Ernst Presman

Optimal feedback production planning in a stochastic N -machine flowshop

We consider a production planning problem in an N-machine flowshop subject to breakdown and repair of machines and to non-negativity constraints on work-in-process. The machine capacities and demand processes are assumed to be finite-state Markov chains. The problem is to choose the rates of production on the N machines over time to minimize the expected discounted cost of production and inventory/backlog over an infinite horizon. It is formulated as a stochastic dynamic programming problem. We show that the value function of the problem is locally Lipschitz and is a solution to a dynamic programming equation together with a certain boundary condition. We provide an interpretation of the boundary condition. We also prove a verification theorem and derive the optimal feedback control policy in terms of the directional derivatives of the value function. Finally, we obtain a deterministic optimal control problem that is equivalent to the stochastic production planning problem under consideration.

Explicit Solution of a General Consumption/Portfolio Problem with Subsistence Consumption and Bankruptcy

This paper solves a general continuous-time single-agent consumption and portfolio decision problem with subsistence consumption in closed form. The analysis allows for general continuously differentiable concave utility functions. The model takes into consideration that consumption must be no smaller than a given subsistence rate and that bankruptcy can occur. Thus the paper generalizes the results of Karatzas, Lehoczky, Sethi, and Shreve (1986).

Risk-Aversion Behavior in Consumption/Investment Problems

In this paper, we study the risk-aversion behavior of an agent in the dynamic framework of consumption/investment decision making that allows the possibility of bankruptcy. Agents consumption utility is assumed to be represented by a strictly increasing, strictly concave, continuously differentiable function in the general case and by a HARA-type function in the special case treated in the paper. Coefficients of absolute and relative risk aversion are defined to be the well-known curvature measures associated with the derived utility of wealth obtained as the value function of the agents optimization problem. Through an analysis of these coefficients, we show how the change in agents risk aversion as his wealth changes depends on his consumption utility and the other problem parameters, including the payment at bankruptcy. Moreover, in the HARA case, we can conclude that the agents relative risk aversion is nondecreasing with wealth, while his absolute risk aversion is decreasing with wealth only if he is sufficiently wealthy. At lower wealth levels, however, the agents absolute risk aversion may increase with wealth in some cases.

Existence of optimal feedback production plans in stochastic flowshops with limited buffers

We consider an N-machine flowshop with unreliable machines and bounds on work-in-process. Machine capacities and demand processes are finite-state Markov chains. The problem is to choose the rates of production on the machines over time to minimize the expected discounted costs of production and inventory/backlog. We show that the value function of the problem is locally Lipschitz and is a solution to a dynamic programming equation with a certain boundary condition. We provide a verification theorem, and derive the optimal feedback control policy in terms of the directional derivatives of the value function.

Optimal Production Control of a Failure-Prone Machine

We consider a problem of optimal production control of a single unreliable machine. The objective is to minimize a discounted convex inventory/backlog cost over an infinite horizon. Using the variational analysis methodology, we develop the necessary conditions of optimality in terms of the co-state dynamics. We show that an inventory-threshold control policy is optimal when the work and repair times are exponentially distributed, and demonstrate how to find the value of the threshold in this case. We consider also a class of distributions concentrated on finite intervals and prove properties of the optimal trajectories, as well as properties of an optimal inventory threshold that is time dependent in this case.

Distribution of Bankruptcy Time in a Consumption/Portfolio Problem

In this note we provide an explicit formula for the probability distribution function of the bankruptcy time in a general consumption/investment problem involving subsistence consumption and bankruptcy penalty.

Average Cost Optimality for an Unreliable Two-Machine Flowshop with Limited Internal Buffer

We consider a production planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to nonnegativity and upper bound constraints on work-in-process. The objective is to choose machine production rates over time to minimize the long-run average inventory/backlog and production costs. For sufficiently large upper bound on the work-in-process, the problem is formulated as a stochastic dynamic program. We then establish a verification theorem and a partial characterization of the optimal control policy if it exists.

On Optimal Stopping of Random Sequences Modulated by Markov Chain

This paper has two main goals: first, to describe a new class of optimal stopping (OS) problems for which the solutions can be found either in an explicit form, or in a finite number of steps, and second, to demonstrate the potential of the state elimination algorithm developed by one of the authors earlier, for the problem of OS of a finite or countable Markov chain.

On Optimality of Stochastic N-Machine Flowshop with Long-Run Average Cost

This paper is concerned with the problem of production planning in a stochastic manufacturing system with serial machines that are subject to break-down and repair. The machine capacities are modeled by a Markov chain. The objective is to choose the input rates at the various machines over time in order to meet the demand for the system’s production at the minimum long-run average cost of production and surplus, while ensuring that the inventories in internal buffers between adjacent machines remain nonnegative. The problem is formulated as a stochastic dynamic program. We prove a verification theorem and derive the optimal feedback control policy in terms of the directional derivatives of the potential function.

The Existence and Uniqueness of Nash Equilibrium Point in an m-player Game “Shoot Later, Shoot First!”

We consider the following “silent duel” of m players with a possible economic interpretation. Each player has one “bullet”, which she can shoot at any time during the time interval [0,1]. The probability that the i-th player hits the “target” at moment t is given by an increasing accuracy function fi(t). The winner is the player who hits the target first. Under natural assumptions on the functions fi(t) we prove the existence and uniqueness of a Nash equilibrium point in this game, and we provide an explicit construction of this equilibrium. This construction allows us to obtain exact solutions for many specific examples. Some of them are presented.