Subject:
Computer Science And Mathematics,
Security Systems
Keywords:
optimization; Lanchester equations; attrition coefficients; differential equation system; numerical iteration
Online: 22 February 2024 (07:11:04 CET)
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.
Subject:
Environmental And Earth Sciences,
Sustainable Science And Technology
Keywords:
sustainable forestry; sustainability; continuous cover forestry; size distribution; forest dynamics; optimization; estimation
Online: 21 February 2024 (23:41:55 CET)
Sustainable continuous cover forestry is defined and analyzed in several ways. The differential
equation representing growth of the basal areas of individual trees, motivated by fundamental
biological production theory by Lohmander (2017a), is analyzed and extended in different directions.
From the solution of the differential equation, the basal area and the tree diameter are obtained as
explicit functions of time. The diameter is a strictly increasing function of time. In the absence of
competitors, the diameter increment is shown to be a strictly decreasing function of time. Hence, the
diameter increment can also be interpreted as a strictly decreasing function of the diameter. Alternative
forms of adjustment of the differential equation, with consideration of competition, are defined. If the
competition is strong, with large trees in the vicinity of a particular tree, then the basal area increment,
and the diameter increment, are reduced. The growth of a large tree is less sensitive than the growth of
a small tree, to competition from other trees. Under strong competition, the basal area increment, and
the diameter increment, are strictly concave functions of the size of the tree. The unique maximum of
the diameter increment occurs at a higher diameter, if the competition increases. In dynamic
equilibrium, the tree size frequency distribution is stationary. If natural tree mortality can be avoided
via the harvest strategy, the tree size frequency distribution is a function of the size and competition
dependent growth function, and the harvest strategy. Empirical tree size frequency data are used to
simultaneously estimate parameters of a size and competition dependent growth function and the
applied harvest strategy, via nonlinear optimization. The properties of the estimated growth function
are consistent with the corresponding properties of the production theoretically motivated hypothetical
function, and the properties of the estimated harvest strategy confirm the corresponding hypotheses.
The R2 of the nonlinear regression exceeds 0.97. With access to an empirically estimated equilibrium
tree size distribution, it is possible to: 1. Estimate size frequency relevant parameters of tree size and
competition dependent growth functions for individual trees. 2. Estimate the applied harvest strategy.
3. Explain and reproduce the empirically estimated tree size equilibrium distribution.
Subject:
Computer Science And Mathematics,
Applied Mathematics
Keywords:
optimal control theory; military strategy; dynamic game theory
Online: 3 November 2022 (01:08:42 CET)
A proxy war, between a coalition of countries, BLUE and a country, RED, is considered. RED wants to increase the size of the RED territory. BLUE wants to involve more regions in trade and other types of cooperation. GREEN is a small and independent nation that wants to become a member of BLUE. RED attacks GREEN and tries to invade. BLUE decides to give optimal arms support to GREEN. This support can help GREEN in the war against RED and simultaneously reduce the military power of RED, which is valuable to BLUE, also outside this proxy war, since RED may confront BLUE also in other regions. The optimal control problem of dynamic arms support, from the BLUE perspective, is defined in general form. The objective function is a weighted sum of the present value of the free GREEN territory and the present value to BLUE of the net loss of military resources in the RED army. The net loss of RED at a particular point in time is a function of the location of the front line and the size of the mobile GREEN forces behind the RED line. First, it is assumed that the expected net loss is proportional to the force ratio behind the red front line. It is proved that the net loss function is a strictly concave quadratic function of x, the location of the front line. It is also proved that the unique maximum of the net loss function occurs at the same front location, also if the net loss function is proportional to the strength ratio behind the RED lines, raised to any strictly positive exponent. From the BLUE perspective, there is an optimal position of the front. This position is a function of the weights in the objective function and all other parameters. Optimal control theory is used to determine the optimal dynamic BLUE strategy, conditional on a RED strategy, which is observed by BLUE military intelligence. The optimal arms support strategy for BLUE is to initially send a large volume of arms support to GREEN, to rapidly move the front to the optimal position. Then, the support should be almost constant during most of the war, keeping the war front location stationary. In the final part of the conflict, when RED will have almost no military resources left and tries to retire from the GREEN territory, BLUE should strongly increase the arms support and make sure that GREEN rapidly can regain the complete territory and end the war