Location of emergency station s as the capacitated p-median problem

1. Location of emergency stationsas the capacitated p-median problem Ľudmila Jánošíková, Michal Žarnay University of Žilina, Slovak Republic Ludmila.Janosikova@fri.uniza.sk

2. Motivation Workload of ambulances is uneven. A more uniform distribution of workload • could improve accessibility, • will be fair with regard to providers, • could be achieved if the number of people served by one ambulance was limited. This leads to a location model with capacity constraints. 2

3. Preliminaries • The number of stations to be located is given (p). • Customer = municipality, the weight of customer j is the number of inhabitants bj. • Criterion: the total travel time needed to reach all potential patients. • Potential locations (candidates) for stations: nodes, where stations are today, and the other municipalities with at least 300 inhabitants. • Capacity limit of one ambulance = 25000 people. • If a city has more than 25000 inhabitants, open there bj div 25000 stations, reduce p by this number and set bj = bj mod 25000. 3

4. Metaheuristics LOPT Taillard, É.D.: Heuristic methods for large centroid clustering problems. Journal of Heuristics 9 (2003), 51–73. If the problem cannot be optimised as a whole, optimise it in parts. 4

5. Metaheuristics LOPT Input: initialpositionofpcentres, parameter r. Labelallcentresastemporary: C= {1, ..., p}. WhileC , repeatthefollowingsteps: 5

6. Randomlyselect a centre i  C. 6

7. Let R be the set of the r closest centres to i (i∈ R). 7

8. Consider the subproblem constructed with the customersallocated to the centres of R and optimize this subproblem with rcentres. • If no improved solution has been found at step 3c),set C = C \ { i }(locationof centre iisfinal),elseC= C R. 8

9. Computational experiments • 2 916 municipalities • 2 282 candidate locations • p = 223 • Initial solution: candidates are current stations and 100 most populated cities (after reduction bj = bj mod 25000) • Optimisation algorithm at step 3c): solverXpress-Optimizerwith the computation time limited to 5 minutes • Number of loops at step 3: 332 • Total computation time: 52 minutes 9

10. Objective function 10

11. Number of temporarily located centres 11

12. Computer simulation 12

13. Conclusions • Capacitatedp-median results in • a shorter response time than current location of stations, • a more uniform distribution of ambulance workload. Thank you for your attention.

14. Capacitatedp-median problem Notation: I the set of candidate locations J the set of municipalities tij the shortest travel time from node i to node j bj the number of inhabitants of municipality j Q the capacity limit of an ambulance 14

15. Capacitatedp-median problem Locationvariables: Allocationvariables: 15

16. Capacitatedp-median problem 16

17. Reduction of the model Eliminate the allocation variables which are less likely to belong to a good or optimal solution. The remaining variables xij are those for which 17

18. a b ab/p 18