1
Topics on " Functional Research" Mar. 2007, IST
Linear Development, an introduction
MIGUEL A. T. CASQUILHO IST AUCH, Universidade Técnica de Lisboa, Ave. Rovisco Pais, IST; 1049001 Lisboa, Portugal Linear Programming is usually presented at an introductory level, mainly through the book simply by Hillier and Lieberman [2005], cut and tailored to suit the objectives of the " Operational Research” program. It starts with segments of its third phase.
Key words: geradlinig programming; simplex method.
I. Fundamentals and scope
Depending on a model example, Geradlinig Programming is definitely presented, and also the simplex method of resolution. This method was first presented by G. B. Dantzig in 1947 [MacTutor, 2007]. The written text is based on the book by Hillier and Lieberman [2005], and begins with segments of the third chapter of the book.
II. Reason of the simplex method a few Introduction to Geradlinig Programming (H& L 25)
The development of thready programming has been ranked among the most important technological advances in the mid20. a century, and must go along with this analysis. Its effects since just 1950 continues to be extraordinary. Today it is a common tool which has saved many thousands or huge amount of money for most companies or businesses of even moderate size in the numerous industrialized countries of the world; as well as use in various other sectors of society has become spreading rapidly.
3. one particular Prototype case
(H& T 26)
Desk 1 Info for the Wyndor Glass Co. issue
Plant A B C Profit per batch Creation time per batch (h) Product one particular 2 one particular 0 zero 2 three or more 2 several 000 a few 000 Creation time offered per week (h) 4 doze 18
x1 = volume of batches of product 1 produced weekly x2 sama dengan number of amounts of product 2 produced per week Z = total profit each week (in $1000) from making these two items
M. Casquilho is Helper Professor in the Department of Chemical and Biological Executive, Instituto Remarkable Técnico, Universidade Técnica para Lisboa. Email address: [email protected] utl. rehabilitation. © 2007 MC, IST: OR File=TopicLP. doc
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MIGUEL CASQUILHO — " Operational Research"
4 Solving Linear Coding problems: the simplex technique
(H& L 103)
4. 2 Creating the Simplex Method
(H& L 108)
Original kind of the unit (" t. t. ”, subject to):
[max] z =
t. t. 1
3 x1 x1 three or more x1
& 5 x2 ≤4 ≤ 12 ≤ 18
a couple of x2 + 2 x2
and x1 ≥ zero, x2 ≥ 0. Increased form of the model:
[max] z =
s. to 2
3 x1 x1 3x1
+ 5 x2
+ 0 x3 + x3
& 0 x4
+ zero x5 =4 = 12 = 18
2 x2 + a couple of x2
+ x4 & x5
and xi ≥ 0, i = 1 .. 5. The brand new variables xi, i = 3.. 5, are the " slack variables”. A basic answer has the subsequent properties: 1 ) Each adjustable is selected as whether nonbasic varying or a basic variable. installment payments on your The number of basic variables equals the number of useful constraints (now equations). Consequently , the number of nonbasic variables means the total number of variables without the number of efficient constraints. three or more. The nonbasic variables are set comparable to zero. 5. The ideals of the basic variables happen to be obtained because the sychronizeds solution of the system of equations (functional limitations in increased form). The set of basic variables is often referred to as the foundation. 5. If the basic variables satisfy the nonnegativity constraints, the essential solution is known as a basic feasible (BF) solution. Adjacent solutions: Two HUBBY solutions will be adjacent in the event that all but among their nonbasic variables are exactly the same (so all but one of all their basic variables also are the same, although maybe with different numerical values). Consequently, moving from your current HUBBY solution to a great adjacent one involves switching one varying from nonbasic to simple and the other way round for one additional variable (and then modifying the beliefs of the basic variables to continue satisfying the machine of equations). It is practical for the simplex approach to rewrite the challenge in the pursuing equivalent approach:
Linear Development, an introduction
several
Maximize Z . subject to
3
(0) (1) (2) (3)...
References: BUESCU, Jorge, 2001, «Dez algoritmos que abalaram o Mundo», Ingenium, Maio, Lisboa, s 40. CASQUILHO, Miguel, 2007, " Cálculos, Calculations” [online], Instituto Superior Técnico (cited on 200703): http://alfa.ist.utl.pt/ ~mcasquil/Calcmenu. html code DESBAZEILLE, Gérard, 1976, «Exercices et Problèmes de Recherche Opérationnelle», 2 . electronic éd., Dunod, Paris. DILWORTH, James B., 1989, " Production and Operations Management (Manufacturing and nonmanufacturing )”, 4. th ed., McGrawHill, New York. ECKER, Joseph G., Michael KUPFERSCHMID, 1988, " Introduction to Procedures Research”, Ruben Wiley & Sons, Nyc. GUERREIRO, Jorge, Alípio MAGALHÃES, Manuel RAMALHETE, 1985, «Progamação Linear», Vol. II, McGrawHill de Italy, Lisboa. HILLIER, Frederick T., Gerald T. LIEBERMAN, 2006, " Introduction to Operations Research”, 8. th ed., McGrawHill, Inc., New york city.
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MIGUEL CASQUILHO — " Operational Research"
KARMARKAR, N. (Narendra), 1984, " A new polynomialtime algorithm to get linear programming”, Combinatorica, 5, pp 37395. MACHOL, Robert E., 1976, " Elementary Systems Math. Linear coding for business and the social sciences”, McGrawHill Kogakusha, Ltd. (Intl. Stud. Impotence. ), Tokyo (Japan). MACTUTOR, 2007, " George Dantzig” [online], Univ. of St . Andrews (cited 200703), http://wwwgroups.dcs.stand.ac.uk/~history/Biographies/ Dantzig_George. html RAMALHETE, Manuel, Jorge GUERREIRO, Alípio MAGALHÃES, 1984, «Programação Linear», Vol. My spouse and i, McGrawHill para Portugal, Lisboa. WAGNER, Harvey M., 1972, " Concepts of Operations Research (with applications to managerial decisions)”, PrenticeHall Foreign, London. WILLIAM, H. P., 1978, " Model Building in Statistical Programming”, John Wiley & Sons, UK.
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