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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

Secure multiparty computation is a basic concept of growing interest in modern cryptography. It allows a set of mutually distrusting parties to perform a computation on their private information in such a way that as little as possible is revealed about each private input. The early results of multiparty computation have only theoretical signi??cance since they are not able to solve computationally complex functions in a reasonable amount of time. Nowadays, e??ciency of secure multiparty computation is an important topic of cryptographic research. As a case study we apply multiparty computation to solve the problem of secure linear programming. The results enable, for example in the context of the EU-FP7 project SecureSCM, collaborative supply chain management. Collaborative supply chain management is about the optimization of the supply and demand con??guration of a supply chain. In order to optimize the total bene??t of the entire chain, parties should collaborate by pooling their sensitive data. With the focus on e??ciency we design protocols that securely solve any linear program using the simplex algorithm. The simplex algorithm is well studied and there are many variants of the simplex algorithm providing a simple and e??cient solution to solving linear programs in practice. However, the cryptographic layer on top of any variant of the simplex algorithm imposes restrictions and new complexity measures. For example, hiding the number of iterations of the simplex algorithm has the consequence that the secure implementations have a worst case number of iterations. Then, since the simplex algorithm has exponentially many iterations in the worst case, the secure implementations have exponentially many iterations in all cases. To give a basis for understanding the restrictions, we review the basic theory behind the simplex algorithm and we provide a set of cryptographic building blocks used to implement secure protocols evaluating basic variants of the simplex algorithm. We show how to balance between privacy and e??ciency; some protocols reveal data about the internal state of the simplex algorithm, such as the number of iterations, in order to improve the expected running times. For the sake of simplicity and e??ciency, the protocols are based on Shamir's secret sharing scheme. We combine and use the results from the literature on secure random number generation, secure circuit evaluation, secure comparison, and secret indexing to construct e??cient building blocks for secure simplex. The solutions for secure linear programming in this thesis can be split into two categories. On the one hand, some protocols evaluate the classical variants of the simplex algorithm in which numbers are truncated, while the other protocols evaluate the variants of the simplex algorithms in which truncation is avoided. On the other hand, the protocols can be separated by the size of the tableaus. Theoretically there is no clear winner that has both the best security properties and the best performance.

Abstract: Linear Programming (LPP) is a mathematical method used to optimize resource allocation systems with linear relationships, it aims to maximize or minimize a linear objective function while adhering to linear constraints. This paper showcase how to optimize linear programming problems through the simplex and graphical methods. The excel environment facilitates efficient modeling and solution generation. The simplex method is executed step – by – step, showcasing its versatility, while the graphical method provides visual insights. The solver add in automates the optimization process, adjusting decision variables within constraints. Comparative analysis reveals insights into the strengths of both the methods used in the process of the study. This research contributes to practical applications of LP, emphasizing the significance of excel and solver for real – world problem solving

A Survey of Lagrangean Techniques for Discrete Optimization

In this paper, we discusses the problem that involving the conflict fuzzy multi-objective in transportation planning which is one of special case in multi-objective linear programming. To find the simultaneously optimal solution of the problem, we use the Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) method. This method can reduce the fuzzy multi-objective linear programming problem into deterministic single objective linear programming which can be solved using simplex method. Beside that , with IFMOLP method, decision maker (DM) can establish interactively the goal from the objective function to produce the pareto optimal solution so that.


 The simplex method is a method used to solve linear programming problems with repeated calculations (iterations) that are performed many times until an optimal solution is reached. In solving linear programming problems, it is necessary to determine the constraints and inequalities contained in linear programming. The simplex method can solve linear programming problems involving many inequalities and many variables. Therefore, the purpose of this research is to optimize the production cost of sandals in the Diona Shoes Home Industry. The steps taken to optimize the production cost of sandals include: observing and collecting data, then making a mathematical model, then optimizing production results using the simplex method (iteration), and finally optimizing production results with the Lindo 16.0 software tool.
 Keywords: Diona Shoes, Linear Programming, Simplex Method, Lindo 16.0.

The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most violated reduced cost. In their seminal work, Klee and Minty showed that this pivot rule takes exponential time in the worst case. We prove two main results on the simplex method. Firstly, we show that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzig's pivot rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a specific variable to enter the basis is PSPACE-complete. We use the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs. We construct MDPs and then show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method.

This paper presents a method, called the convex simplex method, for minimizing a convex objective function subject to linear inequality constraints. The method is a true generalization of Dantzig's linear simplex method both in spirit and in the fact that the same tableau and variable selection techniques are used. With a linear objective function the convex simplex method reduces to the linear simplex method. Moreover, the convex simplex method actually behaves like the linear simplex method whenever it encounters a linear portion of a convex objective function. Many of the sophisticated techniques designed to enhance the efficiency of the linear simplex method are applicable to the convex simplex method. In particular, as an example, a network transportation problem with a convex objective function is solved by using the standard transportation tableau and by only slightly modifying the usual procedure for a linear objective function.

A hybrid particle swarm-genetic algorithm optimization scheme was combined with Nelder and Mead's simplex method to produce an experimental hybrid method that integrates the global search capabilities of evolutionary heuristics with the highly effective localized search ability of the simplex method. This algorithm was applied to various missile design problems where the goal is to design a missile that can match the given flight trajectory. Three different flight trajectories were analyzed for this work: a 117 km trajectory, a 164 km trajectory, and a 221 km trajectory. The efficiency and effectiveness of the performance of the simplex method was compared with the performance of the same algorithm using the pattern search method as the local search routine in lieu of the simplex method. Optimizer performance was evaluated according to two criteria: (1) “fitness function” accuracy (how closely solutions are able to match a given objective function), and (2) convergence speed (how many calls to the objective function are required in order to reach an acceptable solution). The pattern search proved to be a more robust and reliable algorithm for such a large and complex solution space, as it was always able to avoid getting lost in the solution space. Conversely, the simplex method would occasionally become trapped around a bad solution, typically while traversing regions of the solution space sensitive to discontinuities formed by mixed-variable conversions.

Elementary Linear Programming with Applications

Chapter two - Linear programming

Problem-solving in mathematics is a fundamental skill that must be mastered by individuals. However, previous research indicates that students' problem-solving abilities tend to be moderate to low. Many factors contribute to this, including the factor of number sense ability. The objective of this study is to describe how number sense and problem-solving skills are used by mathematics education students with low number sense ability in solving linear programming problems using the simplex method. This research is qualitative descriptive involving 36 third-year mathematics education students as participants. Subjects were selected using purposive sampling technique based on a number sense ability test to identify those with low number sense ability. Participants were given an advanced test involving linear programming problems solved using the simplex method. The results indicate that mathematics education students with low number sense ability struggle to solve problems accurately. Stages of problem-solving that were not performed effectively included gathering information, planning, and implementing the plan. Furthermore, students did not engage in the reviewing stage. Moreover, students were not flexible in manipulating numbers, especially when dealing with large numbers. As a result, more time was required for computations, and the iteration process in the simplex method could not be effectively completed. Keywords: linear programming, number sense, number sense ability, problem solving, simplex method. DOI: https://dx.doi.org/10.23960/jpmipa/v25i1.pp103-115

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.

This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to be a special case of convex programming, but is more substantial, and is really an embodiment of the theory of systems of linear inequalities (as reflected here). This chapter initiates the subject with reference to systems of linear inequalities and natural questions about them, and all LP (linear programming) theorems are encountered simply in pursuing those. Theorems about linear inequalities that have uses directly on their own are also derived (and are illustrated in many places in this book). The eight sections of the chapter are: linear inequalities; separation theorems; theorems of alternatives; polyhedra and polytopes; LP Duality Theorem; the pivot operation; the Simplex Algorithm; and BASIC program.

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.