1 Introduction

In order to face future information war, a new military doctrine called Air Sea Battle is introduced, whose main realization form is aircraft carrier battle. Aircraft carrier battle is a confrontation between aircraft carrier battle groups, including aircraft carriers, submarines, and escorts, etc. During the combat process, the commander needs to deal with a lot of information and make a decision quickly and accurately^[1]. However, the decision cannot be made only according to the commander's experiences and judgments^[2]. To avoid the subjectivity and miscalculation, aid decision making system is introduced to provide an aid for the commander. Furthermore, one of special notes is that the prediction about the dynamic process of a battle is the precondition of aid decision making system^[3].

During the past decades, analysis and prediction on complex dynamic processes and their results have drawn much attention of scholars ^[4-7]. In Ref.[4], fuzzy neural network was applied to predict the occurrence of river ice. In Ref.[6], the prediction about the dynamic process of coal demands was studied based on Markov residual correction theory. In practice, the dynamic processes often have uncertain and incomplete information^[8-10], especially for the military field. Therefore, a few contributions are available for uncertain dynamic processes by using the advantages of Bayesian networks. In Ref.[7], Niu et al. put forward a score predication model of NBA and player skill evaluation based on Bayesian network. However, with respect to aircraft carrier battle, the research results are relatively few.

Fortunately, it can be found that the dynamic process and winning probability estimation of aircraft carrier battle are similar to that of football match, due to that they all are carried out with both offensive and defensive sides through teamwork. The deference mainly lies in that the ability assessment method of match's player is not suitable for combat units. To solve the assessment problem of combat units, BP neural network is considered here due to its advantages in nonlinear data processing^[11-12]. In Ref.[13], the threat assessments of a combat were obtained by using BP neural network and the improved glowworm swarm optimization, where the abilities of combat units were assessed through the characteristics of their missions. However, achieving ability values of combat units are only the first step of assessment. To assess the winning probability of the battle, the ability values should be compared with both combat sides. Very recently, Bradley-Terry model was developed for comparison of high-dimensional sparse data in Ref.[14]. However, the general Bradley-Terry model is inadequate for the comparison of large populations of unites due to the intractable computation ^[14-16], furthermore, based on which, the defender advantages cannot be considered. Therefore, to deal with the assessment problem of aircraft carrier battle, the general Bradley-Terry model should be modified.

As discussed above, the whole assessment process is divided into three steps. Firstly, the armed forces of aircraft carrier are classified into three types. Then, each type of armed forces is estimated to obtain its attack ability value and defense ability value. Next, an improved Bradley-Terry model and a Bayesian network are applied to the unit comparison and global assessment respectively, based on which, the winning probabilities of both combat sides are achieved. Finally, the simulation results indicate the effectiveness of the proposed scheme.

2 Problem Statements

The research framework of the winning probability estimation for a combat dynamic process has four parts, which can be seen in Fig. 1.

From Fig. 1, it can be seen that the first part of the winning probability estimation for a combat dynamic process is the classification of combat units. Then, by using BP neural network, the second part, the assessment of combat units is obtained. Next, through an improved Bradley-Terry model, the third part, the comparison analysis of combat units is achieved. Finally, the winning probability results are acquired by using Bayesian network.

3 Main Results 3.1 Classification of Combat Units

Aircraft carrier battle group is a basic combat formation of naval forces, which includes aircraft, surface ships, and submarines. Without loss of generality, it is assumed that there are two battle groups, Army A (as the defender) and Army B (as the attacker), will fight each other during the battle.

According to the units' characteristics, each group can be divided into three subsets. The classification of the units of Army A is shown in Table 1. where UAV means unmanned aerial vehicle.

By using similar method, the units of Army B can be classified into f_B, c_B and s_B.

3.2 The Assessment of Combat Units

Because the combat unit of aircraft carrier group has limited practical experience in peace time, the traditional experience-dependent assessment method cannot be applied. Furthermore, the ability of combat unit is greatly influenced by the operator, and it will change with time. By applying the reasoning features and dynamic characteristics of BP neural network, the ability of combat unit can be obtained.

Due to the method for assessing each subset is similar, here only the aircraft is regarded. Generally, aircraft can attack aircraft and ship, on the contrary, it can be attacked by aircraft and ship. Therefore, the ability of aircraft is classified into four kinds: ability to attack aircraft (α_f→f), ability to attack ship (α_f→c), ability to defense aircraft (β_f→f) and ability to defense ship (β_c→f). By analogy, all combat units have 12 kinds of ability. Due to similarities of evaluation method, the ability assessment of aircraft attack aircraft is only taken as example (See Fig. 2).

From Fig. 2, it can be seen that three-layer neural network is applied to assess the ability. In the input layer, there are five nodes, which are aircraft types, firepower strength, mobility, anti-jamming ability and pilot's flight time. In the middle layer, there may be three to nine nodes, which is dependent on training speed and accuracy requirement.At the end, the ability α_fA→fB can be obtained from the network.

In the input layer, some parameters, aircraft types, firepower strength, anti-jamming and mobility, are needed to be quantified before training, while the parameter, pilot's flight time, can be used as a numerical value directly. The detailed disposal way is given as follows.

On aircraft carrier, fighters and bombers are two major types of aircraft, which can be quantified by using Eq.(1).

${\mu _{type}} = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\rm{fighters}}}\\ {1}&{{\rm{bombers}}}\\ {2}&{{\rm{others}}} \end{array}} \right.$

(1)

Due to the other three important parameters showing the aircraft surviving ability, firepower strength, mobility and anti-jamming, are mainly related to the combat performance. Then, these aircraft characteristics are classified into three grades, as shown in Eq. (2).

${\mu _{{\rm{survial}}}} = \left\{ {\begin{array}{*{20}{c}} {0.3,}&{{\rm{Weak}}}\\ {0.5,}&{{\rm{Medium}}}\\ {0.7,}&{{\rm{Strong}}} \end{array}} \right.$

(2)

With the artificial neural network tool box of Matlab, the training result of α_fA→fB can be seen in Table 2, which includes four aircrafts.

In Table 2, the sample data for training the network is given from the second row to the fifth row. After 22 times iteration, the network achieves the required precision, and the assessed ability values α_fA→fB are obtained as shown in the last row of Table 2. Obviously, it can be seen that the assessed values of α_fA→fB by using the network are very close to the expert estimation values. Therefore, the constructed neural network can be used to assess the ability of combat unit.

3.3 The Comparison of Combat Units

After obtaining the attack ability values and defense ability values of combat unit, we compared the values each other to achieve the winning probability. For convenience, only the ability comparison of aircraft f_A attack aircraft f_B is regarded here.

The general Bradley-Terry model for paired comparisons has been broadly applied in many areas^[9-12], and it can be described as

${p_{i,{\rm{ }}j}} = \frac{1}{{1 + {e^{\left( {{\pi _j} - {\pi _i}} \right)}}}}$

(3)

where p_{i, j} means the probability that the ith unit is better than the jth unit, and π_i is the total skill of the ith unit.

Supposed that there are two groups of aircraft, which are indexed by f_Ai(i=1, 2, …, m) and f_Bj(j=1, 2, …, n), the ability α_fAi→fBj denoted as α_Ai means f_Ai attacks f_Bj, the ability β_fAi←fBj denoted as β_Ai means f_Ai defenses f_Bj, and the ability α_fBj→f_Ai denoted as α_Bj means f_Bj attacks f_Ai, the ability β_fBj→fAi denoted as β_Bj means f_Bj defenses f_Ai.According to Eq.(3), the probability of f_Ai attacks f_Bj and the probability of f_Ai defenses f_Bj can be obtained by using Eqs.(4) and (5).

${p_{{f_{{A_i}}} \to {f_{{B_j}}}}} = \frac{1}{{1 + {e^{{\beta _{{B_j}}} - {\alpha _{{A_i}}}}}}}$

(4)

${p_{{f_{{A_i}}} \to {f_{{B_j}}}}} = \frac{1}{{1 + {e^{{\beta _{{A_j}}} - {\alpha _{{B_i}}}}}}}$

(5)

However, the general Bradley-Terry model (see as Eqs.(4) and (5)) cannot be applied to group comparison^[15], especially for the comparison problem of aircraft carrier groups, which should be improved in the following two aspects.

1) Group comparison.

Generally, an aircraft carrier group has hundreds of aircraft, which will lead to a huge computation to achieve the comparison value for each aircraft unit. For the convenience of engineering computation, the aircraft group can be considered as a whole group instead of hundreds of units. Therefore, an improved Bradley-Terry model is proposed as follows:

${p_{{f_A} \to {f_B}}} = \frac{1}{{1 + {e^{\left( {\sum\limits_{j \in {f_B}} {{\beta _j}} - \sum\limits_{i \in {f_A}} {{\alpha _i}} } \right)}}}}$

(6)

where ${\sum\limits_{i \in {f_A}} {{\alpha _i}} }$ is the total ability value of f_A attacks f_B, and ${\sum\limits_{j \in {f_B}} {{\beta _j}} }$ is the total ability value of f_A defenses f_B.

2) Weight value analysis.

During the aircraft carrier battle, weather and terrain environment may influence the result of combat^[16]. Therefore, a weight parameter θ is considered to indicate the effect of combat environment, and then Eq.(6) can be rewritten as:

${p_{{f_A} \to {f_B}}} = \frac{1}{{1 + {e^{\left( {\sum\limits_{j \in {f_B}} {\left( {1 - \theta } \right){\beta _j} - } \sum\limits_{i \in {f_A}} {\theta {\alpha _i}} } \right)}}}}\left( {0 \le \theta \le 1} \right)$

(7)

From Eq.(7), we can know that, when the parameter θ is bigger than 0.5, the combat environment is better for Army A, while θ=0.5, the combat environment has equal influence on both sides. In further study, the parameter θ can be calculated by using ANN to embody the change of combat environment.

3.4 Global Assessment of Aircraft Carrier Battle

Based on the above steps, Bayesian network^[17-19]is considered to achieve the global assessment of aircraft carrier battle. Inspired by Ref.[1], a Bayesian network including the ability values of units and comparison results is established as shown in Fig. 3.

From Fig. 3, the Bayesian model structure has three layers, which are the input layer, the middle layer and the output layer. The detailed introduction is given as follows.

(1) The input layer.

There are 12 nodes in the input layer. Among these nodes, "→" means attack ability, so P_fA→fB means the probability of f_A attacks f_B. "←" means defense ability, so P_fA←fB means the probability of f_A defenses f_B. The probability of P_fA→fB can be obtained through Eq.(7), and the other nodes can be obtained by the similar method.

(2) The middle layer.

There are six nodes in the middle layer, which are introduced as follows.

P_fAS: Aircraft survive probability of Army A;

P_cAS: Ship survive probability of Army A;

P_sAS: Submarine survive probability of Army A;

P_fAK: Aircraft killing probability of Army A;

P_cAK: Ship killing probability of Army A;

P_sAK: Submarine killing probability of Army A.

The values of the nodes in the middle layer are obtained through the node values of the input layer. The calculation method of node P_fASis given as an example (see Table 3 and Fig. 4).

In Table 3 and Fig. 4, it is the conditional probability table attached to the variable P_fAS. The conditional probability is given through expert knowledge.

(3) The output layer.

There is only one node in this layer, which is the global assessment value P_AW to describe the winning probability of Army A. Obviously, this node is related to the five nodes in the middle layer, and its calculation method of P_AW is similar to the calculation of P_fAS as shown in Table 3 and Fig. 4.

4 Case Analysis

To illustrate the effectiveness of the proposed method, World War Ⅱ Battle of Midway, the largest naval battle is simulated as an example.

Step 1:Classification of the combat units

The classification of combat units is shown below.

(1) The U.S. Military.

f_A= {wildcat fighters (79), dive bombers (112), and torpedo bombers (42) };

c_A= {heavy cruisers (6), light cruisers (1), and destroyers (16) };

s_A= {submarines (15) }.

(2) The Japanese Army.

f_B= {seaplanes (140), aircraft (395) };

c_B= {light cruisers (9), heavy cruisers (13), destroyers (66), and battleships (11) };

s_B= {submarines (22) }

Step 2:Assessment of the combat units

According to Section 3.2, the assessments of fighters, dive bombers, and torpedo bombers are given in Table 4.

Step 3:Comparison of the combat units

According to Section 3.3, by using ANN techniques, the value of parameter θ is obtained as 0.65.

Combining with the improved Bradley-Terry model Eq.(7), four probabilities are achieved, which are P_fA→fB=0.489, P_fB→fA=0.263, P_fA→cB=0.648, P_cB→fA=0.324.

Step 4:Global simulation diagram

Based on the four probabilities and Table 4, the simulation diagramof Bayesian network by using Netica software is established as shown in Fig. 5.

To verify that the improved Bradley-Terry model is reliable, the comparisons of the simulation results are given in Table 5, which includes the results by using the proposed method and expert experiences.

From Table 5, it can be seen that: on the one side, the proposed method based on the improved Bradley-Terry model is effective due to the results are more close to the actual values than the expert judgments. On the other side, the American Army has higher winning probability which is consistent with the actual battle result, because the winner has more aircraft than the loser does. It also indicates that the aircraft as a weapon is the key point of the battle, because it has more powerful fighting force than other combat units.

5 Conclusions

This paper presents a novel scheme to study the winning probability estimation problem for an aircraft carrier battle. The main contribution lies in an improved model is put forward to deal with the situation of group comparison which cannot be solved by the general Bradley-Terry model. The simulation result shows that the proposed method can obtain more precise assessed values than the expert judgments.In addition, the research work extends a new technique not only for the result prediction of a complicated battle, but also for the prediction of other complex nonlinear dynamical processes.