Bi-objective dynamic weapon-target assignment problem with stability measure

Abstract

In this paper, we develop a new bi-objective model for dynamic weapon-target assignment problem. We consider that the initial weapon assignment plan of defense is disrupted during engagement because of a destroyed air target, breakdown of a weapon system or a new incoming air target. The objective functions are defined as the maximization of probability of no-leaker and the maximization of stability in engagement order of weapon systems. Stability is defined as assigning same air target in sequence in engagement order of a weapon system so that reacquisition and re-tracking of air target are not required by sensors. We propose a new solution procedure to generate updated assignment plans by maximizing efficiency of defense while maximizing stability through swapping weapon engagement orders. The proposed solution procedure generates non-dominated solutions from which defense can quickly choose the most-favored course of action. We solve a set of representative problems with different sizes and present computational results to evaluate effectiveness of the proposed approach.

Appendix A: Problem data generation

The problem data generation steps are as follows:

• Step 1. Choosing the weapon systems

  Generate a number $rn$ between (0,1). If $rn < 0.5$, choose a self-air defense system from Table 3. Install this weapon system into a defense unit. If $rn \ge 0.5$, choose both a self-air defense system and an area-air defense system from Table 3. Install these two weapon systems into another defense unit. So, each defensive unit has either self-air defense system or one self-air and one area-air defense system. Randomly set the number of rounds $d$ each weapon systems have. Repeat this step until $n$ number of weapon systems are chosen. This step also determines the number of defense units.

  Table 3 Features of weapon systems and air targets

• Step 2. Choosing the air targets.

  Choose randomly an air target from Table 3. Select randomly the defense unit that this air target will aim to destroy. Repeat this step until $m$ air targets are chosen. Create the valid engagement combination matrix between weapons and targets $\left( {i,j} \right) \in V$ accordingly. Randomly set the present distance of air targets between (60-80 km.).

• Step 3. Set time durations.

  Set unit duration of each time slot as 1 s. Set the set-up time of each engagement as $\Delta_{c} = 9$ s.

We present following example only for a weapon system and a target to present the calculations. Assume weapon system 5 and target 2 is chosen from Table 3. Consider surveillance sensors observe the present distance of air target as 70 km. So, the time of air target reaches to defense unit is $70{*}1000/306 = 229$ s.

The setup time of an engagement is 9 s. According to the velocities, the minimum and the maximum effective ranges of weapon system and given the formulas in Sect. 2, the $q_{ij}$ earliest beginning time of the first engagement and $r_{ij}$ is the latest ending time of the last engagement is calculated as $q_{ij} = 40$ time slot and $r_{ij} = 213$ time slot.

The engagement duration, $\Delta_{ijk}$ are calculated in each time slots. The engagement durations depend on the starting time slot since the air target gets closer to the defense unit continuously. For instance, at $k = 40$, the engagement duration $\Delta_{ijk} = 65$ s and at $k = 150$ the engagement duration $\Delta_{ijk} = 30$ s.

We generate $S_{ij}$ for this example and $S_{ij} = \left\{ {40,41, \ldots \ldots \ldots \ldots \ldots ,195} \right\}$. So, an engagement between weapon system 5 and air target 2 can start at the beginning of any time slot between 40 and 195. Since we use SLS tactic, for instance if an engagement starts at $k = 40$ and the engagement duration $\Delta_{ijk} = 65$ s. This engagement blocks all time slots from 40 to 105 for other weapon systems. Similarly, the maximum number of engagements between the weapon system and the target is calculated as $\mu_{ij} = 4$ by dividing the engageability interval in accordance with SLS firing policy. The first engagement starts at $k = 40$ ends at $k = 105$. The second engagement starts at $k = 106$ ends at $k = 150$. The third engagement starts at $k = 151$ ends at $k = 181$. The fourth engagement starts at $k = 182$ ends at $k = 203$.