Techniques for automatic tolerance control in linear programming

Omah Jenang is one of the industries engaged in the production of traditional snacks. Currently Omah Jenang does not yet have guidelines for realizing the formation of the composition of production quantities correctly based on calculations that still use forecasting methods. Therefore Omah Jenang often experiences difficulties in providing raw materials and determining maximum profits. The purpose of this study was to de-termine the optimum yield of profits at Omah Jenang by using the simplex method linear program with the help of Lindo Software. The data used is food production data at the Omah Jenang Home Industry in 2022. The method used in this study is the Linear Pro-gramming simplex method and the Lindo Software application. The results of optimization calculations using the linear programming method and the help of Lindo Software are that the maximum profit in a month is IDR 20,000,000 in producing 800 packages of madumongso. Based on the calculation results it can be concluded that the benefits obtained by using the simplex method with the help of Lindo Software can be obtained maximum results.

A basis-deficiency-allowing variation of the simplex method for linear programming

Chamfer distances on the isometric grid are considered. A new method to compute the chamfer distances based on linear optimization is presented. In the LP model the starting pixel is the Origin, that is a triangle of the grid having co-ordinates (0, 0, 0). The co-ordinates of the end pixel of the path give the right-hand side of the model. The variables are the used numbers of the elementary steps. Each type of an elementary step has a uniquely defined weight. Our operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. We give directed graphs with feasible bases as nodes and arcs with conditions on the used weights such that the simplex method of linear programming may step from one feasible basis to another feasible basis. Thus, the possible course of the simplex method can be followed and the optimal bases can easily be captured. Thus, the final result of the analysis is an O(1) checking of the feasibility and optimality conditions. The optimal bases are summarized in a theorem which is the consequence of the general theory of linear programming. The method can be applied for other grids, but it needs to be adjusted for the particular grid.

The simplex method for linear programming is known to be highly efficient in practice, and understanding its performance from a theoretical perspective is an active research topic. The framework of smoothed analysis, first introduced by Spielman and Teng (JACM ’04) for this purpose, defines the smoothed complexity of solving a linear program with d variables and n constraints as the expected running time when Gaussian noise of variance σ2 is added to the LP data. We prove that the smoothed complexity of the simplex method is O(σ−3/2 d13/4log7/4 n), improving the dependence on 1/σ compared to the previous bound of O(σ−2 d2√logn). We accomplish this through a new analysis of the shadow bound, key to earlier analyses as well. Illustrating the power of our new method, we use our method to prove a nearly tight upper bound on the smoothed complexity of two-dimensional polygons.

Given a directed network with two integer weights, cost and delay, associated with each link, quality-of-service (QoS) routing requires the determination of a minimum cost path from one node to another node such that the delay of the path is bounded by a specified integer value. This problem, also known as the constrained shortest path problem (CSP), admits an integer linear programming (ILP) formulation. Due to the integrality constraints, the problem is NP-hard. So, approximation algorithms have been presented in the literature. Among these, the LARAC algorithm, based on the dual of the LP relaxation of the CSP problem, is very efficient. In contrast to most of the currently available approaches, we study this problem from a primal perspective. Several issues relating to efficient implementations of our approach are discussed. We present two algorithms of pseudopolynomial time complexity. One of these allows degenerate pivots and uses an anticycling strategy and the other, called the NBS algorithm, is based on a novel strategy which avoids degenerate pivots. Experimental results comparing the NBS algorithm, the LARAC algorithm, and general purpose LP solvers are presented. In all cases, the NBS algorithm compares favorably with others and beats them on dense networks.

Time Preference, Firm Growth, and Programming Models††Research on this topic was supported in part by a grant from the Halliburton Education Foundation.

On solutions and distribution problems of the linear programming with fuzzy random variable coefficients

A simplex-like method to compute the eigenvalue of an irreducible [formula omitted]-system

In the simplex method for linear programming the algorithmic step of checking the reduced costs of nonbasic variables is called the “pricing” step. If these reduced costs are all of the “right sign” the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable for inclusion in the basis. While theoretically the choice of any profitable vector will lead to a finite termination (provided degeneracy is handled properly) but the number of iterations until termination depends very heavily on the actual choice (which is defined by the selection rule applied). Pricing has long been an area of heuristics to help make better selection. As a result, many different and sophisticated pricing strategies have been developed, implemented and tested. So far none of them is known to be dominating all others in all cases. Therefore, advanced simplex solvers need to be equipped with many strategies so that the most suitable one can be activated for each individual problem instance. In this paper we present a general pricing scheme. It creates a large flexibility in pricing. It is controlled by three parameters. With different settings of the parameters many of the known strategies can be reproduced as special cases. At the same time, the framework makes it possible to define new strategies or variants of them. The scheme is equally applicable to general and network simplex algorithms.

The k-means method is a widely used technique for clustering points in Euclidean space. While it is extremely fast in practice, its worst-case running time is exponential in the number of data points. We prove that the k-means method can implicitly solve PSPACE-complete problems, providing a complexity-theoretic explanation for its worst-case running time. Our result parallels recent work on the complexity of the simplex method for linear programming.

This paper describes a one-dimensional compactor which simultaneously compacts the contents of all cells of a layout hierarchy without changing the hierarchy. The compactor performs both compaction and wire length minimization hierarchically using the power of the Simplex method for linear programs. Compaction of arrays, compaction of overlapping cells, and symmetry preserving compaction are also handled, since these are special cases of layout hierarchies. Using dedicated Simplex algorithms for compaction and wire length minimization, a globally optimum result is produced quickly and efficiently without the use of protection frames or domains and terminals. The compactor corrects design rule violations, preserves wire widths, and maintains terminal connections automatically. It does not yet introduce jogs in wires automatically. Results are provided for a few CMOS modules, including a ROM and an SRAM core.

Five problems of finding efficient vectors as a subset of a finite set of vectors are shown to be related and a common methodology based on the simplex method of linear programming is developed for solving all of them. Randomly generated problems for one of the five types are solved using the method, and the implications regarding computational requirements are discussed.

1. The use of the method of Latin squares, and specifying or excluding those alloying elements whose optimum content is known and is largely independent of the concentrations of the principal alloying elements makes it possible to reduce the number of investigated experimental melts in working out the composition of steel with the required properties. 2. We obtained multiple regression equations type "chemical composition-properties" for weldable low carbon alloy steel, heat-treated to the same strength (σ0 · 2 ⩾ 750 MPa). 3. The simplex method of linear programming as used to optimize the composition of steel with σ0 · 2 ⩾ 750 MPa according to the criteria of ductility, impact toughness, and crack resistance, with constraints on the chemical composition for obtaining satisfactory weldability. The devised weldable steel in sections up to 250 mm after hardening in water and temnering has good ductility, impact toughness, and crack resistance.

Agriculture in India is gradually tending to be less interesting and lucrative profession due to high risks and less reliability. In order to increase net return from agriculture, proper planning and consideration of prevailing constraints is essential. Bhagwanpur distributary command area-based problem, consisting of maximization of net return with available land, water, labour, energy and other affinity constraints was formulated and solved employing modified Simplex method of linear programming. The area allocated under different crops, land productivity and water productivity for various combinations of canal water and ground water use were computed. When more area was allocated under remunerative crops like maize and potato with 100 % tube well water and canal water use by the crops, maximum net return of `2.31× 108 could be obtained. Maximum land productivity for the command area, was 63826 `.ha-1 when 85 % canal water and 100 % ground water were used by the crops in total area of 3426.9 ha; while maximum water productivity of 25.66 `.m-3 was obtained when 100 % canal water and ground water were used. The results may be useful in suggesting farmers on allocation of their agricultural land under profitable interventions to maximize net return, water productivity and land productivity.

I describe a C++ program for computing the smallest enclosing ball of a point set in d-dimensional space, using floating-point arithmetic only. The program is very fast for d ? 20, robust and simple (about 300 lines of code, excluding prototype definitions). Its new features are a pivoting approach resembling the simplex method for linear programming, and a robust update scheme for intermediate solutions. The program with complete documentation following the literate programming paradigm [3] is available on the Web.