Minimum Long-run Average Cost Research Articles

Online Risk-Averse Resource Allocation in Queuing Networks

In this article, we address the online resource allocation problem in service queuing systems under uncertainty. In particular, the optimal control policy is derived by using the real-time data, i.e., without fully knowing information about the system state. We employ the conditional value-at-risk to achieve the minimum long-run average cost subject to the constraints on the risk of instability. Then, we can ensure the service ability of the system at a high level under such on-the-fly uncertainty. We show that the proposed model naturally leads to a minmax saddle point optimization problem. We first present an intuitive offline primal–dual learning method, which can achieve a desirable convergence rate. Then, we further improve the algorithm by learning the optimal Lagrange multiplier concerning the instantaneous system state, i.e., online primal–dual learning algorithm, to achieve the same convergence rate as the offline algorithm. Besides, we demonstrate that the proposed method outperforms classical models in providing the lower average delay under uncertainty, which means it can improve the stability and provide a better quality of service under the ambiguity of uncertainty. A real case of inpatient bed allocation for hospital operations is presented to show the performance in applications. The results show that our model can reduce the average wait with certain total available beds under uncertain arrivals.

Optimal Condition-Based Maintenance Strategy for Multi-Component Systems under Degradation Failures

This paper proposes a condition-based maintenance strategy for multi-component systems under degradation failures. The maintenance decision is based on the minimum long-run average cost rate (LACR) and the maximum residual useful lifetime (RUL), respectively. The aim of this paper is to determine the optimal monitoring interval and critical level for multi-component systems under different optimization objectives. A preventive maintenance (PM) is triggered when the degradation of component exceeds the corresponding critical level. Afterwards, the paper discusses the relationship between the critical level and the monitoring interval with regards to the LACR and RUL. Methods are also proposed to determine the optimal monitoring interval and the critical level under two decision models. Finally, the impact of maintenance decision variables on the LACR and RUL is discussed through a case study. A comparison with conventional maintenance policy shows an outstanding performance of the new model.

On the Minimum Pair Approach for Average Cost Markov Decision Processes with Countable Discrete Action Spaces and Strictly Unbounded Costs

We consider average-cost Markov decision processes (MDPs) with Borel state spaces, countable, discrete action spaces, and strictly unbounded one-stage costs. For the minimum pair approach, we introduce a new majorization condition on the state transition stochastic kernel, in place of the commonly required continuity conditions on the MDP model. We combine this majorization condition with Lusin's theorem to prove the existence of a stationary minimum pair, i.e., a stationary policy paired with an invariant probability measure induced on the state space, with the property that the pair attains the minimum long-run average cost over all policies and initial distributions. We also establish other optimality properties of a stationary minimum pair, and for the stationary policy in such a pair, under additional recurrence or regularity conditions, we prove its pathwise optimality and strong optimality. Our results can be applied to a class of countable action space MDPs in which the dynamics and one-stage costs are discontinuous with respect to the state variable.

Availability analysis and inspection optimisation for a competing-risk k-out-of-n: G system

The availability and cost rate for a maintained k-out-of-n: G system encountering multiple failure modes undergoing periodic inspection are studied in this paper. Explicitly, a k-out-of-n: G system with a functional state and N failure modes is taken into account. Failure time of each failure mode is random. At each inspection and on failure detection, perfect repairs are carried out. Some theorems on the point availability, limiting availability and long-run average cost rate are obtained in the study. A condition for determining the optimal inspection period for minimum long-run average cost rate is also studied. An illustration is taken to explain the results obtained in the proposed work.

Modeling and Optimization of Stochastic Joint Replenishment and Delivery Scheduling Problem with Uncertain Costs

The stochastic joint replenishment and delivery scheduling (JRD) problem is a key issue in supply chain management and is a major concern for companies. So far, all of the work on stochastic JRDs is under explicit environment. However, the decision makers often have to face vague operational conditions. We develop a practical JRD model with stochastic demand under fuzzy backlogging cost, fuzzy minor ordering cost, and fuzzy inventory holding cost. The problem is to determine procedures for inventory management and vehicle routing simultaneously so that the warehouse may satisfy demand at a minimum long-run average cost. Subsequently, the fuzzy total cost is defuzzified by the graded mean integration representation and centroid approaches to rank fuzzy numbers. To find optimal coordinated decisions, a modified adaptive differential evolution algorithm (MADE) is utilized to find the minimum long-run average total cost. Results of numerical examples indicate that the proposed JRD model can be used to simulate fuzzy environment efficiently, and the MADE outperforms genetic algorithm with a lower total cost and higher convergence rate. The proposed methods can be applied to many industries and can help obtaining optimal decisions under uncertain environment.

Optimal Production Planning in a Multi-Product StochasticManufacturing System with Long-Run Average Cost

This paper is concerned with the problem of production planning in a flexible manufacturing system consisting of a single or parallel failure-prone machines producing a number of different products. The objective is to choose the rates of production of the various products over time in order to meet their demands at the minimum long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach for the average cost problem that the Hamilton-Jacobi-Bellman equation in terms of directional derivatives has a solution consisting of the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem, and in specifying an optimal control policy in terms of the potential function. The results settle a hitherto open problem as well as generalize known results.

A generalized cost model for stochastic clearing systems

A stochastic clearing system is a stochastic input-output system in which the output occurs periodically in the form of clearing operations that instantaneously clear the total content of the system. In this paper a generalized cost model for the system is assumed in which the cost of clearing the system is C 1 + C 2 q , where q is the quantity cleared, and C 1 and C 2 are positive constants. An equation is derived whose unique solution yields the optimal clearing level, and the resulting solutions are demonstrated using three example systems with renewal, compound Poisson and gamma input processes. The minimum long-run average cost of the stochastic clearing system is shown to be less than the optimal cost of an approximating EOQ deterministic system. Finally, the system is analyzed using a Brownian motion input process with positive drift under this cost model and an equation for the optimal clearing level is derived.