Development of a New Municipal Solid Waste Management System: Multi-Objective Programming for a Merged Metropolis

Multiobjective modeling of MSW system

Most waste disposal systems perform three basic operations: collection, intermediate processing, and final disposal. In this study, a multi-objective model was developed to illustrate the trade-off between cost efficiency and equity in locating waste disposal facilities for a waste disposal system. Here, the equity objective was to minimize the total metric ton-kilometers of waste being transported. To achieve such an objective, a waste disposal facility would need to be located near the areas where waste is generated.

Although transportation costs are minimized in a highly decentralized configuration of disposal facilities, capital investment and unit operation costs are higher because of the increased number of disposal sites that do not achieve economics of scale. For this reason, minimizing waste transportation (metric ton-kilometers) does not necessarily mean that the total cost of the system has been minimized. Thus, a multi-objective optimization is warranted.

Overall, our problems can be modeled by minimizing the total cost under some constraints, which can be described by the following 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*} {\rm Minimize\ Z} = \sum_{i = 1}^m \sum_{t = 1}^r \lambda_t G_{it} (P_{it}) + \sum_{t = 1}^r \sum_{j = 1}^n \sum_{i = 1}^m \lambda_t CL_{ijt} \ X_{ijt} \\ + \sum_{t = 1}^r \sum_{h = 1}^{m - a} \sum_{g = 1}^a \lambda_t \ C_{ght} \ T_{ght} + \sum_{t = 1}^r \sum_{i = 1}^m \lambda_t F_{it} (Y_{it} ) \tag{1} \end{align*} \end{document}

which is subject to the following 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*}\sum_{t = 1}^r \sum_{j = 1}^n \sum_{i = 1}^m d_{ij} \ X_{ijt} + \sum_{t = 1}^r \sum_{h = 1}^{m - a} \sum_{g = 1}^a d_{gh} \ T_{ght} \leq\ A \tag{2}\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*}\sum_{i = 1}^m X_{ijt} \geq S_{jt} \ \forall \ j , t \tag{3}\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*}\sum_{j = 1}^n X_{ijt} \leq { P}_{it} \ \forall \ i , t \tag{4}\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*}\sum_{t = 1}^r P_{gt} - \sum_{t = 1}^r Y_{gt} - q_g \leq 0 \ \forall \ g \tag{5}\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*}\sum_{t = 1}^r (P_{ht} - Y_{ht}) - q_h \leq 0 \ \forall \ h \tag{6}\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*}\sum_{j = 1}^n X_{ijt} - {\rm LB}_{it} \geq 0 \ \forall \ i , t \tag{7}\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*}\alpha_g P_{gt} - \sum_{h = 1}^{m - a} T_{ght} \leq \ 0 \ \forall \ g , t \tag{8}\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*}\sum_{t = 1}^r Y_{it} \leq R_i \ \forall \ i \tag{9}\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*} \begin{split} X_{ijt,} \ T_{ght,}\ Y_{it,}\ P_{it,}\ P_{gt,}\ P_{ht}\ \geq\ 0\ \\ \quad \hbox {\rm [non-negative decision variables]}\end{split} \tag {10}\end{align*} \end{document}

where:

g=index of intermediate treatment facilities, g=1,2,…,a (a=8)

The multi-objective programming model developed above reflects the trade-off between cost efficiency and equity in locating waste disposal facilities, which are in conflict with each other. In the present study, the two objective functions were treated by the constraints method (Cohon, 1978; Wu, 2009), in which the minimum system cost in Eq. (1) was listed as the major target and the other one was considered as the secondary target to be lowered as a constraint equation shown as Eq. (2). In Eq. (1), the total costs of processing, transportation, and facility expansion costs over all time periods were considered. Furthermore, the transportation cost included the costs of moving waste from where it is generated to intermediate treatment facilities and then to the landfills.

The equity objective function for solving the multi-objective models was to minimize the total number of metric ton-kilometers for transporting solid waste. Note that the equity objective function was transformed into a constraint in Eq. (2) to simplify the multi-objective programming into a single objective function. To be clearer, the total number of metric ton-kilometers for transporting solid waste was designated to be less than A, referring to the maximum allowable metric ton-kilometers of waste to be shipped, a figure assigned by the policy makers of the new Kaohsiung Municipality. In other word, constraint in Eq. (2) gives the problem its multi-objective character.

Furthermore, the constraints in Eqs. (3) and (4) require that all waste be removed from the solid waste generation areas, and be treated in the same manner. Equations (5) and (6) require that the processing capacities of intermediate treatment facilities and landfills not be exceeded. Equation (7) requires that the quantity of solid waste transported from the waste generation area j to the facility location i must not be less than the required minimum treatment capacity of the facility at i. Equation (8) insures that all the remaining waste after the processing operations is removed from intermediate treatment facilities. Finally, Eq. (9) limits the total expansion of each facility.

Moreover, based on the knowledge of urban planning, the MSW management model should be deterministic and not include any uncertainties to ensure sublevel plans (optimal MSW collection route, optimal collection points, optimal dispatch of manpower, etc.) can proceed smoothly without unexpected events.

Case study and overview of studied system

Figure 1 shows the geographic location of the new Kaohsiung Municipality and location of existing and potential facilities, where G1–G4 are the four existing incineration plants; G5–G8 are the four proposed transfer stations to be decided in the model whether to install or not to install; H1–H10 are the 10 existing final disposal sanitary landfills. Thus, in the model of this study, the facility location i=18 is the combination of the intermediate treatment facilities (G1–G8) and the final disposal landfills (H1–H10). Also shown on Fig. 1 are the 33 solid waste generation areas (indicated by the solid dots), where J1–J22 (Table 1) are located in the region which was formerly Kaohsiung County and J23–J33 are in the territory formerly belonging to Kaohsiung City.

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FIG. 1.

Map of new Kaohsiung Municipality and location of existing and potential facilities, marked with intermediate treatment facilities (□, G), landfill sites (△, H), and solid waste generation areas (●).

These solid waste generation areas, intermediate treatment facilities, and final disposal sanitary landfills are the key targets to be modeled and optimized in this study. Thus, their influential background parameters are overviewed as follows:

Numerical solution and model assumptions

The multiple-objective mathematical programming model could be solved using various available software programs, including Excel, Lingo, Matlab, ILOG CPLEX, and GAMS. However, Lingo (2008) was chosen because it can be used conveniently, practically, and quickly to solve nonlinear problems. In this case study, the following assumptions are made to facilitate computer manipulations and reveal the model significance:

1. The actual distance (d_ij and d_gh) between a solid waste generation area and an intermediate treatment facility or a final disposal landfill is calculated as 1.2 times the linear distance.

Compromise of conflicting objectives

Figure 2 shows the 10-year compromise curve of the two conflicting objective functions, minimizing the total system cost and providing maximum equity. If the A value in Eq. (2), maximum allowable metric ton-kilometers of waste shipped, is adjusted, various solutions for minimizing the cost can be obtained and used to plot the compromise curve. According to Fig. 2, if the amount of solid waste transportation increased (equity minimized), the system cost decreased. On the other hand, if the amount of solid waste transportation decreased (equity maximized), the system cost increased. In general, the best compromise solution should be around the turning point. The Kaohsiung Department of Environmental Bureau has therefore selected an A value at (33.9 billion NTD, 90.3 million ton-kilometers) that corresponds to 3125 tons/day of waste processed to represent the best compromise solution.

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FIG. 2.

The 10-year trade-off curve of equity and total cost.

In obtaining an optimal solution for a given A value, we had to try various combinations in allocating different treatment facilities. For the best compromise solution, we reached the optimal operation decision in allocating the intermediate treatment facilities, as shown on Table 3. According to the arrangement shown on Table 3, by leaving two incinerators (out of four) in operation at the Southern Kaohsiung Incineration Plant, two out of three at the Central Kaohsiung Incineration Plant, two out of three at the Renwu Incineration Plant, and one out of three at the Kangshan Incineration Plant, an optimized solution was reached. However, the overall operating capacity of 2850 tons/day for all four plants will be less than the daily generated solid waste quantity of 3128 tons. The extra 278 tons must be transported to the sanitary landfill for disposal.

Also shown on Table 3, as proposed by the model, the Tianliao solid waste transfer station (G5) would be constructed, and the other three (G6, G7, and G8) would not be installed. It is worth mentioning that during the modeling processes to reach the above results, the quantity of solid waste generated was shown to greatly influence the results of the model.

Optimal solution for the decision variable X_ijt

One of the influential variables to be decided by the multi-objective programming model is the amount of waste to be transported from waste generation area j to facility location i (mainly intermediate treatment facilities here) in time period t, and is denoted as X_ijt in the model. Table 4 clearly displays the first-year optimal solution for X_ijt. For example, X(1,1,1) suggests Fengshan City (J1) can transport 122,713 tons of solid wastes to the Southern Kaohsiung Incineration Plant (facility location I1) during the first year (T1), while X(1,3,1) suggests Daliao Township (J3) can ship 45,676 tons to location I1 during time period T1, and so forth.

Please note that we had included time as a variable in this model, thus making possible a time-dependent dynamic annual solution for each year of the 10-year study period. In other words, the practical operational outputs for the second year to the 10th year can also be produced in this model. Owing to the limit of the manuscript length, these results are not included. Nowadays, engineers are facing challenges to deal with the complexity of society; thus, the inclusion of the time variable in this model has fulfilled the practical needs of the current dynamics of solid waste generation and management.

Optimal solutions for decision variable T_ght

After solid waste was processed at the intermediate treatment facilities, the resulting residue or waste were transported to sanitary landfills for final disposal. Thus, the amount of waste to be transported each year (T_ght) from intermediate treatment facility G to landfill H has a direct effect on the total system operation cost. Table 5 lists the quantity of solid waste to be transported from intermediate treatment facilities to the final disposal landfills for the first year; for example, T(1,1,1) suggests 98,500 tons/year of residue to be shipped from the Southern Kaohsiung Incineration Plant (G1) to the Linyuan Township Sanitary Landfill (H1), and 65,700 tons/year to be shipped from Central Kaohsiung Incineration Plant (G2) to the Niaosong Township Sanitary Landfill (H2). Similarly, the practical operational outputs for the second year to the 10th year can also been found.

Optimal solution for decision variable P_it

Another important waste management variable to be decided by the administrator is to allocate the optimized amount of waste to be processed (P_it) at the facility location “i” in time period “t.” Table 6 lists the model output for the first year, for example, P(1,1) suggests 328,500 tons/year of waste to be incinerated at the Southern Kaohsiung Incineration Plant, and 219,000 tons/year at the Central Kaohsiung Incineration Plant. In addition, the model can suggest the practical operational outputs for the second year to the 10th year.

Optimal solution for facility expansion, Y_it

Finally, to compromise the above optimized suggestion, the current capacity of some of the existing facilities may need to expand, or installation of a new facility may also be necessary. Our model suggested that the capacities of all facilities need not be expanded for the coming 10 years because the Y_it values, which represent the incremental capacity for all existing facilities, remain unchanged based on the A value of the model solution, providing that the Tianliao Transfer Station is newly installed.

The results obtained in this research indicate that changing the solid waste transportation A value (metric ton-kilometers) will affect the cost such that the solid waste route must be modified accordingly. Among various A values and their corresponding optimal solutions, the decision maker is suggested to accept the cost around the turning point of the compromise curve. This solution should be the compromise optimal solution between the cost and equity and fulfill all the constraints at the same time.

Future models will become more complex by taking consideration of manpower, budget, and machinery, increasing intermediate treatment facilities, forecasting waste quantity changes, social and environmental objectives, and adding more constraints. Current software may not be capable of solving such complex mathematical models. Therefore, the MPOS software (e.g., MPOS-BBMIP) recently developed by Northwestern University, or similar system software, will need to be used.