Tests of Optimality in Constrained Optimization

Sufficiency and necessity theorems are described which testthe optimality of a vector when it is required to minimize agiven function of variables which are subject to inequalities. The paper starts with an exposition of theorems of alternatives,which are fundamental when the functions considered are linear,and proceeds to a systematic formulation of various constraintqualifications, which make it possible to extend results whichhold in linear programming to the non-linear case.     

The input to and output from risk evaluation models

The work which has been done on analytic approaches to the evaluation of the risk in major capital investment opportunities suggests the following hypotheses: (a) The distributions of NPV and IRR which are output from risk evaluation models are often approximately normal. (b) The only important features of the distributions of the variables which are input to risk evaluation models are their means and standard deviations. The research which is described in this paper tests these hypotheses using five risk simulation case studies.     

Curve and Surface Fitting

The starting point of this paper is the problem of fitting apolynomial to a set of unequally spaced data. Five alternativeschemes which have been proposed for the solution of this problemwith an automatic computer are mentioned, and the one whichseems best suited to serve as a basis for a general purposeprogram is discussed in some detail. A practical assessmentof the accuracy of the computational processes involved in thismethod is included. The problem in which the data are weightedis also considered, as is the problem in which constraints areimposed upon the fitting function. The procedures describedfor the solution of these two cases are shown to be formallyequivalent. The method favoured for curve fitting is shown to possess stillmore advantages in solving certain types of surface fittingproblems. The general problem, in which data are scattered withina finite plane area, is not discussed in this paper, which concentrateson the case, of great practical importance, in which the dataare situated on parallel straight lines. The most general caseconsidered has a "quasi-rectangular" boundary, composed of apair of parallel sides and a pair of smooth curves. The applicationof constraints is again described; the procedures appropriatefor curve fitting are readily generalized. A practical exampleof the application of the techniques for finding a constrainedsurface is described. Finally there is a brief review of the advantages of these proceduresin common practical situations.     

On Local Coefficients for Non-generic Representations of Some Classical Groups

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. The local coefficients are related to the Plancherel measures, and their zeroes are shown to be among the poles of the standard intertwining operators.     

Some solutions of karman''s equation describing the transonic flow past a corner point on a profile with a curvilinear generatrix

Transonic flows in the neighbourhood of a corner point on a profile are investigated in a class of self-similar solutions of Karman's equation. The corner point is formed by the intersection of two smooth curves, the tangents to which make a convex angle. The generatrix, which lies in the subsonic part of the flow is assumed to be curvilinear and to vary according to a power law. Values of the self-similarity index are found for which transonic flows are possible either with a free streamline or with a rarefaction wave.     

On the effect of centralisation on expected profits in a multi-location Newsboy problem

This paper is a multi-location, single-period, single-product inventory problem with an opportunity for centralisation. The decentralised model in which a separate inventory is kept at every location is compared to the centralised system in which the demands are satisfied from one central warehouse. The two systems are then compared when excess demands are penalised and lost, and in the more general case where a portion or all of the excess demand at a location may be reallocated among other locations with the remaining inventory. The revenue, salvage, cost, and penalty functions at each location are assumed to be identical. Expected profits are evaluated and conditions under which the centralised model outperforms the decentralised one are derived.     

A Chart for Evaluating Product Research and Development Projects

A method of evaluating research and development projects is described. The method is based on the use of an evaluation chart. The main purpose of the chart is to calculate a project score which is a measure of a selected evaluation criterion. A method of constructing a chart is illustrated using a project index as the evaluation criterion. The chart contains a list of twelve questions, the answers to which are assumed to be the main determinants of the variables in the formula of the index. Each question is followed by a set of answers from which a choice is to be made, and underneath each answer is a numerical score. The numerical scores are logarithmic functions of the answers to which they relate, and when added together give a project score which is a logarithmic function of the index. Various methods of using the chart, and a suggested evaluation procedure, are also described.     

Coherent updating of non-additive measures

The conditions under which a 2-monotone lower prevision can be uniquely updated (in the sense of focusing) to a conditional lower prevision are determined. Then a number of particular cases are investigated: completely monotone lower previsions, for which equivalent conditions in terms of the focal elements of the associated belief function are established; random sets, for which some conditions in terms of the measurable selections can be given; and minitive lower previsions, which are shown to correspond to the particular case of vacuous lower previsions.     

带策略约束的区间数双矩阵博弈的双线性规划求解方法

传统区间数双矩阵博弈理论研究局中人支付值为区间数的策略选择问题,但没有考虑局中人策略选择可能受到各种约束.创建一种求解局中人策略选择受约束且支付值为区间数的双矩阵博弈（简称带策略约束的区间数双矩阵博弈）的简单、有效的双线性规划求解方法.首先,将局中人的博弈支付看作支付值区间中数值的函数.通过证明这种函数具有单调性,据此利用支付值区间的上、下界,构造了一对辅助双线性规划模型,可分别用于显式地计算任意带策略约束的区间数双矩阵博弈中局中人区间数博弈支付的上、下界及其相应的最优策略.最后,利用考虑策略约束条件下企业和政府针对发展低碳经济策略问题的算例,通过比较其与不考虑策略约束情形下的结果,说明了提出的模型和方法的有效性、优越性及可应用性.     

An Interactive Approach for Project Selection

This paper describes an interactive method for presenting a sequence of feasible sets of indivisible projects to a decision-maker. For each set as a whole, the decision-maker evaluates its utilities with respect to each of several attributes; the utilities are then combined to give a single utility for the set.A sequence of zero-one programmes is used to ensure that the only sets presented are those which are feasible and which are not contained in larger feasible sets. The sets are presented in descending order of size, and the presentation can be terminated by the decision-maker or by supplementary rules.The method for presenting sets can be used in other contexts in which complete enumeration of possible sets would otherwise be required.     

Non-linear oscillations of DNA base pairs

Non-linear oscillations of the nitrogenous bases of DNA, which form pairs: adenine – thymine or guanine – cytosine, are considered. An approach is used in which this biophysical problem can be reduced to a mechanical problem of the non-linear oscillations of two coupled dissimilar pendulums, which oscillate in a plane orthogonal to the principal axis of the molecule. The dynamics of such a model system are analysed. Singular points are obtained in phase space. The eigenvalues of the Jacobi matrix are calculated. The behaviour of the model system close to the singular points is described. The results are illustrated in the form of graphs of the solutions of the corresponding model equations and graphs of the trajectories of motion in configuration space.     

An Interactive Aid to Financial Planning

The problem of producing financial plans for the subsidiary of a multinational corporation is one which is critical, especially in uncertain times. In addition there is always a pressing time constraint. This paper describes a suite of models which are in use by Caltex Oil South Africa to assist in this area. The approach adopted was to provide simple models which work interactively. The financial planning analysts were able to be very involved in the project because a high level modelling language was used which they could easily comprehend. The tremendous benefits of this type of approach in terms of user acceptance and in other areas are examined.     

TOTAL GENERALIZED MINIMUM BACKWARD ERROR ALGORITHM FOR SOLVING NONSYMMETRIC LINEAR SYSTEMS

1.IntroductionAnimportantaspectofanyiterativemethodforapproximatingthesolutionofalinearsystemAx~b,(1.1)whereAisannxnrealnonsymmetricmatrixandbisann-vector,istodecideatwhatpointtostoptheiteration.Wecustomarilyusetheresidualerrorasastoppingcondition.Theresidualerrorre=b--AxmcanbeviewedasaperturbationtothevectorbsuchthattheapproximatesolutionisanexactsolutionoftheperturbedlinearsystemAx=b 5,inwhichchangesarepermittedtothevectorbonly.TheGMRESalgorithmisbasedonclassicalKrylovsubspacetechniquesa…     

A Multi-attribute Value Model for the Study of UK Energy Policy

A multi-attribute value model has been developed to study UK energy-policy options. The model consists of four main stages: attribute selection, in which users identify up to 15 key attributes of an energy policy; attribute rating, in which a quantitative assessment is made of the relative importance of each of the attributes selected; option evaluation, in which users are asked to score five energy-policy options for the UK (drawn from a range of published sources) on each of the attributes; and experimentation, in which users are encouraged to explore and modify a tabular and graphical representation of the results. The model has been used in a series of workshops at which representatives from different organizations have been invited to use the model to explore each other's viewpoints. User reactions to the model are discussed, and the potential of such models for addressing controversial social issues is considered.     

The choice of constitutive relations for an ice cover

The constitutive relations between the internal stresses and the deformation parameters of a sea ice cover, which are used in the AIDJEX elastoplastic model and Hibler's non-linearly viscous model, are investigated. It is shown that the structural instability of the ice cover with respect to plastic shear deformations is a consequence of the associated flow rule used in these models. The use of constitutive relations which violate the associated flow rule, but which are in good agreement with the physical properties of granular media, is suggested. An ice cover damage parameter and an empirical equation which describes the change in this parameter are introduced into the treatment. Energy relations are investigated.     

Scheduling a repairman in a finite source system

The scheduling of a single server in a finite source model is considered. TheN customers in the system have different failure and repair rates. Also the costs depend on the customers which are broken down. We give a condition under which the average costs are minimized by a simple list policy, and with a counterexample we show that in the general case no optimal list policy may exist. This motivates us to derive policies which are optimal under low and high traffic conditions. They are again list policies, which behave well numerically.     

The limiting steady rotations of a body on a string with a suspension point on the axis of symmetry

The steady motions of an axisymmetrical rigid body suspended from a fixed base by a weightless undeformable rod or a non-twisting inextensible string are investigated. The case when the rod is fastened to the body at a point situated on its axis of dynamic symmetry is considered. All types of limiting equilibrium configurations which are possible when there is an unlimited increase in the angular velocity of rotation of the system about the vertical are analysed. Domains in which each type of limiting regular precession and permanent rotation can exist are constructed in the space of dimensionless parameters, and the nature of their asymptotic behaviour when the angular velocity increases is determined. The limiting motions which are possible in the case of suspension on a rod and impossible in the case of suspension on a string are investigated.     

The GOWA operator based approach to multiattribute decision making using intuitionistic fuzzy sets

The aim of this paper is to develop a methodology for solving multiattribute decision making (MADM) problems in which weights of attributes and ratings of alternatives on qualitative and quantitative attributes are expressed with intuitionistic fuzzy sets (IFSs). In this methodology, relative membership/satisfaction and non-membership/non-satisfaction degrees are formulated to construct IFSs for numerical values of alternatives on quantitative attributes. Alternatives on qualitative attributes are evaluated using linguistic variables and semantics which are parameterized by IFSs. Hereby, weights and ratings of alternatives on both qualitative attributes and quantitative attributes may be expressed with IFSs in a unified way. The generalized ordered weighted averaging (GOWA) operator is further extended to the situations in which the argument values are IFSs and thus a methodology is developed to solve MADM problems using IFSs. Validity and applicability of the proposed methodology in this paper are illustrated with a real numerical example.     

Voting for voters: the unanimity case

We present a simplified model of the evolution of a society which is regulated by a formal unanimity voting procedure. We examine several protocols, which depend on whether admission or expulsion are permissible, and on the order with which they are implemented. Conditions which ensure the existence of pure-strategy perfect equilibrium profiles for some voting protocols, and counter examples for the existence of such profiles in other protocols are presented. Finally, we prove that, if the agents insist on perfect equilibrium strategy profiles in a one-stage play, the original founders would prefer a protocol in which expulsion precedes admission to protocols in which either admission precedes expulsion, or the two are treated simultaneously. The paper concludes with an overview and a discussion on the results and suggestions for further research. Received: December 2001 RID="*" ID="*"  The authors thank Salvador Barberà for his critical reading and his many suggestions. The authors are also grateful to the anonymous referee for many very useful suggestions. Support by a Natural Sciences and Engineering Research Council of Canada grant and by the Center for Rationality and Interactive Decision Theory of the Hebrew University of Jerusalem is gratefully acknowledged.     

Information in the operations research process

The article deals with information in the OR process. The value of an ‘optimal’ model solution which is not implemented is not very high. Many errors are due to a too narrow formulation of the problem, often a formulation which has focused on purely technical and economic aspects, or a formulation which has been adapted for a standard model. The concepts and models treated in the article take their point of departure in the actual problem. The models can be used as a frame of reference by various individuals and groups which are involved in OR work.