Optimal Attack in the Case of No
Defense

Joseph George Caldwell

July 17, 2001

Copyright © 2001.
Joseph George Caldwell. All
rights reserved.

This article determines an optimal allocation of weapons to
targets in the case of no defense. It is
assumed that the expected damage to the target is described by the
"exponential" damage function (described below). Let X denote the total number of weapons, t
the number of targets, and x_{i} the number of weapons allocated to the
i-th target. The attacker's strategy is
hence specified by vector

__x__' = (x_{1}, x_{2},..., x_{t})

where each x_{i} __>__ 0 and

Σ(i:1,t) x_{i} = X ,

where Σ(i:1,t) denotes the summation operator over the
index i from i=1 to i=t.

We assume that the exponential damage function adequately
describes the expected damage at each target; that is, the expected damage is

D_{i}(z_{i}) = P_{i}(1
- exp(-β_{i}z_{i}))

where z_{i} is the number of weapons reaching the
i-th target, P_{i} denotes the value of the i-th target, and β_{i}
is the damage function parameter for the i-th target. The exponential damage function works well
for point targets (e.g., geographically compact military or industrial targets
or small cities). For area targets, such
as large metropolitan areas, it may be desirable to use a sum of two or three
exponential functions to obtain a good fit (to the damage caused to the target
as a function of number of weapons deployed on the target).

Since the attacker wishes to maximize the damage, he will
choose a strategy __x__* such that

H(__x__*) = max(__x__) H(__x__)

where H(__x__) represents the expected damage
corresponding to the strategy __x__ and max(__x__) denotes maximization
with respect to __x__.

The problem, hence, is to determine __x__* such that

H(__x__*) = max(__x__) H(x)

where

H(__x__) = Σ(i:1,t) D_{i}(x_{i})

subject to the constraints

Σ(i:1,t) x_{i} = X

and x_{i} __>__ 0. The function H(__x__) is additively
separable over the targets, and so the Everett generalized Lagrange multiplier
(GLM) technique (Reference 1) can be used to find a solution. The problem becomes one of finding x_{i}
corresponding to

max(x_{i}) (D_{i} -
λx_{i})

where λ (a Lagrange multiplier) is adjusted so that the
constraint

Σ(i:1,t) x_{i} = X

is satisfied.

The optimal solution is determined by adjusting the value of
the Lagrange multiplier so that the constraints are satisfied. The "beauty" of the GLM method in
the case of a separable payoff function is that the problem of determining a
global optimum is reduced to determining an optimal solution at each target,
independently of the other targets.

Since all of the target damage functions are monotonically
increasing in the number of weapons, the solution is readily determined by the
bisection root-finding method. The
Lagrange multiplier is positive, so a lower bound for λ is λ =
0. An upper bound is the maximum value,
over the set of all targets, of the slope, P_{i}β_{i}, of
the individual-target payoff functions.

In the optimal attacks considered in *Can America Survive?*,
the full value of a target was credited if a weapon was placed on a
target. This corresponds to using a
large value for β_{i}, say 10, for each target. In this case, targets are selected in order
of their value, up to the number of weapons available. For example, if the targets are cities and
the payoff function is city population, and the total attack size is X=1,000,
then the largest 1,000 cities are attacked.

1. Hugh Everett, III,
"Generalized Lagrange Multiplier Method for Solving Problems of Optimum
Allocation of Resources," __Operations Research__, 11:399-411, 1963.