Grid computing is now in widespread use, integrating geographical computing resources across multiple virtual organizations to achieve high performance computing. A single grid does not often provide a vast resource because virtual organizations have inadequate computing resource restrictions for management on an organizational scale. This paper presents a new grid architecture named Multi-Grid, which integrates multiple computational grids from different virtual organizations. This study builds a resource broker on multiple grid environments, integrating a number of single grids from different virtual organizations without the limit of organizations. The purpose of the multiple-grid resource is to avoid wasting resources. In addition, this study proposes a Multi-Grid Resource Selection Strategy (MRGSS) for the resource broker to better allocate resources before submitting jobs, to avoid network congestion that consequently causes a decrease in performance.