A Revised Grey Relational Analysis Method for Multicriteria GroupDecision-Making with Expected Utility Theory for Oil Spill Emergency Management

+e substantial importance and complexity of oil spill emergency management have been confirmed by the massive, disastrous oil spill accidents that have occurred in recent years. +e goal of this study is to research and develop an effective new model to accurately respond to oil spill emergencies for the purpose of disaster risk reduction. In this model, firstly, grey relational analysis is used to treat the variable and complex environment of an oil spill. Secondly, a revised grey relational analysis method based on the concept of angle between two vectors is presented to optimize assessment accuracy by measuring the similarity between the reference series and alternative series. +irdly, by incorporating expected utility theory, a revised grey relational analysis for facilitating consensus is developed for multicriteria group decision-making. Finally, an empirical study on oil spill emergency management is discussed to demonstrate the feasibility and effectiveness of the proposed model for oil spill emergency management in a complex environment to reduce disaster risk.


Introduction
In recent years, major emergencies have occurred with increasing frequency, including the 9/11 attack in the United erefore, emergency management during a complex environment has become a global concern.
Accidental oil spill emergency is a significant event that can affect environmentally, ecologically, and economically sensitive marine areas. Massive, disastrous oil spill incidents have also caused a great amount of property losses and casualties [1,2]. After an oil spill occurs, it is necessary to accurately respond within the shortest possible time to protect marine species and minimize cleanup and damage costs [3]. question of how to effectively respond to oil spill emergencies in a complex environment for the purpose of disaster risk reduction and formulation of crisis response strategies has become a challenging issue worldwide. It has also promoted the theoretical research and technology development to deal with emergency problems during a complex environment. Lin et al. [4] developed multiattribute group decision-making methods under probabilistic uncertain linguistic complex environment. Qiyas et al. [5] proposed the concept of Yager operators with the picture fuzzy set environment for emergency program selection. Lin et al. [6] applied MULTIMOORA-based MCDM model for site selection of car-sharing station under picture fuzzy and complexity environment. Garg and Arora [7] proposed a TOPSIS method based on correlation coefficient for solving complexity decision-making problems with intuitionistic fuzzy soft set information. Wu et al. [8] proposed and developed a DMSECA model to evaluate clustering algorithms by merging expert wisdom in order to reconcile differences in their evaluation performance for information fusion during a complex decision-making process.
e objective of the present study is to research and develop an effective model to aid decision-makers involved in oil spill emergency and crisis response planning for a complex environment. It is necessary to minimize the damage and costs caused by potential oil spills and to devise a contingency plan for emergency situations and crisis management [9][10][11]. However, the design and development of an optimal emergency plan is very difficult to achieve because it depends on many variables within a dynamic and complex environment comprising the changing nature of the ocean environment, type of oil, spilled volume, location of the oil spill, and weather. Hence, this can be considered a multicriteria decision-making (MCDM) problem. Grey relational analysis, one of the classical MCDM methods, is a basic method of grey system theory [12][13][14][15][16] that is used to analyze the data changes in the trend curve by comparing the similarity degree of each data sequence with the reference sequence [17][18][19]. is method can process inaccurate and vague information associated with a variable; it can work with an uncertain and ambiguous environment; and it only requires a reasonable amount of sample data. us, in this study, grey relational analysis is applied to address the dynamics and complexities of the variables associated with the environment in oil spill emergency management. is paper simulates oil spill scenarios using the computational package OILMAP [20].
Following oil spill emergencies, many entities-such as environmental agencies, oil companies, and nongovernmental organizations-are involved in the decision-making process to advance their own benefits and interests [9]. Consequently, there is an inevitable conflict among these institutions in the decision-making process with respect to choosing the best emergency response alternatives. In this situation, a consensus decision that increases the level of satisfaction among multiple stakeholders is of considerable importance. Hence, a multicriteria group decision-making (MCGDM) approach based on group wisdom is proposed to model the oil spill emergency response to reduce disaster risk.
Besides, this paper takes into account the evaluation accuracy for an effective and rapid emergency response and introduces the mathematical concept of angle between two vectors to improve the similarity characterization process of grey relational analysis by measuring the similarity between the reference series and alternative series. In the MCGDM process, expected utility theory is applied to aggregate individual preferences into a group's consensus.
Considering all of these factors, the motivation of this work is to research and develop an effective model to aid decision-makers involved in oil spill emergency and crisis response planning for a complex environment. In this paper, a revised grey relational analysis method using MCGDM with expected utility theory is proposed to respond to oil spill emergencies for disaster risk reduction when a variety of stakeholders and a variable and complex oil spill environment are involved. e main contributions of this study are as follows: (1) A revised grey relational analysis for group decisionmaking technique is proposed to evaluate and implement an emergency plan for responding to unexpected events related to an oil spill. (2) In this proposed technique, grey relational analysis is firstly applied to the variable and complex environment. Secondly, based on the concept of angle between two vectors, a revised grey relational analysis method is presented to optimize assessment accuracy. irdly, by incorporating expected utility theory, a revised grey relational analysis for facilitating consensus is developed for multicriteria group decision-making. e rest of this study is organized as follows. Section 2 reviews related research. Section 3 introduces the foundations of grey relational analysis, angle between two vectors, and expected utility theory. In Section 4, a revised grey relational analysis method for MCGDM is proposed. In Section 5, an empirical study on oil spill emergency management in a complex environment is conducted to verify the proposed model. Section 6 concludes the study.

Related Works
Emergencies have a complex evolutionary process, and emergency decision-making is difficult to solve with traditional decision theory. Tufekci and Wallace [21] suggested that emergency management is essentially a complex multicriteria optimization problem. As defined by the International Society on Multiple Criteria Decision Making, MCDM is the study of methods and procedures by which concerns about multiple conflicting criteria can be formally 2 Mathematical Problems in Engineering incorporated into the management planning process [5,[22][23][24][25][26][27]. Over the past 40 years, MCDM has made remarkable progress and evolved into a mature discipline [15,[28][29][30]. It has been used in a wide range of practical applications, such as information science, engineering systems, and decision-making [8,[31][32][33][34][35][36].
In oil spill emergency management for disaster risk reduction, the benefits and interests of environmental agencies, oil companies, and nongovernmental organizations should be considered. Conflicts among these varied interests will inevitably arise in the decision-making process for choosing the best emergency plan. In this context, the process of evaluating and choosing the best emergency plan involves multiple criteria, multiple stakeholders, and a finite number of alternatives [49]. erefore, oil spill emergency management for disaster risk reduction can be interpreted as a multicriteria group decision-making problem, which combines MCDM and GDM methods. MCGDM has been proven to be an effective technique for increasing the level of overall satisfaction of group members with respect to the final decision [50][51][52][53][54].
GRA, which is one of the best-known MCDM techniques, is a type of quantitative analysis that is used to measure the similarity between the reference series and an alternative series [16,17,[55][56][57]. e alternative series with the closest similarity to the reference series is the best alternative for the problem at hand [58][59][60][61]. GRA is a part of grey theory, and it is based on the concept of grey space. It can process the inaccurate and vague information in grey systems under a variable environment [43]. e GRA method is a widely applied technique in data processing, system analysis, control systems, and decision-making [19,55,[62][63][64][65][66][67]. Compared with conventional methods that require massive amounts of data, GRA possesses the following advantages [43,[68][69][70]: (1) It uses simple and easy calculations. (2) It only requires a reasonable amount of sample data. (3) It does not require a typical distribution of samples. (4) e quantified outcomes from the grey relational grade do not result in conclusions that are contradictory to the qualitative analysis. (5) e grey relational grade model is a transfer functional model, which is suitable and effective for dealing with discrete data [43,57,68,69]. It is difficult to design and develop the best oil spill emergency plan to reduce disaster risk because the process depends on a multivariable, changing, and complex environment that involves factors such as the dynamic nature of the ocean environment, type of oil, spilled volume, location of the oil spill, and weather. In this study, based on GRA's advantages and the concept of grey space, GRA is applied to address the complexities of the changing oil spill environment in order to reduce disaster risk.
GRA aims to measure the similarity between the reference series and alternative series. erefore, in this study, the mathematical concept of the angle between two vectors is applied to improve the conventional GRA method. In geometry, an angle is defined as the figure formed by two rays (called the sides of the angle) sharing a common endpoint (called the vertex of the angle) [71]. e angle between two vectors is in the range 0 ≤ cos(α, β) ≤ 1. e higher the value of cos(α, β) is, the higher the similarity between the directions of vectors α and β is. In contrast, the smaller the value of cos(α, β) is, the higher the dissimilarity between the directions of vectors α and β is. us, the angle between two vectors can be used to measure the similarity between the reference series and alternative series. In this study, the reference series and alternative series are regarded as vectors.
e group decision-making problem arises in many realworld decision-making situations [72][73][74][75]. In MCGDM problems, the essence is how to effectively aggregate individual preferences into a group consensus. Ishizaka and Labib [76] noted that there are four ways to combine individual preferences into a group consensus [77,78]. In this study, expected utility theory is used to aggregate individual preferences.
Expected utility theory is a formal mathematical tool that was designed to aid decision-makers in choosing among many possible choices to balance risk and reward [79,80]. It is widely used to address risk and uncertainty decisions in economics, finance, insurance, and management science [81][82][83]. Arrow and Lind [84] indicated that when decisionmakers are in a risk-neutral state, it is possible for them to reach a consensus within a group of stakeholders [85,86]. Expected utility theory holds that decision-makers choose among risky and uncertain prospects by comparing their expected utility values, which are the weighted sums obtained by adding the utility values of outcomes multiplied by their respective probabilities [87]. In this study, expected utility theory is used to facilitate consensus in a group decision-support system.

Preliminaries
MCGDM is a decision-making procedure that is used to find the best satisfactory solution that increases the level of overall satisfaction of the group members with the final decision. MCGDM has the virtue of drawing on the wisdom of masses [73]. In an MCGDM problem, a group of decisionmakers is formed to help rank a finite set of alternatives more rationally by aggregating the individual preferences considering the decision-makers' respective knowledge, experience, and expertise for the decision space [88,89]. In the following subsections, this paper describes some MCDM techniques such as GRA, present the basic concepts of an angle between two vectors, and summarize expected utility theory.

Grey Relational Analysis. Grey theory was developed by
Deng [43] to determine the degree of relation among various alternatives in an MCDM problem [69]. GRA is a part of grey theory. It has been used to handle imprecise and vague information in grey systems under variable factors and a changing environment [55]. GRA aims to measure the degree of similarity between the reference series and alternative series, which is suitable for solving problems with complicated interrelationships among multiple factors and multiple variables [19,90]. is only requires a reasonable amount of sample data, and it uses a simple calculation. e specific steps are presented as follows [91]. e decision matrix R of a multicriteria problem with m alternatives and n criteria is given as in the following equation: (1) Standardize the raw matrix R: e standardization process is as follows: (a) e Larger-the-Better (LB): the larger objective value is better (e.g., the benefit), and it can be expressed by (b) e Smaller-the-Better (SB): the smaller objective value is better (e.g., the cost and defects), and it can be represented as follows: (c) e Nominal-the-Better (NB): the value closer to the objective value x ob is better, and it can be represented as follows: (2) Construct the normalized matrix R ′ : (3) Generate the reference series x′(0): where x 1j ′ (0) is the reference value in relation to the jth factor. It is determined by the largest and normalized value of each factor.
(4) Calculate all differences Δ ij (0) between all the normalized alternative series and the reference series x ′ (0): (5) Compute the grey relational coefficient c ij (0) between all the normalized alternative series and the reference series: where δ is a distinguished coefficient. e value of δ is usually set to 0.5 to provide moderate distinguishing effects and good stability. (6) Get the grey relational degree Γ i : In this study, the weight vector w j of the semantic opinions for different stakeholders is applied to calculate the normalized weight of criteria.

Angle between Two Vectors.
In Section 2, the definition of the angle between two vectors is introduced, which is the cosine formula related to two vectors. We know that the greater the value of cos(α, β) is, the closer the direction of vector α is to that of β. We also know that a vector is composed of two parts: direction and norm. Now, this paper introduces the concept of the norm of a vector and the projection. e norm of a vector is a function that assigns a strictly positive length or size to each vector in a vector space, rather than zero vector. Suppose that vector α � (α 1 , α 2 , . . . , α n ). e norm of vector α can be computed as follows [90]: e projection represents the closeness degree of two vectors by considering the norm magnitude and angle between two vectors together, as the angle between two vectors reflects the similarity between the directions of the two vectors [92]. To measure the degree of similarity between the vectors from a global point of view, this paper uses the following formula for projection.
Suppose that α � (α 1 , α 2 , . . . , α n ) and β � (β 1 , β 2 , . . . , β n ) are two vectors. e projection of vector α onto vector β can be defined as follows [93]: Vector β will be closer to vector α when the value of p(α) is larger. Hence, the projection of vector α onto vector β can measure the degree of similarity between the vectors from a global point of view by considering the two parts of a vector: direction and norm.

Expected Utility eory.
Expected utility theory is a useful model for risk aversion, and it is widely applied in theoretical and practical analysis [79,81,82]. e underlying principle is that the decision-maker has prior knowledge of the probabilities of all the events that can occur. e decision-makers assign a value (e.g., a sum of money) to each alternative [85]. Expected utility theory aims to help decision-makers to choose among various possible choices to balance risk and reward using a formal, mathematical function. Arrow and Lind [84] argued that expected utility theory can help reach a decision-making consensus within a group of stakeholders [85]. e von Neumann-Morgenstern expected utility over a set of outcomes can be expressed as follows: where U(x) is the utility of all sets of possible outcomes; u(x i ) is the utility of an outcome x i (i � 1, 2, . . . n); and p(x i ) is the probability of an outcome x i (i � 1, 2, . . . , n), where p(x i ) ≥ 0 for all i � 1, 2, . . . , n and n i�1 p(x i ) � 1.

Evaluation Model
In this section, the developed model is proposed, which is a revised GRA method for facilitating consensus. e model is composed of two parts: a revised GRA method and MCGDM with expected utility theory.

Revised GRA Method.
e GRA method analyzes the data change in the trend curve by measuring the similarity between the reference series and alternative series. e concept of an angle between two vectors can reflect the similarity between the directions of two vectors [93]. We know that the direction of vector α will be closer to that of β when the value of cos(α, β) is larger. e projection of vector α onto vector β can reflect and measure the degree of similarity between the vectors from the global point of view by considering the two parts of a vector: direction and norm.
Hence, the projection of the angle between two vectors can be applied to express the closeness of the relationship between each alternative series and the reference series [93]. e detailed processes of the revised GRA method are as follows: e decision matrix R with m alternatives and n criteria is as follows: (1) Standardize the raw matrix R: the standardization process is described as follows: (a) e Larger-the-Better (LB): the larger objective value is better (e.g., the benefit), which can be expressed by (b) e Smaller-the-Better (SB): the smaller objective value is better (e.g., the cost and defects), which can be represented as follows: (c) e Nominal-the-Better (NB): the value closer to the objective value x ob is better, which can be represented as follows: (2) Construct the normalized matrix R ′ : (3) Generate the reference series x ′ (0): (18) where x 1j ′ (0) is the reference value in relation to the jth factor. It is determined by the largest and normalized value of each factor.
(4) Calculate all the differences Δ ij (0) between all the normalized alternative series and the reference series x ′ (0):

Mathematical Problems in Engineering
(20) (5) Compute the grey relational coefficient c ij (0) between all the normalized alternative series and the reference series: where δ is a distinguished coefficient. e value of δ is usually set to 0.5 to provide moderate distinguishing effects and good stability. (6) Generate a grey correlation coefficient matrix A between the normalized alternative series and the reference series: e normalized weight of the criterion is determined by the weight vector w j of the semantic opinions for different stakeholders. (8) Calculate the grey relational degree ε. e goal of grey relational degree is to measure the similarity between the reference series and alternative series, which indicates the projection of the normalized alternative series of each alternative onto the reference series. e concept of an angle between two vectors can reflect the similarity between two vectors. e projection of vector α onto vector β can measure the degree of similarity between the vectors from a global point of view by considering the two parts of a vector: direction and norm. Hence, the projection of the angle between two vectors can be used to indicate the degree of similarity between the normalized alternative series and the reference series. Let vector α be the weighted grey correlation coefficient values of the normalized alternative series of each alternative, and let vector β be the weighted grey correlation coefficient values of the reference series separately. So, the grey relational degree ε can be calculated as follows: en, this study can easily obtain the grey relational degree of each alternative onto the reference series. e normalized alternative series α will be closer to the reference series β when the value of the grey relational degree is larger, and the alternative will be better.

MCGDM.
MCGDM considers the problem of evaluating or selecting the best alternative(s) from a set of feasible alternatives according to the preferences provided by a group of experts, which are associated with incommensurate and conflicting criteria [78,[94][95][96]. e aim is to find the best satisfactory solution that increases the level of overall satisfaction with the final decision for a group of decisionmakers. A challenging problem that must be addressed when dealing with MCGDM problems is how to effectively aggregate individual preferences.
In this study, expected utility theory is used to aggregate individual preferences to form a group's consensus.
is study first uses the utility function to measure personal preferences. en, this paper uses expected utility theory to aggregate individual preferences per the von Neumann-Morgenstern utility theory. It is assumed that the decision-makers have a complete, reflexive, transitive, and continuous evaluation. In other words, the individual preferences can be described with the von Neumann-Morgenstern utility function that was expressed in Section 3.3, which can be restated as follows [85]: � u x 1 p x 1 + u x 2 p x 2 + · · · + u x i p x i + · · · + u x n p x n . (25) e expected utility is calculated by taking the weighted average of all possible outcomes. Each group member can be assigned a weight p(x i ) ranging from 0 to 1, which denotes her/his power in the group decision-making [85]. In this study, the utility function is defined as the revised GRA method. us, group decision-making with expected utility theory entails the sufficiency of reliability and belief in the process of aggregating multiple members' utilities. e aggregated utilities denote the group preferences over all criteria and alternatives.

Experiment
In this section, a simulation experiment of an oil spill emergency management is designed to test and verify the revised GRA method and consensus-facilitating technique with expected utility theory. e aim is to propose and develop an effective new technique for crisis management in oil spill emergencies in a complex environment to reduce disaster risk.

Description of the
Problem. An oil spill emergency can have a major impact on environmentally, ecologically, and economically sensitive marine areas and shorelines. To minimize the damage and pollution caused by oil spill accidents, it is necessary to develop an emergency plan for crisis management [9]. However, evaluating the best oil spill emergency plan is difficult because the plan depends on several variables within a changing and complex oil spill environment, such as quantity and type of oil spilled and location of oil spill, among other factors [1,9]. Moreover, the benefits and interests of the stakeholders (e.g., environmental agencies, oil companies, and nongovernmental organizations) involved in accidental oil spill incidents inevitably conflict in the decision-making process of choosing the best emergency alternative [9]. ese factors motivate this research to find a new technique that can effectively handle variable factors with conflicting and incommensurate criteria. Hence, in this study, a revised GRA method for facilitating consensus is proposed and developed to aid the decision-makers who are involved in emergency planning for oil spill incidents.

Data and Criteria.
Emergency response alternatives to oil spill accidents have been researched by simulating the trajectories of oil spills based on information about the quantity and type of the oil spill, the location of the oil spill, the spilled volume, the weather, and the oceanic conditions using the computational package OILMAP [9,20]. e main parameters of the oil spill response scenarios were defined in Ferreira [97] as the amount of oil at the coast (C 1 ) and the amount of oil intercepted representing the amount of oil collected plus the amount of oil dispersed (C 2 ) [9,97]. e process of establishing a model for the MCGDM problem is composed of alternatives and criteria, which form the decision matrix. e decision matrix, consisting of 10 emergency response alternatives and two criteria, is shown in Table 1 [9].

Empirical Process.
To demonstrate the feasibility and effectiveness of the proposed model, this section presents a simulation of oil spill emergency management to evaluate and select the best alternative to aid stakeholders involved in the emergency response for disaster risk reduction.
In the decision-making process during an oil spill emergency response, the opinions of the stakeholders (environmental agency: DM1; oil company: DM2; nongovernmental organization: DM3) on each criterion will usually differ. erefore, a weight vector is expressed to define the importance of each criterion considering the opinions of the stakeholders. Nine levels of semantic importance for each criterion, which originated from the 1-9 scale of the analytic hierarchy process method [42], are assigned and defined as shown in Table 2. e stakeholders' semantic importance preferences are presented in the form of weights in Table 3. e following procedures illustrate the feasibility and effectiveness of the proposed revised GRA method for group decision-making within a variable and complex oil spill environment with incommensurate and conflicting criteria and a group of stakeholders.

Standardize the Raw Matrix.
e raw decision matrix, which consists of 10 emergency alternatives and two criteria, is presented in Table 1. e criterion of oil at cost, C 1 , is an SB criterion. e criterion of oil intercepted, C 2 , is an LB criterion. Hence, the normalized decision matrix can be calculated according to formulas (14)- (17). It is shown in Table 4. Tables 2 and 3, this paper can determine that the weight vector for DM 1 is w 1 � (w 11 , w 12 ) � (0.5, 0.5); the weight vector for DM 2 is w 2 � (w 21 , w 22 ) � (0.1, 0.9); and the weight vector for DM 3 is w 3 � (w 31 , w 32 ) � (0.8, 0.2). e grey relational degree for the three stakeholders can then be calculated using formulas (18)- (24). e results for DM 1 , DM 2 , and DM 3 are given in Tables 5, 6, and 7, respectively.

Group Decision-Making with Expected Utility eory.
e von Neumann-Morgenstern utility theory assumes that the decision-makers have a complete, reflexive, transitive, and continuous evaluation. e weight vector of the three stakeholders in the simulation, namely, DM 1 , DM 2 , and DM 3 , is w DM � (0.3333, 0.3333, 0.3333), and it is assumed that the weight of expert opinion for the three stakeholders is equally important. In this study, the utility function is assumed as the function of the revised GRA method. Hence, the aggregated utilities, which denote the group preference over all alternatives and criteria, can be calculated by formula 25. e results are shown in Table 8. Table 8, this study can see that the ranking of the 10 emergency alternatives is 8, 9, 10, 7, 3, 6, 5, 1, 2, and 4. e best emergency alternative is Alt. 8, followed by Alt. 9, Alt. 5, Alt. 10, Alt. 7, Alt. 6, Alt. 4, Alt. 1, Alt. 2, and Alt. 3. A visual analysis was also conducted, presented in Figure 1.

Analysis and Discussion of the Results. In
In a previous study, fuzzy TOPSIS for group decisionmaking was used to determine the best oil spill emergency alternative, with promising results [9]. e research results are the same as those of Krohling and Campanharo [9] in two respects: the best oil spill emergency alternative is the same, and the ranking of the 10 emergency alternatives is     also the same. In addition, the visual analysis that this paper presents in Figure 1 is the same as that in the work of Krohling and Campanharo [9], which verifies the feasibility and effectiveness of the proposed model for emergency response in the context of oil spill accidents.

Conclusion
Emergency management in response to oil spill accidents is a complex decision-making problem. How to effectively evaluate oil spill emergency alternatives for a complex environment to formulate crisis response strategies for disaster risk reduction has become a challenging issue worldwide. is study proposes and develops an effective technique, which is a revised GRA for facilitating a consensus response to an oil spill emergency. e advantages of this study are summarized as follows: Firstly, a revised GRA for group decision-making model is proposed to evaluate an emergency plan for responding to unexpected events related to an oil spill. Secondly, in this model, the concept of an angle    between two vectors is applied to improve and optimize the GRA method by measuring the similarity between the reference series and alternative series. irdly, the expected utility theory is used to aggregate individual preferences and facilitate group consensus. Fourthly, the simulation experiment verified the feasibility and effectiveness of the proposed model by comparative analysis and visual analysis.
Finally, in addition to oil spill scenarios, the proposed model can be used to solve other complex real-world problems that have a finite number of alternatives, multiple criteria, semantically stated benefits, multiple stakeholders' interests, and the need for emergency decisions under a variable and complex environment. e main limitation of the proposed model is that there are few data samples and indicators. In future work, big data analysis methods will be further developed and proposed for emergency decisions under a variable and complex environment in order to reduce the impact of small sample sizes.

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

Conflicts of Interest
ere are no conflicts of interest regarding the publication of this paper.