This manual provides comprehensive guidance on using the combat simulation spreadsheet tool, designed to estimate casualties based on factors such as force size, weapon effectiveness, terrain, and surprise. The tool integrates mathematical models to help planners and analysts predict battlefield outcomes in various scenarios, offering insights into attrition dynamics. Key mathematical frameworks employed include Lanchester’s Squared Law and Dupuy’s Quantified Judgment Model. These frameworks will be explicated below.
###2.1 Lanchester’s Squared Law in Combat Modeling Lanchester’s Squared Law provides a mathematical basis for modeling attrition in combat where the effectiveness of each force’s size is a squared function. This law is particularly effective for scenarios with ranged weaponry where each unit’s impact on attrition depends heavily on the opposing force’s size. In essence, it suggests that a larger force has a disproportionately greater power due to the “squared” relationship.
####Lanchester’s Equations for Opposing Forces For two forces, Friendly (F) and Enemy (E), the attrition rate for each side can be represented by differential equations:
\[ \frac{dF}{dt} = -k_E \cdot E \]
\[ \frac{dE}{dt} = -k_F \cdot F \] where:
F and E represent the number of troops for Friendly and Enemy forces, respectively. \(k_E\) and \(k_F%\) are the firing coefficients or “effectiveness constants” that quantify how effectively each side inflicts casualties on the other. Higher values reflect superior firepower, accuracy, or other factors that increase effectiveness.
The squared law implies that the outcome of combat is heavily influenced by initial troop numbers and firing effectiveness, leading to an exponential advantage for larger forces.
The integrated form of Lanchester’s equations reveals that: \[ F^2-E^2=F_0^2-E_0^2 \] where \(F_0\) and \(E_0\) are the initial troop counts of Friendly and Enemy forces. This outcome suggests that, over time, the force with a larger initial squared number (accounting for firing effectiveness) has a significantly better chance of prevailing, even if the difference in size or effectiveness is relatively small.
Two opposing forces are engaged in battle, with the following initial conditions:
Friendly Force (F): 500 troops with a firing effectiveness of \(k_F=0.02\). Enemy Force (E): 400 troops with a firing effectiveness of \(k_E=0.025\).
Using Lanchester’s law, the expected outcome can be calculated based on these initial numbers.
\(F_0^2={500}^2=250,000\) \(\ E_0^2={400}^2=160,000\)
Effective Friendly Strength=\(250,000\times k_F=5,000\) Effective Enemy Strength=\(160,000\times k_E=4,000\)
Since the effective squared strength of the Friendly force (5,000) is greater than that of the Enemy force (4,000), Lanchester’s law predicts that the Friendly force will have a decisive advantage. This advantage will increase as time progresses, leading to a higher rate of attrition for the Enemy force.
If we want to calculate the specific attrition rates per day: Friendly Attrition: \(\frac{dF}{dt}={-k}_E\bullet E=-0.025\ \times400=\ -10\)
Enemy Attrition: \(\frac{dE}{dt}={-k}_F\bullet F=-0.020\ \times400=\ -10\)
Thus, on a given day, both sides might lose 10 troops each, but since the Friendly force started with more troops, it can sustain losses longer, solidifying its advantage over time.
In the ACE model, Lanchester’s Squared Law is used to estimate the daily attrition rates by updating troop numbers and recalculating based on the adjusted opposition factor (which accounts for factors like surprise and terrain). This squared relationship ensures that the model accurately reflects how larger forces have a compounding advantage, aligning with real-world combat dynamics.
Military operations historically result in significant casualties, encompassing both direct battlefield injuries (BI) and disease/non-battle injuries (DNBI). Understanding the statistical patterns behind these casualties is vital for optimizing operational planning, medical response, and resource allocation. This paper synthesizes data from various sources, presenting an integrated view of casualty dynamics, focusing on trends, causal factors, and mitigation strategies.
Table 1 presents a consolidated view of BI data extracted from the analyzed sources. The data reflects the frequency, types, and causes of battlefield injuries across different conflicts.
## Warning: package 'knitr' was built under R version 4.4.2## Warning: package 'kableExtra' was built under R version 4.4.2| Conflict | WIA | KIA | KIA.Rate | Source |
|---|---|---|---|---|
| World War I (1917-1918) | 204,002 | 116,516 | 36.3% | DoD Casualty Report (WWI) |
| World War II (1941-1945) | 565,861 | 318,274 | 36.0% | DoD Casualty Report (WWII) |
| Korean War (1950-1953) | 103,284 | 33,739 | 32.7% | DoD Casualty Report (Korea) |
| Vietnam (1965-1975) | 153,303 | 47,434 | 30.9% | DoD Casualty Report (Vietnam) |
| Six-Day War (1967) | ~1,000 | ~776 | 43.7% | Gawrych, G. (2000). The Arab-Israeli Wars |
| Yom Kippur War (1973) | ~7,250 | ~2,656 | 26.8% | Rabinovich, A. (2004). The Yom Kippur War |
| Gulf War (Desert Storm, 1990-1991) | 467 | 147 | 23.9% | DoD Casualty Report (Gulf War) |
| Iraq (OIF, 2003-2011) | 31,994 | 2,481 | 7.8% | DoD Casualty Report (OIF) |
| Afghanistan (OEF, 2001-2021) | 20,149 | 1,845 | 9.2% | DoD Casualty Report (OEF) |
| Ukraine (2022-present, estimate) | 100,000 - 120,000 | 70,000 | ~38% | Congressional Report |
| Russia (2022-present, estimate) | 170,000 - 180,000 | 120,000 | ~41% | Congressional Report |
This table was used to provide dropdown options for the KIA percentage for casualty estimation.
From the table, it is evident that casualty rates vary significantly across conflicts, influenced by factors such as terrain, enemy tactics, medical advancements, and strategic planning. For instance, the Vietnam War exhibited a high casualty rate due to intense guerrilla warfare and challenging terrain.
Table 2 details the DNBI admission rates generated by a 1996 Army Medical Department panel based on real-world data from different military campaigns (World War II, Korea, Vietnam, Operation Desert Shield / Desert Storm). These admission rates do not indicate the entirety of the DNBI because they necessarily omit the many, many presentations which are returned to duty. Still, these rates are informative and completed using qualitative and quantitative assessments by leaders in the field. (Borden Institute, MILITARY PREVENTIVE MEDICINE: MOBILIZATION AND DEPLOYMENT Volume 1 Section 2: National Mobilization and Training Table 11.4, 2003, Editor DE Lounsbury).
| Area of Operations | Intensity | Division Rate | Corps Rate | Theater Rate |
|---|---|---|---|---|
| Disease East | None | 0.60 | 0.59 | 0.45 |
| Light | 1.62 | 1.32 | 0.50 | |
| Moderate | 2.13 | 1.69 | 0.53 | |
| Heavy | 2.51 | 1.96 | 0.56 | |
| Intense | 2.89 | 2.15 | 0.59 | |
| Disease West | None | 0.73 | 0.68 | 0.45 |
| Light | 1.68 | 1.35 | 0.49 | |
| Moderate | 2.16 | 1.69 | 0.51 | |
| Heavy | 2.59 | 2.04 | 0.53 | |
| Intense | 3.02 | 2.38 | 0.55 | |
| NBI Both | None | 0.15 | 0.15 | 0.13 |
| Light | 0.32 | 0.25 | 0.13 | |
| Moderate | 0.65 | 0.50 | 0.14 | |
| Heavy | 0.80 | 0.60 | 0.15 | |
| Intense | 1.00 | 0.70 | 0.16 |
Dupuy’s Quantified Judgment Model (QJM) expands on Lanchester’s foundational equations by incorporating real-world factors that affect combat outcomes beyond mere force size and firepower. While Lanchester’s model focuses on a mathematical interpretation of attrition, Dupuy’s model attempts to capture the complexity of battlefield dynamics through quantified adjustments based on historical analysis.
The core of Dupuy’s QJM includes a constant (0.04) used in the formula to estimate daily casualty rates. This constant was derived from empirical data based on historical engagements, designed to approximate average losses as a percentage of force size in combat conditions. The constant of 0.04 represents the expected daily casualty rate under standard combat conditions, though it is adjusted by the various effectiveness factors in Dupuy’s model.
The QJM achieves this by introducing “combat effectiveness” scores for factors such as terrain, posture, and weather. These factors are translated into a multiplier called the opposition factor, which adjusts the effective combat capability of each side on a day-by-day basis. The model is therefore dynamic, recalculating the combat effectiveness daily, influenced by environmental and tactical factors that affect each side’s relative advantage. The formula follows.
\({CAS}_F=0.04\times Opposition\ Factor\ \times{Strength}_E\) \({CAS}_E=0.04\times\frac{1}{Opposition\ Factor}\ \times{Strength}_F\)
where: 0.04 is the historical casualty estimation constant. Opposition Factor: Adjusts the casualty estimate based on tactical advantage, which can vary each day. Strength: The remaining active troop count for each side (Friendly or Enemy)
More specifically, battle injuries are calculated as follows. \({BI}_{t,f}=0.04\ x\ {force\ size}_{t,f}\times{personnel\ strength\ modifier\ for\ unit\ size}_{t,f}\times{terrain\ factor}_{t,f} \times\) \({\ weather\ factor}_{t,f}\times{posture\ factor}_{t,f}\times{opposition\ factor}_{t,f}\times\ {surprise\ factor}_{t,f}\times\ {sophistication\ factor}_{t,f}\)
where t is in the index for the day and f is the index for either friendly or enemy. If all factors are 1.0 (e.g., divisional casualties in ‘average’ posture), then the casualty rate is 0.04 on that day. This formula estimates the firepower coefficient for the Lanchester’s Squared Law. More details on the factors follow.
\(BI(t,f)\): Number of battle injuries.
Force Size \((t,f)\): Enemy or friendly force size with a minimum of 500 and a maximum of 200,000.
Strength Factor \((t,f)\): Table converted to equation. Smaller forces suffer more personnel casualties than larger forces (range: 0.3 to 8.0).
Terrain Factor: Table lookup for terrain type (range: 0.3 to 1.0).
Weather Factor: Table adjustment for weather (range: 0.3 to 1.0).
Posture Factor \((f)\): Table adjustment for posture (range: 0.8 to 1.0).
Opposition Factor: Firepower equivalent; formula range: 0.4 to 2.5. More on this factor follows.
Surprise Factor \((t,f)\): Accounts for surprise impact, especially in the initial days.
Sophistication Factor \((f=i)\): Applied to the casualty calculations of the under-sophisticated force (range: 1.0 to 1.7).
The final \(BI(t,f)\) receives a uniform distribution multiplier of \(U(0.95, 1.05)\) for planning uncertainty.
In Dupuy’s model, the opposition factor is calculated using quantified values of various METT-TC (Mission, Enemy, Terrain and Weather, Troops and Support Available, Time Available, and Civil Considerations) parameters. The purpose of calculating this opposition factor is to ensure that the model accurately reflects the complex conditions of the battlefield. By adjusting casualty estimates according to these factors, the ACE model can simulate realistic scenarios where advantages due to posture, weather, or surprise are short-lived or countered by enemy sophistication or mobility. The dynamic nature of the opposition factor recalculations each day allows the model to capture the evolving context of combat. This is implemented in ACE as follows.
In the ACE model, the opposition factor is calculated daily using a combination of multiple battlefield factors that impact combat effectiveness. This model goes beyond force size and firepower, factoring in both environmental and tactical variables. The specific elements involved in calculating the opposition factor are:
Posture Factor: This factor reflects the offensive or defensive stance of the forces. An attacking force might face higher risk but also have potential for greater impact, while a defensive posture could reduce casualties.
Weather Factor: Environmental conditions can affect visibility, movement, and weapon effectiveness. Adverse weather may reduce both forces’ capabilities, but the effect might differ depending on equipment and preparedness.
Terrain Factor: The nature of the terrain impacts mobility and protection. For example, rough terrain or urban areas might favor the defending force, providing cover and hindering movement.
CEV (Combat Effectiveness Value) Factor: This value quantifies each side’s combat proficiency, often based on training, morale, and historical performance. A higher CEV factor for one side indicates superior fighting ability.
Mobility Factor: This measures each force’s ability to maneuver, which can affect positioning and responsiveness to the enemy’s movements. Higher mobility can provide tactical advantages.
Sophistication Factor: This reflects the technological level and quality of the equipment, which can influence accuracy, range, and overall effectiveness.
Surprise Factor (Days 1-3): Surprise can provide a temporary boost to combat effectiveness, particularly in the initial days of engagement. In this model, surprise is applied explicitly for Days 1 through 3 of each phase, capturing the early advantage that an unexpected offensive can bring.
The opposition factor is a combination of these individual factors, adjusted daily. Mathematically, it can be expressed as the product of the following factors.
\({Opposition\ Factor}_{Friendly}=\min\funcapply(\max{\left(0.9241\times\frac{{Power}_F}{{Power}_E},\ 0.4\right)},\ 2.5)\) \({Opposition\ Factor}_{Enemy}=\min\funcapply(\max{\left(0.9241\times\frac{{Power}_E}{{Power}_F},\ 0.4\right)},\ 2.5)\)
The power function for friendly and enemy are the products of posture, weather, terrain, combat effectiveness, mobility, sophistication, and surprise factors for the friendly forces and enemy forces, separately.They modify the Lanchester’s firepower coefficients from the base (perfectly matched forces ) of 1.0. The min/max statements constrain the firepower estimates to between 0.4 and 2.5. These two values are reciprocals of each other.
This calculated opposition factor dynamically adjusts each side’s casualty rates and attrition based on changing battlefield conditions, providing a nuanced view of how tactical and environmental factors influence combat outcomes.
Consider a scenario where: Friendly Strength: 600 troops attacking Enemy Strength: 500 troops defending
Opposition Factor: Calculated based on relative strength and other METT-TC factors, yielding 1.2 and 0.83 for attacker and defender, respectively. The Opposition Factor of 1.2 means the enemy defender has a slight advantage based on the day’s METT-TC conditions, reflected in the higher casualty rate for Friendly forces.
Using Dupuy’s QJM formula with the 0.04 casualty constant: Friendly Casualties: \({CAS}_F=0.04\times Opposition\ Factor\ \times{Strength}_E=0.04\times1.2\times600=28.8\)
Friendly forces are estimated to lose 29 troops on this specific day. Recall, that the attrition calculation changes daily and with changes in the opposition factor by phase. Enemy Casualties: \({CAS}_E=0.04\times\frac{1}{Opposition\ Factor}\ \times{Strength}_F=0.04\times0.83\times500=16.6\)
Enemy forces are estimated to lose 17 troops on this specific day.
In the ACE model, Dupuy’s formula and daily opposition factor adjustments help simulate a fluid and realistic combat environment. By combining Lanchester’s attrition approach with the quantified adjustments from Dupuy’s model, ACE dynamically updates daily combat outcomes based on evolving conditions, making the model adaptable to various battlefield scenarios.
Wounded in Action (WIA) and Disease and Non-Battle Injury (DNBI) admissions are computed through a multi-step process involving battle injury calculations, opposition factor adjustments, and DNBI rate tables.
The number of WIA for each day (battlecas) is pulled from the Results Sheet (Column J), which stores the wounded count from each phase. The WIA count is calculated within SimulationForPhase, where casualty multipliers (including surprise, terrain, weather, and posture factors) determine the extent of injuries. The force strength (fsF and fsE) is reduced daily based on the expected battle injury rate. This rate is scaled using the opposition factor (opF), which is computed using a power law function dependent on relative force strengths.
Unlike WIA, DNBI casualties arise from environmental conditions, hygiene, and stress-related illnesses rather than direct combat.
DNBI rates are predefined in lookup tables, with rates varying by intensity of operations, theater conditions (East vs. West), and force level (Division, Corps, or Theater).
The DNBI calculation uses daily DNBI rates per 1,000 troops to estimate the number of personnel requiring treatment each day.
Once WIA and DNBI values are determined, they are classified into admitted vs. non-admitted cases based on injury severity, hospital capacity, and triage considerations.
The LOS (length of stay) for admitted casualties is determined from an empirical Barell matrix from recent Middle East enagements, which categorizes injuries based on mechanism of injury, cause, nature, and body region affected. The simulation follows these steps:
Each casualty is assigned a mechanism of injury (e.g., penetrating, blunt, explosion, burn, or other), based on pre-defined probability distributions.
The cause of injury is then derived from the mechanism using another probability lookup. Nature of Injury & Body Region:
Using a secondary lookup table based on cause, the nature of the injury is assigned (e.g., fractures, open wounds, burns, internal organ damage).
The body region affected (head, torso, extremities, etc.) is selected based on another distribution linked to the injury type.
Each injury type is mapped to an Abbreviated Injury Score (AIS), which ranges from 1 (minor) to 6 (unsurvivable). The Injury Severity Score (ISS) is computed as the sum of the squares of the three most severe AIS values, reflecting the overall trauma impact.
The Barell matrix maps injury types and ISS values to expected hospital stays.
For example: Fractures might result in 2-10 days of hospitalization, depending on body region. Burns often have longer stays (7-30 days), depending on severity. Internal organ damage may require surgery and intensive care, leading to 10-45 days in the hospital. Each patient’s LOS is then randomly drawn from the empirical distribution associated with their injury type and ISS.
The decision on whether a casualty requires surgery follows a hierarchical classification based on the nature and severity of injuries.
Certain injury types (e.g., amputations, severe internal organ damage) are always surgical. If a casualty has multiple injuries, the highest AIS score determines surgical status. Probability-Based Surgical Cases:
For open wounds, fractures, and burns, surgery is required only if ISS is above a threshold. For dislocations, crush injuries, and severe contusions, surgery is probabilistic, depending on injury severity. Surgical Hours Calculation:
Once designated as requiring surgery, a casualty is assigned a required number of surgical hours based on the procedure. This number ranges from 1-2 hours for minor repairs to 12+ hours for complex internal organ surgeries. This surgical status directly affects hospital resource allocation, including operating room availability and patient turnover rates.
Once casualties are classified as admitted or non-admitted, the simulation tracks the daily burden on medical resources.
Each admitted casualty occupies a hospital bed for the duration of their LOS. A daily census is maintained, updating the total number of occupied beds as new patients enter and existing patients are discharged.
If a patient’s ISS is high (16+), requires major surgery, or has a prolonged LOS, they are flagged for evacuation. Evacuations occur in three stages:
-Primary Evacuation: Immediate transfer from the battlefield to a field hospital. -Secondary Evacuation: Transfer to a higher-echelon medical facility. -Tertiary Evacuation: If long-term care is required, patients are evacuated to specialized hospitals.
The final aeromedical evacuation determination introduces random variation (±10%) to reflect real-world operational constraints and prioritization decisions.
The total required surgical hours per day is compared to available operating tables. Each surgical table operates for 8 hours per day—if demand exceeds this limit, delays occur, potentially affecting survival rates.
The code assigns randomized surgical durations within defined ranges based on ISS:
The final surgical hour assignment introduces random variation (±20%) to reflect real-world unpredictability.
Daily medical supply needs (bandages, IV fluids, ventilators) will be estimated based on the number and type of admissions. Medical staffing requirements (surgeons, nurses, medics) are computed to ensure an adequate provider-to-patient ratio.
The simulation integrates multiple empirical models and probability-driven calculations to estimate WIA, DNBI, hospital admissions, LOS, surgical needs, and medical resource allocation. The Barell matrix, ISS scoring, and battlefield casualty models ensure that the system reflects realistic injury outcomes and medical burdens. By dynamically updating force strength, hospital capacity, and surgical throughput, the model allows for a data-driven approach to military casualty management.
MPT-K CREstT offers a robust attrition model that uses ridge regression and quantitative estimates of battle injuries (BI). This model is used by combatant commands for force planning rather than Army casualty estimation, and MPT-K includes a suite of tools that generate admissions, logistics requirements, etc.
ACE is not meant to be a replacement or a competitor for MPT-K. Instead, the tool provides a widely accessible method for estimating Army casualties based on both enemy and friendly force strength (incorporating Lanchester’s Squared Equations with Dupuy’s assessment of ‘firepower’ coefficients). While CREstT leverages regularized regression to forecast casualties based on some but not all of Dupuy’s factors (e.g., surprise and mobility are not modeled), ACE estimates both enemy and friendly casualties based on the complete array of Dupuy’s factors using Lanchester’s Squared Law.
The spreadsheet contains Main, Results, Friendly Graphs, Enemy Graphs, Force Strength Graphs, and Phase I worksheets. The detailed flowcharts follow.
## Warning: package 'DiagrammeR' was built under R version 4.4.3The “Main” tab on the workbook provides four buttons: Build Phase Sheets, Run all Phases. On the tab, the number of phases for modeling should be input into the yellow highlighted cell. In general, the light yellow color indicates that some input is needed. The exception to this is the starting PAR for enemy and friendly forces. These values should only be populated on the Phase 1 sheet, as the rest of the phases will use generated data with replacements (if any).
After entering the number of phases, the button “Build Phase Sheets” should be executed. This will initialize all phases with information.
For each phase, planners must enter the appropriate data for the operation. All required data are shown in yellow. See
Figure 2. Phase Worksheet
This document describes each worksheet (tab) in the simulation, including when they are generated, how they are populated, and their purpose.
numphases, which dictates the number of
Phase Sheets.numphases from the Main sheet.B3,
C3, B5, C5) are auto-adjusted for
continuity between phases.B5 and C5, updating Friendly and Enemy
Strength.GenerateMechanismCauseNatureAndBodyRegion()
runs.This structured workflow ensures validity, continuity, and visualization of the entire battle simulation process.