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Abstract We study the damage probability when M weapons are used against a unitary target. We use the Carleton damage function to model the distribution of damage probability caused by each weapon. The deviation of the impact point from the aimpoint is attributed to both the dependent error and independent errors. The dependent error is one random variable affecting M weapons the same way while independent errors are associated with individual weapons and are independent of each other. We consider the case where the dependent error is significant, non-negligible relative to independent errors. We first derive an explicit exact solution for the damage probability caused by M weapons for any M. Based on the exact solution, we find the optimal aimpoint distribution of M weapons to maximize the damage probability in several cases where the aimpoint distribution is constrained geometrically with a few free parameters, including uniform distributions around a circle or around an ellipse. Then, we perform unconstrained optimization to obtain the overall optimal aimpoint distribution and the overall maximum damage probability, which is carried out for different values of M, up to 20 weapons. Finally, we derive a phenomenological approximate expression for the damage probability vs. M, the number of weapons, for the parameters studied here.

The probability of killing or damaging a target depends heavily on how close a weapon is delivered to the target. This delivery accuracy of a weapon may be affected by many components. In general, the errors are usually divided into two main groups: the dependent error and independent errors. The dependent error is related to the aiming error that results from a miscalculation of latitude, longitude, distance, wind effect, or uncertainty in locating the target position. The dependent error results in the armament impacting away from the desired target point and it affects all weapons the same way. The independent errors refer to ballistic dispersion errors, which may result from variations in bullet shape, variations in gun barrels, or variations in amount of explosive used inside each bullet [

Due to many uncertainties in the field of weapon effectiveness, Monte Carlo simulations have been widely employed to estimate the probability of target damage [

The remainder of this paper will progress as follows. Section 2 provides the detailed mathematical formulation and explicit exact solution for the kill probability. Section 3 considers the performances of various aimpoint distributions. Finally, Section 4 presents conclusions and future work.

We consider a single point target in the two dimensional space. We establish the coordinate system such that the target is located at the origin point

Let

・

・

・

The impact point of weapon j is given by

We model the dependent error

where

Further, we assume that the independent errors of individual weapons

We use the mathematical fact that the sum of two independent normal random variables is a normal random variable. Suppose

The probability density functions of U and V are given by

In terms of the probability density functions, we write Equation (1) as

Applying a change of variables

We rewrite the equation above in terms of expected values:

Here the notation

We use the Carleton damage function to model the probability of killing by an individual weapon. Let

This is called the Carleton damage function or the diffuse Gaussian damage function [

We calculate the probability of the target being killed averaged over independent errors

Since

Each term in the product is an average of the form on the left hand side of (2). Applying Equation (2), we write each average as

Substituting this result into Equation (5), we obtain

Next we average over the dependent error

Thus, the overall average of

Similarly, the overall average of

The probability of target being killed, averaged over independent errors and dependent error, is called kill probability, and is denoted by

where

After the completion of the above derivation, we discovered that similar approaches had been taken separately by von Neumann [

Now we apply the exact solution to examine the kill probability corresponding to various distributions of the aimpoints of M weapons.

Let

The aspect ratio of the weapon radii of the Carleton damage function

where

Once the lethal area

For all the cases considered in this paper, we choose

We first consider the case of M weapons with aimpoints uniformly distributed on a circle as formulated below

where r is the radius and

For each value of M, we maximize the kill probability with respect to

Note that the Carleton damage function we use is not isotropic. It has different effective radii in the range and deflection directions. To accommodate this anisotropic property of the Carleton damage function, we consider the case of M weapons with aimpoints distributed on an ellipse as formulated below

where

From

M | |||
---|---|---|---|

1 | 0 | *** | 0.27597 |

2 | 16.246 | 0 | 0.43690 |

3 | 22.960 | 0.53834 | |

4 | 26.948 | 0 | 0.62291 |

5 | 29.192 | 0.68212 | |

6 | 31.086 | 0.72869 | |

7 | 32.529 | 0.76474 | |

8 | 33.731 | 0 | 0.79360 |

9 | 34.747 | 0.81702 | |

10 | 35.63 | 0.83635 | |

11 | 36.409 | 0.85251 | |

12 | 37.105 | 0 | 0.86617 |

The asterisks reflect that when r = 0, θ is not meaningful, meaning that θ is arbitrary and irrelevant.

We should point out that parameter

For each value of M, we maximize the kill probability with respect to

In the above, we calculated the performance of placing the aimpoints of M weapons along a circle or an ellipse. We now examine the case of aiming one weapon at the center and aiming the rest

For each value of M, we maximize the kill probability with respect to

Next, we fully optimize the distribution of M aimpoints without constraining them

M | (major axis)_{opt} | (minor axis)_{opt} | ||
---|---|---|---|---|

1 | 0 | 0 | *** | 0.27597 |

2 | 16.246 | 16.246 | 0 | 0.43690 |

3 | 25.637 | 17.639 | 0.53989 | |

4 | 29.621 | 23.068 | 0 | 0.62477 |

5 | 30.411 | 27.235 | 0.68264 | |

6 | 32.859 | 28.548 | 0.72958 | |

7 | 34.292 | 30.095 | 0.76560 | |

8 | 35.848 | 30.967 | 0 | 0.79469 |

9 | 37.135 | 31.75 | 0.81829 | |

10 | 38.342 | 32.349 | 0.83784 | |

11 | 39.436 | 32.861 | 0.85420 | |

12 | 40.457 | 33.29 | 0 | 0.86806 |

The asterisks reflect that when r = 0, θ is not meaningful, meaning that θ is arbitrary and irrelevant.

M | (major axis)_{opt} | (minor axis)_{opt} | ||
---|---|---|---|---|

1 | *** | *** | *** | 0.27597 |

2 | 22.161 | 22.161 | 0 | 0.40957 |

3 | 25.412 | 25.412 | 0 | 0.53737 |

4 | 32.918 | 23.814 | 0.60947 | |

5 | 34.369 | 30.581 | 0.67798 | |

6 | 36.213 | 33.451 | 0.73052 | |

7 | 38.374 | 34.765 | 0 | 0.77123 |

8 | 39.45 | 36.17 | 0.80221 | |

9 | 40.859 | 36.86 | 0.8274 | |

10 | 41.814 | 37.655 | 0.84766 | |

11 | 42.838 | 38.163 | 0 | 0.86449 |

12 | 43.709 | 38.648 | 0.87853 |

When M = 1, there is only one aim-point at the center. The ellipse does not exist in this case. So the asterisks simply indicate that the values are irrelevant.

on a circle or an ellipse. We represent the M aimpoints in polar coordinates.

The optimal solutions for

The optimal solutions for

1 | 0 | 16.246 | 0 | 23.975 | 29.621 | 0 | ||

2 | 16.246 | 23.975 | 23.068 | |||||

3 | 17.411 | 29.621 | ||||||

4 | 23.068 |

1 | 33.839 | 38.334 | 0 | 40.69 | 0 | 43.43 | 0 | |

2 | 24.734 | 27.589 | 34.456 | 36.016 | ||||

3 | 33.839 | 27.589 | 34.456 | 34.191 | ||||

4 | 26.353 | 38.334 | 40.69 | 40.507 | ||||

5 | 26.353 | 27.589 | 34.456 | 40.507 | ||||

6 | 27.589 | 34.456 | 34.191 | |||||

7 | 0 | 36.016 | ||||||

8 | 0.6587 | 0 |

The optimal solutions for

As M (the number of weapons) increases, the optimal distribution of aimpoints has more layers, covering a larger area with a more uniform distribution over the area. In

Next, we study the optimal kill probability as a function of M. Let

j | ||||||||

1 | 46.336 | 0 | 48.534 | 0 | 50.078 | 49.091 | ||

2 | 36.994 | 42.116 | 42.093 | 41.676 | ||||

3 | 35.587 | 38.162 | 38.939 | 41.675 | ||||

4 | 36.994 | 42.116 | 42.093 | 49.089 | ||||

5 | 46.336 | 48.534 | 50.078 | 52.524 | ||||

6 | 36.994 | 42.116 | 45.328 | 45.409 | ||||

7 | 35.587 | 38.162 | 38.882 | 40.554 | ||||

8 | 36.994 | 42.116 | 38.882 | 45.407 | ||||

9 | 0 | 12.452 | 0 | 45.328 | 52.523 | |||

10 | 12.452 | 12.535 | 21.097 | |||||

11 | 12.535 | 0.44808 | ||||||

12 | 21.097 |

In the presence of dependent error, however, the situation is completely different. The same dependent error affects all M weapons. The outcomes of individual weapons are no longer independent of each other. As a matter of fact, when the M weapons are all aimed at the same position, the outcomes of individual weapons are highly correlated with each other. As an example, we examine the case of aiming all M weapons at the origin. The averages of

The kill probability is

In the absence of dependent error, we have

In the presence of dependent error, to simplify the analysis, we assume that the independent errors are zero

For the first few values of M, we obtain

Using mathematical induction, we can prove that

Clearly, when all M weapons are aimed at the same positon,

With the optimal distribution of aimpoints for M weapons, we may expect that

Even with the optimal distribution of aimpoints, however, the log survival probability,

After excluding the geometric decay, we explore the possibility of a power law decay for the survival probability. Specifically we examine whether or not the survival probability obeys the power law

In the right panel of

strates clearly that the survival probability does not follow a power law decay.

To find a phenomenological fitting to the decay of survival probability as a function of M, we consider the form of

In the left panel of

We have considered the damage probability caused by multiple weapons against a single target. Explicit exact solution was derived for the damage probability in the case of M weapons with both dependent error and independent errors. Then we applied the explicit exact solution to maximize the damage probability and find the corresponding optimal distribution of aimpoints. We observed that in the presence of significant dependent error, the decay of the survival probability corresponding to the optimal aimpoints distribution (i.e., 1 - optimal damage probability) is slower than the exponential decay with respect to M, the number of weapons. This observation demonstrates that increasing M is much less effective in overcoming the dependent error than in overcoming independent errors. We find that phenomenologically the survival probability decays exponentially with respect to a fractional power of M. Presumably, the fraction power varies with the parameter values of the problem. The mathematics behind this phenomenological expression and the dependence of the fraction power on the parameter values will be investigated in future studies.

H. Zhou would like to thank TRAC-M for supporting this work. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

Wang, H.Y., Moten, C., Driels, M., Grundel, D. and Zhou, H. (2016) Explicit Exact Solution of Damage Probability for Multiple Weapons against a Unitary Target. American Journal of Operations Research, 6, 450-467. http://dx.doi.org/10.4236/ajor.2016.66042