A branch and bound algorithm for computing optimal replacement policies in consecutive k‐out‐of‐n‐systems

This paper presents a branch and bound algorithm for computing optimal replacement policies in a discrete‐time, infinite‐horizon, dynamic programming model of a binary coherent system with n statistically independent components, and then specializes the algorithm to consecutive k‐out‐of‐n systems. The objective is to minimize the long‐run expected average undiscounted cost per period. (Costs arise when the system fails and when failed components are replaced.) An earlier paper established the optimality of following a critical component policy (CCP), i.e., a policy specified by a critical component set and the rule: Replace a component if and only if it is failed and in the critical component set. Computing an optimal CCP is a optimization problem with n binary variables and a nonlinear objective function. Our branch and bound algorithm for solving this problem has memory storage requirement O(n) for consecutive k‐out‐of‐n systems. Extensive computational experiments on such systems involving over 350,000 test problems with n ranging from 10 to 150 find this algorithm to be effective when n ≤ 40 or k is near n. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 288–302, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10017     

A minmax search for the critical level of a system: The asymmetric case

We consider a system composed of k components, each of which is subject to failure if temperature is above a critical level. The failure of one component causes the failure of the system as a whole (a serially connected system). If z_i is the critical temperature of the ith component then z^* = min{z_i: i = 1,2,…, k} is the critical level of the system. The components may be tested individually at different temperature levels, if the temperature is below the critical level the cost is $1, otherwise the test is destructive and the cost is m > 1 dollars. The purpose of this article is to construct, under a budgetary constraint, an efficient (in a minmax sense) testing procedure which will locate the critical level of the system with maximal accuracy.     

IFR results for repairable systems

Consider a k-out-of-n system with independent repairable components. Assume that the repair and failure distributions are exponential with parameters {μ₁, ?,μ_n} and {λ₁, ?,λ_n}, respectively. In this article we show that if λ_i – μ_i = Δ for all i then the life distribution of the system is increasing failure rate (IFR).     

A note on a stochastic production-maximizing transportation problem

A stochastic production-maximizing problem with transportation constraints is considered where the production rates, R_ij, of man i — job j combinations are random variables rather than constants. It is shown that for the family of Weibull distributions (of which the Exponential is a special case) with scale parameters λ_ij and shape parameter β, the plan that maximizes the expected rate of the entire line is obtained by solving a deterministic fixed charge transportation problem with no linear costs and with “set-up” cost matrix ‖λ_ij‖.     

Asymptotically optimal component assembly plans in repairable systems and server allocation in parallel multiserver queues

T identical exponential lifetime components out of which G are initially functioning (and B are not) are to be allocated to N subsystems, which are connected either in parallel or in series. Subsystem i, i = 1,…, N, functions when at least K_i of its components function and the whole system is maintained by a single repairman. Component repair times are identical independent exponentials and repaired components are as good as new. The problem of the determination of the assembly plan that will maximize the system reliability at any (arbitrary) time instant t is solved when the component failure rate is sufficiently small. For the parallel configuration, the optimal assembly plan allocates as many components as possible to the subsystem with the smallest K_i and allocates functioning components to subsystems in increasing order of the K_i's. For the series configuration, the optimal assembly plan allocates both the surplus and the functioning components equally to all subsystems whenever possible, and when not possible it favors subsystems in decreasing order of the K_i's. The solution is interpreted in the context of the optimal allocation of processors and an initial number of jobs in a problem of routing time consuming jobs to parallel multiprocessor queues. © John Wiley & Sons, Inc. Naval Research Logistics 48: 732–746, 2001     

Scheduling multiple products on parallel machines with setup costs

We consider a class of production scheduling models with m identical machines in parallel and k different product types. It takes a time p_i to produce one unit of product type i on any one of the machines. There is a demand stream for product type i consisting of n_i units with each unit having a given due date. Before a machine starts with the production of a batch of products of type i a setup cost c is incurred. We consider several different objective functions. Each one of the objective functions has three components, namely a total setup cost, a total earliness cost, and a total tardiness cost. In our class of problems we find a relatively large number of problems that can be solved either in polynomial time or in pseudo‐polynomial time. The polynomiality or pseudo‐polynomiality is achieved under certain special conditions that may be of practical interest; for example, a regularity pattern in the string of due dates combined with earliness and tardiness costs that are similar for different types of products. The class of models we consider includes as special cases discrete counterparts of a number of inventory models that have been considered in the literature before, e.g., Wagner and Whitin (Manage Sci 5 (1958), 89–96) and Zangwill (Oper Res 14 (1966), 486–507; Manage Sci 15 (1969), 506–527). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

On the first time a separately maintained parallel system has been down for a fixed time

Consider a system consisting of n separately maintained independent components where the components alternate between intervals in which they are “up” and in which they are “down”. When the i^th component goes up [down] then, independent of the past, it remains up [down] for a random length of time, having distribution F_i[G_i], and then goes down [up]. We say that component i is failed at time t if it has been “down” at all time points s ?[t-A.t]: otherwise it is said to be working. Thus, a component is failed if it is down and has been down for the previous A time units. Assuming that all components initially start “up,” let T denote the first time they are all failed, at which point we say the system is failed. We obtain the moment-generating function of T when n = l, for general F and G, thus generalizing previous results which assumed that at least one of these distributions be exponential. In addition, we present a condition under which T is an NBU (new better than used) random variable. Finally we assume that all the up and down distributions F_i and G_i i = l,….n, are exponential, and we obtain an exact expression for E(T) for general n; in addition we obtain bounds for all higher moments of T by showing that T is NBU.     

Rare‐event component importance for the consecutive‐k system

Various indices of component importance with respect to system reliability have been proposed. The most popular one is the Birnbaum importance. In particular, a special case called uniform Birnbaum importance in which all components have the same reliability p has been widely studied for the consecutive‐k system. Since it is not easy to compare uniform Birnbaum importance, the literature has looked into the case p = ½, p → 1, or p ≥ ½. In this paper, we look into the case p → 0 to complete the spectrum of examining Birnbaum importance over the whole range of p. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 159–166, 2002; DOI 10.1002/nav.10001     

On the application and extension of system signatures in engineering reliability

Following a review of the basic ideas in structural reliability, including signature‐based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes m < n with exchangeable component lifetimes, it is shown that the distribution of an m‐component system's lifetime can be written as a mixture of the distributions of k‐out‐of‐n systems. When the system has n components, the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R–34 (1985) 69–72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Deterministic and stochastic scheduling with teamwork tasks

We study a class of new scheduling problems which involve types of teamwork tasks. Each teamwork task consists of several components, and requires a team of processors to complete, with each team member to process a particular component of the task. Once the processor completes its work on the task, it will be available immediately to work on the next task regardless of whether the other components of the last task have been completed or not. Thus, the processors in a team neither have to start, nor have to finish, at the same time as they process a task. A task is completed only when all of its components have been processed. The problem is to find an optimal schedule to process all tasks, under a given objective measure. We consider both deterministic and stochastic models. For the deterministic model, we find that the optimal schedule exhibits the pattern that all processors must adopt the same sequence to process the tasks, even under a general objective function GC = F(f₁(C₁), f₂(C₂), … , f_n(C_n)), where f_i(C_i) is a general, nondecreasing function of the completion time C_i of task i. We show that the optimal sequence to minimize the maximum cost MC = max f_i(C_i) can be derived by a simple rule if there exists an order f₁(t) ≤ … ≤ f_n(t) for all t between the functions {f_i(t)}. We further show that the optimal sequence to minimize the total cost TC = ∑ f_i(C_i) can be constructed by a dynamic programming algorithm. For the stochastic model, we study three optimization criteria: (A) almost sure minimization; (B) stochastic ordering; and (C) expected cost minimization. For criterion (A), we show that the results for the corresponding deterministic model can be easily generalized. However, stochastic problems with criteria (B) and (C) become quite difficult. Conditions under which the optimal solutions can be found for these two criteria are derived. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Dynamic signatures and their use in comparing the reliability of new and used systems

The signature of a system with independent and identically distributed (i.i.d.) component lifetimes is a vector whose ith element is the probability that the ith component failure is fatal to the system. System signatures have been found to be quite useful tools in the study and comparison of engineered systems. In this article, the theory of system signatures is extended to versions of signatures applicable in dynamic reliability settings. It is shown that, when a working used system is inspected at time t and it is noted that precisely k failures have occurred, the vector s [0,1]^n‐k whose jth element is the probability that the (k + j)th component failure is fatal to the system, for j = 1,2,2026;,n ‐ k, is a distribution‐free measure of the design of the residual system. Next, known representation and preservation theorems for system signatures are generalized to dynamic versions. Two additional applications of dynamic signatures are studied in detail. The well‐known “new better than used” (NBU) property of aging systems is extended to a uniform (UNBU) version, which compares systems when new and when used, conditional on the known number of failures. Sufficient conditions are given for a system to have the UNBU property. The application of dynamic signatures to the engineering practice of “burn‐in” is also treated. Specifically, we consider the comparison of new systems with working used systems burned‐in to a given ordered component failure time. In a reliability economics framework, we illustrate how one might compare a new system to one successfully burned‐in to the kth component failure, and we identify circumstances in which burn‐in is inferior (or is superior) to the fielding of a new system. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Single‐warehouse multi‐retailer inventory systems with full truckload shipments

We consider a multi‐stage inventory system composed of a single warehouse that receives a single product from a single supplier and replenishes the inventory of n retailers through direct shipments. Fixed costs are incurred for each truck dispatched and all trucks have the same capacity limit. Costs are stationary, or more generally monotone as in Lippman (Management Sci 16, 1969, 118–138). Demands for the n retailers over a planning horizon of T periods are given. The objective is to find the shipment quantities over the planning horizon to satisfy all demands at minimum system‐wide inventory and transportation costs without backlogging. Using the structural properties of optimal solutions, we develop (1) an O(T²) algorithm for the single‐stage dynamic lot sizing problem; (2) an O(T³) algorithm for the case of a single‐warehouse single‐retailer system; and (3) a nested shortest‐path algorithm for the single‐warehouse multi‐retailer problem that runs in polynomial time for a given number of retailers. To overcome the computational burden when the number of retailers is large, we propose aggregated and disaggregated Lagrangian decomposition methods that make use of the structural properties and the efficient single‐stage algorithm. Computational experiments show the effectiveness of these algorithms and the gains associated with coordinated versus decentralized systems. Finally, we show that the decentralized solution is asymptotically optimal. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r,s)‐out‐of‐(m,n):F system

A 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system consists of m × n components, and fails if and only if k or more components fail in an r × s submatrix. This system can be treated as a reliability model for TFT liquid crystal displays, wireless communication networks, etc. Although an effective method has been developed for evaluating the exact system reliability of small or medium‐sized systems, that method needs extremely high computing time and memory capacity when applied to larger systems. Therefore, developing upper and lower bounds and accurate approximations for system reliability is useful for large systems. In this paper, first, we propose new upper and lower bounds for the reliability of a 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system. Secondly, we propose two limit theorems for that system. With these theorems we can obtain accurate approximations for system reliabilities when the system is large and component reliabilities are close to one. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

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     

Algorithms for the multi‐item multi‐vehicles dynamic lot sizing problem

We consider a two‐stage supply chain, in which multi‐items are shipped from a manufacturing facility or a central warehouse to a downstream retailer that faces deterministic external demand for each of the items over a finite planning horizon. The items are shipped through identical capacitated vehicles, each incurring a fixed cost per trip. In addition, there exist item‐dependent variable shipping costs and inventory holding costs at the retailer for items stored at the end of the period; these costs are constant over time. The sum of all costs must be minimized while satisfying the external demand without backlogging. In this paper we develop a search algorithm to solve the problem optimally. Our search algorithm, although exponential in the worst case, is very efficient empirically due to new properties of the optimal solution that we found, which allow us to restrict the number of solutions examined. Second, we perform a computational study that compares the empirical running time of our search methods to other available exact solution methods to the problem. Finally, we characterize the conditions under which each of the solution methods is likely to be faster than the others and suggest efficient heuristic solutions that we recommend using when the problem is large in all dimensions. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006.     

Economic lot sizing with constant capacities and concave inventory costs

This article studies the classical single‐item economic lot‐sizing problem with constant capacities, fixed‐plus‐linear order costs, and concave inventory costs, where backlogging is allowed. We propose an O(T³) optimal algorithm for the problem, which improves upon the O(T⁴) running time of the famous algorithm developed by Florian and Klein (Manage Sci18 (1971) 12–20). Instead of using the standard dynamic programming approach by predetermining the minimal cost for every possible subplan, we develop a backward dynamic programming algorithm to obtain a more efficient implementation. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

A diffusion model for the control of a multipurpose reservoir system

This paper develops a methodology for optimizing operation of a multipurpose reservoir with a finite capacity V. The input of water into the reservoir is a Wiener process with positive drift. There are n purposes for which water is demanded. Water may be released from the reservoir at any rate, and the release rate can be increased or decreased instantaneously with zero cost. In addition to the reservoir, a supplementary source of water can supply an unlimited amount of water demanded during any period of time. There is a cost of C_i dollars per unit of demand supplied by the supplementary source to the i^th purpose (i = 1, 2, …, n). At any time, the demand rate R_i associated with the i^th purpose (i = 1, 2, …, n) must be supplied. A controller must continually decide the amount of water to be supplied by the reservoir for each purpose, while the remaining demand will be supplied through the supplementary source with the appropriate costs. We consider the problem of specifying an output policy which minimizes the long run average cost per unit time.     

Approximation schemes for ordered vector packing problems

In this paper we deal with the d‐dimensional vector packing problem, which is a generalization of the classical bin packing problem in which each item has d distinct weights and each bin has d corresponding capacities. We address the case in which the vectors of weights associated with the items are totally ordered, i.e., given any two weight vectors a_i, a_j, either a_i is componentwise not smaller than a_j or a_j is componentwise not smaller than a_i. An asymptotic polynomial‐time approximation scheme is constructed for this case. As a corollary, we also obtain such a scheme for the bin packing problem with cardinality constraint, whose existence was an open question to the best of our knowledge. We also extend the result to instances with constant Dilworth number, i.e., instances where the set of items can be partitioned into a constant number of totally ordered subsets. We use ideas from classical and recent approximation schemes for related problems, as well as a nontrivial procedure to round an LP solution associated with the packing of the small items. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Algorithm to solve a chance‐constrained network capacity design problem with stochastic demands and finite support

We consider the problem of determining the capacity to assign to each arc in a given network, subject to uncertainty in the supply and/or demand of each node. This design problem underlies many real‐world applications, such as the design of power transmission and telecommunications networks. We first consider the case where a set of supply/demand scenarios are provided, and we must determine the minimum‐cost set of arc capacities such that a feasible flow exists for each scenario. We briefly review existing theoretical approaches to solving this problem and explore implementation strategies to reduce run times. With this as a foundation, our primary focus is on a chance‐constrained version of the problem in which α% of the scenarios must be feasible under the chosen capacity, where α is a user‐defined parameter and the specific scenarios to be satisfied are not predetermined. We describe an algorithm which utilizes a separation routine for identifying violated cut‐sets which can solve the problem to optimality, and we present computational results. We also present a novel greedy algorithm, our primary contribution, which can be used to solve for a high quality heuristic solution. We present computational analysis to evaluate the performance of our proposed approaches. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 236–246, 2016     

Parametric inference for component distributions from lifetimes of systems with dependent components

In system reliability analysis, for an n ‐component system, the estimation of the performance of the components in the system is not straightforward in practice, especially when the components are dependent. Here, by assuming the n components in the system to be identically distributed with a common distribution belonging to a scale‐family and the dependence structure between the components being known, we discuss the estimation of the lifetime distributions of the components in the system based on the lifetimes of systems with the same structure. We develop a general framework for inference on the scale parameter of the component lifetime distribution. Specifically, the method of moments estimator (MME) and the maximum likelihood estimator (MLE) are derived for the scale parameter, and the conditions for the existence of the MLE are also discussed. The asymptotic confidence intervals for the scale parameter are also developed based on the MME and the MLE. General simulation procedures for the system lifetime under this model are described. Finally, some examples of two‐ and three‐component systems are presented to illustrate all the inferential procedures developed here. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012