Chance-constrained programming with skewed distributions of matrix coefficients and applications to environmental regulatory activities

This paper provides a theoretical framework for application of Chance-Constrained Programming (CCP) in situations where the coefficient matrix is random and its elements are not normally distributed. Much of the CCP literature proceeds to derive deterministic equivalent in computationally implementable form on the assumption of “normality”. However, in many applications, such as air pollution control, right skewed distributions are more likely to occur. Two types of models are considered in this paper. One assumes an exponential distribution of matrix coefficients, and another one uses an empirical approach. In case of exponential distributions, it is possible to derive exact “deterministic” equivalent to the chance-constrained program. Each row of the coefficient matrix is assumed to consist of independent, exponentially distributed random variables and a simple example illustrates the complexities associated with finding a numerical solution to the associated deterministic equivalent. In our empirical approach, on the other hand, simulated data typically encountered in air pollution control are provided, and the data-driven (empirical) solution to the implicit form of deterministic equivalent is obtained. Post-optimality analyses on model results are performed and risk implications of these decisions are discussed. Conclusions are drawn and directions for future research are indicated.