The optimal composition of cattle feed, which can be formulated as a linear programming problem in the case of certainty, is considered when compositions of inputs vary. In the corresponding linear programming formulation the coefficients of the constraints are not constant but can be considered as stochastic. Reformulating the constraints as chance constraints, a nonlinear programming problem results. For an illustrative example this problem is solved using one of Zoutendijk's methods of feasible directions.

Abstract. In Data Envelopment Analysis, when the number of decision making units is small, the numb e r o f u n i t s o f t h e d o m i n a n t or e cient set is relatively large and the average e ciency is generally high. The high average e ciency is the result of assuming that the units in the e cient set are 100% e cient. If this assumption is not valid, this results in an overestimation of the e ciencies, which will be larger for a smaller number of units. Samples of various sizes are used to nd the related bias in the e ciency estimation. The samples are drawn from a large scale application of DEA to bank branch e ciency. The e ects of di erent assumptions as to returns to scale and the number of inputs and outputs are investigated.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. This article gives a reformulation of the simplex method for quadratic programming having the advantage of generating tableaux with certain symmetry properties. It is proved that this method gives the same sequence of iterations as formulation of the simplex method for quadratic programming given earlier by Dantzig and the authors, which may be called the asymmetric formulation. A proof of convergence to the optimal solution is given, which is much simpler than the corresponding proof for the asymmetric formulation. A second symmetric formulation, which is equivalent to the two other formulations, is indicated. For the usual method for quadratic programming as well as for parametric quadratic programming, similar formulations exist. Wolfe's long form of his method for quadratic programming turns out to be the asymmetric variant for parametric quadratic programming. 1. INTRODUCTION FOR THE SOLUTION of convex quadratic programming problems, a number of efficient methods have been developed.The most well known methods are the simplex method for quadratic programming, discovered by Dantzig [4] and, together with the closely related dual method, further developed by van de Panne and Whinston [6, 7], and methods developed by Beale [1, 2], Houthakker [5], and Wolfe [11]. The authors have shown that the methods by Beale and Houthakker can be considered as variants of the simplex method for quadratic programming or are closely related to it.2Compared with the simplex tableaux used in lifnear programming, quadratic programming tableaux have a larger size. A tableau for a linear programming problem with n variables and m constraints has (m + 1)(n + 1) nontrivial elements, while a simplex tableau for a quadratic programming problem with the same number of variables and constraints has (m + n + 1)2 elements. In the simplex method for quadratic programming, a considerable number of tableaux will be in standard form, which means that the tableau can be divided into symmetric and skew-symmetric parts, so that the number of elements to be computed and stored is reduced by nearly one half. Nonstandard tableaux, however, do not have these symmetry properties, so that all elements of these tableaux must be computed.This article gives a reformulation of the simplex method in which all tableaux are in standard form, so that use can be made of the symmetry properties in every tableau. The number of nontrivial elements in a quadratic simplex tableau is therefore decreased by a factor of two. This formulation also has the advantage of leading to a much shorter and simpler proof of convergence for ...

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