An Indirect Method for the Generalized k-Median Problem Applied to Lock-Box Location

Abstract

The problem of locating lock-boxes (post office boxes operated by banks for corporations) has been formulated as an uncapacitated plant location problem, for which there exist many efficient methods. However, corporations are often reluctant to establish as many lock-boxes as indicated by the optimal solution to this formulation. Hence, it is advisable to find the sequence of solutions containing 1, …, m lock-boxes, where m denotes the optimal number for the uncapacitated problem. This can be done by imposing an additional constraint specifying that the number of lock-boxes to be established be equal to k, and solving the resulting generalized k-median problem parametrically for k = m, m − 1, …, 1. It is very unlikely that an efficient direct algorithm exists for this problem, because it has been shown to be NP-complete. An indirect method is given in this paper, using an algorithm for the uncapacitated problem to solve the k-median problem. This method would be particularly valuable in situations where an uncapacitated algorithm is already in use; by adjusting the existing algorithm as indicated in the method, the time and cost required to implement a new algorithm can be avoided. The method has been successfully applied to some 1,000 real lock-box problems. It seems to be roughly as efficient as the underlying uncapacitated algorithm. The computational experience is discussed.

Discrete facility location with nonlinear facility costs

“…The technique of determining a solution with a prespecified number of established facilities (i. e. solving a "/?-median" problem) by varying the fixed costs of UFLPs has been proposed several times in literature [3,6,11,15,17]. To obtain a certain p value, two papers [11,17] independently suggested a scheme for updating y.…”

“…For this problem, severâl branch-and-bound algorithms have been proposed in literature: [9] is explicitly addressed to a fixed cost /j-median problem, while the solution methods [1,6,8,11] have to be modified slightly by introducing the facility costs f t . However, the reported expérience [11, 15,17] shows that the non-convex case occurs rather seldom when practical data is processed. So we can expect that, in the most situations, NFCLP will already be solved during phase (b).…”

What Lockbox and Disbursement Models Really Do

“…The papers by Kramer [15], Stancill [27], McAdams [19], Kraus, Jansen and McAdams [17], Shanker and Zoltners [26] and Maier and Vander Weide [20] fall in the first category. The papers by Baker, Maier and Vander Weide [1], Bulfin and Unger [3], Ciochetto, Swanson, Lee and Woolsey [5], Corneujols, Fisher and Nemhauser [7], Fielitz and White [11], Levy [18], Mavrides [22], Nauss and Markland [23] and Stone [31] fall in the second.…”

“…We have generally recommended the use of the 1‐for‐1 and 2‐for‐2 interchange heuristics because they provide valuable information on both optimal and close‐to‐optimal solutions. However, we have also employed a Lagrangian Relaxation/Branch‐and‐Bound [22] procedure as a check on the heuristics. In the hundreds of studies we have done over the last seven years, we have never found a single instance in which the more elaborate optimization techniques found a superior solution to the heuristics.…”

Queuing Model Based Edge Placement for Work Offloading in Mobile Cloud Networks

“…The problem is formally defined as an integer programming problem with the aim to minimize the generated traffic. These kinds of problems have been shown to be NP-complete [4], [5]. It is a complex combinatorial problem involving selecting appropriate locations of the edge servers and then allocating the offloading jobs to these edge servers in a proper manner.…”