Linear Programming; Simplex Method

Powerful Essays
1
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

You May Also Find These Documents Helpful

  • Good Essays

    and Simplex Methods of Linear Programming The graphical method is the more popular method to use because they are easy to use and understand. Working with only a few variables at a time they allow operations managers to compare projected demand to existing capacity. The graphical method is a trial and error approach that can be easily done by a manager or even a clerical staff. Since it is trial and error though, it does not necessarily generate the optimal plan. One downside of this method though…

    • 552 Words
    • 3 Pages
    Good Essays
  • Good Essays

    Linear Programming

    • 372 Words
    • 2 Pages

    Linear Programming History of linear programming goes back as far as 1940s. Main motivation for the need of linear programming goes back to the war time when they needed ways to solve many complex planning problems. The simplex method which is used to solve linear programming was developed by George B. Dantzig, in 1947. Dantzig, was one in who did a lot of work on linear programming, he was reconzied by several honours. Dantzig's discovery was through his personal contribution, during WWII when Dantzig…

    • 372 Words
    • 2 Pages
    Good Essays
  • Better Essays

    Simplex Method

    • 977 Words
    • 4 Pages

    The simplex method is used regularly on applied problem involving thousands of variables and problem constrains (Barnett, Ziegler, & Byleen, 2011). This was the method chosen when Wintel Technologies needed to figure out the best way to utilize time and schedule for their field engineers. With a busy schedule and being needed on several different areas in the United States, the simplex method was brought in to decide where and when employees should be spending their time and the company’s money.…

    • 977 Words
    • 4 Pages
    Better Essays
  • Powerful Essays

    Linear Programming

    • 15778 Words
    • 64 Pages

    Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow prices give us directly the marginal worth of an additional unit of any of the resources. Second, when an activity is ‘‘priced out’’ using these shadow prices, the opportunity cost of allocating resources to that activity relative to other activities is determined. Duality in linear programming…

    • 15778 Words
    • 64 Pages
    Powerful Essays
  • Better Essays

    Simplex Method

    • 1287 Words
    • 6 Pages

    The Simplex Method: Learning Team A Mike Smith, Todd Jones Math212/Introduction to Finite Mathematics February 1, 2011 The Simplex Method: Learning Team A Sam’s Hairbows and Accessories is a small company preparing for the next scheduled craft fair. The owners, Sam and Todd, both have full-time jobs in addition to owning the company so they are only able to spend a combined total of 80 hours labor to prepare for the fair in four weeks. Sam’s offers five main product lines: basic bows, elaborate…

    • 1287 Words
    • 6 Pages
    Better Essays
  • Powerful Essays

    Linear Programming

    • 1683 Words
    • 7 Pages

    RESEARCH PAPER ON LINEAR PROGRAMMING Vikas Vasam ID: 100-11-5919 Faculty: Prof. Dr Goran Trajkovski CMP 561: Algorithm Analysis VIRGINIA INTERNATIONAL UNIVERSITY Introduction: One of the section of mathematical programming is linear programming. Methods and linear programming models are widely used in the optimization of processes in all sectors of the economy: the development of the production program of the company…

    • 1683 Words
    • 7 Pages
    Powerful Essays
  • Better Essays

    Simplex Method

    • 1192 Words
    • 5 Pages

    Simplex Method Paper Simplex Method Paper Many people may be wondering exactly what the simplex method is. The simplex method definition is a method for solving linear programming problems. According to Barnett, Byleen, and Karl (2011) the simplex method is used routinely on applied problems involving thousands of variables and problem constraints. George B. Dantzig developed the simplex method in 1947. In this paper the topic of discussion includes how to solve a simplex method problem that…

    • 1192 Words
    • 5 Pages
    Better Essays
  • Powerful Essays

    Linear Programming

    • 10813 Words
    • 44 Pages

    An Introduction to Linear Programming Steven J. Miller∗ March 31, 2007 Mathematics Department Brown University 151 Thayer Street Providence, RI 02912 Abstract We describe Linear Programming, an important generalization of Linear Algebra. Linear Programming is used to successfully model numerous real world situations, ranging from scheduling airline routes to shipping oil from refineries to cities to finding inexpensive diets capable of meeting the minimum daily requirements. In many of these problems…

    • 10813 Words
    • 44 Pages
    Powerful Essays
  • Powerful Essays

    Linear Programming

    • 22881 Words
    • 92 Pages

    12:39 PM Page 1 7 MODULE Linear Programming: The Simplex Method LEARNING OBJECTIVES After completing this chapter, students will be able to: 1. Convert LP constraints to equalities with slack, surplus, and artificial variables. 2. Set up and solve LP problems with simplex tableaus. 3. Interpret the meaning of every number in a simplex tableau. 4. Recognize special cases such as infeasibility, unboundedness and degeneracy. 5. Use the simplex tables to conduct sensitivity analysis…

    • 22881 Words
    • 92 Pages
    Powerful Essays
  • Good Essays

    Linear Programming

    • 11494 Words
    • 46 Pages

    (2003) 1 OPERATIONS RESEARCH: 343 1. LINEAR PROGRAMMING 2. INTEGER PROGRAMMING 3. GAMES Books: Ð3Ñ IntroÞ to OR ÐF.Hillier & J. LiebermanÑ; Ð33Ñ OR ÐH. TahaÑ; Ð333Ñ IntroÞ to Mathematical Prog ÐF.Hillier & J. LiebermanÑ; Ð3@Ñ IntroÞ to OR ÐJ.Eckert & M. KupferschmidÑÞ LP (2003) 2 LINEAR PROGRAMMING (LP) LP is an optimal decision making tool in which the objective is a linear function and the constraints on the decision problem are linear equalities and inequalities. It is a very popular…

    • 11494 Words
    • 46 Pages
    Good Essays