AMBULANCE LOCATION PROBLEM The problem of locating ambulance
AMBULANCE LOCATION PROBLEM
ØThe problem of locating ambulance stations in a region has received considerable attention. ØMany municipalities applied facility location techniques in designing their ambulance systems. ØThere may be different objectives competing with each other. ØThere may be application-specific conditions.
ØIt would be unrealistic to consider every single dwelling in a city separately. Therefore, population is aggregated into zones. ØA zone may consist of a block, or a set of blocks. ØInputs to the problem: Candidate locations for stations Demand at each zone Response time between stations and zones
Example: I = {1, 2, . . . , 25} and J = {1, 2, . . . , 13} Assume travel distance between two adjacent zones is 5 minutes and travel can only take place in horizontal and vertical directions. 1 2 3 4 5 6 7 8 9 10 11 6 12 13 7 14 15 8 16 17 9 18 19 10 20 21 11 22 23 12 24 25 13 1 4 2 5 3
Objective 1 Ø Respond to each call in 10 minutes with a minimum number of ambulance stations. Objective: Subject to:
AMBULANCE LOCATION PROBLEM I = set of city zones, indexed by i. J = set of candidate sites, indexed by j. Xj= 1 if an ambulance station is located at site j, 0 otherwise.
Zone 1 would be covered (response time within 10 min) if there is an ambulance station in at least one of the sites {1, 2, 4, 6} 11 2 32 4 5 3 6 74 8 95 10 116 12 137 14 158 16 179 18 1910 20
• To cover zone 1, X 1+X 2+X 4+X 6 1. • We have to write a similar constraint for each zone.
Example: 1 I = {1, 2, . . . , 25} and J = {1, 2, . . . , 13} 2 3 4 5 6 7 8 9 10 11 6 12 13 7 14 15 8 16 17 9 18 19 10 20 21 11 22 23 12 24 25 13 1 4 2 5 3
Travel Times => A candidate site zone i 1 2 3. . 25 zone i 1 2 3 25 1 2 3 4 . . . 13 0 5 10. . . 40 10 5 0. . . 30 20 15 10. . . 20 10 5 10 . . 30 . . . 40 35 30. . . 0 1 1 1. . . 0 0 0 1. 1 1 1. . . 0 . . . 0 0 0. . . 1
Modeling the Problem Ø aij = 1 0 if travel time between zone i and site j is 10 minutes, otherwise. Ø aij’s are parameters, not decision variables. Ø To ensure that zone i is covered, we have to write constraints of the form: ai 1 X 1+ai 2 X 2+…+ai 13 X 13 1.
Ambulance Location Model 1: Min j Xj s. t. j aij Xj 1, i=1, …, 25 Xj=(0, 1), j=1, …, 13
Remarks on Model 1 Ø The objective of Model 1 is to find the minimum number of ambulance stations to cover all the zones, and the locations of these ambulance stations. Ø It ignores the demands in the zones. Every zone is treated the same. Ø The solution may require an excessive number of ambulance stations, and the city may not have sufficient funds to implement the solution.
Model 2 Ø The economically infeasible requirement of covering every zone is relaxed. Ø We try to locate ambulance stations in such a way so as to maximize the demand that can be covered within 10 minutes. Ø This version of the problem is solved for a number of ambulance stations and study the marginal impact of each added ambulance station.
Demand Data 18 1 22 30 2 18 3 12 41 31 5 18 9 31 4 11 54 6 11 10 21 7 11 21 8 14 9 8 4 11 12 12 32 10 18 3 6 13 portion of demand covered = 230 / 375 = 61. 3 %
Objective 2 Ø Maximize the population covered by locating p ambulance stations. Øn Ø ri = Population in zone i.
Model 2 Max i ri Yi s. t. j aij Xj Yi, i=1, …, 25 j Xj = p Xj = (0, 1), j=1, …, 13 Yi = (0, 1), i =1, …, 25
- Slides: 17