# Linear Programming Fundamentals

• Nolberto Munier
• Eloy Hontoria
• Fernando Jiménez-Sáez
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 275)

## 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.

## References1

1. Cascales M-T, Lamata M-T (2012) On rank reversal and TOPSIS method. Math Comput Model 56(5–6):123–132
2. Cox A (2016) Good decision-making guide for public bodies. http://www.arthurcox.com/wp-content/uploads/2017/10/Good-Decision-Making-Guides-Collection-1-7-booklet.pdf. Accessed 6 May 2018
3. Dantzig G (1948) Linear Programming and extensions. United States Air ForceGoogle Scholar
4. Fylstra D (Solver) https://www.solver.com/. Accessed 5 May 2018
5. Kantorovich L (1939) The best uses of economic resources. Pergamon Press (1965)Google Scholar
6. Munier N (2011) 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, EspañaGoogle Scholar

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Nolberto Munier
• 1
• Eloy Hontoria
• 2
• Fernando Jiménez-Sáez
• 3
1. 1.INGENIO, Polytechnic University of ValenciaKingstonCanada