Multi-attribute decision-making method based on Taylor expansion

Abstract

Determining attribute weights is an indispensable step in multi-attribute decision-making problems, and it is also a top priority in the study of multi-attribute decision-making problems. Existing methods for determining attribute weights do not completely and effectively reflect the decision-maker’s dependency preferences, which will result in unreasonable ranking results for decision-makers. To solve this problem, this article proposes a feature-weighted multi-attribute decision-making method based on Taylor expansion. The method uses the natural base and the eigenvalues of the matrix to construct the feature-weighted coefficients and weights; normalizes all the feature vectors of the matrix; and constructs a new weight vector. Combined with the example to analyze and verify, the method makes reasonable use of all decision information, which saves the decision time of decision-makers.

Introduction

Multi-attribute decision-making is the selection of finite solutions linked to multiple attributes. ¹ It has extensive theoretical and practical application contexts. Multi-objective decision-making problems, also called multi-attribute decision-making, are decisions that involve the complexity of multiple goals. In decision science, the goal is defined as the representation of the state that the decision-maker wants to achieve. According to this definition of the goal, it has the following characteristics:

1. Hierarchy: the decision-makers divide the goals into different levels. The highest level is called the “total goal.” The following are the different types and different categories of goals and sub-goals.

2. Fuzziness: decision-makers’ metrics for goals are often expressed in terms of fuzzy concepts such as “good,”“bad,” and “poor.” This expression is closer to practice.

3. Relative independence: although there are various links between goals, they are often considered independently in decision-making.

For multi-attribute decision-making problems, the smallest decision-making unit is the decision-maker himself. The goal is a statement about the state that a decision-maker of the research question wishes to achieve. In a multi-attribute decision-making problem, there are several statements that express the state that the decision-maker wants to achieve. A multi-attribute decision problem can be represented by a hierarchical structure. The highest level is the total goal. The second level is all the attributes that the decision-maker thinks has a significant correlation with the total goal. The third layer is all the decision-making solutions proposed to solve this problem. The hierarchical structure and the tree structure are both related and different. The tree structure belongs to an incomplete hierarchical structure. That is, each element of the upper layer cannot completely control all the elements of the adjacent lower layer. The hierarchical structure is not necessarily a tree structure. The connection lines of the tree structure are disjoint, that is, the two elements in the adjacent layer do not intersect, and the hierarchical structure generally intersects. In the hierarchical structure, the elements of the lower layer are more explicit, more specific, and easier to calculate than the elements of the upper layer. Some goals do not have one or several obvious indicators to directly measure the extent to which they are achieved, but there may still be one or several attributes that are both easy to measure and indirectly reflect the extent to which the goal is achieved. This attribute is called a surrogate attribute. Decision-making refers to the structure and decision-making environment of this decision-making problem. A known probability distribution and an unknown probability distribution are included in the environmental state of uncertainty. The decision rule is a rule for arranging the order of the solutions, including the optimal rule and the satisfaction rule.

Although more than 40 years of research on multi-attribute decision-making problems has yielded fruitful results, there are still new challenges. With the progress of mankind, the decision-making issues have become more and more complicated. In order to adapt to the complex realities, the development of decision-making methods also shows corresponding diversity. In 1975, the number of multi-criteria/multi-attribute decision-making literature had been more than 500, including hundreds of different decision-making methods. ¹ Multi-attribute decision-making is an important part of modern decision science, systems engineering, and management science. Its theory and method have been widely applied in many fields such as economy, management, engineering, military, and society.^2–4 The essence of multi-attribute decision-making is to use the existing decision-making information to sort and select a group of (or a limited number of) alternatives in a certain way. ⁵ It is mainly composed of two parts: one is the acquisition of decision information and the other is what methods are used to sort and select solutions. ⁶ The decision information generally includes two aspects: attribute weight and attribute value. The main form of attribute value includes three types: real number, interval number, and language. Therefore, the study of multi-attribute decision-making methods mainly focuses on the determination of attribute weights and the research on integration operators. The determination of attribute weights is the most important part in multi-attribute decision-making methods. ⁵

Analytic hierarchy process (AHP) was proposed by the American telecommunicator Satyr in the early 1970s.^7,8 It uses a complex multi-objective decision-making problem as a system to decompose the target into multiple goals or criteria, which are then decomposed into several levels of multiple indicators (or criteria and constraints). It calculates the hierarchical single-order (the weight) and total order by the qualitative index fuzzy quantization method, as the target (multi-indicator) and multi-program optimization decision.

The AHP is to decompose decision-making problems into different hierarchical structures according to the general goal, the sub-goals, the evaluation criteria, and the specific preparation plan. Then it uses the method of solving the eigenvectors of the judgment matrix to find the priority of each element of each level to the element of the upper level. Finally, the method of weighted sum is used to hierarchically merge the final weights of the alternative solutions to the total goal. The final weight is the optimal solution. The AHP is more suitable for the target system with hierarchical and interlaced evaluation indicators, and the target value is difficult to quantitatively describe the decision problem. In essence, it is the formalization of the human understanding of the hierarchical structure in complex problems. It has received extensive attention due to its advantages of practicality, simplicity, and system. Meanwhile, AHP has been rapidly applied to multi-attribute decision-making problems in various fields.^9,10 The key point of AHP in multi-attribute decision-making is to make it possible for decision-makers to visually use attribute hierarchies to construct complex multi-attribute decision-making problems. For complex and large hierarchical hierarchies, AHP is more robust. ¹¹

The AHP can combine qualitative judgments and quantitative analysis of decision-makers (or experts) on each attribute by constructing a hierarchical structure and ratio analysis. The whole process is in line with the requirements of human decision-making thinking activities, which greatly improves the effectiveness and mobility of decision-making.

There are several calculation methods for the weight vectors of the comparative judgment matrix in the AHP, ¹² such as the arithmetic mean method, the geometric mean method, the eigen-root method (EM), the least squares method (LSM), the logistic least squares method (LLSM), and the gradient eigen-root method (GEM). ¹³ Due to the possible inconsistency in the comparative judgment matrix, Saaty recommended the method of eigenvectors. Only the results produced by this method consider the inconsistency. ¹⁴ Chen ¹⁵ proposed the least deviation method (LDM) and the mixed least squares method (MLSM) to reflect the consistency index (C.I.) of judgment matrix deviation. Wang and Fu ¹⁶ proposed the generalized LDM. Zhang and Li ¹⁷ studied the properties of the LDM. Zhang and Cheng ¹⁸ proposed the MLSM.

The article proposes a new method for calculating the weight vector of the judgment matrix. Based on the AHP model, we construct the feature-weighting coefficients by using the natural base and eigenvalues of the decision matrix and transform the coefficients by Taylor expansion. Then we weight and normalize all the feature vectors of the decision matrix, and finally get a new weight vector. Thus, we rank the alternatives in good or bad. In the fourth part of this article, we analyze and verify the method with examples.

Preliminaries

Comparison judgment matrix

We assume that the top-level element $x_0$ is the criterion, and the lower-level elements are $x_1,x_2,\dots ,x_n$ . The weights $w_1,w_2,\dots ,w_n$ of $x_1,x_2,\dots ,x_n$ to the criterion $x_0$ are calculated by comparing with each other. To this end, decision-makers (or experts) have to repeatedly judge which of the two elements, $x_i$ and $x_j$ , in the $L_1$ layer is more important. The two elements are assigned to a relatively important degree (referred to as $a_{\mathrm{ij}}$ ) according to the 1–9 scale shown in Table 1. To the criterion $x_0$ , the lower-level elements constitute a pairwise comparison judgment matrix: $A=(a_{\mathrm{ij}}{)}_{n\times n}$ . In the formula, $a_{\mathrm{ij}}$ represents the 1–9 quantized values of the importance of the elements $x_i$ and $x_j$ relative to $x_0$ . Obviously, the judgment matrix A has the following properties:

1. $\forall i,j\in N$ , if $a_{\mathrm{ij}}>0,a_{\mathrm{ji}}=(1/a_{\mathrm{ij}}),a_{\mathrm{ii}}=1$ , then A is a positive reciprocal matrix.

2. $\forall i,j,k\in N$ , if $a_{\mathrm{ij}}\times a_{\mathrm{jk}}=a_{\mathrm{ik}}$ , then A is a complete consistency matrix.

Table 1.

The basis of the absolute scale.

Importance scale	The definition of the importance of pairwise comparison	Explanation
1	They are equally important	Their contribution to the goal is the same
3	The former is slightly more importantthan the latter	Experience and judgment believe that theformer is more important than the latter
5	The former is obviously more importantthan the latter	Experience and judgment strongly believe thatthe former is more important than the latter
7	The former is greatly more importantthan the latter	It is very strongly believed that the former isimportant to the latter and can be empirical
9	The former is extremely more important thanthe latter	There is the most positive evidence that theformer is much more important than the latter
2, 4, 6, 8	Indicates the middle value of the adjacentjudgment	Their contribution to the goal is the same
Reciprocal	If i has the above scale compared to j,then j is the inverse of the scale valuecompared to i
Rational number	Scale-to-scale conversion	Use n values to obtain the judgment matrix

It is easy to know from property (1) that it is only necessary to give $n(n-1)/2$ elements of the upper (or lower) triangle for the n-order judgment matrix A. Each element in the judgment matrix can be generated by using the absolute scale given in Table 1. Thus, we can calculate the ratio of the importance of all elements. In Table 1, the descriptive interpretation of the 1–9 scale used to quantify the qualitative level indicates that the level differences are equally spaced. When AHP is applied to different occasions, sometimes it is judged from another angle, such as superiority and inferiority, contribution size, and reliability, which can be qualitatively described by other characteristics, such as superiority, instead of importance.

The main reason for selecting the integer between 1 and 9 and its reciprocal as the quantification scale is that it meets the psychological habits of people when making judgments. Many experimental psychological studies have also shown that ordinary people can correctly identify the level of attributes or the number of things generally between 5 and 9, when comparing certain attributes of a group of things at the same time and making the judgment consistent with satisfaction. ¹⁹ Saaty took a discrete number of 1–9 and a difference of 1 as the quantification value of the qualitative level. This method basically obtained social identification, and it was widely used. ¹⁴

The AHP method is a measure for measuring the physical characteristics of a property when it is impossible to use a scale or interval scale. The general-order scale can only represent the order relationship between the attributes of the comparison elements. After the order-preserving transformation, the derived scale of the total ordering of the hierarchy is obtained. However, only the actual meaning of the superior order relationship between the elements is obtained, and the actual physical meaning is not obtained. The weight coefficients obtained by AHP, including single rights and comprehensive rights, do not satisfy the proportional relationship between themselves. If the reliability weight of the solution $A_3$ is $w_3=0.636$ and the reliability weight of the solution $A_2$ is $w_2=0.043$ , the reliability of the $A_3$ cannot be considered to be $(0.636/0.043)=14.8\mathrm{times}$ of the $A_2$ .

Consistency test

Due to the complexity of objective things and the limitations and diversity of the subject’s cognition, the judgment is often accompanied with errors. Generally, it is impossible for the comparison judgment matrix to have complete consistency. This is why the AHP method requires a comparison of the n-order comparison judgment matrix for $n(n-1)/2$ times. If the ordering of the elements is determined only by $n-1$ times, then any one of the judgment errors will result in an unreasonable sorting result. By conducting $n(n-1)/2$ comparison judgments, it is possible to aggregate more information provided by decision-makers from repeated comparisons from different angles, and finally derive a more reasonable ranking that reflects the judgment of decision-makers. This method can play a certain mutual compensation correction for multiple judgments with errors, so that the total sorting result has better order preservation. After Saaty et al.’s theoretical research and social practice, the following method steps of consistency test were summarized. ⁸

• 1. Calculate $C.I.$

  $$C.I.=\frac{{\lambda }_{\max }-n}{n-1}$$ (1)

  where n is the order of the comparison judgment matrix, and ${\lambda }_{\max }$ is the maximum eigenvalue of the comparison judgment matrix.

• 2. Find random index $(R.I.)$ for the corresponding n.

$R.I.$ is the average value obtained by the computer after multiple iterations. Saaty gave an $R.I.$ value of 1–15th order matrices with a sample size of 100–500 in 1980. ⁸ Table 2 shows the $R.I.$ value with a sample size of 1000.

• 3. Calculate consistency ratio $(C.R.)$

  $$C.R.=\frac{C.I.}{R.I.}$$ (2)

When $C.R.<0.1$ , that is, the ratio of the consistency of the decision-maker’s judgment to the consistency of the random generation judgment is less than 10%, it is considered that the C.R. is acceptable. On the other hand, when $C.R.\ge 0.1$ , the comparative judgment matrix should be properly modified to maintain better consistency. For the first- and second-order matrices, formula (1) is always true. At this time, $C.R.=0$ .

Table 2.

Random index.

Order $(n)$	1	2	3	4	5	6	7	8
$R.I.$	0	0	0.52	0.89	1.12	1.26	1.36	1.41
Order $(n)$	9	10	11	12	13	14	15
$R.I.$	1.46	1.49	1.52	1.54	1.56	1.58	1.59

Taylor-based feature-weighted multi-attribute decision-making model

Problem description

Assuming that a multi-attribute decision-making problem includes n evaluation attributes and m alternatives, we divide it into three levels. Its hierarchical structure is shown in Figure 1.

Figure 1.

A hierarchy diagram of a multi-attribute decision-making problem.

The following symbol represents a multi-attribute decision-making problem:

H: the hierarchical hierarchy of a multi-attribute decision-making problem.

$C=\{c\}$ : the total goal of a multi-attribute decision-making problem. It is the first layer in H. It is also called the total goal layer.

$P=\{P_1,P_2,\dots ,P_n\}$ : a set of n properties. It is the second layer in H. It is also called the attribute layer (assuming these attributes are objective and additively independent).

$S=\{S_1,S_2,\dots ,S_m\}$ : a set of m possible solutions. It is the third level in H. It is also called the solution layer.

Calculate the element weight vector

The weight vector is used to reflect the relative importance of each element. The more important the element’s attribute, the larger the corresponding weight value of the element; in contrast, the weight value is smaller. Let the weight vector be $\mathit{w}=(w_1,w_2,\dots ,w_n{)}^T$ , where $\sum _{j=1}^n{w}_j=1,w_j\ge 0,\forall j$ . Considering that each eigenvector of the comparison judgment matrix A reflects the degree of relative preference of the decision-maker for each element attribute, the weight is not represented by the eigenvector corresponding to the maximum eigenvalue. In this article, a feature-weighting method is applied to each eigenvector of the comparison judgment matrix A to update the weight vector. That is, each eigenvector of A is multiplied by a feature-weight coefficient. Then add all the feature-weight vectors and finally normalize them. The specific steps are as follows:

• Step 1: An element $x_0$ of a layer is used as the criterion. The decision-maker repeatedly judges and gives the relative importance $(a_{\mathrm{ij}})$ of the lower-level elements $x_i(i=1,2,\dots ,n)$ and $x_j(i=1,2,\dots ,n)$ and then constructs an n-order comparison judgment matrix $A=(a_{\mathrm{ij}}{)}_{n\times n}$ . The judgment matrix $A=(a_{\mathrm{ij}}{)}_{n\times n}$ is normalized by the column. Therefore, $(a_{\mathrm{ij}}{)}_{n\times n}$ becomes $({\overline{a}}_{\mathrm{ij}}{)}_{n\times n}=(a_{\mathrm{ij}}/\sum _{i=1}^n{a}_{\mathrm{ij}}{)}_{n\times n}$ .

• Step 2: The n eigenvalues ${\lambda }_i(i=1,2,\dots ,n)$ of the judgment matrix $A=(a_{\mathrm{ij}}{)}_{n\times n}$ and the eigenvectors corresponding to ${\lambda }_i(i=1,2,\dots ,n)$ are calculated by formula (3)

  $$\mathit{A}{\mathit{v}}_{\mathit{i}}={\lambda }_i{\mathit{v}}_{\mathit{i}}(i=1,2,\dots ,n)$$ (3)

• Step 3: $e^{i\theta }$ is often used in signal processing research. It is good at tracking signal/frequency changes. That is, we can think of $e^{i\theta }$ as effectively containing all the information in the signal. Using this information, we can track, predict, and restore this signal. Here, we consider matrix A as a signal, and all eigenvalues and eigenvectors of A contain all the information of A. Therefore, we use $e^{\lambda }$ as the coefficient of feature weighting. We can obtain a new weight vector

  $$\mathit{w}=\sum _{i=1}^n{e}^{{\lambda }_i}{\mathit{v}}_{\mathit{i}}$$ (4)

Assume that the function $e^{{\lambda }_i}$ is n + 1 differentiable within a neighborhood of the point ${\lambda }_i=0$ . Within this neighborhood, proceed $e^{{\lambda }_i}$ Taylor around ${\lambda }_i=0$ , and we can obtain

$$e^{{\lambda }_i}=1+{\lambda }_i+\frac{{\lambda }_i^2}{2!}+\cdots +\frac{{\lambda }_i^n}{n!}+R_n\left({\lambda }_i\right)$$ (5)

In formula (5), $R_n({\lambda }_i)=((f^{(n+1)}(\xi ))/(n+1)!){({\lambda }_i)}^{(n+1)}$ , where $\xi$ is a value between ${\lambda }_i$ and 0.

In order to simplify the expression structure, the high-order terms of formula (5) are removed, and the first three terms of their expansion are taken

$$e^{{\lambda }_i}=1+{\lambda }_i+\frac{{\lambda }_i^2}{2!}$$ (6)

Substitute formula (6) into formula (4), and we obtain

$$\mathit{w}=\sum _{i=1}^n(1+{\lambda }_i+\frac{{\lambda }_i^2}{2!}){\mathit{v}}_{\mathit{i}}$$ (7)

by solving equation (7), we have

$$\mathit{w}=\sum _{i=1}^n{\mathit{v}}_{\mathit{i}}+\sum _{i=1}^n{\lambda }_i{\mathit{v}}_{\mathit{i}}+\sum _{i=1}^n\frac{{\lambda }_i^2{\mathit{v}}_{\mathit{i}}}{2!}$$ (8)

Substitute formula (3) into formula (8), and we obtain

$$\mathit{w}=\sum _{i=1}^n{\mathit{v}}_{\mathit{i}}+\sum _{i=1}^{n}A{\mathit{v}}_{\mathit{i}}+\sum _{i=1}^n\frac{A^2{\mathit{v}}_{\mathit{i}}}{2!}$$ (9)

From formula (9), we have

$$\mathit{w}=(I+A+\frac{A^2}{2!})•\sum _{i=1}^n{\mathit{v}}_{\mathit{i}}$$ (10)

• Step 4: We consider that the element $w_i(i=1,2,\dots ,n)$ of the weight vector $\mathit{w}=(w_1,w_2,\dots ,w_n{)}^T$ may have an imaginary number. Therefore, we use modulo value of $w_i(i=1,2,\dots ,n)$ and normalize it, then we have

$$w{\prime }_i=\frac{w_i}{\sum _{i=1}^n\left|w_i\right|},(i=1,2,\dots ,n)$$ (11)

Therefore, we obtain the feature-weighted weight vector

$$\mathit{w}\prime ={({w\prime }_1,{w\prime }_2,\dots ,{w\prime }_n)}^T$$ (12)

Finally, a consistency test is performed according to formulas (1) and (2).

Calculate the combined weight of each layer

According to Steps 1–4, the weight vector of each layer element to the element in the upper layer is calculated. The consistency test is performed according to formulas (1) and (2). From top to bottom, the weight vectors of the single-layer elements are synthesized and calculated. Finally, the combined weight of the solution layer relative to the total goal layer is obtained.

It is known that the weight vector of the sole element c in the total goal layer relative to itself is ${\mathit{w}}^{(1)}=1$ . Suppose the combined vector weight ${\mathit{w}}^{(2)}=(w_1^{(2)},w_2^{(2)},\dots ,w_n^{(2)}{)}^T$ is known. The elements in the solution layer related to the kth elements in the attribute layer are set as the single-weight vector ${\mathit{q}}^k={(q_1^k,q_2^k,\dots ,q_m^k)}^T$ , where the weight of the element is taken as 0, which is not subject to the kth element $(1\le k\le n)$ . $\mathit{q}={({\mathit{q}}^1,{\mathit{q}}^2,\dots ,{\mathit{q}}^n)}_{m\times n}$ represents the combined weights of m elements in the solution layer related to every element in the attribute layer. Then the combined weight vector of the elements in the solution layer for the general criterion $(c)$ is given by

$${\mathit{w}}^{\left(3\right)}={(w_1^{\left(3\right)},w_2^{\left(3\right)},\dots ,w_m^{\left(3\right)})}^T=\mathit{q}{\mathit{w}}^{\left(2\right)}$$ (13)

Similarly, consistency test is performed layer by layer from top to bottom.

If the consistency index $C.I{.}_k^{(3)}$ , the random index $R.I{.}_k^{(3)}$ , and the consistency ratio $C.R{.}_k^{(3)}$ related to the kth element in the attribute layer have been determined, then the comprehensive index $C.I{.}^{(3)}$ and $R.I{.}^{(3)}$ of the solution layer are given by

$$C.I{.}^{\left(3\right)}=(C.I{.}_1^{\left(3\right)},C.I{.}_2^{\left(3\right)},\dots ,C.I{.}_n^{\left(3\right)}){\mathit{w}}^{\left(2\right)}$$ (14)

$$R.I{.}^{\left(3\right)}=(R.I{.}_1^{\left(3\right)},R.I{.}_2^{\left(3\right)},\dots ,R.I{.}_n^{\left(3\right)}){\mathit{w}}^{\left(2\right)}$$ (15)

Substitute formulas (14) and (15) into formula (2), we obtain

$$C.R{.}^{\left(3\right)}=\frac{C.I{.}^{\left(3\right)}}{R.I{.}^{\left(3\right)}}$$ (16)

When $C.R{.}^{(3)}<0.1$ , all judgments of the hierarchical hierarchy above level 3 have total satisfactory consistency.

Example

Description of the problem

Country A decided to purchase jet fighters from country B. Government officials in country B provided the characteristics of the three types of fighters. When deciding to buy one of the fighters, it is often not directly to compare the fighters as a whole, but to select some intermediate indicators for investigation because there are many incomparable factors. The analysis team from Country A thinks that five characteristics should be considered: maximum speed ( $P_1$ /Mach), range ( $P_2$ /n mile), maximum load ( $P_3$ /lb), price ( $P_4$ /million USD), and combat radius ( $P_5$ /km). The five attribute values for each type of fighters (solution) are shown in Table 3.

Table 3.

Fighter selection table.

Options $S_i$	Attributes
	Maximum speed (Mach)	Range (n mile)	Maximum load (lb)	Purchase price (million USD)	Combat radius (km)
$S_1$	2.0	1500	20,000	5.5	1100
$S_2$	2.5	2700	18,000	6.5	2000
$S_3$	2.2	1800	20,000	5.0	1800

The decision-maker then considers the pros and cons of the three types of fighters under the above five feature attributes. With this sort of order, the final decision is made. In the decision-making process, the ranking of the merits of each of the three types of fighters is generally inconsistent. Therefore, the decision-maker must first make an estimate of the importance of the five feature attributes and give a sort. Then the decision-maker finds out the sorting weights of each of the three fighters for each attribute. Finally, the information data are combined to obtain the ranking weights for the total goal, that is, the purchase of fighters. With this weight vector, the decision is easy.

Construct a comparison judgment matrix and determine the weight

First, establish the hierarchical structure of the decision problem. Then, according to the steps of constructing the AHP decision model under the condition of Taylor formula, the decision-maker judges by preference and obtains the comparison judgment matrix of the attribute layer as follows

$$A^{\left(1\right)}=\left(\begin{array}{ccccc}1\hfill & 2\hfill & 7\hfill & 5\hfill & 5\hfill \\ \multicolumn{1}{c}{\frac{1}{2}}& 1\hfill & 4\hfill & 3\hfill & 3\hfill \\ \multicolumn{1}{c}{\frac{1}{7}}& \frac{1}{4}\hfill & 1\hfill & \frac{1}{2}\hfill & \frac{1}{3}\hfill \\ \multicolumn{1}{c}{\frac{1}{5}}& \frac{1}{3}\hfill & 2\hfill & 1\hfill & 1\hfill \\ \multicolumn{1}{c}{\frac{1}{5}}& \frac{1}{3}\hfill & 3\hfill & 1\hfill & 1\hfill \end{array}\right)$$

It is known that ${\mathit{w}}^{(1)}=1$ . By performing consistency tests according to Table 2 and formulas (1) and (2), we can obtain $C.I{.}_1^{(2)}=0.0180$ , $R.I{.}_1^{(2)}=1.12$ , and $C.R{.}_1^{(2)}=(C.I{.}_1^{(2)}/R.I{.}_1^{(2)})=0.0161<0.1$ . It can be seen that the degree of inconsistency of $A^{(1)}$ is acceptable.

By performing consistency tests according to formulas (14)–(16), we can obtain $C.I{.}^{(2)}=C.I{.}_1^{(2)}•{\mathit{w}}^{(1)}=0.0180$ , $R.I{.}^{(2)}=R.I{.}_1^{(2)}•{\mathit{w}}^{(1)}=1.12$ , and $C.R{.}^{(2)}=(C.I{.}^{(2)}/R.I{.}^{(2)})$ $=0.0161<0.1$ . We can see that the total hierarchical structure of the fighter’s selection problem has total satisfactory consistency at all levels above the second level.

According to Steps 1–4, the weighted feature-weight vector for the attribute layer element is calculated as

$${\mathit{w}}^{\left(2\right)}={(0.4601,0.2966,0.0592,0.0975,0.0866)}^T$$

This weight vector can be seen as an estimate of the importance of the five feature attributes by the decision-maker.

Then the decision-maker judges by preference and obtains the comparative judgment matrix of the solution layer as follows

$$\begin{array}{c}{B}^{\left(1\right)}=\left(\begin{array}{ccc}1\hfill & \frac{1}{3}\hfill & \frac{1}{8}\hfill \\ \multicolumn{1}{c}{3}& 1\hfill & \frac{1}{3}\hfill \\ \multicolumn{1}{c}{8}& 3\hfill & 1\hfill \end{array}\right),B^{\left(2\right)}=\left(\begin{array}{ccc}1\hfill & 2\hfill & 5\hfill \\ \multicolumn{1}{c}{\frac{1}{2}}& 1\hfill & 2\hfill \\ \multicolumn{1}{c}{\frac{1}{5}}& \frac{1}{2}\hfill & 1\hfill \end{array}\right)\hfill \\ B^{\left(3\right)}=\left(\begin{array}{ccc}1\hfill & 1\hfill & 3\hfill \\ \multicolumn{1}{c}{1}& 1\hfill & 3\hfill \\ \multicolumn{1}{c}{\frac{1}{3}}& \frac{1}{3}\hfill & 1\hfill \end{array}\right),B^{\left(4\right)}=\left(\begin{array}{ccc}1\hfill & 3\hfill & 4\hfill \\ \multicolumn{1}{c}{\frac{1}{3}}& 1\hfill & 1\hfill \\ \multicolumn{1}{c}{\frac{1}{4}}& 1\hfill & 1\hfill \end{array}\right),\hfill \\ \multicolumn{1}{c}{B^{\left(5\right)}=\left(\begin{array}{ccc}1\hfill & 1\hfill & \frac{1}{4}\hfill \\ \multicolumn{1}{c}{1}& 1\hfill & \frac{1}{4}\hfill \\ \multicolumn{1}{c}{4}& 4\hfill & 1\hfill \end{array}\right)}\end{array}$$

By performing consistency tests according to Table 2 and formulas (1) and (2), we can get the following results

$$\begin{array}{c}\multicolumn{1}{c}{C.I{.}_1^{\left(3\right)}=0.0008,R.I{.}_1^{\left(3\right)}=0.52,\mathrm{and}C.R{.}_1^{\left(3\right)}}\\ \multicolumn{1}{c}{=\frac{C.I{.}_1^{\left(3\right)}}{R.I{.}_1^{\left(3\right)}}=0.0015<0.1}\\ \multicolumn{1}{c}{C.I{.}_2^{\left(3\right)}=0.0028,R.I{.}_2^{\left(3\right)}=0.52,\mathrm{and}C.R{.}_2^{\left(3\right)}}\\ \multicolumn{1}{c}{=\frac{C.I{.}_2^{\left(3\right)}}{R.I{.}_2^{\left(3\right)}}=0.0054<0.1}\\ \multicolumn{1}{c}{C.I{.}_3^{\left(3\right)}=0,R.I{.}_3^{\left(3\right)}=0.52,\mathrm{and}C.R{.}_3^{\left(3\right)}}\\ \multicolumn{1}{c}{=\frac{C.I{.}_3^{\left(3\right)}}{R.I{.}_3^{\left(3\right)}}=0<0.1}\\ \multicolumn{1}{c}{C.I{.}_4^{\left(3\right)}=0.0046,R.I{.}_4^{\left(3\right)}=0.52,\mathrm{and}C.R{.}_4^{\left(3\right)}}\\ \multicolumn{1}{c}{=\frac{C.I{.}_4^{\left(3\right)}}{R.I{.}_4^{\left(3\right)}}=0.0088<0.1}\\ \multicolumn{1}{c}{C.I{.}_5^{\left(3\right)}=0,R.I{.}_5^{\left(3\right)}=0.52,\mathrm{and}C.R{.}_5^{\left(3\right)}}\\ \multicolumn{1}{c}{=\frac{C.I{.}_5^{\left(3\right)}}{R.I{.}_5^{\left(3\right)}}=0<0.1}\end{array}$$

It can be seen that the degrees of inconsistency of $B^{(1)}$ , $B^{(2)}$ , $B^{(3)}$ , $B^{(4)}$ , and $B^{(5)}$ are acceptable.

By performing consistency tests according to formulas (14)–(16), we can get the following results

$$\begin{array}{c}\multicolumn{1}{c}{C.I{.}^{\left(3\right)}=(C.I{.}_1^{\left(3\right)},C.I{.}_2^{\left(3\right)},C.I{.}_3^{\left(3\right)},C.I{.}_4^{\left(3\right)},C.I{.}_5^{\left(3\right)}){\mathit{w}}^{\left(2\right)}}\\ \multicolumn{1}{c}{=0.0016}\\ \multicolumn{1}{c}{R.I{.}^{\left(3\right)}=(R.I{.}_1^{\left(3\right)},R.I{.}_2^{\left(3\right)},R.I{.}_3^{\left(3\right)},R.I{.}_4^{\left(3\right)},R.I{.}_5^{\left(3\right)}){\mathit{w}}^{\left(2\right)}}\\ \multicolumn{1}{c}{=0.52}\\ \multicolumn{1}{c}{C.R{.}^{\left(3\right)}=\frac{C.I{.}^{\left(3\right)}}{R.I{.}^{\left(3\right)}}=0.0031<0.1}\end{array}$$

We can see that the total hierarchical structure of the fighter’s selection problem has total satisfactory consistency at all levels above the third level.

According to Steps 1–4, the weighted feature-weight vectors for the solution layer elements are calculated as

$$\begin{array}{c}{\mathit{q}}^1={(0.1028,0.3039,0.5933)}^T\hfill \\ {\mathit{q}}^2={(0.6699,0.2297,0.1005)}^T\hfill \\ {\mathit{q}}^3={(0.3773,0.4270,0.1957)}^T\hfill \\ {\mathit{q}}^4={(0.7059,0.1597,0.1345)}^T\hfill \\ {\mathit{q}}^5={(0.1169,0.1477,0.7354)}^T\hfill \end{array}$$

They can be regarded as the scores of the three types of fighters in the five attribute values.

Sorting of solutions

Integrate all the information data and calculate the total score of the three types of fighter according to formula (13). That is $\mathit{q}=({\mathit{q}}^1,{\mathit{q}}^2,{\mathit{q}}^3,{\mathit{q}}^4,{\mathit{q}}^5)=$ $\left(\begin{array}{ccccc}0.1028\hfill & 0.6699\hfill & 0.3773\hfill & 0.7059\hfill & 0.1169\hfill \\ \multicolumn{1}{c}{0.3039}& 0.2297\hfill & 0.4270\hfill & 0.1597\hfill & 0.1477\hfill \\ \multicolumn{1}{c}{0.5933}& 0.1005\hfill & 0.1957\hfill & 0.1345\hfill & 0.7354\hfill \end{array}\right)$ . We can obtain ${\mathit{w}}^{(3)}=\mathit{q}{\mathit{w}}^{(2)}=(\begin{array}{ccc}0.3473\hfill & 0.2616\hfill & 0.3912\hfill \end{array}{)}^T$ . We call this as the total score. The results are shown in Table 4.

Table 4.

Fighter plan table.

	$P_1$ 0.4601	$P_2$ 0.2966	$P_3$ 0.0592	$P_4$ 0.0975	$P_5$ 0.0866	Total score
$S_1$	0.1028	0.6699	0.3773	0.7059	0.1169	0.3473
$S_2$	0.3039	0.2297	0.4270	0.1597	0.1477	0.2616
$S_3$	0.5933	0.1005	0.1957	0.1345	0.7354	0.3912

According to Table 3, we can know that the ranking is $S_3\succ S_1\succ S_2$ . After comparison, the third type of fighter is the best choice. This is the Taylor-based feature-weighted multi-attribute decision-making method proposed in this article.

Conclusion

Aiming at the problems of classical multi-attribute decision-making, this article proposes a feature-weighted multi-attribute decision-making method based on Taylor expansion. Based on the AHP model, we have established a new feature-weighted decision model. We construct the feature-weighted coefficients by using the natural base and the eigenvalues of the decision matrix, and transform the coefficients by Taylor expansion. Then we weight and normalize all the feature vectors of the decision matrix, and finally get a new weight vector and sort results. Combined with the example, the weight vector weighted by the method not only comprehensively reflects the objective situation of the attribute information but also completely reflects the subjective will of the decision-maker (expert). Therefore, the method avoids the loss of effective decision information. Since the method does not change the elements and structure of the original comparison judgment matrix, it is not necessary to re-perform a conformance test on the comparison judgment matrix. Compared with the AHP, this method saves the decision time of the decision-maker (expert).