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.

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.

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.

Global Lipschitz Stability in an Inverse Problem for the Non-stationary Transport Equation

An inverse problem of determining the source term of the non-stationary transport equation was considered.The extra lateral boundary condition was chosen as the observation data.Based on a Carleman estimate on the non-stationary transport equation,a global Lipschitz stability for the inverse problem was proved.

Efficient Multiobjective Genetic Algorithm for Solving Transportation, Assignment, and Transshipment Problems

This paper presents an efficient genetic algorithm for solving multiobjective transportation problem, assignment, and transshipment Problems. The proposed approach integrates the merits of both genetic algorithm (GA) and local search (LS) scheme. The algorithm maintains a finite-sized archive of non-dominated solutions which gets iteratively updated in the presence of new solutions based on clustering algorithm. The use clustering algorithm makes the algorithms practical by allowing a decision maker to control the resolution of the Pareto set approximation. To increase GAs’ problem solution power, local search technique is implemented as neighborhood search engine where it intends to explore the less-crowded area in the current archive to possibly obtain more nondominated solutions. The inclusion of local search and clustering algorithm speeds-up the search process and also helps in obtaining a fine-grained value for the objective functions. Finally, we report numerical results in order to establish the actual computational burden of the proposed algorithm and to assess its performances with respect to classical approaches for solving MOTP.

On the Quadratic Transportation Problem

We present a direct analytical algorithm for solving transportation problems with quadratic function cost coefficients. The algorithm uses the concept of absolute points developed by the authors in earlier works. The versatility of the proposed algorithm is evidenced by the fact that quadratic functions are often used as approximations for other functions, as in, for example, regression analysis. As compared with the earlier international methods for quadratic transportation problem (QTP) which are based on the Lagrangian relaxation approach, the proposed algorithm helps to understand the structure of the QTP better and can guide in managerial decisions. We present a numerical example to illustrate the application of the proposed 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.

Linear Plus Linear Fractional Capacitated Transportation Problem with Restricted Flow

In this paper, a transportation problem with an objective function as the sum of a linear and fractional function is considered. The linear function represents the total transportation cost incurred when the goods are shipped from various sources to the destinations and the fractional function gives the ratio of sales tax to the total public expenditure. Our objective is to determine the transportation schedule which minimizes the sum of total transportation cost and ratio of total sales tax paid to the total public expenditure. Sometimes, situations arise where either reserve stocks have to be kept at the supply points, for emergencies or there may be extra demand in the markets. In such situations, the total flow needs to be controlled or enhanced. In this paper, a special class of transportation problems is studied where in the total transportation flow is restricted to a known specified level. A related transportation problem is formulated and it is shown that to each basic feasible solution which is called corner feasible solution to related transportation problem, there is a corresponding feasible solution to this restricted flow problem. The optimal solution to restricted flow problem may be obtained from the optimal solution to related transportation problem. An algorithm is presented to solve a capacitated linear plus linear fractional transportation problem with restricted flow. The algorithm is supported by a real life example of a manufacturing company.

Numerical Approach of Network Problems in Optimal Mass Transportation

In this paper, we focus on the theoretical and numerical aspects of network problems. For an illustration, we consider the urban traffic problems. And our effort is concentrated on the numerical questions to locate the optimal network in a given domain (for example a town). Mainly, our aim is to find the network so as the distance between the population position and the network is minimized. Another problem that we are interested is to give an numerical approach of the Monge and Kantorovitch problems. In the literature, many formulations (see for example [1-4]) have not yet practical applications which deal with the permutation of points. Let us mention interesting numerical works due to E. Oudet begun since at least in 2002. He used genetic algorithms to identify optimal network (see [5]). In this paper we introduce a new reformulation of the problem by introducing permutations . And some examples, based on realistic scenarios, are solved.

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.

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.

New Procedure of Finding an Initial Basic Feasible Solution of the Time Minimizing Transportation Problems

Minimization of transportation time is a great concern of the transportation problems like the cost minimizing transportation problems. In this writing, a transportation algorithm is developed and applied to obtain an Initial Basic Feasible Solution (IBFS) of transportation problems in minimizing transportation time. The developed method has also been illustrated numerically to test the efficiency of the method where it is observed that the proposed method yields a better result.

Obtaining Optimal Solution by Using Very Good Non-Basic Feasible Solution of the Transportation and Linear Programming Problem

For the transportation problem, Sharma and Sharma [1] have given a very computationally efficient heuristic (runs in O(c*n2) time) to give very good dual solution to transportation problem. Sharma and Prasad [2] have given an efficient heuristic (complexity O(n3) procedure to give a very good primal solution (that is generally non-basic feasible solution) to transportation problem by using the very good dual solution given by Sharma and Sharma [2]. In this paper we use the solution given by Sharma and Prasad [2] to get a very good Basic Feasible Solution to transportation problem, so that network simplex (worst case complexity (O(n3*(log(n))) can be used to reach the optimal solution to transportation problem. In the second part of this paper, we give a simple heuristic procedure to get a very good BFS to linear programming problem from the solution given by Karmarkar [3] (that generally produces a very good non-basic feasible solution in polynomial time (O(n5.5)). We give a procedure to obtain a good BFS for LP by starting from the solution given by Karmarkar [3]. We note that this procedure (given here) is significantly different from the procedure given in [4].

Localisation Inverse Problem and Dirichlet-to-Neumann Operator for Absorbing Laplacian Transport

We study Laplacian transport by the Dirichlet-to-Neumann formalism in isotropic media (γ = I). Our main results concern the solution of the localisation inverse problem of absorbing domains and its relative Dirichlet-to-Neumann operator . In this paper, we define explicitly operator , and we show that Green-Ostrogradski theorem is adopted to this type of problem in three dimensional case.

Localisation Inverse Problem of Absorbing Laplacian Transport

We study the localisation inverse problem corresponding to Laplacian transport of absorbing cell. Our main goal is to find sufficient Dirichelet-to-Neumann conditions insuring that this inverse problem is uniquely soluble. In this paper, we show that the conformal mapping technique is adopted to this type of problem in the two dimensional case.

APPLICATION OF SUBPLEX OPTIMIZATION TO SOLUTE-TRANSPORT PROBLEM

Determining parameters,such as interphase exchange rate and dispersivity,in multiphase solute transport problem has always been an interesting issue.These parameters are usually not available because they are too difficult or too expensive to measure although they are necessary as input data or parameters for numerical modeling.To overcome this problem,inverse techniques have been developed.Recently,the subplex optimization approach,which considers reflection,expansion,contraction,and shrinkage as basic components in seeking the minimization point and which uses the subspace concept in search space,has been incorporated into our coupled multiphase fluid－ flow and solute－ transport simulator.In the application of the finite element model to multiphase infiltration and solute transport problem,physical variables,which are easy to observe(such as solute concentrations),are used as constraints in minimizing the differences between computed output and measured data.Therefore,modeling results provide optimized parameter estimates in addition to comparison with field data.Our numerical－ simulation example on interphase－ exchange coefficient as well as water and gas dispersivities shows optimized parameters approaching the same values specified in the forward simulation used to generate the synthetic constrained data.This provides an implication of possible application to the fields of earch sciences,including geotectonics and metallogeny.

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.

Operator Splitting Method for Coupled Problems:Transport and Maxwell Equations

In this article a new approach is considered for implementing operator splitting methods for transport problems, influenced by electric fields. Our motivation came to model PE-CVD (plasma-enhanced chemical vapor deposition) processes, means the flow of species to a gas-phase, which are influenced by an electric field. Such a field we can model by wave equations. The main contributions are to improve the standard discretization schemes of each part of the coupling equation. So we discuss an improvement with implicit Runge- Kutta methods instead of the Yee’s algorithm. Further we balance the solver method between the Maxwell and Transport equation.

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.

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.