The hnbue and hnwue classes of life distributions

Different properties of the HNBUE (HNWUE) class of life distributions (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx\, \le \,(\ge)\,\mu }\]$\end{document} exp(?t/μ) for t ≥ 0, where μ = \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx}$\end{document} are presented. For instance we characterize the HNBUE (HNWUE) property by using the Laplace transform and present some bounds on the survival function of a HNBUE (HNWUE) life distribution. We also examine whether the HNBUE (HNWUE) property is preserved under the reliability operations (i) formation of coherent structure, (ii) convolution and (iii) mixture. The class of distributions with the discrete HNBUE (discrete HNWUE) property (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=k}^\infty {\mathop{\mathop P\limits^-_{j\,\,\,}\, \le(\ge)\,\mu(1 - 1/\mu)^{^k }}\limits^{}} $\end{document} for k = 0, 1, 2, ?, where μ =\documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=0}^\infty {\mathop {\mathop P\limits^- _{j\,\,\,\,\,}and\mathop P\limits^ - _{j\,\,\,\,\,}=}\limits^{}}\,\,\sum\limits_{k=j+1}^\infty {P_k)}$\end{document} is also studied.     

Testing fit with nuisance location and scale parameters

For each n, X₁(n),…, X_n(n) are independent and identically distributed random variables, each with cumulative distribution function F(x) which is known to be absolutely continuous but is otherwise unknown. The problem is to test the hypothesis that \documentclass{article}\pagestyle{empty}\begin{document}$ F(x) = G\left( {{\textstyle{{x - \theta _1 } \over {\theta _2 }}}} \right) $\end{document}, where the cumulative distribution function G_x is completely specified and satisfies certain regularity conditions, and the parameters θ₁, θ₂ are unknown and unspecified, except that the scale parameter θ₂, is positive. Y₁ (n) ≦ Y₂ (n) ≦ … ≦ Y_n (n)are the ordered values of X₁(n),…, X_n(n). A test based on a certain subset of {Y_i(n)} is proposed, is shown to have asymptotically a normal distribution when the hypothesis is true, and is shown to be consistent against all alternatives satisfying a mild regularity condition.     

Readiness and the optimal redeployment of resources

This paper considers the problem of the optimal redeployment of a resource among different geographical locations. Initially, it is assumed that at each location i, i = 1,…, n, the level of availability of the resource is given by a₁ ≧ 0. At time t > 0, requirements R_f(t) ≧ 0 are imposed on each location which, in general, will differ from the a₁. The resource can be transported from any one location to any other in magnitudes which will depend on t and the distance between these locations. It is assumed that ΣR_j > Σa_t The objective function consideis, in addition to transportation costs incurred by reallocation, the degree to which the resource availabilities after redeployment differ from the requirements. We shall associate the unavailabilities at the locations with the unreadiness of the system and discuss the optimal redeployment in terms of the minimization of the following functional forms: \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj) + } $\end{document} transportation costs, Max \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {Max}\limits_j \,[kj(Rj - yj)] + $\end{document} transportation costs, and \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj)^2 + } $\end{document} transportation costs. The variables y_j represent the final amount of the resource available at location j. No benefits are assumed to accrue at any location if y_j > R_j. A numerical three location example is given and solved for the linear objective.     

The knapsack problem: A survey

A unifying survey of the literature related to the knapsack problem; that is, maximize \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_i {v_i x_{i,} } $\end{document}, subject to \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_j {w_i x_i W} $\end{document} and x_i ? 0, integer; where v_i, w_i and W are known integers, and w_i (i = 1, 2, …, N) and W are positive. Various uses, including those in group theory and in other integer programming algorithms, as well as applications from the literature, are discussed. Dynamic programming, branch and bound, search enumeration, heuristic methods, and other solution techniques are presented. Computational experience, and extensions of the knapsack problem, such as to the multi-dimensional case, are also considered.     

On some stochastic inequalities involving minimum of random variables

Let X_i be independent IFR random variables and let Y_i be independent exponential random variables such that E[X_i]=E[Y_i] for all i=1, 2, ? n. Then it is well known that E[min (X_i)] ≥E[min (X_i)]. Nevertheless, for 1≤i≤n exponentially distributed X_i's and for a decreasing convex function ?(.). it is shown that .     

Optimal allocation of unreliable components for maximizing expected profit over time

In the present paper, we solve the following problem: Determine the optimum redundancy level to maximize the expected profit of a system bringing constant returns over a time period T; i. e., maximize the expression \documentclass{article}\pagestyle{empty}\begin{document}$ P\int_0^T {Rdt - C} $\end{document}, where P is the return of the system per unit of time, R the reliability of this system, C its cost, and T the period for which the system is supposed to work We present theoretical results so as to permit the application of a branch and bound algorithm to solve the problem. We also define the notion of consistency, thereby determining the distinction of two cases and the simplification of the algorithm for one of them.