Optimization Concepts and Applications in Engineering

Introduction

Linear programming (LP) is the term used for defining a wide range of optimization problems, in which the objective function to be minimized or maximized is linear in the unknown variables and the constraints are a combination of linear equalities and inequalities. LP problems occur in many real-life economic situations where profits are to be maximized or costs minimized with constraint limits on resources. While the simplex method introduced in the following can be used for hand solution of LP problems, computer use becomes necessary even for a small number of variables. Problems involving diet decisions, transportation, production and manufacturing, product mix, engineering limit analysis in design, airline scheduling, and so on, are solved using computers. Linear programming also has applications in nonlinear programming (NLP). Successive linearization of a nonlinear problem leads to a sequence of LP problems that can be solved efficiently.

Practical understanding and geometric concepts of LP problems including computer solutions with EXCEL SOLVER and MATLAB, and output interpretation are presented in Sections 4.1–4.5. Thus, even if the focus is on nonlinear programming, the student is urged to understand these sections. Subsequently, algorithms based on Simplex (Tableau-, Revised-, Dual-) and Interior methods, and Sensitivity Analysis, are presented.