Single Facility Goal Location Problems with Symmetric and Asymmetric Penalty Functions

Location theory is an interstice field of optimization and operations research‎. ‎In the classic location models‎, ‎the goal is finding the location of one or more facilities such that some criteria such as transportation cost‎, ‎the sum of distances passed by clients‎, ‎total service time, and cost of servicing are minimized‎. ‎The goal Weber location problem is a special case of location models that have been considered recently by some researchers‎. ‎In this problem, the ideal is locating the facility in the distance $r_i$‎, ‎from the $i$-th client‎. ‎However‎, ‎in most instances‎, ‎the solution to this problem doesn't exist‎. ‎Therefore‎, ‎the minimizing sum of errors is considered‎. ‎In the previous versions of the goal location problem, the penalty functions have been considered by some symmetric functions such as square and absolute errors of distances between clients and ideal point‎. ‎In this paper‎, ‎we consider the asymmetric linex function as the error function‎. ‎We consider the case that the distances are measured by $L_p$ norm‎. ‎Some iterative methods are used to solve the problem and the results are compared with some previously examined methods.