Article Contents
Article Contents

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

• * Corresponding author: Saber Shiripour
• 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.

Mathematics Subject Classification: Primary: 90-10; Secondary: 90C59.

 Citation:

• Figure 1.  Two probabilistic line barriers on the plane.

Figure 2.  Visibility and effectiveness conditions.

Figure 3.  Examples for $(x_{s1},x_{s2})\in \psi _1$.

Figure 4.  Examples for $(x_{s1},x_{s2})\in \psi _2$.

Figure 5.  Examples for $(x_{s1},x_{s2})\in \psi _3$.

Figure 6.  An example for $(x_{s1},x_{s2})\in \psi _4$.

Figure 7.  An example for $(x_{s1},x_{s2})\in \psi _5$.

Figure 8.  Presentation of the location problems of type $1/\mathbb{R}^2/\mathcal{B} = 2ProbL/d_1/\sum$ and $1/\mathbb{R}^2/\mathcal{R}:\mathcal{B} = 2ProbL/d_1/\sum$.

Figure 9.  General steps of the proposed algorithm.

Figure 10.  Gaps in terms of sample problems.

Figure 11.  Solution times in terms of sample problems.

Figure 12.  Representation of the empirical study and the obtained results.

Figure 13.  Impact of lengths of the trains.

Table 1.  Literature review of facility location problems with probabilistic barriers

Table 2.  Invisible regions and their corresponding distance functions for $i\in \mathcal{I}_2$

Table 3.  Results for small and medium problems

Table 4.  Results for large problems

Table 5.  The Cartesian coordinates of the barrier routes

Figures(13)

Tables(5)