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DOCUMENT DE TRAVAIL 1999-003

THE SIR METHOD : A SUPERIORITY AND INFERIORITY

RANKING METHOD FOR MULTIPLE CRITERIA DECISION

MAKING

Xiaozhan Xu

Centre de Recherche sur l’Aide à l’Évaluation et à laDécision dans les Organisations (CRAEDO)

Version originale :Original manuscript :Version original :

ISBN – 2-89524-073-6ISBN -ISBN -

Série électronique mise à jour :One-line publication updated :Seria electrónica, puesta al dia

01-1999

* This work was supported by China Scholarship Council. Email : xiaozhanx@hotmail.com

The SIR method: A Superiority and Inferiority Ranking Method for Multiple

Criteria Decision Making*

Xiaozhan Xu

Department of Applied Mathematics, Sichuan Union University, Chengdu, Sichuan, 610065, China

Abstract

In this paper a Superiority and Inferiority Ranking (SIR) method is proposed. This new method uses two

types of information, the superiority and the inferiority information, to derive two types of flows, the

superiority flow and the inferiority flow, by which the set of alternatives are ranked partially or completely.

Relationships between the SIR method and some of the classical MCDM methods (such as SAW, TOPSIS

and PROMETHEE) are explored. It is proved that the SIR method is a significant extension of the well-

known PROMETHEE method.

Keywords: multiple criteria decision making (MCDM); superiority; inferiority; flow; SAW; TOPSIS;

PROMETHEE

1. Introduction

Two multiple criteria decision making methods based on the theory of fuzzy bags

were proposed by Rebai (1993; 1994). In the two methods, three scores, the superiority,

inferiority and noninferiority scores, were introduced via the comparison between criteria

values. One of the main features of these methods is that they can deal with noncardinal

data as well as cardinal data. However, when dealing with cardinal criteria, the above

scores are obtained according to the so-called “true-criteria”, i.e., the following

preference structure { P, I } is used : given two alternatives A and A' and a criterion g,

A P A' (A is preferred to A' ) iff g(A) > g(A'),

A I A' (A is indifferent to A' ) iff g(A) = g(A'),

where g(A) and g(A') are the criteria values of A and A' on criterion g. The true-criteria

preference structure was questioned by the authors of outranking methods (see Brans and

Mareschal, 1990; Roy et al., 1992). In their opinion, when comparing two criteria values,

one should not only consider which of them is larger but also take their difference (or

�

amplitude of deviation) into account. Due to the effects of imprecision, indetermination

and uncertainty in the evaluation of criteria values, small difference between criteria

values does not eligibly imply a strict preference of one alternative over another. In other

words, small difference between criteria values may not differentiate two alternatives

(They may be indifferent. ). One alternative is preferred to another only if its value is

larger enough than that of the latter (Ostanello, 1985).

The purpose of the outranking methods is to enrich the “poor” true-criteria preference

structure (Brans and Mareschal, 1990). In ELECTRE III (Roy, 1978), some thresholds

were introduced to describe the ranges for indifference, weak preference and strict

preference. The PROMETHEE methods (Brans et al., 1986), as offshoots of the

outranking methods, take differences between criteria values into full account via

generalized criteria. The value differences are accumulated into two types of flows, the

leaving flow and the entering flow, by which the alternatives are ranked. In the

outranking methods, all cardinal criteria are treated as pseudo-criteria, leaving the true-

criteria as the special case (Roy, 1978; Brans et al., 1986). Compared to ELECTRE III,

the process of the PROMETHEE methods seems easier to be understood by the decision-

maker and simpler to be managed by the analyst.

In this paper, we first generalize the notions of superiority and inferiority scores

defined by Rebai (1993) by taking the differences between criteria values into account as

what was done in the first step of the PROMETHEE methods, thus enriching the relevant

true-criteria preference structure. For this purpose, in Step 1 of our method, the

generalized criteria are carefully chosen by the decision-maker and the analyst. Then the

superiority matrix S, made up of the superiority indexes, and the inferiority matrix I,

made up of the inferiority indexes, are built from the original decision matrix. In Step 2,

we employ some aggregation procedure to derive two types of flows, the superiority flow>(·) and the inferiority flow <(·). Since different aggregation procedure produces

different kind of flows, the SIR method is, in fact, not a single method. It represents a

family of methods. When using Simple Additive Weighting as the aggregation procedure

in this step, our method coincides with the second step of PROMETHEE methods, i.e.,

the derived superiority flow >(·) and the inferiority flow <(·) are exactly the leaving

flow +(·) and the entering flow �(·), respectively, defined by Brans et al.(1986).

�

However, we have more choices here. Some other aggregation procedures can be used in

this step. It is in this sense, that our method can be thought as a further extension of the

PROMETHEE methods. In Step 3, the superiority and inferiority flows >(·) and <(·) are

used to derive two complete rankings, > and <, of the alternatives and the two complete

rankings are then combined into a final partial ranking as the intersection of the two: =

> < in Step 4. Like the PROMETHEE methods, the SIR method also allows a

complete ranking of alternatives. When a complete ranking is requested by the decision-

maker, some synthesizing flows, such as n (·)= >(·)� <(·) (like the net flow in

PROMETHEE) and r(·)=>(·)¼( >(·)+ <(·)) (like the relative distance in TOPSIS), can

be used to derive a complete ranking. The derived ranking (partial or complete) is then

proposed to the decision-maker for further exploitation before a final decision is made.

This paper is organized as follows. In Section 2, the superiority and inferiority

matrixes are built from the original decision matrix via generalized criteria. Section 3

discusses different aggregation procedures in order to derive superiority and inferiority

flows from the above two matrixes. In Section 4, we discuss how to use the flows to rank

the alternatives partially or completely. A numerical example is given to illustrate the SIR

method in Section 5. In Section 6, a comparison between the SIR method and several

MCDM methods is made and some relationships are revealed.

2. The superiority and inferiority matrixes

Let A1, A2 , ···, Am be m alternatives and g1, g2 , ···, gn be n criteria. We suppose that all

criteria are cardinal ( numerical or quantitative) criteria and let gj(Ai) be the criteria value

(performance) of the ith alternative Ai with respect to the jth criterion gj, where gj(·) is a

real-valued function (i=1,2, ···, m; j=1,2, ···, n). These criteria values form a decision

matrix D:

�

g1(A1) ····· gj(A1) ····· gn(A1)

····· ····· ····· ····· ·····

D = g1(Ai) ····· gj(Ai) ····· gn(Ai)

····· ····· ····· ····· ·····

g1(Am) ····· gj(Am) ····· gn(Am)

Without loss of generality, we will assume that all criteria are to be maximized.

Now we will compare the criteria values on each criterion. Given two alternatives A

and A' and a criterion g, let g(A) and g(A') be the criteria values of A and A' with respect

to g. In Rebai (1993; 1994), the comparison proceeds as follows: if g(A) > g(A'), then

one point is assigned to the superiority score of A and to the inferiority score of A'

respectively on the criterion g. In this comparison, the difference d = g(A) � g(A') is not

considered. In order to take the difference into consideration, an appropriate generalized

criterion function f(d) should be introduced to express the intensity of the preference

(denoted P(A, A')) of A over A' on g:

P(A, A') = f(g(A) � g(A')) = f(d),

where f(d) is a nondecreasing function from R (the real numbers) to [0, 1] such that f(d) =

0 for d �����L�H���g(A) ��g(A')). Such a function is called a generalized criterion. In Brans

et al.(1986), six such generalized criteria were introduced (Table 1):

Table 1. Generalized criteria

�

Type 1 True-criterion:

1 if d > 0f(d) = 0 if d ���

Type 2 Quasi-criterion:

1 if d > qf(d) =

0 if d ��q

Type 3 Criterion with linear preference:

1 if d > p f(d) = d¼p if 0<d�p 0 if d ���

Type 4 Level criterion:

1 if d > p

f(d) = 1¼2 if 0<d�p 0 if d ���

Type 5 Criterion with linear preference and indifference area:

1 if d > p

f(d)= (d-q)¼(p-q) if q<d�p 0 if d ��q

Type 6 Gaussian criterion:

1-exp(-d2¼2 2) if d>0f(d)= 0 if d���

The parameters p and q in the above formulas are preference and indifference

thresholds respectively.

Note: Type 1 is a special case of Type 2 (q=0) and Type 3 is a special case of Type 5

(q=0).

Clearly, different generalized criterion (with different shape) represents different

attitude towards preference structure and the intensity of preference. It was observed by

Brans and Mareschal (1990) that Gaussian criterion has been mostly selected by users for

practical applications followed by the criterion with linear preference and indifference

area. In both criteria (like the criterion with linear preference), the intensity of preference

changes gradually from 0 to 1, while in the other three criteria ( true-criterion, quasi-

criterion and level criterion), there are some sudden changes of the intensity of

preference.

The above six types of generalized criterion are not exhaustive. Some other shapes can

be considered to best meet the decision-maker’s preference attitude.

Let fj be a generalized criterion adopted by the jth criterion gj (j = 1,2, ···, n). For each

pair of alternatives Ai and Ak, let Pj(Ai, Ak) = fj(gj(Ai) � gj(Ak)) represents the intensity of

preference or superiority of Ai over Ak with respect to the jth criterion. It also represents

the intensity of inferiority of Ak to Ai with respect to the jth criterion.

�

For each alternative Ai, we define its superiority index Sj(Ai) and inferiority index Ij(Ai)

with respect to the jth criterion by the following formulas:

Sj(Ai) = ∑=

m

1k

Pj(Ai, Ak) =∑=

m

1k

fj(gj(Ai) � gj(Ak)) (1)

and

Ij(Ai)= ∑=

m

1k

Pj(Ak, Ai) =∑=

m

1k

fj(gj(Ak) � gj(Ai)). (2)

Since fj(d) = 0 for d����IRUPXODV�����DQG�����FDQ�EH�UHVWDWHG�E\

Sj(Ai) =∑ { fj(gj(Ai) � gj(Ak)) | gj(Ai) > gj(Ak) } (3)

and

Ij(Ai) = ∑ { fj(gj(Ak) � gj(Ai)) | gj(Ai) < gj(Ak) } (4)

Therefore, if all fj in (3) and (4) are true-criteria, then Sj(Ai) and Ij(Ai) are exactly the

superiority score and the inferiority score respectively defined by Rebai (1993; 1994).

Now, the superiority indexes and the inferiority indexes constitute the following two

types of matrixes:

the Superiority matrix (S-matrix):

S1(A1) ····· Sj(A1) ····· Sn(A1)

····· ····· ····· ····· ·····

S = S1(Ai) ····· Sj(Ai) ····· Sn(Ai) or S = Sj(Ai) m×n

····· ····· ····· ····· ·····

S1(Am) ····· Sj(Am) ····· Sn(Am)

and the Inferiority matrix (I-matrix):

I1(A1) ····· Ij(A1) ····· In(A1)

····· ····· ····· ····· ·····

I = I1(Ai) ····· Ij(Ai) ····· In(Ai) or I = Ij(Ai) m×n

····· ····· ····· ····· ·····

I1(Am) ····· Ij(Am) ····· In(Am)

�

The two matrixes S and I include better information than the original decision matrix

D because the intensity of superiority and inferiority given by the generalized criteria are

taken into account. Also, the superiority matrix S and the inferiority matrix I convey

different information because they represent different types of comparison results. Matrix

S tells us the information about the intensity of superiority of each alternative on each

criterion and matrix I tells us the information about the intensity of inferiority.

Note: The matrix = S ²�I = Sj(Ai)�² Ij(Ai) m×n is ( up to a normed coefficient) the

matrix composed of the unicriterion flows j(Ai) defined by Mareschal and Brans

(1988). It seems that the S-matrix and the I-matrix contain “finer” (or more accurate)

information than , because the latter contains only the “net” information which can be

derived from the former two matrixes. Also, the indexes Sj(Ai)�and Ij(Ai) enjoy the same

advantages of j(Ai) as described by Mareschal and Brans (1988): (1) all Sj(Ai)�and Ij(Ai)

are in the same unit and are independent of the scales of the criteria; and (2) large (or

small) differences in criteria values will have large (resp. small) contributions to the

indexes Sj(Ai)�and Ij(Ai).

3. Aggregation procedure: the superiority flow and the inferiority flow

With the superiority matrix S and the inferiority matrix I at hand, we can use standard

MCDM aggregation procedures to aggregate the superiority and inferiority indexes into

two types of global preference indexes: the superiority flow (S-flow) >(·)� and the

inferiority flow (I-flow) <(·) , which represent the global intensity of superiority and

inferiority of each alternative.

Let V be the aggregation function, then for each alternative Ai, its superiority flow>(Ai) and inferiority flow <(Ai) are defined as

>(Ai) = V[S1(Ai),···, Sj(Ai),···, Sn(Ai)]� and <(Ai) = V[I1(Ai),···, Ij(Ai),···, In(Ai)].

Clearly, the higher the S-flow >(Ai) and the lower the I-flow <(Ai), the better Ai is.

The choice of an appropriate aggregation procedure should be governed by some

guidelines by taking some important aspects (such as the decision-maker’s attitudes

�

towards compensation, trade-off, the global preference structure (e.g., the types of the

output information)) into consideration (Guitouni and Martel, 1998). As examples, in this

paper, we only consider two aggregation procedures: (1) Simple Additive Weighting

(SAW) and (2) TOPSIS. However, this choice is by no means exhaustive. One can

choose other procedure which best meets his needs.

In the sequel, when some standard MCDM aggregation procedure named XXX is used

in the SIR aggregation step, we will call it a SIR·XXX method.

For most MCDM aggregation procedures, we need to know the weights (relative

importance) of the criteria. Let the weights be wj (j = 1,2, ···, n) and ∑=

n

1j

wj = 1.

(1) SIR·SAW

SAW (Simple Additive Weighting)(i.e., weighted average) is the best known and most

widely used aggregation procedure thanks to its simplicity attraction. If SAW is used as

the aggregation method, the S-flow and the I-flow are given by:

>(Ai) = ∑=

n

1j

wj Sj(Ai) and <(Ai) = ∑=

n

1j

wj Ij(Ai). (5)

Now, we show that in this case the S-flow (I-flow) is the same as the leaving

flow(entering flow) of the PROMETHEE.

Proposition 1. The S-flow and the I-flow of SIR·SAW are, respectively, the leaving flow

and the entering flow of PROMETHEE.

Proof. Since the S-flow

>(Ai) = ∑=

n

1j

wj Sj(Ai) =∑=

n

1j

wj ( ∑=

m

1k

fj(gj(Ai) ��gj(Ak)))

= ∑=

m

1k∑

=

n

1j

wj fj(gj(Ai) ��gj(Ak)) = ∑=

m

1k

(Ai, Ak),

�

where (Ai, Ak)�=∑=

n

1j

wj fj(gj(Ai) ��gj(Ak)) is the multicriteria preference index of Ai over

Ak defined in Brans et al.(1986), it is clear that >(Ai) is exactly the leaving flow +(Ai) of

PROMETHEE methods. Similarly, the I-flow <(Ai) is the entering flow �(Ai).

Proposition 1 indicates that when SAW is used in the aggregation step, our SIR

method coincides with the second step of the PROMETHEE methods. However, it is not

restrictive to consider other aggregation procedures in this step. It is in this sense, the SIR

method introduced in this paper can be thought as a further extension of the

PROMETHEE methods.

Also, the need for choosing other aggregation procedure can be justified by the

following concern about the weighted average used in PROMETHEE. If, for example,

the generalized criteria fj are all chosen to be the criteria with linear preference (Type3),

i.e.,

1 if d > pj pj if d > pj

fj(d) = d¼pj if 0<d�pj = cj¼pj ( cj = d if 0<d�pj ). 0 if d ���������������������������������������������LI�d ���

Then the multicriteria preference index

(Ai, Ak) = ∑=

n

1j

wj fj(gj(Ai) � gj(Ak)) = ∑=

n

1j

wj · (cj¼pj) = ∑=

n

1j

(wj¼pj) · cj = ∑=

n

1j

vj · cj,

where vj = wj¼pj. It is clear that the weight wj and the preference threshold pj are not

independent. They affect each other in an inversely proportional manner and they can be

merged into a single parameter. If the weight wj and the preference threshold pj are

determined by the same decision-maker, he just determines one parameter vj ( = wj¼pj)

instead of two.

The above observation indicates that SAW is not the only or best aggregation

procedure. Other aggregation procedures might be reasonable in some cases.

(2) SIR·TOPSIS

��

The classical TOPSIS (Hwang and Yoon, 1981) and other versions of TOPSIS (such

as BBTOPSIS (Rebai, 1993)) can be applied to derive the superiority and inferiority

flows.

The ideal solution A+

S and the negative-ideal solution A

−

S for the superiority matrix S =

Sj(Ai) m×n is defined by

A+

S=(

imax S1(Ai), ···,

imax Sn(Ai) ) = ( S

+

1, ···, S

+

n�

and

A−

S= (

imin S1(Ai), ···,

imin Sn(Ai) ) = (S

−

1, ···, S

−

n�

The superiority flow by the classical TOPSIS is defined as

>(Ai) = S�(Ai)¼( S�(Ai)�� S�(Ai)), (6)

where

S�(Ai) = ∑=

n

1j

| wj(Sj(Ai)���S+

j)| �������� �����and S�(Ai) = ∑

=

n

1j

| wj(Sj(Ai)���S−

j)|� ������� .

Here, the Minkowski distance

d ( , ) = ∑=

n

1j

|aj ��bj|� �������� ����(1� ���)

between two vectors =(a1, ···, an) and =(b1, ···, bn) is used.

Similarly, the ideal solution A+

I and the negative-ideal solution A

−

I for the inferiority

matrix I = Ij(Ai) m×n is defined by

A+

I= (

imin I1(Ai), ···,

imin In(Ai) ) = ( I

+

1, ···, I

+

n�

and

A−

I = (

imax I1(Ai), ···,

imax In(Ai) ) = ( I

−

1, ···, I

−

n�

The inferiority flow is defined as

����������

<(Ai) = I�(Ai)¼( I�(Ai)�� I�(Ai)), (7)

where

��

I�(Ai) = ∑=

n

1j

| wj(Ij(Ai)���I+

j)|� � ��� �����and I�(Ai) = ∑

=

n

1j

| wj(Ij(Ai)���I−

j)|� ������ .

Let us look at some special and important cases of SIR·TOPSIS.

������L���:KHQ� � ����ZH�DUH�XVLQJ�WKH�Euclidean distance to derive the S-flow and the I-

flow.

������LL���:KHQ� � �����ZH�KDYH�WKH�VR�FDOOHG�block distance:

d1( , ) = ∑=

n

1j

|aj ��bj|.

Some connections between SAW and TOPSIS were mentioned by Hwang and Yoon

(1981). It can be concluded that PROMETHEE and SIR·SAW are special cases of

SIR·TOPSIS using the block distance (see Section 6). The block distance was also

adopted in the BBTOPSIS technique: i.e., d1(·, ·) in Rebai (1993).

(iii) When � ���� the Minkowski distance becomes:

d�( , ) = j

max | aj ��bj |,

which was the distance d4(·, ·) defined in Rebai (1993). In this sense, when using TOPSIS

as the aggregation procedure, our method is partly a generalization of the BBTOPSIS

technique.

4. The SIR ranking

In this section, the superiority flow >(Ai) and the inferiority flow <(Ai) obtained in

Section 3 will be used to rank the alternatives.

According to the constructions, the superiority flow >(Ai) measures how Ai is

globally superior to (or outranks) all the others and the inferiority flow <(Ai) measures

how Ai is globally inferior to (or outranked by) all the others. Therefore, the higher >(Ai)

and the lower <(Ai), the more preferred the alternative Ai is.

Now, according to the descending order of >(Ai), we obtain a complete ranking

(called S-ranking) ! ^P!, I!` of the alternatives:

��

Ai P!Ak iff >(Ai)�! >(Ak) and Ai I!Ak iff

>(Ai)� >(Ak).

Similarly, according to the ascending order of� <(Ai), we obtain another complete

ranking (called I-ranking) �� ^P�, I�` of the alternatives:

Ai P�Ak iff <(Ai)�� <(Ak) and Ai I�Ak iff

<(Ai)� <(Ak).

In general, !� and � are different complete rankings. The two complete ranking

structures ! ^P!, I!`� and � ^P�, I�`� are then combined� into� a partial ranking

structure ^P, I, R`� � > < according to the following intersection principle (Brans

et al., 1986; Roy et al., 1992): given two alternatives A and A’:

A P A’ (A is preferred to A’) iff (A P! A’ and A P��A’) or (A P! A’ and A I��A’) or

(A I!A’ and A P� A’);

A I A’ (A is indifferent to A’) iff A I!�A’ and A I��A’ ;

ARA’ (A is incomparable to A’) iff (A P! A’ and A’ P� A) or ( A’ P! A and A P��A’).

The process of the SIR method (partial ranking) can be outlined in Figure 1.

··· Sj(A1) ··· >(A1)

S= ··· Sj(Ai) ··· >(Ai) ! ^P!, I!`

··· Sj(Am) ··· >(Am)

D= gj(Ai) ^P, I, R̀

··· Ij(A1)··· <(A1)

I= ··· Ij(Ai) ··· <(Ai) � ^P�, I�`

··· Ij(Am)··· <(Am)

Figure 1. The process of the SIR method (partial ranking)

S-matrixand

I-matrix

S-rankingand

I-ranking

Aggregationprocedure :

S-flow and I-flow

SIR ranking(partial)

��

If a complete ranking is requested by the decision-maker, some synthesizing flows,

such as the net flow (n-flow) n(Ai)=>(Ai)�

<(Ai) (like the net flow in PROMETHEE)

and the relative flow (r-flow) r(Ai)=>(Ai)¼(

>(Ai)+<(Ai)) (like the relative distance in

TOPSIS), can be derived in Step 3. Then n(Ai) (or r(Ai)) can be used in Step 4 to obtain

a complete ranking n (or r) of the alternatives.

Note that while n(Ai) can be any number,� r(Ai) is always between 0 and 1.

The process of the SIR method (complete ranking) can be outlined in Figure 2.

··· Sj(A1) ··· >(A1)

S= ··· Sj(Ai) ··· >(Ai)

··· Sj(Am) ··· >(Am)

n(Ai) n

D= gj(Ai)

r(Ai ) r

··· Ij(A1)··· <(A1)

I= ··· Ij(Ai) ··· <(Ai)

··· Ij(Am)··· <(Am)

Figure 2. The process of the SIR method (complete ranking)

Figures 1 and 2 show that the process of the SIR method is clear and simple, and it can

be easily implemented with computers. (For small number of alternatives and criteria, a

PC is sufficient.) Also, the choice of the generalized criteria (the types and the

S-matrixand

I-matrix

n-flowor

r-flow

Aggregationprocedure :

S-flow and I-flow

SIR ranking

(complete)

��

parameters) and the choice of an appropriate aggregation procedure can be supported by a

decision support system based on the SIR method.

The outcome of the SIR method is a (partial or complete) ranking of the alternatives.

This outcome can now be proposed to the decision-maker for further exploitation in order

to reach a final decision. The alternatives ranked in front should be highly recommended.

5. Numerical example

In this section we calculate a numerical example to illustrate the SIR method. In order

to compare our method with the PROMETHEE methods, we will use the same numerical

example in Brans et al. (1986).

Note: All the computations in this example are carried out by a computer program with

p, q, , wj�DQG� �EHLQJ�YDULDEOH�SDUDPHWHUV�

Let A1, A2, …, A6 be six alternatives which are evaluated against six criteria g1, g2, …,

g6. The decision matrix and the type of generalized criterion for each decision criterion gj

are shown in Table 2.

Table 2. The decision matrix

g1

min

g2

max

g3

min

g4

min

g5

min

g6

max

A1 80 90 6 5.4 8 5

A2 65 58 2 9.7 1 1

A3 83 60 4 7.2 4 7

A4 40 80 10 7.5 7 10

A5 52 72 6 2 3 8

A6 94 96 7 3.6 5 6

Type of criterion Type 2 Type 3 Type 5 Type 4 Type 1 Type 6

Parameters q=10

p=30

q=0.5

p=5

q=1

p=6

= 5

��

According to formulas (1) and (2), we can calculate the S-matrix and the I-matrix as

follows:

1 2.933 0.889 1.5 0 0.274

3 0 3.889 0 5 0

S = 1 0.067 2.222 0.5 3 0.610

5 1.667 0 0.5 1 1.711

4 0.867 0.889 3 4 0.886

0 3.533 0.556 2.5 2 0.413

and

3 0.2 1.111 1 5 0.655

2 3.267 0 3.5 0 2.607

I = 3 3.067 0.333 1.5 2 0.185

0 0.867 4.111 1.5 4 0

1 1.667 1.111 0 1 0.077

5 0 1.778 0.5 3 0.371

Now, we will use two aggregation procedures, SAW and TOPSIS, to calculate the S-

flows and the I-flows. Like Brans et al. (1986), we assume the six decision criteria have

equal weights: wj = 1/6 (j=1, …, 6).

(1) SIR·SAW

According to formulas (5), we can calculate the S-flows and the I-flows respectively,

from the S-matrix and the I-matrix, by which the n-flows and r-flows can be obtained

readily. The flows are listed in Table 3.

��

Table 3. The flows of SIR·SAW

S-flows I-flows n-flows r-flows>(A1) = 1.099>(A2) = 1.981>(A3) = 1.233>(A4)=1.646>(A5)=2.274>(A6) =1.500

<(A1)=1.828<(A2)=1.896<(A3)=1.681<(A4)=1.746<(A5)=0.809<(A6)=1.775

n(A1)= -0.728

n(A2)= 0.086

n(A3)=�-0.448

n(A4)=�-0.100

n(A5)= 1.464

n(A6)=�-0.274

r(A1)=0.276

r(A2)=0.511

r(A3)=0.423

r(A4)=0.485

r(A5)=0.738

r(A6)=0.458

The above S-flows and I-flows are exactly the leaving flows +(Ai) and the entering

flows �(Ai) of the PROMETHEE methods (see Brans et al. (1986)). Therefore, the

produced rankings are the same as the PROMETHEE methods.

The two complete rankings are:

! : A5 A2 A4 A6 A3 A1

� : A5 A3 A4 A6 A1 A2

The resulting partial ranking is:

A4 A6

= ! � : A5 A3 A1

A2

This is the partial ranking of PROMETHEE I.

The complete ranking by n-flows is:

n : A5 A2 A4 A6 A3 A1

This is the complete ranking of PROMETHEE II.

��

The complete ranking by r-flows is:

r : A5 A2 A4 A6 A3 A1

(2) SIR·TOPSIS

Using formulas (6) and (7), we can calculate the S-flows and I-flows. In order to make

D�FRPSDULVRQ��ZH�FRQVLGHU�WKUHH�GLVWDQFHV�ZLWK� � ������DQG�����7DEOHV������

7DEOH�����7KH�EORFN�GLVWDQFH�� � ���

S-flows I-flows n-flows r-flows>(A1) = 0.298>(A2) = 0.537>(A3) = 0.334>(A4)=0.446>(A5)=0.616>(A6) =0.407

<(A1)=0.467<(A2)=0.483<(A3)=0.429<(A4)=0.446<(A5)=0.207<(A6)=0.453

n(A1)=�-0.169

n(A2)= 0.053

n(A3)= -0.095

n(A4)= 0.000

n(A5)= 0.410

n(A6)=�-0.047

r(A1)=0.390

r(A2)=0.526

r(A3)=0.438

r(A4)=0.500

r(A5)=0.749

r(A6)=0.473

�����,W�ZDV�PHQWLRQHG�DERYH�WKDW��ZKHQ� � ����6,5Â7236,6�ZLOO�SURGXFH�WKH�VDPH�UDQNLQJV

(partial or complete) as the PROMETHEE (or SIR·SAW). This can be easily verified

from the flows in Table 4.

��

7DEOH�����7KH�(XFOLGHDQ�GLVWDQFH�� � ���

S-flows I-flows n-flows r-flows>(A1) =0.325>(A2) =0.568>(A3) = 0.378>(A4)=0.469>(A5)=0.603>(A6) =0.413

<(A1)=0.515<(A2)=0.449<(A3)=0.450<(A4)=0.479<(A5)= 0.237<(A6)=0.512

n(A1)=�-0.190

n(A2)= 0.119

n(A3)= -0.072

n(A4)=�-0.009

n(A5)= 0.366

n(A6)= -0.099

r(A1)=0.387

r(A2)=0.559

r(A3)=0.456

r(A4)=0.495

r(A5)=0.718

r(A6)=0.446

The S-flows and I-flows derive the following two complete rankings:

! : A5 A2 A4 A6 A3 A1

� : A5 A2 A3 A4 A6 A1

The resulting partial ranking is:

A4 A6

= ! � : A5 A1

A2 A3

The n-flows and r-flows derive the following two complete rankings:

n : A5 A2 A4 A3 A6 A1

and

r : A5 A2 A4 A3 A6 A1.

��

7DEOH�����7KH�GLVWDQFH�ZLWK�D�VXIILFLHQW�ODUJH� ��� � ����

S-flows I-flows n-flows r-flows>(A1) =0.367>(A2) =0.584>(A3)=0.424>(A4)=0.541>(A5)=0.582>(A6) =0.414

<(A1)=0.604<(A2)=0.419<(A3)=0.460<(A4)=0.465<(A5)=0.277<(A6)=0.595

n(A1)= -0.237

n(A2)= 0.164

n(A3)=�-0.036

n(A4)= 0.076

n(A5)= 0.305

n(A6)=�-0.181

r(A1)=0.378

r(A2)=0.582

r(A3)=0.480

r(A4)=0.538

r(A5)=0.677

r(A6)=0.410

The S-flows and I-flows derive the following two complete rankings:

! : A2 A5 A4 A3 A6 A1

� : A5 A2 A3 A4 A6 A1

The resulting partial ranking is:

A2 A3

= ! � : A6 A1

A5 A4

The n-flows and r-flows derive the following two complete rankings:

n : A5 A2 A4 A3 A6 A1

and

r : A5 A2 A4 A3 A6 A1.

�����,W� LV� FKHFNHG� WKDW� IRU� �!����� WKH� UDQNLQJ� LV� WKH� VDPH��7KXV�� 7DEOH� �� UHSUHVHQWV� WKH

result for � ��.

��

In this example, PROMETHEE (or SIR·SAW) and different SIR·TOPSIS (with

GLIIHUHQW� ��UDQN�WKH�DOWHUQDWLYH�VLPLODUO\��,Q�PRVW�FDVHV��A5 is the best choice and A1 is the

worst.

It is not the purpose of this paper to judge the advantage and disadvantage of different

SIR methods. Just as different classical MCDM methods usually lead to different results,

different SIR methods will usually produce different rankings due to different

aggregation procedures employed.

6. A comparison of some MCDM methods

In this section we compare some MCDM methods and reveal their relationships.

By Proposition 1, we readily have:

Theorem 1. PROMETHEE is equivalent to SIR·SAW.

Salminen et al.(1998) have revealed some connection between PROMETHEE and

SAW (called SMART in the paper). We can restate the connection by the following

theorem.

Theorem 2. SAW is a special case of SIR·SAW.

Proof. In SAW, the value function is

(Ai) = ∑=

n

1j

wj gj(Ai).

Now, if all the generalized criteria fj are of Type 3 with a criteria-independent preference

threshold p, such that

p�j,k,i

max | gj(Ai) ² gj(Ak) |,

then

1 if gj(Ai) – gj(Ak) > p

Pj(Ai .Ak)= (gj(Ai) ² gj(Ak))¼p if 0< gj(Ai) – gj(Ak) �p = max{0, (gj(Ai) – gj(Ak))¼p}.

0 if gj(Ai) ��gj(Ak)

��

Now, the net flow of SIR·SAW (or PROMETHEE) is

n(Ai)=>(Ai)�

<(Ai)=∑=

n

1j

wjSj(Ai)��∑=

n

1j

wj Ij(Ai)=∑=

n

1j

wj(Sj(Ai)�Ij(Ai))

=∑=

n

1j

wj( ∑=

m

1k

max{0, (gj(Ai) ² gj(Ak))¼p}� max{0, (gj(Ak) – gj(Ai))¼p})

=∑=

n

1j

wj( ∑=

m

1k

(gj(Ai) ² gj(Ak))¼p)

=∑=

n

1j

wj( ∑=

m

1k

gj(Ai)¼p) ��∑=

n

1j

wj( ∑=

m

1k

gj(Ak)¼p)

=(m¼p)∑=

n

1j

wjgj(Ai) + C = (m¼p) (Ai) + C,

where C = �∑=

n

1j

wj( ∑=

m

1k

gj(Ak)/p) is a constant independent of Ai. Now, it is clear that

the net flow n(Ai) of SIR·SAW and the value function (Ai) of SAW derive the same

complete ranking.

Corollary 1. SAW is a special case of PROMETHEE.

Hwang and Yoon (1981) have concluded that SAW is a special case of TOPSIS using

WKH�EORFN�GLVWDQFH�� � �����:H�FDQ�SURYH�D�SDUDOOHO�UHVXOW�

Theorem 3. SIR·SAW is a special case of SIR·TOPSIS.

Proof. It suffices to show that the S-flow (I�IORZ��RI�6,5Â7236,6�ZLWK�EORFN�GLVWDQFH��

=1) derives the same complete ranking as that derived by the S-flow (I-flow) of

SIR·SAW.

�����6LQFH� � ����ZH�KDYH��E\�GHILQLWLRQ�

S�(Ai) = ∑=

n

1j

wj((S+

j� Sj(Ai)) and S�(Ai) = ∑

=

n

1j

wj(Sj(Ai)�²�S−

j).

��

Thus

S�(Ai) + S�(Ai) =∑=

n

1j

wj(S+

j��S

−

j) = K,

where K is a positive constant independent of Ai. Therefore, by formula (6), the S-flow of

SIR·TOPSIS is

>(Ai)� S�(Ai)/K =(1/K)∑=

n

1j

wjSj(Ai)�–�(1/K)∑=

n

1j

wjS−

j�= (1/K) +(Ai)+K ,

where K �= –�(1/K)∑=

n

1j

wjS−

j� is a constant independent of�Ai and +(Ai) is the S-flow of

SIR·SAW (or leaving flow of PROMETHEE). It is clear that >(Ai) and +(Ai) derive the

same complete ranking.

A similar reasoning shows that <(Ai) (the I-flow of SIR·TOPSIS ) and �(Ai) (the I-

flow of SIR·SAW) derive the same complete ranking.

Corollary 2. PROMETHEE is a special case of SIR·TOPSIS.

The above results show that SAW, SIR·SAW and PROMETHEE can be classified as

members of the SIR·TOPSIS family. Since SIR·TOPSIS is only a specific application of

the SIR method, we can summarize the above relationships in Figure 3.

Figure 3. The relationships of some MCDM methods

SIR

SIR·TOPSIS SIR·SAW PROMETHEE

SAW

��

7. Conclusion

In this paper we propose a new MCDM ranking method: the SIR method. By

generalizing the superiority and inferiority scores (Rebai, 1993), we propose the concept

of superiority and inferiority matrixes (S-matrix and I-matrix) via generalized criteria

introduced in the PROMETHEE methods. The S-matrix and I-matrix contain new

information which reflects the decision-maker’s attitude towards each decision criterion

and describes the intensities of superiority and inferiority of each alternative. By using an

appropriate aggregation procedure (agreed by the decision-maker), we can derive two

types of global preference indexes, the S-flows and the I-flows, from the S-matrix and the

I-matrix respectively. The two types of flows are used to rank the alternatives partially or

completely according to the decision-maker’s needs. When SAW is used as the

aggregation procedure, the SIR method coincides with the PROMETHEE methods.

However, we have more choices at the aggregation step. We have used TOPSIS as an

example to build an important model, SIR·TOPSIS, and used it to compare some MCDM

methods and reveal some of their relationships. It appears that the SIR method is not a

single method. It represents a general MCDM approach because it uses new types of

information extracted from the original decision matrix instead of using the decision

matrix directly like many classical MCDM methods do.

Acknowledgement

This research was completed while the author was a visiting researcher at Université

Laval. The author wishes to thank Professors B.F. Lamond and J.-M. Martel for their

support and helpful opinions.

��

References

Bouyssou, D. (1990), “Building criteria: A prerequisite for MCDA”, in: C.A. Bana e

Costa (ed.), Readings in Multiple Criteria Decision Aid, Sringer-Verlag, Berlin,58-80.

Brans, J.P., and Mareschal, B. (1990), “ The PROMETHEE methods for MCDM; The

PROMCALC, GAIA and BANKADVISER software”, in: C.A. Bana e Costa (ed.),

Readings in Multiple Criteria Decision Aid, Sringer-Verlag, Berlin, 216-252.

Brans, J.P., Vincke, Ph., and Mareschal, B.(1986), “How to select and how to rank

projects: The PROMETHEE method”, European Journal of Operational Research 24,

228-238.

Guitouni, A., and Martel, J.-M. (1998), “Tentative guidelines to help choosing an

appropriate MCDA method”, European Journal of Operational Research 109, 501-

521.

Hwang, C.L., and Yoon, K.L.(1981), Multiple Attribute Decision Making: Methods and

Applications, Springer-Verlag, New York.

Mareschal, B., and Brans, J.P. (1988), “Geometrical representations for MCDA”,

European Journal of Operational Research 34, 69-77.

Ostanello, A.(1985), “Outranking methods”, in: G. Fandel and J. Spronk (eds.), Multiple

Criteria Decision Methods and Applications, Springer-Verlag, Berlin, 41-60.

Rebai, A. (1993), “BBTOPSIS: A bag based technique for order preference by similarity

to ideal solution”, Fuzzy Sets and Systems 60, 143-162.

Rebai, A. (1994), “Canonical fuzzy bags and bag fuzzy measures as a basis for MADM

with mixed non cardinal data”, European Journal of Operational Research 78, 34-48.

Roy, B. (1978), “ELECTRE III : un algorithme de classement fondé sur une

représentation floue des préférences en présence de critères multiples”, Cahiers

Centre Etudes Recherche Opérationnelle 20/1, 3-24.

Roy, B., Slowinski, R., and Treichel, W. (1992), “Multicriteria programming of water

supply systems for rural areas”, Water Resources Bulletin 28/1, 13-31.

Salminen, P., Hokkanen, J., and Lahdelma, R. (1998), “Comparing multicriteria methods

in the context of environmental problems”, European Journal of Operational Research

104, 485-496.