Multiple criteria assessment of pile-columns alternatives/ Daugiatikslis koloniniu poliu alternatyvu vertinimas/ Vairaku kriteriju metode alternativu palu kolonnu novertejumam/ Vaisamba alternatiivide mitmekriteeriumiline hindamine.
Susinskas, Saulius ; Zavadskas, Edmundas Kazimieras ; Turskis, Zenonas 等
1. Introduction
Technological progress and innovation in civil engineering,
management, and conditions of life level yields an enormous influence
over economic activity, employment and growth rates. There is an
increasing complexity and interplay between all issues associated with
property management decisions (Langston et al. 2008). In urban areas,
many high-rise buildings and viaducts are supported by pile foundations
(Zhang et al. 2011).
There are many reasons a geotechnical engineer would recommend a
deep foundation over a shallow foundation, but some of the common
reasons are very large design loads, a poor soil at shallow depth, or
site. A single pile foundation utilizes a single, generally
large-diameter, foundation structural element to support all the loads
(weight, wind, etc.) of a large above-surface structure.
Bridges are the crucial components of highway networks. A pile
bridge is a structure that uses foundations consisting of long poles
(referred to as piles), which are made of wood, concrete or steel and
which are hammered into the soft soils beneath the bridge until the end
of the pile reaches a hard layer of compacted soil or rock. Piles in
such cases are hammered to a depth where the grip or friction of the
pile and the soil surrounding it will support the load of the bridge
deck. Bhattacharya et al. (2005) critically reviewed the current
understanding of pile design under earthquake loading. Tomlinson and
Woodward (2008), Tonias and Zhao (2007) provided a series of simple
examples of the design of piles. Zhao et al. (2007) presented
catastrophic model for stability analysis of high pilecolumn bridge pear
and described a pile-column calculation model (Zhao et al. 2009).
A good example of such structure is the pile-column (also known in
the American practice as extended pile-shaft), where the column is
continued below the ground level as a pile of the same or somewhat
larger diameter (Fig. 1). Obviously, the design of such foundation
requires careful consideration of the flexural strength and ductility
capacity of the pile. An advantage of supporting a column bent on
drilled pile is the cost savings associated with the construction of
large cast-in-drilled-hole piles instead of multiple piles of smaller
diameter. Another advantage of such a design is that localized damage
that could otherwise develop at the column-pile cap joint is avoided by
the pile-column combination, since there is no structural distinction
between the pile and the column other than the presence of a
construction joint at the pile-column interface.
In case of a single pile-column, formation of a plastic hinge in
the pile shaft is the only mechanism by which ductile performance can be
attained. A pile-column bent may first tend to plastify at the
column-beam joint, but the full flexural capacity of the system can only
be obtained through the formation of a secondary plastic hinge,
belowground surface (at least slightly below). Bending moment
distribution varies with height, but diminishes after attaining a max
bending moment below the ground level. A typical depth for max bending
moment, and possibly the location of the plastic hinge, ranges from one
to three or four pile diameters below ground surface, depending on the
above-ground height and soil stiffness.
[FIGURE 1 OMITTED]
Deep mixing/mass stabilization techniques are essentially
variations of in situ reinforcements in the form of piles, blocks or
larger volumes. Cement, lime/quick lime, fly ash, sludge and/or other
binders are mixed into the soil to increase bearing capacity. The result
is not solid as concrete, but should be seen as an improvement of the
bearing capacity of the original soil.
It is difficult to apply probability-based approaches in structural
safety predictions (Kudzys, Kliukas 2010). There is an urgent need for a
systematic methodology for condition assessment of the bridges
structures. It is necessary to learn about criteria determining both
development and downfall of feasible alternatives (Kaplinski 2008). In a
mono-criterion approach, the analyst builds a unique criterion capturing
all the relevant aspects of the problem. Such a one-dimensional approach
is an oversimplification of the actual nature of the problem. All new
ideas and possible variants of decisions in real world must be compared
according to a set of multiple conflicting criteria (Turskis et al.
2009). Sasmal and Ramanjaneyulu (2008) made attempt to develop a
systematic procedure and formulations for condition evaluation of
existing bridges using Analytic Hierarchy Process in a fuzzy
environment. Computer programs have been developed based on the
formulations presented in this paper for evaluating condition of
existing bridges and the details are presented in the investigation.
Classical methods of multiple criteria optimization and
determination of priority and utility function were first applied by
Pareto in 1896 (Pareto 1971). Methods of multiple criteria analysis were
developed to meet the increasing requirements of human society and the
environment.
It was investigated, applied and developed a wide range of multiple
criteria decision-making methods (MCDM): COPRAS--Complex Proportional
Asessment (Zavadskas et al. 2009b), its modification COPRAS-G ( Complex
Proportional Assessment Method with Grey Interval Numbers) (Zavadskas et
al. 2009a), ARAS (Additive Ratio Assessment) method (Zavadskas, Turskis
2010), ARAS-G, ARAS-F (Turskis, Zavadskas 2010a, 2010b) and other
methods.
An alternative in multiple criteria evaluation is usually described
by quantitative and qualitative criteria (Zavadskas et al. 2009b). The
criteria have different units of measurement. Normalization aims at
obtaining comparable scales of the criteria values. Different techniques
of criteria value normalization are used. The impact of the
decision-matrix normalization methods on the decision results has been
investigated by many authors (Peldschus 2009; Zavadskas 1987).
Techniques and planning methods and decision making methods develop
dynamically (Kalibatas, Turskis 2008; Peldschus 2008; Peldschus et al.
2010; Turskis 2008). MCDM researches in civil engineering and management
is dominated in the Lithuanian-German-Polish triangle (Vilnius Gediminas
Technical University, Poznan University of Technology, and Leipzig
University of Applied Science) (Brauers et al. 2010; Brauers, Ginevicius
2009; Maskeliunaite et al. 2010; Podvezko et al. 2010;
Radziszewska-Zielina 2010; Sivilevicius 2011).
2. Case study
Case study presents the process of selection the column-piles
alternative for building which stands on the aquiferous soil (Fig. 2).
The aim of problem is to design and install the piles-columns. The aim
of this study is to show how a decision-maker can find the most
reasonable alternative having the set of certain data.
[FIGURE 2 OMITTED]
Projects of pile-columns are complex systems, and they are quite
difficult to select in practice. For this reason, a decision-maker
should possess a large amount of multidisciplinary knowledge and should
be familiar with multidisciplinary techniques of operations research.
Operations research is based on four main assumptions (Turskis,
Zavadskas 2010a):
[FIGURE 3 OMITTED]
--the problem situations exist as realities and do not depend on a
decision-maker and aims of problem solution;
--the analysis of problem is objective (not related to different
viewpoints of stakeholders, contractors, final user and impact on the
environment) and described only by quantitative data;
--all participants of decision-making seeks optimal solution;
--the solutions are clearly optimal and can be implemented without
complications.
The problem of a decision-maker consists of evaluating a finite set
of alternatives in order to find the best one, to rank them from the
best to the worst, to group them into predefined homogeneous classes, or
to describe how well each alternative meets all the criteria
simultaneously.
A decision-maker first of all must understand and describe the
situation. This stage includes determination and assessment of the
stakeholders, the different alternatives of feasible actions, the large
number of different and important decision criteria, the type and the
quality of the information, etc. It appears to be the key point defining
MCDM as a formal approach (Fig. 2).
The results of the site investigation are presented in a detailed
report, which provides a step-by-step account of the processes
undertaken.
Taking into account the aforementioned suggestions and references
of experts and the aim to install the most effective of pile-columns,
the seven following alternatives were considered (Figs 3, 4).
Criteria, their optimality direction (the max or min value is
preferable), criteria measuring units and their weights [w.sub.j] are
presented in Tables 1 and 2.
[FIGURE 4 OMITTED]
2.1. Entropy-based criteria weights determination
To achieve the goal, first of all criteria weights were determined
by applying Entropy method. The initiator of the method (Shannon 1948)
gave the following equation of Entropy method (1) (quantity of
information in a dataset):
s= 1/N[[SIGMA].sub.j][x.sub.j]ln([x.sub.j]), (1)
where S - entropy matrix; N - number of criteria; [x.sub.j] -
criteria value; j - criteria index (j = 1 ... n).
This method was applied to solve multiple criteria problems in
construction (Zavadskas 1987). Mamtani et al. (2006), You and Zi (2007),
Li (2009), Ye (2010), Taheriyoun et al. (2010), Hsieh et al. (2010), Liu
and Zhang (2011) applied this method to solve different problems. The
block diagram of Entropy method is presented in Fig. 5.
The weights demonstrate which criterion is the most important in
comprising the other criteria. For determining criteria weights, the
criteria are transformed in such a manner that max value of each
criterion would be the best. While preparing initial data for
multi-criteria assessment of feasible alternatives, first of all the
criteria set is determined. These criteria have an impact on the
problems solution results. Further in the article the following criteria
will be analyzed: mass, cost of instalment, labour expenditures,
machinery expenditures, earthwork, instalment tolerance.
[FIGURE 5 OMITTED]
Initial criteria values for evaluation of 7 feasible alternatives
are presented in Table 1.
Further the normalization of the initial matrix (Table 1) was
performed as follows:
[bar.x].sub.ij] = [x.sub.ij]/[max.sub.i][x.sub.ij] (2)
[bar.x].sub.ij] = [min].sub.i][x.sub.ij]/[x.sub.ij] (3)
where [bar.x].sub.ij] - criteria normalized values, [x.sub.ij] -
criteria initial values.
Thus, the normalised decision-making matrix X is obtained.
Entropy level for each criterion [E.sub.j] is determined as
follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
where k = 1/lnm is known as entropy index, which varies by interval
[1, 0], so
0 [less than or equal to] [E.sub.j] [less than or equal to] 1; (j =
[bar.1,n]), (5)
j - index change level in current problem is determined:
[d.sub.j] = 1 - [E.sub.j]; (j = [bar.1,n]) (6)
If all criteria are equally important i.e. there are no subjective
or expert evaluations of their values, weight of j criterion (Table 2)
is determined as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
After determination of criteria weights, the priority order for
considered criteria can be specified (Table 2): machinery expenditures
> cost of instalment labour expenditures > mass > earthwork
> instalment tolerance.
Further ARAS method (Zavadskas, Turskis 2010; Zavadskas et al.
2010; Tup?nait? et al. 2010) is applied to evaluate the priority of each
alternative under investigation.
2.2. The determination of priority and importance of considered
alternatives by ARAS method
According to the ARAS method, a utility function value determining
the complex relative efficiency of a feasible alternative is directly
proportional to the relative effect of values and weights of the main
criteria considered in a project.
The first stage is Decision-Making Matrix (DMM) forming. In the
MCDM of the discrete optimization problem any problem to be solved is
represented by the following DMM of preferences for m feasible
alternatives (rows) rated on n signfull criteria (columns):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
where m - number of alternatives; n - number of criteria describing
each alternative; [x.sub.ij] - value representing the performance value
of the i alternative in terms of the j criterion; [x.sub.0j] - optimal
value of j criterion.
If optimal value of j criterion is unknown, then
[x.sub.0j] = [max.sub.i][x.sub.ij], if [max.sub.i][x.sub.ij] is
preferable, and
[x.sub.0j] = [min.sub.i][x.sub.ij], if [min.sub.i][x.sub.ij] is
preferable. (9)
Usually, the performance values [x.sub.ij] and the criteria weights
[w.sub.j] are viewed as the entries of a DMM. The system of criteria as
well as the values and initial weights of criteria are determined by
experts. Then the determination of the priorities of alternatives is
carried out in several stages.
Usually, the criteria have different dimensions. In order to avoid
the difficulties caused by different dimensions of the criteria, the
ratio to the optimal value is used. The values are mapped either on the
interval [0; 1] by applying the normalization of a DMM.
In the second stage the initial values of all the criteria are
normalized - defining values [bar.x].sub.ij] of normalised
decision-making matrix [bar.X]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
The criteria, whose preferable values are max, are normalized as
follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)
The criteria, whose preferable values are min, are normalized by
applying two-stage procedure:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)
When the dimensionless values of the criteria are known, all the
criteria, originally having different dimensions, can be compared.
The third stage is defining normalized-weighted matrix - [??]. It
is possible to evaluate the criteria with weights Wj. The values of
weight Wj are determined by the entropy method. Normalized-weighted
values of all the criteria are calculated as follows:
[[??].sub.ij] = [[bar.x].sub.ij][w.sub.j]; i = [bar.0,m] (13)
where [w.sub.j] is the weight (importance) of the j criterion and
[bar.x].sub.ij] is the normalized rating of the j criterion.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)
The following task is to determine values of optimality function:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)
where [S.sub.i] - the value of optimality function of i
alternative.
The biggest value is the best, and the least one is the worst.
Taking into account the calculation process, the optimality function
[S.sub.i] has a direct and proportional relationship with the values
[x.sub.ij] and weights [w.sub.j] of the investigated criteria and their
relative influence on the final result. Therefore, the greater the value
of the optimality function [S.sub.i], the more effective the
alternative. The priorities of alternatives can be determined according
to the value Si. Consequently, it is convenient to evaluate and rank
decision alternatives when this method is used.
The degree of the alternative utility is determined by a comparison
of the variant, which is analysed with the ideally best one [S.sub.0].
The Eq used for the calculation of the utility degree [K.sub.i] of an
alternative [A.sub.i] is given below:
[K.sub.i] = [S.sub.i]/[S.sub.0]; i = [bar.0,m] (16)
where [S.sub.i] and [S.sub.0] are the optimality criterion values,
obtained from Eq (15).
The calculated values of [K.sub.i] are in the interval [0, 1] and
can be ordered in an increasing sequence, which is the wanted order of
precedence. The complex relative efficiency of the feasible alternative
can be determined according to the utility function values.
According to the above proposed algorithm of ARAS method the
problem was solved and the results are presented in Table 3 and Fig. 6.
On the basis of results obtained in Table 3 it can be concluded
that according to selected criteria, reflecting the effectiveness of
pile-columns construction and their weights, the most reasonable
alternative according to the calculation results is the third (A3).
[FIGURE 6 OMITTED]
The priority order of the investigated pile-columns instalment
alternatives can be represented as follows (Fig. 6):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)
It means that the worst alternative is the second.
It can be stated that the alternative 3 is 97% of optimal
alternative performance level, and the performance of the worst
alternative 2 is only 76%.
According to the given data on the criteria describing the
pile-columns instalment alternatives, rational solutions about its
construction improvement and cost reduction can be made.
3. Conclusions
Traditional optimization, statistical and econometric analysis
approaches used within the engineering context are often based on the
assumption that the considered problem is well formulated and
decision-makers usually consider the existence of a single objective,
evaluation criterion or point of view that underlies the conducted
analysis. In such a case the solution of engineering problems is easy to
obtain.
According to the proposed ARAS method the utility function value,
determining the complex efficiency of a feasible alternative, is
directly proportional to the relative effect of values and weights of
the main criteria considered in a project.
The degree of the alternative utility is determined by a comparison
of the variant, which is analysed with the ideally best one.
It can be stated that the ratio with an optimal alternative may be
used in cases when it is seeking to rank alternatives and find ways of
improving alternative projects.
In conclusion, the proposed model has a promising future in the
construction engineering field, because it offers a highly
methodological basis for decision support.
doi: 10.3846/bjrbe.2011.19
Received 20 April 2011; accepted 10 May 2011
References
Bhattacharya, S.; Bolton, M. D.; Madabhushi, S. P. G. 2005. A
Reconsideration of the Safety of Piled Bridge Foundations in Liquefiable
Soils, Soils and Foundations 45(4): 13-25.
Brauers, W. K. M.; Ginevicius, R. 2010. The Economy of the Belgian
Regions Tested with MULTIMOOORA, Journal of Business Economics and
Management 11(2): 173-209. doi:10.3846/jbem.2010.09
Brauers, W. K. M.; Ginevicius, R.; Podvezko, V. 2010. Regional
Development in Lithuania Considering Multiple Objectives by the MOORA
Method, Journal of Technological and Economic Development of Economy
16(4): 613-640. doi:10.3846/tede.2010.38
Hsieh, L.-F.; Wang, L.-H.; Huang, Y.-C.; Chen, A. 2010. An
Efficiency and Effectiveness Model for International Tourist Hotels in
Taiwan, The Service Industries Journal 30(13): 21832199.
doi:10.1080/02642060903215030
Kalibatas, D.; Turskis, Z. 2008. Multicriteria Evaluation of Inner
Climate by Using MOORA Method, Information Technology and Control 37(1):
79-83.
Kaplinski, O. 2008. Usefulness and Credibility of Scoring Methods
in Construction Industry, Journal of Civil Engineering and Management
14(1): 21-28. doi:10.3846/1392-3730.2008.14.21-28
Kudzys, A.; Kliukas, R. 2010. Probability-Based Design of Spun
Concrete Beam-Columns, Journal of Civil Engineering and Management
16(4): 451-461. doi:10.3846/jcem.2010.51
Langston, C.; Wong, F. K. W.; Hui, E. C. M.; Shen, L.-Y. 2008.
Strategic Assessment of Building Adaptive Reuse Opportunities in Hong
Kong, Building and Environment 43(10): 17091718.
doi:10.1016/j.buildenv.2007.10.017
Li, X. 2009. Multiobjective Optimization and Multiattribute
Decision Making Study of Ship's Principal Parameters in Conceptual
Design, Journal of Ship Research 53(2): 83-92.
Liu, P.; Zhang, X. 2011. Research on the Supplier Selection of a
Supply Chain Based on Entropy Weight and Improved ELECTRE-III Method,
International Journal of Production Research 49(3): 637-646.
doi:10.1080/00207540903490171
Mamtani, G.; Green, G.; McDonald, S. 2006. Relative Reliability
Risk Assessment Applied to Original Designs During Conceptual Design
Phase, in Proc. of the Institution of Mechanical Engineers, Part B:
Journal of Engineering 220(6): 917-927. doi:10.1243/09544054JEM155
Maskeliunaite, L.; Sivilevicius, H.; Podvezko, V. 2010. Research on
the Quality of Passenger Transportation by Railway, Transport 24(2):
100-112. doi:10.3846/1648-4142.2009.24.100-112
Pareto, V. 1971. Manual of Political Economy. New York: Augustus M.
Kelley. 504 p. ISBN 0678008817
Peldschus, F. 2008. Experience of the Game Theory Application in
Construction Management, Technological and Economic Development of
Economy 14(4): 531-545. doi:10.3846/1392-8619.2008.14.531-545
Peldschus, F. 2009. The Analysis of the Quality of the Results
Obtained with the Methods of Multi-Criteria Decisions, Technological and
Economic Development of Economy 15(4): 580592.
doi:10.3846/1392-8619.2009.15.580-592
Peldschus, F.; Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J.
2010. Sustainable Assessment of Construction Site by Applying Game
Theory, InZinerine Ekonomika - Engineering Economics (3): 223-237.
Podvezko, V.; Mitkus, S.; Trinkuniene, E. 2010. Complex Evaluation
of Contracts for Construction, Journal of Civil Engineering and
Management 16(2): 287-297. doi:10.3846/jcem.2010.33
Radziszewska-Zielina, E. 2010. Methods for Selecting the Best
Partner Construction Enterprise in Terms of Partnering Relations,
Journal of Civil Engineering and Management 16(4): 510-520.
doi:10.3846/jcem.2010.57
Sasmal, S.; Ramanjaneyulu, K. 2008. Condition Evaluation of
Existing Reinforced Concrete Bridges Using Fuzzy Based Analytic
Hierarchy Approach, Journal Expert Systems With Applications 35(3):
1430-1443. doi:10.1016/j.eswa.2007.08.017
Shannon, C. E. 1948. A Mathematical Theory of Communication, The
Bell System Technical Journal 27: 379-423.
Sivilevicius, H. 2011. Application of Expert Evaluation Method to
Determine the Importance of Operating Asphalt Mixing Plant Quality
Criteria and Rank Correlation, The Baltic Journal of Road and Bridge
Engineering 6(1): 48-58. doi:10.3846/bjrbe.2011.07
Taheriyoun, M.; Karamouz, M.; Baghvand, A. 2010. Development of an
Entropy-Based Fuzzy Eutrophication Index for Reservoir Water Quality
Evaluation, Iranian Journal of Environmental Health, Science and
Engineering 7(1): 1-14.
Tomlinson, M. J.; Woodward, J. 2008. Pile Design and Construction
Practice. 5th edition. Taylor & Francis. 568 p. ISBN 9780415385824.
Tonias, D. E.; Zhao, J. J. 2007. Bridge Engineering. 2th edition.
McGraw-Hill. 488 p. ISBN 9780071459037. doi:10.3846/jcem.2010.30
Tupenaite, L.; Zavadskas, E. K.; Kaklauskas, A.; Turskis, Z.;
Seniut, M. 2010. Multiple Criteria Assessment of Alternatives for Built
and Human Environment Renovation, Journal of Civil Engineering and
Management 16(2): 257-266. doi: 10.3846/jcem.2010.30
Turskis, Z. 2008. Multi-Attribute Contractors Ranking Method by
Applying Ordering of Feasible Alternatives of Solutions in Terms of
Preferability Technique, Technological and Economic Development of
Economy 14(2): 224-239. doi:10.3846/1392-8619.2008.14.224-239
Turskis, Z.; Zavadskas, E. K. 2010a. A Novel Method for Multiple
Criteria Analysis: Grey Additive Ratio Assessment (ARAS-G) Method,
Informatica 21(4): 597-610.
Turskis, Z.; Zavadskas, E. K. 2010b. A New Fuzzy Additive Ratio
Assessment Method (ARAS-F). Case Study: the Analysis of Fuzzy Multiple
Criteria in Order to Select the Logistic Centers Location, Transport
25(4): 423-432. doi:10.3846/transport.2010.52
Turskis, Z.; Zavadskas, E. K.; Peldschus, F. 2009. Multi-Criteria
Optimization System for Decision Making in Construction Design and
Management, InZinerine Ekonomika--Engineering Economics (1): 7-17.
Ye, J. 2010. Fuzzy Decision-Making Method Based on the Weighted
Correlation Coefficient under Intuitionistic Fuzzy Environment, European
Journal of Operational Research 205(1): 202-204.
doi:10.1016/j.ejor.2010.01.019
You, T.; Zi, H. M. 2007. The Economic Crisis and Efficiency Change:
Evidence from the Korean Construction Industry, Applied Economics
39(14): 1833-1842. doi:10.1080/00036840600690199
Zavadskas, E. K. 1987. Multiple Criteria Evaluation of
Technological Decisions of Construction. Dissertation of Dr. Sc. Moscow
Civil Engineering Institute, Moscow (in Russian) [3aBagcKac, 3. K. 1986.
[TEXT NOT REPRODUCIBLE IN ASCII.]
Zavadskas, E. K.; Turskis, Z. 2010. A New Additive Ratio Assessment
(ARAS) Method in Multicriteria Decision-Making, Technological and
Economic Development of Economy 16(2): 159-172. doi:10.3846/tede.2010.10
Zavadskas, E. K.; Kaklauskas, A.; Turskis, Z.; Tamosaitiene, J.
2009a. Multi-Attribute Decision-Making Model by Applying Grey Numbers,
Informatica 20(2): 305-320.
Zavadskas, E. K.; Kaklauskas, A.; Vilutiene, T. 2009b.
Multicriteria Evaluation of Apartment Blocks Maintenance Contractors:
Lithuanian Case Study, International Journal of Strategic Property
Management 13(4): 319-338. doi:10.3846/1648-715X.2009.13.319-338
Zavadskas, E. K.; Turskis, Z.; Vilutiene, T. 2010. Multiple
Criteria Analysis of Foundation Instalment Alternatives by Applying
Additive Ratio Assessment (ARAS) Method, Archives of Civil and
Mechanical Engineering 10(3): 123-141.
Zhang, R.; Zheng, J.; Pu, H.; Zhang, L. 2011. Analysis of
Excavation-Induced Responses of Loaded Pile Foundations Considering
Unloading Effect, Tunnelling and Underground Space Technology 26:
320-335. doi:10.1016/j.tust.2010.11.003
Zhao, M.; Jiang, C.; Cao, W.; Liu, J. 2007. Catastrophic Model for
Stability Analysis of High Pile-Column Bridge Pier, Journal of Central
South University of Technology 14(5): 725-729.
doi:10.1007/s11771-007-0138-5
Zhao, M.; Liu, E.; Yang, J. 2009. Analysis of Stability of Pile
Foundation with Higher Pile-Column Bridge Piers, Journal of Highway and
Transportation Research and Development 4(1): 40-44.
Saulius Susinskas (1), Edmundas Kazimieras Zavadskas (2), Zenonas
Turskis (3), ([mail])
(1) Faculty of Technologies, Kaunas University of Technology
Panevezys Institute, S. Daukanto g. 12-138, 37164 Panevezys, Lithuania
(2,3) Dept of Construction Technology and Management, Vilnius
Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius,
Lithuania
E-mails: (1) saulius.susinskas@ktu.lt; (2)
edmundas.zavadskas@vgtu.lt; (3) zenonas.turskis@vgtu.lt
Table 1. Criteria values for evaluation--initial decision-making
matrix [X.sup.*]
Criteria
Alternative Mass, Cost of Labour
kg/[m.sup.3] instalment, expenditures,
[euro] man-hours
[x.sup.*.sub.1] [x,sup.*.sub.2] [x.sup.*.sub.3]
Optim. direction min min min
[A.sub.0] 180 23.2 4.06
(optimal values)
[a.sub.1] 260 23.2 4.15
[A.sub.2] 260 25.8 4.8
[A.sub.3] 180 24.0 4.2
[A.sub.4] 180 24.8 4.06
[A.sub.5] 185 26.2 4.46
[A.sub.6] 190 27.0 5.3
[A.sub.7] 265 25.3 4.85
Alternative Machinery Amount of
expenditures, earthworks,
machine-hours [m.sup.3]
[x.sup.*.sub.4] [x.sup.*.sub.5]
Optim. direction min min
[A.sub.0] 15.1 0.2
(optimal values)
[a.sub.1] 15.5 0.6
[A.sub.2] 16.1 0.6
[A.sub.3] 15.7 0.2
[A.sub.4] 15.95 0.2
[A.sub.5] 16.2 0.4
[A.sub.6] 15.2 0.2
[A.sub.7] 15.1 0.2
Criteria
Alternative Instalment
tolerance,
points
[x.sup.*.sub.6]
Optim. direction min
[A.sub.0] 1
(optimal values)
[a.sub.1] 1
[A.sub.2] 2
[A.sub.3] 1
[A.sub.4] 2
[A.sub.5] 3
[A.sub.6] 2
[A.sub.7] 2
Table 2. Entropy level, j index change level and criteria weights
Criteria Mass Cost of Labour Machinery
instalment expenditures expenditures
[E.sub.j] 0.4367 0.2611 0.3260 0.1270
[d.sub.j] 0.5633 0.7389 0.6740 0.8730
[W.sub.j] 0.1660 0.2177 0.1986 0.2572
Priority order 4 2 3 1
Criteria Earthwork Instalment
tolerance
[E.sub.j] 0.5545 0.9006
[d.sub.j] 0.4455 0.0994
[W.sub.j] 0.1313 0.0293
Priority order 5 6
Table 3. Weighted-normalised criteria values of foundation instalment
alternatives (weighted-normalised DMM [??]) and solution results
Criteria
Alternative [[??].sub.1] [[??].sub.2] [[??].sub.3] [[??].sub.4]
[A.sub.0] 0.0238 0.0292 0.0272 0.0332
[A.sub.1] 0.0165 0.0292 0.0267 0.0323
[A.sub.2] 0.0165 0.0263 0.0230 0.0311
[A.sub.3] 0.0238 0.0282 0.0263 0.0319
[A.sub.4] 0.0238 0.0273 0.0272 0.0314
[A.sub.5] 0.0231 0.0259 0.0248 0.0309
[A.sub.6] 0.0225 0.0251 0.0209 0.0330
[A.sub.7] 0.0161 0.0268 0.0228 0.0332
Criteria Solution results
Alternative [[??].sub.5] [[??].sub.6] S K Rank
[A.sub.0] 0.0212 0.0054 0.1401 1.0000 0
[A.sub.1] 0.0071 0.0054 0.1172 0.8364 5
[A.sub.2] 0.0071 0.0027 0.1067 0.7616 7
[A.sub.3] 0.0212 0.0054 0.1369 0.9775 1
[A.sub.4] 0.0212 0.0027 0.1337 0.9545 2
[A.sub.5] 0.0106 0.0018 0.1172 0.8363 6
[A.sub.6] 0.0212 0.0027 0.1254 0.8953 3
[A.sub.7] 0.0212 0.0027 0.1229 0.8772 4
