The maintainability analysis considering the discard of failed units is discussed. The model includes the following properties of each unit: ordinary failure rate, fatal failure rate, repair rate, mean setting time, unit cost and mean setting time. In this model, a steady state availability is meanningless, therefore, a point-wise availability is discussed. The point-wise availability is obtained by a numerical method. But, in some cases, it is obtained analytically. Then optimal spare units allocation problems are discussed. The problems are formulated as nonlinear integer programming problems. Finally, some numerical examples are presented.

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An inspection policy for a standby unit is considered by taking a standby electric generator as an example. The expected cost by the time of the electric power supply failure is derived by rising renewal-type equations. We discuss an optimum checking time which minimizes the expected cost. We further obtain a checking time such that the probability that a standby generator has failed at the time of the electric power supply failure is not greater than a prespecified value. Two cases where the failure rate is not disturbed by any inspection and a standby generator may fail during the time of the electric power supply failure are considered. Numerical examples are finally presented.

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The multiple-choice continuous knapsack problem is defined as follows: maximize z =Σ^n_<i=1>Σ^<mi>_<j=1>c_<ij>x_<ij> subject to (1) Σ^n_<i=1>Σ^<mi>_<j=1> a_<ij>x_<ij> 〓 b, (2) 0 〓 x_<ij> 〓 1, i = 1 , 2, …, n, j = 1. 2, …, m_i, (3) at most one of x_<ij>(j = 1, 2, …, m_i) is positive for each i = 1 , 2, …, n, where n, m_i are positive integers and a<ij> are nonnegative integers. In this paper, it is proved that this problem is NP-complete (which strongly suggests the computational intractability to obtain exact optimal solutions). Then, starting with an approximate solution obtained from the LP optimal solution by rounding, two approximate algorithms are proposed and analyzed. The first one (called the breadth-1 search method) obtains an approximate solution with the (worst case) relative error less than 25%. The required computation time is o (mN logN), where N = Σ^n_<i=1>m_i and m = max m_i. The second one (called the breadth-K search method) obtains an approximate solution within an arbitrarily specified (worst case) error bound lt;isinsgt; x 100% in 0 (m┌1/4(&i;sins;-&i;sins;^2)┐N┌l/4(&i;sins;-&i;sins;^2)┐)time.

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In this paper, an optimal allocation problem (APQ) with a quadratic objective function, a total resource constraint and an upper and lower bound constraint is considered. The APQ is a very basic and simple model but it can serve as a subproblem in the solution of the generalized allocation problem. Applying the Lagrange relaxation method, an explicit expression of the dual function associated with the APQ and an equation which the optimal dual variable must satisfy are derived first. Then, some properties of the equation are discussed. Finally, three algortihms for solving the equation are proposed, and some computational results for the APQ are given. These results reveal the effectiveness of the algorithm.

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A two-person zero-sum time-sequential game for sums of iid random variables, in which decision for the players at each stage is either to accept or reject sequentially presented r.v.'s, each r.v. is selected only when both players accept it, and each player has a limited number of rejections he can excercise. The algorithm for deriving the optimal strategies is given and a numerical calculation is provided for the case of uniform distribution. The bilateral-game version, in which decisions are made by each side one by one, is also discussed and is shown to possess "reversibility" of the optimal strategies. These results clarify some earlier works in this area.

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