An approach to interval programming problems with left-hand-side stochastic coefficients: An application to environmental decisions analysis
Highlights
► The IPSC model incorporates stochastic coefficients with multivariate normal distributions within an interval parameter linear programming (ILP) framework. ► A left-hand-side chance-constrained programming (LCCP) method is developed to solve the problem. ► The developed IPSC model is applied to a regional air quality management system. ► Interval solutions associated with different violation probability levels and/or different environmental standards have been obtained.
Introduction
Optimization techniques were widely used in the field of environmental management and pollution control (Huang & Chang, 2003). Many innovated optimization techniques were developed to tackle uncertainties existing in environmental systems (Chanas and Zielinski, 2000, Chang and Kashani, 2009, Chang and Wang, 1997, He and Huang, 2008, Huang et al., 1992, Huang et al., 1994, Huang et al., 1995, Huang et al., 1996, Huang et al., 2007, Liu et al., 2009, Luo et al., 2005, Qin and Huang, 2009, Yeomans, 2008, Yeomans and Huang, 2003). Among them, Huang et al. (1992) developed an interval linear programming (ILP) approach for municipal solid waste management system. In ILP, all parameters in objective function and constrains were interval numbers. A two-step method was developed to solve the ILP problem. It has the following advantages: (1) it was applicable to practical problems because it was relatively easy for planners to define fluctuation intervals of uncertain parameters instead of specifying their probability distributions; (2) the two-step method did not lead to high computational requirements; (3) in ILP, interval solutions were obtained which could reflect the inherent system uncertainties and were convenient for the managers to interpret and adjust the decision schemes according to practical situations. However, some parameters do have certain probability distributions which should be incorporated into the optimization process.
It is thus desired that distribution information be integrated into the ILP framework Based on ILP and chance-constrained programming (CCP), Huang introduced an inexact chance-constrained programming method (ICCP) for water quality management within an agricultural system (Huang, 1998). In this method, the left-hand-side (LHS) parameters were interval numbers, and the right-hand-side (RHS) ones were allowed to be stochastic variables. Compared with ILP, ICCP can reflect stochastic coefficients with known probability density distributions in RHS parameters of the constraints. Moreover, solutions of ICCP are intervals under different violation probability levels. However, ICCP cannot deal with problems where the LHS parameters of the constraints are stochastic variables.
Therefore, the objective of this research is to develop a method of interval programming with stochastic coefficients (IPSC) that can address the above problem. In IPSC, uncertainties could be presented as stochastic coefficients in the LHS of the constraints. Meanwhile, intervals will be introduced as the coefficients in both the objective function and constraints. A left-hand side chance-constrained programming (LCCP) method was developed to solve the IPSC problem. The solutions will be presented as intervals which will supply a number of alternatives such that the decision makers could incorporate their preferences and experiences when deciding the final management strategy. The developed model will then be applied to the planning of a regional air quality management system, and the results would be used for helping regional air quality managers to identify desired decision schemes.
Section snippets
Interval programming with stochastic variables (IPSC)
In interval-parameter linear programming (ILP), interval values are allowed to be directly communicated into the optimization process (Huang et al., 1995). An ILP model can be formulated as follows (Huang et al., 1992):where , and R± denotes a set of interval numbers. An interactive solution algorithm was proposed to solve the above problem. The solution for model (1a),
Left-hand-side chance-constrained programming (LCCP)
Suppose that the constraints are independent with each other, and all the parameters () in the objective function are not less than 0. Taking the ith (1 ⩽ i ⩽ l) constraint as an example, a method for handling constraints with stochastic variables in its LHS and intervals in its RHS will be illustrated as follows.
Suppose that ξ = (ξi1, ξi2, …, ξin)T obeys an n-dimensional normal distribution:where μ = (ai1, ai2, …, ain)T is the expectation of ξ and V = (vij)n×n is the variance-covariance matrix of ξ
Overview of the study system
The phenomenon of regional air pollution involves many processes, such as pollutant generation, emission, transport, transformation and removal. A number of impact factors like properties of pollutants, locations of emission sources and receptors, meteorological conditions, and control measures make the management of air quality complicated (Ellis, 1991, Ellis and Bowman, 1994, Fortin and McBean, 1983, Guldmann, 1986, Haith, 1982, Liu et al., 2003). Since it is generally either economically
Result analysis
Table 4 presents the solutions obtained form the IPSC model under different environmental restrictions and violation probabilities. In general, the LWS method should be adopted during the first three years (period 1). In period 2, the methods of LWS and OPA should be used. The amounts of emission as treated by those technologies would vary with the environmental standards and violation probabilities. In period 3, the LWS method would be out of use due to its low efficiency. Instead, both OPA
Conclusions
An interval programming with stochastic variables (IPSC) approach was developed for air quality management under uncertainty. Stochastic variables with multivariate normal distributions were introduced as left-hand-side-parameters. Other parameters and decision variables in the IPSC were expressed as intervals to reflect the uncertainties. A left-hand-side chance-constrained method (LCCM) was developed to solve the IPSC model. Compare with the existing methods, IPSC was advantageous in
Acknowledgments
This research was supported by the Major State Basic Research Development Program (2005CB724200), and the Natural Science and Engineering Research Council of Canada. The authors are grateful to the editor and reviewers for their insightful comments.
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