Kullback-Leibler Distance Based Generalized Grey Target Decision Method With Index and Weight Both Containing Mixed Attribute Values

This paper proposes a generalized grey target decision method (GGTDM) with index and weight both containing mixed attribute values based on Kullback-Leibler (K-L) distance. The proposed approach builds the weight function converting the mixed attribute-based weights into the certain number-based weights and takes the comprehensive weighted K-L distance as the decision-making basis (DMB). The proposed approach conducts its task in the following steps. First, all indices of alternatives are converted into binary connection numbers. Second, the two-tuple (determinacy, uncertainty) numbers originated from index binary connection numbers are obtained. Third, the two-tuple (determinacy, uncertainty) numbers of target center are calculated. Following that the certain number-based weights are obtained by the weight function. Then the comprehensive weighted K-L distance of each alternative and its target center is calculated. And the final decision making is based on the value of comprehensive weighted K-L distance with which the smaller the better. A case study illustrates the proposed approach with its effectiveness of converting the uncertain weights into the certain weights and the accurate results comparing with other decision-making methods.


I. INTRODUCTION
The decision-making basis (DMB) of grey target decision method (GTDM) is referred to as the target center distance (TCD), which is the distance of each alternative and its target center. In certain number-based GTDM, Euclidean distance and Mahalanobis distance are often applied to obtain the TCDs [1], [2]. However, the mixed attribute-based GTDM obtains the TCD in different ways: at first, the conventional Euclidean distance-based method was reported [3]- [5]. Then the equivalent methods including cobweb area and correlation coefficient appeared [6], [7]. Besides, the proximity-based method, entropy-based method and Gini-Simpson index-based method were also investigated, as is named as generalized grey target decision method(GGTDM) [8]- [10]. The GGTDM differs from the The associate editor coordinating the review of this manuscript and approving it for publication was Alba Amato . conventional one in the calculation process, but obeys the same principle [8], [9], [11]- [13]. Given that uncertainty originated from the uncertain number, an effective tool is required to measure it in the mixed attribute-based GTDM. The entropy is often used to measure the uncertainty, and entropy theory has many forms. As an important form, cross-entropy has been used widely. Ioannis and George applied cross-entropy to pattern recognition with intuitive fuzzy information and achieved desirable results [14]. Li and Wu adopted intuitionistic fuzzy cross-entropy to solve preference problem on alternatives [15]. Xia and Xu applied cross-entropy to group decision making under intuitionistic fuzzy environment [16]. Smieja and Geiger used cross-entropy theory to clustering under information restriction [17]. Tang et al. proposed an optimization algorithm based on cross-entropy [18]. While the kullback-Leibler (K-L) distance derived from cross-entropy can well reflect the degree of nearness between the two vectors, which has the VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ meaning of distance [19]. Therefore, the K-L distance can be applied to GGTDM for mixed attributes. The GGTDM with index and weight both containing mixed attribute values has been investigated. However, the previous research mainly adopts the comprehensive weighted proximity(CWP) and comprehensive weighted Gini-Simpson index(CWGSI) as the DMBs [12], [20]. Accordingly, a few of methods converting uncertain weights into certain weights were presented. At first, the proximitybased method [12] and the module-based method [21] were proposed to obtain the certain weights. Later, different weight functions were also built to covert uncertain weights into certain weights [10], [22]. However, this work proposes a K-L distance-based method involving mixed attribute weight values. The decision process is as follows. First, all alternative indices are transformed into binary connection numbers and converted into two-tuple (determinacy, uncertainty) numbers. Second, the two-tuple (determinacy, uncertainty) numbers of target center under all attributes are obtained. Third, the certain number-based weights are determined by the weight function. Then the comprehensive weighted K-L distance of each alternative and its target center is calculated. And the final decision making is based on the value of comprehensive weighted K-L distance with which the smaller the better.

II. BASIC THEORY
A. KULLBACK-LEIBLER DISTANCE Definition 1: Kullback-Leibler distance [14], [19]. Let X and Y be two random variables with their discrete distributions denoted by X = (x 1 , x 2 , · · · , x m ) T and Y = (y 1 , y 2 , · · · , y m ) T respectively, where x j , y j ≥ 0, j = 1, 2, . . . , m, 1 = m j=1 x j ≥ y j , then the Kullback-Leibler distance of X and Y can be obtained as follows: where H (X , Y ) has following properties: When x j = 0, y j = 0, H (X , Y ) → ∞. Therefore, it is necessary to improve the original K-L distance in practice. The improved version of K-L distance is as follows [14], [19]:

B. UNCERTAIN NUMBER
The objective thing could be recognized by human being in an uncertain state due to the complex, uncertain thing itself, and people's knowledge limitation and ambiguous recognition abilities. There are some ways to solve this uncertainty, such as fuzzy mathematics, grey system theory, set-pair analysis theory and rough set theory. And the uncertain number is often used to describe the characteristics, which is opposite to deterministic real number. In practical use, the data error brought by measurement and calculation, and data missing originated from incomplete information, the original data representing the characteristics of object may be uncertain. For this reason, the definitions of interval number (including the extended n-parameter interval number) and the binary connection number which can unify the determinacy and uncertainty of an uncertain number are given.

1) INTERVAL NUMBER AND MULTI-PARAMETER INTERVAL NUMBER
Definition 2: Let R be a real number domain; ifx is an interval number, then it can be represented by [x L , x U ], where x L and x U are the upper and lower limits, respectively, satisfying 0 < x L < x U ∈ R [23], [24]. The interval number is an uncertain number with upper and lower limits. Furthermore, if an interval number contains more than two parameters in the extension of the primary form, then it can be termed an n-parameter interval number (also called multi-parameter interval number). If n = 3 or 4, then it can be named as three-parameter interval number or four-parameter interval number, respectively, and expressed by [ the n-parameter interval number can be expressed as [x 1 , x 2 , . . . x j , . . . x n ], where x j satisfy 0 < x 1 < . . . . . . < x j < . . . . . . < x n ∈ R. For an n-parameter interval number, if n = 3 or 4, it can be read as triangular fuzzy number or trapezoidal fuzzy number, respectively, in fuzzy theory; while it can also be called triangular grey number, or trapezoidal grey number, respectively, in grey theory. In this research, it is spoken of as the n-parameter interval number or multi-parameter interval number.

2) BINARY CONNECTION NUMBER
Definition 3: Let R be a real domain; µ + σ i is a binary connection number, where µ denotes the deterministic term, σ denotes the uncertain term and i is a variable term unifying the determinacy and uncertainty of an uncertain number, µ, σ ∈ R and i ∈ [−1, 1].
Definition 4: Let x and v be the mean value and deviation value of the n(n ≥2) parameters of an interval numberx, respectively, then is called mean value-deviation value connection number, where x, ξ, ψ and v are obtained by using Eqs(4) to (7): where the mean value x can be regarded as relatively deterministic measure of n(n ≥ 2) parameters about x, the standard deviation ξ and the maximum deviation ψ are relatively uncertain measure of n parameters aboutx [8], [25]. Definition 5: The mutual interaction of the mean value x and the deviation value v of the mean value-deviation value connection number u(x, v) can be mapped to the determinacyuncertainty space (D-U space), then (x, v) is supposed to be the micro-vector in the D-U space [23], [24].

III. METHOD FOR CONVERTING MIXED ATTRIBUTE WEIGHTS INTO CERTAIN WEIGHTS A. THE DIFFICULTY OF GENERALIZED GREY TARGET DECISION MAKING WITH INDEX AND WEIGHT BOTH CONTAINING MIXED ATTRIBUTE VALUES
The difficulty of generalized grey target decision making with index and weight both containing mixed attribute values lies in [12]:(1) different types of weight values cannot be aggregated directly with mixed index values;(2) the simple conversion of different types of data into deterministic weights without scientific and reasonable method may lead to large deviation and affect the accuracy of decision;(3) how to address the mixed index weights in a unified, simple and accurate way. Let w t (t = 1, 2, · · · , m) be the attribute value that could be a real number or an uncertain number. When it is an uncertain number, the w t can be expressed as interval number or multi-parameter interval number form as follows: The following relationship is generally established: where ω L t and ω U t represent the lower and upper limits of an uncertain number, respectively, and the real number is the deterioration of the interval number. For example, if the parameters ω L t and ω U t of an interval number are equal with each other, the uncertain number will be a real number.

1) Transformation of attribute weight values into binary connection numbers
First of all, the basic parameters of each type of weight data including mean values, standard deviations and maximum deviations can be calculated by Eqs.(4) to (7). The weight values of various types of data can be converted into binary connection numbers using Eq.(3), and thus the twotuple(determinacy, uncertainty) numbers of weights can be obtained. And the weight vector can be written as 2) Calculation of baseline value of the weight two-tuple (determinacy, uncertainty) number According to the weight two-tuple (determinacy, uncertainty) numbers, the maximum and minimum values of the determinacy and uncertainty of each two-tuple number can be obtained using Eq. (10).
where a max = max a j , a min = min a j , b max = max b j and b min = min b j represent the maximum and minimum values of the determinacy and uncertainty of the weight two-tuple numbers, respectively, expect for a min = b min = 0.

3) Construction of weight function
In order to keep the information of an uncertain weight, the influence of deterministic and uncertain factors on certain weight should be considered. The weight function is built as follows.
where α j is the contribution ratio of deterministic term of the weight two-tuple (determinacy, uncertainty) number for calculating the weight w j , and β j is the contribution ratio of uncertain term; b min is the minimum value of uncertain term. Eq. (11) indicates that the deterministic weight depends on the deterministic and uncertain parts of the two-tuple (determinacy, uncertainty) number for b j = 0;otherwise w j simply relies on the deterministic part.
In Eq.(11), α j can be determined as required. However, α j and β j can also be determined by the uncertain number itself regarding the information contained therein. So, the improved function can be given as follows.
where a j a 2 j +b 2 j and b j a 2 j +b 2 j originated from a binary connection number denote the contribution of deterministic term VOLUME 8, 2020 and that of the uncertain term in a mixed attribute weight respectively.

4) Index weight normalization
The index weight obtained in step (3) is unnormalized, which is different from that the sum of the index weights is one. Therefore, the normalized weights can be obtained by using Eq. (15).

A. TRANSFORMATION OF ALTERNATIVE INDICES INTO BINARY CONNECTION NUMBERS
Different types of index values can be converted into binary connection numbers µ + σ i using Eqs. (4) to (7). The form µ + 0i means that the deterministic term is the real number itself and the uncertain term is 0. The converted index number can be expressed as V st = µ st + σ st i(s = 1, 2, · · · , n; t = 1, 2, · · · , m).

D. CALCULATION OF COMPREHENSIVE WEIGHTED KULLBACK-LEIBLER DISTANCE
Having obtained the vector constituted by two-tuple numbers of alternatives and target center, then the comprehensive weighted K-L distance can be obtained. Definition 6: Comprehensive weighted Kullback-Leibler distance (CWKL). Suppose that S = ((x 1 , y 1 ), (x 2 , y 2 ), . . ., (x m , y m )) T and E = ((p 1 , q 1 ), (p 2 , q 2 ), . . . , (p m , q m )) T are two vectors of two-tuple numbers, and let W = (w 1 , w 2 , · · · , w m ) T be the weight vector, then the comprehensive weighted Kullback-Leibler distance of S and E can be obtained.
The properties of H W (S, E) are as follows: 2) H W (S, E) = 0, when and only, when S = E, which means x j = p j , y j = q j , ∀j; 3) When x j = p j = 0 or y j = q j = 0, then by definition x j ln x j p j = 0 or y j ln y j q j = 0. The situations x j = 0, p j = 0 and y j = 0, q j = 0 may occur in Eq. (16). In order to avoid this, the modified equation can be given as follows.

E. DECISION-MAKING STEPS
The algorithm of GGTDM is as follows. 1) All alternative indices and weight values are converted into binary connection numbers by Eqs.(4) to (7), and the results can be transformed into twotuple(determinacy, uncertainty) numbers.
2) The two-tuple (determinacy, uncertainty) numbers of target center under all attributes can be obtained by Eq. (13).
3) The two-tuple (determinacy, uncertainty) numbers of all indices and target center can be normalized by Eq. (14), then the two-tuple(deterministic degree, uncertainty degree) numbers of all attributes can also be normalized by Eq.(15). 4) The deterministic weights can be obtained by Eq. (10) and Eq.(12).   5) The comprehensive weighted K-L distance of each alternative and target center can be obtained by Eq. (16) and Eq. (17), and the decision is made based on the value with which the smaller, the better.

V. CASE STUDY A. DESCRIPTION OF DECISION PROBLEM
To evaluate a weapon system, six indices including mobility (km h −1 ), reliability, accuracy(km), maintainability, warhead payload(kg) and price(10 6 g) denoted by Z 1 to Z 6 are considered. Among these attributes Z 3 and Z 6 are cost-type indices and the others are benefit-type indices. The attribute weights are given as W = (0.

1) Calculation of the parameters of binary connection numbers for all alternative indices
The parameters of binary connection numbers of all alternatives can be calculated by the data shown in Table 1 using Eqs.(4) to (7). The results are shown in Table 2.
2) Translation of all index values into binary connection numbers All index values can be transformed into binary connection numbers using Eqs.(3) to (7) based on the data shown in Table 2. The results are shown in Table 3.
The two-tuple numbers of alternative indices and target center indices can be normalized using Eq. (15), and the results are shown in Table 5.
In Table 5, the symbols a st and b st represent the deterministic and uncertain terms of the same index, respectively. If the index is real number, the b st is 0. For example, the indices under attributes Z 3 and Z 5 are both real numbers. The symbol NC p denotes the normalized two-tuple number of target center.  (7), the results are shown in Table 6.
Next, the weight values can be expressed as binary connection numbers, the results are shown in Table 7.
The maximum and minimum values of the determined terms in weight two-tuple (determinacy, uncertainty) numbers are calculated as 0. 21

C. COMPARISON AND ANALYSIS 1) COMPARISON OF METHODS FOR OBTAINING THE CERTAIN WEIGHTS
To verify the weight function method transforming mixed attribute weights into certain weights, the proximity-based method discussed in [12] and the module-based method presented in [21] are used to make a comparison.  Table 8 and Figure 1. Seen from Table 8, the certain weights determined by three methods have some differences under all attributes. It should be noted that the result by proximity-based method is almost the same as that by the module-based method. But the result determined by the weight function method is obviously different from the results calculated by the other two methods. Seen from Figure 1, the two curves representing the results that determined by the proximity-based method and that by the module-based method almost overlap with each other, while the curve denoting the result that obtained by the weight function method can be discriminated easily.

2) COMPARISON OF DECISION-MAKING RESULTS
The comprehensive weighted proximity-based method is adopted to make a comparison with the proposed approach. In previous research, the CWP based GGTDM with index and weight both containing mixed attribute values has two ways to fulfill the decision-making task: one way is to obtain the certain weights by proximity-based method and adopt the CWP as the DMB [12]; the other is to arrive at the certain weights by module-based method and also take the CWP as the DMB [21]. The former way could be called proximity-based method-1, while the later could be referred to as proximity-based method-2.
Having obtained the certain weights W P = (0.101, 0.162, 0.182, 0.141, 0.202, 0.212) by the proximity-based method, the comprehensive weighted proximity vector can be calculated as I CWP1 = (0.2681, 0.2894, 0.2405, 0.2021). According to the principle the smaller the proximity the better the alternative, the alternatives ranking by proximity-based method-1 is: T 4 T 3 T 1 T 2 . Given the certain weights W M = (0.1004, 0.1618, 0.1818, 0.1419, 0.2017, 0.2124) determined by module-based method, the comprehensive weighted proximity vector can be obtained as I CWP2 = (0.2695, 0.2928, 0.2354, 0.2023) by proximitybased method-2. Thus the alternatives are ranked as: T 4 T 3 T 1 T 2 . It is obvious that the decision-making result determined by the proposed approach is completely in accordance with the results that by proximity-based method-1 and proximity-based method-2. The decision-making results are listed in Table 9 and shown in Figure 2.

VI. CONCLUSION
This work proposes a novel GGTDM with index and weight both involving mixed attribute values. The novelty of this paper lies in two aspects: the improved K-L distance is adopted as the DMB, and the weight function is built to obtain the certain weights. The principle of the proposed method is that the comprehensive weighted K-L distance reflecting the uncertain measure between the two vectors is applied to fulfil the decision making. Furthermore, the weight function is constructed to transform mixed attribute weights into deterministic weights, which considers the information of mixed attribute values. The proposed method is verified in ranking alternatives effectiveness and accuracy compared with other methods. In the future, more suitable DMB of GGTDM and mixed attribute weights determining method should be investigated to support the decision making accurately and effectively.