Revised Primal Simplex Algorithm

The simplex method is frequently the most efficient method of solving linear programming (LP) problems. This paper reviews previous attempts to parallelise the simplex method in relation to efficient serial simplex techniques and the nature of practical LP problems. For the major challenge of solving general large sparse LP problems, there has been no parallelisation of the simplex method that offers significantly improved performance over a good serial implementation. However, there has been some success in developing parallel solvers for LPs that are dense or have particular structural properties. As an outcome of the review, this paper identifies scope for future work towards the goal of developing parallel implementations of the simplex method that are of practical value.

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.

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.

Linear programming algorithms have been widely used in Decision Support Systems. These systems have incorporated linear programming algorithms for the solution of the given problems. Yet, the special structure of each linear problem may take advantage of different linear programming algorithms or different techniques used in these algorithms. This paper proposes a web-based DSS that assists decision makers in the solution of linear programming problems with a variety of linear programming algorithms and techniques. Two linear programming algorithms have been included in the DSS: (i) revised simplex algorithm and (ii) exterior primal simplex algorithm. Furthermore, ten scaling techniques, five basis update methods and eight pivoting rules have been incorporated in the DSS. All linear programming algorithms and methods have been implemented using MATLAB and converted to Java classes using MATLAB Builder JA, while the web interface of the DSS has been designed using Java Server Pages.

As we have shown, the simplex method refers to the so-called finite methods which allow one to find a solution for any problem of linear programming or to prove its unsolvability performing a finite number of elementary operations (addition, subtraction, multiplication, division, comparison of two real numbers). It stands to reason that the number of operations depends on the dimension (n, m) of the problem (n is the number of variables, m is the number of equality and inequality constraints). The practice of solving linear programming problems has shown that the simplex method and its modifications are very effective. It is accepted as a fact that in the majority of linear programming problems the number of elementary operations which are necessary for their solution is of the order O(n 2 m + m 2 n) [1, 91]. Here and in the sequel we denote by O(a) the quantities for which |O(a)| ≤ C(a), where C is a positive constant independent of a. We have found out that there exist “poor” linear programming problems in which the amount of elementary operations required for their solution by the simplex method is estimated by the number which exponentially depends on n and m.

In the real word, there are many problems which have linear programming models and sometimes it is necessary to formulate these models with parameters of uncertainty. Many numbers from these problems are linear programming problems with fuzzy variables. Some authors considered these problems and have developed various methods for solving these problems. Recently, Mahdavi-Amiri and Nasseri (2007) considered linear programming problems with trapezoidal fuzzy data and/or variables and stated a fuzzy simplex algorithm to solve these problems. Moreover, they developed the duality results in fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables. Here, the authors show that this presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or post optimality) analysis using primal simplex tableaus.

Ramp metering is a motorway control method to avoid onset congestion through limiting the access of ramp inflows into the main road of the motorway. The optimization model of ramp metering is developed based upon cell transmission model (CTM). With the piecewise linear structure of CTM, the corresponding motorway traffic optimization problem can be formulated as a linear programming (LP) problem. It is known that LP problem can be solved by established solution algorithms such as SIMPLEX or interior-point methods for the global optimal solution. The commercial software (CPLEX) is adopted in this study to solve the LP problem within reasonable computational time. The concept is illustrated through a case study of the United Kingdom M25 Motorway. The optimal solution provides useful insights and guidances on how to manage motorway traffic in order to maximize the corresponding efficiency.

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

A characteristic feature of the primal network simplex algorithm (NSA) is that it usually makes a large number of degenerate iterations. Though cycling and even stalling can be avoided by recently introduced pivot rules for NSA, the practical efficiency of these rules is not known yet. For the case when the simplex algorithm is used to solve the continuous linear programming (LP) problem there exists a practical anti-cycling procedure that proved to be efficient. It is based on an expanding relaxation of the individual bound on the variables. In this paper we discuss the adaptation of this method to NSA, taking advantage of the special integer nature of network problems. We also give an account of our experience with these ideas as they are experimentally implemented in the MINET network LP solver. Reductions of CPU time have been achieved on a smaller set of specially structured real-life problems.

In this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

Elementary Linear Programming with Applications

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.