Fuzzy FlowSort: An integration of the FlowSort method and Fuzzy Set Theory for decision making on the basis of inaccurate quantitative data
Introduction
Multicriteria Decision Aid (MCDA) provides a set of powerful decision-support tools and methods to address decision problems involving multiple and usually conflicting criteria. Promethee is one of the most commonly used MCDA methods since it is very intuitive and easy to use.
The Promethee family of methods, consists of Promethee I for partial ranking of the alternatives; Promethee II when a complete ranking from the best to the worst alternative is required [5]; Promethee III for ranking based on intervals; Promethee IV for complete or partial ranking of the alternatives when the set of possible solutions is continuous; Promethee V for a problem with a set of constraints [4]; Promethee VI for ranking when the decision maker (DM) is unable to or does not want to define the weights of the criteria precisely; Promethee GDSS for group decision making; Promethee Tri and FlowSort for dealing with a sorting problem; and Promethee Cluster for nominal classification. PROMETHEE methods have been applied in several contexts, such as in portfolio selection [1], [28], maintenance [8] and water management [25].
In some decision problems it is difficult to quantitatively express the evaluation criteria precisely or precise values are deemed to be inadequate to model the criteria in real life. For these situations, the methods of the Promethee family can be extended by using fuzzy theory, thereby providing mathematical methods able to deal with fuzzy data. These new methods bring additional information to the decision making and can avoid the loss of the original (input) information and thus enable the decision-making process to be improved by creating the conditions needed to deal with real world problems more accurately and easily since the DM does not need to determine a single value for evaluating alternatives.
Fuzzy Set Theory and Promethee were first integrated by [19]. Then, [13] proposed the fuzzy extension of the Promethee II method by taking fuzzy inputs and crisp weights into consideration. Geldermann et al. [12] made improvements and used fuzzy preferences, scores and weights. Wei-xiang and Bang-yi [30] extended Promethee II to fuzzy environments for use during the group decision-making process. In their work, generalized fuzzy numbers are used to determine the weights of the importance of different DMs and to evaluate the criteria weights and the alternative with respect to each criterion. Thus, all these extensions of Promethee involve problems that need to be ranked.
For selection problems, [14] proposed two new methods that integrate two different approaches namely Promethee and 2-tuple. These methods are able to deal with both quantitative and qualitative information in an uncertain context.
However, nowadays, most real-life problems can be modeled as sorting problems and therefore, the study of sorting problems is an active research topic in the MCDA area [27], [33].
In this context and in order to address these sorting problems in which it is not possible to make accurate evaluations of the alternatives or it is too difficult to make them, we propose a sorting method called Fuzzy FlowSort (F-FlowSort) that integrates Fuzzy Set Theory. FlowSort was selected to integrate Fuzzy Set Theory among Promethee-based sorting methods in the literature since the FlowSort sorting result obtained is in line with the Promethee II result. Accordingly, if a is preferred to b according to the ranking provided by Promethee II, alternative b will never be assigned to a better category than the alternative a by FlowSort. With the aim of demonstrating the usefulness and applicability of the F-FlowSort method, a numerical example is considered to illustrate the method when the quantitative input data cannot be determined accurately.
Jansen and Nemery [16] also extended the FlowSort sorting method to the case where the input data are imprecise. However, they do not use fuzzy numbers. Instead, in their work, the performances of the alternatives and the reference profiles are defined by means of intervals. On that basis, their approach assumes that all the values of an interval are given the same importance [16]. Moreover, there is a well-defined boundary between those elements that belong to a set.
By contrast, Fuzzy Set Theory, which is based on sets the elements of which have degrees of membership, permits the gradual assessment of the membership of elements in a set. This gradual assessment of the membership means that no sharply boundary exists, and thus, some elements of the universal set can belong to different sets with different membership degrees [24].
Making an analogy, choosing between these two types of extension of FlowSort is the same thing as choosing a type of probability distribution that best represents the variable of interest (uniform, Gaussian, gamma, …), which in our case is the input data. Thus, [16] assume a uniform weight on a bounded interval, while fuzzy numbers are more flexible. They can attribute variable weights on a bounded interval (e.g., triangular) or an unbounded interval (e.g., Gaussian).
Apart from this approach proposed by [16], another multicriteria method worth mentioning that was developed to deal with imprecision in the problem of sorting is the fuzzy extension of the PROMSORT method, called F-PROMSORT [2].
The paper is organized as follows. In Section 2, Promethee I and II, FlowSort, Fuzzy Set Theory and Operations between two LR type triangular fuzzy numbers are reviewed; in Section 3 the F-FlowSort method is described; in Section 4 a numerical example is presented to clarify the F-Flowsort; Section 5 discusses the results obtained with F-FlowSort and compares these with the results provided by FlowSort and by the algorithm proposed in [16] for the same problem. Moreover, Section 5 verifies if there is any correlation between the results obtained with the F-FlowSort and FlowSort methods and Promethee and F-Promethee methods; and finally, Section 6 outlines conclusions and future work.
Section snippets
Background
In this section some basic information about the main topics involved in this paper is given. Thus, Promethee I and II, FlowSort, Fuzzy Set Theory and Operations between two LR type Triangular Fuzzy Numbers are presented.
Fuzzy Flowsort
Several real life problems may be formulated as a sorting problem. For instance, classifying inventory items into ordered categories [26], sorting the suppliers of a company to identify the suppliers that are strategically important and critical to the business [16], classifying process modeling languages as per the purposes of the modeling [7], identifying the most interesting locations to install sustainable data centers [9], classifying activities in project management [21] and classifying
Numerical example
A numerical example is presented in this section to demonstrate the applicability and performance of the F-FlowSort method. This numerical example is based on the numerical application described in [13].
In [13] four alternatives (scenarios) for the exploitation of the low enthalpy geothermal area of Nea Kessani, a rural community located in Greece, are evaluated. The motivation for the study was explained in the paper of [13] as being the growing need to develop the region and to create new
Discussion
The application of the F-FlowSort method to the model with inaccurate quantitative assessments proved to be simple and easy to use.
The results confirmed the potential and the applicability of the fuzzy logic associated with the FlowSort method for processing vague and imprecise data. The introduction of imprecision in the model by evaluating the alternatives provided a new range of information that enabled the decision-making process to be improved by creating conditions for dealing with real
Conclusion
We propose the integration of the FlowSort method with fuzzy numbers for dealing with decision making involving inaccurate data. The FlowSort sorting method based on Promethee was selected since it incorporates the advantages offered by the Promethee including flexibility, ease of computation and application. Moreover, the sorting result obtained with the FlowSort method is in accordance with the Promethee II ranking result. The use of triangular fuzzy numbers was suggested because they are
Acknowledgments
This research study has been supported by CNPq (the Brazilian Research Council) and CAPES (the Brazilian Government Agency that supports Higher Education Personnel seeking to enhance their academic qualifications) for which the authors are grateful.
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