Chapter 1

Introduction to Multicriteria Decision

Analysis

1.1 Decision Analysis

Decision analysis is the science and art of designing or choosing

the best alternatives based on the goals and preferences of the
decision maker (DM). Making a decision implies that there are
alternative choices to be considered. In such cases, we do not
want to identify only as many of these alternatives as possible but
we want to choose the one that best fits our goals, desires,
lifestyle, values, and so on (Harris 1997). In other words, decision
analysis is the science of choice. For example, selecting the best
technology for urban water supply, developing flood protection
alternatives, or optimizing the operation of a reservoir are all the
problems of choice.
To describe the preferences of a DM we may use one of the
terms of Goals, Objectives, Criteria, and Attributes. However,
there are differences among their meanings. Goals are useful for
clearly identifying a level of achievement to strive toward
(Keeney and Raiffa 1993). Goals relate to desired performance
outcomes in the future, while an objective is something to be
pursued to its fullest level or it may generally indicate the direc-
tion of desired change. Criteria are more specific and measurable
outcomes. A criterion generally indicates the direction in which
we should strive to do better. In all decision problems, we want

M. Zarghami and F. Szidarovszky, Multicriteria Analysis,

DOI 10.1007/978-3-642-17937-2_1, 1
# Springer-Verlag Berlin Heidelberg 2011
2 1 Introduction to Multicriteria Decision Analysis

Fig. 1.1 Relation among

the goals, objectives and
criteria
Goals

Objectives

Criteria

to accomplish or avoid certain things. To what degree we accom-

plish our goals or avoid unfavorable consequences should be
among the criteria. They are either achieved, or surpassed or
not exceeded (Hwang and Yoon 1981). The relation among
these three terms is indicated in Fig. 1.1. The attributes are also
performance parameters, components, factors, characteristics,
and properties. In this book, we will use the term criteria, instead
of objectives and attributes, which is closer to the meaning of
what is usually used by the DMs in water resources and environ-
ment management.

Example 1.1. The Common Agricultural Policy (CAP) absorbs

roughly 45% of the total budget of the European Union. The CAP
is a widely debated policy, in terms of both its budget and the
instruments being used (Gomez-Limon and Atance 2004). The
hierarchy of its goal, main objectives and criteria used to evaluate
the objectives is shown in Table 1.1.
The management of water resources and the environment takes
place in a multicriteria framework when it is necessary to con-
sider the technical, environmental and social implications of
the water resources projects, in addition to the economic criteria
to ensure sustainable decisions and favorable decision outcomes.
The traditional cost–benefit analysis, used for many decades
in water resources planning and environmental management,
1.1 Decision Analysis 3

Table 1.1 Goal, objectives and criteria for the CAP project
Goal Objectives Criteria
Improving the welfare Social 1. To safeguard family agricultural holdings
of residents in objectives 2. To maintain villages and improving the
European countries quality of rural life
3. To conserve traditional agricultural
products (typical local products)
Environmental 1. To encourage agricultural practices
objectives compatible with environmental
conservation
2. To contribute to the maintenance of
natural areas
3. To maintain traditional agricultural
landscapes
Economic 1. To ensure reasonable prices for consumers
objectives 2. To ensure safe and healthy food
3. To encourage competitiveness of farms
4. To provide adequate income for farmers
5. To guarantee national food self-
sufficiency

transformed the different types of impacts into a single monetary

metric. Once that was done, the task was to find the plan or policy
that maximized the difference between the benefits and costs. If
the maximum difference between the benefits and costs was
positive, then the best plan or policy was found. However, not
all system performance criteria can be easily expressed in mone-
tary units. Even if monetary units are used to describe each
objective, then they do not address the distributional issues of
who benefits, who pays, and how much (Loucks and van
Beek 2005). To overcome this inefficiency multicriteria decision
analysis (MCDA) techniques are applied. The most important
advantages of using these methods for water resources manage-
ment are:
• To cope with limited water, financial and human resources
• To allow the combination of multiple criteria instead of a
single criterion
• To avoid opportunity costs of delay in decision-making
• To resolve conflict among stakeholders
• To simplify the administration of the projects
4 1 Introduction to Multicriteria Decision Analysis

1.2 The Components of MCDA Problems

Any MCDA problem has three main components: decision

maker/s (DMs), alternatives and criteria. These three elements
can be shown as the three basis of a triangle (Fig. 1.2).
The classification of an MCDA problem depends on the types
of these elements. The definitions of the three components are as
follow:
• Decision maker/s. The first element is identifying the DMs. For
a particular problem, we might have a single person who is
responsible for deciding what to do or several people or organi-
zations being involved in the decision-making process. In the
first case, we have only one DM; in the second case, we have
multiple DMs. When more than one DM is present, then they
might have different preferences, goals, objectives and criteria,
so no decision outcome is likely to satisfy every decision maker
equally. In such cases, a collective decision has to be made when
the outcome depends on how the different DMs take the inter-
ests of each others into account. In other words, the outcome
depends on their willingness to cooperate with each other. In the
case of multiple decision makers, we might consider the prob-
lem as an MCDA problem, where the criteria of the different
decision makers are considered the criteria of the problem
(Karamouz et al. 2003). In the case of a single DM and one
criterion, we have a single-objective optimization problem. The
applied methods depend on the type of the problem (linear
programming, nonlinear programming, integer or mixed pro-
gramming, dynamic optimization, stochastic programming,
etc.). Typical MCDA problems arise when a single decision
maker considers several criteria simultaneously. In the presence

Decision maker/s

Fig. 1.2 The elements of an

MCDA problem Alternatives Criteria
1.2 The Components of MCDA Problems 5

of multiple DMs the problem can be modeled by MCDA as

mentioned above, or in the case of conflicting priorities and
desires of the DMs, game theory can be used. MCDA is often
considered as the most powerful methodology of solving game
theoretical problems with cooperating players.
• Alternatives. These are the possibilities one has to choose
from. Alternatives can be identified (that is, searched for and
located) or even developed (created where they did not previ-
ously exist). The set of all possible alternatives is called the
decision space. In many cases, the decision space has only a
finite number of elements. For example, selecting a technology
from four possibilities results in a decision space with four
alternatives. In many other cases, the decision alternatives are
characterized by continuous decision variables that represent
certain values about which the decision has to be made. For
example, reservoir capacity can be any real value between the
smallest feasible value and the largest possibility.
• Criteria. These are the characteristics or requirements that each
alternative must possess to a greater or lesser extent. The alter-
natives are usually rated on how well they possess the criteria.
Since we have to make a choice from a given set of feasible
alternatives we need to measure how good those alternatives are.
The goodness of any alternative can be characterized by its evalua-
tions with respect to the criteria. These evaluations can be described
by crisp numbers, linguistic values, random or fuzzy numbers. A
criterion is called positive, if better evaluation is indicated by larger
values. Similarly a criterion is called negative, if better evaluation is
shown by a smaller value. Regarding the types of the alternatives,
we have two major classes of MCDA problems.
Before proceeding further, some comments are in order. In the
case of one criterion the problem is given as

Maximize f ðxÞ
subject to x 2 X :

Here x represents an alternative and X is the set of all feasible

alternatives. All values of f(x) when x runs through the feasible
6 1 Introduction to Multicriteria Decision Analysis

set, X, are located on the real line. The optimal solution has
therefore the following properties:
1. The optimal solution is at least as good as any other solution.
2. There is no better solution than the optimal solution.
3. All optimal solutions are equivalent, i.e., they have the same
objective value.
In the case of one criteria, any two decisions xð1Þ and xð2Þ can be
compared since either f ðxð1Þ Þ>f ðxð2Þ Þ, or f ðxð1Þ Þ ¼ f ðxð2Þ Þ, or
f ðxð1Þ Þ<f ðxð2Þ Þ. In the case of multiple criteria, this is not true.
For example, in the case of two positive criteria the following two
outcomes cannot be compared:
   
1 2
and ;
2 1

since the first outcome is better in the second criterion and worse
in the first criterion.

1.3 Classification of MCDA Problems

1.3.1 Discrete Case

If the decision space is finite, then the construction of the feasible

decision space is very simple. We have to check the feasibility of
each alternative by determining whether or not it satisfies all
restrictions. We can show the discrete alternatives, criteria and
the evaluations of the alternatives with respect to the criteria in a
matrix, called the evaluation or decision matrix. In a decision
matrix, the (i, j) element indicates the evaluation of alternative j
with respect to criteria i, as it will be explained in the following
example.

Example 1.2. Table 1.2 represents an evaluation matrix. The

problem is to choose the best scheme for inter basin water transfer
from five alternatives (what can we do). The four criteria (what
1.3 Classification of MCDA Problems 7

Table 1.2 Evaluation matrix of Example 1.2

Criteria Weights Alternatives
A1 A2 A3 A4 A5
C1 0.2 1.3 1.4 1.1 1.7 1.2
C2 0.1 High Low Medium High Very high
C3 0.4 Easy Easy Difficult Difficult Medium
C4 0.3 70 90 75 40 55

we get) are benefit–cost ratio, environmental sustainability, easy

operation/maintenance, and compliance with former water rights
in the watershed (in subjective judgment on a scale between 0 and
100). These criteria show and indicate the Integrated Water
Resources Management (IWRM) principles.
If we quantify the linguistic values on a 0 through 100 scale,
then the evaluation values of the alternatives with respect to
criteria C2 and C3 might become
80 10 50 80 100
and
90 90 10 10 50.
The decision space of the problem has five elements, the five
alternatives: A1, A2, A3, A4 and A5. The consequence of selecting
any one of the alternatives is characterized by the simultaneous
values of the criteria, which is a four-element vector. So the
objective space consists of five points in the four dimensional
space: (1.3, 80, 90, 70), (1.4, 10, 90, 90), (1.1, 50, 10, 75), (1.7,
80, 10, 40) and (1.2, 100, 50, 55). The decision space shows our
possible choices. That is, it represents what can be done. The
objective space shows the simultaneous criteria values, that is,
what we can get.
If we compare these alternatives, then we see that neither of
them can be improved in all criteria simultaneously by selecting
another alternative. In this case all of these alternatives can be
considered reasonable choices. In order to choose only one of
them which could be considered as the “best”, additional pref-
erence information is needed from the DM. The preferences of
the DM can be represented by many different ways, for example,
by specifying relative importance weights. These values are
shown in the second column of Table 1.2.
8 1 Introduction to Multicriteria Decision Analysis

Figure 1.3 represents the steps of formulating and solving a

mathematical model for a discrete MCDA problem. As it is
shown in this procedure, the decision making process has recur-
sive nature.

1.3.2 Continuous Case

If the decision alternatives are characterized by continuous vari-

ables then the problem is considered to be continuous. In this
case, the alternatives satisfying all constraints are feasible, and
the set of all feasible alternatives is the feasible decision space.
The constraints are usually presented as certain equalities or
inequalities containing the decision variables.
In the classical optimization models, we have only one crite-
rion to optimize. However, in most decision-making problems we
are faced with several criteria that might conflict with each other.
For example, treatment cost and water quality are conflicting
criteria, as better quality requires higher cost. We can assume

Identify the decision problem with its goals it

should achieve

Get the information

and obtain the criteria

Develop alternatives

Feedbacks from
decision maker
Evaluate each alternative
with respect to the criteria

Make the decision after

sensitivity analysis

Fig. 1.3 The steps of formulating and solving a discrete MCDA problem
1.3 Classification of MCDA Problems 9

that all criteria are maximized, otherwise each negative criterion

can be multiplied by 1.

Example 1.3. The water demand of an urban area can be supplied

from two sources, from groundwater and also from surface water.
The decision variables (alternatives) are how much water should
be pumped from the groundwater resource, x1, and how much
water should be transformed from the reservoir, x2. The DM
wants to minimize the total cost of satisfying the demand. The
unit cost of water supply from groundwater and surface water
supply are 3 and 2, respectively. The DM also desires to maxi-
mize the reliability of the supply, which can be identified by a
numerical scale. The groundwater is more reliable than surface
water in this area and then according to the knowledge of an
expert, the reliability can be shown by the numbers of 5 and 3 for
groundwater and surface water, respectively. The minimum
amount of total supplied water should be at least 5 units. The
groundwater can supply at most 4 units/year and the surface water
can supply maximum 3 units/year in average. The corresponding
continuous MCDA problem can be formulated as follows:

Minimize f1 ¼ 3x1 þ 2x2 (1.1)

and

Maximize f2 ¼ 5x1 þ 3x2

subject to x1 þ x2 r5
x 1 b4 (1.2)
x 2 b3
x1 ; x2 r0:

The decision space of this problem is shown in Fig. 1.4. The

decision alternatives should be chosen from this space. So, the set
of alternatives (possible supply designs) allows infinitely many
different choices.
The criteria (f1 and f2) are functions of the decision variables
(x1 and x2). These criteria are clearly in conflict: a low cost water
10 1 Introduction to Multicriteria Decision Analysis

Fig. 1.4 The decision space Surface water, x2

for Example 1.3
x1+x2 ≥ 5
x1 £ 4

(4, 3)
x2 £ 3
(2, 3)
(4, 1)

Groundwater, x1

supply scheme will certainly have low reliability. Figure 1.4

shows the set of feasible alternatives, it shows only what we
can do. In order to see the consequences of the decisions we
have to find and illustrate the set of the feasible criteria values,
which is called the objective space. In order to do this, we have to
express the decision variables as functions of the criteria values
by solving (1.1) and (1.2) for unknowns x1 and x2:

x1 ¼ 3f1 þ 2f2 (1.3)

and

x2 ¼ 5f1  3f2 : (1.4)

By substituting these expressions into the constraints of the

original decision model, the corresponding constraints for the
criteria values become as follows:

2f1  f2 r5
3f1 þ 2f2 b4
5f1  3f2 b3
3f1 þ 2f2 r0
5f1  3f2 r0:

The feasible set of these inequalities is the objective space,

which is shown in Fig. 1.5.
1.3 Classification of MCDA Problems 11

Fig. 1.5 The objective Reliability, f2

space for Example 1.3
(18, 29)
(14, 23)

(12, 19)

Cost, f1

Any point of the broken line with segments connecting the

point (12, 19) with (14, 23) and (14, 23) with (18, 29), shown in
Fig. 1.5, is reasonable since none of the criteria can be improved
without worsening the other. The choice of a single “best” point
from this infinite set should be based on additional preference and
tradeoff information obtained from the DM.
The steps of formulating and solving a mathematical model of
a continuous MCDA problem are presented in Fig. 1.6.
The continuous case is a special case of infinite problems.
There are many decision making problems with infinitely many
alternatives where some of the alternatives cannot be described
by continuous variables. For example this is the case if some
variables have only integer values. Consider the case when we
decide on doing something or not doing it at all. In this case, the
MCDA model has variables with discrete (0 or 1) values and
some other variables with continuous scales. These mixed pro-
blems can be solved by combining discrete and continuous meth-
ods. These types of decision problems are very rare in the water
resources modeling and environmental management problems. In
this book, we restrict our discussions to the purely discrete and
continuous models.
Notice that regardless of the type of the MCDA problem,
decision making is usually an iterative and continuous process.
That is, most decisions are made by moving back and forth
between choosing the criteria and identifying the alternatives
and the preferences of the DM. The available alternative set
often influences the choice of the criteria we use to evaluate
12 1 Introduction to Multicriteria Decision Analysis

Get preferences, tradeoff information from the

DM

Find appropriate model fitting the

above information

Find solution

Present it to DM

No DM modifies
Does DM accept
preferences, tradeoff
solution?
information

Yes

Job done

Fig. 1.6 The steps of formulating and solving a continuous MCDA problem

them, and similarly the criteria set might also influence the
selection of the alternatives. After a computed solution is pre-
sented to the DM, it is either accepted or the DM makes some
changes and modifications in the model. Then the new solution is
computed, which is shown again to the DM. This interactive
process continues until a satisfactory solution is obtained.