1.5 Classification of Optimization Problems 29
and the constraints by
4 x 1 + 8 x 2 + 2 x 3 + 5 x 4 + 3 x 5 ≤ 0002 (E 2 )
9 x 1 + 7 x 2 + 4 x 3 + 3 x 4 + 8 x 5 ≤ 5002 (E 3 )
xi≥ and integral 0 , i= 1 , 2 ,... , 5 (E 4 )
Since xi are constrained to be integers, the problem is an integer programming
problem.
1.5.6 Classification Based on the Deterministic Nature of the Variables
Based on the deterministic nature of the variables involved, optimization problems can
be classified as deterministic and stochastic programming problems.
Stochastic Programming Problem. A stochastic programming problem is an opti-
mization problem in which some or all of the parameters (design variables and/or
preassigned parameters) are probabilistic (nondeterministic or stochastic). According
to this definition, the problems considered in Examples 1.1 to 1.7 are deterministic
programming problems.
Example 1.8 Formulate the problem of designing a minimum-cost rectangular under-
reinforced concrete beam that can carry a bending momentMwith a probability of at
least 0.95. The costs of concrete, steel, and formwork are given byCc= 200 $ /m^3 , Cs=
$ 5000 /m^3 , andCf= 40 $ /m^2 of surface area. The bending momentMis a probabilistic
quantity and varies between 1× 105 and 2 × 105 N-m with a uniform probability. The
strengths of concrete and steel are also uniformly distributed probabilistic quantities
whose lower and upper limits are given by
fc= 5 and 35 MPa 2
fs= 00 and 550 MPa 5
Assume that the area of the reinforcing steel and the cross-sectional dimensions of the
beam are deterministic quantities.
SOLUTION The breadthbin meters, the depthdin meters, and the area of reinforcing
steelAsin square meters are taken as the design variablesx 1 , x 2 , andx 3 , respectively
(Fig. 1.13). The cost of the beam per meter length is given by
f (X)=cost of steet+cost of concrete+cost of formwork
=AsCs+ (bd−As)Cc+ 2 (b+d)Cf (E 1 )
Theresisting moment of the beam section is given by [1.119]
MR=Asfs
(
d− 0. 59
Asfs
fcb
)
