Engineering Optimization: Theory and Practice, Fourth Edition

32 Introduction to Optimization

cjxj/. Thus the objective function (cost of ordering plus storing) can be expressed 2

as

f (X)=

(

a 1 d 1

x 1

+

q 1 c 1 x 1

2

)

+

(

a 2 d 2

x 2

+

q 2 c 2 x 2

2

)

+

(

a 3 d 3

x 3

+

q 3 c 3 x 3

2

)

(E 1 )

where the design vectorXis given by

X=







x 1

x 2

x 3







(E 2 )

Theconstraint on the worth of inventory can be stated as

c 1 x 1 +c 2 x 2 +c 3 x 3 ≤ 54 , 000 (E 3 )

The limitation on the storage area is given by

s 1 x 1 +s 2 x 2 +s 3 x 3 ≤ 09 (E 4 )

Since the design variables cannot be negative, we have

xj≥ 0 , j= 1 , 2 , 3 (E 5 )

Bysubstituting the known data, the optimization problem can be stated as follows:

FindXwhich minimizes

f (X)=

(

40 , 000

x 1

+ 01 x 1

)

+

(

32 , 000

x 2

+ 03 x 2

)

+

(

120 , 000

x 3

+ 02 x 3

)

(E 6 )

subjectto

g 1 ( X)= 40 x 1 + 201 x 2 + 08 x 3 ≤ 54 , 000 (E 7 )

g 2 (X)= 0. 40 (x 1 +x 2 +x 3 ) ≤ 90 (E 8 )

g 3 ( X)=−x 1 ≤ 0 (E 9 )

g 4 ( X)=−x 2 ≤ 0 (E 10 )

g 5 ( X)=−x 3 ≤ 0 (E 11 )

It can be observed that the optimization problem stated in Eqs. (E 6 ) to (E 11 ) is a

separable programming problem.

1.5.8 Classification Based on the Number of Objective Functions

Depending on the number of objective functions to be minimized, optimization prob-

lems can be classified as single- and multiobjective programming problems. According

to this classification, the problems considered in Examples 1.1 to 1.9 are single objective

programming problems.