Fuzzy TOPSIS

Logic, Approaches, and Case Studies

Mohamed El Alaoui

• Publisher: CRC Press
• Year: 2021
• Print ISBN: 9780367767488
• eBook ISBN: 9781000385793
• Edition: 1
• Topic: Mathematics
• Subtopic: Operations
• Index: Mathematics

This book aims to justify the use of fuzzy logic as a logic and as an uncertainty theory in the decision-making context. It also discusses the development of the TOPSIS method (Technique for Order of Preference by Similarity to Ideal Solution) with related examples and MATLAB codes. This is the first book devoted to TOPSIS and its fuzzy versions.

It presents the use of fuzzy logic as a logic and as an uncertainty theory in the decision-making content and discusses the development of the TOPSIS method in classical and fuzzy context. The book justifies the use of fuzzy logic as an uncertainty theory and provides illustrative examples for each fuzzy TOPSIS extension, along with related MATLAB codes and case studies.

This book is for industrial engineers, operations research engineers, systems engineers, and production engineers working in the areas of decision analysis, multi-criteria decision making, and multiple objective optimization.

1 Uncertainty

Ambiguity and Vagueness

“Who speaks of vagueness should himself be vague.”

(Russell, 1923)

“It seems that the only certain aspect of science is that it is uncertain.”

(Ronen, 1988)

“It is the mark of an educated mind to rest satisfied with the degree of precision which the nature of the subject admits and not to seek exactness where only an approximation is possible.”

(Aristotle)

When introducing me to cooking, my mother asked me to add “a little of X.” After I put in the first ingredient, she said, “You add also a little of Y.” While I was trying to put in approximately the same quantity, she intervened: “Not that much.” Trying to be funny, I said, “So a little of X does not equal a little of Y?”

How much of the second ingredient would be too much? The same question can be asked with respect to the lower limit—the amount below which the ingredient used would be insufficient.

It is not because we do not know the definition of “a little” (small amount), nor because we lack balances to weigh quantities down to the gram or even the milligram; this uncertainty is mainly due to the multiplicity of interpretations that can be linked to the situation and the absence of a distinguishable limit that would allow the unambiguous separation of different possible states. According to Peirce’s 1902 definition of vagueness in logic,

…a proposition is vague when there are states of things concerning which it is intrinsically uncertain whether, had they been contemplated by the speaker, he would have regarded them as excluded or allowed by the proposition. By intrinsically uncertain we mean not uncertain in consequence of any ignorance of the interpreter, but because the speaker’s habits of language were indeterminate; so that one day he would regard the proposition as excluding, another as admitting those states of things. Yet this must be understood to have reference to what might be deduced from a perfect knowledge of his state of mind; for it is precisely because these questions never did, or did not frequently, present themselves that his habit remained indeterminate.

(Baldwin, 1902)

Eubulides formulated the sorites paradox (named from the Greek soros, meaning “heap”), also known as the heap paradox, as early as the fourth century BC. It can be approached in two ways. In the first, everyone can say that a single grain of sand cannot be a heap. Adding a second grain does not make it a heap either. Continuing in the same way by adding a grain at a time, it is obvious that the result will necessarily be a heap, since there seems to be no valid reason to stop at one point rather than another (Black, 1963). The second approach, the decreasing one, starts with a heap of sand and states that after removing a single grain, the result will still be a heap, and after removing a second, and so on. The question that can be deduced is: How many grains are needed to have a heap?

Other variants also exist, such as baldness, length, age, wealth, and adulthood (see Chapter 2), where any line arbitrarily drawn between bald and hairy, long and short, old and young, rich and poor, major and minor will be challenged. Frege sees this vagueness in the definitions of these words as a defect in human language that must be eliminated (Ebert & Rossberg, 2016). According to Williamson, such a concrete language is an unachievable ideal. If we tried to establish it, it would be based on vague stipulations that would taint the end result (Williamson, 1994). One can simply notice that the phrase “a pile minus a grain is always a pile” involves a qualitative notion (“pile”) and a quantitative one (“grain”), whose subtraction is either contradictory or nonmodelable (Borislav, 2018).

From Peirce’s point of view, language is and will remain vague. However, this is only harmful if it leaves the borders of the problem to be dealt with too wide. The same vision is shared by Russell (1923), who adds that even though the logical connectors “OR” and “AND” are certainly less vague than other terms of everyday language, they cannot deviate from the rule when they are jointly employed with other vague elements. Russell even attacks scientific units such as the second and the meter. Relevantly, the 26th General Conference on Weights and Measures, held November 13–16, 2018, redefined four of the seven basic units of the International System of Units, knowing that the meter is currently based on Planck’s constant, which is measured by two methods—one using a watt balance (also called a Kibble balance) and one involving the Avogadro constant—have provided slightly different results. (It should be noted that this small difference can only affect research, especially in nanotechnology.)

Russell also points to the discrepancy between words and their meanings. As long some words may have several meanings, there is ambiguity in the overall meaning of a statement. While admitting the impossibility of defining a language devoid of ambiguity, Russell advocates the use of semantics in which each word or couple of words has a unique meaning. There are certainly bald people, there are certainly hairy people, and there are people whom you cannot call either. The principle of the excluded middle cannot be applied for vague terms: the words are vague and that will not change. However, words derive their sense from context. Thus, the juxtaposition of vague words can lead to a meaning that is less vague. Russell (1923) concludes, “Vagueness, clearly, is a question of degree, depending on the extent of the possible disparities between the different systems evoked by the same representation. Precision, on the other hand, is an ideal limit.”

Uncertainty, which is the generic term (Raskin & Taylor, 2014), can be divided into two main branches (Klir, 1987):

• Ambiguity means the existence of several possible alternatives which are left unspecified. Zhang (1998) points out that an expression is ambiguous if it has several paraphrases that are not paraphrases of one another.

• Vagueness refers to situations where clear boundaries cannot be defined.

While fuzzy sets represent a basic mathematical framework for dealing with vagueness, fuzzy measures are a general framework for dealing with ambiguity (Klir, 1987). A similar classification of fuzzy linear programming has been proposed by Inuiguchi and Ramı́k (2000).

Before showing the need for fuzzy logic, do we need to model uncertainty? In trying to create a new collaboration with a third-party research team, the obvious question is: What more can everyone bring? That is why I proposed modeling uncertainty based on fuzzy logic. While I expected to justify the choice of fuzzy logic rather than a probabilistic model, the answer was formal: there is no uncertainty in our model. That is why I asked myself, is there an uncertainty free model?

Uncertainty in Exact Sciences

Avoiding philosophical quarrels on the subject (Sanford, 1995), uncertainty is everywhere: chemistry (Tchougréeff, 2016), economics (Köhn, 2017), even where you do not expect it (Deemter, 2010). While its existence may go unnoticed in management (Nicolai & Dautwiz, 2010) and music (Venrooij & Schmutz, 2018), it can pose ethical problems in justice (Braun, Schickl, & Dabrock, 2018; Keil & Poscher, 2016). In a world where one has the illusion of mastering certain concepts to the last comma, people assume—without proof—that exact sciences, such as mathematics, would not be tainted with uncertainty (Hoyt, 1941). However, the definition of exact science is itself vague.

Far from the noble structure the term “exact science” may imply, it is not based on any universally valid principle (Planck, 1949); since all sciences are based on a number of axioms, which are by definition accepted without demonstration. Can we build science without presuppositions? According to Planck, scientific thought must relate to something. However, no uniformly acceptable worldview has been defined. Hence, it is impossible to place an exact science in a fixed and inclusive content (Planck, 1949). An extensive discussion on the meaning of presuppositions in science is given by Kattsoff (1957).

The term “exact science” has excessively been linked to mathematics (Grant, 2007; Neugebauer, 1969), what pushes the disciples of many fields of thought, including literary ones, try to make their respective specialties as mathematical as possible so that the results obtained can be considered reliable. Working in an inexact discipline can seem pejorative, whereas the appearance of exactitude is perceived as a form of nobility, as in economics (Nemchinov, 1962), biology (Smart, 1959), and meteorology (Sutton, 1954). According to Dompere (2013), the characteristics of an exact science are:

• Exact measurements and quantities

• Observation, experimental testing, and verification

• Sensory data as a material entry into the exact reasoning

• Reliable logical laws, mathematical representation, and mathematics based on exact logic

• Axioms ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface
• Author
• Introduction
• Chapter 1 Uncertainty
• Chapter 2 Nonclassical Logics
• Chapter 3 Fuzzy Logic
• Chapter 4 Frequently Used Multicriteria Decision-Making Methods
• Chapter 5 TOPSIS Methodology and Limits
• Chapter 6 Fuzzy TOPSIS
• Chapter 7 Intuitionistic Fuzzy TOPSIS
• Chapter 8 Other Fuzzy TOPSIS Approaches
• Index