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Top1. Introduction
Multi-criteria decision making (MCDM) is considered as an essential part of modern decision science and operational research. It is the process of finding the best compromise among the feasible alternatives. It provides a wide variety of methodologies and techniques that enable the systematic treatment of decision problems under multiple criteria. The MCDA methods can be applied to four different kinds of analyses that can be performed in order to provide significant support to decision-makers (Remadi and Frikha, 2019). These are: (1) the choice of the best alternative, (2) the ranking of the set of the alternatives from the best to the worst, (3) the description of the features of the alternatives and (4) the classification of the alternatives into predefined homogenous groups.
In this paper, we studied the ordinal classification problem, also called sorting problem. It consists to orienting the decision problem to an assignment of alternatives to one of the predefined, ordered and homogenous categories or classes. The class is a set of alternatives with similar properties or even values for the same properties, when compared to the alternatives of the other classes. Multiple methods have been proposed during the previous decades. Among these, we can mention the well-known sorting methods, the ELECTRE-TRI (Shen et al. 2016), the THESEUS (Fernandez and Avarro, 2011), etc. Relying on the PROMETHEE (Brans et al. 1984) methodology, several authors proposed the PROMETHEE-TRI (Figueira et al. 2004), the PROMSORT (Araz and Ozkarahan, 2007) and the FlowSort (Nemery and Lamboray, 2007). In fact, the PROMETHEE is one of the most famous MCDM methods since it is easy to use, simple to process and uses less parameter than the other MCDM methods like ELECTRE (Govindan and Jepsen, 2016). Figueira et al. were pioneers in the PROMETHEE-TRI (Figueira et al. 2004) method, extending it to the sorting context, but it rather used incompletely ordered categories. In 2007, Araz & Ozkarahan proposed the PROMSORT (Araz and Ozkarahan, 2007) method which uses completely ordered categories but the assignment of the alternatives was not independent.
Developed by Nemery and Lamboray in 2007, FlowSort (Nemery and Lamboray, 2007) was proposed for assigning actions to completely ordered categories defined by limiting profiles or central profiles. It solves the drawbacks of PROMETHEE-TRI (Figueira et al. 2004) and PROMSORT (Araz and Ozkarahan 2007) and treats the problematic sorting issue for independent assignments and completely ordered categories. The alternatives and preference parameters evaluation of the FlowSort method are defined as crisp values. But, in a real-world situation, decisional problems are multidimensional and ambiguous in nature. So, it is difficult to express the evaluation criteria precisely. Multiple extensions of the FlowSort method have been developed to solve these problems. Indeed, Janssen and Nemery (2012) proposed an extension of the FlowSort sorting method to the case of input data imprecision. Moreover, Campos et al. (2015) extended FlowSort method to introduce a fuzzy sorting method called Fuzzy FlowSort (F-FlowSort). For a simplified FlowSort version, Assche and De Smet (2016) found the parameters of a sorting model using classification examples in the context of traditional sorting and interval sorting. Moreover, Pelissari et al. (2019) suggested a new multicriteria method SMAA-Fuzzy-FlowSort for sorting problems under uncertainty through applying the Stochastic Acceptability Analysis to the Fuzzy-FlowSort method.