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.

A New Method for Achieving an Initial Regular Solution of a Linear Programming

A method is provided for finding an initial regular solution of a linear programming in this paper. The key to this method is to solve an auxiliary linear programming instead of to introduce any artificial variable or constraint. Compared with the traditional method of achieving the regular solution by introducing an artificial constraint, it has advantages of saving the memories and little computational efforts.

An Effective Approach to Determine an Initial Basic Feasible Solution: A TOCM-MEDM Approach

Transportation of products from sources to destinations with minimal total cost plays an important role in logistics and supply chain management. In this article, a new and effective algorithm is introduced for finding an initial basic feasible solution of a balanced transportation problem. Number of numerical illustration is introduced and optimality of the result is also checked. Comparison of findings obtained by the new heuristic and the existing heuristics show that the method presented herein gives a better result.

A Comparative Study of Initial Basic Feasible Solution by a Least Cost Mean Method (LCMM) of Transportation Problem

Transportation Problems (TP), as is known, are a basic network problem which can be formulated as a Linear Programming Problem (LPP). Transportation networks are built up in order to save transportation cost. In the solution procedure of a TP, finding an Initial Basic Feasible Solution (IBFS) is necessary to obtain the optimal solution. Optimality gives us the optimal route that prompts either most extreme benefit or least aggregate cost whichever is required. In this research paper, a new method named Least Cost Mean Method is proposed to obtain a better IBFS where row penalty and column penalty is brought out by the mean of lowest and next lowest cost of each row and each column of the cost matrix. The method is illustrated with numerical examples. To verify the performance of the proposed method, a comparative study is also carried out and observed that it is computationally easier and yielding comparatively better solution.

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.

Large Solutions to the Generalized Debye-Hückel System in Fourier-Besov Spaces

The special structure in some coupled equations makes it possible to drop partial smallness assumption of the initial data to gain the global well-posedness.In this paper,we study the Cauchy problem for generalized Debye-Hückel system in Fourier-Besov spaces.Under more generalized index range,we obtain the global solution with small initial data and local solution with arbitrary initial.Besides,by constructing some weighted function,we prove that the global well-posedness still holds under the small assumption of the charge of initial data.Thus we show that although the initial densities and the hole in electrolytes are large,the equation is still global well-posedness.

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.

Two-phase Method without Any Artificial Variable

A method is provided to achieve an initial basic feasible solution of a linear programming in this paper. This method dose not need introducing any artificial variable, but needs only solving an auxiliary linear programming. Compared with the traditional two-phase method, it has advantages of saving the memories and reducing the computational efforts.

Incessant Allocation Method for Solving Transportation Problems

Industries require planning in transporting their products from production centres to the users end with minimal transporting cost to maximize profit. This process is known as Transportation Problem which is used to analyze and minimize transportation cost. This problem is well discussed in operation research for its wide application in various fields, such as scheduling, personnel assignment, product mix problems and many others, so that this problem is really not confined to transportation or distribution only. In the solution procedure of a transportation problem, finding an initial basic feasible solution is the prerequisite to obtain the optimal solution. Again, development is a continuous and endless process to find the best among the bests. The growing complexity of management calls for development of sound methods and techniques for solution of the problems. Considering these factors, this research aims to propose an algorithm “Incessant Allocation Method” to obtain an initial basic feasible solution for the transportation problems. Several numbers of numerical problems are also solved to justify the method. Obtained results show that the proposed algorithm is effective in solving transportation problems.

Existence and Uniqueness of Positive Solutions to Sublinear-Linear Initial Value Problems

Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and the main results of H.G.Kaper and M.K.Kwong~[4].

STRESS RATE INTEGRAL EQUATIONS OF ELASTOPLASTICITY

The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.

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.

A Note on a Generalized Network Flow Model for Manufacturing Process

Manufacturing network flow （MNF） is a generalized network model that overcomes the limitation of an ordinary network flow in modeling more complicated manufacturing scenarios, in particular the synthesis of different materials into one product and/or the distilling of one type of material into many different products. Though a network simplex method for solving a simplified version of MNF has been outlined in the literature, more research work is still needed to give a complete answer whether some classical duality and optimality results of the classical network flow problem can be extended in MNF. In this paper, we propose an algorithmic method for obtaining an initial basic feasible solution to start the existing network simplex algorithm, and present a network-based approach to checking the dual feasibility conditions. These results are an extension of those of the ordinary network flow problem.

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L（x, u, t）, and considers the following cost function： c（x, y） = inf x（0）=x x（1）=y u∈U∫0^1 L（x（s）, u（x（s）, s）, s）ds, where U is a control set, and x satisfies the ordinary equation x（s） = f（x（s）, u（x（s）, s））.It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation：Vt（t, x） ＋ sup u∈U = 0,V（0, x） = Φ0（x）.