E. Mu and M. Pereyra-Rojas,Practical Decision Making,SpringerBriefs in Operations Research, DOI 10.1007/978-3-319-33861-3107
Using the scale from Fig.B.2we will ask questions such as: With respect to thepurpose of this decision, which is more important criterion“C3”or“C2”? If weconsider thatC3is moderately more important thanC2we are mathematicallystatingC3/C2= 3 (using the scale from Fig.B.3). Notice that this judgment auto-matically implies that the comparison ofC2withC3will yield the ratioC2/C3= 1/3.This constitutes the reciprocity rule that can be expressed mathematically asCij= 1/Cjiwhere i and j are any element (i corresponds to the row and j refers to thecolumn) in the comparison matrix.Fig. B.1Basic AHP model exampleRelative IntensityImportanceExplanation1EqualBoth criteria are equally important3ModeratelyOne criterion is moderately more important than the other5StrongOne criterion is strongly more important than the other7Very StrongOne criterion is very strongly more important than the other9ExtremeOne criterion is extremely more important than the other2,4,6,8IntermediateValuesCompromise is neededFig. B.2Intensity scale for criteria pairwise comparison108Appendix B: AHP Basic Theory
These judgments are recorded in a comparison matrix as shown in Fig.B.3.Notice that the judgment diagonal, given that the importance of a criterion com-pared with itself (Cij/Cij), will always be equal and is 1 in the comparison matrix.Also, only the comparisons thatfill in the upper part of the matrix (shaded area) areneeded. The judgments in the lower part of the comparison matrix are the recip-rocals of the values in the upper part, as shown in Fig.B.3.Another important consideration when completing the comparison matrix is theextent to which it respects the transitivity rule. If the importance ofC1/C2= 1/5,andtheimportanceofC2/C3= 1/3,thenitisexpectedthatC1/C3= (1/5)±(1/3) = 1/15. In other words,Cij=Cik±CkjwhereCijis the com-parison of criteria i and j. However, this is not the case in Fig.B.3whereC1/C3= 1as indicated by the decision-maker. This means there is some inconsistency in thismatrix of judgment as will be explained next.Checking Consistency of JudgmentsAny comparison matrix that fulfills the reciprocity and transitivity rules is said to beconsistent. The reciprocity rule is relatively easy to respect, whenever you elicit thejudgmentCijyou make a point of recording the judgmentCjias the reciprocal valuein the comparison. However, it is much harder to comply with the transitivity rulebecause of the use of English language verbal comparisons from Fig.B.2such as“strongly more important than,” “very strongly more important than,” “extremelymore important than,”and so forth.
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- Fall '18
- Zia Ul Rehman
- Decision making software, comparison matrix, AHP, Analytic Hierarchy Process, AHP Basic Theory