Revisiting inconsistent judgments for incomplete fuzzy linguistic preference relations: Algorithms to identify and rectify ordinal inconsistencies
Introduction
Decision making usually tries to find the best alternative(s) from a set of given alternatives [1]. A decision maker (DM) generally compares a set of alternatives with respect to a single criterion and gives his pairwise comparisons, which are used to constitute a preference relation. There are different preference relation formats, e.g., fuzzy, linguistic, multiplicative, and other extensions.
In the real-life world, there are many decision making situations in which the information cannot be assessed precisely in quantitative form but may be accessed in qualitative form. For example, when evaluating the “comfort” or “design” of a car, terms like “good”, “medium”, “bad” are usually used, and evaluating a car’s speed, terms like “very fast”, “fast”, “slow” can be used instead of numeric values. Fuzzy linguistic preference relations (FLPRs) [2], [3] appeared when it was not easy for experts to offer precise numerical values, and they could only provide qualitative descriptions of every pair of alternatives for the comparison. However, sometimes, a DM may provide a preference relation with some values missing. It may be that an expert does not have enough knowledge for a specific problem, or because the expert does not have the ability to discriminate the degree to which some alternatives are better than others. In these situations, experts are only able to give incomplete preference relations [4], [5], [6], [7], [8] in which some values of the entries are missing or unknown. Over the past few decades, incomplete FLPRs [6], [7], [9] have received more attention.
Studying the consistency of preference relations is an important topic, because inconsistent preferences can lead to wrong decisions being made. Ordinal consistency is related to the transitivity property. Transitivity is a fundamental notion in decision theory. In fact, weak transitivity has been regarded as helpful for a logical and consistent expert to avoid giving self-contradictory judgments. Therefore, it is a minimum condition that a consistent preference relation should have [10]. Weak transitivity is acyclic [11] when ranking alternatives, i.e., if an alternative is better or equivalent to , and is better or equivalent to , then must be better or equivalent to . The weak transitivity assumption can be utilized to check whether a DM’s preferences are ordinally consistent. If a DM provides a preference relation that does not have weak transitivity [12], the priority result for alternatives is questionable [13], [14]. Saaty [15] first announced the consistency issue in AHP. Saaty [15] also developed the notion of acceptable consistency. Later this was extended to the fuzzy reciprocal preference relations [14], [16], [17], [18], incomplete fuzzy preference relations [5], [19] and others [20].
In real decision-making, perfect additive or multiplicative consistency is difficult to achieve, because DM’s judgments rarely conform to an exact mathematical formula. Therefore, the consistency level is measured by a consistency index CI. However, Kwiesielewicz and Van Uden [21] showed that even if a matrix successfully passes a consistency test for the multiplicative positive matrices, it may still have contradictory judgments. Ordinal consistency is the minimum condition necessary for an expert not to give self-contradictory judgments.
Ordinal consistency [22], [23] is almost universally assumed in disciplines of decision theory and generally accepted as a principle of rationality, and ordinal inconsistency has a direct impact on the ranking alternatives. In addition, lack of ordinal consistency often results in inconsistent opinions.
There are some challenges for the ordinal consistency problems of incomplete FLPRs which need to be tackled:
What is the concept of ordinal consistency for incomplete FLPRs?
How can inconsistent judgments in the incomplete FLPRs be identified?
If an incomplete FLPR is inconsistent, how can it be modified to be ordinally consistent and maintain the DM’s original judgments as much as possible?
The aim of the paper is to tackle the above problems. The main studies and innovations of the present paper are:
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The ordinal consistency of FLPRs is revisited. In some cases, if an incomplete FLPR is ordinally consistent, the alternatives can be ranked and the best alternative(s) can be found directly. The proposed method can simplify the computational process, while the other methods [6], [7], [24] firstly estimate the unknown values and then obtain the ranking of the alternatives. These methods are very complicated and require huge computational effort.
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Three algorithms are devised. The first algorithm (Algorithm I) is designed to judge whether an incomplete FLPR is ordinally consistent. If an incomplete FLPR is not ordinally consistent, the algorithm can also identify the inconsistent judgments,as well as all cycles of length 3 to n in the digraph of incomplete FLPR. The proposed method can rectify the ordinally inconsistent incomplete FLPR quickly and find the best alternative(s) accurately. Although Xu, Gupta and Wang [25] proposed an algorithm to search cycles of length m for an incomplete reciprocal preference relation, in this paper, we have developed a new algorithm to search cycles for incomplete FLPRs. It shows that the proposed new algorithm requires less computational complexity. Furthermore, other methods do not judge whether the DM’s preference information is consistent or not. And if the preference relation is ordinally inconsistent, then the results would be unreliable. Specifically, if an FLPR is complete, Algorithm II is devised to identify all the 3-cycles in its digraph.
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Two rules are introduced, and then Algorithm III is developed to rectify the ordinal inconsistencies. It is also shown that this method can not only improve the ordinal consistency of an incomplete FLPR, but also tries to keep the original judgment information as much as possible. We do not change the comparison intensity between different alternatives, we only reverse the direction of the arcs which cause the cycles, and the inconsistent incomplete FLPR is rectified to be ordinally consistent.
The rest paper of the paper is structured in the following way. Section 2 gives the basic concepts of the 2-tuple FLPR and incomplete FLPR. Section 3 proposes the definition of ordinal consistency and adjacency matrix of an incomplete FLPR. Section 4, an algorithm (Algorithm I) is developed to judge the ordinal inconsistency of an incomplete FLPR, which is to find judgments of cycles for various lengths in its incomplete digraph. Specifically, an algorithm (Algorithm II) which identifies the ordinal inconsistency of the complete FLPR is also provided, which only needs to find and locate all the cycles of length 3. An algorithm (Algorithm III) for rectifying the ordinal inconsistency of an incomplete FLPR is developed in Section 5. Section 6 first gives two numerical examples to demonstrate the validity and effectiveness of the proposed method, and then provides the comparison with the existing methods to show the advantages of the proposed method. Finally, Section 7 concludes this paper and points out directions for future research.
Section snippets
Preliminaries
This section introduces some preliminary concepts of FLPRs and the 2-tuple fuzzy linguistic model which is used to carry out the computations.
For simplicity, we denote . Let () be a set of given alternatives, where denotes the ith alternative. And the DM needs to rank the alternatives according to the given preference values. Suppose that is a finite and totally ordered discrete term set, where represents a possible value for a
The definition of ordinal inconsistency of an incomplete FLPR
The ordinal consistency is known as a transitivity condition between three comparison values in complete preference relations. However, for an incomplete FLPR, the ordinal consistency cannot be easily judged only by the transitivity between any three elements as some of the elements might be missing. Therefore, what the ordinal consistency of an incomplete FLPR is, and how to judge its ordinal consistency are the first tasks that need to be solved.
In this section, we first present the
An algorithm to judge the ordinal consistency of an incomplete FLPR
In this section, we will apply graph theory to develop an algorithm (Algorithm I) to judge the ordinal inconsistency of the incomplete FLPR, and find inconsistent entries associated with different length cycles in its incomplete digraph. Specially, we develop another algorithm (Algorithm II) to judge the ordinal consistency of a complete FLPR.
Obviously, each different length cycle can be divided into two different connected links. For example, a 5-cycle () can be divided
A method for rectifying the ordinal inconsistency of an incomplete FLPR
In the former section, we have developed an algorithm to search all various cycles of an incomplete FLPR. When we have found all the cycles, we can decide the illogical judgments that destroy the ordinal consistency of the incomplete FLPR, and finally, to rectify them. Next, we will put forward another algorithm to rectify the ordinal inconsistency of an incomplete FLPR.
Rectification of inconsistent entries can be achieved by removing cycles and revising a minimal number of entries in the
Numerical examples and comparative analysis
In this section, we first give some illustrative examples, and then we give detailed comparisons with the existing methods.
Conclusions
In this paper, we have revisited the ordinal consistency of an incomplete FLPR. Based on the graph theory, the ordinal consistency of an incomplete FLPR can be measured by 3 to n-lengths cycles in its digraph. Therefore, the ordinal consistency problem is transformed to eliminate the different cycles. In order to do this, we have developed three algorithms:
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Algorithm I is devised to judge the ordinal inconsistency of the incomplete FLPR, by finding cycles of various lengths in its incomplete
Acknowledgments
The authors are very grateful to Editor-in-Chief Hamido Fujita and the anonymous reviewers for their constructive comments and suggestions that have helped to improve the quality of this paper. This paper has been revised by the proofreader Rebecca Craig ( [email protected]). This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants (No. 71471056, 71871085), Qing Lan Project of Jiangsu Province.
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