Uncertain rank correlation analysis based on normal interval value

Abstract

To handle the randomness of individual judgments, a novel method, called uncertain rank correlation analysis (RCA), is proposed. Conventional RCA cannot effectively deal with the uncertainty in the comprehensive evaluation. However, normal distribution interval number is often used to express the uncertainty. Adopting normal cloud theory, this paper extends conventional RCA into the field of uncertainty, via the transformation and operation of the normal distribution interval-valued number. An example with uncertainties is given to verify the effectiveness of the proposed method.

Keywords

AHP comprehensive evaluation rank correlation interval value cloud theory

1 Introduction

Analytic hierarchy process (AHP) [9 , 12] is a widely applied comprehensive evaluation approach, and the construction of the consistent judgment matrix is a crucial step of this method [9].

In practice, however, the judgment matrix is often inconsistent. Saaty [14] proposed by the consistency ratio (CR) <0.1 as a consistent criterion for judging, but its reliability and validity have been questioned.

In response to this difficulty, on the one hand, many scholars have dedicated to rectify the consistency of a judgment matrix [1 , 17]. However, there exists many shortcomings, such as the results are not unique, the computational complexity increased. Moreover, the judgment matrix rectified is inconsistent with the original one.

On the other hand, inspired by the AHP method, many new methods have been put forward, wherein an influential method is the rank correlation analysis (RCA) [2 , 18], also known as G1 method. The basic principle of the RCA is to order all the indexes according to their importance degree, and then compare the two adjacent indexes, to avoid inconsistency of pairwise comparisons between any two indexes. Because of its concise, practical and scientific advantages, RCA has been applied in many fields.

In addition, on the other hand, the comparison of the two indicators usually involves uncertainty. In many cases, the use of precise information to describe complex problems is also impractical. Furthermore, the precise description does not exist objectively. In practical decision making, it is difficult to give accurate values, but it is easy to estimate the upper and lower bounds of information [8, 13].

In [2], conventional RCA is expanded to uniformly distributed interval value. Using interval endpoints, the interval value is converted into the real number. The method is simple and contains decision makers’ uncertain factors. In order to simplify the computations, the method assumes that the interval value is uniformly distributed and treats it as a real number. In fact, by the central limit theorem, property values given by decision makers are not only stability but also random. So, it is more realistic that the interval values are normally distributed. Meanwhile, interval judgments are more natural for human beings than real numbers. In the interval judgment, people also have a higher sense of control and flexibility.

So, the RCA, as well as its improved forms, to a certain extent, avoids the complexity of conventional AHP in constructing the consistent judgment. However, the most important uncertainty in human cognition comes from randomness or ambiguity. The inherent cognitive limitations of humans often lead to inconsistencies in the results of the RCA approach, or lead to the failure of the RCA method to be effectively solved. Interval value is an effective approach to model subjective uncertainty. The conventional RCA cannot handle the decision making problem with interval value. Therefore, decision making based on the RCA is often of no practical significance.

The normal distribution is used to represent the randomness or fuzziness of the interval value, so that the original RCA method is effectively extended to the uncertain domain. We develop novel theorems and finally promote an improved uncertain RCA method. Meanwhile, as the adoption of the interval value, direct interval arithmetic is somehow complex in the process. As to promote the practicability, we adopt the Cloud model [6, 7] to express the interval value. The cloud model mainly reflects the uncertainty of objective things and fuzzy ideas in human decision-making. We find that the interval numbers that satisfy the normal distribution have properties of fuzziness and randomness, which are consistent with the cloud model. And, the normal cloud owns a simple, consistent algorithm. Hence, using the cloud model, the normal distribution interval value can be converted into a normal cloud with fewer parameters, to simplify operations of interval weights.

2 Preliminaries

Some basic definitions are reviewed in this section.

Definition 1. Suppose that $\tilde{a} = [a^{L}, a^{U}] = {x | a^{L} \leq x \leq a^{U}, a^{L}, a^{U} \in R}$ is an interval-valued number, if x ∼ N (μ, σ²), then $\tilde{a}$ is a normal distribution interval-valued number.

Definition 2. [6] Suppose that U is a domain, and C is a qualitative concept in U. x belongs U and is a random concept in C. The certainty degree of x in U is expressed as: μ : U → [0, 1], ∀x ∈ U, x → μ (x). So x is a normal cloud distribution in U, each (x, μ (x)) is a Cloud drop.

Definition 3. [6] The three parameters (Ex, En, He) represents the uncertainty of a cloud drop, where Ex represents the mathematical expectation of a concept in a particular domain, En means the uncertainty, that is, the degree of randomness, discreteness, and ambiguity of the first order, He represents the higher-order uncertainty degree of En, which is function equivalent to the entropy of entropy.

Definition 4. [6, 16] The assumptions of U, C, x and μ (x) are the same as before. If x ∼ N (Ex, |y|), where y ∼ N (En, He), and μ (x) = exp(- (xEx) ²/2y²), then in domain U, the distribution of x is named a normal Cloud, (x, μ (x)) is a normal Cloud drop.

3 Conventional RCA

Conventional RCA comprises three steps [2]:

Step 1. Establishment of the ordinal relation

Definition 5. Based on the specific evaluation criteria, if the importance of X_i is greater than X_j, then this relationship is expressed by X_i ≻ X_j.

Definition 6. If the indexes of X₁, …, X_m has relationship $X_{i} ≻ X_{j} ≻ \dots ≻ X_{k} i, j, k = 1, 2, \dots, m .$ (1)

Then it means that according to the symbol ≻, the ordinal relation of the indexes is determined.

Step 2.m times comparison of the relative importance between adjacent indexes

The importance of adjacent indexes between X_k-1 and X_k can be expressed as follows: $a_{k} = ω_{k - 1} / ω_{k} k = m, m - 1, \dots, 2$ (2)

The relative importance of each index can be calculated according to the ordinal relation between the indexes. The values of a_k are shown in Table 1.

Table 1

The values of a_k

a _k	Value Inscription
1	X_k-1 and X_k are equally important
1.2	X_k-1 is slightly more important than X_k
1.4	X_k-1 is more important than X_k
1.6	X_k-1 is much more important than X_k
1.8	X_k-1 is extremely more important than X_k

There is the following theorem for the numerical constraint on the a_k.

Theorem 1.IfX₁, …, X_mhas ordinal relation (1), then $a_{k - 1} > 1 / a_{k} k = m, \dots, 2$ (3)

Theorem 2. If a _k satisfies Theorem 1, the ω _m is calculated as: $ω_{m} = {(1 + \sum_{k = 2}^{m} \prod_{i = k}^{m} a_{i})}^{- 1}$ (4)

4 Uncertain RCA

4.1 Interval assignment ratios

In practice, due to the uncertainty of human thinking, decision maker usually cannot give an accurate ratio number of a_k, but will be easy to give its uncertain interval value of ${\tilde{a}}_{k}$ . If the adjacent indexes’ comparison is uncertain, random variables will be used on quantitative judgment, and that is a random variable interval-valued ratio.

Interval-valued ratios satisfy normal distribution and meet with the general interval arithmetic rule. In consistent with the traditional RCA’ ratio, assuming that the upper and lower range of the ratios are respectively 1.9 and 1/1.9. Interval-valued rational assignments are given in Table 2.

Table 2
The values of ${\tilde{a}}_{k}$

${\tilde{a}}_{k}$ Value Inscription

1 X_k-1 and X_k are equally important

(1.0, 1.3] X_k-1 is slightly more important than X_k

(1.3, 1.5] X_k-1 is more important than X_k

(1.5, 1.7] X_k-1 is much more important than X_k

(1.7, 1.9] X_k-1 is extremely more important than X_k

${\tilde{a}}_{k}$	Value Inscription
1	X_k-1 and X_k are equally important
(1.0, 1.3]	X_k-1 is slightly more important than X_k
(1.3, 1.5]	X_k-1 is more important than X_k
(1.5, 1.7]	X_k-1 is much more important than X_k
(1.7, 1.9]	X_k-1 is extremely more important than X_k

4.2 Interval-valued judgment matrix and feature vector

Definition 7. If 1) ${\tilde{a}}_{ii} = 1 (\forall i)$ ; 2) ${\tilde{a}}_{ij} (\forall i, j)$ is the interval-valued number; 3) ${\tilde{a}}_{ij} = 1 / {\tilde{a}}_{ji} (\forall i, j)$ , then $A = ({\tilde{a}}_{ij})$ is called an interval-valued judgment matrix, which can be writing as $A = ([a_{ij}^{L}, a_{ij}^{U}])$ .

Definition 8. If A ω = λ_max ω , then ω is the interval -valued feature vector, where $ω = ({\tilde{ω}}_{1}, \dots, {\tilde{ω}}_{n})^{T} = ([ω_{1}^{L}, ω_{1}^{U}], \dots, [ω_{n}^{L}, ω_{n}^{U}]) (i = 1, \dots, n)$ .

Notably, in the conventional RCA, the Equation (4) is calculated according to ω₁ + ω₂ + … + ω_m = 1. Obviously, it is no longer adapted to the interval-valued feature or weight vector; therefore, we should propose a new method to calculate interval-valued weights. In addition, traditional RCA uses only m indexes to judge and contrast. Consequently, with a combination of interval arithmetic rules, filling the rest m² - (m - 1) elements of the interval-valued judgment matrix is needed.

Filling rules of elements are as follows:

Rule 1. If ${\tilde{a}}_{ij} = [a_{ij}^{L}, a_{ij}^{U}]$ is given, then ${\tilde{a}}_{ji} = [1 / a_{ij}^{U}, 1 / a_{ij}^{L}] (\forall i, j)$ .

Proof. Based on the interval-valued algorithms, we have ${\tilde{ω}}_{i} / {\tilde{ω}}_{j} = [\frac{ω_{i}^{L}}{ω_{j}^{U}}, \frac{ω_{i}^{U}}{ω_{j}^{L}}]$ , ${\tilde{ω}}_{j} / {\tilde{ω}}_{i} = [\frac{ω_{j}^{L}}{ω_{i}^{U}}, \frac{ω_{j}^{U}}{ω_{i}^{L}}]$ .

According to the actual meaning of the judgment matrix, we should have ${\tilde{a}}_{ij} \in [\frac{ω_{i}^{L}}{ω_{j}^{U}}, \frac{ω_{i}^{U}}{ω_{j}^{L}}]$ , ${\tilde{a}}_{ji} \in [\frac{ω_{j}^{L}}{ω_{i}^{U}}, \frac{ω_{j}^{U}}{ω_{i}^{L}}]$ .

Rule 2. If ${\tilde{a}}_{ik} = [a_{ik}^{L}, a_{ik}^{U}]$ and ${\tilde{a}}_{kj} = [a_{kj}^{L}, a_{kj}^{U}]$ are given, then ${\tilde{a}}_{ij} = [a_{ik}^{L} a_{kj}^{L}, a_{ik}^{U} a_{kj}^{U}]$ , ∀ i, j, k, k ≠ i, j.

Proof. According to the principle of transfer and interval algebra, it is easy to prove.

Rule 3. ${\tilde{a}}_{ii} = [1, 1] ≜ 1, \forall i$ .

Proof. In Rule 2, let i = j, and then in conjunction with Rule 1, the proof ends.

Rule 4. If $min {a_{ij}^{L}, a_{ij}^{U}} < 1 / 1.9$ , then the lower bound of ${\tilde{a}}_{ij}$ is assigned to 1/1.9; If $max {a_{ij}^{L}, a_{ij}^{U}} > 1.9$ , then the upper bound of ${\tilde{a}}_{ij}$ is assigned to 1.9.

In Rule 4, considering the complexity of things, and the diversity of human knowledge, if a filled-element does not belong to [1/1.9, 1.9], that is to say, beyond the assignment ratios, then it will be amended.

4.3 Interval-valued matrix converted to cloud matrix

According to Rules 1∼4, we can obtain the interval-valued judgment matrix. However, the traditional random sampling interval estimation [10], interval arithmetic and many other methods are too complicated to handle it. This will result in a large amount of computing resource requirements. With fewer parameters, the cloud model is simple. What’s more, the normal cloud is uniform in meaning with interval values. Therefore, we convert interval-valued judgment matrix to normal cloud judgment matrix. The algorithm is:

Theorem 3.For ${\tilde{a}}_{ij} = [a_{ij}^{L}, a_{ij}^{U}]$ , its corresponding cloud numerical parametersC_ij = (Ex_ij, En_ij, He_ij) is calculated as [16]: ${\begin{matrix} {Ex}_{ij} = (a_{ij}^{L} + a_{ij}^{U}) / 2 \\ {En}_{ij} = (a_{ij}^{U} - a_{ij}^{L}) / 6 \\ {He}_{ij} = max | {En}_{ijk} - {En}_{ij} | / 3 \\ = max | a_{ik}^{U} a_{kj}^{U} - a_{ik}^{L} a_{kj}^{L} + a_{ij}^{L} - a_{ij}^{U} | / 18, k \neq i, j \end{matrix}$ (5)

The calculation algorithm for cloud parameters is explained as follows [7]: Ex is the expression of interval expectations. In a normally distributed case, the expectation is equivalent to the middle point of the interval, which is easy to understand. According to the 3En rule of the normal distribution, basically, the interval [Ex - 3En, Ex + 3En] is considered as the actual possible range of the random variables (The probability of falling outside of the interval is considered to be less than 3/1000 in practical problems). Therefore, the upper bound and the fuzziness of interval numbers can be expressed by 6En. He is computed by the judgment matrix.

In a same universe of discourse, two Clouds C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂) are given. Then they can be calculated according to the following rules in Table 3 [3 , 16]. The arithmetic operation result of C₁ and C₂ is C and its digital characters is C (Ex, En, He). The operational rules are defined as follows.

Table 3

The algorithm of Cloud [3 , 16]

Operator	Ex	En	He
+	Ex₁ + Ex₂	$\sqrt{{En}_{1}^{2} + {En}_{2}^{2}}$	$\sqrt{{He}_{1}^{2} + {He}_{2}^{2}}$
−	Ex₁ - Ex₂	$\sqrt{{En}_{1}^{2} + {En}_{2}^{2}}$	$\sqrt{{He}_{1}^{2} + {He}_{2}^{2}}$
×	Ex₁ × Ex₂	$\sqrt{({Ex}_{2} {En}_{1})^{2} + ({Ex}_{1} {En}_{2})^{2}}$	$\sqrt{({Ex}_{2} {He}_{1})^{2} + ({Ex}_{1} {He}_{2})^{2}}$
÷	$\frac{{Ex}_{1}}{{Ex}_{2}}$	$\sqrt{({En}_{1} / {Ex}_{2})^{2} + ({En}_{2} / {Ex}_{1})^{2}}$	$\sqrt{({He}_{1} / {Ex}_{2})^{2} + ({He}_{2} / {Ex}_{1})^{2}}$
^∧ n	${Ex}_{1}^{n}$	$\sqrt{n} {Ex}_{1}^{n - 1} {En}_{1}$	$\sqrt{n} {Ex}_{1}^{n - 1} {He}_{1}$

Thus, for a given normal cloud judgment matrix $\tilde{A}$ , it can be wrote as:

$\begin{matrix} \tilde{A} = [\begin{matrix} C_{11} ({Ex}_{11}, {En}_{11}, {He}_{11}) & C_{12} ({Ex}_{12}, {En}_{12}, {He}_{12}) & \dots & C_{1 m} ({Ex}_{1 m}, {En}_{1 m}, {He}_{1 m}) \\ C_{21} ({Ex}_{21}, {En}_{21}, {He}_{21}) & C_{22} ({Ex}_{22}, {En}_{22}, {He}_{22}) & \dots & C_{2 m} ({Ex}_{2 m}, {En}_{2 m}, {He}_{2 m}) \\ ⋮ & ⋮ & ⋮ \\ C_{m 1} ({Ex}_{m 1}, {En}_{m 1}, {He}_{m 1}) & C_{m 2} ({Ex}_{m 2}, {En}_{m 2}, {He}_{m 2}) & \dots & C_{mm} ({Ex}_{mm}, {En}_{mm}, {He}_{mm}) \end{matrix}] \end{matrix}$

4.4 Cloud weights

Using the geometric mean method, normal cloud weights (interval-valued feature vectors) are calculated as: ${\tilde{ω}}_{i} = {(\prod_{j = 1}^{m} C_{ij})}^{1 / n} / \sum_{i = 1}^{m} {(\prod_{j = 1}^{m} C_{ij})}^{1 / n}$ (6)

4.5 Process of uncertain RCA

In summary, the uncertain RCA, based on normal interval value, is following these steps:

Step 1. Determine the order relation between indexes, considering the general situation, it can be referred to X₁ ≻ X₂ ≻ … ≻ X_m.

Step 2. According to ratios in Table 2, in X₁ ≻ X₂ ≻ … ≻ X_m, decisition-maker or expert give importance value between adjacent indexes.

Step 3. According to Rule 1 to 4, we fill and construct the interval-valued judgment matrix.

Step 4. According to Theorem 3 and the algorithm of Cloud, the interval-valued judgment matrix is converted to cloud judgment matrix; then according to the Formula 6, the interval-valued feature vectors (normal cloud weights) are obtained.

5 A numerical example

From the previous discussion, it can be seen that the traditional RCA is only a special case of the method proposed in this paper. Next, we mainly demonstrate how to apply the uncertain RCA.

There is an index-set {X₁, X₂, X₃, X₄}, the sequence has been determined as $X_{2} ≻ X_{1} ≻ X_{3} ≻ X_{4}$

Interval-valued ratios of adjacent indexes are $\begin{matrix} {\tilde{a}}_{21} & = & [1.0, 1.2], {\tilde{a}}_{13} = [1.2, 1.3], \\ {\tilde{a}}_{34} & = & [1.4, 1.6] \end{matrix}$

Fill it and the interval-valued judgment matrix is as follows: $\tilde{A} = [\begin{matrix} [1, 1] & [1 / 1.2, 1] & [1.2, 1.3] & [1.68, 1.9] \\ [1, 1.2] & [1, 1] & [1.2, 1.56] & [1.68, 1.9] \\ [1 / 1.3, 1 / 1.2] & [1 / 1.56, 1 / 1.2] & [1, 1] & [1.4, 1.6] \\ [1 / 1.9, 1 / 1.68] & [1 / 1.9, 1 / 1.68] & [1 / 1.6, 1 / 1.4] & [1, 1] \end{matrix}]$

As discussed in 4.3, we convert it into a cloud matrix. Then we calculate the weights of the cloud. The results are, $[\begin{matrix} ω_{1} \\ ω_{2} \\ ω_{3} \\ ω_{4} \end{matrix}] = [\begin{matrix} (0.2885, 0.0129, 0.0098) \\ (0.3095, 0.0191, 0.0011) \\ (0.2355, 0.0089, 0.0019) \\ (0.1666, 0.0033, 0.0008) \end{matrix}]$

Utilizing normal cloud weights obtained, we can get the comprehensive evaluation value, combined with the property values of the indexes.

It is worth noting that property values of the indexes can be both real and interval-valued number. However, traditional RCA can neither handle the situation of normal interval-valued ratios, nor can deal with the comprehensive evaluation that the index values are interval-valued numbers.

Furthermore, when each interval is degraded to one point, or when each interval is converted to an exact value, the decision matrix becomes an ordinary decision matrix, which is the case where the traditional RCA can handle it. So, the weight vectors obtained by traditional RCA are all “point” vectors, that is, the components of the vector are “point” data. In the actual decision-making problem, when the decision maker’s judgment is not completely determined and consistent, the weight vector derived by the above method can only approximate the sorting result. Therefore, the decision maker may wish to get a more “flexible” result. That is, a range of changes in the weight of each program. With the method proposed in this paper, the normal interval results, to represent the range of changes. This also adapts to the development trend of modern decision-making from “rigid” to “flexible”, from “pure qualitative or quantitative” to “integration of the qualitative and quantitative”.

In conclusion, compared with the traditional RCA method, this paper extends the point value judgment to the normal distribution interval, which is in accordance with the characteristics of human decision and avoids the loss of decision information. It makes the decision more accurate and reasonable.

6 Conclusion

In this paper, we propose a simple approach to model uncertainty in RCA. By using the normal distribution interval number, the method proposed in this paper expands the conventional certainty RCA into the field of uncertainty. The proposed method has the following advantages: 1) Considering the normal interval-valued ratios, it’s more realistic in applications and human decision-making habits; 2) Compared with the traditional RCA, there is a wider field of application for this proposed method. Future research will be concerned with group decision making.

Footnotes

Acknowledgments

This work was supported by the National Key Laboratory of Human Factors Engineering (Grant Nos. SYFD160051812, SYFD13006181, and 614222201060217), and the Equipment Pre-research on Common Technology (Grant No. 41402060101).

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Uncertain rank correlation analysis based on normal interval value

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Conventional RCA

4.1 Interval assignment ratios

Table 2 The values of a ˜ k a ˜ k Value Inscription 1 Xk-1 and X k are equally important (1.0, 1.3] Xk-1 is slightly more important than X k (1.3, 1.5] Xk-1 is more important than X k (1.5, 1.7] Xk-1 is much more important than X k (1.7, 1.9] Xk-1 is extremely more important than X k

4.3 Interval-valued matrix converted to cloud matrix

5 A numerical example

6 Conclusion

Footnotes

Acknowledgments

References