Supply chain management begins with supplier evaluation and selection. The supplier selection deals with various criteria with different contexts which makes it a complex multi-criteria decision-making (MCDM) method. In this paper, a novel MCDM method, called the alternative ranking process by alternatives’ stability scores (ARPASS), is proposed to solve supplier selection problems. ARPASS considers each alternative as a system that is constructed on integrated components. To perform properly, a system requires high integrity and stability. ARPASS utilizes the stability of alternatives as an effective element for ranking the alternatives. The ARPASS is developed in two forms, ARPASS and ARPASS*. The new method utilizes standard deviations and Shannon’s entropy to compute the alternatives’ stabilities. In this paper, in addition to the new MCDM methods, a new method called the grey equilibrium product (GEP) is introduced to convert grey linguistic variables into crisp values, using decision makers’ subjective perceptions and judgments. To highlight and validate the novel methods’ performance, they are applied to two sustainable supplier selection problems. For evaluation of the reliability of ARPASS and ARPASS*, their results were compared with the results of the popular MCDM methods. We compared the methods in terms of calculation time, simplicity, transparency, and information type.