Optimal maintenance center inventories for fault-tolerant repairable systems

In many military and commercial contexts, complex equipment which is expected to perform very reliably is often designed to be fault-tolerant, that is, able to function although some of the parts have failed. A popular fault-tolerant design is the m-out-of-n system, where there are n identical parts, at least m of which must be functional for machine operation. Complex equipment of this type often undergoes scheduled maintenance overhauls at regular intervals during which all failed components are replaced. Failure to have replacements on hand for failed parts requires emergency measures at premium cost. When repairable parts are highly reliable and expensive, both holding and shortage costs are high. A reasonable objective is to choose initial spares inventory to minimize the sum of holding costs and expected shortage costs.We first develop a model to determine the optimal repairable parts inventory for a maintenance center servicing machines containing a single m-out-of-n system. The model is then extended to handle a related problem, finding optimal maintenance center inventories for machines containing several m-out-of-n systems of different parts, minimizing total expected costs subject to a constraint on total inventory investment.We assume that there is a fleet of machines, which experience identical workloads. There is a cycle time of T days between overhauls for an individual machine. A machine arrives at the maintenance center for overhaul each day. At the overhaul, all failed parts are removed and sent to a repair shop, from which they eventually return to the maintenance center to be used again as spares. The total number of spares undergoing repair and on hand is a constant. There are no backorders; if the number on-hand spares is insufficient to meet demand at an overhaul, a shortage penalty is assessed which depends on the number and type of spares required.While computing holding costs is straightforward, computing expected shortage costs is more complex. Expected shortage costs are dependent upon several factors, including component failure rates, the values of m and n, part repair rates, and the initial number of spares on hand. We assume that the system of interest is well specified, so that the parameters of the model are known except for the number of initial spares of each type, which are the decision variables. We model the on-hand inventory of each type of part as a Markov chain with the number of spares on hand at the end of each day as the states, under the assumptions that failure rates are constant and repair times follow independent exponential distributions. We then calculate the steady-state probabilities of stockout of various numbers of spares, as a function of the initial spares inventory. The expected shortage costs for a given type of spare may then be calculated by finding the product of the penalty cost for lacking p spares and the probability of lacking p spares and summing over all possible p values.Solutions to the problem of finding optimal initial inventory level for a machine containing a single m-out-of-n system may be found easily by enumeration. Solutions to the constrained problem where the machine contains several independent m-out-of-n systems, may be found by dynamic programming. Sensitivity analysis of costs to changes in the inventory investment constraint is clear, and computational effort is reasonable. A simple example is included to illustrate the solution method for both problems.     

Managing inventories for maximum benefit.

In many companies it is not feasible to go to just-in-time (JIT) inventory management practices. In these instances, an aggressive inventory-monitoring program is a must. This article describes a program that resulted in a 75-percent reduction of inventory over a five-year interval. The topics covered will be applicable to those companies where JIT is not the answer.     

Dispersive ordering of fail-safe systems with heterogeneous exponential components

Let X ₁, . . . , X _n be independent exponential random variables with respective hazard rates λ₁, . . . , λ _n , and Y ₁, . . . , Y _n be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X _2:n , the second order statistic from X ₁, . . . , X _n , is larger than Y _2:n , the second order statistic from Y ₁, . . . , Y _n , in terms of the dispersive order if and only if

$\lambda\geq \sqrt{\frac{1}{{n\choose 2}}\sum_{1\leq i < j\leq n}\lambda_i\lambda_j}.$

We also show that X _2:n is smaller than Y _2:n in terms of the dispersive order if and only if

$ \lambda\le\frac{\sum^{n}_{i=1} \lambda_i-{\rm max}_{1\leq i\leq n} \lambda_i}{n-1}. $

Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993–1997, 2008), Zhao et al. (J Multivar Anal 100:952–962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027–3037, 2009), respectively.

     

On the residual lifetime of coherent systems with heterogeneous components

The residual lifetime is of significant interest in reliability and survival analysis. In this article, we obtain a mixture representation for the reliability function of the residual lifetime of a coherent system with heterogeneous components in terms of the reliability functions of residual lifetimes of order statistics. Some stochastic comparisons are made on the residual lifetimes of the systems. Some examples are also given to illustrate the main results.     

Stochastic comparisons of distorted distributions,coherent systems and mixtures with ordered components

A distribution function F is a generalized distorted distribution of the distribution functions \(F_1,\ldots ,F_n\) if \(F=Q(F_1,\ldots ,F_n)\) for an increasing continuous distortion function Q such that \(Q(0,\ldots ,0)=0\) and \(Q(1,\ldots ,1)=1\). In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions \(F_1,\ldots ,F_n\) are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1–3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures.     

Optimal design for interacting blocks with OAVS incidence

Optimal design is studied for multiway cross classifications in which the blocking factors exhibit interaction and in which the incidence structure of the blocking factors describes an orthogonal array. Assuming the orthogonal array to be of sufficient strength (enough to allow orthogonal estimation of all of the interaction terms), easily used forms of the information matrix for treatment estimation are derived, and optimality conditions are stated. Some illustrative construction examples are given.     

Optimal investment for executive stockholders with exponential utility

The scope of this paper is to enhance the model for the own-company stockholder (Desmettre et?al. in Math Methods Oper Res 72(3):347?C378, 2010), who can voluntarily performance-link his personal wealth to his management success by acquiring stocks in the own-company whose value he can directly influence via spending work effort. The executive is thereby characterized by a parameter of risk aversion and the two work effectiveness parameters inverse work productivity and disutility stress. We extend the model to a constant absolute risk aversion framework using an exponential utility/disutility setup. A closed-form solution is given for the optimal work effort an executive will apply and we derive the optimal investment strategies of the executive. Furthermore, we determine an up-front fair cash compensation applying an indifference utility rationale. Our study shows to a large extent that the results previously obtained are robust under the choice of the utility/disutility setup.     

Optimal inference for instrumental variables regression with non-Gaussian errors

This paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor, nonrandom exogenous regressors and instruments, and i.i.d. errors whose distribution is unknown. It is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are “nearly” efficient in the sense of Andrews et al. (2006) when identification is weak and consistent and asymptotically optimal when identification is strong. In addition, an estimator is presented which can be used in the usual way to construct valid (indeed, optimal) confidence intervals when identification is strong. The estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal.     

Optimal procurement mechanisms for divisible goods with capacitated suppliers

Procurement auction literature typically assumes that the suppliers are uncapacitated [see, e.g. Dasgupta and Spulber in Inf Econ Policy 4:5–29, 1990 and Che in Rand J Econ 24(4):668–680, 1993]. Consequently, the auction mechanisms award the contract to a single supplier. We study mechanism design in a model where suppliers have limited production capacity, and both the marginal costs and the production capacities are private information. We provide a closed-form solution for the revenue maximizing direct mechanism when the distribution of the cost and production capacities satisfies a modified regularity condition [Myerson in Math Oper Res 6(1):58–73, 1981]. We also present a sealed low bid implementation of the optimal direct mechanism for the special case of identical suppliers. The results in this paper extend to other principle-agent mechanism design problems where the agents have a privately known upper bound on allocation. The authors would like to thank the anonymous referees for valuable suggestions and comments.     

随机需求下一类变质性产品的生产库存最优策略

建立了计划期需求量是随机变量、生产系统不稳定的变质性产品生产库存模型.通过计划期总成本最小化来寻求最优生产时间,最后给出了相应的算例.     

Optimal strategy for a fund manager with option compensation

I consider the problem of portfolio optimization for a manager whose compensation is given by the sum of a constant and a variable term. The constant term is a fixed percentage of the managed funds that is payed to the manager independently of his performance. The variable term is a premium that is proportional to the profit earned by the manager over a benchmark at a certain evaluation date. I find the optimal strategy and the optimal portfolio value in the Black–Scholes setting when the benchmark is a linear combination of the risky asset and the money market account.     

Optimal designs for generalized linear models with biased response

This paper is concerned with the search for locally optimal designs when the observations of the response variable arise from a weighted distribution in the exponential family. Locally optimal designs are derived for regression models in which the response follows a weighted version of Normal, Gamma, Inverse Gaussian, Poisson or Binomial distributions. Some conditions are given under which the optimal designs for the weighted and original (non-weighted) distributions are the same. An efficiency study is performed to find out the behavior of the D-optimal designs for the original distribution when they are used to estimate models with weighted distributions.     

需求依赖卖场库存下变质性产品的最优补充策略

讨论了需求是卖场库存函数的变质性产品库存模型,假设卖场的库存容量有限,需求依赖卖场库存,建立了该库存模型,证明了最优补充策略的存在性,并给出了相应的算法和算例。     

Simultaneous equations with error components

In this paper we derive a limited as well as a full information estimator for the structural parameters of a simultaneous equations model with error components. Under this model, the gain in efficiency by performing these estimators rather than the classical two-stage and three-stage least squares procedures is demonstrated. It is shown that the full information estimator will reduce to the limited information estimator when the disturbances of different structural equations are uncorrelated with each other but not necessarily when all structural equations are just identified. This is different from the analogous situation in the classical case.     

Optimal designs for models with block-block resp. treatment-treatment correlations

Summary In this paper we study optimal designs assuming two special covariance structures of the observations, namely that the covariance between the observations depends only on the blocks resp. the treatments. We show that the weighted least squares estimator equals the ordinary least squares estimator. Then we prove that block-block correlations resp. treatment-treatment correlations do not have any influence on the A- and MV-optimality resp. A-optimality. For the study of the MV-optimality in the case of treatment-treatment correlations we use the idea of invariance to find optimal C-matrices.     

Optimal consumption choice with intolerance for declining standard of living

We reconsider the optimal consumption choice of investors who do not tolerate any decline in their consumption rate. We connect the demand behavior of such agents to the behavior of standard time-additive agents. With consumption ratcheting, the investor demands the running maximum of the optimal plan a conventional time-additive investor with lower initial wealth would choose. The analysis is carried out for Lévy processes, in particular for jump-diffusions. We obtain explicit solutions for all Bernoulli utility functions, not only those of CRRA type.