This article was downloaded by: [Universidad Autonoma de Barcelona]On: 28 October 2014, At: 02:37Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

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Dynamic Analysis for Solid Waste ManagementSystems: An Inexact Multistage IntegerProgramming ApproachYongping Li a & Guohe Huang b ca College of Urban and Environmental Sciences , Peking University , Beijing , People’sRepublic of Chinab Chinese Research Academy of Environmental Science , Beijing Normal University ,Beijing , Regina , Saskatchewan, Canadac Faculty of Engineering, University of Regina , Regina , Saskatchewan , CanadaPublished online: 24 Jan 2012.

To cite this article: Yongping Li & Guohe Huang (2009) Dynamic Analysis for Solid Waste Management Systems: AnInexact Multistage Integer Programming Approach, Journal of the Air & Waste Management Association, 59:3, 279-292

To link to this article: http://dx.doi.org/10.3155/1047-3289.59.3.279

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Dynamic Analysis for Solid Waste Management Systems: AnInexact Multistage Integer Programming Approach

Yongping LiCollege of Urban and Environmental Sciences, Peking University, Beijing, People’s Republicof China

Guohe HuangChinese Research Academy of Environmental Science, Beijing Normal University, Beijing, People’sRepublic of China; and Faculty of Engineering, University of Regina, Regina, Saskatchewan,Canada

ABSTRACTIn this study, a dynamic analysis approach based on aninexact multistage integer programming (IMIP) model isdeveloped for supporting municipal solid waste (MSW)management under uncertainty. Techniques of interval-parameter programming and multistage stochastic pro-gramming are incorporated within an integer-program-ming framework. The developed IMIP can deal withuncertainties expressed as probability distributions andinterval numbers, and can reflect the dynamics in termsof decisions for waste-flow allocation and facility-capacityexpansion over a multistage context. Moreover, the IMIPcan be used for analyzing various policy scenarios that areassociated with different levels of economic conse-quences. The developed method is applied to a case studyof long-term waste-management planning. The resultsindicate that reasonable solutions have been generatedfor binary and continuous variables. They can help gen-erate desired decisions of system-capacity expansion andwaste-flow allocation with a minimized system cost andmaximized system reliability.

INTRODUCTIONEffective planning for municipal solid waste (MSW) man-agement is important for facilitating sustainable socioeco-nomic development in urban communities throughout

the world. However, such a planning effort is complicatedwith a variety of uncertainties as well as their interactions.Such uncertainties may exist in many system componentsand impact factors and will affect the related decisionprocesses. These uncertainties may be further multipliedbecause many activities and system components are ofmultiperiod, multilayer, and multi-objective features;they are also associated with economic penalties if thepreregulated policies are violated. Moreover, an inte-grated MSW management system may involve severalwaste-management facilities to meet the overall demandfor waste processing, treatment, and disposal. These facil-ities have overall cumulative or daily operating capacitylimits. Conflicts between increasing waste-generationrates and decreasing waste-treatment/disposal capacitiesare being further intensified because of the growing pop-ulation and the boosting economic development. There-fore, incorporation of various uncertain and dynamiccomplexities within a general mathematical program-ming framework is desired for evaluating integrated pol-icy effects and improving the management.

During the past decades, inexact optimization tech-niques were developed to deal with uncertainties in theMSW management; they included fuzzy, stochastic, andinterval mathematical programming (FMP, SMP, andIMP, respectively).1–17 For example, Chang and Wang9

developed a fuzzy goal programming model to evaluatethe compatibility of waste recycling and incineration inthe planning of MSW management systems. Huang et al.11

proposed a violation analysis approach for the planning ofsolid waste-management systems under uncertainty, on thebasis of methods of interval-parameter fuzzy integer pro-gramming and regret analysis. However, the main limita-tions of IMP and FMP remained in their difficulties tack-ling uncertainties expressed as probability distributionsand in providing a linkage between the preregulated pol-icies and the associated economic implications.

Two-stage stochastic programming (TSP) is effectivefor handling optimization problems in which an analysisof policy scenarios is desired while the model’s right-hand-side uncertainties are expressed as probability dis-tributions. The fundamental idea behind TSP is the con-cept of recourse, which is the ability to take corrective

IMPLICATIONSConflicts between increased waste-generation rates anddecreased waste-treatment capacities are being intensifiedbecause of the growing population and the boosting eco-nomic development. Capacity expansion has become acrucial issue in planning MSW management systems. Thedeveloped IMIP method can facilitate dynamic analysis ofcapacity-expansion planning for waste-management facil-ities within a multistage context under uncertainties. Theuncertain and dynamic information can be incorporatedwithin a multilayer scenario tree; revised decisions are per-mitted in each time period on the basis of the realizedvalues of uncertain events. This can help minimize theeconomic penalties and/or capital costs associated withthe MSW management systems.

TECHNICAL PAPER ISSN:1047-3289 J. Air & Waste Manage. Assoc. 59:279–292DOI:10.3155/1047-3289.59.3.279Copyright 2009 Air & Waste Management Association

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actions after a random event has taken place.18 Recently,a few researchers considered uncertainties in the MSWsystem through the TSP approaches. For example, Maq-sood and Huang12 explored an inexact two-stage pro-gramming model for planning solid waste managementunder uncertainty, in which interval parameters were in-corporated within a two-stage stochastic optimizationframework; however, this method was unable to reflectdynamic complexities in waste-management systems,such as timing, sizing, and siting decisions in terms ofcapacity-expansion schemes for waste-management facil-ities. Li et al.16 developed an inexact two-stage stochasticmixed integer linear programming model for capacityexpansion of MSW management systems under stochasticconditions; however, the expansion scheme must havebeen determined at the first stage before the uncertaintieswere realized, whereas recourse actions could only beundertaken at the second stage to correct any infeasibil-ity. Therefore, TSP could hardly reflect dynamic variationsof system conditions, especially for large-scale multistageproblems with a sequential structure.18–20

Multistage stochastic programming (MSP) methodsextend the TSP approaches by permitting revised deci-sions in each time stage on the basis of the sequentiallyrealized uncertain events. The uncertain information in aMSP model is often modeled through a multilayer sce-nario tree.18 In the past decades, several researchers dealtwith capacity-expansion issues under stochastic condi-tions through development of multistage stochastic inte-ger programming (MSIP) models, in which an expansionscheme that hedged against the multilayer scenario treecould be generated.21–24 Unfortunately, no application ofMSIP to waste management was reported. The primaryadvantage of such a scenario-based stochastic program-ming method was the flexibility it offered in modeling thedecision processes and defining the scenarios, particularlywhen the state dimension was high.25,26 However, MSIPrequired probabilistic specifications for uncertain param-eters, whereas in many practical problems the quality ofinformation that could be obtained was mostly not satis-factory enough to be presented as probabilities. In addi-tion, even if such distributions were available, reflectionof them and their interactions in large-scale MSP modelscould be extremely challenging.19 The IMP methods wereeffective for handling uncertain parameters that couldnot be quantified as distribution functions in the model’sleft- and/or right-hand sides as well as in the objectivefunction.5 Consequently, one potential approach for bet-ter accounting for the uncertainties, economic penalties,and system dynamics within a multistage context is toincorporate the IMP and MSIP within a general optimiza-tion framework.

Therefore, the objective of this study is to develop aninexact multistage integer programming (IMIP) methodfor waste management through incorporating techniquesof IMP and MSP within an integer-programming (IP)framework. The IMIP method will be able to tackle uncer-tainties expressed as probability distributions and discreteintervals, and to reflect dynamics in terms of decisions forwaste-flow allocation and facility-capacity expansionthrough constructing a multilayer scenario tree. Then, acase study will be provided for demonstrating how the

developed model will support dynamic analysis for MSWmanagement under uncertainty. The results will be usedfor generating a range of decision alternatives under var-ious system conditions, and thus for helping MSW man-agers to identify desired waste-management policies.

METHODOLOGYConsider a waste-management system wherein a manageris responsible for allocating waste flows from multipledistricts to multiple facilities within multiple periods, asillustrated in Figure 1. On the basis of the local waste-management policies, a projected preregulated waste-flowlevel from each district must be provided before furtherinformation of initial system uncertainties is revealed. Ifthis level is not exceeded, it will result in a regular cost tothe system. However, if it is exceeded, the surplus wasteflow will have to be disposed with a higher cost, resultingin an excess cost to the system; this implies penalties interms of raised transportation and operation costs. Undersuch a situation, the total waste-flow amount will be thesum of both fixed preregulated and probabilistic surplusflows.

The goal is to generate an optimal plan of waste-flowallocation with a minimized system cost. The waste-gen-eration amounts from the districts are uncertain (ex-pressed as random variables); moreover, the relevantwaste-flow allocation plan would be of dynamic feature,in which decisions must be made at discrete points intime and discrete probability levels for several uncertainvariables. Thus, the problem of waste-flow allocation canbe formulated as a scenario-based MSP model with re-course. Uncertainties in the MSP can be conceptualizedinto a multilayer scenario tree, with a one-to-one corre-spondence between the previous random variable andone of the nodes (states of the system) in each time stage(t). Furthermore, because the planning problem underconsideration includes multiple periods, a discount factoris needed for each period to derive the present cost orrevenue values. Therefore, a multistage stochastic waste-management model can be formulated as follows:

Min f � �j�1

J �t�1

T

LtT1 jt�t�TR1 jt � OP1t� � �i�2

I �j�1

J �t�1

T

LtTijt�t�TRijt

� OPit � FEi�FTit � OP1t� � REit� (1a)

Figure 1. The study system.

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� �j�1

J �t�1

T �k�1

Kt(j)

LtpjtkM1 jtk�t�DR1 jt � DP1t� (1a cont)

� �i�2

I �j�1

J �t�1

T �k�1

Kt(j)

LtpjtkMijtk�t�DRijt � DPit

� FEi�DTit � DP1t� � RMit]

subject to:

�j�1

J �t�1

T

Lt�(T1 jt � M1 jtk) � �i�2

I

FEi(Tijt � Mijtk)� � LC,

k � 1, 2, · · · , Kt�j�

(1b)

�j�1

J

�Tijt � Mijtk� � TCi,(1c)

� h, t; i � 2, 3, · · · , I; k � 1, 2, · · · , Kt�j�

�i�1

I

�Tijt � Mijtk� � WGjtk, � j, t; k � 1, 2, · · · , Kt�j� (1d)

�j�1

J

�T1 jt � M1 jtk� � DG1t �j � 1

J

WGjtk,(1e)

� t; k � 1, 2, · · · , Kt�j�

Tijt min � Tijt � Tijt max, � i, j, t (1f)

0 � Mijtk � Tijt, � i, j, t; k � 1, 2, · · · , Kt�j� (1g)

where:

f � expected net system cost ($);i � type of waste management facility, with i � 1

for landfill and i � 2, 3, …, I for waste treat-ment/diversion facilities (e.g., recycling, incin-erating, and composting);

j � name of district, j � 1, 2,. . . , J;t � time period, t � 1, 2, …, T;

Lt � length of time period t (days);DPit � operating cost of facility i for excess waste flow

during period t ($/t), where DPit � OPit and i �1, 2, …, I;

DPijt � collection and transportation cost for excesswaste flow from district j to facility i duringperiod t ($/t), where DPijt � TRijt and i � 1, 2,…, I;

DTit � transportation cost for excess waste residuefrom facility i to the landfill during period t($/t), where DTit � FTit and i � 2, 3, …, I;

FEi � residue flow rate from facility i to the landfill(% of incoming mass to facility i), i � 2, 3, …,I;

FTit � transportation cost for preregulated residueflow from facility i to the landfill during periodt ($/t), i � 2, 3, …, I;

LC � existing landfill capacity (t);Mijtk � amount by which the preregulated waste-flow

level (Tijt) is exceeded when the waste-genera-tion rate is WGjtk with probability pjtk underscenario k (t/day);

OPit � operating cost of facility i for preregulatedwaste flow during period t ($/t);

pjtk � probability of occurrence for waste generationin district j in period t under scenario k, withpjtk � 0 and ¥

k�1Kt

(j)pjtk � 1;

Kt(j) � number of waste-generation scenarios in dis-

trict j in period t, with the total number ofscenarios (for each district) being K(j)

� ¥t�1T Kt

(j);�t � discount factor for waste transportation and

operation costs in period t, where �t � 1/[1 �(i g)]t, with i and g being interest and infla-tion rates in period t, respectively;

REit � revenue from waste-treatment facility i duringperiod t ($/t), i � 2, 3, …, I;

RMit � revenue from waste-treatment facility i be-cause of excess flow during period t ($/t), i � 2,3, …, I;

TCi � existing capacity of waste diversion facility i(t/day), i � 2, 3, …, I;

TRijt � collection and transportation cost for preregu-lated waste flow from district j to facility iduring period t ($/t);

Tijt � preregulated waste flow from district j to facil-ity i during period t (t/day);

Tijt max � maximum preregulated waste flow from dis-trict j to facility i during period t (t/day);

Tijt min � minimum preregulated waste flow from dis-trict j to facility i during period t (t/day);

WGjtk � amount of waste generated in district j in pe-riod t under scenario k (t/day).

The objective is to minimize the expected systemcost with a desired plan for waste-flow allocation overthe entire planning horizon. The system cost will coverexpenses for handling preregulated and excess wasteflows minus revenues from the waste-management fa-cilities. The constraints define the interrelationshipsamong the decision variables and the waste generationand management conditions. In detail, constraint 1bdenotes that the total waste flows to the landfill mustnot exceed its existing capacity; constraint 1c meansthat the actual waste flows handled by each facilitymust not exceed its existing capacity; constraint 1ddenotes that the waste flows handled by the waste-management facilities should not be less than the totalwaste generation amount; constraint 1e denotes thatthe waste flows handled by the landfill should meet thewaste-diversion goals as preregulated by the authority;constraint 1f regulates that each preregulated wasteflow must between the minimum and maximum pre-regulated levels; and constraint 1g denotes that theexcess waste flow to each facility should not exceed thepreregulated flow level.

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Model 1 can reflect uncertainties in waste-generationrates presented as random variables when the left-handsides and cost coefficients are deterministic. However, it isincapable of reflecting dynamic complexities such as thetiming, sizing, and siting of waste-management facilities.From a long-term planning point of view, MSW genera-tion rates may keep increasing because of population in-crease and economic development. This tendency couldoften result in insufficient capacities of waste-manage-ment facilities to meet the overall waste disposal demand.Capacity expansion for the facilities becomes a crucialissue, in which a related optimization analysis will typi-cally require the use of integer variables to indicatewhether a particular facility development or expansionoption needs to be undertaken. Consequently, IP can beintroduced into the above multistage model for dealingwith such a capacity-planning problem to generate de-sired plans for waste-management facilities under sto-chastic conditions. In addition, fixed-charge cost func-tions will be used to reflect the economies of scale in theexpansion cost.23,27,28 Therefore, a multistage integer op-timization model can be formulated as follows:

Min f � �j�1

J �t�1

T

LtT1 jt�t�TR1 jt � OP1t�

� �i�2

I �j�1

J �t�1

T

LtTijt�t�TRijt � OPit � FEi�FTit � OP1t�

� REit� � �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkM1 jtk�t�DR1 jt � DP1t�

� �i�2

I �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkMijtk�t�DRijt � DPit

� FEi�DTit � DP1t� � RMit�

� �j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t�FLC1tY1tk � VLC1tX1tk�

� �i�2

I ��j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t(FTCitYitk � VTCitXitk)�

(2a)

subject to:

�j�1

J �t�1

t

Lt�(T1 jt � M1 jtk) � �i�2

I

FEi(Tijt � Mijtk)�� LC � �

t�1

t

X1tk, t � 1, 2, · · · , T; k � 1, 2, · · · , Kt�j�

(2b)

�j�1

J

�Tijt � Mijtk� � TCi � �t�1

t

Xitk,(2c)

t � 1, 2, · · · , T; i � 2, 3, · · · , I; k � 1, 2, · · · , Kt�j�

�i�1

I

�Tijt � Mijtk� � WGjtk, � j, t; k � 1, 2, · · · , Kt�j� (2d)

�j�1

J

�T1 jt � M1 jtk� � DG1t �j�1

J

WGjtk,

(2e)

� t; k � 1, 2, · · · , Kt�j�

Tijt min � Tijt � Tijt max, � i, j, t (2f)

0 � Mijtk � Tijt, � i, j, t; k � 1, 2, · · · , Kt�j� (2g)

Yitk� � 1, if capacity expansion is undertaken� 0, if otherwise ,

� i, t; k � 1, 2, · · · , Kt�j�

(2h)

Xitk � NitkYitk, � i, t; k � 1, 2, · · · , Kt�j� (2i)

Xitk � 0, � i, t; k � 1, 2, · · · , Kt�j� (2j)

where FLC1t and VLC1t are the fixed-charge and variablecosts for landfill expansion in period t; FTCit and VTCit arethe fixed-charge and variable costs for treatment facility iexpansion in period t; Xitk is the continuous variable, denot-ing facility i capacity to be expanded in period t underscenario k; Yitk is the binary variable, identifying whether afacility-expansion action needs to be undertaken in period tunder scenario k; Nitk is the variable upper bound for theexpanded capacity in period t under scenario k; and �t

is thediscount factor for facility-expansion cost in period t (therelevant expansion project is assumed to be completed bythe end of the previous period if the system requires addi-tional capacity at the beginning of a particular period). Ac-tivities for waste management involve investment frommultiple sources, leading to varied interest rates and thusvaried discount factors for the transportation, operation,and capital costs.

In model 2, the decision variables can be sorted intotwo categories: continuous and binary. The continuous vari-ables represent district-to-facility waste flows and capacity-expansion levels, whereas the binary ones indicate whetherindividual capacity-expansion actions need to be carriedout. Obviously, model 2 can address expansions of wastemanagement facilities and uncertainties in waste-genera-tion rates presented as probability distributions, on the basisof a multilayer scenario tree. However, uncertainties mayexist in many system components such as cost and revenuedata, preregulated waste flows, facility capacities, and wastediversion goals. For example, activities for waste manage-ment may involve capitals from multiple sources, leading tovarious interest rates; moreover, from a long-term planningpoint of view, the interest and inflation rates may both keepfluctuating because of the effects from many socioeconomic,technical, legislational, institutional, and political factors.

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Thus, the generating costs for waste collection, transporta-tion, and disposal; the capital costs for facility expansionand development; and the revenues from the relevantwaste-processing products may not be available as determin-istic values. In addition, it may often be difficult for a plan-ner to promise a deterministic waste-allocation allowance(Tijt) to each district when its waste-generation level is un-certain. Furthermore, uncertainties may also exist in capacitiesof waste-management facilities because of variations in work-ing hours, requirements for system maintenance, and incon-sistent manners among workers in operating the facilities.Consequently, the IMP method can be introduced to model2 to facilitate communication of the uncertainties into theoptimization process. This leads to an IMIP model as follows:

Min f � � �j�1

J �t�1

T

LtT1 jt� �t�TR1 jt

� � OP1t� �

� �i�2

I �j�1

J �t�1

T

LtTijt� �t�TRijt

� � OPit� � FEi�FTit

�

� OP1t� � � REit

� � � �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkM1 jtk� �t�DR1 jt

�

� DP1t� � � �

i�2

I �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkMijtk� �t�DRijt

�

� DPit� � FEi�DTit

� � DP1t� � � RMit

� �

� �j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t�FLC1t� Y1tk

� � VLC1t� X1tk

� �

� �i�2

I ��j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t(FTCit� Yitk

� � VTCit� Xitk

� )�

(3a)

subject to:

�j�1

J �t�1

t

Lt�(T1 jt� � M1 jtk

� ) � �i�2

I

FEi(Tijt� � Mijtk

� )� (3b)

� LC � � �t�1

t

X1tk� , t � 1, 2, · · · , T; k � 1, 2, · · · , Kt

�j�

�j�1

J

�Tijt� � Mijtk

� � � TCi� � �

t�1

t

Xitk� ,

(3c)

t � 1, 2, · · · , T; it � 2, 3, · · · , I; k � 1, 2, · · · , Kt�j�

�i�1

I

�Tijt� � Mijtk

� � � WGjtk� , � j, t; k � 1, 2, · · · , Kt

�j� (3d)

�j�1

J

�T1 jt� � M1 jtk

� � � DG1t� �

j�1

J

WGjtk� ,

(3e)

� t; k � 1, 2, · · · , Kt�j�

Tijt min � Tijt� � Tijt max, � i, j, t (3f)

0 � Mijtk� � Tijt

� , � i, j, t; k � 1, 2, · · · , Kt�j� (3g)

Yitk� � � 1, if capacity expansion is undertaken

� 0, if otherwise ,(3h)

� i, t; k � 1, 2, · · · , Kt�j�

Xitk� � NitkYitk

� , � i, t; k � 1, 2, · · · , Kt�j� (3i)

Xitk� � 0, � i, t; k � 1, 2, · · · , Kt

�j� (3j)

where Tijt� , Mijtk

� , Xitk� , and Yitk

� are interval variables; TRijt� ,

OPit�, DRijt

� , DPit�, REit

�, RMit�, LC�, and TCi

� are intervalparameters; WGjtk

� are interval-random variables (i.e., theuncertainties in waste-generation rates may be expressedin terms of probability distributions and, at the sametime, some individual random events can only be quan-tified as intervals; these lead to dual uncertainties); the ‘’and ‘�’ superscripts represent lower and upper bounds ofthe parameters/variables, respectively. An interval is de-fined as a number with known lower and upper boundsbut unknown distribution information.5 For example, let-ting Tijt

and Tijt� be lower and upper bounds of Tijt

� , wehave Tijt

� � [Tijt , Tijt

� ]. When Tijt � Tijt

� , Tijt� becomes a

deterministic number.In model 3, uncertainties expressed as random variables

and interval values can be tackled through a multilayerscenario tree, whereas desired policies for waste-flow alloca-tion and capacity-expansion planning would hedge againstthis tree. The objective is to minimize the sum of regularcosts for preregulated waste flows, probabilistic penalties forexcess flows, and probabilistic expansion costs under uncer-tainty. The IMIP model can be transformed into two deter-ministic submodels that correspond to the lower and upperbounds of the desired objective. Interval solutions can thenbe obtained by solving the two submodels sequentially,which can be further interpreted for generating multipledecision alternatives.5 The submodel corresponding to f

can be firstly formulated as follows:

Min f � � �j�1

J �t�1

T

LtT1 jt� �t�TR1 jt

� � OP1t� �

� �i�2

I �j�1

J �t�1

T

LtTijt� �t�TRijt

� � OPit� � FEi�FTit

�

� OP1t� � � REit

� � � �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkM1 jtk� �t�DR1 jt

�

� DP1t� � � �

i�2

I �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkMijtk� �t�DRijt

� (4a)

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� DPit� � FEi�DTit

� � DP1t� � � RMit

� ]

� �j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t�FLC1t� Y1tk

� � VLC1t� X1tk

� �

� �i�2

I ��j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t(FTCit� Yitk

� � VTCit� Xitk

� )�subject to:

�j�1

J �t�1

t

Lt�(T1 jt� � M1 jtk

� ) � �i�2

I

FEi(Tijt� � Mijtk

� )�(4b)

� LC � � �t�1

t

X1tk� , t � 1, 2, · · · , T; k �1, 2, · · · , Kt

(j)

�j�1

J

�Tijt� � Mijtk

� � � TCi� � �

t�1

t

Xitk� ,

(4c)

t � 1, 2, · · · , T; i � 2, 3, · · · , I; k � 1, 2, · · · , Kt�j�

�i�1

I

�Tijt� � Mijtk

� � � WGjtk� , � j, t; k � 1, 2, · · · , Kt

�j� (4d)

�j�1

J

�T1 jt� � M1 jtk

� � � DG1t� �

j�1

J

WGjtk� ,

(4e)� t; k � 1, 2, · · · , Kt

�j�

Tijt min � Tijt� � Tijt max, � i, j, t (4f)

0 � Mijtk� � Tijt

� , � i, j, t; k � 1, 2, · · · , Kt�j� (4g)

Yitk� � � 1, if capacity expansion is undertaken

� 0, if otherwise ,

� i, t; k � 1, 2, · · · , Kt�j�

(4h)

0 � Xitk� � NitkYitk

� , � i, t; k � 1, 2, · · · , Kt�j� (4i)

where Tijt , Mijtk

, Xitk , and Yitk

are decision variables. LetT

ijt opt

, Mijtk opt , Xitk opt

, and Yitk opt be the solutions of

submodel 4. Then, the submodel corresponding to f� canbe formulated as follows:

Min f � � �j�1

J �t�1

T

LtT1 jt� �t�TR1 jt

� � OP1t� �

� �i�2

I �j�1

J �t�1

T

LtTijt� �t�TRijt

� � OPit�

� FEi�FTit� � OP1t

� � � REit� ]

� �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkM1 jtk� �t�DR1 jt

� � DP1t� �

(5a)

� �i�2

I �j�1

J �t�1

T �k�1

Kt�j�

LtpjtkMijtk� �1�DRijt

�

� DPit� � FEi�DTit

� � DP1t� � � RMit

� ]

� �j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t�FLC1t� Y1tk

� � VLC1t� X1tk

� �

� �i�2

I ��j�1

J

pjtk �t�1

T �k�1

Kt�j�

�t(FTCit� Yitk

� � VTCit� Xitk

� )�subject to:

�j�1

J �t�1

t

Lt�(T1 jt� � M1 jtk

� ) � �i � 2

I

FEi(Tijt� � Mijtk

� )�(5b)

� LC � � �t�1

t

X1tk� , t � 1, 2, · · · , T; k � 1, 2, · · · , Kt

�j�

�j�1

J

�Tijt� � Mijtk

� � � TCi� � �

t�1

t

Xitk� , t �1, 2, · · · ,

(5c)

T; i � 2, 3, · · · , I; k � 1, 2, · · · , Kt�j�

�i�1

I

�Tijt� � Mijtk

� � � WGjtk� , � j, t; k � 1, 2, · · · , Kt

�j� (5d)

�j�1

J

�T1 jt� � M1 jtk

� � � DG1t� �

j�1

J

WGjtk� ,

(5e)

� t; k � 1, 2, · · · , Kt�j�

Mijtk opt� � Mijtk

� � Tijt� , � i, j, t; k � 1, 2, · · · , Kt

�j� (5f)

Tijt opt� � Tijt

� � Tijt max, � i, j, t (5g)

Yitk� � � 1, if capacity expansion is undertaken

� 0, if otherwise ,(5h)

� i, t; k � 1, 2, · · · , Kt�j�

Xitk opt� � Xitk

� � NitkYitk� , � i, t; k � 1, 2, · · · , Kt

�j� (5i)

Yitk� � Yitk opt

� , � i, t; k � 1, 2, · · · , Kt�j� (5j)

where Tijt� , Mijtk

� , Xitk� , and Yitk

� are decision variables. LetT

ijt opt

� , Mijtk opt� , Xitk opt

� , and Yitk opt� be the solutions of

submodel 5. Solutions of primal model 3 can then beobtained through integration of the solutions from sub-models 4 and 5:

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Tijt opt� � �Tijt opt

� , Tijt opt� �, � i, j, t (6a)

Mijtk opt� � �Mijtk opt

� , Mijtk opt� �, � i, j, t; k � 1, 2, · · · , Kt

�j� (6b)

Xitk opt� � �Xitk opt

� , Xitk opt� �, � i, t; k � 1, 2, · · · , Kt

�j� (6c)

Yitk opt� � �Yitk opt

� , Yitk opt� �, � i, t; k � 1, 2, · · · , Kt

�j� (6d)

fopt� � �fopt

� , fopt� � (6e)

Consequently, the optimized waste-flow allocation pat-tern is:

Aijtk opt� � Tijt opt

� � Mijtk opt� , � i, j, t; k � 1, 2, · · · , Kt

�j� (6f)

In comparison with the interval-parameter two-stage sto-chastic IP (ITIP) method as advanced by Li et al.,16 IMIP isadvantageous in reflecting dual dynamics in terms of bothwaste-flow allocation and facility-capacity expansion un-der uncertainty. In ITIP, the capacity-expansion schemefor the entire planning horizon is determined at the firststage, whereas recourse actions are taken at the secondstage to correct any infeasibility. Such recourse actionscan be interpreted as outsourcing additional capacities.16

In IMIP, as a comparison, dynamics of waste-generationrates and excess flows can be taken into account throughthe multilayer scenario tree; moreover, desired facility-expansion schemes that hedge against this tree can begenerated. This implies that, in IMIP, all potential facility-expansion schemes would be considered as modeling sce-narios with associated probability levels. Generally, IMIPhas two advantages over ITIP. Firstly, it can deal withuncertainties expressed as interval values and randomvariables through generating scenarios of future events;these scenarios correspond to different effects of variedwaste-flow allocation and facility-expansion schemes onthe economic objective. Secondly, IMIP can reflect dualdynamics in terms of waste-flow allocation and facility-capacity expansion within a multistage context. A deci-sion at the second or later stage can be made dynamicallyon the basis of both the progressively acquired informa-tion about the actual realizations of the random variablesand the decisions in the previous stages; thus, both con-straint-violation penalties and facility-expansion costscould be potentially minimized.

Furthermore, compared with the existing multistagestochastic IP methods, IMIP can incorporate more uncer-tain information within its modeling framework. It candeal with uncertainties presented as both probabilitiesand intervals. Within a multidistrict, multifacility, multi-period, and multiwaste-generation-level context, solu-tions of IMIP can provide desired waste-flow patterns andfacility-expansion plans with a minimized cost and max-imized system feasibility. The solutions are presented as acombination of deterministic, interval, and distributionalinformation and can thus facilitate the communicationsfor multiple forms of uncertainties as well as the genera-tion of multiple decision alternatives.

CASE STUDYIn the study case, a manager is responsible for allocatingwaste flows from one city to three treatment facilities overa 15-yr planning horizon (with three 5-yr periods). Thecity’s population is approximately 200,000, and thehouseholds generate residential waste of approximately90,000 t/yr. An existing landfill, an incinerator, and acomposting facility are available to serve the MSW treat-ment/disposal needs. The landfill is located in the north-eastern part of the city, occupying 97 ha with an actuallandfilling area of 60 ha. The landfill is used directly tosatisfy waste disposal demand or alternatively to providecapacity for accepting the other facilities’ residues; it hasan overall cumulative capacity of 1.1 to 1.2 million t. Theincinerator generates residues of approximately 30% on amass basis of the incoming waste streams, and the com-posting facility generates residues of approximately 10%of the inflow waste. The incinerator has a treatment ca-pacity of 100–120 t/day; the composting facility can treatapproximately 80 t waste/day. The revenue from the in-cinerator is approximately $15–25/t of waste combusted.The compost product can be sold by $11–17/t. The MSWtypically includes paper, yard waste, food waste, plastics,metals, glass, wood, and others. Among them, the yardand organic wastes (compostable materials) approxi-mately amount 30,000 t/yr, accounting for around 30% ofthe total waste stream.

In fact, a MSW management system involves severalprocesses with socioeconomic and environmental impli-cations, such as waste generation, transportation, treat-ment, and disposal.29 In this study, waste-generation rateis uncertain in nature because it is affected directly byseveral factors, such as economic development, popula-tion growth, and human activities. Moreover, complexi-ties in socioeconomic conditions, solid waste characteris-tics, and geographical conditions may result inuncertainties in estimating the levels of waste generation.For example, several unusual events (e.g., continuinglarge amounts of waste generated during a festival) maylead to a peak of waste-generation rate; moreover, whenthe delivery vehicles arrive late in a previous day but earlyin the following day, a peak of waste treatment will occur.In general, from a long-term planning of point view, thecity’s waste generation rate could fluctuate with random

Table 1. Waste-generation rates and preregulated waste-flow levels.

Time Period

t � 1 t � 2 t � 3

Waste-generation rate, WGtk� (t/day):

L (probability � 20%) �210, 240� �245, 280� �285, 315�M medium (probability � 60%) �240, 280� �280, 330� �315, 365�H high (probability � 20%) �280, 320� �330, 380� �365, 415�

Maximum preregulated waste flow, Tit max (t/day):To landfill 130 130 130To incinerator 75 95 115To composting facility 60 65 70

Minimum preregulated waste flow, Tit min (t/day):To landfill 80 80 80To incinerator 45 60 70To composting facility 40 40 40

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feature because of the existence of many uncertaintiesand complexities. Moreover, the relevant waste-flow allo-cation plan will involve a dynamic feature, in whichdecisions must be made at discrete points in time anddiscrete probability levels for several uncertain variables.Table 1 provides the waste-generation rates under differ-ent probability levels and the preregulated waste flowsfrom the city to the three facilities. The minimum andmaximum preregulated waste flows are used for identify-ing an optimal preregulated flow level. Variations in theminimum and maximum preregulated flows would leadto different optimal preregulated waste flow and thuscorrespond to different policies for managing waste gen-eration and management. Excess waste flows will be gen-erated if the preregulated flow levels are exceeded; thetotal waste flow will then be the sum of both preregulatedand probabilistic excess flows.

Table 2 presents the discounted fixed and variablecosts for incinerator-capacity expansions. Because of theenvironmental impacts from landfill sites, the scarcity ofland near urban centers, and the growing public opposi-tion, the city is implementing waste diversion and creat-ing an integrated MSW management system to reduce thewaste flows disposed at the landfill. It is projected that theproportion of waste flow to the landfill should decreasefrom 50% in period 1 to 40% in period 3. However, theMSW generation rate may keep increasing because ofpopulation increase and economic development. Thesetwo facts imply that the city should consider expandingits incinerating facility, whereas the composting capacitycould hardly be expanded because of the limited com-postable material and compost demand.

Table 3 contains the regular transportation costs forpreregulated waste flows from the city to the three facili-ties, the operating costs of the three facilities, and therevenues from the incinerating and composting facilitiesover the planning horizon. These cost and revenue dataare expressed in present values based on model 3. Obvi-ously, the transportation costs for residues (from the in-cinerator and composting facility to the landfill) are lowerthan those for waste flows from the city to the waste-management facilities. This is mainly because the costsfor residue transportation only contain shipping ex-penses, whereas those for MSW transportation involveboth curbside collection and shipping expenses. In fact,the curbside collection cost depends on the type of vehi-cle used (and costs associated with it) and the efficiency ofcollection. The transportation costs are related to vehiclemovements from the collection areas to the facilities,which include several segments: (1) from garage to collec-tion area at the beginning of a working day, (2) from

collection area to discharge site, (3) from discharge site tocollection area, and (4) from discharge site to garage. Trips2 and 3 may be repeated when necessary.

Table 4 presents the penalty levels for excess wasteflows when the decision-makers underestimate the waste-generation amounts. It is indicated that the penalty ismuch higher than the regular cost. The surplus waste flow(i.e., when the projected preregulated waste level is ex-ceeded) will be disposed of at a premium, resulting in araised cost (penalty) to the system. The raised cost forexcess waste is mainly because of more expensive laborand facility operation (for the collection and transporta-tion of waste, and operation of the facility). In detail, suchraised costs (economic penalties) are associated with: (1)increased collection cost for excess waste (i.e., longer timeand more workers are required for collecting the surpluswaste), (2) increased transportation cost for shipping theexcess waste to more remote facilities (when the capaci-ties of local facilities are exhausted), (3) increased operat-ing costs for waste-management facilities (i.e., extended

Table 2. Discounted fixed and variable costs for capacity expansion ofthe incinerator.

Economic Data

Time Period

t � 1 t � 2 t � 3

Fixed investment cost,FTCt

� ($106)�5.0, 6.5� �4.8, 6.2� �4.6, 6.0�

Variable cost, VTCt� ($/t�yr) �730.0, 912.5� �702.0, 877.4� �675.0, 843.7�

Table 3. Regular costs for preregulated waste flows (present values).

Time Period

t � 1 t � 2 t � 3

Transportation cost, TRit� ($/t):

To landfill �12.8, 17.1� �11.4, 16.1� �10.9, 14.6�To incinerator �9.6, 12.8� �9.0, 12.1� �8.2, 10.9�To composting facility �12.1, 16.1� �11.4, 15.2� �10.3, 13.7�

Operation cost, OPit� ($/t):

Landfill �30, 45� �28.3, 42.4� �25.6, 38.4�Incinerator �60, 80� �56.5, 75.5� �51.2, 68.3�Composting facility �50, 70� �47.1, 66.0� �42.7, 59.8�

Residue transportation cost, FTit� ($/t):

From incinerator �4.7, 6.3� �4.4, 5.9� �4.0, 5.4�From composting facility �8.8, 11.7� �8.3, 11.0� �7.5, 10.0�

Revenue, REit� ($/t):

From incinerator �15, 25� �14.1, 23.5� �12.8, 21.3�From composting facility �10, 15� �9.4, 14.1� �8.5, 12.8�

Table 4. Raised costs for excess waste flows (present values).

Time Period

t � 1 t � 2 t � 3

Transportation cost for excess waste, DRit� ($/t):

To landfill �19.2, 25.6� �18.1, 24.1� �16.4, 21.8�To incinerator �15.2, 20.6� �14.3, 19.4� �13.0, 17.6�To composting facility �18.1, 25.2� �17.1, 23.8� �15.4, 21.5�

Operation cost for excess waste, DPit� ($/t):

Landfill �45, 60� �42.4, 56.6� �38.4, 51.2�Incinerator �90, 120� �84.8, 113.1� �76.8, 102.4�Composting facility �75, 105� �70.7, 99.0� �64.0, 89.6�

Transportation cost for excess residue, DTit� ($/t):

Incinerator �7.1, 9.5� �6.7, 9.0� �6.1, 8.1�Composting facility �13.2, 17.6� �12.4, 16.6� �11.3, 15.0�

Revenue from excess waste, RMit� ($/t):

From incinerator �15, 25� �14.1, 23.5� �12.8, 21.3�From composting facility �10, 15� �9.4, 14.1� �8.5, 12.8�

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working hours, more workers, and more expensive facili-ties), and (4) extra expenses and/or fines caused by con-tingent events.

For the revenue (listed in Tables 3 and 4), the reve-nues generated by preregulated and excess waste flows areconsidered identical in this study. This is mainly becausethe MSW management is costly (i.e., the revenue gener-ated from waste management is far less than the expensefor waste collection, transportation, and disposal). More-over, because the incinerating and composting facilitieshave higher operating costs than the landfill, allotment ofwaste to the landfill would be considered more econom-ical. In the case of excess waste, the allotment to thelandfill would be assigned initially, and then to the incin-erating and composting facilities. Consequently, only asmall portion of excess waste would be shipped to theincinerator and composting facility, resulting in a lowrevenue level from the treatment of excess waste. There-fore, the revenue per unit of treated excess waste is con-sidered to be the same as that for the preregulated waste.

In this case, uncertainties exist in many of the systemcomponents (provided as intervals for economic data andfacility capacities, as well as probability distributions forwaste-generation rates). The problems under consider-ation include: (1) how to allocate waste flows to multiplefacilities, (2) how to identify desired waste-managementpolicies with a minimized risk of system disruption, and(3) how to generate an optimized facility-expansionscheme with sound timing and sizing considerations. Thedeveloped IMIP method is considered to be a suitableapproach for dealing with these problems. Consequently,random variables (waste-generation rates) with probabil-ities ptk can be handled through constructing three sce-nario trees (for waste flows to three facilities) with eachhaving a branching structure of 1-3-3-3. All of the sce-nario trees have the same structure with one initial nodeat time 0 (the present) and three succeeding nodes inperiod 1; each node in period 1 corresponds to threesucceeding ones in period 2, and so on for each node inperiod 3; these result in 27 nodes (scenarios) in period 3.

RESULTS AND DISCUSSIONIn this study, an IMIP model is developed for supportingdecisions of long-term MSW management. The IMIP so-lutions are useful for (1) identification of desired capacity-expansion schemes for waste-management facilities, (2)allocation of waste flows to suitable facilities, and (3)analysis of the tradeoff between the cost of waste man-agement and the risk of system disruption. Table 5 pre-sents the solutions of incinerator expansion schemes ob-tained through the IMIP model under all of the scenarios.The results indicate that no expansion is needed in period1. In period 2, there would be three possible expansionschemes. For example, when the waste-generation ratesare low in period 1 and high in period 2 with a jointprobability of 4% (i.e., under three scenarios of LHL,LHM, and LHH, where L is low, M is medium, and H is ahigh waste-generation rate in periods 1, 2, or 3 whenperiod 1 is listed first in abbreviation, period 2 listedsecond, and period 3 listed third), no expansion is neededunder advantageous conditions, and an expansion of 39t/day would be required under demanding conditions. In

period 3, the incinerator would be expanded under mostof the scenarios (there would only be a few exceptions inwhich the incinerator would be expanded in period 2instead of period 3). For example, when waste-generationrate is low in period 1 and high in periods 2 and 3 with ajoint probability of 0.8% (i.e., under scenario LHH), itwould be appropriate to expand the incinerator by [0,39]t/day in period 2 followed by another[37,60] t/day expan-sion in period 3; consequently, under this scenario, theincinerator would be expended twice and the total ex-panded capacity would be[37,99] t/day.

Figure 2 shows the optimal expansion schemes forthe incinerator under all waste-generation scenarios overthe planning horizon. Scenario 1 denotes that the waste-generation rates are low in the three periods; scenario 27corresponds to high waste-generation rates in the threeperiods. In general, dynamics of capacity expansion canbe taken into account through constructing a multilayerscenario tree. For all of the scenarios under consideration,a decision for facility expansion can be made at each stagein a real-time manner on the basis of the progressivelyacquired information about the actual realizations of therandom variables and the decisions in the previous stages;this allows corrective actions to be taken dynamicallysuch that the relevant expansion and/or developmentcosts can be minimized.

Table 5. Solution of IMIP model for incinerator-capacity expansion.

Waste-GenerationScenario

JointProbability

(%)

Optimized Expansion Scheme(t/day)

t � 1 t � 2 t � 3

LLL 0.8 0 0 �0, 35�LLM 2.4 0 0 �7, 65�LLH 0.8 0 0 �37, 95�LML 2.4 0 0 �0, 35�LMM 7.2 0 0 �0, 65�LMH 2.4 0 0 �30, 95�LHL 0.8 0 �0, 39� 0LHM 2.4 0 �0, 39� 0LHH 0.8 0 �0, 39� �37, 60�MLL 2.4 0 0 �0, 35�MLM 7.2 0 0 �0, 65�MLH 2.4 0 0 �29, 95�MML 7.2 0 0 �0, 35�MMM 21.6 0 0 �0, 65�MMH 7.2 0 0 �25, 95�MHL 2.4 0 �0, 65� 0MHM 7.2 0 �0, 65� 0MHH 2.4 0 �0, 65� 30HLL 0.8 0 0 �0, 28�HLM 2.4 0 0 �0, 58�HLH 0.8 0 0 �30, 88�HML 2.4 0 0 �0, 33�HMM 7.2 0 0 �0, 63�HMH 2.4 0 0 �26, 93�HHL 0.8 0 �0, 63� 0HHM 2.4 0 �0, 63� 0HHH 0.8 0 �0, 63� �26, 30�

Notes: LMH � the waste-generation rates are low in period 1, medium inperiod 2, and high in period 3.

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Table 6 presents the solutions of waste-flow alloca-tion under different waste-generation scenarios. An excessflow will be generated if the preregulated waste flow isexceeded (i.e., excess flow � generated flow optimalpreregulated flow). In the case of excess waste, the allot-ment to the landfill should be assigned initially, and thento the incinerating and composting facilities (the incin-erating and composting facilities have higher operatingcosts and constraint-violation penalties). The waste-flowallocation patterns would vary under different scenariosbecause of the temporal and spatial variations of wastegeneration/management conditions. Analyses of wasteflows allocated to the landfill, incinerator, and compost-ing facility when the waste-generation rates are mediumin all of the three periods (with a joint probability of21.6%) are provided below. The solutions under the otherwaste-generation conditions can be similarly interpretedbased on the results presented in Table 6. In detail, wehave:

• For the landfill, the optimal preregulated flowswould be [100,104] t/day in period 1, [100,130]t/day in period 2, and 96 t/day in period 3. Theexcess flows would be [4,16] t/day in period 1,[10,20] t/day in period 2, and 24 t/day in period3. Thus the optimized total flows to this facilitywould be [104,120], [110,150], and 120 t/day inperiods 1, 2, and 3, respectively (i.e., Aijtk

� � Tijt� �

Mijtk� ).

• For the incinerator, the optimal preregulatedflows would be [56,75] t/day in period 1, [90,95]t/day in period 2, and 115 t/day in period 3. Theexcess flows would be [0,5] t/day in period 1, [0,5]t/day in period 2, and [0,50] t/day in period 3.Thus the optimized total flows to this facilitywould be [56,80], [90,100], and[115,165] t/day inperiods 1 to 3, respectively.

• For the composting facility, the optimal preregulated-waste flows would be 54 t/day in period 1, 55 t/dayin period 2, and 70 t/day in period 3. The excessflows would be 26 t/day in period 1, 25 t/day inperiod 2, and 10 t/day in period 3. Thus the opti-mized total flow to this facility would be 80 t/day ineach of the three periods.

Figures 3–5 provide the optimized waste-flow allocationpatterns (including preregulated and excess flows) from

the city to the landfill, incinerator, and composting facil-ity over the planning horizon. Under different waste-generation scenarios, the total waste flows to the threefacilities would fluctuate dynamically. Moreover, the ex-panded capacities for the incinerator in periods 2 and 3would result in changes in available waste-managementcapacities and thus variations of waste-flow allocationpatterns.

The expected net system cost would be fopt� � $[76.4,

146.7] � 106, which covers (1) expenses for handlingfixed preregulated and probabilistic excess flows, (2) rev-enues from incinerating and composting facilities, and (3)probabilistic expansion cost for the incinerator. In detail,the regular cost for handling preregulated waste flowswould be $[60.85, 104.38] � 106; the penalty for handlingexcess flows would be $[14.41, 36.48] � 106; the probabi-listic expenses for incinerator expansion would be $[1.17,5.88] � 106. Solution of the objective function value (fopt

� )provides two extremes of net system cost over the plan-ning horizon. As the actual value of each variable varieswithin its lower and upper bounds, the net system costwould change correspondingly between fopt

and fopt� . A

plan with a lower cost is based on an anticipation of lowerwaste-generation rate but, at the same time, it may resultin a higher risk of violating environmental requirements.Conversely, a plan with a higher system cost would betterresist environmental risk (i.e., lower constraint-violationrisk) because it is based on an anticipation of higherwaste-generation rate; moreover, it corresponds to a con-servative estimation towards the economic effects (i.e.,lower revenues and upper costs). These demonstrate theexistence of a tradeoff between system cost and environ-mental risk.

The problem can also be solved through the conven-tional two-stage stochastic integer linear programming(ITIP) method that is not associated with the multilayerscenario tree.16 The results obtained through ITIP aresignificantly different from those through IMIP. BecauseITIP does not have a tree structure, the uncertain waste-generation levels in each stage (period) are considered tobe discrete and mutually independent. For the incinera-tor, there would be two expansion options over the plan-ning horizon. When the decision scheme tends towardsf under advantageous conditions, the incinerator wouldbe expanded by 19 t/day in period 3; when the scheme

Figure 2. Expansion schemes for the incinerator under different waste-generation scenarios.

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Table 6. Solution of IMIP model for waste-flow allocation.

Scenario

JointProbability

(%) Facility

Optimized Excess Flow Mijtk opt� (t/day) Optimized Total Waste Allocation Aijtk opt

� (t/day)

t � 1 t � 2 t � 3 t � 1 t � 2 t � 3

LLL 0.8 Landfill 0 0 �0, 4� �100, 104� �100, 130� �96, 100�Incinerator 0 0 �0, 20� �56, 75� �90, 95� �115, 135�Composting �0, 7� 0 �4, 10� �54, 61� 55 �74, 80�

LLM 2.4 Landfill 0 0 �12, 24� �100, 104� �100, 130� �108, 120�Incinerator 0 0 �12, 50� �56, 75� �90, 95� �127, 165�Composting �0, 7� 0 10 �54, 61� 55 80

LLH 0.8 Landfill 0 0 �32, 44� �100, 104� �100, 130� �128, 140�Incinerator 0 0 �42, 80� �56, 75� �90, 95� �157, 195�Composting �0, 7� 0 10 �54, 61� 55 80

LML 2.4 Landfill 0 �7, 20� 4 �100, 104� �107, 150� 100Incinerator 0 �3, 5� �0, 20� �56, 75� �93, 100� �115, 135�Composting �0, 7� 25 �0, 10� �54, 61� 80 �70, 80�

LMM 7.2 Landfill 0 �7, 20� �19, 24� �100, 104� �107, 150� �115, 120�Incinerator 0 �3, 5� �5, 50� �56, 75� �93, 100� �120, 165�Composting �0, 7� 25 10 �54, 61� 80 80

LMH 2.4 Landfill 0 �7, 20� �39, 44� �100, 104� �107, 150� �135, 140�Incinerator 0 �3, 5� �35, 80� �56, 75� �93, 100� �150, 195�Composting �0, 7� 25 10 �54, 61� 80 80

LHL 0.8 Landfill 0 �34, 44� 0 �100, 104� �134, 174� 96Incinerator 0 �26, 31� �0, 24� �56, 75� �116, 125� �115, 139�Composting �0, 7� 25 �4, 10� �54, 61� 80 �74, 80�

LHM 2.4 Landfill 0 �34, 44� �12, 20� �100, 104� �134, 174� �108, 116�Incinerator 0 �26, 31� �12, 54� �56, 75� �116, 125� �127, 169�Composting �0, 7� 25 10 �54, 61� 80 80

LHH 0.8 Landfill 0 �34, 44� �32, 40� �100, 104� �134, 174� �128, 136�Incinerator 0 �26, 31� �42, 84� �56, 75� �116, 125� �157, 199�Composting �0, 7� 25 10 �54, 61� 80 80

MLL 2.4 Landfill �4, 16� 0 4 �104, 120� �100, 130� 100Incinerator �0, 5� 0 �0, 20� �56, 80� �90, 95� �115, 135�Composting 26 0 �0, 10� 80 55 �70, 80�

MLM 7.2 Landfill �4, 16� 0 �20, 24� �104, 120� �100, 130� �116, 120�Incinerator �0, 5� 0 �4, 50� �56, 80� �90, 95� �119, 165�Composting 26 0 10 80 55 80

MLH 2.4 Landfill �4, 16� 0 �40, 44� �104, 120� �100, 130� �136, 140�Incinerator �0, 5� 0 �34, 80� �56, 80� �90, 95� �149, 195�Composting 26 0 10 80 55 80

MML 7.2 Landfill �4, 16� �10, 20� 4 �104, 120� �110, 150� 100Incinerator �0, 5� �0, 5� �0, 20� �56, 80� �90, 100� �115, 135�Composting 26 25 �0, 10� 80 80 �70, 80�

MMM 21.6 Landfill �4, 16� �10, 20� 24 �104, 120� �110, 150� 120Incinerator �0, 5� �0, 5� �0, 50� �56, 80� �90, 100� �115, 165�Composting 26 25 10 80 80 80

MMH 7.2 Landfill �4, 16� �10, 20� 44 �104, 120� �110, 150� 140Incinerator �0, 5� �0, 5� �30, 80� �56, 80� �90, 100� �145, 195�Composting 26 25 10 80 80 80

MHL 2.4 Landfill �4, 16� �35, 40� 4 �104, 120� �135, 170� 100Incinerator �0, 5� �25, 35� �0, 20� �56, 80� �115, 120� �115, 135�Composting 26 25 �0, 10� 80 80 �70, 80�

MHM 7.2 Landfill �4, 16� �35, 40� �19, 24� �104, 120� �135, 170� �115, 120�Incinerator �0, 5� �25, 35� �5, 50� �56, 80� �115, 120� �120, 165�Composting 26 25 10 80 80 80

MHH 2.4 Landfill �4, 16� �35, 40� �39, 44� �104, 120� �135, 170� �135, 140�Incinerator �0, 5� �25, 35� �35, 80� �56, 80� �115, 120� �150, 195�Composting 26 25 10 80 80 80

HLL 0.8 Landfill �21, 25� 0 �4, 11� �121, 129� �100, 130� �100, 105�Incinerator �23, 36� 0 �0, 13� �78, 111� �90, 95� �115, 128�Composting 26 0 �0, 10� 80 55 �70, 80�

HLM 2.4 Landfill �21, 25� 0 �19, 31� �121, 129� �100, 130� �105, 127�Incinerator �23, 36� 0 �5, 43� �78, 111� �90, 95� �120, 158�Composting 26 0 10 80 55 80

HLH 0.8 Landfill �21, 25� 0 �39, 51� �121, 129� �100, 130� �135, 147�Incinerator �23, 36� 0 �35, 73� �78, 111� �90, 95� �150, 188�Composting 26 0 10 80 55 80

HML 2.4 Landfill �21, 25� �10, 25� �4, 6� �121, 129� �110, 155� �100, 102�Incinerator �23, 36� 0 �0, 18� �78, 111� �90, 95� �115, 133�Composting 26 25 �0, 10� 80 80 �70, 80�

HMM 7.2 Landfill �21, 25� �10, 25� �23, 26� �121, 129� �110, 155� �119, 122�Incinerator �23, 36� 0 �1, 48� �78, 111� �90, 95� �116, 163�Composting 26 25 10 80 80 80

HMH 2.4 Landfill �21, 25� �10, 25� �43, 46� �121, 129� �110, 155� �139, 142�Incinerator �23, 36� 0 �31, 78� �78, 111� �90, 95� �146, 193�Composting 26 25 10 80 80 80

HHL 0.8 Landfill �21, 25� �30, 45� �4, 6� �121, 129� �130, 175� �100, 102�Incinerator �23, 36� 30 �0, 18� �78, 111� �120, 125� �115, 133�Composting 26 25 �0, 10� 80 80 �70, 80�

HHM 2.4 Landfill �21, 25� �30, 45� �23, 26� �121, 129� �130, 175� �119, 122�Incinerator �23, 36� 30 �1, 48� �78, 111� �120, 125� �116, 163�Composting 26 25 10 80 80 80

HHH 0.8 Landfill �21, 25� �30, 45� �43, 46� �121, 129� �130, 175� �139, 142�Incinerator �23, 36� 30 �31, 78� �78, 111� �120, 125� �146, 193�Composting 26 25 10 80 80 80

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tends towards f� under demanding conditions, it wouldbe expanded by 29 t/day in period 2 followed by another40 t/day expansion in period 3. Table 7 provides solutionsof waste-flow allocation through the ITIP model. It isindicated that, because of its simplification of uncertain-ties into several discrete scenarios, the ITIP is unable toexploit the sequential effects in the second and laterstages. The solution of objective function value is fopt

� �$[77, 142.9] � 106, demonstrating a shrunk solution in-terval (and thus a potential loss of the related informa-tion) as compared with the IMIP solution. Therefore,

compared with the ITIP, the IMIP can incorporate moreuncertain information within its modeling framework. Itcan also reflect dual dynamics in terms of decisions forboth waste-flow allocation and facility-capacity expan-sion over a multistage context.

CONCLUSIONSIn this study, a dynamic optimization approach based onthe IMIP method has been developed for MSW manage-ment under uncertainty. The developed IMIP can dealwith uncertainties expressed as probability distributions

Figure 4. Waste flows to the incinerator under different waste-generation scenarios.

Figure 3. Waste flows to the landfill under different waste-generation scenarios.

Figure 5. Waste flows to the composting facility under different waste-generation scenarios.

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and interval values, and can reflect dynamics in terms ofdecisions for waste-flow allocation and facility-capacityexpansion, through transactions at discrete points of acomplete scenario set over a multistage context. Further-more, it can reflect the effects of economies of scale inexpansion costs through introduction of the fixed-chargecost functions. The IMIP can be transformed into two deter-ministic submodels that correspond to the lower and upperbounds of the objective-function value, on the basis of aninteractive algorithm. Interval solutions, which are usefulfor generating multiple decision alternatives, can then beobtained by sequentially solving the two submodels.

The IMIP method has been applied to a case study oflong-term MSW management planning based on a mul-tilayer scenario tree that is associated with a finite groupof scenarios. The results indicate that reasonable solutionshave been generated for both binary and continuous vari-ables. They are useful for supporting decisions of waste-flow allocation and system-capacity expansion. Comparedwith the conventional inexact two-stage programmingmethod, the IMIP can incorporate more uncertain informa-tion within its modeling framework. It can also reflect dualdynamics in terms of decisions for both waste-flow alloca-tion and facility-capacity expansion within a multistagecontext. Although this study is the first attempt for waste-management systems through the IMIP approach, the re-sults suggest that the developed method is applicable forother planning problems that are associated with dynamiccomplexities within multistage contexts as well as uncer-tainties in multiple formats.

ACKNOWLEDGMENTSThis research has been supported by the Major State BasicResearch Development Program of MOST (2005CB724207),and the Natural Sciences and Engineering Research Coun-cil of Canada. The authors thank the editor and theanonymous reviewers for their insightful comments andsuggestions.

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Table 7. Solution of ITIP model for waste-flow allocation.

Symbol Facility PeriodLevel of Waste

GenerationProbability

(%)Preregulated Waste Flow

(t/day)Optimized Excess Flow

(t/day)Optimized Total Waste

(t/day)

111 Landfill 1 Low 20 �105, 120� 0 �105, 120�112 Landfill 1 Medium 60 �105, 120� �0, 20� �105, 140�113 Landfill 1 High 20 �105, 120� �35, 40� �140, 160�211 Incinerator 1 Low 20 75 0 75212 Incinerator 1 Medium 60 75 0 75213 Incinerator 1 High 20 75 �0, 5� �75, 80�311 Composting 1 Low 20 60 0 60312 Composting 1 Medium 60 60 �0, 5� �60, 65�313 Composting 1 High 20 60 �5, 20� �65, 80�121 Landfill 2 Low 20 �110.3, 126� 0 �110.3, 126�122 Landfill 2 Medium 60 �110.3, 126� �9.8, 22.5� �120.1, 148.5�123 Landfill 2 High 20 �110.3, 126� �38.3, 45.0� �148.6, 171.0�221 Incinerator 2 Low 20 95 0 95222 Incinerator 2 Medium 60 95 �0, 6.5� �95, 101.5�223 Incinerator 2 High 20 95 �6.5, 34� �101.5, 129�321 Composting 2 Low 20 65 0 65322 Composting 2 Medium 60 65 �0, 15� �65, 80�323 Composting 2 High 20 65 15 80131 Landfill 3 Low 20 114 �0, 12� �114, 126�132 Landfill 3 Medium 60 114 �12, 32� �126, 146�133 Landfill 3 High 20 114 �32, 52� �146, 166�231 Incinerator 3 Low 20 115 0 115232 Incinerator 3 Medium 60 115 �0, 24� �115, 139�233 Incinerator 3 High 20 115 �24, 54� �139, 169�331 Composting 3 Low 20 70 �0, 4� �70, 74�332 Composting 3 Medium 60 70 �4, 10� �74, 80�333 Composting 3 High 20 70 10 80

Notes: Expected net system cost ($106): fopt� � �77.0, 142.8�.

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24. Lulli, G.; Sen, S. A Branch-and-Price Algorithm for Multistage Stochas-tic Integer Programming with Application to Stochastic Batch-SizingProblems; Manage. Sci. 2004, 50, 786-796.

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About the AuthorsYongping Li is a professor in the College of Urban andEnvironmental Sciences at Peking University, People’s Re-public of China. Guohe Huang is a professor and CanadaResearch Chair in the Faculty of Engineering at the Univer-sity of Regina. Please address correspondence to: Dr. Y.P.Li, College of Urban and Environmental Sciences, PekingUniversity, Beijing 100871, People’s Republic of China;phone: �86-10-6275-6137; fax: �86-10-6275-6137;e-mail: yongping.li@urban.pku.edu.cn; or Dr. G.H. Huang,Faculty of Engineering, University of Regina, Regina,Saskatchewan, Canada S4S 0A2; phone: �1-306-585-4095; fax: �1-306-585-4855; e-mail: gordon.huang@uregina.ca.

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