Optimal Resource Allocation In Lanchester Attrition Model Based Conflicts
Force deployment and optimal resource allocation has been an area of considerable research interest in conventional warfare. In the modern scenario, with significant advances in technology related to communication and computation, sophisticated decision-making in these situations has become feasible. This has generated renewed interest in formulating decision-making problems in these areas and seeking optimal solutions to them. This thesis addresses one such problem in which the defending forces need to optimally Partition their resources between several attacking forces of differing strengths. The basic model considered for resource allocation is Lanchester attrition models. Lanchester models are deterministic differential equations that model attrition to forces in convict. In this thesis we address a resource allocation problem where the resource allocation is done using different approaches. First, we developed a (2,1) model using the Lanchester square law model for attrition. For this model we assumed that the attacking force consists of two types of forces and the defending force consists of only one type of force. The objective is to optimally partition the defending force against the two attacking forces so as to maximize the surviving defending force strength and to minimize the attacking force strength. The objective function considered in this thesis is the weighted sum of the surviving defending force strength and the destroyed attacking force strength. We considered a resource allocation problem in which allocation of resources are done using four different approaches. The simplest is the case when allocation is done initially and no further action is taken Iv Abstract v (Time Zero Allocation (TZA)). For the TZA allocation scheme, when any of attacking forces gets destroyed, the corresponding defending force which was engaging that attacking force will stop interacting further. This situation rarely happens in reality. Hence to make this scenario more realistic, we considered another allocation scheme in which allocation is followed by redistribution of resources depending on certain decisive events (Time Zero Allocation with Redistribution (TZAR)). In TZA and TZAR schemes, the allocation of defending force is done only at the initial time. Deviating from this assumption, we considered another allocation scheme in which a constant allocation ratio is used continuously over time till the end of the convict (Continuous Constant Allocation (CCA). To account for the redistribution of resources we extended this allocation scheme to the case in which continuous constant allocation is followed by redistribution of the resources (Continuous Constant Allocation with Redistribution (CCAR)). In each of these formulations we define the conditions for an optimal resource partitioning and allocation. We were able to obtain analytical expression for resource partitioning in almost all of these cases. Next, in order to consider situations in which area fire is required, we developed a (2,1) model using Lanchester linear law model for attrition. Here we considered a resource allocation problem in which the resource allocation is done using ideas similar to the square law case. In the Linear law, the resources will get destroyed completely only at infinite time, hence a situation for redistribution of resources does not arise for this law. We considered Time Zero Allocation and Continuous Constant Allocation schemes for this law. We obtained analytical results for the TZA scheme. For the CCA scheme, closed form solutions are difficult to obtain but numerical solutions were obtained. The above schemes were extended to an (n, 1) model for resource allocation using Lanchester square and linear laws. Here the defending forces have to determine an optimal partitioning of available resources to counter attacks from an adversary from n different fronts. For the square law model, we considered both TZA and CCA schemes for resource allocation. As the number of force types increases, the equations becomes much more complicated and the analytical solutions are difficult to obtain. We were able to obtain analytical solutions for some of the situations that occurs during the conflict. For the linear law, we considered only the TZA scheme since, even for the simpler (2,1) model, the analytical solutions are difficult to obtain for the CCA scheme. The resource allocation strategies developed in this thesis contribute to the growing research in the field of conflicts. The thesis concludes with a discussion on some future Extensions of this work.