22-10-1 America's Missile Defense Problem - Poly > .
The salvo combat model provides a mathematical representation of anti-ship missile battles between modern warships. It was developed by Wayne Hughes at the U.S. Naval Postgraduate School in Monterey and published in 1995. The salvo model describes the basic elements of modern missile combat in a very simple manner. This is similar to how Lanchester's Square Law provides a simple model of modern gun combat.
Suppose that two naval forces, Red and Blue, are engaging each other in combat. The battle begins with Red firing a salvo of missiles at Blue. The Blue ships try to shoot down those incoming missiles. Simultaneously, Blue launches a salvo that Red tries to intercept.
This exchange of missile fire can be modeled as follows. Let symbol A represent the number of combat units (warships or other weapon platforms) in the Red force at the beginning of the battle. Each one has offensive firepower α, which is the number of offensive missiles accurately fired per salvo at the enemy. Each one also has defensive firepower y, which is the number of incoming enemy missiles intercepted per salvo by its active defenses. Each ship has staying power w, which is the number of enemy missile hits required to put it out of action. Equivalently, one could say that each attacking missile can cause damage equal to a fraction u=1/w of a Red ship.
The Blue force is represented in a similar manner. Blue has B units, each with offensive firepower β, defensive firepower z, and staying power x. Each missile that hits will cause damage v=1/x.
The salvo combat model calculates the number of ships lost on each side using the following pair of equations. Here, ΔA represents the change in the number of Red's ships from one salvo, while ΔB represents the change in the number of Blue ships.
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Salvo Combat Model ..This exchange of missile fire can be modeled as follows. Let symbol A represent the number of combat units (warships or other weapon platforms) in the Red force at the beginning of the battle. Each one has offensive firepower α, which is the number of offensive missiles accurately fired per salvo at the enemy. Each one also has defensive firepower y, which is the number of incoming enemy missiles intercepted per salvo by its active defenses. Each ship has staying power w, which is the number of enemy missile hits required to put it out of action. Equivalently, one could say that each attacking missile can cause damage equal to a fraction u=1/w of a Red ship.
The Blue force is represented in a similar manner. Blue has B units, each with offensive firepower β, defensive firepower z, and staying power x. Each missile that hits will cause damage v=1/x.
The salvo combat model calculates the number of ships lost on each side using the following pair of equations. Here, ΔA represents the change in the number of Red's ships from one salvo, while ΔB represents the change in the number of Blue ships.
ΔA = -(βB - yA)u, subject to 0 ≤ -ΔA ≤ A
ΔB = -(αA - zB)v, subject to 0 ≤ -ΔB ≤ B
Each equation starts by calculating the total number of offensive missiles being launched by the attacker. It then subtracts the total number of interceptions by the defender. The number of remaining (non-intercepted) offensive missiles is multiplied by the amount of damage caused per missile to get the total amount of damage. If there are more defensive interceptions than offensive missiles, then the total damage is zero; it cannot be negative.
These equations assume that each side is using aimed fire; that is, a force knows the location of its target and can aim its missiles at it. If however a force knows only the approximate location of its target (e.g., somewhere within a fog bank), then it may spread its fire across a wide area, with the hope that at least some of its missiles will find the target. A different version of the salvo equations is required for such area fire.
Mathematically, the salvo equations can be thought of as difference equations or recurrence relations. They are also an example of operations research.
A stochastic (or probabilistic) version of the model also exists. In this version, the ship parameters listed above are random variables instead of constants. This means that the result of each salvo also varies randomly. The stochastic model can be incorporated into a computer spreadsheet and used instead of the Monte Carlo method of computer simulation. An alternative version of this model exists for situations where one side attacks first, and then the survivors (if any) on the other side counter-attack, such as at the Battle of Midway.
Each equation starts by calculating the total number of offensive missiles being launched by the attacker. It then subtracts the total number of interceptions by the defender. The number of remaining (non-intercepted) offensive missiles is multiplied by the amount of damage caused per missile to get the total amount of damage. If there are more defensive interceptions than offensive missiles, then the total damage is zero; it cannot be negative.
These equations assume that each side is using aimed fire; that is, a force knows the location of its target and can aim its missiles at it. If however a force knows only the approximate location of its target (e.g., somewhere within a fog bank), then it may spread its fire across a wide area, with the hope that at least some of its missiles will find the target. A different version of the salvo equations is required for such area fire.
Mathematically, the salvo equations can be thought of as difference equations or recurrence relations. They are also an example of operations research.
A stochastic (or probabilistic) version of the model also exists. In this version, the ship parameters listed above are random variables instead of constants. This means that the result of each salvo also varies randomly. The stochastic model can be incorporated into a computer spreadsheet and used instead of the Monte Carlo method of computer simulation. An alternative version of this model exists for situations where one side attacks first, and then the survivors (if any) on the other side counter-attack, such as at the Battle of Midway.
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