An Improved Multi-Objective Programming with Augmented ε-Constraint Method for Hazardous Waste Location-Routing Problems

. 2016 May 31;13(6):548. doi: 10.3390/ijerph13060548

Set:
N = (V, S):	Set of transportation network formed by nodes V and arcs S
C = (1, …, c):	Set of hazardous waste collection points, C∈V
P = (1, …, p):	Set of candidate locations of treatment facility, P∈V
R = (1, …, r):	Set of candidate locations of recycling facility, R∈V
U = (1, …, u):	Set of candidate locations of disposal facility, U∈V
TP = (1, …, tp):	Set of types of hazardous waste
TE = (1, …, te):	Set of treatment technologies of hazardous waste
Parameters:
$U c_{a, b}^{s 1}$ :	Unit transportation cost of hazardous waste from collection point a∈C to treatment facility b∈P, (a, b)∈S
$U c_{a, b}^{s 2}$ :	Unit transportation cost of hazardous waste from collection point a∈C to recycling facility b∈R, (a, b)∈S
$U c_{a, b}^{s 3}$ :	Unit transportation cost of waste components from treatment facility a∈P to recycling facility b∈R, (a, b)∈S
$U c_{a, b}^{s 4}$ :	Unit transportation cost of waste residue from treatment facility a∈P to disposal facility b∈U, (a, b)∈S
$U c_{a, b}^{s 4}$ :	Unit transportation cost of waste residue from recycling facility a∈R to disposal facility b∈U, (a, b)∈S
$F c_{a, t e}^{v 1}$ :	Fixed facility cost for opening a treatment facility at candidate point a∈P with technology te∈TE
$F c_{a}^{v 2}$ :	Fixed facility cost for opening a recycling facility at candidate point a∈R
$F c_{a}^{v 3}$ :	Fixed facility cost for opening a disposal facility at candidate point a∈U
$P c_{a, t e}^{v 1}$ :	Unit processing cost at treatment facility a∈P with technology te∈TE
$P c_{a}^{v 2}$ :	Unit processing cost at recycling facility a∈R
$P c_{a}^{v 3}$ :	Unit processing cost at disposal facility a∈U
$R S P_{a, t e}$ :	Risk level of processing technology te∈TE at treatment facility a∈P
$R S U_{a}$ :	Risk level of disposal facility a∈U
$R S W_{a, b, t p}$ :	Risk level of hazardous waste type tp∈TP transported from collection point a∈C to treatment facility b∈P
$R S R_{a, b}$ :	Risk level of post-treatment waste residue transported from treatment facility a∈P to disposal facility b∈U
$P E P_{a}^{v 1}$ :	Population exposed to treatment facility a∈P
$P E P_{a}^{v 3}$ :	Population exposed to disposal facility a∈U
$P E P_{a, b}^{s 1}$ :	Population exposed along the transportation route from collection point a∈C to treatment facility b∈P, (a, b)∈S
$P E P_{a, b}^{s 4}$ :	Population exposed along the transportation route from treatment facility a∈P to disposal facility b∈U, (a, b)∈S
$H W G_{a, t p}$ :	Hazardous waste type tp∈TP collected at collection point a∈C
$τ_{a, t e}^{v 1}$ :	Conversion rate of hazardous waste at treatment facility a∈P with technology te∈TE
$τ_{a}^{v 2}$ :	Conversion rate of hazardous waste at recycling facility a∈R
$C a p t_{a, t e}^{v 1}$ :	Capacity of treatment facility a∈P with technology te∈TE
$C a p t_{a}^{v 2}$ :	Capacity of recycling facility a∈R
$C a p t_{a}^{v 3}$ :	Capacity of disposal facility a∈U
$R e R_{a, t e}$ :	Required recycling rate of post-treatment residue at treatment facility a∈P with technology te∈TE
$F r_{a, t e}$ :	Recyclable fraction of post-treatment residue at treatment facility a∈P with technology te∈TE
$I F N$ :	An infinitive large number
$C o m p_{t p, t e}^{v 1}$ :	Binary compatibility factor determines if the type of waste tp∈TP is compatible with the treatment technology te∈TE
$C o m p_{t p, b}^{v 1}$ :	Binary compatibility factor determines if the type of waste tp∈TP is compatible with direct recycling at b∈R
$R S_{x}$ :	Compensation faction of risk level, $R S_{x} \in {R S P_{a, t e}, R S U_{a}, R S W_{a, b, t p}, R S R_{a, b}}$
$C o m p e n s a t i o n_{C o n s e q u e n c e}$ :	Compensation factor of consequence of accident
$C o m p e n s a t i o n_{P r o b a b i l i t y}$ :	Compensation factor of probability of accident
$P r o b a b i l i t y_{G e n e r a l}$ :	General probability of catastrophic accident
$P r o b a b i l i t y_{L o c a l i t y}$ :	Locality factor
$C o n s e q u e n c e_{t y p e}$ :	Compensation factor for type of hazardous waste
$C o n s e q u e n c e_{t e c h n o l o g y}$ :	Compensation factor for technology applied at treatment facility
Decision variables:
$Q a_{a, b, t p, t e}^{s 1}$ :	Amount of hazardous waste type tp∈TP from collection point a∈C to treatment facility b∈P with technology te∈TE, (a, b)∈S
$Q a_{a, b, t p}^{s 2}$ :	Amount of hazardous waste type tp∈TP from collection point a∈C to recycling facility b∈R, (a, b)∈S
$Q a_{a, b, t e}^{s 3}$ :	Amount of waste components from treatment facility a∈P with technology te∈TE to recycling facility b∈R, (a, b)∈S
$Q a_{a, b, t e}^{s 4}$ :	Amount of waste residue from treatment facility a∈P with technology te∈TE to disposal facility b∈U, (a, b)∈S
$Q a_{a, b}^{s 3}$ :	Amount of waste components from treatment facility a∈P to recycling facility b∈R, (a, b)∈S
$Q a_{a, b}^{s 4}$ :	Amount of waste residue from treatment facility a∈P to disposal facility b∈U, (a, b)∈S
$Q a_{a, b}^{s 5}$ :	Amount of waste residue from recycling facility a∈R to disposal facility b∈U, (a, b)∈S
$Q a_{a, t e}^{v 1} / Q a_{b, t e}^{v 1}$ :	Amount of hazardous waste received at treatment facility a∈P, b∈P with technology te∈TE
$Q a_{a}^{v 2} / Q a_{b}^{v 2}$ :	Amount of hazardous waste received at recycling facility a∈R, b∈R
$Q a_{a}^{v 3} / Q a_{b}^{v 3}$ :	Amount of hazardous waste received at disposal facility a∈U, b∈U
$D e_{a, t e}^{v 1} / D e_{b, t e}^{v 1}$ :	Binary decision variable determines if a treatment facility is opened at candidate point a∈P, b∈P with technology tp∈TP
$D e_{a}^{v 2} / D e_{b}^{v 2}$ :	Binary decision variable determines if a recycling facility is opened at candidate point a∈R, b∈R
$D e_{a}^{v 3} / D e_{b}^{v 3}$ :	Binary decision variable determines if a disposal facility is opened at candidate point a∈U, b∈U