On the analytic hierarchy process and decision support based on fuzzy-linguistic preference structures
Introduction
Decision support methodologies need to take into account the different aspects or dimensions that are relevant for representing and solving common real-life problems, where a given set of alternatives, criteria or some objects of interest have to be evaluated, aggregated and exploited in order to identify priorities and arrive at the most attractive solutions. A popular methodology for establishing such priorities is Saaty’s Analytic Hierarchy Process (AHP) [30], [34], [35]. This technique allows aggregating expert preference judgments made over pairs of objects, which are gathered under the form of a comparison matrix. Traditionally, the AHP elicitation of preferences is based on a valuation scale where one linguistic label agrees with a crisp value or precise number, while the fuzzy AHP allows general type of evaluations taking the form of fuzzy sets.
Two traditional general lines of research can be found in literature concerning the fuzzy methodology for the AHP (see e.g. [28], [45], [46]). On the one hand, the Extent Analysis Method (EAM) was introduced in 1996 [4], where crisp weights are obtained from the fuzzy comparison matrix. On the other hand, the logarithmic least squares method (LLSM) was firstly proposed in 1983 [20], being later extended and modified (as in [1], [45]), estimating fuzzy weights from the respective fuzzy judgments. For a detailed overview on the number of methods found in literature for handling fuzzy comparison matrices based on the AHP, see [45], [46].
Different applications for the traditional fuzzy AHP exist, evaluating expert opinions for arriving at decisions that take into account the natural complexity and uncertainty of real-world problems. Many examples on the application of the EAM can be found in literature (see e.g. [3], [12], [17], [18], [36], [41]), while fewer examples can be found for the LLSM approach (see e.g. [19], [43]). Despite its popularity, the EAM has been subject of various criticisms showing that its misapplication may lead to wrong decisions [46], [50], [51], on the contrary to the modified LLSM-AHP model and its constrained nonlinear optimization model [45] which show better generalization results.
On the other hand, some criticism has been given to the unquestioned fuzzification of the AHP [33]. Here, it is acknowledged that the AHP, as it has been originally proposed [30], makes use of a linguistic valuation scale that enables the use of precise numbers to handle linguistic terms for comparing pairs of objects (as it has been pointed out in [11], the selection of a particular scale is an open problem). For example [24], [31], [32], consider the traditional valuation scale with crisp numbers from 1 to 9, where the numbers {1, 3, 5, 7, 9} respectively agree with the predicates “not more dominant”, “moderately more dominant”, “strongly more dominant”, “very strongly more dominant” and “extremely more dominant”, while the numbers in between express compromise between those terms. Hence, the words and predicates that experts use for making their evaluations are understood by means of a one-to-one correspondence with crisp numbers, although it can generally be accepted that linguistic characterizations are less precise than numerical ones.
The justification for using fuzzy logic jointly with the AHP is grounded on the use of linguistic assessments for comparing and valuing the relationship between pairs of objects (see e.g. [5]). In this sense, based on the Computing with Words paradigm [21], [29], [48], [49], fuzzy logic allows examining the way in which computing with words can be developed (see e.g. [8], [23], [47]), while maintaining the numerical approach of the AHP [30]. Therefore, the objective of this paper is to examine the AHP and the fuzzy LLSM-AHP under a general framework for handling linguistic evaluations, which are represented by fuzzy sets (or even representing more complexity, from both a linguistic and computational viewpoint, by interval or type-2 fuzzy sets [2], [10], [39], [40]). In this way, experts can express their evaluations by means of words and natural language, based on our proposal for fuzzy-linguistic preference structures, so a priority order can be assigned on the set of objects according to their estimated fuzzy weights.
In order to do so, this paper is organized as follows. In Section 2, we review the original AHP and the fuzzy LLSM-AHP proposals. In Section 3, linguistic-preference structures are used to represent experts’ evaluations under the AHP approach, focusing on the design of the linguistic values by means of fuzzy sets. Section 4 introduces the proposal for decision support based on the linguistic-fuzzy AHP algorithm, and in Section 5, an ordinal ranking procedure for decision support is introduced based on the estimated fuzzy weights. We conclude with some final comments for future research.
Section snippets
AHP and fuzzy LLSM-AHP overview
The AHP offers a solid numerical methodology for decision support, where objects, such as criteria or alternatives, are compared in a pairwise manner for estimating their importance through a vector of decision weights [30], [34]. In the following, we review the AHP original proposal and the fuzzy LLSM approach to the AHP.
Computing with linguistic knowledge
Computing with linguistic data presents great challenges on how to deal with human experts’ evaluations for decision making [21]. In order to value such information under a general and appropriate framework, words have to refer to a structure that allows organizing and giving meaning to them. Here, with the objective of examining the way in human experts manage language for making preference judgments, we explore preference structures according to the decomposition of the preference predicate
The fuzzy-linguistic AHP
Based on the ordinal interval framework presented above for representing expert’s fuzzy judgments, the fuzzy-linguistic AHP can be formulated in the following way. Consider the comparison matrix,whose elements are given byHere, represents the fuzzy-linguistic terms with which experts value the degree of preference between criteria, such that for every
Ranking procedure for the fuzzy-linguistic AHP
Once the fuzzy weights are estimated, objects/criteria have to be ranked taking into account the linguistic character of information. That is, recalling that the classical AHP methodology offers crisp estimations, priorities between objects can be directly assigned, such that two objects are considered to be equivalent only if their weights are equal. Now, fuzzy weights offer different information both from a qualitative and quantitative view, as it is explored below.
Final comments
The fuzzy-linguistic model for the AHP has been presented as a general framework for decision support, where words are represented according to the appropriate design of their meaning. This design process uses specific building blocks which allow obtaining fuzzy-linguistic preference structures for understanding and exploiting experts’ knowledge, as expressed by words and predicates. This approach remains to be further explored, examining different prototypes/characterizations for LM and
Acknowledgements
Financial support from the Center for research in the Foundations of Electronic Markets (CFEM), funded by the Danish Council for Strategic Research is gratefully acknowledged.
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