# Inventory Control with Pricing Optimization in Continuous Time

## Abstract

Companies have always sought to find methods to optimize their business platform. An optimal strategy that can achieve these goals is therefore needed. To find the optimal strategy, a constrained optimization problem is introduced. In this dissertation, we study an inventory control problem in continuous time with no fixed cost associated with replenishment. We seek to find the optimal strategy that will maximize the profit functional while having control over the price and the replenishment of the product. We show, through theoretical results, that the policy which optimizes the problem statement is the continuous analog of the discrete time case, known as the Base Stock List Price policy. Firstly, we provide the general setup and theoretical results to the inventory control problem. We set up an analytical problem known as quasi-variational inequalities in a strong format. We show, through transformations, how we arrive at a nonlinear two point boundary value problem, of which the solution satisfies the quasi-variational inequalities. From the solution to the nonlinear two point boundary value problem we build the optimal strategy. Through a verification argument, we establish that the optimal strategy is indeed the value function which constitutes the optimal profit. In order to study the nonlinear two point boundary value problem, we introduce the associated epsilon problem. By use of a limiting process, we show convergence of the epsilon solution to the solution of the original nonlinear two point boundary value problem. The original nonlinear two point boundary value problem will be referenced through the dissertation as the base problem. Secondly, we provide a numerical methodology that solves the nonlinear two point boundary value problem on a semi-infinite domain. We show how theoretical results provide guidance in finding the numerical solution as well as circumvent issues presented. We solve the epsilon problem and show convergence to the base problem when epsilon decreases. The MATLAB solver bvp5c is used to provide the solution which constitutes the optimal strategy. Finally, we study a specific numerical case. We study the case where the average demand function decreases with respect to the price as a power function. We provide the Base Stock List Price policy for our case study as well as numerical results for the epsilon problem which confirms our theoretical findings. Based on our findings, we are able to draw economic conclusions and give guidance as to how a company should maximize their profit margins.