We introduce dominance measuring methods to derive a ranking of alternatives to deal with incomplete information in multi-criteria decision making problems on the basis of Multi-Attribute Utility Theory (MAUT). We consider the situation where the alternative performances are represented by uniformly distributed intervals, and there exists imprecision concerning the decision-makers¿ preferences, by means of classes of individual utility functions and imprecise weights represented by weight intervals or fuzzy weights, respectively. An additive multi-attribute utility model is used to evaluate the alternatives under consideration, which is considered a valid approach in most practical cases. The approaches we propose are based on the dominance values between pairs of alternatives that can be computed by linear programming, which are then transformed into dominance intensities from which a dominance intensity measure is derived. The methods proposed are compared with other existing dominance measuring methods and other methodologies by Monte Carlo simulation techniques. The performance is analyzed in terms of two measures of efficacy: hit ratio, the proportion of all cases in which the method selects the same best alternative as in the TRUE ranking, and the Rank-order correlation, which represents how similar the overall rank structures of alternatives are in the TRUE ranking and in the ranking derived from the method. The approaches are illustrated with an example consisting on the selection of intervention strategies to restore an aquatic ecosystem contaminated by radionuclides

Jiménez Martín, Antonio

Mateos Caballero, Alfonso

Sabio, Pilar

English

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Ranking Alternatives on the Basis of Intensity of 
Dominance and Fuzzy Logic within MAUT 
Antonio Jimenez1 , Alfonso Mateos1 and Pilar Sabio2 
Departamento de Inteligencia Artificial, Facultad de Informatica, 
Universidad Politecnica de Madrid, Campus de Montegancedo s/n, 28660 Madrid, Spain. 
Departmento de Economi'a Aplicada, Facultad de Ciencias Legales y Sociales, 
Universidad Rey Juan Carlos, 28933 Madrid, Spain 
E-mail: ajimenez@fi.upm.es, amateos@fi.upm.es, pilar.sabio@urjc.es 
Abstract . We introduce a dominance measuring method to derive a ranking of al-
ternatives to deal with incomplete information in multi-criteria decision-making problems 
on the basis of multi-attribute utility theory (MAUT) and fuzzy sets theory. We consider 
the situation where the alternative performances are represented by intervals and there 
exists imprecision concerning the decision-makers' preferences, leading to classes of com-
ponent utility functions and trapezoidal fuzzy weights. An additive multi-attribute utility 
model is used to evaluate the alternatives under consideration. The approach we propose 
is based on the dominance values between pairs of alternatives that can be computed by 
linear programming. These values are then transformed into dominance intensities from 
which a dominance intensity measure is derived. The performance of the proposed method 
is analyzed using MonteCarlo simulation techniques. 
Keywords: Decision Analysis, Fuzzy Sets, Environmental Studies. 
I N T R O D U C T I O N 
The additive model is considered a valid approximation in most real decision-
making problems (Raiffa, 1982; Stewart, 1996). The functional form of the additive 
model is 
n 
where x^ is the performance in the attribute Xj of alternative Ai; the utility 
associated with value x^, and Wj are the weights of each attribute, representing their 
relative importance in decision-making. 
Most complex decision-making problems involve imprecise information. For ex-
ample, it is impossible to predict the exact performance of each alternative under 
consideration, being derived from statistical methods. At the same time, it is often 
not easy to elicit precise weights, which are represented by intervals. DMs may 
find it difficult to compare criteria or not want to reveal their preferences in public. 
Furthermore, the decision may be taken within a group, where the imprecision of 
the preferences is the result of a negotiation process. 
A lot of work on MAUT has dealt with incomplete information. Sarabando 
and Dias (2009) give a brief overview of approaches proposed by different authors 
within the MAUT and MAVT (multi-attribute value theory) framework to deal with 
incomplete information. 
Eum et al (2001) provided linear programming characterizations of dominance 
and potential optimality for decision alternatives when information about perfor-
mances and/or weights is incomplete, extended the approach to hierarchical struc-
tures (Lee et al, 2002), and developed the concepts of weak and strong potential 
optimality (Park, 2004). More recently, Mateos and Jimenez (2009) and Mateos et 
al (2009) considered the more general case where imprecision, described by means of 
fixed bounds, appears in alternative performances, as well as in weights and utilities. 
At the same time, a number of studies have been conducted concerning impre-
cision using fuzzy sets theory. These studies feed off the advances of research into 
arithmetic and logical operators of fuzzy numbers, like Tran and Duckstein's study 
proposing the comparison of fuzzy numbers by a fuzzy measure of distance (Tran 
and Duckstein, 2002). 
Following this research line, we consider a decision-making problem with m al-
ternatives, Ai,i = 1,... ,m, and n attributes, Xj,j = 1 , . . . ,n, where incomplete 
information about input parameters has been incorporated into the decision-making 
process: 
• Alternative performances are described under uncertainty [x^, xfA, i = 1 , . . . , m, 
j = 1 , . . . , n, where xf, and xf, are the lower and the upper performances of 
the attribute Xj for the alternative Ai, respectively. 
• Component utilities are described by functions «(• ) , which belong to classes 
of functions of utility [UJ(»), uj (•)], j = 1 , . . . , n, where Uj(») and uj (•) are 
the lower and the upper utility functions of the attribute Xj. 
• Imprecise weights are represented by trapezoidal fuzzy numbers Wj, j = 1 , . . . , n. 
One possibility described in the literature for dealing with imprecision attempts 
to eliminate inferior alternatives based on the concept of pairwise dominance. Given 
two alternatives Ak and Ai, the alternative Ak dominates Ai if Dki > 0, Dki being 
the optimum value of the optimization problem: 
n n 
DM = min {u{Ak) - u{Ai)} = J ] WjUj(xkj) - J ] WjUj{xij) 
xkj — Xkj S xkj i 3 = J- j • • • j n 
s.t. 
xfj < xtj <xfp j = l,...,n 
Uj(xkj) < Uj(xkj) < uf(xkj),j = l,...,n 
u
LAxi3) < Uj{xij) < uuAxi3),j = 1 , . . . ,n . 
The objective function in the above optimization problem can also be represented 
n 
by Dki = m i n Yl Wj [uj(xkj) — Uj(xij)]. Therefore, its resolution is equivalent to 
3=1 
solving (Mateos et al, 2007) 
n 
Dki = ^jWjZ*klj, (1) 
3=1 
where z*kl • is the optimum value of the following optimization problem 
mm zklj = Uj (xkj) - Uj {xLj, 
s.t. (2) 
xkj — Xkj S xkj i 3 — J-i • • • i n 
xfj < xij < zg, j = l,...,n 
Uj(xkj) < Uj(xkj) < v%(xkj),j = 1 , . . . ,n 
Uj(xij) < Uj{xij) < uu-{xi3),3 = 1 , . . . ,n . 
The optimal solution of problem (2) can be determined in a very simple way 
for certain types of utility functions (Mateos et al, 2007). If the utility function 
is monotonically increasing or decreasing, then z^- = u^{xh) — uH(x^,) or z*k 
u1(xki) ~ uf(xii)-> respectively. 
'klj ~ j K^kj) j K-^ljJ ^L ^klj 
A recent approach is to use information about each alternative's intensity of 
dominance, known as dominance measuring methods. Ahn and Park (2008) compute 
both dominating and dominated measures, from which they derive a net dominance. 
This is used as a measure of the strength of preference in the sense that a greater 
net value is better. 
In the next section we introduce a dominance measuring method accounting for 
fuzzy weights. Section 3 analyzes the performance of the proposed method using 
MonteCarlo simulation techniques. Finally, some conclusions are provided in Section 
4. 
M E A S U R E M E N T M E T H O D F O R F U Z Z Y D O M I N A N C E 
The proposed dominance measuring method adapts the proposal by Mateos and 
Jimenez (2009) and Mateos et al (2009) to account for fuzzy weights making use 
of work by Tran and Duckstein (2002) on distances between fuzzy numbers based 
on the generalization of the left and right fuzzy numbers (GLRFN) (Dubois and 
Prade, 1980; Bardossy and Duckstein, 1995). 
A fuzzy set a = (a1; a^ 03,04) is called GLRFN when its membership function 
is defined as 
f^(x) = < 
L 
R 
0 
0,2 — X 
Obi — (l\ 
X — (I3 
G>4 — Cl3 
if d\ < x < a<i 
if a<i<x < a3 
if 03 < x < 0,4 
otherwise, 
where L and R are strictly decreasing functions defined in [0,1] and satisfying the 
conditions: 
L(x) = R(x) = 1 if x < 0 
L(x) = R(x) = 0 if x > 0. 
For a<i = 03, you have Dubois and Prade classic definition of right and left 
fuzzy numbers (Dubois and Prade, 1980). The trapezoidal fuzzy numbers are a 
special case of GLRFN with L(x) = R(x) = 1 — x. A GLRFN is denoted as 
a = (ai, <22, 03, a-iJLa-Ra a n d a ct-cut of a is defined as 
a(a) = (aL(a),aR(a)) = (a2 - (a2 - a1)a3L^1(a),a3 - (a4 - a3)a3R^1(a),). 
(Tran and Duckstein, 2002) define the distance between two GLFRN fuzzy num-
bers a and bas 
D2(a,b,f) 
aL(a)+aR(a) _ bL(a)+bR(a) 
2 2 
2 / a L (a )+a f l (a ) 
I 2 
6L(a)+6 f l(a) 
2 
x f(a)(da) > / / f(a)(da 
The function / ( a ) , which serves as a function of weights, is positive continuous 
in [0,1], the distance being computed as a weighted sum of distances between two 
intervals along all of the a-cuts from 0 to 1. The presence of function / permits 
the DM to participate in a flexible way. For example, when the DM is risk-neutral, 
f{a) = a seems to be reasonable. A risk-averse DM might want to put more weight 
on information at a higher a level by using other functions, such as f{a) = a2 or a 
higher power of a. A constant ( / ( a ) = 1), or even a decreasing function / , can be 
utilized for a risk-prone DM. 
For the particular case of the distance of a trapezoidal fuzzy number a = 
(ai, a2, a3, a4) to a constant (specifically 0), we have: 
1. I f / ( a ) = a t h e n d ( a , 0 ) 2 = f 2 3 J + - f 2 J [(a4 - a3) - (a2 - ai)] + 
2 1 03 — G>2 \ 1 / Q>3 — G>2 \ r / s / s-| J- r/ \2 / \ 2 l 
3 ( g J + 9 I 2 J K a 4 - a 3 ) + ( a 2 - a i ) J + — |>4 - as) + («2 - «i) J-
— [(a2 - ai) (a4 - a3)] . l o 
2. I f / ( a ) = 1 then d(a, 0)2 = ( ^ ^ ) +\ (^±^-\ [(a4 - a3) - (a2 - ffll)] + 
[(a4 - a3) + (a2 - «i)] + - [(«4 - «3)2 + («2 - «i)2] • 
1 fa3 - a 2 \ 1 / a 3 - a 2 \ , , ^ 1 r, N2 , , ^ 
- [(a2 - ai) (a4 - a 3 ) ] . 
y 
As trapezoidal fuzzy numbers are used to represent weights, the objective func-
tion in (1) can be now represented by 
n n 
DM = dkl =^2 ™jzkij = ^2(wji>wji> wj3, wjdzlij = (dkii,dm, dkl3, dm). 
3=1 3=1 
The first step in the proposed method, then, consists of computing the above 
trapezoidal fuzzy numbers. Consequently, the strength of dominance of alternative 
Ak can be defined as 
n n n n 
dk = (dki, dk2, dk3, dkA) = 2_^ DM = 2_^ ^MI , 2_^ dMi, /_^ ^kl3•> /_^ ^MA 
\=} \ 1=1 1=1 1=1 1=1 , 
l
^
k
 \l^k l^k l^k l^k ) 
Next, a dominance intensity, DIk, for each alternative Ak is computed as the 
proportion of the positive part of the fuzzy number dk by the distance of the fuzzy 
number to zero. Specifically, the dominance intensity for alternative Ak is computed 
according to the location of dk as follows: 
1. If dk is completely located at the left of zero, then DIk is minus the distance 
of dk to zero, because there is no positive part in dk. 
2. If 4 is completely located at the right of zero, then DIk is the distance of 4 
to zero, because there is no negative part existing in 4-
3. If 4 includes the zero in its base, then the fuzzy number will have a part on 
the right of zero that we denote d^ and another part on the left of zero that 
we denote d\. DIk is the proportion that represents d^ with respect to 4 by 
the distance of 4 to zero less the proportion that represents d\ with respect 
to 4 by the distance of 4 to zero. 
Next, we analyze each one of these cases in more detail. 
• If 44 < 0, see Figure la), then the dominance intensity of alternative Ak is 
defined as DIk = — < (^4, 0, / ) . 
- Figure 1 -
• If 4 i > 0, see Figure lb), then the dominance intensity of alternative Ak is 
defined as DIk = d(dk, 0, / ) . 
• If 4 i < 0 and dki > 0, see Figure lc), the corresponding trapezoidal fuzzy 
number is divided by the vertical axis (at zero) into two parts. The left part 
d\ represents the proportion 
—4i 
-d fci 42 — 4 l
 2 
2 (dkl) 
44 + 43 - 42 - 4 i (44 + 43 - 42 - 4i) (42 - 4 i ) ' 
2 
whereas the right part d^ represents the proportion 
dkA + dk3 — dk2 — dk\ (—4i)(—4i) 
2 2(42 -
44 + 43 — 42 — 4 i 
2 
dk2{ — dk2 + 43 + 44) -
- 4 i ) 
- 4 i (43 + 44) 
(42 — 4i) (44 + 43 — 42 — d klj 
The dominance intensity of alternative Ak is defined as 
n r dk2(—dk2 + dk3 + dkA) — dki(dk3 + dk4:) u j f. U1k= — 7 1 , x / , — — , ~, -;—,—44, U,/J 
\Ct-k2 — Ctkl) \akA + «fc3 — «fc2 _ Ctkl) 
-JT^Ti }dkl)] M, T~A^ °> /)• 
[flkA + «fc3 — «fc2 — Ctkl) \ak2 ~ Ctkl) 
If 4 3 < 0 and 4 4 > 0, Figure Id) , the corresponding trapezoidal fuzzy number 
is again divided by the vertical axis into two parts, d\ and d^, represented by 
the proportions 
, 4 4 
dkA 
dkA 
+ dk3 — dk2 -
2 
dkA + dk3 
dkA {dkA -
{dkA 
dkA 
dkA — dk3 
2 
- dki 
— dk2 
2 
- dk2 — 
+ 43" 
u>kA , 
Cl-kA 
2 
— dki 
dki) ~ 
- dk2 — 
— dk3 
dk3 {dk3 — 
dki) {dkA ~ 
{dkA)' 
dk2 — 
- dk3) 
i 
dki) 
, 
and 
dkA + 4)3 - dk2 - dki {dkA - dk3) {dkA + dk3 - dk2 - 4 i ) ' 
2 
respectively, and the dominance intensity of alternative Ak is 
Dh
= J7l rl Mrl{dkfrl 7, ^A^Af) 
[dkA — Clk3) [Ct-kA + «fc3 — «fc2 _ Ctkl) 
dkA {dkA — dk2 — dki) — dk3 {dk3 — dk2 — 4 i ) , / T n f\ 
—a{ak,(J,j). {dkA + dk3 — dk2 — dki) {dkA — d k3 
If dk2 < 0 and 4 3 > 0, see Figure le) , dj^ and d^ are 
dkA — dk3 -, 
2 + dk3 dkA + 4 3 
dkA + 4 3 — 4 2 — 4 i dkA + 4 3 — 4 2 — 4 l 
2 
and 
4 2 — 4 i dk2
 -4 2 - 4i 
dkA + 4 3 — 4 2 — 4 i dkA + 4 3 — 4 2 — 4 l ' 
2 
respectively, and the dominance intensity of alternative Ak is 
P>T ~42 - 4l ,/T
 n r\ 44 + 43 ,/T n r\ DIk =
 1—77i 1 1—44, 0, /) - - — — —44, 0, /). 
akA T ak3 — ak2 — akl akA T «fc3 _ ak2 ~ akl 
Once the dominance intensity, DIk, has been computed for each alternative Ak, 
the alternatives are ranked, where the best (rank 1) is the alternative with greatest 
DIk and the worst is the alternative with the least DIk-
P E R F O R M A N C E A N A L Y S I S B A S E D O N M O N T E C A R L O SIMU-
LATION T E C H N I Q U E S 
We carried out a simulation study of the above method to analyze its perfor-
mance. Four different levels of alternatives (m = 3, 5, 7,10) and five different levels 
of attributes (n = 3, 5, 7,10,15) were considered in order to validate the results. 
Also, 5000 trials were run for each of the 20 design elements. 
We used two measures of efficacy, hit ratio and rank-order correlation (Ahn and 
Park, 2008; Mateos et al, 2009). The hit ratio is the proportion of all cases in which 
the method selects the same best alternative as in the TRUE ranking. Rank-order 
correlation represents how similar the overall structures ranking alternatives are in 
the TRUE ranking and in the ranking derived from the method. It is calculated 
using Kendall's r (Winkler and Hays, 1985): r = 1 — 2 x (number of pairwise 
preference violations)/(total number of pair preferences). 
First, component utilities for each alternative in each attribute from a uniform 
distribution in (0,1) are randomly generated, leading to an m x n matrix. The 
columns in this matrix are normalized to make the smallest value 0 and the largest 
1, and dominated alternatives are removed. Next, attribute weights representing 
their relative importance are generated. Note that these weights are the TRUE 
weights and the derived ranking of alternatives will be denoted as the TRUE ranking. 
Those weights are then transformed into triangular fuzzy numbers applying a 5% 
deviation from the original weight. Finally, the ranking of alternatives is computed 
and compared with the TRUE ranking. Table 1 exhibits the measures of efficacy for 
each of the 20 design elements for the cases of a risk-prone and a risk-neutral DM. 
- Table 1 -
We can conclude that the hit ratio and the correlation coefficient are very similar 
for the cases of a risk-prone and a risk-neutral DM; the hit ratio is greater than 78% 
for all the design elements, whereas Kendall's r is greater than 86%. Both are 
increasing against the number of attributes. Whereas the hit ratio decreases when 
the number of alternatives is increased, Kendall's r increases. 
C O N C L U S I O N S 
Dominance measuring methods are becoming widely used in a decision-making 
context with incomplete information and have been proved to outperform other 
approaches, like most surrogate weighting methods or the modification of classical 
decision rules to encompass an imprecise decision context. 
In this paper a new dominance measuring method has been proposed, in which 
weights representing the relative importance of decision-making criteria are de-
scribed by fuzzy numbers. The method is based on the distance of a fuzzy number 
to a constant, where the generalization of the left and right fuzzy numbers, GLRFN, 
is used. 
The application of Monte Carlo simulation techniques has demonstrated that 
the proposed method performs well in terms of two measures of efficacy: hit ratio 
and rank-order correlation. 
Acknowledgments . This paper was supported by the Spanish Ministry of Ed-
ucation and Innovation project TIN 2008-06796-C04K)2 and by Madrid Regional 
Goverment project S2009/ESP-1594. 
Refe rences . 
Ahn BS and Park KS (2008). Comparing methods for multiattribute decision 
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Bardossy A and Duckstein L (1995). Fuzzy rule-based modeling with applications 
to geophysical, biological and engineering systems. CRC Press: Boca Raton. 
Dubois D and Prade H (1980). Fuzzy sets and systems: theory and applications. 
Academic Press: New York. 
Eum Y, Park KS and Kim H (2001). Establishing dominance and potential 
optimality in multi-criteria analysis with imprecise weights and values. Comput. 
Operations Res. 28: 397-409. 
Lee K, Park KS and Kim H (2002). Dominance, potential optimality, imprecise 
information, and hierarchical structure in multi-criteria analysis. Comput. Opera-
tions Res. 29: 1267-1281. 
Mateos A and Jimenez A (2009). A trapezoidal fuzzy numbers-based approach 
for aggregating group preferences and ranking decision alternatives in MCDM. In: 
Evolutionary Multi-Criterion Optimization, LNCS 5467: 365-379. 
Mateos A, Jimenez A and Blanco J F (2009). Ranking methods based on domi-
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5783: 328 339. 
Mateos A, Rios-Insua S and Jimenez A (2007). Dominance, potential optimality 
and alternative ranking in imprecise decision making. J. Oper. Res. Soc. 58: 
326^336. 
Park K (2004). Mathematical programming models for characterizing dominance 
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Multi-Criteria Decision Anal. 5: 301-309. 
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Holt, Rinehart & Winston: New York. 
a) 
ft 
•
 b > . 
3"* 
H h 
^ 
c) • 
ft 
^*2 " B "ifc4 >*2 ^ 3 "S:4 
H h 
4 l < 2^ ^ 3 S^:4 
d) 
ft 
^ J r 1 
e) 
^ 
ft 
<4l d i 2 ^B ^ 4 "M ^£2 ^ B ^ M 
Figure 1 
Table 1 
Alternatives 
3 
3 
3 
3 
3 
5 
5 
5 
5 
5 
7 
7 
7 
7 
7 
10 
10 
10 
10 
10 
Attributes 
3 
5 
7 
10 
15 
3 
5 
7 
10 
15 
3 
5 
7 
10 
15 
3 
5 
7 
10 
15 
Hit ratio 
risk-prone 
90.78 
91.42 
91.02 
90.9 
89.78 
85.02 
85.8 
85.16 
83.64 
83.74 
84.4 
83.72 
83.48 
82.98 
81.14 
85.22 
83.08 
82.28 
82.74 
78.36 
risk-neutral 
90.72 
91.34 
91 
90.96 
89.78 
84.98 
85.8 
85.12 
83.66 
83.66 
84.38 
83.66 
83.34 
82.94 
81.16 
85.08 
82.96 
82.18 
82.66 
78.54 
Kendall's r 
risk-prone 
88.90 
88.77 
87.97 
87.08 
86.24 
89.8 
89.60 
89.29 
88.11 
86.72 
92.16 
91.48 
91.07 
90.21 
88.66 
93.89 
93.51 
93.01 
92.08 
90.41 
risk-neutral 
88.85 
88.66 
87.97 
87.15 
86.27 
89.79 
89.59 
89.28 
88.18 
86.70 
92.15 
91.47 
91.04 
90.26 
88.67 
93.87 
93.49 
93.01 
92.12 
90.45 
Captions of Figures and Tables 
Figure 1: Computing dominance intensities 
Table 1. Measures of efficacy 


Ranking Alternatives on the Basis of the Intensity of Dominance and Fuzzy Logic within MAUT

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