Modelling and Solving Anti-aircraft Mission Planning for Defensive Missile Battalions

Abstract

The theater defense distribution is an important problem in the military that determines strategies against a sequence of offensive attacks in order to protect his targets. This study focuses on developing mathematical models for two defense problems that generate anti-aircraft mission plans for a group of missile battalions. While the Anti-aircraft Mission Planning problem maximizes the defender’s effectiveness using all his available battalions, the Inverse Anti-aircraft Mission Planning problem computes necessary weapon resources (battalions and their missiles) to obtain a given defensive effectiveness value. The proposed formulations are Mixed Integer Programs that describe not only the positions of missile battalions, but also engage battalions to fleets of attacking aircrafts. We additionally prove that these problems are NP-hard. A comprehensive set of experiments is then evaluated to show that these proposed programs can be applied to solve fast real-life instances to optimality.

A Appendix

1.1 A.1 Compute e(b, f, t)

Suppose that a battalion $b \in B$ plan to launch t missiles to fleet $f \in F$ that has n(f) aircrafts. We are given coefficient corresponding to each missile battalion $b \in B$, $c^b = c^b_t c^b_c c^b_d$, where $c^b_t$ is technical coefficient, $c^b_c$ is control coefficient and $c^b_d$ is combat complex coefficient. The probability of kill of each missile launched from battalion b to fleet f is known as p ($p \in [0,1]$). Based on defensive mode, we consider following situations:

• 1. Disperse mode: Suppose that each time a battalion decides to launch 2 missiles to an aircraft of a fleet. Since the probability of kill is $p(b,f) =p$ for all $b \in B, f \in F$, expected number of killed aircraft is $e(b,f,t)= t(1- (1-p)^{2} ).$

• 2. Focus mode: Suppose that battalion b launches t times focusing on fleet f, where $t = n(f) t_1 + t_2$, then probability of kill on each aircraft in $t_1$ launches is $1 - (1-p)^{t_1}$. Battalion b has $t_2 (t_2 < n(f))$ launches left, inferring probability of kill on one aircraft in each launch is $p(1-p)^{n(f) -1}$. Then, expected number of killed aircraft can be estimated by $e{b,f,t}= n(f) (1 - (1-p)^{t_1} ) + t_2 p(1-p)^{n(f) -1}.$

• 3. Random mode: Let $X_i$ where $i=1,2,..., n(f)$, be random variables defined by $X_i = {\left\{ \begin{array}{ll} 1 &{} \text {if aircraft} \; i \in f \; \text{ is } \text{ killed }\\ 0 &{} \text {otherwise} \end{array}\right. }$

  While probability of kill on aircraft i in fleet f is $1 - (1 - \frac{p}{n(f)})^{t}$, expected number of killed aircraft can be approximated as $e(b,f,t)= E(\sum _{i=1}^{n(f)} X_i )= n(f) E(X_i) = n(f)(1 - (1 - \frac{p}{n(f)})^t).$

1.2 A.2 Compute t(b, l, f)

Value t(b, l, f) is maximum number of launches that a battalion b located at location l can launch to fleet f. This number depends on following quantities. For a fleet f, we let v(f), h(f) and l(f) be its velocity, height and length, respectively. In an attack, fleet brings different type of bomb that can be verified as $tb(f) =1$ if fleet f brings nuclear bomb and $tb(f) =0$ if fleet f brings regular bomb. For a battalion b, we denote $d_{max}$ and $r_{b}$ by long range of missile on battalion b and distance between that battalion and the target, respectively. We suppose that the shortest time between two consecutive launches, $t_{as}$, as well as obscured coefficient, $\delta$, are known. Furthermore, angle of battalion location, $\alpha _b$, and angle of in-coming fleet, $\alpha _f$, are parameters. Function t(b, l, f) can be computed as follows:

• 1. Compute critical radius $r_s = 5000tb(f) + v(f) \sqrt{\frac{2h(f)}{g} }- \varDelta$ where 5000 m is active radius of nuclear bomb, $g \approx 9.8$ m/s$^2$ is gravity acceleration, $\varDelta = 0.25h(f)$ if $v(f) \le 300$ m/s, $\varDelta = 0.4h(f)$ if $v(f) > 300$ m/s.

• 2. Compute shape time of fleet $t_{fs}$: $t_{fs} = \frac{l(f)}{v(f)}$

• 3. Compute launching time of battalion $t_{bs}$:

  • (a) Angle between battalion’s location and fleet $\varphi$: $\varphi = |\alpha _f - \alpha _b|$.

  • (b) If ($r_b + r_s > d_{max}$ and $r_b + d_{max} > r_s$ and $r_s + d_{max} > r_b$ ) then

    • i. If $\varphi > \varphi ^*$ then $t(b,l,f) =0$ where $\varphi * = \arccos (\frac{r^2_b + r^2_s - d^2_{max}}{2r_b r_s})$.

    • ii. If $\varphi \le \varphi ^*$ then $t_{bs} = \frac{ x - r_s }{v(f)}$ where x is root of equation $x^2 + r^2_b - d^2_{max} = 2 x r_b \cos \varphi$.

  • (c) If ($d_{max} \ge r_b + r_s$ ) then $t_{bs} = \frac{y - r_s}{v_f}$ where y is root of equation $y^2 + r^2_b - d^2_{max} = 2 y r_b \cos \varphi$.

  • (d) If ($r_s \ge d_{max} + r_b$) then $t(b,l,f) =0$.

  • (e) If ($r_b \ge d_{max} + r_s$) then

    • i. If $\varphi > \varphi ^*$ then $t(b,l,f) =0$ where $\varphi * = \arcsin (\frac{d_{max}}{r_b})$.

    • ii. If $\varphi \le \varphi ^*$ then $t_{bs} = \frac{2 \sqrt{d^2_{max} - r^2_b \sin ^2 \varphi }}{v_f}$.

• 4. Compute $t(b,l,f) = 1 + \frac{ \delta t_{bs} + t_{fs}}{t_{as}}.$