Revised Simplex Method Research Articles

A station location design problem in a bike-sharing system with both conventional and electric shared bikes considering bike users’ roaming delay costs

Bike-sharing systems (BSSs) have emerged in many cities worldwide. One key issue regarding the strategic design of BSSs is the deployment of bike stations. Innovations in technology have enabled new types of bikes, such as shared e-bikes, to work alongside conventional shared bikes. However, existing studies on bike station location design mainly focus on single bike type, and there is a lack of a theoretical model to determine the optimal bike station locations for a BSS where both conventional shared bikes and e-bikes are considered. This study investigates the station location design problem in a BSS with conventional shared bikes and e-bikes. The design problem is formulated as a bi-level optimization problem. The upper-level problem is to determine the optimal station locations with the objective of maximizing social welfare, and the lower-level problem is a multi-period multi-modal network equilibrium problem with pick-up and drop-off constraints. The upper-level problem is solved using the Genetic Algorithm, while the rolling horizon method is used to decompose the lower-level problem into multiple period-specific subproblems. Each subproblem is solved via a block Gauss-Seidel decomposition approach coupled with the revised simplex method and column generation. Numerical examples are given to demonstrate the properties of the problem, illustrate the performance of the solution algorithm, and offer key insights into the planning of BSSs.

Integer Programming Application to Optimize Profit of Samarinda Special Souvenir Production


 
 
 
 Yulia's Amplang shop is one of the businesses engaged in the culinary field in the form of souvenirs typical of Samarinda City, namely Amplang. The amplang variants produced by this store include flat fish (original), shrimp, crab, squid and seaweed. Limitations and rising raw material prices make stores often experience difficulties in optimizing production profits. This study aims to determine the number of packages in one production process for each variant produced so as to obtain maximum profit. This study used the revised simplex method and then continued with the Branch and Bound Method, which is a method used to solve linear programming problems that produce solutions in integer form. From the analysis using this method, the optimal amount was obtained as many as 317 packages per production, including 144 flat fish flavors (original), 52 shrimp flavors, 34 crab flavors, 54 squid flavors and 33 seaweed flavors with the maximum profit obtained. namely Rp. 4,125,635. The application of this method resulted in an increase in profits of 5.53% or Rp. 216,103 in one production process compared to the previous company's profits.
 
 
 

An Optimal Model and Production Planning in Upholstery by Linear Integer Programming

This research effort used an upholstery factory as a case study to apply the idea of integer programming for an ideal model. In order to allocate limited and readily available material resources to competing products (Bed (6x6ft), Bed (6x4ft), Wardrobe (6x6ft), Wardrobe (6x4ft), Side Drawer, and Shoe Rack) in the furniture factory with a view to maximizing profit, the Revised Simplex method of solving linear programming and the branch and bound method of solving integer programming models were used in this study. The analysis was carried out with the help of the LINDO software version 6.1 using data containing the availability of materials, sales volume for each of the product types, machine and time it takes to complete the manufacturing process of each product type, as well as profit per unit of the product as collected from the factory. The outcome indicated that 38 units of the Bed (6x6ft), 10 units of the Wardrobe (6x4ft), and 0 units of the other products needed to be manufactured to make a monthly profit of no more than ₦1711700. From the result, it was observed that the Wardrobe (6x4ft) and Bed (6x6ft) contributed majorly to the profit, hence the recommendation that more of the Wardrobe (6x4ft) should be produced and sold for optimum profit.

Using Revised Simplex Modelling to Determine the Optimal Loan Portfolio

Purpose: The main aim of this study is to develop Linear Programming (LP) model to help ADB, Sunyani branch in the Brong Ahafo Region to allocate their funds to prospective loan seekers in order to maximize profits. Methodology: To achieve this aim, a secondary data were extracted from the annual reports and financial statements of the bank. Based on these empirical data, LP model was formulated. A LP Solver based on Revised Simplex Algorithm (a computerized software application) was used to solve the problem. Findings: The results from the model showed that ADB, Sunyani branch would be making annual profit of GH¢ 476732.00 on loans alone as against GH¢ 190693.00 in 2011 if they stick to the model. Recommendation: From the study, it was realized that the Scientific method used to develop the proposed model can have a significant increase in the Bank’s profit margin if put into use.

Penyelesaian Goal Programming menggunakan Metode Simpleks Direvisi dalam Memaksimalkan Keuntungan pada Home Industri Upik Padang Panjang, Sumatera Barat

Upik Home Industry is one of the businesses engaged in the chips industry. Problems in Upik Home Industry that are often faced in decision making are problems that contain several goals in it. To overcome these problems, the Goal Programming approach is used. Goal Programming is an extension of linear programming which is used to solve linear equations with more than one objective function. The purpose of this study was to determine the completion of Goal Programming using the revised simplex method in maximizing profits at Upik Home Industry Padang Panjang, West Sumatra. The method used in this study is the revised simplex method. The revised simplex method uses an inverse basis to perform calculations in determining the incoming and outgoing variables. Based on the results of research, the goal is to minimize working times, minimize raw materials for cassava, oil, salt and maximize sales quotas for sanjai balado and is not achieved. And goals for maximizing profit, goals for minimizing raw material for garlic and minimizing the sales quota for sanjai original, sanjai kunyit dan tusuk gigi is achieved. So that the total production for original sanjai ( as much 30,67 kg, sanjai kunyit ( as much 18 kg and tusuk gigi ( as much 18 kg. Keywords: Goal Programming, Home industry, Revised Simplex.

Optimization of a complex hydropower development operation using HEC-ResSim

In this paper HEC-ResSim is applied for a complex hydropower development formed by five reservoirs and related hydropower plants. There were considered characteristics of five existing hydropower developments in Romania, for which three reservoirs are with annual regulation and two with daily regulation. The objective function was the realization of a planned energy generation for one year (the mean hydrological year). Obtained results are very close to those obtained applying linear programming, a revised simplex algorithm.

Application of Linear Programming for Decision Making to Business in Whao Beverages Nig. Ltd

Linear Programming is a scientific method used for obtaining and determine the most optimal solution for a problem with given constraints. This technique is useful for guidance of a quantitative decision-making in different fields such as business planning, industrial engineering, production firm and to the extent of social and physical sciences. This research examines the application of linear programming to determine the product(s) that maximize(s) the profit of WHAO Beverages Nig. Ltd. The data used for this research is a secondary data obtained from the company. The method used to analyze the data is a revised simplex method of a linear programming. The findings in this research was able to formulate mathematical model for the research problem; determine the product that will yield the maximum profit for the company; and know the quantity of this product that will give this profit. Data obtained from the company undergoes two iterations, whose analysis shows that the squash drinks maximize the profit of the company at the second iteration. The quantity of SQUASH Drinks that gave the maximum profit of 264 units produced and the maximum profit of the company (WHAO Beverages Nig. Ltd.) was #2200.83.

Application of Linear Programming Technique for Optimal Production in Profit Maximization

In this paper, we are concerned with applying linear programming technique to determine optimum product mix of Gusau Sweet Factory Limited to maximize profit. Data on the quantity of major raw materials used in the production of the three different sweets: Toffee. Lollipop and Candy, cost and selling prices and the profit of each product were collected from the extract of the financial record of the company. The problem was formulated as a linear programming problem (LPP) and solved using the revised simplex algorithm. The results show that Toffee and lollipop should be produced daily for the company to realize an optimum profit.

Implementation of an interior point method with basis preconditioning

The implementation of a linear programming interior point solver is described that is based on iterative linear algebra. The linear systems are preconditioned by a basis matrix, which is updated from one interior point iteration to the next to bound the entries in a certain tableau matrix. The update scheme is based on simplex-type pivot operations and is implemented using linear algebra techniques from the revised simplex method. An initial basis is constructed by a crash procedure after a few interior point iterations. The basis at the end of the interior point solve provides the starting basis for a crossover method which recovers a basic solution to the linear program. Results of a computational study on a diverse set of medium to large-scale problems are discussed.

Accelerating Revised Simplex Method Using GPU-Based Basis Update

Optimization problems lie at the core of scientific and engineering endeavors. Solutions to these problems are often compute-intensive. To fulfill their compute-resource requirements, graphics processing unit (GPU) technology is considered a great opportunity. To this end, we focus on linear programming (LP) problem solving on GPUs using revised simplex method (RSM). This method has potentially GPU-friendly tasks, when applied to large dense problems. Basis update (BU) is one such task, which is performed in every iteration to update a matrix called basis-inverse matrix. The contribution of this paper is two-fold. Firstly, we experimentally analyzed the performance of existing GPU-based BU techniques. We discovered that the performance of a relatively old technique, in which each GPU thread computed one element of the basis-inverse matrix, could be significantly improved by introducing a vector-copy operation to its implementation with a sophisticated programming framework. Second, we extended the adapted element-wise technique to develop a new BU technique by using three inexpensive vector operations. This allowed us to reduce the number of floating-point operations and conditional processing performed by GPU threads. A comparison of BU techniques implemented in double precision showed that our proposed technique achieved 17.4% and 13.3% average speed-up over its closest competitor for randomly generated and well-known sets of problems, respectively. Furthermore, the new technique successfully updated basis-inverse matrix in relatively large problems, which the competitor was unable to update. These results strongly indicate that our proposed BU technique is not only efficient for dense RSM implementations but is also scalable.

Revised simplex algorithm for linear programming on GPUs with CUDA

The revised simplex algorithm (RSA) is a typical algorithm for solving linear programming problems. Many theoretical modifications have been done to make the algorithm more efficient, but almost all of them were based on single-instruction single-data architecture processors (CPUs), which could not make full use of the inherent parallel characteristics of RSAs. We propose a novel single-instruction multiple-data architecture processor (GPU) based on the RSA in this paper. The intensive matrix manipulations of a traditional RSA are offloaded to the GPU, which helps to make full use of its powerful parallel processing ability. We implemented the GPU-based RSA on compute unified device architecture (CUDA). Numerical experiments on randomly generated linear programs show that the GPU-based RSA can not only find the correct optimal solutions, but can also reach a speed of up to 100 times as fast as that of a CPU-based RSA: it also runs 3 to 11 times as fast as MATLAB.

A Compromise Solution for the Fully Fuzzy Multiobjective Linear Programming Problems

A new approach is undertaken to solve the fully fuzzy multiobjective linear programming (FFMLP) problem. The coefficients of the objective functions, constraints, right-hand-side parameters, and variables are of the triangular fuzzy number (T r FN)s. A solution strategy, called compromise solution algorithm (CSA), is presented using a three-step procedure. First, a revised simplex method together with Gaussian elimination in the environment of the linear ranking function is used to convert the FFMLP problems partially into semi fully fuzzy multiobjective linear programming (SFFMLP) problems. Then, the obtained SFFMLP problems are gathered together as a single problem. Finally, the gathered problem is solved by one of four different methods to find a fuzzy compromise solution for the FFMLP problems. The CSA is then numerically applied to a FFMLP problem to illustrate the practicability of the proposed procedure.

Parallelizing the dual revised simplex method

This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational results show that the performance of PAMI is superior to that of the leading open-source simplex solver, and that SIP complements PAMI in achieving speedup when PAMI results in slowdown. One of the authors has implemented the techniques underlying PAMI within the FICO Xpress simplex solver and this paper presents computational results demonstrating their value. In developing the first parallel revised simplex solver of general utility, this work represents a significant achievement in computational optimization.

A revised simplex method of solving degenerate fuzzy number linear programming problems

A revised simplex method of solving degenerate fuzzy number linear programming problems

Efficient GPU-based implementations of simplex type algorithms

Recent hardware advances have made it possible to solve large scale Linear Programming problems in a short amount of time. Graphical Processing Units (GPUs) have gained a lot of popularity and have been applied to linear programming algorithms. In this paper, we propose two efficient GPU-based implementations of the Revised Simplex Algorithm and a Primal–Dual Exterior Point Simplex Algorithm. Both parallel algorithms have been implemented in MATLAB using MATLAB’s Parallel Computing Toolbox. Computational results on randomly generated optimal sparse and dense linear programming problems and on a set of benchmark problems (netlib, kennington, Mészáros) are also presented. The results show that the proposed GPU implementations outperform MATLAB’s interior point method.

The Shortlist Method for fast computation of the Earth Mover's Distance and finding optimal solutions to transportation problems.

Finding solutions to the classical transportation problem is of great importance, since this optimization problem arises in many engineering and computer science applications. Especially the Earth Mover's Distance is used in a plethora of applications ranging from content-based image retrieval, shape matching, fingerprint recognition, object tracking and phishing web page detection to computing color differences in linguistics and biology. Our starting point is the well-known revised simplex algorithm, which iteratively improves a feasible solution to optimality. The Shortlist Method that we propose substantially reduces the number of candidates inspected for improving the solution, while at the same time balancing the number of pivots required. Tests on simulated benchmarks demonstrate a considerable reduction in computation time for the new method as compared to the usual revised simplex algorithm implemented with state-of-the-art initialization and pivot strategies. As a consequence, the Shortlist Method facilitates the computation of large scale transportation problems in viable time. In addition we describe a novel method for finding an initial feasible solution which we coin Modified Russell's Method.

Elastoplastic analysis of plane stress/strain structures via restricted basis linear programming

In this paper, elastoplastic analyses of plane-stress and plane-strain structures are addressed. The traditional von Mises yield surface is assumed in the material constitutive model and its piecewise-linear (PWL) approximation is derived. An associated flow rule is adopted and a consistent linear mixed hardening rule is developed in the so called mixed formulation. The relevant mathematical programming (MP) problem, which aims at the maximization of the linear combination of plastic multipliers, is constructed. Restricted basis linear programming (RBLP) is used to solve the MP problem, while special provisions are accounted for to obtain nonholonomic solutions. In order to reduce the required storage space, the Revised Simplex method and a sifting technique are employed. The proposed algorithm is validated and its performance is illustrated by solving some classical nonlinear problems.

Novel update techniques for the revised simplex method

This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple Forrest–Tomlin updates to be performed. Computational results show that one of the product form variants is significantly more efficient than the traditional approach, with its performance approaching that of the Forrest–Tomlin update for some problems. The other is less efficient, but valuable in the context of the dual revised simplex method with suboptimization. Results show that the multiple Forrest–Tomlin updates are performed with no loss of serial efficiency.

A KKT Simplex Method for Efficiently Solving Linear Programs for Grasp Analysis Based on the Identification of Nonbinding Constraints

A one-phase efficient method to solve linear programming (LP) problems for grasp analysis of robotic hands is proposed. Our method, named as KKT Simplex method, processes free variables directly while choosing the entering and leaving variables, which makes it a one-phase method able to start at any point of the set of feasible solutions. Besides, the proposed method lowers the number of simplex steps by an angular pricing strategy to choose the entering variable. Moreover, the method reduces the size of an LP problem by the identification of nonbinding constraints that preserves the Karush-Kuhn-Tucker (KKT) cone. We developed the KKT Simplex method by incorporating to the well-known revised simplex method the following components: a method to process free variables, a pricing strategy, and an identification method. We solve LP problems of grasp analysis to test the efficiency and the one-phase nature of the proposed method.

A KKT Simplex Method for Efficiently Solving Linear Programs for Grasp Analysis Based on the Identification of Nonbinding Constraints

A one-phase efficient method to solve linear programming (LP) problems for grasp analysis of robotic hands is proposed. Our method, named as KKT Simplex method, processes free variables directly while choosing the entering and leaving variables, which makes it a one-phase method able to start at any point of the set of feasible solutions. Besides, the proposed method lowers the number of simplex steps by an angular pricing strategy to choose the entering variable. Moreover, the method reduces the size of an LP problem by the identification of nonbinding constraints that preserves the Karush-Kuhn-Tucker (KKT) cone. We developed the KKT Simplex method by incorporating to the well-known revised simplex method the following components: a method to process free variables, a pricing strategy, and an identification method. We solve LP problems of grasp analysis to test the efficiency and the one-phase nature of the proposed method.