An integrated two-stage optimization model for the development of long-term waste-management strategies

S CIE N CE OF T H E TOT AL E N V I RO N ME N T 3 9 2 ( 2 00 8 ) 1 7 5–1 86

a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / s c i t o t e n v

An integrated two-stage optimization model for the development of long-term waste-management strategies Y.P. Li a,b,⁎, G.H. Huang b,c,⁎, Z.F. Yang a , S.L. Nie d a

School of Environment, Beijing Normal University, Beijing 100875, China Environmental Systems Engineering Program, University of Regina, Regina, SK S4S 0A2, Canada c Chinese Research Academy of Environmental Science, North China Electric Power University, Beijing 100012-102206, China d School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China b

AR TIC LE I N FO

ABS TR ACT

Article history:

In this study, an integrated two-stage optimization model (ITOM) is developed for the

Received 21 July 2007

planning of municipal solid waste (MSW) management in the City of Regina, Canada. The

Received in revised form

ITOM improves upon the existing optimization approaches with advantages in uncertainty

15 November 2007

reflection, dynamic analysis, policy investigation, and risk assessment. It can help analyze

Accepted 24 November 2007

various policy scenarios that are associated with different levels of economic penalties when

Keywords:

effective diversion program that targets on the prolongation of the existing landfill. Moreover,

Decision making

violations for capacity and diversion constraints are allowed under a range of significance

the promised policy targets are violated, and address issues concerning planning for a cost–

Environment

levels, which reflect the tradeoffs between system-cost and constraint–violation risk. The

Management

modeling results are useful for generating a range of decision alternatives under various

Solid waste

environmental, socio-economic, and system-reliability conditions. They are valuable for

Stochastic programming

supporting the adjustment (or justification) of the existing waste-management practices, the

Two-stage optimization

long-term capacity planning for the city's waste-management system, and the identification of desired policies regarding waste generation and management. © 2007 Elsevier B.V. All rights reserved.

1.

Introduction

The conventional waste collection and disposal system in North America consists of garbage trucks and a landfill. However, the trend for disposal of municipal solid waste (MSW) is towards implementing waste diversion and creating an integrated MSW management system (Environment Canada, 2000; Chang et al., 2005). Thus, the conventional single-choice management, i.e. reliance on landfill for waste disposal, is inadequate. Complexities exist in such a diversion effort, including collection techniques to be used, levels of service to be offered, and facilities to be adopted; moreover, many related processes and/or factors are complex with interactive, dynamic and uncertain features. Therefore, a

systems approach for analyzing waste management is desired, in order to support decisions of short-term wastemanagement operation and long-term strategic planning. A large number of deterministic optimization techniques have been developed for supporting decisions of MSW management and evaluating relevant operation and investment policies since the 1960s (Anderson, 1968); they involved linear, dynamic, integer and multi-objective programming methods, as well as a number of cities in the world. For example, Kirca and Erkip (1988) formulated a linear programming model for determining transfer station locations in the MSW management system of Istanbul, Turkey. Or and Curi (1993) applied a mixed integer linear programming model for improving solid waste collection and transportation system in

⁎ Corresponding authors. Environmental Systems Engineering Program, University of Regina, Regina, SK S4S 0A2, Canada. Tel.: +1 306 585 4095; fax: +1 306 585 4855. E-mail addresses: [email protected] (Y.P. Li), [email protected] (G.H. Huang). 0048-9697/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.scitotenv.2007.11.028

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the City of Izmir, Turkey, in order to minimize the city's total solid waste collection and transportation costs. Chang et al. (1997) explored the cost–effective and workload-balancing operation in the regional solid waste-management system of the Taipei City, where linear and integer programming methods were applied within a two-stage analytical framework. More recently, Fiorucci et al. (2003) developed a decision-support system (DSS) for solid waste-management planning in the City of Genova, Italy, where waste-management-facility numbers as well as waste-flow patterns were examined, with the objective of minimizing the recycling, transportation and maintenance costs. Chang et al. (2005) utilized integer programming method to support the decisions of location and capacity for a material-recovery facility in the City of San Antonio, Texas, USA. However, in the real-world MSW management problems, uncertainties may exist in the related costs, impact factors and objectives, which will affect the optimization processes and the decision schemes generated (Huang et al., 1992). The complexities could be further compounded not only by interactions among the uncertain parameters but also through additional economic implications. Such complexities have placed many MSW management problems beyond deterministic programming methods. Various methods dealing with uncertainties were developed for MSW management and planning, such as fuzzy, stochastic and interval mathematical programming (abbreviated as FMP, SMP and IMP, respectively). For example, Koo et al. (1991) proposed a framework using WRAP (Waste Resources Allocation Program) and fuzzy set theory to address tradeoffs among the objectives of economic efficiency, environmental quality, and administrative efficiency, such that an optimal site for a new hazardous waste treatment facility in southwestern Korea could be determined. Arey et al. (1993) applied a mixed optimization and probabilistic-analysis approach for determining daily wastemanagement practices in the Municipalities of Hamilton and St. Catharines, Ontario, Canada. Huang et al. (1997, 1998) proposed interval-parameter programming techniques for planning solid waste management and capacity expansion in the Regional Municipality of Hamilton-Wentworth, Ontario, Canada; the developed techniques could readily process interval data, thereby avoiding problems encountered by other optimization methods when faced with complexities in parameter uncertainties and solution algorithms. Chen and Chang (2000) formulated a grey fuzzy dynamic model for the prediction of solid waste generation in the City of Tainan, Taiwan, where the technique of fuzzy goal regression was employed to minimize the discrepancy between the predicted and observed values. More recently, Davila et al. (2005) proposed a grey integer programming-based game theory (GIP-based game theory) for system optimization and cost–benefit analysis at two competing landfills in the Lower Rio Grande Valley, Texas, USA. Although many studies of waste management under uncertainty were conducted for a number of cities in the world, there have been no previous studies focusing on proposing an integrated two-stage optimization model (ITOM) for a real study of MSW management under uncertainty. Therefore, the objective of this study is to develop such an ITOM for the planning of MSW management in the City of Regina, Canada. In ITOM, approaches of stochastic mathematic programming (SMP) and interval mathematical pro-

gramming (IMP) will be incorporated within an integer programming (IP) framework for accounting for multiple uncertainties, the relevant economic penalties, system reliabilities, and capacity-expansion decision issues.

2.

Methodology

Two-stage stochastic programming (TSP) method is effective for problems where an analysis of policy scenarios is desired and the related data are mostly uncertain. In TSP, decision variables are divided into two subsets: those that must be determined before random variables are disclosed, and those (recourse variables) that will be determined after the uncertainties are disclosed. A general TSP model can be formulated as follows (Birge and Louveaux, 1997): z ¼ min CT X þ ExaX ½Q ðX; xÞ subject to: ð1aÞ

xaX with Q ðx; xÞ ¼ min f ðxÞT y subject to: DðxÞyzhðxÞ þ TðxÞx

ð1bÞ

yaY where XpRn1 , CaRn1 , and YpRn2 . Here, ω is a random variable from space (Ω, F, P) with Ω ⊆ Rk, f : XYRn2 , h : XYRm2 , D : XYRm2 n2 , and T : XYRm2 n1 . By letting random variables (i.e. ω) take discrete values wh with probability levels ph (h = 1, 2,…, v and ∑ ph = 1), the above TSP can be equivalently formulated as a linear programming model as follows (Ahmed et al., 2004; Li et al., 2007): Min f ¼ CT1 X þ

v X

ph DT2 Y

ð2aÞ

h¼1

subject to: Ar X V Br ;

r ¼ 1; 2; N ; m1

At X þ A tVY z wh ; xj z 0; yjh z 0;

t ¼ 1; 2; N ; m2 ; h ¼ 1; 2; N ; v

xj aX; j ¼ 1; 2; N ; n1 yjh aY; j ¼ 1; 2; N ; n2 ; h ¼ 1; 2; N ; v:

ð2bÞ ð2cÞ ð2dÞ ð2eÞ

Obviously, model (2) can deal with uncertainties in the right-hand sides presented as probability distributions when coefficients in the left-hand sides and in the objective function are deterministic. However, in real-world optimization problems, the quality of information that can be obtained is mostly not satisfactory enough to be presented as probabilities (Li et al., 2006). Such complexities cannot be solved through model (2). IMP is effective in tackling uncertainties expressed as interval values with known lower and upper

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bounds but unknown distribution functions (Huang et al., 1992). Therefore, through incorporating IMP and TSP within a general optimization framework, a hybrid two-stage programming linear model can be formulated as follows: Min f

F

¼

F CF T1 X

þ

v X

F ph DF T2 Y

ð3aÞ

h¼1

subject to: F F AF r X V Br ;

r ¼ 1; 2; N ; m1

F F F F AF t X þ A tV Y z wh ;

ð3bÞ

t ¼ 1; 2; N ; m2 ; h ¼ 1; 2; N ; v

ð3cÞ

xF j z 0;

F xF j aX ; j ¼ 1; 2; N ; n1

ð3dÞ

yF jh z 0;

F yF jh aY ; j ¼ 1; 2; N ; n2 ; h ¼ 1; 2; N ; v

ð3eÞ

m n

m n

ðqs Þ F F F AF ; s X þ A sV Y V Bs ðtÞ

ð5dÞ

xF j z 0;

F xF j aX ; j ¼ 1; 2; N ; n1

ð5eÞ

yF jh z 0;

F yF jh aY ; j ¼ 1; 2; N ; n2 ; h ¼ 1; 2; N ; v:

ð5fÞ

In model (5), decision variables can be sorted into two categories: continuous and binary. Model (5) can be transformed into two deterministic submodels that correspond to the lower and upper bounds of desired objective function value. This transformation process is based on an interactive algorithm, which is different from the best/worst case analysis (Huang, et al., 1992). Interval solutions associated with varying levels of constraint–violation risk can then be obtained by solving the two submodels sequentially. The submodel corresponding to the lower-bound objective function value ( f −) can be firstly formulated as follows (assume that B± N 0, and f ± N 0):

m 1

1n1 1 1 1 2 2 F F F where AF , AF , BF , CF , r afR g t afR g r afR g T1 afRg 1n2 ± F F n1 1 F F n2 1 a R , X a R , Y a R and {R }denote a set of DF f g f g f g T1 interval parameters and/or variables; superscripts ‘−’ and ‘+’ represent lower and upper bounds of the interval values, respectively. In a real-world MSW management problem, randomness in other right-hand-side parameters such as available waste-management capacities, also need to be reflected. For example, the waste-management capacity may be fixed with a level of probability, i.e. qs ∈ [0, 1] (significance level), which represents the admissible risk of violating the uncertain capacity constraint. The chance-constrained programming (CCP) method can be used for dealing with this type of uncertainty. In CCP, it is required that the constraints be satisfied under given probabilities. A general CCP formulation can be expressed as (Charnes et al., 1972):

s ¼ 1; 2; N ; m3

Min

f ¼

Xk1

c x j¼1 j j

þ

þ

Xn1

c  xþ j¼k1 þ1 j j

v X

Xn2 j¼k2 þ1

þ

v Xk2 X j¼1

 pjh d j yjh

h¼1

þ pjh d j yjh

ð6aÞ

h¼1

subject to: Xk1 j¼1

  Xn1  jarj jþ Sign aþ rj xj þ j¼k

1 þ1

  þ þ jarj j Sign a rj xj Vbr ; 8r

ð6bÞ

    Xn1  þ jatj jþ Sign aþ ja j Sign a tj xj tj xj þ j¼k1 þ1 tj     Xk2 Xn2 þ   y yþ þ j¼1 ja tVj jþ Sign aV jaVj Sign aV jh þ tj tj jh zwh ; 8t; h j¼k þ1 tj

Xk1

j¼1

2

ð6cÞ     Xn1  þ jasj jþ Sign aþ ja j Sign a sj xj sj xj þ j¼k1 þ1 sj     Xk2 X  þ ðqs Þ n2 þ þ j¼1 jaV Vþ y ja Vj Sign a sjV yþ ; 8s; h sj j Sign a sj jh þ jh V bs j¼k þ1 sj

Xk1

j¼1

Min cT x

ð4aÞ

2

ð6dÞ

subject to: Pr½fAs ðtÞX V bs ðtÞg z 1  qs ; As ðtÞaAðtÞ; s ¼ 1; 2; N ; m

ð4bÞ

x z 0;

ð4cÞ

Model (4) can be converted into deterministic versions through: (i) fixing a certain level of probability qs for uncertain constraint s, and (ii) imposing the condition that the constraint should be satisfied with at least a probability level of 1 − qs (Huang, 1998). Therefore, to deal with uncertainties presented in terms of interval values and random variables and to reflect the reliability of satisfying (or risk of violating) system constraints, the approaches of TSP, IMP and CCP can be incorporated within an integer programming (IP) framework. This leads to an integrated two-stage optimization model (ITOM) as follows: F Min f F ¼ CF T1 X þ

v X

F ph DF T2 Y

ð5aÞ

h¼1

x j z 0; j ¼ 1; 2; N ; k1

ð6eÞ

xþ j z 0; j ¼ k1 þ 1; k1 þ 2; N ; n1

ð6fÞ

y jh z 0; 8h; j ¼ 1; 2; N ; k2

ð6gÞ

yþ jh z 0; 8h; j ¼ k2 þ 1; k2 þ 2; N ; n2

ð6hÞ

where x±j , j = 1, 2,…, k1, are interval variables with positive coefficients in the objective function; x±j , j = k1 + 1, k1 + 2,…, n1, are interval variables with negative coefficients; y±jh, j = 1, 2,…, k2 and h = 1, 2,…, v are random variables with positive coefficients in the objective function; y±jh, j = k2 + 1, k2 + 2,…, n2 and h = 1, 2,…, v, are random variables with negative coefficients. Solutions of x−j opt( j = 1, 2,…, k1), xj +opt( j = k1 + 1, k1 + 2,…, n1), y−jh opt( j = 1, 2,…, k2), and y+jh opt( j = k2 + 1, k2 + 2,…, n2) can be obtained through submodel (6). Based on the above solutions, the second submodel for f+ can be formulated as follows:

subject to: F F AF r X V Br ;

r ¼ 1; 2; N ; m1

F F F F AF t X þ A tV Y z wh ;

t ¼ 1; 2; N ; m2 ; h ¼ 1; 2; N ; v

ð5bÞ ð5cÞ

Min f þ ¼

Xk1

cþ xþ j¼1 j j

þ

Xn2 j¼k2 þ1

þ

Xn1

v X h¼1

c þ x j¼k1 þ1 j j

 pjh dþ j yjh

þ

v Xk2 X j¼1

þ pjh dþ j yjh

h¼1

ð7aÞ

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subject to:   Xn1 Xk1  þ ja j Sign a rj rj xj þ j¼1 j¼k

1

  þ   ja j Sign aþ rj rj xj Vbr ; 8r þ1

ð7bÞ

    Xn1 þ  jatj j Sign a ja jþ Sign aþ tj xj þ tj xj j¼k1 þ1 tj     Xk2 Xn2   þ þ j¼1 jaV ja Vjþ Sign atjVþ y yþ tj j Sign aV rj jh zwh ; 8t; h jh þ j¼k þ1 t j

Xk1

j¼1

2

ð7cÞ     Xn1 þ  ja j Sign a ja jþ Sign aþ sj xj þ sj xj j¼1 sj j¼k1 þ1 sj     Xk2 X   ðqs Þ n2   þ þ j¼1 jaV jaVjþ Sign aV ; 8s; h yþ y sj j Sign aV sj jh V bs jh þ sj j¼k þ1 sj

Xk1

2

ð7dÞ  xþ j z xj opt ; j ¼ 1; 2; N ; k1

ð7eÞ

þ 0Vx j V xj opt ; j ¼ k1 þ 1; k1 þ 2; N ; n1

ð7fÞ

 yþ jh z yjh opt ; 8h; j ¼ 1; 2; N ; k2

ð7gÞ

þ 0 V y jh V yjh opt ; 8h; j ¼ k2 þ 1; k2 þ 2; N ; n2 :

ð7hÞ

Solutions of xj+ opt( j = 1, 2,…, k1), xj− opt(j = k1 + 1, k1 + 2,…, n1), − 2,…, k2), and yjh opt( j = k2 + 1, k2 + 2,…, n2) can be obtained through submodel (7). Through integrating solutions of submodels (6) and (7), interval solution for model (5) under a set of qs (s = 1, 2,…, m3) levels can be expressed as follows: y+jh opt( j = 1,

h i  þ xF j opt ¼ xj opt ; xj opt ; 8j

ð8aÞ

h i  þ yF jh opt ¼ yjh opt ; yjh opt ; 8j; h

ð8bÞ

h i F  þ fopt : ¼ fopt ; fopt

ð8cÞ

Generally, four special characteristics of the developed ITOM will make it unique compared with the existing optimization techniques: (i) it can handle uncertainties in the model's left-hand sides presented as intervals and those in the right-hand sides as probability distributions; (ii) it can facilitate dynamic analysis for capacity-expansion planning within a multi-stage context; (iii) it can facilitate analyses of policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated; (iv) it can help examine the reliability of satisfying the system constraints under uncertainty.

3.

The study system

The City of Regina is the capital of the Province of Saskatchewan, and is located in the heart of western Canada. Its population is about 195,000, where the households generate residential wastes of 71,000 to 82,000 tonnes/year (Statistics Canada, 2006; City of Regina, 2006). Solid waste management in the city covers many areas, ranging from garbage collection to environmental protection. It involves the provision of specific and personal services to most of the residents (through waste collection and recovery) as well as indirect services to the entire community (through waste disposal and recovery). The generated solid wastes typically include paper,

yard waste, food waste, plastics, metals, glass, wood and other items. Consistent with many communities in western Canada, the city relies mostly on a sanitary landfill for disposing of its MSW. The landfill is located in the northeastern part of the city, occupying 97 ha with an actual landfilling area of 60 ha. Approximately 65,000 tonnes/year of MSW generated from the residential sector were buried at the landfill (nearly 90% of the total waste generated by households). The existing landfill is expected to be able to accept waste till 2011 or 2012 (City of Regina, 2005). Due to the scarcity of land around the urban center and the growing opposition from the public with regard to landfill operation, the city is making efforts on waste diversion through the integrated solid waste-management (ISWM) approach to change the current practice of relying solely on landfill for its waste disposal. The study time horizon is 25 years (from 2006 to 2030), which is further divided into five 5-year periods. Tables 1 and 2 contain regular collection and transportation costs for allowable-waste flows, operating costs for waste-management-facilities, penalties for surplus waste flows, and revenues from recycling and composting facilities over the five planning periods. A projected allowable-waste-flow level (quota of waste flow) to each facility is pre-regulated by the Municipal Engineering Department of the City of Regina. If this level is not exceeded, a regular (normal) cost will apply to the system. However, if it is exceeded, an excess waste flow will be generated (i.e., excess flow = generated flow − assigned quota). The surplus flow will be disposed of at a premium, resulting in excess costs (penalties) to the system. The penalties are associated with (i) operating cost for excess waste flows as shipped locally to alternative and more expensive facilities, (ii) collection and transportation costs for excess flows to more remote facilities, and (iii) extra expenses and/or fines caused by the improper policies. Thus, the plan for allowable-waste-flow allocation at the beginning is named the first-stage decision, and that for the recourse actions is called the second-stage decision. The first-stage decision has to be made before further waste-generation information is disclosed, whereas the second-stage one is to adapt to the previous decision based on further information of waste-generation levels. Under such a situation, the total waste flow will be the sum of both fixed allowable and probabilistic surplus flows. Table 3 presents the waste-generation rates and the associated probabilities of occurrence in the five planning periods, indicating that the waste-generation amounts are highly uncertain, presented by intervals with the associated probabilities varying dynamically. Table 4 shows the capacityexpansion options and the relevant capital costs for different waste-management-facilities. For example, for the landfill development issue, there are three options being taken into account by the city: (a) developing a new landfill at the southern, (b) developing a new landfill at the north, and (c) expanding the existing landfill. Moreover, the Regina Round Table on Solid Waste Management (RRTSWM) developed a 20-year solid waste-management strategy (City of Regina, 2005). The city is putting efforts on increasing waste-diversion rate through developing an integrated solid waste-management approach. Generally, many challenging questions will arise in the study system, such that (i) what is the least-cost means for meeting the reduction and/or diversion goals? (ii) if a new landfill is needed, where should it be located? (iii) what new facilities should be

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Table 1 – Costs and revenues for allowable-waste flows Planning period k=1

k=2

k=3

k=4

Operating costs of waste-management-facilities, OP±ik Landfill [9, [6.9, [5.83, 17] 13.02] 11.02] Composting [21, [16.09, [13.61, 26] 19.92] 16.85] Recycling [61, [46.74, [39.54, 67.8] 51.95] 43.95]

k=5

($/tonne): [4.94, 9.32] [11.52, 14.26] [33.45, 37.18]

[4.18, 7.89] [9.74, 12.06] [28.30, 31.46]

[17.55, 20.29] 40.03 55.39

[14.85, 17.17] 33.87 46.86

Revenues from waste-management-facilities, RE±ik ($/tonne): Composting [5.0, [3.83, [3.24, [2.74, 10.0] 7.66] 6.48] 5.48] Recycling [45.0, [34.48, [29.17, [24.68, 55.0] 42.14] 35.65] 30.16]

[2.32, 4.64] [20.88, 25.52]

Costs of residue disposal at the landfill, FT±ik ($/tonne): From composting [1.68, [1.287, [1.089, [0.92, 2.1] 1.609] 1.36] 1.152] From recycling [1.47, [1.126, [0.953, [0.806, 1.68] 1.287] 1.089] 0.92]

[0.779, 0.974] [0.68, 0.779]

Collection and transportation To landfill [32, 37] To composting 73 To recycling 101

costs, TR±ik [24.52, 28.35] 55.93 77.38

($/tonne): [20.74, 23.98] 47.32 65.47

developed or expanded? (iv) what capacity of expansion or development should be undertaken? and (v) in each period, how to effectively use the facilities and allocate the relevant waste

Table 2 – Costs and revenues for excess waste flows Planning period k=1

k=2

k=3

k=4

k=5

DP±ik

Operating costs of waste-management-facilities, ($/tonne): Landfill [18, 34] [13.79, [11.67, [9.87, 26.05] 22.04] 18.65] Composting [34, 42] [26.05, [22.04, [18.65, 32.18] 27.23] 23.03] Recycling [104, [79.69, [67.41, [57.04, 115.3] 88.34] 74.74] 63.23] Collection and transportation costs, To landfill [48, 55.5] To composting 109.5 To recycling 151.5

DR±ik ($/tonne): [36.77, [31.11, 42.52] 35.98] 83.89 70.98 116.07 98.20

[8.35, 15.78] [15.78, 19.49] [48.25, 53.50]

[26.32, 30.27] 60.05 83.08

[22.27, 25.75] 50.81 70.29

Revenues from waste-management-facilities, RM±ik ($/tonne): Composting [5.0, [3.83, [3.24, [2.74, 10.0] 7.66] 6.48] 5.48] Recycling [45.0, [34.48, [29.17, [24.68, 55.0] 42.14] 35.65] 30.16]

[2.32, 4.64] [20.88, 25.52]

Costs of residue disposal at the landfill, DT±ik ($/tonne): From composting [2.52, [1.93, [1.63, 3.15] 2.41] 2.04] From [2.21, [1.69, [1.43, recycling 2.52] 1.93] 1.63]

[1.17, 1.46] [1.03, 1.17]

[1.38, 1.73] [1.21, 1.38]

Table 3 – Waste-generation rates and the associated probabilities Level of Probability waste generation h = 1 (low)

0.1

h = 2 (lowmedium) h=3 (medium) h=4 (mediumhigh) h = 5 (high)

0.2 0.4 0.2

0.1

Waste-generation rate, w±kh (tonne/week) k=1

k=2

k=3

k=4

k=5

[1373, 1423] [1423, 1478] [1478, 1543] [1543, 1598]

[1418, 1469] [1469, 1525] [1525, 1595] [1595, 1651]

[1465, 1517] [1517, 1574] [1574, 1649] [1649, 1706]

[1513, 1567] [1567, 1624] [1624, 1705] [1705, 1762]

[1563, 1619] [1619, 1678] [1678, 1761] [1761, 1820]

[1598, 1648]

[1651, 1702]

[1706, 1758]

[1762, 1816]

[1820, 1876]

flows? Thus, the proposed method will be applied to the city's long-term solid waste-management planning. Issues concerning planning for cost–effective diversion and prolongation of the landfill will be addressed. The modeling results will be used for supporting decisions of the city's long-term MSW management, such as (a) identification of desired capacity-expansion schemes for waste-management-facilities, (b) allocation of waste flows to suitable facilities, and (c) analysis of the tradeoff between the cost of waste management and the risk of system disruption. The modeling formulation (based on the developed ITOM method) for the city's MSW management is presented in the Appendix. The model includes continuous and binary decision variables. The binary variables represent the development or expansion options for waste-management-facilities in different periods (i.e. Y±ik and Z±imk); their solutions can be used for answering the questions related to timing, sizing and siting for wastemanagement-facility development and/or expansion under uncertainty. The continuous variables represent the optimized waste flows from the city to the waste-management-facilities. Furthermore, the continuous variables include two subsets: those ± ) that must be determined before the (the first-stage ones, Tik random variables (i.e. waste-generation rates) are disclosed, and ± ) that will be determined after those (the second-stage ones, Xikh the random variables are disclosed. A set of chance constraints on waste-management capacities and waste-diversion rates are considered, which can help investigate the risks of violating the capacity and diversion constraints. Interval solutions under given significance levels can be obtained based on solution method as described in Section 2. They will be useful for generating a range of decision alternatives under various environmental, socioeconomic, and system-reliability conditions.

4.

Result and discussion

In this study, a set of chance constraints on waste-management capacities and waste-diversion rates are considered, which can help investigate the risks of violating the capacity and diversion constraints. Table 5 presents the solutions of facility expansion schemes under different qi levels (i.e. significance levels, which represent the admissible levels of constraint–violation risk). Under qi = 0.01, 0.05 and 0.10, the results indicate that the landfill

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Table 4 – Capacity-expansion options and the relevant capital costs Expansion option

Total expanded capacity

Expansion cost (06)

Expanded capacity for residential

k=1

k=2

k=3

k=4

k=5

28.5 ha 28.5 ha 28.5 ha

6.32 [9.60, 10.45] [6.44, 14.94]

5.72 [8.69, 9.46] [5.83, 13.53]

4.84 [7.35, 8.01] [4.93, 11.45]

4.10 [6.22, 6.77] [4.17, 9.69]

3.46 [5.26, 5.73] [3.53, 8.20]

Composting facility (ΔTC±4mk): Option 1 (m = 1) 189 tonnes/week Option 2 (m = 2) 385 tonnes/week Option 3 (m = 3) 483 tonnes/week

132 tonnes/week 270 tonnes/week 338 tonnes/week

[2.12, 2.54] [4.23, 5.08] [5.29, 6.35]

[1.92, 2.30] [3.83, 4.60] [4.79, 5.75]

[1.62, 1.95] [3.24, 3.89] [4.05, 4.86]

[1.37, 1.65] [2.74, 3.29] [3.43, 4.12]

[1.16, 1.39] [2.32, 2.79] [2.90, 3.48]

Recycling facility (ΔTC±5mk): Option 1 (m = 1) 350 tonnes/week Option 2 (m = 2) 700 tonnes/week Option 3 (m = 3) 875 tonnes/week

140 tonnes/week 280 tonnes/week 350 tonnes/week

[4.51, 5.00] [9.02, 10.01] [10.86, 12.51]

[4.08, 4.53] [8.17, 9.06] [9.83, 11.33]

[3.46, 3.83] [6.91, 7.67] [8.32, 9.58]

[2.92, 3.24] [5.85, 6.49] [7.04, 8.11]

[2.47, 2.74] [4.95, 5.49] [5.96, 6.86]

Landfill (ΔLC±i ): Expansion (i = 1) North site (i = 2) South site (i = 3)

71.3 ha 71.3 ha 71.3 ha

would be expanded (based on the existing landfill) at the start of period 2 (the start of 2010) with an incremental capacity of 71.3 ha (i.e.Y±12 opt = [1, 1]). However, only 28.5 ha (approximately 40% of the total) would be dedicated to the city's residential waste, since the IC&I (industrial, commercial and institutional) and rural sectors were not included within the scope of this study. In comparison, the expansion schemes for composting and recycling facilities would vary under different qi levels. When qi = 0.01, a centralized composting facility should be developed with a capacity of 385 tonnes/week at the start of period 1, with 70% of the capacity (i.e. 270 tonnes/week) being dedicated to the residential waste; then this facility should be expanded by an increment of 189 tonnes/week at the starts of periods 3 and 4, respectively (i.e. Z±421 opt = [1, 1], Z±413 opt = [1, 1], and Z±414 opt = [1, 1]). When qi = 0.05 and 0.10, a centralized composting facility would be developed with a capacity of 189 tonnes/week at the start of period 1; then this facility should be expanded by an increment of 189 tonnes/week at the starts of periods 3 and 4, respectively. Consequently, the total expanded capacities for the composting facility would be 763, 567 and 567 tonnes/week under qi = 0.01, 0.05 and 0.10, respectively. The recycling facility would be expanded twice (each with an increment of 350 tonnes/week) at the starts of periods 1 and 3 when qi = 0.01, 0.05 and 0.10 (i.e. Z±511 opt = [1, 1] and Z±513 opt = [1, 1]); then this facility would be expanded by 350 tonnes/week at the

start of period 4 when qi = 0.01 and 0.05 (i.e. Z±514 opt = [1, 1]), and expanded by 350 tonnes/week at the start of period 5 when qi = 0.10 (Z±515 opt = [1, 1]). Generally, an increased qi level means a raised risk of violating the modeling constraints but, at the same time, it leads to a decreased strictness for the capacity constraints and thus results in a lower capacity-expansion amount and a lower capital cost, and vice versa. Table 5 also provide the results for expanded capacity of composting and recycling facilities being 861 and 1050 tonnes/week when qi = 0, further demonstrating that more diversion capacity increments would be needed if no constraint violation is allowed. Table 6 shows the solutions of waste-flow allocation during periods 1 to 5 under different qi levels. An excess flow will be generated if the allowable-waste-flow level is exceeded (i.e. excess flow = generated flow − assigned quota). The waste-flow patterns (including allowable and excess flows) would vary dynamically due to temporal and spatial variations of wastegeneration/management conditions. Analyses of the flows allocated to the landfilling, composting and recycling facilities (when the waste-generation rate is high in period 1) indicate: (a) for the landfill, the optimized allowable flows would be [1068, 1093] tonnes/week; the optimized excess flows would be [146.6, 159.6] tonnes/week under qi = 0.01,

Table 5 – Solutions of capacity development/expansion under different qi levels Facility

Landfill Composting

Recycling

Expansion cost ($106)

Symbol

Y±12 opt Z±411 opt Z±421 opt Z±431 opt Z±413 opt Z±414 opt Z±511 opt Z±513 opt Z±514 opt Z±515 opt

Total expanded capacity

Expanded capacity for residential

Period

71.3 ha 189 tonnes/week 385 tonnes/week 483 tonnes/week 189 tonnes/week 189 tonnes/week 350 tonnes/week 350 tonnes/week 350 tonnes/week 350 tonnes/week

28.5 ha 132 tonnes/week 270 tonnes/week 338 tonnes/week 132 tonnes/week 132 tonnes/week 140 tonnes/week 140 tonnes/week 140 tonnes/week 140 tonnes/week

2 1 1 1 3 4 1 3 4 5

Capacity-expansion solution q=0

q = 0.01

q = 0.05

q = 0.10

[1, 1] 0 0 [1, 1] [1, 1] [1, 1] [1, 1] [1, 1] [1, 1] 0 [12.44, 14.08]

[1, 1] 0 [1, 1] 0 [1, 1] [1, 1] [1, 1] [1, 1] [1, 1] 0 [11.70, 13.19]

[1, 1] [1, 1] 0 0 [1, 1] [1, 1] [1, 1] [1, 1] [1, 1] 0 [11.56, 13.02]

[1, 1] [1, 1] 0 0 [1, 1] [1, 1] [1, 1] [1, 1] 0 [1, 1] [11.38, 12.82]

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Table 6 – Solution of the ITOM for continuous variables Period

k=1

k=2

k=2

k=3

k=4

Facility (i)

Level of Probability Allowable- q = 0.01 (tonnes/week) q = 0.05 (tonnes/week) waste (%) waste flow Excess Optimized Excess Optimized generation (tonnes/ waste waste flow waste waste flow week) flow flow

1 2 3 1

L L L L–M

10 10 10 20

[1068, 1093] [143, 146] [214, 219] [1068, 1093]

0 0 0 [0.0, 20.0]

2 3 1

L–M L–M M

20 20 40

[143, 146] [214, 219] [1068, 1093]

0 0 [53.0, 79.8]

2 3 1

M M M–H

40 40 20

[143, 146] [214, 219] [1068, 1093]

2 3 1

M–H M–H H

20 20 10

[143, 146] [214, 219] [1068, 1093]

2 3 1 2 3 1 2 3 1

H H L L L L–M L–M L–M M

10 10 10 10 10 20 20 20 40

[143, [214, [925, [250, [294, [925, [250, [294, [925,

[0.0, 5.2] 0 [104.7, 121.6] [13.2, 18.4] 0 [146.6, 159.6] [26.4, 30.4] 0 0 0 0 [0.0, 21.0] 0 0 [51.1, 73.9]

2 3 1

M M M–H

40 40 20

[250, 256] [294, 301] [925, 947]

2 3 1

M–H M–H H

20 20 10

[250, 256] [294, 301] [925, 947]

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

H H L L L L–M L–M L–M M M M M–H

10 10 20 20 20 20 20 20 40 40 40 20

[250, [294, [759, [380, [380, [759, [380, [380, [759, [380, [380, [759,

2 3 1

M–H M–H H

20 20 10

[380, 388] [380, 388] [759, 776]

2 3 1 2 3 1 2 3

H H L L L L–M L–M L–M

10 10 10 10 10 20 20 20

[380, [380, [580, [518, [471, [580, [518, [471,

146] 219] 947] 256] 301] 947] 256] 301] 947]

256] 301] 776] 388] 388] 776] 388] 388] 776] 388] 388] 776]

388] 388] 592] 529] 481] 592] 529] 481]

[4.9, 17.1] 0 [95.9, 109.7] [30.1, 37.3] 0 [131.7, 142.4] [50.3, 55.6] 0 0 0 0 [0.0, 22.0] 0 0 [43.8, 65.1] [11.2, 31.9] 0 [82.1, 94.1] [47.9, 59.9] 0 [111.1, 120.7] [75.9, 85.3] 0 0 0 0 [0.0, 19.8] [0.0, 2.2] 0

[1068, 1093] [143, 146] [214, 219] [1068.0, 1113.0] [143, 146] [214, 219] [1121.0, 1172.8] [143.0, 151.2] [214, 219] [1172.7, 1214.6] [156.2, 164.4] [214, 219] [1214.6, 1252.6] [169.4, 176.4] [214.0, 219.0] [925, 947] [250, 256] [294, 301] [925.0, 968.0] [250, 256] [294, 301] [976.1, 1020.9] [254.9, 273.1] [294, 301] [1020.9, 1056.7] [280.1, 293.3] [294, 301] [1056.7, 1089.4] [300.3, 311.6] [294, 301] [759, 776] [380, 388] [380, 388] [759.0, 798.0] [380, 388] [380, 388] [802.8, 841.1] [391.2, 419.9] [380, 388] [841.1, 870.1] [427.9, 447.9] [380, 388] [870.1, 896.7] [455.9, 473.3] [380, 388] [580, 592] [518, 529] [471, 481] [580.0, 611.8] [518.0, 531.2] [471, 481]

0 0 0 [0.0, 20.0] 0 0 [53.0, 85.0] 0 0 [118.0, 138.0] 0 0 [163.0, 179.5] [10.0, 13.5] 0 0 0 0 [0.0, 21.0] 0 0 [56.0, 90.3] [0.0, 0.7] 0 [112.3, 126.7] [13.7, 20.3] 0 [148.7, 159.9] [33.3, 38.1] 0 0 0 0 [0.0, 22.0] 0 0 [55.0, 82.0] [0.0, 15.0] 0 [99.0, 111.7] [31.0, 42.3] 0 [128.7, 138.7] [58.3, 67.3] 0 0 0 0 [0.0, 22.0] 0 0

[1068, 1093] [143, 146] [214, 219] [1068.0, 1113.0] [143, 146] [214, 219] [1121.0, 1178.0] [143, 146] [214, 219] [1186.0, 1231.0] [143, 146] [214, 219] [1231.0, 1272.5] [153.0, 159.5] [214.0, 219.0] [925, 947] [250, 256] [294, 301] [925.0, 968.0] [250, 256] [294, 301] [981.0, 1037.3] [250.0, 256.7] [294, 301] [1037.3, 1073.7] [263.7, 276.3] [294, 301] [1073.7, 1106.9] [283.3, 294.1] [294, 301] [759, 776] [380, 388] [380, 388] [759.0, 798.0] [380, 388] [380, 388] [814.0, 858.0] [380.0, 403.0] [380, 388] [858.0, 887.7] [411.0, 430.3] [380, 388] [887.7, 914.7] [438.3, 455.3] [380, 388] [580, 592] [518, 529] [471, 481] [580.0, 614.0] [518, 529] [471, 481]

q = 0.1 (tonnes/week) Excess waste flow

Optimized waste flow

0 0 0 [0.0, 20.0]

[1068, 1093] [143, 146] [214, 219] [1068.0, 1113.0] [143, 146] [214, 219] [1121.0, 1178.0] [143, 146] [214, 219] [1186.0, 1233.0] [143, 146] [214, 219] [1239.7, 1278.5] [144.3, 159.5] [214.0, 219.0] [925, 947] [250, 256] [294, 301] [925.0, 968.0] [250, 256] [294, 301] [981.0, 1038.0] [250, 256] [294, 301] [1046.0, 1082.7] [255.0, 267.3] [294, 301] [1082.7, 1116.2] [274.3, 284.8] [294, 301] [759, 776] [380, 388] [380, 388] [759.0, 798.0] [380, 388] [380, 388] [814.0, 867.0] [380.0, 394.0] [380, 388] [867.0, 897.0]

0 0 [53.0, 85.0] 0 0 [118.0, 140.0] 0 0 [171.7, 185.5] [1.3, 4.5] 0 0 0 0 [0.0, 21.0] 0 0 [56.0, 91.0] 0 0 [121.0, 135.7] [5.0, 11.3] 0 [157.7, 169.2] [24.3, 28.8] 0 0 0 0 [0.0, 22.0] 0 0 [55.0, 91.0] [0.0, 6.0] 0 [108.0, 121.0] [22.0, 33.0] 0 [138.0, 148.4] [49.0, 57.6] 0 0 0 0 [0.0, 22.0] 0 0

[402.0, 421.0] [380, 388] [897.0, 924.4] [429.0, 445.6] [380, 388] [580, 592] [518, 529] [471, 481] [580.0, 614.0] [518, 529] [471, 481]

(continued on next page) (continued on next page)

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Table 6 (continued) Period

k=4

k=5

Expected system cost ($):

Facility (i)

Level of Probability Allowable- q = 0.01 (tonnes/week) q = 0.05 (tonnes/week) waste (%) waste flow Excess Optimized Excess Optimized generation (tonnes/ waste waste flow waste waste flow week) flow flow

1 2 3 1 2 3 1

M M M M–H M–H M–H H

40 40 40 20 20 20 10

[580, [518, [471, [580, [518, [471, [580,

592] 529] 481] 592] 529] 481] 592]

2

H

10

[518, 529]

74.2

[42.9, 62.0] [12.1, 41.0] 0 [73.9, 83.8] [62.0, 76.2] 0 [95.8, 104.5] [592.2, 603.2] 92.6

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

H L L L L–M L–M L–M M M M M–H M–H M–H H

10 10 10 10 20 20 20 40 40 40 20 20 20 10

[471, [567, [535, [519, [567, [535, [518, [567, [535, [518, [567, [535, [518, [567,

[35.1, 47.7] 0 0 0 [0.0, 19.5] [0.0, 4.5] 0 [31.5, 49.1] [25.5, 57.2] [0.0, 0.7] [61.1, 70.2] 57.2 [21.7, 38.6] [82.2, 90.2]

[506.1, 528.7] [567, 579] [535, 546] [519, 529] [567.0, 598.5] [535.0, 550.5] [518, 529] [598.5, 628.1] [560.5, 603.2] [518.0, 529.7] [628.1, 649.2] [592.2, 603.2] [539.7, 567.6] [649.2, 669.2]

2 3

H H

10 10

[535, 546] [519, 529]

481] 579] 546] 529] 579] 546] 529] 579] 546] 529] 579] 546] 529] 579]

[31.8, 50.3] [23.2, 52.7] 0 [62.3, 71.7] [73.7, 74.2] [0.0, 14.1] [83.7, 92.1]

[611.8, 642.3] [541.2, 581.7] [471, 481] [642.3, 663.7] [591.7, 603.2] [471.0, 495.1] [663.7, 684.1]

[622.9, 654.0] [530.1, 570.0] [471, 481] [653.9, 675.8] [580.0, 605.2] [471, 481] [675.8, 696.5] [610.6, 621.6]

[4.6, 16.9] [475.6, 497.9] 0 [567, 579] 0 [535, 546] 0 [519, 529] [0.0, 24.0] [567.0, 603.0] 0 [535, 546] 0 [518, 529] [43.0, 61.2] [610.0, 640.2] [14.0, 45.8] [549.0, 591.8] 0 [518, 529] [73.2, 82.7] [640.2, 661.7] [66.8, 75.6] [601.8, 621.6] [0.0, 7.7] [518.0, 536.7] [94.7, [661.7, 682.0] 103.0] 57.2 [592.2, 603.2] 75.6 [610.6, 621.6] [59.6, 74.6] [578.6, 603.6] [28.7, 43.4] [547.7, 572.4] [104.58, 130.48] × 106 [105.23, 131.29] × 106

[163.0, 179.5] tonnes/week under qi = 0.05, and [171.7, 185.5] tonnes/week under qi = 0.1; thus the total flows to the landfill would be [1214.6, 1252.6], [1231.0, 1272.5], and [1239.7, 1278.5] tonnes/week under qi = 0.01, 0.05 and 0.10, respectively (i.e. A±ijtk = T±ijt = M±ijtk); (b) for the composting facility, the optimized allowable flows would be [143, 146] tonnes/week; the optimized excess flows would be [26.4, 30.4] tonnes/week under qi = 0.01, [10.0, 13.5] tonnes/week under qi = 0.05, and [1.3, 4.5] tonnes/week under qi = 0.1; thus the total flows to the composting facility would be [169.4, 176.4], [153.0, 159.5], and [144.3, 159.5] tonnes/week under qi = 0.01, 0.05 and 0.10, respectively; (c) for the recycling facility, the optimized allowable-waste flows would be [214, 219] tonnes/week; there would be no excess waste to this facility because it has the highest operating costs and penalties. The solutions under the other waste-generation conditions can be similarly interpreted based on the results presented in Table 6. Generally, the planned flows to the landfilling, composting and recycling facilities over the five periods can be summarized as follows: (i) The probabilistic excess flows to the landfill would be [57.68, 78.66] × 10 3 , [67.34, 91.55] × 10 3 and [71.25,

q = 0.1 (tonnes/week) Excess waste flow

Optimized waste flow

[48.8, 68.2] [628.8, 660.2] [6.2, 34.8] [524.2, 563.8] 0 [471, 481] [80.2, 90.2] [660.2, 682.2] [55.8, 69.8] [573.8, 598.8] 0 [471, 481] [102.2, [682.2, 703.2] 111.2] [90.8, [608.8, 631.8] 102.8] 0 [471, 481] 0 [567, 579] 0 [535, 546] 0 [519, 529] [0.0, 24.0] [567.0, 603.0] 0 [535, 546] 0 [518, 529] [49.2, 67.6] [616.2, 646.6] [7.8, 39.4] [542.8, 585.4] 0 [518, 529] [79.6, 89.3] [646.6, 668.3] [60.4, 76.7] [595.4, 622.7] 0 [518, 529] [101.3, [668.3, 688.9] 109.9] 89.1 [624.1, 635.1] [8.6, 23.0] [527.6, 552.0] [104.22, 130.04] × 106

98.61] × 103 tonnes when the significance levels (qi) are 0.01, 0.05 and 0.10, respectively; thus, the optimized waste flows allocated to the landfill (including allowable and excess flows) would be [1074.19, 1118.12] × 103, [1083.85, 1131.01] × 103, and [1087.76, 1138.07] × 103 tonnes when qi = 0.01, 0.05 and 0.10, respectively. (ii) There would be [25.74, 38.23] × 103, [18.80, 29.34] × 103 and [15.56, 25.69] × 103 tonnes of probabilistic excess flows diverted to the composting facility when qi levels are 0.01, 0.05 and 0.10, respectively; thus, the optimized waste flows to this facility would be [501.80, 524.46] × 103, [494.86, 515.57] × 103, and [491.62, 511.92] × 103 tonnes when qi = 0.01, 0.05 and 0.10, respectively. (iii) The excess flows diverted to the recycling facility would be [3.60, 6.01] ×103, [0.87, 1.97] × 103, and [0.22, 0.60] × 103 tonnes when qi levels are 0.01, 0.05 and 0.10; thus, the optimized waste flows to this facility would be [493.22, 506.07] × 103, [490.49, 502.03] × 103, and [489.84, 500.66] ×103 tonnes when qi = 0.01, 0.05 and 0.10, respectively. Figs. 1 and 2 summarize the excess and total waste flows to the landfilling, composting and recycling facilities under different constraint–violation-risk levels during the entire planning horizon. The excess and total flows to the landfill would increase with the qi level, while the excess and total flows to the composting and recycling facilities would decrease

S CIE N CE OF T H E TOT AL E N V I RO N ME N T 3 9 2 ( 2 00 8 ) 1 7 5–1 86

Fig. 1 – Excess waste flows to the landfill, composting and recycling facilities under different qi levels. with qi level. The diversion rates (i.e. waste flows to the composting and recycling facilities) would be [47.80, 48.25]% when qi = 0.01, [47.28, 47.70]% when qi = 0.05, and [47.04, 47.47]% when qi = 0.10. The diversion rate would keep decreasing when qi level is increased. The remaining landfill capacities at the end of the planning horizon would be [837.95, 938.16] × 103, [826.20, 929.23] × 103 and [819.46, 925.26] × 103 tonnes when qi = 0.01, 0.05 and 0.10, respectively, demonstrating a decreasing tendency along with an increasing qi level. This is due to the following facts: (i) an increased qi level means a relaxed diversion-rate constraint and thus a raised risk in violating the diversion requirements, leading to more excess wastes to the landfill; (ii) since the landfill has lower regular and penalty costs than the composting and recycling facilities, it would often be the desired target for accepting the excess wastes. The results also indicate that only a small portion of excess wastes would be diverted to the recycling facility since this facility has a high penalty for excess flows. Variations in the qi levels also correspond to the decision makers' preferences regarding the tradeoff between the system-cost and the constraint–violation risk. The resulting

Fig. 2 – Total waste flows to the landfill, composting and recycling facilities under different qi levels.

183

system costs ( f ± ) are $[105.23, 131.29] × 10 6 , $[104.58, 130.48] × 106 and $[104.22, 130.04] × 106 under qi = 0.01, 0.05 and 0.10, respectively (Table 6). As the actual values of the variables and/or parameters vary within their two bounds, the expected system cost would change correspondingly between f −opt and f +opt with different reliability levels. Fig. 3 shows the trend of system-cost variations under different qi levels. It is indicated that a lower qi level corresponds to a higher system reliability and, meanwhile, leads to a higher system cost; in comparison, a higher qi level would result in a lower cost but a higher constraint–violation risk. The system cost includes expenses for expansions/developments of landfilling, composting and recycling facilities, disposal of wastes at the landfill, and diversion wastes to the composting and recycling facilities. Fig. 4 presents the detailed costs for waste management under different significance levels. The costs for facility expansions would be $[11.70, 13.19] × 106 (or [10.0, 11.1]% of the total system cost) when qi = 0.01, $[11.56, 13.02] × 106 (or [10.0, 11.1]% of the total) when qi = 0.05, and $[11.38, 12.82] × 106 (or [9.9, 10.9]% of the total) when qi = 0.10; this demonstrates that a raised qi level would lead to a reduced expansion cost. The costs for landfilling wastes are $[36.38, 49.73] × 106 (i.e. [34.6, 37.9]% of the total) when qi = 0.01, $[36.8, 50.51] × 106 (i.e. [35.2, 39.0]% of the total) when qi = 0.05, and $[36.96, 50.79] × 106 (i.e. [35.5, 39.0]% of the total) when qi = 0.10; this indicates that a raised qi level would result an increased operating cost at the landfill. The diversion costs for wastes treated at both composting and recycling facilities are $[57.15, 68.37] × 106 (or [52.1, 54.3]% of the total) when qi = 0.01, $[56.22, 66.95] × 106 (or [51.3, 53.8]% of the total) when qi = 0.05, and $[55.88, 66.43] × 106 (or [51.1, 53.6]% of the total) when qi = 0.10; the corresponding diversion rates would be 48.1% when qi = 0.01, 47.5% when qi = 0.05, and 47.2% when qi = 0.10, implying that a higher qi level would sacrifice the environmental objective but correspond to reduced diversion costs. Therefore, decisions with a lower qi level would possess a higher system reliability but a higher system cost as well; conversely, a desire for reducing the system cost would result in an increased risk of violating the constraints.

Fig. 3 – Relationships between system cost and qi level.

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SC IE N CE OF T H E TOT AL E N V I RO N ME N T 3 9 2 ( 2 00 8 ) 1 7 5–1 86

policies for managing waste generation, reduction and recycling, as well as waste diversion and disposal. For example, if the allowable-waste flows are regulated at too low levels, then high penalties may have to be paid when the allowances are violated; conversely, if the allowable-wasteflow levels are too high, then the planned capacities for waste management would become too large, leading to waste of capital for capacity expansion and/or development. From a long-term planning point of view, municipal waste-generation rates would keep rising due to population increase and economic development, and the available capacities of wastemanagement-facilities may also vary among different time periods. Thus, another complexity is associated with the dynamic variations of system capacity due to the expansions and/or developments of waste-management-facilities.

5.

Fig. 4 – Costs for waste-management activities under different qi levels. On the other hand, the operating costs include expenses for handling fixed allowable and probabilistic excess waste flows. The regular cost for the disposal of allowable-waste flows would be $[88.36, 108.98] × 106 under all significance levels. The penalty costs for handling excess flows would be $[5.17, 9.12] × 106 (qi = 0.01), $[4.66, 8.48] × 106 (qi = 0.05) and $[4.48, 8.24] × 106 (qi = 0.10), demonstrating a decreasing tendency along with the increasing qi level. This is because more excess flows would be shipped to the landfill when the qi level is raised (i.e. relaxed diversion constraints), while the landfill has lower penalty costs for handling the excess flows than the composting and recycling facilities. The developed model could provide a linkage to the preregulated policies that have to be respected when a modeling effort is undertaken. The complexity associated with the allowable-waste-flow levels is mainly caused by the limited capacity for waste disposal and the increasing waste-generation amount. Therefore, variations in the values of X±ik (allowable-waste flow to facility i during planning period k) would lead to multiple scenarios corresponding to different

Conclusions

A solid waste decision-support system based on an integrated two-stage optimization model (ITOM) has been developed for the City of Regina, Canada. The ITOM incorporates the approaches of two-stage stochastic programming, chanceconstrained programming, and interval mathematical programming within an integer programming framework. It can reflect the dynamic, interactive and uncertain characteristics of the solid waste-management system in the study city, and can address issues concerning waste diversion and landfill prolongation as deemed critical by the local authority. Violations for capacity and diversion constraints are allowed under a range of significance levels, which reflect the tradeoffs between system-cost and constraint–violation risk. Reasonable solutions for both binary and continuous variables have been generated. They represent decisions of facility expansion and waste-flow allocation. These inexact solutions can be used to generate multiple decision alternatives, such that desired policies under various environmental, economic, and system-reliability conditions can eventually be identified. Decisions at a lower significance level would lead to a higher system reliability and, at the same time, a higher system cost; conversely, a desire for reducing the system cost would result in an increased risk of violating the constraints. Owing to the complex nature of the waste-management system, the data required for defining different scenarios were extensive. Although most of the obtained data are relatively accurate, others are less so. Therefore, increasing the certainty of the data sets through further investigation and verification would help increase the certainty of the generated solutions. Moreover, extension of the ITOM to a wide scope of waste-management problem for both residential and non-residential sectors would be an interesting topic that deserves future research efforts.

Acknowledgments This research has been supported by the Major State Basic Research Development Program (2005CB724200, 2006CB403303, and 2006CB403307), and the Natural Science Foundation (50675074, 50775081, and 2006AA09Z238) of China, and the

S CIE N CE OF T H E TOT AL E N V I RO N ME N T 3 9 2 ( 2 00 8 ) 1 7 5–1 86

Natural Science and Engineering Research Council of Canada. The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.

Appendix A. Modeling formulation for MSW management Based on the city's waste-management policies, an allowablewaste-flow level from the city to each facility is pre-regulated. If this level is not exceeded, it will result in a regular cost to the system. If it is exceeded, however, the surplus waste has to be handled through more expensive manners, resulting in an excess cost (or penalty) to the system (i.e., excess flow = generated flow− assigned quota). The excess cost may be represented in terms of raised collection, transportation and/or operation costs. The study time horizon is 25 years, consisting of five 5year periods. Consequently, the objective for the entire system is to minimize the net total cost for facility expansion/ development and waste disposal over a long-term planning horizon, subject to various constraints related to waste generation, disposal capacity, and diversion policy. Thus, we have: Min f F ¼

5 X 5 X

 F  F Lk Tik TRik þ OPF ik þ

i¼1 k¼1 5 X 5 X 5 X

þ 

i¼1 k¼1 h¼1 5 X 5 X 5 X

5 X 5 X

F F Lk Tik FEF i FTik 

i¼4 k¼1

ph Lk XF ikh



DRF ik

þ

F ph Lk XF ikh RMik þ

i¼4 k¼1 h¼1

DPF ik



þ

3 X 5 X

5 X 5 X 5 X

5 X 5 X

F Lk Tik REF ik

i¼1 k¼1

[Recycling capacity constraints] 3  X   F qi F F Tik þ XF wkh ; 8k; h; 8i ¼ 1; 2; 3 ikh V DGik i¼1 [Constraints of waste flow to the landfill] 5  X  F F Tik þ XF ikh zwkh ;

8k; h

ða8Þ

ða9Þ

i¼1

[Waste disposal demand constraints] F 0 V XF ikh V Tik V Tik

max ; 8i; k; h

ða10Þ

[Non-negativity and technical constraints] F Yik



¼ 1; if landfill development or expansion is undertaken ; i ¼ 1; 2; 3; 8m ¼ 0; if otherwise

ða11Þ

ZF imk



¼ 1; if capacity development=expansion is undertaken ; i ¼ 4; 5; 8m; k ¼ 0; if otherwise

ða12Þ [Binary constraints] 3 X 5 X

F Yik V1

ða13Þ

i¼1 k¼1

i¼4 k¼1

[Landfill expansion/development may only be considered once in the planning horizon]

F F ph Lk XF ikh FEi DTik

i¼4 k¼1 h¼1 F FLCF k Yik þ

185

5 X 3 X 5 X

F FTCF imk Zimk

i¼4 m¼1 k¼1

ða1Þ

3 X

ZF imk V 1 i ¼ 4; 5 8k:

ða14Þ

m¼1

subject to: Xk V

L k¼1 k

& 3  X

’ 5  X  F   F qi F F F Tik þ XF FE T þ X þ V LC1 ikh i ik ikh

i¼1

i¼4

3 X X qi F kV  þ DLCF Yik ; 8h; kV¼ 1; 2; 3; 4; 5; i ¼ 1; 2; 3 i k¼1

[Expansions for composting and recycling facilities may occur in any given time period] ða2Þ

i¼1

[Total landfill capacity constraints] Xk V k¼1

 F   F q1 Xk V  q1 F Lk T1k þ XF þ DLCF Y1k ; 8h; 1 1kh V LC1 k¼1

ða3Þ

kV¼ 1; 2; 3; 4; 5; i ¼ 1 [Existing landfill capacity constraints] Xk V k¼1

 F  Xk V  q2 F Lk T2k þ XF DLCF Y2k ; 8h; k V¼ 1; 2; 3; 4; 5; i ¼ 2 2 2kh V k¼1 ða4Þ

[Capacity constraints of the northern-site landfill] Xk V k¼1

 F  Xk V  q3 F Lk T3k þ XF DLCF Y3k ; 8h; 3 3kh V k¼1

ða5Þ

kV¼ 1; 2; 3; 4; 5; i ¼ 3 [Capacity constraints of the southern-site landfill] 3 X  F q4 X q4 F kV  F þ XF þ DTCF Z4mk ; 8h; T4k 4kh V TC4 4mk k¼1

ða6Þ

m¼1

k V¼ 1; 2; 3; 4; 5; i ¼ 4

[Composting capacity constraints] 3 X  F q5 X q5 F kV  F þ XF þ DTCF Z5mk ; 8h; T5k 5kh V TC5 5mk k¼1 m¼1

k V¼ 1; 2; 3; 4; 5; i ¼ 5

ða7Þ

where f ± is expected system cost ($); Lk is length of time period k (week); k is time period, k = 1, 2, 3, 4, 5; m is name of expansion option for composting and recycling facilities, m = 1, 2, 3; h is level of waste generation, where h = 1, 2, 3, 4 and 5 when the waste-generation rate is low, low-medium, medium, medium-high and high, respectively; i is type of wastemanagement-facility, i = 1, 2, 3, 4, 5, where i = 1 for the existing landfill, 2 and 3 for two new landfill options, 4 for composting ± facility, and 5 for recycling facility; (DGik)qi is regulated wasteflow rate to the landfill in period k under significance level qi (%), i = 1, 2, 3; DP±ik is operating cost of facility i for excess waste flow during period k ($/tonne) (the second-stage cost parameter), where DP±ik ≥ OP±ik, i = 1, 2, 3, 4, 5; DR±ik is cost of collecting and transporting excess waste from the city to facility i during period k ($/tonne) (the second-stage cost parameter), where ± is disposal cost for excess waste DR±ik ≥ TR±ik, i = 1, 2, 3, 4, 5; DTik residues generated by the composting or recycling facility during period k ($/tonne) (the second-stage cost parameter), where DTik± ≥ FTik± , i = 4, 5; FE±i is residue flow from facility i to the landfill (% of incoming mass to facility i), i = 4, 5; FLC±ik is capital cost for landfill expansion/development in period k ($/tonne), ± is disposal cost for allowable residues generated i = 1, 2, 3; FTik by facility i during period k ($/tonne), i = 4, 5; FTC±imk is capital cost of expanding composting or recycling facility by option m in period k ($/tonne), i = 4, 5; (LC±)qi is existing landfill capacity corresponding to significance level q1 (tonne); (ΔLC±i )qiis expanded/developed capacity of the landfill corresponding

186

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to significance level qi (tonne), i = 1, 2, 3; OP±ik is regular operating cost of waste-management-facility i during period k ($/tonne), i = 1, 2, 3, 4, 5; ph is probability of waste-generation rate (w±kh) with level h (%); qi is probability of violating constraint i (%); RE±ik is revenue generated by processing allowable-waste flows in composting or recycling facility during period k ($/tonne), i = 4, 5; RM±ik is revenue generated by processing excess waste flows in composting or recycling facility during period k ($/tonne) (the second-stage revenue parameter), i = 4, 5; Tik± is allowable-waste flow to facility i during period k (tonne/week) (the first-stage decision variable), i = 1, 2, 3, 4, 5; Tik maxis maximum allowable-waste flow from the city to facility i during period k (tonne/week); (TC±i )qi is existing capacity of composting or recycling facility corresponding to significance level qi (tonne/week), i = 4, 5; (ΔTC±imk)qi is available expanded/developed capacity for composting or recycling facility with expansion option m in period k corresponding to significance level qi (tonne/week), i = 4, 5; TR±ik is cost of collection and transportation for allowablewaste flow to facility i during period k ($/tonne), i = 1, 2, 3, 4, 5; w±kh is residential waste-generation rate with probability ph in period k (tonne/week); X±ikh is amount by which the allowablewaste level is exceeded when the waste-generation rate is w±kh with probability ph (tonne/week) (the second-stage decision variable); Y±ik is binary decision variable for landfill expansion/ development at the start of period k, i = 1, 2, 3; Z±imk is binary decision variable for expanding composting or recycling facility with option m at the start of period k, i = 4, 5.

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