Optimal burn‐in procedure for periodically inspected systems

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we study burn‐in procedure for a system that is maintained under periodic inspection and perfect repair policy. Assuming that the underlying lifetime distribution of a system has an initially decreasing and/or eventually increasing failure rate function, we derive upper and lower bounds for the optimal burn‐in time, which maximizes the system availability. Furthermore, adopting an age replacement policy, we derive upper and lower bounds for the optimal age parameter of the replacement policy for each fixed burn‐in time and a uniform upper bound for the optimal burn‐in time given the age replacement policy. These results can be used to reduce the numerical work for determining both optimal burn‐in time and optimal replacement policy. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Bounding the optimal burn‐in time for a system with two types of failure

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we consider the problem of determining bounds to the optimal burn‐in time and optimal replacement policy maximizing the steady state availability of a repairable system. It is assumed that two types of system failures may occur: One is Type I failure (minor failure), which can be removed by a minimal repair, and the other is Type II failure (catastrophic failure), which can be removed only by a complete repair. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, upper and lower bounds for the optimal burn‐in time are provided. Furthermore, some other applications of optimal burn‐in are also considered. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Bounds on optimal cost for a replacement problem with partial observations

This paper characterizes the structure of optimal strategies for a replacement problem for two special cases of observation quality. It is shown that when the state of the system is either completely observed or completely unobserved at every decision epoch by the controller, reasonable assumptions are sufficient for the existence of optimal replacement strategies composed of policies having a generalized, control-limit form. These structural results are of particular interest since the optimal cost functions for the two special cases represent bounds on the optimal cost function for the general partially observed case, significant computational simplification can result for the two special cases due to their optimal strategy structure, and optimal strategies possessing a control-limit structure do not necessarily exist for the general partially observed case.     

Combination setwise-Bonferroni-type bounds

We consider three classes of lower bounds to P(c) = P (X₁ ≤ c₁,…, X_n ≤ c); Bonferroni-type bounds, product-type bounds and setwise bounds. Setwise probability inequalities are shown to be a compromise between product-type and Bonferroni-type probability inequalities. Bonferroni-type inequalities always hold. Product-type inequalities require positive dependence conditions, but are superior to the Bonferroni-type and setwise bounds when these conditions are satisfied. Setwise inequalities require less stringent positive dependence bound conditions than the product-type bounds. Neither setwise nor Bonferroni-type bounds dominate the other. Optimized setwise bounds are developed. Results pertaining to the nesting of setwise bounds are obtained. Combination setwise-Bonferroni-type bounds are developed in which high dimensional setwise bounds are applied and second and third order Bonferroni-type bounds are applied within each subvector of the setwise bounds. These new combination bounds, which are applicable for associated random variables, are shown to be superior to Bonferroni-type and setwise bounds for moving averages and runs probabilities. Recently proposed upper bounds to P(c) are reviewed. The lower and upper bounds are tabulated for various classes of multivariate normal distributions with banded covariance matrices. The bounds are shown to be surprisingly accurate and are much easier to compute than the inclusion-exclusion bounds. A strategy for employing the bounds is developed. © 1996 John Wiley & Sons, Inc.     

Optimal spares for stochastically failing equipment

A mathematical model is formulated for determining the number of spare components to purchase when components stochastically fail according to a known life distribution function and there is a cost incurred when a component is replaced. Bounds are determined for the optimal inventory which indicate that the inclusion of the replacement cost lowers the optimal inventory. Since these bounds are no easier to calculate than the optimal spares level, the theory is specialized to components with exponentially distributed time to failure. Procedures are given for calculating the optimal spares level, and numerical examples are provided.     

Computing bounds for the optimal value in linear programming

Consider a standard linear programming problem and suppose that there are bounds available for the decision variables such that those bounds are not violated at an optimal solution of the problem (but they may be violated at some other feasible solutions of the problem). Thus, these bounds may not appear explicitly in the problem, but rather they may have been derived from some prior knowledge about an optimal solution or from the explicit constraints of the problem. In this paper, the bounds on variables are used to compute bounds on the optimal value when the problem is being solved by the simplex method. The latter bounds may then be used as a termination criteria for the simples iterations for the purpose of finding a “sufficiently good” near optimal solution. The bounds proposed are such that the computational effort in evaluating them is insignificant compared to that involved in the simplex iterations. A numerical example is given to demonstrate their performance.     

Preventive maintenance and replacement under additive damage

A system deteriorates due to shocks received at random times, each shock causing a random amount of damage which accumulates over time and may result in a system failure. Replacement of a failed system is mandatory, while an operable one may also be replaced. In addition, the shock process causing system deterioration may be controlled by continuous preventive maintenance expenditures. The joint problem of optimal maintenance and replacement is analyzed and it is shown that, under reasonable conditions, optimal maintenance rate is decreasing in the cumulative damage level and that beyond a certain critical level the system should be replaced. Meaningful bounds are established on the optimal policies and an illustrative example is provided.     

Probabilistic solution and bounds for serial inventory systems with discounted and average costs

We consider the infinite horizon serial inventory system with both average cost and discounted cost criteria. The optimal echelon base‐stock levels are obtained in terms of only probability distributions of leadtime demands. This analysis yields a novel approach for developing bounds and heuristics for optimal inventory control polices. In addition to deriving the known bounds in literature, we develop several new upper bounds for both average cost and discounted cost models. Numerical studies show that the bounds and heuristic are very close to optimal.© 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Reliability bounds for the NBUE distributions

In this paper, we study upper and lower bounds on the reliability in new better than used in expectation (NBUE) life distribution class with fixed first two moments. By a constructive proof, we determine the upper bounds on the reliability analytically in different regions and show that these bounds are sharp. For the lower bounds, similar results are obtained except in one region. For that region, a conjecture is given for further study. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 781–797, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10035     

On sequencing with earliest starts and due dates with application to computing bounds for the (n/m/G/Fmax) problem

Recent efforts to improve lower bounds in implicit enumeration algorithms for the general (n/m/G/F_max) sequencing problem have been directed to the solution of an auxiliary single machine problem that results from the relaxation of some of the interference constraints. We develop an algorithm that obtains optimal and near optimal solutions for this relaxed problem with relatively little computational effort. We report on computational results achieved when this method is used to obtain lower bounds for the general problem. Finally, we show the equivalence of this problem to a single machine sequencing problem with earliest start and due date constraints where the objective is to minimize the maximum lateness.     

Scheduling tasks with sequence-dependent processing times

In this article we consider the problem of minimizing the maximum completion time of a sequence of n jobs on a single machine. Nonzero ready times and sequence-dependent processing times are allowed. Upper bounds, lower bounds, and dominance criteria are proposed and exploited in a branch-and-bound algorithm. Computational results are given.     

Sharp bounds for the reliability of systems and mixtures with ordered components

In this article, we study how to derive bounds for the reliability and the expected lifetime of a coherent system with heterogeneous ordered components. These bounds can be used to compare the performance of the systems obtained by permuting the components at a given system structure, that is, to study where we should place the different components at a system structure to get the most reliable system. Moreover, a similar procedure is applied to get bounds for mixtures and for the generalized proportional hazard rate model when the baseline populations are ordered. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 108–116, 2017     

Distribution free bounds for service constrained (Q,r) inventory systems

A classical and important problem in stochastic inventory theory is to determine the order quantity (Q) and the reorder level (r) to minimize inventory holding and backorder costs subject to a service constraint that the fill rate, i.e., the fraction of demand satisfied by inventory in stock, is at least equal to a desired value. This problem is often hard to solve because the fill rate constraint is not convex in (Q, r) unless additional assumptions are made about the distribution of demand during the lead‐time. As a consequence, there are no known algorithms, other than exhaustive search, that are available for solving this problem in its full generality. Our paper derives the first known bounds to the fill‐rate constrained (Q, r) inventory problem. We derive upper and lower bounds for the optimal values of the order quantity and the reorder level for this problem that are independent of the distribution of demand during the lead time and its variance. We show that the classical economic order quantity is a lower bound on the optimal ordering quantity. We present an efficient solution procedure that exploits these bounds and has a guaranteed bound on the error. When the Lagrangian of the fill rate constraint is convex or when the fill rate constraint does not exist, our bounds can be used to enhance the efficiency of existing algorithms. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 635–656, 2000     

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

In this article, we analyze a discrete‐time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady‐state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state‐dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions—from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.     

Managing waiting times of backordered demands in single‐stage (Q,r) inventory systems

We present a service constrained (Q, r) model that minimizes expected holding and ordering costs subject to an upper bound on the expected waiting time of demands that are actually backordered. We show that, after optimizing over r, the average cost is quasiconvex in Q for logconcave continuous lead time demand distributions. For logconcave discrete lead time demand distributions we find a single‐pass efficient algorithm based on a novel search stopping criterion. The algorithm also allows for bounds on the variability of the service measure. A brief numerical study indicates how the bounds on service impact the optimal average cost and the optimal (Q, r) choice. The discrete case algorithm can be readily adapted to provide a single pass algorithm for the traditional model that bounds the expected waiting time of all demands (backordered or not). © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 557–573, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10028     

A note on the “value” of bounds on EVPI in stochastic programming

The existing literature concentrates on determining sharp upper bounds for EVPI in stochastic programming problems. This seems to be a problem without an application. Lower bounds, which we view as having an important application, are only the incidental subject of study and in the few instances that are available are obtained at an extremely high cost. In order to suggest a rethinking of the course of this research, we analyze the need for bounds on EVPI in the context of its significance in decision problems.     

The parallel replacement problem with demand and capital budgeting constraints

A generalized parallel replacement problem is considered with both fixed and variable replacement costs, capital budgeting, and demand constraints. The demand constraints specify that a number of assets, which may vary over time, are required each period over a finite horizon. A deterministic, integer programming formulation is presented as replacement decisions must be integer. However, the linear programming relaxation is shown to have integer extreme points if the economies of scale binary variables are fixed. This allows for the efficient computation of large parallel replacement problems as only a limited number of 0–1 variables are required. Examples are presented to provide insight into replacement rules, such as the “no‐splitting‐rule” from previous research, under various demand scenarios. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 40–56, 2000     

A bounded dual (all-integer) integer programming algorithm with an objective cut

In this article, we describe a new algorithm for solving all-integer, integer programming problems. We generate upper bounds on the decision variables, and use these bounds to create an advanced starting point for a dual all-integer cutting plane algorithm. In addition, we use a constraint derived from the objective function to speed progress toward the optimal solution. Our basic vehicle is the dual all-integer algorithm of Gomory, but we incorporate certain row- and column-selection criteria which partially avoid the problem of dual-degenerate iterations. We present the results of computational testing.     

Optimal machine capacity expansions with nested limitations under stochastic demand

This paper studies capacity expansions for a production facility that faces uncertain customer demand for a single product family. The capacity of the facility is modeled in three tiers, as follows. The first tier consists of a set of upper bounds on production that correspond to different resource types (e.g., machine types, categories of manpower, etc.). These upper bounds are augmented in increments of fixed size (e.g., by purchasing machines of standard types). There is a second‐tier resource that constrains the first‐tier bounds (e.g., clean room floor space). The third‐tier resource bounds the availability of the second‐tier resource (e.g., the total floor space enclosed by the building, land, etc.). The second and third‐tier resources are expanded at various times in various amounts. The cost of capacity expansion at each tier has both fixed and proportional elements. The lost sales cost is used as a measure for the level of customer service. The paper presents a polynomial time algorithm (FIFEX) to minimize the total cost by computing optimal expansion times and amounts for all three types of capacity jointly. It accommodates positive lead times for each type. Demand is assumed to be nondecreasing in a “weak” sense. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.