A smart grid system can be considered as a multi-layered network with power network in one layer and communication network in the other. The entities in both the layers exhibit complex intra-and-interdependencies between them. A reliable decision making by the smart grid operator is contingent upon correct analysis of such dependencies between its entities and also on accurate identification of the most critical entities in the system. The Modified Implicative Interdependency Model (MIIM)  successfully captures such dependencies using multi-valued Boolean Logic based equations called Interdependency Relations (IDRs) after most of the existing models made failed attempts in doing that. In this paper, for any given integer K, this model is used to identify the K-most vulnerable entities in a smart grid, failure of which can maximize the network damage. Owing to the problem being NP complete, an Integer Linear Programming (ILP) based solution is given here. Validation of the model  and the results of the ILP based solution is done by simulating a smart grid system of IEEE 14-Bus using MATPOWER and Java Network Simulator (JNS). Simulation results prove that not only the model MIIM  is correct but also it can predict the network damage for failure of K-most vulnerable entities more accurately than its predecessor Implicative Interdependency Model (IIM) .