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The uncapacitated facility location problem in .NET Add 2d Data Matrix barcode in .NET The uncapacitated facility location problem




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The uncapacitated facility location problem using .net vs 2010 tomake ecc200 on asp.net web,windows application GS1 DataBar Theorem 5.19: Algor Visual Studio .NET DataMatrix ithm 5.

1 is a randomized 3-approximation algorithm for the uncapacitated facility location problem. Proof. In an iteration k, the expected cost of the facility opened is fi x k fi yi , ij.

i N (jk ) i N (jk ) using the LP constraint x k yi . As we argued in Section 4.5, the neighborhoods N (jk ) form ij a partition of a subset of the facilities so that the overall expected cost of facilities opened is at most fi yi fi yi .

k i N (jk ) i F. We now x an iterat barcode data matrix for .NET ion k and let j denote the client jk selected and let i denote the facility ik opened. The expected cost of assigning j to i is cij x = Cj .

ij. i N (j). As can be seen from gs1 datamatrix barcode for .NET Figure 5.5, the expected cost of assigning an unassigned client l N 2 (j) to i, where the client l neighbors facility h which neighbors client j is at most chl + chj + cij x = chl + chj + Cj .

ij. i N (j) By Le mma 4.11, chl vl and chj vh , so that this cost is at most vl + vj + Cj . Then since + C among all unassigned clients, we know that v + C v + C .

we chose j to minimize vj j j j l l Hence the expected cost of assigning l to i is at most vl + vj + Cj 2vl + Cl .. Thus we have that o ECC200 for .NET ur total expected cost is no more than fi yi + (2vj + Cj ) = fi yi + cij x + 2 vj ij. i F j D i F i F,j D j D 3 OPT . Note that we were a ble to reduce the performance guarantee from 4 to 3 because the random choice of facility allows us to include the assignment cost Cj in the analysis; instead of bounding only the facility cost by OPT, we can bound both the facility cost and part of the assignment cost by OPT. One can imagine a di erent type of randomized rounding algorithm: suppose we obtain an optimal LP solution (x , y ) and open each facility i F with probability yi . Given the open facilities, we then assign each client to the closest open facility.

This algorithm has the nice feature that the expected facility cost is i F fi yi . However, this simple algorithm clearly has the di culty that with non-zero probability, the algorithm opens no facilities at all, and hence the expected assignment cost is unbounded. We consider a modi ed version of this algorithm later in the book, in Section 12.

1. Electronic web edition. Copyright 2010 by David P.

Williamson and David B. Shmoys. To be published by Cambridge University Press.

Random sampling and randomized rounding of linear programs Scheduling a single machine with release dates In this section, we Data Matrix barcode for .NET return to the problem considered in Section 4.2 of scheduling a single machine with release dates so as to minimize the weighted sum of completion times.

Recall that we are given as input n jobs, each of which has a processing time pj > 0, weight wj 0, and release date rj 0. The values pj , rj , and wj are all nonnegative integers. We must construct a schedule for these jobs on a single machine such that at most one job is processed at any point in time, no job is processed before its release date, and once a job begins to be processed, it must be processed nonpreemptively; that is, it must be processed completely before any other job can be scheduled.

If Cj denotes the time at which job j is nished processing, then the goal is to nd the schedule that minimizes n wj Cj . j=1 In Section 4.2, we gave a linear programming relaxation of the problem.

In order to apply randomized rounding, we will use a di erent integer programming formulation of this problem. In fact, we will not use an integer programming formulation of the problem, but an integer programming relaxation. Solutions in which jobs are scheduled preemptively are feasible; however, the contribution of job j to the objective function is less than wj Cj unless job j is scheduled nonpreemptively.

Thus the integer program is a relaxation since for any solution corresponding to a nonpreemptive schedule, the objective function value is equal to the sum of weighted completion times of the schedule. Furthermore, although this relaxation is an integer program and has a number of constraints and variables exponential in the size of the problem instance, we will be able to nd a solution to it in polynomial time. We now give the integer programming relaxation.

Let T equal maxj rj + n pj , which is j=1 the latest possible time any job can be processed in any schedule that processes a job nonpreemptively whenever it can. We introduce variables yjt for j = 1, . .

. , n, t = 1, . .

. , T , where { 1 if job j is processed in time [t 1, t) yjt = 0 otherwise We derive a series of constraints for the integer program to capture the constraints of the scheduling problem. Since at most one job can be processed at any point in time, for each time t = 1, .

. . , T we impose the constraint.

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