A first course in stochastic processes

Control: A perspective

This book is devoted to the study of maximum principles in partial differential equations. I t contains a wealth of material much of which is presented for the first time in a book form. An attractive feature of the book is that it is completely elementary and thus accessible to a wide audience of readers. The book has four chapters. Chapter I deals with the one dimensional maximum principle. The discussion of this very simple model of a maximum principle forms a good introduction to the general theory. Various applications of the principle are given to show that it is a very useful tool even in the study of ordinary differential equations. As an example, it is shown that many oscillation and comparison results in the Sturm-Liouville theory could be deduced most easily by a maximum principle argument. The proper discussion of maximum principles in partial differential equations begins in Chapter II . This chapter, which is the backbone of the book, is devoted to elliptic equations. The material covered in this chapter includes the E. Hopf maximum principle and its generalizations; the Phragmèn-Lindelöf principle for solutions of elliptic equations; Serrin's version of the Harnack inequality for solutions of general elliptic equations in two variables (this is probably the most difficult result discussed in the book) ; various versions of the Hadamard three circles theorems for solutions of elliptic equations; applications of the maximum principle to nonlinear equations and to problems of fluid flow. Chapter III is devoted to parabolic equations. The plan of this chapter parallels that of Chapter II . The topics discussed include the L. Nirenberg strong maximum principle; a three curves theorem with an interesting application to the Tikhonov uniqueness theorem; a Phragmèn-Lindelöf principle for parabolic equations with applications to uniqueness results; nonlinear operators; a maximum principle for certain parabolic systems. The fourth and the last chapter is devoted to hyperbolic equations. The results in this chapter are somewhat special since a maximum principle in the proper sense does not hold for solutions of hyperbolic equations. Nevertheless, solutions of certain hyperbolic equations

Maximum Principles In Differential Equations

Optimal pricing, production, and inventory for deteriorating items under demand uncertainty: The finite horizon case

Integrated demand and procurement portfolio management with spot market volatility and option contracts

In this paper we study a stochastic production/inventory system with finite production capacity and random demand. The cumulative production and demand are modeled by a two-dimensional Brownian motion process. There is a setup cost for switching on the production and a convex holding and shortage cost, and our objective is to find the optimal production/inventory control that minimizes the average cost. Both lost-sales and backlog cases are studied. For the lost-sales model we show that, within a large class of policies, the optimal production strategy is either to produce according to an s, S policy, or never turn on the machine at all thus it is optimal for the firm to not enter the business; whereas for the backlog model, we prove that the optimal production policy is always of the s, S type. Our approach first develops a lower bound for the average cost among a large class of nonanticipating policies and then shows that the value function of the desired policy reaches the lower bound. The results offer insights on the structure of the optimal control policies as well as the interplays between system parameters.

Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion

In this paper we consider an inventory system with increasing concave ordering cost and average cost optimization criterion. The demand process is modeled as a Brownian motion. Porteus (1971) studied a discrete-time version of this problem and under the strong condition that the demand distribution belongs to the class of densities that are finite convolutions of uniform and/or exponential densities (note that normal density does not belong to this class), an optimal control policy is a generalized (s, S) policy consisting of a sequence of (si , Si ). Using a lower bound approach, we show that an optimal control policy for the Brownian inventory model is determined by a single pair (s, S).

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Optimal control policy for a Brownian inventory system with concave ordering cost

In this paper we study a class of selective newsvendor problems, where a decision maker has a set of raw materials each of which can be customized shortly before satisfying demand. The goal is then to select which subset of customizations maximizes expected profit. We show that certain multi-period and multi-product selective newsvendor problems fall within our problem class. Under the assumption that the demands are independent and normally, but not necessarily identically, distributed we show that some problem instances from our class can be solved efficiently using an attractive sorting property that was also established in the literature for some related problems. For our general model we use the KKT conditions to develop an exact algorithm that is efficient in the number of raw materials. In addition, we develop a class of heuristic algorithms. In a numerical study, we compare the performance of the algorithms, and the heuristics are shown to have excellent performance and running times as compared to available commercial solvers.

Exact and heuristic methods for a class of selective newsvendor problems with normally distributed demands

Techniques for determining a decision to acquire units of an item to be inventoried may be provided. For example, a demand for an item may be simulated to determine a consumption of a capacity for inventorying the item. A discrepancy between the consumption of the capacity and the capacity may be determined. An opportunity cost associated with the capacity may be updated based at least in part on determining that the discrepancy fails a convergence criterion. The opportunity cost may indicate a value associated with using the capacity. The consumption of the capacity may be simulated based at least in part on the updated opportunity cost. A resulting discrepancy may be determined. If the resulting discrepancy meets the convergence criterion, the decision to acquire the units of the item may be generated based at least in part on the updated opportunity cost.

Adaptive control of an item inventory plan

We consider a stochastic inventory system with general piecewise linear ordering cost. The cumulative demand is modeled as a Brownian motion process. The ordering cost function is neither convex nor concave; it may not be monotone; and it is not even necessarily continuous, and it includes most ordering cost functions studied in the literature, e.g., economies-of-scale or dis-economies of scale, all-unit discount or incremental discount, and multiple setup costs, as special cases. In addition to ordering cost, the system incurs the usual holding/shortage cost, and the objective is to minimize the average system cost per unit of time. Despite the complexity in the ordering cost function, we show that an optimal control policy is very simple: it is either an $(s,S)$ policy or a one-sided singular control policy.

Jingchen Wu

Papers

Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion

Optimal control policy for a Brownian inventory system with concave ordering cost

Exact and heuristic methods for a class of selective newsvendor problems with normally distributed demands

Adaptive control of an item inventory plan

Optimal Policies for Brownian Inventory Systems With a Piecewise Linear Ordering Cost