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ISMISIP: an inexact stochastic mixed integer linear semi-infinite programming approach for solid waste management and planning under uncertainty

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Abstract

An inexact stochastic mixed integer linear semi-infinite programming (ISMISIP) model is developed for municipal solid waste (MSW) management under uncertainty. By incorporating stochastic programming (SP), integer programming and interval semi-infinite programming (ISIP) within a general waste management problem, the model can simultaneously handle programming problems with coefficients expressed as probability distribution functions, intervals and functional intervals. Compared with those inexact programming models without introducing functional interval coefficients, the ISMISIP model has the following advantages that: (1) since parameters are represented as functional intervals, the parameter’s dynamic feature (i.e., the constraint should be satisfied under all possible levels within its range) can be reflected, and (2) it is applicable to practical problems as the solution method does not generate more complicated intermediate models (He and Huang, Technical Report, 2004; He et al. J Air Waste Manage Assoc, 2007). Moreover, the ISMISIP model is proposed upon the previous inexact mixed integer linear semi-infinite programming (IMISIP) model by assuming capacities of the landfill, WTE and composting facilities to be stochastic. Thus it has the improved capabilities in (1) identifying schemes regarding to the waste allocation and facility expansions with a minimized system cost and (2) addressing tradeoffs among environmental, economic and system reliability level.

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Abbreviations

A ± ∈{R ±}m ×  n :

where R ± denotes a set of interval numbers

A ± (s) ∈{R ±}m ×  n :

where R ± denotes a set of functional interval numbers

a ± ij :

the ith row and jth line element of A ±

a ± ij (s i ):

the ith row and jth line element of A ±(s i )

B ±∈{R ±}m × 1 :

where R ± denotes a set of interval numbers

B ±(s)∈{R ±}n × 1 :

where R ± denotes a set of functional interval numbers

b ± i :

the ith element of B ±

b ± i (s i ):

the ith element of B ±(s i )

C ±∈{R ±}1×n :

where R ± denotes a set of interval numbers

c ± j :

the jth element of C ±

f ± :

system cost

f +opt :

the most desirable system cost

FLC ± k :

unit capacity expansion cost for the landfill in period k ($/tonne)

FT ± jk :

residue transportation costs from WTE facility (j = 1) and composting facility (j = 2) to the landfill in period k ($/tonne)

FTC ± mk :

unit cost of capacity expansion type m for composting facility in period k ($/tonne)

FTW ± mk :

unit cost of capacity expansion type m for WTE facility in period k ($/tonne)

i :

index for cities (i = 1, 2, 3)

j :

index for facilities (j = 1, 2, 3)

k :

index for periods (k = 1, 2, 3)

m :

index for facility capacity expansion type (m = 1, 2, 3)

OP ± jk :

operational costs of facility j in period k ($/tonne)

p i :

probability of violating constraint i

RE ± jk :

revenue of unit municipal solid wastes disposed by WTE facility (j = 1) and composting facility (j = 2) in period k ($/tonne)

RF j :

residue flow rate from WTE (j = 1) and composting (j = 2) facilities to the landfill

s :

time, an independent variable in the range of 0 and 5

TC± :

existing capacity of composting facility

\({\rm TC}^{{(p_{{\rm TC}})}}_{j}\) :

cumulative distribution function of TC ± j

TL± :

existing capacity of the landfill (tonnes)

\({\rm TL}^{{(p_{{\rm TL}})}}\) :

cumulative distribution function of TL±

TR ± ijk :

waste transportation costs from city i to facility j in period k ($/tonne)

TW± :

existing capacity of WTE facility

\({\rm TW}^{{(p_{{\rm TW}})}}_{j}\) :

cumulative distribution function of TW± j

WG ± ik (s):

 daily generation amount of solid waste in city i during period k ($/tonne) (functional intervals)

X ±∈{R ±}n ×1 :

where R ± denotes a set of interval numbers

x ± ijk :

solid waste stream from city i to facility j in period k ($/tonne)

x ± ijk,opt :

optimized waste stream from city i to facility j in period k ($/tonne)

YC ±mk :

integer variable for the mth expansion of composting facility in period k

YW ± mk :

integer variable for the mth expansion of WTE facility in period k

Z ± k :

integer variable for expansion of the landfill in period k

ΔTC ± m :

the amount of the mth type of expansion capacity for composting facility (tonnes/day)

ΔTL±(s):

total amount of expansion capacity for the landfill (tonnes) (functional intervals)

ΔTW ± m :

the amount of the mth type of expansion capacity for WTE facility (tonnes/day).

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Acknowledgments

This research was supported by the Major State Basic Research Development Program (2005CB724200 and 2006CB403307).

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Authors and Affiliations

  1. Environmental Systems Engineering Program, University of Regina, Regina, SK, S4S 0A2, Canada

    P. Guo

  2. Chinese Research Academy of Environmental Science, North China Electric Power University, Beijing, 100012-102206, China

    G. H. Huang

  3. Center for Studies in Energy and Environment, University of Regina, Regina, SK, S4S 0A2, Canada

    L. He

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  1. P. Guo
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  2. G. H. Huang
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  3. L. He
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Correspondence to G. H. Huang.

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Guo, P., Huang, G.H. & He, L. ISMISIP: an inexact stochastic mixed integer linear semi-infinite programming approach for solid waste management and planning under uncertainty. Stoch Environ Res Risk Assess 22, 759–775 (2008). https://doi.org/10.1007/s00477-007-0185-3

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