NORMALIZATION PROCEDURES ON MULTICRITERIA

DECISION MAKING

An Example on Environmental Problems

Ethel Mokotoff, Estefanía García and Joaquín Pérez

Departamento de Fundamentos de Economía e Hª Eª, Universidad de Alcalá, Alcalá de Henares, Spain

Keywords: Strategic Decision Support System, Multicriterio Decision Making, Normalization Procedures, Sustainable

Development.

Abstract: In Multicriteria Decision Making, a normalization procedure is required to conduct the aggregation process.

Even though this methodology is widely applied in strategic decision support systems, scarce published

papers detail this specific question. In this paper, we analyze the results of the influence of normalization

procedures in the weight sum aggregation in Multicriteria Decision problems devoted to sustainable

development.

1 INTRODUCTION

Multicriteria Decision Making (MDM) methodology

has reached a high level of maturity and its

applications pervade nowadays to almost every field

of human activity. In fact, there is a growing demand

for systems specifically designed to support such a

kind of analysis, even by casual users who do not

have a deep understanding of its theoretical

foundations.

This situation is especially true concerning the so

called Discrete Multicriteria Decision Making

problem, i.e. the branch of MDM devoted to

problems where there are a finite, and usually small,

number of alternatives competing for one to be

finally selected or which have to be ranked.

Problems of this kind are everywhere: selection of

research projects, biddings to a public contest,

candidates for a job, locations of a new facility,

investments, etc. In the last few years, the increasing

concern on environmental problems has created a

need of including these issues and developing more

effective decision tools.

Whatever the problem in question is, the criteria

used to evaluate alternatives usually respond to

different issues. In particular, dealing with

sustainable development problems, the selected

criteria respond to questions not just economic but

also social and ecological. Thus, original data are

not measurable by the same units for the whole set

of criteria, and a normalization procedure (that

converts all the criteria values into non-dimensional,

i.e. comparable quantities) is required to make

possible the aggregation procedure. MDM exercises

use sometimes a particular normalization procedure

regardless the influence of this procedure on the

results. Therefore, the objective of this article is to

point out the fact that prior normalization of data is

not neutral, and, more important, the final ranking of

alternatives may well depend on the normalization

procedure used.

We have conducted an experiment comparing the

results obtained by varying the normalization

procedure in a real environmental application

(Pasanen, et al., 2005, and Hiltunen, et al., 2009).

We have employed the weighted sum method

included in the SMC package (Barba-Romero and

Mokotoff, 1998).

This paper is organized as follows. Section 2

briefly introduces the theoretical foundations of

MDM and the normalization procedures. Section 3

presents the example and the experiment that serves

to thoroughly illustrate the influence of the

normalization procedure. Section 4 provides some

concluding remarks.

206

Mokotoff E., García-Vázquez E. and Pérez Navarro J. (2010).

NORMALIZATION PROCEDURES ON MULTICRITERIA DECISION MAKING - An Example on Environmental Problems.

In Proceedings of the 12th International Conference on Enterprise Information Systems - Artiﬁcial Intelligence and Decision Support Systems, pages

206-211

DOI: 10.5220/0002896102060211

 SciTePress

2 MDM: THEORETICAL BASIS

2.1 MDM Matrix

In any multicriterion analysis, the first step is the

information gathering phase that will apply to the

whole of the problem at hand, involving a survey of

the criteria and possible alternatives. Then, the

design phase consists of constructing the choice sets,

i.e. the alternatives, a finite and discrete set, in this

case.

Let us suppose there are m alternatives,

constituting the choice set A = {A

, A

,…, A

} and n

criteria, C

, C

,…, C

. Thus, an mxn matrix of

evaluations, [a

], characterizes a MCDM instance.

Each line of the matrix expresses the performance of

the alternative A

according to the n criteria, while

each column, C

, expresses the evaluations of all the

alternatives according to the criterion C

Leaving aside the problem of how to construct

the criteria proper, it is not an easy matter to

evaluate each alternative A

relative to a given

criterion C

, to obtain a coefficient a

. In this paper,

we assume that these evaluations are known with

certainty.

Each of the referred criteria is originally

measured in its inherent unit (even we can have not

only numerical attributes). Thus, the MDM matrix

may presents evaluations of different nature. These

evaluations must be aggregated, taking into account

the preferences of the decision maker, to achieve a

global evaluation value for each alternative, on

which the overall ranking is based.

In our study, we consider the most widely known

aggregation method, i.e. the weighted (linear) sum,

also known as simple additive weight, whose main

advantage is that it is both, intuitive and simple to

apply. Although the method is very simple, we must

nevertheless carefully specify the starting data and

the transformation it undergo.

Weighted sum is a compensatory method.

Compensatory aggregation methods require the

different criteria evaluations and weights to be

settled down in compatible scale. This means that a

normalization procedure has to be executed to

transform figures of the matrix on a comparable

scale.

We suppose that the evaluations, a

, result from

the nature of the attribute that criterion C

measures

on a numerical scale. We also assume that the

decision maker’s preferences can be stated by means

of positive weights, w

, which should be associated

to each criterion, C

For weights there is no problem because the

normalization is achieved making their sum equal to

1, dividing each weight, w

, by

∑

With respect to the evaluations, two classical

normalization procedures are considered for

comparison in the present study: proportionality

preservation and natural thresholds, which are

briefly described below.

For a given criterion C

, a normalization

procedure transforms the evaluations of the m

alternatives, (a

, a

,…, a

), into a new vector,

, v

,…, v

), where v

is the normalization of a

Without loss of generality, we assume that all

criteria are going to be maximized and that values of

are strictly positive (since, in the example of this

paper indeed it is).

2.2 MDM Matrix Normalization

Procedures

2.2.1 Proportionality Preservation

This procedure transforms the evaluation vector,

, a

,…, a

), of each criterion, C

, into a

normalized one by making

∀= ,

max

(1)

Therefore, for each criterion, C

, the normalized

value of the best alternative is 1, and all the rest are

percentages of the maximum value, resulting in the

interval 0<v

≤1, (we assume a

>0).

It is the most widely used normalization

procedure. The main advantage of the method is that

the original proportion existing between the

evaluations of every pair of alternatives is preserved

after normalization, i.e. a

i´j

is equal to v

i´j

. This

is a very desirable property in many circumstances,

but it is not trivial to obtain, especially when dealing

with minimizing criteria. Even it may be impossible

to apply when there are evaluations with a zero

value or with different signs, because proportionality

is then not defined. The drawback of the result

vector is that the evaluations obtained by this

procedure are not forced to cover the complete

interval [0, 1].

2.2.2 Natural Thresholds

This procedure transforms the evaluation vector,

, a

,…, a

), of each criterion, C

, into a

normalized one by making

NORMALIZATION PROCEDURES ON MULTICRITERIA DECISION MAKING - An Example on Environmental

Problems

207

∀

−

= ,

minmax

min

(2)

Therefore, for each criterion, C

, the normalized

value of the best alternative is 1, while the

normalized value of the worst alternative is 0. The

rest ones take values 0 ≤ v

≤ 1, (if a

should take the

same value ∀i, then v

=1, ∀i).

The main advantage of the method is that it

ensures that the evaluations cover the entire range

[0, 1], through a simple linear interpolation between

the extreme points. If the criterion is to minimize,

the transformation is inverse, in an obvious way.

This procedure respects cardinality but it does not

preserve proportionality.

2.3 Aggregation

Once coefficients and weights have been

normalized, for each alternative, A

, the global

evaluation is computed as follows

∑

ijji

vwAGE

)(

(3)

Alternatives are then ranked in descending order of

their global evaluation values. In case of ties, the

rank average of Kendall is applied.

3 ILLUSTRATIVE EXAMPLE OF

THE INFLUENCE OF THE

NORMALIZATION

PROCEDURES

3.1 Model

To illustrate the normalization problem we have

chosen the example presented by Pasanen et al.

(2005), named MESTA, to provide support to

landowners in the forest planning process. The

model considers four alternative forest plans for 10

years:

 A

: Status Quo

 A

: Cuttings

 A

: Recreation

 A

: Nature Protection

The forest owners have different objectives as

regards forest utilisation. Their individual owner

goals will be not only including economic, but

ecological, and social aims too. The model proposes

the following five criteria to evaluate the alternative

plans:

 C

: Old forest area (%): Percentage of land

preserved to old-growth forest conservation. It

captures biodiversity values including

endangered species protection.

 C

: Cutting removal (1000 m

): Timber

extraction.

 C

: Scenery forests (ha): Land area preserved to

landscape and recreation activities.

 C

: Job opportunities (men/years): Local

employment in any of the alternative plans.

 C

: Turnover (mill €): Monetary return from the

various activities in the area.

Clearly, C

and C

are economical criteria. C

is a

pure ecological criterion. C

is a social criterion, and

social sustainability aspects can also be found in the

recreation and landscape criterion, C

Table 1 presents the decision matrix with the

corresponding evaluations that establishes the

correspondence between alternative plans and

criteria. (The matrix data has been extracted from

www.metla.fi/hanke/3292/metsauunnittelu/). It is

easy to realize, from the figures on this matrix, that

starting from the status quo alternative, A

, the

different values in corresponding evaluations are

based on the objectives each plans pursued. Thus, A

can be named as an economic option, A

as a social

one, and A

as a conservative or ecological plan.

Table 1: MESTA Decision Matrix.

ALT/CRIT C

27.2 749 138504 350 33

26 984 122213 440 42

29.3 535 138504 269 25

32 156 138110 124 10

3.2 Experiment

For a better understanding, we have organized the

criteria into three different groups: Environmental

Conservation, Economic and Social criteria. This

allows us to make a balanced allocation of weights

among these three "super-criteria", assigning 0.33 to

each of them. Within each group, weights had been

distributed the way we subjectively consider most

appropriate. (We decided not to include the

sensitivity analysis of weights in this paper, by

limited extension thereof). These model data are

then completely determined (see Table 2).

Obviously, these evaluations must be converted to

comparable units in order to get the final

aggregation result.

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At this point, we have employed the SMC because it

offers the possibility to choose the normalization

procedure and the aggregation method. We have

chosen proportionality preservation and natural

threshold procedures because of their automatism,

without necessity to set up any parameters. As

aggregation method, we have chosen the weighted

sum method because it probably is the best to clearly

show the essence of evaluation in MDM.

3.2.1 Proportionality Preservation

This normalization procedure converts evaluations

into numbers in the (0, 1] interval. Table 3 shows the

normalized evaluations for the model we present.

When preserving proportionality, transformed

figures maintain the original dispersion. The best

alternative always presents 1, while 0 does not

appears (unless an original evaluation is null).

Figure 1 shows results after computing global

evaluations by the weighted sum method. The final

order of the alternatives turned out to be, A

> A

Figure 1: Ranking and Global Evaluations computed by

Proportionality Preservation.

3.2.2 Natural Thresholds

Table 4 shows the normalized evaluations using

Natural Threshold. We can observe that, regardless

of the dispersion of the original figures, each

criterion numbers are distributed along the closed

interval [0, 1]. There are 0 and 1 evaluation values,

corresponding to the worst and best alternatives, for

all criteria.

Aggregated evaluations give the results showed in

Figure 2. Alternatives are now ordered as A

> A

, though there is no ties, we can realize that

differences between A

and A

are negligible, even

is quite close by A

and A

, however, A

notably the most underprivileged.

In the example the social and ecological alternatives,

and A

, respectively, present relatively good

evaluations with respect to C

and C

When

Figure 2: Ranking and Global Evaluations computed by

Natural Thresholds.

proportionality is preserved for all criteria, the social

and ecological alternatives will never be well ranked

because, keeping proportionality on values from

criteria C

and C

, where data are sparsely dispersed,

makes negligible the differences between the values

of these attributes.

After proving the great influence of the

normalization procedure on results (global

evaluations and ranking of the alternatives), we have

essayed two other possible normalization schemes,

which emerged from the analysis of each criterion

and the corresponding figures.

3.2.3 Normalization Procedures According

to Each Criterion

In this model, an alternative is a possible plan to be

carried out by the owner of a small portion of land.

Although each alternative plan that one individual

owner can implement will directly generate a certain

amount of cuttings, income and job opportunities,

the incidence of her own decision on the global

environment can be quite small. That is why, Nature

Protection plan, A

, is not significantly differentiated

from the rest (not even Cuttings option, A

) when

considering Environment Protection criteria.

Something similar occurs with the Recreation

option, A

, and the Scenery Forests criterion.

To prevent loss of discrimination between

different plans in Old Forest Area and Scenery

Forests, we have applied the Natural Threshold

procedure, while the other three criteria have been

normalized by the Proportionality Preservation

procedure.

Results for this model are shown in Table 5 and

Figure 3. Alternatives are now ordered as A

> A

, We can realize that global evaluations are

still less disperse. Even with this normalization, A

notably the most underprivileged. A

and A

are now

ranked in better position than before.

NORMALIZATION PROCEDURES ON MULTICRITERIA DECISION MAKING - An Example on Environmental

Problems

209

Table 2: Model with original evaluation and weights.

Weights 0,33 0,33 0,06 0,27 0,33 0,22

0,11

0,33

ALT/CRIT C

Environment C

Economics C

Social

27,2 - 749 33 - 350 138504 -

26,0 - 984 42 - 440 122213 -

29,3 - 535 25 - 269 138504 -

32,0 - 156 10 - 124 138110 -

Table 3: Model with evaluations normalized by Proportionality Preservation.

Weights 0,33 0,33 0,06 0,27 0,33 0,22

0,11

0,33

ALT/CRIT C

Environment C

Economics C

Social

0,850 - 0,761 0,786 - 0,796 1,000 -

0,813

1,000 1,000

1,000 0,882

0,916 - 0,544 0,595 - 0,611 1,000 -

1,000 - 0,159 0,238 - 0,282 0,997 -

Table 4: Model with evaluations normalized by Natural Thresholds.

Weights 0,33 0,33 0,06 0,27 0,33 0,22

0,11

0,33

ALT/CRIT C1 Environment C

Economics C

Social

0,200 - 0,716 0,719 - 0,715 1,000 -

0,000 - 1,000 1,000 - 1,000 0,000 -

0,550 - 0,458 0,469 - 0,459 1,000 -

1,000 - 0,000 0,000 - 0,000 0,977 -

Table 5: Model with C

and C

normalized by Natural Thresholds, and C

, C

and C

by Proportionality Preservation.

Weights 0,33 0,33 0,06 0,27 0,33 0,22

0,11

0,33

ALT/CRIT C1 Environment C

Economics C

Social

A1 0,200 - 0,761 0,786 - 0,796 1,000 -

A2 0,000 - 1,000 1,000 - 1,000 0,000 -

A3 0,550 - 0,544 0,595 - 0,611 1,000 -

A4 1,000 - 0,159 0,238 - 0,282 0,976 -

Table 6: Model with evaluations normalized by Satiation Thresholds.

Weights 0,33 0,33 0,06 0,27 0,33 0,22

0,11

0,33

ALT/CRIT C1 Environment C

Economics C

Social

0,272

0,749 0,330

0,350 0,770

0,260

0,984 0,420

0,440 0,444

0,293 - 0,535 0,250 - 0,269 0,770 -

0,320 - 0,156 0,100 - 0,124 0,762 -

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3.2.4 Satiation Thresholds

In this procedure, thresholds are not automatically

determined, but rather we have to set them up.

Indeed, it has been originally developed to avoid the

“irrelevant alternatives dependence” effect (Barba-

Romero and Mokotoff, 1998). Thresholds can be

fixed independently of the evaluations values. In this

case, we have essayed settling down a wide range.

This way, there are neither 0 nor 1 evaluation values,

as we can see in Table 6. Results show (Figure 4)

the same ranking as when proportionality

preservation is applied.

Figure 4: Ranking and Global Evaluations computed by

Satiation Thresholds.

4 CONCLUSIONS

Regarding normalization procedures, there is no

doubt about the relevance of preserving the original

proportion existing between the evaluations of every

pair of alternatives. However, we have observed

that, when the evaluation values for a criterion are

not widely dispersed, maintaining proportionality

(by applying proportionality preservation as

normalization procedure) implies that the

normalized evaluation vectors remain with the same

dispersion and, therefore, it does not help to

differentiate alternatives. When evaluation values of

different alternatives are very close together, it is

possible to gain dispersion, applying natural

thresholds normalization. In this way, the

normalization procedure helps to distinguish

alternatives with apparently similar attribute values.

We can conclude that the decision maker may

miss the importance of choice of a criterion under

certain normalization procedures. It is unrealistic to

hope for general normalization procedures that

performs equally well for different type of criterion.

The analysis of the results obtained by this

experiment give support to the hypothesis which

states that the normalization procedure should be

specially chosen in accordance with every criterion

in a MCDM model.

Concerning to the preferences, we can claim that,

to request the decision maker to express the criterion

weights, disregarding the normalized evaluation

values, makes the decision process not valid.

ACKNOWLEDGEMENTS

This research was supported by the Research

Projects ECO2008-05895-C02-02, Ministerio de

Ciencia e Innovación, Spain.

REFERENCES

Barba-Romero, S., and Mokotoff, E. (1998). A System to

Support Discrete MultiCriteria Evaluations and

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International Multiconference. Computational

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Hammamet, Túnez.

Hiltunen, V., Kurttila, M., Leskinen, P., Pasanen, K., and

Pykäläinenet, J. (2009). Mesta: An internet-based

decision-support application for participatory

strategic-level natural resources planning. Forest

Policy and Economics 11, 1–9.

Pasanen, K., Kurttila, M., Pykäläinen, J., Kangas, J., and

Leskinen, P. (2005). Mesta: nonindustrial private

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