It is known that Pareto dominance has its own weaknesses as the selection criterion in evolutionary multiobjective optimization. Algorithms based on Pareto criterion (PC) can suffer from problems such as slow convergence to the optimal front and inferior performance on problems with many objectives. Non-Pareto criterion (NPC), such as decomposition-based criterion and indicator-based criterion, has already shown promising results in this regard, but its high selection pressure may lead to the algorithm to prefer some specific areas of the problem's Pareto front, especially when the front is highly irregular. In this paper, we propose a bi-criterion evolution (BCE) framework of the PC and NPC, which attempts to make use of their strengths and compensates for each other's weaknesses. The proposed framework consists of two parts: PC evolution and NPC evolution. The two parts work collaboratively, with an abundant exchange of information to facilitate each other's evolution. Specifically, the NPC evolution leads the PC evolution forward and the PC evolution compensates the possible diversity loss of the NPC evolution. The proposed framework keeps the freedom on the implementation of the NPC evolution part, thus making it applicable for any non-Pareto-based algorithm. In the PC evolution, two operations, population maintenance and individual exploration, are presented. The former is to maintain a set of representative nondominated individuals and the latter is to explore some promising areas that are undeveloped (or not well-developed) in the NPC evolution. Experimental results have shown the effectiveness of the proposed framework. The BCE works well on seven groups of 42 test problems with various characteristics, including those in which Pareto-based algorithms or non-Pareto-based algorithms struggle.