Mathematical model of air-to-air combat and loses

Here is mathematical study of air-to-air combat done by USAF Major Ronald E. Gilbert in 2011. using Lancaster model . Especially interesting is his prediction what would happen in potential China vs US conflict . He basically concluded that US would need to employ more then 100 fighters (mixed force of F-15 and F-22) to defeat then available Chinese force . Since that time , things were progressing for China , so using his methods you could calculate what would be happening now Interesting part starts at page 55 of 84 (marked 45 in document)

A significant disparity is noticed immediately with the output of the ARENA attrition model for China in comparison to the other two case studies. Based on the sizeable initial force of 3rd and 4 th
generation fighters and the increased technological capability that presents itself in greater EA degradation, China is able to defeat the entire blue force of 64 F-15Cs and 32 F-22s (initial force and reinforcements). This result leads to running the simulation again with increased initial blue forces and results in similar output. The second set of data highlights total destruction of the US fighters once more; however, the increased blue numbers delay the outcome allowing more reinforcements to arrive, while destroying almost twice as many Chinese aircraft. This iteration of blue initial strength is repeated to determine the point of inflection, representing the smaller blue force attrition rate as demonstrated in Section III. Captured below in Table 16 are the average results of each of the five separate tests, run ten times each to account for the stochastic effect of maintenance variation, each representing a different blue force initial strength ...

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thunderchief said:

Here is mathematical study of air-to-air combat done by USAF Major Ronald E. Gilbert in 2011. using Lancaster model . Especially interesting is his prediction what would happen in potential China vs US conflict . He basically concluded that US would need to employ more then 100 fighters (mixed force of F-15 and F-22) to defeat then available Chinese force . Since that time , things were progressing for China , so using his methods you could calculate what would be happening now Interesting part starts at page 55 of 84 (marked 45 in document)

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This was an interesting read. Thanks.

My takeaway from this paper is basically that the US military's trade-off of technology over quantity is unsuitable for maintaining US hegemony in a multi-polar world that we are heading into. So if the US is really serious about potential conflicts in the APAC region with near-peer competitors, then it is better to rely on a larger number medium tech than it is to rely on a smaller number of high tech.

This paper doesn't mention a word about the domestic political fallout that will surely arise from such a conflict with China of course. But it is reasonable to assume that the US will never be the same again after this conflict happens.

Thanks for posting it Thunderchief, it'll take me awhile to read most of it at later time.

That's so freaky thread that I decided to follow it I just downloaded the paper ...

while i generally agree with the paper, they did rely on one not so realistic way of doing large scale combat. please correct me, but i got the impression the paper said all the missiles fired in their simulations were fired one at a time per target. (multiple targets yes, but not multiple missiles). meaning, one missile would be fired at a target, then fighter would wait for confirmation, then he'd fire another missile if necessary.

in reality, against neer peer adversaries, salvos of two or perhaps even three missiles would be more efficient. that being said, it'd also mean less targets fired at. but firing one per target and saving missiles for later might lead to a situation where the merge distances get to wvr all bvr missiles are expended.

plus with shown historical statistics of missile effectiveness, two missiles in a salvo is really the minimum if someone wants to have a half-decent chance of hitting the enemy.

Totoro said:

while i generally agree with the paper, they did rely on one not so realistic way of doing large scale combat. please correct me, but i got the impression the paper said all the missiles fired in their simulations were fired one at a time per target. (multiple targets yes, but not multiple missiles). meaning, one missile would be fired at a target, then fighter would wait for confirmation, then he'd fire another missile if necessary.

in reality, against neer peer adversaries, salvos of two or perhaps even three missiles would be more efficient. that being said, it'd also mean less targets fired at. but firing one per target and saving missiles for later might lead to a situation where the merge distances get to wvr all bvr missiles are expended.

plus with shown historical statistics of missile effectiveness, two missiles in a salvo is really the minimum if someone wants to have a half-decent chance of hitting the enemy.

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Actually , is not such a big problem for the model . Model relies on single aimed shoots on target . But what is a single shoot ? It could be one missile fired ,or two missiles , burst of cannon fire etc ... Only thing that matters is probability of kill or Pk . In other words , how probable is that your one missile , or salvo of three missiles would down enemy aircraft .

There are studies showing that you don't increase much your Pk if you fire two same missiles at the target instead of one . Soviets had a bit different doctrine , they would usually fire two missiles , but one was SARH and other was IR .

thunderchief said:

Actually , is not such a big problem for the model . Model relies on single aimed shoots on target . But what is a single shoot ? It could be one missile fired ,or two missiles , burst of cannon fire etc ... Only thing that matters is probability of kill or Pk . In other words , how probable is that your one missile , or salvo of three missiles would down enemy aircraft .

There are studies showing that you don't increase much your Pk if you fire two same missiles at the target instead of one . Soviets had a bit different doctrine , they would usually fire two missiles , but one was SARH and other was IR .

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Yes it is, and it can have a big impact on the outcome.

The one missile per target assumption the model made favours a smaller force with more missiles per plane than a larger force with more planes but fewer missiles carried by each.

If you only fire one at a time, a plane with say 10 missiles could potentially kill 10 enemies before moving onto guns. If he was salvo firing, that number becomes a fraction of the number of missiles. So instead of each American plane being able to kill up to 10 Chinese planes, solve firing drops that theoretical max number down or 5 or 3 depending on how many missiles they salvo.

And contrary to your idea that only PK matters, whether you fire a single missile of a salvo directly impacts on PK. With the general unsupported propensity in the west of automatically assume a higher PK for western missiles compared to Chinese ones, only firing a single missile again favours the western aircraft in the model's outcome.

For example, if they assumed a western missile had a hit rate of 0.9 and a Chinese one had one of 0.8, firing a single missile would thus result in a 90% PK for the western missile and only 80% for the Chinese plane. If both were to fire a pair of missiles, the PK for the western missile goes up to 99% and it goes up to 96% for the Chinese missile. See the difference?

If you combine the effects of the two effects described above, the impact on the outcome could be multiplied.

The detailed attack order within the model is also important when it comes to determining the outcome. How does it work when a plane with 10 missiles comes against one with 6 in this model? Does the model assume there are only 6 rounds of missile combat or 10? 6 rounds means American planes are entering WRV with surplus missiles but 10 rounds means the American planes are getting in effect 4 rounds of free shots before going I to WVR combat. Neither are very likely scenarios in real life.

plawolf said:

Yes it is, and it can have a big impact on the outcome.

The one missile per target assumption the model made favours a smaller force with more missiles per plane than a larger force with more planes but fewer missiles carried by each.

If you only fire one at a time, a plane with say 10 missiles could potentially kill 10 enemies before moving onto guns. If he was salvo firing, that number becomes a fraction of the number of missiles. So instead of each American plane being able to kill up to 10 Chinese planes, solve firing drops that theoretical max number down or 5 or 3 depending on how many missiles they salvo.

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Not necessarily . First of all , this is not a model of single air combat ( i.e. every US and every Chinese fighter go to certain location and slug it out until one side is wiped out) . It is a simplified model of air campaign , with reloading of weapons and fuel , even with reinforcements .

plawolf said:

And contrary to your idea that only PK matters, whether you fire a single missile of a salvo directly impacts on PK. With the general unsupported propensity in the west of automatically assume a higher PK for western missiles compared to Chinese ones, only firing a single missile again favours the western aircraft in the model's outcome.

For example, if they assumed a western missile had a hit rate of 0.9 and a Chinese one had one of 0.8, firing a single missile would thus result in a 90% PK for the western missile and only 80% for the Chinese plane. If both were to fire a pair of missiles, the PK for the western missile goes up to 99% and it goes up to 96% for the Chinese missile. See the difference?

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Problem with that model is that you assume that two missiles are independent of each other , and therefore you use this simplified formula : P2k = 1 - (1-P1k) * (1-P1k) where P2k is probability of kill of two missiles , and P1k probability of kill of a single missile .

In real world , two missiles in a salvo are not independent of each other as shown by American and Soviet studies . In other words , if you fire two missiles at the target at the same time , if first missiles misses most likely other will miss too . And if first one hits , the other one will hit two . Therefore , if probability of kill of one missile is P1k = 50% , probability of kill of salvo of two missiles will not be P2k = 75% as per formula , but somewhere in range of 50-55% .

plawolf said:

The detailed attack order within the model is also important when it comes to determining the outcome. How does it work when a plane with 10 missiles comes against one with 6 in this model? Does the model assume there are only 6 rounds of missile combat or 10? 6 rounds means American planes are entering WRV with surplus missiles but 10 rounds means the American planes are getting in effect 4 rounds of free shots before going I to WVR combat. Neither are very likely scenarios in real life.

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As I understand the logic behind model , it doesn't assume that one side will run out of missiles , or get advantage because it has more missiles per plane .

What a thread!

I just looked into the system of ODEs described by equations (5), (6), (7) and attrition coefficients from Table 4; the Symbolic Math Toolbox is able able to handle it symbolically and for

syms x(t) y(t) z(t)
[x(t), y(t), z(t)] = dsolve(diff(x) == -0.8*y, diff(y) == -2.4*x - 3.3*z, diff(z) == -0.4*y, x(0) == 12, y(0) == 20, z(0) == 6)

I got

x(t) =
(exp(-(9*t)/5)*((112*exp((18*t)/5))/9 + 752/9))/8

y(t) =
-(9*exp(-(9*t)/5)*((7*exp((18*t)/5))/9 - 47/9))/2

z(t) =
exp(-(9*t)/5)*((7*exp((18*t)/5))/9 + 47/9)

This solution describes the time-dependence of the number of F-15s, Su-27s and F-22s, respectively, on the assumption that initially 12 F-15s and 6 F-22s met 20 Su-27 Let's solve y(t) for zero (I call this value "win" because at that point in time all Su-27s would be shot down) and see how many F-15s and F-22s this would cost:

win=solve(y,t)

>> eval(x(win))

ans =

8.061...

>> eval(z(win))

ans =

4.030...

meaning 4 F-15s and 2 F-22s (out of the initial 12 and 6, respectively) were lost ... What do you think?

P. S. The Pk data (Fig. 5) are not included!

Graphical presentation of the stuff I described above (but this time the solution was obtained by solving the system numerically):





Isn't the Blue Force winning too easily?