Linear Programming Fundamentals

Abstract

This chapter refers to linear programming (LP) Kantorovich (The best uses of economic resources, 1939), Dantzig (Linear Programming and extensions. United States Air Force, 1948). Fylstra (Solver) https://www.solver.com/. Accessed 5 May 2018), which is fundamental to understanding the SIMUS method (Sequential Interactive Method for Urban Systems), Munier (Tesis Doctoral – Procedimiento fundamentado en la Programación Lineal para la selección de alternativas en proyectos de naturaleza compleja y con objetivos múltiples- Universidad Politécnica de Valencia, España, 2011).

SIMUS is explained in Chap. 7, and it is the method used in this book to illustrate how most of the aspects related to real-world scenarios can be incorporated in the modelling and solved. It constitutes a hybrid system since it utilizes linear programming, as well as weighted sum and outranking procedures.

It employs the Simplex linear programming algorithm to produce a Pareto efficient matrix and thus obtaining an optimal solution (scores) for each objective. This matrix is then analysed vertically and used to weigh the sums of scores, to find the total score for each alternative. Later, it examines the matrix horizontally by means of the outranking concept, which also produces a score for each alternative. However, even when scores are different for the two procedures, the selection of the best alternative and the corresponding ranking are always the same for both; this is equivalent to solving the same problem through two different methods and getting the same result.

LP, because of its mathematical structure, as per these authors’ opinion, offers unmatched ability to reproduce actual scenarios more realistically than present-day methods.

However, as commented, some real problems cannot be addressed by LP, because it works with only one objective and quantitative criteria.