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

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

Problem Formulation (PCIP)

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

Example Formation of Set Covering

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

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.

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)

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

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

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

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

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…

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.  428-436.  Minieka, E., 1970, The m-Center Problem , SIAM Review, 12:1. , pp. 138-139.