Engineering Optimization: Theory and Practice, Fourth Edition

122 Linear Programming I: Simplex Method

whereαis a constant indicating the weight per unit length of the member with a

unit plastic moment capacity. Since a constant multiplication factor does not affect the

result,f can be taken as

f= 2 lMb+ 2 hMc= 02 Mb+ 61 Mc (E 1 )

The constraints(U≥E)from the four collapse mechanisms can be expressed as

Mc≥ 6

Mb≥ 2. 5

2 Mb+Mc≥ 71

Mb+Mc≥ 21 (E 2 )

3.3 Standard Form of a Linear Programming Problem

The general linear programming problem can be stated in the following standard

forms:

Scalar Form

Minimizef (x 1 , x 2 ,... , xn)=c 1 x 1 +c 2 x 2 + · · · +cnxn (3.1a)

subject to the constraints

a 11 x 1 +a 12 x 2 + · · · +a 1 nxn =b 1

a 21 x 1 +a 22 x 2 + · · · +a 2 nxn =b 2

..

.

am 1 x 1 +am 2 x 2 + · · · +amnxn =bm

(3.2a)

x 1 ≥ 0

x 2 ≥ 0

..

.

xn ≥ 0

(3.3a)

wherecj, bj, andaij(i = 1 , 2 ,... , m;j= 1 , 2 ,... , n)are known constants, andxj

are the decision variables.

Matrix Form

Minimizef (X)=cTX (3.1b)

subject to the constraints

aX=b (3.2b)

X≥ 0 (3.3b)