Research on Combat Composition of Ship-to-Air Missile Formation

— In view of the quantitative determination of the combat composition number and relative positions of ship-to-air missile formation, the factor of the horizontal required azimuth range of the ship-to-air missile formation to cover a single ship, and the factors of the horizontal fire circle of short-range ship-to-air missile ships, medium-range ship-to-air missile ships, long-range ship-to-air missile ships were considered. A combat composition model of ship-to-air missile formation was proposed, which can quantitatively determine the number and relative positions of short-range, medium-range, and long-range ship-to-air missile ships, and provide a model basis for the combat composition of ship-to-air missile formation.


I. INTRODUCTION
Combat composition of ship-to-air missile formation includes the number and relative positions of ship-to-air missile ships. The ship-to-air missile weapons are usually classified from near to long distance, including short-range ship-to-air missiles, medium-range ship-to-air missiles, and long-range ship-to-air missile weapons.
When the ship-to-air missile formation covers a single ship, under the condition that the ship-to-air missile formation covers a single ship with the horizontal required azimuth range, the ship-to-air missile combat composition needs to determine the number and relative positions of short-range, medium-range, and long-range ship-to-air missile ships to cover a single ship.
At present, the relevant research is mainly as follows: The multi-attribute decision-making method of fuzzy set theory was used to evaluate the air defense combat composition of an aircraft carrier formation. The quantitative static and dynamic models, the multi-objective decision-making, and a parallel simulation method were used to assess combat composition. The anti-ship combat composition scheme of the warship formation was optimized. The mathematical description of the task-based force composition cutting problem was constructed. The fuzzy comprehensive evaluation and effectiveness evaluation of the formation of the island and reef warning force was constructed 7 . The composition optimization model based on a multi-attribute decision was constructed. The formation and command relationship of the amphibious combat force was qualitatively analyzed. The test analysis method was used to study the composition problem. The multi-objective decision-making method was used to study the aircraft carrier formation problem. The clustering algorithm was used to study the composition problem.
The above-mentioned research results are scattered, and no research has been conducted on the combat composition of ship-to-air missile formations, and factors such as the horizontal required azimuth range of a ship-to-air missile formation to cover a single ship, as well as factors such as the horizontal fire circle of a short-range ship-to-air missile, medium-range ship-to-air missile, long-range ship-to-air missile ships, have not been taken into account. Therefore, a combat composition model of ship-to-air missile formation is proposed in this paper, which can quantitatively determine the number and relative positions of short-range ship-to-air missile ships to cover a single ship, the number and relative positions of medium-range ship-to-air missile ships to cover a single ship, and the number and relative positions of longrange ship-to-air missile ships to cover a single ship, and provide a composition model basis for ship-to-air missile formation.

II. COMBAT COMPOSITION PROCESS OF SHIP-TO-AIR MISSILE SHIP FORMATION
Assume that the ship-to-air missile combat composition of the ship-to-air missile formation covers a single ship. The steps of the composition process of ship-to-air missile formation are as follows.

A. Step 1
Determining the position of a single ship covered by the ship-to-air missile formation, and the horizontal required azimuth range of the ship-to-air missile formation to cover a single ship.

B. Step 2
According to the range of the ship-to-air missile, the formation ship-to-air missile weapons are classified into formation short-range ship-to-air missile, formation mediumrange ship-to-air missile, and formation long-range ship-toair missile weapons.

Research on Combat Composition of Ship-to-Air Missile Formation
Jihua Dou, Yuliang Sun, and Jiaqi Zou a single ship, based on the position of a single ship covered by the ship-to-air missile formation, and the maximum horizontal range of the ship-to-air missile weapon covering a single ship.

D. Step 4
Determine the horizontal fire circle of the formation shortrange ship-to-air missile based on the horizontal fire circle of the covered single ship. The ship-to-air missile ship's horizontal fire circle refers to a circle with the ship's position as the center and the radius as the distance from the ship-toair missile horizontal launch area. The far range of the horizontal launching area of the ship-to-air missile refers to the maximum horizontal distance of the ship-to-air missile intercepting aircraft and other air targets.

E. Step 5
Determine the horizontal fire circle of the formation medium-range ship-to-air missile based on the horizontal fire circle of the formation short-range ship-to-air missile.

F. Step 6
Determine the horizontal fire circle of the formation longrange ship-to-air missile based on the horizontal fire circle of the formation medium-range ship-to-air missile.

G. Step 7
Determine the ship-to-air missile combat composition based on the covered single ship's horizontal fire circle, the required azimuth range of the ship-to-air missile formation covering the single ship, the short-range ship-to-air missile's horizontal fire circle, the medium-range ship-to-air missile's horizontal fire circle, and the long-range ship-to-air missile's horizontal fire circle. The ship-to-air missile combat composition includes the number and relative positions of the formation's short-range ship-to-air missile ships, mediumrange ship-to-air missile ships, and long-range ship-to-air missile ships.

III. DETERMINATION MODEL OF THE NUMBER AND POSITIONS OF THE FORMATION SHORT-RANGE SHIP-TO-AIR MISSILE SHIPS
The schematic diagram of the combat composition of a ship-to-air missile formation is shown in Fig. 1.

A. Determination of the Horizontal Fire Circle of the Formation Short-range Ship-to-air Missile
In Fig. 1, the single ship is a U ship. The horizontal fire circle of the short-range ship-to-air missile is determined on the basis of the ship-to-air missile horizontal fire circle of the U ship. The Y circle is the ship-to-air missile horizontal fire circle of a U ship, which is a circle with point O as the center and radius E. E is the distance from the ship-to-air missile horizontal launching area of the U ship. The intersections of the Y circle and the Yj1 circle are point C1 and point C4. Point Aj1 is also the position of a single short-range ship-to-air missile ship, and Aj1 circle is also the horizontal fire circle of single short-range ship-to-air missile ship, which is a circle with point Aj1 as the center and radius Ej. The intersection of Yj1 circle tangent to line OJ2 is point C1. Rj is the distance between point O and point Aj1, which is given in (1).
Point Aj2 is also the position of another short-range shipto-air missile ship, and Yj2 circle is also the horizontal fire circle of another short-range ship-to-air missile ship, which is a circle with point Aj2 as the center and radius Ej. The Yj2 circle is tangent to the straight line OJ3, and the tangent intersection point is point C4.

B. Determination of the Number and Positions of the Formation Short-range Ship-to-air Missile Ships
In Fig. 1, on the basis of the horizontal fire circle of the formation short-range ship-to-air missile, the number Nj of the formation short-range ship-to-air missile ships is determined, and the positions of the Nj short-range ship-to-air missile ships are determined.
When the formation of short-range ship-to-air missile ships covers the U ship, they need to be horizontally configured in Fig. 1. That is, the formation of short-range ship-to-air missile ships can be laterally arranged on the arc line S1 with the point O as the center and the radius Rj. The lateral arrangement range is the arc section from the point Aj1 to the point Aj2 with the point O as the center and the radius Rj, which constitutes the shielding horizontal closed area formed by the line segment OC4, the line segment OC1 and the horizontal fire circle of the short-range ship-to-air missile ships when the short-range ship-to-air missile ships cover the U ship in this arc segment. This shielding horizontal enclosed area is the horizontal enclosed area when the short-range ship-to-air missile ships cover the U ship in the horizontal required azimuth range angle H.
Calculation of the maximum horizontal azimuth range angle Fjd of a single short-range ship-to-air missile ship covering U ship. Take the point O as the starting point for the ray OD1. The ray OD1 is tangent to the Yj1 circle, and the tangent point is the point D1. The maximum horizontal azimuth range angle Fjd for a single short-range ship-to-air missile ship to cover U ship, which is given in (2).
Calculate the number of short-range ship-to-air missile ships in the formation and determine the location of the Nj medium-range ship-to-air missile ships. There are three situations.
In case 1, when H≤Fjd, Nj=1. The position of the one shortrange ship-to-air missile ship is position point Aj1. The distance of point Aj1 from point O is Rj, and the relative horizontal angle !$ $ is given in (3).
In case 2, when Fjd<H≤2Fjd, Nj=2. The positions of the two short-range ship-to-air missile ships are point Aj1 and point Aj2. The distance of point Aj1 from point O is Rj, and the relative horizontal angle is !$ $ . The distance between point Aj2 and point O is Rj, and the relative horizontal angle !" $ is given in (4).
The positions of the Nj short-range ship-to-air missile ships are position points Aj1, Aj2, and Nj-2 ships. The positions of the Nj-2 ships are, the distance between the ith ship and the relative point O is Rj, and the relative horizontal angle !% $ is given in equation (6). 1≤i≤Nj-2.

A. Determination of the Horizontal Fire Circle of the Formation Medium-range Ship-to-air Missile
In Fig. 1, the horizontal fire circle of the formation medium-range ship-to-air missile is determined on the basis of the short-range ship-to-air missile ship configuration arc line S1. The intersection of the arc line S1 and the straight line OJ2 is the point C2, the intersection of the arc line S1 and the straight line OJ3 is the point C5, the point Az1 is also the position of a single medium-range ship-to-air missile ship, and the Yz1 circle is also the horizontal fire circle of a single medium-range ship-to-air missile ship, which is a circle with the point Az1 as the center and the radius Ez. The intersection of Yz1 circle tangent to line OJ2 is point C2, and Rz is the distance between point O and point Az1, Rz is given in (7).
Point Az2 is also the position of another medium-range ship-to-air missile ship, and Yz2 circle is also the horizontal fire circle of another medium-range ship-to-air missile ship, which is a circle with point Az2 as the center and radius Ez. The Yz2 circle is tangent to the straight line OJ3, and the tangent intersection point is point C5.

B. Determination of the Number and Positions of the Formation Medium-range Ship-to-air Missile Ships
In Fig. 1, on the basis of the horizontal fire circle of the formation medium-range ship-to-air missile, the number Nz of the formation medium-range ship-to-air missile ships are determined, and the positions of the Nz medium-range shipto-air missile ships are determined.
When the formation of medium-range ship-to-air missile ships cover the U ship, they need to be horizontally configuredd in Fig. 1. That is, the formation of medium-range ship-to-air missile ships can be horizontally arranged on the arc S2 with the point O as the center and the radius Rz. The horizontal arrangement range is the arc segment from the point Az1 to the point Az2 with the point O as the center and the radius Rz, which constitutes the shielding horizontal closed area formed by the line segment OC2, the line segment OC5 and the horizontal fire circle of the medium-range shipto-air missile ships when the medium-range ship-to-air missile ships cover the U ship in this arc segment. This cover horizontal enclosed area is the horizontal enclosed area when the medium-range ship-to-air missile ship can cover the U ship in the horizontal required azimuth range angle H.
Calculation of the maximum horizontal azimuth range angle Fzd of a single medium-range ship-to-air missile ship covering U ship. Take the point O as the starting point for the ray OD2. The ray OD2 is tangent to the Yz1 circle, and the tangent point is the point D2. The maximum horizontal azimuth range angle Fzd for a single medium-range ship-toair missile ship to cover U ship is given in (8).
Calculate the number of medium-range ship-to-air missile ships in the formation and determine the location of the Nz medium-range ship-to-air missile ships. There are three situations.
In case 4, when H≤Fzd, Nz=1. The position of the one medium-range ship-to-air missile ship is point Az1, the distance between point Az1 and point O is Rz, and the relative horizontal angle is *$ $ is given in (9).
In case 5, when Fzd<H≤2Fzd, Nz=2. The positions of the two medium-range ship-to-air missile ships are pointAz1 and point Az2. The distance of point Az1 from point O is Rz, and the relative horizontal angle is *$ $ . The distance between point Az2 and point O is Rz, and the relative horizontal angle *" $ is given in (10).
The positions of the Nz medium-range ship-to-air missile ships are position points Az1, Az2, and Nz-2 ships. The positions of the Nz-2 ships are, the distance between the jth ship and the relative point O is Rz, and the relative horizontal angle *! $ is given in (12), where 1≤j≤Nz-2.  Fig. 1, the horizontal fire circle of the formation's longrange ship-to-air missile is determined on the basis of the arc line S2 of the medium-range ship-to-air missile ship configuration. The intersection of the arc line S2 and the straight line OJ2 is point C3, the intersection of the arc line S2 and the straight line OJ3 is point C6, the point Ay1 is also the position of a single long-range ship-to-air missile ship, and the Yy1 circle is also the horizontal fire circle of a single longrange ship-to-air missile ship, which is a circle with the point Ay1 as the center and the radius Ey. The intersection of the circle Yy1 tangent to the straight line OJ2 is the point C3, and Ry is the distance between the point O and the point Ay1, Ry is given in equation (13).
Point Ay2 is also the position of another long-range shipto-air missile ship, and the Yy2 circle is also the horizontal fire circle of another long-range ship-to-air missile ship, which is a circle with point Ay2 as the center and radius Ey. The circle Yy2 is tangent to the straight line OJ3, and the tangent intersection point is point C6.

B. Determination of the Number and Positions of the Formation Long-range Ship-to-air Missile Ships
In Fig. 1, based on the horizontal fire circle of the formation of long-range ship-to-air missiles, the number Ny of the formation of long-range ship-to-air missile ships is determined, and the positions of the Ny long-range ship-to-air missile ships are determined.
When the formation of long-range ship-to-air missile ships covers the U ship, they need to be laterally configured at the far side in Fig. 1. That is, the formation of long-range shipto-air missile ships can be horizontally arranged on the arc S3 with the point O as the center and the radius Ry. The horizontal arrangement range is the arc segment from the point Ay1 to the point Ay2 with the point O as the center and the radius Ry, which constitutes the shielding horizontal closed area formed by the line segment OC5, the line segment OC6 and the horizontal fire circle of the long-range ship-toair missile ships when the long-range ship-to-air missile ships cover the U ship in this arc segment. This cover horizontal enclosed area is the horizontal enclosed area when the longrange ship-to-air missile ship can cover the U ship in the horizontal required azimuth range angle H.
Calculation of the maximum horizontal azimuth range angle Fyd of a single long-range ship-to-air missile ship covering U ship. Take point O as the starting point for the ray OD3. The ray OD3 is tangent to the Yy1 circle, and the tangent point is the point D3. The maximum horizontal azimuth range angle of a single long-range ship-to-air missile ship covering U ship is Fyd, which is given in (14).
Calculate the number of long-range ship-to-air missile ships in the formation and determine the positions of Ny longrange ship-to-air missile ships. There are three situations.
In case 7, when H≤Fyd, Ny=1. The position of the one longrange ship-to-air missile ship is point Ay1. The distance of point Ay1 from point O is Ry, and the relative horizontal angle +$ $ is given in (15).
In case 8, when Fyd<H≤2Fyd, Ny=2. The positions of the two long-range ship-to-air missile ships are point Ay1 and point Ay2. The distance of point Ay1 from point O is Ry, and the relative horizontal angle is +$ $ . The distance between point Ay2 and point O is Ry, and the relative horizontal angle +" $ is given in (16).
The positions of the Ny medium-range ship-to-air missile ships are points Ay1, Ay2, and Ny-2 ships. The positions of the Ny-2 ships are, the distance between the kth ship and the relative point O is Ry= , and the relative horizontal angle +, $ is given in (18), where 1≤k≤Ny-2.

VI. CONCLUSION
The combat composition model of ship-to-air missile formation is proposed in this paper, which takes into account the factors of the horizontal required azimuth range of shipto-air missile formation to cover a single ship and the factors of the horizontal firepower circle of short-range ship-to-air missile, medium-range ship-to-air missile, long-range shipto-air missile ships, quantitatively determines the number and relative positions of short-range, medium-range, long-range ship-to-air missile ships for protecting a single ship, and provides a composition model basis for ship-to-air missile formation.