Safety Stock Determination with Serially Correlated Demand in a Periodic-Review Inventory System

We consider a periodic-review inventory replenishment model with an order-up-to-R operating doctrine for the case of deterministic lead times and a covariance-stationary stochastic demand process. A method is derived for setting the inventory safety stock to achieve an exact desired stockout probability when the autocovariance function for Gaussian demand is known. Because the method does not require that parametric time-series models be fit to the data, it is easily implemented in practice. Moreover, the method is shown to be asymptotically valid when the autocovariance function of demand is estimated from historical data. The effects on the stockout rate of various levels of autocorrelated demand are demonstrated for situations in which autocorrelation in demand goes undetected or is ignored by the inventory manager. Similarly, the changes to the required level of safety stock are demonstrated for varying levels of autocorrelation.     

V-positivity,poverses and the economic global unicity theorems of Gale and Nikaido

Summary Strong theorems concerning globally unique solutions to nonlinear inequalities have been obtained byGale andNikaido via P-matrix characterizations of theJacobians of the mappings involved. We introduce two new concepts here: v-positivity and the poverse. These permit us to state and prove significant generalizations of the theorems just mentioned and, equally important, provide access to preliminary results concerning linear inequalities by the powerful direct techniques of linear programming theory.

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Zusammenfassung Gale undNikaido haben starke Sätze über globale, eindeutige Lösungen nichtlinearer Ungleichungen erhalten für den Fall, daß dieJacobi- Matrix der auftretenden Abbildung eineP-Matrix ist. Darauf aufbauend werden zwei neue Konzepte vorgestellt: v-positivity und die poverse. Diese erlauben es, Verallgemeinerungen der erwähnten Sätze aufzustellen und zu beweisen und, was von gleicher Bedeutung ist, die Verbindung zu früheren Ergebnissen über lineare Ungleichungen durch die wirkungsvollen direkten Verfahren der Theorie der Linearen Optimierung herzustellen.

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This Research was partly supported by Project No. NR 047-021, ONR Contracts NO 0014-67-A-0126-0008 and NO 0014-67-A-0126-0009 with the Center for Cybernetic Studies, The University of Texas.     

Stability of efficiency evaluations in data envelopment analysis

Efficiency evaluations in data envelopment analysis are shown to be stable for arbitrary perturbations in the convex hulls of input and output data. Also, the corresponding restricted Lagrange multiplier functions are shown to be continuous. The results are proved using point-to-set mappings and a particular region of stability from input optimization.Research partly supported by National Science Foundation Grants, Office of Naval Research Grant, and by the Natural Sciences and Engineering Council of Canada.     

An effective non-Archimedean anti-degeneracy/cycling linear programming method especially for data envelopment analysis and like models

A non-Archimedean effective anti-degeneracy/cycling method for linear programming models, especially Data Envelopment Analysis (DEA), processing networks and advertising media mix models is herein developed. It has given a tenfold speed increase plus elimination of cycling difficulties over conventional Marsden or Kennington/Ali LP software modules in a 1000 LP DEA application.     

Mollifiers for games in normal form and the Harsanyi-Selten valuation function

The concepts of disruption and mollifiers ofCharnes/Rousseau/Seiford [1978] for games in characteristic function form are here extended to games in normal form. We show for a large class of games that theHarsanyi-Selten [1959] modification ofvon Neumann /Morgenstern's [1953] construction of a characteristic function for games in normal form, to take better account of “disruption” or “threat” possibilities, yields a constant mollifier. In general, it can be non-superadditive when the von Neumann-Morgenstern function is superadditive, and it also fails to take account of coalitional sizes. Our extended “homomollifier” concept does, and always yields a superadditive constant sum characteristic function.     

Constrainedn-person games

The usual properties of a characteristic function game were derived byvon Neumann andMorgenstern from the properties of a game in normal form. In this paper we give a linear programming principle for the calculation of the characteristic function. The principle is a direct application ofCharnes' linear programming method for the calculation of the optimal strategies and the value of a two-person zero-sum game. The linear programming principle gives another method for proving the standard properties of a characteristic function when it is derived from a game in normal form. Using an idea originated byCharnes for two person games, we develop the concept of a constrainedn-person game as a simple, practical extension of ann-person game. However the characteristic function for a constrainedn-person game may not satisfy properties, such as superadditivity, usually associated with a characteristic function.     

Preface to topics in data envelopment analysis

This paper serves as an introduction to a series of three papers which are directed to different aspects of DEA (Data Envelopment Analysis) as follows: (1) uses and extensions of window analyses' to study DEA efficiency measures with an illustrative applications to maintenance activities for U.S. Air Force fighter wings, (2) a comparison of DEA and regression approaches to identifying and estimating, sources of inefficiency by means of artificially generated data, and (3) an extension of ordinary (linear programming) sensitivity analyses to deal with special features that require attention in DEA. Background is supplied in this introductory paper with accompanying proofs and explanations to facilitate understanding of what DEA provides in the way of underpinning for the papers that follow. An attempt is made to bring readers abreast of recent progress in DEA research and uses. A synoptic history is presented along with brief references to related work, and problems requiring attention are also indicated and possible research approaches also suggested.This research was partly supported by the National Science Foundation and USARI Contract MDA 903-83-K0312, with the Center for Cybernetic Studies, the University of Texas at Austin. It was also partly supported by the IC² Institute at the University of Texas at Austin. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Polyextremal principles and separably-infinite programs

As a direct extension of Charnes' characterization of two-person zero-sum constrained games by linear programming, we show how a general class of saddle value problems can be reduced to a pair of uniextremal dual separably-infinite programs. These programs have an infinite number of variables and an infinite number of constraints, but only a finite number of variables appear in an infinite number of constraints and only a finite number of constraints have an infinite number of variables. The conditions under which the characterization holds are among the more general ones appearing in the literature sufficient to guarantee the existence of a saddle point of a concave-convex function.The key construction involves augmenting a given player's original set of variables by generalized finite sequences determined by the other player's constraint set and objective function. A duality theory is developed which includes complementarity conditions, thereby making contact with the numerical treatment of semi-infinite programming.

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Zusammenfassung Als eine direkte Erweiterung von Charnes' Charakterisierung von Zweipersonen-Nullsummenspielen durch lineare Programme wird gezeigt, daß eine allgemeine Klasse von Sattelpunktproblemen auf ein Paar dualer separabel-infiniter Programme zurückgeführt werden kann. Diese Programme haben unendlich viele Variablen und unendlich viele Nebenbedingungen, wobei nur endlich viele Variablen in unendlich vielen Restriktionen vorkommen und nur endlich viele Nebenbedingungen unendlich viele Variablen enthalten. Es wird eine Dualitätstheorie entwickelt, die Komplementaritätsbedingungen einschließt, wobei auf die numerische Behandlung semi-infiniter Optimierungsprobleme Bezug genommen wird.

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This research was partially supported by Project NR047-021, ONR Contract N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, and by the National Science Foundation Grant NSF ENG-7825488 with Carnegie-Mellon University. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Measuring the efficiency of decision making units

A nonlinear (nonconvex) programming model provides a new definition of efficiency for use in evaluating activities of not-for-profit entities participating in public programs. A scalar measure of the efficiency of each participating unit is thereby provided, along with methods for objectively determining weights by reference to the observational data for the multiple outputs and multiple inputs that characterize such programs. Equivalences are established to ordinary linear programming models for effecting computations. The duals to these linear programming models provide a new way for estimating extremal relations from observational data. Connections between engineering and economic approaches to efficiency are delineated along with new interpretations and ways of using them in evaluating and controlling managerial behavior in public programs.