Engineering Optimization: Theory and Practice, Fourth Edition

10.3 Gomory’s Cutting Plane Method 603 SOLUTION Step 1: Solve the LP problem by simplex method by neglecting the integer require ...

604 Integer Programming When this constraint is added to the tableau above, we obtain the following: Basic Coefficients of varia ...

10.4 Balas’ Algorithm for Zero–One Programming Problems 605 method by introducing the additional constraint that all the variabl ...

606 Integer Programming Initial Solution. An initial solution for the problem stated in Eqs. (10.28) can be taken as f 0 = 0 xi= ...

10.5 Integer Polynomial Programming 607 in two stages. In the first stage we see how an integer variable,xi, can be represented ...

608 Integer Programming Method of Findingq 0 , q 1 , q 2 ,.... LetMbe the given positive integer. To find its binary representat ...

10.6 Branch-and-Bound Method 609 thekth term of the polynomial simply becomesckyk. However, we need to add the following constra ...

610 Integer Programming Xsatisfies constraints (10.46) and (10.47). A design vectorXthat satisfies all the constraints, Eqs. (10 ...

10.6 Branch-and-Bound Method 611 It can be seen that a node can be fathomed if any of the following conditions are true: 1.The c ...

612 Integer Programming x 1 Figure 10.4 Graphical solution of problem (E 3 ). Step 3: The next branching process, with integer b ...

10.6 Branch-and-Bound Method 613 Figure 10.5 Graphical solution of problem (E 4 ). The solutions of problems (E 5 ) and (E 6 ) a ...

614 Integer Programming Figure 10.6 Graphical solution of problem (E 5 ). Next, the branching problems, with integer bounds onx ...

10.7 Sequential Linear Discrete Programming 615 Figure 10.7 Graphical solution of problem (E 6 ). where the firstn 0 design vari ...

616 Integer Programming Figure 10.8 Solution of the welded beam problem using branch-and-bound method. [10.25] The problem state ...

10.7 Sequential Linear Discrete Programming 617 subject to gj(X)≈gj(X^0 )+ ∑n^0 i= 1 ∂gi ∂xi (n 0 ∑ l= 1 yildil−xi^0 ) + ∑n i=n ...

618 Integer Programming adjacent lower value—for simplifying the computations. UsingX^0 = { 1. 2 1. 1 } ,we have f (X^0 ) = 6. 5 ...

10.8 Generalized Penalty Function Method 619 10.8 Generalized Penalty Function Method The solution of an integer nonlinear progr ...

620 Integer Programming the point always remains in the feasible region. The termskQk(Xd) an be consideredc as a penalty term wi ...

10.8 Generalized Penalty Function Method 621 Figure 10.10 Solution of a single-variable integer problem by penalty function meth ...

622 Integer Programming where ∇Pk=      ∂Pk/∂x 1 ∂Pk/∂x 2 .. . ∂Pk/∂xn      (10.82) The initial value ofs 1 , ...