1. An Efficient P-center Algorithm
M. Çelebi Pınar
Taylan İlhan
Bilkent University
Industrial Engineering Department
Ankara, Turkey

2. Problem Definition & Notation
Definition: Locating p facilities and assigning clients to them in order to minimize the maximum distance between a client and the facility to which it is assigned
Notation:
W : Set of clients
V : Set of facilities
xij : binary variable, it is 1 if client i is assigned to facility j, 0 otherwise
yj : binary variable, it is 1 if facility j is opened, 0 otherwise
wi : binary variable, it is 1 if client i is assigned to a facility, 0 otherwise
dij : distance between client i and facility j

3. Problem Formulation
(PCIP)

4. Literature (Exact ones)
Minieka(1970): solving a series of minimal set covering problems
Daskin(2000): solving a series of maximal set covering problems

5. Example Formation of Set Covering

6. The proposed Algorithm
We basically solve the p-center problem by solving following feasibility problem:
(IP)

7. The proposed Algorithm (cont’d.)
Our Algorithm has two parts:
PART I (LP feasibility):
Set l to minimum and u to maximum of all distance values
Threshold distance = (u-l)/2
Define aij as follows 
Solve the LP feasibility problem(relaxing binary constraint in IP)
If the problem is feasible set u = Threshold distance else set l = Threshold distance
If (u-l) < 1 go to PART 2 else go to step 1.

8. The proposed Algorithm (cont’d.)
PART II (IP feasibility):
If the LP formulation is not feasible then set Threshold distance = u, else Threshold distance = l
Define aij as follows 
Solve the IP feasibility problem
If the problem is not feasible set new threshold distance = minimum of all distances which is greater than threshold value, and go to step 1.
Else, Stop. (Optimal Solution is found)

9. Computational Results
Comparison of PCIP Formulation on CPLEX 5 with
Proposed Algorithm over p-median data

10. Computational Results(Cont’d)
Deviations of PCIP Formulation, its LP relaxation, and LP part of the algorithm from the optimal solution for each p-median problem instance

11. Computational Results(Cont’d)
Results of the experiments with ORLIB p-median data

12. Computational Results(Cont’d)
Results of the proposed algorithm with feasibility subproblem
and maximal set covering subproblem

13. Future Research
Elloumi, Labbe’ and Pochet (2001) introduced a new algorithm
It is similar to the present work. Comparisons should be made..
Extensions to more general p-center problems…

14. References
Daskin, M.S., 2000, A New Approach to Solving the Vertex P-Center Problem to Optimality: Algorithm and Computational Results, Communications of the Operations Research Society of Japan, 45:9, pp.   
Minieka, E., 1970, The m-Center Problem , SIAM Review, 12:1. , pp