Xianyi Wu

Optimal acquisition and production policy in a hybrid manufacturing/remanufacturing system with core acquisition at different quality levels

We study the acquisition and production planning problem for a hybrid manufacturing/remanufacturing system with core acquisition at two (high and low) quality conditions. We model the problem as a stochastic dynamic programming, derive the optimal dynamic acquisition pricing and production policy, and analyze the influences of system parameters on the acquisition prices and production quantities. The production cost differences among remanufacturing high- and low-quality cores and manufacturing new products are found to be critical for the optimal production and acquisition pricing policy: the acquisition price of high-quality cores is increasing in manufacturing and remanufacturing cost differences, while the acquisition price of low-quality cores is decreasing in the remanufacturing cost difference between high- and low-quality cores and increasing in manufacturing and remanufacturing cost differences; the optimal remanufacturing/manufacturing policy follows a base-on-stock pattern, which is characterized by some crucial parameters dependent on these cost differences.

Stochastic scheduling to minimize expected maximum lateness

This paper is concerned with the problems in scheduling a set of jobs associated with random due dates on a single machine so as to minimize the expected maximum lateness in stochastic environment. This is a difficult problem and few efforts have been reported on its solution in the literature. In this paper, we first derive a deterministic equivalent to the expected maximum lateness and then propose a dynamic programming algorithm to obtain the optimal solutions. The procedures to compute optimal solutions are initially developed in the case of deterministic processing times, and then extended to stochastic processing times following arbitrary probability distributions. Moreover, several heuristic rules are suggested to compute near-optimal solutions, which are shown to be highly efficient and accurate by computer-based experiments.

Dynamically optimal policies for stochastic scheduling subject to preemptive-repeat machine breakdowns

We consider the problem of finding a dynamically optimal policy to process n jobs on a single machine subject to stochastic breakdowns. We study the preemptive-repeat breakdown model, i.e., if a machine breaks down during the processing of a job, the work done on the job prior to the breakdown is lost and the job will have to be started over again. Our study is built on a general setting, which allows: 1) the uptimes and downtimes of the machine to follow general probability distributions, not necessarily independent of each other; 2) the breakdown process to depend upon the job being processed; and 3) the processing times of the jobs to be random variables following arbitrary distributions. We consider two possible cases for the processing time of a job interrupted by a breakdown: a) it is resampled according to its probability distribution or b) it is the same random variable as that before the breakdown. We introduce the concept of occupying time and find its Laplace and integral transforms. For the problem with resampled processing times, we establish a general optimality equation on the optimal dynamic policy under a unified objective measure. We deduce from the optimality equation the optimal dynamic policies for several problems with well-known criteria, including weighted discounted reward, weighted flowtime, truncated cost, number of tardy jobs under stochastic order, and maximum holding cost. For the problem with same random processing time, we develop the optimal dynamic policy via the theory of bandit process. A set of Gittins indices are derived that give the optimal dynamic policies under the criteria of weighted discounted reward and weighted flowtime. Note to Practitioners-It is common in practice that a machine is subject to breakdowns, which may severely interrupt the job it is processing. In such a situation, there may be limited information on the breakdown patterns of the machine and the processing requirements of the jobs. A great challenge faced by the decision-maker is how to utilize the information available to make a right decision. Stochastic scheduling considering stochastic machine breakdowns aims to determine the optimal policies in these situations. In this paper, we study the problem within the preemptive-repeat breakdown framework, to address the practical situations where a job will have to be re-started again if a machine breakdown occurs when it is being processed. Problems of such can be found in many industrial applications. Examples include refining metal in a refinery factory, running a program on a computer, performing a reliability test on a facility, etc. Generally, if a job must be continuously processed with no interruption until it is totally completed, then the preemptive-repeat breakdown formulation should be used. Our research in this paper focuses on optimal dynamic policies, which aim to utilize real-time information to dynamically adjust/improve a decision. We consider two types of models, depending on whether the processing time of the job interrupted by a breakdown must be resampled or not. For the problem with resampled processing times, we establish a general optimality equation under a unified objective measure. We further deduce the optimal dynamic policies under a number of well-known criteria. For the problem without resampled processing times, we develop the optimal dynamic policies, under the criteria of weighted discounted reward and weighted flowtime. Broadly speaking, our findings can be applied in any situations where it is desirable to derive the best dynamic decisions to tackle the problem with stochastic machine breakdowns and preemptive-repeat jobs.

Stochastic Scheduling Subject to Preemptive-Repeat Breakdowns with Incomplete Information

This paper considers the problem of scheduling a set of jobs on a single machine subject to stochastic breakdowns with incomplete information on the probability distributions involved in the decision process. We focus on the preemptive-repeat discipline, under which a machine breakdown leads to the loss of the work done on the job being processed. The breakdown process of the machine is allowed to depend on the job it is processing. The processing times required to complete the jobs, and the machine uptimes and downtimes, are random variables with incomplete information on their probability distributions characterized by unknown parameters. We establish the preemptive-repeat model with incomplete information and investigate its probabilistic characteristics. We show that optimal static policies can be obtained for a wide range of performance measures, which are determined by the prior distributions of the unknown parameters. We derive optimal dynamic policies via Gittins indices represented by the posterior distributions, which are updated adaptively based on processing histories. Under appropriate conditions, the optimal dynamic policies can be calculated by one-step reward rates in a closed form. As a by-product, we also show that our incomplete information model subsumes the traditional preemptive-repeat models with complete information as extreme cases.

Scheduling deteriorating jobs on a single machine subject to breakdowns

We investigate the problem of scheduling a set of jobs to minimize the expected makespan or the variance of the makespan. The jobs are subject to deteriorations which are expressed as linear increments of the processing requirements. The machine is subject to preemptive-resume breakdowns with exponentially distributed uptimes and downtimes. It has been well known in the classical models that the expectation and variance of the makespan of deteriorating jobs can be minimized analytically by an index policy if no machine breakdowns are involved. Such basic features, however, change dramatically when breakdowns and deteriorations are present together. In this paper, we derive conditions for jobs to be processible in the sense that they will be eventually completed, and the characteristics of the time that a job occupies the machine. We further find that the expected makespan can still be minimized by a simple index policy that is independent of the breakdown process, but this is no longer the case for the variance of the makespan.

ON CHARACTERIZATION OF DISTORTION PREMIUM PRINCIPLE

In this paper, based on the additive measure integral representation of a nonadditive measure integral, it is shown that any comonotonically additive premium principle can be represented as an integral of the distorted decumulative distribution function of the insurance risk. Furthermore, a sufficient and necessary condition that a premium principle is a distortion premium principle is given.

Stochastic scheduling problems with general position-based learning effects and stochastic breakdowns

The focus of this study is to analyze position-based learning effects in single-machine stochastic scheduling problems. The optimal permutation policies for the stochastic scheduling problems with and without machine breakdowns are examined, where the performance measures are the expectation and variance of the makespan, the expected total completion time, the expected total weighted completion time, the expected weighted sum of the discounted completion times, the maximum lateness and the maximum tardiness.

The Credibility Estimator with General Dependence Structure Over Risks

In almost all credibility models considered previously, the claims are assumed to be independent over risks. However, from the practical point of view, it is not always the case, and hence, in this article, we investigate the credibility premiums when risks are allowed to be generally dependent. Firstly, we re-build the credibility estimators for Bühlmann and Bühlmann–Straub credibility models under general dependence structure over risks. The methods are then extended to the regression credibility models.

Asymptotic behaviors of stochastic reserving : aggregate versus individual models

In this paper, we investigate the asymptotic behaviors of the loss reservings computed by individual data method and its aggregate data versions by Chain-Ladder (CL) and Bornhuetter–Ferguson (BF) algorithms. It is shown that all deviations of the three reservings from the individual loss reserve (the projection of the outstanding liability on the individual data) converge weakly to a zero-mean normal distribution at the nrate. The analytical forms of the asymptotic variances are derived and compared by both analytical and numerical examples. The results show that the individual method has the smallest asymptotic variance, followed by the BF algorithm, and the CL algorithm has the largest asymptotic variance.

Credibility Estimation of Distribution Functions with Applications to Experience Rating in General Insurance

This article presents a new credibility estimation of the probability distributions of risks under Bayes settings in a completely nonparametric framework. In contrast to the Fergusons Bayesian nonparametric method, it does not need to specify a mathematical form of the prior distribution (such as a Dirichlet process). We then show the applications of the method in general insurance premium pricing, a procedure commonly known as experience rating, which utilizes the insureds claim experience to calculate a proper premium under a given premium principle (referred to as a risk measure). As this method estimates the probability distributions of losses, not just the means and variances, it provides a unified nonparametric framework to experience rating for arbitrary premium principles. This encompasses the advantages of the well-known Bühlmanns and Fergusons approaches, while it overcomes their drawbacks. We first establish a linear Bayes method and prove its strong consistency in nonparametric settings that require only knowledge of the first two moments of the loss distributions considered as a stochastic process. Then an empirical Bayes method is developed for the more general situation where a portfolio of risks is observed but no knowledge is available or assumed on their loss and prior distributions, including their moments. It is shown to be asymptotically optimal. The performance of our estimates in comparison with traditional methods is also evaluated through theoretical analysis and numerical studies, which show that our approach produces premium estimates close to the optima.