Engineering Optimization: Theory and Practice, Fourth Edition
3.3 Standard Form of a Linear Programming Problem 123 where X= x 1 x 2 .. . xn , b= b ...
124 Linear Programming I: Simplex Method Similarly, if the constraint is in the form of a “greater than or equal to” type of ine ...
3.4 Geometry of Linear Programming Problems 125 on the various machines are given by 10 x+ 5 y≤ 2500 (E 1 ) 4 x+ 10 y≤ 2000 (E 2 ...
126 Linear Programming I: Simplex Method Figure 3.4 Contours of objective function. In some cases, the optimum solution may not ...
3.5 Definitions and Theorems 127 Figure 3.6 Unbounded solution. can be taken as an optimum solution with a profit value of $20,0 ...
128 Linear Programming I: Simplex Method Definitions 1.Point inn-dimensional space.A pointXin ann-dimensional space is char- act ...
3.5 Definitions and Theorems 129 Figure 3.8 Hyperplane in two dimensions. 4.Convex set.A convex set is a collection of points su ...
130 Linear Programming I: Simplex Method Figure 3.11 Convex polytopes in two and three dimensions(a, b)and convex polyhedra in t ...
3.5 Definitions and Theorems 131 10.Basic feasible solution.This is a basic solution that satisfies the nonnegativity conditions ...
132 Linear Programming I: Simplex Method Proof: The feasible regionSof a standard linear programming problem is defined as S= {X ...
3.6 Solution of a System of Linear Simultaneous Equations 133 a feasible solution andfC= λfA+ ( 1 −λ)fB. In this case, the value ...
134 Linear Programming I: Simplex Method the equationkEr, where kis a nonzero constant, and (2) any equationEris replaced by the ...
3.7 Pivotal Reduction of a General System of Equations 135 0 x 1 + 0 x 2 + 1 x 3 + · · · + 0 xn=b′′ 3 (3.16) 0 x 1 + 0 x 2 + 0 x ...
136 Linear Programming I: Simplex Method One special solution that can always be deduced from the system of Eqs. (3.19) is xi= { ...
3.7 Pivotal Reduction of a General System of Equations 137 Finally we pivot ona′ 33 to obtain the required canonical form as x 1 ...
138 Linear Programming I: Simplex Method into the current basis in place ofx 2. Thus we have to pivota′′ 23 in Eq. (II 4 ) This ...
3.9 Simplex Algorithm 139 solutions and pick the one that is feasible and corresponds to the optimal value of the objective func ...
140 Linear Programming I: Simplex Method minimizes the functionf (X)and satisfies the equations: 1 x 1 + 0 x 2 + · · · + 0 xm+a ...
3.9 Simplex Algorithm 141 Since the variablesxm+ 1 , xm+ 2 ,... , xnare presently zero and are constrained to be nonnegative,the ...
142 Linear Programming I: Simplex Method xm=b′′m−a′′msxs, b′′m≥ 0 f=f 0 ′′+c′′sxs, c′′s< 0 (3.28) Sincec′′s < , Eq. (3.28) ...
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