Optimal maintenance of a production-inventory system with idle periods

In this paper we consider a production-inventory system in which an input generating installation supplies a buffer with a raw material and a production unit pulls the raw material from the buffer with constant rate. The installation deteriorates in time and the problem of its optimal preventive maintenance is considered. It is assumed that the installation after the completion of its maintenance remains idle until the buffer is evacuated. Under a suitable cost structure it is shown that the average-cost optimal policy for fixed buffer content is of control-limit type, i.e. it prescribes a preventive maintenance of the installation if and only if its degree of deterioration is greater than or equal to a critical level. Using the usual regenerative argument, the average cost of a control-limit policy is computed exactly and then, the optimal control-limit policy is determined. Furthermore, the stationary probabilities of the system under the optimal policy are computed.     

Computation of the Optimal Policy for the Control of a Compound Immigration Process through Total Catastrophes

In this paper we consider a Markov decision model introduced by Economou (2003), in which it was proved that the optimal policy in the problem of controlling a compound immigration process through total catastrophes is of control-limit type. We show that the average cost of a control-limit policy is unimodal as a function of the critical point. This result enables us to design very efficient algorithms for the computation of the optimal policy as the bisection procedure and a special-purpose policy iteration algorithm that operates on the class of control-limit policies.AMS 2000 Subject Classification: Primary 9OC40; Secondary 6OJ25     

Optimal preventive maintenance of a production system with an intermediate buffer

In this paper we consider a model consisting of a deteriorating installation that transfers a raw material to a production unit and a buffer which has been built between the installation and the production unit. The deterioration process of the installation is considered to be nonstationary, i.e. the transition probabilities may depend not only on the working conditions of the installation but on its age as well. The problem of the optimal preventive maintenance of the installation is considered. Under a suitable cost structure it is shown that, for fixed age of the installation and fixed buffer level, the optimal policy is of control-limit type. When the deterioration process is stationary, an efficient Markov decision algorithm operating on the class of control-limit policies is developed. There is strong numerical evidence that the algorithm converges to the optimal policy. Two generalizations of this model are also discussed.     

Joint optimisation of maintenance and buffer size in a manufacturing system

The objective of this work is to jointly optimise the maintenance of a capacity-constrained resource, its feed machine/operation and inlet buffer size. The maintenance of the capacity-constrained resource is condition-based, whilst that of the inlet machine/operation is time-based. The joint optimisation is achieved by the development of a mixed integer linear programming model. A machining example is used to illustrate the application of the model.     

Optimal admission control in queues with abandonments

We consider a multi-sever Markovian queueing system with abandonments where admitted customers pay a reward either at the time of arrival or service completion. There is a cost associated with abandonments and a holding cost associated with customers in the system. We prove that the policy that maximizes the long-run average reward is of threshold type and completely characterize the optimal thresholds. We conclude with a comparison of various characteristics of the two variants of the model.     

Optimal control in light traffic Markov decision processes

We consider Markov Decision Processes under light traffic conditions. We develop an algorithm to obtain asymptotically optimal policies for both the total discounted and the average cost criterion. This gives a general framework for several light traffic results in the literature. We illustrate the method by deriving the asymptotically optimal control of a simple ATM network.     

Optimal replacement policies for two-unit machines with running costs: II

In a previous paper [1] we have proved the optimality of a control limit policy for certain two-unit systems with running costs. Under such a policy when either of theunits fails we also replace the other unit if its age exceeds a predetermined control limit.In this work we develop a procedure for computing various important characteristics for any control limit policy including, of course, the optimal one. Some of the resulting characteristics can then be used for a restricted optimization within the class of control limit policies. The fact that this restricted optimization leads to the global optimal is implied by the previous paper. In this sense the two approaches are complementary.     

Optimal control of an inventory-production system with state-dependent random yield

In this paper we study a single stage, periodic-review inventory problem for a single item with stochastic demand. The inventory manager determines order sizes according to an order-up-to logic and observes a random yield due to quality problems in the production. We distinguish between two different states of the production process combined with different probabilities to produce a defective unit. In order to improve the production process, periodic inspections are conducted and in case of a failure the machine is repaired. Approximations are developed to evaluate the average cost for a given order-up-to level and a given inspection interval and we illustrate the existence of optimal policy parameters. The approximations are tested in a simulation study and reveal an excellent performance as they lead to near optimal policy parameters. Moreover, we decompose the problem and test different methods to compute the policy parameters either sequentially or separately. Our results show that a joint optimization of the inventory and maintenance policy leads to a better system performance and reduced costs.     

Optimal preventive maintenance for equipment with two quality states and general failure time distributions

We present an economic model for the optimization of preventive maintenance in a production process with two quality states. The equipment starts its operation in the in-control state but it may shift to the out-of-control state before failure or scheduled preventive maintenance. The time of shift and the time of failure are generally distributed random variables. The two states are characterized by different failure rates and revenues. We first derive the structure of the optimal maintenance policy, which is defined by two critical values of the equipment age that determine when to perform preventive maintenance depending on the actual (observable) state of the process. We then provide properties of the optimal solution and show how to determine the optimal values of the two critical maintenance times accurately and efficiently. The proposed model and, in particular, the behavior of the optimal solution as the model parameters and the shift and failure time distributions change are illustrated through numerical examples.     

On the optimality of a maintenance queueing system

We consider a finite single-server maintenance queue with multiple types of customers. The difference between customers' types is defined by the offered rewards. We show that the optimal admission control policy for maximizing the long-run average reward per unit time has a trunk reservation structure. Meanwhile, if the equipment is off, there exists a threshold of the queue length, above which the optimal repair speed is increasing in the queue length and below which the optimal repair speed is 0.     

Average cost Markov decision processes under the hypothesis of Doeblin

Average cost Markov decision processes (MDPs) with compact state and action spaces and bounded lower semicontinuous cost functions are considered. Kurano [7] has treated the general case in which several ergodic classes and a transient set are permitted for the Markov process induced by any randomized stationary policy under the hypothesis of Doeblin and showed the existence of a minimum pair of state and policy. This paper considers the same case as that discussed in Kurano [7] and proves some new results which give the existence theorem of an optimal stationary policy under some reasonable conditions.     

Optimal threshold policies in a workload model with a variable number of service phases per job

We consider an M|E _N |1 queue in which there are two essential on-line decisions that have to be taken. The first one is the decision to either accept or reject new jobs. The second one is the decision to either continue or abort the service of a job. We show that under certain regularity conditions, there exist optimal threshold policies for these two types of decisions.     

Optimal threshold policies in a two-class preemptive priority queue with admission and termination control

We consider a two-class 1 preemptive priority queue in which there are two essential, on-line decisions that have to be taken. The first is the decision to either accept or reject new type-1 or type-2 jobs. The second is the decision to abort jobs, i.e., to remove any type-1 or type-2 jobs from the system. We show that there exist optimal threshold policies for these two types of, decisions.     

Optimal corrective maintenance contract planning for aging multi‐state system

This paper considers an aging multi‐state system, where the system failure rate varies with time. After any failure, maintenance is performed by an external repair team. Repair rate and cost of each repair are determined by a corresponding corrective maintenance contract with a repair team. The service market can provide different kinds of maintenance contracts to the system owner, which also can be changed after each specified time period. The owner of the system would like to determine a series of repair contracts during the system life cycle in order to minimize the total expected cost while satisfying the system availability. Operating cost, repair cost and penalty cost for system failures should be taken into account. The paper proposes a method for determining such optimal series of maintenance contracts. The method is based on the piecewise constant approximation for an increasing failure rate function in order to assess lower and upper bounds of the total expected cost and system availability by using Markov models. The genetic algorithm is used as the optimization technique. Numerical example is presented to illustrate the approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Risk-sensitive average continuous-time Markov decision processes with unbounded rates

Abstract

In this paper we study the risk-sensitive average cost criterion for continuous-time Markov decision processes in the class of all randomized Markov policies. The state space is a denumerable set, and the cost and transition rates are allowed to be unbounded. Under the suitable conditions, we establish the optimality equation of the auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function. Then by a technique of constructing the appropriate approximating sequences of the cost and transition rates and employing the results on the auxiliary optimization problem, we show the existence of a solution to the risk-sensitive average optimality inequality and develop a new approach called the risk-sensitive average optimality inequality approach to prove the existence of an optimal deterministic stationary policy. Furthermore, we give some sufficient conditions for the verification of the simultaneous Doeblin condition, use a controlled birth and death system to illustrate our conditions and provide an example for which the risk-sensitive average optimality strict inequality occurs.     

On computing average cost optimal policies with application to routing to parallel queues

The Approximating Sequence Method for computation of average cost optimal stationary policies in denumerahle state Markov decision chains, introduced in Sennott (1994), is reviewed. New methods for verifying the assumptions are given. These are useful for models with multidimensional state spaces that satisfy certain mild structural properties. The results are applied to four problems in the optimal routing of packets to parallel queues. Numerical results are given for one of the models.     

Optimal control of price and production in an assemble-to-order system

We study the optimal control of an assembly system that produces one assembled-to-order final product with multiple made-to-stock components and sells it at variable price. It is shown that a threshold control on component production, product price, and product orders maximizes total discounted profit over an infinite horizon.     

On the optimal control of arrivals to a single queue with arbitrary feedback delay

We consider a problem of admission control to a single queue in discrete time. The controller has access to k step old queue lengths only, where k can be arbitrary. The problem is motivated, in particular, by recent advances in high-speed networking where information delays have become prominent. We formulate the problem in the framework of Completely Observable Controlled Markov Chains, in terms of a multi-dimensional state variable. Exploiting the structure of the problem, we show that under appropriate conditions, the multi-dimensional Dynamic Programming Equation (DPE) can be reduced to a unidimensional one. We then provide simple computable upper and lower bounds to the optimal value function corresponding to the reduced unidimensional DPE. These upper and lower bounds, along with a certain relationship among the parameters of the problem, enable us to deduce partially the structural features of the optimal policy. Our approach enables us to recover simply, in part, the recent results of Altman and Stidham, who have shown that a multiple-threshold-type policy is optimal for this problem. Further, under the same relationship among the parameters of the problem, we provide easily computable upper bounds to the multiple thresholds and show the existence of simple relationships among these upper bounds. These relationships allow us to gain very useful insights into the nature of the optimal policy. In particular, the insights obtained are of great importance for the problem of actually computing an optimal policy because they reduce the search space enormously. This revised version was published online in June 2006 with corrections to the Cover Date.     

非一致有界费用MDP的强平均最优性条件

研究可数状态空间任意行动空间非一致性有界费用马氏决策过程(MDP)的强平均最优,给出了使得每个常用的平均最优策略也是强平均最优的条件,并实质性的推广了Cavazos-Cadena和Fernandez-Gaucheran(Math. Meth. Oper. Res., 1996, 43: 281-300)的主要结果.