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In this paper we introduce a new version of single facility location problem. Let

We consider a special single facility location problem such that each point

It is necessary to note that this problem differs from the covering problem which asks to find the minimum number of facilities such that the distance from any point

In what follows at first the problem formulation is presented in Section

Let

As an application of this problem consider finding the location of a company in the vicinities of some cities with respect to the establishing and transportation costs. Suppose that cost of establishing a facility in the regions that are farther than a given distance

In this section we pose some properties of the problem. Suppose that the distances in the plane

Consider the point

Note in the case that

Since (

Let

Let

Let

As we showed that the objective function of problem (

The big square small square (BSSS) method is a geometrical branch and bound algorithm originally suggested by Hansen et al. [

By Lemma

The rectangle

For a usual implementation process of the BSSS algorithm the distance between points inside a rectangle is taken to be zero (see McGarvey and Cavalier [

(a) The error is positive. (b) The error is zero.

To provide convincing explanation note that the rectangle which is employed to calculate lower bound originally can be considered as an extensive facility. And locating a facility on the boundary of a circle is ideal because the error does not occur. So in Figure

Now let

(a) The rectangle

The outline of the BSSS algorithm is described below. It is in essence similar to the one presented by McGarvey and Cavalier [

(6) Set

In this section we show the efficiency of BSSS algorithm by giving four examples. The first example is small and contains just four points. We give this example to compare the results of BSSS algorithm with those obtained by LINGO software and show that while the NLP solver software may trapped on a local optimum, the BSSS method could be an appropriate method. The second and third examples are presented with the coordinates, weights, and radius of points to make it possible to compare the obtained results of this method with other methods in the future works. And finally the last one which contains more points is presented to become the CPU time of BSSS method comparable with other methods.

The above algorithm was written in MATLAB and run on a PC with

The first example contains four points

As the results show that the solutions in the second and third cases do not lie on the convex hull of existing points, however as we expected, they are inside the extended rectangular hull of these points. Figures

The results of Example

BSSS | LINGO | ||||||

case | radius | CPU/sec | |||||

1 | 0.33 | 0.34 | 0.34 | 1.06 | |||

2 | ( | 0.00 | 0.00 | 0.05 | 0.00 | ||

3 | ( | 0.91 | 0.93 | 2.53 | 1.02 |

Two different perspectives of objective function for first case of Example

Two different perspectives of objective function for third case of Example

The second example contains 18 points which are randomly generated and are presented in Table

The results of Example

radius | CPU/sec | |||
---|---|---|---|---|

case 1 | 275.06 | 275.76 | 3.62 | |

case 2 | 181.40 | 181.94 | 3.73 | |

case 3 | 63.55 | 63.89 | 3.68 |

As the third example we consider a problem with 30 points. The relevant data and results are given in Tables

The results of Example

radius | CPU/sec | |||
---|---|---|---|---|

case 1 | 1635.40 | 1638.20 | 5.74 | |

case 2 | 1158.60 | 1161.43 | 4.91 | |

case 3 | 753.11 | 755.39 | 3.97 |

Finally we consider the problems with

The results of Example

CPU/sec | ||||
---|---|---|---|---|

50 | 33657.91 | 33674.61 | 31.54 | |

100 | 73774.99 | 73813.48 | 62.91 | |

200 | 151267.25 | 151348.92 | 129.43 | |

300 | 228371.47 | 228498.30 | 184.41 | |

400 | 300070.07 | 300238.98 | 255.33 | |

500 | 381561.67 | 381777.87 | 287.45 |

In order to be able to make a better judgment about the efficiency of the proposed algorithm we have tried to solve the problems in Example

The results of Example

Objective function | ||

Center of gravity | BSSS | |

50 | 51202.22 | 51202.23 |

100 | 111032.93 | 111032.93 |

200 | 225490.50 | 225490.50 |

300 | 346396.00 | 346396.05 |

400 | 456804.57 | 456804.59 |

500 | 578703.13 | 578703.26 |

Data for 18-point problem.

Radius | Radius | Radius | |||
---|---|---|---|---|---|

Data for 30-point problem.

Radius | Radius | Radius | |||
---|---|---|---|---|---|

We considered the problem of finding the location of a single facility such that the sum of the weighted square errors over all points is minimized. We showed that the problem in general is nonconvex and then proved that the optimal solution lies in an extended rectangular hull of the existing points. Based on this finding then a big square small square (BSSS) was applied to solve the problem.

Other nonconvex solver methods such as big triangle Small triangle (BTST) method of Drezner and Suzuki [

See Tables

The authors would like to express their sincere gratitude to the anonymous referees for their careful reading of the manuscript and their constructive comments which resulted in the improvement of the paper.