INTRODUCTION
In the present world, most of the problems given to the managers and
even our everyday life problems have various dimensions which are formulized
with multiple variables. In other words, the final decision cannot be
made with optimization of just one variable. So, it is natural that solving
of these problems becomes more complicated especially when the variables
are in contradiction and increasing the desirability of one variable may
require decreasing the desirability of other one. So Multi-Criteria Decision
Making (MCDM) and especially Multi Attribute Decision Making (MADM) have
been developed for solving such kind of problems. A Multi Attribute Decision
Making (MADM) problem can be summarized in the form of a matrix in which
the rows are different alternatives and the columns are the criteria specifying
the properties of the alternatives (Hwang and Masud, 1981). Also the matrix
cells show the position of the row alternative in relation to the correspondent
column criterion. Until now, we just presented the problem. Now to prioritize
the alternatives, a decision technique is required that specifies an alternative
of higher rank by interchanging and trading off of the different criteria
(Hwang and Yoon, 1995).
Due to the nature of present data, the ELECTRE method must be developed
and applied in this method. Since in the real world we are confronted
with data having interval nature like temperature, energy-related problems
and etc, the decision methods are required to be developed and matched
to these kinds of data. The current method is presented to develop the
ELECTRE method.
BACKGROUND
As a ranking method, ELECTRE has its own uses in energy planning and
has been used in several researches. For instance, this method has been
used in determining and defining an accessible penetration level for renewable
energy resources in an isolated decentralized system for generating electricity
(Papadopoulos and Karagiannidis, 2008). Also, it has been used in another
research to assess an action plan for diffusing the renewable energy technologies.
This method helps the energy department to choose the most suitable innovative
technology (Beccali et al., 2003).
Almeida (2007) has used the ELECTRE method to choose the contractor for
outsourcing the activities. In this study, the value and the cost of the
service quality are considered and each criterion is evaluated against
a utility function. Roy and Figueira (2002) proposed a method based on
ELECTRE method and Simo`s approach to determine the criteria. Combined
together, ELECTRE and similar branch and bound techniques can be used
to determine the non-dominated answers in a multiple objective mixed integer
linear problem (Lourenco and Costa, 2003).
ELECTRE method can be used in facility layout problem. For instance in
the research of Aiello et al. (2006) the Pareto-optimal solutions
are determined by employing a multi-objective constrained genetic algorithm
and the subsequent selection of the optimal solution is carried out by
means of ELECTRE method which is one of the multi criteria decision methods.
The transportation costs and the distance between the departments are
some of the criteria used in the analysis.
Group decision is one of the issues that deserves considerable attention.
When multiple decision makers are present, the final decision will be
the result of interaction of different ideas interaction, since each decision
maker has his own information and values systems. ELECTRE method enhancement
can help a decision group to achieve a consensus on a set of possible
alternatives (Leyva-Lopez and Fernandez-Gonzalez, 2003).
In all the above mentioned problems, ELECTRE method is used as a tool
for ranking and determining the best alternative. So, after reviewing
literature and conducting researches, it seems necessary to do a comprehensive
study on the development of this method as a fundamental study. In fact,
this is one of the objectives pursued in the present study which has been
little noticed so far. ELECTRE method is one of the effective methods
for multi attribute decision making with qualitative and quantitative
features. So, the development of this method for increasing the capability
of such decision making and satisfying the exact requests of the decision
maker is a very important task. Besides, in many cases we face with data
of interval nature such as temperature, time, date and energy-related
computations. In the real world, many problems conditions are impossible
to be described using absolute numerical data. Indeed, fuzzy, interval
and bounded data can often express the information well. In some situations,
we can not use absolute and exact data because of the data collecting
conditions. So, we must use interval data. For instance, for locating
gas stations and first aid stations on roads or locating a fire station
or hospital in a city, an exact location can not be usually candidate
and these locations have to be considered as interval. One of the most
important researches conducted in this area, in which an algorithmic method
is presented to develop TOPSIS for the data having interval nature (Jahanshahloo
et al., 2006).
In the above research, the existing steps in TOPSIS method (technique
for order-preference by similarity to ideal solution) have been changed
and a new structure has been developed for options ranking. Finally, a
numerical example showing the necessity of developing the method has been
solved. However, since many problems such as locating and energy problems
are analyzed using ELECTRE methods and are expressed by interval data,
we try to meet such needs in the present study. Since classic and fuzzy
methods cannot provide exact and suitable answers to many problems, we
present a new algorithmic method for ELECTRE method with interval data
for ranking.
THE PRINCIPLES OF DECISION METHODS
Generally, Multi-Criteria Decision making (MCDM) models have been proposed
for complicated decision having multiple optimization assessment criteria.
Some uses of MCDM in engineering include flexible manufacturing systems
(Cambron and Evans, 1991), layout design (Putrus, 1990), integrated manufacturing
systems (Boucher and Macstravic, 1991) and the evaluation of technology
investment decisions (Wanga and Triantaphylloub, 2008; Wabalickis, 1988.).
These models are generally classified into two categories (Hwang and Masud,
1981; Hwang and Yoon, 1995):
| • |
Multi Objective Decision (MODM) |
| • |
Multi Attribute Decision Making (MADM) |
Here, the main emphasis is on multi attribute decision making. The MADM
models are choosers and are used to choose the most suitable alternative
among m criteria and as we know, MODM models are used for designing. Any
MADM problem has multiple attributes that the decision maker has to exactly
specify them in the problem. The number of the criteria depends on the
nature of the problem. In Multi-Criteria Decision making, multiple criteria
which are sometimes in contradiction are considered and this usually happens
in daily life. For instance in the personal life factors such as job selection,
job prestige, work place, salary and wages, progress opportunities, work
conditions and etc, are considered as criteria and could be of high importance
for a person. In organizational problems, when strategy selection of an
organization is considered, the criteria such as obtained income in a
period of time, stock value of an organization, market share, organization
image in society (key money) and etc can be important. In governmental
problems, the transportation sector has to design the transportation system
in a way that the travel time, delays, transportation cost and etc have
become minimized. Multi attribute decision making is normally formularized
like the following matrix:
where, Ai indicates the ith alternative and rij
indicates the value of jth criteria for ith alternative.
MADM models can be analyzed using different information processing techniques
based on the criteria provided by the decision maker. To this aim, the
MADM data are classified into two general categories (Hwang and Masud,
1981; Hwang and Yoon, 1995):
| • |
Compensatory models: In these models, interchange among criteria
is allowed. This indicates indicating that changing of a criterion
is compensated by an opposite change (in converse direction) in the
other attribute(s). Compensatory models include simple weight mean,
TOPSIS (technique for order-preference by similarity to ideal solution),
ELECTRE (elimination et choice translating reality), linear assignment,
AHP and etc |
| • |
Non-compensatory models: In these models, interchange among
criteria is not allowed. This indicates that the weakness of an criterion
is not compensated by the strength of another one, but each criterion
is separately considered as the assessment basis of competing alternatives.
The important advantage of these models is their simplicity that is
compatible with the behavior and information limitation of the decision
maker. Non-compensatory methods include dominance method, lexicography,
elimination, maximin, minimin, conjunctive-satisfying-method and disjunctive-satisfying-method |
Compensatory models are themselves classified into three main subgroups:
| • |
Scoring sub-model: This model aims at finding a desirability
function for each alternative and choosing the alternative with highest
desirability. The related methods include simple additive weighting
method (SAW), interactive simple average weighting method and hierarchical
additive weighting method |
| • |
Compromising subgroup: In this subgroup, the option nearest
to the ideal option is chosen. The methods belongs to this subgroup
include TOPSIS, MRS (marginal rate of substitution of attributes),
MDS (multidimensional-scaling-with ideal-point) and LINMAP (linear
programming for multidimensional analysis of preferences) |
| • |
Concordance subgroup: In these models, the output is a set
of ranks such that the necessary concordance will be provided in the
most suitable way. This subgroup includes ELECTRE methods and linear
assignment (Hwang and Masud, 1981; Hwang and Yoon, 1995) |
THE REASON OF ELECTRE METHOD SELECTION BY CONSIDERING INTERVAL DATA
This study aims at analyzing a problem with a suitable MADM model. Having
this fact in mind that in compensatory models the interchange among the
criteria is permitted, the criteria are not viewed separately but are
analyzed as a whole set and the interactions and relations among the criteria
are taken into account. So, when there is such kind of relations among
the criteria, using the compensatory models is recommended. Besides, in
real world we face with a lot of problems which are of compensatory models
kind.
In scoring-sub model, there is always the need for finding a desirability
function and this in turn requires the complete recognition of the problem
and high experience. This is a weakness of this subgroup. So, the problem
is how to find the multi attribute desirability function. Also, the compromising
subgroup selects the option nearest to the optimized answer. This may
not be free of errors and not be usable practically in the problem. In
the present study, we chose the ELECTRE method of the concordance subgroup,
because we need to find an order and in fact, rank the existing alternatives
(Hwang and Masud, 1981; Hwang and Yoon, 1995).
ELECTRE METHOD
ELECTRE is a multi attribute decision making method for ranking multiple
alternatives based on some criteria. This method is a very effective assessment
solution that equips the decision activities with quantitative and qualitative
features (Huang and Chen, 2005). As mentioned before, ELECTRE is one of
the most important compensatory methods (Hwang and Masud, 1981).
In this method, the output is a set of ranks such that the necessary
concordance will be provided in the most appropriate form. ELECTRE uses
a new concept known as outranking. For example Ak→Al
indicates that although k and l options do not have any priority to each
other mathematically, the decision maker accepts that Ak is
better than Al (Hwang and Masud, 1981). In this method, all
the alternatives are assessed using the outranking comparisons and the
non-effective alternatives are omitted. Pair comparisons performed based
on agreement rank of weights (Wj) and difference rank from
weighting assessment values (Vij) and are tested simultaneously
for alternatives assessment. All these steps are planned according to
a concordant and a discordant set that is known as concordance analysis.
ELECTRE METHOD WITH INTERVAL DATA
In the present study, we have tried to develop the ELECTRE model by using
the interval data. As mentioned earlier, in many problems the output data
are not exact values and fluctuate or due to the nature of the problem
can be expressed as interval. We can mention time, date and temperature
are some examples.
Decision matrix based on the interval data is shown below. In this matrix,
Ai s are analyzed alternatives and Cj s are selection
criteria:
where, Wj is the weight of criterion Cj.
The developed ELECTRE algorithm is described as follows.
Step 1
Computing dimensionless decision making matrix: Here, we need a new
method of unscaling, because interval data are present. In this method,
the upper and lower limits of the numbers are unscaled separately. In
this step, the decision matrix is transformed into a normalized matrix
by using the following relationships:
Step 2
Creating a weighted dimensionless matrix (V): Here, a weighted dimensionless
matrix is calculated by using a known vector W.
W = {w1,w2,...,wn}≈ |
Assumed from DM |
Such that 
are the data of ND matrix that their criterion scores are dimensionless
and comparable. Wn*n is a diagonal matrix obtained from weighting
criteria.
Step 3: Specifying the concordance and discordance sets for each
pair of alternatives l ≠ k; k,l = 1,2,3,...,n. In this step, the existing
attributes set J = {j|j = 1,2,...,n} is divided by distinguished concordant
(Sk,l) and discordant (Dk,l) subsets.
The concordant set (Sk,l) will include Ak and Al
with criteria such that Ak will be preferred to Al.
On the other hand, due to interval nature of data, if two concordance
and discordance sets are presented in ELECTRE classic algorithm, there
will be some difficulties in data distribution. In fact, in this case
it is not clear which of the upper or lower limits has to be used for
formation of concordant and discordant sets. So, the best possible method
is dividing each concordance and discordance sets by upper and lower limit
sets. In this procedure, the concordance set includes two sets 
and the discordance set includes two sets 
In other words, at first the lower limits of the decision matrix is analyzed
like a classic ELECTRE problem and two concordance and discordance sets
are formed. After that, the same procedure is repeated for upper limits.
(rij is assumed to have increasing desirability). So, concordance
set for upper and lower limits is as follows:
where, I and J indicate the positive (profit) and the negative (cost)
sets, respectively.
In order to find the upper and lower limits of the discordance sets,
the following formulas are used:
Step 4
Computing concordance matrix: The concordance criterion Ik,l
is equal to the average of 
values. Since, here the concordance set includes two subsets 
these two matrices are separately determined and finally the average of
two values is calculated in order to form the concordance matrix. The
concordance matrix 
is
the weight sum (Wj) of criteria forming the 
set.
So the concordance criterion 
between
Ak and Al is as follows:
And the concordance matrix 
is the weight sum (Wj) of criteria forming the 
set. The concordance criterion 
between Ak and Al is as follows:
Finally, the concordance matrix Ik,l must be created:
The concordance criterion Ik,l reflects the relative importance
of Ak in comparison with Al such that 0≤Ik,l≤1.
Higher value of Ik,l means that the priority of Ak
over Al is more concordant. So, we have to find only one matrix
and that is why the mentioned values are averaged.
Hence, the sequential values of Ik,l criteria (k,l = 1,2,...,m,
k ≠ 1) form the asymmetric concordance matrix (I) as follows:
Step 5
Computing discordance matrix: In contrast to Ik,l criterion,
the discordance criterion shows how much Ak assessment is worse
than Al. Because of the presence of interval data, two values
are defined like the concordance matrix; it means that this criterion
for each discordance sets 
is calculated by using the V matrix elements (weighted scores) for the
correspondent discordant set and finally the average value of 
is used for computing discordance matrix NIk,l as follows:
o, matrix NIk,l is computed as follows:
Hence, the discordance matrix for each pair comparison of the alternatives
will be as follows:
It is clear that the information present in I and NI matrixes are considerably
different and yet complement each other. Matrix I shows the weights wj
of concordant criteria and the asymmetrical NI matrix indicates the biggest
relative difference from Vij = nij.wj
for discordant criteria.
Step 6
Specifying the effective concordant matrix: Ik,l values
of concordance matrix have to be assessed in relation to a threshold value
so, that the priority chance of Ak over Al can be
better judged. This chance will increase if Ik,l exceeds a
minimum threshold (Ī):
For instance, Ī (optional) can be calculated by averaging concordance
criteria as follows:
Then according to the Ī, a Boolean F matrix (having 0s and 1s) is
created such that:
Then, each element of matrix F (the effective concordant matrix) will
show an effective alternative and will be dominant alternative.
Step 7
Specifying the effective discordant matrix: The Nik,l elements
of discordant matrix also have to be assessed against a threshold value
as in step 6. For instance, the threshold value NĪ can be calculated
as follows:
Then a Boolean G matrix (known as the effective discordant matrix) is
created such that:
Elements of matrix G having the value of 1 show the dominance relations
among the alternatives.
Step 8
Specifying the global and effective matrix: The common elements (hk,l)
form the global matrix (h) for decision through peer to peer multiplication
of matrices F and G.
Step 9
Neglecting non-effective alternatives: The global matrix h shows the
relative priority order of the alternatives. It means that hk,l
= 1 indicates that Ak is preferred to Al from the
viewpoint of concordance and discordance criteria, but Ak may
be still dominated by other alternatives. Therefore, In order to have
Ak as an effective alternative when using the ELECTRE method,
the following conditions must be met:
Meeting both of these two conditions at the same time may be rarely occurs.
But the effective elements can simply be distinguished in matrix h in a way
that each column in h having at least one 1 element can be omitted, since this
column is dominated by one or more rows.
One way for determining ranking is using graphs. In such graphs, all
the alternatives are displayed using nodes. fk,l = 1 shows
the path (arc) between two nodes k and l and fk,l = 0 shows
that there is no path between k and l. In this graph the alternative having
the most output is selected as the best option. Also the options can be
sorted according to their output numbers.
NUMERICAL EXAMPLE
Here, the problem is solved based on data provided in (Jahanshahloo et
al., 2006) which are interval data of 15 bank branches (A1,
A2,...A15) in Iran. The method used in this example
for calculating numerical values by the developed ELECTRE method is presented
below.
Step 1: Dimensionless decision matrix computation (using the formulas
(1) and (2)). The interval decision matrix and interval normalized decision
matrix are shown in Table 1 and 2,
respectively.
Step 2: Creating a weighted dimensionless matrix (V) (using the
formulas (3 and 4)). The interval weighted normalized decision matrix
is presented in Table 3.
Step 3: Specifying the concordance and discordance sets for each
pair of alternatives (using the formulas (6) and (7) for concordance set
and formulas (8) and (9) for discordance sets). The upper and lower limits
of concordance set and discordance sets are presented in Table
4 and 5, respectively.
Step 4: Computing concordance matrix: (using formulas (10-12)).
The concordance matrix is as following matrix:
| Table 1: |
The interval decision matrix |
|
| Table 2: |
The interval normalized decision matrix |
|
| Table 3: |
The interval weighted normalized decision matrix |
|
Step 5: Computing discordance matrix (using formulas (14) and
(15)). The discordance matrix shown below is a big matrix. So, we assign
one matrix to the lower limit and another one to the upper limit (as shown
in matrices (29) and (30), respectively).
| Table 4: |
The concordance sets for 15 existing alternatives (upper
and lower limits) |
|
| Table 5: |
The discordance sets for 15 existing alternatives (upper
and lower limits) |
|
Then the final discordance matrix is calculated by considering lower
and upper limits using formula (16) as follows:
Step 6: Specifying the effective concordant matrix (using formulas
(18), (20) and (21)). The average value of the concordance matrix (using
formula (19)) and the effective matrix are shown in (32) and (33), respectively.
Step 7: Specifying the effective discordant matrix (using formulas
(23), (24)). The average value of the discordant matrix (using formula
(22)) and the effective matrix are shown in (34) and (35), respectively.
Step 8: Specifying the global and effective matrix (using the
formula (25)) is shown in (36):
| Table 6: |
Ranking the alternatives |
|
Step 9: Neglecting non-effective alternatives (using formulas
(26) and (27)) and ranking the alternatives. Based on matrix h, the ranking
can be determined without drawing any graph. Drawing the graph in the
current problem is complicated, because there are many alternatives. In
fact, the number of outputs for each node can be found by using matrix
h. In the case that the numbers of outputs for two alternatives are the
same, the option having fewer inputs is preferred for sorting purposes,
since it will be less dominated by other options. Ranking the alternatives
is shown in Table 6.
CONCLUSION
Everyday, we need to make decisions both in the personal and professional
activities. Therefore, increasing the power of decision making using the
various decision methods is highly paid attention. The decision method
used in the current research is the ELECTRE method. This method was developed
to be used when the data are of interval nature. Considering the nature
of the data, the existing steps in ELECTRE method were reviewed and matched
to the current conditions. Then a numerical example was solved to prove
the correctness of the proposed method. The results show the ranking of
15 bank branches. Also the comparison of the results with the ones of
the TOPSIS method shows that both methods are extremely similar and prove
each other. This is a rare situation and proves the accuracy of the proposed
method.
Also, the contribution of the present study is to use and develop ELECTRE
method which is one of the multi attribute decision making method in ranking
various alternatives for deterministic and exact data. Therefore, when
we face with interval data and it is impossible to use existing methods
for uncertain, inexact and fuzzy data (for data like a time interval,
etc.). It is needed to use a new method for ranking. Since, the ELECTRE
method is based on deterministic and exact data. The method presented
in this study will allow the users to rank their existing alternatives
more efficiency and easily.
ACKNOWLEDGMENT
The authors of the present study are grateful through to the staff of
selected banks for their help to provide them with data and other facilities.
