Engineering Optimization: Theory and Practice, Fourth Edition

1.5 Classification of Optimization Problems 31

Separable Programming Problem.

Definition A functionf(X) is said to beseparableif it can be expressed as the sum
ofnsingle-variable functions,f 1 (x 1 ), f 2 (x 2 ),... , fn(xn) that is,,

f (X)=

∑n

i= 1

fi(xi) (1.11)

A separable programming problem is one in which the objective function and the
constraints are separable and can be expressed in standard form as

FindXwhich minimizesf (X)=

∑n

i= 1

fi(xi) (1.12)

subject to

gj(X)=

∑n

i= 1

gij(xi)≤bj, j= 1 , 2 ,... , m

wherebjis a constant.

Example 1.9 A retail store stocks and sells three different models of TV sets. The
store cannot afford to have an inventory worth more than $45,000 at any time. The
TV sets are ordered in lots. It costs $ajfor the store whenever a lot of TV modelj
is ordered. The cost of one TV set of modeljiscj. The demand rate of TV model
jisdjunits per year. The rate at which the inventory costs accumulate is known to
be proportional to the investment in inventory at any time, withqj= 0. 5 , denoting
the constant of proportionality for TV modelj. Each TV set occupies an area of
sj= 0. 4 0 m^2 and the maximum storage space available is 90 m^2. The data known from
the past experience are given below.

TV modelj

1 2 3

Ordering cost,aj($) 50 80 100

Unit cost,cj($) 40 120 80

Demand rate,dj 800 400 1200

Formulate the problem of minimizing the average annual cost of ordering and storing
the TV sets.

SOLUTION Letxj denote the number of TV sets of modelj ordered in each lot
(j= 1 , 2 ,3). Since the demand rate per year of modelj isdj, the number of times
the TV modeljneeds to be ordered isdj/xj. The cost of ordering TV modeljper
year is thusajdj/xj, j = 1 , 2 , 3. The cost of storing TV sets of modeljper year is
qjcjxj/ since the average level of inventory at any time during the year is equal to 2