Pseudo-monotonic interval programming

A pseudo-monotonic interval program is a problem of maximizing f(x) subject to x ε X = {x ε Rⁿ | a < Ax < b, a, b ε R^m} where f is a pseudomonotonic function on X, the set defined by the linear interval constraints. In this paper, an algorithm to solve the above program is proposed. The algorithm is based on solving a finite number of linear interval programs whose solutions techniques are well known. These optimal solutions then yield an optimal solution of the proposed pseudo-monotonic interval program.     

Estimating value in a uniform auction

Consider an auction in which increasing bids are made in sequence on an object whose value θ is known to each bidder. Suppose n bids are received, and the distribution of each bid is conditionally uniform. More specifically, suppose the first bid X₁ is uniformly distributed on [0, θ], and the i^th bid is uniformly distributed on [X_i?1, θ] for i = 2, …?, n. A scenario in which this auction model is appropriate is described. We assume that the value θ is un known to the statistician and must be esimated from the sample X₁, X₂, …?, X_n. The best linear unbiased estimate of θ is derived. The invariance of the estimation problem under scale transformations in noted, and the best invariant estimation problem under scale transformations is noted, and the best invariant estimate of θ under loss L(θ, a) = [(a/θ) ? 1]² is derived. It is shown that this best invariant estimate has uniformly smaller mean-squared error than the best linear unbiased estimate, and the ratio of the mean-squared errors is estimated from simulation experiments. A Bayesian formulation of the estimation problem is also considered, and a class of Bayes estimates is explicitly derived.     

Refined prediction for linear regression models

Adequate prediction of a response variable using a multiple linear regression model is shown in this article to be related to the presence of multicollinearities among the predictor variables. If strong multicollinearities are present in the data, this information can be used to determine when prediction is likely to be accurate. A region of prediction, R, is proposed as a guide for prediction purposes. This region is related to a prediction interval when the matrix of predictor variables is of full column rank, but it can also be used when the sample is undersized. The Gorman-Toman ten-variable data is used to illustrate the effectiveness of the region R.     

A note on the “value” of bounds on EVPI in stochastic programming

The existing literature concentrates on determining sharp upper bounds for EVPI in stochastic programming problems. This seems to be a problem without an application. Lower bounds, which we view as having an important application, are only the incidental subject of study and in the few instances that are available are obtained at an extremely high cost. In order to suggest a rethinking of the course of this research, we analyze the need for bounds on EVPI in the context of its significance in decision problems.     

Evaluation of force structures under uncertainty

A model, for assessing the effectiveness of alternative force structures in an uncertain future conflict, is presented and exemplified. The methodology is appropriate to forces (e.g., the attack submarine force) where alternative unit types may be employed, albeit at differing effectiveness, in the same set of missions. Procurement trade-offs, and in particular the desirability of special purpose units in place of some (presumably more expensive) general purpose units, can be addressed by this model. Example calculations indicate an increase in the effectiveness of a force composed of general purpose units, relative to various mixed forces, with increase in the uncertainty regarding future conflicts.     

On the optimal allocation of crewmen aboard a warship

In the temporarily isolated situation in which a warship finds itself during a mission. not only spare parts, but also “spare” crewmen in various categories of specialization must he on board. Mathematical models for the probabilities of mission survival for personnel and for personnel and materiel jointly are proposed. A practical example is worked out: the optimal allocation of spare crewmen to different categories of specialization is calculated.     

A simulation–optimization method: Its convergence and utility

An optimization method is given for solving problems where a portion of the explicit mathematical form is unknown but can be evaluated. The solution scheme is an iterative process utilizing optimization and subsystem evaluation (such as via simulation). Conditions for the convergence of the iterative process are given. Several published application articles are noted as using this basic methodology. The method is superior to most other numerical optimization procedures. However, the class of problems for which the method is applicable is restricted to problems with enough known structure to generate a convergent iterative procedure. Three numerical examples are given and comparisons made with several other methods of optimizing unknown systems.     

Parameter estimation for the EOQ lot-size model: Minimax and expected value choices

This article provides formulas for estimating the parameters to be used in the basic EOQ lot-size model. The analysis assumes that the true values of these parameters are unknown over known ranges and perhaps nonstationary over time. Two measures of estimator “goodness” are derived from EOQ sensitivity analysis. Formulas are given for computing the minimax choice and the minimum expected value choice for the parameter estimates using both measures of estimator “goodness”. A numerical example is included.