چکیده:
The Analytic Hierarchy Process (AHP) is one of the valid and powerful multi-criteria decision-making techniques. In this article, a mathematical approach to AHP is discussed. In 1997, a researcher named Jonathan Barزیلای considered the AHP model to lack a proper mathematical framework for hierarchical analysis in decision-making. In Barزیلای's critique article regarding AHP, it is addressed that the criticisms are mainly as follows: 1) Value functions are usually non-linear; AHP creates multiple value functions through constant substitution rates, which leads to the creation of multiple linear value functions. 2) In AHP, estimates of substitution rates are used to compensate for the lack of substitution rates, which may not be consistent. However, the original substitution rates are always consistent; in fact, the model coefficients are obtained by estimating inconsistent weight ratios through the matrix normalization method. 3) The use of the matrix normalization method and standardization of coefficient vectors in AHP leads to incorrect modeling, and in fact, equivalent tree decomposition may lead to the categorization of non-equivalent value functions.
خلاصه ماشینی:
(2) Comparison of equivalent alternatives (Refer to the page image) In single-level problems, all criteria are determined by the AHP matrix method; in these cases, the input data are estimates of the final marginal rates of substitution of the criteria (which have been discussed in Section 5).
As a result, it can be said that the number of constraint equations is incorrect and the final weights depend on how the criteria are grouped; in other words, an equivalent tree decomposition may lead to non-equivalent classification of utility functions and rankings.
(13) U1(x)-x1+x2+2x3+x4+x5 (For convenience, we have multiplied both sides of the equation by 6) The second manager, who gave priority to the east, performed the decomposition using the method shown in Figure 3, and similarly, the third senior manager performed his decomposition as shown in Figure 4, and the utility functions for each are as follows: (14) U2(x)-2x1+2x2+2x3+3x4+3x5 (15) U2(x)-3x1+3x2+2x3+2x4+2x5 (Refer to the page image) Figure 2- Decomposition by manager number 1 (Refer to the page image) Figure 3- Decomposition by manager number (Refer to the page image) Figure (4)- Decomposition by manager number 3 Figures 2, 3, and 4: Decision making regarding marketing strategy It is clear that the functions (12-15) are different from each other and we have: U3(p)<U3(Q) and U2(p)>U2(q) and U1(p)-U1(Q) As you can observe, with this different spectrum of priorities that we see, it becomes clear that the AHP approach in equivalent hierarchical decomposition (which are either equivalent models or problem descriptions) has ranked the functions in a non-equivalent (heterogeneous) manner.
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