Asymptotic Optimality Of Hierarchical Controls In Stochastic Manufacturing Systems

Most manufacturing systems are large and complex and operate in an uncertain environment. One approach to managing such systems is that of hierarchical decomposition. This paper reviews the research devoted to proving that a hierarchy based on the frequencies of occurrence of different types of events in the system results in decisions that are asymptotically optimal as the rates of some events become large compared to those of others. Manufacturing systems that are addressed include single machine systems, flowshops, and jobshops producing multiple products, incorporate random production capacity and demands, and involve such decisions as production rates, capacity expansion, and promotional campaigns. The paper concludes with a review of computational results and areas of applications.

This paper presents an extension of earlier research on hierarchical control of stochastic manufacturing systems with long-run average cost in which a positive inventory deterioration/cancellation rate for each product is assumed. Here we drop the assumption of the positive inventory deterioration/cancellation rate for each product, and give an asymptotic analysis of the manufacturing systems as the rates of change of the machine states approach infinity. We obtain a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. We use a near optimal control of the limiting problem to construct nearly asymptotically optimal open-loop piecewise deterministic controls for the original problem.

One of the most important methods in dealing with the optimization of large, complex systems is that of hierarchical decomposition. The idea is to reduce the overall complex problem into manageable approximate problems or subproblems, to solve these problems, and to construct a solution of the original problem from the solutions of these simpler prob lems. Development of such approaches for large complex systems has been identified as a particularly fruitful area by the Committee on the Next Decade in Operations Research (1988) [42] as well as by the Panel on Future Directions in Control Theory (1988) [65]. Most manufacturing firms are complex systems characterized by sev eral decision subsystems, such as finance, personnel, marketing, and op erations. They may have several plants and warehouses and a wide variety of machines and equipment devoted to producing a large number of different products. Moreover, they are subject to deterministic as well as stochastic discrete events, such as purchasing new equipment, hiring and layoff of personnel, and machine setups, failures, and repairs.

This paper presents an asymptotic analysis of hierarchical manufacturing systems with stochastic demand and machines subject to breakdown and repair as the rate of change in machine states approaches infinity. This situation gives rise to a limiting problem in which the stochastic machine availability is replaced by the equilibrium mean availability. The value function for the original problem converges to the value function of the limiting problem. Moreover, a control for the original problem can be constructed from the optimal control of the limiting problem in a way which guarantees its asymptotic optimality. Asymptotic properties of the system trajectories are analyzed for both feedback and open loop controls in the system. The convergence rate of the value function for the original problem is found. This helps in providing an error estimate for the constructed asymptotically optimal control.

Most manufacturing systems are large and complex and operate in an uncertain environment. One approach to managing such systems is that of hierarchical decomposition. This paper reviews the research devoted to proving that a hierarchy based on the frequencies of occurrence of different types of events in the systems results in decisions that are asymptotically optimal as the rates of some events become large compared to those of others. The paper also reviews the research on stochastic optimal control problems associated with manufacturing systems, their dynamic programming equations, existence of solutions of these equations, and verification theorems of optimality for the systems. Manufacturing systems that are addressed include single machine systems, dynamic fowshops, and dynamic jobshops producing multiple products. These systems may also incorporate random production capacity and demands, and decisions such as production rates, capacity expansion, and promotional campaigns are also presented.

This paper presents an asymptotic analysis of a stochastic manufacturing system consisting of parallel machines subject to breakdown and repair and facing a constant demand, as the rates of change of the machine states approach infinity. This situation gives rise to a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The long-run average cost for the original problem converges to the long-run average cost of the limiting problem. Open-loop and feedback controls for the original problem are constructed from optimal controls of the limiting problem in a way that guarantees their asymptotic optimality. The convergence rate of the long-run average cost for the original problem to that of the limiting problem is established. This helps in providing an error estimate for the constructed open-loop asymptotic optimal control.

This paper deals with an asymptotic analysis of hierarchical production planning in stochastic manufacturing systems consisting of a single or parallel failure-prone machines producing a number of different products without attrition. The objective is to choose production rates over time in order to minimize the long-run average expected cost of production and surplus. As the rate of machine break-down and repair approaches infinity, the analysis results in a limiting problem in which the stochatic machine capacity is replaced by the equilibrium mean capacity. The optimal value for the original problem is proved to converge to the optimal value of the limiting problem. This suggests a heuristic to construct an open-loop control for the original stochastic problem from the open-loop control of the limiting deterministic problem. We as well as obtain error bound estimates for constructed open-loop controls.

This paper presents an extension of earlier research on heirarchical control of stochastic manufacturing systems with linear production costs. A new method is introduced to construct asymptotically optimal open-loop and feedback controls for manufacturing systems in which the rates of machine breakdown and repair are much larger than the rate of fluctuation in demand and rate of discounting of cost. This new approach allows us to carry out an asymptotic analysis on manufacturing systems with convex inventory/backlog and production costs as well as obtain error bound estimates for constructed open loop controls. Under appropriate conditions, an asymptotically optimal Lipschitz feedback control law is obtained.

This paper is concerned with near-optimal control of manufacturing systems consisting of two unreliable machines in tandem and having the objective of minimizing the total discounted cost of inventories/shortages over an infinite horizon. Asymptotic optimal feedback controls are constructed with respect to the rate of machine breakdown/repair as compared to the given discount rate. Performance of these controls, known as hierarchical controls, is compared with the optimal cost (when possible) and the costs obtained with two well-known heuristics, known as Kanban controls and two boundary controls. It is shown that hierarchical controls perform better than Kanban controls in some cases and no worse in others. Costs of hierarchical and two boundary controls are not significantly different, although the former is a simpler policy than the latter. Also examined computationally is the asymptotic nature of hierarchical controls.

This book articulates a new theory that shows that hierarchical decision making can in fact lead to a near optimization of system goals. The material in the book cuts across disciplines. It will appeal to graduate students and researchers in applied mathematics, operations management, operations research, and system and control theory.