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Data Inconsistency and Incompleteness Processing Model in Decision Matrix

Daji ERGU1, Gang KOU2
1 School of Management and Economics,
University of Electronic Science and Technology of China,
Chengdu, 611731, China
2 School of Business Administration,
Southwest University of Finance and Economics of China,
Chengdu, 610074, China

Abstract: Data Inconsistency and incompleteness issues of pairwise comparison matrix (PCM) are hot research topics in multi-criteria decision making (MCDM). The goal of this paper is to propose a simple approach to identify and adjust the inconsistent data while estimate the missing data in a PCM. Specifically, an arithmetic mean matrix is induced to identify the most inconsistent data efficiently while preserving most of the original information in a PCM, and then we adapt it to estimate the missing data in an incomplete PCM. The proposed model is only dependent on the data in the original matrix, and can effectively process the most inconsistent or missing data in a PCM. The correctness of the proposed method is proved mathematically. Two numerical examples are used to illustrate the proposed method. The result shows that the proposed method is accurate and efficient when processing the inconsistent or missing data to satisfy the consistency requirements of PCM.

Keywords: Inconsistency data, Missing data, Pairwise comparison matrix, Arithmetic mean induced bias matrix.

>>Full text
Daji ERGU, Gang KOU, Data Inconsistency and Incompleteness Processing Model in Decision Matrix, Studies in Informatics and Control, ISSN 1220-1766, vol. 22 (4), pp. 359-368, 2013.

  1. Introduction

Over the past few decades, the pairwise comparison technique, originally proposed by Thurstone [1], is extensively used to deal with tangible and intangible criteria in multi-criteria decision making (MCDM) methods, especially in the Analytical Hierarchy Process (AHP) and the Analytical Network Process (ANP) [2-9]. All results of n numbers of being compared criteria or alternatives are arranged in a comparison matrix A=(aij)nxn, where aij>0, aij =1/aji and popularly termed pairwise comparison matrix (PCM hereinafter) in literature. The PCM is built to assign criteria weights or scores of alternatives, and is composed of data expressed on a numerical scale (e.g. Saaty’s fundamental 9-point scale) and given by decision makers or surveyed experts based on their experiences and expertise. A PCM is said to be perfectly consistent if the expression aij=aikakj holds for all i, j and k. However, as the surveyed experts are often biased in their subjective comparisons, a PCM is usually difficult to satisfy the perfectly consistency condition, indicating that the inconsistent comparisons of preference judgment may exist in a PCM. Therefore, the inconsistency issue in a PCM has been widely studied, and a number of approaches and models are proposed and developed [10-15]. Currently, the consistency ratio (CR) proposed by Saaty [16] is widely used to test the consistency of a PCM.

If CR<0.1, then the PCM is said to be of acceptable consistency, otherwise, the inconsistent entries should be revised.

In addition to the inconsistent issue, a PCM could also be incomplete due to the large number of criteria being compared (or alternatives), time pressure, lack of the expertise or incomplete information as well as the complexity nature of the decision problem [17]. Therefore, the issue of processing missing data in a PCM has been another hot research topic in the study of multi-criteria decision making (MCDM), and many models are proposed to handle this issue [18-20]

To identify the inconsistent elements simply and accurately while preserving most of the original comparison information in the PCM, an induced bias matrix (IBM) model, which is only based on the original comparison matrix, is proposed in [15]. Ergu and Kou [17] extended the IBM model to process the missing data in a PCM. In this paper, we borrow the concept of the IBM model and propose an arithmetic mean induced bias matrix (AMIBM) model to identify and adjust the most inconsistent elements. Different from the IBM model proposed in [15] and [17], the most inconsistent data can easily be identified by observing the largest negative entry in the AMIBM C. Again, we also extend the proposed AMIBM to process the missing data in a PCM, and the missing data are estimated by optimization method.

The rest of this paper is organized as follows. In Section 2, the theorems and corollaries of AMIBM model for inconsistency is proposed and proved mathematically.

The theorem of AMIBM is extended to estimate the missing data. The processes of inconsistency identification and the procedures of missing data estimation are further proposed in this section. Two numerical examples are used to illustrate the proposed method for inconsistency identification and missing comparisons estimation in Section 3. A brief conclusion is presented in Section 4.


  1. THURSTONE, L., Law of Comparative Judgment, Psychological Review, vol. 34, no. 4, 1927, pp. 273-273.
  2. SAATY, T. L., Axiomatic Foundation of the Analytic Hierarchy Process, Management Science, vol. 32, No. 7, 1986, pp. 841-855.
  3. SAATY, T.L., How to Make a Decision: The Analytic Hierarchy Process, European Journal of Operational Research, vol. 48, no. 1, 1990, pp. 9-26.
  4. SAATY, T. L., Some Mathematical Concepts of the Analytic Hierarchy Process, Behaviormetrika, vol. 29, 1991, pp. 1-9.
  5. SAATY, T. L., How to Make a Decision: The Analytic Hierarchy Process, Interfaces, vol. 24, no. 6, 1994, pp. 19-43.
  6. ERGU, D., G. KOU, Y. PENG, Y. SHI, Y. SHI, The Analytic Hierarchy Process: Task Scheduling and Resource Allocation in Cloud Computing Environment, DOI: 10.1007/s11227-011-0625-1, The Journal of Supercomputing, 2011.
  7. PENG, Y., G. KOU, G. WANG, Y. SHI, FAMCDM: A Fusion Approach of MCDM Methods to Rank Multiclass Classification Algorithms, Omega, vol. 39, no. 6, 2011, pp. 677-689.
  8. PENG, Y., G. KOU, G. WANG, W. WU, Y. SHI, Ensemble of Software Defect Predictors: An AHP-based Evaluation Method, International Journal of Information Technology & Decision Making, vol. 10, no. 1, 2011, pp. 187-206.
  9. ERGU, D., G. KOU, Y. PENG, Y. SHI, Y. SHI, Analytic Network Process in Risk Assessment and Decision Analysis, Computers & Operations Research, vol. 42, 2014, pp. 58-74.
  10. XU, Z. S., C. P. WEI, A Consistency Improving Method in the Analytic Hierarchy Process, European Journal of Operational Research, vol. 116, no. 2, 1999, pp. 443-449.
  11. ISHIZAKA, A., M. LUSTI, An Expert Module to Improve the Consistency of AHP Matrices, International Transactions in Operational Research, vol. 11, no. 1, 2004, pp. 97-105.
  12. LI, H., L. MA, Detecting and Adjusting Ordinal and Cardinal Inconsistencies through a Graphical and Optimal Approach in AHP Models, Computers & Operations Research, vol. 34, no. 3, 2007, pp. 780-798.
  13. CAO, D., L. C. LEUNG, J. S. LAW, Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach, Decision Support Systems, vol. 44, no. 4, 2008, pp. 944-953.
  14. IIDA, Y., Ordinality Consistency Test About Items And Notation Of A Pairwise Comparison Matrix in AHP, Proceedings of the International Symposium on the Analytic Hierarchy Process, 2009.
  15. ERGU, D., G. KOU, Y. PENG, Y. SHI, A Simple Method to Improve the Consistency Ratio of the Pair-wise Comparison Matrix in ANP, European Journal of Operational Research, vol. 213, no. 1, 2011, pp. 246-259.
  16. SAATY, T. L., The Analytical Hierarchy Process, New York: McGraw-Hill, 1980.
  17. ERGU, D., G. KOU, Questionnaire Design Improvement and Missing Item Scores Estimation for Rapid and Efficient Decision Making, Annals of Operations Research, 2011, DOI: 10.1007/s10479-011-0922-3
  18. FEDRIZZI, M., S. GIOVE, Incomplete Pairwise Comparison and Consistency Optimization, European Journal of Operational Research, vol. 183, no. 1, 2007, pp. 303-313.
  19. GOMEZ-RUIZ, J. A., M. KARANIK, J. I. PELÁEZ, Estimation of Missing Judgments in AHP Pairwise Matrices Using a Neural Network-based Model, Applied Mathematics and Computation, vol. 216, 2010, pp. 2959-2975.
  20. BOZÓKI, S., J. FÜLÖP, L. RÓNYAI, On Optimal Completion of Incomplete Pairwise Comparison Matrices, Mathematical and Computer Modelling, vol. 52, no. 1-2, 2010, pp. 318-333.
  21. HARKER, P. T., Alternative Modes of Questioning in the Analytic Hierarchy Process, Mathematical Modelling, vol. 9, no. 3-4, 1987, pp. 335-360.
  22. SIRAJ, S., Preference Elicitation from Pairwise Comparisons in Multi-criteria Decision Making, Doctoral dissertation, University of Manchester, 2011.