Integration of ordinal and cardinal information in multi-criteria ranking with imperfect compensation

European Journal of Operational Research 158 (2004) 317–338 www.elsevier.com/locate/dsw

Integration of ordinal and cardinal information in multi-criteria ranking with imperfect compensation Edwin Hinloopen a b

a,*

, Peter Nijkamp b, Piet Rietveld

b

Department of Finance, Nederlandse Spoorwegen, P.O. Box 2025, NL 3500 HA Utrecht, The Netherlands Department of Economics, Free University, De Boelelaan 1105, NL 1081 HV Amsterdam, The Netherlands Received 1 April 2002; accepted 1 June 2003 Available online 8 October 2003

Abstract The method presented in this paper is a MCDA method, which outranks a certain number of choice options that are evaluated on a mixture of cardinal and ordinal judgement criteria. Characteristics of the method are a theoretically sound integration of cardinal and ordinal information and a decision-makersÕ preference structure that allows for less than perfect compensation between criteria. The method is based on a pairwise comparison of choice-options. It belongs to the family of decomposed scaling methods: the assessment of the relative importance of the criteria on one hand and the assessment of the scores (cardinal criteria) and rankings (ordinal criteria) on the other hand are performed separately. The assessment of the scores and rankings is realised by means of (marginal) value functions in such a way that the ordinal information is fully comparable with the cardinal information. Essentially, the ordinal information is translated into stochastical information without imposing any assumption on the probability distributions attached to the ordinal data. The relative importance of the judgement criteria is represented by a set of cardinal weights or by a ranking of weights. The final ranking of the choice options is based upon an overall value function that is some kind of weighed aggregation of the marginal value functions. In order to allow for less-than-perfect-compensation between criteria, the method uses a convex combination of two different CES-type value functions. The paper ends with an empirical application, on the planning of public transport systems in The Netherlands.
2003 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Decision support systems; Paired comparison; Ranking; Transportation; Uncertainty modelling; Utility theory

1. Introduction Suppose a decision-maker faces the problem of ranking I choice options or alternatives i (i ¼ 1; . . . ; I) and J attributes or judgement criteria j (j ¼ 1; . . . ; J ). The information available is presented in a so-called effect or consequence matrix S: *

Corresponding author. E-mail addresses: [email protected] (E. Hinloopen), [email protected] (P. Nijkamp), [email protected] (P. Rietveld).

0377-2217/$ - see front matter
2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.06.007

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E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

0

1 S11 . . . S1J B C .. S¼@ A: . SI1 . . . SIJ

ð1:1Þ

The entry Sij (i ¼ 1; . . . ; I; j ¼ 1; . . . ; J ) may represent the (cardinal) consequence of alternative i according to criterion j (for instance, the fuel consumption for the choice of a car) or may represent the (ordinal) rank order of alternative i with respect to criterion j (for instance, the safety of a car). In this paper, we assume that the rankings of the alternatives with respect to the individual criteria are not identical. This means that for every alternative i its corresponding profile (Si1 ; . . . ; SiJ ) has to be assessed. Two categories of assessment techniques can be distinguished: holistic scaling and decomposed scaling. Holistic scaling is based on an overall assessment of profiles. In decomposed scaling, the assessment of the relative importance of the criteria and the assessment of the consequences of the alternatives for all criteria are performed separately. Both techniques have their specific pros and cons (Beinat, 1995). In case of a decomposed scaling method, the information about the relative importance of the judgement criteria is given by means of preference weights. These preference weights are presented in a so-called weight vector k: k ¼ ðk1 ; . . . ; kJ Þ:

ð1:2Þ

The entry kj (j ¼ 1; . . . ; J ) may represent the (cardinal) weight of criterion j or may represent the (ordinal) rank order of criterion j in relation to the other criteria. Based on the information of the consequence matrix S and the weight vector k, a final outranking of the alternatives is created. In this paper a technique based upon decomposed scaling is presented. It builds on an earlier MCDA method, called the Regime Method that was developed to deal with the situation that all consequences and weights are measured on an ordinal scare (see Hinloopen et al., 1983; Hinloopen and Nijkamp, 1986). The method can be characterized as an ordinal variant of the concordance analysis. As described in Hinloopen and Nijkamp (1990), it was adapted to deal with cardinal information. This was done by translating the ordinal information into stochastic information, following a specific probability distribution function. Until now, the overall value function used was the additive value function. In this paper, two developments are reported. First, the use of the concept of stochastic dominance in order to generalise the method in dealing with a mixture of cardinal and ordinal consequences. Second, the use of non-linear overall value functions in order to deal with imperfect compensation. As many other decomposed scaling methods, the method consists of the following steps: (1) standardisation; (2) aggregation and comparison; (3) ranking. In the literature, steps 1 and 2 are elements of the so-called ‘‘construction step’’. Step 3 is the ‘‘exploitation step’’ (Bouyssou and Vincke, 1997). The following of this paper describes these steps. As a case study, the choice of an automated people mover in the city of Nijmegen (Netherlands) is used. 2. Standardisation (step 1) 2.1. Introduction The first step is to make the information of the consequence matrix comparable for the various criteria by means of a standardisation procedure. Various ways of standardisation have been developed (Voogd, 1983). One of these standardisation methods is the concept of value functions (Keeney and Raiffa, 1976). A value function translates consequences into the decision-makerÕs judgements. A value function associates a real number to the consequences of the alternatives considered by the decision-maker. The shape of the value function is determined by the decision-makersÕ preferences.

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Depending on the problem considered, value functions may have one argument (single attribute value functions) or more than one (multiple attribute value functions). In case of a multi-criteria ranking problem, a multiple value function has to be used. The method is based on the concept of decomposed scaling: the multiple attribute value function (‘‘overall value function’’) is some kind of aggregation of a number of single attribute value functions (‘‘marginal value functions’’) (Beinat, 1995). The shape of the marginal value functions is determined by the decision-makersÕ preferences, with the restriction that all marginal value functions have the same value range (usually ½0; 1). In this section, we discuss marginal value functions for cardinal data and for ordinal data separately. 2.2. Marginal value functions for cardinal criteria As a ‘‘default’’ situation, the method assumes that the consequences indicate either a benefit criterion or a cost criterion: the corresponding marginal value functions are either strictly non-decreasing or nonincreasing functions. In this paper, we assume that all criteria are benefit criteria: if two alternatives are compared with each other, then the following holds: Sij P Si0 j () Vj ðSij Þ P Vj ðSi0 j Þ

ð2:1Þ

with Vj ð Þ the marginal value function of the consequences with respect to criterion j. Second, the method assumes that only the two threshold values are known and that no other information about the shape of the marginal value function is available. In this situation, linear value functions are used: 8 9 if Sij < Minj <0 = Vj ðSij Þ ¼ ðSij Minj Þ=ðMaxj Minj Þ if Minj 6 Sij 6 Maxj : ð2:2Þ : ; 1 if Sij > Maxj The threshold values Max and Min are determined independently of the consequences Sij . For example, the criterion ‘‘maximum velocity’’ of a car may have threshold values 120 and 160 km/hour. This means that the value of the consequences with respect to the criterion ‘‘maximum velocity’’ equals 0 for all cars with a maximum velocity less than 120 km/hour and equals 1 for all cars with a maximum velocity above 160 km/ hour. The lower bound threshold value can also be used as a selection criterion for alternatives to be considered. As formulated above, the ‘‘bounded linear function’’ is used as the default function. If, however, there is additional information about the shape of the marginal value functions, this information can be used to derive more appropriate value functions (Keeney and Raiffa, 1976; Beinat, 1995). For example, the criterion ‘‘length of the car’’ will usually have a value function that has its maximum value at a finite length (the optimal length), which suggests a concave value function. 2.3. Marginal value functions for ordinal criteria 2.3.1. Basic concept One of the main problems in MCDM is the integration of cardinal and ordinal information. Examples of methods that deal with cardinal as well as ordinal information are EVAMIX (Voogd, 1983) and QUALIFLEX (Paelinck, 1978; Ancot and Paelinck, 1982). QUALIFLEX is a method that investigates all possible permutations of alternatives with respect to the consequences of all criteria. The advantage of this method is the correct treatment of cardinal and ordinal information. Drawbacks of the method are that with a substantial (20 or more) alternatives, the number of permutations to be investigated explodes, so one has to use a heuristic in order to arrive at a final ranking of the alternatives. Another drawback is that ties in the final ranking are excluded.

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EVAMIX is a method that is based on pairwise comparison of alternatives. Typical for this method is that cardinal and ordinal information is processed separately: for each pair of alternatives, a dominance score for the cardinal criteria is calculated and a dominance score for the ordinal criteria. These dominance scores are then combined into an overall dominance score. An advantage of this method is that cardinal and ordinal information is processed correctly. A drawback of this method is the separation of cardinal and ordinal criteria into two groups. This separation introduces a utility tree that is not based upon the preference structure of the decision-maker, but is based upon the level of information of the criteria. In order to integrate cardinal and ordinal information, we interpret ordinal information as stochastic information. As in the framework of stochastic dominance (Rietveld and Ouwersloot, 1992) ordinal consequences are interpreted as random variables following an unknown probability distribution function (pdf). The main idea of the stochastic interpretation of ordinal data is that behind the ordinal data there are latent (unobserved) cardinal data. Any set of cardinal data that is not in conflict with the ordinal information is allowed. As an example (see Section 5): by the selection of public transport techniques, one of the criteria is the ‘‘comfort in the vehicle’’. This criterion is measured on an ordinal scale: the alternatives can be ranked according to this criterion, but no numerical values are known that describe the alternativesÕ ‘‘comfort in the vehicle’’. We assume that these numerical values do exist but that they are, apart from their ranking, unknown. This means that to every ordinal consequence Sij , a random variable S ij is attached, following an unknown pdf Fj ð Þ. This distribution function is used as the marginal value function. This can be motivated as follows. Analogous to the cardinal criteria, we assume that if two alternatives are compared with each other, the following holds: S ij P S i0 j () Fj ðS ij Þ P Fj ðS i0 j Þ

ð2:3Þ

with Fj ð Þ the marginal value function of the consequences with respect to criterion j. The distribution function F is a suitable candidate for the marginal value function because of its resemblance with the marginal value functions for cardinal data: • both types of marginal value functions have domain R2 and range ½0; 1; • both types of marginal value functions are strictly non-decreasing. Notice that, irrespective of the distribution of S ij , Fj ðS ij Þ itself is a random variable following a uniform distribution (Mood et al., 1974, p. 202, Theorem 12). This can be formulated as follows: ProbðFj ðS ij Þ 6 zÞ ¼ ProbðS ij 6 Fj 1 ðzÞÞ ¼ Fj ðFj 1 ðzÞÞ ¼ z

for 0 6 z 6 1:

ð2:4Þ

The interpretation of (2.4) is as follows. The probability that alternative i belongs with respect to ordinal criterion j to the worst 80% (represented by a z-value of 0.8) of the alternatives, equals 0.8. Further, notice that E½Fj ðS ij Þ ¼ 0:5 for all Sij :

ð2:5Þ

As (2.5) indicates, no meaningful use of (2.4) can be made when one wants to derive the expected value of an individual criterion outcome. However, as will be shown below, when alternatives are compared with each other, sensible use of (2.4) can be made. Consider alternatives i and i0 with ordinal consequences Sij and Si0 j . Assume that Sij > Si0 j . Consider two independent random variables S ij and S i0 j . Then, to Sij the random variable T 2j ðS ij ; S i0 j Þ ¼ MaxfFj ðS ij Þ; Fj ðS i0 j Þg is attached. Similarly, to Si0 j the random variable T 1j ðS ij ; S i0 j Þ ¼ MinfFj ðS ij Þ; Fj ðS i0 j Þg is attached. In this way the ordinal numbers Sij and Si0 j are translated into the stochastic cardinal numbers T 1j ðS ij ; S i0 j Þ and T 2j ðS ij ; S i0 j Þ. Then T 1j ð ; Þ and T 2j ð ; Þ are used as marginal value functions. The distributions of these marginal value functions are formulated in Theorem 1.

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Theorem 1. If S ij and S i0 j are independent random variables following a certain pdf Fj ð Þ and if T 1j ðS ij ; S i0 j Þ ¼ MinfFj ðS ij Þ; Fj ðS i0 j Þg and T 2j ðS ij ; S i0 j Þ ¼ MaxfFj ðS ij Þ; Fj ðS i0 j Þg, then ProbðT 2j ðS ij ; S i0 j Þ 6 zÞ ¼ z2

ð2:6Þ

for 0 6 z 6 1

and ProbðT 1j ðS ij ; S i0 j Þ 6 zÞ ¼ 1 ð1 zÞ2

ð2:7Þ

for 0 6 z 6 1:

The proof of this theorem is given in Appendix A. It shows the role of (2.4) in order to arrive at (2.6) and (2.7). The interpretation of Theorem 1 is illustrated by the following example. If Sij > Si0 j then the probability that alternative i belongs with respect to ordinal criterion j to the worst 60% (z ¼ 0:6) of the alternatives, equals 0.36 and the probability that alternative i0 belongs to the worst 60% of the alternatives, equals 0.84. Furthermore, it follows from Appendix A that E½T 2j ðS ij ; S i0 j Þ ¼ 2=3

for all Sij ; Si0 j with Sij > Si0 j ;

ð2:8Þ

E½T 1j ðS ij ; S i0 j Þ ¼ 1=3

for all Sij ; Si0 j with Sij > Si0 j :

ð2:9Þ

Now, based on Theorem 1, we can conclude that when two alternatives are compared, there is a relation between the ordinal consequences and the random variables attached. This relation can be formulated as follows. If Sij > Si0 j , then for alternative i (2.6) holds and for alternative i0 (2.7) holds (and, of course, vice versa if Si0 j > Sij ). Finally we arrive at the relation with the concept of stochastic dominance. Consider ProbðT 2j ðS ij ; S i0 j Þ 6 zÞ ProbððT ij ðS ij ; S i0 j Þ 6 zÞ. We see that ProbðT 2j ðS ij ; S i0 j Þ 6 zÞ ProbðT 1j ðS ij ; S i0 j Þ 6 zÞ ¼ z2 1 ð1 zÞ2 ¼ 2zð1 zÞ

for 0 6 z 6 1:

ð2:10Þ

Notice that ProbðT 2j ðS ij ; S i0 j Þ 6 zÞ ProbðT 1j ðS ij ; S i0 j Þ 6 zÞ is non-positive for all z and negative for some z. This means that, by definition, ProbðT 2j ðS ij ; S i0 j Þ 6 zÞ dominates ProbðT 1j ðS ij ; S i0 j Þ 6 zÞ by the first degree of stochastic dominance. 2.3.2. Extension of the basic concept As shown in Appendix A, the marginal value functions formulated in Theorem 1 are Beta functions with parameters 2 and 1 (2.6) and with parameters 1 and 2 (2.7). This holds for every ordinal criterion and for every pair of alternatives considered. For the cardinal criteria, the corresponding variables are U2j ðSij ; Si0 j Þ ¼ MaxfVj ðSij Þ; Vj ðSi0 j Þg and U1j ðSij ; Si0 j Þ ¼ MinfVj ðSij Þ; Vj ðSi0 j Þg. Generally, U2j ðSij ; Si0 j Þ 6¼ 2=3 and U1j ðSij ; Si0 j Þ 6¼ 1=3. As a consequence, the importance of the criteria is not only reflected by the weights of the criteria, but also by the scale of measurement. In order to avoid this, the parameters of the Beta functions will be adjusted in such a way that the expected values equal the average value of the ‘‘best’’ cardinal scores and the average value of the ‘‘worst’’ cardinal scores. This can be formalised as follows. Define X Maxii0 ¼ 1=Jc  MaxfVj ðSij Þ; Vj ðSi0 j Þg; ð2:11Þ j2C

Minii0 ¼ 1=Jc 

X

MinfVj ðSij Þ; Vj ðSi0 j Þg

j2C

(C is the set of cardinal criteria, Jc is the number of cardinal criteria).

ð2:12Þ

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Then, when E½T 2j ðS ij ; S i0 j Þ ¼ Maxii0 and E½T 1j ðS ij ; S i0 j Þ ¼ Minii0 , the influence of the scale of measurement to the importance of the criteria is eliminated. As stated above, this will be realised by generalising the probability distribution function Fj ð Þ. Consider an ordinal criterion j and two alternatives i and i0 . Then T 2j and T 1j can be seen as order statistics having ranks 2 and 1 corresponding to a random sample of size 2 from the uniform distribution. However, when T 2j and T 1j are interpreted as order statistics having ranks b and a (b > a) corresponding to a random sample of size N from the uniform distribution, the probability distribution function is sufficiently generalised, see Theorem 2. Theorem 2. If T bj and T aj are order statistics having ranks b and a (b > a) corresponding to a random sample of size N from the uniform distribution, then Z z Bðt; b; N þ 1 bÞ dt for 0 6 z 6 1 ð2:13Þ ProbðT bj 6 zÞ ¼ 0

and ProbðT aj 6 zÞ ¼

Z

z

Bðt; a; N þ 1 aÞ dt

for 0 6 z 6 1

ð2:14Þ

0

with Bðx; a; bÞ the Beta distribution with parameters a and b. The proof is given in Appendix A. (It can be shown (David, 1981) that all N order statistics corresponding to a random sample of size N from the uniform distribution follow Beta distributions with parameter n and N þ 1 n, for n ¼ 1; . . . ; N .) Based on Theorem 2, we know that E½T bj  ¼ b=ðN þ 1Þ;

ð2:15Þ

E½T aj  ¼ a=ðN þ 1Þ:

ð2:16Þ

As stated above, E½T bj  must be equal to Maxii0 and E½T aj  must be equal to Minii0 . It follows that, a, b and N have to meet the following restrictions: b=ðN þ 1Þ ¼ Maxii0 ;

ð2:17Þ

a=ðN þ 1Þ ¼ Minii0 :

ð2:18Þ

If all criteria are ordinal, no Maxii0 or Minii0 are known. In this case, the basic model will be used (a ¼ 1, b ¼ 2, N ¼ 2). If at least one criterion is cardinal, then the deviation with respect to the basic model will be minimised. This means that N will be minimised. The minimisation problem is formulated as follows: ð2:19Þ

Minimise:

N

subject to:

b=ðN þ 1Þ ¼ Maxii0

ðiÞ;

a=ðN þ 1Þ ¼ Minii0

ðiiÞ;

b6N

ðiiiÞ;

aP1

ðivÞ;

b aP1

ðvÞ:

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Elimination of a and b lead to the following reformulation: Minimise: subject to:

ð2:20Þ

N 0

N P 1 þ 1=ð1 Maxii0 Þ N P 1 þ 1=Minii0

ðiii Þ; ðiv0 Þ;

N P 1 þ 1=ðMaxii0 Minii0 Þ

ðv0 Þ:

The solution is as follows: N  ¼ Maxf1=ð1 Maxii0 Þ; 1=Minii0 ; 1=ðMaxii0 Minii0 Þg 1;

ð2:21Þ

a ¼ Maxii0  ðN  þ 1Þ;

ð2:22Þ

b ¼ Minii0  ðN  þ 1Þ:

ð2:23Þ

Example. Suppose that Maxii0 ¼ 0:6 and Minmii0 ¼ 0:4. Then: N  ¼ Maxf1=ð1 0:6Þ; 1=0:6; 1= ð0:6 0:4Þg

1 ¼ 4, a ¼ 0:4  5 ¼ 2 and b ¼ 0:6  5 ¼ 3. Interpretation of this result is that T 2j and T 1j are order statistics with ranks 3 and 2 corresponding to a random sample of size 4 from the uniform distribution. Additionally, ProbðT bj 6 zÞ is Beta distributed with parameters 3 and 2 and ProbðT aj 6 zÞ is Beta distributed with parameters 2 and 3. 3. Aggregation and comparison (step 2) 3.1. Introduction Since more than one criterion is considered, and usually the rankings of the alternatives with respect to the individual criteria are not identical, an aggregation over the criteria is necessary. This aggregation is based on the marginal value functions and results in an overall value function. The construction of this function is the second step of the method. The arguments of this function are the corresponding marginal value functions and weights reflecting the relative importance of the criteria. In the next subsection, the construction of the overall value function is restricted to the cardinal criteria. In Section 3.2, the overall value function is extended with the ordinal criteria. Since only sensible use of ordinal information can be made when alternatives are compared with each other, the aggregation function is developed within the framework of paired comparison of alternatives. 3.2. Cardinal criteria: Aggregation If we consider a situation with only cardinal criteria, the overall value function of alternative i can be formulated as follows: Wi ¼ W ðVj ðSij Þ; j ¼ 1; . . . ; J ; k1 ; . . . ; kJ Þ:

ð3:1Þ

We assume that the weights can be measured on a cardinal scale as well on an ordinal scale. In the situation that the weights are measured on an ordinal scale, the weights are interpreted as random variables, following a certain probability distribution function. If no additional information is available about the shape of this distribution (default assumption), the decision-maker is faced with a decision under uncertainty. In this situation, according to the Laplace criterion, the Regime Method uses the uniform distribution (see Taha, 1976). With stochastic weights, the overall value function can be formulated as follows: W i ¼ W ðVj ðSij Þ; j ¼ 1; . . . ; J ; k1 ; . . . ; kJ Þ:

ð3:2Þ

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In Sections 3 and 4, only underscored kÕs are used in the formulas: weights are stochastic variables. In this context, the situation of cardinal weights is nothing more than a specific set of cardinal values of the stochastic variables. A widely used overall value function is the additive or linear value function. A characteristic of this value function is its elasticity of substitution being equal to infinity. In order to allow for a finite elasticity of substitution, some curved overall value function has to be used. The overall value function we propose is the CES-function (see Keller, 1976). We are aware that other approaches exist to address the issue of compensation. The choice for the CES-function is based on two arguments. First, the CES-function is a relatively simple flexible value function in the sense that the elasticity of substitution is described by only one parameter (r) and that this parameter that can take any non-negative value: ( ) 1=q X

q Wi ¼ kj  Vj ðSij Þ ; ð3:3Þ j

with q ¼ ð1 rÞ=r; r > 0:

ð3:4Þ

Second, three appealing types of overall value functions are nothing more than three limiting cases of the CES-function: the CES-function with r ! 0 (Leontief function), the CES-function with r ! 1 (Cobb– Douglas function) and the CES-function with r ! 1 (Additive function). We see that if r ! 0, then W i ¼ MinfVj ðSij Þg; j

if r ! 1, then Y kj Wi ¼ Vj ðSij Þ ;

ð3:5Þ

ð3:6Þ

j

if r ! 1, then X Wi ¼ kj  Vj ðSij Þ:

ð3:7Þ

j

Eq. (3.5) demonstrates a shortcoming of the CES-function: in the Leontief case, weights are excluded from the overall value function. Keller (1976) solved this problem by specifying a modified CES-function (MCES-function) ( ) 1=q X 1þq

q Wi ¼ kj  Vj ðSij Þ : ð3:8Þ j

Using the MCES-function, we see that if r ! 0, then W i ¼ MinfVj ðSij Þ=kj g; j

if r ! 1, then Y kj Wi ¼ a Vj ðSij Þ

ð3:9Þ

ð3:10Þ

j

Q kj with a ¼ j kj if r ! 1, then X Wi ¼ Vj ðSij Þ: j

ð3:11Þ

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As we can see, the MCES-function eliminates the shortcoming of the CES-function in the Leontief situation (3.9). However, in the situation of perfect substitution (r ! 1), the MCES-function is simply a nonweighted linear aggregation function. The question is to determine which specification is the most favourable one within the multi-criteria framework. The above mentioned limiting cases should not be regarded as only hypothetical situations, on the contrary, in order to investigate the consequences of different values of the elasticity of substitution, these values of r are of great importance. The question now is to specify a CES-type function, which results in a weighted Leontief function when r ! 0, and which results in a weighted additive function when r ! 1. We meet these demands by specifying a convex combination of the CES-function and the MCESfunction: ( ) 1=q ( ) 1=q X X 1þq

q

q W i ¼ GðrÞ  kj  Vj ðSij Þ þ ð1 GðrÞÞ  kj  Vj ðSij Þ ð3:12Þ j

j

with GðrÞ a real valued function Rþ ! ½0; 1, with Gð0Þ ¼ 0 and limr!1 GðrÞ ¼ 1. It is easily seen that if r ! 0, then W i ¼ MinfVj ðSij Þ=kj g

ð3:13Þ

j

and if r ! 1, then X Wi ¼ kj  Vj ðSij Þ;

ð3:14Þ

j

which are the desired results. The issue that remains is to specify GðrÞ. Before answering this, the question how to determine the value of r itself will be dealt with. If the decision-maker knows his elasticity of substitution, this value can be given to r. However, often the value of r is unknown. In this situation, it is assumed that r is an exponentially distributed variable on the interval ½0; 1Þ: F ðrÞ ¼ 1 expð e  rÞ;

ð3:15Þ

r P 0:

In order to choose a value for e, the elasticity of substitution of 1 (r ¼ 1) is seen as a ‘‘neutral’’ value. The idea of neutrality is translated into the demand that F ð0Þ ¼ 0:5. This means that the value of the parameter e is to be set at )ln(0.5): F ðrÞ ¼ 1 0:5r ;

ð3:16Þ

r P 0:

Now, let us return to the specification of GðrÞ. A promising candidate is of course F ðrÞ. It meets all restrictions imposed on GðrÞ. Besides, F ðrÞ is a strictly monotonic function in r. Inserting (3.16) and (3.5) in (3.12) results in ( ) 1=q ( ) 1=q X X 1þq

q

q W i ¼ ð1 0:5r Þ  kj  Vj ðSij Þ þ 0:5r  kj  Vj ðSij Þ : ð3:17Þ j

j

The Cobb–Douglas function corresponding to (3.17) for r ! 1 has the following specification: Y W i ¼ a0  Vj ðSij Þk j

ð3:18Þ

j

Q kj with a0 ¼ 1=2  ð1 þ j kj Þ. We are aware that other approaches exist to address the issue of compensation. We do not claim that our approach––the use of a CES-like function––is superior, but we think it is an attractive approach for the two arguments mentioned above.

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Notice that the use of the CES-function implies identical elasticities of substitution between all criteria. This characteristic also holds for the CES-function according to the specification of (3.17). If this characteristic is too restrictive, a nested CES-function can be used (Keller, 1976). A thorough treatment of this type of value function is beyond the scope of this paper. Roughly spoken, the CES-type function (3.17) is used at the level of groups of individual criteria. The results of these CES-functions are used as arguments of the CES-function of the next higher level, and so on. Within each CES-function, the elasticities of substitution are equal and different CES-functions may have different elasticities of substitution. This paper, however, is restricted to the one level CES-function. It should be emphasized that the meaning of weights is dependent on the way the value functions are formulated and the aggregation takes place (e.g., Roy and Mousseau, 1996). For example, when decision makers or their representatives express their views on weights, they should be informed about the way the standardisation and the aggregation carried out. Under ideal circumstances where there is enough time to interact with decision makers, interactive methods can be used to obtain estimates of value functions and of aggregation methods (see for example Beinat, 1995). In the present context we assume that weights, or at least a (partial) ranking of weights, have been expressed by the decision maker. We also assume that there is no information about the specific aggregation rule to be applied. Under this condition, on cannot escape the necessity to investigate the sensitivity of the final results with respect to the assumptions about the aggregation rules. 3.3. Mixed criteria: Aggregation and comparison If we consider the situation of a mixture of cardinal and ordinal criteria, an overall value function of an alternative only has a meaning when it is compared with another alternative. The overall value function of alternative i with respect to alternative i0 is formulated as follows: ( ) 1=q X X

q

q r W ii0 ¼ ð1 0:5 Þ  kj  Vj ðSij Þ þ kj  Fj ðT ii0 j Þ j2C

( r

þ 0:5 

X

kj1þq

j2O

 Vj ðSij Þ

q

þ

j2C

X

) 1=q k1þq j

 Fj ðT ii0 j Þ

q

ð3:19Þ

j2O

with  T ii0 j ¼

T2j T 1j

if Sij > Si0 j ; if Sij < Si0 j :

ð3:20Þ

T 2j and T 1j are order statistics having ranks b and a (b > a) corresponding to a random sample of size N from the uniform distribution (see Section 2.3, Theorem 2). Additionally, the overall value function of alternative i0 with respect to alternative i is formulated as follows: ( ) 1=q X X

q

q r W i0 i ¼ ð1 0:5 Þ  kj  Vj ðSi0 j Þ þ kj  Fj ðT i0 ij Þ j2C

( r

þ 0:5 

X j2C

kj1þq

j2O

 Vj ðSi0 j Þ

q

þ

X

) 1=q kj1þq

 Fj ðT i0 ij Þ

q

:

ð3:21Þ

j2O

Finally, the comparison of two alternatives is made by means of a value difference function. Let Dii0 be the overall value difference function of alternatives i and i0 , then

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

Dii0 ¼ W ii W i0 i :

327

ð3:22Þ

The pairwise comparison of alternatives and the resulting final ranking of alternatives (Section 4) are highly related to the comparison and ranking rules applied in tournaments of, for instance, football teams. As the measure of distinction between alternatives the probability that the overall value difference function is greater than 0 is used: ProbðDii0 > 0Þ ¼ ProbððW ii W i0 i Þ > 0Þ:

ð3:23Þ

Since the distributions of all stochastical variables of W ii and W i0 i are known, by means of a Monte Carlo procedure, ProbðDii0 > 0Þ can be calculated. The resemblance with a football match is the following. ProbðDii0 > 0Þ describes the probability that team i beats team i0 .

4. Ranking (step 3) Based on the overall value difference functions, the final ranking is determined. In the situation of at least one ordinal criterion or in the situation that the weights are measured on an ordinal scale, Dii0 is a stochastic variable. As stated in Section 3.3, we use the probability that alternative i wins a paired comparison from alternative i0 as the measure of distinction between alternatives i and i0 . In other words: the probability that Dii0 > 0 is used as the measure of distinction between alternatives i and i0 . Let Pii0 ¼ ProbðDii0 > 0Þ:

ð4:1Þ 0

Then Pii0 , the probability that alternative i dominates alternative i , is the measure of distinction between alternatives i and i0 . Based on the Pii0 , the final ranking of alternatives is established. The final ranking is determined by the average probability that an alternative wins a paired comparison from a (randomly chosen) other alternative. Let X Pii0 : ð4:2Þ Pi ¼ 1=ðI 1Þ  i0 6¼i

Then Pi is the average probability that an alternative wins a paired comparison from a (randomly chosen) other alternative. The resemblance with a football competition may be clear: Pi describes the average probability that team i beats a (randomly chosen) other team. The measure Pi is a valued relation on the set of all alternatives considered. This relation corresponds to the well-known rule of Borda and to the so-called valued net flow rule (Bouyssou and Vincke, 1997). This means that a ranking based on Pi satisfies the following properties. • Neutrality. The ranking does not discriminate between alternatives just because of their label. • Continuity. The ranking will not be ‘‘dramatically’’ changed due to ‘‘small’’ changes in a valued relation. • Strict monotonicity. When an alternative is improved sufficiently, its position in the ranking will be improved. • Faithfulness. If the ranking rule is applied to a weak order, then this weak order is preserved. For a more formal treatment of these (and other) properties, the reader is referred to Bouyssou (1996) and Bouyssou and Vincke (1997). A different axiomatic treatment of ranking rules can be found in Brink (1994). An appealing interpretation of the Pii0 and the Pi may be the following. Suppose one wants to outrank a number of soccer teams based on a number of competitions played in the past. One can proceed as follows. First, consider all individual matches of two specific teams A and B. The percentage of matches won by A is

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the estimation of the probability Pii0 that A beats B (draws are excluded). This probability is the measure of distinction between soccer teams. Secondly, the final ranking is determined by the probabilities that a team beats a (randomly chosen) other team. An interesting quantity is the probability that a certain soccer team beats in a competition a certain number of other teams. Let Pik the probability that an alternative dominates k alternatives. Then, also on this quantity a final ranking of alternatives can be established. Let Ei ¼

I 1 X

k  Pik :

ð4:3Þ

k¼0

Then Ei is the expected number of alternatives that is dominated by alternative i. In other words: Ei is the expected number of times a soccer team beats some other team. An interesting result is that Pi and Ei establish the same final ranking. This is formulated in the following theorem. Theorem 3. Let Pi ¼ 1=ðI 1Þ 

X

ð4:4Þ

Pii0 :

i0 6¼i

Let Ei ¼

I 1 X

k  Pik :

ð4:5Þ

k¼0

Then Ei ¼ ðI 1Þ  Pi :

ð4:6Þ

Corollary. Pi and Ei lead to the same final ranking. The proof of this theorem is given in Appendix B. The proof of the corollary is rather trivial.

5. Case study: Public transport in Nijmegen (Netherlands) Hague Consulting Group (a Dutch consulting company in the field of traffic and transportation) used the Regime Method (Hague Consulting Group, 1995) in order to rank a number of transport modes for the Table 1 Alternatives in public transport case study No.

Name

1 2 3 4 5 6 7 8 9 10

Bus Skytrain: train-like automatic people mover (‘‘steel on steel’’) SK: automatic people mover drawn by a cable SPM: Small sized automatic people mover (‘‘rubber on concrete’’) ‘‘Traditional’’ monorail H-Bahn: hanging monorail Cable tram Bus with 1 100% free lane Trolley bus with a 100% free lane Tram with a 100% free lane

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329

Table 2 Criteria in public transport case study No.

Name

Weight (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Result of a cost–benefit analysis PassengerÕs probability of having a seat Punctuality Comfort in vehicle Comfort in waiting location Is it easy to enter or leave the vehicle? Image of the transportation system Accessibility by disabled passengers VehicleÕs safety Social safety Spatial impact: land use Spatial impact: creation of a physical barrier Spatial impact: interference with other means of transport Local emissions Global emissions Noise emission Visual impact Impact on local employment Impact on local ground prices Impact on population density Technical complexity Flexibility in upgrading the transportation system Flexibility in the level of service Organisational complexity of operating the system Flexibility in avoiding traffic jam Impact on the environment during construction

47.20 2.16 6.12 0.96 0.96 0.96 3.96 1.08 1.80 1.80 3.06 3.06 3.06 2.07 1.98 2.97 1.98 2.04 1.87 1.87 2.03 0.98 2.03 0.98 0.98 2.04

city of Nijmegen (Hague Consulting Group, 1997). The specific project considered is a public transport link between the centre of the city and a large expansion of the city at the other side of the Waal river. The selection of the transport modes, the definition of the criteria, the evaluation of the transport modes with respect to the criteria and the determination of the weights have been done by the department Transport Research Centre (AVV) of the Dutch Ministry of Transport, Public Works and Water Management. Ten public transport techniques were evaluated (Table 1) according to 26 criteria (Table 2). The following tables give the names of the alternatives and the names of the criteria. For a detailed description of the alternatives and a thorough treatment of the criteria, the reader is referred to Hague Consulting Group (1995). A striking aspect of the weights vector is that the cost–benefit result has a weight that is almost as large as the sum of the weights of the other 25 criteria. The background is that decision making in this policy domain in the Netherlands is rather strongly dominated by cost– benefit approaches (see for a recent illustration Eigenraam et al., 2000). This leads to the danger of neglecting aspects that are difficult to assign monetary values. In order to overcome this problem, sometimes a combination of cost–benefit analysis and multi-criteria decision analysis is carried out (see for example, Van Pelt, 1993). The present case study is an example of this. The consequence matrix has the following information (row numbers indicate the numbers of the transport modes, column numbers indicate the numbers of the criteria). Transport mode no. 1 (the traditional bus) is the reference option. Its net cost–benefit value is set at zero and the other net cost–benefit values are related to this. For instance, reduction of travelling time (one of the elements of the cost–benefit analysis) is related to the travelling time by bus. The upper bound of the net cost–benefit value is set at 2 and the lower value at )1.

330

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

1 2 3 4 5 6 7 8 9 10

0.000 0.129 0.394 0.358 )0.160 0.241 1.457 1.889 1.427 1.043

0 1 1 1 1 1 1 1 1 1

0 2 2 2 2 2 2 1 1 1

0 3 2 3 3 3 1 1 1 2

0 1 1 1 1 1 1 1 1 1

0 2 1 2 2 2 1 0 0 2

0 3 2 3 3 3 2 1 1 2

0 3 3 3 3 3 3 1 1 2

0 3 3 3 3 3 3 1 1 2

1 0 0 0 0 0 0 1 1 1

3 1 1 1 1 1 2 0 0 0

3 0 0 0 0 0 1 2 2 2

0 2 2 2 2 2 2 1 1 1

0 2 2 3 3 3 3 1 3 3

0 1 1 2 2 2 3 3 3 2

0 1 2 2 5 5 4 2 4 3

5 1 2 2 0 1 3 4 4 4

0 2 1 2 2 2 1 0 0 1

0 2 1 2 2 2 1 0 0 1

0 2 1 2 2 2 1 0 0 1

2 0 1 0 0 0 1 2 2 2

0 1 0 0 0 0 0 1 1 2

0 2 2 2 2 2 2 0 0 1

3 0 0 0 0 0 0 2 2 1

3 0 0 0 0 0 0 2 1 0

2 0 0 0 0 0 1 1 1 1

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

Table 3 Consequence matrix (row numbers: alternatives; column numbers: criteria)

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

331

Table 4 Final ranking (Pi ) with r unknown Alternatives

Pairwise victories

No.

Name

Estimated average

95% confidence interval Minimum

Maximum

Max–Min

7 8 9 10 3 4 6 2 5 1

Cable tram Bus lane Trolley Tram SK SPM H-Bahn Sky train Monorail Bus

0.911 0.896 0.697 0.646 0.513 0.488 0.403 0.247 0.113 0.086

0.905 0.890 0.687 0.636 0.503 0.478 0.393 0.239 0.107 0.080

0.917 0.902 0.706 0.655 0.523 0.498 0.413 0.256 0.120 0.091

0.011 0.012 0.018 0.019 0.020 0.020 0.020 0.017 0.013 0.011

Table 5 Relative frequencies of number of pairwise victories (Pik ) No.

9

8

7

6

5

4

3

2

1

0

7 8 9 10 3 4 6 2 5 1

0.207 0.675 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.787 0.065 0.000 0.026 0.000 0.002 0.000 0.000 0.000 0.000

0.006 0.074 0.558 0.095 0.001 0.022 0.017 0.000 0.000 0.000

0.000 0.074 0.243 0.603 0.029 0.075 0.067 0.000 0.000 0.000

0.000 0.070 0.129 0.221 0.714 0.169 0.112 0.002 0.010 0.000

0.000 0.031 0.054 0.050 0.137 0.731 0.136 0.022 0.046 0.000

0.000 0.009 0.014 0.005 0.085 0.001 0.668 0.179 0.095 0.000

0.000 0.001 0.002 0.000 0.031 0.000 0.000 0.796 0.128 0.023

0.000 0.000 0.000 0.000 0.003 0.000 0.000 0.002 0.243 0.725

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.478 0.252

All ordinal information is formulated as ‘‘the higher, the better’’. Ties are treated like ‘‘equals’’. For the sake of convenience, for each ordinal criterion, the worst value for each criterion is set at 0. This means that, although alternative 1 (bus) is the reference alternative, the corresponding consequences may be unequal to 0. The results of the evaluation method do not depend on these scaling issues. Since no information is known about the elasticity of substitution, is treated as a stochastic variable. The results are given in Tables 4–6. The column ÔEstimated averageÕ in Table 4 contains the values of Pi . We see that the final ranking of the alternatives is almost identical to the ranking according to the results of the cost–benefit analysis (see Table 3, column 1). Only the best two alternatives and the worst two alternatives have a reversed ranking compared to the ranking of the net cost–benefit analysis. The cells of Table 5 contain the values of Pik , the probabilities that a certain alternative (rows) ‘‘beats’’ a certain number (column) simultaneously. We see in Table 5 that the maximum row values (bold) are usually situated in the diagonal cells. The only exceptions are the best two alternatives and the worst two alternatives. This may seem to be a surprising result. On the other hand, if we look at the off-diagonal values, we see that the best alternative always dominates at least seven alternatives simultaneously and that the second best alternatives is sometimes dominated by seven alternatives simultaneously. The same holds for the

332

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

Table 6 Relative frequencies of number of pairwise victories of row alternatives (Pii0 ) No.

7

7 8 9 10 3 4 6 2 5 1

0.791 0.000 0.004 0.000 0.002 0.001 0.001 0.001 0.001

8

9

10

3

4

6

2

5

1

0.209

1.000 1.000

0.996 0.719 0.883

1.000 0.947 0.946 0.993

0.998 0.946 0.926 0.884 0.852

0.999 0.848 0.851 0.845 0.829 1.000

0.999 0.971 0.973 0.864 0.964 1.000 1.000

0.999 0.843 0.692 0.844 0.860 0.999 1,000 0.996

0.999 1.000 0.999 0.979 1.000 1.000 1.000 1.000 0.252

0.000 0.281 0.053 0.054 0.152 0.029 0.157 0.000

0.117 0.054 0.074 0.149 0.027 0.308 0.001

0.007 0.116 0.155 0.136 0.156 0.021

0.148 0.171 0.036 0.140 0.000

0.000 0.000 0.001 0.001

0.000 0.000 0.000

0.004 0.001

0.748

worst two alternatives. The worst alternative is always dominated by a least seven alternatives simultaneously. The worst-but-one alternative sometimes dominates five alternatives simultaneously. The cells of Table 6 contain the values of Pii0 , the probabilities that a certain row alternative ‘‘beats’’ a certain column alternative. We see that the final ranking is in concordance with almost all paired comparison of the alternatives. Only the paired comparison of the best two alternatives and the worst two alternatives is in discordance with the final ranking. The reason for this is that the best alternative dominates the worst eight alternatives stronger than the second best alternative. Analogously, the worst-but-one alternative is weaker dominated by the best eight alternatives than the worst alternative.

1.000

0.900

0.800

0.700

Bus Lane Cable Tram

0.600

Trolley Tram

0.500

SK SPM

0.400

H-Bahn Skytrain

0.300

Bus Monorail

0.200

0.100

0

2

4

6

8

0

2

4

6

8

0

3.

3.

3.

3.

3.

4.

4.

4.

4.

4.

5.

2 2.

8

0 2.

2.

8 1.

6

6 1.

4

4 1.

2.

2 1.

2.

0 1.

0.000

Fig. 1. Final scores with r varying from 1.0 to 5.0.

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

333

The reader may verify that the weighted summation of the row elements of Table 5 (weights equal to the corresponding number of alternatives that are dominated simultaneously) and the summation of the row elements of Table 6 lead to the same results. This is an illustration of Theorem 3 (Section 4). If one criterion fully determines the final ranking, than the cells of the upper right triangle of Table 6 will all have value 1 and the cells of the lower left triangle will have value 0. The consequences for Table 5 are that the diagonal cells will have value 1 and that all other cells will have value 0. In this example, the results of the cost–benefit analysis do not completely determine the final ranking. This is of course due to the fact that that its weight is a little below 50%. However, the value of r also appears to be of importance. To illustrate the influence of the value of r on the final ranking, the method is run for a number of different values of r. Fig. 1 is restricted to the r-range 1.0–5.0. When r ranges from 0 to 1, the final scores are identical to the final scores with r equals 1. In other words: when the decision makersÕ overall value function is a Leontieftype, a Cobb–Douglas-type or Ôsomething in betweenÕ, the final ranking is always identical to the ranking order of the net cost–benefit values. Apparently, as long as 6 1.3, the final ranking is never ÔoverruledÕ by ranking orders with respect to the other criteria. The main reason, of course, is the fact that weight attached to the net cost–benefit value is almost 50%. When the elasticity of substitution is sufficiently low, the overall value (difference) function is very close to the Leontief model, which is similar to a weighted max–min-rule. Therefore, when one of the weights is almost 50%, the final ranking will be very close to (in this situation: identical to) the ranking according to the most important criterion. Between r-values 1.4 and 4.6, the final ranking varies with the value of r. For instance, when r P 1:5 and r 6 4:3, the ‘‘Cable Tram’’ is to on top of the ranking. When r P 4:7, the final ranking is identical to the one belonging to r-values between 0 and 1.4. The main reason is again the fact that the weight attached to the net cost–benefit value is almost 50%. When the elasticity of substitution is sufficiently high, the overall value (difference) function is very close to the additive model, which is similar to a weighted summation. Therefore, when one of the weights is almost 50%, again the final ranking will be very close to the ranking according to the most important criterion. The issue that remains is to find an explanation for the final ranking when r varies from 1.4 to 4.6. In this situation, the influence of the 25 less important criteria is apparently sufficiently high in order to change the final ranking. In other words, in this situation all criteria play a role in determining the final ranking. Since the CES-function is a convex function, it is likely to expect alternatives that that have ‘‘reasonable results’’ on all criteria are ranked higher than alternatives that perform very well on some criteria and perform very poor on some other criteria. If all scores were cardinal, one would expect that, comparing alternatives with equal average scores, the alternative with the lowest standard deviation of the scores would be on top of the ranking. This idea will be used to find an explanation for the final ranking when r varies from 1.4 to 4.6. In order to compare the ordinal data with each other, we calculate for each alternative the net value of the number of other alternatives that are ranked lower minus the number of other alternatives that are ranked higher, see Table 7. Now, for every alternative, the standard deviation of the row values is calculated. These results are compared with the final ranking belonging to r ¼ 2:5, the most ‘‘distorting’’ r-value, see Table 8. What we see is that the final ranking of the best six alternatives is (almost) identical to the ranking of their standard deviations. We see that the alternatives with relatively low standard deviations (Cable Tram, Tram, SPM and H-Bahn) have climbed in the final ranking and that alternatives with relatively high standard deviations (Bus Lane and Trolley) have lost some positions. In addition, ‘‘Monorail’’, ‘‘Skytrain’’ and ‘‘Bus’’ are now ranked according to increasing standard deviations, which is the complete reverse order with respect to final ranking when r 6 1:3 or r P 4:7. These changes in final ranking are what we have expected. On the other hand, the standard deviation is of course not fully responsible for changes in the final ranking, also the weights have influence. This is illustrated by the position changes of ‘‘SK’’. Despite the fact that its standard deviation is the lowest of all alternatives, it still loses two places. What remains

334

No.

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

1 2 3 4 5 6 7 8 9 10

)9 1 1 1 1 1 1 1 1 1

)9 4 4 4 4 4 4 )5 )5 )5

)9 6 0 6 6 6 )5 )5 )5 0

)9 1 1 1 1 1 1 1 1 1

)7 5 )2 5 5 5 )2 )7 )7 5

)9 6 )1 6 6 6 )1 )6 )6 )1

)9 4 4 4 4 4 4 )6 )6 )3

)9 4 4 4 4 4 4 )6 )6 )3

6 )4 )4 )4 )4 )4 )4 6 6 6

9 1 1 1 1 1 7 )7 )7 )7

9 )5 )5 )5 )5 )5 1 5 5 5

)9 4 4 4 4 4 4 )5 )5 )5

)9 )4 )4 4 4 4 4 )7 4 4

)9 )6 )6 0 0 0 7 7 7 0

)9 )7 )3 )3 8 8 4 )3 4 1

9 )6 )2 )2 )9 )6 1 5 5 5

)7 6 )1 6 6 6 )1 )7 )7 )1

)7 6 )1 6 6 6 )1 )7 )7 )1

)7 6 )1 6 6 6 )1 )7 )7 )1

6 )6 0 )6 )6 )6 0 6 6 6

)4 5 )4 )4 )4 )4 )4 5 5 9

)7 4 4 4 4 4 4 )7 )7 )3

9 )4 )4 )4 )4 )4 )4 6 6 3

9 )3 )3 )3 )3 )3 )3 7 5 )3

9 )5 )5 )5 )5 )5 4 4 4 4

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

Table 7 Number of alternatives ranked higher minus number of alternatives ranked lower (criteria 2–26)

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

335

Table 8 Final ranking with r ¼ 2:5 and the standard deviation of the row values of Table 7 No.

Name

Pi with r ¼ 2:5

S.D.

Average

7 10 4 6 8 9 3 5 2 1

Cable tram Tram SPM H-Bahn Bus lane Trolley SK Monorail Skytrain Bus

0.968 0.671 0.607 0.599 0.551 0.547 0.401 0.331 0.293 0.032

3.52 4.08 4.25 4.58 5.75 5.68 3.17 4.81 4.87 7.91

1.2 0.8 1.0 1.2 )0.9 )0.4 )0.8 0.8 0.3 )3.0

however, is the idea that when there is a relatively high (but finite) elasticity of substitution (we consider a r of 1 as ‘‘neutral’’), alternatives that perform relatively well on all criteria are ranked relatively high. This is what we expected. 6. Some concluding remarks Applications of multiple criteria analysis often relate to cases where part of the criteria is cardinal whereas the other part is ordinal. This calls for an approach where the two data types are treated in a compatible way. In the present paper we discussed such a method specifically designed for mixed quantitative/qualitative data, based on paired comparisons of choice options. In order to arrive at a compatible measure for value differences between cardinal data and ordinal data, we apply a stochastic interpretation to the ordinal data. The method described can be used in the case of cardinal criterion weights as well as in the case of ordinal weights. The method described also allows for the use of non-linear overall value functions. The case study shows that the final ranking of alternatives depends on the elasticity of substitution between criteria, even in the situation that one criterion dominates all others. We indicate that there are various ways to arrive at final rankings based on paired comparison matrices. The procedure chosen here is based on a net flow score and is closely related to the Copeland Score or the Borda Rule. An interesting question is whether other score rules also have the attractive characteristic of having two different ways to arrive at the same final ranking. Further research has to be done in order to take account of different elasticities of substitution. Incorporation of a nested CES-type function seems a promising approach to realise this extra flexibility. Appendix A Let s1j ; s2j ; . . . ; sNj be a random sample from a probability distribution Fj ð Þ. Let xnj ¼ Fj ðsnj Þ, n ¼ 1; . . . ; N and let y1j < y2j < < yNj be the corresponding order statistic to x1j ; x2j ; . . . ; xNj . Then fY a ðya Þ ¼

N!  ½F ðya Þa 1 ½1 F ðya ÞN a f ðya Þ; ða 1Þ!ðN aÞ!

for 0 6 ya 6 1;

ðA:1Þ

fY b ðyb Þ ¼

N! b 1 N b  ½F ðyb Þ ½1 F ðyb Þ f ðyb Þ ðb 1Þ!ðN bÞ!

for 0 6 yb 6 1

ðA:2Þ

(Mood et al., 1974, p. 254, Theorem 12).

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E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

Notice that the joint probability density function of x1 ; x2 ; . . . ; xN is the uniform density (Mood et al., 1974, p. 202, Theorem 12). It follows that fY a ðya Þ ¼

N!  y a 1 ð1 ya ÞN a ða 1Þ!ðN aÞ! a

for 0 6 ya 6 1

ðA:3Þ

fY b ðyb Þ ¼

N! N b  y b 1 ð1 yb Þ ðb 1Þ!ðN bÞ! b

for 0 6 yb 6 1;

ðA:4Þ

and

which are Beta density functions. It follows that: Z z If Sij P Si0 j ; then ProbðSii0 j 6 zÞ ¼ Bðt; b; N þ 1 bÞ dt with 0 6 z 6 1:

ðA:5Þ

0

If Sij 6 Si0 j ; then ProbðSii0 j 6 zÞ ¼

Z

z

Bðt; a; N þ 1 aÞ dt with 0 6 z 6 1:

ðA:6Þ

0

Theorem 1 is simply applying formulas (A.5) and (A.6) with a ¼ 1, b ¼ 2 and N ¼ 2.

Appendix B Consider the paired comparison of alternative i with all other alternatives. Define Bi as the set of alternatives that are compared with alternative i: Bi ¼ fA1 ; . . . ; Ai 1 ; Aiþ1 ; . . . ; AI g:

ðB:1Þ

Define Gi as the family of all subsets of Bi : Gi ¼ fDi j Di  Bi g

ðB:2Þ

with Di some subset of Bi . Gi is called the power set of Bi . Since Bi contains I 1 elements, Gi contains 2I 1 distinct subsets. Two specific examples of elements of Gi are Ø and Bi . For the sake of convenience, the subsets Di will be numbered from 1 to 2I 1 , so Dni represents subset n (n ¼ 1; . . . ; 2I 1 ) of Gi . Define Pin as the probability that alternative i dominates all alternatives of Dni simultaneously. Notice that 2I 1 X

Pin ¼ 1:

ðB:3Þ

n¼1

Define Pii0 as the probability that alternative i dominates alternative i0 . Then Pii0 ¼

2I 1 X

ðPin j Ai0 2 Dni Þ

ðB:4Þ

n¼1

with ðPin j Ai0

2

Dni Þ

 ¼

Pin 0

if Ai0 2 Dni ; if Ai0 2 6 Dni :

ðB:5Þ

E. Hinloopen et al. / European Journal of Operational Research 158 (2004) 317–338

337

Define Pi as the probability that alternative i dominates a (randomly chosen) other alternative. Then X Pi ¼ 1=ðI 1Þ  Pii0 : ðB:6Þ i0 6¼i

Combination of (B.4) and (B.6) results in Pi ¼ 1=ðI 1Þ 

2I 1 XX i0 6¼i

ðPin jAi0 2 Dni Þ ¼ 1=ðI 1Þ 

n¼1

2I 1 X X n¼1

ðPin jAi0 2 Dni Þ:

ðB:7Þ

i0 6¼i

Define k n as the number of elements (alternatives) in Dni . Then X ðPin jAi0 2 Dni Þ ¼ k n  Pin :

ðB:8Þ

i0 6¼i

Combination of (B.7) and (B.8) results in Pi ¼ 1=ðI 1Þ 

2I 1 X X n¼1

ðPin jAi0 2 Dni Þ ¼ 1=ðI 1Þ 

i0 6¼i

2I 1 X

k n  Pin :

ðB:9Þ

n¼1

Define Pik as the probability that alternative i dominates k alternatives simultaneously. Notice that I 1 X

Pik ¼ 1:

ðB:10Þ

k¼0

Define Ei as the expected number of alternatives that is simultaneously dominated by alternative i. Then Ei ¼

I 1 X

k  Pik :

ðB:11Þ

k¼1

Notice that 2I 1 X

k n  Pin ¼

n¼1

I 1 X

k  Pik :

ðB:12Þ

k¼1

Eq. (B.12) is the key to the proof, since combination (B.9), (B.11) and (B.12) results in Pi ¼ 1=ðI 1Þ  Ei ;

ðB:13Þ

which completes the proof of Theorem 3.

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