To comprehensively assess the operational performance of an integrated energy system, this paper introduces a multi-objective optimal operational method that takes into account the variable efficiency of energy conversion devices. First, the operational characteristic curves of energy conversion devices with variable efficiency are segmented and linearized. Second, a multi-objective optimal scheduling model is established considering the economic costs, carbon emissions and comprehensive energy utilization efficiency. The non-convex nonlinear problem with comprehensive energy utilization efficiency is solved by the Charnes-Cooper transformation. Finally, to solve the multi-objective model, the normalized normal constraint method is employed to generate a Pareto optimal set with uniform distribution. Following this, the improved entropy weight double-base point method is used to identify the compromise optimal solution. The modified IEEE 33-bus power distribution network and 8-bus heating network coupled integrated energy system are simulated and analyzed. The results show that the optimized model and algorithm are capable of obtaining an operational scheme that considers economy, low-carbon, and high efficiency of an integrated energy system. This offers decision makers scientific references for decision-making.