Simplex Technique Research Articles

Simplex techniques for nonlinear optimization

Simplex techniques for nonlinear optimization

Simplex optimization of yield of sec-butylbenzene in a Friedel-Crafts alkylation: A special undergraduate project in analytical chemistry

A series of experiments using the two-dimensional sequential simplex technique to effect yield optimization for a chemical system.

Transition state determination by the X-method

A simple direct procedure to locate saddle points (transition states) on energy surfaces is described. The advantage of the method is that it may utilize effective unconstrained optimization techniques while convergence may occur only to saddle points and not to minima. Thus no further tests are needed to decide the nature of the critical point located. Both the simplex and conjugate gradients optimization techniques were applied within the framework of the proposed method (the X-method) and numerical tests were carried out on both 'mathematical' and 'chemical' model surfaces.

Application of factor analysis and simplex technique to the optimization of a phosphate determination via molybdenum blue

This article discusses the optimization of a phosphate analysis via molybdenum blue, using both factor analysis and the simplex technique. As a result of this study, a final recommended procedure for the photometric determination of phosphate has been given.

Evaluation of Trinder's Glucose Oxidase Method for Measuring Glucose in Serum and Urine

Trinder's method for glucose has nearly all the attributes of an ideal automated colorimetric glucose oxidase procedure. The chemicals used in the color reaction with peroxidase are readily available, the solutions are stable and can be prepared by the user, the method is highly specific and largely free of interferences, the sensitivity can be adjusted by the user to cover a wide range of glucose concentrations, and the reagents are not hazardous. We found very good agreement between results by this method and by the hexokinase and Beckman Glucose Analyzer methods. The method has been modified and adapted to the AutoAnalyzer I and SMA 6/60 (Technicon) with manifolds that give very little interaction between specimens. A study of the method by the simplex technique revealed that the glucose oxidase activity in the reagent is the most critical variable.

The optimisation of information obtained by multicomponent chromatographic separation using the simplex technique

This article discusses the optimisation of multicomponent chromatographic techniques with the Simplex technique. The most important difficulty is the choice of the criterion, which must be optimised. The informing power is proposed for this purpose. The technique is applied to the cation exchange separation of cadmium, zinc, copper, manganese and calcium ions in a mixed organic-aqueous solvent. The applicability of the method for chromatography in general is discussed and the limitations are given.

The Structure of Integer Programs under the Hermitian Normal Form

This paper discusses the structure of integer programming under the Hermitian normal form. It shows that this formulation is akin to the simplex technique for linear programming in that there is a basis and a basic solution associated with each Hermitian normal matrix, and that this Hermitian basis forms a set of natural cutting planes; these cutting planes are strong in that they provide facets for at least one of the corner polyhedra associated with a linear programming basis B. In addition, the cofactors of the Hermitian basis are group elements under a homomorphism of the original group structure. The Hermitian basis also allows one to characterize the values of the right-hand side for which the present solution is feasible. Finally, for an optimal Hermitian basis, one can perform sensitivity analysis on the cost coefficients.

Use of the Simplex Method to Optimize Analytical Conditions in Clinical Chemistry

Abstract The optimum analytical conditions with regard to two or more variables can be determined in an unambiguous way by use of the simplex method. The method is particularly applicable to problems involving interacting variables. We describe the application of the simplex technique to the optimization of two procedures on a continuous-flow analyzer (AutoAnalyzer) and to the optimization of chemical variables in the lactate dehydrogenase-catalyzed pyruvate-to-lactate reaction and certain instrumental variables on a centrifugal analyzer (the ENI-GEMSAEC).

ASCIM—an accelerated constrained simplex technique

In many design studies similar optimization problems have to be solved. In these problems a non-linear objective,function, defined in terms of a small number of variables, has to be minimized subject to the additional requirement that the variables must satisfy non-linear and/or linear inequality constrainss. There are still relatively few algorithms that claim to solve this type of problem, especially when analytical expression for the first derivatives of the function and constraints are not available. In the author's experience, these derivatives are rarely available and not always continuous over all the feasible region and the algorithm described here is initially a combination of the simplex technique with the ideas of projection and hemstitching. A switch to quadratic hillclimbing techniques is made when the computer senses that the region of the minimum has been achieved, thus enabling rapid ultimate convergence to be obtained.

Indefinite quadratic fractional functional programming

The present paper suggests a finite iteration technique for findinglocal minimum of a special type quasi-concave quadratic fractional functional subject to linear inequalities. The procedure adopted is exactly similar to “Simplex Technique” in linear programming and the problem has been attacked directly starting with a basic feasible solution and finding conditions under which the solution can be subjected to improvement. A numerical example has been given to illustrate the procedure.

Letter to the Editor—Linear Fractional Functionals Programming

The present paper deals with the problem of maximizing the ratio of two linear functions subject to a set of linear equalities and nonnegativity constraints on the variables. The problem is attacked directly, beginning with a basic feasible solution and showing the conditions under which the solution can be improved. Conditions for optimality criteria are established. The method followed is similar to “simplex technique” in linear programming.

Some Experiences in Price Mapping

Price mapping is one of the first generalizations to arise from pure linear programming. A resource map is a dual version of the same type of calculation. The process of calculating a price map has two stages. The first stage is the determination of the normal optimum solution of a linear program by the usual simplex process for linear programming. The complete matrix forming this solution is called the optimum plan. Every one of the constraints in a linear program holds over a given range of values of any one selected variable, and the end points of such a range of values can be calculated quickly. In two dimensions, when two of the constraining variables are allowed to vary, the optimum plan is seen to hold over a given region in a plane, and this region is bounded by linear segments. It is the object of the first stage of a price mapping program to calculate, not only the optimum plan, but also the coefficients which determine these boundary lines of the optimum region, for two selected variables, together with the co-ordinates of their points of intersection. The second stage of a price mapping program consists of a seeking process to determine the boundaries of the regions in which all the next plans hold. These regions surround the optimum region. Later repetitions of this second stage of the program substitute every one of these next plans in turn for the optimum plan, and use these new plans to seek further solutions, which, in their turn, are valid over regions neighbouring those regions already found. In this way a complete price map is built up, showing every possible plan and the region over which it is valid. It can be proved rigorously that this seeking process is a finite one and that it must come to an end sometime. The normal simplex process can be thought of, in economic terms, as that of calculating the maximum profit on a given investment of money and other resources, such as labour, land and materials. The solution consists of a matrix, the optimum plan; the elements of the first column vector in this matrix are the amounts of the various resources required to produce this best possible plan. Such models are very crude, even when the resulting matrix exceeds 100 by 100 in size. As well as the practical difficulties of performing the calculations on such large matrices, there are also difficulties in the actual measurement of the observed coefficients, namely the resources. These models can, at best, provide only a coarse outline picture of this best possible plan. The picture could be made much more detailed if we could measure subjective elements, such as managerial preferences and workers' prejudices in money terms. But this would introduce still more elements into our matrix. Instead, it is often more acceptable to the person who wants the answers, to be given a price map showing the region in which the best possible plan holds, and also all those other regions in which the next plans hold, plans resulting from the relaxation of every one of the boundary restrictions in turn. The outlines of the simplex technique are well known, but little has been written about the computational details of a really efficient price mapping program for a fast electronic computer. It is the object of this paper to fill this gap. Two difficulties were encountered in practice. First it is necessary to design the process of price mapping in such a way that the calculation comes to an end sooner rather than later. Most early programs (see Cran, 1961) were aimed primarily at satisfying this requirement, particularly those designed for small or slow computers. However, it was soon found that most of the methods (see Heady and Candler, 1960) proposed in the standard text books omitted a few of the possible next solutions altogether. Mr. E. Evans, now at Aberystwyth, worked out part of one of these price maps by using a desk calculator, so that the fault in the seeking process, which was then in use, could be traced and corrected. The final outcome was the present program, developed for use on the Cambridge University electronic computer, EDSAC 2. We shall assume in this paper that the first part of the problem, that of collecting the data and of setting it up in a form suitable for the application of a price mapping program, has already been carried out by the economists, and that only the calculations remain to be worked out.

Analytical Methods in Transportation: Left-Turn Penalties in Traffic Assignment Models

An examination of the use of linear programming is presented for the assignment of traffic to routes in a network when the origins and destinations of the trips are known. The formulation of the traffic assignment problem as a linear programming problem is given. The simplex computational technique and an equivalent geometric algorithm are presented side-by-side in the solution of a sample problem so that the relationship of the two methods can be seen. Also mentioned is an intersection mode that permits time penalties to be assigned to individual turning maneuvers within the intersection. Essentially the intersection node is broken into several sub-nodes so that, for visualization purposes, a spacial separation can be substituted for the temporal one. Adaptations of the model for several types of intersections are presented. The intersection model can be applied equally well to the linear programming approach to assignment or to presently used techniques.

New Methods in Mathematical Programming—Optimal Flow Through Networks with Gains

The special class of linear programs described as network flow problems includes many of the special structure optimization problems in such areas as assignment, transportation, catering, warehousing, production scheduling, etc. Besides the physical motivation of the flow description, this class of problems possesses conceptually simple, and computationally elegant, solution techniques originated by Dantzig, Ford, Fulkerson, et al. This paper describes a generalization of this class of problems to “process-flow networks,” or “flow with gains,” in which flow in any branches of the network may be multiplied by an arbitrary constant, called the branch gain, before leaving the branch and flowing into the remainder of the network. This generalization permits the description of networks in which different kinds of flow may be converted one to another, without “constant returns to scale.” Among other interesting problems, this model includes the metal-processing problem, the machine-loading problem, financial budgeting, aircraft routing, warehousing with “breeding” or “evaporation,” the two-equation capacitated linear program, etc. The solution method here described and proved is a natural extension of Ford and Fulkerson's technique (or specialization of the primal-dual method) and retains much of the physical motivation and easy computational aspect of their technique, a direct starting solution is usually available, and optimization of the imbedded linear program (the restricted primal problem) can be accomplished without the use of the simplex technique. Conceptually, the solution procedure consists of the following steps Find the incrementally cheapest loop in the network that will absorb flow Establish as much additional flow into the network along this route as possible Repeat Steps (1) and (2) until the desired flow is established, or until infeasibility is discovered The solution technique includes a maximal-flow procedure and produces a “min-cut equals max-flow” theorem for networks with gains. Additional features, such as piece-wise linear convex costs, parametric studies, network “tearing,” mixed boundary conditions, etc., may be easily incorporated in the algorithm.

A Two-Phase Method for the Simplex Tableau

A version of a two-phase simplex technique is given for manually solving those linear-programming problems in which artificial vectors are introduced and subsequently driven out. The first phase of the method determines feasibility, provided it exists, the second phase, which follows, searches for optimality.

Blending Aviation Gasolines--A Study in Programming Interdependent Activities in an Integrated Oil Company

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