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Towards a stochastic programming modeling framework for districting

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A Correction to this article was published on 08 June 2020

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Abstract

In this paper a stochastic districting problem is investigated. Demand is assumed to be represented by a random vector with a given joint probability distribution function. A two-stage mixed-integer stochastic programming model is proposed. The first stage comprises the decision about the initial territory design: the districts are defined and all the territory units assigned to one and exactly one of them. In the second stage, i.e., after demand becomes known, balancing requirements are to be met. This is ensured by means of two recourse actions: outsourcing and reassignment of territory units. The objective function accounts for the total expected cost that includes the cost for the first-stage territory design plus the expected cost incurred at the second stage by outsourcing and reassignment. The (re)assignment costs are associated with the distances between territory units, i.e., the focus is put on the compactness of the solution. The model is then extended in different ways to account for aspects of practical relevance such as a maximum desirable dispersion, reallocation constraints, or similarity of the second-stage solution w.r.t. the first-stage one. The new modeling framework proposed is tested computationally using instances built using real geographical data.

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Change history

  • 08 June 2020

    Third authors’ name should appear as: Francisco Saldanha-da-Gama.

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Acknowledgements

This work was partially supported by National Funding from FCT - Fundação para a Ciância e a Tecnologia, under the project: UIDB/04561/2020. The authors would like to thank the anonymous reviewer for his/her detailed comments on our work, which helped us improving the manuscript.

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Authors and Affiliations

  1. Department of Industrial Engineering (DII), Università degli Studi di Napoli Federico II, Piazzale Tecchio, 80, 80125, Naples, Italy

    Antonio Diglio

  2. Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Stefan Nickel

  3. Department of Logistics and Supply Chain Optimization, Research Center for Information Technology (FZI), Karlsruhe, Germany

    Stefan Nickel

  4. Dept. Estatística e Investigação Operacional, Faculdade de Ciências da Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Francisco Saldanha-da-Gama

  5. Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências da Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Francisco Saldanha-da-Gama

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  1. Antonio Diglio
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Correspondence to Antonio Diglio.

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The research of the third author was supported by the Portuguese Science Foundation (FCT—Fundação para a Ciência e Tecnologia) under the Projects UID/MAT/ 04561/2013 (CMAF-CIO/FCUL) and SFRH/BSAB/130291/2017.

Appendix

Appendix

In this Appendix we details the information concerning the test data used to realize our computational experiments and we present the detailed results that were reported in Sect. 6.

For each TU, Table 5 reports the generated demand vector and the coordinates of the respective centroid. The original code assigned by ISTAT to uniquely identify TUs (i.e. PRO_COM) is also reported. TUs constituting each of the considered instances are indicated in Table 6.

Table 5 Demands and coordinates of the centroids (Coordinate Reference System: ED50 / UTM Zone 32N EPSG:23032)
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Table 6 TUs that are part of each instance
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Table 7 Computational results for the 88-TU instances, \(p=4\)
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Table 8 Computational results for the 88-TU instances, \(p=6\)
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Table 9 CPU time (seconds)—for all the instances with \(p=4\)
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Table 10 CPU time (seconds)—for all the instances with \(p=6\)
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For each tested instance, Tables 7 and 8 contain the realtive values of VSS and EVPI w.r.t. SP as well as the CPU time in seconds required by the general purpose solver to solve the instance to optimality. Tables 7 refers to the instances with 4 districts and Tables 8 to instances with 6 districts.

Finally, Tables 9 and 10 contain the CPU time (seconds) required to solve the instances to proven optimalty for different values of |I| (40, 60, 88, and 120).

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Diglio, A., Nickel, S. & Saldanha-da-Gama, F. Towards a stochastic programming modeling framework for districting. Ann Oper Res 292, 249–285 (2020). https://doi.org/10.1007/s10479-020-03631-7

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