1. Introduction
In recent years, the problem of energy exhaustion and environmental pollution has become more and more serious. New energy power generation has become a priority power project for all countries. Wind power generation has also become a priority clean energy for all countries. Since offshore wind farms are far away from the coast and have less visual interference, they are also rich in resources, have high annual utilization hours, stable wind speed, no land resources and less impact on ecological environment. They have developed rapidly in developed areas such as Europe and the United States.
With the increasing scale of offshore wind power, large-capacity and long-distance offshore wind power will be a trend in the future development of wind power. Compared with onshore wind power transmission, offshore wind power transmission is more difficult and costly. The submarine structure is not as stable as the land, which poses some obstacles to the installation of offshore wind power plant structure. At the same time, due to the high humidity and high salinity of the sea air and seawater, wind power equipment and submarine cable are required to have a higher corrosion resistance [
1]. Traditional high voltage alternating current (HVAC) transmission technology is mature and widely used in built offshore wind farms, but its loss increases with the increase of transmission distance. Therefore, it is of great practical significance and urgency to select suitable transmission mode to realize long-distance transmission of large-capacity offshore wind power. With the development of science and technology, the technologies of voltage source converter based on high voltage direct current (VSC-HVDC) transmission and hybrid high voltage direct current(Hybrid HVDC) transmission are gradually developed, which provides a guarantee for long-distance offshore power transmission.
At present, there is abundant research on transmission technology, and a variety of transmission technologies have been applied to offshore wind power transmission process. Many scholars have studied VSC-HVDC based on a voltage source converter (VSC). The main components, topologies, technical performance and control strategies of the VSC-HVDC system are introduced. Korompili, Wu and Zhao gave the converter control system and control strategies, and focused on the control method adopted by VSC-HVDC to meet the requirements of low-voltage ride through (LVRT) and frequency regulation of network coding [
2]. Wu, Wang, Xu and Wang introduced the structure, control methods and application of modular multi-level converter (MMC), discussed the control strategy of VSC-HVDC transmission system, and focused on the double closed-loop vector control and direct power control, the control methods for unbalanced grid voltage, deadbeat control, etc. [
3]. Qiu, Fan, Cheng and Qiao proposed a flexible control strategy to realize offshore power transmission. AC voltage control method was used for rectifier, DC voltage control method was used for inverter, and the performance of this strategy was carried out in emulation and analysis [
4]. Moawwad, El Moursi and Xiao proposed a management scheme based on VSC and a new control strategy. By setting up an experimental device for the control strategy and management scheme, the feasibility and effectiveness of the strategy and scheme were verified by simulation [
5,
6]. With the continuous development of electronic devices and the wide application of HVDC transmission technology, hybrid HVDC transmission system has become a research hotspot. Tang and Xu proposed a topology structure with DC line fault clearance capability based on the study on the hybrid HVDC transmission of a line commutated converter(LCC) and modular multilevel converter(MMC), using power systems computer aided design (PSCAD) to model the hybrid HVDC system, which verifies the validity of the control system and start-up scheme [
7]. Torres-Olguin, Molinas and Undeland studied the feasibility of Hybrid HVDC transmission that is combined by the traditional LCC and VSC. Under various conditions, the proposed control strategy was tested [
8]. Jung, Cui, Lee, & Sul proposed a new topological structure of hybrid HVDC transmission, it can reduce the system cost and operation loss under the existing conditions, but it can still maintain the capability to cope with a DC short-circuit fault [
9]. In order to improve the dynamic performance of the system, Zhou, Yang and Wang modeled, analyzed and controlled the offshore wind power that adopts hybrid HVDC transmission. The corresponding mathematical model is derived by using state function. Considering different control modes, the dynamic analysis of the AC current of the rectifier is carried out, the control scheme is designed according to the analysis results, and the superiority of the control scheme is verified by examples, which lays a foundation for the hybrid direct current transmission as an offshore transmission mode [
10].
There are differences in economy, loss, stability and fault response among different transmission modes. Comparing and analyzing the transmission modes in many aspects to determine the advantages and disadvantages of each mode is more conducive to the selection of transmission modes in wind farms. By means of technical and economic analysis, Ruddy, Meere and O’Donnell compared and analyzed the power loss, reliability and investment cost of low Frequency AC transmission and VSC-HVDC [
11]. From the perspective of economic value, Eeckhout, Hertem, Reza, Srivastava and Belmans and Sousa, Dos Santos, Jardini, Casolari and Nicola considered the relevant economic benefits of transmission modes, established models to observe the impact of different voltage grades, transmission distances and transmission power on costs, analyzed and compared the HVAC transmission mode and VSC-HVDC transmission [
12,
13]. Negra, Todorovic and Ackermann compared transmission losses of HVAC and VSC-HVDC transmission under different voltage levels and distances and drew the conclusion that transmission power of VSC-HVDC transmission is lower than that of HVAC transmission [
14]. Wang and Nguyen This made a comparative analysis of the stability of large offshore wind farms based on HVAC and HVDC transmission. The eigenvalue technique and root-loci plot were used to analyze the reaction of three-phase short-circuit fault in these two ways. The system method is used to evaluate them, and a conclusion is drawn that the stability of HVDC transmission mode is better [
15]. Chou, Wu, Han and Lee synthetically analyzed the frequency response of the transient voltage of a large offshore wind farm in Taiwan and simulated its HVAC and HVDC transmission side by using simulation method. The simulation results show that the HVDC can provide better voltage and frequency response and reduce the impact of grid faults on wind farms, so it has higher reliability [
16].
Currently, emerging transmission technology has been gradually applied to practical projects. An, Tang and Wang summarized the key technologies of VSC-HVDC and the key technologies and equipment of DC grid, introduced the HVDC transmission projects built or planned to be built in China, and provided valuable reference for the development and application of VSC-HVDC transmission technology [
17]. Gustavsen and Mo proposed a method of adjusting the operating voltage of cables to reduce the high loss caused by capacitor charging current under long-term high voltage AC connection. It was applied to a 200 km and 320 MW offshore wind power plant. The result showed that the annual transmission loss is reduced by 9% in this way [
18]. Benasla, Allaoui, Brahami, Denaï and Sood briefly introduced the HVDC transmission lines between North Africa and Europe, as well as other large-scale projects applying HVAC and HVDC transmission, pointed out some problems existing in the operation and control of HVDC transmission lines, and proposed solutions to these problems [
19]. Ingemansson, Wheeler, Macleod, Gallon and Ruiton introduced a high voltage transmission line in Sweden, it used VSC-HVDC transmission to transfer electric energy from Arkeryd to Hurva. The transformer designed on this line based on VSC technology was described, and another Hybrid HVDC transmission line which will be designed and built soon was introduced [
20]. Rao introduced the application of VSC-HVDC transmission mode in Nan-ao islands in China, and gave the corresponding system configurations and control strategies, which indicated this technology has great significance for the transmission of offshore wind power [
21].
At present, in the selection and research of offshore wind power transmission modes, most of them are considered from the economic point of view, such as Eeckhout et al. and Serrano, Burgos and Riquelme, although they considered the line losses, transmission capacity, reliability and other factors of transmission modes in the analysis, in actual modeling and analysis, because the fuzzy information and uncertain data of these indexes cannot be effectively processed, the cost model can only be established based on the economic indicators such as equipment cost, maintenance cost and construction cost, which led to the transmission mode of the offshore wind farm being selected [
12,
22]. Due to the excessive emphasis on cost and neglect of other factors, this choice will inevitably result in a large amount of information loss and one-sided decision-making. The selection of offshore wind power transmission mode is a multiple attribute decision-making problem involving technology, economy, society and other aspects, which requires comprehensive evaluation and selection from multiple perspectives and attributes. In order to solve the above problems, this paper adopts VIKOR method to select the mode of offshore wind power transmission. This method can not only effectively deal with uncertain data and fuzzy information in the decision-making process, fully consider the impact of multi-attribute indexes on the decision-making results, but also make group decision based on many other experts’ opinions to obtain a compromise.
The VIKOR method has great advantages in solving decision-making problems with conflicting and non-generable standards. To evaluate alternatives according to all established standards and to determine compromise that is closest to the ideal. More and more scholars are studying the VIKOR method to solve the related decision-making problems. Wan, Yuan and Dong proposed a linguistic hesitant fuzzy algorithm to deal with multi-criteria decision-making problems, put forward a new order relationship and a series of new distance measures, and defined some desirable properties of distance, established optimization models and deduced objectively criteria weights. Finally, an example of the intelligent transportation system is analyzed to verify the feasibility and validity of this method [
23]. Prakash and Barua proposed a composition model based on the fuzzy analytic hierarchy process (FAHP) and VIKOR to evaluate and rank the selection criteria. Taking the selection decision-making of logistics partners of an Indian company as an example, the sensitivity analysis is carried out by using this composition model, and the robustness of the method is verified [
24]. You, You, Liu and Zhen proposed the ITL-VIKOR method for multi-criteria supplier selection based on interval 2-tuple linguistic information. The feasibility and practicability of this method were verified by an example of supplier selection. The result showed that this method is more suitable for the supplier selection problem under fuzzy, uncertain and incomplete information environment [
25]. Gupta used the VIKOR method to rank airlines according to their reliability, safety, ticket price and other attributes. Through extensive literature review and analysis, it is concluded that quality of service attributes is of great significance for airline managers to solve problems [
26]. Shojaei, Seyed Haeri and Mohammadi used the VIKOR method to establish an effective evaluation model for airport performance worldwide [
27]. Seyed Haeri and Alireza used the VIKOR method to select materials for sustainable development products [
28]. Rajesh used the VIKOR method to build a model to measure the elastic barriers of supply chain [
29]. Ali, Razi, Felice, Sabir and Petrillo assessed the social, environmental and economic impacts of smog based on VIKOR method [
30]. Mateusz, Danuta, Małgorzata, Mariusz and Kesra used technique for order preference by similarity to an ideal solution (TOPSIS) and VIKOR to compare and analyze the sustainable development capacity of European countries, which had the important value in operability evaluation of concept of sustainable development [
31]. The VIKOR method is widely used in all aspects of technical and economic life, but no scholar has used this method to evaluate and select transmission modes.
In this paper, group decision-making method based on the VIKOR method is introduced into the comprehensive evaluation and selection of offshore wind power transmission modes. At the same time, innovations are made in the method. Firstly, the entropy method is used to calculate the index weight for different types of attribute values information. Secondly, the expert trust function is established to give experts weight. Thirdly, all indexes are finally summed as intuitionistic fuzzy numbers, which are aggregating operators, and the final calculation results are obtained. This method solves the disadvantage of uncertain index data and has certain pioneering significance.
The VIKOR method is widely used in various selection evaluation problems, such as renewable energy evaluation, project maintenance grade evaluation, construction risk evaluation, etc. It is seldom used in transmission mode evaluation and selection for offshore wind power. This paper verifies the applicability of this method in this field.
The structure and content of this paper are as follows:
Section 2 studies the components, transmission principles, advantages, disadvantages and applications of three types of offshore wind power transmission modes: HVAC, VSC-HVDC and hybrid HVDC;
Section 3 expounds the basic principles of the VIKOR method, and introduces the interval number, intuitionistic fuzzy number, linguistic variables and other operational rules. Based on the principle of entropy, the calculation methods of attribute objective weights, expert objective weights and corresponding algorithm steps are proposed.
Section 4 constructs the index model of offshore wind farm transmission, and explains the relevance of each index to the problem.
Section 5 uses VIKOR method to analyze an example of offshore wind farm, and obtains the corresponding results and compares them with the actual situation;
Section 6 summarizes the foregoing content of this paper and looks forward to the future research of related technologies.
3. Selection of Evaluation Indexes for Offshore Wind Power Transmission
The transmission of electric energy is closely related to the realization of offshore wind power projects. The evaluation and selection of transmission mode is the key to realize the transmission of electric energy, and the selection of indexes is the focus of evaluation and selection of transmission mode. Different indexes reflect the different characteristics of transmission modes. When choosing transmission modes, appropriate indexes can provide better measurement standards to make the selected transmission modes more in line with the actual situation of offshore wind farms, which is more conducive to the continuous progress of the project. According to the existing literature and the selection requirements of offshore wind farm transmission mode, from technical value, economic value and social value, the following indexes are selected to construct the evaluation index model of offshore wind farm transmission [
13,
14,
17]. The evaluation indexes system for offshore wind power transmission is shown in
Figure 4.
3.1. Indexes of Technical Value
This paper mainly chose the indexes from two aspects: technical advancement and technical applicability, including voltage level, transmission distance, transmission capacity, line loss, reliability, technology maturity, etc.
Voltage level is the highest voltage level that can be reached on power lines. In order to reduce the loss of the transmission line in the long distance transmission line, transmission current should be reduced; and transmission voltage should be increased in order to ensure certain amount of power to users at the same time. The voltage of modern long-distance transmission is very high. Currently, the common voltage of high-voltage transmission lines is 500 kV, AC lines can reach up to 765 kV, and DC lines can reach up to 800 kV [
45]. The voltage level is limited by the insulation performance of the transmission line. The higher the voltage is, the higher the insulation performance requirements of the transmission line will be, and the higher the cost of line construction will be. The higher the transmission voltage, the higher the voltage on the transformer, and the higher the requirements for transformers. Under technical and economic constraints, the higher the voltage, the smaller the loss and the better the transmission performance. When transporting electric energy in offshore wind farms, various factors should be considered comprehensively, such as the size of transmission power, distance, technical and economic requirements, etc. In the controllable range, the appropriate transmission voltage should be selected according to different conditions.
Transmission distance refers to the distance from the power collection system of the wind power plant to an onshore power grid. From the point of view of the built and being built offshore wind farms in the world, the development trend is toward the far sea and the deep sea, especially in Britain and Germany, where the development level of offshore wind power is relatively high, their development of deep sea wind farms has begun. Denmark, the Netherlands and Germany are currently studying the wind farm built 100 kilometers offshore from the UK [
45]. As far as China is concerned, the wind energy resources in the coastal zone are slightly worse than those in the European sea area. In order to reduce the roughness of the sea surface and improve the quality of the wind resources, the trend of developing the deep sea from intertidal zones to inshore has been gradually developed [
1]. Brook no delay. Therefore, transmission distance is an important factor to be considered in the selection of transmission technology. Technologies that provide longer transmission distances are more suitable for deep-sea wind power projects.
Transmission capacity refers to the maximum power allowed to transmit under normal conditions in power lines, which is usually higher than the installed capacity of wind farms. Since the transmission power transmitted by power lines is related to the power quality, power loss, the allowable temperature of the wire and the stability of the power system, this limit is determined by comprehensive judgment of many factors such as economy, technology and so on. Due to the different length of the line and the different position of the line in the system, the emphasis of consideration is also different. In order to improve the transmission capacity of transmission lines, the ways of increasing voltage and reducing current are usually adopted to reduce losses, but the improvement of voltage is limited by the development of insulation technology. It can be expected that with the improvement of insulation technology, the voltage of transmission lines can be further improved, the loss of transmission lines can be further reduced and the transmission capacity of lines can be further increased.
Line losses refer to the energy loss caused by the transmission of electric energy through transmission lines. In addition to power transmission lines, other transmission and transformation equipment in power network, such as transformers, also produce power loss. In AC transmission system, capacitive current will be generated during the transmission of electric energy, resulting in a significant reduction in the transmission performance of cables and an increase in losses. In contrast, the loss rate of DC cables is slightly lower, but it also exists in the process of converter commutation. Hybrid HVDC uses the rectifier side LCC to reduce losses.
Reliability refers to the ability of technology to fulfill specified functions under specified conditions and within specified time. Reliability of transmission mode refers to its ability to operate safely, stably and continuously live. The reliability of transmission mode mainly involves the reliability of submarine cables and related equipment. At the same time, adopting certain technical measures can also improve the overall reliability of transmission mode. Compared with VSC-HVDC and hybrid HVDC, HVAC lacks some technical measures to guarantee its reliability, such as black start and dynamic voltage control, therefore, its reliability is low. Hybrid HVDC technology combines the advantages of flexible HVDC technology and traditional HVDC technology, it has high reliability.
Technology maturity refers to the industrialization practical level of technological achievements in terms of the technology level, technological processes, supporting resources and technical life cycle. It also refers to the development status of technology relative to a specific system or project, which reflects the degree to which technology satisfies the expected goal of the project. The purpose of technology maturity evaluation is to accurately evaluate the state of technology development, improve the scientificity of management decisions. Mature technology is more conducive to controlling the project cost, time limit and quality, and reduce the application risk of technology in engineering. Offshore wind power is developing rapidly in China. Advanced technology is gradually applied to offshore wind power projects. At present, China’s offshore wind power projects have basically adopted high-voltage AC transmission technology. Flexible direct current transmission technology has been successfully applied to offshore wind power in some developed countries, but China has not yet used it. Hybrid direct current transmission technology as a new technology has a low maturity.
3.2. Indexes of Economic Value
The indexes of economic value mainly refer to the index of investment, cost and time cost of technology, including project cost and construction difficulty degree.
Project cost refers to the construction price of a project. It refers to the anticipated or actual total cost required to complete the construction of a project, which generally includes the purchase cost of equipment, construction cost, installation cost and other expenses. In investment decision, project cost is usually the most intuitive economic index. For offshore wind power transmission system, it mainly involves the purchase and installation costs of cable, booster stations, converters and other equipment. Within other constraints, investors are more likely to choose lower-cost options to reduce risk.
The construction difficulty degree is related to the cost, quality and construction period of the project. The laying of submarine cables and the installation and construction of necessary equipment are the main design of offshore transmission line construction. Compared with land transmission lines, offshore wind power lines are difficult to design and construct because of their special geographical environment, which has a great impact on investment decisions of power companies. For the three transmission modes of high voltage AC, flexible DC and hybrid DC, there are differences in construction difficulty due to different technical characteristics and composition structure. The measurement of construction difficulty is the construction period. The longer the construction period is, the more difficult the construction is, and vice versa.
3.3. Indexes of Social Value
Choices are made mainly from a social perspective, including noise and radiation effects.
Noise and radiation effects are environmental problems that should be considered in any project while creating economic benefits. Due to the large scale of offshore wind farms, the development, implementation and operation of offshore wind farms will inevitably affect the ecological environment of the engineering sea area, such as marine fishery, sound environment, seabed topography and electromagnetic radiation during the laying of submarine cables and the construction of equipment. When choosing the scheme, we should try our best to adopt the scheme which has little impact on the environment so as to reduce the damage to the ecological environment.
4. The VIKOR Method and Hybrid Multiple Attribute Group Decision-Making
4.1. Basic Principles of the VIKOR Method
The VIKOR [
46] method is a compromise sorting method based on the ideal point method proposed by Professor Opricovic of Yugoslavia in 1998. It is characterized by considering the maximization of group benefits, the minimization of individual regrets and the integration of subjective preferences of decision makers. The VIKOR method is a compromise sorting method to solve the hybrid multiple attribute decision-making problem.
The evaluation principle of the VIKOR method is to calculate the distance between alternatives and ideal points according to the Lp-metric aggregate function.
In Equation (1), is the weight of target , is the attribute value of target in alternative , is the positive ideal point of target , choosing the optimum value of each alternative in target and is the negative ideal point of target and chooses the worst value of each alternative in target . The meaning of represents the weighted combination deviation of each target from the ideal point. When , denotes 1 norm; , denotes 2 norm and , denotes infinite norm. Since denotes the geometrical distance in a general sense, when , and denote the bound of the distance, namely . By minimizing and to solve the optimal solution, these two optimal solutions are generally different, and they are all Pareto solutions. When , all Pareto solutions lie between these two Pareto solutions.
The VIKOR method is to select a Pareto solution between the two Pareto solutions as a compromise. Since the group benefit value of each alternative is equal to , and the individual regret value is equal to , the corresponding compromise alternative can be obtained by setting the ratio coefficients between group benefit and individual regret value, which reflects the preference of decision makers. This compromise alternative can be accepted by decision makers.
Taking two objectives as an example, the compromise solution determined by the VIKOR method can be represented in the target space by the following
Figure 5:
is the ideal point,
and
are the optimal values in the first objective and the second objective attribute values respectively and
is the closest compromise solution to the optimal solution
in the Pareto solution determined by 1 norm and infinite norm. The compromise solution of
is the result of compromise and concession between the two objectives. The corresponding amount of compromise is
and
, respectively.
4.2. Hybrid Decision Information
4.2.1. Concepts of Interval Numbers and Intuitionistic Fuzzy Numbers
is called an interval number on , is the lower bound of the interval, is the upper bound of the interval and is the real field. When is called number of positive closed intervals; when , the interval number degenerates to real number. Interval number is a set of all real numbers on closed intervals.
Suppose there are any two interval numbers
and
, then the distance between
and
is.
is called an intuitionistic fuzzy set in
.
and
are membership and non-membership of element
belonging to
in
,
,
,
,
. In addition,
is called hesitancy degree or uncertainty degree of element
belonging to
in
. An ordered pair composed of the membership and non-membership of element
in
belongs to
is called the intuitionistic fuzzy number [
47].
Generally, intuitionistic fuzzy numbers are denoted as , , and .
Suppose there are any two intuitionistic fuzzy numbers
and
, then the hamming distance between
and
is,
4.2.2. Processing of Language Variables
The language variable [
48] is expressed in the form of the natural language. Expert group members use familiar simple language phrases to judge alternatives qualitatively, which is more in line with the individual initiative of individual experts. The data of linguistic variable types are usually processed in a transformational way, where the information of semantic evaluation is transformed into intuitionistic fuzzy numbers. The linguistic variables standard and the corresponding intuitionistic fuzzy numbers are shown in
Table 1.
4.3. Decision Matrix Normalization
Considering the multi-attribute group decision-making problem, suppose alternatives set as , attributes set as and decision makers set as . The attribute value of many data types given by experts for each objective is . Assume that attribute weights information is known and , . The weight information of the expert group is known, and , . The hybrid decision matrix of subjective and objective evaluation of information of expert group is constructed as .
Suppose normalized decision matrix as . The most common attributes are benefit indexes and cost indexes. Make represent subscript set of benefit and cost respectively, and , there are.
For real data, the normalized data are:
For interval numbers, the normalized data are:
4.4. Evaluation of Attributes and Objective Weights of Individual Expert
For the objective weights of evaluation attributes, the entropy method [
49] can be used to calculate the weight, and the entropy formulas of each data type can be used to determine the weight to reflect the importance of evaluation attributes more objectively.
Under attribute set
, the entropy of real numbers is,
Under attribute set
, the entropy of interval numbers is,
Among them, , is the element in the deviation degree matrix , , is the ideal value of benefit or cost attributes under the attribute of the normalized decision matrix .
Under the attribute set
, the entropy of intuitionistic fuzzy numbers is,
In the above formula, is the subscript set for real data attributes, is the subscript set for interval data attributes, and is the subscript set for intuitionistic fuzzy data attributes.
The weight
of each attribute
is,
The objective weights of individual experts can be determined by the hesitancy degree in the attribute evaluation information given by experts to alternatives. For an expert, the degree of hesitation in the evaluation information given by the expert reflects the degree of uncertainty of the expert to the evaluation object. The less hesitation, the more information experts have on the evaluation problem, and the higher the reliability of the evaluation results, the greater the weight given to the attribute. Therefore, the expert trust function can be established according to the principle of entropy to determine the objective weight of experts.
The expert trust function is defined as
, and the objective weight of the individual expert is
, there are,
In the formula, is the hesitation value of alternative given by the decision maker in converting the evaluation information of attribute into intuitionistic fuzzy numbers.
4.5. Constructing Intuitionistic Fuzzy Number Group Decision Matrix
In the group decision-making analysis, it is necessary to aggregate the decision information of expert group into group decision matrix
. To achieve this goal, the intuitionistic fuzzy weighted averaging (IFWA) can be used to aggregate [
50].
Let
be a set of intuitionistic fuzzy numbers,
be the weight vector of
,
,
,
, then we call,
as the intuitionistic fuzzy weighted averaging. Among them,
.
4.6. Group Decision Making Based on the VIKOR Method
(1) According to group decision information
, the positive ideal solution
and negative ideal solution
are determined. There are different types of attributes, which are generally divided into efficiency and cost.
In the formula, is the attribute value of alternative about attribute and and are the subsets of benefit and cost objectives respectively.
(2) The group benefits
, individual regrets
and compromise value
of the alternatives.
In the formula, , , , , the smaller the values of , and , the better. is the decision mechanism coefficient, is the tendency of decision makers to group opinions and individual regrets. If , which means that the decision-maker prefers to make decisions according to the way of maximizing the proportion of group benefits, if , which means that he or she prefers to make decisions by minimizing individual regrets, and means that the decision-maker does not have obvious preferences and adopts balanced compromise. In the VIKOR method, is generally taken as an equilibrium compromise to maximize group benefits and minimize individual regrets.
(3) Determining the compromise solution
When the following two conditions are satisfied, the alternative can be sorted according to the value of . The smaller the value, the better the alternative. Let the alternative corresponding to the minimum be . If the minimum satisfies both the following conditions and , then the alternative is the best compromise solution.
Condition : Acceptability advantage . In the formula, and are the minimum and sub-minimum combined values sorted by respectively, and is the number of alternatives.
Condition : In order , the value or value corresponding to the best alternative is also the smallest.
If the above two conditions are not valid at the same time, the obtained compromise solution can be divided into two kinds: if condition is not satisfied, then is the compromise solution, in which the maximum value of is determined by , and these solutions are close to ideal alternative; if condition is not satisfied, alternatives and are the compromise solution.
4.7. Algorithm Steps
To normalize objective attribute values, and transform the linguistic variables into intuitionistic fuzzy numbers, calculate the objective weight of experts by Equations (12) and (13);
To aggregate the individual information of experts by using the Equation (14) intuitionistic fuzzy number weighted average operator (IFWA) to form a complete group decision matrix ;
According to the group decision matrix, to calculate the attribute weight by using the entropy value method (8)–(11);
To determine the ideal solution and negative ideal solution ;
To calculate the group benefits value , individual regrets value and compromise value of alternatives by Equations (17)–(19);
According to the value of , and , they are sorted in descending order. The smaller the value, the better the alternative.
Determine a compromise.