Models, Gaming, and Simulation Session 6

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Models, Gaming, and Simulation - Session 6

• Aggregated Attrition Models -
• Non-Lanchester Approaches

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Topics

• Force Ratio models of attrition
• Firepower Scores / WEI/WUVs
• Correlation of Forces
• Potential-Antipotential Method
• Hierarchical Attrition Algorithms
• Next Session Review

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Force Ratio Attrition Models

• CONCEPT
• Summarize effectiveness in combat with a single
  scalar measure of combat power for each unit.
• When combat occurs, use the ratio of attacker's
  to defender's measures to determine the outcome
• Measures can be very subjective, or based on
  summary characteristics of unit equipment, or
  based on a weighted combination of weapon
  firepower, mobility, vulnerability, etc.

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Force Ratio Approach - Firepower Scores

• CONCEPT Assign a firepower score to each weapon
  system and sum these scores for each weapon
  system on hand in a unit
• DEFINITIONS
• n number of distinct types of weapon systems in
  a unit
• Xi number of systems of type i (i1,2,...,n) in
  a unit
• Si firepower score for each weapon of type i

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Force Ratio Approach - Limitations of Firepower
Scores

• Weapon system interactions ("synergisms") are not
  represented
• FPI is additive across weapon types (e.g.,
  FPI100 derived from 15 tanks, 30 infantry
  fighting vehicles and 6 mortars equals FPI100
  derived from 30 tanks).
• FPI is linear in the number of weapons - cannot
  directly represent
• Minimum unit size required for effectiveness.
• Diminishing returns with large numbers of
  systems.
• Weapon system types become submerged in the FPI
  formula
• Illogical battles can occur (e.g., Arty Bn FPI80
  defeats Tank Co FPI30).
• Validity is suspect because
• Specification of firepower scores is arbitrary
• Force-Ratio predictive power is not validated by
  empirical historical research.
• Then why are we learning about Firepower Scores?
  
• Because they match the rule-of-thumb approach
  often used by military planners.
• Because you need to know enough about them to
  note when they are misused.

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Force Ratio Approach - Determining Firepower
Scores

• Alternate Approaches
• Use historical data about combat performance
  (1950s analysis)
• Use technical measures of weapon firepower
  (ATLAS)
• Use situationally-dependent technical firepower
  measures
• Use weighted combinations of firepower, mobility,
  vulnerability, reliability, etc., where weights
  were assigned by "Delphi" analysis (consensus of
  "experts"). This approach uses the term WEI/WUV
  (pronounced "wee-wuv")
• WEI Weapon Effectiveness Index
• Weapon scores are given without regard to
  opposing target types
• WUV Weighted Unit Value
• Units are given a base value by summing the WEIs
  of all their weapons
• Unit values are weighted by subjective ratings
  (e.g., for training, morale, C2 systems)
• Simple, but very subjective
• Use a measure of what the weapon can kill
  (potential/antipotential method)

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Force Ratio Approaches - Correlation of Forces

• Used primarily in planning, not in combat models
• Used in Soviet-style operations planning to
  allocate forces to a subordinate for a combat
  mission.
• Similar approach used in U.S. operations planning
  to estimate whether various parts of a plan are
  feasible.
• Could be used in a model where attrition is not
  the focus.

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Force Ratio Attrition Models - U.S. Army
Capability Analysis

• A base type unit is selected and given the combat
  value 1.0
• Units are assigned subjective values based on
  their perceived strength compared with base unit
  type, considering
• Primary type of equipment in unit (e.g., M1A1
  versus M1 versus M60A3),
• Percent strength authorized, and
• Modifications subjectively based on intel
  estimate and current status of own unit.
• Ratio of friendly to enemy estimated combat power
  is used to infer the capabilities of the friendly
  force.
• Example U.S. Division planning for an operation
  against a threat division using Soviet-style
  equipment and organization.

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Force Ratio Attrition Models - U.S. Army
Capability Analysis Example

• U.S. Comparison Values (with respect to a BTR
  Battalion)
• Maneuver
• M113 Bn 1.5
• M2 Bn 2.0
• M1A1 Bn 3.15
• M1 Bn 3.0
• M60A3 Bn 2.25
• ACR Sqdn 2.75
• Div Cav Sqdn 1.5
• Div Cav Sqdn(h) 2.0
• Atk Hel Bn (AH64) 4.0
• Atk Hel Bn (AH1) 3.0
• Artillery
• MLRS Battery 2.0
• 155mm SP Bn 2.0

NOTE 1 All units are given an estimated value
with respect to a BTR Battalion NOTE 2 The unit
commander and operations officer are advised to
develop their own table to allow them to take
into consideration local knowledge about the
situation. This table does not contain real
planning data, but only notional data. (Example
taken from US Army CGSC ST 100-9)
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Force Ratio Attrition Models - U.S. Army
Capability Analysis Example

• Planning Rules of Thumb for Force Capabilities

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Force Ratio Attrition Models - U.S. Army
Capability Analysis Example

• Threat Comparison Values
• Maneuver
• BTR Bn 1.0 (Baseline unit)
• BMP Bn 1.5
• Tk Bn ITR TR ITB MRR
• T-80 2.42 1.56 2.0 2.0
• T-64 2.23 1.44 1.86 .86
• T-72 1.86 1.20 1.55 1.55
• T-62 1.24 .80 1.0 1.0
• T-55 1.0 .64 .83 .83
• AT Bn 1.0
• Atk Hel Sqdn 2.0
• Artillery
• 122mm or 152mm Bn 2.0
• MRL Battery 1.0

ITR Independent Tank Regiment TR Tank
Regiment ITB Independant Tank Battalion MRR
Motorized Rifle Regiment
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Force Ratio Attrition Models - U.S. Army
Capability Analysis Example

• 53 Mech Division
• Maneuver
• TYPE BN VALUE TOTAL
• M113 4 1.5 6.0
• M2 1 2.0 2.0
• M1 2 3.0 6.0
• M60 3 2.25 6.8
• Cav 1 1.5 1 .5
• Atk Hel 1 3.0 3.0
• TOTAL 25.3
• X Strength X .9
• Relative Cbt Power 22.8
• Ratio for Maneuver 22.8 22.2 11
• Artillery
• TYPE BN VALUE TOTAL
• 155(SP) 3 2.0 6.0
• MLRS 1 2.0 2.0
• TOTAL 8.0

27 Gds Motorized Rifle Division
Maneuver TYPE BN VALUE TOTAL BTR
6 x 1.0 6.0 BMP 3
x 1.5 4.5 T64(MRR) 3 x 1.8
5.4 T64 (TR) 3 x 1.4 4.2 T64
(ITB) 1 x 2.0 2.0 Div Recon
1 x 1.6 1.6 AT Bn 1 x 1.0
1.0 Atk Hel 1 x 3.0
3.0 TOTAL 27.7 X Strength
X .8 Relative Cbt Power
22.2 Artillery TYPE BN VALUE
TOTAL 122(SP) 2 x 2.0
4.0 122(T) 4 x 2.0
8.0 152(SP) 1 x 2.0
2.0 MRL Btry 3 x 1.0
3.0 TOTAL 17.0 X Strength
X .8 Relative Cbt Power
13.6
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Attrition Coefficient Calculation - Potential /
Anti-potential ("Eigenvalue") Method

• A way to assign firepower scores.
• Avoids the problem of assigning scores based only
  on weapon's own characteristics, not on opposing
  targets' characteristics.
• Avoids some problems of subjectivity.
• CONCEPT Let the value of a weapon system be
  directly proportional to the rate at which it
  destroys the value of enemy weapon systems.
• NOTE Problem reduces to system of simultaneous
  linear eqns.

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Attrition Coefficient Calculation - Potential /
Anti-potential Method

• DEFINITIONS
• Force X fights Force Y
• Xi weapons of type i in X force, i 1,2,...,m
• Yj weapons of type j in Y force, j 1,2,...,n
• SXi value of one type i weapon in X force
• SYj value of one type j weapon in Y force
• Kij rate at which one Xi system kills Yj
  systems
• i1,2,...,m j1,2,...,n
• Lji rate at which one Yj system kills Xi
  systems
• i1,2,...,m j1,2,...,n
• NOTE Kij and Lji are determined from
  killer-victim scoreboards from a high-resolution
  model. This implies great dependency on the
  high-res scenario, e.g., force structures,
  missions, terrain, etc.
• So FPIX ?i SXi Xi and FPIY ?j SYj Yj

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Attrition Coefficient Calculation - Potential /
Anti-potential Method

• The above concept and definitions lead to a
  system of simultaneous equations
• for all j, cY ??SYj ?iLji SXi
• for all i, cX ??SXi ?jKij SYj
• This is a system of mn equations in mn2
  unknowns, since the two proportionality constants
  cX and cY are not specified.
• In matrix notation
• (1) cY ?SY L SX
• (2) cX ?SX K SY

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Attrition Coefficient Calculation - Potential /
Anti-potential Method

• Solving for SY in eqn (1) and substituting in eqn
  (2)
• cY cX?SX K L SX
• Similarly for SX
• cX cY?SY L K SY
• Let cX cY? ?, then
• ? SX K L SX (eqn 3)
• ? SY L K SY (eqn 4)
• This is a pair of eigenvalue problems for K L
  (mxm) and L K (nxn) where the eigenvalue is ? and
  eigenvectors are SX and SY .

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Attrition Coefficient Calculation - Potential /
Anti-potential Method

• Problem Solutions are not unique (mn2
  unknowns, with mn equations). In fact, they are
  only unique to a scaling factors MX for X and MY
  for Y.
• A Solution Let and set SXi 1.0 for some
  weapon Xi which is a major contributor to killing
  many Yj systems. This can be shown not to change
  the overall Force Ratio as Mx and MY vary, so
  the solution is appealing.

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Matrix Algebra Review Eigenvalues and
Eigenvectors

• GIVEN a set of linear equations in unknown
  variables xij such that ? I X A X
  for some constant ?
• DEFINE ? as an eigenvalue and X as an
  eigenvector.
• THEOREM due to Frobenius guarantees that
• 1. There exists a real, non-negative, largest
  eigenvalue ?, and
• 2. There exist non-negative eigenvectors (SX and
  SY in our formulation) which are unique up to a
  scale factor.

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Matrix Algebra Review Eigenvalues and
Eigenvectors

• SOLUTION PROCESS for 2x2 matrix
• 1. Solve for ?
• Characteristic polynomial is
• Let d1 c11c22 and d0 c11c22 - c12c21, then
• 2. Substitute for ? in equations (3) and (4) and
  solve for Xij.

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Attrition Coefficient Calculation - Potential /
Anti-potential Example

• GIVEN
• X force of 2 systems, strengths X1 200, X2
  150
• Y force of 2 systems, strengths Y1 75, Y2 100
• X1 kills Y1 at rate .03, Y1 kills X1 at rate .04
• X1 kills Y2 at rate .05, Y1 kills X2 at rate .02
• X2 kills Y1 at rate .03, Y2 kills X1 at rate .04
• X2 kills Y2 at rate .02, Y2 kills X2 at rate .01
• So
• for ?? cX cY , solve for ? in ?SX K L SX

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Attrition Coefficient Calculation - Potential /
Anti-potential Example

• d1 .0032 .0008 .004, and
• d0 (.0032 .0008) - (.0011 .0020)
  .00000036
• ? .5(.004 sqrt(.0042 - 4 .00000036) )
• ? max(?1, ?2) .003907878
• Let SX1 1.0
• Let
• Since ? SX K L SX, then
• ( K L - ??I ) SX 0
• Since SX1 1.0, we can solve for SX2 in two
  ways
• -.000707878SX1 .0011SX2 0
• or .002SX1 - .003107878SX2 0
• Either way gives SX2 .64353

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Attrition Coefficient Calculation - Potential /
Anti-potential Example

• Now solve for SY1 and SY2 using eqn (1) cY SY
  L SX
• so,
• Now compute FPIX and FPIY, and then FR
• Now to assess actual casualties lost by the X and
  Y forces, enter a table indexed by FR (and
  perhaps other factors) to find a percent of force
  lost. Apply this percentage to each weapon type.

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Attrition Coefficient Calculation - Potential /
Anti-potential Comments

• Solutions are scenario-dependent
• Kill rates must be generated with respect to X
  and Y posture and actions (e.g., X attacking Y in
  a prepared defensive position, versus X delaying
  against Y's attack)
• Changes in kill rates are hard to understand.
• In fact, scores sometimes vary greatly with small
  changes to a few inputs, and occasionally the
  score of a major weapon system will be zero.
• This method has the advantage of automatic
  adjustment of weapon scores, preventing poor
  judgement from influencing the outcome, but the
  related disadvantage is that its computations are
  obscured by the eigenvalue method. But also,
  remember that historical analysis shows little or
  no relation of force ratios to battle outcome.

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Hierarchical Attrition Algorithms

• Most combat models have the characteristic that
  higher-resolution models feed them with data.
• High-resolution models need PH's and PKHs which
  are usually supplied by AMSAA and ARL engineering
  models.
• Medium and low-resolution models usually depend
  on kill rates from high-resolution models or
  occasionally on engineering models.
• Some low-resolution models depend on kill rates
  generated by medium-resolution models, e.g.,
  ATCAL (Attrition Calibration) links the
  medium-resolution COSAGE with the low-resolution
  CEM.

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Hierarchical Attrition Algorithms

• A hierarchy of ground combat models was proposed
  and partially built by the Army in the 1970s and
  1980s, but never reached completion because of
  enormous technical difficulties
• The vision was a semi-automated way of getting
  input data for low-res models from high-res
  models, so low-res modelers would not have to
  request special runs from high-res modelers
  (usually in different organizations).
• Too much variation in scenarios, force
  structures, weapons, and postures meant that
  libraries of high-res results were too hard to
  build.
• Additionally, organizations resisted getting
  "answers" from "black-box" models they could not
  influence.
• ATCAL is one of the few examples of an existing
  formal link between models of differing
  resolution for the purpose of generating input
  data from the high-res model for the low-res
  model. Both models were built and are run by the
  same organization, contrary to what was
  envisioned.

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Hierarchical Attrition Algorithms - ATCAL

• CONCEPT a set of equations is used to compute
  attrition in a low-res model if a set of input
  parameters are known (provided by a medium-res
  model). The same set of equations can be used
  "backwards" when the medium-res model is run to
  generate the parameters it will provide in
  accordance with its attrition results.
• For each scenario which the low-res model needs,
  the medium-res model is run in nine different
  "vignettes", or shorter, narrower-scope scenarios
  which fit with the larger scenario. These nine
  vignettes vary the posture of the two opposing
  sides and calculate results for each
• Blue attack, Red prepared defense
• Blue attack, Red hasty defense
• Blue attack, Red delay
• Meeting engagement
• Static

• Red attack, Blue delay
• Red attack, Blue prep defense
• Red attack, Blue hasty defense
• Reserve

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Hierarchical Attrition Algorithms - ATCAL

• Medium-resolution model generates three
  parameters (for point fire)
• Pij Probability of kill of target j by firer i
• Aij Availability of target j to firer i
• RATEi shots per unit time by firer i
• Care must be taken to ensure that all Blue firer
  types have an opportunity to fire at all Red
  target types and vice versa, so that all Pij and
  Aij have values.