Optimizing Solution Procedures for the Transshipment Problem with Mixed Constraints

Abstract

The whole thesis is divided into six chapters newlineIn the first chapter we have introduced the problem starting with classical transportation problem and shaped it in Mathematical model. It is shown that the model helps us to reduce the optimal transportation time of transportation problem with mixed constraints by allowing all the sources and destinations to act as trans-shippers in the transportation problem with mixed constraints. All the related problems which help directly or indirectly in solving it are also introduced together with its existence in the literature. newlineIn the second chapter a primal method has been developed for solving the transshipment problem with mixed constraints. Methodology adopted here is to create an equivalent transportation problem with mixed constraints, then it is reconverted into a classical transportation problem. A span method has been developed for getting its solution. Through the solution of classical transportation problem, an optimal solution of the transshipment problem with mixed constraints is obtained. newlineThird chapter is devoted to develop a method for the optimal solution of transshipment problem with mixed constraints .Transshipment problem is converted into an equivalent transportation problem with mixed constraints. Since the transportation problem with mixed constraints is a linear programming problem and every linear programming problem have their dual linear programming problem, so it is then converted into an equivalent dual transportation problem with mixed constraints. Its basic solution is obtained and improved to reach optimality. With the help of this optimal solution, the optimal solution of transshipment problem with mixed constraints is achieved. newlineChapter four focuses on the development of a tabular method for solving the problem via converting its constraints into required constraints of Fourier Elimination Method form. The newlinexiv newlineobjective value is obtained by eliminating all the variables one by one using Fourier Elimination Method, values of the variables are obtained by back substitution method. Iterations are performed from table to table instead of set of equation to equation. newlineIn chapter five, a zero suffix method has been developed for transshipment problem with mixed constraints. Transshipment problem with mixed constraints is directly converted into classical transportation problem. In the method, basic cells of classical transportation problem are located through the possible minimum cost cell for allotment. Minimum cost cell is located by subtracting the least element of all the rows and columns from all the elements of the corresponding rows and columns of the transportation table. At least one cells with zeros entries are created in each row and each column of transportation table. Suffix value of each zero is obtained. The cell with greatest suffix value is assigned and the row/column having exhausted rim requirements are deleted to get the reduced table. Continuing this process, optimal solution of classical transportation table is obtained. With the help of this optimal solution, the optimal solution of transshipment problem is achieved. newlineChapter six deals with searching the possibility of reducing the optimal cost of transshipment problem with mixed constraints by increasing the rim requirements. If possible, then the corresponding solution is termed as more-for-less (more-for-same) solution. The situation is termed as more-for-less paradoxical situation. A method has been developed for finding more-for-less solution. In the process parameters corresponding to equal to type constraints are added on RHS and all the other constraints are converted into less than equal to type constraints, if needed multiplying by - sign. Introducing slack variables corresponding to each less than equal to type constraints and performing dual simplex iteration through simplex table gives the more-for- less (more-for-same) solution. newline