Optimal Basic Solution Research Articles

Comparative view of the modified Vogel’s approximation methods in transportation problem

Recently, the whole world is experiencing economic hardship and one fast road to ply is to cut out some expenses. Xlife group is equally affected and so the study proposes a means to minimize its transportation cost if effected. A comparative surgery on Vogel’s approximation with its modified versions were conducted on minimizing the cost of transporting manufactured goods from (Xlife group’s) points of supply to certain destinations was undertaken. The Vogel's Approximation Method (VAM) and the Modified Vogel's Approximation Method based on standard deviation, distribution indicators and average cost were applied to determine an initial basic feasible solution and optimal solution for certain transportation problems. All methods yield identical minimum transportation cost except at an instance.

Advanced method for solving the transportation problem

Transportation problem is one of the predominant areas of operations research. It is a type of linear programming problem designed to minimize the cost of distributing a single product from m sources to n destinations. Transportation problem is widely used as a decision making tool in different fields such as engineering, business management and many others. These types of problems can be solved by general network methods. In this paper The Advanced Method for Solving the Transportation Problem (AMSTP) is proposed to find initial basic feasible solution and optimal solution of transportation problem and compared its results with other existing methods.

Research on fuel injection characteristics of common rail system based on high pressure pipeline integration and matching

AbstractThe modular prediction model of high pressure common rail system was established to study the influence of high‐pressure pipeline integration matching on the fuel injection characteristics of common rail system. The influence of common rail volume on pressure building and pressure fluctuation in common rail tubes was analyzed. The influence of common rail volume on the matching of fuel injection frequency (number of fuel injectors) and fuel supply frequency (number of plunger pumps) was studied by using the control variable method, and the basic optimal solution of the system was obtained. The influence of the common rail structure on the fuel flow characteristics of the system was also studied. The matching relationship between high pressure tubing and common rail system is studied by introducing dimensionless numbers such as length‐diameter ratio of pipeline. The results demonstrate that the increase of the storage space of the accumulator can effectively reduce the fluctuation in the system, so that the injection volume increases slowly and then decreases rapidly, with a slope of −5.70 × 10−3 mm3. The increase of the diameter of the common rail makes the fuel flow characteristics become good first and then bad, and the increase of the length of the common rail can soothe the pressure fluctuations. The increase of the diameter of the link tube (LT) makes the fuel injection volume of the injector increase first and then decrease. The optimal VR solution for the LT is 0.0249 and the optimal ratio of the length to diameter of a pipeline solution is 55.10. The structure of the distribution tube and its matching to the common rail have little effect on the fuel flow characteristics. When the diameter and length of the injection tube are too large, the influence on the injection pressure of the injector is negatively correlated.

An application of generalized symmetric type-2 intuitionistic fuzzy variables to a transportation problem with the effect of a new ranking function

The primary objective of this study is to establish and introduce generalized symmetric type-2 intuitionistic fuzzy set, a type-2 fuzzy set extension. The major goal of such an extension is to enable the concept of fuzzy outranking relation which is known as foot print of uncertainty (FOU) to be employed in generalized symmetric type-2 intuitionistic fuzzy number (GST2IFN) for the first time. This is interesting to deal this GST2IFNs and it is complicated to find the equivalent crisp quantities due to this special structure. Hence, we have introduced a new ranking function using the mean level set of membership and non-membership functions. Besides, the arithmetic operations of the GST2IFNs are discussed. To express the effectiveness and application of the proposed GST2IF variables, a transportation problem (TP) is formulated with all cost parameters are GST2IFNs. In addition, few different algorithmic approaches are applied to find the initial basic feasible solution (IBFS) and optimal solution of the TP. Finally, the feasibility and richness of the obtained optimal solution is discussed.

New Algorithm Permitting the Construction of an Effective Spanning Tree

Abstract In this paper, we have done a rapid and very simple algorithm that resolves the multiple objective combinatorial optimization problem. This, by determining a basic optimal solution, which is a strong spanning tree constructed, according to a well-chosen criterion. Consequently, our algorithm uses notions of Bellman’s algorithm to determine the best path of the network, and Ford Fulkerson’s algorithm to maximise the flow value. The Simplex Network Method that permits to reach the optimality conditions manipulates the two algorithms. In short, the interest of our work is the optimization of many criteria taking into account the strong spanning tree, which represents the central angular stone of the network. To illustrate that, we propose to optimize a bi-objective distribution problem.

Model Optimasi Produksi Kue Menggunakan Metode Simpleks Pada Toko Kue Rosalina Cabang Jatake

The continuity and development of the cake production business needs to be maintained with steps to be able to allocate raw materials and increase profits. Therefore, an effort is needed using a method in determining the right combination of production factors from the products made and the combination of products produced. This problem can be overcome by creating an optimization model using the simplex method. The simplex method is a method that mathematically starts from a basic feasible solution to another basic feasible solution and is done repeatedly (iterative) so that finally an optimal basic solution is obtained. The optimum value is obtained by analyzing whether the resources have been fully used (scarce) or excessively (Abundant) used with the optimal simplex table approach. Based on the results of iteration 3, it is obtained that line Z no longer has a negative value so that the solution obtained is optimal, meaning that Rosalina Bakery cake shop if you want to get maximum profit, it produces brownie cakes (x1) as much as 3.3333; Bolu Pandan (x2) as much as 8.6667. The maximum profit will be achieved at Rp 93,999.9 rounded up to Rp 94.000,- per day. Based on the results of the analysis, it shows that the cake production optimization model can help maximize profits from limited resources..Keywords: model, simplex, optimization, production, profit, maximum

Limit laws for empirical optimal solutions in random linear programs

We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. For the corresponding random linear program, we characterize under general assumptions the random fluctuations of the empirical optimal solutions around their population quantities after standardization by a distributional limit theorem. Our approach is geometric in nature and further relies on duality and the collection of dual feasible basic solutions. The limiting random variables are driven by the amount of degeneracy inherent in linear programming. In particular, if the corresponding dual linear program is degenerate the asymptotic limit law might not be unique and is determined from the way the empirical optimal solution is chosen. Furthermore, we include consistency and convergence rates of the Hausdorff distance between the empirical and the true optimality sets as well as a limit law for the empirical optimal value involving the set of all dual optimal basic solutions. Our analysis is motivated from statistical optimal transport that is of particular interest here and distributional limit laws for empirical optimal transport plans follow by a simple application of our general theory. The corresponding limit distribution is usually non-Gaussian which stands in strong contrast to recent finding for empirical entropy regularized optimal transport solutions.

Use of Excel Solver for Minimizing the Total Transportation Cost at a Recently Opened Clothing Boutique

Optimization is the basis of any business. It can be of maximization type or minimization type. When the optimization will be of maximization type, the focus will be on Profit maximization or Sales maximization and when it will be of minimization type, main focus will be on cost minimization, waste minimization or resource minimization. Be it any kind of optimization, business always focuses on the costing part. Reduction in cost can lead to the maximization of profit. In business, there are different kinds of costs involved, like production cost, transportation cost, packing cost, marketing cost and many more. Transportation cost is not the direct cost, which is involved in manufacturing of product, but it consumes too much of finance. If any business wants to gain competitive advantage over its competitors, transportation cost is the important factor, which needs to be considered. There are different transportation problem solution methods are available like Least cost method, North West corner method, Vogel’s Approximation method, Modified distribution method and Stepping Stone method. Though these methods can provide the initial basic feasible solution and optimal solution also, but all are manual method. In this research paper, researcher has presented how the simple excel spreadsheet can be used for getting the optimal transportation cost. A budding clothing boutique has been operating for two years, selling different clothing garments wants to increase its area of operation. The store does not have any transportation software to focus on the shipping cost. Most of the big companies use different kind of software’s to minimize the transportation cost. But in this research paper, researcher has presented, how the clothing boutique has been benefited with the spreadsheet and how the minimum transportation cost model has been achieved.

Wind vibration control of stay cables using an evolutionary algorithm

In steel cable bridges, the use of magnetorheological (MR) dampers between butt cables is constantly increasing to dampen vibrations caused by rain and wind. The biggest problem in the actual applications of those devices is to launch a kind of appropriate algorithm that can effectively and efficiently suppress the perturbation of the tie through basic calculations and optimal solutions. This article discusses the optimal evolutionary design based on a linear and quadratic regulator (hereafter LQR) to lessen the perturbation of the bridges with cables. The control numerical algorithms are expected to effectively and efficiently decrease the possible risks of the structural response in amplification owing to the feedback force in the direction of the MR attenuator. In addition, these numerical algorithms approximate those optimal linear quadratic regulator control forces through the corresponding damping and stiffness, which significantly lessens the work of calculating the significant and optimal control forces. Therefore, it has been shown that it plays an important and significant role in the practical application design of semiactive MR control power systems. In the present proposed novel evolutionary parallel distributed compensator scheme, the vibrational control problem with a simulated demonstration is used to evaluate the numerical algorithmic performance and effectiveness. The results show that these semiactive MR control numerical algorithms which are present proposed in the present paper has better performance than the optimal and the passive control, which is almost reaching the levels of linear quadratic regulator controls with minimal feedback requirements.

Wind vibration control of stay cables using an evolutionary algorithm

In steel cable bridges, the use of magnetorheological (MR) dampers between butt cables is constantly increasing to dampen vibrations caused by rain and wind. The biggest problem in the actual applications of those devices is to launch a kind of appropriate algorithm that can effectively and efficiently suppress the perturbation of the tie through basic calculations and optimal solutions. This article discusses the optimal evolutionary design based on a linear and quadratic regulator (hereafter LQR) to lessen the perturbation of the bridges with cables. The control numerical algorithms are expected to effectively and efficiently decrease the possible risks of the structural response in amplification owing to the feedback force in the direction of the MR attenuator. In addition, these numerical algorithms approximate those optimal linear quadratic regulator control forces through the corresponding damping and stiffness, which significantly lessens the work of calculating the significant and optimal control forces. Therefore, it has been shown that it plays an important and significant role in the practical application design of semiactive MR control power systems. In the present proposed novel evolutionary parallel distributed compensator scheme, the vibrational control problem with a simulated demonstration is used to evaluate the numerical algorithmic performance and effectiveness. The results show that these semiactive MR control numerical algorithms which are present proposed in the present paper has better performance than the optimal and the passive control, which is almost reaching the levels of linear quadratic regulator controls with minimal feedback requirements.

A novel method to obtain initial basic solution and optimal solution of pentagonal fuzzy transportation problem

A novel method to obtain initial basic solution and optimal solution of pentagonal fuzzy transportation problem

Improving a primal–dual simplex-type algorithm using interior point methods

ABSTRACTInterior point methods and simplex-type algorithms are the most widely-used algorithms for solving linear programming problems. The simplex algorithm has many important applications. Hence, even small improvements in simplex-type algorithms could result in noticeable practical impact. This paper presents a hybrid algorithm that combines the strengths of interior point methods and exterior point simplex algorithms. It applies an interior point method for a few iterations leading to significant improvement of the objective function value. At this point, the proposed algorithm uses an exterior point simplex algorithm to find an optimal solution. A crucial point is the selection of the interior point that will be used by the exterior point simplex algorithm to calculate a direction to the feasible region. The goal of the proposed implementation is twofold: (i) improve the performance of the exterior point algorithm, and (ii) find an optimal basic solution starting from an interior point (purification process). The latter goal is very important since an optimal basic solution can be used to solve closely related linear programming problems (warm-start) and linear programming relaxations of integer programming problems. Computational results on a set of benchmark problems (Netlib, Kennington, Mészáros) are presented to demonstrate the efficiency of the proposed hybrid algorithm. The results show that the proposed algorithm is faster than the exterior point simplex algorithm.

A New Technique for Solving Transportation Problems by Using Decomposition-Based Pricing and its Implementation in Real Life

In this paper, we develop a new technique for solving transportation problems (TP) and develop a computer code by using mathematical programming language AMPL. There are many existing techniques for solving TP problems in use. By these techniques one has to determine initial basic feasible solution at first then improve this solution to determine optimal solution by another method. But this process is very lengthy and time consuming. By our technique we can determine optimal solution directly without determining initial basic feasible solution and optimal solution separately and we hope that this technique will provide an easier way than that of the other methods. We use the idea of decomposition based pricing (DBP) method to develop our technique. To our knowledge, there is no other paper which used DBP to solve TP. We demonstrate our technique by solving real life models developed by collecting data from a business organization of Bangladesh.Dhaka Univ. J. Sci. 64(1): 45-50, 2016 (January)

On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers

In this paper we introduce Hexagonal intuitionistic fuzzy number with its membership and non membership functions. The main objective of this paper is to introduce an Intuitionistic Fuzzy Transportation problem with hexagonal intuitionistic fuzzy number. The arithmetic operations on hexagonal intuitionistic fuzzy numbers are performed. Based on this new intuitionistic fuzzy number, we obtain a initial basic feasible solution and optimal solution of intuitionistic fuzzy transportation problem. The solutions are illustrated with suitable example.

Dual relaxations of the time-indexed ILP formulation for min–sum scheduling problems

Linear programming (LP)-based relaxations have proven to be useful in enumerative solution procedures for \(\mathbf {NP}\)-hard min–sum scheduling problems. We take a dual viewpoint of the time-indexed integer linear programming (ILP) formulation for these problems. Previously proposed Lagrangian relaxation methods and a time decomposition method are interpreted and synthesized under this view. Our new results aim to find optimal or near-optimal dual solutions to the LP relaxation of the time-indexed formulation, as recent advancements made in solving this ILP problem indicate the utility of dual information. Specifically, we develop a procedure to compute optimal dual solutions using the solution information from Dantzig–Wolfe decomposition and column generation methods, whose solutions are generally nonbasic. As a byproduct, we also obtain, in some sense, a crossover method that produces optimal basic primal solutions. Furthermore, the dual view naturally leads us to propose a new polynomial-sized relaxation that is applicable to both integer and real-valued problems. The obtained dual solutions are incorporated in branch-and-bound for solving the total weighted tardiness scheduling problem, and their efficacy is evaluated and compared through computational experiments involving test problems from OR-Library.

Optimal renewable energy systems for industries in rural regions

BackgroundEnergy systems for smart regions will require on the one hand a much higher share of renewable energy sources for heat and electricity and on the other hand a high standard integration of industry and utilities supplying households and businesses. As energy options become more diverse, integration of industry into the energy provision for settlements and vice versa become important from an economic and ecological point of view. A particular problem is the supply of bio-energy to large industrial complexes as this requires considerable efforts to manage resource provision as well as constructing smart systems to utilise available resources without posing direct competition to the supply of food and feed.MethodsThis paper will discuss the application of Process Network Synthesis (PNS) using the P-Graph method as a tool to generate optimal energy systems supplying industry with energy based on bio-resources in the context of a rural region and small urban centre. Application of these methods to a real life case study in an industrial city in rural context (Weiz in Austria) will be discussed from the viewpoint of general lessons to be learned from the design of complex energy systems linking different energy provision sources to supply industry and residential areas in a systemic way with energy.ResultsThe synthesis resulted in a basic optimal solution containing the economically most feasible regional energy-technology network. Scenarios show differences in resource use and the creation of revenue. Anaerobic digestion, CHP and wood gasification appear for most of the scenarios.ConclusionsThe case study shows that a support of large consumers with sustainable regional energy systems can already be realised with current market prices. Heat demand outside industry critically influences the economic success of regional technology networks. Feed-in tariffs remain as a key factor for the creation of revenue. As regional parameters like resource availability, prices and locations where technologies could be situated change, the economic success also changes. Price limits and stable technologies can be identified.

Efficient computation of tolerances in the weighted independent set problem for some classes of graphs

After an optimal solution of a combinatorial opti� mization problem (COP) is obtained, the next natural step is an analysis of its sensitivity. The sensitivity anal� ysis of an optimal solution of COP consists in deter� mining the dependence of this solution on the varia� tion of the initial data. The simplest sensitivity analysis studies the perturbations of only one element of an optimal solution. The purpose of the study of such perturbations is the determination of tolerances, that is, the maximum variation of an individual cost (such as weight, time, etc.) under which the optimality of a given solution is preserved, provided that the remain� ing COP’s data are fixed. The interest in tolerances is caused by that the min� imum tolerance of elements of an optimal solution of a problem is a lower bound for the stability radius of this optimal solution and serves as a basis for develop� ing and improving algorithms for some COPs. The first implicit algorithmic application of tolerances appeared in Vogel’s method [12] for finding a solution closest to an optimal basic solution in the simplex method for solving the transportation problem. Among the cases where tolerances were successfully

Interpretation of Different Approaches to Sensitivity Analysis in Cash Flow Problem

Cash is the driving power of all business and cash-flow statement is one of the major issues of institutions, especially in crisis. Optimal cash-flow plan of a company could be one of the most important indicators of that business's financial health and can be considered as its financial analysts' ability and skill. Linear optimization (LO) is one of the mathematical tools in modeling the cash-flow problem and its rich literature helps analysts to device the optimal one when the situation satisfies the requirements of the LO model. However, the situation is always due to variation and the optimal solution arisen from the LO model have to be analyzed according to measurable variation of input data. Sensitivity analysis and parametric programming is the tool to this analysis. Using the Simplex method to find a basic optimal solution and having multiple optimal solutions is one of the reasons that different solvers lead to different optimal solutions. In these situations, sensitivity analysis may produce confusing results. Moreover, there are different points of views to sensitivity analysis such as optimal basis invariancy, optimal partition invariancy, support set invariancy to name some examples. Here, we briefly review different approaches to sensitivity analysis in LO and a short term cash-flow problem of a dummy institution is modeled as an LO problem. It is shown that the problem has multiple optimal solutions which are degenerate, the situation that usually occurs in practice and causes of ambiguous and unclear results. The confusing results in analyzing the sensitivity of these solutions are highlighted in this example. Then, a strictly complementary optimal solution is provided and its useful interpretation in sensitivity analysis is mentioned in a nutshell. In the sequel, the concept of the results arising in different points of views to sensitivity analysis is analyzed. Keywords: Cashflow Problem; Linear Optimization; Sensitivity Analysis. 2010 Mathematics Subject Classification: 62P05

Parallel distributed-memory simplex for large-scale stochastic LP problems

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.

Type II sensitivity analysis of cost coefficients in the degenerate transportation problem

This paper focuses on sensitivity analysis of the degenerate transportation problem (DTP) when perturbation occurs on one cost coefficient. The conventional Type I sensitivity analysis of the transportation problem (TP) determines the perturbation ranges for the invariant optimal basis. Due to different degenerate optimal basic solutions yielding different Type I ranges, the Type I range is misleading for the DTP. Type II sensitivity analysis, which determines the perturbation ranges for the invariant shipping pattern, is more practical for the DTP. However, it is too tedious to obtain Type II ranges by enumerating all optimal basic solutions and all primal optimal basic solutions while getting the union of each corresponding Type I ranges. Here, we propose two labeling algorithms to determine the Type II ranges of the cost coefficient. Besides, three lemmas are provided for obtaining the upper bound or lower bound of the Type II ranges of the cost coefficient directly under specific conditions of the DTP. A numerical example is given to demonstrate the procedure of the proposed labeling algorithms and computational results have been provided.