3.1. Notations
In this section, we formulate the multi-period network design in regional hazardous waste management systems as a bi-objective optimization model. The notations for the sets, parameters and decision variables to be used in the model are defined and explained as follows.
Sets:
: set of existing recycling (treatment, disposal) centers.
: set of candidate new recycling (treatment, disposal) centers.
: set of existing and new recycling (treatment, disposal) centers indexed by r (t or d ).
L: set of treatment technologies indexed by l.
H: set of hazardous wastes indexed by h.
W: set of types of hazardous wastes indexed by w.
P: set of discretized periods indexed by p in the planning horizon.
: set of effective operation periods for waste facilities which are closed at the end of period or opened at the beginning of period . According to the problem assumptions, for an existing waste facility, if it is closed at the end of period p, it has been operating in and before period p, that is . For a new waste facility, if it is opened at the beginning of period p, it should keep in operation in and after period p, that is, .
: set of relevant periods for waste facilities indicating that the facilities are operated in period . If an existing waste facility is closed at the end of period p or another period after p, it is operated in period p. Thus, we have . If a new waste facility is opened at the beginning of period p or another period before p, it is operated in period p. It holds .
Parameters:
, and : cost of transporting one unit of hazardous waste from the generation node to recycling center , treatment center and disposal center in period , respectively.
and : cost of transporting one unit of waste residues from treatment center to recycling center and disposal center in period , respectively.
: cost of transporting one unit of waste residues from recycling center to disposal center in period .
, and : risk of transporting one unit of hazardous waste from the generation node to recycling center , treatment center and disposal center in period , respectively.
and : risk of transporting one unit of waste residues from treatment center to recycling center and disposal center in period , respectively.
: risk of transporting one unit of waste residues from recycling center to disposal center in period .
, , and : fixed opening/closing cost, fixed operating cost, unit recycling cost and unit recycling risk at recycling center in period , respectively.
, , and : fixed opening/closing cost, fixed operating cost, unit treatment cost and unit treatment risk at treatment center with technology in period , respectively.
, , and : fixed opening/closing cost, fixed operating cost, unit disposal cost and unit disposal risk at disposal center in period , respectively. The fixed closing cost of existing waste facilities in the last period of the planning horizon is set to 0 for the modeling purpose.
Note that there are many methods in the literature to measure the transportation risk of hazardous wastes and the location risk of waste facilities, please refer to Erkut et al. [
2] for discussion. Here, the transportation and location risk are both measured by the traditional risk method, which has been widely used in existing studies, such as Nema and Gupta [
15], Alumur and Kara [
16], Samanlioglu [
17] and Zhao et al. [
20]. The unit of the risk is people · ton.
: average recycling rate of hazardous wastes and waste residues at recycling center in period .
: percentage of residues in waste type after being treated by technology in period .
and : recyclable rate and disposable rate of residues in waste type after being treated by technology in period , respectively.
: 1 if waste type is comparable with technology , 0 otherwise.
: type of hazardous waste .
, , and : amount, recyclable rate, treatable rate and disposable rate of hazardous waste in period , respectively.
and : minimal recycling workload and maximum recycling capacity of recycling center in period , respectively.
and : minimal treatment workload and maximum treatment capacity of treatment center with technology in period , respectively.
and : minimal disposal workload and maximum disposal capacity of disposal center in period , respectively.
: maximum capacity of disposal center in its life cycle. The life cycle capacity of existing waste facilities should subtract its used capacity in the last planning horizon.
Decision variables:
, and : amount of recyclable, treatable and disposable portions in hazardous waste recycled at recycling center , treated at treatment center with technology , and disposed at disposal center in period , respectively.
and : amount of recyclable and disposable waste residues in treatment center recycled at recycling center and disposed at disposal center in period , respectively.
: amount of disposable waste residues in recycling center disposed at disposal center in period .
: amount of hazardous wastes and waste residues recycled at recycling center in period .
: amount of hazardous wastes in type treated at treatment center with technology in period .
: amount of all types of hazardous wastes treated at treatment center with technology in period .
: amount of waste residues disposed at disposal center in period .
: 0–1 variables, 1 if existing recycling center is closed at the end of period or new recycling center is opened at the beginning of period , 0 otherwise.
: 0–1 variables, 1 if existing treatment center using technology is closed at the end of period or new treatment center using technology is opened at the beginning of period , 0 otherwise.
: 0–1 variables, 1 if existing disposal center is closed at the end of period or new disposal center is opened at the beginning of period , 0 otherwise.
3.2. Model Formulation
Using the above notations, the multi-period network design problem in regional hazardous waste management systems (MPND-RHWMS) can be formulated as a bi-objective optimization model as follows:
(MPND-RHWMS)
Objective (
1) including twelve parts minimizes the total cost in the transportation and location procedures of the whole planning horizon. The first three parts are the transportation cost of hazardous wastes from generation nodes to recycling centers, treatment centers and disposal centers, respectively. The transportation cost of waste residues from treatment centers to recycling centers and disposal centers are given by the fourth and fifth parts, respectively. The sixth part provides the transportation cost of waste residues from recycling centers to disposal centers. The seventh to ninth parts calculate the fixed cost of locating and operating recycling centers, treatment centers and disposal centers, respectively. Based on our modeling strategy, the fixed cost of both existing and new waste facilities can be correctly counted using the same expression with set
. The last three parts determine the variable process cost at recycling centers, treatment centers and disposal centers, respectively.
Objective (
2) is to minimize the total risk in the regional hazardous waste management system. This objective can be understood analogously following Objective (
1). There are two notices. The one is that there is only variable process risk at recycling centers, treatment centers and disposal centers depending on the workload at these waste facilities. The another is that the waste residues disposed at a disposal center in period
p will be remained at the center, incurring risk at the center in the following periods of the planning horizon, see the last part of Objective (
2).
Constraints (
3) to (
5) are the flow conservation for the recyclable portion, treatable portion and disposable portion of hazardous wastes among recycling centers, treatment centers and disposal centers in each period, respectively. Constraints (
6) and (
7) specify the flow conservation for the recyclable portion and disposable portion of waste residues at treatment centers among recycling centers and disposal centers in each period, respectively. Constraints (
8) give the flow conservation for the disposable waste residues at recycling centers among disposal centers in each period. Constraints (
9) determine the workload in each period at recycling centers where the wastes are from both generation nodes and treatment centers. Constraints (
10) calculate the workload per waste type in each period at treatment centers, while Constraints (
11) count the total workload in each period at treatment centers. Constraints (
12) compute the workload in each period at disposal centers at which the wastes from generation nodes, treatment centers and recycling centers are all disposed.
Constraints (
13) guarantee that the workload in each period at recycling centers is not less than the minimal recycling workload and is not greater than the maximum recycling capacity. Constraints (
14) and (
15) are similar constraints for treatment centers and disposal centers, respectively. With the definition of set
, the minimal workload and maximum capacity constraints of both existing and new waste facilities can be modeled uniformly. Constraints (
16) ensure that the total workload in the planning horizon at disposal centers should be less than the life cycle capacity. Constraints (
17) and (
18) are the location consistence of recycling centers. Constraints (
17) assure that existing recycling centers are operated by at least one period and they cannot be reopened once they are closed. Constraints (
18) make sure that new recycling centers can be opened by at most once and they should be operated until the end of the planning horizon once they are opened. Constraints (
19) and (
20) are the location consistence of treatment centers, while Constraints (
21) and (
22) are those for disposal centers. Constraints (
23) and (
24) are the waste-technology comparability requirement with which hazardous wastes can only be treated by comparable technologies. Constraints (
25) to (
27) specify the integer and non-negative requirements of decision variables.