## 1 Introduction
A well-known example is the probability for randomness behavior which is based on a density function, and fuzzy sets for fuzzy problems according to a membership function for ambiguity. The key differences between probabilistic and fuzzy sorting are that precise classifications for the number of alternatives each class has is given by fuzzy sorting and the percentage chances that an alternative is held by each class is given by probability sorting [13]; however, the probability statistic approach requires large sample data quantities to determine the probability density function. By way of illustration, [PERSON] and [PERSON] [14] showed how large data quantities were required for triage as an agile sorting strategy in extreme core arrival scenarios. Obtaining large data quantities can be time-consuming and is often technically difficult to perform. On the other hand, the fuzzy mathematical method depends on experience and cognitive aspects to develop a fuzzy membership function. The quality sorting problem for incoming cores can be subjectively or objectively uncertain depending on the facts. For example, the classification of physical condition based on damage level as determined by visual inspection is subjective, as it can change from one inspector to another. In contrast, the classification of cores according to their usage conditions according to the frequency of use is objective. As a result, there remains a need for an efficient method that can handle quality uncertainty with small data quantities in the sorting problem of incoming cores for remanufacturing. Quality sorting problems with incoming cores may have multiple and conflicting criteria as per the technological, physical, and usage conditions. To overcome the conflicting criteria, a multi-criteria decision-making methodology was constructed using an analytical hierarchy process (AHP). The primary benefit of the use of an AHP is that it can be used for criteria which are quantitative and qualitative, and additionally the capacity to consider the subjective views of decision-makers. In 1982, grey system theory (GST) was first proposed by Professor [PERSON] from Huazhong University of Science and Technology as a quantitative model for limited and incomplete data [15]. Incompleteness and inadequacy of data is a basic feature of uncertain systems [16]. GST has been widely used in various fields for decision-making problems with uncertainty. This paper sets out to investigate the usefulness of grey systems for handling uncertain quality information for sorting incoming cores in a remanufacturing system. The concern with grey system theory is that uncertainty issues arising from limited or missing data are difficult to address with probability theory. In grey systems, grey sets employ the basic concept of grey numbers and deal with the characteristic function values of a set as grey numbers. A grey number is a number that has clear upper and lower limits, but which has an unknown location within the limits. Moreover, grey clustering is a method that may be defined as the branch of grey system theory, which is concerned with the classification of observation indices or observation objects into definable groups using grey incidence matrices or grey possibility functions [17]. Grey clustering evaluation models using possibility functions have been extensively studied for uncertain systems analysis. If a grey number's value data are known to some degree, we may use the possibility function to explain the possibility of the potential values that should be taken up by the grey number. Many recent studies [18; 19; 20] have shown that a method based on grey clustering was helpful for classification problems in remanufacturing operations with uncertain conditions because grey classification has moderate computation complexity. Difficulties arise when an attempt is made to consider multi-level criteria when making decisions regarding grey sorting problems.