Linear programming is a mathematical technique used to select the best choice among alternatives by finding the best use of limited resources in an industry. It is applied in production system engineering to optimize problems such as product mix, plant location, and job assignment. Linear programming problems are solved using methods such as graphical, simplex, index distribution, and modified distribution. Queuing theory is the study of waiting lines, while value analysis is a cost-preventive technique that aims to reduce costs and increase profits without compromising quality.