Management Science 461

1. Management Science 461 Lecture 3 – Covering Models September 23, 2008

2. Covering Models • We want to locate facilities within a certain distance of customers • Each facility has positive cost, so we need to cover with minimum # of facilities • Easy “upper bound” for these problems. What is it?

3. Defining Coverage • Geographic distance • Euclidean or rectilinear – distance metrics • Time metric • Network distance • Shortest Paths • Coverage is usually binary: either node i is covered by node j or it isn’t • A potential midterm question would be to relax this assumption…

4. 14 A B 13 10 E 17 23 16 C 12 D Network example

5. 14 A B 13 10 E 17 23 16 C 12 D Network example • If coverage distance is 15 km, a facility at node A covers which nodes?

6. 14 A B 13 10 E 17 23 16 C 12 D Example Network (cont.) • When D = 22km, what is the coverage set of node A?

7. Algebraic formulation • Assume cost of locating is the same for each facility (again – possible HW / midterm relaxation) • The objective function becomes … • (Set of facility locations – J; set of customers – I)

8. 14 13 A B 10 E 23 17 16 C 12 D Example – D = 15

9. 14 Example – D = 15 13 A B 10 E 23 17 16 C 12 D

10. 14 Example – D = 15 13 A B 10 E 23 17 16 C 12 D

11. 14 13 A B 10 E 23 17 16 C 12 D Complete Model

12. Algebraic formulation • More generally, we can define • The value of aij does not change for a given model run. • We can include cost of opening a facility

13. General Formulation Cost of covering all nodes Each node covered Integrality

14. The Maximal Covering Problem • Locate P facilities to maximize total demand covered; full coverage not required • Extensions: • Can we use less than P facilities? • Each facility can have a fixed cost • Main decision variable remains whether to locate at node j or not

15. 250 100 14 A B 13 150 10 E 17 23 16 C 12 Demand 200 D 125 The Maximal Covering Problem

16. 250 100 14 A B 13 150 10 E 17 23 16 C 12 Demand 200 D 125 Max Covering Solution for P=1 Locate at __ which covers nodes ___ for a total covered demand of ___ . Distance coverage: 15 Km

17. Modeling Max Cover • If we use a similar model to set cover, we might double- and triple-count coverage. • To avoid this and still keep linearity, we need another set of binary variables • Zi = 1 if node i is covered, 0 if not • Linking constraints needed to restrict the model

18. 14 A B 13 10 E 17 23 16 C 12 D Max Cover Formulation (D=15) Total covered demand Linkage constraints Locate P sites Integrality

19. Max Covering Formulation Covered demands Node i not covered unless we locate at a node covering it Locate P sites Integrality

20. Max Covering – Typical Results 150 cities Dc= 250 Decreasing marginal coverage Last few facilities cover relatively little demand ~ 90% coverage with ~ 50% of facilities

21. Problem Extensions • The Max Expected Covering Problem • Facility subject to congestion or being busy • Application: in locating ambulances, we need to know that one of the nearby ambulances is available when we call for service • Scenario planning • Data shifts (over time, cycles, etc) force multiple data sets – solve at once