# Two-Stage Stochastic Facility Location Model with Quantile Criterion and Choosing Reliability Level

S.V. Ivanov, V.N. AkmaevaA two-stage discrete model for the location of facilities is considered. At the first stage, a set of facilities to be opened is selected. At the second stage, additional facilities may be opened due to the realization of random demand for products. Customers preferences are taken into account in choosing the facility in which they will be served. The quantile of losses (income with the opposite sign) is used as a criterion function of the model. Several optimization problems are stated. In the first problem, a set of facilities to be opened is selected for a given value of the reliability level. In the second problem, along with the set of facilities to be opened, the reliability level of the quantile criterion is selected. At the same time, restrictions on the level of reliability and the value of the quantile criterion are introduced. Two approaches to setting these constraints are proposed. To solve the problems stated, the method of sample approximations is used. A theorem on sufficient conditions for the convergence of the proposed method is proved. We formulate mathematical programming problems, the solutions of which under certain conditions are solutions to the obtained approximating problems. Numerical results are presented.Full text

- Keywords
- facility location; stochastic programming; quantile criterion; sample approximation.
- References
- 1. Kibzun A.I., Kan Yu.S. Zadachi stohasticheskogo programmirovaniya s veroyatnostnymi kriteriyami. [Stochastic Programming Problems with Probabilistic Criterions]. Мoscow, Fizmatlit, 2009. (in Russian)

2. Laporte G., Nickel S., Saldanha da Gama F. Location Science, Cham, Springer, 2019. DOI: 10.1007/978-3-030-32177-2

3. Snyder L.V. Facility Location Under Uncertainty: a Review. IIE Transactions, 2006, vol. 38, no. 7, pp. 547-564. DOI: 10.1080/07408170500216480

4. Daskin M.S., Hesse S.M., Revella C.S. alpha-Reliable p-Minimax Regret: a New Model for Strategic Facility Location Modeling. Location Science, 1997, vol. 5, no. 4, pp. 227-246. DOI: 10.1016/S0966-8349(98)00036-9

5. Gang Chen, Daskin M.S., Zuo-Jun Max Shen, Uryasev S. The alpha-Reliable Mean-Excess Regret Model for Stochastic Facility Location Modeling. Naval Research Logistics, 2006, vol. 53, no. 7, pp. 617-626. DOI: 10.1002/nav.20180

6. Shaopeng Zhong, Rong Cheng, Yu Jiang, Zhong Wang, Larsen A., Nielsen O.A. Risk-Averse Optimization of Disaster Relief Facility Location and Vehicle Routing Under Stochastic Demand. Transportation Research Part E: Logistics and Transportation Review, 2020, vol. 141, pp. 102-115. DOI: 10.1016/j.tre.2020.102015

7. Ivanov S.V., Morozova M.V. Stochastic Problem of Competitive Location of Facilities with Quantile Criterion. Automation and Remote Control, 2016, vol. 77, no. 3, pp. 45-461. DOI: 10.1134/S0005117916030073

8. Melnikov A., Beresnev V. Upper Bound for the Competitive Facility Location Problem with Quantile Criterion. Lecture Notes in Computer Science, 2016, vol. 9869, pp. 373-387. DOI: 10.1007/978-3-319-44914-2_30

9. Beresnev V., Melnikov A. varepsilon-Constraint Method for Bi-Objective Competitive Facility Location Problem with Uncertain Demand Scenario. EURO Journal on Computational Optimization, 2020, vol. 8, pp. 33-59. DOI: 10.1007/s13675-019-00117-5

10. Shapiro A., Dentcheva D., Ruszczy'nski A. Lectures on Stochastic Programming. Modeling and Theory. Philadelphia, SIAM, 2014. DOI: 10.1137/1.9781611973433

11. Louveaux F.V., Peeters D. A Dual-Based Procedure for Stochastic Facility Location. Operations Research, vol. 40, no. 3, pp. 564-573. DOI: 10.1287/opre.40.3.564

12. Santoso T., Ahmed S., Goetschalckx M., Shapiro A. A Stochastic Programming Approach for Supply Chain Network Design Under Uncertainty. European Journal of Operational Research, 2005, vol. 167, pp. 96-115. DOI: 10.1016/j.ejor.2004.01.046

13. Ivanov S.V., Kibzun A.I. Sample Average Approximation in a Two-Stage Stochastic Linear Program with Quantile Criterion. Proceedings of the Steklov Institute of Mathematics, 2018, vol. 303, no. 1, pp. 115-123. DOI: 10.1134/S0081543818090122

14. Ivanov S.V., Kibzun A.I. On the Convergence of Sample Approximations for Stochastic Programming Problems with Probabilistic Criteria. Automation and Remote Control, 2018, vol. 79, no. 2, pp. 216-228. DOI: 10.1134/S0005117918020029

15. Norkin V.I., Kibzun A.I., Naumov A.V. Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems. Cybernetics and Systems Analysis, 2014, vol. 50, no. 5, pp. 679-692. DOI: 10.1007/s10559-014-9658-9