LEVI 3.0 – A Program For Multiple Criteria Evaluation
Peldschus, F., Leipzig University of Applied Science
Zavadskas, E.K., Vilnius Gediminas Technical University
Ustinovičius, L., Vilnius Gediminas Technical University
1. General description of the program
The program LEVI 3.0 is a result of the co-operation between
the Vilnius Gediminas Technical University and the Leipzig
University of Applied Sciences. Both institutions have done
research into the problem of multiple criteria evaluation for
several years. Due to the different approaches different
methods were used by the two institutions. The common aim has
always been the evaluation of the optimal variant for
production processes in the building sector. With this new
software it is possible to find solutions for a task using
different methods and to compare the solutions.
2. Structure of the program
2.1 Input
Every problem to be solved is represented by a matrix, which
contains variants (rows) and criteria (columns). The variants
represent a set of situations for a problem that really exist.
All considered variants are evaluated using the same criteria.
The results of the evaluation are put in a matrix. Usually the
criteria have different dimensions. That is why their
effectiveness can not be compared directly. An exception is
the application of evaluation numbers without any dimensions
according to a points system. This, however, involves
subjective influences to a great extent. Hence, it should only
be used in exceptional cases.
In order to avoid the difficulties due to different dimensions
of the criteria, the ratio to the optimal value is used. That
way the discrepancy between the different dimensions of the
optimal values is also eliminated. There are various theories
about the ratio to the optimal value. Note that the decision
for a theory may affect the solution. However, the values are
mapped either on the interval [0;1] or on the interval
[0,infinity) by the transformation.
Only those well-known theories of transformation are used that
are appropriate for both problems of maximisation and
minimisation.
Transformation through normalization of vectors is applied
mainly to the method of solution “distance to the ideal
point”, but may also be used for other methods. The ratio of
the values remains constant for this transformation to the
interval [0;1].
aij
bij 
(1)
m
a
i 1
2
ij
The linear transformation uses a scale of the existing values.
The calculated values are dependent on the size of the
interval [a(io);a(iu)] and thus change if the interval is
altered.
bij 
aij  aiu
if bij should be maximised
aio  aiu
(2)
bij 
aio  aij
if bij should be minimised
aio  aiu
aio – maximum value, aiu – minimum value
The calculation of the relative deviation is a well performing
linear transformation. The application of this transformation
is limited to an interval (0..2 Min).
a j  aij
*
bij  1 
aj
(3)
*
a(j*) – optimal value for the criterion
Using the non-linear transformation the calculation of the
results is not dependent on the size of the interval. The
values, however, are diminished more than by using the other
methods.
 Min aij

bij   i
 aij






3
if Min
i
a ij is desirable
(4)

 aij
bij  
 Max aij
i

2


a ij is desirable
 if Max
i


After the transformation it is possible to evaluate the
criteria with factors 0 < q(j) < 1. The sum of the weighting
factors for the different criteria should equal 1, otherwise
errors might occur in the solution. Only well-founded
weighting factors should be used because the weighting factors
are always subjective and influence the solution.
2.2 Methods of solution
A distinction is made between one-sided and two-sided problems
for the methods of solution.
The one-sided problems are solved using various well-known
methods of the selection of variants and the determination of
an order of precedence.
Using the Game Theory, the two-sided question aims at finding
the equilibrium as a result of the rational behaviour of two
parties having opposite interests or at the equilibrium in a
game against nature.
2.2.1 One-sided problems
For one-sided problems only the method of solution “distance
to the ideal point” is considered in the actual version. A
completion is planned.
Using the method “distance to the ideal point” an order of
precedence according to the deviation from the ideal variant
is determined. The solutions consists of the following steps:
The input matrix is transformed according to formula (1). With
these dimension-less values the set a+ (ideal variant) and a(negatively ideal variant) are calculated.
 max f / j  J , min f
  min f / j  J , max f
a 
a
ij
i
i
ij
i
i
ij
ij


/ j  J   / i  1, m f




(5)



(6)
/ j  J  / i  1, m  f 1 , f 2 ,..., f n

1
, f 2 ,..., f n
where J is the set of the problems of maximisation and J’ is
the set of the problems of minimisation.
With the distances L and L

Li 

Li 
f
 fj
 2
ij
f
 fj
 2
ij
n
j 1
n
j 1


;  i; i  1, m
; i; i  1, m
(7)
(8)
the relative proximity to the ideal variant is determined.
Ki 
L1


Li  Li

,  i ; i  1, m
(9)
The calculated values K(i) are in the interval [0;1] and can
be ordered in a decreasing sequence, which is the wanted order
of precedence.
The limits of the interval are obtained from:
1 , if ai  a  ;
Ki  
0 , if ai  a  .
(10)
2.2.2 Two-sided problems
For two-sided problems a distinction is made between games
with rational behaviour and games against nature.
2.2.2.1 Games with rational behaviour
The solutions for problems with rational behaviour are found
in the ideal case as a saddle point solution (simple min-max
principle) or as a combination of strategies (extended min-max
principle).
Simple min-max principle
  max min aij
j
i
  min max aij
j
(11)
i
If      the solution with pure strategies is a saddle point
(only one optimal strategy for each player)- trivial solution.
Extended min-max principle
Calculation of a point of equilibrium with mixed strategies
(combination of strategies)


max min A( s1 , s2 )  min max A s1 , s2   A s1 , s2 
j
i
i
j
*
*
(12)
2.2.2 Games against nature
Wald’s rule: This method searches for the best of the worse
solutions. The decision-maker acts according to the worst
situation occurring – pessimistic attitude.

S1  S1i / S1i  S1  max min aij
*
i
j

(13)
Savage criterion: The aim is the minimization of the loss of
appropriateness, which is the difference between the greatest
and the achieved benefit.


 
S1  S1i / S1i  S1  min max cij  cij  max a rs  a rs
*
i
j
r
(14)
r = 1, m ; s = 1, n
Disadvantage of the method: the presence of non-optimal
strategies affects the solution.
Hurwicz’s rule: The optimal strategy is based on the best and
the worst result. These values, calculated from the row
minimum and row maximum, are unified to a weighted average
using optimism parameters.


S1  S1i / S1i  S1  max hi  hi  min aij  1    max aij  0    1
*
i
i
j
(15)
The value  =1 gives the most pessimistic solution (Wald’s
rule).For the value  =0 only the maximal values are
considered, greatest risk.
Laplace’s rule: The solution is calculated under the
condition, that all probabilities for the strategies of the
opponent are equal.
n
*



S1  S1i / S1i  S1  max 1 / n aij 
i
i 1



(16)
Bayes’s rule: If the probabilities for the strategies of the
opponent are given, the maximum for the expected value can be
used.
n


 n

*
S1  S1i / S1i  max   q j aij    q j  1
i


 j 1
 j 1
(17)
Hodges-Lehmann-rule: With this rule confidence in the
knowledge of the probabilities of the strategies of the
opponent can be expressed by the parameter  .


 n

*
S1  S1i / S1i  S1  max   q j aij  1    min aij   0    1
(18)
j
i


 j 1

 =0 (no confidence) gives the solution according to Wald’s
rule.  =1 (great confidence) gives the solution according to
Bayes’s rule.
2.2.3 Solutions for series of production
The optimal variant for production of various products with
different number of pieces is calculated using a weighted
matrix. A requirement is that all product can be manufactured
with the same variants of production. For each product a
matrix for the considered variants of production is made and
evaluated using the selected criteria. The sum of all these
matrices, which are multiplied with the number of units of the
corresponding product, is divided by the total number of
units. Thus the modified matrix, which is processed using the
described methods of solution, is created. The optimal variant
of production is determined using the described methods of
solution.
3. Applications
The program LEVI was developed for the calculation of
production processes in the building sector. Following the
static equilibrium the equilibrium in the game theory has a
particular significance. Comparing calculations with other
methods of solution are necessary because the application of
the equilibrium in the game theory to the building process is
not always possible.
Apart from the practical use of the program LEVI the
scientific interest should be mentioned. Up to now it is not
possible to evaluate the effects of the different methods of
transformation on the numerical result. Another problem is the
variety of the solutions. This causes difficulties for the
practical user.
These problems are to be solved through the application of the
program LEVI.