Engineering Optimization: Theory and Practice, Fourth Edition

616 Integer Programming

Figure 10.8 Solution of the welded beam problem using branch-and-bound method. [10.25]

The problem stated in Eqs. (10.57) to (10.62) cannot be solved using mixed-integer

linear programming techniques since some of the design variables are discrete and

noninteger. The discrete variables are redefined as [10.26]

xi=yi 1 di 1 +yi 2 di 2 + · · · +yiqdiq=

∑q

j= 1

yijdij, i= 1 , 2 ,... , n 0 (10.63)

with

yi 1 +yi 2 + · · · +yiq=

∑q

j= 1

yij= 1 (10.64)

yij= or 1 0 , i= 1 , 2 ,... , n 0 , j= 1 , 2 ,... , q (10.65)

Using Eqs. (10.63) to (10.65) in Eqs. (10.57) to (10.62), we obtain

Minimizef (X)≈f (X^0 )+

∑n^0

i= 1

∂f

∂xi





∑q

j= 1

yijdij−x^0 i





+

∑n

i=n 0 + 1

∂f

∂xi

(xi−xi^0 ) (10.66)