Michael H. Veatch

Optimal Control of a Two-Station Tandem Production/Inventory System

A manufacturing facility consisting of two stations in tandem operates in a make-to-stock mode: After production, items are placed in a finished goods inventory that services an exogenous Poisson demand. Demand that cannot be met from inventory is backordered. Each station is modeled as a queue with controllable production rate and exponential service times. The problem is to control these rates to minimize inventory holding and backordering costs. Optimal controls are computed using dynamic programming and compared with the kanban, basestock and buffer control mechanisms that have been proposed for manufacturing facilities. Conditions are found under which certain simple controls are optimal using stochastic coupling arguments. Insights are gained into when to hold work-in-process and finished goods inventory, comparable to previous studies of production lines in make-to-order and unlimited demand environments.

Monotone control of queueing networks

This paper uses submodularity to obtain monotonicity results for a class of Markovian queueing network service rate control problems. Nonlinear costs of queueing and service are allowed. In contrast to Weber and Stidham [14], our monotonicity theorem considers arbitrary directions in the state space (not just control directions), arrival routing problems, and certain uncontrolled service rates. We also show that, without service costs, transition-monotone controls can be described by simple control regions and switching functions. The theory is applied to queueing networks that arise in a manufacturing system that produces to a forecast of customer demand, and also to assembly and disassembly networks.

Dynamic allocation of reconfigurable resources ina two-stage Tandem queueing system with reliability considerations

Consider a two-stage tandem queueing system, with dedicated machines in each stage. Additional reconfigurable resources can be assigned to one of these two stations without setup cost and time. In a clearing system (without external arrivals) both with and without machine failures, we show the existence of an optimal monotone policy. Moreover, when all of the machines are reliable, the switching curve defined by this policy has slope greater than or equal to -1. This continues to hold true when the holding cost rate is higher at the first stage and machine failures are considered.

Inspection strategies for multistage production systems with time-varying quality

An economic, cost of quality model is formulated for a production system with multiple inspection locations and sampling. The model is applied to a thermal printer for digital photographs to determine when inspection of incoming material is appropriate. A simple dynamic model of quality is used, allowing for common-cause defects and the evaluation of sampling plans. The opportunity to correct quality problems more quickly when they are detected earlier is also considered. The cost structure used allows the costs of quality improvement, and not just that of defective units, to be considered.

Optimal average cost manufacturing flow controllers: convexity and differentiability

The authors consider the control of a production facility consisting of a single workstation with multiple failure modes and part types using a continuous flow control model. Technical issues concerning the convexity and differentiability of the differential cost function are investigated. It is proven that under an optimal control policy the differential cost is C/sup 1/ on attractive control switching boundaries.

Approximate Linear Programming for Average Cost MDPs

We consider the linear programming approach to approximate dynamic programming with an average cost objective and a finite state space. Using a Lagrangian form of the linear program LP, the average cost error is shown to be a multiple of the best fit differential cost error. This result is analogous to previous error bounds for a discounted cost objective. Second, bounds are derived for average cost error and performance of the policy generated from the LP that involve the mixing time of the Markov decision process MDP under this policy or the optimal policy. These results improve on a previous performance bound involving mixing times.

Fluid analysis of arrival routing

In Hajeks arrival routing problem (1984), customers are routed to one of n queues to minimize average holding cost. Interarrival and service times are exponentially distributed. We solve the associated fluid model. The optimal fluid policy tells us the asymptotic slopes of the switching surfaces in the original problem when the queues are large. If these slopes are nonzero, then numerical tests indicate that the fluid policy performs well in the original stochastic network. The fluid policy also indicates the approximate path that will be taken to recover from large queues: Routing only switches to queues with larger holding cost and once a large queue empties it will remain approximately empty.

Optimal manufacturing flow controllers: zero-inventory policies and control switching sets

A continuous-flow control model of a single workstation with multiple failure modes and part types is considered. Although the general model is intractable, properties of the optimal control policy are identified that can be used to help formulate heuristic policies. Control switching sets are described and shown to have a threshold form. For a single machine with two part-types, conditions are found under which no inventory is held, analogous to the single-part-type result of Bielecki and Kumar (1988).

Reliability of Periodic, Coherent, Binary Systems

A new phased-mission analysis considers periodic systems without repair. Periodic systems arise in applications such as aviation where a sequence of operating phases is repeated during each flight. A theorem relates such systems to a traditional single-phase system. The single-phase system is useful for approximating the reliability and mean life of the periodic system and is much simpler to analyze than exact transformations to a single-phase system.

Using Fluid Solutions in Dynamic Scheduling

We review some recent work on fluid approximation of queueing network control problems and present new results. The relationship between the optimal control of a queueing network and the optimal control of the associated fluid model is investigated. A general theory, a number of examples, and a few numerical results are presented. The fluid model replaces stochastic counting processes by their mean rates. Surprisingly, although the fluid model is deterministic and transient, it often contains valuable information about the original problem. We show that, under some technical assumptions, the asymptotic slope (normal vector in higher dimensions) of the switching surfaces for large queue lengths can be found from the fluid model. The implications of the fluid policy for designing manufacturing scheduling and control policies are discussed. Unlike the queueing network control problem, which is usually an intractable dynamic program, very efficient algorithms exist to solve the fluid problem from a given initial buffer state. Fluid solutions are presented for a number of examples, some of them new.