Abstract
The objective of territorial design for a distribution company is the definition of geographic areas that group customers. These geographic areas, usually called districts or territories, should comply with operational rules while maximizing potential sales and minimizing incurred costs. Consequently, territorial design can be seen as a clustering problem in which clients are geographically grouped according to certain criteria which usually vary according to specific objectives and requirements (e.g. costs, delivery times, workload, number of clients, etc.). In this work, we provide a novel hybrid approach for territorial design by means of combining a K-means-based approach for clustering construction with an optimization framework. The K-means approach incorporates the novelty of using tour length approximation techniques to satisfy the conditions of a pork and poultry distributor based in the region of Valparaíso in Chile. The resulting method proves to be robust in the experiments performed, and the Valparaíso case study shows significant savings when compared to the original solution used by the company.
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References
Achterberg, T., & Wunderling, R. (2013). Mixed integer programming: Analyzing 12 years of progress. In M. Jünger & G. Reinelt (Eds.), Facets of combinatorial optimization (pp. 449–482). Heidelberg: Springer. https://doi.org/10.1007/978-3-642-38189-818.
Altman, M. (1997). Is automation the answer: The computational complexity of automated redistricting. Rutgers Computer and Law Technology Journal, 23, 81–141.
Assis, L., Franca, P., & Usberti, F. (2014). A redistricting problem applied to meter reading in power distribution networks. Computers and Operations Research, 41(1), 65–75. https://doi.org/10.1016/j.cor.2013.08.002.
Baçao, F., Lobo, V., & Painho, M. (2005). Applying genetic algorithms to zone design. Soft Computing, 9(5), 341–348. https://doi.org/10.1007/s00500-004-0413-4.
Bard, J., & Jarrah, A. (2009). Large-scale constrained clustering for rationalizing pickup and delivery operations. Transportation Research Part B: Methodological, 43(5), 542–561. https://doi.org/10.1016/j.trb.2008.10.003.
Barnhart, C., Johnson, E., Nemhauser, G., Savelsbergh, M., & Vance, P. (1998). Branch-and-price: Column generation for solving huge integer programs. Operations Research, 46(3), 316–329. https://doi.org/10.1287/opre.46.3.316.
Beardwood, J., Halton, J., & Hammersley, J. (1959). The shortest path through many points. Mathematical Proceedings of the Cambridge Philosophical Society, 55, 299–327. https://doi.org/10.1017/S0305004100034095.
Bodin, L., & Levy, L. (1991). The arc partitioning problem. European Journal of Operational Research, 53, 393–401. https://doi.org/10.1016/0377-2217(91)90072-4.
Bonomo, F., Delle, D., Durán, G., & Marenco, J. (2013). Automatic dwelling segmentation of the Buenos Aires Province for the 2010 Argentinian census. Interfaces, 43(4), 373–384. https://doi.org/10.1287/inte.2013.0685.
Bozkaya, B., Erkut, E., & Laporte, G. (2003). A tabu search heuristic and adaptive memory procedure for political districting. European Journal of Operational Research, 144(1), 12–26. https://doi.org/10.1016/S0377-2217(01)00380-0.
Butsch, A., Kalcsics, J., & Laporte, G. (2014). Districting for Arc routing. INFORMS Journal on Computing, 26, 809–824. https://doi.org/10.1287/ijoc.2014.0600.
Camacho-Collados, M., Liberatore, F., & Angulo, J. (2015). A multi-criteria Police Districting Problem for the efficient and effective design of patrol sector. European Journal of Operational Research, 246(2), 674–684. https://doi.org/10.1016/j.ejor.2015.05.023.
Caprara, A., Toth, P., & Fischetti, M. (2000). Algorithms for the set covering problem. Annals of Operations Research, 98(1), 353–371. https://doi.org/10.1023/A:1019225027893.
Caro, F., Shirabe, T., Guignard, M., & Weintraub, A. (2004). School redistricting: Embedding GIS tools with integer programming. Journal of the Operational Research Society, 55, 836–849. https://doi.org/10.1057/palgrave.jors.2601729.
Çavdar, B., & Sokol, J. (2015). A distribution-free tsp tour length estimation model for random graphs. European Journal of Operational Research, 243(2), 588–598. https://doi.org/10.1016/j.ejor.2014.12.020.
Chaves, A., & Nogueira, L. (2010). Clustering search algorithm for the capacitated centered clustering problem. Computers and Operation Research, 37(3), 552–558. https://doi.org/10.1016/j.cor.2008.09.011.
Chaves, A., & Nogueira, L. (2011). Hybrid evolutionary algorithm for the capacitated centered clustering problem. Expert Systems with Applications, 38(5), 5013–5018. https://doi.org/10.1016/j.eswa.2010.09.149.
Chien, T. (1992). Operational estimators for the length of a traveling salesman tour. Computers and Operations Research, 19(6), 469–478. https://doi.org/10.1016/0305-0548(92)90002-M.
Daganzo, C. F. (1984). The distance traveled to visit n points with a maximum of c stops per vehicle: An analytic model and an application. Transportation Science, 18(4), 331–350. https://doi.org/10.1287/trsc.18.4.331.
Davidson I (2005) Clustering with constraints: Feasibility issues and the K-Means algorithm. In: Proceedings of the Fifth SIAM International Conference (pp. 138–149). https://doi.org/10.1137/1.9781611972757.13
Duque, J., Ramos, R., & Suriñach, J. (2007). Supervised regionalization methods: A aurvey. International Regional Science Review, 30(3), 195–220. https://doi.org/10.1177/0160017607301605.
Duque, J., Anselin, L., & Rey, S. (2012). The max-p-regions problem. Journal of Regional Science, 52(3), 397–419. https://doi.org/10.1111/j.1467-9787.2011.00743.x.
Figliozzi, M. A. (2010). The impacts of congestion on commercial vehicle tour characteristics and costs. Transportation Research Part E: Logistics and Transportation Review, 46(4), 496–506. https://doi.org/10.1016/j.tre.2009.04.005.
Fleischmann, B., & Paraschis, J. (1988). Solving a large scale districting problem: A case report. Computers and Operations Research, 15(6), 521–533. https://doi.org/10.1016/0305-0548(88)90048-2.
Friedman, J., Tibshirani, R., & Hastie, T. (2009). The elements of statistical learning: Data mining, inference, and prediction. Berlin: Springer. https://doi.org/10.1111/j.1751-5823.2009.00095_18.x.
Galvao, L., Novaes, A., Souza De Cursi, J., & Souza, J. (2006). A multiplicatively-weighted voronoi diagram approach to logistics districting. Computers and Operations Research, 33(1), 93–114. https://doi.org/10.1016/j.cor.2004.07.001.
García-Ayala, G., González-Velarde, J., Ríos-Mercado, R., & Fernández, E. (2016). A novel model for arc territory design: Promoting Eulerian districts. International Transactions in Operational Research, 23(3), 433–458. https://doi.org/10.1111/itor.12219.
Garfinkel, R., & Nemhauser, G. (1970). Optimal political districting by implicit enumeration techniques. Management Science, 16(8), 495–508. https://doi.org/10.1287/mnsc.16.8.B495.
Hansen, P., Labb, M., & Schindl, D. (2009). Set covering and packing formulations of graph coloring: Algorithms and first polyhedral results. Discrete Optimization, 6(2), 135–147. https://doi.org/10.1016/j.disopt.2008.10.004.
Hess, S., & Samuels, S. (1971). Experiences with a sales districting model: Criteria and implementation. Management Science, 18(4–part—-ii), 41–54. https://doi.org/10.1287/mnsc.18.4.P41.
Hess, S., Weaver, J., Siegfeldt, H., Whelan, J., & Zitlau, P. (1965). Nonpartisan political redistricting by computer. Operations Research, 13, 998–1008. https://doi.org/10.1287/opre.13.6.998.
Hoffman, K., & Padberg, M. (2009). Set covering, packing and partitioning problems. In C. A. Floudas & P. M. Pardalos (Eds.), Encyclopedia of optimization (pp. 3482–3486). Boston: Springer. https://doi.org/10.1007/978-0-387-74759-0599.
Holguín-Veras, J., Ozbay, K., Kornhauser, A., Brom, M., Iyer, S., Yushimito, W., et al. (2011). Overall impacts of off-hour delivery programs in New York city metropolitan area. Transportation Research Record: Journal of the Transportation Research Board, 2238, 68–76. https://doi.org/10.3141/2238-09.
Hu, Z., Ding, Y., & Shao, Q. (2009). Immune co-evolutionary algorithm based partition balancing optimization for tobacco distribution system. Expert Systems with Applications, 36(3 PART 1), 5248–5255. https://doi.org/10.1016/j.eswa.2008.06.074.
Jarrah, A., & Bard, J. (2012). Large-scale pickup and delivery work area design. Computers and Operations Research, 39(12), 3102–3118. https://doi.org/10.1016/j.cor.2012.03.014.
Kalcsics, J. (2015). Districting problems. In G. Laporte & S. Nickel (Eds.), Location science. Berlin: Springer. https://doi.org/10.1007/978-3-319-13111-5.
Kalcsics, J., Nickel, S., & Schröder, M. (2005). Towards a unified territorial design approach-applications, algorithms and GIS Integration. Top, 13(1), 1.
Kwon, O., Golden, B., & Wasil, E. (1995). Estimating the length of the optimal tsp tour: An empirical study using regression and neural networks. Computers and Operations Research, 22(10), 1039–1046. https://doi.org/10.1016/0305-0548(94)00093-N.
Langevin, A., Mbaraga, P., & Campbell, J. (1996). Continuous approximation models in freight distribution: An overview. Transportation Research Part B: Methodological, 30(3 PART B), 163–188. https://doi.org/10.1016/0191-2615(95)00035-6.
Lei, H., Laporte, G., Liu, Y., & Zhang, T. (2015). Dynamic design of sales territories. Computers and Operations Research, 56, 84–92. https://doi.org/10.1016/j.cor.2014.11.008.
Li, W., Church, R., & Goodchild, M. (2014). The p-compact-regions problem. Geographical Analysis, 46(3), 250–273. https://doi.org/10.1111/gean.12038.
Lloyd, S. (2006). Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2), 129–137. https://doi.org/10.1109/TIT.1982.1056489.
Mehrotra, A., Johnson, E., & Nemhauser, G. (1998). An optimization based heuristic for political districting. Management Science, 44(8), 1100–1114. https://doi.org/10.1287/mnsc.44.8.1100.
Negreiros, M., & Palhano, A. (2006). The capacitated centred clustering problem. Computers and Operations Research, 33(6), 1639–1663. https://doi.org/10.1016/j.cor.2004.11.011.
New York City Department of Transportation (2016) New york city mobility report. Technical report, Washington, DC, http://www.nyc.gov/html/dot/downloads/pdf/mobility-report-2016-print.pdf
Nock, R., & Nielsen, F. (2006). On weighting clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(8), 1223–1235. https://doi.org/10.1109/TPAMI.2006.168.
Novaes, A., & Graciolli, O. (1999). Designing multi-vehicle delivery tours in a grid-cell format. European Journal of Operational Research, 119(3), 613–634. https://doi.org/10.1016/S0377-2217(98)00344-0.
Novaes, A., de Cursi, J. E., & Graciolli, O. (2000). A continuous approach to the design of physical distribution systems. Computers and Operations Research, 27, 877–893. https://doi.org/10.1016/S0305-0548(99)00063-5.
Ong, H., & Huang, H. (1989). Asymptotic expected performance of some tsp heuristics: An empirical evaluation. European Journal of Operational Research, 43(2), 231–238. https://doi.org/10.1016/0377-2217(89)90217-8.
O’Rourke, J. (1998). Computational Geometry in C (2nd ed.). New York: Cambridge University Press. https://doi.org/10.1017/CBO9780511804120.
Ouyang, Y., & Daganzo, C. (2006). Discretization and validation of the continuum approximation scheme for terminal system design. Transportation Science, 40(1), 89–98. https://doi.org/10.2307/25769285.
Pereira, J., & Averbakh, I. (2013). The robust set covering problem with interval data. Annals of Operations Research, 207(1), 217–235. https://doi.org/10.1007/s10479-011-0876-5.
Ricca, F., & Simeone, B. (2008). Local search algorithms for political districting. European Journal of Operational Research, 189(3), 1409–1426. https://doi.org/10.1016/j.ejor.2006.08.065.
Ricca, F., Scozzari, A., & Simeone, B. (2013). Political districting: From classical models to recent approaches. Annals of Operations Research, 204(1), 271–299. https://doi.org/10.1007/s10479-012-1267-2.
Ríos-Mercado, R., & Fernández, E. (2009). A reactive GRASP for a commercial territory design problem with multiple balancing requirements. Computers and Operations Research, 36(3), 755–776. https://doi.org/10.1016/j.cor.2007.10.024.
Salazar-Aguilar, M., Ríos-Mercado, R., & Cabrera-Ríos, M. (2011a). New models for commercial territory design. Networks and Spatial Economics, 11(3), 487–507. https://doi.org/10.1007/s11067-010-9151-6.
Salazar-Aguilar, M., Ríos-Mercado, R., & González-Velarde, J. (2011b). A bi-objective programming model for designing compact and balanced territories in commercial districting. Transportation Research Part C: Emerging Technologies, 19(5), 885–895. https://doi.org/10.1016/j.trc.2010.09.011.
Salazar-Aguilar, M., Ríos-Mercado, R., González-Velarde, J., & Molina, J. (2012). Multiobjective scatter search for a commercial territory design problem. Annals of Operations Research, 199(1), 343–360. https://doi.org/10.1007/s10479-011-1045-6.
Salazar-Aguilar, M., Ríos-Mercado, R., & González-Velarde, J. (2013). Grasp strategies for a bi-objective commercial territory design problem. Journal of Heuristics, 19(2), 179–200. https://doi.org/10.1007/s10732-011-9160-8.
Stein, D. (1978). An asymptotic, probabilistic analysis of a routing problem. Mathematics of Operations Research, 3(2), 89–101. https://doi.org/10.1287/moor.3.2.89.
Tavares-Pereira, F., Figueira, J., Mousseau, V., & Roy, B. (2007). Multiple criteria districting problems: The public transportation network pricing system of the Paris region. Annals of Operations Research, 154(1), 69–92. https://doi.org/10.1007/s10479-007-0181-5.
Tavares-Pereira, F., Figueira, J., Mousseau, V., & Roy, B. (2009). Comparing two territory partitions in districting problems: Indices and practical issues. Socio-Economic Planning Sciences, 43(1), 72–88. https://doi.org/10.1016/j.seps.2007.04.001.
Transportation Research Board. (2010). Highway capacity manual (Vol. 5). Washington, DC: National Sciences Foundation.
Wang, J., Kwan, M., & Ma, L. (2014). Delimiting service area using adaptive crystal-growth Voronoi diagrams based on weighted planes: A case study in Haizhu District of Guangzhou in China. Applied Geography, 50, 108–119. https://doi.org/10.1016/j.apgeog.2014.03.001.
Zoltners, A., & Sinha, P. (2005). Sales territory design: Thirty years of modeling and implementation. Marketing Science, 24(3), 313–331. https://doi.org/10.1287/mksc.1050.0133.
Acknowledgements
Sebastián Moreno acknowledges the support of “CONICYT + PAI/ Concurso nacional de apoyo al retorno de investigadores/as desde el extranjero, convocatoria 2014 + folio 82140043”.
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Moreno, S., Pereira, J. & Yushimito, W. A hybrid K-means and integer programming method for commercial territory design: a case study in meat distribution. Ann Oper Res 286, 87–117 (2020). https://doi.org/10.1007/s10479-017-2742-6
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DOI: https://doi.org/10.1007/s10479-017-2742-6