A decomposition approach for the p -median problem on disconnected graphs

Abstract

The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of p k , and the global solution is obtained by finding the best combination of p k medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time.

A three‐stage p‐median based exact method for the optimal diversity management problem

“…In most sub‐PMP s on the third stage a lot variables are fixed at 0 by the variable fixing rules embedded into in the Lagrangian heuristic, while the structure of ODMP implies the frequent fulfillment of the condition (18). Actually, the considered variable fixing rules and aggregation play a very important role in the process of finding optimal solutions to the third stage problems, which is evidently illustrated by the results of computational experiments compared with those presented in .…”

“…Our test bed consists of some known ODMP instances from the car industry: , , taken from and , taken from . Coherently with the real data, we also generated 10 new problem instances named from to .…”

“…The huge computation time required by a commercial MIP solver to find the optimal solution of all sub‐PMP s pushed us to develop a method that makes an intelligent use of different heuristic and exact algorithms. We propose a method adapting several known ideas about greedy and Lagrangian heuristics , dynamic programming , simulated annealing , variable fixing and probing , variable aggregation , and algorithm parallelization . The proposed solution method consists of three stages.…”

“…In the authors noted that a lot of sub‐PMP s can be excluded from further consideration by checking whether the optimal solution of the initial problem contains the solution of the corresponding sub‐PMP . Let be an optimal value of MCKP over the lower bounds and the additional constraint .…”

“…A decomposition approach was also proposed in where ‐median heuristics and a MCKP dynamic programming algorithm were developed, taking into account the particularity of ODMP instances. Recently, the same authors proposed an exact approach using LP relaxation bounds for sub‐PMP s and variable fixing in MCKP to reduce the number of sub‐PMP s to be solved . Note that the proposed approach has a limited use due to the very large computational time required to solve the tackled instances.…”