Let X_1,X_2,…, X_n,…be mutually independent random variables with a common cdf F, which is unknown but belongs to some class F of cdf's. The class F = F(μ,σ^2,M) is the set of all cdf's whose mean, variance and domain are -∞ < μ < ∞,0 < σ^2 < ∞, and [μ - M, μ + M] respectively. It is assumed that they are known. Under an observation cost c,0 < c < ∞, we consider a stopping problem φ(x, F) as a two-person zero-sum game in which the. player I decides his stopping set {X > x}, x ∈ R, and the player 2 chooses her cdf F in F. We analyze the upper bound problem φ^δ = sup_<x∈R>sup_<F∈F>φ(x, F) and the game problem φ^δ = value_<x∈R>_<F∈F>φ(x, F) to derive a simple and meaningful solution with the parameters c,μ,σ and M.

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In this paper, we deal with a stochastic resource allocation model. Suppose that we have resource X_n at time n, and allocate A_<n+1>X_n Out of X_n for production and (1-A<n+1>)X_n for consumption at time n, where An+1 is the proportion of the resource that is allocated for production at time n. Then utility U_<n+1>f((1-A<n+1>)X_n) is obtained and resource X<n+1> = R_<n+1>g(A<n+1>X_n) is occurred at time n + 1. where U_<n+1> and R<n+1> are random parameters, f is the utility function, and g is the production function that is an extension of g(x) = x in [7]. We are interested in the way of allocation that maximizes the sum of the expected utilities. We show necessary and sufficient conditions for an allocation to be optimal in this model. The sufficient condition is obtained via the technique of dynamic programming in the same way as one in [7]. Although the necessary condition is shown by means of the similar way used in [9], the sigma additive set function is substituted for the measure as the utility function that we treat in this paper takes more generally a real value. The necessary and sufficient conditions are represented via a martingale: the supremum of the conditional expected utilities forms a supermartingale if we arbitrarily allocate the resource, however it is necessary and sufficient for an allocation to be optimal that the supremum is a martingale. Further, we have obtained an optimal allocation in the resource allocation model with a logarithmic utility function and a nonlinear production function g(x) = x^P(p > 0), utilizing the sufficient condition for an allocation to be optimal obtained above. The optimal allocation includes one obtained in [7].

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We define the utility of the specified up state of the system, ΣX_i, consisting of n independent multistate items. A measure of importance of an item and that of a given state of an item are discussed with respect to the expected utility function. It is shown that, for non-decreasing utility functions, the perfect state of an item yields maximum contribution to the expected utility function. However, such a choice of a state is not obvious when the utility function is non-monotonic. In this case, we propose the use of the linear programming technique to decide the availabilities of states of an item so that its contribution to the expected utility of the system is maximum. Further, we derive sufficient conditions to compare the overall and the state-wise impact of any two items on the expected utility of the system. A numerical example is given to illustrate the procedure.

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A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was first considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new theorems, proofs and algorithms in oriented matroids whose specializations to the linear case are also new. For this, the notion of sufficiency of square matrices, introduced by Cottle, Pang and Venkateswaran, is extended to oriented matroids. Then, we prove a sort of duality theorem for oriented matroids, which roughly states: exactly one of the primal and the dual system has a complementary solution if the associated oriented matroid satisfies "weak" sufficiency. We give two different proofs for this theorem, an elementary inductive proof and an algorithmic proof using the criss-cross method which solves one of the primal or dual problem by using surprisingly simple pivot rules (without any perturbation of the original problem).

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An algorithm strategy is proposed for use with the assurance region (AR) approach in data envelopment analysis (DEA). The strategy addressed in this study characterizes and classifies all decision making units (DMUs) into several subsets, using the revised simplex method of linear programming. Then, each DMU subset is solved by a different algorithm. Experimental studies consisting of randomly generated data sets have confirmed that the proposed algorithm outperforms the conventional DEA use of the revised simplex method. An important feature related to the DEA/AR algorithm is that it can deal effectively with large data sets.

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This paper gives numerical validation of a couple of interpolation approximations for the mean waiting time in a GI/G/s queue, which are provided by a unified approach similar to that in Kimura (1991). Both approximations are represented as certain combinations of the mean waiting times for the GI/M/s and GI/D/s queues in which the arrival processes and the me_an service times are the same as in the approximating GI/G/s queue. To let these approximations be more tractable, we further provide simple interpolation approximations for the mean waiting times in GI/M/s and GI/D/s queues with low variable interarrival times. The quality of the approximations is tested by comparing them with exact solutions and previous two-moment approximations for a variety of cases. Extensive numerical comparisons indicate that our approximations are more accurate than the two-moment approximations and that the relative percentage errors are in the order of 5% in moderate traffic and in the order of 1% in heavy traffic.

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A fast algorithm for solving large scale MV (mean-variance) portfolio optimization problems is proposed. It is shown that by using T independent data representing the rate of return of the assets, the MV model consisting of n assets can be put into a quadratic program with n + T variables, T linear constraints and T quadratic terms in the objective function. As a result, the computation time required to solve this problem would increase very mildly as a function of n. This implies that a very large scale MV model can now be solved in a practical amount of time.

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