An Efficient Approach for Transforming Unbalanced Transportation Problems into Balanced Problems in Order to Find Optimal Solutions

In operations research, the transportation problem (TP) is among the earliest and most effective applications of the linear programming problem. Unbalanced transportation problems reflect the reality of supply chain and logistics situations where the available supply of goods may not precisely match the demand at different locations. To deal with an unbalanced transportation problem (UTP), it is essential first to convert it into a balanced transportation problem (BTP) to find an initial basic feasible solution (IBFS) and hence the optimal solution. The present paper is concerned with introducing a new approach to convert an unbalanced transportation problem into a balanced one and as a consequence to obtain optimum total transportation cost. Numerical examples are provided to demonstrate the suggested method.

Analysis of the Impact of Optimal Solutions to the Transportation Problems for Variations in Cost Using Two Reliable Approaches

In this paper, we have used two reliable approaches (theorems) to find the optimal solutions to transportation problems, using variations in costs. In real-life scenarios, transportation costs can fluctuate due to different factors. Finding optimal solutions to the transportation problem in the context of variations in cost is vital for ensuring cost efficiency, resource allocation, customer satisfaction, competitive advantage, environmental responsibility, risk mitigation, and operational fortitude in practical situations. This paper opens up new directions for the solution of transportation problems by introducing two key theorems. By using these theorems, we can develop an algorithm for identifying the optimal solution attributes and permitting accurate quantification of changes in overall transportation costs through the addition or subtraction of constants to specific rows or columns, as well as multiplication by constants inside the cost matrix. It is anticipated that the two reliable techniques presented in this study will provide theoretical insights and practical solutions to enhance the efficiency and cost-effectiveness of transportation systems. Finally, numerical illustrations are presented to verify the proposed approaches.

An Artificial Intelligence Approach for Solving Stochastic Transportation Problems

Recent years witness a great deal of interest in artificial intelligence(AI)tools in the area of optimization.AI has developed a large number of tools to solve themost difficult search-and-optimization problems in computer science and operations research.Indeed,metaheuristic-based algorithms are a sub-field of AI.This study presents the use of themetaheuristic algorithm,that is,water cycle algorithm(WCA),in the transportation problem.A stochastic transportation problem is considered in which the parameters supply and demand are considered as random variables that follow the Weibull distribution.Since the parameters are stochastic,the corresponding constraints are probabilistic.They are converted into deterministic constraints using the stochastic programming approach.In this study,we propose evolutionary algorithms to handle the difficulties of the complex high-dimensional optimization problems.WCA is influenced by the water cycle process of how streams and rivers flow toward the sea(optimal solution).WCA is applied to the stochastic transportation problem,and obtained results are compared with that of the new metaheuristic optimization algorithm,namely the neural network algorithm which is inspired by the biological nervous system.It is concluded that WCA presents better results when compared with the neural network algorithm.

A New Approach to Solve Transportation Problems

Finding an initial basic feasible solution is the prime requirement to obtain an optimal solution for the transportation problems. In this article, a new approach is proposed to find an initial basic feasible solution for the transportation problems. The method is also illustrated with numerical examples.

New Approach to Find Initial Basic Feasible Solution (IBFS) for Optimal Solution in Transportation Problem

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 LCM, northwest, Vogel’s approximation and so on. This method has been illustrated by taking an example;afterwards, it compares basic initial feasible solution with other methods IBF and optimal dictate solutions such as MODI and Steppingstone method.

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

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.

Gaining-Sharing Knowledge Based Algorithm for Solving Stochastic Programming Problems

This paper presents a novel application of metaheuristic algorithmsfor solving stochastic programming problems using a recently developed gaining sharing knowledge based optimization (GSK) algorithm. The algorithmis based on human behavior in which people gain and share their knowledgewith others. Different types of stochastic fractional programming problemsare considered in this study. The augmented Lagrangian method (ALM)is used to handle these constrained optimization problems by convertingthem into unconstrained optimization problems. Three examples from theliterature are considered and transformed into their deterministic form usingthe chance-constrained technique. The transformed problems are solved usingGSK algorithm and the results are compared with eight other state-of-the-artmetaheuristic algorithms. The obtained results are also compared with theoptimal global solution and the results quoted in the literature. To investigatethe performance of the GSK algorithm on a real-world problem, a solidstochastic fixed charge transportation problem is examined, in which theparameters of the problem are considered as random variables. The obtainedresults show that the GSK algorithm outperforms other algorithms in termsof convergence, robustness, computational time, and quality of obtainedsolutions.

Optimal Solution of Fuzzy Transportation Problem Using Octagonal Fuzzy Numbers

In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics to solve a system of linear equations in many applications in science.Systems of concurrent linear equations play a vital major role in operational research.The main perspective of this research paper is to find out the minimum amount of transportation cost of some supplies through a capacitated network formerly the availability and the demand notes are octagonal fuzzy numbers.Octagonal fuzzy numbers are used and showed a membership function.To illustrate this method,a fuzzy transportation problem is solved by using octagonal fuzzy numbers using the ranking technique.It is shown that it is the best optimal solution and it is demonstrated with a numerical example.

TRANSPORTATION NETWORKS:OLD AND NEW

This paper shows a number of Problems in pure and applied mathematicsthat are solved by constructing transportation networks.Moreover,it also shows thatall the solutions are characterized by forbidden configurations which are not minors.However,all the characterizations are much related to the graphic method which wasfound by Chinese for solving a kind of the transportation problem in the fifties.

Minimum Cost of Capacity Expansion for Time-Limited Transportation Problem On-Demand

The minimum cost of capacity expansion for time-limited transportation problem on-demand (MCCETLTPD) is to find such a practicable capacity expansion transportation scheme satisfying the time-limited T along with all origins’ supply and all destinations’ demands as well as the expanding cost is minimum. Actually, MCCETLTPD is a balance transportation problem and a variant problem of minimum cost maximum flow problem. In this paper, by creating a mathematical model and constructing a network with lower and upper arc capacities, MCCETLTPD is transformed into searching feasible flow in the constructed network, and consequently, an algorithm MCCETLTPD-A is developed as MCCETLTPD’s solution method basing minimum cost maximum flow algorithm. Computational study validates that the MCCETLTPD-A algorithm is an efficient approach to solving the MCCETLTPD.

A Heuristic on Risk Management System in Goods Transportation Model Using Multi-Optimality by MODI Method

Transport risk management is one of the predominant issues to any industry for supplying their goods safely and in time to their beneficiaries. Damaging goods or delaying the shipping both make penalty to the company and also reduce the goodwill of the company. Every way of transportation routes has to be comfy which can make sure the supplies will attain without damaging goods and in time and additionally cost efficiently. In this paper, we find a few not unusual risks which might be concerned about all types of way of routes which include Highway, Waterway, Airway, Railway and so forth. Additionally, we proposed a technique to attain multiple optimal solutions by using Modified Distribution Method (MODI) of a transportation problem. Finally, we reduce the risks by minimizing the possible number of transportation routes using multi-optimality technique of the transportation problem.

Study on the Transportation Problem of Petrol Secondary Distribution with Considering Shortage Cost

Petrol is a kind of strategic natural resources. Provide legitimate transportation plans for the petrol secondary distribution are the key links to guarantee the petrol provision. If the total supply is insufficient, some petrol stations can’t avoid shortage because their demands could not be met. So the shortage cost will appear. This paper studies the problem of how to arrange the transportation plan in order to minimize the total cost when the total volume of supply is insufficient. Given the storage volume, the sales rate and the unit shortage cost of every petrol station, considering the full loading constraints of the compartment vehicle, a mixed integer programming model for minimizing the total cost of petrol secondary distribution is established. A Lingo program is compiled for solving the model. Finally, simulation on an example has been done and a reasonable transportation plan is obtained. The model and algorithm in this paper can provide a theoretical basis for dispatching department to make transportation plan.

An Algorithm for Solving a Class of Transportation Problems

In this paper, we present an algorithm for solving a class of transportation problems and show that it is globally convergent. The proposed algorithm is simple to implement and easy to parallelize. The numerical results indicate that the proposed algorithm is effective for this class of problems.

Mathematical Methods Applied to Economy and Sustainable Development Goals

Mathematics is a key factor in achieving the Sustainable Development Goals (SDGs), because of its applicability to real situations. To achieve the set goals in SDG, this paper suggests some mathematical methods that will be useful for solving real situations in relation to goals 2 and 12 of SDGs approved by UN when modeled mathematically. The Northwest Corner Method (NWCM), Least Cost Method (LCM), and Vogel Approximation Method (VAM), which are the initial solution methods were examined to ascertain the ideal route of transporting commodities from production facilities to requirement destination while the optimal solution methods involve Stepping Stone Method (SSM), and Modified Distribution Method (MDM), that give the feasible solution which will enhance minimum transportation cost were also thoroughly defined. Subsequent research shall focus on application of the methods in relation to SDGs problems in comparison with other existing methods.

Outlier Robust Feature Correspondence by Learning Based Matching Process

Feature correspondence is a crucial aspect of various computer vision and robot vision tasks.Unlike traditional optimization-based matching techniques,researchers have recently adopted a learning-based approach for matching,but these methods face challenges in dealing with outlier features.This paper presents an outlier robust feature correspondence method that employs a pruned attentional graph neural network and a matching layer to address the outlier issue.Additionally,the authors introduce a modified cross-entropy matching loss to handle the outlier problem.As a result,the proposed method significantly enhances the performance of learning-based matching algorithms in the presence of outlier features.Benchmark experiments confirm the effectiveness of the proposed approach.

Transportation Problem with Multi-choice Cost and Demand and Stochastic Supply

This paper analyzes the multi-choice stochastic transportation problem where the cost coefficients of the objective function and the demand parameters of the constraints follow multi-choice parameters.Assume that the supply parameters of the constraints in a transportation problem(TP)follow logistic distribution.The main objective of this paper is to select an appropriate choice from the multi-choices for the cost coefficients of the objective function and the demand of the constraints in the TP by introducing Lagrange’s interpolating polynomial in such a way that the total cost is minimized and satisfies the required demand.Using stochastic programming,the stochastic supply constraints of the TP are transformed into deterministic constraints.Finally,a non-linear deterministic model is formulated.Using Lingo software,the optimal solution of the proposed problem is derived.To illustrate the methodology,a real-life problem on the TP is considered.

Power Coal Transportation and Storage:A Programming Analysis of Road and Rail Options

In this study, the transportation and storage problems for regional power coal allocation planning are formulated as transportation and storage problems to realize the minimization of the regional transportation and storage cost. An effective optimiza- tion model is proposed to solve transportation and storage prob- lems for regional power coal allocation planning, which has inter- active effect on multiple participants, such as regional power plants, coal transportation companies, logistics centers, and coal storage centers. A case study illustrates that the model and algo- rithm are more reasonable compared with the classic transporta- tion model, and the sensitivity analysis improves transportation and storage strategies for regional power coal allocation planning. Results demonstrate that this model can not only satisfy more of the actual requirements of the integral system but also offer more information to the decision-makers （DMs） for reference in favor of exalting decision-making quality.

Trapezoidal fuzzy model to optimize transportation problem

The main aim of this paper is to develop an approach based on trapezoidal fuzzy numbers to optimize transportation problem in fuzzy environment.The present algorithm has representation of availability,demand and transportation cost as trapezoidal fuzzy numbers.This algorithm is found quicker in terms of runtime as comparison to fuzzy VAM discussed in[Kaur A.,Kumar A.,A new method for solving fuzzy transportation problem using ranking function,Appl.Math.Model.35:5652–5661,2011;Ismail Mohideen S.,Senthil Kumar P.,A comparative study on transportation problem in fuzzy environment,Int.J.Math.Res.2:151–158,2010].On the other hand this technique gives much better results than some classical methods like north-west corner and least cost method.Another benefit of this algorithm is that for certain transportation problems it directly gives optimal solution.It is one of the simplest methods to apply and perceive.Practical usefulness of the new method over other existing methods is demonstrated with two numerical examples.

Breakable Fuzzy Multi-stage Transportation Problem

In this paper,we investigate two new transportation models with breakability and restriction on transportation.Sometime in transportation process the items which are transported,have got damaged due to bad conditions of the road and vehicle.Here we consider the problem that there are so many plants and customers and the goods are transported in n-stages.We formulate two transportationmodels under crisp and fuzzy environment where we consider the transportation parameters are crisp and fuzzy in nature,respectively.We also consider the breakability(takes the deterministic value for the respectivemodels)at each stages.For the fuzzy model,generalized triangular fuzzy number and mean ofα-cut method are considered.Numerical illustration is provided to illustrate the developed models.

A time-space network based international transportation scheduling problem incorporating CO_2 emission levels

Environmental problems have received a great deal of attention in recent years.In particular,CO2 emissions worsen global warming and other environmental problems.The transport sector accounts for 20% of the total CO2 emissions.Therefore,the CO2 emission reduction of the transport sector is of great importance.In order to reduce emissions effectively,it is necessary to change the distribution and transportation processes.The purpose of this study is to minimize both the transportation costs and CO2 emissions during transportation.Our model considers a transportation scheduling problem in which loads are transported from an overseas production base to three domestic demand centers.The need for time-space networks arises naturally to improve the model.It is possible to know the distance carriers are moving,and also consider the timetables of carriers during transportation.Carrier choice,less-than carrier load,and domestic transportation among demand centers are considered as the three target areas to reduce CO2 emissions during the distribution process.The research model was formulated as a mixed integer programming (MIP) problem.It achieves cost reduction,and will contribute to improvement of the natural environment.