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This appendix is based on work done for the committee by Paul Chrzanowski of the Lawrence Livermore National Laboratory; it will appear in an extended form in an LLNL report now in preparation. We very much appreciate his assistance.

The outcome of a strike against a set of targets depends not only on the capability of the weapons but also on the allocation of weapons to targets, which, in turn, depends on preattack planning factors. These factors include variables such as the reliability of weapons and delivery systems, the effectiveness of defenses in preventing weapons from reaching the target area, and the probability of damage to a target given that the weapon arrives in the target area. These factors can be combined into a single quantity *P** _{s}*, which represents the survivability of a target if a single weapon is assigned to attack it. The “single-shot probability of kill” (SSPK) is 1 −

A simple model is used to demonstrate that strike results are not very sensitive to misassumptions about attack planning factors. A specific case is considered in this appendix, where it is possible to derive analytic formulas for the optimal attack tactics, the value damaged, and the variance in value damaged. The case is one where values can be assigned to targets and the distribution of target value obeys a simple power law: *V*_{cum} = (*x/T*)^{Î±}*, *where *V*_{cum} is the cumulative value of the targets, *T *is the total number of targets, *x *is target number with installations ranked in order of decreasing value, and Î± is in the range 0 < Î± < 1.

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Appendix C
The Sensitivity of Strike Results to Preattack Planning Factors 1
This appendix is based on work done for the committee by Paul Chrzanowski of the Lawrence Livermore National Laboratory; it will appear in an extended form in an LLNL report now in preparation. We very much appreciate his assistance.
The outcome of a strike against a set of targets depends not only on the capability of the weapons but also on the allocation of weapons to targets, which, in turn, depends on preattack planning factors. These factors include variables such as the reliability of weapons and delivery systems, the effectiveness of defenses in preventing weapons from reaching the target area, and the probability of damage to a target given that the weapon arrives in the target area. These factors can be combined into a single quantity Ps, which represents the survivability of a target if a single weapon is assigned to attack it. The “single-shot probability of kill” (SSPK) is 1 − Ps. The purpose of this appendix is to explore the sensitivity of strike results to prestrike assumptions about Ps.
A simple model is used to demonstrate that strike results are not very sensitive to misassumptions about attack planning factors. A specific case is considered in this appendix, where it is possible to derive analytic formulas for the optimal attack tactics, the value damaged, and the variance in value damaged. The case is one where values can be assigned to targets and the distribution of target value obeys a simple power law: Vcum = (x/T)α, where Vcum is the cumulative value of the targets, T is the total number of targets, x is target number with installations ranked in order of decreasing value, and α is in the range 0 < α < 1.

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For example, if one has α = 0.3, half of the value of the target base resides in the most valuable 10 percent of the targets [Vcum = 1/2 = (0.1)0.3]. Sample results for this value of α are plotted in
Figure C-1 and
Figure C-2. In the figures, Qs is the anticipated value of the single-shot probability of target survival, whereas Ps is the actual value. For various values of Qs, attack efficiency is shown as a function of Ps in
Figure C-1.
Figure C-2 shows the damage extracted as a function of attack size with perfect planning (Qs = Ps). Notice that the total target damage depends strongly on Ps (how well weapons perform) but that for a given value of Ps the results are fairly insensitive to Qs (the preattack assumption about Ps). In short, accurate attack planning assumptions (Qs = Ps) are important for understanding how well the strike will succeed but do not help one to devise a much more effective plan.
~ enlarge ~
FIGURE C-1 Efficiency of attack plans with imperfect estimates of single-shot target probability of survival.
NOTE: Qs is the prestrike estimate of Ps (and Ps is the actual value). Efficiency (attack damage/attack damage with perfect planning [Qs = Ps]) is shown for the case of a many weapon attack (W≫1).

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~ enlarge ~
FIGURE C-2 Attack effectiveness as a function of attack size in the case of perfect planning (Qs = Ps).
NOTE
1. For a set of 1,000 targets with cumulative value growing as the 0.3 power of target number, and with perfectly planned allocation of weapons to targets (but without bomb-damage assessment or shoot-look-shoot), the fraction destroyed of total target set value may be calculated exactly:
Case
SSPK
Weapons used
Fractional value destroyed
1
0.90
187
0.556
1a
0.20*
18
0.171
2
0.20
187
0.299
3
0.90
1,075
0.942
3a
0.20*
1,075
0.294
4
0.90
22
0.294
Case 1a is the allocation of Case 1, but with the single-shot probability of kill degraded to the equivalent of 0.20 by random destruction before launch, for example. Case 2 shows the potential benefit over 1a of statistical reassignment (factor 1.75 increase in damage realized). Case 3a illustrates the much larger initial force required (without statistical reassignment) to provide damage comparable with the statistical reassignment Case 2. For comparison, Case 4 shows that the ability to reallocate specific surviving weapons to the most valuable 21 targets would enable damage to be maintained with 22 surviving weapons rather than the 40+ of Cases 1a or 2 or the 220+ of Case 3a.

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