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)