5th International Vilnius Conference
EURO Mini Conference
“Knowledge-Based Technologies and OR Methodologies
for Strategic Decisions of Sustainable Development”
(KORSD-2009)
September 30–October 3, 2009, Vilnius, LITHUANIA
ISBN 978-9955-28-482-6
M. Grasserbauer, L. Sakalauskas,
E. K. Zavadskas (Eds.): KORSD-2009
Selected papers. Vilnius, 2009, pp. 154–158
© Institute of Mathematics and Informatics, 2009
© Vilnius Gediminas Technical University, 2009
The Values of the Equilibrium Point for the Sustainable
Decision Making in Construction
Edmundas Kazimieras Zavadskas1, Friedel Peldschus2,
Zenonas Turskis3, Jolanta Tamosaitiene4
1, 3, 4
Vilnius Gediminas Technical University,
Sauletekio av. 11, LT-2040 Vilnius, Lithuania
2
Leipzig University of Applied Sciences,
Karl-Liebkneckt 132, 04277 Leipzig, Germany
E-mail: [email protected]; [email protected];
3
[email protected]; [email protected];
Abstract: Different methods for the solution of decisive duties are known. Inclusively one can
make a distinction for the numerical solution in a one-sided and a two-sided problem. Besides,
the calculation of an equilibrium point with a play-theoretical solution counts to the two-side
problem. This solution differs from the one-sided question by bigger information content.
While one is content with the one-sided problem with the assessment of the variations, an assessment of the criteria also occurs with the two- side problem. This statement can deliver
valuable tips for appraisal of a complicated job. The numerical calculation occurs with the theory of the matrix plays.
Keywords: Game theory, equilibrium point, building site, assessment.
1
Introduction
2
Game theory
For the solution of decisive duties different methods are known. Inclusively one can make
a distinction for the numerical solution in a one-sided and a two-sided problem. Besides, the
calculation of a equilibrium point with a play-theoretical solution counts to the two- side problem. This solution differs from the one-sided problem by bigger information content. While one
is content with the one-sided question with the assessment of the variations, an assessment of the
criteria also occurs with the two-side problem. This statement can deliver valuable tips for appraisal of a complicated job. The numerical calculation occurs with the theory of the matrix
plays.
The game theory can by used for sustainable decision making. In this case is very important to find the equilibrium point for the sustainable decision making.
Game theory analysed many researches: Peldschus et al. (1983). Peldschus (1986, 2007a,
b) analysed the effectiveness of assessments in multi-criteria decision. Zavadskas et al. (2003)
develop the software for multiple criteria evaluation. Peldschus and Zavadskas (2005) developed
fuzzy matrix games. Zavadskas and Turskis (2008) developed a new logarithmic normalization
method in games theory. Zavadskas et al. (2008) developed multi-criteria optimization software
154
THE VALUES OF THE EQUILIBRIUM POINT FOR THE SUSTAINABLE DECISION…
LEVI-4.0 based on the game theory; multi-criteria optimization system for decision making in
construction design and management analysed Turskis et al. (2009).
Game theory was applying in many fields: Zavadskas et al. (2004) analysed the building
technology and management problem, Su et al. (2007) applied the game theory model of urban
public traffic networks; Sun and Gao (2007) applied an equilibrium model for urban transit assignment; Homburg and Scherpereel (2008) analysed the cost of joint risk capital be allocated
for performance measurement; Motchenkova (2008) analysed the determination of optimal penalties for antitrust violations in a dynamic setting; Peldschus (2008) review the experience of the
game theory application in construction management fields; Schotanus et al. (2008) analysed
unfair allocation of gains under the equal price allocation method in purchasing groups; Tamošaitienė et al. (2008) applied the game theory for the modelling of contractor selection taking
into account different risk level; Westergaard 2008 analysed the game-theoretic approach to
visualisation problems; Levitin and Hausken (2009) analysed the false targets efficiency in defence strategy.
Matrix plays are limited by two-person-zero sum plays (Hollert 2006), (Peldschus, Zavadskas 1997). For the game:
Γ = (S1, S 2 , A)
(1)
In the formula S1i for i = 1,...,m strategies of the first player and S 2 j is for j = 1,...,n strategies
of the second player and the pay-off function for the first and second player: or these games ideally a saddle point solution (simple min-max principle (Arrow et al. 1949)) or a strategy combination (extended min-max principle) is obtained.
S11
a11
M
M
S12
S1m
S 22
S 21
a21
am1
S 2n
a12
a22
...
...
...
a1n
am 2
...
amn
a2 n
Simple min – max principle is appraisal by:
α = max min aij , β = min max aij ;
i
j
j
(2)
i
If α = β = γ , a saddle point with pure strategies (one optimal strategy for each player) is
obtained as solution – trivial solution.
Extended min – max principle. An equilibrium point with mixed strategies is calculated
(combination of strategies):
(
)
max min A( s1, s2 ) = min max A (s1, s2 ) = A s1* , s2* =ν
i
3
j
i
j
Applications
(3)
The main application of matrix games is the selection of alternative, which is a problem of
multi-criteria decisions. For the description of the problem the alternatives are assigned to the
strategies of player first and the criteria are assigned to the strategies of player second. For the
pay-off function a dimensionless evaluation numbers are used in simple cases. Such dimensionless evaluation numbers describe the situation only coarsely. It is therefore sensible to use
real characteristic values. As such characteristic values have different dimensions their effec155
E. K. Zavadskas, F. Peldschus, Z. Turskis, J. Tamosaitiene
tiveness is not comparable. In order to allow a comparability of the characteristic values they are
mapped on the interval [1;0] or [1;~0]. Depending on the kind of problem there are several options for the transformation of the characteristic values. Generally, a distinction can be made
between linear and non-linear transformations (Peldschus 2007a).
4
Calculation of the equilibrium point for practical example:
construction building site assessment
For building site assessment applying game theory is calculated the equilibrium point. For
the decision making is construct the matrix of the alternative. For the practical example was assessed five different constructions building site. She is in different site of Vilnius city. For the
construction building site assessment was ranking by eighth criteria:
x1 – price, [mln.Euro/1a.];
x2
x3
x4
x5
x6
x7
x8
– site area, [a.];
– soak density [%];
– site technical, [point];
– site ecological, [point];
– social facility, [point];
– appeal district, [point];
– green expansion, [point].
The weight of the criteria was establisch applying questioning of the ten experts. The initial
decision making matrix, the weight of the criteria and the criteria optimization direction is presented in the Table 1.
Table 1. Initial decision making matrix
Criteria
Optimization
direction
Weight
Alternative 1
Alternative 2
Alternative 3
Alternative 4
Alternative 5
x1
x2
x3
x4
x5
x6
x7
x8
min
min
max
max
max
max
max
max
0.21
0.039
0.035
0.042
0.050
0.048
0.16
10
20
40
60
80
0.15
45
43
41
38
44
0.05
12
46
38
67
24
0.05
5
2
6
2
4
0.11
2
5
3
8
6
0.12
4
5
3
2
4
0.07
8
7
9
2
4
As a result one receives the values α = 0.02 and β = 0.05. This means: there exists no equilibrium point. Therefore, the calculation with extended min – max principle occurs.
As a result one receives the following vectors: for the first player S1* = (0.00; 0.00; 0.60;
0.00; 0.40); and the second player S 2* = (0.00; 0.00; 0.00; 0.00; 0.69; 0.31; 0.00; 0.00).
The calculation of the equilibrium point states that the alternative 3 receives an assessment
with 60% and the alternative 5 an assessment with 40%. The alternatives 1, 2 and 4 are not involved in the calculation of the equilibrium point because her functional influence is lower and
is dominated, therefore, by the other influence.
156
THE VALUES OF THE EQUILIBRIUM POINT FOR THE SUSTAINABLE DECISION…
As a specific feature with the calculation of the equilibrium point still the assessment of the
criteria can be given here. This is represented by the optimum strategy for the second player.
Then is valued the criterion 5 (ecology) with 69% and the criterion 6 (social infrastructure) with
31%. The criteria 1, 2, 3, 4, 7 and 8 are not involved in the calculation of the equilibrium point,
because they are dominated by the criteria 5 and 6.
5
Results
The calculation of an equilibrium point delivers more information. Since beside the assessment of the alternatives an assessment of the criteria also occurs.
With the calculation of a equilibrium point the same assessment is lifted for all criteria. The
meaning of the criteria comes thus into the result as her effectiveness appears in the transformed
matrix. That is the effectiveness of the criteria is depending on the toe-in of the initial values.
Besides, a big toe-in means a big influence and a low toe-in a small influence. Such an explanation seems logical and plausible. An insinuation for the same effectiveness of the criteria cannot
be founded unambiguously and stands also in the contradiction to the use of important factor. If
one is ready by important factor the meaning of the criteria to change, then one must also accept
that her effectiveness on the solution is different.
With the calculated solution one also has to carry out the possibility special investigations
for the assessment of a complicated problem. If another assessment of the alternative is wished,
one can work on the criteria involved in the equilibrium point according to her effectiveness
specifically.
The calculation of a equilibrium point is to be carried out an other possibility multiattribute decisions for complicated duties.
References
Arrow, K. J.; Blackwell, D.; Girshick, M. A. 1949. Bayes and Minimax Solutions of Sequential Decision
Problems, Econometrica 17: 213–243.
Hollert, M. J. 2006. Einführung in die Spieltheorie. Springer Verlag Berlin: 429.
Homburg, C.; Scherpereel, P. 2008. How should the cost of joint risk capital be allocated for performance
measurement? European Journal of Operational Research 187(1): 294–312.
Levitin, G.; Hausken, K. 2009. False Targets Efficiency in Defense Strategy, European Journal of Operational Research 194(1): 155–162.
Motchenkova, E. 2008. Determination of optimal penalties for antitrust violations in a dynamic setting,
European Journal of Operational Research 189(1): 269–291.
Peldschus, F.; Vaigauskas, E.; Zavadskas, E. K. 1983. Technologische Entscheidungen bei der
Berücksichtigung mehrerer Ziehle, Bauplanung–Bautechnik 37(4): 173–175.
Peldschus, F. 1986. Zur Anwendung der Theorie der Spiele für Aufgaben der Bautechnologie. Diss. B.
Technologie. Diss. B. Technische Hochschule Leipzig.
Peldschus, F.; Zavadskas, E. K. 1997. Matrix Games in Building Technology and Management. Vilnius:
Technika (in Lithuanian).
Peldschus, F.; Zavadskas, E. K. 2005. Fuzzy Matrix Games Multi-Criteria Model for Decision-Making in
Engineering, Informatica 16(1): 107–120.
Peldschus, F. 2007a. The effectiveness of assessments in multi-criteria decision, International Journal
Management and Decision Making 8(2/3): 193–200.
Peldschus, F. 2007b. Games-theory solutions in construction operation, In Modern Building Materials,
Structures and Techniques, in M. J. Skibniewski; P. Vainiūnas; E. K. Zavadskas (Eds.). The 9th International Conference, May 16–18, 2007. Vilnius, Lithuania, Selected Papers, 348–353.
157
E. K. Zavadskas, F. Peldschus, Z. Turskis, J. Tamosaitiene
Peldschus, F. 2008. Experience of the game theory application in construction management, Technological
and Economic Development of Economy 14(4): 531–545.
Schotanus, F.; Telgen, J. de Boer, L. 2008. Unfair allocation of gains under the equal price allocation
method in purchasing groups, European Journal of Operational Research 187(1): 162–176.
Su, B. B.; Chang, H.; Chen, Y.-Z.; He, D. R. 2007. A game theory model of urban public traffic networks,
Physica A: Statistical Mechanics and its Applications 379(1): 291–297.
Sun, L.-J.; Gao, Z.-Y. 2007. An equilibrium model for urban transit assignment based on game theory,
European Journal of Operational Research 181(1): 305–314.
Tamošaitienė, J.; Turskis, Z.; Zavadskas, E. K. 2008. Modelling of contractor selection taking into account
different risk level, In 25th International Symposium on Automation and Robotics in Construction
ISARC-2008, in E. K. Zavadskas; A. Kaklauskas; M. J. Skibniewski (Eds.). The 25th International
Symposium on Automation and Robotics in Construction, Jun 26–29, 2008 Vilnius Lithuania, Selected Papers, 676–681.
Turskis, Z.; Zavadskas, E. K.; Peldschus, F. 2009. Multi-criteria optimization system for decision making
in construction design and management, Inzinerine Ekonomika–Engineering Economics 1(61): 7–17.
Westergaard, M. 2008. A Game-theoretic approach to behavioural visualisation, Electronic Notes in Theoretical Computer Science 208: 113–129.
Zavadskas, E. K.; Ustinovichius, L.; Peldschus, F. 2003. Development of software for multiple criteria
evaluation, Informatica 14(2): 259–272.
Zavadskas, E. K.; Peldschus, F.; Ustinovichius, L.; Turskis, Z. 2004. Game Theory in Building Technology
and Management. Vilnius: Technika (in Lithuanian).
Zavadskas, E. K.; Peldschus, F.; Turskis, Z. 2008. Multi-criteria optimization software LEVI-4.0 - A tool
to support design and management in construction, in 25th International Symposium on Automation
and Robotics in Construction ISARC-2008, in E. K. Zavadskas; A. Kaklauskas; M. J. Skibniewski
(Eds.). The 25th International Symposium on Automation and Robotics in Construction, Jun 26–29,
2008 Vilnius Lithuania, Selected Papers, 731–736.
Zavadskas, E. K.; Turskis, Z. 2008. A new logarithmic normalization method in games theory, Informatica
19(2): 303–314.
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