A basis-deficiency-allowing variation of the simplex method for 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.

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.

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.

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.

In this technical note, the numerical steps for the simplex method of linear programming are reviewed and the tolerances needed in the numerical procedure are defined. Objective criteria are given for accomplishing the numerical steps of the method and the calculation of necessary tolerances.

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

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

The need for cake as a complementary food at various events causes the demand to increase. This increase has an impact on the production level that will be produced with the aim of increasing profits. The production process is related to the allocation of the right raw materials to be efficient, so that maximum profits will be obtained. This study aims to analyze the maximum profit of the Maesa Cake and Bakery Shop, Permata Legenda Branch. There are two types of products sold in this shop, namely Banana Bolu (X1) and Banana Bolen (X2). The research method used is linear programming which is solved by the simplex method and the use of the POM For Windows application. The data obtained is observational data by meeting the shop owner directly to be interviewed. The process of calculating the simplex method is carried out using Ms. Excel with several completion steps, starting from creating a simplex table, entering the coefficients of the objective function and constraints, looking for key columns, key rows, and key numbers, until the last step is achieving positive coefficients generated in the objective function row. The final result obtained is the process of the iterations carried out. Apart from using Ms. Excel, researchers also use the POM for Windows application. Use both software to find the maximum profit to be achieved. So it is found that the maximum profit that can be obtained in one day by the Maesa Cake and Bakery Branch of Permata Legenda is IDR 270,000.00; by producing 5 units of Banana Bolu and 50 units of Banana Bolen.

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.