The Problem of the Adequacy of the Analytic Hierarchy Process and Its Solution

Abstract

The Analytic Hierarchy Process (AHP) is a popular method for solving multi-criteria problems. Many researchers emphasize the simplicity and naturalness of the AHP procedure for evaluating alternatives. However, many scholars believe that the AHP is flawed and therefore cannot be applied in practice. Such a scattering of opinions requires an explanation. The method of AHP uses pairwise comparisons of alternatives. And it is based on the assumption that the alternatives can be measured on the scale of relations. Fechner’s psychophysical law is used as justification for the existence of method of measurement with ratio scales. But there are not one but two psychophysical laws. The existence of two psychophysical laws is a problem of psychophysics. This problem has been solved quite recently. It was shown that the basic psychophysical laws are equivalent. In order to solve this problem, an adjustment of Stevens’ direct measurement model was required. It is suggested that a direct measurement model be used to overcome the shortcomings of the Analytic Hierarchy Process. In this case, the fundamental AHP measurement scale can be used. The adjusted AHP method contains a direct measurement model and a built-in mechanism for checking the adequacy of measurement results. An example of direct measurement of alternatives is analyzed.

1. Introduction

The Analytic Hierarchy Process (AHP) evaluates alternatives using pairwise comparisons based on expert judgments [1, 2]. The AHP methodology involves the measurement of alternatives and the transformation of measurement results [3]. Using AHP methods, it is possible to construct a mathematical model of decision-making [4, 5]. Since the AHP method does not have a strict mathematical justification, there are various modifications to the method [6, 7, 8]. The Multiattribute Utility Theory (MAUT) [9, 10, 11] and specially created methods [12, 13] can be used instead of the AHP method.

In the monograph [14], numerous examples of AHP applications were considered. The authors of the monograph concluded that, despite its popularity, AHP is incapable of solving complex problems. The popularity of AHP is because the AHP method gives the researcher the feeling that he or she is actually solving a complex problem based on his or her preferences. Therefore, many AHP users consider that this method can be applied to any scenario. The authors of the monograph believe that, even when solving trivial problems, AHP is based on questionable procedures. There are references to the works of more than 100 scientists who support this view.

J. Barzilai is the author of a New Theory of measurement [15]. J. Barzilai set the conditions for using the mathematical operations of linear algebra and calculus. The failure to meet the conditions for the application of arithmetic operations in the mathematical foundations of measurement theory, utility theory, and decision theory caused fundamental errors [16]. J. Barzilai believes that T. Saaty did not define what was meant by the terms “importance of criteria” or “relative importance” of criteria. Moreover, criterion importance coefficients cannot be interpreted as a measure of the relative importance of criteria. “In fact, the AHP is plagued by many flaws, and these flaws are fundamental” [17]. There are many examples of the method not working correctly [18]. However, the AHP method is still popular. This can be explained by the fact that the AHP contains a direct measurement model, which is absent in axiomatic theories of measurement. In addition, the AHP measurement model considers the psychological features of the person.

The purpose of this paper is to propose a modification of the AHP that retains the AHP measurement scale. In this case, the decision-making model is free from fundamental errors. The possibility of such modification is explained by the fact that the justification of the AHP method derives from Fechner’s psychophysical law. It should be emphasized that there are not one but two psychophysical laws in psychophysics: Fechner’s law and Stevens’ law. The existence of two psychophysical laws was until recently considered a problem [19, 20, 21]. A solution to this problem was obtained in [22, 23, 24] through the application of the Stevens measurement model. In this paper, this measurement model is used to adjust the AHP method. For a specialist, the AHP correction comes down to small changes in the calculation scheme.

At the beginning of the article, the AHP measurement model, which is the basis of the AHP method, is briefly described. An analysis of the measurement model indicates that the model needs to be adjusted. The Stevens measurement model is then discussed, and the multifactor model is justified. The application of the new measurement model is then described. As an illustration, the paper considers an example of a proper evaluation of alternatives using the AHP scale. Finally, the advantages of the proposed approach are presented.

2. A critique of the analytic hierarchy process as a method of measurement with ratio scales

The measurement model was defined by T. Saaty using the results of Fechner’s works [1]. Fechner (1860) investigated the reactions that arise when paired comparisons of stimuli are made. For example, in an experiment, participants are offered two objects of a certain weight. The weight of one object is changed until the participants notice a difference. Fechner called such a difference “just noticeable.”

Fechner believed that for a sequence of “just noticeable differences” in stimuli, the relation $v_i=v_{i-1}\mathrm{\alpha }$ is satisfied, where $v_i$ is the value of the stimulus and $v_0$ is the numerical value of the first stimulus, and $\mathrm{\alpha }$ is a constant. Then the equality $v_i=v_0{\mathrm{\alpha }}^i,$ and $v_j=v_0{\mathrm{\alpha }}^j$ are fulfilled, and equality is obtained

$$ln\left(\mathrm{\alpha }\right)\left(u_i-u_j\right)=ln\left(v_i/v_j\right),$$E1

where $\left(u_i-u_j\right)=\left(i-j\right).$ Equality (1) demonstrates that the ratio of values of the physical quantity $v_i/v_j$ can be found by subjectively measuring the difference in values of $\left(u_i-u_j\right)$.

If the result of the measurement is the difference of values, the values are determined to be the additive constant. Assume $u_1-u_0=1,u_2-u_1=2$. Then $u_0=С,u_1=1+С,u_2=2+С,$ where $С$ is a constant. Only one arithmetic operation is defined for the values $u_0,$ $u_1,$ and $u_2:$ the difference of values. The ratio of these values is undefined. T. Saaty believed that if the constant $С$ is set to zero, the values are uniquely defined: $u_0=0$, $u_1=1$, and $u_2=2$, and the division operation is defined. Indeed, in this case, the division operation is formally defined. But this operation does not make empirical sense. For example, there is no reason to suppose that the value of $u_2$ is twice the value of $u_1$. Thus, the justification of the AHP method looks unconvincing.

A similar modeling error is observed when forming a matrix of pairwise comparisons using the AHP method. A matrix of pairwise comparisons is the basic element of the AHP [1]. Consider a set of alternatives $A_1,A_2,\dots ,A_n.$ Weights of alternatives $w_j=w\left(A_j\right)$ are found as a result of pairwise comparison of alternatives. The results of pairwise comparisons are represented as a square matrix $V_{\mathit{ij}}=\left[v_{\mathit{ij}}\right],$ $i,j=1,2,\dots ,n,$ where $v_{\mathit{ij}}=w_i/w_j$ is an estimate of the relative importance of alternative $A_i$ compared to alternative $A_j$ and $v_{\mathit{ij}}=1/v_{\mathit{ji}}.$ The weights of the alternatives are unknown and are determined by the results of pairwise comparisons. In the hierarchy analysis method, the results of pairwise comparisons are chosen from an integer “fundamental” scale of {1, 2, ..., 9}.

If a matrix of pairwise comparisons is given, the weights of the alternatives are not uniquely defined. For example, let $v_{21}=2,$ $v_{32}=2,$ and $v_{31}=4$. Then the system of equations has the form: $w_2/w_1=2,w_3/w_2=2,$ $w_3/w_1=4$. This system of equations can be solved as follows, where $C$ is an unknown constant: $ln\left(w_1\right)=C,$ $ln\left(w_2\right)=C+ln\left(2\right),$ $ln\left(w_3\right)=C+ln\left(4\right).$Hence, the weights are defined on log-intervals scale. If the constant $C$ is fixed, then a division and addition operation is formally defined for the values of the quantity. For example, if $C=0,$ then $w_1+w_2=3,$but this operation has no empirical justification. However, the sum of weights is used in the AHP. Therefore, the AHP uses arithmetic operations without proper justification. Since arithmetic operations do not correspond to empirical operations, it violates the principle of reflection [15].

The aim of the work is to modify the AHP method based on the correct model for measuring alternatives. To this end, the paper considers a mathematical model of the empirical system (J. Bazilai [15]) and a model of direct measurement.

3. The correction of Stevens’ scales of measurement (direct measurement model)

Let the empirical system be a straight line with a set of points and a set of vectors [15]. For any ordered pair of points $A_1$ and $A_2$ there is a unique vector $\left(A_1,A_2\right).$ Points on a straight line are measurement objects (Figure 1). A vector is the empirical result of a measurement that characterizes the difference in the position of two points on a straight line.

Figure 1.

Empirical system. Measurement objects $A_1,A_2,\dots ,A_n$.

Thus, the model of the empirical system is a one-dimensional affine space. J. Barzilai calls this space homogeneous because, in this case, it is possible to compare vectors with each other [15]. For example, according to Figure 1, the vector $\left(A_1,A_2\right),$ is two times smaller than the vector $\left(A_1,A_3\right)$.

Following A. Friedman (1922), let us axiomatically define an “exceptional group of objects” admitting a special estimation [25]. Let us assume that objects $A_1,A_2,\dots ,A_n$ are arranged in increasing order of the measured value and the value of objects changes uniformly. This means that the vectors $\left(A_1,A_2\right),\left(A_2,A_3\right),\dots ,$ $\left(A_{n-1},A_n\right)$ are equal to each other. The founder of the representative theory of measurement, S. S. Stevens, used a similar model for direct measurement of value. Let the value of the quantity $u_i$ corresponds to object $A_i$. Consider that successive value differences are equal to one another:

$$u_2-u_1=u_3-u_2=\dots =u_n-u_{n-1}.$$E2

A.A. Friedman called such special estimation “measurement.” Then

$$u_i-u_j={\mathrm{\lambda }}_1\left(i-j\right),$$E3

when ${\mathrm{\lambda }}_1$ is an unknown constant, ${\mathrm{\lambda }}_1>0.$ This implies that the values of the measured quantity are defined with an accuracy of a linear transformation, that is, in the interval scale. Assign the values of the quantity $v_i$ to each object $A_i,$ and assume that the successive ratios are equal

$$v_2/v_1=v_3/v_2=\dots =v_n/v_{n-1}.$$E4

The equality

$$ln\left(v_i/v_j\right)={\mathrm{\lambda }}_2\left(i-j\right),$$E5

is then satisfied, where ${\mathrm{\lambda }}_2$ is an unknown constant, ${\mathrm{\lambda }}_2>0$. The values of the logarithms of the measured quantity are determined with an accuracy of a linear transformation, i.e., on the log-interval scale [26].

Thus, two ways of measuring the value are obtained. In the first case, the result of the measurement is the difference $\left(u_i-u_j\right)$, and in the second case, the ratio of the values $\left(v_i/v_j\right)$. The values of a quantity are measured on a scale of intervals (3) or log intervals (5) [26, 27].

Similar models were used by C. S. Stevens to classify measurement scales. The choice of four measurement scales was made by S.S. Stevens back in 1946 [26]. S.S. Stevens later added a fifth scale to them, the scale of logarithmic intervals, but it was later recognized as useless [27]. At first glance, Stevens’ concept of measurement looks convincing, and only the presence of an “extra” fifth scale violates the logic of the presentation. According to S.S. Stevens, the scale of logarithmic intervals is mathematically interesting but, like many mathematical models, empirically useless. Let us use an example to demonstrate why such a claim is controversial. To do this, let us measure a non-additive quantity using the Stevens model.

Density is an example of a non-additive value. For example, if the density values of two samples are 3 kg/m³ and 2 kg/m³, it is not clear what the sum of these values would mean. But the division operation is defined for density; specifically, 3 kg/m³ is 1.5 times greater than 2 kg/m³. Let the densities of five samples $A_1,A_2,A_3,A_4,A_5$ change uniformly, and for clarity, let the ratio of the densities of two consecutive samples be two. Density values can be compared in two ways. To calculate the difference in values, use formula (3): $u_i-u_j=\left(i-j\right),$ where $u_i$ is the density value. Using formula (5), the ratio $v_i/v_j=2^{i-j},$ is obtained, where$i,j=1,2,\dots ,5$.

The density values $v_i$ are determined by the accuracy of the multiplier; values$u_i$ are determined up to the additive constant. In this particular case, the values are given in Table 1. The values have a natural interpretation. For example, the density of the third sample is four times greater than the first, or two orders of magnitude greater than the first.

$u_i$	1	2	3	4	5
$v_i$	2	2^2	2^3	2^4	2^5

Table 1.

Density values.

$u_i$ are “values” in the interval scale,

$v_i$ are values in the log-interval scale.

The example confirms that for arbitrary objects $A_1,A_2,\dots ,A_n$, the value of which changes uniformly, it is reasonable to consider two measurement scales: the scale of intervals and the scale of log intervals.

The ratio scale is the highest level of measurement in Stevens’ classification of levels of measurement (nominal, ordinal, interval, and ratio) [26, 27, 28]. The ratio scale is invariant over transformations in which the numerals on the scale are multiplied by a constant. From the analyzed example, it follows that the result of direct measurement is the scale of intervals or the scale of log intervals (Table 1). Therefore, the ratio scale is not a scale of direct measurement. But S.S. Stevens believed that the log-interval scale was useless [27] and carried out direct measurements on the ratio scale. This point of view was considered correct for a long time and was the cause of numerous errors, including those in the AHP method.

4. Adjusted model of direct measurement

From equalities (3) and (5), it follows that the values of the quantity on the scale of intervals and log intervals are related by the formula

$$\left(u_i-u_j\right)=\mathrm{\lambda ln}\left(v_i/v_j\right).$$E6

where $i,j=1,2,\dots ,n;$ $u_i$, $v_i$ are values of the quantity and $\mathrm{\lambda }={\mathrm{\lambda }}_1/{\mathrm{\lambda }}_2.$ Equality (6) holds true for the values $u_i$ and $v_i$ in Table 1, if $\mathrm{\lambda }=1/ln\left(2\right)$.

Equality (6) means that the mapping $u=ln\left(v\right)$ preserves the operation of measurement: the ratio of values maps to the difference of values. In addition, there is a one-to-one correspondence between the values of $u_i$ and $v_i.$ The mapping $u=ln\left(v\right)$ is an isomorphism of two algebraic structures: the set of all positive numbers with the operation of division, and the set of all real numbers with the operation of subtraction. In algebra, isomorphic structures are not distinguished; they are equivalent [29].

Let in the process of measurement, each pair of objects is assigned a difference $u_i-u_j$ or ratio $v_i/v_j$ of values. In order to consider two methods of measuring simultaneously, let us denote the left and right parts of equality (6) by the symbol $R_{\mathit{ij}}$ and define two mappings

$$R_{\mathit{ij}}={\mathrm{\lambda }}_1\left(u_i-u_j\right),$$E7

$$R_{\mathit{ij}}={\mathrm{\lambda }}_{2}ln\left(v_i/v_j\right),$$E8

where $R_{\mathit{ij}}$ is the values of the rating; $i,j=1,2,\dots ,n;$ $u_i$ and $v_i$ are the values of quantity, and ${\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2$ are positive constants. For objects whose value changes uniformly, the rating is defined by the formula $R_{\mathit{ij}}=\mathrm{\lambda }\left(i-j\right),\mathrm{\lambda }>0.$ This definition of rating is called “classical” [22]. The rating does not depend on the choice of measurement method (7) or (8).

It is possible to check directly that the rating values $R_{\mathit{ij}}$ satisfy the compatibility condition of the form

$$R_{\mathit{ij}}=R_{\mathit{ik}}+R_{\mathit{kj}},$$E9

In [22, 23, 24], the compatibility condition (9) is axiomatically defined, and the theoretical model of direct measurement is formulated.

The values of a quantity are obtained on a scale of intervals if they are the solution of the system of Eq. (7), and on a scale of log intervals if they are the solution of the system of Eq. (8). For example, if the respondent believes that the criterion $A_3$ exceeds the criterion $A_1$ by six conditional units and the criterion $A_2$ exceeds criterion $A_1$ by three conditional units, then the equations $u_3-u_1=6$ and $u_2-u_1=3$ are true and values of $u_1,u_2,$ and $u_3$ are determined on an interval scale. If the respondent believes that criterion $A_3$ is four times more important than criterion $A_1$ and criterion $A_2$ is two times more important than criterion $A_1,$ then the system of equations looks like this: $v_3/v_1=4$ и $v_2/v_1=2.$ In this case, the values are determined using a logarithmic interval scale. The values on the interval scale and the log-interval scale must be related by the equivalence condition (6).

Various measurement models have been widely used for a long time, but the direct measurement model includes isomorphism (equivalence condition (6)) of scales. From the equivalence condition, follow Fechner’s law in the form of paired comparisons [22], Stevens’ law in the form of paired comparisons [22], and Rush’s model [30, 31].

Stevens’ experimental law (1947) was proposed instead of Fechner’s experimental law (1848). There is now a paradoxical contradiction in psychophysics between Fechner’s and Stevens’ laws, in that the two basic laws contradict each other. The harmonization of these two laws has been the subject of much discussion, but a solution that would satisfy all involved has never been found [19]. The fact that the direct measurement model under consideration solves the complex problem of psychophysics confirms its theoretical and practical importance. Using the direct measurement model, it is possible to introduce the notion of independent variables.

5. Independence of variables

Let $A$ and $B$ be two sets of real numbers and let $D$ be the Cartesian product of sets $A$ and $B,D=A\times B.$ Each pair of values $\left(a,b\right)$ from $D$ corresponds to the value of the function $u=u\left(a,b\right).$ Let $M_0\left(a_0,b_0\right)$ is point from the set $D$. The total increment of the function $u=u\left(a,b\right)$ at the point $M_0\left(a_0,b_0\right)$ is the difference

$$\mathrm{\Delta }u=u\left(a,b\right)-u\left(a_0,b_0\right).$$E10

For a fixed point, $M_0\left(a_0,b_0\right)$ the total increment $z$ is a function of the variables (a, b), $\mathrm{\Delta }u=\mathrm{\Delta }u\left(a,b\right).$ The partial increment of the function $u=u\left(a,b\right)$ by the variable is the difference

$${\mathrm{\Delta }}_{a}u\left(a,b\right)=u\left(a,b\right)-u\left(a_0,b\right),$$E11

$${\mathrm{\Delta }}_{b}u\left(a,b\right)=u\left(a,b\right)-u\left(a,b_0\right).$$E12

Definition. Two variables are called independent if the partial increment of each of them does not depend on what value the other variable has taken:

$${\mathrm{\Delta }}_{a}u\left(a,b\right)={\mathrm{\Delta }}_{a}u\left(a\right),$$E13

$${\mathrm{\Delta }}_{b}u\left(a,b\right)={\mathrm{\Delta }}_{b}u\left(b\right).$$E14

The problem of finding the total increment of a function based on partial increments has no solution in the general case. In the special case where the variables are independent, the problem is solved quite simply. In this case, the total increment of a function is equal to the sum of the partial increments

$$\mathrm{\Delta }u\left(a,b\right)=\mathrm{\Delta }{u}_a\left(a\right)+\mathrm{\Delta }{u}_b\left(b\right).$$E15

Let $\mathrm{\Delta }{u}_a\left(a\right)=k_{1}x$ and $\mathrm{\Delta }{u}_b\left(b\right)=k_{2}y$then equality (15) can be written as

$$\mathrm{\Delta }u\left(x,y\right)=k_{1}x+k_{2}y,$$E16

where $\mathrm{\Delta }u\left(x,y\right)=u\left(x,y\right)-u\left(x_0,y_0\right);k_1$ and $k_2$ are scale constants. Similarly, the definition of independent variables can be formulated on the basis of a direct measurement of the ratio of values. In this case equality

$$\mathrm{\delta }u\left(x,y\right)=k_{1}x+k_{2}y$$E17

is satisfied, where $\mathrm{\delta }u\left(x,y\right)=ln\left(v\left(x,y\right)/v\left(x_0,y_0\right)\right).$ Modified AHP method uses an additive representation of the form (16) or (17).

6. Modified analytical hierarchy process (example)

Let three main factors influence the favorable sociopolitical development of the state: $x$, economy; $y$, ecology; $z$, security. Each factor has a certain number of discrete levels. (The example is partially taken from the monograph [1]).

The project is a point $\left(x,y,z\right)$ in three-dimensional space, the coordinates of which correspond to the level of development of the economy, ecology, and security. Let the value of the function $U\left(x,y,z\right)$ be a numerical characteristic of the state’s development. Factors $x,y,$ and $z$ can be considered mutually independent. By analogy with the representation (16), let us choose an additive model to assess the development of the state

$$U-U_0=k_{1}x+k_{2}y+k_{3}z,$$E18

where $k_1,k_2$_, and $k_3$ are coefficients of influence of factors; $x,y$_, and $z$ are variables (factors). Let us denote the lower and upper levels of each factor as 0 and 1, $0\le x\le 1,0\le y\le 1$, and $0\le z\le 1$.

6.1 Influence of factors

The projects that need to be considered here are $A_0\left(0,0,0\right),$ $A_1\left(1,0,0\right),$ $A_2\left(0,1,0\right),$ and $A_3\left(0,0,1\right).$ Let us introduce the notation $U_i=U\left(A_i\right),$ $i=0,1,2,3;$ $U_i$ is a numerical characteristic of state development. The value of $U_0$ corresponds to the current level of the State, and the value of $U_1$ corresponds to the project with the maximum development of only the economy. The decision maker performs the pairwise comparison of projects. The results of the pairwise comparisons are recorded in the form of a matrix. The matrix of pairwise comparisons according to the AHP method [1] is presented in Table 2.

$U_i/U_j$	$U_1$	$U_2$	$U_2$
U1	1	5	3
U2	1/5	1	1/2
U3	1/3	2	1

Table 2.

Matrix of pairwise comparisons (Т. Saaty).

The first row of Table 2 contains the number 5. This means that the equality $U_1/U_2=5$ is satisfied, i.e., the influence of the first factor is five times greater than the influence of the second factor. In the corrected version of the AHP method, it is necessary to replace the ratios in this matrix with differences. For example, if the number 5 appears in the pairwise comparison matrix in Table 2, replace it with the difference (5–1). Similarly, if the ratio is 1/ 5, replace it with a difference of 1 and 5. As a result, $U_1-U_2=4$ and $U_2-U_1=-4$ are obtained. Equations $U_3-U_2=1$ and $U_1-U_3=2$ are obtained similarly.

To find the influence coefficients $k_1,k_2,$ and $k_3$ in the formula (18), the data in Table 2 is not enough. Therefore, the expert must additionally evaluate the influence of each factor using the fundamental scale of the AHP [1]. Let the equalities $U_1-U_0=6,U_2-U_0=2,U_3-U_0=2$ be obtained with the help of an expert. This means that the expert estimates the influence of the first factor to be four units and the influence of the second and third factors to be two units. The result is an extended matrix of pairwise comparisons (Table 3).

$U_{\mathit{ij}}$	$U_0$	$U_1$	$U_2$	$U_3$
$U_0$	0	−6	−2	−2
$U_1$	6	0	4	2
U2	2	−4	0	−1
$U_3$	2	−2	1	0

Table 3.

Extended matrix of pairwise comparisons.

The result of the pairwise comparison of projects $A_i$ and$A_j$ is the difference in values (rating): $U_{\mathit{ij}}=U_i-U_j,U_{\mathit{ij}}=-U_{\mathit{ji}}$.

Columns $U^k$ of Table 4 are obtained from columns $U_k$ of Table 3 by subtracting the first element from each element of the column by the formula $U_i^k=U_{\mathit{ik}}-U_{0k},$ $i,k=0,1,2,3$.

$U_i-U_0$	$U^0$	$U^1$	$U^2$	$U^3$	$\overline{U}$
$U_0$	0	0	0	0	0.00
$U_1$	6	6	6	4	5.50
$U_2$	2	2	2	1	1.75
$U_3$	2	4	3	2	2.75

Table 4.

Rating of factors.

$\overline{U}$ is Average Rating.

If the rating estimates $U_{\mathit{ij}}$ are found exactly, then all columns $U^0,$ $U^1,U^2,$ and $U^3$ in Table 4 must match. Equality in some sense of the columns can be used as a criterion for the adequacy of the measurement results. The vector $\overline{U}$ are found as the arithmetic mean of the columns $U^0,$ $U^1,U^2,$ and $U^3$ in Table 4. The correlation coefficient ${\rho }_k=\rho \left(U^k,\overline{U}\right),$ is denoted by ${\rho }_k$. In this case, correlation coefficients ${\rho }_0=0.979,{\rho }_1=0.982,{\rho }_2=1.000,{\rho }_3=0.997$are significant at the 0.05 significance level. Therefore, the hypothesis that the mathematical model is adequate for the measurement results is accepted.

6.2 Influence of factor levels

Let $x_0,x_1$_, and $x_2$ be the levels of the first factor $x$; $y_0,y_1$ and $y_2$ be the levels of the second factor$y$; and, $z_0$ and $z_1$ be the levels of the third factor$z.$ For example, level $x_0,$ corresponds to the current state of the economy. Let the result of pairwise comparisons be the difference in values of factor levels $x_{\mathit{ij}}=x_i-x_j$. The matrix of pairwise comparisons of levels of the economy (Table 5) is made with the fundamental scale AHP.

$x_{\mathit{ij}}$	$x_0$	$x_1$	$x_2$
$x_0$	0	−1	−3
$x_1$	1	0	−2
$x_2$	4	2	0

Table 5.

Pairwise comparisons of Economic Levels.

The number 4 in the first column of the data in Table 5 means that the level of $x_2$ is four units higher than the current level of $x_0$.

Columns $x^0{x}^1,$ and $x^2$ of Table 6 are obtained from columns $x_0,x_1,$ and $x_2$ of Table 5 by subtracting the first element from each column element using the formula $x_i^k=x_{\mathit{ik}}-x_{0k}$.

$x$	$x^0$	$x^1$	$x^2$	$\overline{x}$	$\widehat{x}$
$x_0$	0	0	0	0.00	0.00
$x_1$	1	1	1	1.00	0.30
$x_2$	4	3	3	3.33	1.00

Table 6.

Rating of Economic Levels.

$\overline{x}$ is Average Rating

$\widehat{x}$ is Normalized Rating.

The correlation coefficient is denoted by ${\rho }_k=\rho \left(\overline{x},x^k\right).$ In this case, correlation coefficients ${\rho }_0=0.999,{\rho }_1=0.999,$ and ${\rho }_2=0.999,$ are significant at the 0.05 significance level. Therefore, there is reason to believe that the values of the levels correspond to the results of the measurement. Similarly, levels of ecological development are compared to the current level (Table 7).

$y_{\mathit{ij}}$	$y_0$	$y_1$	$y_2$
$y_0$	0	−4	−8
$y_1$	4	0	−4
$y_2$	8	4	0

Table 7.

Pairwise comparison of Ecological Levels.

Columns $y^0{y}^1,$ and $y^2$ of Table 8 are obtained from columns $y_0,y_1,$ and $y_2$ of Table 7 using the formula $y_i^k=y_{\mathit{ik}}-y_{0k}$.

$y$	$y^0$	$y^1$	$y^2$	$\overline{y}$	$\widehat{y}$
$y_0$	0	0	0	0	0,0
$y_1$	4	4	4	4	0,5
$y_2$	8	8	8	8	1,0

Table 8.

Rating of Ecological Levels.

$\overline{y}$ is Average Rating

$\tilde{y}$ is Normalized Rating.

In this case, all three vectors $y^0,y^1,$ and $y^2$ coincide (Table 8). Therefore, the hypothesis that a normalized rating vector is adequate for the measurement results is accepted. The safety factor $z$ has only two levels. So, let $z_0=0$ and $z_1=1$.

6.3 Project assessment

The linear model (18) allows us to evaluate various social development projects. Let $A_m$ be the project; $m$ be the project number; $m=0,1,\dots ,7;$ $\left(x_i,y_j,z_k\right)$be the levels of factors in accordance with Table 9. The values of the function $U=U\left(A_m\right)$ are estimates of the impact of the projects on the state. The function $U=U\left(A_m\right)$ is defined as the linear transformation. To calculate the values of the function, it is convenient to assume that $U\left(A_0\right)=0$ and $U\left(A_7\right)=8$. The rating of the projects (Table 8) has been calculated by formula (18).

$A_m$	xi	yj	zk	$U\left(A_m\right)$
$A_0$	0.00	0.00	0.00	0.00
$A_1$	1.00	0.00	0.00	4.40
$A_2$	0.00	1.00	0.00	1.40
$A_3$	0.00	0.00	1.00	2.20
$A_4$	0.30	1.00	1.00	4.92
$A_5$	1.00	0.50	0.00	5.10
$A_6$	1.00	0.50	1.00	7.30
$A_7$	1.00	1.00	1.00	8.00

Table 9.

Rating of projects.

The outcome of measuring a project’s impact on society is its project rating. Rating values, differences, and ratios of rating values have an empirical meaning. In contrast to the AHP method, the rating has a quantitative structure. For example, the task is to choose the best project among $A_4$,$A_5$, or $A_6$. From the analysis of Table 9, it follows that the $A_6$ project should be considered first. And project $A_6$ has a greater influence than project $A_5$, by 43%.

7. Conclusions and future research directions

The theoretical foundations of the AHP are currently being criticized, in particular, the correctness of the mathematical model. This paper proposes an adjustment to the mathematical model of the AHP using the Stevens direct measurement model. This allows the AHP method’s measurement scale to be used to evaluate alternatives. Statistical criteria can be applied to check the adequacy of the measurement results. The corrected algorithm is no more complicated than the AHP method.

The AHP method uses a variant of the pairwise comparison method. At present, the theoretical foundations of the AHP are being criticized, and in particular, the possibility of measuring preferences on the ratio scale is being questioned. Moreover, the criticism of the AHP method looks quite reasonable. For example, the method of pairwise comparisons has long been used by psychologists, but the values are found to be on an interval scale. The paper shows why the method of pairwise comparisons cannot be used to find values on the ratio scale.

The direct measurement model of psychologist S. Stevens was proposed to measure preferences. The article considers two methods of measurement. The first way is to find the difference between values, and the second way is to find the ratio of values. The algorithm for processing measurement results does not change significantly in this case. In this paper, only the first method, which uses the AHP scale, was considered in detail. The values of the quantity in the interval scale were discovered by the method of pairwise comparisons. This approach has certain advantages. In this case, you can use the measurement scale of the AHP method.

The method of pairwise comparisons considered in this paper is fully compatible with the modern theory of measurement by J. Barzilai. Moreover, standard statistical criteria can be used to check the adequacy of the model. As a result, many of the comments made by opponents of the AHP method have been removed. Measurement results obtained by the method of pairwise comparisons can be used to find coefficients of the multiple linear regression model.

It is relevant to continue further research on the corrected AHP method. Future research could include the use of two measurement methods as well as a test of the regression equation’s adequacy with measurement results.