# Linear Programming; Simplex Method

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Topics on "Operational Research" Mar. 2007, IST

Linear Programming, an introduction
MIGUEL A. S. CASQUILHO IST, Universidade Técnica de Lisboa, Ave. Rovisco Pais, IST; 1049-001 Lisboa, Portugal
Linear Programming is presented at an introductory level, mainly from the book by Hillier and Lieberman [2005], abridged and adapted to suit the objectives of the “Operational Research” course. It begins with segments of its third chapter.

Key words: linear programming; simplex method.

I. Fundamentals and scope
Based on a prototype example, Linear Programming is presented, as well as the simplex method of resolution. This method was first presented by G. B. Dantzig in 1947 [MacTutor, 2007]. The text is based on the book by Hillier and Lieberman [2005], and begins with segments of the third chapter of the book.

II. Explanation of the simplex method 3 Introduction to Linear Programming
(H&L 25)

The development of linear programming has been ranked among the most important scientific advances in the mid-20.th century, and we must agree with this assessment. Its impact since just 1950 has been extraordinary. Today it is a standard tool that has saved many thousands or millions of dollars for most companies or businesses of even moderate size in the various industrialized countries of the world; and its use in other sectors of society has been spreading rapidly.

3.1 Prototype example
(H&L 26)

Table 1 Data for the Wyndor Glass Co. problem
Plant A B C Profit per batch Production time per batch (h) Product 1 2 1 0 0 2 3 2 3 000 5 000 Production time available per week (h) 4 12 18

x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Z = total profit per week (in \$1000) from producing these two products

M. Casquilho is Assistant Professor in the Department of Chemical and Biological Engineering, Instituto Superior Técnico, Universidade Técnica de Lisboa. E-mail address: mcasquilho@ist.utl.pt.

References: BUESCU, Jorge, 2001, «Dez algoritmos que abalaram o Mundo», Ingenium, Maio, Lisboa, p 40. CASQUILHO, Miguel, 2007, “Cálculos, Calculations” [online], Instituto Superior Técnico (cited on 2007-03): http://alfa.ist.utl.pt/ ~mcasquil/Calcmenu.html DESBAZEILLE, Gérard, 1976, «Exercices et Problèmes de Recherche Opérationnelle», 2.e éd., Dunod, Paris. DILWORTH, James B., 1989, “Production and Operations Management (Manufacturing and Nonmanufacturing)”, 4.th ed., McGraw-Hill, New York. ECKER, Joseph G., Michael KUPFERSCHMID, 1988, “Introduction to Operations Research”, John Wiley & Sons, New York. GUERREIRO, Jorge, Alípio MAGALHÃES, Manuel RAMALHETE, 1985, «Progamação Linear», Vol. II, McGraw-Hill de Portugal, Lisboa. HILLIER, Frederick S., Gerald J. LIEBERMAN, 2005, “Introduction to Operations Research”, 8.th ed., McGraw-Hill, Inc., New York. 12 MIGUEL CASQUILHO — "Operational Research" KARMARKAR, N. (Narendra), 1984, “A new polynomial-time algorithm for linear programming”, Combinatorica, 4, pp 373-95. MACHOL, Robert E., 1976, “Elementary Systems Mathematics. Linear programming for business and the social sciences”, McGraw-Hill Kogakusha, Ltd. (Intl. Stud. Ed.), Tokyo (Japan). MACTUTOR, 2007, “George Dantzig” [online], Univ. of St. Andrews (cited 2007-03), http://www-groups.dcs.st-and.ac.uk/~history/Biographies/ Dantzig_George.html RAMALHETE, Manuel, Jorge GUERREIRO, Alípio MAGALHÃES, 1984, «Programação Linear», Vol. I, McGraw-Hill de Portugal, Lisboa. WAGNER, Harvey M., 1972, “Principles of Operations Research (with applications to managerial decisions)”, Prentice-Hall International, London. WILLIAM, H. P., 1978, “Model Building in Mathematical Programming”, John Wiley & Sons, UK. v

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