By Gang Kou, Daji Ergu, Yi Peng, Yong Shi
The optimistic reciprocal pairwise comparability matrix (PCM) is among the key elements that's used to quantify the qualitative and/or intangible attributes into measurable amounts. This ebook examines six understudied problems with PCM, i.e. consistency attempt, inconsistent information identity and adjustment, information assortment, lacking or doubtful facts estimation, and sensitivity research of rank reversal. the utmost eigenvalue threshold process is proposed because the new consistency index for the AHP/ANP. An prompted bias matrix version (IBMM) is proposed to spot and modify the inconsistent info, and estimate the lacking or doubtful facts. functions of IBMM together with chance evaluate and selection research, job scheduling and source allocation in cloud computing surroundings, are brought to demonstrate the proposed IBMM.
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Additional info for Data Processing for the AHP/ANP
The decision maker may revise his judgments once the deviation elements are identified. 2, the correctness of the IBMM for consistency case is proved mathematically. 2 and the above analysis, in the following, the critical theorem of IBMM for inconsistency identification is proposed and proved mathematically by maximum eigenvalue method and by contradiction method. 3. There must be some inconsistent elements in the induced bias matrix (IBM) C deviating far away from zero if the pairwise matrix is inconsistent.
For instance, suppose cij is such an element in matrix C and the location is i t h row and j t h column. Step II: Identify the potential inconsistent elements by the bias identifying vector. a1j ; a2j ; ; anj /T , where cjT is the transpose vector of column vector cj . 42 3 IBMM for Inconsistent Data Identification and Adjustment in the AHP/ANP Step 4: Calculate the scalar product of the vectors ri and ciT in n dimension. ai1 ; ai2 ; ; ai n / a1j ; a2j ; ; anj D ai1 a1j ; ai2 a2j ; ; ai n anj Step 5: Compute the deviation elements which are far away from aij in vector b by the following formula.
I; j; k D 1; 2; ; n/. Moreover, it can be shown that if A is inconsistent, then for any i there exist j and k such that aij ¤ ai k akj , see Corollary 2 in Boz´oki et al. (2011). Now assume a PCM A is inconsistent, but the i-th row of the induced bias matrix C contains only nonpositive elements. Then aij ¤ ai k akj with some j and k, and ci1 Ä 0;ci 2 Ä 0;: : : ;ci n Ä 0. 26) holds if and only kj aij if akj aaiijk D 1, namely, aij D ai k aj k for all j and k. However, this result contradicts the previous assumption that aij ¤ ai k akj for some j and k.
Data Processing for the AHP/ANP by Gang Kou, Daji Ergu, Yi Peng, Yong Shi