Linear Transportation Problem Research Articles

Navigating uncertain distribution problem: a new approach for resolution optimization of transportation with several objectives under uncertainty

Amidst uncertainty, decision-making in manufacturing becomes a central focus due to its complexity. This study explores complex transportation constraints and uses novel ways to guide manufacturers. The Multi-objective Stochastic Linear Fractional Transportation Problem (MOSLFTP) is a crucial tool for managing supply chains, manufacturing operations, energy distribution, emergency routes, healthcare logistics, and other related areas. It adeptly addresses uncertainty, transforming efficiency and effectiveness in several domains. Stochastic programming is the process of converting theoretical probabilities into concrete certainties. The artistic compromise programming technique acts as a proficient mediator, reconciling opposing objectives and enabling equitable decision-making. This novel approach also addresses the Multi-objective Stochastic Linear plus Linear Fractional Transportation Problem (MOSLPLFTP), which involves two interconnected issues. The effectiveness of these principles is clearly shown with the help of the LINGO® 18 optimization solver. This study uses a ranking method to compare the similar methods to solve the current problems. A meticulously designed example acts as a significant achievement, shedding light on our method in a practical setting. It serves as a distinctive instrument, leading manufacturers through the maze of uncertainty and assisting them in determining the most advantageous course of action. This journey involves subtle interactions between complexity and simplicity, uncertainty is overcome by decisiveness, and invention is predominant.

Study Of Multi-Objective Linear Transportation Problem And Its Optimal Solution Using Maximum Divide And Minimum Allotment (MDMA) Method

Logistics, supply chain management, and resource allocation are just a few of the many real-world contexts in which the Multi-Objective Linear Transportation Problem presents itself as a formidable optimization issue. In this paper, we seek to solve a multi-objective, linear and non-linear transportation problem by suggesting the most efficient solution possible. The unique Maximum divide and minimum allotment (MDMA) approach was proposed to find a workable answer to a particular transportation issue. It also calculates the solution to the problem using a number of different transportation methods and compares the result to MDMA methods. It is determined that the MDMA approach offers a practical algorithm and solution for the designated transportation problem.

An aspect of bilevel interval linear fractional transportation problem with disparate flows: a fuzzy programming approach

An aspect of bilevel interval linear fractional transportation problem with disparate flows: a fuzzy programming approach

On Fuzzy Linear Fractional Programming Problems via α-Cut-Based Method with Application in Transportation Sector

The article provides an α-cut-based method that solves linear fractional programming problems with fuzzy variables and unrestricted parameters. The parameters and variables are considered as asymmetric triangular fuzzy numbers, which is a generalization of the symmetric case. The problem is solved by using α-cut of fuzzy numbers wherein the α- and r-cut are applied to the objective function and constraints, respectively. This reduces the problem into an equivalent biobjective model which leads to the upper and lower bounds of the given problem. Afterwards, the membership functions corresponding to various values of r∈(0,1] are obtained using the optimal values of the biobjective model. The proposed method is illustrated by taking an example from the literature to highlight the fallacy of an existing approach. Finally, a fuzzy linear fractional transportation problem is modelled and solved using the aforementioned technique.

A new solution procedure for multi-objective linear fractional transportation problem with rough parameters

In this article, we are discussing a rough multi-objective linear fractional transportation problem (RMOLFTP). In Day-to-day life, we frequently encounter many situations where the transportation problems are represented by a rough set. The main aim of our research article is to solve the RMOLFTP where the transportation cost and profit, of the product as rough interval coefficients. Here a new logic is introduced by us to solve RMOLFTP that diversifies the form of Nomani et. al., (2016). To demonstrate the recommended algorithm, a rough numeric problem with two objective functions is offered. A weighted model is presented in this approach for the solutions.

An improved solution approach for solving multi-objective linear and linear fractional transportation problem

This work shows a transportation problem (TP) that can be thought of as the sum of a linear function and a linear fractional function. Here, the linear function shows the total transportation cost, and the linear fractional function shows the proportion of damaged objects and the transport profit. We aim to set up a transportation plan that reduces the total transportation cost and the proportion of total damage to the object to the total profit. Here we are discussing three types of objective functions as: linear plus linear fractional, linearfractional, and linear separately. We formulated a TP using the three objective functions for the solution. We solve all three objective functions for different weights with the proposed method and get a compromise solution.

An approach for solving fully fuzzy linear fractional transportation problem with the using of splitting technique

An approach for solving fully fuzzy linear fractional transportation problem with the using of splitting technique

Hybrid particle swarm optimization for pure integer linear solid transportation problem

This paper appends three proposed heuristic algorithms, viz. Initialization Algorithm, Fraction Repair Algorithm and Negative Repair Algorithm, to Particle Swarm Optimization (PSO) and extends its application to Pure Integer Linear Solid Transportation Problem (STP). The chief contribution of the paper is that the proposed Hybrid Particle Swarm Optimization (HPSO) algorithm discretizes the continuous search space of PSO so that it can operate in the discrete solution space of STP with any of the valid parameter settings. The Initialization Algorithm is developed to generate the sufficient number of random solutions for initiating HPSO with a diverse population of given size. Whereas Fraction Repair and Negative Repair Algorithms are developed to ensure that each successive population obtained with HPSO remains in the solution space of STP. The validity of HPSO is examined and illustrated by solving a numerical problem with certain parameter settings. Further, the HPSO demonstrates its ability to obtain an optimal or near optimal solution along with the alternate optimal solution, if it exists. It searches for the solutions even without adhering to the rigid conditions of traditional methods that every solution of STP must have a fixed number of non-zero decision variables. Moreover, the performance of HPSO is tested with three different parameter settings and compared amongst each other to analyze their significance in solving the problem. The performance of HPSO and the variants of parameters are statistically validated through an extensive experimental design.

A Game Theory-based Approach to Fuzzy Linear Transportation Problem

Background: Transport models have wide application areas in the real world and play an important role in reducing transportation costs, increasing service quality, etc. These models may have uncertain transportation costs and supply or demand capacities of the product. Hence, it would be effective to model the vagueness of customer demands, economic conditions, and technical or non-technical uncertainties because of uncontrollable factors. Therefore, we focus on developing a mathematical solution approach to the fuzzy transportation problems. Objective: In this paper, an integrated approach is proposed for the solution of the fuzzy linear transportation problem that has fuzzy cost coefficients in the objective function. Since transportation problem is encountered frequently in the national and international environment, it is considered that proposing a new solution method to this problem will be useful. Methods: Fuzzy cost coefficients are taken as trapezoidal fuzzy numbers due to their widespread use in the literature. Firstly, the fuzziness is removed by converting the original single objective fuzzy transportation problem into a crisp Multi-Objective Linear Programming Problem (MOLPP). After the classical payoff matrix is constructed, ratio matrices are obtained to scale the objectives. Then, an approach based on game theory is implemented to solve the MOLPP, which is handled as a zero-sum game. Results: Creating different ratio matrices in the game theory part of the approach can generate compromise solutions for the decision-makers. To demonstrate the effectiveness of the proposed approach, two numerical examples from the literature are solved. While the same solution is obtained in one of the examples, a different compromise solution set is generated, which could be presented to the decision-maker in the other example. Conclusion: In this paper, we developed a novel game theory-based approach to the fuzzy transportation problem. The proposed approach overcomes the non-linear structure due to the uncertainty in the cost coefficients. The greatest advantage of the proposed approach is that it can generate more than one optimal solution for the decision-maker.

Mehar approach to solve fuzzy linear fractional transportation problems

Mehar approach to solve fuzzy linear fractional transportation problems

A novel iterative method to solve a linear fractional transportation problem

The linear fractional transportation problem (LFTP) is widely encountered as a particular type of transportation problem (TP) in real-life. In this paper, a novel algorithm, based on the traditional definition of continuity, is presented to solve the LFTP. An iterative constraint is constructed by combining the objective function of the LFTP and the supply-demand condition since the fractional objective function is continuous at every point of the feasible region. By this constraint obtained, LFTP is converted into an iterative linear programming (LP) problem to reach the optimum solution. In this study, the case of asymptotic solution for LFTP is discussed for the first time in the literature. The numerical examples are performed for the linear and asymptotic cases to illustrate the method, and the approach proposed is compared with the other existing methods to demonstrate the efficiency of the algorithm. Also, an application had environmentalist objective is solved by proposed mathematical method using the software general algebraic modeling system (GAMS) with data set of the real case. Finally, some computational results from tests performed on randomly generated large-scale transportation problems are provided.

Time variant multi-objective linear fractional interval-valued transportation problem

Time variant multi-objective linear fractional interval-valued transportation problem

Bilevel transportation problem in neutrosophic environment

In the current times of the predominance of COVID-19, almost all the countries are conducting inoculation drives. Given the market’s inability to compute how much to manufacture, how to transport and the frequently changing demand, the cost of safely and timely transporting the vaccines from factory to syringe is currently indeterminate. In this paper, we formulate this situation using a bilevel transportation problem with neutrosophic numbers (BLTP-NN). The problem comes from a vaccine manufacturing company where the vaccine is produced and then transported to different distribution centres from where it is further transported to various health centres for the conduction of their vaccination drive. The authors have tried to perceive this situation from two perspectives by formulating two different problems. The first problem is a bilevel linear fractional transportation problem which aims at minimizing the transportation cost in proportion to per unit maximization of quantity transported. The second problem is a bilevel indefinite quadratic transportation problem which aims at minimizing the transportation cost and depreciation cost. In both problems, cost coefficients are neutrosophic numbers along with availabilities and demands in the constraint set. These formulated bilevel transportation problems in neutrosophic environment are solved using goal programming strategy to arrive at a satisfactory solution. The relevance of this work is to help the decision makers in budgeting their finances related to the transportation by strategic disbursement leading to a smooth administration of vaccination program.

Solving multi-objective linear fractional transportation problem under neutrosophic environment

We are discussing multi-objective linear fractional transportation problem (MOLFTP) under neutrosophic environment in the present paper. In proposed methodology, we characterize each objective of our MOLFTP in three different ways such that truth membership; indeterminacy membership and falsity membership functions. Proposed methodology gives us best neutrosophic compromise programming model under the given membership functions. In the given approach, we convert MOLFTP model into the non-linear integer programming model using proposed approach and solve it using MATLAB. To support the theory a numerical example is also discussed.

Solving Fuzzy Transportation Problem Using ASM Method and Zero Suffix Method

The transportation problem is a special case for linear programming. Sometimes, the amount of demand and supply in transportation problems can change from time to time, and thus it is justified to classify the transportation problem as a fuzzy problem. This article seeks to solve the Fuzzy transportation problem by converting the fuzzy number into crisp number by ranking the fuzzy number. There are many applicable methods to solve linear transportation problems. This article discusses the method to solve transportation problems without requiring an initial feasible solution using the ASM method and the Zero Suffix method. The best solution for Fuzzy transportation problems with triangular sets using the ASM method was IDR 21,356,787.50, while the optimal solution using the Zero Suffix method was IDR 21,501,225.00. Received February 5, 2021Revised April 16, 2021Accepted April 22, 2021

On Linear and Quadratic Two-Stage Transportation Problem

Introduction. When formulating the classical two-stage transportation problem, it is assumed that the product is transported from suppliers to consumers through intermediate points. Intermediary firms and various kinds of storage facilities (warehouses) can act as intermediate points. The article discusses two mathematical models for two-stage transportation problem (linear programming problem and quadratic programming problem) and a fairly universal way to solve them using modern software. It uses the description of the problem in the modeling language AMPL (A Mathematical Programming Language) and depends on which of the known programs is chosen to solve the problem of linear or quadratic programming. The purpose of the article is to propose the use of AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems, to formulate a mathematical model of a quadratic programming two-stage transportation problem and to investigate its properties. Results. The properties of two variants of a two-stage transportation problem are described: a linear programming problem and a quadratic programming problem. An AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems is given. The results of the calculation using Gurobi program for a linear programming two-stage transportation problem, which has many solutions, are presented and analyzed. A quadratic programming two-stage transportation problem was formulated and conditions were found under which it has unique solution. Conclusions. The developed AMPL-code for a linear programming two-stage transportation problem and its modification for a quadratic programming two-stage transportation problem can be used to solve various logistics transportation problems using modern software for solving mathematical programming problems. The developed AMPL code can be easily adapted to take into account the lower and upper bounds for the quantity of products transported from suppliers to intermediate points and from intermediate points to consumers. Keywords: transportation problem, linear programming problem, AMPL modeling language, Gurobi program, quadratic programming problem.

Solving Multi-Objective Linear Fractional Stochastic Transportation Problems Involving Normal Distribution using Simulation-Based Genetic Algorithm

In real-life situations, we human beings faced with multi-objective problems that are conflicting and non-commensurable with each other. Especially, when goods are transported from source to locations with a goal to keep exact relationships between a few parameters, those parameters of such problems might also arise in the form of fractions which are linear in nature such as; actual transportation fee/total transportation cost, delivery fee/desired path, total return/total investment, etc. Due to the uncertainty of nature, such a relationship is not deterministic. Mathematically such kinds of mathematical problems are characterized as a multi-objective linear fractional stochastic transportation problem. However, it is difficult to handle such types of mathematical problems. It can't be solved directly using mathematical programming approaches. In this paper, a solution procedure is proposed for the above problem using a stochastic Genetic Algorithm based simulation. The parameters in the constraint of the above problem follow a normal distribution. The probabilistic constraints are handled by stochastic simulation-based GA for the solution procedure of the proposed problem. The feasibility of probability constraints is checked by the stochastic programming through the Genetic Algorithm approach, without finding the equivalent deterministic model. The feasibility is maintained all-over the problem. The stochastic simulation-based Genetic Algorithm is considered to generate non-dominated solutions for the given problem. Then, a numerical case study is provided to illustrate the method.

Genetic algorithm based fuzzy programming approach for multi-objective linear fractional stochastic transportation problem involving four-parameter Burr distribution

In real life situations, it is difficult to handle multi-objective linear fractional stochastic transportation problem. It can’t be solved directly using mathematical programming approaches. In this paper, a solution procedure is proposed for the above problem using a genetic algorithm based fuzzy programming method. The supply and demand parameters of the said problem follow four-parameters Burr distribution. The proposed approach omits the derivation of deterministic equivalent form as required in case of classical approach. In the given methodology, initially the probabilistic constraints present in the problem are tackled using stochastic programming combining the strategy adopted in genetic algorithm. Throughout the problem, feasibility criteria is maintained. Then after, the non-dominated solution are obtained using genetic algorithm based fuzzy programming approach. In the proposed approach, the concept of fuzzy programming approach is inserted in the genetic algorithm cycle. The proposed algorithm has been compared with fuzzy programming approach and implemented on two examples. The result shows the efficacy of the proposed algorithm over fuzzy programming approach.

Decomposition Algorithm for the Linear Three-Index Transportation Problem

The method of sequential modification of an objective function, which was earlier used for the classical transportation problem, is extended to the case of three indices. In the iterative process, the problem with three constraints and one binding variable is solved. Then, three independent problems with one constraint in which the coefficients for the binding variable are changed are considered. Using the suggested algorithm, a sequence of pseudosolutions with a monotonic growth of the objective function that converges to the optimum is constructed. The degeneracies are analyzed.

English

English