Basic Feasible 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.

A NEW DUAL HEURISTIC ALGORITHM FOR FINDING THE INITIAL BASIC FEASIBLE SOLUTION FOR A TRANSPORTATION PROBLEM

Because determining the best initial basic feasible solution (IBFS) for a transportation problem is so crucial, numerous authors have expended a great deal of energy developing effective algorithms that will result in the lowest possible cost of moving products from a given source to a destination. The goal of this work was to develop an efficient dual algorithm for finding an initial basic feasible solution to a transportation problem (TP). Two distinct algorithms that produce the same IBFS make up our suggested approach. Compared to some popular methods in the literature, Using four numerical examples, the Row Minimum Method (RMM), Column Minimum Method (CMM), Least Cost Method (LCM), Extremum Difference Method (EDM), Northwest Corner Method (NWCM), Vogel's Approximation Method (VAM), etc. In comparison to the other heuristic techniques compared with the optimal dictate solution modified distribution (MODI), the proposed heuristic approach (PS-DESPAN) approximation was shown to provide a better starting solution (a solution that is extremely close to the optimal solution)

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.

An efficient computational approach for basic feasible solution of fuzzy transportation problems

An efficient computational approach for basic feasible solution of fuzzy transportation problems

To Solve Pythagorean Fuzzy Decagonal Transportation Problem Using Bipartite Graph with Optimal Solution

Objectives: This study aims to solve the Pythagorean fuzzy transportation problem by means of providing a novel approach. Methods: To determine the optimum decision-making, we applied the MODI approach. To illustrate the proposed approach, we applied several types of numerical issues. An algorithm for solving the suggested technique is described. Findings: This study presents a bipartite graph-based strategy to the Pythagorean fuzzy transportation problem. Applying TORA statistical software, the data analysis discovered that the crisp model becomes linear when the membership function is applied. Novelty: This study has paved a way for better optimal results to obtain when the Pythagorean fuzzy set is converted using the score function instead of the standard techniques for Defuzzification. Keywords: Score Function, Pythagorean Fuzzy, Defuzzification, Pythagorean Fuzzy Transportation Problem, Decagonal Fuzzy Number, Ranking Method, Initial Basic Feasible Solution, and Optimal Solution

Transportation Problem via Median Approximatiom Technique for IBFS and Revised Modi’s Technique

The work sought to improve the modified distribution method (MODI) and propose an approach for estimating the initial basic feasible solution (IBFS) for a transportation problem. The Median Approximation Technique (MAT), an explicit and effective approach for locating an IBFS to a transportation problem, was created and successfully tested on 10 problem sets. The created methodology, MAT, was compared to the five existing approaches for obtaining the IBFS: Vogel Approximation Method{VAM}; North West Corner Method {NWCM}; Least Cost Method{LCM}; Column Minimum Method{CMM}; and Row Minimum Method{RMM} in terms of iterations’ number required to reach the optimum stage. Conclusively, the MAT under development provided a better IBFS across the board, as it produced optimum solutions for around 60% of the solved problem cases, as opposed to the IBFS created by the five existing approaches employed in this study, which gave optimum solutions for roughly 0%, 20%, 20%, 10%, and 10% of the examples, respectively.

A branch-and-price-and-cut algorithm for the unmanned aerial vehicle delivery with battery swapping

In the UAV-based urban delivery setting, the battery capacity serves as one of the key factors affecting the delivery range of drones. For a typical multi-rotor unmanned aerial vehicle (UAV), the maximum effective working range of a lithium battery for a single flight leg is approximately 20 km. In practice, the lithium battery is replaced each time the UAV completes a delivery, even though the remaining power of the replaced lithium battery is still capable of supporting the following flight legs. Therefore, the one-order-one-swapping strategy significantly brings down the battery utilisation as well as the efficiency of the delivery system. In the present paper, the UAV-based delivery problem with battery swapping decision is considered, where multiple en route pickup and deliveries are allowed. To obtain optimal decisions in terms of flight route, time and location of each battery swapping, a branch-and-price-and-cut algorithm is introduced. First we formulate this problem as a mixed integer linear programming (MILP) model, and then decompose it into a master problem and a pricing sub-problem, where the master problem is considered as a set covering model, and the sub-problem as an elementary shortest path problem with resource constraints (ESPPRC). To speed up the process of column generation, efficient heuristics are also integrated to provide basic feasible solutions. In the pricing sub-problem, a combination of heuristic algorithm and dynamic programming is implemented to improve the computational efficiency. In the numerical experiments, the proposed algorithm is tested on various scales of UAV-based urban delivery instances, the performance of which is analysed and compared with the off-the-shelf commercial solver.

Finding a Basic Feasible Solution for Neutrosophic Linear Programming Models: Case Studies, Analysis, and Improvements


 Since the inception of operations research, linear programming has received the attention of researchers in this field due to the many areas of its use. The focus was on the methods used to find the optimal solution for linear models. The direct simplex method, with its three basic stages, begins by writing the linear model in standard form and then finding a basic solution that is improved according to the simplex steps until We get the optimal solution, but we encounter many linear models that do not give us a basic solution after we put it in a standard form, and here we need to solve a rule through which we reach the optimal solution. For these models, researchers and scholars in the field of operations research introduced the simplex method with an artificial basis, which helped to Find the optimal solution for linear models, given the importance of this method and as a complement to the previous research we presented using the concepts of neutrosophic science. In this research, we will reformulate the simplex algorithm with an artificial basis using concepts of neutrosophic science.

Optimal value determination using traditional and newly developed method based on using initial basic feasible solution of a transportation problem using northwest and Russell method

Optimal value determination using traditional and newly developed method based on using initial basic feasible solution of a transportation problem using northwest and Russell method

A new approach for finding an initial basic feasible solution to a transportation problem

A new approach for finding an initial basic feasible solution to a transportation problem

Region-based flexibility quantification in distribution systems: An analytical approach considering spatio-temporal coupling

Operational flexibility is to depict the ability of a power system in guaranteeing a secure and high-performance system operation, which is essential for distribution systems (DSs) with high renewable energy generation (REG) penetration. However, due to the dispersed integration of energy storage systems and uncertain REGs, the available regulation capability exhibits spatio-temporal coupling. It poses a challenge to quantify flexibility considering multiple locations and time periods. This paper proposes a novel analytical region-based method to characterize the flexibility considering spatio-temporal coupling, where the total dispatch budget and carbon emission constraints are considered to guarantee economic and environmental benefits. Leveraging the basic feasible solution theory, we derive an analytical formulation of the flexibility region, expressed as a set of linear inequalities. Note that no inner or outer approximation techniques are employed in the deduction procedure. The effectiveness and scalability of the proposed method are validated on test systems. Case studies demonstrate that the proposed method enables precise quantification and visualization of operational flexibility for DSs. The derived flexibility region is a closed polyhedron in a high-dimensional space, signifying the maximum ranges of uncertainties that can be accommodated.

New Methods to Solve and Find Initial Basic Feasible Solution of Transportation Problems

In this paper to solve the initial basic solution for transportation problem and Minimizing transportation time and getting optimal solutions are always considered as important factors while solving transportation problem. This paper shows a new approach for finding initial basic solution for transportation problem which reduces cost of transportation more than any transportation method such as Northwest Corner Method(NCM), Least Cost Method(LCM), northwest, Vogel’s approximation, Total Opportunity Cost Matrix-Minimal Total (TOCM-MT) and so on. This method has been illustrated by taking an Numerical example.

Pythagorean fuzzy transportation problem: New way of ranking for Pythagorean fuzzy sets and mean square approach

The predominant domain for optimization in the current situation is the transportation problem (TP). In the majority of cases, accurate data have been employed, yet in reality, the values are vague and imprecise. In any decision-making process, imprecision is a significant issue. To deal with the ambiguous setting of collective decision-making, many tools and methods have been established. The Pythagorean fuzzy set is an extension of fuzzy sets that successfully handles ambiguity and fuzziness. To overcome the shortcomings of intuitionistic fuzzy context, Pythagorean fuzzy sets are considered to be the most recent tools. This study proposes a new method for addressing the uncertain Pythagorean transportation issue. In this study, we created a novel sorting technique for Pythagorean fuzzy sets that converts uncertain quantities into crisp numbers. We developed an innovative mean square strategy for obtaining the initial basic feasible solution (IBFS) for a Pythagorean Fuzzy Transit Issue (PyFTP) of three types (I, II, III) wherein the requirement, availability, and unit of transportation expenses are all in Pythagorean uncertainty. In addition, we used the MODI technique to find the best option. To demonstrate the suggested strategy, we used numerical problems of three distinct kinds. A comparison table with the results of the previous strategy and the suggested method is created to state the benefits of the ranking methodology with the proposed algorithm. The discussion of future research and conclusions is the final step.

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.

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

An Efficient Method to Obtain Initial Basic Feasible Solution of a Transportation Problem

Transportation Problem (TP) is the study of strategy for shipping the product from the origin (supplier) to the destination (consumers) and the optimal shipment cost of the TP obtained from the initial basic feasible solution (IBFS). IBFS is the prime element of TP which helps to optimize the TP objective function. Existing methods for finding IBFS could not give the nearest value to minimal cost everywhere. Hence an innovative technique Known as TOCM-MT provides IBFS, which is the closest to an optimal solution. Many researchers have used this TOCM-MT technique through thousands of real-life TP problems, in every time got a better result than VAM, JHM, and TDM1 techniques. Hence, this technique is best for obtaining IBFS. This research is illustrated the TOCM-MT approach through time minimizing, transshipment, and profit maximization problems with some numerical examples.

A NEW APPROACH FOR FINDING MINIMIZATION COST IN TRANSPORTATION PROBLEM – AN ALGORITHM BY USING PYTHON LANGUAGE

In this paper the package Numpy from the programing language Python is presented as a computational tool that can help to solve the initial basic feasible solution involving transportation problem. This is also extension of our previous paper. The objective of this paper is to find how to minimize the transportation cost (TC) and to save a significant amount of time.

Investigating the Pitfalls of the Least Cost and Vogel’s Approximate Methods: Understanding the Impact of Cost Matrix Patterns

To obtain an optimal solution for any Transportation Problem (TP), the first step is to find an Initial Basic Feasible Solution (IBFS). A better IBFS requires fewer iterations to reach an optimal solution. The Least Cost Method (LCM) and Vogel’s Approximate Method (VAM) are commonly used approaches to finding IBFS due to their ease of implementation. Researchers frequently propose various methods to discover IBFS, but most of them are modifications of LCM or VAM. While VAM generally performs better, there are instances where it produces worse results compared to LCM and other approaches. Additionally, although researchers have developed new approaches, mainly modified versions of VAM, and demonstrated improved solutions with a few numerical instances, they have not yet identified the causes behind these results. The reasons for LCM and VAM's inability to obtain better IBFS have not been fully determined. This article aims to uncover the causes of pitfalls and the mechanisms of the allocation flow in both LCM and VAM through hypothetical and experimental domains. Several typical numerical instances have been conducted to demonstrate the causes of pitfalls and the allocation flow mechanisms of these methods. Journal of Engineering Science 14(1), 2023, 123-135

On solving fully intuitionistic fuzzy transportation problem via branch and bound technique

In dealing with real world transportation problems, several issues are frequently encountered as a result of uncontrollable factors. To deal this uncertainties, many authors have suggested transportation problems with intuitionistic fuzzy parameters. In this article, fully intuitionistic fuzzy transportation problem (InFTP) is considered in which the parameters are triangular intuitionistic fuzzy numbers. To solve this, initially intuitionistic fuzzy branch and bound technique are applied to get the initial basic feasible solution and then intuitionistic fuzzy modified distribution (InFMODI) method is applied to acquire the optimal solution of the fully InFTP. Also, a new ordering is developed here in which some properties of compensation, linearity, additive and partial order relations are satisfied. The optimal solution obtained from the proposed method satisfies the condition of optimality and feasibility. Finally, two numerical examples are provided to show the effectiveness of the proposed method.

Revisiting degeneracy, strict feasibility, stability, in linear programming

Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs, LPs. Unlike general conic programs, LPs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is seldom a concern, even though strict feasibility is equivalent to stability and a compact dual optimal set. This lack of concern is also true for other types of degeneracy of basic feasible solutions in LP. In this paper we discuss that the specific degeneracy that arises from lack of strict feasibility necessarily causes difficulties in both simplex and interior point methods. In particular, we show that the lack of strict feasibility implies that every basic feasible solution, BFS, is degenerate; thus conversely, the existence of a nondegenerate BFS implies that strict feasibility (regularity) holds. We prove the results using facial reduction and simple linear algebra. In particular, the facially reduced system reveals the implicit non-surjectivity of the linear map of the equality constraint system. As a consequence, we emphasize that facial reduction involves two steps where, the first guarantees strict feasibility, and the second recovers full row rank of the constraint matrix. This illustrates the implicit singularity of problems where strict feasibility fails, and also helps in obtaining new efficient techniques for preproccessing. We include an efficient preprocessing method that can be performed as an extension of phase-I of the two-phase simplex method. We show that this can be used to avoid the loss of precision for many well known problem sets in the literature, e.g., the NETLIB problem set.