37
4
Compromise Solution
and TOPSIS
In this chapter, the compromise solutions and TOPSIS (technique for order prefer-
ence by similarity to ideal solution) for MODM are introduced. Both methods involve
the concept of the Lp-norm and nd the optimal solutions based on reference points.
4.1 COMPROMISE SOLUTION
In a multiple objective programming (MOP) problem, an ideal (or utopian) point is
usually not attainable if trade-offs between objectives exist. Hence, Yu (1973) pro-
posed the compromise solutions to determine the optimal solution closest to the ideal
point among Pareto solutions based on the L
p
-norm distance. Figure4.1 depicts the
concept. The L
p
-norm distance between a point and an ideal point can be dened as:
…=−
=∞
∗
dffp
,1,,
p
p
(4.1)
In a generalized optimal problem, the distance measured by the L
p
-norm between
a point and the ideal point can be presented as shown in Figure4.2. The lower left
square belongs to the maximized problems (maximize all the objective functions) and
is the case covered in this chapter. From Figure4.2, it can be seen that the shape of
p = 1 is a square diamond, p = 2 is a circle, and p = ∞ is a square. The different shapes
of the L
p
-norm may result in a different result due to the optimal solution. Other kinds
of the L
p
-norm are less discussed because they have no concrete meanings in practice.
The procedures of the compromise solutions can be demonstrated by considering
a multiple objective programming (MOP) problem as follows:
…=zx xx xzz zmax()[(),(), ,()]
n12
(4.2)
≤ Ax bxbst g.. () (orwriting≤),
x ≥ 0.
The rst step of the compromise solution is to determine the ideal point of each
objective. This can be done by optimizing each objective as follows:
z xmax()
i
(4.3)
xbst.. ()≤
x ≥ 0.
Then we can obtain the ideal point as
z zz z(,,,
)
n12
…=
∗∗
∗∗
. Next, we want to determine
which point located on the Pareto solutions is closest to the ideal point as the optimal
38 Fuzzy Multiple Objective Decision Making
solution. We can use the concept of the L
p
-norm to measure the distance between objec-
tive values and the ideal point and formulate a compromise solution method (Yu, 1973):
…
∑
=−
=∞
∗
=
xxdwzz pmin [()()]
,1
,,
p
i
p
ii
p
i
n
p
1
1
(4.4)
≤ Ax bxbst g.. () (orwriting≤
x ≥ 0.
where w
i
denotes the importance of the ith objective. Besides using the traditional
L
p
-norm, Duckstein (1984) proposed the normalized L
p
-norm and formulated the
compromise solutions as:
…
∑
=
−
−
=∞
∗
∗−
=
xx
xx
dw
zz
zz
pmin
() ()
() ()
,1,,
p
j
p
ii
ii
p
i
n
p
1
1
(4.5)
≤ Ax bxbst g.. () (orwriting ≤)
x ≥ 0.
where
xz ()
i
*
and
−
xz ()
i
denote the maximal value (or aspiration level, or positive
ideal point) and the minimum value (or the worst value, or negative ideal point) of
the ith goal respectively.
f
p
2p
1p
FIGURE 4.2 Concept of L
p
-norm distance.
Flexible
Solutions
f
(Ideal point)
1
f
2
f
1
f
2
f
Pareto Solutions
FIGURE 4.1 Concept of compromise solutions.
39Compromise Solution and TOPSIS
Next, we depict a two-objective case, as shown in Figure 4.3, to illustrate the
concepts of the PIS and the NIS. If ‘max’ is better and/or ‘min’ is better in each
objective, we can rst dene the PIS and the NIS, respectively, as:
Vector f* = PIS
= {max f
i
(x), ∀i, and/or min f
j
(x), ∀j; or setting the aspiration level of each objective}
and
Vector f
–
= NIS
= {min f
i
(x), ∀
i
, and/or max f
j
(x), ∀j; or setting the worst value of each objective}
Example 4.1 Consider a two-objective programming problem as follows:
−+xx
max35
12
−xx
max7 4
12
+≤
st xx.. 23 30,
12
+≤
xx53 45,
12
+≥
xx
26
,
12
≥
xx
,0
.
12
The first step of the compromise solution is to determine an ideal point. This can
be done by optimizing each objective separately. Hence, the ideal point of the
above problem can be calculated as =
∗
z (50,63) and =− −
−
z (27, 40
)
. Then, if
we set p = 2, we can formulate the following compromise solution programming:
=×
−− +
−−
+
−−
−−
=
d
xx xx
min0.5
50 (3 5)
50 (27)
63 (7 4)
63 (40)
p 2
12
2
12
2
12
+≤st xx.. 23 30,
12
+≤xx53 45,
12
+≥xx26,
12
≥xx,0.
12
Finally, we can obtain
=
x 5.25
1
,
=
x 6.25
2
, and
=
=
d
0.6696
p 2
. The objective
values are
=
xf () 15.48
1
and
=
xf ()
11.78
2
, respectively. On the other hand, if we
set p = ∞, we can also formulate the min–max compromise solution programming:
vmin
−− +
−−
≤st
xx
v..
50 (3 5)
50 (27)
,
12
−−
−−
≤
xx
v
63 (7 4)
63 (40)
,
12
+≤
xx23 30,
12

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