2.1 The extended GRA method used to obtain the weights

Grey correlation analysis is a derivative method based on grey system theory [Reference Ju-Long13] and has little requirements on the sample size and is suitable for the problem of a small sample with, a scarce amount of information, which is difficult to be solved by fuzzy mathematics. Therefore, it is more suitable for determining the attribute weight based on a small amount of decision information in the context of equipment optimal selection.

Literature [Reference Zhang, Liu and Zhai17] proposes an extended GRA method of interval triangular fuzzy numbers. The single membership degree of the traditional fuzzy set can express the degree of support and further obtain the degree of opposition [Reference Zadeh18], but it cannot express the degree of neither support nor opposition. Intuitionistic fuzzy number introduces the concept of non-membership degree [Reference Atanassov15], and is more in line with the actual situation of people’s voting in reality. In this paper, Zhang’s method is extended to intuitionistic fuzzy numbers with the following steps:

Step 1: Determining the ideal sequence.

The ideal sequence is generally consisting of the best value of each attribute of each alternative. In this paper, we use the maximum (or minimum) of the intuitionistic fuzzy number as the ideal sequence. Whether the maximum or minimum value is used depends on whether the decision target is the best or the worst.

The ideal sequence is denoted as:

(1) \begin{align}{R_0} = \left\{ {{r_{01}},{r_{02}}, \ldots, {r_{0n}}} \right\},{r_{0j}} = (1,0,0)\;or\;(0,1,0),j = 1,2, \ldots, n\end{align}

Step 2: Calculating the distance between the attribute value sequence and the ideal sequence.

The distance between the jth attribute of the ith alternative and the jth attribute of the ideal sequence is calculated as follows:

(2) \begin{align}{\Delta _{ij}} = d({r_{0j}},{x_{ij}}) = {1 \mathord{\left/ {\vphantom {1 2}} \right.} 2} \cdot (\left| {{\mu _{{r_{0j}}}} - {\mu _{{x_{ij}}}}} \right| + \left| {{\nu _{{r_{0j}}}} - {\nu _{{x_{ij}}}}} \right| + \left| {{\pi _{{r_{0j}}}} - {\pi _{{x_{ij}}}}} \right|)\end{align}

where ${r_{0j}} = ({\mu _{{r_{0j}}}},{\nu _{{r_{0j}}}},{\pi _{{r_{0j}}}})$ and ${x_{ij}} = ({\mu _{{x_{ij}}}},{\nu _{{x_{ij}}}},{\pi _{{x_{ij}}}})$ are intuitionistic fuzzy numbers, and μ, v and π represent the degree of membership, the degree of non-membership and the degree of hesitation in the intuitionistic fuzzy number.

Step 3: Calculating the grey relational coefficient.

Step 4: Constructing a multi-objective programming equation.

Since the alternative with a greater grey correlation degree with the ideal sequence is closer to the ideal sequence, the multi-objective programming equation which is constructed with the grey correlation degree as the programming objective is as follows:

(3) \begin{align}\left\{ \begin{array}{l}\max {\gamma _i} = \sum\limits_{j = 1}^n {{\omega _j}{\xi _{ij}}}, i = 1,2, \ldots, m.\\s.t.:\sum\limits_{j = 1}^n {\omega _j^{} = 1}, {\omega _j} \geqslant 0,j = 1,2, \ldots, m.\end{array} \right.\end{align}

where ${\xi _{ij}}$ represents the grey relational coefficient between the $j{\rm{th}}$ attribute of the $i{\rm{th}}$ alternative and the $j{\rm{th}}$ attribute of the ideal sequence.

Step 5: Transforming the equation and solving the equation.

Considering that there is no preference relationship between alternatives and fair competition, the problem can be transformed into a single-objective programming problem by taking the sum of the grey correlation degrees of each alternative as the planning objective. The conditional extremum is obtained by the Lagrange multiplier method, and the weight expression is finally obtained:

(4) \begin{align}{\omega _j} = {{\sum\limits_{i = 1}^m {{\xi _{ij}}} } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^m {{\xi _{ij}}} } {\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{\xi _{ij}}} } }}} \right.} {\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{\xi _{ij}}} } }}\end{align}

2.2 The VIKOR method used on IFN

VIKOR method [Reference Opricovic14] is a compromise ranking method based on ideal points. It is more suitable for situations where the decision maker cannot express the preference accurately, the measurement units used are different, or there is a conflict between evaluation criteria. In addition, VIKOR method can obtain a compromise solution with priority, which can provide decision-makers with more comprehensive and accurate decision aids.

The steps of VIKOR based on intuitionistic fuzzy numbers are as follows:

Step 1: Determining the positive ideal solutions and negative ideal solutions.

The two represent the most and least expected solutions, respectively. They can be denoted as ${r^ + }$ and ${r^ - }$ .

Step 2: Calculating group effect value and individual regret value.

The group effect value S and individual regret value R of the ith alternative is calculated using the following formula:

(5) \begin{align}{S_i} = \sum\limits_{j = 1}^n {{\omega _j}{{(r_j^ + - {r_{ij}})} \mathord{\left/ {\vphantom {{(r_j^ + - {r_{ij}})} {(r_j^ + - r_j^ - )}}} \right.} {(r_j^ + - r_j^ - )}}}, i = 1,2, \ldots, m;\;\; j = 1,2, \ldots, n\end{align}

(6) \begin{align}{R_i} = \mathop {\max }\limits_j {\omega _j}{{(r_j^ + - {r_{ij}})} \mathord{\left/ {\vphantom {{(r_j^ + - {r_{ij}})} {(r_j^ + - r_j^ - )}}} \right.} {(r_j^ + - r_j^ - )}},i = 1,2, \ldots, m;\;\; j = 1,2, \ldots, n\end{align}

where j represents the jth attribute. $r_j^ +, r_j^ -, {r_{ij}}$ are intuitionistic fuzzy numbers, so subtracting two intuitionistic fuzzy numbers means calculating the distance between them:

(7) \begin{align}\alpha - \beta = d(\alpha, \beta ) = {1 \mathord{\left/ {\vphantom {1 2}} \right.} 2} \cdot (\left| {{\mu _\alpha } - {\mu _\beta }} \right| + \left| {{\nu _\alpha } - {\nu _\beta }} \right| + \left| {{\pi _\alpha } - {\pi _\beta }} \right|)\end{align}

Step 3: Calculating comprehensive evaluation value Q.

Step 4: Sorting the alternatives.

S, R and, Q are sorted in ascending order respectively, and the final alternative is determined according to the evaluation criteria.

3.1 Criteria

Tactics for a battlefield commander include not only the planning of weapons and equipment but also the planning of various resources such as troops and maintenance resources [Reference Özdemir and Kurtz2].

At present, excluding the temporary formation, the equipment of the forces of various countries is in the minimum operational formation, such as flying squadrons, missile battalions, naval formations, etc., and the equipment for different purposes is usually composed of only one type of equipment. In the combat organisation, in addition to the combat equipment to carry out combat tasks, there are also support forces to meet support needs, such as the fourth station company of the flying squadron, the maintenance squadron, the combat support battalion, and the comprehensive support battalion in the missile brigade, etc. In addition to the selection of models and support forces, the commander also needs to carry out the planning of troop arrangements and individual assaults.

To sum up, this paper adopts the sequential decision-making method to construct an intelligent optimal selection system. Firstly, the model selection should be completed through effectiveness evaluation to determine the equipment model for the combat mission. Then, in the units equipped with this type of equipment, the support capacity of each unit is evaluated to determine the troops to carry out the combat mission. Finally, when it comes to specific equipment, it is necessary to determine it in the selected troops based on the state of the equipment through health assessment. The determination logic of sequential evaluation criteria is shown in Fig. 1.

Figure 1. The logic of the determination of the sequential criteria.

3.2 Index

The three types of evaluation all need to establish a corresponding index system for evaluation calculation. This section determines the index system by analysing the characteristics of each evaluation.

Without considering the enemy factors, the combat effectiveness of model selection is mainly affected by the equipment’s performance, useability, survivability and other aspects. Therefore, the indicators can be divided into the performance parameters and geometric parameters of the equipment. For example, performance parameters such as thrust-weight ratio and maximum flight speed of fighter aircraft will affect the performance and use of aircraft, while geometric parameters such as wing length and head-on radar reflection area will affect the survivability of fighter aircraft [Reference Zhu Baoliu and Xiaofei19]. However, studies involving maintenance strategies and support schemes [Reference Lin, Luo and Zhong7, Reference Ge, Xiao, Yang and Hou20] usually only focus on the selection of maintenance support schemes and support resources of equipment, which are often separated from the optimal selection of equipment in practical use.

The implementation of safeguards requires a decision based on the specific circumstances of the support force in which the equipment is located. The common indexes in effectiveness and support evaluation can be seen in Table 1. To sum up, support capability can be generally summarised as support personnel index, support facilities index, support management index and support design index.

Table 1. Common indexes in effectiveness and support evaluation

For health assessment, the data of each sub-system of the equipment is tested by sensors and other detection devices according to concepts such as health state-based maintenance (CBM) [Reference Pedersen and Vatn8] and fault diagnosis prediction and health management (PHM) [Reference Vogl, Weiss and Helu24]. Data is transmitted back to the management system for condition monitoring, fault diagnosis and life prediction to determine when and how equipment should be repaired. Simplifying the above ideas, use language evaluation which is given by the experts according to the test data of each subsystem as the decision information for the health evaluation of equipment optimal selection.

The final general index system of sequential decision-making is shown in Fig. 2. As it is based on multi-faceted research on the optimal models of typical equipment in various fields such as fighters, missiles and so on, the index system is applicable for not only fighter optimal selection but can be transformed to other equipment optimal selection quickly. For fighters, with the central maintenance computer architecture for reference, the fighter’s key systems can be divided into: engine systems, aircraft health monitoring facilities by recording analysis mechanical and electrical systems (fuel, power supply, hydraulic, environmental control system) and avionics system (such as navigation, communication, weather radar system) of the state information [Reference Ranasinghe, Sabatini, Gardi, Bijjahalli, Kapoor, Fahey and Thangavel25].

Figure 2. The general index system.

3.3 Decision-making with uncertainty

The indicators of support evaluation generally need to be given according to the actual situation of the formation, which often has complexity and uncertainty and is difficult to express by a definite value. In this paper, the attribute value is obtained by the language evaluation method given by experts. In health evaluation, in order to reduce the algorithm complexity and improve the decision-making efficiency, this paper adopts the same method to get the attribute value. Therefore, for support assessment and health assessment, this paper uses the hybrid MCDM method based on IFNs. The process of the hybrid MCDM method is shown in Fig. 3.

Figure 3. Steps of the hybrid MCDM method.

Step 1: the support index of each support force and the health index of each piece of equipment will be evaluated by experts in a seven-level system. The relations between linguistic variables and IFNs are shown in Table 2 [Reference Jing, He, Ma, Xie, Zhou, Gao and Jiang26].

Step 2: It is transformed into intuitionistic fuzzy numbers according to Table 2, and the attribute value matrix composed of intuitionistic fuzzy numbers is obtained.

Step 3: The attribute value matrix was substituted into the extended GRA model to obtain the attribute weight.

Step 4: The attribute weight and attribute value matrix are brought into the VIKOR method to obtain the evaluation ranking results of each alternative, and finally the result scheme is obtained.

3.4 Sequential decision-making framework

Based on the determined index system and sequential decision-making method, the universal decision model is constructed as follows:

1. (1) Integrated decision-making model

   \begin{align*} Optimization = sequential(E,Support,Health). \end{align*}

E represents the evaluation result of combat effectiveness, Support represents the evaluation result of support capability and Health represents the evaluation result of health status.

Equipment optimal selection needs to be carried out sequentially by efficiency evaluation, support evaluation and health evaluation. Input the decision information of the three types of evaluation, and the comprehensive evaluation results of the three can be obtained, to complete the optimal selection from model level, to support force level and then to specific equipment level.

1. (2) Effectiveness evaluation model

The efficiency evaluation equation is used for effectiveness evaluation, as the research on performance evaluation has a history of several decades, and the theory and application have been more comprehensive and reliable.

\begin{align*}E = efficiency(Per,Geo).\end{align*}

In efficiency evaluation, the parameters of equipment performance index Per and geometric index Geo should be taken as input. After normalisation, the evaluation results are calculated by the evaluation equation, and the sorting is completed to obtain the final selected equipment model.

In this paper, the composite index E of the survivability Geo and the performance index Per is used to measure the effectiveness of fighter aircraft. The calculation equation is as follows [Reference Dong Yanfei27–Reference Huang Guoqing and Zang29]:

(8) \begin{align}Geo = {\omega _{s1}} \cdot {(5/RCS)^{0.25}} + {\omega _{s2}} \cdot {(Sw \times {L_{all}})^{ - 1}}\end{align}

(9) \begin{align}Per = {\omega _{B1}} \cdot {P_{SEP}} + {\omega _{B2}} \cdot {n_{y,cir}} + {\omega _{B3}} \cdot {T_R} + {\omega _{B4}} \cdot Ma + {\omega _{B5}} \cdot {H_{PC}}\end{align}

(10) \begin{align}E = {\omega _{E1}} \cdot Per + {\omega _{E2}} \cdot Geo\end{align}

where RSC represents the average head-on radar reflection interface, Sw represents the area of wing, ${L_{all}}$ represents the full length of the fighter. ${P_{SEP}}$ indicates the remaining power per unit weight, ${n_{y,cir}}$ indicates the maximum hover overload, ${T_R}$ is the thrust-weight ratio, Ma is maximum speed, ${H_{pc}}$ is service ceiling.

Table 2. The relations between linguistic variables and IFNs

According to the literature [Reference Dong Yanfei27, Reference Huang Guoqing and Zang29], it can be determined that ${\omega _{B1}} = 0.35$ , ${\omega _{B2}} = 0.25$ , ${\omega _{B3}} = 0.15$ , ${\omega _{B4}} = 0.15$ , ${\omega _{B5}} = 0.10$ . Refers to the comprehensive evaluation of effectiveness, which can determine the main influencing factors of energy based on references. However, since geometric indexes affect the survivability of fighter aircraft, they also have a certain weight [Reference Dong, Wang and Zhang28], so it is determined as ${\omega _{E1}} = 0.75$ , ${\omega _{E2}} = 0.25$ .

1. (3) Support assessment model

   \begin{align*} Support = support(val,v,\rho ). \end{align*}

In the sub-model of support evaluation, experts should give the language evaluation val of four kinds of support evaluation indexes for the alternative support forces, and input the decision mechanism coefficient v and discrimination coefficient $\rho$ according to the actual decision tendency. In the model, the evaluation value is transformed into the attribute value matrix of intuitionistic fuzzy number, and the weight is obtained by the weight solving algorithm and the optimal result of the final support force is obtained by the scheme ranking algorithm.

1. (4) Health assessment model

   \begin{align*} Health = health(val,v,\rho ).\end{align*}

Health assessment requires experts to give the language evaluation val of the health status of the subsystem according to the parameter data of each key subsystem for each alternative equipment, and input the decision mechanism coefficient v and discrimination coefficient $\rho$ according to the actual decision tendency. The optimal selection procedure is the same as that of the support evaluation model.

In summary, the overall process of sequential equipment optimal selection can be obtained in Fig. 4.

Figure 4. Steps of sequential MCDM method.

4.1 Sensitivity analysis

4.1.1 Resolution coefficient $\rho$

Resolution coefficient $\rho$ is the coefficient used to measure the resolution ability of the algorithm in grey correlation analysis [Reference Wang, Meng, Zhai and Zhu30]. Therefore, this paper takes support assessment as an example, and selects several representative values in the interval of [0,1], to observe the change of attribute weight results. The result is shown in Fig. 5.

Figure 5. Sensitivity analysis result of ρ.

It can be observed from the above figure that the resolution coefficient $\rho$ has a significant influence on the calculated weight. The fluctuation of the weights of different attributes is different. When ρ = 0, the weight fluctuation is the largest; when ρ = 1, the weight fluctuation is the smallest. Thus, it can be seen that the value of $\rho$ will affect the resolution of the weight, a higher value will lead to a decline in resolution, which means that even if the two sequences are in the worst correlation situation, they will get a higher correlation coefficient, which is inconsistent with people’s actual cognition.

Referring to the literature [Reference Shen Maoxing and Zhang31], according to the ‘3σ rule’, the implementation range can cover 0.95, which can be regarded as a significant result. With the distinguishing coefficient $\xi $ as the benchmark, when the range [0.05, 1] can be obtained, the resolution ability can be considered significant. Literature [Reference Shen Maoxing and Zhang31] gives the calculation method to ensure high resolution: when ${\Delta _{ij}} = {\Delta _{\max }}$ , ${\xi _{ij}}$ gets the minimum value:

(11) \begin{align}{\xi _{ij}} = ({{{\Delta _{\min }} + \rho {\Delta _{\max }})} \mathord{\left/ {\vphantom {{{\Delta _{\min }} + \rho {\Delta _{\max }})} {((1 + \rho ){\Delta _{\max }}}}} \right.} {((1 + \rho ){\Delta _{\max }}}})\end{align}

When ${\Delta _{\min }} = 0$ , ${\xi _{ij}}$ gets the infimum:

(12) \begin{align}\inf {\xi _{ij}} = \rho /(1 + \rho )\end{align}

Let $\inf {\xi _{ij}} = \rho /(1 + \rho ) = 0.05$ , it can be obtained that ρ = 0.0526, and because the infimum of ${\xi _{ij}}$ is increasing with ρ, it is obtained that when ρ ≤ 0.0526, the resolution is strong. In the decision-making, the commander can take ρ = 0.0526, to ensure that the final weight of the solution results relative gap is larger and more realistic.

4.1.2 Decision-making mechanism coefficient v

The decision-making mechanism coefficient v in the VIKOR method represents the degree of emphasis of decision-makers on group effect and individual regret. In this paper, several representative values are selected in the interval of [0, 1] to observe the change in ranking results. Since the ranking of the compromise value itself can approximately represent the ranking result of the scheme, the two criteria are not considered in the analysis, and the compromise value is used as the ranking basis.

It can be observed in Fig. 6 that the comprehensive evaluation value Q will change significantly with the value change of v. Sometimes it may even influence the result of the plan ranking. It can be seen that the value of the decision-making mechanism coefficient also has a significant impact on the ranking results. When the decision-maker focuses on maximising the group effect, it can make v > 0.5; when the decision-maker focuses on minimising individual regrets, it can make v < 0.5; when decision-makers want to balance group effect and individual regret, they can make v = 0.5.

Figure 6. Sensitivity analysis result of v.

4.2 Example of the application of sequential MCDM method

4.2.1 Effectiveness assessment

Ten typical models in service are taken as alternative fighter aircraft for model selection (letters A-J are used to represent actual models in this paper), and their performance indexes and geometric indexes are shown in Tables 3 and 4, with data from the literature [Reference Zhu Baoliu and Xiaofei19].

Table 3. Performance indexes

Table 4. Geometric indexes

Step 1: Normalising the attribute value, and denote the normalised decision matrix obtained as $\widetilde {\rm{R}} = {({\widetilde r_{ij}})_{m \times n}}$ .

Step 2: The normalised attribute values are substituted into Equation (8), Equation (9), Equation (10) for calculation, and the ranking results of efficiency are shown in Fig. 7. According to the results, the effectiveness of model E fighter is the best.

Figure 7. Result of effectiveness assessment.

4.2.2 Support assessment

Step 1: Assuming that the E-type aircraft obtained in the optimal selection of the previous level has five alternative support forces, the language evaluation of each alternative force is given by the experts according to the indicators, and the language evaluation matrix is shown in Table 5.

Step 2: The support evaluation model transforms the input language variable matrix into the attribute value matrix composed of intuitionistic fuzzy numbers, as shown in Table 6.

Step 3: Input the value matrix in Step 2 into the weight solving model and the weights were obtained:

\begin{align*}\omega = [{\rm{0}}{\rm{.2351, 0}}{\rm{.2616, 0}}{\rm{.2533, 0}}{\rm{.2500}}],\end{align*}

where the resolution coefficient ρ is chosen as 0.5.

Step 4: Input the attribute weight and attribute value matrix into the VIKOR model, select the decision mechanism coefficient v as 0.5, and the obtained values S, R, and Q are respectively:

\begin{align*}S = [{\rm{0}}{\rm{.6989}}, {\rm{0.5927}}, {\rm{0.2573}}, {\rm{0.2669}}, {\rm{0.2262}}],\end{align*}

\begin{align*}R = [{\rm{0}}{\rm{.2153}}, {\rm{0.2125}}, {\rm{0.1176}}, {\rm{0.1308}}, {\rm{0.0875}}],\end{align*}

\begin{align*}Q = [{\rm{1}}{\rm{.0000}}, {\rm{0.8765}}, {\rm{0.1506}}, {\rm{0.2125}}, 0].\end{align*}

The visualisation of the sorted results is shown in Fig. 8.

Step 5: According to the acceptable advantage criterion, and the acceptable stability criterion, there are three schemes in the compromise scheme set, and the compromise alternative scheme set is finally obtained {troop5, troop3, troop4}.

4.2.3 Health assessment

Step 1: Assuming that the troop5 obtained in the optimal selection of the previous level has seven alternative aircrafts, the language evaluation of each alternative force is given by the experts according to the key system test data of each alternative sortie, and the language evaluation matrix is shown in Table 7.

Step 2: The health evaluation model transforms the input language variable matrix into the attribute value matrix composed of intuitionistic fuzzy numbers, as shown in Table 8.

Step 3: Input the value matrix in Step 2 into the weight solving model and the weights were obtained:

\begin{align*}\omega = [{\rm{0}}{\rm{.1187}}, {\rm{0.1396}}, {\rm{0.1191}}, {\rm{0.1133}}, {\rm{0.1215}}, {\rm{0.1358}}, {\rm{0.1258}}],\end{align*}

where the resolution coefficient ρ is chosen as 0.5.

Step 4: Input the attribute weight and attribute value matrix into the VIKOR model, select the decision mechanism coefficient v as 0.5, and the obtained values S, R, and Q are respectively:

\begin{align*}S = [{\rm{0}}{\rm{.4419}}, {\rm{0.4363}}, {\rm{0.4179}}, {\rm{0.6494}}, {\rm{0.6195}}, {\rm{0.5727 }}, 0.4609],\end{align*}

\begin{align*}R = [{\rm{0}}{\rm{.1069}},{\rm{0.0963}},{\rm{0.0771}},{\rm{0.1154}},{\rm{0.1069}},{\rm{0.1154 }}, 0.0947],\end{align*}

\begin{align*}Q = [{\rm{0}}{\rm{.4413}},{\rm{0.2902}},0, {\rm{1.0000}},{\rm{0.8247}},{\rm{0.8344}},{\rm{0.3220}}].\end{align*}

Table 5. Linguistic evaluation of support forces

Table 6. Attribute value matrix of support forces

Figure 8. Sorting result of support assessment.

Table 7. Linguistic evaluation of alternative aircraft

Table 8. Attribute value matrix of support forces

The visualisation of the sorted results is shown in Fig. 9.

Step 5: According to the acceptable advantage criterion, and the acceptable stability criterion, the VIKOR ranking results of the health assessment are in line with the judgement criteria 1 and 2, and the fighter with the optimal health status of Fighter3 can be directly obtained.

4.3 Comparison analysis

4.3.1 Comparison experiment

Due to the complexity of the GRA-VIKOR-IFNs method, this paper mainly performs the comparison experiment of this method. The comparison experiments are conducted from both qualitative and quantitative perspectives and are divided into two parts, attribute weight solving methods and sort methods. For the former, five methods are selected including the full consistency method (FUCOM) [Reference Pamučar, Stević and Sremac32, Reference Ranjan, Kumar, Bhavani, Padmavathi and Bakka33], the level-based weight assessment (LBWA) [Reference Žižovic and Pamucar34], ordinal priority approach (OPA) [Reference Ataei, Mahmoudi, Feylizadeh and Li35], and entropy weight method [Reference Li, Luo, Wang, Fan and Zhang36]. Table 9 shows a qualitative comparison of the above methods. According to the analysis, the objective empowerment method process is much simpler and does not require subjective knowledge input, and expert knowledge collection. The extended GRA method is applicable for the small simple scenario which is in line with the research scenario of this paper.

Figure 9. Sorting result of health assessment.

Further, a quantitative comparison is made in this paper. Since subjective assignment methods require human subjective input and are not easily accessible, this paper compares two objective assignment methods, entropy weighting method and GRA method. This paper indirectly evaluates the two weight methods by assessing the quality of the decision results obtained using the generated weights, the decision-making method uses the VIKOR method which is applied in this paper, and the quality of the results is expressed using the resolution of the ranked results. The higher the degree of identification between the results, the easier it is for the commander to make a choice based on the ranking results. In Ref. (Reference Mingxing37), the definition of the sort resolution ${R_v}$ is given to calculate the discernibility of the sort result. The formula is as follows:

(13) \begin{align}{R_v} = {{\mathop {\min }\limits_{i,j;i \ne j} \left\{ {\left| {{v_i} - {v_j}} \right|} \right\}} \mathord{\left/ {\vphantom {{\mathop {\min }\limits_{i,j;i \ne j} \left\{ {\left| {{v_i} - {v_j}} \right|} \right\}} {(\mathop {\max }\limits_i {v_i} - \mathop {\min }\limits_i {v_i})}}} \right.} {(\mathop {\max }\limits_i {v_i} - \mathop {\min }\limits_i {v_i})}}\end{align}

where $v = ({v_1},{v_2}, \ldots, {v_n})$ represents the sorting vector, satisfying ${v_i} \in [0,1],\sum\limits_{i = 1}^m {{v_i}} = 1$ . In this paper, the sorting results of each method are normalised as sorting vectors. The methods are utilised to evaluate and optimise the support force of E-type fighter and the comparison result is shown in Table 10. It can be seen that the decision result of the two methods is consistent and the sort resolution of the extended GRA method is bigger than the entropy weight method, which indicates that the weight method in this paper is effective and can provide more identifiable sorting results. Because the grey correlation analysis method is suitable for solving the problem of ‘small data’, it has high applicability for the situation where the decision-making needs to be selected from several or more than ten kinds of alternatives.

For the sort method, this paper selects five methods to compare with the utilised VIKOR method in this paper, including TOPSIS method [Reference Boran, Genç, Kurt and Akay38], GRA method, multi-attributive border approximation area comparison (MABAC) [Reference Pamučar and Ćirović39, Reference Wang, Wei, Wei and Wei40], multi-attributive ideal-real comparative analysis (MAIRCA) [Reference Pamučar, Lukovac, Božanić and Komazec41, Reference Pamucar, Tarle and Parezanovic42]. The qualitative analysis is in Table 11. Each method has its characteristics and applicability, and it is impossible to determine which method is the best. For this paper, the VIKOR method was chosen mainly because of its combined consideration of group utility values and individual regret values. For fighter optimal selection, especially in exercise scenarios, commanders sometimes do not consider using the equipment closest to the ideal solution, but rather synthesise conflicting criteria. In this case, the subjective preference of the decision maker can be satisfied by adjusting the coefficients of the decision mechanism, so the VIKOR method is chosen in this paper, which is more suitable for the fighter aircraft preference problem.

Table 9. The qualitative comparison of weight methods

Table 10. The comparison results of weight methods

Table 11. The qualitative comparison of sort methods

As for the quantitative comparison part of the sort methods, this paper utilises the extended GRA method as the weight method. Since MAIRCA requires the generation of weighting results in the form of IFNs for the subsequent calculation of the gap between the theoretical rating matrix and the real rating matrix, this method will not be considered in this part of the comparison. The result is illustrated in Table 12. This paper compares the different methods in terms of two aspects: the validity of the results and the resolution of the results. It is shown that each method gets the same ranking results, which proves that the results have some credibility and validity. According to the sorting resolution, the fuzzy MCDM method adopted in this research has a higher sorting resolution and more identifiable sorting results, which can provide more reasonable and reliable decision support for commanders. The VIKOR method can also obtain a compromise solution set when the sorting result is not highly identified, providing more comprehensive and reliable decision support for the commander. Therefore, the research methods used in this paper are more flexible and more applicable than other methods.

Table 12. The comparison results of sort methods

4.3.2 Usage of intuitionistic fuzzy numbers (IFNs)

Finally, this paper analyses the necessity of using IFNs by comparing them with the way of expressing decision information by deterministic values. As an example of a support assessment comparison, experts first give the score of each alternative force. The score can be seen in Table 13.

Table 13. Attribute value matrix of support assessment

The weight obtained by extended GRA is:

\begin{align*}\omega = [{\rm{0}}{\rm{.2210, 0}}{\rm{.2632, 0}}{\rm{.2479, 0}}{\rm{.2679}}].\end{align*}

S, R and Q values obtained by VIKOR method are:

\begin{align*}S = [{\rm{0}}{\rm{.5063, 0}}{\rm{.5490, 0}}{\rm{.1412, 0}}{\rm{.1331, 0}}{\rm{.0770}}],\end{align*}

\begin{align*}R = [{\rm{0}}{\rm{.1918, 0}}{\rm{.2348, 0}}{\rm{.0949, 0}}{\rm{.1038, 0}}{\rm{.0474}}],\end{align*}

\begin{align*}Q = [{\rm{0}}{\rm{.8401, 1}}{\rm{.0000, 0}}{\rm{.1948, 0}}{\rm{.2099, 0]}}.\end{align*}

Visualisation results are shown in Fig. 10.

Figure 10. Sorting result of support assessment.

According to the acceptable advantage criterion, and the acceptable stability criterion, there are three schemes in the compromise scheme set, and the compromise alternative scheme set is finally obtained $\left\{ {troop5,troop3,troop4} \right\}$ .

The ranking result is: troop5 > troop3 > troop4 > troop2 > troop1, and sort resolution ${R_v}{\rm{ = }}{{(0.0935 - 0.0868)} \mathord{\left/ {\vphantom {{(0.0935 - 0.0868)} {(0.4455 - 0)}}} \right.} {(0.4455 - 0)}}{\rm{ = 0}}{\rm{.0150}}$ .

In summary, there is a certain difference between the ranking results and the original method when the decision information is expressed by the determined value, and the ranking resolution is less than the multi-criteria decision method based on intuitionistic fuzzy numbers. This is because the determined value will cause problems such as information loss when describing vague information. In addition, experts are more inclined to fill in the language evaluation when performing the scoring, while the hesitation and deviation are greater when filling in the deterministic value. Therefore, the sorting results are not accurate enough and the resolution is low.