Modeling Municipal Solid Waste Management System under Uncertainty

Journal of the Air & Waste Management Association ISSN: 1096-2247 (Print) 2162-2906 (Online) Journal homepage: http://www.tandfonline.com/loi/uawm20 Modeling Municipal Solid Waste Management System under Uncertainty Yongping Li & Guohe Huang o cite this article: Yongping Li & Guohe Huang (2010) Modeling Municipal Solid Waste Management System under Uncertainty, Journal of the Air & Waste Management Association, 60:4, 49-45, DOI: 10.155/1047-289.60.4.49 o link to this article: https://doi.org/10.155/1047-289.60.4.49 Published online: 22 Feb 2012. Submit your article to this journal Article views: 199 View related articles Citing articles: 6 View citing articles Full erms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalinformation?journalcode=uawm20 Download by: [7.44.194.16] Date: 0 January 2018, At: 16:55

ECHNICAL PAPER ISSN:1047-289 J. Air & Waste Manage. Assoc. 60:49 45 DOI:10.155/1047-289.60.4.49 Copyright 2010 Air & Waste Management Association Modeling Municipal Solid Waste Management System under Uncertainty Yongping Li and Guohe Huang Research Academy of Energy and Environmental Studies, North China Electric Power University, Beijing 102206, People s Republic of China Downloaded by [7.44.194.16] at 16:55 0 January 2018 ABSRAC In this study, a dynamic inexact waste management (DIWM) model is developed for identifying optimal waste-flow-allocation and facility-capacity-expansion strategies under uncertainty and is based on an inexact scenario-based probabilistic programming (ISPP) approach. he DIWM model can handle uncertainties presented as interval values and probability distributions, and it can support assessing the risk of violating system constraints. Several violation levels for facility-capacity and waste-diversion constraints are examined. Solutions associated with different risks of constraint violation were generated. he modeling results are valuable for supporting the planning of the study city s municipal solid waste (MSW) management practices, the long-term capacity expansion for waste management system, and the identification of desired policies regarding waste diversion. Sensitivity analyses are also undertaken to demonstrate that the violations of different constraints have varied effects on the planning of waste-flow allocation, facility expansion, and waste management cost. INRODUCION Many optimization techniques have been developed for supporting decisions of municipal solid waste (MSW) management and evaluating relevant operation and investment policies since the 1960s 1 ; they have involved linear, dynamic, integer, and multiobjective programming methods. For example, Kirca and Erkip 2 formulated IMPLICAIONS In MSW management systems, uncertainties exist in various parameters, impact factors, and waste disposal processes that lead to difficulties in identifying desired alternatives for managing waste under economic objectives and environmental concerns. he developed DIWM model has advantages in uncertainty reflection, dynamic facilitation, policy analysis, and risk assessment. In addition, it can address issues concerning planning for a cost-effective diversion program that targets reducing the MSW amount disposed of by landfills. he results obtained can be used to help MSW managers identify desired policies under various environmental, economic, and system-reliability conditions. a linear programming model for determining transfer station locations in the MSW management system of Istanbul, urkey. Baetz used dynamic programming to determine the optimal sizing and timing for landfills and waste-to-energy facilities. Or and Curi 4 used a mixed integer linear programming (MILP) model for improving solid waste collection and transportation system in the city of Izmir, urkey, to minimize the city s total solid waste collection and transportation costs. Kulcar 5 used a linear programming method for optimizing collection strategies under minimized waste transportation costs in a major urban area. Fiorucci et al. 6 developed a decision support system (DSS) for solid waste management planning in the city of Genova, Italy. Solano et al. 7,8 developed an integrated solid waste management (ISWM) model to assist in identifying desired solid waste management strategies that could satisfy cost, energy, and environmental emission objectives. Chang et al. 9 utilized an integer programming method to support the decisions of location and capacity for a material recovery facility in the city of San Antonio, X. However, in real-world MSW management problems, uncertainties exist in the related costs, impact factors, and objectives that may affect the optimization processes and the decision schemes generated. 10 he complexities could be further amplified by interactions among the uncertain parameters and through additional economic implications. Such complexities have placed many MSW management problems beyond deterministic programming methods. Previously, several efforts were made in developing various methods for dealing with uncertainties in MSW management systems. For example, Koo et al. 11 proposed a framework using a waste resources allocation program and fuzzy set theory to address tradeoffs among the objectives of economic efficiency, environmental quality, and administrative efficiency, such that an optimal site for a new hazardous waste treatment facility in southwestern Korea could be determined. Arey et al. 12 used a mixed optimization and probabilistic analysis approach for determining daily waste management practices in the municipalities of Hamilton and St. Catharines, Ontario, Canada. Chang et al. 1 proposed a fuzzy interval mixed integer multiobjective linear programming model for the evaluation of management strategies for solid waste management in a metropolitan region, demonstrating that uncertainties could be quantified by specific membership functions and interval numbers within a multiobjective analytical framework. Wilson and Baetz 14,15 developed a Volume 60 April 2010 Journal of the Air & Waste Management Association 49

Downloaded by [7.44.194.16] at 16:55 0 January 2018 derived probability model for curbside waste collection activities that allowed for analyzing stochastic information in MSW management systems. Huang et al. 16 advanced an inexact fuzzy-stochastic MILP model for waste management planning in which fuzzy flexible programming was introduced into an interval-parameter chanceconstrained programming framework. Maqsood and Huang 17 explored an inexact two-stage programming model for planning solid waste management under uncertainty in which interval parameters were incorporated within a twostage stochastic optimization framework. Davila et al. 18 proposed a gray integer programming-based game theory for system optimization and cost-benefit analysis at two competing landfills in the Lower Rio Grande Valley, X. Li and Huang 19 developed an interval parameter minimax regret programming method for the planning of MSW management systems in which uncertainties were expressed as interval values and random variables without knowing their probability distributions. Although many previous efforts of MSW management and planning under uncertainty have been conducted, there are few studies focusing on a scenario-based multistage stochastic programming (MSP) method for waste management and planning. In fact, MSP was effective for handling uncertainties expressed as probability distributions and permitting revised decisions in each time stage on the basis of the information of sequentially realized uncertain events. 20 22 In MSP, decision variables are divided into two subsets: those that must be determined before the realizations of random variables are disclosed and those (recourse variables) that can be determined after the random variable values are available. In the past decades, several MSP methods were developed and applied to environmental management and energy systems planning 22 25 ; unfortunately, few applications of MSP to waste management were reported. he chanceconstrained programming (CCP) method can effectively reflect the reliability of satisfying (or risk of violating) system constraints under uncertainty. It can provide information on the tradeoffs among the objective function value, tolerance values of the constraint, and the prescribed levels of probability that could be valuable to decision-makers for MSW management and planning. Interval-parameter chance-constrained programming (ICCP) is useful for dealing with uncertainties expressed as intervals and probabilities by introducing an interval-parameter programming (IPP) approach into the CCP framework. herefore, one potential approach for better accounting for uncertainties, risks, and policies for planning MSW management systems is to develop an inexact scenariobased probabilistic programming (ISPP) method on the basis of MSP and ICCP techniques. he objective of this study aims to develop a dynamic inexact waste management (DIWM) model for identification of optimal MSW management strategies under multiple uncertainties. he DIWM model is based on ISPP and MILP techniques. he developed DIWM model can (1) handle uncertainties (presented as intervals and probabilities) by constructing a set of scenarios that are representative for the universe of possible outcomes; (2) reflect risk levels of violating system constraints over a multistage context; and () facilitate dynamic analysis for decisions of timing, sizing, and siting in planning capacity expansion for waste management facilities under uncertainty. he modeling results will be useful for generating a range of decision alternatives under various environmental, socioeconomic, and system-reliability conditions. MEHODOLOGY First, a scenario-based MSP model can be formulated as follows 26 : Min f c 1 x 1 E 2 minc 2 x 2 E 2,..., 1 min x c x x 2 subject to A 1 x 1 b 1, 2 x 1 W 2 x 2 b 2, x 1 W x b, x 1, x 2,, x 0. (1a) (1b) where t (b t, c t, t, W t ), t 2,, and are random vectors of appropriate dimension. Assume that the probability associated with each realization of the random vector is known in advance and equal to p i (and i 1, 2,, m), the problem can be equivalently formulated as a linear program by assuming discrete distributions for the uncertain parameters. Convexity properties of the recourse value functions can be effectively used in the decomposition-based solution strategy. 25 28 In model 1, uncertainties can be conceptualized into the scenario tree, with a one-to-one correspondence between the previous random variable and one of the nodes (state of the system) in each stage. 22 However, randomness in other parameters (e.g., waste management facility capacities) also needs to be reflected. he CCP method can be used for dealing with this type of uncertainty and analyzing the risk of violating the uncertain constraints. 29 Consider a general probabilistic linear problem as follows: subject to Min CtX AtX Bt X 0 (2a) (2b) (2c) where X is a vector of decision variables, and A(t), B(t), and C(t) are sets with random elements defined on a probability space, t 29,0. CCP solves the above model by converting it into a deterministic version through (1) fixing a certain level of probability q i (q i [0,1]) for uncertain constraint i, which represents the admissible risk of violating constraint i; and (2) imposing 440 Journal of the Air & Waste Management Association Volume 60 April 2010

Downloaded by [7.44.194.16] at 16:55 0 January 2018 the condition that the constraint should be satisfied with at least a probability level of 1 q i. he feasible solution set is thus subject to the following constraints 10,1 : PrA i tx b i t 1 q i, A i t At, i 1,2,...,m (a) Constraint a is generally nonlinear, and the set of feasible constraints is only convex for some particular cases, one of which is when elements of A i (t) are deterministic and b i (t) are random (for all q i values). Constraint a can be converted into a linear one as follows: A i tx b i t qi, i (b) where b i (t) qi F i 1 (q i ), given the cumulative distribution function (CDF) of b i (i.e., F i (b i )) and the probability of violating constraint i (i.e., q i ). he problem with constraint b is that linear constraints can only reflect the case when the left-hand-side coefficients (A) are deterministic. If both left- and right-hand sides (A and B) are uncertain, the set of feasible constraints may become more complicated; this brings about considerable difficulty to its practical application, particularly for largescale problems. 10,1,2 On the other hand, in many realworld situations, the distribution information may hardly be known, whereas only two bounds of each imprecise parameter can be identified as intervals. herefore, for reflecting uncertainties presented as intervals (in A and C), an extended consideration is the introduction of IPP technique into the CCP framework. his leads to an ICCP model as follows: subject to Min f C X PrA i tx b i t 1 q i, A i t A t, i 1,2,...,m (4a) (4b) x j 0, x j X, j 1,2,...,n (4c) where A {R } m n, C {R } 1 n, X {R } n 1, and R denote a set of interval numbers. An interval value can be defined as a number with known lower and upper bounds but unknown distribution information. 1 hen, model 4 can be converted into an equivalent deterministic version as follows: subject to Min f C X A i tx B t q, A i t A t, i 1,2,...,m (5a) (5b) x j 0, x j X, j 1,2,...,n (5c) where B (t) q {b i (t) qi i 1, 2,, m}. hen, to better reflect (1) uncertainties as presented in probabilities, intervals, and their combinations; (2) dynamics of the uncertainties as well as the relevant decisions; and () reliability of satisfying (or risk of violating) the system constraints under uncertainty, ICCP can be introduced into the above MSP framework. his leads to an ISPP model as follows: subject to Min f C t X t A rt X t B rt, r 1,2,...,m 1 ;,2,..., A it X t A itk Y tk w, itk i 1,2,...,m2 ;,2,...,; k 1,2,..., p tkd tk Y tk (6a) k 1 A st X t A st Y tk B s t qs, s 1,2,...,m ;,2,...,; k 1,2,..., x jt 0, x jt X t, j 1,2,...,n 1 ;,2,..., y jtk 0, y jtk Y tk, j 1,2,...,n 2 ;,2,...,; k 1,2,..., (6b) (6c) (6d) (6e) (6f) where p tk is probability of occurrence for scenario k in period t, with p tk 0 and k p 1 tk 1; is the number of t scenarios in period t, with the total being K t K 1 t ; X t are first-stage variables that must be determined before the realizations of random variables are disclosed; and Y tk are recourse variables that can be determined after the realized random variable values are available. A twostep method is proposed for solving model 6. he submodel for f can be formulated in the first step when the system objective is to be minimized; the other submodel (corresponding to f ) can then be formulated based on the solution of the first submodel. hrough combining solutions of the two submodels, solutions for the ISPP model under a set of q s (s 1, 2,, m ) levels can be obtained. Generally, the ISPP method can deal with uncertainties presented in terms of probabilities and intervals, as well as their combinations, through constructing a multilayer scenario tree. Moreover, the ISSP can reflect dynamics of the uncertainties and the relevant decisions. For all scenarios under consideration, a decision must be made at each stage on the basis of information about the actual realizations of the random variables and the earlier decisions; this allows corrective actions to be taken dynamically for the related policies and can thus help maximize the system benefit. Furthermore, it can be used for Volume 60 April 2010 Journal of the Air & Waste Management Association 441

Downloaded by [7.44.194.16] at 16:55 0 January 2018 examining the reliability of satisfying (or the risk of violating) the system constraints under uncertainty; a range of violations for constraints is allowed that is related to tradeoffs between the system cost and the constraintviolation risk. CASE SUDY Statement of Problems In real-world MSW management problems, many impact factors and their interactions (e.g., uncertainties in economic and technical data, dynamic variation in system components, randomness in waste generation rates, policy analysis in waste-flow allocation, and limited resources and capacities) must be systematically evaluated in planning an integrated MSW system (as shown in Figure 1). he complexities could be further compounded by interactions among the uncertain and dynamic parameters and through additional economic implications caused by improper policies. For example, the MSW generation rates may vary among different consumer behaviors, different communities, and different periods; moreover, waste generation rate from a community could vary temporally, with the detailed level at a given time period being uncertain. Vehicle types, collection efficiencies, oil prices, and collection routes can also affect waste collection; the operation cost may be related to labor fees, equipment prices, energy prices, and management expenses that can result in uncertain waste management cost in the future. Furthermore, the decision for desired waste treatment approach is related to multiple criteria (e.g., environmental, social, and economic objectives) with the detailed practices (e.g., landfill, incineration, composting, and/or recycling) being interrelated and interactive to each other. herefore, incorporation of various uncertain and dynamic complexities within a general mathematical programming framework for improving the regional MSW management is desired. he city of Regina is located in western Canada. Its population is approximately 195,000, and the households generate residential wastes of 71,000 82,000 t/yr.,4 Figure 1. Interrelationships among various system components. Solid waste management in the city covers many areas, ranging from garbage collection to environmental protection. It involves the provision of specific and personal services to most of the residents (through waste collection and recovery) and indirect services to the entire community (through waste disposal and recovery). he generated solid wastes typically include paper, yard waste, food waste, plastics, metals, glass, wood, and other items. Consistent with many communities in western Canada, the city relies mostly on a sanitary landfill for disposal of its MSW. he landfill is located in the northeastern part of the city, occupying 97 ha with an actual landfilling area of 60 ha. Approximately 65,000 t/yr of MSW generated from the residential sector were buried at the landfill (nearly 90% of the total waste generated by households). he existing landfill is expected to be able to accept waste until 2011 or 2012. 5 Because of the scarcity of land around the urban center and the growing opposition from the public with regard to landfill operation, the city is making efforts to divert waste with the ISWM approach to change the current practice of relying solely on the landfill for its waste disposal. Nevertheless, various complexities exist in such a diversion effort that can affect the detailed plans for the city s MSW management. On the basis of the city s waste management policy, a projected waste-flow level is preregulated. If this level is not exceeded, a regular (normal) cost to the system will result. However, if it is exceeded, the surplus waste flow will be disposed of at a premium, resulting in an excess cost (penalty) to the system (i.e., excess flow generated waste assigned quota). In addition, from a long-term planning point of view, the waste generation rate in the city will keep increasing because of the population increase and economic development; the waste management facilities will face problems of insufficiency in their capacities to meet the city s overall waste disposal demand in the future. Generally, the challenge for decision-makers is how to address the above uncertainties and dynamic complexities because of a lack of knowledge founded on an incomplete characterization, understanding, or measurement of the study system. 442 Journal of the Air & Waste Management Association Volume 60 April 2010

Downloaded by [7.44.194.16] at 16:55 0 January 2018 able 1. echnical data. Waste generation levels and the associated probabilities (t/wk) Waste generation rate in period 1 Waste generation rate in period 2 Waste generation rate in period 17, 14 (0.125), 144, 1494 (0.280), 1495, 1577 (0.404), 1578, 1648 (0.191) 1418, 1510 (0.19), 1511, 1605 (0.575), 1606, 1702 (0.22) 1465, 1555 (0.185), 1556, 1648 (0.605), 1649, 1759 (0.210) Lower and upper bounds of regular waste flow (t/wk) o landfill 700, 950 (), 600, 850 (t 2), 500, 750 (t ) o composting facility 100, 200 (), 250, 00 (t 2), 00, 400 (t ) o recycling facility 200, 00 (), 00, 400 (t 2), 50, 450 (t ) Waste-diversion rate 75% (), 6% (t 2), 50% (t ) In detail, the main problems include the following questions: (1) What collection techniques should be used to enhance waste diversion activities? (2) What facilities should be adopted to meet the overall demand for waste disposal? () What is the cost-effective means for satisfying waste disposal and diversion requirements? and (4) What capacity for waste diversion should be expanded and/or developed? able 1 presents the waste generation rates and the associated probabilities, the relevant waste diversion able 2. Costs and revenues for waste management ($/t). goals, and the regular waste levels. he study time horizon is 15 yr, consisting of three 5-yr periods. As required by the authorities, 50% diversion of residential waste landfilled would be achievable within the planning horizon. able 2 provides collection and transportation costs for allowable and excess waste flows from the city to the three facilities, operating costs of the three facilities, penalties for surplus flows, and revenues from the composting and recycling facilities. Costs for waste collection and transportation are estimated based on the existing conditions in the collection areas, the average container size, collection frequency, collection mode (automatic and manual), and collection time (per load). he economic penalties are associated with (1) operating cost for excess waste flows (i.e., disposed locally to alternative and more expensive facilities), (2) transportation cost for excess flows to more remote facilities, and () extra expenses and/or fines caused by the improper policies. Consequently, the penalties are significantly higher than the regular costs. able presents the fixed and variable costs for capacity expansions of the three facilities. he cost and revenue data listed in ables 2 and are expressed in present values. he modeling parameters are mainly identified or assumed based on representative data from governmental reports and literatures. 9 Modeling Formulation In the study system, many factors and related processes are complex with multiperiod, multilayer, and multiuncertainty features. he developed ISPP method is capable Cost/Revenue t 2 t Normal cost for shipping regular waste o landfill 2.0, 7.0 24.52, 28.5 20.74, 2.98 o composting facility 68.0, 78.2 52.09, 59.90 44.06, 50.67 o recycling facility 9.0, 108.5 71.24, 8.11 60.26, 70.1 Normal cost for treating regular waste Landfill 9.0, 17.0 6.90, 1.02 5.8, 11.02 Composting facility 21.0, 26.0 16.09, 19.92 1.61, 16.85 Recycling facility 61.0, 67.8 46.74, 51.95 9.54, 4.95 Normal cost for shipping regular residue Composting facility 1.68, 2.10 1.29, 1.61 1.09, 1.6 Recycling facility 1.47, 1.68 1.1, 1.29 0.95, 1.09 Normal revenue generated by regular waste Composting facility 5.0, 10.0.8, 7.66.24, 6.48 Recycling facility 45.0, 55.0 4.48, 42.14 29.17, 5.65 Penalty cost for shipping excess waste o landfill 48.0, 55.5 6.77, 42.52 1.11, 5.98 o composting facility 102.5, 118.0 78.52, 90.9 66.42, 76.46 o recycling facility 141.0, 162.5 108.01, 124.48 91.7, 105. Penalty cost for treating excess waste Landfill 18.0, 4.0 1.79, 26.05 11.67, 22.04 Composting facility 4.0, 42.0 26.05, 2.18 22.04, 27.2 Recycling facility 104.0, 115. 79.69, 88.4 67.41, 74.74 Penalty cost for shipping excess residue Composting facility 2.52,.15 1.9, 2.41 1.6, 2.04 Recycling facility 2.21, 2.52 1.69, 1.9 1.4, 1.6 Excess revenue generated by excess waste Composting facility 5.0, 10.0.8, 7.66.24, 6.48 Recycling facility 45.0, 55.0 4.48, 42.14 29.17, 5.65 Volume 60 April 2010 Journal of the Air & Waste Management Association 44

able. Capital parameters for facility expansion. Planning Period Parameter t 2 t Cost for landfill expansion Fixed cost ($10 6 ) 4.04,4.85.66,4.9.09,.72 Variable cost ($/t) 2.47,.20 2.24,2.90 1.89,2.45 Cost for composting facility development/expansion Fixed cost ($10 6 ) 2.10,2.52 1.90,2.28 1.61,1.9 Variable cost ($/t) 2.00,2.58 1.81,2. 1.5,1.98 Cost for recycling facility expansion Fixed cost ($10 6 ) 2.78,.08 2.52,2.79 2.1,2.6 Variable cost ($/t) 6.5,7.69 5.75,6.96 4.86,5.89 Downloaded by [7.44.194.16] at 16:55 0 January 2018 of dealing with this type of problem. Moreover, MILP technique will be introduced into the ISPP framework to facilitate dynamic analysis for decisions of timing, sizing, and siting in planning capacity expansions for waste management facilities through constructing a multilayer scenario tree. Decisions for capacity expansion can then be made based on the progressively acquired information about the actual realizations of the random variables and the decisions in the previous stages; this allows corrective actions to be undertaken dynamically such that the relevant expansion costs can be minimized. hus, based on techniques of ISPP and MILP, a DIWM model can be formulated as follows: General Objective Function. (1) Cost for regular waste transportation and operation i 1 L t it R it OP it ] (7a) (2) Cost for excess waste transportation and operation i 1 L t p tkm itk DR it DP it (7b) k 1 () Cost for regular residue transportation and operation i 2 p tkflc 1t Y 1tk VLC 1t X 1tk k 1 p tkfc it Y itk VC it X itk k 1 (7e) (6) Revenue from composting and recycling facilities i 2 L t it RE it i 2 L t p tkm itk RM it (7f) k 1 Constraints. (1) Constraints of waste management facility capacity Pr t L t 1t M 1tk FE i it M itk i 2 t LC X 1tk, t 1,2,...,; k 1,2,..., 1q (7g) t it M itk Pr C i X itk, (7h) t 1,2,...,; i 2, ; k 1,2,...,1q (2) Constraints of waste flow to the landfill i 2 L t it FE i F it OP 1t (7c) Pr 1t M 1tk DG 1t WG, tk t; k 1,2,..., 1 q (7i) (4) Cost for excess residue transportation and operation i 2 L t p tkm itk FE i D it DP 1t (7d) k 1 (5) Capital cost for facility expansion () Constraints of waste disposal demand it M itk WG, tk t; i 1 k 1,2,..., (4) Constraints of regular waste flow (7j) 444 Journal of the Air & Waste Management Association Volume 60 April 2010

Downloaded by [7.44.194.16] at 16:55 0 January 2018 Y itk it l it it u, i, t (7k) (5) Constraints of excess waste flow 0 M itk it, i, t; k 1,2,..., (7l) (6) Constraints of waste management facility expansion 1, 0, if capacity expansion is undertaken if otherwise, i, t; k 1,2,..., t Y 1tk 1, t 1,2,...,; k 1,2,..., (7m) (7n) X itk N itk Y itk, i, t; k 1,2,..., (7o) X itk 0, i, t; k 1,2,..., (7p) In the DIWM model, the objective function covers (1) expense for handling regular and probabilistic excess flows, (2) revenue from composting and recycling facilities, and () probabilistic expansion cost for the three facilities. he model includes continuous and binary decision variables. he binary variables represent the development or expansion options for waste management facilities in different periods (i.e., Y itk ); their solutions can be used for answering the questions related to timing, sizing, and siting for waste management facility development and/or expansion under uncertainty. he continuous variables represent the optimized waste flows from the city to the waste management facilities. Furthermore, the continuous variables include two subsets: those (the first-stage variables, it ) that must be determined before the random variables (i.e., waste generation rates) are known, and those (the recourse variables, M itk and X itk ) that will be determined after the random variables are disclosed. A set of chance constraints on waste management capacities and waste-diversion rates are considered that can help investigate the risks of violating the capacity and diversion constraints. In the DIWM model, fixed-charge cost functions (i.e., eq 7e) are used to reflect economies-of-scale (EOS) effects on the capacity-expansion costs. 25,28 he fixed cost is linked to the capacity expansion activity, whereas the variable cost is linked to the capacity expansion size. he total cost for facility expansion would increase with the capacity expanded; however, because of the fixed-charged cost function, the unit cost for facility expansion would be decreased with the amount of capacity expanded, which reflects the EOS effect on the expansion cost. For example, the unit cost for the landfill expansion in period 1 would be $[6.51, 8.05]/t when the landfill capacity expanded is 1 million t; in comparison, the unit cost for the landfill expansion in period 1 would be $[4.49, 5.6]/t when the capacity expanded is 2 million t. Several assumptions are made when formulating and solving the DIWM model, including (1) on the basis of the local waste management policies, an allowable waste-flow level from the city to each facility is preregulated (violation of this limit will lead to penalties in terms of raised transportation and operation costs); (2) all solid waste flows have to be shipped to a disposal site within a certain period after their generation and no mass loss is incurred in the transportation process; () construction or expansion of any facility should be completed within the time period during which it was initiated; (4) in this study, waste collection and disposal serve only the city s residential sector; (5) uncertainties associated with waste management facility capacity and waste-diversion rate are characterized as random variables with normal probability distributions; and (6) the models are converted into a linear programming model by assuming discrete distributions for the random variables. hen, the DIWM model can be transformed into two deterministic submodels that correspond to the lower and upper bounds of the desired objective. Interval solutions can then be obtained by sequentially solving the two submodels, which can be further interpreted for generating multiple decision alternatives for MSW management under uncertainty. Submodel 1: Min f L t it R it OP it i 1 subject to t i 1 L t p tkm itk DR it DP it k 1 L t it FE i F it OP 1t i 2 i 2 L t p tkm itk FE i D it DP 1t k 1 p tkflc 1t Y 1tk VLC 1t X 1tk k 1 i 2 i 2 p tkfc it Y itk VC it X itk k 1 L t it RE it i 2 (8a) L t p tkm itk RM it k 1 t L 1t M 1tk FE i it M itk LC q i 2 (8b) t X 1tk, t 1,2,...,; k 1,2,..., Volume 60 April 2010 Journal of the Air & Waste Management Association 445

t it M itk C i q X itk, i 2, ; k 1,2,..., t 1,2,...,; (8c) 1t M 1tk DG 1t q WG, tk t; k 1,2,...,Kt (8d) it M itk WG, tk t; k 1,2,...,Kt (8e) i 1 it l it it u, i, t (8f) 0 M itk it, i, t; k 1,2,..., (8g) subject to t L t 1topt M 1tk FE i itopt M itk i 2 t LC q X 1tk, t 1,2,...,; k 1,2,..., t itopt M itk C i q X itk, t 1,2,...,; i 2, ; k 1,2,..., (9b) (9c) Downloaded by [7.44.194.16] at 16:55 0 January 2018 Y itk 1, 0, if capacity expansion is undertaken if otherwise, (8h) i, t; k 1,2,..., t Y 1tk 1, t 1,2,...,; k 1,2,..., (8i) Submodel 2: 0 X itk N itk Y itk, i, t; k 1,2,..., (8j) Min f L t itopt R it OP it i 1 i 1 L t p tkm itk DR it DP it k 1 L t itopt FE i F it OP 1t i 2 i 2 L t p tkm itk FE i D it DP 1t k 1 p tkflc 1t Y 1tk VLC 1t X 1tk k 1 i 2 p tkfc it Y itk VC it X itk k 1 L t itopt RE it i 2 i 2 L t p tkm itk RM it k 1 (9a) itopt M 1tk DG 1t q WG, tk t; k 1,2,...,Kt (9d) itopt M itk WG, tk t; k 1,2,...,Kt (9e) i 1 M itkopt M itk itopt, i, t; k 1,2,..., (9f) 1, if capacity expansion is undertaken Y itk 0, if otherwise, (9g) Y itk Y itkopt i, t; k 1,2,..., t Y 1tk 1, t 1,2,...,; k 1,2,..., (9h) X itkopt X itk N itk Y itk, i, t; k 1,2,..., (9i) where it opt, M itk opt, X itk opt, and Y itk opt are the solutions of submodel 1; M itk opt, X itk opt, and Y itk opt are the solutions of submodel 1. Solutions of DIWM model can then be obtained through integration of the solutions of submodels 1 and 2. he modeling results can be used for supporting decisions of the city s long-term MSW management, such as (1) identification of desired capacity expansion schemes for waste management facilities, (2) allocation of waste flows to suitable facilities, and () analysis of the tradeoff between the cost of waste management and the risk of system disruption. RESULS AND DISCUSSION Figure 2 presents the solutions for waste-flow-allocation patterns to the landfill (including the allowable and excess waste flows) under different waste-generation scenarios and constraint-violation levels (i.e., q levels). Scenario 1 (denoted as symbol LLL) means that the waste generation rates are low in all of the three periods; scenario 6 (denoted as symbol HHH) corresponds to high waste generation rates in the three periods. Moreover, excess flows would be generated if the allowable waste-flow levels as 446 Journal of the Air & Waste Management Association Volume 60 April 2010

Downloaded by [7.44.194.16] at 16:55 0 January 2018 Figure 2. Waste flows to the landfill under different q levels for the (a) lower and (b) upper bounds. preregulated by the authority are exceeded; under such a condition, the total waste flows are the sum of allowable waste and probabilistic excess waste. he results indicate that the waste-flow-allocation patterns would vary with waste generation scenarios and constraint-violation levels (Figure 2). For example, under waste generation scenario MMM (i.e., when the city s waste generation rates are medium in all of the three periods with a joint probability of 14.05%), the total waste disposed of by the landfill would be [516.1, 5622.] 10 t when q 0.01, [5229.9, 5718.4] 10 t when q 0.05, and [514.4, 5800] 10 t when q 0.1. In comparison, under waste generation scenario HHH (i.e., when the city s waste generation rates are high in all of the three periods, with a joint probability of 0.9%), the waste allocated to the landfill would be [5625.8, 598.9] 10 t when q 0.01, [5724.6, 6106.7] 10 t when q 0.05, and [5809, 6199.9] 10 t when q 0.1. he solutions under the other waste-generation scenarios can be similarly interpreted based on the results as presented in Figure 2. In summary, waste flow disposed of by the landfill would increase as the q level raises; this is because an increased q level leads to a softened landfill capacity and a relaxed diversion requirement, whereas the landfill possesses the relatively low costs (regular and penalty costs) for disposing of waste compared with the other two facilities. he total waste flows (obtained from the DIWM model) diverted to the composting and recycling facilities are summarized in Figure. he waste-flow-allocation patterns would vary under different scenarios and q levels because of the temporal and spatial variations of waste generation scenarios and management conditions. For example, under scenario HHH, the optimized waste flows (including allowable and excess flows) treated by the composting facility would be [164.2, 182.9] 10, [1460.7, 1582.] 10, and [1460, 1572.7] 10 t under q 0.01, 0.05, and 0.1, respectively; the optimized flows diverted to the recycling facility would be [1551., 1552.1] 10, 164.9 10, and 1551. 10 t under q 0.01, 0.05, and 0.1, respectively. Correspondingly, the total flows diverted would be [194.5, 85] 10 t under q 0.01, [095.6, 217.2] 10 t under q 0.05, and [02.7, 124] 10 t under q 0.1. In general, waste flows to the composting and recycling facilities should increase as the q level is raised because of the relaxed capacities of the two facilities; however, the results indicate that the waste flows to the two facilities would decrease with the q level. his is because an increased q level can also lead to a relaxed waste-diversion requirement, such that less waste flows would be treated by the two facilities because of their high collection and operation costs. In this study, the increased q level means a raised risk of violating the constraints of waste-management-facility capacities and waste-diversion rates (i.e., relaxed constraints), leading to different waste-management-facility expansion plans in the city. he results indicate that, under all q levels, the landfill would be expanded in period 2, whereas no expansion would be undertaken in periods 1 and. However, different waste generation rates and q levels would lead to varied expansion schemes for the landfill, as shown in Figure 4. For example, when the city s waste generation rates are low in all three periods, its landfill would be expanded with increments of [11.2, 40.7] 10 t under q 0.01, [06.7, 429.6] 10 t under q 0.05, and [08, 4.1] 10 t under q 0.1. In comparison, when the city s waste generation rates are high in all three periods, this facility would be expanded with increments of [41, 511.1] 10, [409.2, 519.7] 10, and [411.2, 524.4] 10 t under q 0.01, 0.05, and 0.1, respectively. Figure 5 presents the solutions for composting facility expansion schemes obtained from the DIWM model. he expansion plans for this facility would also be changed under different q levels. he results indicate that when q 0.01 this facility would be expanded under most of the scenarios over the planning horizon; in comparison, when q 0.05 and 0.1, this facility would only be expanded in period 1 under advantageous conditions, whereas no expansion would be undertaken in periods 2 and. Figure 6 provides the optimal expansion schemes for the recycling facility under q 0.01, 0.05, and 0.1. he results indicate that the recycling facility would be expanded with different Figure. otal waste flows diverted to the composting and recycling facilities for the (a) lower and (b) upper bounds. Volume 60 April 2010 Journal of the Air & Waste Management Association 447

Downloaded by [7.44.194.16] at 16:55 0 January 2018 Figure 4. Expansion schemes for the landfill for the (a) lower and (b) upper bounds. increments in period 1 under the three q levels, whereas no expansion would be needed in periods 2 and. Figure 7 presents the total expanded diversion capacities (for composting and recycling facilities) under different q levels. he total expanded capacities for waste diversion would decrease as the q level is increased. In fact, an increased q level leads Figure 6. Expansion schemes for the recycling facility for the (a) lower and (b) upper bounds. to a decreased strictness for capacity constraint and a relaxed requirement for waste diversion synchronously, such that a lower diversion capacity-expansion level could be generated. Variations in q level also correspond to the decisionmakers preferences regarding the tradeoff between the total cost and the constraint-violation risk. Figure 8, a and b, shows the results for total cost from the DIWM model Figure 5. Expansion schemes for the composting facility for q (a) 0.01, (b) 0.05, and (c) 0.1. Figure 7. Expansion schemes for total diversion capacities for the (a) lower and (b) upper bounds. 448 Journal of the Air & Waste Management Association Volume 60 April 2010

Downloaded by [7.44.194.16] at 16:55 0 January 2018 Figure 8. Costs for waste management activities under different q levels: (a) lower- and (b) upper-bound total cost, (c) lower and (d) upper cost for waste disposal, and (e) lower and (f) upper expansion cost. under a range of q levels (including the lower- and upperbound values, f ). he total costs for the city s waste management would be $[70.25, 96.97] 10 6 under q 0 (i.e., no violation on any constraint), $[69.80, 95.5] 10 6 under q 0.01, $[68.04, 9.49] 10 6 under q 0.05, and $[67.67, 9.0] 10 6 under q 0.10. heir midvalues (i.e., f mid [f f ]/2) would be $8.61 10 6, $81.58 10 6, $80.77 10 6, and $80.5 10 6 when q 0, 0.10, 0.05, and 0.10, respectively. Figure 8, c and d, shows the results for waste disposal cost under different q levels. he costs for directly disposing of waste would be $[57.92, 80.84] 10 6 under q 0, $[57.55, 80.29] 10 6 under q 0.01, $[57.58, 80.40] 10 6 under q 0.05, and $[57.2, 79.99] 10 6 under q 0.10. Moreover, the expanses for waste-management-facility expansion would be $[12.2, 15.1] 10 6 (q 0), $[12.25, 15.06] 10 6 (q 0.01), $[10.46, 1.09] 10 6 (q 0.05) and $[10.4, 1.04] 10 6 (q 0.10), as shown in Figure 8, e and f. he results indicate that the total cost would decrease as q level is raised. his is because (1) an increased q level leads to a decreased strictness for the facility-capacity constraints and thus results in a lower capacity-expansion amount and a lower capital cost; and (2) an increased q level corresponds to a lower waste-diversion requirement, so that more waste flows would be allocated to the landfill and bring about a lower disposal of cost. In this study, two sensitivity analyses were undertaken to identify the effects of variations among different constraints on the objective value. Figure 9, a and b, show the results of the first sensitivity analysis (denoted as SA1) for examining the effects of facility-capacity constraint variation on the system cost under a range of q levels. In SA1, the waste-diversion rates were considered as a set of deterministic values (i.e., no violation on the wastediversion constraint). he total cost obtained would be $[70.14, 95.86] 10 6, $[68.98, 95.19] 10 6 and $[68.95, 95.10] 10 6 under q 0.01, 0.05, and 0.1, respectively. he corresponding mid-values of the total costs would be $8 10 6, $82.08 10 6 and $82.0 10 6 when q 0.10, 0.05, and 0.10, respectively. he mid-value of total cost would be $8.61 10 6 when q 0. Consequently, the mid-cost variations would be $0.61 10 6 (q 0.01), $1.5 10 6 (q 0.05), and $1.58 10 6 (q 0.1) when a set of chance constraints on waste-management-facility capacities were enforced. Figure 9, c and d, presents the results of the second sensitivity analysis (denoted as SA2) for the effects of waste-diversion variation on the objective value, through considering the facility-capacity constraints as a set of interval values (i.e., no violation on the waste-managementfacility constraint). he total cost obtained from SA2 would be $[69.89, 95.47] 10 6 (q 0.01), $[69.62, 95.17] 10 6 (q 0.05), and $[69.51, 95.10] 10 6 (q 0.1); correspondingly, their mid-values would be $82.68 10 6, $82.40 10 6, and $82.1 10 6 when q 0.01, 0.05, and 0.10, respectively. he variations would be $0.9 10 6 (q 0.01), $1.21 10 6 (q 0.05), and $1.0 10 6 (q 0.10), compared with the result under q 0. However, because of the potential for groundwater contamination, the scarcity of land near the urban center, and the growing opposition from the public with regard to landfill disposal, solid waste management is becoming an increasingly complex issue for Volume 60 April 2010 Journal of the Air & Waste Management Association 449

Figure 9. Effects of violating different constraints on total cost: (a) lower- and (b) upper-bound under SA1, and (c) lower- and (d) upper-bound under SA2. Downloaded by [7.44.194.16] at 16:55 0 January 2018 the city. he traditional aim of the local waste managers (i.e., to provide a reliable and low-cost waste removal service for residents through operating solely a landfill) is being required to alter. herefore, from a long-term planning point view, the city should develop an ISWM approach that highlights waste diversion through various reduction, reuse, recycling, and composting programs. Figure 10 presents the relative difference of total costs (denoted as R [(f 0 f*) 100/f 0 ]%) among DIWM, SA1, SA2 and under q 0, where f* are mid-costs obtained from the DIWM, SA1, and SA2 under several q levels, and f 0 is the mid-cost obtained when q 0. he maximum relative difference (of f* and f 0 ) would be.2% obtained through DIWM under q 0.1 (i.e., [(8.11 80.5)/ 8.11] 100%); this implies that the system would achieve the lowest cost when approximately 10% of the violations against capacity and diversion constraints are allowed. However, this would be linked to a potentially high system-failure risk. Summarily, the results illustrate that (1) the system cost (obtained from DIWM, SA1, and SA2) are all lower than that under q 0, (2) the cost obtained from DIWM under a range of q levels are lower than those obtained from SA1 and SA2 because more relaxation on system constraints is allowed in the DIWM model, and () violation of different constraints has different effects on the system cost. Figure 10. conditions. Relative difference of system costs under different Figure 11 presents the plans for the city s landfill expansion under SA1 and SA2. Although the landfill would be expanded in period 2 under SA1 and SA2, their expansion plans would be different from each other. For example, when waste generation rates are low over the planning horizon, under SA1, the landfill would be expanded with increments of [00.8, 419.2] 10 t when q 0.01, [281.4, 404.9] 10 t when q 0.05, and [270.5, 95.7] 10 t when q 0.1; in comparison, under SA2, this facility would be expanded with increments of [28.7, 444.8] 10, [4.7, 459.7] 10, and [52.9, 468.7] 10 t under q 0.01, 0.05, and 0.1, respectively. Generally, under all of waste-generation scenarios, the amount of landfill expansion under SA1 would be lower than that under SA2. his is because, in SA1, a set of chance constraints on waste-management-facility capacities were considered; this could lead to a decreased strictness for the landfill-capacity constraints and thus result in a lower capacity-expansion amount. On the other hand, in SA2, chance constraints on waste-diversion rates were undertaken; this allowed more waste flows being allocated to the landfill, leading to a higher expansion requirement for this facility. Figure 12 shows the results for the expansion schemes of total diversion capacities (including recycling and composting facilities) under SA1 and SA2, which are also different from each other. For example, under scenario MMM, the total diversion capacities expanded under SA1 would be [505.5, 55.4], [402., 450], and [99.6, 450] t/wk when q 0.01, 0.05, and 0.1, respectively; in comparison, the total diversion capacities expanded under SA2 would be [519, 548.], [456.9, 481.9], and [419.1, 444] t/wk under q 0.01, 0.05, and 0.1, respectively. However, when chance constraints on waste management capacities and waste-diversion rates were considered simultaneously (i.e., in DIWM model), the expanded amount of diversion capacity would be lower than those under SA1 and SA2 (see Figure 8). his is because the DIWM model is associated with a relaxed capacity constraint (i.e., leading to a reduced capacity expansion) and a decreased requirement for waste diversion (i.e., allowing more waste disposed of at the landfill); the two facts could 450 Journal of the Air & Waste Management Association Volume 60 April 2010

Downloaded by [7.44.194.16] at 16:55 0 January 2018 Figure 11. Expansion schemes for the landfill under SA1 and SA2: (a) lower- and (b) upper-bound under q 0.01, (c) lower- and (d) upper-bound under q 0.05, and (e) lower- and (f) upper-bound under q 0.1. lead to a lower diversion capacity-expansion level obtained from the DIWM model. CONCLUSIONS In this study, a DIWM model has been developed for planning a MSW management system under uncertainty that is based on an ISPP method. ISPP integrates MSP and ICCP into a general framework such that ISPP can handle uncertainties presented as interval values and probability distributions and support assessing the risk of violating various constraints. Moreover, the developed DIWM model can reflect the dynamics of system uncertainties and decision processes under a representative set of waste generation scenarios. he DIWM model has been applied to a real case of planning a long-term MSW management system. he results demonstrate that the DIWM model can help tackle the dynamic, interactive, and uncertain characteristics of MSW management systems. he results indicate that optimized solutions have been generated for binary and continuous decision variables under various conditions. hey can be used for answering several questions such as (1) how to plan capacity expansion/development for MSW management facilities under multiple uncertainties, (2) how to allocate waste flows to meet a projected diversion goal within a multistage context, and () how to achieve a compromise between a minimized system cost and a maximized system reliability? Generally, decisions at a lower risk level would lead to an increased reliability in fulfilling system requirements but with a higher cost; conversely, a desire for reducing the cost could result in an increased risk of violating system constraints. In the DIWM model, only the EOS effect on the expansion cost was reflected by fixed-charge cost functions; however, the EOS effects on waste transportation and treatment costs (for regulated and excess waste flows) were both neglected, such that the DIWM model can be solved through linear programming method. If the EOS effects on waste transportation and treatment costs are considered, uncertainties and nonlinearities may exist in many system components. A direct nonlinear algorithm is ineffective in dealing with such complexities. In addition, multiple local optima may exist in a nonlinear program, leading to difficulties in identifying the global optimum. Such difficulties could be further intensified because of the existence of interval and stochastic uncertainties. herefore, one of the main challenges with the nonlinear DIWM model is the identification of uncertain relationships between the objective function and the related decision variables. Further research for nonlinear DIWM will be desired for more robustly reflecting the complexities in the MSW management systems. On the other hand, because of the complex nature of the waste management system, the data required for defining different scenarios were extensive. Although most of the obtained data are relatively accurate, others are less so. herefore, increasing the certainty of the datasets through further investigation and verification would help increase the certainty of the generated solutions. Moreover, in practical applications, the solutions from the proposed model are suitable for a preliminary evaluation of various alternatives and for identifying the important data requirement. he model results would be more applicable Volume 60 April 2010 Journal of the Air & Waste Management Association 451

Downloaded by [7.44.194.16] at 16:55 0 January 2018 Figure 12. Expansion schemes for total diversion capacities under SA1 and SA2 for q (a) 0.01, (b) 0.05, and (c) 0.1. for practical situations, if postoptimality analysis methods (e.g., multicriteria decision analysis, group decisionmaking, and public survey) can be performed. NOMENCLAURE i type of waste management facility, with i 1 for landfill, i 2 for composting facility, and i for recycling facility t time period,, 2,, L t length of time period t (wk) DP it operating cost of facility i for excess waste flow during period t ($/t) DG 1t waste diversion rate (%), which represents waste to the landfill (in each period) that should not exceed the level regulated by the authority DR it collection and transportation cost for excess waste flow to facility i during period t ($/t) D it transportation cost for excess residue from facility i to the landfill during period t ($/t) FE i residue flow rate from facility i to the landfill (% of incoming mass to facility i) F it transportation cost for residue flow from facility i to the landfill during period t ($/t) FLC 1t fixed-charge cost for landfill expansion in period t ($10 6 ) FC it fixed-charge cost for the development and/or expansion of composting and recycling facilities in period t ($10 6 ) LC existing landfill capacity (t) M itk amount by which the preregulated waste-flow level is exceeded when the waste generation rate is WG tk with probability p tk under scenario k (t/wk) N itk variable upper bound for the expanded capacity in period t under scenario k (t or t/wk) OP it operating cost of facility i for preregulated waste flow during period t ($/t) p tk probability of occurrence for waste generation in period t under scenario k number of waste generation scenarios in period t q admissible risk of violating capacity and diversion constraints, and q [0, 1] RE it revenue generated from composting and recycling facilities during period t ($/t) RM it revenue from composting and recycling facilities because of excess flow during period t ($/t) C i existing capacity of composting and recycling facilities (t/wk) R it collection and transportation cost for preregulated waste to facility i during period t ($/t) 452 Journal of the Air & Waste Management Association Volume 60 April 2010