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Course: CPTS 580, Fall 2009
School: Midwestern State...
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Programming Standard Linear Form Objective function n maximize j=1 cj xj n j=1 Constraints subject to Nonnegativity constraints aij xj bi for i = 1, 2, . . . , m for j = 1, 2, . . . , n xj 0 Polynomial time solution in n, m Non-Standard Form n maximize j=1 n cj xj subject to j=1 aij xj bi for i = 1, 2, . . . , m for j = 1, 2, . . . , n xj 0 Converting into Standard Form (1) maximize -2x1 + 3x2...

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Programming Standard Linear Form Objective function n maximize j=1 cj xj n j=1 Constraints subject to Nonnegativity constraints aij xj bi for i = 1, 2, . . . , m for j = 1, 2, . . . , n xj 0 Polynomial time solution in n, m Non-Standard Form n maximize j=1 n cj xj subject to j=1 aij xj bi for i = 1, 2, . . . , m for j = 1, 2, . . . , n xj 0 Converting into Standard Form (1) maximize -2x1 + 3x2 subject to x1 + x2 x1 - 2x1 x1 = minimize 2x1 - 3x2 7 4 0 Converting into Standard Form (2) minimize 2x1 - 3x2 subject to x1 + x2 = 7 x1 - 2x1 x1 4 0 2x1 - 3x2 + 3x2 x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 = 7 4 0 x2 = x2 - x2 Converting into Standard Form (3) x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 = 7 4 0 x1 + x2 - x2 x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 7 7 4 0 Converting into Standard Form (4) x1 + x2 - x2 x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 7 7 4 0 x1 + x2 - x2 -x1 - x2 + x2 x1 - 2x2 + 2x2 x1 , x2 , x2 7 -7 4 0 Optimal Routes Topology G = (V, E) K -- set of origin-destination flows k K, s -- source, t -- destination, k k dk -- demand Set of given link weights {w | (i, j) E} X -- Fraction of flow k going over (i, j) ij k ij minimize kK (i,j)E k wij Xij subject to k K, i = sk , tk , j:(i,j)E k Xij - k Xji = 0 j:(j,i)E k, k X sk j = 1 j:(sk ,j)E k, k Xjtk = 1 j:(j,tk )E k i, j, k, Xij 0 minimize kK (i,j)E k wij Xij ... 1 wij Xij (i,j)E (i,j)E 2 wij Xij ... k wij Xij (i,j)E ... subject to i = sk , tk , j:(i,j)E k Xij - k Xji = 0 j:(j,i)E k Xij 0 k X sk j = 1 j:(sk ,j)E j:(j,tk )E k Xjtk = 1 Interpretation k X ij Optimal solutions {0, 1} {0, 1} k X ij k X ij is the shortest path k X ij Multiple shortest paths exist n minimize j=1 n cj xj aij xj bi subject to j=1 for i = 1, 2, . . . , m for j = 1, 2, . . . , n xj 0 minimize subject to cT x Ax = b x0 x0 is feasible if Ax0 = b and x0 0 Dual Problem Primal maximize Problem cT x subject to Ax b x0 Dual Problem maximize yT b subject to y T A cT y0 If {xi} and {yi} are optimal, then the objective functions are equal y i bi = i i ci x i Properties If x and y are feasible and if c x = y b, If x and y are feasible, then cTx yTb T T then x and y are optimal If either problem has finite solution, so does other; if either has unbounded solution, so does other k wij Xij kK (i,j)E ... 1 wij Xij (i,j)E (i,j)E 2 wij Xij minimize ... k wij Xij (i,j)E ... subject to i = sk , tk , j:(i,j)E k Xij - k Xji = 0 j:(j,i)E k Xij 0 k X sk j = 1 j:(sk ,j)E j:(j,tk )E k Xjtk = 1 Example: Primal minimize (i,j)E k wij Xij subject to 0 k k 1 Xij - Xji = -1 j:(i,j)E j:(i,j)E k Xij 0 i = sk , tk i = sk i = tk minimize (i,j)E k wij Xij subject to 0 k k 1 Xij - Xji = -1 j:(i,j)E j:(i,j)E k Xij 0 i = sk , tk i = sk i = tk Primal Problem maximize cT x subject to Ax b x0 Dual Problem maximize yT b subject to y T A cT y0 minimize (i,j)E k wij Xij subject to 0 k k 1 Xij - Xji = -1 j:(i,j)E j:(i,j)E k Xij 0 i = sk , tk i = sk i = tk Primal Problem maximize ...

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