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 Network example 14 A E D C B

5 Network example If coverage distance is 15 km, a facility at node A covers which nodes? 14 A E D C B

6 Example Network (cont.) When D = 22km, what is the coverage set of node A? 14 A E D C B

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 Example – D = A E D C B

9 Example – D = A E D C B

10 Example – D = A E D C B

11 Complete Model 14 A E D C B

12 Algebraic formulation More generally, we can define The value of a ij 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 The Maximal Covering Problem 14 A E D C B Demand

16 Max Covering Solution for P=1 Locate at __ which covers nodes ___ for a total covered demand of ___. Distance coverage: 15 Km 14 A E D C B Demand

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 Z i = 1 if node i is covered, 0 if not Linking constraints needed to restrict the model

18 Max Cover Formulation (D=15) Total covered demand Linkage constraints Locate P sites Integrality 14 A E D C B

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 D c = 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