We study the capacitated k-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number k. It costs f i to open facility i, and c ij for facility i to serve one unit of demand from client j. The objective is to open at most k facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs.In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated k-facility location problem. Then, we show that the capacitated k-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated k-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening cost.
A weapon system of systems (WSoS) is a collection of interdependent systems providing multiple interdependent capabilities to support multiple tasks. In this paper, we propose a WSoS scheme generation method based on a granular analysis of a WSoS supernetwork. More precisely, we establish a WSoS supernetwork model based on a "network-centered" combat mode. Next, based on the theory of quotient space, a granular analysis is presented to reduce the complexity of the WSoS scheme generation. Then, the WSoS supernetwork is converted into a directed weighted network with multiple granularities, and an algorithm for the WSoS scheme generation based on the directed weighted network is presented. Consequently, each WSoS scheme corresponds to a path in the directed weighted network, and the capability of supporting the realization of a task by a WSoS can be analyzed based on the schemes. Finally, the feasibility of the proposed approach is demonstrated using an illustrative example.
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