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A new synthesis method of structural, technological and safety decisions (SyMAD-3)/Naujas konstrukciniu, technologiniu ir saugos sprendimu sintezes metodas (SyMAD-3).

1. Introduction

The abundance of technological processes provides opportunities for various decisions on structural engineering. Using a variety of construction materials and applying certain work processes, an eye piercing aggregate of structural elements has been produced. The following questions may arise when observing technological progress, the architectural complexity and height of buildings and analysing accidents and occupational diseases: Is work safety always ensured in the technological process? How much attention is devoted to ensuring a safe working procedure?

The developed countries pay much attention to the working conditions of employees, i.e. they are aimed at making sure that working conditions do not jeopardise human health or pose a risk to their life (Hola 2009; Liaudanskiene et al. 2009; Reinhold, Tilt 2009; Perera et al. 2009; Kazlauskaite, Buciuniene 2008; Hernaus et al. 2008; Reinhold et al. 2008; Grybait?, Tvaronavi?ien? 2008; Zou et al. 2007). Occupational health and the problem of safety are especially relevant, because any violation system foundation causes not only moral damage, but can also frequently lead to health problems, and sometimes even risk to the life of employees (McCabe et al. 2008).

In order to prevent accidents and occupational diseases, improve productivity and job satisfaction of employees, it is necessary to take measures ensuring safety on construction sites (Giretti et al. 2009; Stankuvien? et al. 2008; Idoro 2008; Enshassi et al. 2007). In the course of various construction processes, safety at work can be ensured not only using collective and individual protective equipment, occupational risk assessment and coaching staff on health and safety issues-accidents can also be prevented by ensuring the proper organisation of work and working conditions (Sawacha et al. 1999; Jorgensen et al. 2007), which is often the case that the organisation of work directly depends on the decision regarding structural and technological solutions. Thus, one way to help with reaching a decision in the construction sector is to combine all structural, technological, and safety decisions. Then, the focus would be on one object consisting of the elements of three main areas, namely, the structural elements of a building, the technology of construction processes and safety solutions to construction processes.

In the case where a set of a possible alternative to a problem is known in advance and information about the attributes is provided in quantitative measurements, it is recommended to use multiple attribute decision-making (MADM) methods providing a quantitative evaluation of each alternative on the basis of which ranking alternatives is carried out to solve the problem. These methods are widely applied for analyzing various types of construction problems (Edalat et al. 2010; Zavadskas et al. 2008a, b; Tupenaite et at. 2010; Liaudanskiene et al. 2009) and assessing real estate investment projects (Ginevicius et al. 2009).

There are quite a few research papers in which the methods based on quantitative measurements are used for multiple criteria decision-making. Some works describe the use of only one method (Liaudanskiene et al. 2009; Zavadskas et al. 2008a, etc.), whereas others compare the results obtained using several methods (Tupenaite et al. 2010; Ustinovichius 2007; Zavadskas et al. 2010, etc.). Decision models (Kaklauskas et al. 2011; Sarka et al. 2008; Marzouk et al. 2011; Vasilecas et al. 2011, etc.) and decision support systems (Zavadskas et al. 2008b) have been developed or are being developed for solving engineering and investment problems of various construction projects. Multiple criteria decision-making methods are used for developing similar models in cases of certainty (when quantitative methods are used) and under uncertainty (when game theory methods are used).

Given the fact that the accuracy of some attributes in construction investment projects may vary (Popov et al. 2010; Zavadskas et al. 2008a) and that each decision-making method has its own sensitivity with respect to fluctuations in the input data (Simanaviciene, Ustinovichius 2010) and with respect to the normalisation of the decision matrix (Zavadskas et al. 2007), the authors of the paper propose the application of several rather than one decision-making method in order to increase the reliability of the outcomes of the multiple criteria decision (to get a prioritised list of alternatives). To achieve the aim of this work, questionnaires containing the questions related to the evaluation of structural, technological and safety decisions were distributed to respondents.

This paper provides a new synthesis method of multiple attribute decisions--SyMAD-3 (Synthesis of Multiple Attribute Decisions using three methods)--intended for combining multi-stage and multiple attribute decisions into a single common decision. To increase the reliability of the decision, three multiple attribute decision-making methods based on quantitative measurements were applied. The algorithm of the proposed method uses methods for identifying the integrated signifycances of attributes (Ustinovichius 2007) and those for multiple attribute decision-making (SAW--Simple Additive Weighting, TOPSIS Technique for Order Preference by Similarity to Ideal Solution, and COPRAS--COmplex PRoportional ASsessment) (Tupenaite et al. 2010; Ustinovicius et al. 2007; Zavadskas et al. 2010, etc.). The aim of the proposed method is the synthesis of multi-stage and multiple attribute decisions through the application of multiple attribute decision methods. This paper presents a practical application of the method for selecting the external wall of a building from possible project alternatives considering the main elements of the construction project: structural elements of the building, technology for construction processes and safety solutions to construction processes. The selected alternative decision, regarding structural elements, should comply with the requirements for the technological process that would ensure the quality of work and work safety requirements.

2. The synthesis method of multi-stage and multiple attribute decisions applying three methods (SyMAD-3), intended for identifying the efficiency of the proposed alternatives presented in the form of a decision tree

Decision synthesis is a decision that links multiple decisions into a joint project in concordance with decision tables of the alternatives. To perform decision synthesis, a decision tree showing all possible combinations of decision alternatives is made. Multiple attribute design often needs a decision to be made analysing and combining several problems into one. The primary idea of creating the synthesis method of multiple attribute decisions was mentioned in literature by Zavadskas (Zavadskas 1991). A revised method of multiple attribute decision synthesis was developed, practically tested (Sarka et al. 2008) and applied for determining the most effective decisions on construction problems.

The synthesis of multi-stage and multiple attribute decisions provides the possibility of making an efficient decision when there is a need to evaluate multiple, often conflicting, situations. The first point that needs to be discussed dealing with this problem is that it is usually not possible to identify one decision, judgement or action that would be optimal in all respects. Unlike classical methods for the study on relationships with alternatives, multiple attribute methods do not require objectively best decisions (Sarka et al. 2008). The essence of the method is the synthesis of several inter-related technical decisions by selecting only two by default (or more) best alternatives at each stage. Thus, it is recommended that this method should be applied in case there are more than three decision stages. However, by selecting only two alternatives at each stage of a decision, the possibility of observing the results of potential combinations is lost. Yet, if we retain all potential alternative combinations, a very large number of alternative combinations will be obtained, and therefore will be difficult to assess using the above described method.

In order not to lose interim information about possible decisions, the authors of the article propose a new synthesis method of multiple attribute decisions based on the decision tree diagram used for establishing a problem analysis model by integrating structural, technological and safety decisions. With reference to data in the decision tree diagram, a new decision matrix (Y) is produced and employed for ranking alternative decision combinations in light of rationality.

The key principle of the provided decision synthesis method is as follows: using three quantitative multiple attribute methods (TOPSIS, SAW and COPRAS) and those for identifying the significance of attributes, to assess the rationality of construction choices in constructing an external wall of a building in terms of the proposed structural design, construction techniques and safety requirements recommended for implementing the selected project.

Based on the model of the analysis of multi-stage decisions extrapolated in the previous research done by the authors, the current article presents a new synthesis method of multi-stage and multiple attribute decisions (SyMAD-3). The provided multi-stage decision tree model shows the structure of the analysis of decisions that belong to different stages. The diagram of the decision analysis tree is described below using the following notation technique:

1) A set of stages in decision analysis K = {k}, (k = 1, 2, c), k--the number of the stage, c--the quantity of stages;

2) The quantity of decision tree nodes--[m.sub.k] (k = 1, 2, c) at each stage are determined depending on the number of decision tables;

3) The quantity of the paths of the model decision tree connecting the root node with the terminal node (called leaf): z = [m.sub.c], where mc is the quantity of nodes at the final decision tree stage. The quantity of paths in the tree is the number of alternative combinations.

Once the decision analysis model and the data described according to tree notation are available, the rational decision must be sought, i.e. by selecting an appropriate algorithm, the alternatives can be ranked by rationality. The authors propose the synthesis method of multi-stage and multiple attribute decisions applying three decisions--SyMAD-3. The algorithm of the proposed method consists of two stages:

Decision stage 1 (Fig. 1) is intended for the formulation of the problem, the preparation of evaluation data and performing preliminary alternative evaluation. The stage consists of the following six steps:

1) The identification of the quantity of decision stages and the formulation of an attribute system for each stage k (k = 1, 2,...,c) of the decision tree. Filling decision tables ([A.sub.t], t = [bar.1, [m.sub.k]]);

2) By using these data tables, decision matrixes are subsequently formed:

[X.sub.t] = [[x.sup.t.sub.ij]], (t = [bar.1,[m.sub.k]]; i = [bar.1,[a.sub.t]], j = [bar.1,[n.sub.k]]), (1)

where: t is the number of a decision table, [a.sub.t] is the quantity of alternatives in t-th decision table, [n.sub.k] is the quantity of alternatives at the k-th stage;

3) Filling (expert) pair wise comparison matrixes used for identifying the significance of attributes. E = {p}, p = 1, 2,...,[e.sub.p], E--expert set, p--the number of an expert, [e.sub.p]--the quantity of experts;

4) The identification of the consistency of pair wise comparison matrixes. For this purpose, the consistency degree S of each matrix is calculated:

S = [S.sub.I]/[S.sub.A], (2)

where: [S.sub.1] is the matrix consistency index, SA is the average random index. If S < 0.1, matrix consistency is sufficient and the matrix is used for identifying the subjective significance of attributes; in case it is not, matrix data is not used for further calculations (Saaty 1990);

5) The identification of the significance of attributes for the k-th stage applying pair wise attribute comparison matrixes completed by experts and the method of least squares for identifying the subjective significance of attributes:

[[bar.q].sub.kj](j = [bar.1,n] k = [bar.1,c]). (3)

Using the subjective significance of attributes, the degree of an agreement of expert estimates can be determined applying W. Kendall's concordance coefficient (Ustinovichius et al. 2007). If the agreement of expert estimates are sufficient, the integrated significances of attributes

[q.sup.*.sub.kj](j = [bar.1,[n.sub.k]], k = [bar.1,c]) (4)

are calculated, otherwise the group of experts is reconsidered;

6) Using decision matrixes (1) and integrated significances of attributes (4), the rationality of alternatives is identified employing three methods: TOPSIS, SAW and COPRAS;

7) Once calculations are completed using all three methods, the results are provided in the form of relative significance criteria according to TOPSIS, SAW and COPRAS methods without adding them up:

([A.sup.i.sub.k](Topsis,Saw,Copras)) = ([R.sup.i.sub.kT], [R.sup.i.sub.kS], [R.sup.i.sub.kC]), (5)

where: (k = [bar.1,c], i = [bar.1,[m.sub.k]]).

Decision stage 2 (Fig. 2) is intended for the formulation of alternative combinations and evaluation of their rationality. Using the alternatives produced at Decision stage 1 and described in decision tables [A.sub.t] as well as rationality evaluation results and the decision tree model provided in the previous research of the authors, alternative combinations

[B.sub.s], (s = [bar.1.z]) (6)

are completed.

The following actions are carried out at Decision stage 2:

1) Data on alternative combinations are provided in the form of vectors:

[B.sub.s] = [([R.sup.i.sub.1,T], [R.sup.i.sub.1,S], [R.sup.i.sub.1,C]),...,([R.sup.i.sub.c,T], [R.sup.i.sub.c,S], [R.sup.i.sub.C,C]), (7)

where: (i = [bar.1,[m.sub.k]], s = [bar.1,z]);

2) The obtained alternative combinations are entered into the decision table (Table 1) the data of which will be used for further calculations;

3) When using TOPSIS, SAW and COPRAS methods, the evaluation of the table of alternative combinations (Table 1) is made:

1. Using data on made decisions (Table 1), the decision matrix is completed:

Y = [[y.sub.sl]], (s = [bar.1,z], l = [bar.1,k x mt]), (8)

where: mt is the number of the methods applied (in our case mt = 3), k is the number of the stage (k = 1, 2,...,c). In this case, s is the quantity of rows in matrix Y and / is the quantity of columns in matrix Y.

([y.sub.sl]) = ([R.sup.i.sub.k,M]), (s = [bar.1,z], l = [bar.1,k x mt], M = [bar.1,mt]), (9)

where: (i = [bar.1,[m.sub.k]], k = [bar.1,c], M = [bar.1,mt]), M is the number of the method, mt--the quantity of methods;

2. A set of attributes required for evaluating the alternatives provided in matrix Y is provided:

R = {[R.sub.1]}, (l = 1, 2,...,k x mt). These attributes are maximised, whereas their significance values are the same because they are not affected either by subjective or objective factors. The significance values of the attributes must satisfy the equation:

[k+mt.summation over (j=1)] [w.sub.j] = 1, (10)

where: k is the quantity of stages, mt is the quantity of methods;

3. The performed evaluation of alternative combinations using the above methods, rationality evaluation and ranking alternative combinations are given in a form of a table.

The algorithm for the synthesis method of multistage and multiple attribute decisions is provided in two flow charts below (Fig. 1 and Fig. 2). The proposed method may be used for solving various multi-stage and multiple attribute decision-making problems when information about the attributes is provided in a quantitative form.

3. Methods applied for calculations

Based on the judgement of each expert, the subjective, objective and integrated values of the significances of the attributes are determined.

The subjective values of criteria significance are determined based on expert pair wise comparison. The values of [bar.[q.sub.j]] (j = [bar.1,n]) are found by solving the optimization problem:

min{[n.summation over (i=1)][n.summation over (j=1)]([b.sub.ij][bar.[q.sub.j]] - [bar.[q.sub.i]])}(11)

when, the unknown values of [bar.[q.sub.j]] (j = 1,n) satisfy constraints:

[n.summation over (j=1)][bar.[q.sub.j]] = 1, [bar.[q.sub.i]] > 0, (j = [bar.1,n])(12)

Group evaluation may be considered to be reliable only if the estimates elicited from various experts or the members of a cooperative decision making group are consistent. The level of the agreement of expert estimates can be determined using W. Kendall's concordance coefficient (Saaty 1990).

The next step is the calculation of the objective significance values of the criteria using the Entropy method (Ustinovichius 2007).

Values [q.sup.*.sub.j] (the significance of integrated attributes)

are determined according to the formula:

[bar.[q.sub.j]] [n.summation over (j=1)] [q.sup.*.sub.j][q.sub.j] - [q.sup.*.sub.j][q.sub.j] = 0, (j = l,2,...,n), (13)

when [bar.[q.sub.j]] (the significance of subjective attributes found making a pair wise comparison) and [q.sub.j] (objective significance found employing the Entropy method) are known (Ustinovichius 2007).

To identify the rationality of alternatives, three multiple attribute decision-making methods--TOPSIS, SAW and COPRAS--are applied based on quantitative calculations.

Mathematically, Simple Additive Weighting (SAW) method can be stated as follows: suppose the decision maker assigns a set of importance weights to attributes [bar.q] = {[q.sub.1],[q.sub.2],...,[q.sub.n]}. Then, the most preferred alternative [A.sup.*] is selected such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where: [x.sub.ij] is the outcome of the ith alternative about the jth attribute with a numerically comparable scale. The weights are usually normalized so that:

[n.summation over (j=1)][q.sub.j] = 1 (15)

Simple Additive Weighting method requires a comparable scale for all elements in the decision matrix. The comparable scale is obtained using equation:

[[ba.x].sub.ij] = [x.sub.ij]/[x.sup.max.sub.j] (16)

for benefit criteria and equality:

[[bar.x].sub.ij] = [x.sup.min.sub.j]/[x.sub.ij] (17)

for cost criteria.

Method TOPSIS was developed by Hwang and Yoon (1981). The technique is based on the idea that the optimal alternative is the most similar one to an ideal solution (being closest to it and at the longest distance from the negatively ideal solution). This method is known as TOPSIS--Technique for Order Preference by Similarity to Ideal Solution.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

A relative distance of any ith variant from the ideal one is obtained by the formula:

[K.sub.BIT] = [L.sup.-.sub.i]/[L.sup.+.sub.i] + [L.sup.-.sub.i], i = [bar.1,m], (18)

where: [K.sub.BIT][0,1], [L.sup.+.sub.i] is the distance between the compared i-th variant and the ideal one; [L.sup.-.sub.i] is the distance between the compared ith variant and the negatively ideal alternative. The nearer to one is [K.sub.BIT] value, the closer is the ith variant to [a.sub.+], i.e. an optimal variant is the one that has the highest value of [K.sub.BIT].

Method COPRAS consists of several stages of calculation. At stage 1, the normalisation of the elements of the decision matrix is conducted using the formula:

[d.sub.ij] = [x.sub.ij] x [q.sup.*.sub.j]/[m.summation over (i=1)][x.sub.ij], i = [bar.1,m], j = [bar.1,n], (19)

where: [x.sub.ij] is the value of the attribute j of alternative i; m is the number of alternatives; n is the number of attributes; [q.sup.*.sub.j] is the integrated significance value of the jth attribute.

At stage 2, the sums of minimising [S.sub.-i] and maximizing [S.sub.+i] evaluating the normalised attributes of each alternative are calculated. The following formulas are used:

[S.sub.+i] = [n.summation over (j=1)][d.sub.+ij], (20)

[S.sub.-i] = [n.summation over (j=1)][d.sub.-ij], i = [bar.1,m], j = [bar.1,n] (21)

At stage 3, the relative significance of comparable alternatives is identified on the basis of positive [S.sub.-i] and negative [S.sub.+i], the characteristics that describe the alternatives. The relative significance (rationality) of each alternative [Q.sub.i] is identified using the formula:

[Q.sub.i] = [S.sub.+i] + [S.sub.-min] x [m.summation over (i=1)] [S.sub.-i]/[S.sub.-i] x [m.summation over (i=1)][[S.sub.-min]/[S.sub.-i]], i = [bar.1,m] (22)

The higher is [Q.sub.i] value, the more the alternative complies with the needs (preferences) of a decision-making person (Zavadskas et al. 2010; Tupenaite et al. 2010).

4. The synthesis of structural, technological and safety decisions using SyMAD-3

Construction work must be organised so that the safety of employees should be ensured during the entire construction process (construction of a building) to prevent/reduce the number of accidents and occupational diseases. In the course of construction, such technological procedures shall apply to ensure the quality of work and safety as well as to observe technology requirements set out in the technology project.

In order to improve working conditions and the quality of work, structural, technological and safety decisions of construction processes should be integrated into a whole. Then, the focus should be on one object, the elements of which would include three main areas, namely, structural elements, technology and safety decisions of construction processes.

The multi-stage decision tree model is used for the above purpose, with the help of which, the analysis of finding a solution to the problem is conducted and a set of possible alternatives is modelled. This model is used for analysing the possible construction variants of the wall by combining structural, technological and security decisions into a single object of a decision. The overall decision tree model using the synthesis method of multi-stage and multiple attribute decisions SyMAD-3 is shown in Fig. 3.

5. Case study

A multiple attribute decision problem the decision analysis model of which was described in the previous research done by the authors has been formulated. The overall decision tree model is made and presented in Fig. 3. The use of the SyMAD-3 method solves the multi-stage and multiple attribute decision problems.

In order to identify coherence among structural, technological and safety decisions, three types of external wall variants were selected: Masonry structure No. 1 Arko calcium silicate blocks; Masonry structure No. 2 Ventilated facade; Masonry structure No. 3 Insulated solid masonry wall.

[FIGURE 3 OMITTED]

The variants of external wall structure are identified in this article as three alternatives: [A.sub.1], [A.sub.2] and [A.sub.3]. The priority and significance of each variant directly and proportionally depend on the system of attributes characteristic to each alternative, their values and significance. For this reason, 12 attributes were selected: [R.sub.C1], [R.sub.C2], [R.sub.C3], [R.sub.C4], [R.sub.T1], [R.sub.T2], [R.sub.T3], [R.sub.T4], [R.sub.T5], [R.sub.T6], [R.sub.T7] and [R.sub.T8]. These attributes were divided into three separate groups: structure, technology and work safety. Four attributes were allocated to each structural group: wall resistance to cold (wall longevity, years), wall heat transfer coefficient (W/[m.sup.2]K), the weight of external walls ([m.sup.2] kg) and material cost per 1 [m.sup.2] of wall installation (LTL/[m.sup.2]). The attributes of the technological group cover labour cost (man-hour/[m.sup.2], employee qualification category (score), mechanism demand (mechanism-hour/[m.sup.2])and construction process labour cost (LTL/[m.sup.2]). The attributes of the work safety group consist of the level of risk at the work place (score), protective equipment cost (LTL/[m.sup.2]) labour cost to ensure safety (man-hour/[m.sup.2])and mechanism demand to ensure safety (mechanism-hour/[m.sup.2]). The values of the alternative attributes are provided in Table 2. Steps 1 and 2 of the first decision stage have been completed.

To identify attribute significance, expert judgement is required (steps 3-4 of stage 1).

The survey involved 33 respondents. The evaluation of structural, technological and work safety decisions was provided by a group of people that involved construction company directors, construction technical supervisors, construction managers, occupational health and safety professionals, researchers, employees of the State Labour Inspectorate, engineers and construction workers.

Table 3 provides the subjective values of attribute significance on the basis of which the degree of the consistency of expert judgment and the coefficient of the pair wise matrix consistency are determined. Following the procedure of determining two kinds of consistency, the following findings were obtained:

1) To verify the consistency of expert judgement, the value of the significance of concordance coefficient [chi square] is calculated and compared with table distribution value [chi square](0.05; 32). The made calculations showed that expert judgement on the significance values of structural and technological attributes were sufficiently consistent; however, judgement on the significance of work safety attributes is of insufficient consistency;

2) After verifying the consistency of pair wise comparison matrixes completed by experts, consistency coefficient S was calculated. 33 experts found that the consistency coefficient of the matrixes was greater than 0.1 in nearly 50%, which indicates that attribute ranking done by experts does not satisfy transitivity property;

3) Pair wise comparison matrixes, the consistency of which is sufficient (S < 0.1), were used for further calculations.

Four steps of stage 1 of the SyMAD-3 method were completed and 3 decision stages was identified; a set of attributes was formed and decision tables were completed --3; finally, calculations were made to determine the significance of attributes. The subjective and integrated values of significance at all stages are provided in Table 4. To evaluate the rationality of alternatives, the integrated values of attribute significance will be used.

In step 5 of stage 1, the evaluation of the rationality of all alternatives at all stages is carried out applying three methods: TOPSIS, SAW and COPRAS. Calculation results are provided in Table 5.

After evaluation using three methods, the following results were obtained:

1. [T.sub.1] [??] [T.sub.3] [??] [T.sub.2]. It can be maintained that the rationality value of alternative [T.sub.1] is the highest;

2. [K.sub.2] [??] [K.sub.3] [??] [K.sub.1]. It can be maintained that the rationality value of alternative [K.sub.2] is the highest;

3. [D.sub.2] [??] [D.sub.1] [??] [D.sub.3]. It can be maintained that the rationality value of alternative [D.sub.2] is the highest.

On the basis of the conclusions provided above, we cannot establish which alternative is the most rational one with respect to structural, technological and safety decisions.

The data displayed in Table 5 are used for completing alternative combinations (see Table 6).

The data set in Table 6 will be used for calculations, whereas the values of attribute significance are equal to [w.sub.j] = 0.111, ([w.sub.j] = 1, 2,...,9). All attributes are maximised.

Having applied the algorithm of the synthesis method SyMAD-3, the following calculation results were obtained (Table 7).

The calculated results presented above show that alternative [B.sup.2]--ventilated facade is the most rational alternative in light of structural, technological and work safety aspects.

It order to compare calculation results obtained by applying the new method SyMAD-3 when alternatives are separately evaluated from the point of view of structural, technological and work safety requirements with the results using the method when these groups of attributes are not segregated, the authors selected the expert method identifying the significance of attributes (Zavadskas1991).

The significance of twelve attributes obtained using the expert method is provided in Table 8. The verification of expert judgement consistency finds it sufficient. The values of attribute significance were calculated using the dataset of 33 experts.

The values of attribute significance, including structural ([R.sub.Cj], j = 1, 2, 3, 4), technological ([R.sub.Tj], j = 1, 2, 3, 4) and work safety ([R.sub.Sj], j = 1, 2, 3, 4) aspects are provided in Table 8.

Using the significance (Table 8) and values of the attributes (Table 2), the rationality of alternatives A., A2 and [A.sub.3] using three methods TOPSIS, SAW and COPRAS can be calculated. The results of calculation present the most rational alternative [A.sub.1] which is Arko calcium silicate blocks. Calculation results [A.sub.1] [??] [A.sub.2] [??] [A.sub.3] are provided in Table 9.

A comparison of these two methods shows the existing difference. Multiple attribute decision problems are solved by identifying the stages of the problem and by carrying out calculations at each stage. Finally, the results of the made calculations are summarized, which allows the analysis and assessment of a more detailed multiple attribute decision making problem than in the case where the decision making problem does not fall into smaller segments.

6. Conclusions

The analysis of related work demonstrates that multiple attribute decision analysis and synthesis allow a more detailed approach to decision making in terms of which the authors of the previous research provided a multistage decision model. Then, the problems of a multi-stage and multiple attribute decision-making were provided in the form of this model. However, the authors of the current article present a new multi-stage and multiple attribute decision synthesis method--SyMAD-3 for solving multiple attribute decision-making problems. If qualitative input data for a decision-making problem are provided for solving the problem, it is more convenient to use such multiple attribute qualitative methods as TOPSIS, SAW and COPRAS. Since each method has its own premises, the authors suggested that these three methods for decision-making should be combined into a single method. Combining methods increases decision reliability, because a decision-maker has an opportunity to see the results of rationality evaluation considering each alternative to all three methods.

The new synthesis method of multiple attribute and multi-stage decisions using three methods (SyMAD-3) complies with the following main requirements imposed in this paper:

--completing all possible alternative combinations increases the level of result details;

--exists a possibility of noting the impact of individual decision stages on the rationality of alternative combinations in the course of calculations;

--considering different sensitivity of multiple attribute decision-making methods with respect to input data, using the synthesis method of multiple attribute decisions, three multiple attribute decision-making methods are combined into a single system thus increasing the reliability of the made decision.

This method allows combining such elements of the construction process as structural, technological and work safety decisions into a single complex decision.

The authors have evaluated the complexity of the algorithm of the proposed synthesis decision method (SyMAD-3) and compared it with the complexities of the algorithms of multi-stage synthesis methods proposed by other authors. As a result, they make an assumption that the method described in the article is more efficient time-wise. The complexity of the algorithms used in the method take linear time O(n.), because the basis of this method is vector algebra for arrays.

Apart from construction projects, the proposed method can also be applied to other problems related to decision making. In the future, the authors are planning to apply the SyMAD-3 method to solve multiple attribute decision-making problems encountered in other areas where input data rely on quantitative estimates.

doi: 10.3846/13923730.2012.666504

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NAUJAS KONSTRUKCINI?, TECHNOLOGINI? IR SAUGOS SPRENDIM? SINTEZ?S METODAS (SYMAD-3)

Ruta Simanaviciene (1), Rita Liaudanskiene (2), Leonas Ustinovichius (3)

(1,3) Vilnius Gediminas Technical University, Saul?tekio al. 11, LT-10223 Vilnius, Lithuania

(2) Kaunas University of Technology, Student? g. 48, LT-51367 Kaunas, Lithuania

E-mails: (1) ruta.simanaviciene@vgtu.lt (corresponding author); (2) rita2290@gmail.com; (3) leonas.ustinovicius@vgtu.lt

Received 19 Aug. 2011; accepted 04 Oct. 2011

Ruta SIMANAVICIENE. A PhD student at the Department of Information Systems, the Faculty of Fundamental Sciences, Vilnius Gediminas Technical University. Studies in Mathematics and Informatics at Vilnius Pedagogical University (BA in Mathematics), MA in Mathematics and school teacher's degree (2002). Research interests: applied mathematics, multicriteria decision making methods and decision support systems.

Rita LIAUDANSKIENE. A graduate from the Department of Construction Technology, the Faculty of Civil Engineering and Architecture, Kaunas University of Technology. Studies in Civil Engineering at the Faculty of Civil Engineering and Architecture in 2002 (BA in Civil Engineering). MA in Civil Engineering and professional engineer's qualification (2004). Research interests: construction technology and organization, ensuring worker health and safety in the construction sector.

Leonas USTINOVICHIUS. Prof, Dr Habil, the chairman of the laboratory of Construction Technology and Management. Vilnius Gediminas Technical University. Dr (1989), Dr Habil (2002). Publications: more than 150 scientific papers. Research interests: building technology and management, decision-making theory, automation in design, expert systems.
Table 1. A decision table of alternative combinations

Stages:                             stage 1

Alternatives\   [R.sub[1]]          [R.sub.[2]]
  Attributes
B1              [R.sup.1.sub.c,T]   [R.sup.1.sub.c,S]
B2              [R.sup.1.sub.c,T]   [R.sup.1.sub.c,S]
...             ...                 ...
[B.sub.z]       R.sup.n1.sub.c,T    R.sup.n1.sub.c,S
Min/max         Max                 Max

Stages:                             ...   C-th stage

Alternatives\   [R.sub.[3]]         ...   [R.sub.[7]]
  Attributes
B1              [R.sup.1.sub.c,C]   ...   [R.sup.1.sub.s,T]
B2              [R.sup.1.sub.c,C]   ...   [R.sup.2.sub.s,T]
...             ...                 ...   ...
[B.sub.z]       R.sup.n1.sub.c,C    ...   R.sup.n3.sub.s,T
Min/max         Max                       Max

Stages:                    C-th stage

Alternatives\   [R.sub.[8]]         [R.sub.[9]]
  Attributes
B1              [R.sup.1.sub.s,S]   [R.sup.1.sub.s,C]
B2              [R.sup.2.sub.s,S]   [R.sup.2.sub.s,C]
...             ...                 ...
[B.sub.z]       R.sup.n3.sub.s,S    R.sup.n3.sub.s,C
Min/max         Max                 Max

Table 2. The values of the attributes of evaluated external wall
construction variants

              Attributes                         Masonry
                                                structure
                                                  No. 1
                                                  Arko
                                                 calcium
                                                silicate
                                                 blocks

Technology    Labour cost                 Min     3.76
                (man-hour/[m.sup.2)
              Employee qualification      Max     4.17
                category (score)
              Mechanism demand            Min     0.56
                (mechanism-hour/
                [m.sup.2])
              Construction process        Min      84
                labour cost
                (LTL/[m.sup.2])
Structure     Wall resistance to          Max      50
                cold (cycles)
              Wall heat transfer          Min     0.223
                coefficient
                (W/[m.sup.2]K)
              The weight of external      Min      573
                walls ([m.sup.2]kg)
              Material cost per 1         Min      181
                [m.sup.2] of wall
                installation
                (LTL/[m.sup.2])
Work safety   The level of risk at        Min       3
                the work place (score)
              Protective equipment cost   Min      177
                (LTL/[m.sup.2])
              Labour cost to              Min     0.146
                ensure safety
                (man-hour/[m.sup.2])
              Mechanism demand to         Min     0.021
                ensure safety
                (mechanism-hour/
                [m.sup.2])

              Attributes                      Masonry      Masonry
                                             structure    structure
                                               No. 2        No. 3
                                             Ventilated   Insulated
                                               facade       solid
                                                           masonry
                                                            wall

Technology    Labour cost                 Min   4.61        6.06
                (man-hour/[m.sup.2)
              Employee qualification      Max    4          4.17
                category (score)
              Mechanism demand            Min  0.955        1.107
                (mechanism-hour/
                [m.sup.2])
              Construction process        Min    96          135
                labour cost
                (LTL/[m.sup.2])
Structure     Wall resistance to          Max    75          50
                cold (cycles)
              Wall heat transfer          Min  0.217        0.222
                coefficient
                (W/[m.sup.2]K)
              The weight of external      Min   635          672
                walls ([m.sup.2]kg)
              Material cost per 1         Min   269          193
                [m.sup.2] of wall
                installation
                (LTL/[m.sup.2])
Work safety   The level of risk at        Min    4            3
                the work place (score)
              Protective equipment cost   Min   162          185
                (LTL/[m.sup.2])
              Labour cost to              Min   0.14        0.154
                ensure safety
                (man-hour/[m.sup.2])
              Mechanism demand to         Min  0.021        0.022
                ensure safety
                (mechanism-hour/
                [m.sup.2])

Table 3. The subjective values of attribute significance using data
obtained by 33 experts

Expert No. Expert 1     [R.sub.T1]      [R.sub.T2]      [R.sub.T3]

                        0.1718          0.0685          0.1177
Expert 2                0.175           0.0752          0.1055
Expert 3                0.0674          0.0882          0.1045
Expert 4                0.4831          0.1311          0.2682
Expert 5                0.0566          0.7582          0.1287
Expert 6                0.0939          0.1323          0.0881
Expert 7                0.4151          0.1321          0.0853
Expert 8                0.0765          0.1159          0.0703
Expert 9                0.0655          0.6766          0.1595
Expert 10               0.0899          0.1145          0.0682
Expert 11               0.0799          0.3647          0.3467
Expert 12               0.0722          0.6692          0.1608
Expert 13               0.0516          0.7033          0.1625
Expert 14               0.1             0.1087          0.0857
Expert 15               0.0832          0.6414          0.0552
Expert 16               0.2375          0.2966          0.3593
Expert 17               0.5673          0.0827          0.2196
Expert 18               0.5807          0.0678          0.22
Expert 19               0.0374          0.2171          0.0874
Expert 20               0.0785          0.2452          0.0612
Expert 21               0.1676          0.4326          0.063
Expert 22               0.1201          0.3587          0.0643
Expert 23               0.0959          0.443           0.1907
Expert 24               0.1746          0.1213          0.5828
Expert 25               0.2104          0.379           0.1778
Expert 26               0.098           0.1049          0.1823
Expert 27               0.5995          0.0849          0.1515
Expert 28               0.1803          0.0477          0.0572
Expert 29               0.2441          0.3258          0.1131
Expert 30               0.3115          0.1202          0.3115
Expert 31               0.3051          0.4382          0.0362
Expert 32               0.3328          0.144           0.1195
Expert 33               0.1123          0.0637          0.0838
Expert                  Consistent

Expert No. Expert 1     [R.sub.T4]      [R.sub.K1]      [R.sub.K2]

                        0.642           0.07            0.4368
Expert 2                0.6443          0.0503          0.2625
Expert 3                0.74            0.0558          0.3486
Expert 4                0.1176          0.2438          0.1371
Expert 5                0.0566          0.7871          0.1039
Expert 6                0.6857          0.0831          0.2414
Expert 7                0.3675          0.0513          0.3725
Expert 8                0.7373          0.0528          0.6208
Expert 9                0.0984          0.5161          0.1225
Expert 10               0.7275          0.1031          0.7706
Expert 11               0.2086          0.1631          0.3846
Expert 12               0.0978          0.4784          0.1546
Expert 13               0.0827          0.5868          0.2233
Expert 14               0.7056          0.1917          0.1583
Expert 15               0.2202          0.1317          0.0855
Expert 16               0.1067          0.043           0.1337
Expert 17               0.1305          0.4228          0.3091
Expert 18               0.1315          0.4396          0.3074
Expert 19               0.658           0.1021          0.6828
Expert 20               0.6151          0.0711          0.6035
Expert 21               0.3368          0.0608          0.6709
Expert 22               0.4569          0.0953          0.2088
Expert 23               0.2704          0.126           0.3956
Expert 24               0.1213          0.1461          0.1066
Expert 25               0.2328          0.0899          0.7089
Expert 26               0.6148          0.148           0.6418
Expert 27               0.1641          0.1827          0.066
Expert 28               0.7147          0.343           0.1997
Expert 29               0.3169          0.4468          0.276
Expert 30               0.2568          0.5128          0.1923
Expert 31               0.2204          0.4346          0.302
Expert 32               0.4037          0.1522          0.1535
Expert 33               0.7402          0.0869          0.1052
Expert                                  Consistent

Expert No. Expert 1     [R.sub.K3]    [R.sub.K4]     [R.sub.D1]

                        0.0445        0.4487         0.4091
Expert 2                0.0646        0.6226         0.2351
Expert 3                0.0504        0.5451         0.633
Expert 4                0.4755        0.1437         0.178
Expert 5                0.0455        0.0636         0.7925
Expert 6                0.0694        0.6062         0.386
Expert 7                0.2069        0.3694         0.25
Expert 8                0.0541        0.2723         0.6454
Expert 9                0.2587        0.1028         0.0908
Expert 10               0.0546        0.0717         0.5586
Expert 11               0.0559        0.3965         0.5801
Expert 12               0.243         0.124          0.1015
Expert 13               0.1166        0.0733         0.0968
Expert 14               0.0711        0.5789         0.5051
Expert 15               0.2668        0.516          0.1774
Expert 16               0.0649        0.7584         0.0394
Expert 17               0.1828        0.0853         0.4257
Expert 18               0.1834        0.0696         0.4257
Expert 19               0.0663        0.1488         0.1393
Expert 20               0.0525        0.2729         0.5188
Expert 21               0.0514        0.217          0.4475
Expert 22               0.27          0.4259         0.2603
Expert 23               0.1627        0.3156         0.0681
Expert 24               0.0623        0.685          0.6608
Expert 25               0.098         0.1032         0.0773
Expert 26               0.1155        0.0947         0.6288
Expert 27               0.6536        0.0977         0.2231
Expert 28               0.0621        0.3952         0.6527
Expert 29               0.0919        0.1853         0.4
Expert 30               0.1026        0.1923         0.5102
Expert 31               0.0946        0.1688         0.4206
Expert 32               0.1214        0.573          0.1183
Expert 33               0.0681        0.7397         0.1972
Expert                                               Inconsistent

Expert No. Expert 1     [R.sub.D2]    [R.sub.D3]    [R.sub.D4]

                        0.37          0.11          0.111
Expert 2                0.606         0.078         0.081
Expert 3                0.252         0.057         0.059
Expert 4                0.058         0.139         0.626
Expert 5                0.045         0.06          0.102
Expert 6                0.433         0.085         0.096
Expert 7                0.25          0.25          0.25
Expert 8                0.216         0.071         0.068
Expert 9                0.129         0.672         0.108
Expert 10               0.303         0.075         0.063
Expert 11               0.128         0.155         0.136
Expert 12               0.149         0.634         0.116
Expert 13               0.121         0.694         0.089
Expert 14               0.101         0.238         0.156
Expert 15               0.268         0.209         0.346
Expert 16               0.061         0.117         0.783
Expert 17               0.312         0.166         0.097
Expert 18               0.312         0.166         0.097
Expert 19               0.064         0.076         0.721
Expert 20               0.296         0.093         0.093
Expert 21               0.448         0.05          0.055
Expert 22               0.157         0.322         0.26
Expert 23               0.406         0.381         0.145
Expert 24               0.068         0.136         0.136
Expert 25               0.719         0.148         0.056
Expert 26               0.071         0.086         0.215
Expert 27               0.104         0.289         0.384
Expert 28               0.086         0.108         0.153
Expert 29               0.2           0.2           0.2
Expert 30               0.125         0.172         0.193
Expert 31               0.096         0.285         0.199
Expert 32               0.29          0.504         0.089
Expert 33               0.557         0.183         0.064
Expert
  judgement

Table 4. The values of attribute significance

Structural attributes      [R.sub.C1]   [R.sub.C2]   [R.sub.C3]

Subjective values of         0.2129       0.3144       0.0965
  significance
Integrated values of         0.0024       0.9817       0.0098
  attribute significance
Ranking significance           4            1            2

Technological attributes   [R.sub.S1]   [R.sub.T2]   [R.sub.T3]

Subjective values of         0.3745       0.1051       0.2628
  significance
Integrated values of         0.0344       0.9323       0.0097
  attribute significance
Ranking significance           2            1            4

Work safety attributes     [R.sub.S1]   [R.sub.S2]   [R.sub.S3]

Subjective values of         0.4559       0.2603       0.1463
  significance
Integrated values of         0.0537       0.1698       0.1944
  attribute significance
Ranking significance           4            3            2

Structural attributes      [R.sub.C4]

Subjective values of         0.3763
  significance
Integrated values of         0.0061
  attribute significance
Ranking significance           3

Technological attributes   [R.sub.T4]

Subjective values of         0.2576
  significance
Integrated values of         0.0236
  attribute significance
Ranking significance           3

Work safety attributes     [R.sub.S4]

Subjective values of         0.1376
  significance
Integrated values of         0.5821
  attribute significance
Ranking significance           1

Table 5. Rationality indicators of structural, technological and
safety decisions

Technology   Alternatives    [T.sub.1]   [T.sub.2]   [T.sub.3]

             Rationality     1           0.2546      0.6525
               value acc.
               to TOPSIS
             Ranking acc.    1           3           2
               to TOPSIS
             Rationality     1           0.949       0.973
               value acc.
               to SAW
             Ranking acc.    1           3           2
               to SAW
             Rationality     0.3432      0.3247      0.33209
               values acc.
               to COPRAS
             Ranking acc.    1           3           2
               COPRAS

Structure    Alternatives    [K.sub.1]   [K.sub.2]   [K.sub.3]

             Rationality     0.0981      0.91        0.1809
               value acc.
               to TOPSIS
             Ranking acc.    3           1           2
               to TOPSIS
             Rationality     0.973       0.997       0.975
               value acc.
               to SAW
             Ranking acc.    3           1           2
               SAW
             Rationality     0.3303      0.3385      0.3312
               values acc.
               to COPRAS
             Ranking acc.    3           1           1
               COPRAS

Safety       Alternatives    [D.sub.1]   [D.sub.2]   [D.sub.3]

             Rationality     0.6735      0.7142      0.2858
               value acc.
               to TOPSIS
             Ranking acc.    2           1           3
               to TOPSIS
             Rationality     0.978       0.987       0.935
               value acc.
               to SAW
             Ranking acc.    2           1           3
               SAW
             Rationality     0.3376      0.3398      0.3226
               values acc.
               to COPRAS
             Ranking acc.    2           1           3
               COPRAS

Table 6. A dataset of alternative combinations

Types of attributes               Structure               Technology

Alternative\          [R.sup.T]   [R.sup.S]   [R.sup.C]   [R.sup.T]
  Attributes          [1]         [2]         [3]         [4]

[B.sup.1]             0.0981      0.973       0.33034     1
[B.sup.2]             0.91        0.997       0.33847     0.2546
[B.sup.3]             0.1809      0.975       0.33118     0.6525
max/min               Max         Max         Max         Max

Types of attributes                           Safety
Alternative\          [R.sup.S]   [R.sup.C]   [R.sup.T]   [R.sup.S]
  Attributes          [5]         [6]         [7]         [8]

[B.sup.1]             1           0.3432      0.6735      0.978
[B.sup.2]             0.949       0.3247      0.7142      0.987
[B.sup.3]             0.973       0.3321      0.2858      0.935
max/min               Max         Max         Max         Max

Types of attributes  Technology

Alternative\          [R.sup.C]
  Attributes          [9]

[B.sup.1]             0.3375
[B.sup.2]             0.3397
[B.sup.3]             0.3226
max/min               Max

Table 7. The rationality and ranking of alternative combinations

                                  Rationality               Ranking

Method\Combinations   [B.sub.1]    [B.sub.2]    [B.sub.3]   [B.sub.1]
  of alternatives

TOPSIS                 0.4528       0.6124       0.2662         2
SAW                     0.887        0.906        0.783         2
COPRAS                 0.3367       0.37025      0.29305        2

                       Ranking

Method\Combinations   [B.sub.2]   [B.sub.3]
  of alternatives

TOPSIS                    1           3
SAW                       1           3
COPRAS                    1           3

Table 8. The significance of attributes applying the expert method

Types of the attributes                      Structure

Attribute                       [R.sub.c1]   [R.sub.c2]   [R.sub.c3]

Significance of the attribute      0.08        0.083        0.079
max/min                            min          max          min

Types of the attributes        Structure          Technology

Attribute                      [R.sub.c4]   [R.sub.t1]   [R.sub.t2]

Significance of the attribute     0.083        0.085        0.086
max/min                            min          max          min

Types of the attributes              Technology            Safety

Attribute                      [R.sub.t3]   [R.sub.t4]   [R.sub.s1]

Significance of the attribute     0.084        0.086        0.085
max/min                            min          min          min

Types of the attributes                       Safety

Attribute                      [R.sub.s2]   [R.sub.s3]   [R.sub.s4]

Significance of the attribute     0.083        0.082        0.083
max/min                            min          min          min

Table 9. The rationality of the external wall alternatives according
to technological, structural and work safety attributes using the
significance of attributes determined by the expert method

                                     Rationality

Method\                  [A.sub.1]     [A.sub.2]   [A.sub.3]
  Alternatives

TOPSIS                   0.7185        0.4877      0.3098
SAW                      0.961         0.879       0.826
COPRAS                   0.36296       0.3306      0.30646

                                      Ranking

Method\                  [A.sub.1]   [A.sub.2]   [A.sub.3]
  Alternatives

TOPSIS                   1           2           3
SAW                      1           2           3
COPRAS                   1           2           3
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Author:Simanaviciene, Ruta; Liaudanskiene, Rita; Ustinovichius, Leonas
Publication:Journal of Civil Engineering and Management
Article Type:Report
Geographic Code:4EXLT
Date:Apr 1, 2012
Words:8969
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