Sensitivity Analysis

We conducted a sensitivity analysis to examine the effect of several model pa- rameters on solution quality and required computational effort. When modeling the MmEP, there are several important modeling parameters that can affect the minimal exposure objective. We chose to investigate the respective effect of (1) the separa- tion distance between potential SAM battery locations and (2) the exposure weights assigned to each SAM battery type on the minimal exposure objective value. Specifi- cally, we considered potential SAM battery separations of 25, 30, 40, and 50 km, and

we examined exposure weights wt ∈ (0, 1] for each SAM battery type.

Although each SAM battery spacing alternative yields a unique layout, several trends emerged for this test instance. The long-range SAM batteries remained cen-

trally located in the border region, whereas the medium-range and short-range SAM batteries were most often dispersed along the northern and southern edges of the bor- der region. Perhaps of greater interest, however, are the effects on solution quality and computation time, as reported in Table 6. A decrease in the distance between potential SAM battery locations from 40 km to 30 km (i.e., an increase in granular- ity of the hexagonal grid) yields a relative increase of 9.5% in the minimal exposure for an intruder. However, this result does not portend a monotonically increasing relationship, as an additional decrease to 25 km corresponds to a relative decrease of 12.7% from the exposure attained from a 30 km tessellation. We postulate but do not examine further herein that such relative decreases with increased granularity result merely from the altered feasible set of locations for SAM battery sites, S, specific to a given instance.

Table 2. Effect of potential SAM battery location spacing on minimal exposure and computation times

Distance Between Number of Number Minimal Computation Potential SAM Potential SAM of Exposure Time Batteries (km) Battery Locations Arcs (min) (sec)

25 600 1825 2.4507 28,558.5

30 420 1281 2.8075 107.1

40 210 645 2.5641 7.8

50 156 481 1.6165 4.0

Moreover, the 25 km spacing instance required nearly 8 hours to solve, whereas the 30 km instance required under 2 minutes. While further testing may be conducted to determine if this trend holds in general, there is indeed a practical spacing value to maximize the minimal exposure objective. For this instance, we recommend using a 30 km spacing of potential SAM batteries. One may even be able to construct a general spacing rule that results in a superlative solution found with reasonable computational effort.

a feasible solution. For example, in the 50 km instance, the medium-range SAM batteries were no longer capable of satisfying the high-value asset coverage constraints because their effective ranges were too small. One could remedy this situation by increasing the number of potential SAM battery locations for a set hexagon size or by reducing the minimum high-value asset and/or SAM battery coverage requirements. The 50 km instance, for example, required reducing the coverage requirements by 10% in order to obtain a feasible solution.

Although the model specifies exact locations for each SAM battery, in real-world applications, commanders of air defense units should be given the latitude to adjust the prescribed locations of their specific SAM batteries within the associated hexagon region while still satisfying the coverage requirements. For example, there may be local terrain restrictions or other aspects indiscernible from a high-level modeling standpoint that need to be considered during implementation of a specific IADS layout solution.

If a defender prefers to engage an enemy aircraft using certain SAM batteries over others, our model allows for the specification of exposure weights (wt_{) to capture these}

preferences. Instead of equally weighting the exposure values (i.e., wt_{= [1, 1, 1]) as in}

the baseline solution shown in Figure 3, the defender may prefer to assign exposure weights of wt _{= [1, 0.5, 0.2], for example. That is, the defender is half as effective at}

employing the medium-range SAM battery against an enemy aircraft as compared to using the long-range SAM battery. Likewise, the defender considers their forces to be five times more effective at employing a long-range SAM battery than a short-range SAM battery. Table 3 details the change in exposure values for each of the alternative intrusion paths, using the exposure weights wt_{= [1, 0.5, 0.2].}

Compared to the baseline solution in Figure 3, the minimal weighted exposure solution locates all the medium-range SAM batteries on the southern edge of the

Table 3. Exposure values for the weighted exposure (wt_{= [1, 0.5, 0.2]) solution}

Intrusion Path Type Exposure Percent

(min) Change (%)* Minimal Weighted Exposure Path 2.3898 -14.9

Maximal Breach Path 12.5862 17.9

Maximal Weighted Breach Path 2.4656 -23.1 Maximum Probability of Survival Path 2.3898 -15.7 *compared to the baseline instance solution with wt= [1, 1, 1]

border region and transfers an additional short-range SAM battery near the northern edge of the border region. As a result, the locations of the intrusion paths also change. More importantly, this layout produces a minimal exposure value of 2.3898 minutes. This represents a 14.9% decrease in the minimal exposure, as compared to the baseline solution. Even though the defender may prefer using one SAM battery over another, the solution is actually worse if the model is forced to comply with the defender’s additional exposure weights; the model produces a better solution in terms of minimal exposure when allowed to determine SAM battery placement using equally weighted exposure values.

To further examine the effect of exposure weighting, we fixed the long-range ex- posure weight at 1 and systematically decreased the medium-range and short-range

exposure weights at the same rate (i.e., w1 _{= 1 and w}2 _{= w}3 _{: 1 → 0). Our results}

confirmed that the equally weighted, baseline exposure model results in an optimal IADS layout that maximizes the minimal exposure. Additional exposure weights imposed by the defender (i.e., weights less than 1) produced suboptimal minimal exposure objective values for all test instances analyzed. However, if the defender prefers a non-equal weighting, the model does offer such flexibility and will identify the optimal solution corresponding to such user-imposed weights. For example, the defender may choose to implement a weighted exposure scheme to account for differ- ences in crew training or expertise between various SAM battery types. Although the overall exposure decreases in the weighted instance, Table 4 indeed shows that the

IADS solution using weighted exposures wt_{= [1, 0.5, 0.2] produces an increase in the}

long-range SAM battery exposure and a decrease in the medium-range SAM battery exposure, as desired by the defender.

Table 4. Differences in exposure for the equally and unequally weighted exposure instances

Exposure Total Exposure Exposure by SAM Battery Type (seconds) Weights (minutes) Long-range Medium-range Short-range

wt_{= [1, 1, 1] 2.8075 75.2 93.3 7.4E-05}

wt_{= [1, 0.5, 0.2] 2.3898 143.4 1.1E-80 7.4E-05}