European Journal of Operational Research 160 (2005) 676–695

www.elsevier.com/locate/dsw

Stability analysis of tree structured decision functions

Anik�o Ek�art *, S.Z. N�emeth

Computer and Automation Research Institute, Hungarian Academy of Sciences, 1518 Budapest, P.O. Box 63, Hungary

Received 14 January 2002; accepted 15 June 2003

Available online 10 December 2003

Abstract

In multicriteria decision problems many values must be assigned, such as the importance of the different criteria and

the values of the alternatives with respect to subjective criteria. Since these assignments are approximate, it is very

important to analyze the sensitivity of results when small modifications of the assignments are made. When solving a

multicriteria decision problem, it is desirable to choose a decision function that leads to a solution as stable as possible.

We propose here a method based on genetic programming that produces better decision functions than the commonly

used ones. The theoretical expectations are validated by case studies.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Decision support systems; Evolutionary computation; Stability analysis; Decision functions

1. Introduction

In most real life problems, if we want to make a decision then generally we have to take into consid-

eration more than one criterion. For example, if we want to buy a TV set, our criteria could be the cost, the

quality of image, the physical aspect etc. In general, the criteria have different importance. In the case of TV

set we would probably prefer the quality of image to the physical aspect. If a possible alternative has values

not worse for all criteria and better value for at least one criterion than another alternative (i.e., the

alternatives are Pareto ordered) then it is easy to make a decision. We would definitely choose the first

alternative. Unfortunately, in most of the cases the criteria are conflicting. In particular, a TV set withbetter image could have higher cost.

In the economical world there are many multicriteria problems where a cardinal method is preferred

to an ordinal one. For example, when deciding what product to buy, the director of a company could

be interested not only in the importance order of the products. Sometimes they want to know some kind

of efficiency numbers of the products. One would not be satisfied with theoretical explanations concern-

ing the lack of exact mathematical solution for this problem. Since there is a high demand for solv-

ing problems of this type, we should think of as good numerical methods as possible. Our paper is an

* Corresponding author.

E-mail addresses: ekart@sztaki.hu (A. Ek�art), snemeth@sztaki.hu (S.Z. N�emeth).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2003.10.007

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 677

attempt to find such a method. We consider an infinite number of models described by multiparameterdecision functions. We measure the quality of a model as the stability of the corresponding decision

functions. This paper is only a first attempt to compare different models. Hopefully, inspired by this

(probably far from perfect) approach a new trend will arise in multicriteria decision making. Throughout

our paper we consider only the cardinal approach of multicriteria decision problems. This does not mean

that we are not aware of the high importance of the ordinal methods, and that there are problems of ordinal

nature where ordinal methods should be considered instead of cardinal ones. Of course, comparing the

ordinal methods (beyond the scope of our paper) is also a sound issue worth studying. However, we think

that it is no point in comparing cardinal methods with ordinal ones since they handle problems of differentnature.

We are aware of the fact that our method has its limitations and may not be affordable for problems

where the number of criteria and/or alternatives is very large. If the number of criteria is too large then a

possibility could be to choose the most important criteria and neglect the others. If the number of alter-

natives is too large then they might be divided into groups and a two step analysis can be made: first the

groups and then the individuals within the groups are ordered. The general global stability index described

in the conclusion part of our paper can be adapted to this method, too. This kind of two step analysis is not

unfamiliar in large real world applications. However, the nature of the problem will decide which decisionmethod should be used. We believe that it is not possible to give a decision method, which works for every

problem.

In real life problems we must make many assignments, such as importance of criteria and values of

alternatives with respect to subjective criteria. For example, in the case of the TV set, quality of image is a

subjective criterion, it is difficult to assign an exact value to it. When slightly perturbing the assigned values

(i.e., weights and values of alternatives with respect to subjective criteria) we might obtain a different order

of the alternatives.

By the sensitivity analysis of a multicriteria decision problem we mean the analysis of the results, whenwe make small perturbations to the weights and/or values of alternatives with respect to criteria. There

are many different types of sensitivity analysis used by the known multicriteria methodologies [2,6,16–18].

A sensitivity analysis with respect to more than one parameter was given in the Promethee methodology [2].

The WINGDSS methodology [4] presents a sensitivity analysis with respect to all parameters (both

alternative values and weights). Here we consider a similar sensitivity analysis (but either only with respect

to alternative values or only with respect to weights). The originality of our approach is that we con-

sider more complicated decision functions (represented in form of trees) rather than decision functions

of linear fractional type only. Hence, instead of solving linear fractional programming problems we haveto solve much more difficult global optimization problems. Also, the problem of searching for stable

decision functions seems to be a new idea in the area of multicriteria decision making. As far as we can

see, the classical methods of optimization are not appropriate in this case, this problem seems too com-

plicated for classical methods, and therefore we apply a novel genetic programming based optimization

method.

Multicriteria decision problems may have different goals. Usually the goal of a multicriteria problem

is (1) to eliminate a number of worst alternatives, or (2) to choose a number of best alternatives, or (3)

to rank the alternatives. In the problem of elimination or choice, the order between the eliminated orchosen alternatives could be also important. In this case we have a mixed problem of (4) choice and

ranking.

Since the assignment of weights and alternative values with respect to subjective criteria is approximate,

it would be always preferable either to get the same solution (i.e., to have a stable solution) or to obtain very

close solutions (i.e., to have a solution with reasonable stability). For the beginning, we consider the

problem (3) of full ranking of the alternatives. In order to rank the alternatives, we shall use some decision

functions, which is a function of the values of an alternative with respect to the different criteria. The

678 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

decision function must take into consideration the value on each criterion with the correspondingimportance. In the future we shall consider the problems (1), (2) and (4), as well.

In this paper the stability index is given for the full ranking problem. It is possible to give stability

indexes of similar nature for the problems (1), (2) and (4), too. First we consider the local stability of n� 1

subproblems where n is the number of alternatives. These subproblems are the stability of the order of an

alternative and the next alternative in the ranking. If all of these subproblems are stable then the ranking of

the alternatives is stable. We shall introduce local stability indexes for these problems. The local stability

index which takes values between 0 and 1 measures the stability of these n� 1 orders (n� 1 and not n since

the last alternative has no follower).We shall introduce a global index called the stability index, which takes values between 0 and 1 and

measures the stability of a solution (here full ranking). If the stability index is 1 then the solution is stable.

The closer is the index to 1, the more stable is a solution. The stability index is a function of the local

stability indexes such that the global stability index is 1 if and only if all the local stability indexes are 1.

This function can be a power mean with power a 2 R [ f�1;þ1g (or other kind of mean, see for example

[7]) of the local stability indexes. Our idea was to penalize the solutions, which have a local stability index

zero or very close to zero. We would like the stability index to be zero if one of the stability indexes is zero.

There are two power means which satisfy this condition: the a-power mean with power a ¼ �1 (theminimum function) and the a-power mean with power a ¼ 0 (geometric mean). Although we want to

penalize the solutions, which have a stability index close to zero we should give an average stability index to

solutions, which have both small and large stability indexes. The minimum function does not satisfy this

condition. So, we chose the geometric mean.

The goal of this paper is to find a decision function such that––for a fixed problem––the solution be as

stable as possible. It is almost a hopeless task to search for a solution on the whole function space of

possible decision functions. Therefore, we shall restrict the search to a subclass of these functions.

We call decision function a function, which satisfies some natural axioms. The most general class offunctions satisfying these axioms that we could find can be represented in form of trees. The leaves of

these trees are decision functions of weighted power mean type. On the upper level of the tree are power

means. The value of the function is in the root of the tree (There exist other kinds of decision func-

tions where the importance of criteria is given by capacities instead of weights [5,6,10,11]. It would be

interesting to see whether our method could be extended to that type of decision functions.). This value is

calculated recursively, so that the value of each non-leaf node is the corresponding power mean of its

children. We shall use genetic programming [12] for finding decision functions (of the above types) as stable

as possible for certain multicriteria decision problems. The individuals in the genetic programming systemare decision functions represented in the form of trees. Formally, the subtrees of such a tree are decision

functions. However, no clear practical meaning can be given to a subtree. If the problem were a group

decision problem we could think of the group decision function as the aggregation of the individual

decision functions of the decision makers which are subtrees of the tree representing the group decision

function. In that case the associativity of the aggregation would be an important issue [9]. In our framework

we considered multicriteria decision problems with only one decision maker. In this case we could not

think of a reason for the decision maker to consider more than one decision function and to aggregate

them. Hence, we think that in our case the issue of the associativity is not important and we would not dealwith it.

The paper is structured as follows: In Section 2 we shall describe the methodology of multicriteria

decision problems: decision tables and decision functions. In Sections 3 and 4 we shall make a sensitivity

analysis with respect to alternatives and weights, respectively. In both cases we shall introduce a global

stability index, which measures the stability of the order of alternatives with respect to small perturbations

of alternative values and weights, respectively. In Section 5 we shall give experimental results and finally, in

the last section we shall draw the conclusions and present the future perspectives.

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 679

2. Preliminaries

First let us define a multicriteria decision problem.

Definition 1. In a multicriteria decision problem n alternatives A1; . . . ;An must be ranked by using m cri-

teria C1; . . . ;Cm (generally) of different importance expressed by using the positive numbers w1; . . . ;wm

called the weights of criteria C1; . . . ;Cm, respectively. The values of the alternatives on each criterion can be

organized in the following table:

1 Th

subject

In this table aij > 0 denotes the value of the jth alternative in the ith criterion and xj denotes the aggregatedvalue of the jth alternative taking into account all the criteria. fx1; . . . ; xng is called solution of the mul-ticriteria decision problem. The alternatives are ranked by the values xj, i.e., the larger is xj, the better is Aj.

If the values of the alternatives with respect to the different criteria are not in the same scale range a

scaling should be made. It is not clear yet how this scaling should be made, since different scaling could

produce different results. However, many multicriteria decision methodologies use scaling. Sometimes the

nature of a criterion could give a hint for an appropriate scaling. This topic is not clarified yet and we do

not want to enter in details since the scaling is not the central issue of our paper. Our scaling is based on a

utility function similar to the utility functions used in the Promethee methodology (see [2]). Of course, ourmethod can be used with a different scaling, too. We used this scaling only for the simplicity of the ideas. In

our paper, for calculating xj we shall make the replacement:

aij :¼ 8aij �minfaik : k ¼ 1; . . . ; ng

maxfaik : k ¼ 1; . . . ; ng �minfaik : k ¼ 1; . . . ; ng þ 1;

if the larger values and

aij :¼ 8maxfaik : k ¼ 1; . . . ; ng � aij

maxfaik : k ¼ 1; . . . ; ng �minfaik : k ¼ 1; . . . ; ng þ 1;

if the smaller values are preferred with respect to the ith criterion, respectively. In this way the values of thealternatives with respect to different criteria are scaled in the same interval [1,9]. 1 If the larger values are

preferred, the smallest value with respect to a criterion Ci takes the value 1 and the largest value with respect

to Ci takes value 9. If the smaller values are preferred the situation is the opposite. All the other values of the

alternatives with respect to Ci are disposed proportionally in the [1,9] interval. The scaling is important for

comparing the values of the alternatives with respect to different criteria. With this replacement, the larger is

aij, the better is the alternative Aj with respect to criterion Ci. The aggregated value xj will be a function of the

weights wi and the values aij, where i ¼ 1; . . . ;m. The same function must be taken for every alternative.

e scaling interval [1,9] borrowed from the AHP methodology [17] was proved by psychologists to be the best scaling interval for

ive decisions.

680 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

This function will be called decision function. In order to have a feasible solution, this function shouldsatisfy some natural conditions. These conditions will be the axioms of a decision function and are listed in

the next definition. In Definition 2 f is a function of the weights� vector w ¼ ðw1; . . . ;wnÞ and the alter-

natives� vector a ¼ ða1; . . . ; anÞ.

Definition 2. A function f : Rmþ � Rm

þ ! Rþ is called a decision function if the following axioms are satisfied:

A1. f ¼ f ðw; aÞ satisfies the following homogeneity conditions: f ðkw; aÞ ¼ f ðw; aÞ and f ðw; kaÞ ¼ kf ðw; aÞfor all w; a 2 Rm

þ and k 2 Rþ.A2.

f ðwrð1Þ; . . . ;wrðmÞ; arð1Þ; . . . ; arðmÞÞ ¼ f ðw1; . . . ;wm; a1; . . . ; amÞ;

for all w1; . . . ;wm; a1; . . . ; am 2 Rþ and permutation r of the set f1; . . . ;mg.A3. f is strictly increasing in each variable ai, i ¼ 1; . . . ; nA4.

mini¼1;...;m

ai 6 f ðw1; . . . ;wm; a1; . . . ; amÞ6 maxi¼1;...;m

ai:

Hence, xj will be calculated by the following formula:

xj ¼ f ðw1; . . . ;wm; a1j; . . . ; amjÞ;

where f is a decision function.

One natural class of decision functions are the a power means given by

maðw1; . . . ;wm; a1; . . . ; amÞ ¼Pm

i¼1 wiaaiPm

i¼1 wi

� �1=a

; aP 0:

For example in AHP the arithmetic mean (a ¼ 1) is employed [17].

When subjective values are present in a multicriteria decision problem, it is very important to choose a

decision function that conducts to a fairly stable solution.

Definition 3. By the sensitivity analysis of a multicriteria decision problem we mean the analysis of the

results, when we make small perturbations to the weights wi and/or values aij of a decision table. We call a

solution {x1; . . . ; xn} stable with respect to these perturbations if the decreasing order of the new solution

{x01; . . . ; x0n} coincide with the decreasing order of {x1; . . . ; xn}. This means that there is a permutation r such

that

xr1 P � � � P xrn

and

x0r1 P � � � P x0rn:

3. Sensitivity with respect to alternatives

3.1. General setting

The decreasing order of the solution is not altered by the perturbations if the pairwise order of alter-

natives remains unchanged. The allowable perturbations may differ for the different pairs of consecutive

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 681

alternatives. The stability of the order of two consecutive alternatives is a measure of the allowable per-turbation. The notion of stability of order was introduced in the multicriteria group decision software

WINGDSS 4.1 [3,4,15] developed at the Laboratory of Operations Research and Decision Systems,

Computer and Automation Research Institute, Hungarian Academy of Sciences.

In the case of sensitivity analysis with respect to alternatives, for each i; j 2 f1; 2; . . . ; ng the value of the

jth alternative with respect to the ith criterion will be the interval [aij � eaij; aij þ eaij], where e > 0 and aij isonly an estimated value of the jth alternative with respect to the ith criterion, which is allowed to change etimes its original value. By using interval arithmetics the aggregated value of the jth alternative will be an

interval [x�j ðeÞ, xþj ðeÞ]. The exact mathematical details are presented in Definition 4. For simplicity weconsider only the case where e does not depend of the values i, j. A similar analysis could be done when e isa function of i and j. However, the notations would become rather cumbersome and we try to avoid this.

We must also emphasize that the choice of e is of key importance in our following method. Henceforth the

dependence of the sensitivity analysis on the value e is a very important topic and seems to be a tough but

challenging subject of new research.

Definition 4. Let e > 0. Then for each alternative Aj

2 If

x�j ðeÞ ¼ min f ðw; ~aÞ : ~a 2Y

i¼1;...;m

½aij

(� eaij; aij þ eaij�

)

and

xþj ðeÞ ¼ max f ðw; ~aÞ : ~a 2Y

i¼1;...;m

½aij

(� eaij; aij þ eaij�

)

are the minimum and, respectively, the maximum aggregated value for the allowed perturbations. By

sensitivity analysis with respect to alternative values we mean the computation of these extreme values.

Proposition 1. For any decision function the minimal (maximal) value x�j ðeÞ (xþj ðeÞ) is attained for~a ¼ ðaij � eaijÞi¼1;...;m (~a ¼ ðaij þ eaijÞi¼1;...;m).

2

Proof. The proof follows directly from the strict monotonicity of decision functions (axiom A3). h

With the help of the previously defined stability values we can measure the stability of a solution

(decreasing order of the alternatives).

Definition 5. Let us order the alternatives in decreasing order of the aggregated values xj, given by the

permutation r such that xrð1Þ P � � � P xrðnÞ. The stability of the order xrðjÞ P xrðjþ1Þ is the greatest dj 2 ½0; 1�for which x�rðjÞðdjeÞP xþrðjþ1ÞðdjeÞ. That is, if we allow perturbations of at most dje of the alternative values,the order of alternatives ArðjÞ and Arðjþ1Þ remains unchanged. Hence, a stability value of 1 means that the

ordering is stable with respect to perturbations of at most e. We shall call stability index of a decision

function f (for the fixed decision table) the value

SðeÞ ¼Yn�1

j¼1

dj

! 1n�1

:

any value falls outside the [1,9] interval, it is rounded to the closest endpoint.

682 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

A stability index of 1 means that the solution is stable with respect to perturbations of at most e, and astability index of 0 means the solution is not stable at all. The closer the stability index is to 1, the more

stable is the solution. The reasons for introducing this stability index and the intuitive ideas behind it can be

found in the Introduction (Section 1).

3.2. Using the genetic programming paradigm

We would like to find the most stable decision function corresponding to any given multicriteria decision

problem. This is in fact a special function regression problem: we are looking for some function that (1)satisfies the axioms of a decision function and (2) conducts to a stable order of the alternatives for the given

multicriteria decision problem.

Since we do not know in advance the exact structure of the function we are looking for, the genetic

programming paradigm [1,12] is especially well suited for this task:

• as an evolutionary method, it conducts a multidirectional search in the program space;

• the structure of individuals evolves together with their content;

• if the proper representation and evaluation method are chosen, it is likely to discover solutions that arebetter than the manually created solutions [13].

Genetic programming is an extension of genetic algorithms, where the structures undergoing adaptation

are not strings but hierarchical computer programs of dynamically varying size and shape [1,12].

Genetic programs are built from functions and terminals, referred together as nodes. By definition

[1], the terminal set consists of the inputs of the genetic program, the constants and the eventual

functions with no explicit arguments. At the end of each branch of a genetic tree there is some termi-

nal. The function set consists of the statements, operators, and functions available to the geneticprogramming system. Generally, the set of all possible constructs of any programming language is

very large, thus it is advisable to restrict the function set to functions specific to the domain of the

problem.

Thus, the search space of genetic programming is the space of all trees that can be formed using the

chosen function set and terminal set.

The algorithm of genetic programming systems is the following:

1. Generate an initial population of individuals (computer programs consisting of random compositions offunctions and terminals from the function and terminal set).

2. Execute each program of the population on the fitness cases and assign it a fitness value, using the fitness

measure (i.e., evaluate the genetic program).

3. Create a new population of individuals by using the genetic operators:

• reproduction––copying an individual from the old population into the new population,

• crossover––generating two offspring from two parents by exchanging two random subtrees of the

parents, and

• mutation––randomly changing a node (point mutation) or a subtree (subtree mutation) of the genetictree.

4. Iterate through steps 2–3 until the termination criterion is satisfied (i.e., a solution is found) or a prede-

fined number of generations elapsed. The fittest individual that appeared in any generation is designated

as the result of genetic programming.

Before starting the implementation of any genetic programming system, one should decide what rep-

resentation to use and how to evaluate the genetic programs [12].

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 683

When choosing the representation, we have to take into consideration the fact that only those func-tions of the function space constitute feasible solutions, which satisfy the axioms of a decision func-

tion. Generally, in the cases when only a fraction of the search space represent feasible solutions, one can

select

1. a general representation––where additional correcting operators are needed for the transformation of

any evolved non-feasible individual into the closest feasible one; or

2. a representation encoding only feasible solutions––that excludes the possibility of evolving any non-

feasible individual.

We decided for the second alternative. We construct valid decision function starting from elementary

decision functions, i.e., a power means (a P 0, for a ¼ 0 we obtain in limit the weighted geometric mean).

We use tree based genetic programming, where the rules for constructing a tree are the following:

• Any leave of a tree is an a power mean (aP 0) applied to an alternative�s values (for the given multicri-

teria decision problem).

• Any internal node of a tree is an a power mean (aP 0) with equal weights applied to its descendants.

Proposition 2. If f and g are decision functions then, for every a P 0

3 Th

ha ¼f a þ ga

2

� �1a

is also a decision function.

Proof. The axioms A1 and A2 of a decision function are trivially satisfied for ha. The axioms A3 and A4follow from the component-wise strict monotonicity of the function

R2þ 3 ðx; yÞ7! xa þ ya

2

� �1a

:

Using Proposition 2 recursively, it is easy to see that all the individuals of our tree based genetic pro-

gramming system are decision functions.

On the other hand, by using this representation, we do not cover all possible decision functions. Even so

we can obtain decision functions that are more general than the used elementary decision functions.

The evaluation method follows directly from the sensitivity analysis of multicriteria decision problems.For computing the fitness measure of the evolved decision functions we apply the following algorithm:

1. for each alternative Aj, compute xj, x�j ðeÞ, xþj ðeÞ;2. order the alternatives in decreasing order of the aggregated values:

xrð1Þ P xrð2Þ P � � � P xrðnÞ;

3. for each pair of neighboring alternatives ArðjÞ and Arðjþ1Þ compute the stability value dj;3

4. compute fitness as the stability index Fitness ¼ SðeÞ.

e stability value is found by binary search in the [0, 1] interval with a predefined precision.

684 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

With the defined representation and fitness evaluation method our genetic programming system iscapable of finding decision functions that are more stable than the classical ones (i.e., arithmetic and

geometric mean). The experimental results in Section 5 prove this claim. h

4. Sensitivity with respect to weights

In the previous section we supposed that the weights of the different criteria are constant. Here we

suppose that the values of alternatives with respect to the criteria are constant (scaled into the interval [1,9]similar to the previous section) and the weights are subject to change. We perform a stability analysis with

respect to weights.

Definition 6. For each alternative Aj

x�j ðeÞ ¼ min f ð~w; aÞ : ~w 2Y

i¼1;...;m

½wi

(� ewi;wi þ ewi�

)

and

xþj ðeÞ ¼ max f ð~w; aÞ : ~w 2Y

i¼1;...;m

½wi

(� ewi;wi þ ewi�

)

are the minimum and, respectively, the maximum aggregated value for the allowed perturbations. The

sensitivity analysis with respect to weights consists of finding the box [x�ðeÞ, xþðeÞ], where x�ðeÞ ¼ðx�1 ðeÞ; . . . ; x�n ðeÞÞ and xþðeÞ ¼ ðxþ1 ðeÞ; . . . ; xþn ðeÞÞ.

Definition 7. Let us order the alternatives in decreasing order of the aggregated values xj, given by thepermutation r such that xrð1Þ P � � � P xrðnÞ. Let w ¼ ðw1; . . . ;wmÞ, v ¼ ðv1; . . . ; vmÞ and aj ¼ ða1j; . . . ; amjÞ.The stability of the order xrðjÞ P xrðjþ1Þ is the greatest dj 2 ½0; 1� for which

f ðv; arðjÞÞ � f ðv; arðjþ1ÞÞP 0;

where v 2 ½w � djew;w þ djew�. That is, if we allow perturbations of at most dje of the weights, the order ofalternatives ArðjÞ and Arðjþ1Þ remains unchanged. Hence, a stability value of 1 means that the ordering isstable with respect to perturbations of at most e. We shall call stability index of a decision function f (for

the fixed decision table) the value

SðeÞ ¼Yn�1

j¼1

dj

! 1n�1

:

First consider the sensitivity analysis. By the homogeneity condition with respect to weights the decision

functions are not monotone functions of the weights. Hence we cannot state a proposition for x�j ðeÞ andxþj ðeÞ similar to the one in the previous chapter. However, if our decision function is given recursively as in

Section 3.2 we can use a branch and bound technique for calculating these values [14].

We should emphasize here that we switched to this methodology in the case of weights because we have a

non-linear optimization problem here, contrarily to the case of alternatives where we had a rather

straightforward linear optimization problem on a box. The algorithm for non-linear optimization is much

slower and at the moment it is inefficient for inclusion into the genetic programming system.The idea of the algorithm is to reduce the arising global optimization problem to a monotone optimi-

zation problem which can be solved by branch and bound techniques. Since our decision functions are

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 685

‘‘monotone aggregations’’ of weighted power means, this can be done by a radial projection of the opti-mization domain to the simplex S ¼

Pmi¼1 wi ¼ 1.

Let us fix a 2 Rn and consider the restriction of f ðw; aÞ to the simplex S ¼Pm

i¼1 wi ¼ 1. It is easy to see

that this function can be extended to a monotone function of w on Rm. Denote this function by Fa. Let Ue be

the radial projection (with respect to the origin) of the box We ¼ ½w � ew;w þ ew� to the simplex S. Hence

the sensitivity analysis is reduced to finding the minimum and maximum values of Fa on the set Ue.

We devised an algorithm, which solves these optimization problems. Since Fa is monotone, for each m-dimensional simplex it is easy to construct an upper bound (lower bound) of Fa. If si ¼ ðs1i; . . . ; smiÞ;i ¼ 1; . . . ;m are the vertexes of the simplex, then T ¼ FaðsÞ is an upper bound (lower bound) of Fa on thesimplex where s ¼ ðs1; . . . ; smÞ and sp ¼ maxfspq : q ¼ 1; . . . ;mg (sp ¼ minfspq : q ¼ 1; . . . ;mg); p ¼ 1; . . . ;m.First consider the maximization problem. Through the branch and bound technique we recursively extend a

list of simplexes Sk until a sufficiently precise maximum of Fa on any of these simplexes is found. We sketch

the main steps of the algorithm: 4

1. Let S0 ¼ S and T0 be the upper bound of Fa on the simplex S. Let g be the desired precision for the max-

imum. Let k ¼ 0.

2. Consider that Sl; l ¼ 0; 1; . . . ; k are already constructed so that Sl \ Ue 6¼ /. Let Tl be the upper bound ofFa on Sl. The simplex list is ordered such that T0 6 T1 6 � � � 6 Tk. For each simplex Sl, compute the value

of Fa taken in a point Pl 2 Sl \ Ue. Let Fmax be the maximum of these values.

3. If the found maximum value is close enough to the upper bound, that is, Tk � Fmax < g then the algo-

rithm ends and the maximum is Fmax with precision g. Otherwise the search is continued with the next

step.

4. Divide simplex Sk (i.e., the simplex with the largest upper bound) along the longest edge in two simplexes

S0k and S00

k . Simplex S0k is obtained from simplex Sk by replacing one endpoint of the longest edge with the

middle point of this edge. Simplex S00k is obtained similarly, by replacing the other endpoint of the longest

edge with the middle point of this edge. The upper bounds of Fa on S0k and S00

k are T 0k and T 00

k , respectively.

5. Remove simplex Sk from the simplex list.

6. Add the newly created simplexes to the simplex list according to their intersection with Ue:

(a) If both S0k \ Ue 6¼ / and S00

k \ Ue 6¼ /, insert S0k and S00

k in the list S0; . . . ; Sk�1 in the proper positions

according to the values of the upper bounds T 0k and T 00

k . The simplex list will contain k þ 1 elements

with the corresponding upper bounds T0 6 T1 6 � � � 6 Tk 6 Tkþ1.

(b) If only S0k \ Ue 6¼ / or S00

k \ Ue 6¼ /, denote by S000k the simplex, which has non-empty intersection with

Ue. Then insert S000k in the list S0; . . . ; Sk�1 in the proper position in accordance with the upper bound

T 000k . The simplex list will contain k elements with the corresponding upper bounds T0 6 T1 6 � � � 6 Tk.

7. Repeat steps 2-6 until the termination criterion specified at step 3 is satisfied.

The case of the minimization problem is similar. We only have to change in the above algorithm the

maxima into minima and the upper bounds into lower bounds.

Consider now the stability problem. In this case we have to maximize the function

4 Th

f ðv; arðjþ1ÞÞ � f ðv; arðjÞÞ

on the set Ude and find the largest d for which the maximum is non-positive. If the maximum is found, the

largest d can be calculated using binary search. For solving this maximization problem we can use a similar

algorithm to the presented one. Everything becomes straightforward if we can give an upper bound of

FarðjÞ � Farðjþ1Þ on a simplex. If si ¼ ðs1i; . . . ; smiÞ; i ¼ 1; . . . ;m are the vertexes of the simplex, then it is easy to

e geometrical details are beyond the scope of this article.

686 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

see that T ¼ FarðjÞ ðsÞ � Farðjþ1Þ ðs0Þ is an upper bound where s ¼ ðs1; . . . ; smÞ, s0 ¼ ðs01; . . . ; s0mÞ, sp ¼ maxfspq :q ¼ 1; . . . ;mg and s0p ¼ minfs0pq : q ¼ 1; . . . ;mg for p ¼ 1; . . . ;m.

These algorithms can be fastened by taking random points on the feasible set. Consider the case of

maximization (the minimization case is similar). If an upper bound is smaller than the maximal value of the

function (of the types above) taken in these points, then the simplex corresponding to this upper bound

cannot contain the maximum, thus it is removed from the simplex list.

We have implemented and tested this algorithm. The current version is rather time-consuming and we

are working now on acceleration issues. When adopting the genetic programming system for the stabilityanalysis with respect to weights, this algorithm is incorporated into the evaluation method. Since evaluation

is performed for each individual program in each generation, 5 the genetic programming system can be

adopted only if the algorithm is fairly efficient.

5. Experimental results

In the following we present our findings regarding the stability analysis with respect to alternative values.We conducted experiments on three multicriteria decision problems. In all cases we allowed variations of

up to e ¼ 10% of the alternative values. The experimental parameter setting including control parameters is

summarized in Table 1.

Example 1. Let us consider the problem of buying a TV set. We would like to get the best TV set of certain

dimensions for a given price, but there are 10 different offers. We base our decision on the following

subjective criteria (in increasing order of importance):

C1 aesthetics––The physical aspect influences us: we might like a box with rounded forms, with the speak-

ers on the sides, or a brown, almost cubic box with the speakers at the bottom. We might prefer a col-

orful or artistically designed remote controller to a simple one.

C2 type of offered warranty––It is a subjective decision whether a two years full warranty, a one year full

warranty followed by two years free service, or a one year full warranty followed by two years warranty

on components is more preferable.

C3 confidence in trade-name––The preference for the product of one company over the product of another

company is also subjective.C4 quality of image––The perceived image (brightness, sharpness, colors, etc.) represents the most impor-

tant criterion.

This is a multicriteria decision problem with four criteria and 10 alternatives with the following decision

table (after scaling):

5 E.

perfor

g., for a run of the genetic programming system consisting of 40 generations on a population of 500 individuals, evaluation is

med 20,000 times.

Table 1

The genetic programming parameter setting

Objective Evolve the decision function with greatest stability

Terminal set Random reals a 2 ½0; 40�, being evaluated as the a power mean of an alternative�s valuesFunction set a power means with equal weights,a where a 2 ½0; 40�Fitness cases The alternatives of the multicriteria decision problem

Fitness The stability index computed for the given multicriteria decision problem

Population size 500, 1000, 2000

Crossover probability 90%

Probability of point mutation 10%

Selection method Tournament selection, size 5

Termination criterion None

Maximum number of generations 40, 50

Maximum depth of tree after crossover 10

Initialization method Grow

aWe used equal weights for the simplicity of representation. As we will see in the case of the second example, different weights can

evolve in the form of multiple occurrence of the same subtree (or leaf).

Table 2

Some results given by GP for Example 1

# Stability Size Generation Population size

1 0.163 3 7 500

2 0.164 7 12 500

3 0.167 11 9 500

4 0.168 15 22 500

5 0.167 23 17 1000

6 0.167 19 8 1000

7 0.169 19 17 1000

8 0.166 19 19 2000

9 0.169 11 19 2000

10 0.216 5 12 2000

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 687

In Table 2 we show several results produced by our genetic programming system. Each line shows thecharacteristics of one decision function created by genetic programming in terms of (1) the stability index,

(2) the size given as the number of nodes in the genetic tree, (3) the generation at which the function

emerged and (4) the population size used for the run. With larger populations we obtained slightly more

stable solutions. We performed several runs with population size 4000, but the results were not better than

for population size 2000. The best decision function has stability index Sgpð0:1Þ ¼ 0:216, meaning that the

solution can be stable if the perturbations are reduced on average to 21.6% of the initially allowed per-

turbation e ¼ 0:1.The structure of the best decision functions we found is shown in Fig. 1. Each internal node is evaluated

to the corresponding (equal-weight) a power mean of its descendants. Each leaf is evaluated to the cor-

responding a power mean of the alternative�s values. Thus, the encoded decision function is

f ¼m20:6

þm20:1

2

12

� �10

þ m100:1

2

0B@

1CA

110

:

10

0.6 0.1

0.12

Fig. 1. The most stable decision function found by GP for Example 1.

Table 3

Comparison between the solution given by GP and the solution given by the arithmetic mean for Example 1

Best GP Order A1 A7 A6 A10 A4 A9 A8 A2 A5 A3

xj 4.796 4.602 4.377 4.312 4.277 4.06 3.914 3.754 3.385 3.204

dj 0.213 0.252 0.092 0.06 0.287 0.197 0.24 0.678 0.357

Arithmetic

mean

Order A1 A4 A10 A6 A7 A2 A5 A9 A8 A3

xj 5.4 5.4 5.1 5 4.9 4.7 4.7 4.5 4.3 4

dj 0 0.389 0.1 0.1 0.256 0 0.225 0.252 0.404

688 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

The solution given by this decision function is not very stable (stability would have been achieved forSðeÞ ¼ 1). At the same time, for the same problem the arithmetic mean gives an unstable solution,

Sarithð0:1Þ ¼ 0. In Table 3 the two solutions are shown. The alternatives are listed in decreasing order of the

aggregated values xj. For each solution, we also show the stability value dj for each pair of consecutive

alternatives. In the solution given by the arithmetic mean there are two pairs of alternatives whose ordering

is totally unstable: A1, A4 and A2, A5, respectively.

Thus, the evolved decision function suggests buying TV set A1: not aesthetic, but best trade-name and

above average image quality. The arithmetic mean could not differentiate between TV sets A1 and A4. From

practical considerations, nobody would buy a no-name TV set with seemingly better image quality (A4):over the years the technical parameters of the TV set may drastically get worse, the unknown manufacturer

may disappear, etc. Instead, the product of the preferred reliable company (A1) would be chosen even if

image quality is not so good at first glance.

Example 2. The second example is a multicriteria decision problem with three criteria and six alternatives:

The best decision function produced by genetic programming (in generation 38 of a run with population

size 2000) has stability index Sgpð0:1Þ ¼ 0:81, meaning that the solution can be stable if the perturbations

are reduced on average to 81% of the initially allowed e ¼ 0:1. The evolved tree contains 21 nodes, its

structure is shown in Fig. 2. The multiple occurrence of the same leaf indicates the formation of differentweights for the inner nodes of the tree. In the future we plan to introduce the different weights in the

representation of inner nodes of trees. By doing so we expect genetic programming to produce shorter (and

more comprehensible) trees with similar performance in earlier generations.

For this problem, the stability index of the solution produced by the arithmetic mean is Sarithð0:1Þ ¼0:574. Thus, the decision function given by genetic programming leads to a considerably more stable

0

0

0

0

0

0

0

0

0

35

38

38

30

39

11

19

10

12

11

1.1 0

Fig. 2. The most stable decision function found by GP for Example 2.

Table 4

Comparison between the solution given by GP and the solution given by the arithmetic mean for Example 2

Best GP Order A6 A5 A3 A1 A2 A4

xj 8.069 7.114 5.125 4.519 3.33 3.123

dj 0.697 1 1 1 0.498

Arithmetic mean Order A6 A5 A3 A1 A2 A4

xj 8.1 7.3 6.4 6.2 4.7 4.4

dj 0.588 0.818 0.252 1 0.51

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 689

solution. In Table 4 the two solutions are shown. The alternatives are listed in decreasing order of theaggregated values xj. Although the order of the alternatives is the same in the two solutions, the stability

values are higher for our solution. The evolved decision function relies strongly on the geometric mean, but

its stability is higher, namely Sgpð0:1Þ ¼ 0:81 > Sgeomð0:1Þ ¼ 0:768.

Example 3. The third example is related to buying a complete desktop computer configuration, with both

objective and subjective criteria. The values of alternatives with respect to the objective criteria were fixed

and the stability analysis was performed only with respect to the values of alternatives with respect to the

subjective criteria. However, we have to note that the values of alternatives with respect to the objectivecriteria also influenced stability, since they took part in the aggregation process. We collected 15 alterna-

tives of interest from the world wide web starting from the European shopping search engine http://

www.kelkoo.co.uk. Table 5 contains these alternatives, described by brand name, processor type and speed,

memory capacity, hard-disk capacity, included extras (such as CD-RW, DVD, modem, network card, etc.),

display size and type, operating system, and warranty. Since we do not want to advertise the products of

any specific company, we do not disclose brand name, processor type and operating system of configu-

rations.

Table 5

The alternative configurations for the desktop computer example

Alternative Meaning

A1 Brand Name 1, Processor Type 1––1.7 GHz, 256 MB RAM, 40 GB HDD, 52� CD-ROM, Ethernet card, 1700 CRT

display, Operating System 1, 3 years warranty

A2 Brand Name 1, Processor Type 2––2 GHz, 256 MB RAM, 120 GB HDD, 48� CD-RW, network card, 1900 CRT

display, Operating System 2, 3 years warranty

A3 Brand Name 1, Processor Type 2––2.4 GHz, 512 MB RAM, 40 GB HDD, 16� DVD, network card, 1500 LCD flat

panel display, Operating System 1, 3 years warranty

A4 Brand Name 2, Processor Type 1––1.7 GHz, 128 MB RAM, 20 GB HDD, integrated network interface, 1700 CRT

display, Operating System 2, 3 years warranty

A5 Brand Name 2, Processor Type 2––2.4 GHz, 128 MB RAM, 40 GB HDD, integrated network card, 1700 CRT display,

Operating System 2, 1 year warranty

A6 Brand Name 2, Processor Type 2––2.5 GHz, 256 MB RAM, 80 GB HDD, 1700 CRT display, DVD/CD-RW, modem,

Operating System 2, 3 years warranty

A7 Brand Name 2, Processor Type 2––1.8 GHz, 128 MB RAM, 20 GB HDD, 1700 CRT display, 40� CD-RW, Operating

System 1, 3 years warranty

A8 Brand Name 3, Processor Type 3––1.5 GHz, 256 MB memory, 40 GB HDD, 1500 flat panel display

A9 Brand Name 3, Processor Type 2––2 GHz, 256 MB memory, 40 GB HDD, 1500 LCD display

A10 Brand Name 4, Processor Type 2––2.5 GHz, 512 MB memory, 80 GB HDD, 1700 CRT display, DVD, DVD and CD-

RW, Operating System 3

A11 Brand Name 4, Processor Type 1––1.4 GHz, 256 MB memory, 40 GB HDD, 1500 CRT display, 16� DVD, modem,

Operating System 3, 1 year warranty

A12 Brand Name 5, Processor Type 4––700 MHz, 128 MB memory, 40 GB HDD, 1700 built in CRT display, DVD, CD-

RW, modem, Operating System 4, 2 years warranty

A13 Brand Name 5, Processor Type 4––700 MHz, 128 MB memory, 40 GB HDD, 1500 LCD flat panel display, CD-RW,

modem, Operating System 4, 1 year warranty

A14 Brand Name 5, Processor Type 4––700 MHz, 128 MB memory, 40 GB HDD, 1700 built in CRT display, DVD, CD-

RW, modem, Operating System 4, 2 years warranty

A15 Brand Name 5, Processor Type 4––700 MHz, 128 MB memory, 40 GB HDD, 1500 LCD flat panel display, CD-RW,

modem, Operating System 4, 1 year warranty

690 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

For ranking the alternatives we considered 15 criteria, as shown in Table 6. Each criterion is giventogether with its type, i.e., either objective or subjective.

Table 6

The criteria for the desktop computer buying example

Criterion Meaning Type

C1 Cost (in pounds, including VAT) Objective

C2 Communication: e-mail, tel., store Subjective

C3 Payment: card, cash, cheque Subjective

C4 Warranty Subjective

C5 Availability: now, in 3 days, in one week, on request, etc. Subjective

C6 Brand name Subjective

C7 Physical appearance Subjective

C8 Processor type Subjective

C9 Processor speed Objective

C10 Memory capacity Objective

C11 HDD capacity Objective

C12 Extras: CD-RW, DVD, network card, modem, etc. Subjective

C13 Operating system Subjective

C14 Display size Objective

C15 Display type: CRT, LCD, flat-screen Subjective

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 691

The evaluation of alternatives with respect to the subjective criteria was made by using the followingassignment procedure:

C2 Communication––Depending on the possibilities for contacting the companies selling the configurations

(i.e., alternatives) we gave the following values:

• e-mail: 1

• phone: 2

• e-mail and phone: 5

• shop: 5• e-mail, phone, and shop: 9.

C3 Payment––Depending on the payment possibilities we gave the following values:

• by card on internet: 1

• by card on internet or phone: 3

• by card, cash or cheque: 9.

C4 Warranty––We assigned the following values to different warranty types:

• no information: 1

• one year: 3• two years: 6

• three years: 9.

C5 Availability––We assigned the following values for availability:

• longer than two weeks or no information: 1

• on request: 2

• in two weeks: 3

• in one week: 5

• in three days: 7• now: 9.

C6 Brand name––We gave the following values for the different brand names:

• Brand Name 1: 2

• Brand Name 5: 6

• Brand Name 4: 7

• Brand Name 3: 8

• Brand Name 2: 9.

C8 Processor type––We rated the processor types as follows:• Processor Type 1: 3

• Processor Type 4: 5

• Processor Type 3: 7

• Processor Type 2: 9.

C12 Extras––We gave the following values for the individual extras:

• modem: 2

• network card: 2

• CD-RW: 2• DVD and CD-ROM writer: 3

• DVD: 4.

In the case of a configuration with more than one extra component we added the corresponding values.

C13 Operating system––The operating systems were rated as:

• no operating system provided or no information: 1

• Operating System 2: 5

• Operating System 3: 5

692 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

• Operating System 4: 7

• Operating System 1: 9.

C15 Display type––Our preference for display types was quantified as:

• CRT: 3

• flat-screen CRT: 5

• LCD: 7

• flat-screen LCD: 9.

By using these assignments and by scaling the values of alternatives with respect to the objective criteria

in the interval [1,9] we obtained the following values of alternatives with respect to the considered criteria

(at the left of each criterion its corresponding weight is shown):

The best decision function produced by genetic programming (in generation 50 of a run with population

size 2000, with little improvement being made after generation 15) has stability index Sgpð0:1Þ ¼ 0:322,meaning that the solution can be stable if the perturbations are reduced on average to 32.2% of the initially

allowed e ¼ 0:1. The evolved tree contains 7 nodes, its structure is shown in Fig. 3. The encoded decision

function is

f ¼

m0 þ m20þ

m60þm6

0:32

16

� �2

2

0@

1A

12

2:

For this problem, the stability index of the solution produced by the arithmetic mean is Sarithð0:1Þ ¼ 0:18.Thus, the decision function given by genetic programming leads to a more stable solution. In Table 7 thetwo solutions are shown. The alternatives are listed in decreasing order of the aggregated values xj. Theevolved decision function relies strongly on the geometric mean, but its stability is higher, namely

Sgpð0:1Þ ¼ 0:322 > Sgeomð0:1Þ ¼ 0:302.For this example, we performed a number of runs with tree sizes in the initial population larger than in

the previous experiments. We noticed that over the generations, the trees tended to become smaller without

any control of tree size growth. This seems to be in contrast with the general experience in genetic pro-

Table 7

Comparison between the solution given by GP and the solution given by the arithmetic mean for Example 3

Best GP Order A6 A3 A2 A11 A10 A5 A1 A7 A14 A4 A12 A15 A8 A9 A13

xj 5.41 5.17 4.96 4.73 4.48 4.20 4.16 4.09 3.74 3.62 3.40 3.38 3.22 3.20 3.07

dj 0.53 0.42 0.48 0.50 0.69 0.10 0.21 0.74 0.29 0.56 0.06 0.44 0.10 0.49

Arithme-

tic mean

Order A3 A6 A2 A10 A11 A7 A5 A1 A14 A9 A12 A4 A8 A15 A13

xj 6.13 5.86 5.6 5.54 5.31 5.09 4.97 4.83 4.45 4.44 4.43 4.39 4.25 4.11 4.09

dj 0.47 0.54 0.11 0.47 0.47 0.22 0.34 0.63 0.02 0.02 0.08 0.29 0.27 0.04

0

0

1

6

0 0.3

2

Fig. 3. The most stable decision function found by GP for Example 3.

A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695 693

gramming that trees tend to grow over generations. The explanation is that after a few generations, somesmall individual was much better than the larger individuals, therefore got selected, and gained authority in

the population. All individuals in the population became similar to this individual, i.e., population diversity

was lost. Our recent results [8] suggest that diversity of populations should be maintained at different stages

of evolution:

• in the initial phase, diversity should be kept at a high level, in order to find good starting points in the

program space;

• in the middle phase, as fitness improves, diversity may be allowed to lower; and• in the final phase, a much lower diversity should be allowed, since this is the time for convergence.

We plan to investigate the effects of diversity maintenance on the present problem.

6. Conclusions

We presented here novel methods for the sensitivity and stability analysis of multicriteria decisionproblems with respect to alternatives and weights. In the case of alternative values, we created a genetic

programming system for generating new decision functions that are more stable than the classical ones.

We think that in the future stability analysis should play a more important role in multicriteria decisions

making. Small changes in the assignments of decision makers should not be allowed to crucially change the

outcome of a problem. Therefore the decision makers should be bound to give the weights and values for

subjective criteria. They should not interfere in the choice of decision functions. At the moment it seems

that the process of choosing a good decision function is a very complicated matter, beyond human

capabilities. That is why we chose an automated process driven by the power of genetic programming. Wemust emphasize that we are dealing with very complex problems, whose irresponsible simplifications could

lead to very dangerous outcome in real world applications. Despite that at the moment the stability analysis

694 A. Ek�art, S.Z. N�emeth / European Journal of Operational Research 160 (2005) 676–695

for weights is too slow to be introduced in the genetic programming system, we think that it can beaccelerated to a reasonable extent. The increasing speed of computer processors (in every few years the

speed grows with an order of magnitude) gives even more support to our hopes.

We also emphasize that there are projects where the precision of results is more important than the speed

(in many cases the evaluation is allowed to take days, weeks, or even months).

In the case of weights, we devised a special algorithm for computing stability. The current implemen-

tation needs to be accelerated before it can be included in the genetic programming system.

We further plan to realize a joint stability analysis for alternative values and weights. There is no lim-

itation of such a joint stability analysis, it is only a matter of technicality. For fractional linear functions ajoint sensitivity analysis was made in the WINGDSS 4.1 software [4].

We also plan to make a more general stability analysis where only a number of alternatives are of

interest.

Moreover we propose to introduce a very general global stability index which, as far as we can see,

includes all interesting situations for stability analysis. This index will be a weighted a-power mean of the

local stability indexes of each pair of alternatives. For a ¼ 0 and equal weights, we obtain the global

stability index defined in this paper. By putting zero weights in the suitable places we can also obtain the

cases where only a number of alternatives are considered.Another important issue is finding the a power mean with the best stability among all the a power means.

It seems that genetic programming is able to develop decision functions that are more stable than this apower mean.

We compared the stability of decision functions generated by genetic programming with that of the

arithmetic mean. By doing so, we actually compared our methodology with the classical analytic hierarchy

process (AHP) methodology. Our methodology can be viewed as an extension of AHP where the aggregation

is performed through a tree structured function. The case studies show that this generalization gives con-

siderably better stability and has the potential for further improvements. The Promethee methodology, al-though containing aggregations based on the arithmetic mean, is very different from AHP. We need to

analyze whether the Promethee methodology remains sound if we exchange the arithmetic mean with our tree

structured decision function. Comparisons and extensions to other methodologies might also be possible.

It would be also interesting to study more general decision functions (i.e., aggregating functions). Our

aim is to solve the open problems for the class of functions presented here. Afterward we could move on to

study more general functions.

Acknowledgements

The authors acknowledge the support of the School of Computer Science at the University of Bir-

mingham, where a substantial part of this work was carried out. This work was supported by grant no.

T029572 of the National Research Foundation of Hungary. S.Z. N�emeth was supported by the Bolyai

J�anos Research Fellowship. The authors are grateful to J�anos F€ul€op and Csaba M�esz�aros for many helpful

discussions. Many thanks are due to the anonymous reviewers whose important comments helped

improving the clarity of the paper.

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