Multimoora optimization used to decide on a bank loan to buy property/Optimizavimas multimoora metodu imant banko paskola nekilnojamam turtui pirkti.
Brauers, Willem Karel M. ; Zavadskas, Edmundas Kazimieras
The Problem
Just as an example: suppose a single person decides to buy a small
apartment consisting of two rooms in, let us say, Lithuania. The cost
price of the apartment is 30,025 [euro]. The person has sufficient
income to pay the initial down payment including all kinds of fees and
commissions and to guarantee the monthly loan payments, the payment of
interests and of insurance. 2011 Volume 17(1): 174-188
The client is reluctant for the obligation to take a special life
insurance. In addition he has to face the following obligations:
-- The initial payment in [euro].
-- Loan repayment and payment of interest in [euro] per month.
-- Hypothecation bond registration fee in [euro].
-- A one off loan administration fee in [euro],
-- Life insurance in [euro] per year.
-- Insurance of the apartment to be purchased in [euro] per year.
-- Commission for currency exchange in [euro].
-- Monopolization of all bank activities and the wage
administration of the client in % (0%, 50% or 100%) (1).
Even if two obligations are expressed in [euro] the same [euro] is
not necessary meant. For instance, a Euro of initial payment can not be
substituted by a Euro for the payment of interest. We limit us to a
simulation exercise. Contrary to a lot of other definitions, simulation
is defined here in a rather broad sense. Gordon, Enzer and Rochberg
(1970) give the most complete description of simulation as mechanical,
metaphorical, game or mathematical analogs. They conclude: "are
used where experimentation with an actual system is too costly, is
morally impossible, or involves the study of problems which are so
complex that analytical solution appears impractical".
Three banks are assumed in the simulation exercise, one foreign and
two domestic banks A and B as shown in the following Table 1.
Two different methods under the name of MOORA, namely a Ratio
System and a Reference Point Approach, will try to make optimal the
content of the table. By adding the Full Multiplicative Form three
methods will compose MULTIMOORA, whereas the three methods will control
each other (for more information, see also: Brauers, Zavadskas 2010).
The following diagram (Fig. I) clarifies the combination of the three
different methods of MULTIMOORA (Balezentis et al. 2010).
[FIGURE 1 OMITTED]
The figures between brackets refer to the basic equations involved
as shown in the following pages.
Ratio System of MOORA
We go for a ratio system in which each response of an alternative
on an objective is compared to a denominator, which is representative
for all alternatives concerning that objective (2):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
with: [x.sub.ij] = response of alternative j on objective I; j =
1,2, ..., m; m the number of alternatives; i = 1,2, ... n; n the number
of objectives; [x.sup.*.sub.ij] = this time a dimensionless number
representing the response of alternative j on objective i (3).
For optimization, these responses are added in case of maximization
and subtracted in case of minimization:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
with: i = 1,2,... ,g as the objectives to be maximized; i = g+1,
g+2,... , n as the objectives to be minimized; [y.sup.*.sub.j] = the
total assessment of alternative j with respect to all objectives;
[y.sup.*.sub.j] can be positive or negative depending of the totals of
its maxima and minima.
An ordinal ranking of the [y.sup.*.sub.j] shows the final
preference. Indeed, cardinal scales can be compared in an ordinal
ranking after Arrow (1974): "Obviously, a cardinal utility implies
an ordinal preference but not vice versa".
The Reference Point Approach as a part of MOORA
The Reference Point Approach will go out from the ratios found in
formula (1), whereby a Maximal Objective Reference Point is also
deduced. The Maximal Objective Reference Point approach is called
realistic and non-subjective as the co-ordinates ([r.sub.i]), which are
selected for the reference point, are realized in one of the candidate
alternatives. In the example, A (10;100), B (100;20) and C (50;50), the
maximal objective reference point [R.sub.m] results in: (100;100). The
Maximal Objective Vector is self-evident, if the alternatives are well
defined, as for projects in Project Analysis and Project Planning.
Given the dimensionless number representing the normalized response
of alternative j on objective i, namely [x.sup.*.sub.ij] of formula (1)
and in this way arriving to:
([r.sub.i] - [x.sup.*.sub.ij]),
with: i = 1, 2,... , n as the attributes; j = 1, 2,... , m as the
alternatives; ri = the ith co-ordinate of the reference point;
[x.sup.*.sub.ij] = the normalized attribute i of alternative j then this
matrix is subject to the Metric of Tchebycheff (Karlin and Studden 1966)
(4):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
[absolute value of [r.sub.i] - [x.sup.*.sub.ij]]--means the
absolute value if [x.sub.ij] is larger than [r.sub.i] for instance by
minimization.
Concerning the use of the maximal objective reference point
approach as a part of MOORA some reserves can be made in connection with
consumer sovereignty. Consumer sovereignty is measured with the
community indifference locus map of the consumers (Brauers 2008: 92-94).
Given its definition the maximal objective reference point can be pushed
in the non- allowed non-convex zone of the highest community
indifference locus and will try to pull the highest ranked alternatives
in the non-allowed non-convex zone too (Brauers, Zavadskas 2006:
460-461). Therefore an aspiration objective vector can be preferred,
which moderates the aspirations by choosing smaller co-ordinates than in
the maximal objective vector and consequently can be situated in the
convex zone of the highest community indifference locus. Indeed
stakeholders may be more moderate in their expectations. The
co-ordinates qi of an aspiration objective vector are formed as:
[q.sub.i] < [r.sub.i],
([r.sub.i]-[q.sub.i]) being a subjective element we don't like
to introduce subjectivity in that way again. Instead, a test shows that
the min-max metric of Tchebycheff delivers points inside the convex zone
of the highest community indifference locus (Brauers 2008: 98-103).
The Full Multiplicative Form
The following n-power form for multi-objectives is called from now
on a full-multiplicative form in order to distinguish it from the mixed
forms:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
with: j = 1,2,... , m; m the number of alternatives; i = 1,2,...
,n; n being the number of objectives; [x.sub.ij] = response of
alternative j on objective I; [U.sub.j] = overall utility of alternative
j.
The overall utilities (Uj), obtained by multiplication of different
units of measurement, become dimensionless.
How is it possible to combine a minimization problem with the
maximization of the other objectives? Therefore, the objectives to be
minimized are denominators in the formula:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4')
j = 1,2,... ,m; m the number of alternatives; i = the number of
objectives to be maximized,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
n-i = the number of objectives to be minimized,
with: [U'.sub.j] = the utility of alternative j with
objectives to be maximized and objectives to be minimized.
If one of the [x.sub.ij] = 0 it would mean that an objective is not
present in an alternative. In that case a foregoing filtering stage can
prescribe that an alternative with a missing objective to be an optimum
is withdrawn from the beginning. Otherwise for the calculation of a
maximum the zero factor is just left out.
A zero in a minimization problem is much more complicated. A real
zero factor, like in the case of the absence of pollution, has to
maintain its influence. Therefore the zero factor will receive an
extremely low symbolic value like 0.01. If the zero factor means missing
information then the situation is different and will ask for a serious
correction. A correction factor has to be introduced being a bit larger
than all corresponding factors of the other alternatives, for instance
next ten, next hundred etc. With factors 8 and 11 next ten will be 20.
With factors 80 and 110 next hundred will be 200 etc.
MULTIMOORA
MULTIMOORA is composed of MOORA and of the Full Multiplicative Form
of Multiple Objectives and in this way up till now no other approach is
known including three or more methods, in this way MULTIMOORA becomes
the most robust system of multiple objectives optimization
The Importance given to an Objective
The method of multiple objectives which uses non-subjective
dimensionless measures without normalization like in MULTIMOORA is more
robust than this one which uses for normalization subjective weights
(weights were already introduced by Churchman et al. in 1954 and 1957)
or subjective non-additive scores like in the traditional reference
point theory (Brauers 2004: 158-159).
The Additive Weighting Procedure (MacCrimmon 1968: 29-33), which
was called SAW, Simple Additive Weighting Method by Hwang and Yoon
(1981: 99) starts from the following formula:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)
[U.sub.j] = overall utility of alternative j with j = 1, 2,.... ,
m, m the number of alternatives; [w.sub.i] = weight of attribute i
indicates as well as normalization as the level of importance of an
objective
i = 1, 2,..., n; n the number of attributes and objective; xij =
response of alternative j on attribute i.
In addition, weights adding up to one create a new artificial
super-objective, denying any form of multiple objectivity.
Reference Point Theory is non linear with, this time, non-additive
scores replacing weights. The non-additive scores take care of
normalization.
With weights and scores importance of objectives is mixed with
normalization. Indeed weights and scores are mixtures of normalization
of different units and of importance coefficients. On the contrary the
dimensionless measures of MULTIMOORA do not need external normalization.
However the problem of importance remains.
In the Ratio System to give more importance to an objective its
response on an alternative under the form of a dimensionless number
could be multiplied with a Significance Coefficient:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
with: i = 1, 2,... , g as the objectives to be maximized; i = g+1,
g+2, ..., n as the objectives to be minimized; [s.sub.i] = the
significance coefficient of objective I; [[??].sup.*.sub.j] .. = the
total assessment with significance coefficients of alternative j with
respect to all objectives.
For the Reference Point Approach the formula would be:
[absolute value of [s.sub.i] [r.sub.i] - [s.sub.i][x.sup.*.sub.ij].
(7)
If the Full Multiplicative Form has to stress the importance of an
objective a [alpha]-term is added or an exponent is allocated on
condition that it is done with unanimity or at least with a strong
convergence in opinion of all stakeholders concerned.
As the stakeholders concern in the banking example only a group of
candidate mortgagees a representative sample of them will be sufficient.
The attribution of sub-objectives represents another solution. Take
the example of the purchase of fighter planes (Brauers 2002). For
economics, the objectives concerning the fighter planes are threefold:
price, employment and balance of payments, but there is also military
effectiveness. In order to give more importance to military defense,
effectiveness is broken down in, for instance, the maximum speed, the
power of the engines and the maximum range of the plane. Anyway, the
attribution method is more refined than that a significance coefficient
method could be as the attribution method succeeds in characterizing an
objective better. For instance, for employment two sub-objectives
replace a significance coefficient of two and in this way characterize
the direct and indirect side of employment. In the banking example for
instance some criteria could be split for the older or for the younger
generations.
In addition the problem is raised of the subjective choice of
objectives in general, or could we call it robustness of a choice? The
Ameliorated Nominal Group Technique, as explained in Brauers (2004:
44-60), will gather a representative sample of all the candidate
mortgagees to determine the objectives in a non-subjective and anonymous
way. The original Nominal Group Technique of Van De Ven and Delbecq
(1971) was less robust as the Ameliorated Version, as this one excludes
subjective wishes of the group. Indeed in the Ameliorated Nominal Group
Technique the group is questioned about the probability of occurrence of
an event. In this way the experts become more critical even about their
own ideas. The probability of the group is found as the median of the
individual probabilities. Finally, the group rating (R) is multiplied
with the group probability (P) in order to obtain the effectiveness rate
of the event (E). The events are translated into objectives and selected
in a robust way by the Ameliorated Nominal Group Technique (for an
example, see Brauers. Lepkova (2003), also Brauers, Zavadskas (2010:
18-19).
Given the absence of a representative sample of all mortgagees in
the banking simulation all criteria are considered to have the same
importance.
Be careful with Rank Correlation in MULTIMOORA
If we would use for MULTIMOORA the total of the ranks of the ratio
system, the reference point and the multiplicative form at that moment
we would work ordinal and arrive in the rank correlation method (Kendall
1948). The most robust multi-objective method has to satisfy the
following condition: the method of multiple objectives based on cardinal
numbers is more robust than this one based on ordinal numbers. "An
ordinal number is one that indicates order or position in a series, like
first, second, etc." (Kendall and Gibbons 1990: 1). Robustness of
cardinal numbers is based first on the saying of Arrow (1974: 256):
"Obviously, a cardinal utility implies an ordinal preference but
not vice versa" and second on the fact that the four essential
operations of arithmetic: adding, subtracting, multiplication and
division are only reserved for cardinal numbers (see Brauers 2007 and
also: Brauers and Ginevicius 2009: 137-138).
Axioms on Ordinal and Cardinal Scales
1. A deduction of an Ordinal Scale, a ranking, from cardinal data
is always possible (Arrow).
2. An Ordinal Scale can never produce a series of cardinal numbers
(Arrow).
3. An Ordinal Scale of a certain kind, a ranking, can be translated
in an ordinal scale of another kind.
In application of axiom 3 we shall translate the ordinal scale of
the three methods of MULTIMOORA in another one based on Dominance, being
Dominated, Transitivity and Equability.
Dominance, being Dominated, Transitiveness and Equability
The three methods of MULTIMOORA are assumed to have the same
importance. Stakeholders or their representatives like experts may have
a different importance in ranking but this is not the case with the
three methods of MULTIMOORA. These three methods represent all existing
methods with dimensionless measures in multi-objective optimization and
all the three have the same important significance.
Dominance
Absolute Dominance means that an alternative, solution or project
is dominating in ranking all other alternatives, solutions or projects
which are all being dominated. This absolute dominance shows as rankings
for MULTIMOORA: (1-1-1).
General Dominance in two of the three methods is of the form with a
< b < c <d:
(d-a-a) is generally dominating (c-b-b);
(a-d-a) is generally dominating (b-c-b);
(a-a-d) is generally dominating (b-b-c),
and further transitiveness plays fully.
Transitiveness
If a dominates b and b dominates c than also a will dominate c.
Overall Dominance of one alternative on another
For instance (a-a-a) is overall dominating (b-b-b) which is overall
being dominated.
Equability
Absolute Equability has for instance the form: (e-e-e) for 2
alternatives.
Partial Equability of 2 on 3 exists e. g. (5-e-7) and (6-e-3).
Circular Reasoning
Despite all distinctions in classification some contradictions
remain possible in a kind of
Circular Reasoning.
We can cite the case of:
Object A (11-20-14) dominates generally object B. (14-16-15);
Object B. (14-16-15) dominates generally Object C (15-19-12); but Object
C (15-19-12) dominates generally Object A (11-20-14).
In such a case the same ranking is given to the three objects.
Application on the Banking Example
The different criteria as shown in table 1 are grouped in the
following way:
1) The Initial Payment:
-- The initial payment;
-- Hypothecation bond registration fee;
-- A one off loan administration fee;
-- Commission for currency exchange.
2) Regular Payments:
-- Loan repayment and payment of interest;
-- Insurance of the apartment to be purchased.
3) Life Insurance.
4) Monopolization of all bank activities and the wage
administration for the client.
Following Table 2 shows the new grouping.
Annexes A and B gives the details of the MULTIMOORA calculations.
Table 3 shows the results.
The Foreign Bank shows an ABSOLUTE DOMINANCE on the other banks.
Bank A shows a GENERAL DOMINANCE OF TWO ON THREE MULTIMOORA
rankings against Bank B.
Conclusion
MOORA and MULTIMOORA present strong instruments to measure an
optimum in economic calculus. MOORA is composed of a Ratio System,
producing dimensionless numbers. In addition, the ratio system creates
the opportunity to use its ratios as a starting point for a second
approach: a non-subjective Reference Point Theory. The two approaches
form a control on each other. The choice of the objectives is even more
non-subjective if the opinion of all stakeholders interested in the
issue are involved by the use of the Ameliorated Nominal Group and
Delphi Techniques. The overall theory is called MOORA (Multi-Objective
Optimization by Ratio Analysis). The results are still even more
convincing when a Full Multiplicative Form is added to MOORA under the
name of MULTIMOORA.
However at that moment we are in the ordinal sphere, the
combination of three rankings. Which is the next step as summation is
not allowed? Indeed addition belongs to the cardinal and not to the
ordinal sphere. The following axioms are rather to be respected:
1. A deduction of an Ordinal Scale, a ranking, from cardinal data
is always possible.
2. An Ordinal Scale can never produce a series of cardinal numbers.
3. An Ordinal Scale of a certain kind, a ranking, can be translated
in an ordinal scale of another kind.
On the one side we have three separate rankings belonging to the
ordinal sphere and on the other side we will replace them by another
ranking decided by a theory around dominance.
Absolute Dominance means that an alternative, solution or project
is dominating in ranking all other alternatives, solutions or projects
which are all being dominated. This absolute dominance shows as rankings
for MULTIMOORA: (1-1-1).
General Dominance in two of the three methods is of the form:
(d-a-a) is generally dominating (c-b-b);
(a-d-a) is generally dominating (b-c-b);
(a-a-d) is generally dominating (b-b-c).
Whereas Transitiveness means that if a dominates b and b dominates
c then also a will dominate c.
Applied on the loan demand for an apartment from a bank in a
simulation exercise a Foreign Bank shows an ABSOLUTE DOMINANCE on the
other banks. A second Bank shows a GENERAL DOMINANCE OF TWO ON THREE
MULTIMOORA rankings against a third Bank.
doi: 10.3846/13928619.2011.560632
Received 8 September 2010; accepted 27 January 2011
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Willem Karel M. BRAUERS was graduated as: Ph.D. in economics (Un.
of Leuven), Master of Arts (in economics) of Columbia Un. (New York),
Master in Management and Financial Sciences, in Political and Diplomatic
Sciences and Bachelor in Philosophy (Un. of Leuven). He is professor at
the Faculty of Applied Economics and at the Institute for Development
Policy and Management of the University of Antwerp. Previously, he was
professor at the University of Leuven, the Military Staff College, the
School of Military Administrators, and the Antwerp Business School. He
was a research fellow in several American institutions like Rand
Corporation, the Pentagon, the Institute for the Future, the Futures
Group and extraordinary advisor to the Center for Economic Studies of
the University of Leuven. He was consultant in the public sector, such
as the Belgian Department of National Defense, the Department of
Industry in Thailand, the project for the construction of a new port in
Algeria (the port of Arzew) and in the private sector such as the
international seaport of Antwerp and in electrical works. He was
Chairman of the Board of Directors of SORCA Ltd. Brussels, Management
Consultants for Developing Countries, linked to the world-wide group of
ARCADIS and Chairman of the Board of Directors of MARESCO Ltd. Antwerp,
Marketing Consultants. At the moment he is General Manager of
CONSULTING, Systems Engineering Consultants. Brauers is member of many
international scientific organizations. His specialization covers:
Optimizing Techniques with Several Objectives, Forecasting Techniques
and Public Sector Economics such as for National Defense and for
Regional Sub-optimization and Input-Output Techniques. His scientific
publications consist of seventeen books and hundreds of articles and
reports.
Edmundas Kazimieras ZAVADSKAS is Principal Vice-Rector of Vilnius
Gediminas Technical University, and Head of the Dept of Construction
Technology and Management at Vilnius Gediminas Technical University,
Vilnius, Lithuania. He has a PhD in Building Structures (1973) and Dr
Sc. (1987) in Building Technology and Management. He is a member of the
Lithuanian and several foreign Academies of Sciences. He is Doctore
Honoris Causa at Poznan, Saint-Petersburg, and Kiev. He is a member of
international organisations and has been a member of steering and
programme committees at many international conferences. E. K. Zavadskas
is a member of editorial boards of several research journals. He is
author and co-author of more than 400 papers and a number of monographs
in Lithuanian, English, German and Russian. Research interests are:
building technology and management, decision making theory, automation
in design and decision-support systems.
(1) For these items Zavadskas et al. 2004 inspired this article. In
addition we are grateful for a discussion with Prof. Banaitis, one of
the other authors.
(2) Brauers and Zavadskas (2006) prove that the most robust choice
for this denominator is the square root of the sum of squares of each
alternative per objective.
(3) Dimensionless Numbers, having no specific unit of measurement,
are obtained for instance by multiplication or division. The normalized
responses of the alternatives on the objectives belong to the interval
[0; 1]. However, sometimes the interval could be [-1; 1]. Indeed, for
instance in the case of productivity growth some sectors, regions or
countries may show a decrease instead of an increase in productivity
i.e. a negative dimensionless number. Instead of a normal increase in
productivity growth a decrease remains possible. At that moment the
interval becomes [-1, 1]. Take the example of productivity, which has to
increase (positive). Consequently, we look for a maximization of
productivity eg. in European and American countries. What if the
opposite does occur? For instance, take the original transition from the
USSR to Russia. Contrary to the other European countries productivity
decreased. It means that in formula (1) the numerator for Russia was
negative with the whole ratio becoming negative. Consequently, the
interval changes to: [-1, +1] instead of [0, 1].
(4) Brauers (2008) proves that the Min-Max metric is the most
robust choice between all the possible metrics of reference point
theory.
Willem Karel M. Brauers1, Edmundas Kazimieras Zavadskas (2)
(1) Vilnius Gediminas Technical University, Sauletekio al. 11,
10223 Vilnius, Lithuania
(2) Department of Construction Technology and Management, Vilnius
Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius,
Lithuania
E-mails: (1) willem.brauers@ua.ac.be (corresponding author); (2)
edmundas.zavadskas@vgtu.lt
Table 1. Conditions for a Bank Loan of 30,025 [euro]
initial monthly registration administration
payment repayment
MIN. MIN. MIN. MIN.
Foreign 5617.6 331.31 115.44 103.9
Bank
A 6926.41 317.46 129.87 115.44
B 6926.41 346.32 129.87 101.01
life insurance exchange bank
insurance deposits
MIN. MIN. MIN. MIN.
Foreign 0 46.18 115.44 0
Bank
A 69.26 57.72 0 0.5
B 69.26 57.72 0 1
Table 2. Criteria for Loan Payment
1+3+4+7 2+6/12 5 8
MIN. MIN. MIN.
FB 5952.38 335.5 0 0
A 7171.72 322.27 69.26 0.5
B 7157.29 351.13 69.26 1
Table 3. MULTIMOORA results for the Banking Example
RS RP MF MULTIMOORA
FB 1 1 1 1
A 2 2 3 2
B 3 3 2 3
Table 4. MOORA applied on 3 banks Lt with 8 conditions 4a-Matrix of
Responses of Alternatives on Objectives: ([x.sub.ij])
monthly registration
initial payment repayment Administration
MIN. MIN. MIN. MIN.
FB 5617.6 331.31 115.44 103.9
A 6926.41 317.46 129.87 115.44
B 6926.41 346.32 129.87 101.01
4b-Sum of squares and their square roots
FB 31557429.76 109766.3161 13326.3936 10795.21
A 47975155.49 100780.8516 16866.2169 13326.394
B 47975155.49 119937.5424 16866.2169 10203.020
[SIGMA] 127507740.7 330484.7101 47058.8274 34324.624
root 11291.93255 574.8779958 216.9304667 185.26906
4c-Objectives divided by their square roots and MOORA
FB 0.497487917 0.576313587 0.532152084 0.5608060
A 0.61339456 0.552221519 0.598671095 0.6230938
B 0.61339456 0.602423475 0.598671095 0.5452071
4d-Reference Point Theory with Ratios: co-ordinates of the reference
point equal to the maximal objective values
[r.sub.i] 0.4975 0.5522 0.5322 0.5452
4e-Reference Point Theory: Deviations from the reference point
FB 0 0.0241 0 0.0156
A 0.1159 0 0.0665 0.0779
B 0.1159 0.0502 0.0665 0.0000
life bank
insurance insurance exchange deposits
MIN. MIN. MIN. MIN.
FB 0 46.18 115.44 0
A 69.26 57.72 0 0.5
B 69.26 57.72 0 1
4b-Sum of squares and their square roots
FB 0 2132.5924 13326.3936 0
A 4796.9476 3331.5984 0 0.25
B 4796.9476 3331.5984 0 1
[SIGMA] 9594 8795.789 13326.394 1.2500
root 97.948431 93.785869 115.44 1.118034
4c-Objectives divided by their square roots and MOORA
FB 0 0.4923983 1 0.000
A 0.7071068 0.6154445 0 0.4472136
B 0.7071068 0.6154445 0 0.8944272
4d-Reference Point Theory with Ratios: co-ordinates of the reference
point equal to the maximal objective values
[r.sub.i] 0 0.4923983 0 0.000
4e-Reference Point Theory: Deviations from the reference point
FB 0 0.0000 1 0.000
A 0.7071068 0.1230 0 0.4472136
B 0.7071068 0.1230 0 0.894
FB
A
B
4b-Sum of squares and their square roots
FB
A
B
[SIGMA]
root
4c-Objectisum
FB -2.6591579 [a.sub.1] 1
A -1.8485051 [a.sub.2] 2
B -1.3736068 [a.sub.3] 3
4d-Reference Point Theory with Ratios: co-ordinates of the reference
point equal to the maximal objective values
[r.sub.i]
4e-Reference max. min.
FB 1.0000000 [a.sub.1] 3
A 0.7071068 [a.sub.2] 1
B 0.8944272 [a.sub.3] 2
Table 4bis
4a'-Matrix of Responses of Alternatives on Objectives: ([x.sub.ij])
1+3+4+7 2+6/12 5 8
MIN MIN. MIN.
FB 5952.38 335.158333 0 0
A 7171.72 322.27 69.26 0.5
B 7157.29 351.13 69.26 1
4b'-Sum of squares and their square roots
FB 35430827.66 112331.1084 0 0
A 51433567.76 103857.9529 4796.9476 0.25
B 51226800.14 123292.2769 4796.9476 1
[SIGMA] 138091195.6 339481.3382 9593.89520 1.2500
root 11751.22102 582.6502709 97.94843133 1.118034
4c'-Objectives divided by their square roots and MOORA
FB 0.506532895 0.575230717 0 0
A 0.610295729 0.55311053 0.707106781 0.4472136
B 0.609067771 0.602642816 0.707106781 0.8944272
4d'-Reference Point Theory with Ratios: co-ordinates of the
reference point equal to the maximal objective values
[r.sub.i] 0.5065 0.5531 0.0000 0.0000
4e'-Reference Point Theory: Deviations from the reference point
FB 0.0000 0.0221 0.0000 0.0000
A 0.1038 0.0000 0.7071 0.4472
B 0.1025 0.0495 0.7071 0.8944
FB
A
B
4b'-Sum of squares and their square roots
FB
A
B
[SIGMA]
root
4c'-Objectives divided by their square roots and MOORA
sum min.
FB 1.0817636 1
A 2.3177266 2
B 2.8132446 3
4d'-Reference Point Theory with Ratios: co-ordinates of the
reference point equal to the maximal objective values
[r.sub.i]
4e'-Reference Point Theory: Deviations from the reference point
max min.
FB 0.0221202 1
A 0.7071068 2
B 0.8944272 3
Table 5. Full multplicative method applied on 3 banks Lt with 8
conditions
1 3 4
MIN. MIN. MIN. MIN.
FB 5617.6 331.31 16.955721 115.44 0.146879 103.9
A 6926.4 317.46 21.818213 129.87 0.168000 115.44
B 6926.4 346.32 20.000029 129.87 0.154000 101.01
5 6 7
MIN. MIN. MIN.
FB 0.0014137 100.0 1.41366E-05 46.18 3.06119E-07 115.44
A 0.0014553 69.26 2.10122E-05 57.72 3.64037E-07 200
B 0.0015246 69.26 2.20128E-05 57.72 3.81372E-07 200
8
MIN.
FB 2.65176E-09 2 1.3259E-09 1
A 1.82018E-09 0.5 3.6404E-09 3
B 1.90686E-09 1 1.9069E-09 2
4a-Matrix of Responses of Alternatives on Objectives: ([x.sub.ij])
1 2 3 4 5
initial monthly registration administration life
payment repay insurance
MIN. MIN. MIN. MIN. MIN.
FB 5617.6 331.31 115.44 103.9 0
A 6926.41 317.46 129.87 115.44 69.26
B 6926.41 346.32 129.87 101.01 69.26
6 7 8
insurance exchange deposit
MIN. MIN. MIN.
FB 46.18 115.44 2
A 57.72 0 0.5
B 57.72 0 1
Table 5bis. Full multplicative method applied on 3 banks Lt with 4
conditions
3+4+7+1 2+6/12 5
MIN. MIN. MIN. MIN.
FB 4952.38 335.1583333 14.77624009 100 0.1477624 2
A 7171.72 322.27 22.25376237 69.26 0.3213076 0.5
B 7096.68 351.13 20.21097599 69.26 0.2918131 1
8 and results
FB 0.0738812 1
A 0.64261514 3
B 0.29181311 2
SUMMARY OF THE 3 METHODS OF MULTIMOORA
RS RP MF MULTIMOORA
FB 1 1 1 1 ABSOLUTE DOMINANCE
A 2 2 3 2 GENERAL DOMINANCE
OF TWO ON THREE RANKINGS
B 3 3 2 3
