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In Grid Computing, harnessing the power of idle resources in a distributed environment is one of the important features. However, to fully benefit from this computing model, an appropriated resource allocation method needs to be carefully chosen and deployed. A number of studies have been done on this area and one of the promising approaches is to adopt a marketing scheme called the auction model, which has been drawing much attention during past several years. In this paper, we propose a new utility-aware resource allocation protocol to make external scheduling decision in Grid. Users and service providers specify one or more weight values, and then, an auctioneer uses these values for calculating both userspsila and service providerspsila utility values which reflect preference upon the matched members in the different group. Then, we map the scheduling problem with these utility values into the problem in a weighted bipartite graph, and propose a new matching algorithm based on the existing SMP (Stable Marriage Problem) matching algorithm. Finally, the performance of this auctionpsilas awarding technique is evaluated.