Fuzzy Multiple Objective Decision Making

25

3

Goal Programming

The purpose of multiple objective decision making (MODM) is to achieve the ef-

cient frontier of multiple objective programming (MOP). Traditionally, the weight-

ing method and ε-constraint method have been used widely. The weighting method

transforms a set of objectives into a single objective by multiplying each objective

with a user-supplied weight. On the other hand, the ε-constraint method keeps only

one of the objectives and restricts the rest of the objectives within some user-specied

values to derive the efcient frontier. Cohon (1978) developed an algorithm to gen-

erate the efcient set systematically. However, goal programming (GP) is the best

known method for dealing with MODM problems.

GP is an analytical approach devised to address decision making problems in

which targets have been assigned to all the attributes and the decision maker is inter-

ested in minimizing the non-achievements of the corresponding goals (Romero,

2001). Generally, goal programming deals with the following MOP problems:

=…

≤

≥

fx

in

st gx

x

max(), 1, ,

.. () 0,

0,

i

where

xf ()

i

denotes the ith objective function.

3.1 GOAL SETTING

Three kinds of goal settings are utilized in multi-objective optimization problems:

(1) minimize all the objective functions; (2) maximize all the objective functions;

and (3) minimize some and maximize others. However, we can use duality principle

(Reklaitis et al., 1983; Rao, 1984) to convert a minimization problem into a maximiza-

tion problem by multiplying the objective function by –1. Hence, we can simplify mixed

types of objectives and include all types of objectives into maximization problems.

Unlike single-objective problems, multi-objective programming problems do not

usually allow all objectives to be optimized due to trade-offs or conicts among

objectives. Hence, there exists a set of non-dominated solutions (the efcient set)

such that all points belonging to non-dominated solutions are regarded as indifferent.

If we want to determine an optimal solution from the set of non-dominated solutions,

we must measure which point is nearest to the ideal. Assume a two-objective pro-

gramming maximization problem:

≤

≥

Ax b

xx

x

ff

st g

x

max(), ()

.. () 0(or write

≤)

,

0.

12

(3.1)

Figure3.1 illustrates these concepts in detail.

26 Fuzzy Multiple Objective Decision Making

The ideal point is composed of individual optimal objective values. In this exam-

ple, =

∗∗

∗

y

ff

(,

)

12

. However, the ideal point vector usually corresponds to a non-existent

solution because of the trade-off between objectives. Therefore, the problem of opti-

mization in multi-objective programming is transformed into nding a feasible solu-

tion (location on the non-dominated set) that is nearest to the ideal points.

Three kinds of targets or goal settings (the one-lower upper goal, one-sided

upper goal, and two-sided goal) can be consider in goal programming. A one-sided

lower goal sets a lower limit that decision makers do not want to fall under, that is,

+=

−

x

fd

T()

iii

, where

−

d

i

is the underachievement derivational variable of the ith

objective and

T

i

denotes the target of the ith objective. A one-sided upper goal sets

an upper limit that decision makers do not want to exceed, that is,

−=

+

x

fd

T()

ii

i

,

where, where

+

d

i

is the overachievement derivational variable of the ith objective. A

two-sided goal sets an exact target that decision makers do not want to miss on either

side, that is, +−=

−+

xfd

dT

()

i

iii

.

Initially conceived as an application of single objective linear programming by

Charnes and Cooper (1955, 1961), goal programming gained popularity in the 1960s

and 1970s based on the works of Ijiri (1965), Lee (1972), and Ignizio (1976). A key

element of a GP model is the achievement function that represents a mathematical

expression of the unwanted deviation variables. Each type of achievement function

leads to a different GP variant.

Tamiz and others (1995) show that around 65% of GP applications reported used

lexicographic achievement functions, 21% utilized weighted achievement functions,

and the remaining applications involved other types of achievement functions such

as a min–max structure in which maximum deviation is minimized.

The next section introduces three kinds of goal programming: weighted goal pro-

gramming, lexicographic goal programming, and min–max goal programming, to

deal with multi-objective programming problems.

3.2 WEIGHTED GOAL PROGRAMMING

Goal programming was proposed by Charnes and Cooper (1961) to deal with linear

multi-objective programming problems. The idea of goal programming is to seek a solu-

tion that is nearest to the ideal point by considering the relative weights of importance

Outcome

Space

Ideal point

1

f

2

f

Non-dominated

Solutions

1

f

2

f

A

B

y* = (ff

1

,

2

*)

*

*

*

FIGURE 3.1 Concepts of multi-objective programming.