A heuristic algorithm for constrained multi-source Weber problem – The variational inequality approach

For solving the well-known multi-source Weber problem (MWP), each iteration of the heuristic alternate location–allocation algorithm consists of a location phase and an allocation phase. The task of the location phase is to solve finitely many single-source Weber problems (SWP), which are reduced by the heuristic of nearest center reclassification for the customers in the previous allocation phase. This paper considers the more general and practical case – the MWP with constraints (CMWP). In particular, a variational inequality approach is contributed to solving the involved constrained SWP (CSWP), and thus a new heuristic algorithm for CMWP is presented. The involved CSWP in the location phases are reformulated into some linear variational inequalities, whose special structures lead to a new projection–contraction (PC) method. Global convergence of the PC method is proved under mild assumptions. The new heuristic algorithm using the PC method in the location phases approaches to the heuristic solution of CMWP efficiently, which is verified by the preliminary numerical results reported in this paper.