Force concentration
Mass of decision: Force concentration became integral to the Prussian military operational doctrine of the mass of decision, which aimed to cause disproportionate losses on the enemy and therefore destroy the enemy's ability to fight.
From an empirical examination of past battles, the Prussian military theorist Carl von Clausewitz (1780–1831) concluded:

[...] we may infer, that it is very difficult in the present state of Europe, for the most talented General to gain a victory over an enemy double his strength. Now if we see double numbers prove such a weight in the scale against the greatest Generals, we may be sure, that in ordinary cases, in small as well as great combats, an important superiority of numbers, but which need not be over two to one, will be sufficient to ensure the victory, however disadvantageous other circumstances may be.

Lanchester's laws: During the First World War Frederick W. Lanchester formulated Lanchester's laws that calculated that the combat power of a military force is the square of the number of members of that unit so that the advantage a larger force has is the difference of the squares of the two forces, i.e.

If force A has say 2 units and force B has 3 units, then the advantage force B has is 3²−2² or 5.
If force A still has 2 units and force B has 4 units then the advantage force B has is 4²−2² or 12.
If force A still has 2 units and force B has 5 units then the advantage force B has is 5²−2² or 21.
So a two to one advantage in units will quadruple the firepower and inflict four times the punishment, three times as many units will have nine times the combat ability and so on. Basically the greater the numerical superiority that one side has, the greater the damage he can inflict on the other side and the smaller the cost to himself.

Mathematical model: There is no battlefield where battle tactics can be reduced to a pure race of delivering damage while ignoring all other circumstances. However, in some types of warfare, such as a battle for air superiority, confrontation of armoured forces in World War II or battleship-based naval battles, the ratio of armed forces could become the dominant factor. In that case, equations stated in Lanchester's laws model the potential outcome of the conflict fairly well. Balance between the two opponent forces incline to the side of superior force by the factor of 
  
    
      
        
          
            (
            
              
                
                  P
                  o
                  w
                  e
                  
                    r
                    
                      2
                    
                  
                
                
                  P
                  o
                  w
                  e
                  
                    r
                    
                      1
                    
                  
                
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle \left({\frac {Power_{2}}{Power_{1}}}\right)^{2}}
  
. For example, two tanks against one tank are superior by a factor of four.
This result could be understood if the rate of damage (considered as the only relevant factor in the model) is solved as a system of differential equations. The rate in which each army delivers damage to the opponent is proportional to the number of units – in the model each unit shoots at a given rate – and to the ability or effectiveness of each surviving unit to kill the enemy. The sizes of both armies decrease at different rates depending on the size of the other, and casualties of the superior army approach zero as the size of the inferior army approaches zero. This can be written in equations:

  
    
      
        
          
            d
            
              d
              t
            
          
        
        
          N
          
            1
          
        
        =
        −
        
          c
          
            2
          
        
        
          N
          
            2
          
        
      
    
    {\displaystyle {\frac {d}{dt}}N_{1}=-c_{2}N_{2}}
  

  
    
      
        
          
            d
            
              d
              t
            
          
        
        
          N
          
            2
          
        
        =
        −
        
          c
          
            1
          
        
        
          N
          
            1
          
        
      
    
    {\displaystyle {\frac {d}{dt}}N_{2}=-c_{1}N_{1}}
  

  
    
      
        
          N
          
            1
          
        
      
    
    {\displaystyle N_{1}}
  
 is the number of units in the first army

  
    
      
        
          
            d
            
              d
              t
            
          
        
        
          N
          
            2
          
        
      
    
    {\displaystyle {\frac {d}{dt}}N_{2}}
  
 is the rate in which army 1 damages army 2 (affected by unit quality or other advantage)

  
    
      
        
          c
          
            1
          
        
      
    
    {\displaystyle c_{1}}
  
 is a coefficient which describes army 1's ability to inflict damage per unit per time.
The above equations result in the following homogeneous second-order linear ordinary differential equations:

  
    
      
        
          
            
              d
              
                2
              
            
            
              d
              
                t
                
                  2
                
              
            
          
        
        
          N
          
            1
          
        
        =
        
          c
          
            2
          
        
        
          c
          
            1
          
        
        
          N
          
            1
          
        
      
    
    {\displaystyle {\frac {d^{2}}{dt^{2}}}N_{1}=c_{2}c_{1}N_{1}}
  

  
    
      
        
          
            
              d
              
                2
              
            
            
              d
              
                t
                
                  2
                
              
            
          
        
        
          N
          
            2
          
        
        =
        
          c
          
            2
          
        
        
          c
          
            1
          
        
        
          N
          
            2
          
        
      
    
    {\displaystyle {\frac {d^{2}}{dt^{2}}}N_{2}=c_{2}c_{1}N_{2}}
  

To determine the time evolution of 
  
    
      
        
          N
          
            1
          
        
      
    
    {\displaystyle N_{1}}
  
 and 
  
    
      
        
          N
          
            2
          
        
      
    
    {\displaystyle N_{2}}
  
, these equations need to be solved using the known initial conditions (the initial size of the two armies prior to combat).
This model clearly demonstrates (see picture) that an inferior force can suffer devastating losses even when the superior force is only slightly larger, in case of equal per-unit qualitative capabilities: in the first example (see picture, top plot) the superior force starts only 40% larger, yet it brings about the total annihilation of the inferior force while suffering only 40% losses. Quality of the force may outweigh the quantitative inferiority of the force (middle plot) when it comes to battle outcomes.

Business strategy: In the 1960s, Lanchester's laws were popularised by the business consultant Nobuo Taoka and found favour with a segment of the Japanese business community. The laws were used to formulate plans and strategies to attack market share. The "Canon–Xerox copier battle" in the UK, for example, reads like a classic people's war campaign. In this case, the laws supported Canon's establishment of a "revolutionary base area" by concentrating resources on a single geographical area until dominance could be achieved, in this case in Scotland. After this, they carefully defined regions to be individually attacked again with a more focused allocation of resources. The sales and distribution forces built up to support these regions in turn were used in the final "determined push in London with a numerically larger salesforce".

Hypothetical example: Imagine two equally matched sides each with two infantry and two armoured divisions. Now visualize a straight defensive line with the two infantry and two armoured divisions, deployed equally along the length of the line. Hypothetically the attacker can win by concentrating his armour at one point (with his infantry holding the rest of the line).
Traditionally it is accepted that a defending force has a 3:1 advantage over an attacker. In other words, a defending force can hold off three times its own number of attackers. Imagine, then, that the defensive line is four units in length, so that each portion of the line can be held by a single defending division. Assume that they can take on the oncoming armour on equal terms (with ATGWs, pre-prepared artillery fireplans etc.) and that they have had time to dig in. This single unit should be able to hold off 3 times its own number. With the attacking force having only two armoured units, the defenders should have the advantage.
However, as the defensive line increases from the imaginary four units in length, the advantage slips from the defender to the attacker. The longer the line to be held, the thinner the defenders will be spread. With the defender having sacrificed his mobility to dig in, the attacker can choose where and when to attack. Either penetrating the line or turning a flank and thus being able to destroy the enemy in detail. Thus, concentrating two divisions and attacking at a single point generates a far greater force than is achieved by spreading two divisions into a line and pushing forward on a broad front.
Concentration of force in this scenario requires mobility (to permit rapid concentration) and power (to be effective in combat once concentrated). The tank embodies these two properties and for the past seventy years has been seen as the primary weapon of conventional warfare.