Data Processing Theories And Methodologies In Pair-Wise Comparison Decision Matrices

Decision matrices are a list of values in rows and columns that allow decision makers to systematically identify, analyze, and evaluate the performance of qualitative or quantitative relationships between sets of decision elements. Among them, pair-wise comparison matrices(PCMs) are the critical components of multi-criteria decision making methods, especially the analytical hierarchy process(AHP) and analytical network process(ANP) methods. However, the PCMs often involve inconsistency, uncertainty and/or missing data due to unavailable or asymmetric information, prejudice, limited expertise and the complicated decision problems in nature etc, which could result in invalid even wrong decision making. Furthermore, when an(or more) alternative or a criterion is added or removed rank reversal may take place, such that a less preferred alternative becomes more preferred. Therefore, effective methods for data processing in decision matrices become important. This dissertation focuses on these issues and conducted in-depth research, the specific research and innovation results are concluded as follows:In ANP implementation, there exists a large number of PCMs with same orders. In addition, when the number of comparison matrices increases, the inconsistency test of comparison matrices both in the AHP and ANP becomes complicated. Based on consistency ratio(CR) index, a maximum eigenvalue threshold is proposed as a new consistency index to test the consistency of a PCM. The proposed index is mathematically equivalent to the consistency ratio(CR) index, but it simplifies the process of consistency test. If the consistency of a PCM fails, then its inconsistent elements should be identified and adjusted. Currently, a number of methods and models have been proposed to deal with this issue. However, the existing methods and models are dependent on the priority vectors, but there are more than 20 methods available for deriving the priority weights in a PCM. For an inconsistent PCM, priority weights derived from different methods could differ significantly. Therefore, based on the cardinal consistency condition and matrix multiplication theory, an induced bias matrix(IBM) model is proposed to amplify and identify the most inconsistent element in a PCM. The proposed IBM model are then extended to several other models such as arithmetic mean induced bias matrix(AMIBM) model, geometric mean induced bias matrix(GMIBM) model and logarithmic mean induced bias matrix(LMIBM) model,thus several induced bias matrix models that only require the original matrix information are proposed.To deal with the ordinal inconsistency issue in a PCM, a Hadamard product operator is first introduced to build Hadamard product induced bias matrix(HPIBM) model for cardinal inconsistency identification. Then, several possible three-way cycles are analyzed and proposed to build Hadamard product induced bias matrix(HPIBM) model for ordinal inconsistency identification. The total number of three-way cycles that are present in a PCM can be determined by the HPIBM and eliminated by using the HPIBM for cardinal inconsistency model. Therefore, the proposed HPIBM model is capable of simultaneously dealing with both cardinal and ordinal inconsistency issues, and is independent of the methods chosen to derive the priority vectors, which fill in the gap that one model could not deal with both inconsistency simultaneously.In addition to the cardinal and ordinal inconsistency issues, a PCM may be incomplete due to limited expertise or unwillingness to judge. Based on the above proposed IBM models, a “Quasi-complete matrix” is constructed by filling in the missing comparisons with unknown variables. The linear or nonlinear equations or optimization problems can be built by minimizing the bias elements in the induced bias matrix. Thus the optimal solutions of unknown variables can be found by solving the systems of equations or optimization problems. Besides, two case studies are conducted to show the applications of the proposed missing comparisons processing models, i.e. questionnaire design improvement and emergency decision making.For the rank reversal issue, this dissertation focuses on the sensitive analysis of rank reversal when a new alternative is added. Simply speaking, a new row and a column with unknown variables are added to the existing PCM to denote the newly added alternative, then we adapt the induced bias matrix(IBM) model to the generated new PCM, and determine the relationships of unknown variables under the constrains of 9-point scale. The sensitivity of rank reversal can be explored and analyzed by fixing the unknown variables within their plausible interval values. Finally the sensitive reversal points can be recognized when rank reversal will take place. The proposed method is validated by a case study that a Healthcare Device Corporation selects suppliers. The results show that the sensitive reversal points can be derived from the boundary values of the feasible regions of the remaining variables if one variable is fixed. The results show that the final rank reversal depends on whether any of the reversal points derived in each comparison matrix for the suppliers is encountered.Finally, simulation experiments are conducted to validate the effectiveness and correctness of the proposed several IBM models by randomly generating PCM with orders 3 to 15. As examples, in this simulation experiments, only IBM and HPIBM are used to process the random inconsistent matrices. The results show the both models can effectively identify and adjust the most inconsistent elements in a random decision matrix.