ICQSWM: An inexact chance-constrained quadratic solid waste management model

Resources, Conservation and Recycling 54 (2010) 641–657

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Resources, Conservation and Recycling journal homepage: www.elsevier.com/locate/resconrec

ICQSWM: An inexact chance-constrained quadratic solid waste management model Y. Sun a , G.H. Huang b,∗ , Y.P. Li c a b c

Sino-Canada Center of Energy and Environmental Research, North China Electric Power University, Beijing 102206, China Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 College of Urban and Environmental Sciences, Peking University, Beijing 100871, China

a r t i c l e

i n f o

Article history: Received 23 March 2009 Received in revised form 30 October 2009 Accepted 4 November 2009 Keywords: Chance-constrained programming Quadratic Inexact Environment Optimization Solid waste management Uncertainty

a b s t r a c t An inexact chance-constrained quadratic solid waste management (ICQSWM) model is developed for planning municipal solid waste (MSW) management systems. The ICQSWM improves upon existing methods of inexact two-stage chance-constrained linear programming (ITCLP) and interval quadratic programming (IQP). It can tackle uncertainties presented as both probability distributions and discrete intervals; moreover, it can be used for analyzing various policy scenarios which are associated with different levels of economic penalties when the promised targets are violated. It can also reflect the effects of economies of scale under uncertainty. The developed method is applied to a case study of wasteflow-allocation planning within a MSW management system. The results indicate that the ICQSWM model not only provides an effective tool for reflecting system cost variations but also generates decision alternatives; the solutions obtained can thus help managers to identify desired policies under various environmental and system-reliability constraints and economic objectives. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In municipal solid waste (MSW) management, many impact factors and system objectives are influenced by uncertainty, which may affect the relevant optimization analyses of system activities and thus the associated decision making (Huang et al., 1993; Yeomans and Huang, 2003; Li et al., 2007). Recently, various methods dealing with uncertainties have been developed for planning MSW management systems. Most of them can be grouped into fuzzy, stochastic and interval mathematical programming methods (abbreviated as FMP, SMP and IMP) (Kirca and Erkip, 1988; Zhu and Revelle, 1993; Chang and Wang, 1994, 1995, 1997; Leimbach, 1996; Chang and Lu, 1997; Chang et al., 1997; Huang et al., 1992, 1993, 1994, 1995a,b, 2001; Chanas and Zielinski, 2000; Huang et al., 2002; Maqsood and Huang, 2003; Zeng and Trauth, 2005; Li and Huang, 2006; Li et al., 2007; Li et al., 2008, 2009). For example, Huang et al. (2001) developed an integrated fuzzy-stochastic linear programming method and applied it to MSW management, in which chance-constrained programming and fuzzy linear programming were incorporated within a general interval-parameter mixed integer linear programming framework. Li et al. (2007) developed an inexact two-stage chance-constrained linear pro-

∗ Corresponding author. Tel.: +1 306 585 4095; fax: +1 306 585 4855. E-mail address: [email protected] (G.H. Huang). 0921-3449/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.resconrec.2009.11.004

gramming (ITCLP) method through incorporating techniques of two-stage stochastic programming (TSP) and chance-constrained programming (CCP) within a general interval-optimization framework. The ITCLP method could not only reflect the risk of constraint violations and facilitate the analysis of system dynamics but also tackle uncertainties expressed as discrete intervals and random variables with known probabilities. In general, TSP is effective for problems where an analysis of policy scenarios is desired and when the right-hand side coefficients are random with known probability distributions (Li et al., 2006); CCP can effectively reflect the reliability of satisfying (or risk of violating) system constraints under uncertainty (Li et al., 2007). However, the above methods are based on an assumption that the cost functions are linear. In fact, in real-world decision making problems, nonlinear relations may exist in many system components. For example, economies of scale may affect the cost-coefficients in a mathematical programming problem, and make the relevant objective function nonlinear. Since a global-optimum solution for a nonlinear programming problem under uncertainty was potentially difficult to determine, a grey quadratic programming (IQP) method was proposed by Huang et al. (1995c) through introducing grey quadratic variables into the objective function to approximate the effects of economies of scale). The IQP method was applied to realworld optimization problems that have optimal solutions, such as those that satisfied the Kuhn–Tucker conditions (Kuhn and Tucker, 1951). However, for many complicated problems, a large number of

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Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

possibilities would have to be examined, leading to difficulties in the IQP’s practical application (Huang et al., 1995c). Then, Chen and Huang (2001) developed a derivative algorithm (DAM) for solving the IQP problems. The DAM required much lower computational efforts than the previous algorithms, which was especially meaningful for the IQP’s application to large-scale problems. Therefore, to better reflect economic penalties, system reliabilities, economies of scale effects, and multiple uncertainties, one potential approach is to incorporate the concepts of ITCLP and IQP within a general optimization framework in the solid waste management system. This will then lead to an inexact chanceconstrained quadratic solid waste management (ICQSWM) model. The objective of this study is to develop such an ICQSWM model and apply it to MSW management. The solutions obtained will be analyzed and interpreted for generating decision alternatives and for identifying significant factors that affect the system’s performance. The paper is organized as follows: Section 2 describes the formulation and solution of the ICQSWM model; Section 3 provides an application of the proposed ICQSWM model into a hypothetical study; Section 4 discusses the results and analysis of the case study; Section 5 presents conclusions of the work; Appendix depicts the detailed methodology of the proposed model.

± ± = ˛± X ± + ˇijk , (TRijk ijk ijk ± ± ± DRijk = ijk Mijkh + ± , ijk



J K  

± ± (X1jk + M1jkh )+

Lk j=1

k=1

J 

J I K   

i=1

+Lk

+Lk

Minimize f ±

=

J K I   

i=1 I

j=1 J

+

j=1 J

± Lk Xijk FEi (FTik±

+

I 

k=1 K

j=1

k=1

J K I   

−

i=2

j=1

k=1

(1c)

± ± ± (Xijk + Mijkh ) ≥ wjkh ,

∀j, k, h

(1d)

i=1 ± Xijk

± ± ≥ Xijk ≥ Mijkh ≥ 0,

max

∀i, j, k, h

(1e)

The objective function can be reformulated as follows: Minimize f ±

=Lk

J K I   

i=1

+Lk

j=1

j=1

+Lk

2

± ± ± ± [˛± (Xijk ) +Xijk × (ˇijk +OPik )] ijk

k=1

J K I   

2

± ± ± ± ± [ik FEi (Xijk ) +Xijk × [(FEi × (ı± +OP1k )) − REik ]] ik

k=1

J K H I    

i=1

j=1

k=1

h=1

J K H I    

i=2

j=1

k=1

(2) 2

± ± ± ± [ijk pjh (Mijkh ) +Mijkh ×pjh ×(± + DPik )] ijk

± [± FEi pjh (Mijkh ) ik

2

h=1

± ± ± +Mijkh × [(FEi pjh × (± + DP1k )) − pjh RMik ]] ijk

When the coefficients of objective function have different signs, a derivative algorithm proposed by Chen and Huang (2001) can be employed to solve the ICQSWM model. By letting all leftand/or right-hand-side coefficients be equal to their mid-values, a mid-value quadratic programming submodel can be firstly formulated as: 2

2

[ik mv FEi (Xijk mv ) + Xijk mv × [(FEi × (ıik mv + OP1k mv )) − REik mv ]]

k=1

J K I H    

i=1 I

j=1 J

k=1 K

h=1 H

i=2

j=1

k=1

h=1



(3a) 2

[ijk mv pjh (Mijkh mv ) + Mijkh mv × pjh × (ijk mv + DPik mv )]

2

[ik FEi pjh (Mijkh mv ) + Mijkh mv × [(FEi pjh × (ijk mv + DP1k mv )) − pjh RMik mv ]]

J K  

+

± DPik )

(X1jk

mv

+ M1jkh

mv ) +

 FEi (Xijk

mv

+ Mijkh

J  ± ± Lk pjh Mijkh FEi Xijk (DTik±

+

± DP1k )

j=1

k=1

(3b)

(Xijk

mv

+ Mijkh

mv )

≤ (TCik )qi ,

∀h, k, i = 2, 3, · · ·, I

(3c)

(Xijk

mv

+ Mijkh

mv )

≥ wjkh

∀j, k, h

(3d)

j=1

J K H I    

i=2

mv )

i=2

(1a)

h=1 H

± ± Lk Xijk REik −

Lk

I 

≤ (LCk )q1 , ∀h

± ± ± Lk pjh Mijkh Xijk (DRijk

h=1



j=1 k=1

± OP1k )

k=1 K

 i=2

+

(1b)

∀h, k, i = 2, 3, · · ·, I

subject to:

+

j=1 J

∀h

[˛ijk mv (Xijk mv ) + Xijk mv × (ˇijk mv + OPik mv )]

± ± ± Lk Xijk (TRijk + OPik )

H 

i=1 I

q

≤ (LCk ) 1 ,

j=1

k=1 K

 i=2 I

j=1

± ± FEi (Xijk + Mijkh )

k=1

J K I   

i=2

+Lk

j=1



± ± (Xijk + Mijkh ) ≤ (TCik )qi ,

i=2

= Lk

I 

i=2

+Lk

Minimize fmv

± DTik = ± M ± + ± ) ik ijkh ijk

subject to

2. Model development Consider a MSW management system wherein a manager is responsible for allocating waste flows from multiple districts to multiple facilities within multiple periods; the waste treatment options include landfill, incinerating, composting, and recycling. Considering the transportation costs for municipality-to-facility waste flows and incinerator-to-landfill residue flows are approximated as interval linear functions of waste-flow to reflect the EOS effects (Li et al., 2008). The goal is to achieve optimal planning of waste-flow allocation with a minimized system cost. Therefore, based on the techniques of the ITCLP and IQP methods (as provides in Appendix), an ICQSWM model can be formulated as follows:

± ± ± FTik = ik Xijk + ı± , ik

± ± Lk pjh Mijkh RMik

I 

h=1

i=1

mv ,

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Xijk

max

≥ Xijk

mv

≥ Mijkh

mv

≥ 0,

∀i, j, k, h

(3e)

According to the lemma in Chen and Huang (2001), calcu+ + + + late 2dijk (xijk )mv opt + cijk values. When 2d111 (x111 )mv opt + c111 > 0,

+ + fijk (xijk

opt

+ − ) ≥ fijk (xijk

opt

), ∀i, j, k, according to criterion in Chen and

Huang (2001). This means that all

+ xijk

corresponding to

+ fijk

(and

− − thus all xijk corresponding to fijk ). Based on this criterion, the ICQSWM model can be transformed into two deterministic submodels corresponding to lower and upper bounds of f± according to an interactive algorithm and a derivative algorithm. Submodel (1)

Minimize f −

= Lk

J K I   

I 

+ + + (Xijk + Mijkh ) ≥ wjkh ,

643

∀j, k, h

(5d)

i=1 + Xijk

max

+ + ≥ Xijk ≥ Mijkh ≥ 0,

+ − ≥ Mijkh Mijkh

opt

,

∀i, j, k, h

∀i, j, k, h

(5e) (5f)

In comparison, when the coefficients of objective function have same sign, the ICQSWM model can be transformed into two deterministic submodels corresponding to lower and upper bounds of the objective function value (i.e., submodels (4) and (5). Fig. 1 shows the framework of the ICQSWM model. The detailed solution process

2

− − − [˛− (X − ) + Xijk × (ˇijk + OPik )] ijk ijk

i=1 j=1 k=1

+Lk

J K I   

2

− − − − + [ik FEi− (Xijk ) + Xijk × [(FEi− × (ı− + OP1k )) − REik ]] ik

i=2 j=1 k=1

+Lk

J K H I    

(4a) 2 − [ijk pjh (M − )

+M

−

× pjh × (− ijk

− + DPik )]

i=1 j=1 k=1 h=1 J K H I

+Lk



2

− + [− FEi− pjh (M − ) + M − × [(FEi− pjh × (− + DP1k )) − pjh RMik ]] ik ijk

i=2 j=1 k=1 h=1

subject to: J K  

 − (X1jk

Lk

− + M1jkh )+

j=1 k=1

I 

can be summarized as follows (Huang et al., 1992; Huang, 1998; Chen and Huang, 2001; Maqsood and Huang, 2003; Li et al., 2007):

 − FEi− (Xijk

− + Mijkh )

≤ (LCk )q1 ,

∀h

i=2

(4b)

J 

− − (Xijk + Mijkh ) ≤ (TCik )qi ,

∀h, k, i = 2, 3, · · ·, I

(4c)

j=1 I 

− − − (Xijk + Mijkh ) ≥ wjkh ,

∀j, k, h

(4d)

i=1 − Xijk

max

− − ≥ Xijk ≥ Mijkh ≥ 0,

∀i, j, k, h

(4e)

= Lk

+ ple, if 2d111 (x111 )mv + − fijk (xijk opt ),

opt

+ + + + c111 > 0, we can judge that fijk (xijk

opt

)≥

∀i, j, k, according to criterion in Chen and Huang

+ (2001). This means that all xijk corresponding to f+ (and thus all

− xijk corresponding to f− ).

Submodel (2) Minimize f +

• Step 1: Acquire distribution information for the system constraints and the interval linear functions to reflect the economy of scale effects (e.g., capacities of waste-management facilities, rates of waste-generation and the functions of waste-flow). • Step 2: Formulate the ICQSWM models (1) and (2). • Step 3: When the coefficients of objective function have different signs, we have to solve the following mid-value ICQSWM programming submodel (i.e., submodel (3)). • Step 4: According to Chen and Huang (2001), calculate + + 2dijk (xijk )mv opt + cijk values. And give the judgments. For exam-

J K I   

2

+ + + [˛+ (X + ) + Xijk × (ˇijk + OPik )] ijk ijk

i=1 j=1 k=1

+Lk

J K I   

2

+ + + + − [ik FEi+ (Xijk ) + Xijk × [(FEi+ × (ı+ + OP1k )) − REik ]] ik

i=2 j=1 k=1

+Lk

J K H I    

(5a) + + 2 [ijk pjh (Mijkh )

+ + Mijkh

× pjh × (+ ijk

+ + DPik )]

i=1 j=1 k=1 h=1

+Lk

J K H I    

2

+ + + − [+ FEi+ pjh (Mijkh ) + Mijkh × [(FEi+ pjh × (+ + DP1k )) − pjh RMik ]] ik ijk

i=2 j=1 k=1 h=1

subject to:

 J



K

Lk

+ + (X1jk + M1jkh )+

j=1 k=1

 I

 + + FEi+ (Xijk + Mijkh )

≤ (LCk )q1 ,

∀h

i=2

(5b)

J  j=1

• Step 5: We can then formulate two submodels corresponding to lower and upper bounds of the objective function value according to an interactive algorithm (Huang et al., 1994; Huang, 1996). • Step 6: Solve the f− submodel and obtain M − under different ijkh opt qi levels. − − • Step 7: Calculate the total waste flows A− = Xijk + Mijkh ijkh opt opt − (here we assume Xijk corresponding to f− ) and the objective func-

+ + (Xijk + Mijkh ) ≤ (TCik )qi ,

∀h, k, i = 2, 3, · · ·, I

(5c)

− tion value fopt (most optimistic decision option). • Step 8: Formulate f+ submodel, including the objective function and the relevant constraints.

644

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Fig. 1. Framework of the ICQSWM.

• Step 9: Solve the f+ submodel and obtain M + under different ijkh opt qi levels. + + • Step 10: Calculate the total waste flows A+ = Xijk + Mijkh ijkh opt opt + (here we assume Xijk corresponding to f+ ) and the objective func-

+ tion value fopt (most conservative decision option). − + • Step 11: Thus, we have M ± = [Mijkh , Mijkh ], ijkh opt opt opt

A± ijkh opt

=

± Xijk

± + Mijkh opt

and

± fopt

=

− [fopt ,

+ fopt ],

∀i, j, k, h,

under each given

contain transportation costs for allowable waste flows from the cities to the three facilities, operating costs of the three facilities, penalty costs for surplus waste flows, and revenues from the incinerating and composting facilities over the planning horizon. It is indicated that the regular and penalty costs for the incinerating and composting facilities are both higher than those for the landfill. Table 4 presents the distribution information of uncertain capacities for the three facilities. It is indicated that these capaci-

qi level.

3. Case study

Table 1 Waste-generation rates under different probability levels. Level of waste-generation probability

Consider a hypothetical case wherein a solid waste manager is responsible for allocating waste-flow from two cities to three facilities over a 15-year planning horizon (with three 5-year periods). The existing landfill, incinerator and composting facility are available to serve the MSW disposal needs. The MSW generation rates vary among different periods. Table 1 shows the waste-generation rates and their associated probabilities of occurrence. Tables 2 and 3

h=1 h=1 h=2 h=2 h=3 h=3

j=1 j=2 j=1 j=2 j=1 j=2

(L = low) (L = low) (M = medium) (M = medium) (H = high) (H = high)

0.2 0.2 0.6 0.6 0.2 0.2

± Waste-generation rate, wjkh (t/d)

k=1

k=2

k=3

[210,240] [210,240] [240,280] [240,280] [280,330] [280,330]

[240,270] [240,270] [270,310] [270,310] [310,360] [310,360]

[270,300] [270,300] [300,340] [300,340] [340,390] [340,390]

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

645

Table 2 Regular transportation and operation costs. Time period k=1

k=2

k=3

− TR111 = −0.0129X + 15.30 + = −0.0172X + 20.49 TR111

− TR112 = −0.0141X + 16.83 + TR112 = −0.0189X + 22.53

− TR113 = −0.0156X + 18.52 + TR113 = −0.0208X + 24.79

− TR121 = −0.0123X + 14.58 + TR121 = −0.0163X + 19.40

− TR122 = −0.0135X + 16.04 + TR122 = −0.0179X + 21.34

− TR123 = −0.0148X + 17.64 + TR123 = −0.0197X + 23.48

Waste transportation cost (to incinerator) ($/t): ± − TR211 = −0.0106X + 12.65 TR21k + = −0.0142X + 16.87 TR211

− TR212 = −0.0117X + 13.92 + TR212 = −0.0156X + 18.56

− TR213 = −0.0129X + 15.31 + TR213 = −0.0172X + 20.41

− TR221 = −0.0108X + 12.85 + TR221 = −0.0144X + 17.07

− TR222 = −0.0119X + 14.12 + TR222 = −0.0158X + 18.76

− TR223 = −0.0131X + 15.51 + TR223 = −0.0174X + 20.61

Waste transportation cost (to composting facility) ($/t): ± − TR311 = −0.0123X + 14.58 TR31k + TR311 = −0.0163X + 19.40

− TR312 = −0.0135X + 16.04 + TR312 = −0.0179X + 21.34

− TR313 = −0.0148X + 17.64 + TR313 = −0.0197X + 23.48

− TR321 = −0.0129X + 15.30 + TR321 = −0.0172X + 20.49

− TR322 = −0.0141X + 16.83 + TR322 = −0.0189X + 22.53

− TR323 = −0.0156X + 18.52 + TR323 = −0.0208X + 24.79

[30,45] [50,65] [55,70]

[35,50] [60,80] [65,85]

[40,55] [70,90] [70,90]

− FT21 + FT21 − FT31 + FT31

− FT22 + FT22 − FT32 + FT32

− FT23 + FT23 − FT33 + FT33

Waste transportation cost (to landfill) ($/t): ± TR11k ± TR12k

± TR22k

± TR32k

Operation cost ($/t): ± (landfill) OP1k ± OP2k (incinerator) ± OP3k (composting facility) Residue transportation cost ($/t) ± FT2k ± FT3k

Revenue ($/t) ± (incinerator) RE2k ± RE3k (composting facility)

= −0.0048X + 5.71 = −0.0064X + 7.62 = −0.0078X + 10.21 = −0.0103X + 12.36

[15,25] [20,30]

[20,30] [20,30]

ties vary among different periods, and the corresponding operation costs also vary temporally. Table 5 presents the allowable wasteflow levels from the cities to the three facilities predefined by the authorities. In addition, the incinerator generates residues of approximately 30% (on a mass basis) of the incoming waste streams, and the composting facility generates residues of approximately 10% (on a mass basis) of the inflow wastes. Therefore, the problem under consideration is how to effectively allocate waste flows from the city to suitable waste management facilities to minimize the net system cost. Minimize f ±

= −0.0053X + 6.28 = −0.0070X + 8.38 = −0.0096X + 11.45 = −0.0115X + 13.76

=

= −0.0058X + 6.91 = −0.0077X + 9.22 = −0.0107X + 12.74 = −0.0134X + 15.98

[25,35] [20,30]

The decision variables represent probabilistic excess waste flows from the cities j to facility i in period k under waste-generation ± level h (denoted as Mijkh ). The objective is to minimize system costs through effectively allocating the waste flows from the cities to the three facilities. Because uncertainties exist in a variety of the system components and a linkage to the predefined policies as formulated by local authorities is desired, the ICQSWM method is considered to be a suitable approach for tackling this management problem. Consequently, we have:

2 3 3   

± ± ± Lk Xijk (˛± X ± + ˇijk + OPik ) ijk ijk

i=1 j=1 k=1

2 3 3   

+

± ± ± ± Lk Xijk FEi (ik Xijk + ı± + OP1k ) ik

i=2 j=1 k=1

2 3 3 3    

+

± ± ± ± Lk pjh Mijkh (ijk Mijkh + ± + DPik ) ijk

(6a)

i=1 j=1 k=1 h=1 2 3 3 3



+

± ± Lk pjh Mijkh FEi (± M ± + ± + DP1k ) ik ijkh ik

i=2 j=1 k=1 h=1

2 3 2   

−

i=2 j=1 k=1

± ± (TRijk = ˛± X ± + ˇijk , ijk ijk

± ± Lk Xijk REik −

2 3 3 3    

± ± Lk pjh Mijkh RMik

i=2 j=1 k=1 h=1

± ± ± FTijk = ik Xijk + ı± , ik

± ± ± DRijkh = ijk Mijkh + ± , ijk

± DTijk = ± M ± + ± ) ik ijkh ik

646

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Table 3 Transportation and operation costs for excess waste flows. Time period k=1

k=2

k=3

− DR112 = −0.0212X + 25.9 + DR112 = −0.0234X + 30.8

− DR113 = −0.0224X + 27.7 + DR113 = −0.0236X + 33.5

− DR122 = −0.0201X + 24.2 + DR122 = −0.0233X + 29.3

− DR123 = −0.0221X + 26.9 + DR123 = −0.0235X + 32.8

− DR212 = −0.0166X + 19.87 + DR212 = −0.0219X + 26.38

− DR213 = −0.0179X + 20.98 + DR213 = −0.0232X + 28.67

− DR222 = −0.0168X + 20.07 + DR222 = −0.0221X + 26.58

− DR223 = −0.0181X + 21.18 + DR223 = −0.0234X + 28.87

− DR312 = −0.0201X + 24.2 + DR312 = −0.0233X + 29.3

− DR313 = −0.0221X + 26.9 + DR313 = −0.0235X + 32.8

− DR321 = −0.0196X + 23.4 + DR321 = −0.0232X + 28.7

− DR322 = −0.0212X + 25.9 + DR322 = −0.0234X + 30.8

− DR323 = −0.0224X + 27.7 + DR323 = −0.0236X + 33.5

[45,60] [75,95] [110,130]

[50,70] [90,115] [120,140]

[60,85] [105,135] [130,150]

− DT12 + DT12 − DT22 + DT22

− DT13 + DT13 − DT23 + DT23

Transportation cost for excess waste (to landfill) ($/t): ± − DR111 = −0.0196X + 23.4 DR11k + DR111 = −0.0232X + 28.7 ± DR12k

− DR121 = −0.0189X + 22.5 + DR121 = −0.0223X + 27.6

Transportation cost for excess waste (to incinerator) ($/t): ± − DR211 = −0.0155X + 18.50 DR21k + DR211 = −0.0207X + 24.76 ± DR22k

− DR221 = −0.0157X + 18.70 + = −0.0209X + 24.96 DR221

Transportation cost for excess waste (to composting facility) ($/t): ± − DR311 = −0.0189X + 22.5 DR31k + = −0.0223X + 27.6 DR311 ± DR32k

Operation cost ($/t): ± (landfill) OP1k ± OP2k (incinerator) ± OP3k (composting facility)

Transportation cost for residue to landfill ($/t): − DT11 DTik± + DT11 − DT21 + DT21 Revenue ($/t) ± (incinerator) RM2k ± RM3k (composting facility)

= −0.0071X + 8.4 = −0.0078X + 10.4 = −0.0134X + 16.02 = −0.0178X + 21.30

[15,25] [20,30]

= −0.0077X + 9.3 = −0.0096X + 11.5 = −0.0147X + 17.60 = −0.0196X + 23.45

[20,30] [20,30]

= −0.0078X + 10.20 = −0.0103X + 12.34 = −0.0157X + 18.7 = −00211X + 25.8

[25,35] [20,30]

Table 4 Distributional information of uncertain capacities for the three facilities. qi level

0

0.01

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.99

1.00

k=1 Landfill (t/d) Incinerator (t/d) Composting (t/d)

310 270 140

344 304 174

378 338 208

396 356 226

426 386 256

460 420 290

494 454 324

524 484 354

544 502 372

576 536 406

610 570 440

k=2 Landfill (t/d) Incinerator (t/d) Composting (t/d)

330 310 160

364 344 194

398 378 228

416 396 246

446 426 276

480 460 310

514 494 344

544 524 374

562 542 392

596 576 426

630 610 460

k=3 Landfill (t/d) Incinerator (t/d) Composting (t/d)

360 340 190

394 374 224

428 408 258

446 426 276

476 456 306

510 490 340

544 524 374

574 554 404

592 572 422

626 606 456

660 640 490

subject to ± ± ± ± + M11kh + X12k + M12kh )+ (X11k

Table 5 Allowable waste flows from the city to the three facilities.

± ± ± ± FEi (Xi1k + Mi1kh + Xi2k + Mi2kh )

i=2 q1

≤ (LCk ) ,

Time period

Allowable waste flow (t/d): ± (to landfill) X11k ± X12k (to landfill) ± X21k (to incinerator) ± X22k (to incinerator) ± X31k (to composting facility) ± X32k (to composting facility)

3 

k=1

k=2

k=3

[105,110] [105,110] [80,85] [80,85] [50,55] [50,55]

[110,115] [110,115] [95,100] [95,100] [55,60] [55,60]

[120,125] [120,125] [110,115] [110,115] [60,65] [60,65]

∀h

(6b)

± ± ± ± Xi1k + Mi1kh + Xi2k + Mi2kh ≤ (TCik )qi , 3 

± ± ± (Xijk + Mijkh ) ≥ wjkh , ∀j, k, h

∀h, k, i = 2, 3

(6c)

(6d)

i=1 ± Xik

max

± ± ≥ Xik ≥ Mikh ≥ 0,

∀i, k, h

(6e)

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

647

Table 6a Solutions obtained through the ICQSWM and ITQP (for city 1). Period

Facility

k

i

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

± W1kh

L M H L M H L M H L M H L M H L M H L M H L M H L M H

± Xi2kh (t/d)

[105,110] [105,110] [105,110] [110,115] [110,115] [110,115] [120,125] [120,125] [120,125] [80,85] [80,85] [80,85] [95,100] [95,100] [95,100] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

ITQP model (t/d)

ICQSWM model (t/d)

q=0

q = 0.01

q = 0.05

q = 0.10

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

0 [5,8.93] [0,5.67] 0 10 17.14 0 10 30 0 [0,10.54] 45 0 [0,13.85] 32.86 0 [0,13.85] 20 0 [0,10.54] 15 0 [0,14.95] [0,19] 0 [0,14.95] [0,19]

[105,110] [110,118.93] [105,115.67] [110,115] [120,125] [127.14, 132.14] [120,125] [130,135] [150,155] [80,85] [80,95.54] [125,130] [95,100] [95,113.85] [127.86., 132.86] [110,115] [110,128.85] [130,135] [50,55] [50,65.54] [65,70] [55,60] [55,74.95] [55,79] [60,65] [60,79.95] [60,84]

0 [5,30] 25 0 [10,35] 50 0 [10,35] 50 0 0 [20,54] 0 0 [0,59] 0 0 [0,42.1] 0 0 [0,4.67] 0 0 [0,8] 0 0 0

[105,110] [110,140] [130,135] [110,115] [120,150] [160,165] [120,125] [130,160] [170,175] [80,85] [80,85] [100,139] [95,100] [95,100] [95,159] [110,115] [110,115] [110,157.1] [50,55] [50,55] [50,59.67] [55,60] [55,60] [55,68] [60,65] [60,65] [60,65]

0 [5,30] [45,68.57] 0 [10,35] [50,59.79] 0 [10,35] [50,85] 0 0 [0,11.43] 0 0 [0,25.21] 0 0 0 0 0 0 0 0 0 0 0 0

[105,110] [110,140] [150,178.57] [110,115] [120,150] [160,174.79] [120,125] [130,160] [170,210] [80,85] [80,85] [80,96.43] [95,100] [95,100] [95,125.21] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

0 [5,30] [45,80] 0 [10,35] [50,85] 0 [10,35] [50,85] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

[105,110] [110,140] [150,190] [110,115] [120,150] [160,200] [120,125] [130,160] [170,210] [80,85] [80,85] [80,85] [95,100] [95,100] [95,100] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

The detailed definitions for the symbols in model (6) have been provided in Appendix. According to the solution method presented in Section 2, the model can be solved based on the derivative and interactive algorithms. Interval solutions with associated risk levels

of violating constraints can be obtained. They are useful for generating desired decision alternatives for the planning of municipal solid waste management with minimized system cost and constraintviolation risk.

Table 6b Solutions obtained through the ICQSWM and ITQP (for city 2). Period

Facility

k

i

± W1kh

± Xi2kh (t/d)

ITQP model (t/d) q=0 ± Mi2kh

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 ± ($106 ) fopt

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

L M H L M H L M H L M H L M H L M H L M H L M H L M H

[105,110] [105,110] [105,110] [110,115] [110,115] [110,115] [120,125] [120,125] [120,125] [80,85] [80,85] [80,85] [95,100] [95,100] [95,100] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

ICQSWM model (t/d) q = 0.01

A± i2kh

0 [105,110] [5,11.68] [110,121.68] 21.43 [126.43, 131.43] 0 [110,115] 10 [120,125] 0 [110,115] 0 [120,125] 10 [130,135] [0,20] [120,145] 0 [80,85] [0,11.68] [80,96.68] 23.57 [103.57, 108.57] 10 [105,110] [0,13.85] [95,113.85] 50 [145,150] 0 [110,115] [0,13.85] [110,128.85] [0,17] [110,132] 0 [50,55] [0,11.68] [50,66.68] [0,15] [50,70] 0 [55,60] [0,14.95] [55,74.95] [0,19] [55,79] 0 [60,65] [0,14.95] [60,79.95] 60 [120,125] [163.33, 293.40]

± Mi2kh

q = 0.05 A± i2kh

0 [105,110] [5,32.9] [110,142.9] 45 [150,155] 0 [110,115] [10,23.6] [120,138.6] 15.71 [125.71, 130.71] 0 [120,125] [10,23.6] [130,148.6] [0,20] [120,145] 0 [80,85] [0,2.14] [80,87.14] [0,85] [80,170] 0 [95,100] [0,11.4] [95,111.4] [34.29, 85] [129.29, 185] 0 [110,115] [0,11.4] [110,126.4] 0 [110,115] 0 [50,55] 0 [50,55] 0 [50,55] 0 [55,60] 0 [55,60] 0 [55,60] 0 [60,65] 0 [60,65] [60,65] [120,130] [162.71, 290.79]

± Mi2kh

q = 0.10 A± i2kh

0 [105,110] [5,35] [110,145] 45 [150,155] 0 [110,115] [10,35] [120,145] 50 [160,165] 0 [120,125] [10,35] [130,160] [0,20] [120,145] 0 [80,85] 0 [80,85] [0,85] [80,170] 0 [95,100] 0 [95,100] [0,80.5] [95,180.5] 0 [110,115] 0 [110,115] 0 [110,115] 0 [50,55] 0 [50,55] 0 [50,55] 0 [55,60] 0 [55,60] 0 [55,60] 0 [60,65] 0 [60,65] [60,65] [120,130] [162.41, 287.79]

± Mi2kh

A± i2kh

0 [105,110] [5,35] [110,145] 45 [150,155] 0 [110,115] [10,35] [120,145] 50 [160,165] 0 [120,125] [10,35] [130,160] [0,20] [120,145] 0 [80,85] 0 [80,85] [0,70.71] [80,155.71] 0 [95,100] 0 [95,100] [0,80] [95,180] 0 [110,115] 0 [110,115] 0 [110,115] 0 [50,55] 0 [50,55] 0 [50,55] 0 [55,60] 0 [55,60] 0 [55,60] 0 [60,65] 0 [60,65] [60,65] [120,130] [162.41, 286.77]

648

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Fig. 2. Total waste flows from city 1 to the landfill under different qi levels and waste-generation rates.

Fig. 3. Total waste flows from city 2 to the landfill under different qi levels and waste-generation rates.

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Fig. 4. Total waste flows from city 1 to the incinerator under different qi levels and waste-generation rates.

Fig. 5. Total waste flows from city 2 to the incinerator under different qi levels and waste-generation rates.

649

650

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Fig. 6. Total waste flows from city 1 to the composting facility under different qi levels and waste-generation rates.

Fig. 7. Total waste flows from city 2 to the composting facility under different qi levels and waste-generation rates.

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

651

Table 7a Solutions obtained through the ICQSWM and ITCLP (for city 1). ijkh

ICQSWM model (t/d) q = 0.01

1111 1112 1113 1121 1122 1123 1131 1132 1133 2111 2112 2113 2121 2122 2123 2131 2132 2133 3111 3112 3113 3121 3122 3123 3131 3132 3133

ITCLP model (t/d) q = 0.01

ICQSWM model (t/d)

ITCLP model (t/d)

q = 0.05

q = 0.05

ICQSWM model (t/d) q = 0.10

ITCLP model (t/d) q = 0.10

± Mi1kh

A± i1kh

± Mi1kh

A± i1kh

± Mi1kh

A± i1kh

± Mi1kh

A± i1kh

± Mi1kh

A± i1kh

± Mi1kh

A± i1kh

0 [5,30] 25 0 [10,35] 50 0 [10,35] 50 0 0 [20,54] 0 0 [0,59] 0 0 [0,42.1] 0 0 [0,4.67] 0 0 [0,8] 0 0 0

[105,110] [110,140] [130,135] [110,115] [120,150] [160,165] [120,125] [130,160] [170,175] [80,85] [80,85] [100,139] [95,100] [95,100] [95,159] [110,115] [110,115] [110,157.1] [50,55] [50,55] [50,59.67] [55,60] [55,60] [55,68] [60,65] [60,65] [60,65]

0 [5,30] 25 0 [10,35] 45.71 0 [10,35] 50 0 0 [20,54] 0 0 [4.29, 59] 0 0 [0,42.1] 0 0 [0,4.67] 0 0 [0,8] 0 0 0

[105,110] [110,140] [130,135] [110,115] [120,150] [155.71, 160.71] [120,125] [130,160] [170,175] [80,85] [80,85] [100,139] [95,100] [95,100] [99.29, 159] [110,115] [110,115] [110,157.1] [50,55] [50,55] [50,59.67] [55,60] [55,60] [55,68] [60,65] [60,65] [60,65]

0 [5,30] [45,68.57] 0 [10,35] [50,59.79] 0 [10,35] [50,85] 0 0 [0,11.43] 0 0 [0,25.21] 0 0 0 0 0 0 0 0 0 0 0 0

[105,110] [110,140] [150,178.57] [110,115] [120,150] [160,174.79] [120,125] [130,160] [170,210] [80,85] [80,85] [80,96.43] [95,100] [95,100] [95,125.21] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

0 [5,30] [45,68.6] 0 [10,35] [50,59.79] 0 [10,35] [50,85] 0 0 [0,11.43] 0 0 [0,25.21] 0 0 0 0 0 0 0 0 0 0 0 0

[105,110] [110,140] [150,178.57] [110,115] [120,150] [160,174.79] [120,125] [130,160] [170,210] [80,85] [80,85] [80,96.43] [95,100] [95,100] [95,125.21] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

0 [5,30] [45,80] 0 [10,35] [50,85] 0 [10,35] [50,85] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

[105,110] [110,140] [150,190] [110,115] [120,150] [160,200] [120,125] [130,160] [170,210] [80,85] [80,85] [80,85] [95,100] [95,100] [95,100] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

0 [5,30] [45,80] 0 [10,35] [50,85] 0 [10,35] [50,85] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

[105,110] [110,140] [150,190] [110,115] [120,150] [160,200] [120,125] [130,160] [170,210] [80,85] [80,85] [80,85] [95,100] [95,100] [95,100] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [60,65]

4. Results and discussion 4.1. Result analysis Tables 6a and 6b present the solutions obtained through the ICQSWM model under different qi levels. Each optimized waste± = Xijk + flow is a sum of the allowable and excess flows (i.e., A± ijkh

± Mijkh ) from each city to a facility under a given waste-generation condition. The results indicate that any change in qi would yield different waste management capacities and thus result in different waste-flow-allocation patterns. Moreover, in case of excess waste, allotment to the landfill should be assigned firstly then to the incinerating and composting facility mainly to their differences in operation costs. The analysis of the modeling solutions from the two cities to landfill is provided below based on Tables 6a and 6b. The optimized allocation for waste flows (including allowable and excess flows) from the two cities to the landfill under different waste-generation rates and varied risk levels of capacity constraint-violation is presented in Figs. 2 and 3. In period 1, under qi = 0.01, the excess waste flows from city 1 to the landfill would be 0, [5,30] and 25 t/day under low (h = 1), medium (h= 2) and high (h = 3) wastegeneration rates, respectively. Thus, the total waste flows allocated to the landfill would be [105,110], [110,140] and [130,135] t/day under low, medium and high waste-generation rates, respectively. The excess flows from city 2 to the landfill would be 0, [5,32.9], 45 t/day under low, medium and high waste-generation rates, respectively. Thus, the total flows allocated to the landfill would be [105,110], [110,142.9] and [150,155] t/day under low, medium and high waste-generation rates (qi = 0.01), respectively. The solution indicates that the waste flows from city 2 to the landfill are more than those from city 1 due to its lower transportation cost. Secondly, under qi = 0.05, the excess waste flows from city 1 to the landfill would be 0, [5,30] and [45,68.57] t/day under low, medium and high waste-generation rates, respectively; the total waste flows to the landfill would thus be [105,110], [110,140] and

[150,178.57] t/day, respectively. In comparison, the excess waste flows from city 2 to the landfill would be 0, [5,35] and 45 t/day under low, medium and high waste-generation rates, respectively; the total flows to the landfill would thus be [105,110], [110,145] and [150,155] t/day. Under qi = 0.10, the excess waste flows from city 1 to the landfill would be 0, [5,30] and [45,80] t/day under low, medium and high waste-generation rates, respectively; the total waste flows to the landfill would thus become [105,110], [110,140] and [150,159] t/day, respectively. The excess waste flows from city 2 to the landfill would be 0, [5,35] and 45 t/day under low, medium and high waste-generation rates, respectively; the total waste flows to the landfill would thus become [105,110], [110,145] and [150,155] t/day, respectively. The solutions indicate that the waste flows to the landfill would increase with the qi level, and the increment of waste flows from city 1 to the landfill would be more than that from city 2 to the landfill as qi level rises. In fact, (i) an increased qi level means a relaxed landfill-capacity constraint and thus a raised risk in violating this constraint; moreover, since the incinerating and composting facilities have higher regular and penalty costs for their operations, allotment of the excess waste to the landfill would become more economical; (ii) although the transportation cost of city 1 is higher than that of city 2, the effect of the economies of scale takes an important role on waste-flow allocation, such that the increment of flows from city 1 to the landfill would be more than that from city 2 with qi level. The total waste flows from the two cities to the incinerator under different waste-generation rates and varied constraint-violation risk levels are summarized in Figs. 4 and 5. In period 1, for city 1, there would be no excess flow to the incinerator when the waste-generation rates are low and medium under qi = 0.01, 0.05 and 0.10. In comparison, under a high waste-generation rate, the excess waste flows to the incinerator would become [20,54], [0,11.43] and 0 t/day under qi = 0.01, 0.05 and 0.10, respectively; the total waste flows to the incinerator would thus be [100,139],

652

Table 7b Solutions obtained through the ICQSWM and ITCLP (for city 2). ijkh

ICQSWM model (t/d)

ITCLP model (t/d)

q = 0.01

ICQSWM model (t/d)

q = 0.01

q = 0.05

ITCLP model (t/d)

ICQSWM model (t/d)

ITCLP model (t/d)

q = 0.05

q = 0.10

q = 0.10

A± i2kh

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

± Mi2kh

A± i2kh

1211 1212 1213 1221 1222 1223 1231 1232 1233 2211 2212 2213 2221 2222 2223 2231 2232 2233 3211 3212 3213 3221 3222 3223 3231 3232 3233

0 [5,32.9] 45 0 [10,23.6] 15.71 0 [10,23.6] [0,20] 0 [0,2.14] [0,85] 0 [0,11.4] [34.3, 85] 0 [0,11.4] 0 0 0 0 0 0 0 0 0 [60,65]

[105,110] [110,142.9] [150,155] [110,115] [120,138.6] [125.7, 130.7] [120,125] [130,148.6] [120,145] [80,85] [80,87.1] [80,170] [95,100] [95,111.4] [129.3, 185] [110,115] [110,126.4] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [120,130]

0 [5,32.9] 45 0 [10,23.6] 20 0 [10,23.6] [0,20] 0 [0,2.14] [0,85] 0 [0,11.4] [30,85] 0 [0,11.4] 0 0 0 0 0 0 0 0 0 [60,65]

[105,110] [110,142.9] [150,155] [110,115] [120,138.6] [125.71, 130.71] [120,125] [130,148.6] [120,145] [80,85] [80,87.1] [80,170] [95,100] [95,111.4] [129.29, 185] [110,115] [110,126.4] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [120,130]

0 [5,35] 45 0 [10,35] 50 0 [10,35] [0,20] 0 0 [0,85] 0 0 [0,80.5] 0 0 0 0 0 0 0 0 0 0 0 [60,65]

[105,110] [110,145] [150,155] [110,115] [120,145] [160,165] [120,125] [130,160] [120,145] [80,85] [80,85] [80,170] [95,100] [95,100] [95,180.5] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [120,130]

0 [5,35] 45 0 [10,35] 50 0 [10,35] [0,20] 0 0 [0,85] 0 0 [0,80.5] 0 0 0 0 0 0 0 0 0 0 0 [60,65]

[105,110] [110,145] [150,155] [110,115] [120,145] [160,165] [120,125] [130,160] [120,145] [80,85] [80,85] [80,170] [95,100] [95,100] [95,180.5] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [120,130]

0 [5,35] 45 0 [10,35] 50 0 [10,35] [0,20] 0 0 [0,70.7] 0 0 [0,80] 0 0 0 0 0 0 0 0 0 0 0 [60,65]

[105,110] [110,145] [150,155] [110,115] [120,145] [160,165] [120,125] [130,160] [120,145] [80,85] [80,85] [80,155.71] [95,100] [95,100] [95,180] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [120,130]

0 [5,35] 45 0 [10,35] 50 0 [10,35] [0,20] 0 0 [0,70.71] 0 0 [0,80] 0 0 0 0 0 0 0 0 0 0 0 [60,65]

[105,110] [110,145] [150,155] [110,115] [120,145] [160,165] [120,125] [130,160] [120,145] [80,85] [80,85] [80,155.71] [95,100] [95,100] [95,180] [110,115] [110,115] [110,115] [50,55] [50,55] [50,55] [55,60] [55,60] [55,60] [60,65] [60,65] [120,130]

t fopt ($106 )

[162.71, 290.79]

[166.59, 296.78]

[162.41, 287.79]

[166.30, 293.84]

[162.41, 286.77]

[166.30, 292.84]

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

± Mi2kh

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

[80,96.43] and [80,85] t/day, respectively. The results indicate that the excess flow to the incinerator would decrease along with qi level. Although a raised qi would imply an increased capacity for every waste-management facility, the operational cost of the incinerating facility is higher than that of the landfill, the waste flows would be mostly transported to the landfill. For city 2, the similar analysis can be obtained. The solution would be affected by transportation cost, operation cost and the economies of scale effect. The total waste flows from the two cities to the composting facility under different waste-generation rates and risk levels are presented in Figs. 6 and 7. In period 1, for city 1, no excess flow would be allocated to the composting facility under low and medium waste-generation rate under qi = 0.01, 0.05 and 0.10; under high waste-generation rate, the excess waste flows would be [0,4.67], 0, 0 t/day under qi = 0.01, 0.05 and 0.10. In period 1, for city 2, no excess flow would be shipped to the composting facility under low, medium and high waste-generation rate when qi = 0.01, 0.05 and 0.10; This may be due to the following two facts: (i) the composting facility has the highest operating cost for excess flows, so that the majority of such flows would be allocated to the landfill and/or incinerator; (ii) when qi level is low, the landfill and incinerating capacities are both at their lower bounds, leading to the allocation of some flows to the composting facility;(iii) the effects of the economies of scale would reduce the transportation cost, and thus reduce the system cost, then the waste-flow to the landfill is increased. In periods 2 and 3, the allowable and excess waste flows from the two cities to the landfill, incinerating and composting facilities would increase with the waste-generation rate. The results indicate that, with the increasing qi level, more excess waste flows would be allocated to the landfill mainly because of its lower operation cost

653

and the lower effect of economies of scale. However, the excess flows allocated to the incinerating and composting facilities would decrease even though the capacities of the two facilities are relaxed with the increased qi level. For example, for city 1, in period 2, if the waste-generation rate is high, the excess flows to the incinerator would be [0,59], [0,25.21] and 0 t/day under qi = 0.01, 0.05 and 0.10, respectively; at the same time, the excess flows to the composting facility would become [0,8], 0 and 0 t/day, respectively; however, the excess flows to the landfill would become 50, [50,59.79] and [50,85] t/day under qi = 0.01, 0.05 and 0.10, respectively. This fact has been demonstrated in the obtained solutions. The qi levels represent a set of probabilities at which the constraints will be violated (i.e., the admissible risk of violating the constraints). Thus, the relation between f± and qi would demonstrate a tradeoff between system cost and constraint-violation risk. An increased qi means an increased risk of constraint-violation and, at the same time, it would lead to a decreased strictness for the constraints and thus a decreased system cost. Such a decreased cost, however, would be linked to a potentially increased amount of untreated waste and thus an increased risk of constraint-violation. In general, a raised qi level implies a relaxed capacity constraint and would result in an increased capacity for every wastemanagement facility. Table 6b presents the system costs under the different qi levels. The solutions of f± under qi = 0.01, 0.05 and 0.10 are $ [162.71, 290.8] × 106 , [162.4, 287.8] × 106 and $ [162.4, 286.8] × 106 , respectively. It is indicated that, as the actual values of the decision variables vary within their two bounds, the expected system cost would change correspondingly between f− opt and f+ opt associated with different reliability levels. Decisions at a higher qi level would lead to a decreased reliability in fulfilling the system requirements but with a lower cost; in comparison, decisions at a lower qi level would result in a higher cost, but the risk of violating

Fig. 8. The differences of system costs between ICQSWM and ITCLP under different qi level for cities.

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Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

the constraints would be decreased. These demonstrate a tradeoff between the waste-management cost and the system-failure risk due to the dual uncertainties that exist in various system components. 4.2. Comparisons with ITQP and ITCLP approaches Substituting the capacity constraints by deterministic values, the study problem can then be converted into the interval twostage quadratic programming (ITQP) method. Solutions of the ITQP model (Tables 6a and 6b) provide two extremes of the expected ± system cost (i.e., fopt = $ [163.33, 293.40] × 106 ). In practice, decisions for a lower cost may correspond to advantageous system conditions (e.g., lower waste-generation rate), while those with a higher cost correspond to more demanding conditions. The study problem can also be solved through an interval two-stage chance-constrained linear programming (ITCLP) model without considering the effects of economies of scale on waste transportation costs. Solutions of the ITCLP model (Tables 7a and 7b) provide two extremes of the expected sys± ± tem cost fopt = $[166.6, 296.8] × 106 , fopt = $[166.3, 293.8] ×

± = $[166.3, 292.84] × 106 under qi = 0.01, 0.05 and 106 and fopt 0.10 respectively. Compared with ITCLP, the much lower system ± = $[162.7, 290.8] × costs are obtained from the ICQSWM (i.e., fopt

± 106 under qi = 0.01, fopt = $[162.4, 287.8] × 106 under qi = 0.05,

± = $[162.4, 286.8] × 106 under qi = 0.1 respectively). Fig. 8 and fopt shows the differences of the system costs from the ITCLP and ICQSWM models under different qi levels. The results demonstrate that the ICQSWM can not only incorporate more uncertain information within its modeling framework, but also deal with nonlinearities in the objective function and thus reflect the effect of the economies of scale on the waste-management cost.

Acknowledgements This research has been supported by the Major State Basic Research Development Program of MOST (2005CB724200), and the Natural Sciences and Engineering Research Council of Canada. The authors are deeply grateful to the editor and the reviewers for their insightful comments. Appendix A. A.1. (A) Inexact two-stage chance-constrained programming Consider a waste-management system wherein a manager is responsible for allocating waste flows from multiple districts to multiple facilities within multiple periods; the waste treatment options include landfill, incinerating, composting, and recycling. Based on the local waste-management policies, a projected allowable waste-flow level from each district must be provided. If this level is not exceeded, it will result in a regular cost to the system. However, if it is exceeded, the surplus waste-flow will have to be disposed with a higher cost, resulting in an excess cost to the system; this implies penalties in terms of raised transportation and operation costs (Li et al., 2007). Under such a situation, the total waste-flow amount will be the sum of both fixed allowable and probabilistic surplus flows. The goal is to achieve optimal planning of waste-flow allocation with a minimized system cost. The wastegeneration amounts from the districts are uncertain (expressed as discrete random variables), while a plan for the allowable wasteflow levels must be made before the realizations of the random variables. Therefore, a two-stage chance-constrained programming (TCP) model can be formulated as follows:

Minimize f

=

J K I   

i=1 I

j=1 J



5. Conclusions

+

In this study, an ICQSWM model has been developed for planning of municipal solid waste management systems. The developed method improves upon the existing approaches of ITCLP and the inexact quadratic programming (IQP). The ICQSWM can tackle uncertainties presented as both probability distributions and discrete intervals, and can be used for analyzing various policy scenarios which are associated with different levels of economic penalties when the promised policy targets are violated. Moreover, it can reflect the effect of economies of scale on waste-management cost. Furthermore, based on derivative and interactive algorithms, it requires much lower computational efforts than the previous method proposed by Huang et al. (1995c). The developed method has been applied to a case study of planning a municipal solid waste-management system, where violations for capacity constraints are allowed under a range of significance levels. Interval solutions associated different risk levels of constraint-violation have been obtained. They can be used for generating decision alternatives and thus help waste managers to identify desired policies under various environmental, economic, and system-reliability constraints. Compared with the ITQP and ITCLP methods, the ICQSWM can incorporate more uncertain information within its modeling framework, and can deal with nonlinearities in the objective function and thus reflect the effect of the economies of scale in MSW management planning. Although this study is the first attempt for planning waste-management system through developing the ICQSWM approach, the results suggest that this integrated technique is applicable to other environmental problems that involve uncertainties presented in multiple formats.

i=2 I

j=1 J



+

j=1 J

j=1

−

j=1

(A.1a)

(w)

Lk E[Mijk ]FEi (DTik + DP1k )

k=1

J K I   

i=2

(w)

Lk E[Mijk ](DRijk + DPik )

k=1 K

 i=2

Lk Xijk FEi (FTik + OP1k )

k=1 K

+

i=1 I

Lk Xijk (TRijk + OPik )

k=1 K

Lk Xijk REik −

k=1

J K H I    

j=1

k=1

h=1

subject to:

Pr

⎧ J K ⎨  ⎩

(w)

Lk [(X1jk + M1jkh ) +

I 

j=1 k=1

(w)

Lk E[Mijk ]RMik i=2

(w)

⎫ ⎬

FEi (Xijk + Mijkh )]

i=2

⎭

≥ 1 − q1 (A.1b)

(landfill capacity constraint)

Pr

⎧ J ⎨ ⎩

(w)

(Xijk + Mijkh ) ≤ TCi

j=1

⎫ ⎬ ⎭

≥ 1 − qi ,

i = 2, 3, · · ·, I

(A.1c)

(capacity constraints of waste-management facilities) I  i=1

(w)

(Xijk + Mijk ) ≥ WGjk ,

∀j, k

(A.1d)

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

(waste disposal demand constraints)

Xijk

(w)

max

≥ Xijk ≥ Mijk ≥ 0,

∀i, j, k

converted into a linear one (TCLP) as follows:

(A.1e)

Minimize f

=

J K I   

i=1 I

(non-negativity and technical constraints)where f is the expected net system cost ($); i is the type of waste-management facility, where i = 1 for landfill, and i = 2, 3, . . ., I for waste treatment facilities (e.g., recycling, incinerating and composting); j is the name of district, j = 2, 3, . . ., J; k is the time period, k = 2, 3, . . ., K; Lk is the length of time period k (day); DPik is the operating cost of facility i for excess waste-flow during period k (the second-stage cost parameter), where DPik ≥ OPik and i = 2, 3, . . ., I; DRijk is the transportation cost for excess waste-flow from district j to facility i during period k (the second-stage cost parameter), where DRijk ≥ TRijk and i = 2, 3, . . ., I; DTik is the transportation cost for excess waste residue from facility i to the landfill during period k (the second-stage cost parameter), where DTik ≥ FTik and i = 2, 3, . . ., I; E() the expected value of a random variable; FEi is the residue flow rate from facility i to the landfill (% of incoming mass to facility i), i = 2, 3, . . ., I; FTik is the transportation cost for allowable residue flow from facility i to the landfill during period k, i = 2, 3, . . ., I; (w) LC is the existing landfill capacity (tonne); Mijk is the amount by which the allowable waste-flow level (Xijk ) is exceeded when the waste-generation rate in district j during period k is WGjk (t/d) (the second-stage decision variable); OPik is the operating cost of facility i for allowable waste-flow during period k; qi is admissible probability of violating constraint i; REik is the revenue from facility i during period k, i = 2, 3, . . ., I; RMik is the revenue from facility i because of excess waste-flow during period k (the second-stage cost parameter), i = 2, 3, . . ., I; TCi is the existing capacity of facility i, i = 2, 3, . . ., I; TRijk is the transportation cost for allowable wasteflow from district j to facility i during period k (the first-stage cost parameter); WGjk is the random variable of waste-generation rate in district j during period k; Xijk is the allowable waste-flow from district j to facility i during period k (the first-stage variable); Xijkmax is the maximum allowable waste-flow from district j to facility i during period k. However, the above TCP problem can be equivalently formulated as a linear model. In the first place, the distribution of each WGjk can be converted into an equivalent set of discrete values (Li et al., 2007). Let each WGjk take value wijkh with probability pjh (h = 1, 2, . . ., H), where h is denoted as the level of waste-generation rate in district j. Thus, we have:

(w)

E[Mijk ] =

H 

pjh Mijkh ,

∀i, j, k, h

(A.2)

h=1

Secondly, the chance constraints can be converted into deterministic and linear ones through: (i) fixing a certain level of probability qi , qi ∈[0,1], for constraint j, and (ii) imposing the condition that constraint j is satisfied with at least a probability of 1 − qi (Loucks et al., 1981; Huang, 1998; Li et al., 2007). Thus, the chance constraints [Pr(Ai x ≤ bi ) ≥ 1 − qi ] can be specified into: Ai X ≤ bqi i ,

∀i

655

(A.3)

where bqi i = Fi−1 (qi ), given the cumulative distribution function (CDF) of bi [i.e., Fi (bi )] and the probability of violating constraint j (i.e., qi ). Consequently, the above nonlinear TCP model can be

j=1 J



+

+

+

−

Lk Xijk (TRijk + OPik )

k=1 K

Lk Xijk FEi (FTik + OP1k )

i=2 I

j=1 J

k=1 K

i=1 I

j=1 J

k=1 K

h=1 H

i=2 I

j=1 J

k=1 K

h=1

H 

  i=2

j=1

Lk pjh Mijkh Xijk (DRijk + DPik )

(A.4a)

Lk pjh Mijkh FEi (DTik + DP1k )

Lk Xijk REik −

k=1

J K H I    

Lk pjh Mijkh RMik , i=2

j=1

k=1

h=1

subject to J K  

 Lk

(X1jk + M1jkh ) +

j=1 k=1

I 

 FEi (Xijk + Mijkh )

≤ (LCk )q1 ,

∀h

i=2

(A.4b)

J 

(Xijk + Mijkh ) ≤ (TCik )qi ,

∀h, k, i = 2, 3, · · ·, I

(A.4c)

j=1 I 

(Xijk + Mijkh ) ≥ wjkh ,

∀j, k, h

(A.4d)

i=1

Xijk

max

≥ Xijk ≥ Mijkh ≥ 0,

∀i, j, k, h

(A.4e)

where h is the level of waste-generation in district j, h = 1,2,. . .,H; LCqk1 is the cumulative function of landfill capacity in period k with qi the probability of violating constraint (A.4b) being equal to q1 ; TCik is the cumulative function of the capacity for facility i in period k with the probability of violating constraint i being equal to qi ; Mijkh is the amount by which the allowable waste-flow level (Xijk ) is exceeded when the waste-generation rate is wjkh ; pjh is the probability of waste-generation rate in district j being equal to wjkh . The above TCLP model can handle all right-hand side uncertainties expressed as probability distributions. However, the linear constraints only correspond to cases when the left-hand coefficients are deterministic. Although the CCP approach can deal with left-hand-side uncertainties presented as probability density functions, three limitations exist: (i) the resulting nonlinear model would be associated with a number of difficulties in globaloptimum acquisition; (ii) it is unable to handle independent uncertainties in objective coefficients (Birge and Louveaux, 1988; Infanger, 1993; Zare and Daneshmand, 1995); (iii) for many practical problems, the quality of information that can be obtained for these uncertainties is mostly not satisfactory enough to be presented as probability distributions (Li et al., 2007). Thus, for uncertainties in the left-hand-side parameters (e.g., costs of waste transportation and facility operation, and revenues from wastemanagement facilities), an extended consideration would be the introduction of interval parameters into the TCLP framework. This leads to an interval-parameter two-stage chance-constrained

656

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

The meaning of cj± and dj± have the same sign” is that, when

linear programming (ITCLP) model as follows: Minimize f ±

=

J K I   

i=1

j=1

k=1

J K I   

+

i=2

j=1

k=1

i=1 I

j=1 J

k=1 K

± ± Lk Xijk FEi (FTik± + OP1k )

J K H I    

+

+

j=1 J

k=1 K



−

i=2

j=1

± ± ± ± Lk pjh Mijkh Xijk (DRijk + DPik )

h=1

 I

± ± Lk Xijk REik −

k=1

i=2



± ± (X1jk + M1jkh )+

Lk

j=1 k=1

I 

J

j=1

K

k=1

f+ =

H

f− =

 ≤ (LCk )q1 ,

∀h

∀h, k, i = 2, 3, · · ·, I

(A.5c)

j=1 I 

± ± ± (Xijk + Mijkh ) ≥ wjkh ,

∀j, k, h

(A.5d)

i=1 ± Xijk

max

± ± ≥ Xijk ≥ Mijkh ≥ 0,

∀i, j, k, h

(A.5e)

± ± ± ± ± ± ± ± , M ± , TRijk , OPik , DRijk where Xijk , DPik , FEi± , REik , RMik and wjkh are interval parameters and variables; the ‘−’ and ‘+’ superscripts represent lower and upper bounds of the parameters/variables, respectively. This ITCLP model can be transformed into two deterministic submodels that correspond to the lower and upper bounds of the desired objective. Interval solutions, which are feasible and stable in the given decision space, can then be obtained by solving the two submodels (Huang et al., 1992; Li et al., 2007).

2

[cj+ xj− + dj+ (xj− ) ]

(B.7a)

k1 

2

[cj− xj− + dj− (xj− ) ] +

n 

2

[cj− xj+ + dj− (xj+ ) ]

(B.7b)

j=k1 +1

where f+ and f− are to be maximized. (2) When cj± and dj± have different signs Under this situation, it becomes difficult to determine whether xj− or xj+ corresponds to the desired f+ Huang et al. (1995c) proposed that various combinations of the upper and lower bounds of xj± could be examined by formulating a number of objective functions. The relevant quadratic programming submodels should then be solved. Thus, the optimal bound distribution for xj± (i.e., choice of

upper or lower bound for each xj± ) that leads to the highest f+ value could be identified through comparisons of the generated solutions. However, Chen and Huang (2001) proposed a derivative algorithm (DAM) which is especially useful for practical application of IQP to large-scale problems where many {cj± , dj± } pairs have different signs. It requires much lower computational requirement than the method proposed by Huang et al. (1995c) where 2n submodels need to be solved. Consider the IQP objective functions as shown in (B.6a). Letting 2

fj± = cj± xj± + dj± (xj± ) ,

(B.8)

Then, we have: f± =

A.2. (B) Inexact quadratic programming

n  j=k1 +1

j=1

i=2

± ± (Xijk + Mijkh ) ≤ (TCik )qi ,

2

[cj+ xj+ + dj+ (xj+ ) ] +

j=1

h=1

± ± FEi (Xijk + Mijkh )

k1 

± ± Lk pjh Mijkh RMik

(A.5b)

J 

be negative, i.e., cj± < 0 and dj± < 0 (j = k1 + 1, k1 + 2, . . ., n), where k1 + k2 = n. Thus, we can define the bounds for cost-coefficients and decision variables corresponding to the upper and lower bounds of the objective function value as follows:

± ± ± Lk pjh Mijkh FEi Xijk (DTik± + DP1k )

subject to J K  

(A.5a)

h=1 H

 i=2 I

cj± ≥ 0; dj± should also be ≥0, and vice versa. Under this situation, relations between the decision variables and the objective function can be easily defined. We can assume that k1 of the {cj± , dj± } pairs are positive, and k2 are negative. Let the former k1 pairs be positive, i.e., cj± ≥ 0 and dj± ≥ 0 (j = 1, 2, . . ., k1 ), and the latter k2 pairs

± ± ± Lk Xijk (TRijk + OPik )

n 

fj± .

(B.9)

j=1

Consider an inexact quadratic programming (IQP) problem without xj xk (j = / k) terms as follows (Huang et al., 1995c): max f ± =

n 

± Let fopt and xj±opt be optimal solutions of model (B.6), and define 2

2

[cj± xj± + dj± (xj± ) ]

(B.6a)

fj±opt = cj± xj±opt + dj± (xj±opt ) . Then, we have:

j=1

subject to: n 

a± x ± ≤ b± , ij j i

± fopt =

i = 1, 2, . . . , m,

j = 1, 2, . . . , n,

n 

fj±opt =

j=1

(B.6b)

n 

2

[cj± xj±opt + dj± (xj±opt ) ].

(B.11)

j=1

Define the following functions:

j=1

xj± ≥ 0,

(B.10)

(B.6c)

, b± , cj± , dj± and xj± are interval parameters/variables. For where a± ij i example, letting a− and a+ be lower and upper bounds of a± , respecij ij ij ± + tively, we have aij ∈ [aij , aij ]. The main challenge for solving model

(B.6) through Huang’s approach (Huang et al., 1994) is in defining bounds for decision variables corresponding to the upper and lower bounds of the objective function value. Then two situations need to be considered. (1) When cost-coefficients cj± and dj± have the same sign

fj+ (x)

= cj+ x + dj+ x2 ,

(B12)

fj− (x) = cj− x + dj− x2 .

(B.13)

For the purpose of simplification, we eliminate subscript j at this stage. Thus, we have: f + (x) = c + x + d+ x2 , −

−

− 2

f (x) = c x + d x . Then, we have the following lemma:

(B.14) (B.15)

Y. Sun et al. / Resources, Conservation and Recycling 54 (2010) 641–657

Lemma 1. Let x < x and x + x = −c + /d+ . If c± and d± have different signs, then we have: (i)f + (x ) = f + (x ),

(B.16)

f − (x ) > f − (x ).

(B.17)

Corollary 1. Let x < x and x + x = −c + /d+ . If cj± and dj± have different signs, we have f ± (x ) > f ± (x ). Lemma 2.

cj±

If

(B.18) and

dj±

have different signs,

extreme value within open interval (xj−opt , xj+opt ). Lemma 3. (xj−opt , dfj+ dx

fj+ (x)

will have no

Let cj± and dj± have different signs. For open interval

xj+opt ),

we have

= / 0.

(B.19)

Corollary 2. Given x ∈ [xj−opt , xj+opt ], if cj± and dj± have different signs, we have:



(i)fj+ (xj+opt ) ≥ fj+ (xj−opt ),

when

dfj+opt dx

 (ii)fj+ (xj+opt ) ≤ fj+ (xj−opt ),

when



dfj+opt

> 0,

(B.20)

<0

(B.21)

x=xj



dx

x=xj

The equalities in Eqs. (B.20) and (B.21) exist if and only if xj−opt =

xj+opt .

Remark 1. Corollary 2 indicates that, as long as one point within xj±opt = [xj−opt , xj+opt ] can be found, the max {fj+ (xj+opt ), fj+ (xj−opt )} can be identified through calculation of the derivative at the point. Consider the following quadratic programming model: Max fmv =

n 

[(cj )mv (xj )mv + (dj )mv (xj )2mv ]

(B.22a)

j=1

subject to: n 

(aij )mv (xj )mv ≤ (bi )mv ,

i = 1, 2, . . . , m,

(B.22b)

j=1

(xj )mv ≥ 0,

j = 1, 2, . . . , n

(B.22c)

where (cj )mv , (dj )mv , (aij )mv and (bi )mv are mid-values of cj± , dj± , a± ij

and b± [e.g., (cj )mv = (cj− + cj+ )/2]. Assume that the optimal solution j for model (B.22) is Xmv opt = {(xj )mv opt |∀j}. According to Huang et al. (1995c), this solution can be contained in the following inexact solution set for IQP model (B.6): (xj )mv

opt

∈ [xj−opt , xj+opt ] ∀j.

(B.23)

Thus, we have the following lemma: Lemma 4.

If cj± and dj± have different signs, we have

fj+ (xj+opt ) ≥ fj+ (xj−opt ),

when

2dj± (xj )mv

opt

+ cj± > 0,

(B.24a)

fj+ (xj+opt ) ≤ fj+ (xj−opt ),

when

2dj± (xj )mv

opt

+ cj± < 0

(B.24b)

Thus, the optimal bound distribution for xj± that leads to the

highest f+ value can then be identified through a two-step process: (i) solution of submodel (B.22), (ii) judgment through criterion (B.24).

657

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