SUMMARY THE TECHNIQUES of linear programming are here explained in a commercial application-blending aviation gasolines. Blending is critically important to almost all other areas of programming in an integrated oil company. Intelligent programming of production, transportation, manufacturing, or marketing generally requires solution of blending problems as an initial or integral part of the whole process for it is in blending that the final outputs are determined. In the first part of the paper, questions of optimum programming in a given technological and institutional structure are explored. Computations are executed primarily by means of the simplex technique of Dantzig [4]. Because of the presence of multiple degeneracy and the absence of a general method (at the time these computations were made)2 of handling such problems, it was not possible to rest securely on the simplex method. Alternative methods of computation were, therefore, explored and bounding techniques were employed to discover possible divergences from optimality. Finally, to test the validity of the results, calculations undertaken by a company programming official were obtained for purposes of check and comparison. A relatively simple program is first calculated and more than one optimal program obtained. These results are then extended to more complex problems. Finally, the sensitivity of the matrix to possible changes in the coefficients is studied. The problem of linear programming may be sta-ted as follows: A

Abstract The article focuses on a mixed-sequential estimating procedure with application to audit tests in accounting. One of the purposes in external auditing is to develop a valid opinion on the state of control of a business system. To arrive at such an opinion involves detailed system tests, for example, testing invoices against purchase orders with respect to prices and quantities. Because of time and expense, it is desirable to minimize the extent of these tests, consistent with the quality and standards of assurance desired. In this commonplace audit situation, the use of various statistical sampling techniques has been long advocated. In practice, however, few accountants have found existing sampling techniques to be entirely satisfactory in evaluating the state of system control. The purpose of a pure acceptance sampling technique is essentially to make a decision. As applied to a particular control area, acceptance sampling will yield a "yes" or "no" decision as to whether or not the control area meets a pre-specified level of acceptability. In any particular control area, the auditor, however, is rarely interested in making a yes or no decision on acceptability.

R. Byrne, A. Charnes, W. W. Cooper, K. Kortanek, A Chance-Constrained Approach to Capital Budgeting with Portfolio Type Payback and Liquidity Constraints and Horizon Posture Controls, The Journal of Financial and Quantitative Analysis, Vol. 2, No. 4 (Dec., 1967), pp. 339-364

Abstract : This paper shows that control through decentralization can be affected in general if additional information in the form of preemptive goals is delegated to individual units as well as prices. It is shown that the procedure is a robust one and results in small errors in profit. The technique of preemptive goals providing further control advantages in an economy where structural change is taking place over time as reflected by changing technological coefficient matrices of individual units is indicated.

Linear Programming, Budgeting, Breakeven analyses

Abstract The article focuses on some new approaches to risk. A great deal can be said on the subject of risk and how it might be identified, measured and evaluated. It is not the purpose of this article to distinguish rigorously between different categories or dimensions of risk. Rather the author conceptualizes "risk" as emerging from the fact that some of the information, which is pertinent to a decision, can at best be known only in the form of specified probability distributions. The resulting possibility of deviations from any estimate of the events governed by such probability distributions is then the basic phenomenon, which they shall suppose gives rise to risk. Of course, more than one probability distribution may be applicable and a combination of these distributions may then also require consideration prior to effecting choices between investment alternatives. This kind of phenomenon can supposedly be handled, at least in principle, by suitable theorems or algorithms in probability and statistics. At any rate, given this assumption, one version of a more classical approach would next proceed to reduce each alternative to a single-number basis for comparison. In more sophisticated analyses this might be accomplished via a "utility function" approach.

Abstract ABSTRACT: Earlier attempts to extend cost-volume-profit analyses to stochastic sales behavior are reinterpreted in terms of chance-constrained programming formulations to control for risks of failing to break even. The standard types of breakeven analyses usually employed as deterministic systems in accounting are shown also to apply in a surprisingly general manner to situations where risk or uncertainty in sales demands are present. Ways of extending such analyses to allow for new upper as well as lower limits on volume are developed and interpreted in detail. Suggestions and references are also supplied to indicate how such analyses can be extended further to provide contact with probabilistic programming (linear programming under uncertainty as well as stochastic linear programming and chance-constrained programming) approaches to planning and evaluation of portfolios of projects under combinations of liquidity and payback period constraints.

Abstract The article discusses the use of mathematical models across a variety of disciplines and practices. Mathematical models, naturally, are represented by means of mathematical relations. The mathematical models which will be of interest in this article generally proceed via the special kinds of mathematical relations called "functions," which are used to represent some or all of the relations. Double-entry accounting has been used as a basis for planning and control at both economy-wide and individual-enterprise levels. This is to say that double-entry accounting provides a tool of great utility which can be employed in a variety of ways and contexts. Beyond the convenience of moving back and forth between accounting and interindustry analyses, the mathematics associated with this modeling has permitted a variety of other uses and extensions. The examples in this article should make it clear, however, that this is not the end. Still more may be available from further research in model equivalences and related explorations. Indeed the "models" definition we introduced in the first section of the preceding paper is designed to underscore the potential value of such continuing explorations.