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September  2022, 18(5): 3613-3639. doi: 10.3934/jimo.2021128

## Median location problem with two probabilistic line barriers: Extending the Hook and Jeeves algorithm

 1 Faculty of Engineering, University of Garmsar, Garmsar, Iran 2 Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

* Corresponding author: Saber Shiripour

Received  October 2020 Revised  April 2021 Published  September 2022 Early access  August 2021

We consider a median location problem in the presence of two probabilistic line barriers on the plane under rectilinear distance. It is assumed that the two line barriers move on their corresponding horizontal routes uniformly. We first investigate different scenarios for the position of the line barriers on the plane and their corresponding routes, and then define the visibility and invisibility conditions along with their corresponding expected barrier distance functions. The proposed problem is formulated as a mixed-integer nonlinear programming model. Our aim is to locate a new facility on the plane so that the total weighted expected rectilinear barrier distance is minimized. We present efficient lower and upper bounds using the forbidden location problem for the proposed problem. To solve the proposed model, the Hooke and Jeeves algorithm (HJA) is extended. We investigate various sample problems to test the performance of the proposed algorithm and appropriateness of the bounds. Also, an empirical study in Kingston-upon-Thames, England, is conducted to illustrate the behavior and applicability of the proposed model.

Citation: Saber Shiripour, Nezam Mahdavi-Amiri. Median location problem with two probabilistic line barriers: Extending the Hook and Jeeves algorithm. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3613-3639. doi: 10.3934/jimo.2021128
##### References:

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##### References:
Two probabilistic line barriers on the plane.
Visibility and effectiveness conditions.
Examples for $(x_{s1},x_{s2})\in \psi _1$.
Examples for $(x_{s1},x_{s2})\in \psi _2$.
Examples for $(x_{s1},x_{s2})\in \psi _3$.
An example for $(x_{s1},x_{s2})\in \psi _4$.
An example for $(x_{s1},x_{s2})\in \psi _5$.
Presentation of the location problems of type $1/\mathbb{R}^2/\mathcal{B} = 2ProbL/d_1/\sum$ and $1/\mathbb{R}^2/\mathcal{R}:\mathcal{B} = 2ProbL/d_1/\sum$.
General steps of the proposed algorithm.
Gaps in terms of sample problems.
Solution times in terms of sample problems.
Representation of the empirical study and the obtained results.
Impact of lengths of the trains.
Literature review of facility location problems with probabilistic barriers
Invisible regions and their corresponding distance functions for $i\in \mathcal{I}_2$
Results for small and medium problems
Results for large problems
The Cartesian coordinates of the barrier routes
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