Extended Single-Iteration Fuzzy C-Means, and Gustafson-Kessel Algorithms for Medium-Sized (106) Multisource Weber Problem

Abstract

An uncapacitated multisource Weber problem involves finding facility locations for known customers. When this problem is restated as finding locations for additional new facilities, while keeping the current facilities, a new solution approach is needed. In this study, two new and cooperative fuzzy clustering algorithms are developed to solve a variant of the uncapacitated version of a multisource Weber problem (MWP). The first algorithm proposed is the extensive version of the single iteration fuzzy c-means (SIFCM) algorithm. The SIFCM algorithm assigns customers to existing facilities. The new extended SIFCM (ESIFCM), which is first proposed in this study, allocates discrete locations (coordinates) with the SIFCM and locates and allocates continuous locations (coordinates) with the original FCM simultaneously. If the SIFCM and the FCM, show differences between the successive cluster center values are still decreasing, share customer points among facilities. It is simply explained as single-iteration fuzzy c-means with fuzzy c-means. The second algorithm, also proposed here, runs like the ESIFCM. Instead of the FCM, a Gustafson-Kessel (GK) fuzzy clustering algorithm is used under the same framework. This algorithm is based on single-iteration (SIGK) and the GK algorithms. Numerical results are reported using two MWP problems in a class of a medium-size-data (106 bytes). Using clustering algorithms to locate and allocate the new facilities while keeping current facilities is a novel approach. When applied to the big problems, the speed of the proposed algorithms enable to find a solution while mathematical programming solution is not doable due to the great computational costs.