The Values of the Equilibrium Point for the Sustainable Decision Making in Construction

Transcription

1 5 th 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 M. Grasserbauer, L. Sakalauskas, E. K. Zavadskas (Eds.): KORSD-2009 Selected papers. Vilnius, 2009, pp 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 Zavadskas 1, Friedel Peldschus 2, Zenonas Turskis 3, Jolanta Tamosaitiene 4 1, 3, 4 Vilnius Gediminas Technical University, Sauletekio av. 11, LT-2040 Vilnius, Lithuania 2 Leipzig University of Applied Sciences, Karl-Liebkneckt 132, Leipzig, Germany 1 edmundas.zavadskas@adm.vgtu.lt; 2 peldschu@fbb.htwk-leipzig.de; 3 zenonas.turskis@st.vgtu.lt; 4 jolanta.tamosaitiene@st.vgtu.lt; 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 matri plays. Keywords: Game theory, equilibrium point, building site, assessment. 1 Introduction 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 matri 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. 2 Game theory 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 matri games. Zavadskas and Turskis (2008) developed a new logarithmic normalization method in games theory. Zavadskas et al. (2008) developed multi-criteria optimization software 154

2 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 eperience 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. Matri plays are limited by two-person-zero sum plays (Hollert 2006), (Peldschus, Zavadskas 1997). For the game: Γ = ( S 1, S2, A) (1) In the formula S 1 i 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-ma principle (Arrow et al. 1949)) or a strategy combination (etended min-ma principle) is obtained. S S S 2 n S 11 a 11 a a 1 n S 12 a 21 a a 2 n M M S 1 m a m1 a m2... a mn Simple min ma principle is appraisal by: α = ma mina, β = min maa ; (2) i j ij If α = β= γ, a saddle point with pure strategies (one optimal strategy for each player) is obtained as solution trivial solution. Etended min ma principle. An equilibrium point with mied strategies is calculated (combination of strategies): * * ma min A( s1, s2) = min ma A( s1, s2) = A( s1, s2 )ν = (3) i j i j 3 Applications The main application of matri 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 effec- j i ij 155

3 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 eample: construction building site assessment For building site assessment applying game theory is calculated the equilibrium point. For the decision making is construct the matri of the alternative. For the practical eample 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: 1 price, [mln.euro/1a.]; 2 site area, [a.]; 3 soak density [%]; 4 site technical, [point]; site ecological, [point]; 5 6 social facility, [point]; 7 appeal district, [point]; green epansion, [point]. 8 The weight of the criteria was establisch applying questioning of the ten eperts. The initial decision making matri, the weight of the criteria and the criteria optimization direction is presented in the Table 1. Table 1. Initial decision making matri Criteria Optimization direction min min ma ma ma ma ma ma Weight Alternative Alternative Alternative Alternative Alternative As a result one receives the values α = 0.02 and β = This means: there eists no equilibrium point. Therefore, the calculation with etended min ma principle occurs. * As a result one receives the following vectors: for the first player S 1 = (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

4 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 matri. 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 eplanation 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 Bayes and Minima Solutions of Sequential Decision Problems, Econometrica 17: Hollert, M. J Einführung in die Spieltheorie. Springer Verlag Berlin: 429. Homburg, C.; Scherpereel, P How should the cost of joint risk capital be allocated for performance measurement? European Journal of Operational Research 187(1): Levitin, G.; Hausken, K False Targets Efficiency in Defense Strategy, European Journal of Operational Research 194(1): Motchenkova, E Determination of optimal penalties for antitrust violations in a dynamic setting, European Journal of Operational Research 189(1): Peldschus, F.; Vaigauskas, E.; Zavadskas, E. K Technologische Entscheidungen bei der Berücksichtigung mehrerer Ziehle, Bauplanung Bautechnik 37(4): Peldschus, F Zur Anwendung der Theorie der Spiele für Aufgaben der Bautechnologie. Diss. B. Technologie. Diss. B. Technische Hochschule Leipzig. Peldschus, F.; Zavadskas, E. K Matri Games in Building Technology and Management. Vilnius: Technika (in Lithuanian). Peldschus, F.; Zavadskas, E. K Fuzzy Matri Games Multi-Criteria Model for Decision-Making in Engineering, Informatica 16(1): Peldschus, F. 2007a. The effectiveness of assessments in multi-criteria decision, International Journal Management and Decision Making 8(2/3): 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, Vilnius, Lithuania, Selected Papers,

5 E. K. Zavadskas, F. Peldschus, Z. Turskis, J. Tamosaitiene Peldschus, F Eperience of the game theory application in construction management, Technological and Economic Development of Economy 14(4): Schotanus, F.; Telgen, J. de Boer, L Unfair allocation of gains under the equal price allocation method in purchasing groups, European Journal of Operational Research 187(1): Su, B. B.; Chang, H.; Chen, Y.-Z.; He, D. R A game theory model of urban public traffic networks, Physica A: Statistical Mechanics and its Applications 379(1): Sun, L.-J.; Gao, Z.-Y An equilibrium model for urban transit assignment based on game theory, European Journal of Operational Research 181(1): Tamošaitienė, J.; Turskis, Z.; Zavadskas, E. K 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, Turskis, Z.; Zavadskas, E. K.; Peldschus, F Multi-criteria optimization system for decision making in construction design and management, Inzinerine Ekonomika Engineering Economics 1(61): Westergaard, M A Game-theoretic approach to behavioural visualisation, Electronic Notes in Theoretical Computer Science 208: Zavadskas, E. K.; Ustinovichius, L.; Peldschus, F Development of software for multiple criteria evaluation, Informatica 14(2): Zavadskas, E. K.; Peldschus, F.; Ustinovichius, L.; Turskis, Z Game Theory in Building Technology and Management. Vilnius: Technika (in Lithuanian). Zavadskas, E. K.; Peldschus, F.; Turskis, Z Multi-criteria optimization software LEVI 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, Zavadskas, E. K.; Turskis, Z A new logarithmic normalization method in games theory, Informatica 19(2):