Application of the Smooth Approximation of the Probability Function in Some Applied Stochastic Programming Problems

Sobol, Vitaliy; Torishnyy, R.O.

doi:10.14529/mmp210303

Bulletin SUSU MMP

2021

DOI: 10.14529/mmp210303

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Application of the Smooth Approximation of the Probability Function in Some Applied Stochastic Programming Problems

Vitaliy Sobol¹,

R.O. Torishnyy²

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2022

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Cited by 2 publications

References 13 publications

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Software Complex for the Analysis of the Stochastic Programming Problems With Probability Criterion

Torishnyi¹

2022

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This paper describes the software complex for the construction of a smooth approximation of the probability function and its derivatives. The structural parts of the complex, its functionality and mathematical background are described. The software complex constructs an approximation of the probability that some loss function does not exceed a certain level of loss. The program supports lossfunctions defined in a LaTeX format and contains predefined standard functions, variables and mathematical signs. The software supports different variable types, such as constants, control variables, stochastic variables with known distribution and parameters and samples of stochastic variables with unknown distribution. The software complex supports a variety of predefined random distributions and allows to tune the result by setting other service parameters. The implemented approximation is based on the replacement of the Heaviside function inside the probability function expression with the sigmoid function. Next, the approximated probability function and its derivatives are represented as volume integrals. These integrals can be calculated numerically using the Monte-Carlo method. This approach provides a relatively quick and universal method of approximate calculation of the probability function and its derivatives. The software complex has a graphical user interface and produces a graphical representation of approximated functions along with their points data. The program also supports the construction of the surface approximations for the case of the loss function having two control variables. Obtained graphical and point data can be used in the solution of stochastic programming problems with probability criteria. Examples using the software complex as a tool for analyzing stochastic programming problems are given.

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Software Complex for the Analysis of the Stochastic Programming Problems With Probability Criterion

Torishnyi¹

2022

vkit

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Optimization of the Parameters of the Water Supply System of a Desert Region Under a Probabilistic Limit

Shilyaeva¹,

Kibzun²

2022

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The article considers the problem of choosing the parameters of a water supply system in a desert region under a probabilistic constraint. The random activity of the sun is taken into account in the production of water due to the desalination of salt water. A balance equation is derived regarding the cases of company losses when the region's needs for drinking water are not met and regarding the total cost of the system, with the help of which the necessary dependence on the level of probability is subsequently obtained. To which the dichotomy method is then applied to calculate the probability measure. To make sure that the solution is unique, the function is examined for monotonicity with respect to the probability level. Further, to estimate the confidence probability, the problem is solved for average values, where the system cost is estimated and the probability of fulfilling the constraints is calculated. To solve the problem, a confidence method is used, with the help of which the original stochastic poblem is reduced to a linear programming problem. The resulting solution is improved by varying the radius of the confidence ball. One of the possible ways to choose the level of confidence based on the economic balance between the cost of the operation of the water supply system as a whole and the emergency delivery of fresh water is proposed. The choice of the confidence level is made using the initial approximation, which is obtained by solving the problem for average values. An assessment of the reliability of the system used was obtained and a program was implemented to solve this problem.

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Application of the Smooth Approximation of the Probability Function in Some Applied Stochastic Programming Problems

Cited by 2 publications

References 13 publications

Software Complex for the Analysis of the Stochastic Programming Problems With Probability Criterion

Software Complex for the Analysis of the Stochastic Programming Problems With Probability Criterion

Optimization of the Parameters of the Water Supply System of a Desert Region Under a Probabilistic Limit

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