Lanchester models for mixed forces with semi-dynamical target allocation

We consider the three standard Lanchester models of warfare (aimed-fire, unaimed-fire and asymmetric) with heterogeneous (mixed) forces on both sides. We begin by reviewing the homogeneous models, and then construct conserved quantities for the mixed models with separable kill-rates and random target allocation, commenting on the nature and allocation of unit types. Next we consider a more general semi-dynamical target allocation, construct a conserved quantity for the aimed-fire model and prove that the optimal strategy is to annihilate opposing unit types in succession. Finally, we make some comments on the optimal initial allocation of costed unit types in response to this.