Combining fuzzy AHP with MDS in identifying the preference similarity of alternatives
Introduction
Multidimensional scaling (MDS) analysis is used to provide a visual representation of a complex set of relationships (or the pattern of proximities among a set of objects) that can be scanned at a glance. In general, the purpose of an MDS analysis is to detect meaningful underlying dimensions that allow the researcher to explain observed similarities or dissimilarities (distances) between the investigated objects. According to a measure of similarity or distance based on subjects’ direct assessment that has been computed for all pairs of objects, a map or configuration with located objects is developed. However, not each criterion utilized in developing a multidimensional scaling configuration is equally and precisely weighted in the real world.
Developed by Saaty [12], the analytic hierarchy process (AHP) is a decision analysis method that considers both qualitative and quantitative information and combines them by decomposing ill-structured problems into systematic hierarchies to rank alternatives based on a number of criteria. AHP possesses a number of benefits over other multi-attribute decision methods. First, AHP provides a proven, effective means of dealing with complex decision-making and expediting the decision-making process. Second, AHP provides a useful mechanism for checking the consistency of the evaluation measures, which enables the decision-maker to incorporate subjectivity, experience, and knowledge into the decision process in an intuitive and natural way. Finally, AHP computes the weight for each criterion and the final weighted average score for each alternative. This information gives us insights into the elements of the process, thereby giving the analyst a better understanding of the final decision.
When people encounter uncertain or vague decision-making problems in the real world, they often express their thinking and subjective perception in words instead of probability and statistics. But the problem with words is that their meanings are often vague. Furthermore, even when people use the same words, individual judgment of events is invariably subjective and may differ. Moreover, even if the meaning of a word is well defined (e.g., the linguistic comparison labels in the standard AHP questionnaire responses), when we use the word to define a set, the boundary that separates whether an object does or does not belong to the set is often fuzzy or vague. This is why fuzzy numbers and fuzzy sets have been introduced to characterize linguistic variables. The preferences in AHP are essentially human judgments based on one's perception (this is especially true for intangibles), and we believe the fuzzy approach allows for a more accurate description of the decision making process [8], [9].
The primary focus of this article is to combine fuzzy analytic hierarchy process (FAHP) with MDS to identify the similarities and preferences of alternatives in terms of the axes of the space, which represent perceived attributes and characterize those alternatives. By doing so, the visual dimensionality and configuration or pattern of alternatives whose weighted distance structure best fits the input data can be obtained and explained easily. In sum, FAHP plus MDS provides three advantages for decision-makers: (1) a clear snapshot of alternatives could be easily obtained; (2) the reduced dimensions, after clearly explained and labeled, could be treated as a mental shortcut for decision-makers in the future; (3) distinct alternatives clusters could be obtained easily based on the measure of psychological distances.
The remainder of this paper is organized as follows. The notion of fuzzy weights and synthetic utility values for AHP are introduced in Section 2. The method of combining FAHP with MDS for identifying the fuzzy preference similarity of alternatives in terms of a meaningful visual map is proposed in Section 3. We use a real case of expatriate assignment decision-making to demonstrate that the combined method results in a satisfactory and effective visual map in Section 4. Our conclusions are presented in the last section.
Section snippets
Fuzzy weights and synthetic utility values for analytic hierarchy process
Since analytic hierarchy process (AHP) was introduced by Saaty [11], [12] it has become a popular technique that has been employed to model subjective decision-making processes based on multiple criteria. However, the importance of each criterion is not necessarily equal. To resolve this problem, Saaty [12] uses the eigenvector method to determine the relative importance (weights) among the various criteria based on the pairwise comparison matrix in AHP.
If A = [aij] is a positive reciprocal
Combining FAHP with MDS
For adequate analysis, we often reduce the dimensionality of the dataset in order to achieve a balance between parsimony of understanding and retention of sufficient information. Multidimensional scaling (MDS) is a technique for measuring the distances among psychological stimulus, which are represented as points in geometric space. We focus on the measurement of psychological distance, because evaluators often face subjective/perceptive judgments instead of physical phenomena such as distance
Description of a FMCDM problem
Finding the right people for expatriate assignments and helping them stay there for the duration of their assignments within a globalized organization is a challenging task for international human resources management. Multinational companies need to understand a candidate's preferences, as well as a candidate's perception of the similarity and difference between the home country and the host country, so as to enhance the expatriates’ satisfaction and develop appropriate international staffing
Conclusions
This paper uses fuzzy analytic hierarchy process (FAHP) to determine the weighting of subjective judgments and to derive the performance values of each alternative. Furthermore, MDS analysis is conducted to identify similar groups from distances among alternatives based on fuzzy preferences as perceived by the evaluators to obtain a clear visual dimensional map of a multi-criteria decision-making problem. The major advantage for decision makers is to get a clear snapshot of the alternatives.
Acknowledgement
The authors would like to deeply thank the anonymous referees for their valuable comments and suggestions.
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