An Efficient P-center Algorithm

M. Çelebi Pınar

Taylan İlhanBilkent 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

z

subject to

1

,

z

, 0,1 ,

ijj W

ij j

jj W

ij ijj W

ij j

Min

x i V

x y i V j W

y p

d x i V

x y i V j W

(PCIP)

Literature (Exact ones)

• Minieka(1970): solving a series of minimal set covering problems

• Daskin(2000): solving a series of maximal set covering problems

min

subject to

1

0,1

jj W

ij jj W

j

y

a y i V

y j W

max

subject to

, 0,1 ,

ii V

ij j ij W

jj W

j i

w

a y w i V

y p

y w j W i V

1 threshold distance

0 o.w.ij

ij

da

Example Formation of Set Covering

The proposed Algorithm

• We basically solve the p-center problem by solving following feasibility problem:

1

0,1

ij jj W

jj W

j

a y i V

y p

y j W

(IP)

The proposed Algorithm (cont’d.)

• Our Algorithm has two parts:

• PART I (LP feasibility):

1. Set l to minimum and u to maximum of all distance values

2. Threshold distance = (u-l)/2

3. Define aij as follows

4. Solve the LP feasibility problem(relaxing binary constraint in IP)

5. If the problem is feasible set u = Threshold distance else set l = Threshold distance

6. If (u-l) < 1 go to PART 2 else go to step 1.

1 threshold distance

0 o.w.ij

ij

da

The proposed Algorithm (cont’d.)

• PART II (IP feasibility):

1. If the LP formulation is not feasible then set Threshold distance = u, else Threshold distance = l

2. Define aij as follows

3. Solve the IP feasibility problem

4. 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)

1 threshold distance

0 o.w.ij

ij

da

Computational Results

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

file no. size p obj. cpu time obj. cpu time obj. cpu time obj. cpu time1 100 5 90.92 60.98 127 3601.79 121 0.89 127 2.082 100 10 63.35 49.21 110 3601.74 98 0.78 98 0.873 100 10 62.48 82.98 137 3601.44 92 0.69 93 0.834 100 20 41.5 58.66 119 3601.41 73 0.52 74 0.645 100 33 19.12 49.41 60 3601.67 48 0.53 48 0.586 200 5 62.87 2017.63 150 3611.58 83 3.17 84 6.137 200 15 37.14 1654.81 90 3607.54 55 2.03 56 2.78 200 20 33.84 1647.99 202 3605.25 55 1.66 55 1.889 200 40 20.02 747.19 65 3609.69 36 1.51 37 1.74

10 200 67 8.76 524.76 75 3605.49 19 1.2 20 1.37

PCIP Formulation Proposed AlgorithmLP Relaxation PCIP LP part LP+IP

Computational Results(Cont’d)

file no PCIP (%) LP relax(%) LP part1 0 28.41 4.722 12.24 35.36 03 47.31 32.82 1.084 60.81 43.92 1.355 25 60.17 06 78.57 25.15 1.197 60.71 33.68 1.798 267.27 38.47 09 75.68 45.89 2.710 275 56.2 5

Average 90.26 40.01 1.78Max 275 60.17 5

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)

file no N p Obj. Value iter # cpu time Obj. Value iter # cpu time deviation16 400 5 47 8 13.94 47 7 11.93 017 400 10 39 9 13.41 38 7 9.05 2.5618 400 40 28 9 19.42 27 7 5.92 3.5719 400 80 18 9 4.85 17 7 3.85 5.5620 400 133 13 8 4.08 13 7 3.86 021 500 5 40 8 42.34 40 7 19.79 022 500 10 38 9 130.46 37 7 20 2.6323 500 50 22 8 35.81 21 6 7.13 4.5524 500 100 15 9 7.84 14 7 7 6.6725 500 167 11 8 7.06 10 6 6.35 9.0926 600 5 38 10 121.72 36 7 34.09 5.2627 600 10 32 8 73.53 31 6 25.11 3.1328 600 60 18 9 18.16 17 7 13.48 5.5629 600 120 13 7 10.18 13 6 9.67 030 600 200 9 8 9.99 9 7 9.59 031 700 5 30 8 108.22 29 6 33 3.3332 700 10 29 10 460.34 27 7 33.46 6.933 700 70 15 8 32.37 14 6 14.51 6.6734 700 140 11 8 15.56 10 6 14.13 9.0935 800 5 30 7 66.46 30 6 56.39 036 800 10 27 8 342.1 27 7 55.53 037 800 80 15 8 35.18 15 7 33.2 038 900 5 29 8 96.04 28 6 68.05 3.4539 900 10 23 8 536.48 22 6 57.38 4.35

Complete Algorithm LP Part

Results of the experiments with ORLIB p-median data

Computational Results(Cont’d)

file no N p(Feasibility)

cpu time(Max. Cover)

cpu timefactor imp.

(Feasibility) cpu time

(Max. Cover) cpu time

factor imp.

16 400 5 13.94 99.19 6.12 11.93 74.9 5.2817 400 10 13.41 160.09 10.94 9.05 51.6 4.7018 400 40 19.42 66.42 2.42 5.92 34.52 4.8319 400 80 4.85 13.64 1.81 3.85 11.82 2.0720 400 133 4.08 28.4 5.96 3.86 28.09 6.2821 500 5 42.34 185.89 3.39 19.79 129.64 5.5522 500 10 130.46 222.24 1.70 20 85.96 3.3023 500 50 35.81 131.68 2.68 7.13 43.04 5.0424 500 100 7.84 88.37 10.27 7 85.81 11.2625 500 167 7.06 14.01 1.98 6.35 13.08 1.0626 600 5 121.72 619.82 4.09 34.09 399.25 10.7127 600 10 73.53 341.22 3.64 25.11 168.48 5.7128 600 60 18.16 206.49 10.37 13.48 100.61 6.4629 600 120 10.18 139.01 12.66 9.67 123.68 11.7930 600 200 9.99 41.89 3.19 9.59 41.3 3.3131 700 5 108.22 270 1.49 33 205.88 5.2432 700 10 460.34 1080.07 1.35 33.46 211.57 5.3233 700 70 32.37 137.29 3.24 14.51 55.9 2.8534 700 140 15.56 40.43 1.60 14.13 37.57 1.6635 800 5 66.46 1205.75 17.14 56.39 904.45 15.0436 800 10 342.1 802.18 1.34 55.53 314.88 4.67

Complete algorithm LP Part

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.