Engineering Optimization: Theory and Practice, Fourth Edition

26 Introduction to Optimization

As the objective function [Eq. (E 5 )] is a quadratic and the constraints [Eqs. (E 1 ) to

(E 4 )] are linear, the problem is a quadratic programming problem.

Linear Programming Problem. If the objective function and all the constraints in

Eq. (1.1) are linear functions of the design variables, the mathematical programming

problem is called alinear programming(LP)problem. A linear programming problem

is often stated in the following standard form:

FindX=











x 1

x 2

..

.

xn











whichminimizesf (X)=

∑n

i= 1

cixi

subject to the constraints (1.10)

∑n

i= 1

aijxi=bj, j= 1 , 2 ,... , m

xi≥ 0 , i= 1 , 2 ,... , n

whereci, aij, andbjare constants.

Example 1.6 A scaffolding system consists of three beams and six ropes as shown

in Fig. 1.12. Each of the top ropesAandBcan carry a load ofW 1 , each of the

middle ropesCandDcan carry a load ofW 2 , and each of the bottom ropesEand

Fcan carry a load ofW 3. If the loads acting on beams 1, 2, and 3 arex 1 , x 2 , andx 3 ,

respectively,as shown in Fig. 1.12, formulate the problem of finding the maximum

Figure 1.12 Scaffolding system with three beams.