Engineering Optimization: Theory and Practice, Fourth Edition

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 ) are given by

Problem(E 5 ) Fig. 10:. 6 ;(x 1 ∗= 5 ,x 2 ∗= 4 ,f∗= 13 )

Problem(E 6 Fig. 10):. 7 ;(x 1 ∗= 0 ,x 2 ∗= 8 ,f∗= 23 )

Since both the variables assumed integer values, the optimum solution of the

integer LP problem, Eqs. (E 1 ) and (E 2 ), is given by(x∗ 1 = 0 ,x 2 ∗= 8 ,f∗= 23 ).

Example 10.4 Find the solution of the welded beam problem of Section 7.22.3 by
treating it as a mixed-integer nonlinear programming problem by requiringx 3 andx 4
to take integer values.

SOLUTION The solution of this problem using the branch-and-bound method was
reported in Ref. [10.25]. The optimum solution of the continuous variable nonlinear
programming problem is given by

X∗= { 0. 24 , 6. 22 , 8. 29 , 0. 24 }T, f∗= 2. 38