Library / Operational analysis / OPS-01
Operational analysis
Lanchester square law
Surviving strength of force A after a fight to the finish in which both sides bring all their strength to bear.
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- OPS-01
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- Operational analysis
- Result
- Af
A₀ and B₀ are the starting numbers of the two forces, and α and β the rates at which each unit on side A and on side B destroys enemy units. The fight is assumed to run until one side is wiped out. A survives if αA₀² > βB₀², and the survivors are the square root of the difference, weighted by the effectiveness ratio. A shows 0 if it is destroyed.
Surviving strength of force A after a fight to the finish in which both sides bring all their strength to bear. Strength counts as a square: doubling numbers is worth four times the effectiveness.
Calculator
- Ratio of fighting strengths αA₀² / βB₀²1.953
- Starting strength A needs just to win, B₀√(β/α)715.5
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A starts with 25 % more units than B (1000 against 800) and survives with 70 % of its force. Numbers count as a square, so doubling the force quadruples its value.
- Concentration of force pays: two halves of A fought one after another against B can lose where the whole force would win, because each half meets the full strength of B.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| A0 Starting strength of A | 10 to 10,000 | 0 | 9,974 | 9,974 | +10% | +19.6 % |
| B0 Starting strength of B | 10 to 10,000 | 0 | 1,000 | 1,000 | +10% | −11.7 % |
| β Effectiveness of each B unit | 0.001 to 1 | 0 | 993.6 | 993.6 | +10% | −5.39 % |
| α Effectiveness of each A unit | 0.001 to 1 | 0 | 987.1 | 987.1 | +10% | +4.66 % |
Sweep of Starting strength of A
| Starting strength of A | Survivors of A |
|---|---|
| 10 | 0 |
| 23.71 | 0 |
| 56.23 | 0 |
| 133.4 | 0 |
| 316.2 | 0 |
| 749.9 | 224.4 |
| 1,778 | 1,628 |
| 4,217 | 4,156 |
| 10,000 | 9,974 |
Sweep one variable
Variables and how each one contributes
Survivors of A
What the equation gives you. Everything else feeds it.
Starting strength of A
The number of units, or total fighting strength, of force A at the start. Squared: a 10 % increase in numbers is worth about 21 % in effectiveness. A 10 % rise in A₀ raises the survivors by about 20 %.
Starting strength of B
The starting strength of force B. Squared, with the opposite sign: a 10 % rise in B₀ lowers A’s survivors by about 12 %, and near parity can wipe A out.
Effectiveness of each A unit
How many B units one A unit destroys per unit time. Raises A’s survivors, in a weighted form: a 10 % rise in α raises them by about 5 %.
Effectiveness of each B unit
How many A units one B unit destroys per unit time. Lowers A’s survivors: a 10 % rise in β lowers them by about 5 % at the default.
Worked example with the values above
- Fighting strength: αA₀² = 0.05 × 1,000,000 = 50,000; βB₀² = 0.04 × 640,000 = 25,600.
- A₀² − (β/α)B₀² = 1,000,000 − 512,000 = 488,000.
- Af = √488,000 = 698.6.
Assumptions and limits
- Aimed fire: each unit can engage any enemy unit and the rate of kill is proportional to the number of shooters.
- No reinforcement, withdrawal or change of effectiveness during the fight.
Common mistakes
- Using the square law for area fire or for ancient hand-to-hand fighting, where the linear law is more suitable.
- Reading the result as a forecast of an actual engagement.
Statistics and uncertainty
This is a deterministic model, and its inputs are the hardest to estimate. Near parity the survivors depend on a small difference of two large squares, so a 5 % error in α/β can turn a clear win into a loss. Treat it as a way of reasoning about trends, not as a prediction; random effects such as who fires first matter in small fights.
Where it is used
- Operational analysis of attrition and the value of concentrating force.
- Estimating the numerical advantage needed to offset a difference in unit effectiveness.
Maths you need
- Squares and square roots: A₀² and √
- Rates: kill rate per unit