Zahra_mMohaghegh_Feb12

Zahra_mMohaghegh_Feb12 - Engineering Management Systems...

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1 Engineering Management & Systems Engineering The George Washington University EMSE 102 EMSE 102 - - 202 202 Survey of Operations Research Methods Survey of Operations Research Methods Zahra Mohaghegh [email protected] Post-Doctoral Research Associate Center For Risk and Reliability University of Maryland February 12, 2008
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2 Engineering Management & Systems Engineering The George Washington University The Simplex Algorithm The Simplex Algorithm (cont.) (cont.) CREDITS: PARTS OF THIS LECTURE HAVE BEEN DEVELOPED BY PROFESSORS ABELEDO (GWU) AND ORLIN (MIT)
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3 Engineering Management & Systems Engineering The George Washington University Linear Programming: the Simplex Algorithm The Simplex algorithm for solving LP’s requires that all constraints are equations (with exception of sign constraints on the variables) and all variables be nonnegative. An LP in this form is said to be in standard form . Any LP can be converted into an equivalent one in standard form. Example: 12 1 2 max 20 15 s.t. 100 100 50 35 6000 20 15 2000 ,0 zx x x x xx =+ +≤ +≥ 11 22 3 4 s.t. 100 100 50 35 6000 20 15 2000 x xs s e += ++ = +− = all variables 0 s variables are called slack variables e variables are called excess or surplus variables We can also convert any LP with unrestricted or nonpositive variables into standard form.
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4 Engineering Management & Systems Engineering The George Washington University Suppose we have an LP in standard form with n variables and m constraints. For convenience, we label the variables 12 ,,, n x xx K . max (or min) 11 2 2 nn zc x c x c x = ++ + L s.t. 11 1 12 2 1 1 21 1 22 2 2 2 mm m n n m ax ax ax b b b + = += L L M L 0 n x K Let = mn m m n n a a a a a a a a a L M M M L L 2 1 2 22 21 1 12 11 A , 1 2 n x x x = x M , 1 2 m b b b = b M , 0 0 0 ⎡⎤ ⎢⎥ = ⎣⎦ 0 M , and [ ] n cc c = c K then the equation constraints of the LP can be written in matrix form as = Ax b , and the nonnegativity constraints as x0 . Usually, A is called the constraint matrix , b is the right hand side vector, c is the objective function vector, and x is the vector of variables.
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5 Engineering Management & Systems Engineering The George Washington University Using matrix notation we can write a maximization LP in standard form as follows: max .. 0 st = cx Ax b x To present the Simplex method for solving LP’s we need first to review some fundamental definitions and results. Consider a system of equations = Ax b with n variables and m equations, Assumptions nm (i.e., there are more variables than equations). the system of equations is solvable (it has at least one solution). we can choose a set of m “dependent” variables (called basic variables) and express them in terms of the remaining n m “independent” variables (called nonbasic variables).
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Zahra_mMohaghegh_Feb12 - Engineering Management Systems...

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