Annals of Operations Research

, Volume 253, Issue 1, pp 275–305 | Cite as

Facility location under service level constraints for heterogeneous customers

Original Paper

Abstract

We study the problem of locating service facilities to serve heterogeneous customers. Customers requiring service are classified as either high priority or low priority, where high priority customers are always served on a priority basis. The problem is to optimally locate service facilities and allocate their service zones to satisfy the following coverage and service level constraints: (1) each demand zone is served by a service facility within a given coverage radius; (2) at least \(\alpha ^h\) proportion of the high priority customers at any service facility should be served without waiting; (3) at least \(\alpha ^l\) proportion of the low priority cases at any service facility should not have to wait for more than \(\tau ^l\) minutes. For this, we model the network of service facilities as spatially distributed priority queues, whose locations and user allocations need to be determined. The resulting integer programming problem is challenging to solve, especially in absence of any known analytical expression for the service level function of low priority customers. We develop a cutting plane based solution algorithm, exploiting the concavity of the service level function of low priority customers to outer-approximate its non-linearity using supporting planes, determined numerically using matrix geometric method. Using an illustrative example of locating emerging medical service facilities in Austin, Texas, we present computational results and managerial insights.

Keywords

Facility location Congestion Service level Priority queue Cutting plane 

References

  1. Abate, J., & Whitt, W. (1997). Asymptotics for m/g/1 low-priority waiting-time tail probabilities. Queueing Systems, 25(1–4), 173–233.CrossRefGoogle Scholar
  2. Aboolian, R., Berman, O., & Krass, D. (2012). Profit maximizing distributed service system design with congestion and elastic demand. Transportation Science, 46(2), 247–261.CrossRefGoogle Scholar
  3. Amiri, A. (1997). Solution procedures for the service system design problem. Computers & Operations Research, 24(1), 49–60.CrossRefGoogle Scholar
  4. Atlason, J., Epelman, M. A., & Henderson, S. G. (2004). Call center staffing with simulation and cutting plane methods. Annals of Operations Research, 127(1–4), 333–358.CrossRefGoogle Scholar
  5. Baron, O., Berman, O., & Krass, D. (2008). Facility location with stochastic demand and constraints on waiting time. Manufacturing & Service Operations Management, 10(3), 484–505.CrossRefGoogle Scholar
  6. Belotti, P., Labbé, M., Maffioli, F., & Ndiaye, M. M. (2007). A branch-and-cut method for the obnoxious p-median problem. 4OR, 5(4), 299–314.Google Scholar
  7. Berman, O., & Krass, D. (2002). Facility location problems with stochastic demands and congestion. In Z. Drezner & H. Hamacher (Eds.), Facility location: Applications and theory. Berlin: Springer.Google Scholar
  8. Berman, O., & Krass, D. (2015). Stochastic location models with congestion. Location science (pp. 443–486). Berlin: Springer.Google Scholar
  9. Berman, O., Krass, D., & Wang, J. (2006). Locating service facilities to reduce lost demand. IIE Transactions, 38(11), 933–946.CrossRefGoogle Scholar
  10. Boffey, B., Galvao, R., & Espejo, L. (2007). A review of congestion models in the location of facilities with immobile servers. European Journal of Operational Research, 178(3), 643–662.CrossRefGoogle Scholar
  11. Cánovas, L., García, S., Labbé, M., & Marín, A. (2007). A strengthened formulation for the simple plant location problem with order. Operations Research Letters, 35(2), 141–150.CrossRefGoogle Scholar
  12. Castillo, I., Ingolfsson, A., & Sim, T. (2009). Socially optimal location of facilities with fixed servers, stochastic demand and congestion. Production and Operations Management, 18(6), 721–736.CrossRefGoogle Scholar
  13. Church, R. L., & Cohon, J. L. (1976). Multiobjective location analysis of regional energy facility siting problems. Brookhaven National Laboratory, Upton, NY, USA: Tech. rep.CrossRefGoogle Scholar
  14. Daskin, M. S. (1982). Application of an expected covering model to emergency medical service system design. Decision Sciences, 13(3), 416–439.CrossRefGoogle Scholar
  15. Daskin, M. S., & Stern, E. H. (1981). A hierarchical objective set covering model for emergency medical service vehicle deployment. Transportation Science, 15(2), 137–152.CrossRefGoogle Scholar
  16. Dobson, G., & Karmarkar, U. S. (1987). Competitive location on a network. Operations Research, 35(4), 565–574.CrossRefGoogle Scholar
  17. Elhedhli, S. (2006). Service system design with immobile servers, stochastic demand, and congestion. Manufacturing & Service Operations Management, 8(1), 92–97.CrossRefGoogle Scholar
  18. Espejo, I., Marín, A., & Rodríguez-Chía, A. M. (2012). Closest assignment constraints in discrete location problems. European Journal of Operational Research, 219(1), 49–58.CrossRefGoogle Scholar
  19. Gilboy, N., Tanabe, T., Travers, D., & Rosenau, A., (2011). Emergency Severity Index (ESI): A triage tool for emergency department care, version 4. Implementation Handbook 2012 Edition.Google Scholar
  20. Jayaswal, S. (2009). Product differentiation and operations strategy for price and time sensitive markets. PhD thesis. Ontario: Department of Management Sciences, University of Waterloo.Google Scholar
  21. Jayaswal, S., Jewkes, E., & Ray, S. (2011). Product differentiation and operations strategy in a capacitated environment. European Journal of Operational Research, 210(3), 716–728.CrossRefGoogle Scholar
  22. Jayaswal, S., & Jewkes, E. M. (2016). Price and lead time differentiation, capacity strategy and market competition. International Journal of Production Research, 54(9), 2791–2806.CrossRefGoogle Scholar
  23. Kelley, J. E, Jr. (1960). The cutting-plane method for solving convex programs. Journal of the Society for Industrial & Applied Mathematics, 8(4), 703–712.CrossRefGoogle Scholar
  24. Latouche, G., & Ramaswai, V., (1999). Introduction to matrix analytic methods in stochastic modeling. SIAM Series on Statistics and Applied Probability.Google Scholar
  25. Marianov, V., & Serra, D. (1998). Probabilistic, maximal covering locationallocation models for congested systems. Journal of Regional Science, 38(3), 401–424.CrossRefGoogle Scholar
  26. Marianov, V., & Serra, D. (2002). Location-allocation of multiple-server service centers with constrained queues or waiting times. Annals of Operations Research, 111(1–4), 35–50.CrossRefGoogle Scholar
  27. Marín, A. (2011). The discrete facility location problem with balanced allocation of customers. European Journal of Operational Research, 210(1), 27–38.CrossRefGoogle Scholar
  28. Murray, J. M. (2003). The canadian triage and acuity scale: A canadian perspective on emergency department triage. Emergency Medicine, 15(1), 6–10.CrossRefGoogle Scholar
  29. Nair, R., & Miller-Hooks, E. (2009). Evaluation of relocation strategies for emergency medical service vehicles. Transportation Research Record: Journal of the Transportation Research Board, 2137(1), 63–73.CrossRefGoogle Scholar
  30. Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Courier Dover Publications.Google Scholar
  31. Ramaswami, V., & Lucantoni, D. M. (1985). Stationary waiting time distribution in queues with phase type service and in quasi-birth-and-death processes. Communications in Statistics. Stochastic Models, 1(2), 125–136.CrossRefGoogle Scholar
  32. Rojeski, P., & ReVelle, C. (1970). Central facilities location under an investment constraint. Geographical Analysis, 2(4), 343–360.CrossRefGoogle Scholar
  33. Silva, F., & Serra, D. (2008). Locating emergency services with different priorities: The priority queuing covering location problem. Journal of the Operational Research Society, 59(9), 1229–1238.CrossRefGoogle Scholar
  34. Stephan, F. F. (1958). Two queues under preemptive priority with poisson arrival and service rates. Operations research, 6(3), 399–418.CrossRefGoogle Scholar
  35. Vidyarthi, N., & Jayaswal, S. (2014). Efficient solution of a class of location-allocation problems with stochastic demand and congestion. Computers & Operations Research, 48, 20–30.CrossRefGoogle Scholar
  36. Vidyarthi, N., & Kuzgunkaya, O. (2014). The impact of directed choice on the design of preventive healthcare facility network under congestion. Health Care Management Science,. doi:10.1007/s10729-014-9274-2.Google Scholar
  37. Wagner, J., & Falkson, L. (1975). The optimal nodal location of public facilities with price-sensitive demand. Geographical Analysis, 7(1), 69–83.CrossRefGoogle Scholar
  38. Wang, Q., Batta, R., & Rump, C. M. (2002). Algorithms for a facility location problem with stochastic customer demand and immobile servers. Annals of Operations Research, 111(1–4), 17–34.CrossRefGoogle Scholar
  39. Zhang, Y., Berman, O., & Verter, V. (2012). The impact of client choice on preventive healthcare facility network design. OR spectrum, 34(2), 349–370.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Production and Quantitative MethodsIndian Institute of ManagementVastrapur, AhmedabadIndia
  2. 2.Department of Supply Chain and Business Technology Management, John Molson School of BusinessConcordia UniversityMontrealCanada

Personalised recommendations