Given a set P of N objects in a region Q in 2-dimensional space and a rectangle r, the Minimizing Range Sum(MinRS) problem is to find an optimal location of the rectangle r that minimizes the sum of weights of objects covered by r. Each object corresponds to a non-negative weighted point and the size of a rectangle r is fixed. There already exists an in-memory algorithm which is not enough to adapt to a scalable environment. In this paper, we show that the MinRS problem returns a solution in the lower bound in terms of I/O costs. We also propose the All Minimizing Range Sum(AllMinRS) problem, which is to find all optimal locations. We prove how many optimal locations can exist at most in a given region Q, which implies the I/O complexity of the AllMinRS problem.