum\limits_{j = k_{2} + 1}^{n_{2} } {d^{ - }_{j} } }y^{ + }_{{jh}} } \right)} \leqslant f^{ + } - \lambda ^{ + } {\left( {f^{ + } - f^{ - } } \right)}$$

(8b)

$$\sum\limits_{j = 1}^{k_1 } {|a_{rj} |^ + {\text{Sign}}\left( {a_{rj}^ + } \right)x_j^ - } + \sum\limits_{j = k_1 + 1}^{n_1 } {|a_{rj} |^ - {\text{Sign}}\left( {a_{rj}^ - } \right)x_j^ + } \leqslant b_r^ + - \lambda ^ + \left( {b_r^ + - b_r^ - } \right),{\text{ }}\forall r$$

(8c)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{rj} } \right|^ + {\text{Sign}}\left( {a_{rj}^ + } \right)x_j^ - + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{rj} } \right|^ - {\text{Sign}}\left( {a_{rj}^ - } \right)x_j^ + + \sum\limits_{j = 1}^{k_2 } {\left| {a\prime _{rj} } \right|^ + {\text{Sign}}\left( {a\prime _{rj}^ + } \right)y_{jh}^ - + \sum\limits_{j = k_2 + 1}^{n_2 } {\left| {a\prime _{rj} } \right|^ - {\text{Sign}}\left( {a\prime _{rj}^ - } \right)y_{jh}^ + \geqslant \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w} _{rh}^ - + \lambda ^ + \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w} _{rh}^ + - \bar w_{rh}^ - } \right),\;\forall r,h} } } } $$

(8d)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{ij} } \right|^ + } \;{\text{Sign}}\left( {a_{ij}^ + } \right)x_j^ - + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{ij} } \right|^ - {\text{Sign}}\;\left( {a_{ij}^ - } \right)x_j^ + + \sum\limits_{j = 1}^{k_2 } {\left| {a\prime _{ij} } \right|^ + {\text{Sign}}\,\left( {a\prime _{ij}^ + } \right)y_{jh}^ - } + \sum\limits_{j = k_2 + 1}^{n_2 } {\left| {a\prime _{ij} } \right|^ - {\text{Sign}}\left( {a\prime _{_{ij} }^ - } \right)y_{jh}^ + \leqslant b_i^ + - \lambda ^ + \left( {b_i^ + - b_i^ - } \right),\forall i,\,h;\,i \ne r} } $$

(8e)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{sj} } \right|^ + {\text{Sign}}\left( {a_{sj}^ + } \right)x_j^ - } + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{sj} } \right|^ - {\text{Sign}}\left( {a_{sj}^ - } \right)x_j^ + } + \sum\limits_{j = 1}^{k_2 } {\left| {a\prime _{sj} } \right|^ + {\text{Sign}}\left( {a\prime _{sj}^ + } \right)y_{jh}^ - } + \sum\limits_{j = k_2 + 1}^{n_2 } {\left| {a\prime _{sj} } \right|^ - {\text{Sign}}\left( {a\prime _{sj}^ - } \right)y_{jh}^ + } \leqslant \left( {b_s^ + } \right)^{q_s } - \lambda ^ + \left[ {\left( {b_s^ + } \right)^{q_s } - \left( {b_s^ - } \right)^{q_s } } \right],{\text{ }}\forall s,h;{\text{ }}s \ne r;{\text{ }}s \ne i$$

(8f)

$$x_j^ - \geqslant 0,{\text{ }}j = 1,\;2, \cdots ,\;k_1 $$

(8g)

$$x_j^ + \geqslant 0,{\text{ }}j = k_1 + 1,\;k_1 + 2, \cdots ,\;n_1 $$

(8h)

$$y_{jh}^ - \geqslant 0,{\text{ }}\forall h;{\text{ }}j = 1,\;2, \cdots ,\;k_2 $$

(8i)

$$y_{jh}^ + \geqslant 0,{\text{ }}\forall h;{\text{ }}j = k_2 + 1,\;k_2 + 2, \cdots ,\;n_2 $$

(8j)

$$0 \leqslant \lambda ^ + \leqslant 1$$

(8k)

where $\underline w _{\,rh}^ -$ is the lower boundary of lower interval number; $\overline w _{rh}^{\, - }$ is the upper boundary of lower interval number; $\underline w _{\,rh}^ +$ is the lower boundary of upper interval number; $\overline w _{rh}^{\, + }$ is the upper boundary of upper interval number. Assume that there is no intersection between the two bounds of interval numbers (i.e.,$\left[ {\underline w _{\,rh}^ + ,{\text{ }}\overline w _{rh}^{\, + } } \right]$ and $\left[ {\underline w _{rh}^ - ,{\text{ }}\overline w _{rh}^{\, - } } \right]$ ).

In submodels 8a, 8b, 8c, 8d, 8e, 8f, 8g, 8h, 8i, 8j, and 8k, $x_j^ - \left( {j{\text{ }} = {\text{ }}1,{\text{ }}2, \ldots ,{\text{ }}k_1 } \right)$ are interval variables with positive coefficients in the objective function; $x_j^ + \left( {j = k_1 + 1,{\text{ }}k_1 {\text{ }} + {\text{ }}2, \ldots ,{\text{ }}n_1 } \right)$ are interval variables with negative coefficients; $y_{jh}^ -$ (h = 1, 2,…, v and j = 1, 2,…, k ₂) are random variables with positive coefficients in the objective function;$\,y_{jh}^ +$ (h = 1, 2,..., v and j = k ₂ + 1, k ₂ + 2, ...,n ₂) are random variables with negative coefficients. Solutions of $x_{j\,{\text{opt}}}^ - \left( {j = 1,{\text{ }}2, \ldots ,{\text{ }}k_1 } \right)$, $x_{j\,{\text{opt}}}^ + \left( {j = {\text{ }}k_1 {\text{ }} + 1,{\text{ }}k_1 + 2, \ldots ,n_1 } \right)$,$y_{jh\,{\text{opt}}}^ - \left( {j = 1,{\text{ }}2, \ldots ,{\text{ }}k_2 } \right)$, $y_{jh\,{\text{opt}}}^ + \left( {j{\text{ }} = {\text{ }}k_2 + {\text{ }}1,{\text{ }}k_2 + 2, \ldots ,{\text{ }}n_2 } \right)$, and $\lambda _{{\text{opt}}}^ +$ can be obtained through solving submodels 8a, 8b, 8c, 8d, 8e, 8f, 8g, 8h, 8i, 8j, and 8k. Based on the above solutions, the second submodel for λ ⁻ (corresponding to f ⁺) can be formulated as follows:

$${\text{Max}}\quad \lambda ^ - $$

(9a)

subject to:

$$\sum\limits_{j = 1}^{k_1 } {c_j^ + x_j^ + + \sum\limits_{j = k_1 + 1}^{n_1 } {c_j^ + x_j^ - + \sum\limits_{h = 1}^v {p_h \left( {\sum\limits_{j = 1}^{k_2 } {d_j^ + y_{jh}^ + + \sum\limits_{j = k_2 + 1}^{n_2 } {d_j^ + y_{jh}^ - } } } \right) \leqslant f^ + - \lambda ^ - \left( {f^ + - f^ - } \right)} } } $$

(9b)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{rj} } \right|^ - {\text{Sign}}\left( {a_{rj}^ - } \right)x_j^ + } + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{rj} } \right|^ + {\text{Sign}}\left( {a_{rj}^ + } \right)x_j^ - } \leqslant b_r^ + - \lambda ^ - \left( {b_r^ + - b_r^ - } \right),{\text{ }}\forall r$$

(9c)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{rj} } \right|^ - {\text{Sign}}\left( {a_{rj}^ - } \right)x_j^ + } + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{rj} } \right|^ + {\text{Sign}}\left( {a_{rj}^ + } \right)x_j^ - } + \sum\limits_{j = 1}^{k_2 } {\left| {a\prime _{rj} } \right|^ - {\text{Sign}}\left( {a\prime _{rj}^ - } \right)y_{jh}^ + } + \sum\limits_{j = k_2 + 1}^{n_2 } {\left| {a\prime _{rj} } \right|^ + {\text{Sign}}\left( {a\prime _{rj}^ + } \right)y_{jh}^ - } \geqslant \bar w_{rh}^{\, - } + \lambda ^ - (\bar w_{rh}^{\, + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w} _{\,rh}^ - ),{\text{ }}\forall r,h$$

(9d)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{ij} } \right|^ - {\text{Sign}}\left( {a_{ij}^ - } \right)x_j^ + } + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{ij} } \right|^ + {\text{Sign}}\left( {a_{ij}^ + } \right)x_j^ - } + \sum\limits_{j = 1}^{k_2 } {\left| {a\prime _{ij} } \right|^ - {\text{Sign}}\left( {a\prime _{ij}^ - } \right)y_{jh}^ + } + \sum\limits_{j = k_2 + 1}^{n_2 } {\left| {a\prime _{ij} } \right|^ + {\text{Sign}}\left( {a\prime _{ij}^ + } \right)y_{jh}^ - } \leqslant b_i^ + - \lambda ^ - \left( {b_i^ + - b_i^ - } \right),{\text{ }}\forall i,h;{\text{ }}i \ne r$$

(9e)

$$\sum\limits_{j = 1}^{k_1 } {\left| {a_{sj} } \right|^ - {\text{Sign}}\left( {a_{sj}^ - } \right)x_j^ + + \sum\limits_{j = k_1 + 1}^{n_1 } {\left| {a_{sj} } \right|^ + {\text{Sign}}\left( {a_{sj}^ + } \right)x_j^ - + \sum\limits_{j = 1}^{k_2 } {\left| {a\prime _{sj} } \right|^ - {\text{Sign}}\left( {a\prime _{sj}^ - } \right)} \,y_{jh}^ + + \sum\limits_{j = k_2 + 1}^{n_2 } {\left| {a\prime _{sj} } \right|^ + {\text{Sign}}\left( {a\prime _{sj}^ + } \right)y_{jh}^ - \leqslant \left( {b_s^ + } \right)^{q_s } - \lambda ^ - \left[ {\left( {b_s^ + } \right)^{q_s } - \left( {b_s^ - } \right)^{q_s } } \right],\;\forall } s,\,h;\;s \ne r;} \;s \ne i} $$

(9f)

$$x_j^ + \geqslant x_{j\,{\text{opt}}}^ - ,{\text{ }}j = 1,\;2, \cdots ,\;k_1 $$

(9g)

$$0 \leqslant x_j^ - \leqslant x_{j\,{\text{opt}}}^ + ,{\text{ }}j = k_1 + 1,\;k_1 + 2, \cdots ,\;n_1 $$

(9h)

$$y_{jh}^ + \geqslant y_{jh\,{\text{opt}}}^ - ,{\text{ }}\forall h;{\text{ }}j = 1,\;2, \cdots ,\;k_2 $$

(9i)

$$0 \leqslant y_{jh}^ - \leqslant y_{jh\,{\text{opt}}}^ + ,{\text{ }}\forall h;{\text{ }}j = k_2 + 1,\;k_2 + 2, \cdots ,\;n_2 $$

(9j)

$$0 \leqslant \lambda ^ - \leqslant 1$$

(9k)

Solutions of $x_{j\,{\text{opt}}}^ + \left( {j = 1,{\text{ }}2, \ldots ,{\text{ }}k_1 } \right)$, $x_{j\,{\text{opt}}}^ - (j = k_1 + 1,{\text{ }}k_1 + 2, \ldots ,{\text{ }}n_1 )$, $y_{jh\,{\text{opt}}}^ + (j = 1,{\text{ }}2, \ldots ,{\text{ }}k_2 )$, $y_{jh\,{\text{opt}}}^ - \left( {j = k_2 + 1,{\text{ }}k_2 {\text{ }} + {\text{ }}2, \ldots ,{\text{ }}n_2 } \right)$, and $\lambda _{{\text{opt}}}^ -$can be obtained through submodels 9a, 9b, 9c, 9d, 9e, 9f, 9g, 9h, 9i, 9j, and 9k. The general solution for the IFTCIP model can thus be expressed as follows:

$$x_{j\,opt}^ \pm = \left[ {x_{j\,opt}^ - ,{\text{ }}x_{j\,opt}^ + } \right],{\text{ }}\forall j$$

(10)

$$y_{jh\,opt}^ \pm = \left[ {y_{jh\,opt}^ - ,{\text{ }}y_{jh\,opt}^ + } \right],{\text{ }}\forall j,h$$

(11)

$$\lambda _{opt}^ \pm = \left[ {\lambda _{opt}^ - ,{\text{ }}\lambda _{opt}^ + } \right]$$

(12)

Then, the expected objective function value can be calculated as:

$$f_{{\text{opt}}}^ - = \sum\limits_{j = 1}^{k_1 } {c_j^ - } x_{j\,{\text{opt}}}^ - + \sum\limits_{j = k_1 + 1}^{n_1 } {c_j^ - x_{j\,{\text{opt}}}^ + } + \sum\limits_{h = 1}^v {p_h } \left( {\sum\limits_{j = 1}^{k_2 } {d_j^ - } y_{jh\,{\text{opt}}}^ - + \sum\limits_{j = k_2 + 1}^{n_2 } {d_j^ - } y_{jh\,{\text{opt}}}^ + } \right)$$

(13)

$$f_{{\text{opt}}}^ + = \sum\limits_{j = 1}^{k_1 } {c_j^ + } x_{j\,{\text{opt}}}^ + + \sum\limits_{j = k_1 + 1}^{n_1 } {c_j^ + x_{j\,{\text{opt}}}^ - } + \sum\limits_{h = 1}^v {p_h } \left( {\sum\limits_{j = 1}^{k_2 } {d_j^ + } y_{jh\,{\text{opt}}}^ + + \sum\limits_{j = k_2 + 1}^{n_2 } {d_j^ + } y_{jh\,{\text{opt}}}^ - } \right)$$

(14)

Thus, we have $f_{{\text{opt}}}^ \pm = \left[ {f_{{\text{opt}}}^ - ,{\text{ }}f_{{\text{opt}}}^ + } \right]$.

Appendix B

1.1 List of symbols

i	type of waste management facility, and I = 2; where i = 1 for landfill, and i = 2 for incinerator
j	name of city, j = 1, 2
k	planning period, k = 1, 2, 3
L k	length of period k (day), and each period with 5 years
h	level of waste generation rate in city j, h = 1, 2,…, H
m	name of expansion option for waste management facilities, m = 1, 2,…, M
${\text{DG}}_{ik}^ \pm$	regulated diversion rate of waste flow to facility i in period k (%), i = 2
${\text{DP}}_{ik}^ \pm$	operating cost of facility i for excess waste flow during period k (the second-stage cost parameter) ($/ton), where ${\text{DP}}_{ik}^ \pm \geqslant {\text{OP}}_{ik}^ \pm$
${\text{DR}}_{ijk}^ \pm$	transportation cost for excess waste flow from city j to facility i during period k (the second-stage cost parameter) ($/ton), where ${\text{DR}}_{ijk}^ \pm \geqslant {\text{TR}}_{ijk}^ \pm$
${\text{DT}}_{ik}^ \pm$	transportation cost of excess waste residue from waste disposal facility i to the landfill during period k (the second-stage cost parameter) ($/ton), where ${\text{DT}}_{ik}^ \pm \geqslant {\text{FT}}_{ik}^ \pm$ and i = 2
${\text{FE}}_i^ \pm$	residue flow rate from facility i to the landfill (% of incoming mass to facility i), i = 2
${\text{FLC}}_k^ \pm$	capital cost of landfill expansion in period k ($/ton)
${\text{FT}}_{ik}^ \pm$	transportation cost for allowable residue flow from facility i to the landfill during period k ($/ton), i = 2
${\text{FTC}}_{imk}^ \pm$	capital cost of expanding facility i by option m in period k ($/ton), i = 2
${\text{L}}\underline {\text{C}} ^{q_1 }$	lower bound of existing landfill capacity corresponding to significance level (q 1) (ton)
${\text{L}}\overline {\text{C}} ^{q_1 }$	upper-bound existing landfill capacity corresponding to significance level (q 1) (ton)
$\Delta {\text{LC}}_{mk}^ \pm$	level of capacity expansion for the landfill with option m in period k (ton)
$M_{ijkh}^ \pm$	amount by which the allowable waste flow level (X ijk ) is exceeded when the waste-generation rate is w jkh with probability p jh (ton/day)
${\text{OP}}_{ik}^ \pm$	operating cost of facility i for allowable waste flow during period k ($/ton)
$p_{jh}$	probability of waste-generation rate (w jkh ) in city j with generation level h (%)
qi	probability of violating the existing capacity of facility i (%)
${\text{RE}}_{ik}^ \pm$	revenue from allowable waste flow treated by facility i during period k (the first-stage cost parameter) ($/ton), i = 2
${\text{RM}}_{ik}^ \pm$	Revenue from excess flow treated by facility i during period k (the second-stage cost parameter) ($/ton), i = 2
$T_{ijkh}^ \pm$	amount of increased allowance from city j to facility i under waste-generation rate h in period k (the second-stage decision variable) (ton/day);
${\text{T}}\underline {\text{C}} _i^{q_i }$	lower-bound existing capacity of facility i under significance level (q i ) (ton/day), i = 2
${\text{T}}\overline {\text{C}} _i^{q_i }$	upper-bound existing capacity of facility i under significance level (q i ) (ton/day), i = 2
$\Delta {\text{TC}}_{imk}^ \pm$	level of expansion option m for facility i at the start of period k (ton/day), and i = 2
${\text{TR}}_{ijk}^ \pm$	transportation cost for allowable waste flow from city j to facility i during period k (the first-stage cost parameter) ($/ton)
η1	quota of incremental value for allowable flow to the landfill (ton/day)
$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{W} _{jkh}^ -$	lower-bound waste generation rate in city j with probability level h in period k (ton/day)
$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{W} _{jkh}^ +$	upper-bound waste generation rate in city j with probability level h in period k (ton/day)
$X_{ijk}^ \pm$	allowable waste flow from city j to facility i in period k (the first-stage variable) (ton/day)
$Y_{mk}^ \pm$	binary decision variable for landfill expansion with option m at the start of period k
$Z_{imk}^ \pm$	binary decision variable for facility i with expansion option m at the start of period k, i = 2