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Set-covering based p-center problems 155of the current solution z k . Until reaching the optimality, we continue to solve (PC-SC2) byupdating the upper bound D U and the lower bound D L .p-SBsearchInitialization: given p, D 0 , . . . , D K , set L = 0, U = K and ∆ ∗ p = DUStep 1. k=⌊(L+U)/2⌋.Step 2. Solving (PC-SC2) with D k to obtain the optimal solution z k∗ .Step 3. If z k∗ =1, thenStep 3.1. If z k+1 =0, then STOP, set ∆ ∗ p = DL , L ∗ p = L and U p ∗ = U else, let L = k, goto step 1.If z k =0, thenStep 3.2. If z k−1 =0, then STOP, set ∆ ∗ p = D L , L ∗ p = L and Up ∗ = U, else, let U = k, goto step 1.3. Computational Results for p-SBsearchThe procedure was implemented with the code written by C++ and the MIP solver of CPLEX7.1. We use a notebook with 512 MB of RAM and Intel P-M 1.30 GHz of CPU. The time limitof CPLEX is set to 3600 seconds, so the solution of sub-problem stops if no integer solution isfound after one hour of CPU time. The results of the comparison on 40 OR-Lib (Beasley 1990)p-median instances are given in Table 1. The first three columns characterize the instance, andthe optimal radius is in column 4. Columns 5, 6 and 7 are quoted from Elloumi et al. (2004).Column 8 gives the CPU time of our procedure. Even if it is not straightforward to compareCPU times on different machines, we can show the maximum and the average CPU time asindication in Table 1.Table 2 gives the results for TSP-Lib (Reinelt, 1991) instances and makes comparison ofour procedure with Elloumi et al. (2004). The first three columns characterize the instance.Columns 4 through 8 give the results of algorithm Bsearch and BsearchEx (2004). Columns LB ∗and UB ∗ give the lower bound and upper bound obtained by Bsearch, and Column cpu1 isthe CPU time devoted to Bsearch. Column Opt gives the optimal solution or the best foundsolution obtained by BsearchEx, and Column cpu2 is the CPU time devoted to BsearchEx.Columns 9 through 12 give the results of our procedure. There is tradeoff between solutiontime and the preciseness of solution in large scale problem. Based on the updating of maxand min of out procedure, we could set the bound tolerance in advance to obtain a narrowsolution bound in shorter CPU time. If the relative bound tolerance (D max − D min )/D mindoes not exceed 5% for any sub-problem, we stop the procedure and record the current bound.Column 5% Bound gives the results of 5% bound tolerance, and Column cpu is the CPU timeof 5% bound tolerance. Column Opt gives the results of our procedure, and Column cpu2 isthe CPU time of our p-SBsearch procedure. If the optimal solution is not reached in an hour,set z k = 1 and L = k to solve the next sub-problem. When this happens we are no longer surethat our solution is optimal, and then we give the best solved bound in Column Opt.4. ConclusionIn this paper, we introduced a new efficient exact procedure to obtain the globe optimizationof the set-covering-based p-center problems, which have been proved better than the classicalformulation (PC), to solve the large scale problem with the simple repeating procedure.We could maintain the globe optimality in reasonable time limit and without performing thecomplex algorithm of Bsearch and BsearchEx.Moreover, upon the proposed bisecting search method, we introduce the concept of simplificationto the continuous problem with the same distance matrix and increasing p value. Theglobal optimal solution to the original problem still obtainable as the number of sub-problemsis decreased and the total CPU time is shortened; simultaneously.