Optimization

213Multiobjective Optimization

subject to

f

k

(x) + p

k

− n

k

= τ

k

(7.14)

p

k

, n

k

≥ 0 (7.15)

where w

1,k

and w

2,k

are the weights of the kth goal and p

k

and n

k

are the under-

achievement and overachievement for the kth goal. The main advantage of

goal programming is that multiobjectives are transformed into the con-

straints of a single-objective optimization problem.

Let us consider the following multiobjective optimization problem.

Minimize

f x x

1 1

2

2

( )x = +

Minimize

f x x

2 2

2

1

( )x = −

subject to

−5 ≤ x

1

≤ 5, −3 ≤ x

2

≤ 3

Assuming goals for the two objective functions as 1 and 2, the goal pro-

gramming problem can be written as

Minimize

f

1

(x) = w

11

p

1

+ w

12

p

2

+ w

21

n

1

+ w

22

n

2

subject to

x x p n

1

2

2 1 1

1 0+ + − − =

x x p n

2

2

1 2 2

2 0− +