The optimal size of a military force to send to the battle field is determined. The objective function includes 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 an explicit function of the number of deployed troops and the value of a victory with access to a free territory, is 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 the Battle of Iwo Jima, during WW Ⅱ, is analyzed, and the optimal deployment decisions of the attacker, BLUE, are determined under different assumptions. The known attrition coefficients of both armies, BLUE, and the defender, RED, and the initial size of the RED force, are parameters. The optimal solutions are found via Newton- Raphson iteration. Finally, a stochastic version of the optimal deployment problem is defined, where the attrition parameters are considered as stochastic, before the deployment decisions have been made.

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