A Novel Algorithm Based on 2-Additive Measure and Shapley Value and Its Application in Land Pollution Remediation

In this article, a new aggregation operator called the Young–Shapley optimal weight (Y-SOW) operator is proposed to aggregate heterogeneous information in group decision-making. /e Y-SOW operator combines the Shapley value with the Young inequality. Meanwhile, a series of special cases and main properties of the Y-SOWoperator are studied. Furthermore, the dispersion maximization model of the Y-SOW operator is established to obtain the optimal 2-additive measure. In the Shapley value method of the cooperative game, the 2-additive measure not only simplifies the complexity of fuzzy measures but also solves the interaction between attributes./e Shapley value of the 2-additivemeasure is explored to the weight of the Y-SOWoperator. Finally, the Y-SOW operator-based multiattribute group decision (YSMGAD) algorithm is proposed. /e application of the YSMGAD algorithm for land pollution remediation is analyzed.


Introduction
Group decision-making is rapidly developed to an important branch of modern decision sciences, which helps to gather the wisdom of experts from different fields. e transition from individual to group decision-making is a major step forward to cope with increasingly complex human activities. To this end, group decision-making has been recognized and used in economic, military, agricultural, and other fields [1][2][3][4][5][6].
In the state-of-art literature, many methods of aggregation operators and their weight determination have been proposed for multiattribute decision-making. Yager [7] introduced an ordered weighted averaging (OWA) operator, where the input arguments were rearranged in the descending order, and the weight vector was only related to its ordered position. Xu and Da [8,9] developed the ordered weighted geometric averaging (OWGA) operator for multiattribute decision-making. Chen and Liu [10] proposed an extension of the OWA operator called an ordered weighted harmonic mean (OWHA) operator. In 2004, Yager [11] used a generalized mean in the OWA operator and a generalized ordered weighted averaging (GOWA) operator. Based on a minimizing model, Zhou and Chen presented the generalized ordered weighted logarithm averaging (GOWLA) operator [12], generalized ordered weighted harmonic averaging (GOWHA) operator [13], and so on. Merigo et al. expanded the OWA operator and proposed the ordered weighted averaging-weighted average (OWAWA) operator [14], which unified the OWA operator in the same formulation. Other generalizations of the aggregation operators were observed in [12,[15][16][17][18][19][20][21]. e operators proposed above only deal with additive or multiplicative information alone. e existing literature is not sufficient to solve the problem when two kinds of information appear simultaneously in group decision-making.
In 1995, Grabisch [22][23][24][25] proposed the fuzzy measure as an aggregation operator for multiattribute decision-making. However, the fuzzy measure requires a large number of parameters, which is difficult to implement. In order to reduce the computational complexity, various forms and methods of determining the fuzzy measure have been proposed. e choquet integral [26][27][28] is a nonlinear integration operator defined on the basis of fuzzy measures, which can effectively deal with the interaction between decision attributes. e premise of the integral is utilized to determine the fuzzy measure, which is complicated. If there are n attributes, 2 n − 2 parameters are needed. Sugeno [29] proposed a λmeasure that only requires n parameters but cannot fully describe the interaction between attributes. In 1996, Grabisch [30] proposed the k-additive measure. e Shapley value of a single attribute and the interaction among k attributes were determined, the computational complexity was reduced, and the representing ability between the attributes was improved to some extent.
In reality, the attribute value often includes both additive and multiplicative information; however, it is difficult to aggregate precisely in one matrix. Based on the Young inequality and Shapley value, a new optimal operator called the Young-Shapley optimal weight (Y-SOW) operator is proposed. Meanwhile, a series of special cases and the main properties of the Y-SOW operator are studied. Because one certain interaction exists between the attributes, the 2-additive measure is introduced to reduce the complexity of the fuzzy measure. e Shapley value method is the most effective and widely used method in cooperative games. erefore, the dispersion maximization model based on the 2-additive measure and the Shapley value is established to obtain the optimal 2-additive measure. Some formula and programming models are also provided to effectively determine the 2-additive measure. e Shapley value of the 2additive measure is used as the weight of the Y-SOW operator. Finally, the Y-SOW operator-based multiattribute group decision (YSMAGD) algorithm is proposed to effectively aggregate the heterogeneous data. However, the Y-SOW operator can only aggregate the real-type data and has certain limitations for the other types of data, such as interval-type data. e rest of this article is organized as follows. In Section 2, the basic concepts of common aggregation operators, fuzzy measures, and Shapley values are reviewed. In Section 3, the common Y-SOW operator is proposed, and some special cases and ideal characteristics of the Y-SOW operator are also proved. In Section 4, the Y-SOW operator-based multiattribute group decision (YSMAGD) algorithm is established. In Section 5, the YSMAGD algorithm is applied to sequence the remediable location of land pollution. In Section 6, summary is given.

Several Commonly Used Information Aggregation
Operators.
e ordered weighted averaging (OWA) operator was proposed by Yager [7] in 1988 and is widely used in a series of decision problems, as defined below. Definition 1. Let R be the set of real number. AnOWA operator is a mapping, OWA: R n ⟶ R is satisfied: OWA a 1 , . . . , a n � n j�1 w j b j , (1) then OWAis called an ordered weighted averaging operator, where b j is the jth largest of the arguments a 1 , . . . , a n and the weight vector W � (w 1 , . . . , w n ) T satisfies n j�1 w j � 1, w j ∈ [0, 1] and (j � 1, 2, . . . , n). e OWA operator is an effective aggregation method that rearranges the arguments and then weights it according to the sequence position to weaken the adverse effects of the extreme value. It is characterized by considering only the positional relationship of the arguments in the ordering process. e OWA operator has many desirable properties such as monotonicity, boundedness, idempotency, and permutation invariance. When b j � a j , holds for all j � 1, 2, . . . , n, then the OWA operator becomes a weighted averaging (WA) operator [31].
Yager also proposed the BUM function Q [32] to calculate the associated weight of the OWA operator, which satisfies Q(0) � 0, Q(1) � 1, and Q(x) ≤ Q(y) for any 0 ≤ x ≤ y ≤ 1, that is, e BUM function is called the fuzzy semantic quantization operator, and the expression can be stated as follows: where x 1 , x 2 , and x are in the range of [0, 1]. When we choose the pair (0, 0.5), (0.3, 0.8), and (0.5, 1), the fuzzy linguistic representation is "at least half," "more," and "as many as possible," respectively.
Definition 2. An OWG operator [33] is a mapping, OWG: I n ⟶ I and I � x | x ≥ 0 { }, according to the following formula: then OWG is called an ordered weighted geometric operator, where b j is the jth largest of the arguments a 1 , . . . , a n , and the weight vector w � (w 1 , . . . , w n ) T satisfies n j�1 w j � 1 and w j ∈ [0, 1], j � 1, 2, . . . , n. e generalized weighted averaging (GWA) operator was first presented by Yager [11] based on the generalized mean. Assuming that the fusion result is an n-dimensional function f, we can construct a penalty function J � n j�1 w j (f θ − a θ j ) 2 and the minimization problem:

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Let (zJ/zf) � 0, and the aggregation method for obtaining the GWA operator is shown as follows: GWA a 1 , . . . , a n � n j�1 w j a θ If the arguments a 1 , . . . , a n in the GWA operator are arranged in a descending order, the generalized ordered weighted averaging (GOWA) operator can be obtained. e GWA operator also has many desired properties, such as monotonicity, boundedness, idempotency, and permutation invariance. When θ � 1, θ � − 1, and θ ⟶ 0, then the GWA operator reduces to the weighted averaging (WA) operator, the weighted harmonic averaging (WHA) operator, and the weighted geometric averaging (WGA) operator, respectively.

Fuzzy Measure and Shapley Value.
e aggregation operators used in the traditional multiattribute decisionmaking are generally based on the premise that the attributes are independent and do not interact with each other, but the actual situations are often interactive or dependent. To deal with the abovementioned phenomenon, the fuzzy measure [12,[20][21][22][23][24][25][26][27] is introduced as follows.
Definition 3 (see [29]). Let N be the attribute set and P(N) be the power set of N. If set function μ: P(N) ⟶ [0, 1] satisfies the following conditions: en, μ is a fuzzy measure on P(N). Fuzzy measure is difficult to calculate and requires a large number of parameters. Grabisch proposed the 2-additive fuzzy measure based on pseudo-Boolean function and Mobius transform [29].
Definition 4. Let f: 0, 1 { } n ⟶ R be a pseudo-Boolean function [26]. Where 0, 1 { } n denotes the entirety of all n-dimensional Boolean vectors. Let X � x 1 , . . . , x n , ∀A ⊆ N. Any fuzzy measure μ can be seen as a particular case of pseudo-Boolean function denoted by where a T ∈ R, y � (y 1 , y 2 , . . . , y n ) ∈ 0, 1 { } n , and a T � S⊆T (− 1) t− s μ(S) is called the Mobius transform coefficient. Obviously, y i � 1 if and only if i ∈ A. e fuzzy measure μ defined on (X, P(N)) is a k-additive fuzzy measure, and the corresponding pseudo-Boolean function is a k-order linear polynomial, that is,∀T ∈ N, if |T| > k, then a T � 0 and ∃T 0 ∈ P(N), From Definition 4, we get that 1-additive measures are additive measures and n-additive measures are fuzzy measures. Especially, when k � 2, by equation (7) we get a 2additive measure μ.
Definition 5 (see [30]). For a 2-additive measure μ, ∀S ⊆ N, s ≥ 2, then where It is well known that a 2-additive measure requires only n(n + 1)/2 parameters, and it will be simpler to solve a 2additive measure than to solve a usual fuzzy measure.
Definition 6 (see [30]). Let μbe the fuzzy measure on , the 2-additive fuzzy measure satisfies the following conditions: where s and n are the cardinalities of Sand N, respectively. e Shapley value [34] is determined in a grand coalition N based on the marginal contribution of players to obtain the optimal benefit distribution. In order to avoid the irrationality of the average distribution and show certain rationality and fairness, the Shapley value method is widely used for cooperative games. Definition 7. Vector Sh(N, μ) � (Sh 1 (N, μ), Sh 2 (N, μ) . . . , Sh n (N, μ)), and its components are defined by where μ is a fuzzy measure on N and |N| and |T| are the cardinality of set N and T respectively. Let μ be a superadditive fuzzy measure on N, and it follows that It means that Sh(N, μ) can be viewed as a weight vector here, named as the Shapley weight vector.

e Proposed Operator and Its Equivalent Expression
Lemma 1 (Young inequality, see [35]). ∀a, b ≥ 0, λ ∈ [0, 1], it holds that if and only if a � b, and the equation holds.
In the aggregation process, let a 1 , . . . , a n be the nonnegative aggregation arguments and Sh i (N, μ) be a Shapley weighting vector, then function f is strictly monotonically increasing. Construct a penalty function H as follows: where λ is a parameter that satisfies λ ∈ [0, 1]. According to the necessary conditions for the existence of extreme values, taking the partial derivative of H with respect to y, there is Setting (zH/zy) � 0, it follows that Equation (13) can be written as follows: Equation (14) is transformed into the following formula: e function f is strictly monotonically increasing and has a reversible property. us, its inverse function exists as follows: and we called equation (16) as the Young-Shapley optimal weight (Y-SOW) operator. In order to simplify the structure of the Y-SOW operator, the following equivalent expression is introduced in detail. Let then, it holds that n i�1 B i � 1. e Y-SOW operator can be equivalently written as e following theorem shows the monotonicity of B i with respect to a i .
Proof. Taking the derivative of B i with respect to a i , we have Since the function f is monotonically increasing, we get f ′ (a i ) ≥ 0. Moreover, function f is nonnegative and en, B i is monotonically decreasing. □ Theorem 2. Let a i ′ and a i (i � 1, 2, . . . , n) be real numbers, then we have Proof. It can be known from eorem 1, so the proof process is omitted.
e Y-SOW operator reduces to the Shapley weighted harmonic averaging (SWHA) operator.
Equation (23) reduces to quasi-Shapley weighted averaging (QSWA) operator. (iv) When f(x) � x is selected according to the overall risk attitude, the Y-SOW operator reduces to the following operator with respect to risk neutrality: (v) When f(x) � ln x is selected according to the overall risk attitude, the Y-SOW operator is simplified to the following operator in terms of risk aversion: If λ � 0, equation (25) is changed to Equation (26) becomes the Shapley weighted geometric average (SWGA) operator. (vi) When f(x) � e x is selected according to the decision makers overall risk attitude, the Y-SOW operator is reduced to the following operator in terms of risk proneness:

Desirable Property of the Y-SOW Operator.
e Y-SOW operator has many potential properties, such as monotonicity, idempotency, and boundedness.

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It follows that Taking the derivative of the ln f(H) with respect to a i (i � 1, 2, . . . , n), respectively, we get Since erefore, (z(ln f(H)) /za i ) ≥ 0. is means that ln f(H) is a monotonically increasing function. Obviously, f(H) is also a monotonically increasing function. us, we get which can be equivalently expressed as To sum up, we have f(a 1 ′ , . . . , a n ′ ) ≥ f(a 1 , . . . , a n ). erefore, Property 1 is proved.
Proof. Denoting By eorem 1, we have Because λ ∈ [0, 1], we have  It follows that To sum up, we have a min ≤ Y − SOW a 1 , . . . , a n ≤ a max .
e proof of Property 3 is completed.

Deriving the Optimal Weight Vector Based on Dispersion Maximization Method
Aggregation operators play a vital role in multiple fields, such as economics, statistics, and management. In case that the attribute associated with weight information is unknown, we establish the optimal weight vector based on the dispersion maximization method. Suppose A � a 1 , . . . , a m is a finite set of alternatives, C � c 1 , . . . , c n is a set of attributes and D � d 1 , . . . , d t is the set of decision makers and v � (v 1 , . . . , v t ) T is the weight vector of a decision maker, which satisfies the condition v k ∈ [0, 1], t k�1 v k � 1. Assume that A (k) � (a (k) ij ) m×n is the decision matrix given by the decision maker d k , where the estimated value a (k) ij indicates that the alternative a i ∈ A under attributes c j ∈ C is given by the decision maker d k . ω � (ω 1 , ω 2 , · · · , ω t ) T is the weight of the decision maker, satisfying ω i ∈ [0, 1], t i�1 ω i � 1. In multiattribute decision-making, the attributes cannot be aggregated directly due to their different size. e primitive decision maker matrix has to be normalized. e attributes mainly include three types: benefit attribute, cost attribute, and fixed attribute. e benefit attribute refers to the bigger the better index, the cost attribute refers to the smaller the better index, and the fixed attribute refers to how close is it to a fixed value.
Let H i (i � 1, 2, 3) be the subscript set of the above three types of attributes in order [10], and the following formula can be used for normalization: For multiattribute decision-making, if the attribute u j can make a bigger difference in the attribute values of all alternatives, it means that u j plays a bigger role in the ordering of the alternatives and shall be given a bigger weight, vise verse. In particular, if all of the alternatives have no difference in the attribute values under the attribute u j , u j will have no effect on the ordering of the alternatives and u j is zero. Based on that the weight can be determined with the dispersion maximization method.
Because there is a certain interaction between the attributes, in order to simplify the complexity of the fuzzy measure, the Shapley value based on the 2-additive measure is introduced as the weight of the attributes and the traditional Shapley value function has been mentioned in equation (9).
When the fuzzy measure μ is a 2-additive measure, equation (9) is converted to [36][37][38] (47) In Definition 7, Sh i (N, μ) can be regarded as a weight vector. In view of Definition 6 and equation (47), the dispersion maximization model based on the 2-additive measure and Shapley value is established as follows: Model (M − 1): Finally, the optimal 2-additive measure can be obtained, and the optimal 2-additive measure can be reused to calculate the Shapley value in the Y-SOW operator. Using equation (16) to aggregate the Y-SOW operator, the optimal result is obtained. erefore, the Y-SOW operator-based multiattribute group decision (YSMAGD) algorithm is stepped as follows: Step 1. Assuming that A (k) � (a (k) ij ) m×n is the decision matrix given by the decision maker d k (k � 1, 2, . . . , t).
Step 2. By equations (2) and (3), according to the decision makers' preference, the fuzzy linguistic quantifier with the pair (a, b) is selected to compute the Mathematical Problems in Engineering weights of the decision makers w j � (w 1 , w 2 , . . . , w n ) T (j � 1, 2, . . . , n).
Step 3. Based on equations (1) and (4), the OWA operator and the OWG operator are used to aggregate the additive and multiplicative information of the decision matrix A (k) , respectively. Final decision matrix A � (r ij ) m×n is obtained.
Step 4. e final decision matrix A is normalized by equations (44)-(46) to obtain a collective decision matrix R − .
Step 6. μ(i) and μ(i, j) are used to calculate the Shapley value based on equation (47), that is, the weight vector of the Y-SOW operator.
Step 7. Utilize equation (16) to aggregate the normalized matrix R − , and the overall preference value r i (r � 1, . . . , m) of the alternative a i is obtained.
Step 8. Rank all the alternatives a i (i � 1, . . . , m) and select the best one in accordance with the ranking of r i (r � 1, . . . , m).
Step 9. End. Based on the above analysis, the framework of the YSMAGD algorithm is illustrated in Figure 1.

Numerical Example
e economic development brings us livelihood improvement, but with increasing land pollution. It is investigated that among the total 150 million mu of cultivated land in the country, 32.5 million mu is contaminated by sewage, 2 million mu is occupied by solid waste, and 2 million mu is destroyed, accounting for over 20%. Crops accumulate harmful substances from polluted land, which causes diseases and ultimately endanger human future. However, the prevention and control is still weak. Nowadays, distribution and extent of soil pollution in the country are unclear. As a result, the government lacks specific control measures and capital input, and experts on land science research are also difficult to carry out in depth. e first task is to select the polluted location for remediation. ere are many both objective and subjective factors involved. e evaluation index is usually chaotic and miscellaneous. It is a typical problem of multiattribute aggregation with additive and multiplicative information. e proposed Y-SOW operator above is applied to decision analysis.
Consider site selection for land remediation, that is, the assessment of polluted land. Assume that there are eight land experts (d 1 , d 2 , . . . , d 8 ), evaluating the improvement of rural land from six attributes: c 1 pollution area, c 2 remediation potential, c 3 realistic feasibility, c 4 fertilizer and pesticide use rate, c 5  soil pH, and c 6 irrigation guarantee rate. ere exist four lands (A, B, C, D), and eight experts score four lands according to the above six attributes.
Step 1. Because the analytic hierarchy process (AHP) [39][40][41] can easily and flexibly deal with the quantitative problem of decision makers on complex systems, it is a classic weighting method commonly used by decision makers. e core of the method is to compare the degree of importance between the two elements by the 1 to 9 scale method. c 2 and c 3 are multiplicative indicators and cannot be calculated by specific numerical values. erefore, the AHP scale method is used. Suppose there is a sample land E with a scale of 1 for each attribute. Let land E be F j and lands A B C D be F i . e experts compare the four lands A B C D with the sample land E according to the following 1-9 scale (shown in Table 1).

Let us suppose that the decision matrix
Step 2. By equations (2) and (3), and according to the decision makers' preference, the fuzzy linguistic quantifier "more," with the pair (0.3, 0.8) is selected to compute all the weights of the decision makers w j � (w 1 , w 2 , . . . , w n ) T (j � 1, 2, . . . , n).
Step 4. It is clear that c 2 , c 3 , and c 6 are benefit attributes, c 1 and c 4 are cost attributes, and c 5 is the fixed attribute. When F i is as important as F j 3 When F i is slightly more important than F j 5 When F i is more important than F j 7 When F i is much more important than F j 9 When F i is extremely important than F j 2, 4, 6, 8 A compromise of the above degree Mathematical Problems in Engineering 9 Equations (44)-(46) are used for normalization to obtain a collective decision matrix R: R � Step 5. From the above model (M − 1), the dispersion maximization model based on the 2-additive measure and Shapley value is established to obtain the optimal 2additive measure μ(i) and μ(i, j).
ese are the optimal 2-additive measures.
Step 6. From Equation (47), the Shapley value of the Y-SOW operator is calculated as follows: Step 7. Based on equation (16), for convenience, we set f(x) � x, λ � 1, and the overall preference value r i (r � 1, . . . , m) of the alternative a i is obtained. Let r i � Y − SOW(a 1 , . . . , a n ). e overall preference values are obtained as Step 8. e ranking is r 4 > r 1 > r 3 > r 2 . erefore, the best alternative is a 4 . Namely, land D is most needed for remediation.
In this example, we can see that the YSMAGD algorithm can efficiently handle heterogeneous data with the good aggregation. e evaluation index of site selection of polluted land in the above example usually contains objective factors and subjective factors. For example, indicators such as remedial potential and realistic feasibility are multiplicative indicators that cannot be calculated from specific values. Indicators such as pollution area and soil pH are additive indicators that can be obtained through clear numerical measurements. e key to land pollution assessment is to combine both indicators, so decision makers use the Y-SOW operator proposed above to aggregate and use the YSMAGD algorithm for effective land assessment. is is not only beneficial for the government to implement land pollution remediation, but also for experts to conduct land research.

Comparison and Conclusion
Compare the above three cases and use the obtained overall preference values as a scatter plot, as shown in Figure 2.
Obviously, for λ � 0, λ � (1/2), and λ � 1, the results of aggregation are not numerically identical. However, by comparing these results, the same alternative approach can be obtained, namely, land D needs to be repaired the most urgently.
It is clear that the value of penalty function H with respect to the Y-SOW operator is the smallest. From equation (11), we know that the Y-SOW operator is the optimal aggregation for minimizing the penalty function.
is supports the optimality of the Y-SOW operator.

Brief Conclusion.
is paper introduces a new aggregation operator called the Young-Shapley optimal weight (Y-SOW) operator. Some special cases and main properties of the Y-SOW operator are studied. e advantages of this paper are as follows: (i) e Y-SOW operator solves the problem that the additive and the multiplicative decision information appear simultaneously in the group decisionmaking. (ii) In the Shapley value method of cooperative game, the 2-additive measure replaces the original fuzzy measure. is not only can effectively deal with the interaction between decision attributes, but also reduce the computational complexity and improve the representation between attributes. (iii) To solve the problems of multiattribute group decision-making under attribute interaction, we develop the Y-SOW operator-based multiattribute group decision (YSMAGD) algorithm and establish a linear programming model (M − 1).
It is worth noting that we can also apply the Y-SOW operator to the other areas such as deep learning, cluster analysis, and artificial intelligence.
Nevertheless, in this article, we only consider the case where decision information is represented by real numbers. However, in some cases, the decision information may be in other forms, such as interval numbers and fuzzy numbers.
is is an issue that needs further study in the near future.

Data Availability
e data used to support the findings of this study are included within the article.

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