Selection of the effective dwelling house walls by applying attributes values determined at intervals/Daugiabuciu namu isoriniu sienu efektyvumo vertinimas, kai efektyvumo rodikliu reiksmes aprasomos intervalais.

Zavadskas, Edmundas Kazimieras ; Kaklauskas, Arturas ; Turskis, Zenonas 等

Abstract. The higher life quality standards, the changes of habits and new well-being requirements have led to an increase in the demand for housing. Decision-making problems in construction management often involve a complex decision-making process in which multiple requirements and conditions have to be taken into consideration simultaneously. However, not every attribute used in multidimensional scaling is equally and precisely weighted in the real world. Thus quantitative and qualitative assessments are often required to deal with uncertainty, subjective and imprecise data. The accuracy of performance measures in common multi-attribute methods is usually assumed to be accurate. Grey theory is a new technique for performing prediction, relational analysis and decision-making in many areas. This paper considers the application of grey relations methodology for defining the utility of an alternative and is proposed as a method of multiple criteria COmplex PRoportional ASsessment of alternatives with Grey relations (COPRAS-G). In this model parameters of the alternatives are determined by the grey relational grade and are expressed in intervals. A case study of assessing external walls of four alternatives was used to demonstrate the applicability and the effectiveness of the proposed approach. The results show that this method can be implemented as an effective decision aid in multi-attribute selection.

Keywords: walls, alternatives, multi-attribute, selection, COPRAS, COPRAS-G, grey number.

Santrauka

Straipsnio tikslas--pasiulyti alternatyvia daugelio rodikliu vertinimo metodika. Sios metodikos isskirtinumas--rodikliu reiksmes, aprasomos intervalais. Tai labiau atitinka realias aplinkos salygas ir galimas reiksmes. Modelis su intervaluose pateiktomis efektyvumo rodikliu reiksmemis pritaikytas E. K. Zavadsko ir A. Kaklausko sukurtam metodui COPRAS (Complex Proportional Evaluation). Taip gautas naujas metodas COPRAS-G. Sis metodas pritaikytas daugiabuciu namu isoriniu sienu efektyvumo vertinimo uzdaviniui spresti. Sudaryta efektyvumo rodikliu sistema, nustatyta efektyvumo rodikliu reiksme, rodikliu rangas. Pagal sudaryta modeli pritaike COPRAS-G metoda, galime spresti daugeli statybos, vadybos alternatyvu vertinimo uzdaviniu.

Reiksminiai zodziai: sienos, alternatyvos, daugiatikslis parinkimas, vertinimas, COPRAS, COPRAS-G, pilkieji skaiciai.

1. Introduction

The number of residential houses in Lithuania is increasing every year. For a non-insulated building, which could be situated in different climate conditions, these particular heat-losses can vary between 10-20 % (through floors), 25-30 % (through outer walls), 25-30 % (through attic slabs and roof plates) and 30-40 % (through windows) of the total heat loss. In Lithuania nearly a half of all heating losses are caused by low quality walls (Fig. 1). Therefore the thorough and professional selection of an optimal building thermal insulation system represents one of the most important technical and economical goals for both the Designer and the Investor. The selection of an effective variant of external building walls among a vast number of alternatives is an important problem in project management.

Introduction of various thermal insulation systems in the contemporary civil engineering practice is caused by the major expansion of energy resource prices at the world market. As a result, there is a growing need for significant heat-loss reduction during exploitation of buildings, which as a rule could be realized using more or less effective building systems to prevent heat loss through outer walls. Building and exploitation expenses depend on how effective the external wall solution has been chosen. It should be done by establishing the requirements and aims till the expiry of a building. The benefit obtained from effectively heating up the external walls could be defined by indices as shown in Fig. 2. The selection of a building's external walls (Fig. 3) is a decision characterized by multiple attributes. Clients want to minimize the likely costs of the project, but they also want to achieve highest acceptable quality standards as well as to satisfy technological, architectural, and comfort requirements. Other participants of construction process (e.g. designer, contractor) are interested in maximizing profits; they are also concerned with other attributes such as company growth, market share, and the state institutions' interests.

All decisions involve choosing one from several alternatives. Multiple attribute optimization is a process of determining a feasible solution for the decision maker according to the established attributes (e.g. a set of the quantitative and qualitative attributes). Multiple attribute methods are available for evaluation of external walls alternatives. In this paper, the authors present a methodology that allows decision maker to reach a decision by designing alternatives of a building's external walls and to evaluate attributes both qualitative and quantitative contained in the process.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

2. Multi-layered external walls

Facade structures of residential and business buildings are facing following demands:

* Ability to function as bearing or self-bearing walls;

* Possession of high thermal insulation properties;

* Sound insulation;

* Overall hygrothermal performance;

* Frost resistance;

* Air tightness;

* Vapour permeability;

* Sufficient light-weightiness;

* Ecological cleanliness;

* Satisfactory fireproofing;

* Durability.

There is usually not enough attention paid to the fact that multi-layered facade structures are made as composite sections of heterogeneous materials with different physical-mechanical properties, such as:

* Expansion and shrinkage coefficients;

* Compressive and tensile strengths;

* Adhesion properties;

* Behaviour under different types of wind load;

* Behaviour under ultraviolet radiation exposure;

* Difference between strain values in adjacent walls with significant temperature;

* Variation due to different sun exposure and colour of the final facade coating;

* Difference in aging properties of each composite during exploitation;

* Air and vapour permeability values.

Cost-effectiveness in application of multi-layer external walls in Civil engineering--is the most interesting issue for the Investor. Without getting into all the inferior physical, thermo-technical and ecological properties (not to mention the poor durability) of the usually applied facade structures (bricks or blocks insulated with mineral wool or Styrofoam and coated with mineral polymer-cement plaster over glass-fibre net or simply protected with facade bricks), let us analyze the indisputable cost-effectiveness, even possible profit for the Investor calculated per meter of a facade wall built using multi-layer external walls.

For multilayer walls, three basic material configurations were considered: insulation either inside or outside the massive layer, and insulation located between two massive layers. The results of extensive parametric analysis have shown explicitly that walls with the insulation outside always performed better than those with the insulation inside:

* The system covers the entire building wall (except windows and doors). Thus, multi-layered exterior wall systems provide an insulation layer over potential thermal bridges such as wall studs and columns and floor-wall junctions.

* Because the entire exterior wall is covered, building air tightness is improved.

* Because insulation is placed on the building exterior, the building structure is kept warm; this minimizes thermal expansion and contraction.

* Finally, if properly installed, the system avoids a build-up of moisture in the building cladding.

Model of problem

The aim of this investigation is to create a technique for the choice and selection of different and effective versions of the external walls construction. The purpose is to be achieved by using various indicators of effectiveness, which have different dimensions, different significances as well as different directions of optimization (Kendall 1970; Zavadskas 1987; Zavadskas and Vilutiene 2006; Kaklauskas et al. 2006).

The main steps of multiple attributes decisionmaking are as follows:

* Establishing system evaluation attributes that relate system capabilities to goals;

* Developing alternative systems for attaining the goals (generating alternatives);

* Evaluating alternatives in terms of attributes (the values of the attribute functions);

* Applying a normative multiple attributes analysis method;

* Accepting one alternative as "optimal" (preferred);

* If the final solution is not accepted, gather new information and go into the next iteration of multiple attributes optimization.

Alternatives of external wall construction are being formed by using various materials with thermal insulation as well as different kinds of decoration masonry and thin daub layer. A system of indicators for wall construction effectiveness' evaluation has been established (Fig. 3).

The solving of each multi-attribute problem begins with constructing of decision-making matrix (Fig. 4).

In this matrix (Fig. 4) values of the attributes [a.sub.ij] may be:

* Real numbers;

* Intervals;

* Probability distributions;

* Possibility distributions;

* Qualitative labels.

The problem may be:

** Choice-Select the most appropriate (best) alternative;

** Ranking-Draw a complete order of the alternatives from the best to the worst ones;

** Sorting-Select the best k alternatives from the list of n>k.

When we consider a discrete set of alternatives described by some attributes, there are three different types of analyses that can be performed in order to provide a significant support to decision-makers:

* Ensure that the decision-maker follows a "rational" behaviour (Normative option)--Value functions, Utility theory, distance to the Ideal;

* Give some advice based on reasonable (but not indisputable) rules--The French School.

* Find the preferred solution from the partial decision hypothesis--Interactive methods.

Multiple attributes decision aid provides several powerful and effective tools (Hwang and Yoon 1981; Figueira et al. 2005) for confronting sorting problems. There can be used very simplified techniques for the evaluation of a decision support methods base including methods such as the Simple Additive Weighting--SAW; TOPSIS--Technique for Order Preference by Similarity to Ideal Solution (Hwang and Yoon 1981; Zavadskas 1987; Antuchevicien_ 2005; Chang et al. 2005) and methods of the ELECTRE (Elimination and Choice Translating Reality) family, such as ELECTRE and UTA (UTilites Additives, cf.). A variant of the UTA method is the UTADIS method (Utilites Additives DIS-criminantes). The Preference Ranking Organisation MeTHod for Enrichment Evaluations (PROMETHEE) can be applied to the solution too.

For a more detail survey of Multi criteria decisionmaking methods see for applications in the construction context Zavadskas (1987), Zavadskas et al. (2004), Zavadskas et al. (2005), Kaklauskas et al. (2005), Peldschus and Zavadskas (2005), Antuchevicien_ et al. (2006), Su et al. (2006), Saparauskas and Turskis (2006), Turskis et al. (2006), Zagorskas and Turskis (2006), Zavadskas et al. (2006), Kalibatas et al. (2007), Ustinovichius et al. (2007), Zavadskas and Antuchevicien_ (2007), Zavadskas et al. (2007b) and Banaitiene et al. (2008).

The task of the selection of different versions of the effective external walls construction is solved by applying COPRAS method. COPRAS (technique for order preference by similarity to an ideal solution) method is presented by Zavadskas and Kaklauskas (Zavadskas et al. 1994; Zavadskas and Kaklauskas 1996). The COPRAS method determines a solution with the ratio to the ideal solution and the ratio with the ideal-worst solution.

3. Ranking of the alternatives applying COPRAS method

3.1. Determination of the attributes weights

In order to select the best alternative, it is necessary, to have formed the decision matrix, to perform the project' s multiple attributes analysis. MCDM refers to making preference decisions on the alternatives in terms of multiple attributes. Typically, each alternative is evaluated on the established set/system of attributes.

To determine the weights of the attributes, the expert' s judgment method is applied (Kendall 1970) which has been successfully used in research by the authors since 1987 (Zavadskas 1987; Zavadskas and Kaklauskas 1996; Kaklauskas et al. 2006; Zavadskas and Vilutiene 2006; Bardauskiene 2007). In order to establish the importance indicators, a survey has been carried out and 39 experts have been questioned. These experts, basing their answers on their knowledge, experience and intuition, had to rate indicators of effectiveness starting with the most important ones. The rating was done on a scale from 1 to 5, where 5 meant "very important" and 1 "not important at all". The importance of indicators was established according to the rating methods (Zavadskas 1987) of these experts and also demonstrated the priorities of the user (owner).

The significance of the attributes obtained by this method are presented in Table 1.

3.2. A method of multiple criteria complex proportional assessment--COPRAS

In order to evaluate the overall efficiency of a project it is necessary to identify selection attributes, to assess information, relating to these attributes, and to develop methods for evaluating the attributes to meet the participant's needs. Decision analysis is concerned with the situation in which a decision-maker has to choose among several alternatives by considering a common set of attributes. The COPRAS method (Zavadskas et al. 1994; Zavadskas and Kaklauskas 1996) presented here uses a stepwise ranking and evaluating procedure of the alternatives in terms of significance and utility degree. This method was applied to solution of various problems in construction, property management, economics etc. (Zavadskas et al. 1994; Zavadskas and Kaklauskas 1996; Andruskevicius 2005; Malinauskas and Kalibatas 2005; Ziogas and Juociunas 2005; Kaklauskas et al. 2006; Viteikien_ 2006; Zavadskas and Antucheviciene 2006; Zavadskas et al. 2007a; Viteikiene and Zavadskas 2007; Kaklauskas et al. 2007).

The procedure of the COPRAS method consists of the steps as shown in Fig. 5.

3.3. A method of multiple criteria complex proportional assessment with values determined in intervals--COPRAS-G

In many decisions the consequences of the alternative courses of action cannot be predicted with a certainty. A company considering the launch of a new product will be uncertain about how successful the product will be, while an investor on the stock market will generally be unsure about the returns which will be generated, if a particular investment is chosen.

We will first outline a method which assumes that the decision maker is unable, or unwilling, to estimate probabilities for the outcomes of the decision and which, in consequence, makes extremely pessimistic assumptions about these outcomes.

Finally, we will broaden the discussion to consider problems involving both uncertainty and more than one attribute. As we saw in problems involving multiple attributes are often too large for a decision-maker to comprehend in their entirety.

The idea to COPRAS-G method comes from real conditions of decision-making and from applications of the Grey systems theory. This theory was originated by Deng (1982) study of the relation degree among various attributes in an MCDM problem. In 1988 Deng (1988a) presented grey decision-making systems. Grey relational analysis possesses advantages of Deng (1988b): it involves simple calculations; it requires smaller samples; a typical distribution of samples is not needed; the quantified outcomes from the grey relational grade do not result in contradictory conclusions about the qualitative analysis; and the grey relational grade model is a transfer functional model that is effective in dealing with discrete data.

It is useful mathematically when dealing with a system with a limited information. According to this theory, a system whose internal information is completely known is called a white system. On the contrary, a system is defined as a black system if one cannot obtain any information and characteristics about the system. Grey space is thus defined as a system defined between the white and black systems. The grey system has been applied in many fields, such as economics, agriculture, geography, weather, earthquakes, science etc. For example, Wending et al. (2002), Wu and Chang (2003), Du and Sheen (2005) applied the grey model to solution of problems. Haq and Kannan (2007) developed a hybrid normalised multi-criteria decision-making model for evaluating and selecting the vendor using Analytical Hierarchy Process and Fuzzy Analytical Hierarchy Process and an integrated approach of Grey Relational Analysis to a Supply Chain model. Linet et al. (2008) presents an illustrative example of subcontractor selection by applying grey TOPSIS method.

The procedure of the COPRAS method with attributes values expressed in interval COPRAS-G includes the following steps:

1. Selection of the available set of the most important attributes, which describes alternatives;

2. Preparing the decision-making matrix X:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [w.sub.ij]--the least value--lower limit, [b.sub.ij]--the biggest value--upper limit.

3. Determining weights of the attributes [q.sub.j].

4. Normalization of the decision-making matrix [bar.X]. The normalized values of this matrix (Hwang and Yoon 1981; Zavadskas 1987; Migilinskas and Ustinovicius 2007) are calculated as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

In formula (2) [w.sub.ij] is the lower value of the j-th attribute

in the i-th alternative of a solution; [b.sub.ij]--the upper value of the j attribute in the i alternative of a solution; m--the number of attributes; n--the number of the alternatives compared.

After this step we have normalized decision-making matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

5. Calculation of the weighted normalized decision matrix [??]. The weighted normalized values [[??].sub.ij] are calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

In formula (4), [q.sub.j] is significance (weight) of the j-th attribute.

After this step we have weighted normalized decisionmaking matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

6. Sums [P.sub.j] of attributes values which larger values are more preferable (optimization direction is maximization) calculation for each alternative (line of the decisionmaking matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

In formula (6), k is number of attributes which must be maximised (it is assumed that in the decision-making matrix columns first of all are placed attributes with optimization direction maximum and the ones with optimization direction minimum are placed after).

[FIGURE 5 OMITTED]

7. Sums [R.sub.j] of attributes values which smaller values are more preferable (optimization direction is minimization) calculation for each alternative (line of the decision-making matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In formula (7), (m-k) is number of attributes which must be minimized.

8. Determining the minimal value of [R.sub.j]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

9. Calculation of the relative weight of each alternative [Q.sub.j]:

[Q.sub.j] = [P.sub.j] + [R.sub.min][n.summation over (j=1)] [R.sub.j]/[R.sub.j][n.summation over (j=1)] [R.sub.min]/[R.sub.j]. (9)

Formula (9) can to be written as follows:

[Q.sub.j] = [P.sub.j] + [n.summation over (j=1)] [R.sub.j]/[R.sub.j][n.summation over (j=1)] x 1/[R.sub.j]. (9*)

10. Determination of the optimality criterion K:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

11. Determination of the project priority. The greater significance (relative weight of alternative) [Q.sub.j], the higher is the priority (rank) of the project. The relative significance [Q.sub.j] of project j indicates the satisfaction degree of the needs of the project participants. In case of max [Q.sub.max], the satisfaction degree is the highest. The relative significance of other projects is less.

12. Calculation of the utility degree of each alternative. The degree of project utility is determined by comparing the analyzed projects with the best one. The values of the utility degree are from 0 % to 100 % between the worst and the best alternatives. The utility degree [N.sub.j] of each alternative j is calculated as

[N.sub.j] = [Q.sub.j]/[Q.sub.max] 100%, (11)

where [Q.sub.j] and [Q.sub.max] are the significance of projects obtained from Eq. (9*).

The decision approach proposed in this section allows evaluating the direct and proportional dependence of the significance and utility degree of alternatives in a system of attributes, weights and values of the attributes.

4. Selection of the effective dwelling house walls by applying attributes values determined in intervals

The initial data of this problem are taken from Zavadskas et al. (2005) research work. The 39 experts were asked to prioritize the 5 attributes listed in Table 1:

* Durability of walls (frost resistance) (cycles) [x.sub.1] - [[w.sub.i1]; [b.sub.i1]];

* Thermal transmittance (W/m x K) [x.sub.2] - [[w.sub.i2]; [b.sub.i2]];

* The estimated cost of [m.sup.2] walls [x.sub.3] - [[w.sub.i3]; [b.sub.i3]]

* Weight of [m.sup.2] walls (kg) [x.sub.4] - [[w.sub.i4]; [b.sub.i4]];

* Human work expenditures (hour/[m.sup.2]) [x.sub.5] - [[w.sub.i5]; [b.sub.i5]].

Selected attributes do not cover the all important requirements of buildings. The European Council Directive 89/106/EEC has set six essential requirements:

* Mechanical resistance and stability;

* Safety in case of fire;

* Hygiene, health and environment;

* Safety in use;

* Protection against noise;

* Energy economy and heat retention.

The analysis of all of these requirements can be performed according to the factors:

* Quality of components;

* Design level;

* Work execution level;

* Indoor environment;

* Outdoor environment;

* In-use conditions;

* Maintenance levels.

Optimization directions of selected attributes are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Respondents were from one of several kinds of organizations (owners, designers, contractors, scientists). The determination of quantitative attributes values is based on the use of analyzed projects, price-lists, specifications, reference books and recommendations.

According to thermal transmittance data, alternative 3 was first in the list of priorities. According to durability of walls data, alternative 1 (is equal to 2 and 3) was first in the list of priorities, while alternative 4 was the fourth. According to weight of [m.sup.2] walls data, alternative 4 was first in the list of priorities, while alternative 2 was the fourth. According to human expenditures data, alternative 4 was first in the list of priorities, while Alternative 1 was the fourth.

The final choice of external walls was made by COPRAS-G method. In Table 2 the normalized weighted decision-making matrix is given. On the basis of the efficiency priority of alternatives, a rank [R.sub.j] of each alternative is established. According to the calculation results, alternative 1 is the best one (Table 3). The first alternative is also the best in terms of its utility degree that equals 100 %. The second alternative with utility degree 96.9 % has rank 3. The third alternative with utility degree 99.7 % has rank 2. The fourth alternative with the utility degree 82.1 % is the worst and has rank 4. Vector of optimality criterion values [N.sub.j] is:

[N.sub.j] = [100; 96.9; 99.7; 82.1].

According to the [N.sub.j] the alternatives rank as follows: [A.sub.1] > [A.sub.3] > [A.sub.2] > [A.sub.4].

5. Conclusions

In real life multi-attribute modelling of multi-alternative assessment problems some attribute values, which deals with the future, must be expressed in intervals.

For this reason a new method of multiple criteria complex proportional assessment with values determined in intervals--COPRAS-G is developed.

By the analysis of the problem solution results it has been established that silicate brick masonry walls with outer finishing layer are more preferable than three another ones under investigation.

This model and solution results have practical and scientific interests. They allow investors to make decision concerning multiple attributes, when values of initial data are given in the intervals.

This COPRAS-G method can be applied to the solution of wide range discrete multi-attribute assessment problems in construction.

Received 08 Jan 2008, accepted 10 Apr 2008

References

Andruskevicius, A. 2005. Evaluation of contractors by using COPRAS--the multiple criteria method, Technological and Economic Development of Economy 11(3): 158-169.

Antucheviciene, J. 2005. Evaluation of alternatives applying TOPSIS method in a fuzzy environment, Technological and Economic Development of Economy 11(4): 242-247.

Antucheviciene, J.; Turskis, Z.; Zavadskas, E. K. 2006. Modelling renewal of construction objects applying methods of the game theory, Technological and Economic Development of Economy 12(4): 263-268.

Banaitiene, N.; Banaitis, A.; Kaklauskas, A.; Zavadskas, E. K. 2008. Evaluating the life cycle of building: a multivariant and multiple criteria approach, Omega 36(3): 429-441.

Bardauskiene, D. 2007. The expert's estimates application in the preparation of the city general plan, Technological and Economic Development of Economy 13(3): 223-236.

Chang, J.-R.; Chen, D.-H.; Hung, Ch.-T. 2005. Selecting preventive maintenance treatments in Texas. Using the technique for order preference by similarity to the ideal solution for specific pavement Study-3 sites, Transportation Research Record. Journal of the Transportation Research Board 1933: 62-71.

Deng, J. L. 1982. Control problems of grey system, Systems and Control Letters 1(5): 288-294.

Deng, J. L. 1988a. Introduction to Grey System Theory, The Journal of Grey Theory 1: 1-24.

Deng, J. L. 1988b. Properties of relational space for grey systems, in Essential Topics on Grey System--Theory and Applications. China Ocean, 1-13.

Du, J. C. and Sheen, D. H. 2005. Development of paverment permanent deformation prediction model by grey modelling method, Civil Engineering and Environmental Systems 22(2): 109-121.

Council Directive 89/106/EEC of 21 December 1988 on the approximation of laws, regulations and administrative provisions of the Member States relating to construction products (89/106/EEC) (OJ L 40, 11.2.1989).

Figueira, J.; Greco, S.; Ehrgott, M. (Eds.). 2005. Multiple Criteria Decision Analysis: State of the Art Surveys. Springer.

Haq, A. N. and Kannan, G. 2007. A hybrid multicriteria decisionmaking for the vendor selection in a supply chain model, Int. J. Management and Decision-Making 8(5/6): 601-622.

Hwang, C. L. and Yoon, K. 1981. Multiple Attribute Decision Making, in Lecture Notes in Economics and Mathematical Systems 186, Springer-Verlag, Berlin.

Kaklauskas, A.; Zavadskas, E. K.; Raslanas, S. 2005. Multivariant design and multiple criteria analysis of building refurbishments, Energy and Buildings 37: 361-372.

Kaklauskas, A.; Zavadskas, E. K.; Raslanas, S.; Ginevicius, R.; Komka, A.; Malinauskas, P. 2006. Selection of low-e tribute in retrofit of public buildings by applying multiple criteria method COPRAS: A Lithuanian case, Energy and Buildings 38: 454-462.

Kaklauskas, A.; Zavadskas, E. K.; Banaitis, A.; Satkauskas, G. 2007. Defining the utility and market value of real estate a multiple criteria approach, International Journal of Strategic Property Management 11(2): 107-120.

Kalibatas, D.; Krutinis, M.; Viteikiene, M. 2007. Multi-objective evaluation of microclimate in dwelling, Technological and Economic Development of Economy 8(1): 24-31.

Kendall, M. G. 1970. Rank correlation methods (4th ed.). London: Griffin.

Lin, Y.-H.; Lee, P.-C.; Chang, T.-P.; Ting, H.-I. 2008. Multi-attribute group decision-making model under the condition of uncertain information, Automation in Construction 17(6): 792-797.

Malinauskas, P. and Kalibatas, D. 2005. The selection of rational constructional technology processes variants using COPRAS method, Technological and Economic Development of Economy 11(3): 197-205.

Migilinskas, D. and Ustinovichius, L. 2007. Normalisation in the selection of construction alternatives, Int. J. Management and Decision-Making 8(5/6): 623-639.

Peldschus, F. and Zavadskas, E. K. 2005. Fuzzy matrix games multi-criteria model for decision-making in engineering, Informatica 16(1): 107-120.

Su, Ch.-W.; Cheng, M.-Y.; Lin, F.-B. 2006. Simulation-enhanced approach for ranking major transport projects, Journal of Civil Engineering and Management (12)4: 285-291.

Saparauskas, J. and Turskis, Z. 2006. Evaluation of construction sustainability by multiple criteria methods, Technological and Economic Development of Economy 12(4): 321-326.

Turskis, Z.; Zavadskas, E. K; Zagorskas, J. 2006. Sustainable city compactness evaluation on the basis of GIS and Bayes rule, International Journal of Strategic Property Management 10(3): 185-207.

Ustinovichius, L.; Zavadskas, E. K.; Podvezko, V. 2007. Application of a quantitative multiple criteria decision making (MCDM-1) approach to the analysis of investments in construction, Control and Cybernetics 36(1): 251-268.

Viteikiene, M. 2006. Sustainable residental areas evaluation, Technological and Economic Development of Economy 12(2): 152-160.

Viteikiene, M. and Zavadskas, E. K. 2007. Evaluating the sustainability of Vilnius city residential areas, Journal of Civil Engineering and Management 13(2): 149-155.

Wending, H.; Ben, H.; Changzhi, Y. 2002. Building thermal process analysis with grey system method, Building and Environment 37: 599-605.

Wu, C. C. and Chang, N. B. 2003. Grey input-output analysis and its application for environmental cost allocation, European Journal of Operational Research 145: 175-201.

Zagorskas, J. and Turskis, Z. 2006. Multi-attribute model for estimation of retail centres influence of the city structure. Kaunas city case study, Technological and Economic Development of Economy 12(4): 347-352.

Zavadskas, E. K.; Kaklauskas, A.; Sarka, V. 1994. The new method of multicriteria complex proportional assessment of projects, Technological and Economic Development of Economy 1(3): 131-139.

Zavadskas, E. K. and Kaklauskas, A. 1996. Pastatu sistemotechninis ivertinimas [Multiple criteria evaluation of buildings]. Vilnius: Technika.

Zavadskas, E. K.; Kaklauskas, A.; Raslanas, S. 2004. Evaluation of investments into housing renovation, International Journal of Strategic Property Management 8(3): 177-190.

Zavadskas, E. K.; Ustinovicius, L.; Turskis, Z.; Ambrasas, G.; Kutut, V. 2005. Estimation of external walls decisions of multistorey residential buildings applying methods of multicriteria analysis, Technological and Economic Development of Economy 11(1): 59-68.

Zavadskas, E. K. and Antucheviciene, J. 2006. Development of an indicator model and ranking of sustainable revitalization alternatives of derelict property: a Lithuanian case study, Sustainable Development 14: 287-299.

Zavadskas, E. K. and Vilutiene, T. 2006. A multiple criteria evaluation of multi-family apartment block's maintenance contracts: I-Model for maintenance contractor evaluation and determination of its selection criteria, Building and Environment 41: 621-632.

Zavadskas, E. K.; Zakarevicius, A.; Antuchevicien, J. 2006. Evaluation of ranking accuracy in multi-criteria decisions, Informatica 17(4): 601-618.

Zavadskas, E. K. and Antucheviciene, J. 2007. Multiple criteria evaluation of rural building's regeneration alternatives, Building and Environment 42(1): 436-451.

Zavadskas, E. K.; Kaklauskas, A.; Peldschus, F.; Turskis, Z. 2007a. Multi-attribute assessment of road design solution by using the COPRAS method, The Baltic Journal of Road and Bridge Engineering 2(4): 195-203.

Zavadskas, E. K.; Turskis, Z.; Dejus, T.; Viteikiene, M. 2007b. Sensitivity analysis of a simple additive weight method, Int. J. Management and Decision Making 8(5/6): 555-574.

Ziogas, V. and Juociunas, S. 2005. Design and installation peculiarities of monolithic concrete floor, Journal of Civil Engineering and Management 11(2): 153-162.

[TEXT NOT REPRODUCIBLE IN ASCII]. 1987. [TEXT NOT REPRODUCIBLE IN ASCII] [Complex estimation and choice of resource saving decisions in construction]. [TEXT NOT REPRODUCIBLE IN ASCII].

Edmundas Kazimieras Zavadskas (1), Arturas Kaklauskas (2), Zenonas Turskis (3), Jolanta Tamosaitiene (4)

Vilnius Gediminas Technical University, Civil Engineering Faculty, (1,3,4) Dept of Construction Technology and Management, (2) Dept of Construction Economics and Property Management, Sauletekio al. 11, LT-10223 Vilnius, Lithuania E-mail: (1) edmundas.zavadskas@adm.vgtu.lt; (2) arturas.kaklauskas@st.vgtu.lt; (3) zenonas.turskis@st.vgtu.lt; (4) jolanta.tamosaitiene@st.vgtu.lt

Edmundas Kazimieras ZAVADSKAS. Doctor Habil, Professor, Doctor Honoris causa of Poznan, Sankt-Petersburg University and Kiev University, Principal Vice-Rector of Vilnius Gediminas Technical University, Member of Lithuanian Academy of Sciences, President of Lithuanian Operational Research Society, President of Alliance of Experts of Projects and Buildings of Lithuania. Editor-in-Chief of the following journals: "Journal of Civil Engineering and Management", "Technological and Economic Development of Economy", Editor of the International Journal of Strategic, Property Management. His research interests include building technology and management, decision-making theory, automation in design, decision support systems.

Arturas KAKLAUSKAS. Doctor Habil, Professor, the Head of the Department of Construction Economics and Property Management of Vilnius Gediminas Technical University. His domain of research and interests are pollution, sustainable development, analysis, modelling and forecasting of construction and real estate sector, facilities and real estate management, total quality analysis, e-learning, knowledge and decision support systems.

Zenonas TURSKIS. Doctor of Technical Sciences, senior research fellow of Construction Technology and Management laboratory. Author of 25 scientific articles. Research interests: building technology and management, decision-making theory, computer- aided automation in design, expert systems.

Jolanta TAMOSAITIENE. PhD student at the Department of Construction Technology and Management, Vilnius Gediminas Technical University, Lithuania. BSc degree (building technology and management), Vilnius Gediminas Technical University (2000). MSc degree (Building management and economics), Vilnius Gediminas Technical University (2002). Research visits to Leipzig University of Applied Sciences (Germany, 2002). Research interests: construction technology and organisation, management, economics, project administration and evaluation, the multiple attributes of decision-making method.

Table 1. Initial decision-making matrix
with values expressed in intervals

 Thermal
Alternative No. Durability of transmittance
 walls (cycles) (W/m x K)

Optimization direction max min

Attribute weight q 0.21 0.33

 [x.sub.1] [x.sub.2]

 [w.sub.1] [b.sub.1] [w.sub.2] [b.sub.2]

Silicate brick masonry
with masonry outer
finishing layer 75 100 0.22 0.25

Ceramic brick masonry
with masonry outer
finishing layer 75 100 0.22 0.25

Ceramic fenestrate
brick masonry with
masonry outer
finishing layer 75 100 0.21 0.25

Silicate bricks
masonry with outer
plaster finishing
layer 25 25 0.24 0.27

 The estimated Weight of
Alternative No. cost of [m.sup.2] [m.sup.2]
 walls, (LTL) walls, (kg)

Optimization direction min min

Attribute weight q 0.26 0.09

 [x.sub.3] [x.sub.4]

 [w.sub.3] [b.sub.3] [w.sub.4] [b.sub.4]

Silicate brick masonry
with masonry outer
finishing layer 72.08 94.71 590 652

Ceramic brick masonry
with masonry outer
finishing layer 89.01 100.93 596 625

Ceramic fenestrate
brick masonry with
masonry outer
finishing layer 80.32 96.42 581 604

Silicate bricks
masonry with outer
plaster finishing
layer 67.76 98.10 455 479

Alternative No. Human expenditures,
 (hour/[m.sup.2])

Optimization direction min

Attribute weight q 0.11

 [x.sub.5]

 [w.sub.5] [b.sub.5]

Silicate brick masonry
with masonry outer
finishing layer 4.60 4.60

Ceramic brick masonry
with masonry outer
finishing layer 4.60 4.60

Ceramic fenestrate
brick masonry with
masonry outer
finishing layer 4.60 4.60

Silicate bricks
masonry with outer
plaster finishing
layer 4.55 5.01

Table 2. Weighted normalized decision-making
matrix according to a COPRAS-G method

 Weighted normalized values of the attributes
 describing the compared alternatives--matrix [??]
Alternative
No. [[??].sub.1] [[??].sub.1] [[??].sub.2] [[??].sub.2]

1 0.055 0.073 0.076 0.086
2 0.055 0.073 0.076 0.086
3 0.055 0.073 0.073 0.086
4 0.018 0.018 0.083 0.093

 Weighted normalized values of the attributes
 describing the compared alternatives--matrix [??]
Alternative
No. [[??].sub.3] [[??].sub.3] [[??].sub.4]

1 0.054 0.070 0.023
2 0.066 0.075 0.023
3 0.060 0.072 0.023
4 0.050 0.073 0.018

 Weighted normalized values of the attributes
 describing the compared alternatives--matrix [??]
Alternative
No. [[??].sub.4] [[??].sub.5] [[??].sub.5]

1 0.026 0.027 0.027
2 0.025 0. 027 0.027
3 0.024 0.027 0.027
4 0.019 0.027 0.030

Table 3. Decision results according to a COPRAS-G method ([R.sub.j]--
ascending rank of alternatives. The smallest is the best)

 Total sum of maximizing Total sum of minimizing
Alternative normalized indices normalized indices
No [R.sub.j] [P.sub.j]

1 0.390 0.128
2 0.406 0.128
3 0.391 0.128
4 0.393 0.037

 Alternative's Alternative's
Alternative significance degree of Rank
No [Q.sub.j] efficiency [N.sub.j] [R.sub.j]

1 0.528 100 1
2 0.512 96.9 3
3 0.526 99.7 2
4 0.434 82.1 4

Figure 1. Average annual losses of heat in residential
buildings of Lithuania

Ground level floors 48%
External walls 13%
Windows 21%
Overloadings 18%

Note: Table made from pie chart.

Fig. 4. Decision-making matrix for multi-attribute
decision-making problems

Alternatives Attributes

 [C.sub.1] [C.sub.2] ... [C.sub.m]
[A.sub.1] [a.sub.11] [a.sub.12] ... [a.sub.1m]
[A.sub.2] [a.sub.21] [a.sub.22] ... [a.sub.2m]
 ... ... ... ...
[A.sub.n] [a.sub.n1] [a.sub.n2] ... [a.sub.nm]