Properties of a mean residual life function arising from renewal theory

The present article discusses the properties of the mean residual life function in a renewal process. We examine the relationship this function has with the failure rate function and the conventional mean, variance and coefficient of variation of residual life. We also discuss some monotonicity properties of the mean residual life function. A partial order based on the renewal mean residual function is introduced along with its interrelationship with some existing stochastic orders. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

On labeling the vertices of products of complete graphs with distance constraints

Variations of Hale's channel assignment problem, the L(j, k)‐labeling problem and the radio labeling problem require the assignment of integers to the vertices of a graph G subject to various distance constraints. The λ_j,k‐number of G and the radio number of G are respectively the minimum span among all L(j, k)‐labelings, and the minimum span plus 1 of all radio labelings of G (defined in the Introduction). In this paper, we establish the λ_j,k‐number of ∏ K for pairwise relatively prime integers t₁ < t₂ < … < t_q, t₁ ≥ 2. We also show the existence of an infinite class of graphs G with radio number |V(G)| for any diameter d(G). © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Optimal sample size allocation for tests with multiple levels of stress with extreme value regression

In this article, we discuss the optimal allocation problem in a multiple stress levels life‐testing experiment when an extreme value regression model is used for statistical analysis. We derive the maximum likelihood estimators, the Fisher information, and the asymptotic variance–covariance matrix of the maximum likelihood estimators. Three optimality criteria are defined and the optimal allocation of units for two‐ and k‐stress level situations are determined. We demonstrate the efficiency of the optimal allocation of units in a multiple stress levels life‐testing experiment by using real experimental situations discussed earlier by McCool and Nelson and Meeker. Monte Carlo simulations are used to show that the optimality results hold for small sample sizes as well. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Optimal maintenance policies for a safety‐critical system and its deteriorating sensor

We consider the integrated problem of optimally maintaining an imperfect, deteriorating sensor and the safety‐critical system it monitors. The sensor's costless observations of the binary state of the system become less informative over time. A costly full inspection may be conducted to perfectly discern the state of the system, after which the system is replaced if it is in the out‐of‐control state. In addition, a full inspection provides the opportunity to replace the sensor. We formulate the problem of adaptively scheduling full inspections and sensor replacements using a partially observable Markov decision process (POMDP) model. The objective is to minimize the total expected discounted costs associated with system operation, full inspection, system replacement, and sensor replacement. We show that the optimal policy has a threshold structure and demonstrate the value of coordinating system and sensor maintenance via numerical examples. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 399–417, 2017     

A learning tabu search for a truck allocation problem with linear and nonlinear cost components

The two‐level problem studied in this article consists of optimizing the refueling costs of a fleet of locomotives over a railway network. The goal consists of determining: (1) the number of refueling trucks contracted for each yard (truck assignment problem denoted TAP) and (2) the refueling plan of each locomotive (fuel distribution problem denoted FDP). As the FDP can be solved efficiently with existing methods, the focus is put on the TAP only. In a first version of the problem (denoted (P1)), various linear costs (e.g., fuel, fixed cost associated with each refueling, weekly operating costs of trucks) have to be minimized while satisfying a set of constraints (e.g., limited capacities of the locomotives and the trucks). In contrast with the existing literature on this problem, two types of nonlinear cost components will also be considered, based on the following ideas: (1) if several trucks from the same fuel supplier are contracted for the same yard, the supplier is likely to propose discounted prices for that yard (Problem (P2)); (2) if a train stops too often on its route, a penalty is incurred, which represents the dissatisfaction of the clients (Problem (P3)). Even if exact methods based on a mixed integer linear program formulation are available for (P1), they are not appropriate anymore to tackle (P2) and (P3). Various methods are proposed for the TAP: a descent local search, a tabu search, and a learning tabu search (LTS). The latter is a new type of local search algorithm. It involves a learning process relying on a trail system, and it can be applied to any combinatorial optimization problem. Results are reported and discussed for a large set of instances (for (P1), (P2), and (P3)), and show the good performance of LTS. © 2014 Wiley Periodicals, Inc. 62:32–45, 2015     

An algorithm for the solution of a location problem with metric constraints

In many location problems, the solution is constrained to lie within a closed set. In this paper, optimal solutions to a special type of constrained location problem are characterized. In particular, the location problem with the solution constrained to be within a maximum distance of each demand point is considered, and an algorithm for its solution is developed and discussed.     

A two product dynamic economic lot size production model with either-or production constraints

This paper considers the production of two products with known demands over a finite set of periods. The production and inventory carrying costs for each product are assumed to be concave. We seek the minimum cost production schedule meeting all demands, without backlogging, assuming that at most one of the two products can be produced in any period. The optimization problem is first stated as a nonlinear programming problem, which allows the proof of a result permitting the search for the optimal policy to be restricted to those which produce a product only when its inventory level is zero. A dynamic programming formulation is given and the model is then formulated as a shortest route problem in a specially constructed network.     

Covariance of weibull order statistics

Let X₁ < X₂ <… < X_n denote an ordered sample of size n from a Weibull population with cdf F(x) = 1 - exp (?x^p), x > 0. Formulae for computing Cov (X_i, X_j) are well known, but they are difficult to use in practice. A simple approximation to Cov(X_i, X_j) is presented here, and its accuracy is discussed.