Two WAMs were implemented in this thesis, of which only two formed part of the final simula- tion. The two models are an improved SWAP and a modification of the comprehensive dynamic model developed by Du Toit [59].
The implemented WA process functions as follows: The WA process utilises the system threat values obtained from the TE subsystem, as explained in Chapter 5. During every TEWA cycle the status of the GBAD scenario (threats destroyed, DAs destroyed, ammunition expenditure and WSs reloading) is updated. The status of the GBAD scenario forms the input to the sub- sequent iteration of computing. As such, the WAM formulation has to account for the possible changes of states of the simulation entities — i.e. introducing a feedback element and the notion of memory. The problem therefore has an increased complexity when compared to other similar (non air-defense) scheduling/assignment problems. The working of these implementations are described in some detail in the remainder of this section.
6.4.1 Implemented WA Models
The simulation model entities are represented in the simulation as explained in §4.2. How these models interact to achieve the required simulation performance evaluation functionality is detailed in this section. The results of the static and dynamic instances were compared in order to ascertain that the algorithms function as expected and all the implementation bugs have been corrected (verification). Only the dynamic implementation is described here since the static case is, essentially, a simplified version of the dynamic case. Both formulations have the same constraints, with the exception of the notion of a Fire Order (FO) which is lacking for the static case.
The Objective Function
Information required by the WA subsystem includes the data structures of the various WSs, as described in§4.2.3. The data structures of the threats, which indicate their positions throughout the scenario, and their WRLs, are also required. The SSHP values calculated by the flight-path prediction model, together with the current system threat value, are used to minimise the
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accumulated survival probabilities of all the threats, weighted by their respective threat values. The probability of survival of a specific threat is calculated as the product of the complement of the SSHPs of the WSs assigned to the threat. The assumption is made that the events of a threat surviving multiple WS engagements are, indeed, independent events. The objective is therefore to minimise nt X t=1 Λj F X τ=1 ´ c(τ ) nw Y w=1 Qtw(τ )xtwτ, (6.18)
where F denotes the forecasting time-window length. The variables nt and nw denote respec- tively the number of threats and WSs during the start of the current TEWA-cycle, as before. The probability of survival of threat t, when engaged by WS w, is denoted by Qtw. Since a hit is modelled as a kill, as described in §4.2.3, the survivability of the threats may be seen as the complement of the considered WS-threat pair’s SSHP value. The decision variable xtwτ is a binary vector which assumes the value 1 if WS w is scheduled to engage threat t, τ time stages from the current TEWA-cycle, or the value 0 otherwise. The rest of the variables have the same meanings as before. The stage utility function c(τ ) is given by
c(τ ) = ae−bx, (6.19)
where the values of the coefficients a and b are functions of the forecasting time frame F and the required shape of the utility function. This stage utility function was deemed superior to the expected threat priority approach in (6.4), since it provides more flexibility to be tailored to the intuition of the operators. The expected threat priority objective function of Du Toit [59] did not provide any significant advantage when compared to a correctly calibrated stage utility function.
The normalised values ´c(τ ) are, in turn, utilised in objective function (6.18). This normalisation is achieved by dividing the weights for all the time stages in c(τ ) by the total summed weight. This normalised weight then serves as the utility value. An exponential function was used to model the importance rating of selecting earlier engagements, since the relative utility may be seen as decaying exponentially for each interval. This is because later engagements are associated with numerous risks, such as increased uncertainty because of the implemented flight- path prediction model and delaying engagements unnecessarily, thereby providing the threats with more opportunity to engage the DAs. An exponential decaying utility function is therefore seen as more suitable than a linear decaying function.
Constraint Modelling
The WA problem is solved for each iteration by updating the states of the simulation elements — unavailable WSs, expended ammunition and destroyed threats. The forecasting time length during which WSs may be assigned is set by the operators. For each new iteration, the different distances between WSs and threats are recalculated and new corresponding SSHP values are determined. During each iteration, the solution must adhere to the following constraints:
1. A WS may be assigned a pre-specified maximum number of times during the scheduling horizon. This value is fixed beforehand for the scenario.
2. A threat may only be engaged a pre-specified maximum number of times by WSs over the entire scheduling horizon. This is required in order to avoid circumstances where
resources are exploited during the current time stage, thereby preventing the WSs to adapt to changes in the environment (i.e. newly detected threats, threats entering within range of the WSs) because of reloading and/or ammunition constraints — in essence, preventing the occurrence of a so-called overkill5 event.
3. Each time a WS is allocated to a threat, regardless of the success of the engagement, a weapon setup time counter is started. This weapon setup time is WS-specific, and accounts for factors such as operator response time, reloading of ammunition, orientating the WS and, in some circumstances, the time required to compute the interception points before engaging. No new WS engagements are possible until this weapon setup time for the considered WS has passed.
4. Each WS also has a pre-specified amount of ammunition available. An engagement is therefore only possible if the ammunition supply of the WS under consideration has not been depleted.
5. There is a minimum SSHP requirement before a WS can engage a threat. This value there- fore serves as a necessary prerequisite for engagement, and adherence to this prerequisite (or otherwise) has to be determined prior to assigning a WS to a threat. The SSHP value that is used to determine whether this constraint is satisfied, is calculated in conjunction with the flight-path prediction model as explained in §4.2.4. In order to determine the outcome of an engagement, however, the current SSHP value is recalculated, since the “actual” SSHP value will most probably differ from the predicted value.
6. A threat may not be engaged if it has already been destroyed.
The notion of an FO was used to account for constraints (3)–(6) above. An FO is unique for each WS and indicates which threat, and during which time stage, a WS should engage. A binary flag is also set for the specific WS to signify that the WS has an active FO. Constraints (1)–(2) above, on the contrary, do not require any feedback information and are therefore “fixed” throughout the scenario. Since scheduling is included in the modelling approach, the simulation has to, effectively, delay an engagement until the right time stage. When an FO is set for a specific WS, the WS may not receive a new FO until the current FO has been executed. The solutions obtained from running the optimisation routine (i.e. determining the decision variable values) is used so as to generate FOs.
Furthermore, the inclusion of an FO allows the solution to change from one iteration to the next if a significantly improving solution is found before the current FO is executed. This functionality may be achieved by incorporating a minimum fitness function threshold that must be exceeded before an improving solution is selected. For example, if 5% is set as the threshold, an FO is only cancelled and a new one selected if the fitness function increases by 5% or more. This work is, however, left for future performance evaluation studies since the selection of the tolerance value may significantly impact the outcome of a scenario thereby further complicating the identification and interpretation of meaningful results.
The outcome of a WS engagement is determined by using a uniform random distribution on the interval [0, 1] and comparing this value to the SSHP of the WS-threat pair at the current time stage. Recall, from §4.2.3, that the SSHP only depends on the threat-WS stand-off distance. If the generated random number is indeed smaller than the SSHP value, the engagement is deemed
5Overkill, in a military context, refers to the use of excessive force that goes further than what is necessary to achieve a specific goal.
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Figure 6.3: Logic flow of the WA process in the simulation performance evaluation framework developed in this thesis.
as having been successful and the threat destroyed; otherwise, the threat is considered to have survived.
Threat values are assumed to remain constant for the duration of the prediction time frame. The inclusion of a changing threat value will add more uncertainty to the problem and may, if incorrectly predicted, have a detrimental effect on the scenario’s outcome. The increased complexity would therefore be difficult to justify operationally. For short prediction time frames, however, the threat values will not change significantly — the ranking of threats according to threat values should not change and, correspondingly, when using the threat values in a comparative manner, have a minimal impact on the WA output.
This process whereby the objective function is optimised is repeated until either all threats are destroyed, all DAs are destroyed (threats have reached reached their WRL) or the threat paths have ended.
6.4.2 WA Simulation Architecture
Because of the implemented approach, there are many feedback elements present in the WA sub-routine. An effective way to visualise these elements is through the use of a flow diagram. The logic-flow of the reactive WA modelling approach is visualised in Figure 6.3.
6.5 Chapter Summary
This chapter opened in §6.1 with an overview of the different WA problem formulations in the literature. More detail was provided in respect of the working of static WAM formulations (in §6.1.1) and dynamic WAM formulations (in §6.1.2).
Different methods for solving these WAMs were considered in §6.2. The chapter closed in §6.3 with a motivation and an overview of the genetic algorithm implemented in this thesis. After having gained an understanding the characteristics of the different WA formulations, the dynamic WA process implemented in this thesis was introduced in §6.4 with specific reference to the objective function, constraints and simulation framework adopted.
CHAPTER 7
Human-Machine Interface Design
A good solution applied with vigour now is better than a perfect solution applied ten minutes later.
— George Smith Patton
Contents
7.1 Data Fusion within the Decision Support System . . . 116