Hybrid Inexact Optimization Approach with Data Envelopment Analysis for Environment Management and Planning in the City of Beijing,China

Waste generation rate

The total waste generation amount is 18,400 tonne/day in Beijing; moreover, waste generation rate is increased year after year according to statistical yearbook of Beijing (City of Beijing, 2005–2009). Several measures have been taking on waste management system, such as enhancing the treatment facility efficiency and promoting and developing the classified collection system, so the growth rate would slow down in the planning periods. In this article, three levels (low, medium, and high) of waste generation rate are considered, with the probability of 20%, 60%, and 20%, respectively. A sound way for waste treatment should be selected with consideration of waste composition. Table 1 shows the composition of the waste in Beijing from 2000 to 2025 (Zhou et al., 2004). It demonstrates that over 50% of MSW is organic, and it is still increasing gradually, whereas the percentages of paper, plastic, and glass show a downtrend. This may be as well interpreted that the classification collection system improved the recyclable rate and some measures have been taken to improve the recycling system; the growing population has also caused the increasing percentage of organic composition (Ye and Han, 2000; Zhang, 2003). All these show a trend that the waste-manager should pay more attention to composting facilities in the future.

Collection and transportation

The collection and transportation systems in urban districts and suburb districts are different. The waste generated in district is collected by residents firstly and then sent to corresponding transfer station for pretreatment, ultimately transported to treatment facilities for final disposal; in subdistrict, the waste is transported to treatment facilities directly and pretreated in treatment plant by residents or sorting machines (City of Beijing, 2009). The sorted waste is sent to various treatment facilities according to its nature (Fig. 2).

OPEN IN VIEWER

FIG. 2.

Waste flow in the city of Beijing.

Waste treatment facilities

The main approaches for waste treatment include landfill, composting, and incinerator, which will constitute 60%, 15%, and 25%, by the end of 2010, respectively (City of Beijing, 2009, 2010). To simplify the study system, in this article, the comprehensive treatment facility would be regarded as incinerator or composting facility according to its main function (major treatment way incinerating or composting), for example, if a comprehensive treatment plant is made up of screening, incinerating, and composting technique, and the main treatment approach is screening and composting, and the comprehensive plant is regarded as composting plant. It is also assumed that the residue generated by comprehensive facility would be shipped to its adjacent landfill.

Except for Shijingshan district and Chengsiqu district (including Dongcheng district, Xicheng district, Chongwen district, and Xuanwu district), each district/subdistrict has at least one landfill. Each landfill has a cumulative capacity limit and can deal with waste generated in local district/subdistrict or adjacent district/subdistrict, it can also tackle with residue resulted from adjacent composting and incinerator facilities, and one landfill can only expanse one time in one period.

Composting is an effective approach for MSW treatment, especially for organic garbage. Existing composting plants are almost comprehensive treatment plants (except for Nangong composting plant). The main handling techniques are sieving and biological treatment in the study comprehensive treatment plants. The residue (normally 10%–20% volume of original waste) would be finally disposed by landfill (Li et al., 2006). Revenue is obtained through recycling and composting product engendered in biological treatment. Each composting facility has a day capacity limit and can expanse one time in one period.

Waste volume can normally be reduced by 70%–80% through incinerating (Li et al., 2006), depending on the inorganic content of the waste. The combustion heat could be collected to generate power, and the waste heat can be provided to local resident for heat supply. The ultimate residue is disposed of by landfill. Revenue is obtained by recycling and generated power. Each incinerator has a day capacity limit and can expanse one time in one period. Table 2 presents expansion options of various treatment facilities. For example, for composting facility, three expansion options (200, 400, and 600 tonne/day) are provided. Each option has its own expansion cost, and a sound expansion option should be selected by its real situation.

Diversion rate and waste management cost

Diversion rate is defined as the summation percentage of waste treated by incinerators and composting facilities. Currently, landfill is the main processing mode, accounting for about 60%; incineration ranks at the second position, occupying 25%; only 15% of waste is treated by composting facility. According to the city’ s long-term plan for MSW management in the Eleventh Five-Year Plan, the division rate should be increased in the following planning periods (City of Beijing, 2009). The net system costs for waste management in Beijing mainly contains the costs for waste collection, transportation, and processing, as well as the capital costs for developing/expanding waste management facilities. Due to the dynamic of waste management costs, the entire cost value is inexact and presented as an interval value. Table 3 shows operation costs of various treatment facilities for the allowable waste flow and Table 4 for the excess waste flow. For example, for Yongning landfill, the operation cost is US$9.1–9.5 when the allowable waste flow is not exceeded; if not, the penalty would be added. The operation cost will be higher than that of allowable waste flow; it is US$13.65–14.25.

Formulation

The planning horizon is 15 years with three planning periods. In the model, the decision variables are divided into two categories: discrete and continuous. The discrete variables represent the expansion options for waste management facilities in different periods, whereas the continuous ones represent the optimized waste flows from transfer station/collection points to the waste management facilities. The constraints are sorted into five groups, including capacity balance constraints, mass balance constraints, waste residue constraints, facility-expansion constraints, and non-negativity constraints. The model is 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*} \min f^{ \pm} &= \sum \nolimits_{j = 1}^{18} \sum \nolimits_{l = 1}^{16} \sum \nolimits_{k = 1}^3 L_k TL_{jlk}^{ \pm} ( DL_{jl} TR_k^{ \pm} + OL_{lk}^{ \pm} ) \\&\quad+ \sum \nolimits_{j = 1}^{18} \sum \nolimits_{z = 1}^{9} \sum \nolimits_{k = 1}^{3} L_k TZ_{jzk}^{ \pm} ( DZ_{jz} TR_k^{ \pm} + OZ_{zk}^{ \pm} ) \\&\quad + \sum \nolimits_{j = 1}^{18} \sum \nolimits_{i = 1}^{6} \sum \nolimits_{k = 1}^3 L_k TI_{jik}^{ \pm} ( DI_{ji} TR_k^{ \pm} + OI_{ik}^{ \pm} ) \\&\quad+ \sum \nolimits_{z = 1}^{9} \sum \nolimits_{l = 1}^{16} \sum \nolimits_{k = 1}^{3} L_k YZ_{zlk}^{ \pm} ( DZL_{zl} TR_k^{ \pm} + OL_{lk}^{ \pm} ) \\&\quad + \sum \nolimits_{i = 1}^{6} \sum \nolimits_{l = 1}^{16} \sum \nolimits_{k = 1}^3 L_k YI_{ilk}^{ \pm} ( DIL_{il} TR_k^{ \pm} + OL_{ik}^{ \pm} ) \\&\quad+ \sum \nolimits_{j = 1}^{18} \sum \nolimits_{l = 1}^{16} \sum \nolimits_{k = 1}^{3} \sum \nolimits_{h = 1}^3 L_k P_{jh} XL_{jlkh}^{ \pm} ( DL_{jl} DR_k^{ \pm} + DOL_{lk}^{ \pm} ) \\&\quad + \sum \nolimits_{j = 1}^{18} \sum \nolimits_{z = 1}^{9} \sum \nolimits_{k = 1}^3 \sum \nolimits_{h = 1}^3 L_k P_{jh} XZ_{jzkh}^{ \pm} ( DZ_{jz} DR_k^{ \pm} + DOZ_{zk}^{ \pm} ) \\&\quad+ \sum \nolimits_{j = 1}^{18} \sum \nolimits_{i = 1}^{6} \sum \nolimits_{k = 1}^{3} \sum \nolimits_{h = 1}^3 L_k P_{jh} XI_{jikh}^{ \pm} ( DI_{ji} DR_{ik}^{ \pm} + DOI_{ik}^{ \pm} ) \\&\quad + \sum \nolimits_{z = 1}^{9} \sum \nolimits_{l = 1}^{16} \sum \nolimits_{k = 1}^3 L_k DYZ_{zlk}^{ \pm} ( DZL_{zl} DR_k^{ \pm} + DOL_{lk}^{ \pm} ) \\&\quad+ \sum \nolimits_{i = 1}^{6} \sum \nolimits_{l = 1}^{16} \sum \nolimits_{k = 1}^{3} L_k DYI_{ilk}^{ \pm} ( DIL_{il} DR_k^{ \pm} + DOL_{ik}^{ \pm} ) \\&\quad + \sum \nolimits_{l = 1}^{16} \sum \nolimits_{u = 1}^{3} \sum \nolimits_{k = 1}^{3} EL_{luk}^{ \pm} BU_{luk}^{ \pm} \\&\quad+ \sum \nolimits_{z = 1}^{9} \sum \nolimits_{v = 1}^{3} \sum \nolimits_{k = 1}^{3} EZ_{zvk}^{ \pm} BV_{zvk}^{ \pm} \\&\quad+ \sum \nolimits_{i = 1}^{6} \sum \nolimits_{w = 1}^{3} \sum \nolimits_{k = 1}^{3} EI_{iwk}^{ \pm} BW_{iwk}^{ \pm} \\&\quad - \sum \nolimits_{j = 1}^{18} \sum \nolimits_{z = 1}^{9} \sum \nolimits_{k = 1}^{3} L_k TZ_{jzk}^{ \pm} REZ_{zk}^{ \pm} \\&\quad- \sum \nolimits_{j = 1}^{18} \sum \nolimits_{i = 1}^{6} \sum \nolimits_{k = 1}^{3} L_k TI_{jik}^{ \pm} REI_{ik}^{ \pm} \\&\quad - \sum \nolimits_{j = 1}^{18} \sum \nolimits_{z = 1}^{9} \sum \nolimits_{k = 1}^3 \sum \nolimits_{h = 1}^3 L_k P_{jh} XZ_{jzkh}^{ \pm} RMZ_{zk}^{ \pm} \\&\quad- \sum \nolimits_{j = 1}^{18} \sum \nolimits_{i = 1}^{6} \sum \nolimits_{k = 1}^{3} \sum \nolimits_{h = 1}^3 L_k P_{jh} XI_{jikh}^{ \pm} RMI_{ik}^{ \pm} \qquad & (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*} \sum \nolimits_{j = 1}^{18} & \sum \nolimits_{k = 1}^{k^{ \prime}} L_k ( TL_{jik}^{ \pm} + XL_{jikh}^{ \pm} ) + \sum \nolimits_{z = 1}^9 \sum \nolimits_{k = 1}^{k^{ \prime}} L_k ( YZ_{zlk}^{ \pm} + DYZ_{zlk}^{ \pm} ) \\ & + \sum \nolimits_{i = 1}^{6} \sum \nolimits_{k = 1}^{k^{ \prime}}L_k ( YI_{ilk}^{ \pm} + DYI_{ilk}^{ \pm} ) \\ & \leq LC_l + \sum \nolimits_{u = 1}^3 \sum \nolimits_{k = 1}^{k^{ \prime}} \Delta LC_{luk} BU_{luk}^{ \pm} ; \forall h , l;k^{ \prime} = 1 , 2 , 3; \qquad & ( 1{ \rm b} ) \end{align*} \end{document}

[Landfill capacity constraint] \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}\sum \nolimits_{j = 1}^{18} & TZ_{jzk}^{ \pm} + \sum \nolimits_{j = 1}^{18} XZ_{jzkh}^{ \pm} \leq ZC_z \\ & + \sum \nolimits_{v = 1}^3 \sum \nolimits_{k = 1}^{k^{ \prime}} \Delta ZC_{zvk}BV_{zvk}^{ \pm}; \forall h , z;k^{ \prime} = 1 , 2 , 3;\end{split} \qquad & (1{\rm c}) \end{align*} \end{document}

[Composting capacity constraint] \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}\sum \nolimits_{j = 1}^{18} & TI_{jik}^{ \pm} + \sum \nolimits_{j = 1}^{18} XI_{jikh}^{ \pm} \leq IC_i \\ & + \sum \nolimits_{w = 1}^3 \sum \nolimits_{k = 1}^{k^{ \prime}} \Delta IC_{iwk}BW_{iwk}^{ \pm}; \forall h , i;k^{ \prime} = 1 , 2 , 3;\end{split} \qquad & (1{\rm d}) \end{align*} \end{document}

[Incinerator capacity constraint] \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}& \sum \nolimits_{l = 1}^{16} ( TL_{jlk}^{ \pm} + XL_{jlkh}^{ \pm} ) + \sum \nolimits_{z = 1}^9 ( TZ_{jzk}^{ \pm} + XZ_{jzkh}^{ \pm} ) \\ & + \sum \nolimits_{i = 1}^6 ( TI_{jik}^{ \pm} + XI_{jikh}^{ \pm} ) \geq WG_{jkh}^{ \pm}; \forall j , k , h;\end{split} \qquad &(1{\rm e}) \end{align*} \end{document}

[Mass balance constraint] \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 \nolimits_{l = 1}^{16} YZ_{zlk}^{ \pm} = \sum \nolimits_{j = 1}^{18} FZ_z TZ_{jzk}^{ \pm}; \forall z , k; \qquad &(1{\rm f}) \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 \nolimits_{l = 1}^{16} YI_{ilk}^{ \pm} = \sum \nolimits_{j = 1}^{18} FI_i TI_{jik}^{ \pm}; \forall i , h; \qquad &(1{\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*} \sum \nolimits_{l = 1}^{16} DYZ_{zlk}^{ \pm} = \sum \nolimits_{j = 1}^{18} \sum \nolimits_{h = 1}^{3} P_{jh} XZ_{jzkh}^{ \pm}; \forall z , k; \qquad &(1{\rm h}) \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 \nolimits_{l = 1}^{16} DYI_{ilk}^{ \pm} = \sum \nolimits_{j = 1}^{18} \sum \nolimits_{h = 1}^{3} P_{jh} XI_{jikh}^{ \pm}; \forall i , h; \qquad &(1{ \rm i}) \end{align*} \end{document}

[Residue constraint] \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*} TL_{jlk \max}^{ \pm} \geq TL_{jlk}^{ \pm} \geq XL_{jlkh}^{ \pm} \geq 0; \forall j , l , k , h; \qquad &(1{ \rm j}) \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*} TZ_{jzk \max}^{ \pm} \geq TZ_{jzk}^{ \pm} \geq XZ_{jzkh}^{ \pm} \geq 0; \forall j , z , k , h; \qquad &(1{ \rm k}) \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*} TI_{jik \max}^{ \pm} \geq TI_{jik}^{ \pm} \geq XI_{jikh}^{ \pm} \geq 0; \forall j , i , k , h; \qquad &(1{ \rm l}) \end{align*} \end{document}

[Non-negativity constraint] \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 \nolimits_{u = 1}^3 BU_{luk}^{ \pm} \leq 1; \forall l , k; \qquad &(1{ \rm m}) \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 \nolimits_{v = 1}^3 BV_{zvk}^{ \pm} \leq 1; \forall z , k; \qquad &(1{ \rm n}) \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 \nolimits_{w = 1}^3 BW_{iwk}^{ \pm} \leq 1; \forall i , k; \qquad &(1{ \rm o}) \end{align*} \end{document}

[Expansion of facility; may only happen once in any given period] \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*} BU_{luk}^{ \pm} , BV_{zvk}^{ \pm} , BW_{iwk}^{ \pm} \ \hbox{\rm are binary variability;} \qquad &(1{ \rm p}) \end{align*} \end{document}

[Technique constraint]

Assumptions and scenarios

The (TIMIP) model is formulated based on several assumptions. First, it is assumed that each subdistrict has a waste collection points, and all the waste generated in rural area should be collected and transported to the waste collection points, and then to various treatment facilities. Second, costs in each time period can be reduced to the present values according to the cash flow pattern. Third, construction or expansion of any facility should be completed within the period during which it is initiated; for example, if a facility would be demanded in period k, it must be constructed or expanded in period k − 1. MSW is meant to the city's household wastes only in this study.

According to related policies and practice situations, three different scenarios are examined. Scenario 1 is based on the current situation; the proportion of three kinds of treatment facilities keep the existing pattern over the whole planning horizon; more than half of waste will be treated by landfill in the three periods. Consequently, some landfill's lifespan will shrink due to increasing waste generation; to meet the treatment demand, landfill must expand, which can occupy a lot of land resource and bring a series of environmental problems. The total system cost is considered more than the others in scenario 1. Scenario 2 reflects a situation that demands a faster economic development. This aggressive policy was made based on a high diversion rate of 80% by the end of the planning horizon. Under this situation, most incinerators and composting facilities would expand. The economic development is considered more than the others in this scenario. Scenario 3 is based on the long-term point of view that all the Eleventh Five-Year Plan and government policies on MSW management would be achieved in the three periods. According to the government policies, the diversion rate would be 3:3:4 (landfill:incineration:composting) in the end of last period. Environmental friendliness is considered more than the others in this scenario (Fig. 3).

OPEN IN VIEWER

FIG. 3.

Diversion rate in three periods.

Solution method

The model solutions provide stable intervals for the objective function value and the decision variables. The detail solution process can be summarized as follows (Huang, 1998; Maqsood and Huang, 2003; Iskander, 2005; Li et al., 2006, 2007; Qin et al., 2007; Li and Huang, 2008):

Step 1: Acquire the information of intervals (e.g., capacities of waste management facilities and rates of waste generation).

Step 2: Formulate the (TIMIP) model.

Step 3: Transform the (TIMIP) primal model into two submodels, where the submodel corresponding 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} $$f_{opt}^ -$$ \end{document} should be first solved (to obtain the most optimistic decision option within the decision space) if the objective is to minimize f^±.

Step 4: Formulate \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} $$f_{opt}^ -$$ \end{document} submodel, including the objective function and relevant constraints.

Step 5: Solve the \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} $$f_{opt}^ -$$ \end{document} submodel and obtain \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} $${ \rm XL}_{jlkhopt}^ -$$ \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} $${ \rm XZ}_{jzkhopt}^ -$$ \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} $${ \rm XI}_{jikhopt}^ -$$ \end{document} .

Step 6: Calculate the total waste flows \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} $$AL_{jikhopt}^ - = TL_{jlk}^ - + XL_{jlkhopt}^ -$$ \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} $$AZ_{jzkhopt}^ - = TZ_{jzk}^ - + XZ_{jzkhopt}^ -$$ \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} $$AI_{jikhopt}^ - = TI_{jik}^ - + XI_{jikhopt}^ -$$ \end{document} , and the objective function value \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} $$f_{opt}^ -$$ \end{document} (most optimistic decision option).

Step 7: Formulate \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} $$f_{opt}^ +$$ \end{document} submodel, including the objective function and the relevant constrains.

Step 8: Solve the \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} $$f_{opt}^ +$$ \end{document} submodel and obtain \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} $$XL_{jlkhopt}^ +$$ \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} $$XZ_{jzkhopt}^ +$$ \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} $$XI_{jikhopt}^ +$$ \end{document} .

Step 9: Calculate the total waste flows \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} $$AL_{jikhopt}^ + = TL_{jlk}^ + + XL_{jlkhopt}^ +$$ \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} $$AZ_{jzkhopt}^ + = TZ_{jzk}^ + + XZ_{jzkhopt}^ +$$ \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} $$AI_{jikhopt}^ + = TI_{jik}^ + + XI_{jikhopt}^ +$$ \end{document} , and the objective function value \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} $$f_{opt}^ +$$ \end{document} (most conservative decision option).

Step 10: Combine solutions of the two submodels, and obtain optimized interval solutions for the ITMIP model: \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} $$XL_{jlkhopt}^{ \pm} = [ XL_{jlkhopt}^ - , XL_{jikhopt}^ + ] , XZ_{jzkhopt}^{ \pm} = [ XZ_{jzkhopt}^ - , XZ_{jzkhopt}^ + ]$$ \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} $$XI_{jikhopt}^{ \pm} = [ XI_{jikhopt}^ - , XI_{jikhopt}^ + ]$$ \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} $$AL_{jlkhopt}^{ \pm} = TL_{jlk}^{ \pm} + XL_{jlkhopt}^{ \pm}; \forall j , l , k , h$$ \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} $$AZ_{jzkhopt}^{ \pm} = TZ_{jzk}^{ \pm} + XZ_{jzkhopt}^{ \pm}; \forall j , z , k , h$$ \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} $$AI_{jikhopt}^{ \pm} = TI_{jik}^{ \pm} + XI_{jikhopt}^{ \pm}; \forall j , i , k , h$$ \end{document}

The model can be solved through a software lingo.

Solution under scenario 1

Table 5 shows the solution of allowable and excess waste flow shipped to landfill under the scenario 1. Table 6 presents the solution of allowable and excess waste flow shipped to incinerators and composting facilities. For example, 69.2–71.6 tonne waste flow would be transported to landfill 1 (Yongning landfill) during period 1 at low level; there is no excess waste flow, whereas if it is in the condition of medium level, 3.9–4.1 tonnes of excess waste flow would be produced, and this would reach to 7.7–9.9 tonnes under high level. 69.2–71.6 is an interval value, which represents the upper value is 71.6 and the lower value is 69.2. Figure 4 illustrates the surplus capacity of landfill at the end of each planning period. For example, for Xiaozhangjia landfill, as marked landfill 2 in the figure, the surplus capacity is 0.622–0.896×10⁶ tonnes in period 1; however, in period 2, it is 5.727–5.889×10⁶ tonnes; because landfill 2 has expanded with option 2 (1.095×10⁶ tonnes) in this period, the surplus capacity is 3.426–3.471×10⁶ tonnes by the end of period 3. Figure 5 shows the facility-expansion solutions under scenario 1. For example, Xiaozhangjia landfill would expand with option 1 (7.3×10⁵ tonnes) during period 2. Figure 5 illustrates that the capacity of Asuwei landfill, Liulitun landfill, Jiaojiapo landfill, Dadushe landfill, and Gaoantun landfill would be expanded during three periods with different options. The reason is that waste generated in central district all transported to the above landfills (because there is no landfill in central district), besides that the capacity of the above landfills is low and could not meet the treatment demand. Little waste flow would be disposed of by composting and incinerator facilities, and the existing capacity would satisfy the process requirement; it would not need expansion except Liulitun incinerator, which selects the expansion option 1 during the period 2. Under scenario 1, the total system cost is the lowest, meaning that if the government continues mainly relying on landfill for waste disposal in the following three periods, the lowest management cost would be obtained; however, this would (1) shrink the lifespan of landfill, (2) take up a lot of land, and (3) cause second pollution, and the saving from lower system costs could not compensate the losing from environmental penalties.

OPEN IN VIEWER

FIG. 4.

Surplus capacity of various landfills in three periods.

OPEN IN VIEWER

FIG. 5.

Facility expansion under scenario 1.

Solution under scenario 2

Table 7 shows the solution of optimized waste flow allocation during periods 1–3 under scenario 2. In general, the solutions present as interval and probabilistic forms, indicating that the related decision should be sensitive to the uncertain modeling inputs. For example, there would be 6019.7–6166.2 tonnes of waste flows (including residue) per day shipped to the landfill during period 1, including 5915.2–6046.6 tonnes of allowable waste flows and 104.5–119.6 tonnes of excess waste flows. There would be 4561.2–4628.2 tonnes of waste flows transported to incinerator, including 4481.0–4535.0 tonnes of allowable waste flows and 80.2–93.2 tonnes of excess waste flows. About 4560.9–4628.2 tonnes of waste flows would be diverted to the composting facility, including 4481–4535 tonnes of allowable waste flows and 79.9–93.2 tonnes of excess flows. Figure 6 presents the facility-expansion results under scenario 2. It states that expanded landfills are less than that of scenario 1, whereas expanded incinerators and composting facilities are more than that of scenario 1. Waste flow transported to incinerators for treatment is much more than the other two cases, resulting that expanded incinerators are more than the other two cases. The diversion rate has been increased compared with that under scenario 1. Some composting and incinerator facilities are incapable of tackling the waste flow transported from districts and subdistricts completely due to their limited capacity, so several composting and incinerator facilities need to be expanded in certain period. Because the surplus capacity of Asuwei landfill is small and would be sealed in 2012 if no expansion happen, so the landfill must expand during the three periods to meet treatment demand. Under scenario 2, the total system cost is the highest, because of the highest diversion rate and the highest treatment proportion for incinerating, besides that incinerator facility mostly depends on foreign country, and the construction cost and operation cost is too high. However, it would save plenty of land resource and prolong the lifespan of landfill, and the condition of incinerator would be improved with the progress of technology.

OPEN IN VIEWER

FIG. 6.

Facility expansion under scenario 2.

Solution under scenario 3

Table 8 presents the solution of optimized waste flow allocation during periods 1–3 under scenario 3. There would be 6378.4–6471.4 tonnes of waste flows (including residue) diverted to landfill per day in period 2, including 6274.4–6346.4 tonnes of allowable waste flows and 104.0–125.0 tonnes of excess waste flows; there would be 4784.8–4853.4 tonnes of waste flows shipped to incinerator facility, including 4705.8–4759.8 tonnes of allowable waste flows and 79.0–93.6 tonnes of excess waste flows; 4784.8–4853.4 tonnes of waste flows would be transported to composting facility, including 4705.8–4759.8 tonnes of allowable waste flows and 79.0–93.6 tonnes of excess waste flows. Figure 7 shows the facility-expansion results under scenario 3. It demonstrates that more and more waste flow would be tackled by composting facility, and the number of expanded composting facility is largest among three scenarios; this phenomenon would coincide with the decision-maker's view, which would be a trend in the future.

OPEN IN VIEWER

FIG. 7.

Facility expansion under scenario 3.

Compared with the other two scenarios, waste flow tackled by composting facility is increased. It is forecasted the composition of waste will change a lot due to some government policy on waste management (i.e., classification collection program and recycling program), and the organic component is increasing; it implies that waste generated in planning periods would be suitable to be treated by composting facility. Under scenario 3, the total system cost is medium; almost half of the waste would be disposed of through Biological Treatment at the end of planning period, which would generate the least pollutants and prolong the lifespan of landfill in certain degree. The generated product can improve soil properties and bring some revenue. Figure 8 shows the total system cost under three scenarios. It indicates that a high diversion rate can lead to a high system cost. Solutions corresponding to the lower-bound of system costs represent an alternative under the most optimistic waste generation mode, whereas the upper-bound of system cost corresponds to a more conservative consideration.

OPEN IN VIEWER

FIG. 8.

System costs under different scenarios.

Postoptimization analysis

Since each scenario has various expansion options for waste management facility capacity, it is of significance to evaluate the comprehensive benefits of facility-expansion schemes on the decision alternatives. DEA method is the technology to determine the alternative decision schemes by using a ratio of the weighted sum of outputs divided by the weighted sum of inputs (Cooper and Tone, 1997). Assume that there are n decision making units (DMUs) to be evaluated. Each DMU consumes varying amounts of m different inputs to produce r different outputs. In the Charnes, Cooper and Rhodes (CCR) ratio model, multiple inputs and multiple outputs of each DMU are aggregated into a single virtual input and a single virtual output, respectively. The CCR model can be formulated as the following linear program (Charnes et al., 1978; Charnes and Cooper, 1984; Banker et al., 1984): \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 Max} \ h_0 = v^T y_0 \qquad & (2{\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*} u^T x_0 = 1 \qquad & (2{\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*} -u^T X + v^T Y \leq 0 \qquad & (2{\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*} u \geq 0 \qquad & (2{\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*} v \geq 0 \qquad & (2{\rm e}) \end{align*} \end{document}

DMU _i is the ith DMU, DMU₀ is the target DMU, h₀ is the efficiency score of DMU₀, x_i∈R^m×1 is the column vector of inputs consumed by DMU _i ; x₀∈R^m×1 is the column vector of inputs consumed by the target DMU, X∈R^m×n is the matrix of inputs of all DMUs, y_i∈R^r×1 is the column vector of outputs produced by DMU _i ; y₀∈R^r×1 is the column vector of outputs produced by the target DMU, Y∈R^r×n is the matrix of outputs of all DMUs, u∈R^m×1 is the column vector of input weights, and v∈R^r×1 is the column vector of output weights. In general, the column means of cross efficient matrix are utilized to distinguish between good overall performers and poor performers. Therefore, the cross efficiency scores that can be used to rank the alternatives or the candidate DMU with the highest column mean can be selected as the optimal choice. Normally, if cross efficiency score is lower, the decision scheme would probably be eliminated; otherwise, it may mean that the decision scheme could be a better performance (Charnes et al., 1990; Athanassopoulos et al., 1999).

Table 9 shows the input and output of DEA model. In this article, the total system cost generated from the model is considered as the input data, whereas the output are the remaining landfill capacity, the capacity of composting, and incinerator facility at the end of the planning period. Figure 9 presents the cross efficiency scores of the alternative decision schemes for lower bound and upper bound. Scenario 3, which considers more environmental friendliness, is regarded as the optimal alternative based on DEA; the solution is obtained with the consideration of minimum system cost and maximum surplus landfill capacity, and the capacity of composting and incinerator facility.

OPEN IN VIEWER

FIG. 9.

Cross-efficiency scores of three cases.