Determination of weight vector by using a pairwise comparison matrix based on DEA and Shannon entropy

Authors
Hooshyar Azad
Ali Asghar Foroughi
Abstract
Introduction

Analytic hierarchy process (AHP) is a method of multiple criteria decision making (MCDM) that is used to select an alternative from a set of alternatives or to rank a set of alternatives, while data envelopment analysis (DEA) is a nonparametric method that is used based on linear programming to evaluate the performance of decision making units (DMUs) that have multiple inputs and multiple outputs. The relation between methods of MCDM and DEA is a topic of interest to researchers in this part of MCDM, e.g., one of the first works done in this field is the relation between data envelopment analysis and multiple objective linear programming by Golany. Ramanathan proposed a method (DEAHP method) based on DEA for weight generation in the AHP that his method had three main drawbacks: (1) producing irrational weights for inconsistent pairwise comparison matrices; (2) non-use all the information of the inconsistent pairwise comparison matrix; and (3) insensitivity to changing elements in some matrices of pairwise comparison. To solve the problems of DEAHP method, several methods were proposed that each one produces a weight vector in the AHP, e.g., we can mentioned to data envelopment analysis method of wang and chin (DEA method) and data envelopment analysis method with assurance region of wang and et al. (DEA/AR method). In this paper, we propose a new method, which is called E-DEAHP method for short, based on DEA and Shannon entropy, a concept used in information theory, to produce a weight vector in the AHP that does not have the problems of DEAHP method and is different from the mentioned methods.



Material and methods

In this approach, each row of the pairwise comparison matrix is considered as a decision making unit (DMU), so that in the normalized pairwise comparison matrix the arithmetic mean of the ith row and the entropy of ith column is considered as, respectively, output and input of the ith DMU and then with employed data envelopment analysis, we find the local weight vector of the elements (decision criteria or alternatives). Also, to aggregate the obtained local weights, we use the simple additive weighting (SAW) method in multiple criteria decision making.

Results and discussion

It is proved that if a pairwise comparison matrix is perfectly consistent, the entropy of all its columns are the same, so in this case all decision making units will have the same input and the method will produce true weight vector.

The results of the examined numerical examples show that the proposed method of this paper produces perfectly rational weights in comparison with the results of the methods known in the subject literature and can estimate a robust priority (weight) vector for a pairwise comparison matrix. Also, the results of the hierarchical problem survey show that the weights obtained from the method and their aggregation to obtain the global weight vector confirm the potential validity of the method.

Conclusion

In this paper, in relation to E-DEAHP method, we have achieved the following conclusions.

Generating true weight vector for perfectly consistent pairwise comparison matrices.
The method for ranking and selecting alternatives has a high resolution.
The weight vector obtained from this method is robust, In other words, it is not affected by possible errors, unusual and false observations (UFO) that appear because of inaccurate data entry random errors, in the pairwise comparison matrix.
In practice, the E-DEAHP method can be applied without the need to solve linear programming by using a simple relative relation.
Keywords
Multiple criteria decision making
Data envelopment analysis
Analytic hierarchy process
Shannon entropy
Pairwise comparison matrix
Robust estimation
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