# Application of bilevel problem of stochastic programming to optimization of energy saving projects

### Mathematics. Physics. Mechanics

### Аuthors

^{1}

^{*},

^{2}

^{**}

1. Saratov State University named after N. G. Chernyshevsky, 83, Astrakhanskaya str., Saratov, 410012, Russia

2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: evilgraywolf@gmail.com

**e-mail: naumovav@mail.ru

### Abstract

A statement of the bilevel problem of stochastic programming with quantile criterion and several followers is suggested. Random parameters of the problem assumed to have a discrete distribution. In this problem, there are several decision makers: leader and several followers. These decision makers may have different own purposes. The followers choose strategies solving their optimization problems when a leader’s decision is already known. The leader chooses his own strategy taking into account the followers’ optimal strategies and solving his own optimization problem. The followers’ problems are assumed to be linear in follower’s decision variables. Due to this fact, the followers’ problems may be replaced by nonlinear equilibrium constraints using the slackness complementary conditions. The following theorem is proved: the original problem can be reduced to a mixed integer mathematical programming problem. This theorem is used in solving the problem of optimization of energy saving projects. In this problem, the leader is a transport company and the followers are companies carrying out the energy saving projects. The aim of the transport company is to minimize investment costs and expenses for the purchase of energy resources. The aim of a followers is to maximize own profit carrying out their project. In this application, the random parameters are the demands for different energy resources in different planning periods. It is proved that this applied problem can be reduced to a mixed integer linear programming problem. The results of a numerical experiment concerning typical section of the railway are presented. Three projects for saving two energy resources: electrical energy and diesel oil are considered in this experiment.### Keywords:

stochastic programming, bilevel problem, quantile criterion, energy saving### References

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