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     

New product introduction against a predator: A bilevel mixed‐integer programming approach

We consider a scenario with two firms determining which products to develop and introduce to the market. In this problem, there exists a finite set of potential products and market segments. Each market segment has a preference list of products and will buy its most preferred product among those available. The firms play a Stackelberg game in which the leader firm first introduces a set of products, and the follower responds with its own set of products. The leader's goal is to maximize its profit subject to a product introduction budget, assuming that the follower will attempt to minimize the leader's profit using a budget of its own. We formulate this problem as a multistage integer program amenable to decomposition techniques. Using this formulation, we develop three variations of an exact mathematical programming method for solving the multistage problem, along with a family of heuristic procedures for estimating the follower solution. The efficacy of our approaches is demonstrated on randomly generated test instances. This article contributes to the operations research literature a multistage algorithm that directly addresses difficulties posed by degeneracy, and contributes to the product variety literature an exact optimization algorithm for a novel competitive product introduction problem. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

On the (S − 1, S) lost sales inventory model with priority demand classes

In this paper an inventory model with several demand classes, prioritised according to importance, is analysed. We consider a lot‐for‐lot or (S ? 1, S) inventory model with lost sales. For each demand class there is a critical stock level at and below which demand from that class is not satisfied from stock on hand. In this way stock is retained to meet demand from higher priority demand classes. A set of such critical levels determines the stocking policy. For Poisson demand and a generally distributed lead time, we derive expressions for the service levels for each demand class and the average total cost per unit time. Efficient solution methods for obtaining optimal policies, with and without service level constraints, are presented. Numerical experiments in which the solution methods are tested demonstrate that significant cost reductions can be achieved by distinguishing between demand classes. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 593–610, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10032     

Stochastic allocation of inspection capacity to competitive processes

Optimal allocation and control of limited inspection capacity for multiple production processes are considered. The production processes, which operate independently but share inspection capacity, are subject to random failures and are partially observed through inspection. This study proposes an approach of stochastic allocation, using a Markov decision process, to minimize expected total discounted cost over an infinite time horizon. Both an optimal model and a disaggregate approximation model are introduced. The study provides some structural results and establishes that the control policy is of a threshold type. Numerical experiments demonstrate a significantly decreased amount of computational time required for the disaggregate approach when compared to the optimal solution, while generating very good control policies. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 78–94, 2002; DOI 10.1002/nav.1049     

Bilateral phase-type distributions

In this article we define a class of distributions called bilateral phase type (BPH), and study its closure and computational properties. The class of BPH distributions is closed under convolution, negative convolution, and mixtures. The one-sided version of BPH, called generalized phase type (GPH), is also defined. The class of GPH distributions is strictly larger than the class of phase-type distributions introduced by Neuts, and is closed under convolution, negative convolution with nonnegativity condition, mixtures, and formation of coherent systems. We give computational schemes to compute the resulting distributions from the above operations and extend them to analyze queueing processes. In particular, we present efficient algorithms to compute the steady-state and transient waiting times in GPH/GPH/1 queues and a simple algorithm to compute the steady-state waiting time in M/GPH/1 queues.     

Multichannel queueing systems with heterogeneous classes of arrivals

In an integrated telecommunications network, voice and data traffic compete for the same transmission facilities. Assuming Poisson arrivals and exponential service with different rates, analytic expressions are obtained for measures of performance such as blocking probability and average delay under the following operating rule: class 1 traffic behaves as a loss system while class 2 traffic is buffered when all channels are busy. In view of the inordinate amount of computational effort needed when the number of channels is large, simple approximations have been suggested.     

The conjugate gradient technique for certain quadratic network problems

We consider a class of network flow problems with pure quadratic costs and demonstrate that the conjugate gradient technique is highly effective for large-scale versions. It is shown that finding a saddle point for the Lagrangian of an m constraint, n variable network problem requires only the solution of an unconstrained quadratic programming problem with only m variables. It is demonstrated that the number of iterations for the conjugate gradient algorithm is substantially smaller than the number of variables or constraints in the (primal) network problem. Forty quadratic minimum-cost flow problems of various sizes up to 100 nodes are solved. Solution time for the largest problems (4,950 variables and 99 linear constraints) averaged 4 seconds on the CBC Cyber 70 Model 72 computer.