Facility Location Facility Location using Linear using Linear Programming Duality Programming Duality Yinyu Ye Yinyu Ye Department if Management Science Department if Management Science and Engineering and Engineering Stanford University Stanford University
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Facility Location using Linear Programming Duality Yinyu Ye Department if Management Science and Engineering Stanford University.
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Facility Location using Facility Location using Linear Programming DualityLinear Programming Duality
Yinyu YeYinyu YeDepartment if Management Science and Department if Management Science and
EngineeringEngineeringStanford UniversityStanford University
InputInput• A set of clients A set of clients or citiesor cities D D
• A set of facilities A set of facilities F F withwith facility cost facility cost ffii
FLP, ofsolution (integral) feasible a found algorithman Suppose
jDj
j
FifR
FiDjcR
ifDj
ij
ijcijj
(2)
, )1(
.
: have then we0, and 1,constant somefor ** CRFRCF
RR
cfDj
j
ijfc
A bi-factor (Rf,Rc)-approximate algorithm is a max(Rf,Rc)-approximate algorithm
Simple Greedy Algorithm
Introduce a notion of time, such that each event can be associated with the time at which it happened. The algorithm start at time 0. Initially, all facilities are closed; all clients are unconnected; all set to 0. Let C=D
While , increase simultaneously for all , until one of the following events occurs:
(1). For some client , and a open facility , then connect client j to facility i and remove j from C;
(2). For some closed facility i, , then open
facility i, and connect client with to facility i, and remove j from C.
j
C j Cj
Cj ijj ci such that
Cj
iijj fc ),0max(
Cj ijj c
Jain et al [2003]
Time = 0Time = 0
F1=3 F2=4
3 5 4 3 6 4
Time = 1Time = 1
F1=3 F2=4
3 5 4 3 6 4
Time = 2Time = 2
F1=3 F2=4
3 5 4 3 6 4
Time = 3Time = 3
F1=3 F2=4
3 5 4 3 6 4
Time = 4Time = 4
F1=3 F2=4
3 5 4 3 6 4
Time = 5Time = 5
F1=3 F2=4
3 5 4 3 6 4
Time = 5Time = 5
F1=3 F2=4
3 5 4 3 6 4
Open the facility on left, and connect clients “green” and “red” to it.
Open the facility on left, and connect clients “green” and “red” to it.
Time = 6Time = 6
F1=3 F2=4
3 5 4 3 6 4
Continue increase the budget of client “blue”
Continue increase the budget of client “blue”
Time = 6Time = 6
The budget of “blue” now covers its connection cost to an opened facility; connect blue to it.
The budget of “blue” now covers its connection cost to an opened facility; connect blue to it.