Optimal Deployment
Authors: Lohmander Peter 
Published: February 2024

This study focuses on the optimal deployment problem, and determines the optimal size of a
military force to send to the battle field. The decision is optimized, based on an objective
function, that considers the cost of deployment, the cost of the time it takes to win the
battle, and the costs of killed and wounded soldiers with equipment. The cost of deployment
is modeled as an explicit function of the number of deployed troops and the value of a
victory with access to a free territory, is modeled as a function of the length of the time it
takes to win the battle. The cost of lost troops and equipment, is a function of the size of the
reduction of these lives and resources. An objective function, based on these values and
costs, is optimized, under different parameter assumptions. The battle dynamics is modeled
via the Lanchester differential equation system based on the principles of directed fire. First,
the deterministic problem is solved analytically, via derivations and comparative statics
analysis. General mathematical results are reported, including the directions of changes of
the optimal deployment decisions, under the influence of alternative types of parameter
changes. Then, the first order optimum condition from the analytical model, in combination
with numerically specified parameter values, is used to determine optimal values of the
levels of deployment in different situations. A concrete numerical case, based on
documented facts from the Battle of Iwo Jima, during WW II, is analyzed, and the optimal US
deployment decisions are determined under different assumptions. The known attrition
coefficients of both armies, from USA and Japan, and the initial size of the Japanese force,
are parameters. The analysis is also based on some parameters without empirical
documentation, that are necessary to include to make optimization possible. These
parameter values are motivated in the text. The optimal solutions are found via Newton-
Raphson iteration. Finally, a stochastic version of the optimal deployment problem is defined.
The attrition parameters are considered as stochastic, before the deployment decisions have
been made. The attrition parameters of the two armies have the same expected values as in
the deterministic analysis, are independent of each other, have correlation zero, and have
relative standard deviations of 20%. All possible deployment decisions, with 5000 units
intervals, from 0 to 150000 troops, are investigated, and the optimal decisions are selected.
The analytical, and the two numerical, methods, all show that the optimal deployment level
is a decreasing function of the marginal deployment cost, an increasing function of the
marginal cost of the time to win the battle, an increasing function of the marginal cost of
killed and wounded soldiers and lost equipment, an increasing function of the initial size of
the opposing army, an increasing function of the efficiency of the soldiers in the opposing
army and a decreasing function of the efficiency of the soldiers in the deployed army. With
stochastic attrition parameters, the stochastic model also shows that the probability to win
the battle is an increasing function of the size of the deployed army. When the optimal
deployment level is selected, the probability of a victory is usually less than 100%, since it
would be too expensive to guarantee a victory with 100% probability. Some of many results
of relevance to the Battle of Iwo Jima, are the following: In the deterministic Case 0 analysis,
the optimal US deployment level is 66200, the time to win the battle is 30 days and 14000 US
soldiers are killed or wounded. If the marginal cost of the time it takes to win a victory
doubles, the optimal deployment increases to 75400, the time to win a victory is reduced to
26 days, and less than 12000 soldiers are killed or wounded. In the stochastic Case 0 analysis,
the optimal US deployment level is 65000, the expected time to win the battle is 46 days and
almost 25000 US soldiers are expected to be killed or wounded. If the cost per killed or
wounded soldier increases from 0 to 5 M $US, the optimal deployment level increases to
75000. Then, the victory is expected to appear after 35 days and 19900 US soldiers are
expected to be killed or wounded.