This chapter focuses on formulation of the Linear Programming (LP) problems and the solution of problems using the simplex method. It involves general LP problem solution using the graphical approach as well. The computational procedure of the simplex method is discussed in very detail with the concepts involved such as slack and surplus variables, and basic and non-basic variables. In addition, the post-optimality sensitivity analysis with changes in resources and unit cost as well as profit is elaborated on with examples. Every concept and procedure is discussed with illustrative examples.