Shaohui Zheng

Joint Inventory Replenishment and Pricing Control for Systems with Uncertain Yield and Demand

We study the joint inventory replenishment and pricing problem for production systems with random demand and yield. More specifically, we analyze the following single-item, periodic-review model. Demands in consecutive periods are independent random variables and their distributions are price sensitive. The production yield is uncertain so that the quantity received from a replenishment is a random variable whose distribution depends on the production quantity. Stockouts are fully backlogged. Our problem is to characterize the optimal dynamic policy that simultaneously determines the production quantity and the price for each period to maximize the total discounted profit. We show that the optimal replenishment policy is of a threshold type, i.e., it is optimal to produce if and only if the starting inventory in a period is below a threshold value, and that both the optimal production quantity and the optimal price in each period are decreasing in the starting inventory. We further study the operational effects of uncertain yield. We prove that, in the single-period case, the threshold of replenishment is independent of the yield variability, and, in the multiperiod case, it is higher in a system with uncertain yield than in one with certain yield. In addition, the system with uncertain yield always charges a higher price.

Queue Spillovers in Transportation Networks with a Route Choice

This paper examines different traffic phenomena that occur when drivers have to navigate a network in which queues backup past diverge intersections. In particular, it looks at the bottleneck capacity of a network and how reducing it to below a critical level can resume the attractiveness of an alternative route, thus preventing the system from reaching the saturation point. The author concludes that the time-dependent traffic assignment problem with physical queues is chaotic in nature and that it may be impossible to obtain input data with the required accuracy to make reliable predictions of cumulative output flows on severely congested networks.

Optimal Replenishment and Rework with Multiple Unreliable Supply Sources

We study a production-inventory system with multiple unreliable supply sources. Through inspection and rework, the system can improve the quality of the units received from the supply sources. There are two interleaved decisions: the replenishment quantities from the sources and the inspection-rework quantities among the units received. We show the optimal solution to the replenishment decision can be efficiently derived from a greedy algorithm, and inspection-rework is optimally applied to a single source identified by the algorithm. Furthermore, in the case of linear cost functions, it is optimal to place orders from two supply sources, i.e., dual sourcing. The results extend to the infinite-horizon case, where an order-up-to policy is optimal. The model also readily adapts to situations in which the supply imperfection takes the form of a reduced delivery quantity (yield loss).

Quantum Mechanical and Human Violations of Compound Probability Principles: Toward a Generalized Heisenberg Uncertainty Principle

A key central tenet of decision theory is that decomposing an uncertain event into sub-events should not change the overall probability assigned to that uncertain event. As we show, both quantum physics and behavioral decision theory appear to systematically violate this principle in very similar ways. These results suggest that the structuring phase of decision analysis-which specifies how various events are decomposed-helps shape the subjective probabilities which will ultimately be assigned to those events.

Dynamic Capacity Expansion for a Service Firm with Capacity Deterioration and Supply Uncertainty

Motivated by the challenges faced by the telecom industry during the past decade, in this paper we study a dynamic capacity expansion problem for service firms. There is a random demand for the firms capacity in each period: the demand in excess of the capacity is lost, and revenue is generated for the fulfilled demand. At the beginning of each period, the firm might increase its capacity through purchasing equipment for immediate delivery, which is constrained by a random supply limit, or it might sign a future contract for equipment delivery in the following period. We assume that the firms capacity might partially become obsolete due to natural deterioration or technology innovation. We aim at characterizing optimal capacity expansion strategies and comparing the profit functions as well as the optimal control policies of different options. Specifically, we show that the optimal capacity expansion policy for the current period is determined by a base-stock policy. Compared with the case where no future contracts are available, the optimal control parameters of capacity expansion are always smaller. We further show that when the obsolescence rate is deterministic, the optimal policy for capacity expansion through future contracts is also a base-stock type. The results are extended to the cases with stochastically dependent capacity supply limits and stochastically dependent demand processes, which establish the robustness of the optimal policy in various market conditions.

Dynamic release policies for software systems with a reliability constraint

We discuss the optimal release problem of computer software. A conditional non-homogeneous Poisson process model is used to describe the software reliability growth behavior. By formulating with Markov decision programming, we show that, to minimize the total discounted cost, the optimal release policy is threshold-type, which is easy to obtain and to implement. It is then extended to the model with a constraint on the system reliability, for which a similar threshold-type control policy is proved to be optimal too.

Periodic-review inventory systems with random yield and demand: Bounds and heuristics

We revisit the infinite-horizon decision problem of a single-stage single-item periodic-review inventory system under uncertain yield and demand. It is known that under some mild conditions the optimal replenishment policy is of the threshold type: an order is placed if and only if the starting inventory is below a threshold value. Although the structure of the optimal policy is well known, there has been little discussion about the optimal order quantities and the order threshold. In this paper, we construct upper and lower bounds for the optimal threshold value and the optimal order quantities through solving one-period problems with different cost parameters. These bounds provide interesting insights into the impact of yield uncertainty on the optimal policy. Heuristics are developed based on these bounds. Detailed computational studies show that, under some conditions, the performance of the heuristics is very close to that of the optimal solution and better than that of existing heuristics in the literature.

Coordinated quality control in a two-stage system

The authors study a quality control problem in a two-stage system. They assume that at each stage units are processed in batches, and the rates are random variables with known distributions. Final products are supplied to customers under warranties or service contracts, with penalty costs associated with defective units. The focus is on coordinating the inspection procedures at the two stages. Using a stochastic dynamic programming approach, the authors show that the optimal policy at stage 1 is characterized by a sequence of thresholds, and at stage 2, by a priority structure, as well as a threshold structure. The key to optimality is a so-called K-submodularity property, which is a strengthening of the usual notion of submodularity.

Quality Control for Products Supplied with Warranty

A batch of products is to be supplied to customers with warranty. The units in the batch are either defective or nondefective, with different lifetime distributions. The defect rate-the proportion of defects in the batch-is itself a random variable, known only in terms of its distribution. We develop a sequential quality control procedure that exploits the knowledge of the defect distribution gained through inspection, and strikes an optimal balance between the inspection repair cost and the warranty cost. We identify a simple threshold policy, and we prove its optimality for a very general class of warranty cost functions without imposing any restrictions on the type of distributions involved. The key to optimality is that the warranty cost, as a function of the number of inspected units and the conditional defect index, satisfies a so-called K-submodularity property, which is a strengthening of the usual notion of submodularity.

Sequential Inspection Under Capacity Constraints

We study the inspection process in the context of multistage batch manufacturing, focusing on interstage coordination under capacity limits. The problem is formulated as a constrained Markov decision program. We establish the optimality of a sequential policy that is characterized by a sequence of thresholds, with certain randomization at the thresholds. We further show that such an optimal policy can be completely derived through solving a linear program, and that randomization is needed at no more than two threshold values. We discuss an application in semiconductor wafer fabrication, which motivates our study.