A 'Grid' is an infrastructure for resource sharing. It is used for large-scale data processing, many of the applications being scientific ones. The processing of a job on a conventional network takes place on a single computer, whereas the processing of a job on a Grid may be split between several computers. A Grid is, therefore, capable of running applications that are too demanding for a stand-alone computer or a conventional network of computers. The resources making up a Grid need to be managed to provide a good quality of service. Availability is complicated by the fact that a single application may request multiple resource types from multiple locations. Grid resource allocation is regarded as a vital component of a Grid infrastructure. Many methods have been developed for Grid resource allocation. In this paper, we propose a new Grid resource allocation method (Modified Least Cost Method) that assigns tasks to computing nodes in a way that is close to the optimum. The proposed method is an advanced form of the Least Cost Method. In this paper we evaluate Modified Least Cost Method for Grid resource allocation scenario. This paper also proposes a Linear Programming (LP) model for Grid scheduling and includes a comparative performance analysis of our proposed Resource Allocation method with the existing ones. © 2010 IEEE.