# Solutions to Chance-Constrained Programming Problems with Exponential Random Variables by Edgeworth Approximation

## Abstract

This paper introduces three methods for approximating distribution of weighted sum of exponential variates. These methods are useful for transforming chance constraints into their equivalent deterministic constraints when the technologic coefficients are exponential random variables. Hence, the equivalent deterministic constraint is obtained by three methods which are normal approximation and first- and second-term Edgeworth series expansions, respectively. These methods are based on normal approximation related to the central limit theorem (CLT). Furthermore, the exact distribution of weighted sum of exponential variates is presented by using convolution technique. The fourth method is proposed for deriving deterministic equivalent of chance constraint by using this exact distribution. The fifth method is transforming the exponential variates into the chi-squared variates. Illustrative examples are given for the purpose of comparing the solutions of these five methods. Additionally, the optimal solution for Example 1 of Biswal et al. (1998. European Journal of Operational Research 111:589–597) is extended to a global solution by using three methods.

## Keywords

Exponential distribution Edgeworth expansion Central limit theorem Chance-constrained programming Exponential-type chance constraint## Notes

### Acknowledgments

The authors would like to express their deepest gratitude to Dr. Didem Atalay for her insightful comments and considerable encouragements to complete this study.

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