Abstract
Motivated by production systems with nonstationary stochastic demand, we study a double-ended queueing model having back orders and customer abandonment. One side of our model stores back orders, and the other side represents inventory. We assume first-come-first-served instantaneous fulfillment discipline. Our goal is to determine the optimal (nonstationary) production rate over a finite time horizon to minimize the costs incurred by the system. In addition to the inventory-related (holding and perishment) and demand-related (waiting and abandonment) costs, we consider a cost that penalizes rapid fluctuations of production rates. We develop a deterministic fluid-control problem (FCP) that serves as a performance lower bound for the original queueing-control problem (QCP). We further consider a high-volume system for which an upper bound of the gap between the optimal values of the QCP and FCP is characterized and construct an asymptotically optimal production rate for the QCP, under which the FCP lower bound is achieved asymptotically. Demonstrated by numerical examples, the proposed asymptotically optimal production rate successfully captures the time variability of the nonstationary demand.
Highlights
One of the fundamental challenges that precludes desired dynamic behavior of stochastic systems is that of nonstationarity
We develop a deterministic fluid-control problem (FCP) that serves as a performance lower bound for the original queueing-control problem (QCP)
We develop simple effective algorithms based on a linear programming (LP) formulation to numerically solve the FCP and conduct various simulations to demonstrate the effectiveness of the proposed asymptotically optimal production rate for different system sizes
Summary
One of the fundamental challenges that precludes desired dynamic behavior of stochastic systems is that of nonstationarity. Our goal is to obtain an optimal (nonstationary) production rate over a finite-time production-planning horizon to minimize the costs incurred by the system These costs include the production cost; inventoryrelated costs, consisting of holding and perishment costs; demand-related costs, consisting of waiting and abandonment costs; and a cost that penalizes the rapid fluctuations of production rates, measured by the total variation of the production-rate function. We show in Theorem 4.2 that the optimal value of the FCP serves asymptotically as a performance lower bound to the fluid-scaled QCP as a scaling factor n (which measures the “size” of the system—for example, average product demand rate) grows large. We develop simple effective algorithms based on a linear programming (LP) formulation to numerically solve the FCP and conduct various simulations to demonstrate the effectiveness of the proposed asymptotically optimal production rate for different system sizes. There are two streams of literature that are related to our work
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