# A Single-source Weber Problem with Continuous Piecewise Fixed Cost

### Gabriela Iriarte, Pablo Escalona, Alejandro Angulo, Raul Stegmaier

#### Abstract

This paper analyzes the location of a distribution center in an urban area using a single-source Weber problem with continuous piecewise fixed cost to find a global optimal location. The fixed cost is characterized by a Kriging interpolation method. To make the fixed cost tractable, we approximate this interpolation with a continuous piecewise function that is convex in each piece, using Delaunay triangulation. We present a decomposition formulation, a decomposition conic formulation and a conic logarithmic disaggregated convex combination model to optimally solve the single-source Weber problem with continuous piecewise fixed cost. Although our continuous approach does not guarantee the global optimal feasible location, it allows us to delimit a zone where we can intensify the search of feasible points. For instances we tested, computational results show that our continuous approach found better locations than the discrete approach in 23.25% of the instances and that the decomposition formulation is the best one, in terms of CPU time.

#### References

- Anselin, L. and Le Gallo, J. (2006). Interpolation of air quality measures in hedonic house price models: spatial aspects. Spatial Economic Analysis, 1(1):31-52.
- Bongartz, I., Calamai, P. H., and Conn, A. R. (1994). A projection method forl p norm location-allocation problems. Mathematical Programming, 66(1):283-312.
- Brimberg, J., Drezner, Z., Mladenovic, N., and Salhi, S. (2014). A new local search for continuous location problems. European Journal of Operational Research, 232(2):256-265.
- Brimberg, J., Hansen, P., Mladenovic, N., and Salhi, S. (2008). Survey of solution methods for the continuous location-allocation problem. International Journal of Operations Research, 5(1):1-12.
- Brimberg, J., Hansen, P., Mladenovic, N., and Taillard, E. D. (2000). Improvements and comparison of heuristics for solving the uncapacitated multisource weber problem. Operations Research, 48(3):444-460.
- Brimberg, J., Mladenovic, N., and Salhi, S. (2004). The multi-source weber problem with constant opening cost. Journal of the Operational Research Society, 55(6):640-646.
- Brimberg, J. and Salhi, S. (2005). A continuous locationallocation problem with zone-dependent fixed cost. Annals of Operations Research, 136(1):99-115.
- Cellmer, R. (2014). The possibilities and limitations of geostatistical methods in real estate market analyses. Real Estate Management and Valuation, 22(3):54-62.
- Cellmer, R., Belej, M., Zrobek, S., and S?ubic-Kovac?, M. (2014). Urban land value maps a methodological approach. Geodetski vestnik, 58(3):535-551.
- Chen, J.-S., Pan, S., and Ko, C.-H. (2011). A continuation approach for the capacitated multi-facility weber problem based on nonlinear socp reformulation. Journal of Global Optimization, 50(4):713-728.
- Chen, P.-C., Hansen, P., Jaumard, B., and Tuy, H. (1998). Solution of the multisource weber and conditional weber problems by d.-c. programming. Operations Research, 46(4):548-562.
- Cooper, L. (1964). Heuristic methods fot locationallocation problems. SIAM Review, 6(1):37-53.
- Cooper, L. (1972). The transportation-location problem. Operations Research, 20(1):94-108.
- Dolan, E. D. and Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical programming, 91(2):201-213.
- Drezner, Z., Klamroth, K., Schöbel, A., and Wesolowsky, G. O. (2002). 1 the weber problem.
- Fernández-Avilés, G., Minguez, R., and Montero, J.-M. (2012). Geostatistical air pollution indexes in spatial hedonic models: the case of madrid, spain. Journal of Real Estate Research.
- Hansen, P., Mladenovic, N., and Taillard, E. (1998). Heuristic solution of the multisource weber problem as a pmedian problem. Operations Research Letters, 22(2- 3):55-62.
- Helbich, M., Jochem, A., Mücke, W., and Höfle, B. (2013). Boosting the predictive accuracy of urban hedonic house price models through airborne laser scanning. Computers, environment and urban systems, 39:81- 92.
- Hosseininezhad, S. J., Salhi, S., and Jabalameli, M. S. (2015). A cross entropy-based heuristic for the capacitated multi-source weber problem with facility fixed cost. Computers & Industrial Engineering, 83:151- 158.
- Hu, S., Tong, L., Frazier, A. E., and Liu, Y. (2015). Urban boundary extraction and sprawl analysis using landsat images: A case study in wuhan, china. Habitat International, 47:183-195.
- Kuntz, M. and Helbich, M. (2014). Geostatistical mapping of real estate prices: an empirical comparison of kriging and cokriging. International Journal of Geographical Information Science, 28(9):1904-1921.
- Larraz, B. and Poblacin, J. (2013). An online real estate valuation model for control risk taking: A spatial approach. Investment Analysts Journal, 42(78):83-96.
- Liang, H. and Yi, W. (2012). Effect of rent spatial distribution to urban business district planning. In Advanced Materials Research, volume 374, pages 2001-2008. Trans Tech Publ.
- Luis, M., Salhi, S., and Nagy, G. (2015). A constructive method and a guided hybrid grasp for the capacitated multi-source weber problem in the presence of fixed cost. Journal of Algorithms & Computational Technology, 9(2):215-232.
- Luo, J. (2004). Modeling urban land values in a gis environment. University of Wisconsin. Milwaukee, USA.
- Megiddo, N. and Supowit, K. (1984). On the complexity of some common geometric location problems. SIAM Journal on Computing, 13(1):182-196.
- Michelot, C. and Lefebvre, O. (1987). A primal-dual algorithm for the fermat-weber problem involving mixed gauges. Mathematical Programming, 39(3):319-335.
- Mladenovic, N., Brimberg, J., Hansen, P., and MorenoPerez, J. A. (2007). The p-median problem: A survey of metaheuristic approaches. European Journal of Operational Research, 179(3):927-939.
- Oliver, M. A. and Webster, R. (1990). Kriging: a method of interpolation for geographical information systems. International Journal of Geographical Information System, 4(3):313-332.
- Planchart, A. and Hurter, A. P. J. (1975). An efficient algorithm for the solution of the weber problem with mixed norms. SIAM Journal on Control, 13(3):650665.
- Plastria, F. (1987). Solving general continuous single facility location problems by cutting planes. European Journal of Operational Research, 29(1):98-110.
- Sherali, H. D., Al-Loughani, I., and Subramanian, S. (2002). Global optimization procedures for the capacitated euclidean and lp distance multifacility location-allocation problems. Operations Research, 50(3):433-448.
- Vielma, J. P., Ahmed, S., and Nemhauser, G. (2010). Mixed-integer models for nonseparable piecewise linear optimization: Unifying framework and extensions. Operations Research, 58(2):303-315.
- Weber, A. and Friedrich, C. J. (1929). Alfred Weber's Theory of the Location of Industries. The University of Chicago Press, Chicago, Illinois.
- Weiszfeld, E. and Plastria, F. (2009). On the point for which the sum of the distances to n given points is minimum. Annals of Operations Research, 167(1):4-41.
- Xue, G. and Ye, Y. (1997). An efficient algorithm for minimizing a sum of euclidean norms with applications. SIAM Journal on Optimization, 7(4):1017-1036.

#### Paper Citation

#### in Harvard Style

Iriarte G., Escalona P., Angulo A. and Stegmaier R. (2017). **A Single-source Weber Problem with Continuous Piecewise Fixed Cost** . In *Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-218-9, pages 337-344. DOI: 10.5220/0006191003370344

#### in Bibtex Style

@conference{icores17,

author={Gabriela Iriarte and Pablo Escalona and Alejandro Angulo and Raul Stegmaier},

title={A Single-source Weber Problem with Continuous Piecewise Fixed Cost},

booktitle={Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2017},

pages={337-344},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006191003370344},

isbn={978-989-758-218-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - A Single-source Weber Problem with Continuous Piecewise Fixed Cost

SN - 978-989-758-218-9

AU - Iriarte G.

AU - Escalona P.

AU - Angulo A.

AU - Stegmaier R.

PY - 2017

SP - 337

EP - 344

DO - 10.5220/0006191003370344