In this subject, different linear programming techniques are analyzed. First, we see how to proceed in order to formulate a linear model that represents the problem. Next, we introduce some techniques to solve linear models, such as the simplex algorithm, the dual simplex algorithm, the transportation algorithm specially designed to solve the transportation problem, the Hungarian algorithm designed to solve the assignment problem and the branch and bound algorithm to find an optimal solution to an integer linear programming. The sensitivity analysis and the duality are also studied.
Unit 0: Linear Algebra and Convex Sets.
Some basic results from linear algebra and convex sets required for the development of the linear programming theory are reviewed, such as the solution of systems of linear equations, basic solutions, vector spaces and convex sets.
Unit 1: Linear Modeling and Graphical Solution
The first stage in the analysis and solution of a linear programming problem is to formulate the problem by writing a linear model that represents it. In this unit, we present a variety of problems that are solvable by linear programming techniques, and analyze the structure of the linear model and the model-formulation process. We also analyze the graphical solution of linear models with only two variables, and illustrate the different types of solutions that can be found while solving linear models.
Unit 2: The Simplex Method
The linear programming theory is developed, and the simplex algorithm which is used to solve linear models is presented. The Big-M method, the two-phase method and the revised simplex method are also studied. Some linear models are solved, and different kinds of solutions are interpreted.
Unit 3: Duality
The relationships between the primal and dual problems and their solutions are analyzed. By interpreting the dual problem, we make the economic interpretation of the simplex method. The concept of shadow prices is introduced. Finally, the dual simplex algorithm is introduced together with an extension of it: the dual simplex algorithm with artificial constraint. To conclude, some illustrative examples are given.
Unit 4: Sensitivity Analysis
The sensititivy analysis is used to determine the effect a change in the linear model has on the optimal solution. The following five changes in the original linear model are considered: a change in the cost coefficients vector, a change in the right-hand-side vector, a change in a nonbasic column of the constraint matrix, the addition of a new variable to the model and the addition of a new constraint to the model.
Unit 5: The Transportation Problem and The Assignment Problem
The transportation problem was one of the original applications of linear programming. It has a special structure which offers alternative methods of solving transportation problems that are more efficient than the standard simplex algorithm. In this unit, the structure of the transportation problem is analyzed and a solution algorithm is proposed. The assignment problem is a special case of the transportation problem. In this unit, a special algorithm developed for the assignment problem is analyzed.
Unit 6: Integer Programming
An integer programming problem is a linear programming problem in which some or all of the variables are restricted to be integers. A special case is the binary integer programming, in which all the variables are binary. First, some illustrative examples of integer problems are given. Next, the graphical solution of linear models is used to show the set of solutions of the integer problem and to show the process of searching an integer solution by applying the branch and bound algorithm. The general branch and bound algorithm, and a specific one adapted to solve binary integer problems are introduced.