With the advent of cloud computing, computation has become a commodity used by customers to access computing resources with no up-front investment, but as an on-demand and pay-as-you-go basis. Cloud providers make their infrastructure available to public so that anyone can obtain a virtual machine (VM) instance that can be remotely configured and managed. The cloud infrastructure is a large resource pool, allocated to VM instances on demand. In a multi-resource heterogeneous cloud, allocation state of the data center needs to be captured in metrics that can be used by allocation algorithms to make proper assignments of virtual machines to servers. In this paper, we propose two novel metrics reflecting the current state of VM allocation. These metrics can be used by online and offline VM placement algorithms in judging which placement would be better. We also propose multi-dimensional resource allocation heuristic algorithms showing how metrics can be used. We studied the performance of proposed methods and compared them with the methods from the literature. Results show that our metrics perform significantly better than the others and can be used to efficiently place virtual machines with high success rate.