*5.1.
Introduction*

**Multiple criteria decision-making (MCDM) **refers
to making decisions in the presence of multiple - usually conflicting criteria.
Problems for multiple criteria decision making are common occurrences in
everyday life and many examples can be found.

Examples
of MCDM

Decision is understood as choice of one variant
from certain list of real possible variants. This chosen variant is considered
to be optimal but often designation what is optimal is the most difficult
part of problem. We have to take in mind that many criteria are contradictory
and interests of various subject of decision making can be in conflict
(shop owner x customer). Even one person cannot some times make his/her
decision without inner conflict.

Another problem is quantification of available information.
Some input information are quantitative (price of new shoes – we get precise
number expressed in money units) and some kinds of information are not
directly quantitative and must be transferred into numbers (beauty, modishness,
durability etc.) There are many methods for doing that but each has advantages
and disadvantages and certain influence of subjective opinion is not excluded.

Example:
School Grades

Belief in quantitative data generally depends on
local tradition and type of solved problem. Many people still prefer intuition
and experience. The true is that there are no methods (and never will be)
for algorithmisation of whole decision process but using quantitative methods
can lay aside big mistakes (dominated variants – see latter).

The theory on mathematical modelling, but some parts
don’t require deep mathematical knowledge and could be study independently.
Especially exploitation of some software is simple. On the other hand deeper
knowledge of MCDM methods including mathematical background leads to more
serious access and better interpretation of results. This text contains
classification of MCDM methods, characteristic of each class of methods.
Detail description is supplemented only to several best known and widely
used methods.

The aim of studying this text is not to be familiar
with each existing method (impossible due to high number of methods and
continual development of new), but understanding of basic principles and
ability to use some methods (via software) and explain results.

*5.2.
Basic Concepts*

All MCDM problems have the following common characteristics:
**Multiple objectives/attributes**

Each problem has multiple objectives/ attributes.
A decision-maker must generate relevant objectives/attributes for each
problem setting.
**Conflict among criteria**

Multiple criteria usually conflict with each other
(for example the quality and price of goods).
**Incommensurable units**

Each objective/attribute has a different unit of
measurement. In the car selection case, gas mileage is expressed by miles
per gallon (MFG), comfort is by cu ft if fit is measured by passenger space,
safety may be indicated in a nonnumeric way, cost is indicated by dollars,
etc.
**Design/selection**

Solutions to these roblems are either to design
the best alternative or to select the best one among previously specified
finite alternatives.

There exist two alternative sets due to the different
problem settings: one set contains a **finite number of elements** (alternatives),
and the other has an** infinite number**.

The problems of MCDM can be broadly classified into
two categories in this respect:

Multiple Attribute Decision Making - **MODM **(in
this chapter)

Multiple Objective Decision Making – **MADM **(in
following chapter)

Multiple objective decision-making (MODM) is not associated with the problem where the alternatives are predetermined. The thrust of these models is to design the 'best' alternative by considering the various interactions within the design constraints which best satisfy the decision maker by way of attaining some acceptable levels of a set of some quantifiable objectives.

The common characteristics of MODM methods are that
they possess:

1) a set of quantifiable objectives

2) a set of well defined constraints

3) a process of obtaining some trade-off information,
implicit or explicit, between the stated quantifiable objectives and also
between stated or unstated nonquantifiable objectives.

MODM is associated with design problems (in contrast
to selection problems for the MADM).

In MADM, there are usually a limited (and countable) number of predetermined alternatives. The alternatives have associated with them a level of the achievement of the attributes (which may not necessarily be quantifiable) based on which the final decision is to be made. The final selection of the alternative is made with the help of inter- and intra-attribute comparisons. The comparisons may involve explicit or implicit trade-off.

**5.2.1. Terms
and Definitions**

**Criteria -**
A criterion is a measure of effectiveness, basis for evaluation.
**Goals **– A priori values or levels
of aspiration. These are to be either achieved or surpassed or not exceeded.
**Attributes **– Each alternative can
be characterised by a number of attributes = characteristics = parameters
= factors = properties.
**Objectives **- Something to be pursued
to its fullest, generally direction of planned change. For example the
farmer wants to maximise his profit or minimise production costs or minimise
the water pollution.
**Decision matrix **– The MADM problem
can be expressed in a matrix containing criteria in columns and variants
in lines. Elements x_{ij} indicate evaluation of alternative i
with respect to criterion j.

Hence variant a_{i}, i= 1,2,...m

Is denoted by line vector

x_{i}=(x_{i1},x_{i2},…,x_{in})

and column vector

x_{j}=(x_{1j},x_{2j},…,x_{mj})

shows the contrast of each alternative
with respect to attribute f_{j}.

(5.1)

Example:Washing
machines - Decision Matrix

**Transformation to All Maximal Criteria
**-
Many MADM methods suppose that all criteria are maximising. To be in line
with this demand we must transfer minimising criteria to maximising. For
example minimising criterion ”Price” we can transfer to maximising criterion
”Saved money”. The most expensive variant will have zero in the ”Saved
money” column and other – cheaper variants – the value one can save comparing
with the most expensive variant.

Example:
Washing Machines – Criteria Transformation

**Dominated variant** – Variant
is dominating variant

(dominated variant) if

(5.2)

Graphical Presentation of MADM Model

Graphical Presentation of Nondominated Variants

Graphical Presentation of Dominated Variants

Example: Washing Machines – Graphical

These models do not permit trade-offs between attributes. An advantage or favourable value in some other attribute cannot offset a disadvantage or unfavourable value in one attribute.

Hence comparisons are made on an attribute-by-attribute basis. The MADM methods that belong to this model are credited for their simplicity. They are dominance, maximin, maximax, conjunctive constraint method, disjunctive constraint method, and lexicographic method.

Compensatory models permit trade-offs between attributes. That is, in these models changes (perhaps small changes only) in one attribute can be offset by opposing changes in any other attributes. With compensatory models a single number is usually assigned to each multidimensional characterisation representing an alternative. Based upon the principle of calculating this number, compensatory models can be divided into three subgroups.

Example:Washing Machines – Subgroups of MADM Models

**5.2.2.
Normalisation**

A normalisation may be essential for some
methods to facilitate the computational problems inherent to the presence
of different units in decision matrix. The advantage of normalisation is
that all criteria are measured in dimensionless units and inter-attribute
comparison is possible. The normalisation procedure doesn’t lead to equal
minimum and maximum values for all attributes, so straightforward comparison
is still difficult.

There are different methods of normalisation,
some of then will be mentioned together with particular MADM method.
**Vector normalisation**

This procedure implies that each row in
decision matrix is divided by its norm. The elements of normalised matrix
R are calculated:

(5.3)

Another possibility is to divide the outcome of certain criterion by its maximum value:

(5.4)

**5.2.3. Weights
Assessing**

Many methods require information about the relative importance of each attribute (or objective). It is usually given by a set of (preference) weights, which is normalised to sum to 1. In case of n criteria, a set of weights is

(5.5)

There are several techniques for designation the criteria
weights. The main difference is if we collect the judgements from one or
more persons.
**Sequence Method**

The decision-maker should be able to arrange criteria
according their importance to a sequence from most to least important.
A natural numbers from k to 1 are assigned to criteria (most important
criterion number k, second most important k-1 etc.). The weights are calculated
as follows:

*Sum of b _{i} is equal to sum of first
k natural numbers*

(5.6)

This methods counts with judgement of criteria importance by score. The decision-maker evaluates each criterion by certain number of points from chosen scale ( for example 1-100). More important criterion has higher score ( more points). The weight calculations are than made according (5.6).

Example: Washing Machines – Assessing Weights

**Saaty’s Matrix**

The decision-maker is supposed to judge the relative importance of each pair of criteria. The elements of matrix are considered to be an estimation of importance of i-th and j-th criteria. The scale 1,3,5,7,9 is often used.

Verbal Explanation of the Scale

(5.7)

This matrix is called Saaty’s matrix. This is a 'reciprocal
matrix’ that has all positive elements and has the reciprocal (a) and consistency
(b) property:
(5.8)

Saaty introduced a method of scaling ratios using the
principle eingenvector of positive pairwise comparison.
Eingenvector method

(5.9)

Consider the elements sij of Saaty's matrix S in (5.6). It is desired to determine the weights w

The estimations of weights are designated by minimisation of the quadratic form:

(5.10)

Solution of (5.10) is normalised geometrical average
of lines of matrix S:
(5.11)

Numerical
Example Weighted Least Square Method for Assessing Weights
**LINMAP**

LINMAP (Linear Programming Techniques for Multidimensional
Analysis of Preference) is a method for assessing the weights of attributes
as well as for selecting the alternative.

LINMAP

*5.3.
Methods for Multiple Attribute Decision Making*

**5.3.1. Classification
of MADM Methods**

The available input information varies. One may not indicate his preferences at all, or may represent his preference through the form of attribute or alternative. The degree of judgement skill also varies. For instance, we may list the different preference information on attributes by the ascending order of complexity: standard level, ordinal, cardinal, and marginal rate of substitution.

We classify methods for MADM based upon different forms of preference information from a decision-maker.

**1. Class – Methods of No Information Given**

Dominance

Maximin

Maximax

**2. Class – Methods for Information on Attribute
Given**
**2.1. Methods for Standard level of Attribute
Given**

Disjunctive Method

Conjunctive Method

**2.2 Methods for Ordinal of Attribute Given**

Lexicographic Method

Elimination by Aspects

Permutation Method

**2.3 Methods for Cardinal Information of Attribute
Given**

Linear Assignment method

Simple Additive Weighting Method

Hierarchical Additive Weighting Method

ELECTRE

TOPSIS

**2.4 Methods for Marginal Rate of Substitution
of Attribute Given**

Hierarchical Trade-offs Method

**3. Class - Methods for Information on Alternative
Given**

**3.1 Methods for Pairwise Preference Information
Given**

LINMAP

Interactive Simple Additive Weighting Method

**3.2 Method for Order of Pairwise Proximity Information
Given**

Overview
of MADM Methods

Description of some of these methods follows.

If there is no information on criteria importance
we have to use one of the method of 5.4 chapter.

If preference information either on attributes or
alternatives is available, the information can be expressed in various
ways:

1) standard level of each attribute (chapter 5.5)

2) relative importance by ordinal preference (chapter
5.6)

3) relative importance by cardinal preference (chapter
5.7)

4) marginal rate of substitution between attributes

*5.4.
Methods without Preference Information*

**5.4.1. Dominance**

An **alternative is dominated** if there is another
alternative, which excels it in one or more attributes and equals it in
the remainder. Eliminating the dominated ones can reduce the number of
alternatives.** **A set of nondominated solutions is one obtained through
the sieve of dominance method. There are no assumptions so we don’t need
to transfer minimising criteria to maximising. Steps of the method:

- Compare first two alternatives and if one is dominated by other, discard the dominated one.
- Compare the undiscarded alternatives with third alternative and discard any dominated alternative
- The nondominated set is determined after (m-1) comparisons

**5.4.2. Maximin**

Under this procedure only a single **weakest attribute
represents an alternative**; all other (n-1) attributes for a particular
alternative are ignored. If these lowest attribute values come from different
attributes, as they often do, we may be basing our final choice on single
values of attributes that differ from alternative to alternative. Therefore,
the maximin method can be used only when interattribute values are comparable;
that is, all attributes most be measured on a **common scale**; however,
they need not be numerical.

The alternative, A^{+}, is selected such that:

(5.12)

(5.13)

**5.4.3. Maximax**

The maximax method selects an alternative by its
**best
attribute value**. In this case the highest attribute value for each
alternative is identified and these maximal value are compared. The alternative
with highest such value is to be chosen. All criteria must be measured
in the **same scale**, if not use (5.13) or more complicated formula
(5.3) or other formulas designed for criteria transformation.

The alternative, A ^{+}, is selected such
that:

(5.14)

*5.5.
Methods for Standard Level of Attribute Given*

The decision-maker sets the minimal attribute values
(standard levels) he/she will accept for each attribute. Any alternative
that has an attribute value less than standard level will be rejected (is
not acceptable). This procedure is called **conjunctive** method. If
the variant is accepted in the case at least one attribute is over or equal
to standard level the method is called **disjunctive**.

**5.5.1. Conjunctive
Method**

We classify A_{i }as an acceptable variant
only if

(5.15)

An alternative is evaluated on its greatest value of an attribute. For example, professional football players are selected according to the disjunctive method. A player is selected because he can either pass exceptionally, or run exceptionally, or kick exceptionally etc.

We classify A_{i} as
an acceptable alternative only if:

- ,

(5.16)

The most important
information needed is the ordinal interattribute preference information.
The relative importance among attributes determined by **ordinal preference**
is less demanding for the decision-maker to assess than by cardinal preference.

**5.6.1. Lexicographic
Method**

In some decision situations a single attribute seems to predominate. One way of treating this situation is to compare the alternatives on the most important attribute. If one alternative has a higher attribute value than any of the other alternatives, the alternative is chosen and the decision process ends.

However, if some alternatives are tied on the most important attribute, the subset of tied alternatives are then compared on the next most important attribute. The process continues sequentially until a single alternative is chosen or until all n attributes have been considered.

**5.6.2. Permutation
Method**

The method consists of testing each possible ranking of the alternatives against all others. With m alternatives, m! permutation rankings are available, so it is not convenient for big models. The method will identify the best ordering of the alternative rankings, then the dominating alternative. The method can be used also if the cardinal preferences are known.

If there are only ordinal preferences we can define cardinal preferences using for example Sequence Method (see chapter 5.2.3).

**Case of cardinal
preferences**

Suppose a number
of alternatives A_{1},…,A_{m} have to be evaluated according
to a attributes f_{1},…f_{n}. Assume that a set of cardinal
weights v_{1},…,v_{n} be given.

Let’s have three
alternatives: A1, A2, and A3. Then six permutations of the ranking of the
alternatives exist (m! = 3! = 6). They are:

P1=A1,A2,A3 P3=A2,A1,A3 P5=A3,A1,A2

P2=A1,A3,A2 P4=A2,A3,A1 P6=A3,A2,A1

For each permutation
we compare all pairs of variants. If in the partial ranking of variants
A_{k }and A_{l }under j-th criterion is we
put the criterion weight v_{j} into set C_{kl}, if we
are put the v_{j} to the set D_{kl}.

Then the evaluation
criterion of permutation P_{i} is given by:

(5.17)

Example:Fighter
Aircraft Problem – Permutation Method

*5.7.
Methods for Cardinal Preference of Attributes*

The methods in this class require the cardinal preferences of attributes. This (a set of weights for the attributes) is the most common way of expressing the interattribute preference information.

**5.7.1. Linear
Assignment Method**

The linear assignment
method is based on a set of attribute-wise rankings and a set of attribute
weights. The method features a linear compensatory process for attribute
interaction and combination. In the process *only *ordinal data, rather
than cardinal data, are used as the input. This information requirement
is attractive in that we do not need to scale the qualitative attributes.

The method is to
pick the sum of ranks for each alternative and rank them from the lowest
sum to the highest sum. For instance, consider the following attributewise
preferences with equal weight:

Criterion |
|||

Rank |
f1 |
f2 |
f3 |

1-st |
A1 |
A1 |
A2 |

2-nd |
A2 |
A3 |
A1 |

3-rd |
A3 |
A2 |
A3 |

Thus in spite of its apparent simplicity, this method fails to meet our basic linear compensation requirement.

A compensatory model from this simple approach is devised. Let us define a product-attribute matrix n as a square (m x m) nonnegative matrix

whose elements R

The linear assignment method can be written as a following LP model:

(p

(5.18)

**5.7.2. Simple
Additive Weighting Method**

Simple Additive Method
(SAW) is probably more often used method of MADM.

The method is based
on maximal the trade-off of variants. The simplification is established
by considering only linear function of trade-off.

The method requires
all maximal criteria, the same scale for and existing weights’ vectors.

Multiplying the
scale rating for each attribute value by importance weight assigned to
the criterion and then assuming these products over all criteria obtain
total trade-off u (ai) for each alternative.

(5.19)

(5.20)

**5.7.3. TOPSIS**

(Technique for Order Preference by Similarity to Ideal Solution)

TOPSIS takes the cardinal preference information on attributes. The ideal solution is guaranteed to have the longest distance to the negative ideal solution.

TOPSIS assumes that each attribute in the decision matrix takes either monotonically increasing or monotonically decreasing utility. In other words, the larger the attribute outcomes, the greater the preference for the "benefit" criteria and the less the preference for the "cost" criteria. Further, any outcome, which is expressed in a nonnumeric way should be quantified through the appropriate scaling technique. Since all criteria cannot be assumed to be of equal importance, the method receives a set of weights from the decision-maker.

The proposed method will be presented in steps:

**(Assumption: All
maximal criteria see 5.1.1)**

**Step 1. Construct
the normalised decision matrix**

This process tries
to transform the various attribute dimensions into non-dimensional attributes,
which allows comparison across the attributes. An element r_{ij}
of the normalised decision matrix R can be calculated as:

(5.21)

**Step 2. Construct
the weighted normalised decision matrix**

This matrix can
be calculated by multiplying each column of the matrix R with its associated
weight v_{j}. Therefore, the weighted normalised decision matrix
W is equal to:

(5.22)

Let two artificial alternatives H

(5.23)

(5.24)

The separation of each alternative from ideal and negative-ideal solution can be measured by Euclidean distance:

(5.25)

(5.26)

The relative closeness c

(5.27)

**Step 6. Rank the Preference
Order**

The variant that is closest
( c_{i}close to 0) to the negative ideal variant has the highest
distance from ideal variant and is taken as worst variant. Variant with
highest distance from negative ideal variant (c_{i} close to 1)
is closest to ideal variant and can be chosen as best.

A set of alternatives can now
be ranked according to descending order of c_{i}.

Example:Washing Machines – TOPSIS

**5.7.4. Electre**

(Elimination et Choice Translating Reality)

ELECTRE uses the
concept of an 'outranking relationship'. The outranking relationship of says
that even though two alternatives k and l do not dominate each other mathematically,
the decision maker accepts the risk of regarding A_{k} as almost
surely better than A_{l}.

This method consists of a pairwise comparison of alternatives based on the degree to which evaluations of the alternatives and the preference weights confirm or contradict the pairwise dominance relationships between alternatives. It examines both the degree to which the preference weights are in agreement with pairwise dominance relationships and the degree to which weighted evaluations differ from each other. These stages are based on a 'concordance and discordance' set; hence this method is also called concordance analysis.

The ELECTRE method takes the following steps:

**Step 1. Calculate
the normalised decision matrix –see (5.21) TOPSIS**

**Step 2. Calculate
the weighted normalised decision matrix – see (5.22) TOPSIS**

**Step 3. Determine
the concordance and discordance set**

For each pair of
alternatives the set of decision criteria is divided into two distinct
subsets. The **concordance set **C_{kl} for A_{k} and
A_{l} is composed of all criteria for which A_{k }is preferred
to A_{l}. The complementary subset is called **discordance set**.

**Step 4. Calculate
the concordance matrix**

The relative value
of the concordance set is measured by **concordance index,** which is
equal to the sum of weights associated with criteria contained in the set.
The concordance index for normalised weight set is:

(5.28)

The
successive values of concordance idices c_{kl} (k,l = 1,2,…,m;
k? l) form the concordance matrix C (m x m):

**Step 5. Calculate
the discordance matrix**

The concordance
index reflects the relative dominance of a certain alternative over a competing
alternative on the basis of the relative weight attached to the successive
decision criteria. So far no attention has been paid to the degree to which
the evaluations of a certain A_{k} are worse than the evaluations
of competing AR. Therefore a second index, called the **discordance index**,
has to be defined:

(5.29)

Higher value of d_{kl}
implies that, for discordance criteria (=criteria preferring A_{l})
, A_{l} is more favourable that A_{k}; lower value of d_{kl}
A_{l} is less favourable than A_{k}.

The discordance idices
form the discordance matrix D (m x m):

**Step 6. Determine
the concordance dominance matrix**

A_{k} will
only have a chance of dominating A_{l} , if its corresponding concordance
index c_{kl} exceeds at least a certain threshold value i.e. .

The threshold value can be determined as:

(5.30)

On the basis of a threshold
value a Boolean matrix F can be constructed:
(5.31)

Each ”1” in matrix F
represents a dominance of one alternative over another.
**Step 7. Determine
the discordance dominance matrix**

This matrix is constructed
in a way analogous to the F matrix on the basis of a threshold value d
to the discordance indices. The elements of g_{kl} of the discordance
dominance matrix G are calculated as:

(5.32)

Also the unit
elements in the G matrix represent the dominance relationships between
any two alternatives.

**Step 8. Determine
the aggregate dominance matrix**

The next step is
to calculate the intersection of the concordance dominance matrix F and
discordance dominance matrix G. The resulting matrix, called the aggregate
dominance matrix E, is defined by means of its typical elements e_{kl}
as:

(5.33)

**Step 9. Eliminate the
less favourable alternatives**

The aggregate dominance matrix
E gives the partial-preference ordering of the alternatives. If e_{kl}
= 1, then A_{k} is preferred to A_{l} for both the concordance
and discordance criteria, but Ak still has the chance of being dominated
by the other alternatives. Hence the condition that A_{k} is not
dominated by ELECTRE procedure is:

(5.34)

This
condition appears difficult to apply, but the dominated alternatives can
be easily identified in the E matrix. If any column of the E matrix has
at least one element of 1, then this column is 'ELECTREcally' dominated
by the corresponding row(s). Hence we simply eliminate any column(s) which
have an element of 1.

Example: The Fighter Aircraft Problem – ELECTRE

Bibliography:

Ching-Lai Hwang, Kwangsun
Yoon: Multiple Attribute Decisin Making, Springer Verlang Berlin Heidelberg
New York 1981

Fiala P.,Jablonský J., Maňas
M.: Vícekriteriální rozhodování,VŠE Praha 1997

Šubrt T. a kol.: Ekonomicko
matematické metody II, Aplikace a cvičení , ČZU Praha 2000