Dual inexact fuzzy chance-constrained programming for planning waste management systems

Abstract

This study develops a dual inexact fuzzy chance-constrained programming (DIFCCP) method for planning municipal solid waste (MSW) management systems. The concept of random boundary interval (RBI) is introduced to address the high uncertain parameters in the studied system. Fuzzy flexible programming and chance-constrained programming are also introduced to take into account the uncertainties of RBIs and various uncertainties in MSW management system. Compared with the existing methods, the developed method could deal with the uncertainty without simplification and thus is more robust. Moreover, the potential system-failure risks in MSW management system due to the existing uncertainties could be quantified by means of violation levels and satisfaction levels in DIFCCP. The developed method then is applied to a MSW management system. The obtained solutions could be used for generating efficient management schemes. The values of violation and satisfaction levels could help decision makers understand the tradeoffs between system cost and system-failure risk, and identify desired strategy according to the practical economic and environmental situation.

Appendix

1.1 The process of acquiring an RBI

The process of acquiring an RBI is provided as follows. Firstly, samples of the possible intervals for RBI can be acquired:

$$ \left( {u_{1} ,v_{1} } \right),\left( {u_{2} ,v_{2} } \right), \ldots ,\left( {u_{n} ,v_{n} } \right) $$

where (u _i , v _i ) is the value of the ith sample. Secondly, calculate the mean vector and covariance matrix of the samples:

$$ \mu = \left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {u_{i} } ,\,\frac{1}{n}\sum\limits_{i = 1}^{n} {v_{i} } } \right), \quad D = \left( {\begin{array}{*{20}c} {\sigma_{1}^{2} } & {\rho \sigma_{1} \sigma 2} \\ {\rho \sigma_{1} \sigma 2} & {\sigma_{2}^{2} } \\ \end{array} } \right) $$

$$ \sigma_{1}^{2} = \frac{1}{n}\sum\limits_{t = 1}^{n} {\left( {u_{t} - \mu_{1} } \right)^{2} } , \quad \sigma_{2}^{2} = \frac{1}{n}\sum\limits_{t = 1}^{n} {\left( {v_{t} - \mu_{2} } \right)^{2} } ,\,\rho = {\frac{1}{{n\sigma_{1} \sigma_{2} }}}\sum\limits_{t = 1}^{n} {\left( {u_{t} - \mu_{1} } \right)} \left( {v_{t} - \mu_{2} } \right) $$

Thirdly, suppose that the inputs follow a distribution based on the mean vector and covariance matrix. A hypothesis testing can be undertaken. A joint distribution function can then be identified for (B ⁻ _i , B ⁺ _i ) [i.e. = f(s, t)]. Suppose [c, d] to be the range of B ⁺ _i . Finally, the probability density function (PDF) of B ⁻ _i can be generated:

$$ f_{s} \left( s \right) = \int\limits_{c}^{d} {f\left( {s,t} \right)dt} $$

Suppose [a, b] is the range of B ⁻ _i . Similarly, we can get the PDF of B ⁺ _i as follows:

$$ f_{t} \left( t \right) = \int\limits_{a}^{b} {f\left( {s,t} \right)ds} $$