obj .. totCost =e=
sum(mis, prMis*m(mis) + prCan*c(mis));
stat(mis) .. 0.5*c(mis)+m(mis) =l= stations;
minMis(mis) .. m(mis) =g= misReq;
minCan(mis) .. c(mis) =g= canReq;
maxMis .. sum(mis, m(mis)) =l= misAvail;
allFl .. sum(mis, c(mis)) =e= flAvail;
Model minCost /all/;
Solve minCost using mip minimizing totCost;
display m.L;
display c.L;
display totCost.L;
When solving the problem usingGAMS/CPLEX the following is reported:
---- 44 VARIABLE m.L Number of missiles for the i’th mission 1 3.000, 2 3.000, 3 3.000, 4 3.000, 5 3.000
---- 45 VARIABLE c.L Number of canisters for the i’th mission 1 7.000, 2 2.000, 3 2.000, 4 2.000, 5 2.000
---- 46 VARIABLE totCost.L = 15150.000 The total cost
From here it can be seen that exactly three missiles are used for every mission.
For the first mission the aircraft will be equipped with seven canisters of flares.
These will occupy four stations, thus leaving three stations for the missiles.
6.3 The Framework
A mathematical model is developed for finding the optimal use of countermea-sures during a flight. The model is described in Section 6.5. To test the model a number of flight descriptions are needed. For this a framework for designing test scenarios and generating flight descriptions is developed using MATLAB. This is done for two reasons: first of all data about real-world scenarios are difficult to obtain, and second, a threat scenario can be designed to test certain aspects of the model.
A scenario is described in a two-dimensional world. It consists of a number of
6.3 The Framework 117
Figure 6.2: A scenario with two threats. The target is marked with an inversed triangle (O), and each Intermediate Point (IP) is marked with a pentagram (9).
The target is placed in the upper right corner.
ground based radar threats, a target point, and a flight path given by a number of positions, each known as an Intermediate Point (IP). The aircraft flies in straight lines between theIPs in the order given. It is equipped with a number of countermeasures, some of which are limited in their number of deployments.
Figure 6.2 shows a scenario with two threats and twoIPs. The threats are shown as circles indicating the range of each threat. The units on the axes describe distances in metres.
Section 2.4 describes how ground based radar systems can have different modes, and that a radar system will change mode when an aircraft is detected. The ground based radars in this framework are simplified versions of real radar sys-tems. A radar threat is here described by its location and its range/lethal envelope. Three types of radars are used, and they differ only by their ranges and their lethalities. The lethality describes how dangerous a threat is to the aircraft. More details on the lethality measure is given in Section 6.4.1. The lethality of a threat depends on the distance between the aircraft and the threat;
the closer the aircraft is to the threat the larger the lethality. Outside the range of a threat the lethality is set to zero.
The pilot may use one of three countermeasures intended for RFthreats only:
jammer, towed decoy, and chaff. All countermeasures are simplified version of real-world countermeasures, and they are described next:
118 The Mathematical Modelling Approach
The jammer works by obfuscating any nearby threats. It works on a threat if the aircraft is within the range of this threat. The jammer antenna is positioned so it offers the biggest reductions if the threat is placed either in front of or behind the aircraft. When close to the threat the jammer offers no reduction in the threat lethality. The reduction increases as the aircraft gets further away from the threat, and it reaches its peak when the aircraft is at the rim of the range of the threat.
At any time the jammer will be in one of four states: off; on but not yet active;
on and active; and active while turning off. In the model the jammer has to be in one of these states for a given period of time, i.e. for a given number of time steps, before its transition into the next state. No upper limits exist on the time the jammer should be off before it is turned on, or for how long it must be active before being turned off. Turning the jammer on and off takes a preset period of time, so a number of time steps is necessary on these transitional states. There is no limit to the number of times the jammer can be turned on. If it does not offer the best reductions it should not be turned on, since it may attract unnecessary attention from threats.
The towed decoy works basically as a ”jammer on a string”. One of the main differences between the on-board jammer and the towed decoy is the reductions given by the use of the decoy. Since the decoy is towed behind the aircraft it has the highest effect when the aircraft is flying away from the threats. It will be in one of four states: off; on but not yet active; on and active; and severed while the system is settling. It takes time for the decoy to become active since this requires the unreeling of a wire connected to the decoy. The towed decoy will stop jamming as soon as it has been turned off. It is assumed that the jammer may continue to jam as long as it is not severed. While the jammer may be turned on and off as often as necessary the number of towed decoys on-board the aircraft is limited. When it is turned off the wire is cut and the decoy is lost.
When chaff is dispensed it will form a cloud with aRCS comparable to that of the aircraft. Once formed, the cloud will maintain itsRCSfor a while. Chaff can be dispensed even if a chaff cloud is already formed. The frequency of dispensing chaff is limited by a latency period between two contiguous dispensings and by the amount of chaff available.