Application of expert evaluation method to determine the importance of operating asphalt mixing plant quality criteria and rank correlation/Ekspertu vertinimu metodo taikymas veikiancio asfaltbetonio maisytuvo kokybes kriteriju svarbai ir rangu koreliacijai nustatyti/Eksperta novertejuma metodes lietojums nosakot asfaltbetona rupnicas kvalitates kriteriju un rangu korelaciju/Eksperthindamise ... .
Sivilevicius, Henrikas
1. Introduction
Bitumen mixtures are used in the pavement of roads, parking lots,
terminals, airfields, and other trafficked areas. Hot mix asphalt (HMA)
has become the most popular mixture. Its production volumes have
constantly been increasing not only in Europe as could seen from Key
Figures of the European Asphalt Industry in 2009 (European Asphalt
Pavement Association (EAPA)) and other economically well-developed
countries, but in Lithuania as well (Sivilevicius, Sukevicius 2009). HMA
is a mixture of asphalt cement, mineral aggregates and air, and the
properties of this material are significantly influenced by its
components. The asphalt mixture design process generally requires a
balance of various desirable mixture properties with an attempt to
optimize the selection and proportions of these different components (Li
et al. 2009).
HMA mixture is produced in a stationary or a portable asphalt
mixing plant (AMP). According to their inner workings, HMA mixture
manufacturing facilities are classified into batch plants, continuous
mix plants and drum mix plants (ASTM Standart D 995-95b "Standard
Specification for Mixing Plants for Hot-Mixed, Hot-Laid Bituminous
Paving Mixtures"). Structural and technological requirements for
all AMP equipment are presented in the Standard Specification for Mixing
Plants. The builders of roads and other transport infrastructure objects
have to select the operating AMP, capable of producing a suitable HMA
mixture, which complies with the conditions of public procurement tender
and reduces the transportation distance and cost. The best equipment
shall be selected from the operating AMP available within a rational
provision with HMA distance from an infrastructural object (motorway,
road, street) under construction.
AMP produced by different corporations have different structure.
Their clients may require and order additional equipment or reject some
standard original equipment. When exploited, the facilities of AMP wear,
and at the end of the working season are repaired and replaced.
Therefore, operating AMP of not only different but the same company are
of different structure and most probably do not accurately and precisely
perform HMA mixture production technological operations, give different
output and the sequence of technological processes as well as pollute
the environment differently.
AMP is a long-life equipment. Its factual service life from the
beginning of its mounting in the plant (in a separate lot) until its
demounting and replacement by a more upgraded technological equipment is
frequently more than 20-30 years (Sivilevicius 2003). During this AMP
exploitation period requirements set in normative documents to the
properties of the produced HMA mixture, technological operation
parameters, pollutant emissions into the air change a lot. Thus, the
quality parameters of long used AMP frequently do not meet certain
requirements or meet them partially.
AMP belongs to a group of technological equipment producing asphalt
mixtures to lay flexible pavement of transport infrastructure objects.
HMA mixture of optimal composition designed in a laboratory from new and
reclaimed materials through the application of deterministic (Asi 2007;
Doh et al. 2008; Roberts et al. 1991; Sivilevicius et al. 2011;
Widyatmoko 2008) or stochastic methods (Sivilevicius, Vislavicius 2008)
shall be produced in AMP without exceeding component content deviations
(tolerances). The amount of components in HMA mixture and their
deviations from job mix formula (JMF) influence on its physical and
mechanical parameters and the dynamic modulus (Ceylan et al. 2009; Liu,
Cao 2009; Petkevicius et al. 2009; Petkevicius, Sivilevicius 2008). HMA
mixture properties and asphalt concrete structure depend on the mineral
materials and bitumen properties used in its production (Haryanto,
Takahashi 2007; Kim 2009; Lee et al. 2009; Mahmoud et al. 2010; Pan et
al. 2005; Radziszewski 2007).
The amount of components in the produced HMA mixture deviates from
JMF (Braziunas, Sivilevicius 2010) and varies within a certain range
(interval), which is frequently impacted by segregation processes (Brown
et al. 1989; Stroup-Gardiner, Brown 2000). The homogeneity of HMA
mixture produced with reclaimed asphalt pavement (RAP) is reduced by a
huge amount of RAP in it (Aravind, Das 2007; Mucinis et al. 2009). The
content of components in the produced HMA mixture may deviate from JMF,
but not more than it is specified in guidelines Automobiliu keliu dangos
konstrukcijos asfalto sluoksniu irenginio taisykles IT ASFALTAS 08 [The
Installation Rules of the Roads Pavement Asphalt Layers "IT
ASFALTAS 08"].
HMA mixture loaded to the hob storage hopper (storage silo) from
AMP mixer be of a certain temperature, which depends on the type of HMA
mixture as well as the type and mark of bitumen used in it. The actual
temperature of the produced HMA mixture shall meet the requirements
specified in guidelines Automobiliu keliu asfalto misiniu techniniu
reikalavimu aprasas TRA ASFALTAS 08 [The Specification of Technical
Requirements for Automobile Road Asphalt Mixtures "TRA ASFALTAS
08"]. Compliance with this requirement depends not only on the
moisture and temperature of mineral materials, fuel (especialy fuel oil)
quality, air temperature, operator's actions, but on the AMP
structure (burner, drying drum, bitumen system) as well. During the
drying process more energy consumption is required to dry and heat
saturated mineral materials. Mineral particles of different size absorb
different amount of moisture (Ang et al. 1993). Fewer AMP turn offs in
the production of HMA mixture also result in less energy consumption.
When producing HMA mixture, hazardous and environment polluting
materials, such as dust, smoke and combustion product gases, are emitted
during technological processes occurring in AMP equipment (Brock et al.
1995; Dupont et al. 1993; Hobbs 2009; HopafleK 2001). In modern AMP, to
separate dust from air and gas flow pulse-jet filters are used ([TEXT
NOT REPRODUCIBLE IN ASCII] 2009). Pollutant concentration and their
emission may be one of the most important criteria enabling to determine
if AMP is suitable for use (user-friendly).
The cost price of HMA mixture produced in AMP shall be minimal. It
depends on the consumed fuel amount and price, salary, electricity
consumption, materials' price and is frequently calculated per one
ton of HMA mixture.
An important AMP quality indicator is its physical and moral wear
(amortization). The newer the AMP is, the less its equipment structural
parameters change, and the better quality of HMA mixture it can produce.
Frequently, AMP is seasonal technological equipment. When the
working season is over, worn-out elements are replaced (blades of the
mixer (pugmill), sieves, drying drum charging place blades). It requires
additional expenses, which can restore AMP structural parameters from
initial to the condition similar to it.
There are few published research works on the improvement of the
structure of AMP, investigation and evaluation of the technological
parameters and properties of HMA mixture produced in them. Hereby, one
of the reasons for this is the complexity of such research due to a huge
number of samples to be taken and changing technological processes to be
measured. Sometimes when changing HMA mixture production technological
parameters, the process shall be detuned, stopped, deviated from JMF as
well as other materials shall be used or the production process shall be
disturbed.
Divinsky et al. (2003) calculated process capability indices CP and
CPK, index K, quality mark QM values and evaluated the quality of
investigated AMP according to the statistical characteristics of the HMA
mixture density produced in AMP, bitumen content in it, percentages
passing No 4 sieves and percentages passing No 200 sieves.
AMP quality is evaluated by the additive model (Sivi-levicius et
al. 2008) based on 9 criteria, according to which AMP quality
multi-criteria index K is calculated. This article presents the
methodology which enables to estimate the impact of every criterion of
AMP quality on complex index K.
Recently, expert investigation methods have been applied in various
management and engineering areas. The efficiency of buildings' wall
structures is determined (Za-vadskas et al. 2008) and the risk of
construction projects is evaluated (Zavadskas et al. 2010) through the
use of experimental investigation methods. To describe and solve the
task model, TOPSIS (Technique for Order Preference by Similarity to
Ideal Solution) grey and COPRAS-G (COmplex PRoportional ASsessment)
methods are used. Project properties are described by the values of
efficiency indicators defined within intervals. Expert investigation
methods were applied in management by Podvezko (2007); Podvezko et al.
(2010); Zavadskas et al. (2010). ARAS-F (Additive Ratio Assessment)
method was used to select the location of logistic centers (Turskis,
Zavads-kas 2010). In 2009, Maskeliunaite et al. (2009); Sivilevicius,
Maskeliunaite (2010) applied the Analytic Hierarchy Process (AHP)
method, which was proposed by Saaty et al. (2003), to investigate the
importance of quality criteria on passenger carriage by railway. This
method was also applied by Farhan and Fwa (2009) to identify the
priority of road pavement maintenance. The AHP method was used for
another task by Abdelgawad, Fayek 2010; Lin et al. 2008; Medineckiene et
al. 2010; Morkvenas et al. 2008.
The work aims to present the system analysis of operating AMP
quality criteria, the algorithm of identifying their importance through
the use of an expert method, enabling to identify the weights of the
quality criteria required to calculate the AMP quality complex index
according to the additive model.
2. Opinion of experts on the importance of AMP quality criteria and
computation of their correlation
The country's more complex management of the economy and
technological processes requires a comprehensive analysis of the
activity. Frequently, the impact of separate factors on the work
efficiency and production quality improvement shall be determined and
estimated. Sometimes, the help of specialists (experts) is required. The
efficiency of taking solutions influences on the improvement of the
country's economy management.
The essence of the expert evaluation method lies in the rational
organization of the analysis carried out by experts of the quantitative
evaluation of the problem and the processing of findings. The
generalized opinion of the group of experts is taken as a problem
solution (solution result). If the solution shall be taken on the basis
of expert evaluation, the degree of concordance of expert opinions shall
be taken into account (Kendall 1970).
When given a prepared questionnaire, the experts [E.sub.1],
[E.sub.2], ... [E.sub.n] were asked to give quantitative weight values
[X.sub.2], ... [X.sub.m] (points [B.sub.1], [B.sub.2], ..., [B.sub.m])
to AMP quality criteria based on their knowledge, experience and
intuition. The highest point (an integer number) is given to the most
important quality criterion; one point less is given to the next
criterion; and the lowest point is given to the least important
criterion (usually 1 point). The number value of the highest point is
selected depending on number m, which shows AMP quality criteria.
Based on the questionnaires filled in and returned by experts,
weight values (points) given by each expert to AMP quality criteria and
presented in Table 1.
A group of selected n experts give quantitative evaluation of the
operating (not newly purchased) AMP m quality indices (criteria). Rating
by ranks [R.sub.ij] (i = 1, 2, ..., n; j= 1, 2, ..., m) makes up n
number of rows and m number of columns, see Table 2 (matrix) R. Experts
may evaluate the expected value [R.sub.ij] in a different way. Any
evaluation scale may be used, for example, index units, unit parts,
percent, 10 point system or AHP method pair comparison scale (Lin et al.
2008; Podvezko 2009; Saaty 1980, 2003). To calculate the concordance
coefficient, only expert index rating can be used (Podvezko 2005).
If experts' evaluation was presented in any other form, it
shall be preliminary ranked. Ranking is a procedure when the most
important index is given rank equal to 1, next according to its
importance is given rank two, etc. The last index according to its
importance is given rank m; here m is the number of compared indices.
If weight values (points) given by experts to AMP quality criteria
(indices) presented in Table 1 are available, the correlation of their
opinion is determined by computing the Kendall's coefficient of
concordance W. For this reason, first of all, points [B.sub.ij] given to
each criterion shall be replaced by ranks [R.sub.ij] (Table 2), showing
the hierarchy (precedence), inspite of the fact that the same W is
obtained using values (points) instead of ranks. Points [B.sub.ij] may
be replaced by ranks [R.sub.ij] using Eq (1):
[R.sub.ij] = (m + 1) - [B.sub.ij], (1)
where [B.sub.ij] - point (j = 1, 2, ..., m) given by i expert (i =
1, 2, ..., n) to criterion i; n--the number of experts; m--the number of
AMP quality criteria (indices).
For example, ranks [R.sub.ij] of AMP each criterion out of the 9
quality criteria are obtained from Eq [R.sub.ij] = 10 - [B.sub.ij], and
out of 7 quality criteria, from Eq [R.sub.ij] = 8 - [B.sub.ij].
The idea of Kendall's (1970) coefficient of concordance is
related to AMP's each quality criterion (index) rank sum [R.sub.ij]
with respect to all experts:
[R.sub.j] = [n.summation over (i=1)] [R.sub.ij] (j = 1, 2, ..., m),
(2)
to be precise, with values [R.sub.j] deviation from the total mean
[bar.R] square sum S (variance analogue) is:
S = [m.summation over (j=1)] [([R.sub.j] - [bar.R]).sup.2]. (3)
Total mean [bar.R] is calculated according to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)
It is convenient to calculate average rank [[bar.R].sub.j] and
concordance coefficient (Kendall's coefficient of concordance
(Chua, Li 2000)) W by a matrix, the structure of which is presented in
Table 2. Columns refer to quality criteria (j = 1, 2, m), rows to
experts (i = 1, 2, n), and squares to ranks [R.sub.ij] given by experts
to quality criteria.
Average rank [[bar.R].sub.j]; of each criterion is obtained by
dividing the sum of given ranks by the number of experts:
[[bar.R].sub.j] = [n.summation over (i=1)][R.sub.ij]/n (j = 1,2,
..., m), (5)
where [R.sub.ij]--rank given by i expert to j criterion, n--number
of experts.
If S is a real sum of squares, calculated according to Eq (3), the
concordance coefficient, when there are no related ranks, is defined by
the correlation of the obtained S and relevant max [S.sub.max]
(Maskeliunaite et al. 2009; Podvez-ko 2007; [TEXT NOT REPRODUCIBLE IN
ASCII] 1987):
W = 125/[n.sup.2] ([m.sup.3] - m). (6)
It is convenient to calculate the sum S of AMP each criterion ranks
[R.sub.ij] deviations from average rank squares according to:
S=[m.summation over (j=1)][[[n.summation over (i=1)][R.sub.ij] -
1/2 n(m+1)].sup.2], (7)
where m--number of AMP quality criteria (j = 1, 2, ... m);
n--number of experts (i = 1, 2, ... n).
Random value S is calculated by adding values identified for all
quality criteria and presented in Table 2.
The max possible value S, when experts' opinions are
absolutely compatible, i. e. when the evaluations of all experts are the
same. The opposite, the worst (different) ranking would be when
experts' evaluations are contradictory, i.e. all possible ranks
from one to m are used to evaluate each criterion, when the sum of each
criterion ranks is the same and coincides with the total mean. In this
case the value of S is equal to 0, although such result may occur very
rarely in practice and can be treated as theoretical and marginal.
The concordance coefficient may be used in practice if its marginal
value showing when experts' evaluations can be still considered
compatible is identified. Kendall (1970) proved that if the number of
AMP quality criteria (indices) is m > 7, the weight of concordance
coefficient may be identified through the use [chi square] (chikvadrat)
Pearson's criterion.
Random value
[chi square] = n(m-l)W = 12S/nm(m + l) (8)
is distributed according [chi square] to distribution with
[upsilon] = m - 1 degree of freedom. Critical value [[chi
square].sub.v;[alpha]] is found according to the selected level of
significance [alpha] (in practice, [alpha] value is taken 0.05 or even
stricter 0.01) from %2 distribution Table with v = m - 1 degree of
freedom. If [chi square] value calculated according to Eq (8) is higher
than [[chi square].sub.kr], the evaluations of experts shall be
considered as concordant.
When the number of compared AMP quality criteria (indices) n is
from 3 to 7, [chi square] distribution should be applied carefully as
distribution's critical value [[chi square].sub.[upsilon],[alpha]]
may be higher than the calculated one although experts'
compatibility level is still sufficient. In such case, probability
Tables of concordance coefficient or critical values' S Tables
(with 3 [less than or equal to] m [less than or equal to] 7) may be used
(Kendall 1970).
The min concordance coefficient value [W.sub.min] at which it is
stated that opinions of all experts n about the quality of AMP m
compared quality criteria with the determined (required) weight level
[alpha] and the degree of freedom v = m - 1 are concordant, and
calculated according to
[W.sub.min] = [[chi square].sub.v, [alpha]]/n(m - 1), (9)
where [[chi square].sub.v, [alpha]]--critical Pearson's
statistics, the value of which is found in the Table (Montgomery 2009)
when the degree of freedom [upsilon] and weight level [alpha] are taken.
Frequently, in practice in some calculations it is more convenient
to use significance (weight 1, 2, ... 9) indices, the best value of
which is the max value.
When AMP quality is evaluated by the additive mathematical model
(Sivilevicius et al. 2008), according to which its quality complex index
is calculated, which enables to identify its quality by one number and
compare it with the other analogous AMP, it is convenient to use not
quality criteria average ranks [[bar.R].sub.j], which do not show how
one criterion is more important than the other, but their weight indices
[Q.sub.j].
Weight indices in the solved sample in the research paper ([TEXT
NOT REPRODUCIBLE IN ASCII] 1987) were calculated as follows:
object's each criterion average rank [[bar.R].sub.j] is divided by
the value constant for all object's quality criteria, the sum of
ranks [m.summation over (j=1)] [R.sub.j], i.e. value is calculated:
[[bar.q].sub.j] = [[bar.R].sub.j]/[m.summation over (j=1)]
[R.sub.j] (10)
The sum of values [[bar.q].sub.j] calculated according to Eq (10)
is equal to 1.000. Having normalized ranks, the most important quality
criterion is the quality criterion the calculated value of which is the
lowest (min). Final values are identified as follows. In the beginning,
reciprocal quantity [[bar.q].sub.j] is calculated for each quality
criterion.
[d.sub.j] = 1 - [[bar.q].sub.j] = 1 - [[bar.R].sub.j]/[m.summation
over (j=1)] [R.sub.j]. (11)
The sum of all calculated values [d.sub.j] is equal to m - 1.
Finally, quality criteria weight indices [Q.sub.j] are calculated, the
sum of which is also equal to 1:
[Q.sub.j] = [d.sub.j]/[m.summation over (j=1)] [d.sub.j] =
[d.sub.j]/m - 1. (12)
Max weight indices [Q.sub.j] calculated like this are of the most
important quality criteria when calculating additive K. Weight indices
[Q.sub.jmax] calculated according to this methodology slightly differ
from [Q.sub.jmin]. [TEXT NOT REPRODUCIBLE IN ASCII] (1987) in the
presented sample of applying this methodology points out that weight
index [Q.sub.jmax] of the most important criterion identified by 35
respondents on the 5 quality criteria object equals to 0.225, and the
least important [Q.sub.jmin] = 0.176, i. e. it differs only 1.278 times
(27.8%), which makes this index "insensitive".
The importance of AMP quality criteria evaluated by experts by
normalizing them (equating their sum to one) may be identified by
calculating weight index [Q.sub.j] of each quality criterion proposed by
Eq (13):
[Q.sub.j] = (m + 1) - [[bar.R].sub.j]/[m.summation over (j=1)]
[[bar.R].sub.j], (13)
where m--number of quality criteria (indices) showing AMP quality
(properties); [[bar.R].sub.j]--average rank of j criterion, calculated
according to Eq (5).
Control is obtained for j quality criterion given by all experts n
(i = 1, 2 ,... , n) by dividing the sum of given weight values (points)
from AMP for all m quality criteria (j = 1, 2, m) by the sum of all
points given by the same experts, taken from Table 1:
[[??].sub.j] = [n.summation over (i=1)] [B.sub.ij]/[n.summation
over (i=1)] [m.summation over (j=1)] [B.sub.ij] (14)
Weight index [Q.sub.j] enables to determine not only that AMP one
quality criterion is more important than the other (average ranks
[[bar.R].sub.j] prove this as well), but how many times each is more
important than the other.
Independent quality criteria weight indices [Q.sub.j] enable to
calculate K from the additive model:
K = [m.summation over (j=1)] [Q.sub.j] x [x.sub.j] = [Q.sub.1]
[x.sub.1] + [Q.sub.2] [x.sub.2] + ... + [Q.sub.m] [x.sub.m], (15)
where [x.sub.j]--normalized (non-dimensional) variable of AMP j
criterion calculated from factual and permitted or marginal values, the
best value of which is approximate to 1, and the worst is approximate to
0.
3. Numerical illiustration
Forty three specialists (experts) having knowledge of HMA mixture
properties, quality requirements, production technology, AMP structure
and technical parameters of the equipment were given questionnaires on
the evaluation of weight values of 9 indices (quality criteria) showing
the quality of the operating AMP.
The form and structure of the questionnaire as well as the quality
criteria weight points given by the 1st expert to the operating AMP are
presented in Table 3.
According to Eq (1) points [B.sub.ij] of AMP quality indices'
are converted into relevant ranks [R.sub.ij]. For example, the weight
value (point) 9 given by the 1st expert ([E.sub.1]) to the 1st criterion
(C) corresponds to rank 1. Values of all points [B.sub.ij] replaced by
ranks [R.sub.ij], are presented in Table 4.
The sum of ranks [R.sub.j] given by all 43 experts of each
criterion (j = 1, 2, m) is calculated (Eq (2)). The sum of ranks of the
1st criterion is (C) [R.sub.j] = [R.sub.11] = 60, the 2nd criterion (T)
[R.sub.j] = [R.sub.2] = 116, the 3rd criterion (H) [R.sub.j] = [R.sub.2]
= 104, etc.
Constant quantity is found
1/2 n(m + 1) = 1/2 43(9+1) = 215,
which is required when calculating the sum of average rank square S
of criterion rank [R.sub.ij] deviations from average rank squares
according to Eq (7).
The difference between sum [n.summation over (i=1)] [R.sub.ij] of
ranks [R.sub.ij] and constant quantity 1/2n (m + 1) is calculated for
each criterion. For example, this difference of the 1st criterion S is
as follows:
[n.summation over (i=1)] - n(m + 1)/2 = 60 - 43(9 + 1)/2 = - 155.
This calculated difference of other quality criteria is presented
in Table 4. The sum of all 9 quality criteria differences is equal to 0.
The square of the difference between ranks' sum [n.summation
over (i=1)] [R.sub.ij] and constant quantity (n(m + 1)/2 is calculated,
which is presented in the last row of Table 4. For example, the square
of this difference of the 1st criterion C is
[[[n.summation over (i=1)] [R.sub.ij] - 1/2 n (m + 1)].sup.2] =
[[60 - 215].sup.2] = 24 025.
The squares of differences are written in the last row of Table 4
and according to Eq (7) quantity S is summed. Their sum S is solved in
sample 79 818.
Concordance coefficient W is calculated according to Eq (6) when
ranks are not related:
W = 12S/[n.sup.2]([m.sup.3] - m) = 12 x 79 818/[43.sup.2]
([9.sup.3] - 9) = 0.719.
It is larger than 0.5; therefore, it can be approx stated that
experts' opinions are compatible.
As the number of quality criteria in the solved task is m > 7,
the weight of the concordance coefficient is determined through the use
of criterion [chi square], for which according to Eq (8) the random
quantity is calculated
[chi square] = n(m - 1)W = 12S/nm(m+1) = 43(9 - 1)0.719 = 12 x 79
818/43 x 9(9 + 1) = 247.5.
The number of the degrees of freedom v = m - 1 = 9 - 1 = 8 is
calculated for the number of experts n = 43 and the number of compared
quality criteria m = 9 and a rather strict importance level [alpha] =
0.01 is selected. Critical value [[chi square].sub.[upsilon],[alpha]] is
found from the statistical Table (Montgomery 2009; [TEXT NOT
REPRODUCIBLE IN ASCII] 2001), which corresponds with the number of
degrees of freedom and the selected importance level [[chi
square].sub.v,[alpha]], which equals to 20.0902, i.e. it is much larger
than the calculated value [chi square], which is equal to 247.5. As
value [chi square] calculated according to Eq (11) is larger than [[chi
square].sub.[upsilon],[alpha]], it can be stated that the opinions of
all experts are concordant when evaluating the weight of AMP quality
criteria, and the the calculated average ranks show a common opinion.
Min value of the concordance coefficient [W.sub.min] is calculated
from formula (9), at the presence of which with the significance level
[alpha] = 0.01 and the number of the degree of freedom v = m - 1 = 9 - 1
= 8, it could be still stated that experts' opinions are
concordant:
[W.sub.min] = [[chi square].sub.v,[alpha]]/n(m - 1) =
20.0902/43(9-1) = 0.0584 [much less than] 0.719.
Column diagram of all 9 quality criteria average ranks
[[bar.R].sub.j] calculated values of AMP is drawn (Fig. 1) and values n,
W, [chi square] and [[chi square].sub.v,[alpha]] are presented.
The data of the carried out research show that the evaluations of
43 experts of AMP 9 quality criteria weight correlate and may be justly
taken as their generalized opinion.
Calculated average ranks [[bar.R].sub.j] of AMP quality criteria
show that index C is more important than H, T, E, U, P, W, R and B, i.
e. the following hierarchy is obtained: C > H > T > E > U
> P > W > R > B. The calculated [[bar.R].sub.j] do not show
how each of them is more important than the other.
When applying the methodology of Zavadskas ([TEXT NOT REPRODUCIBLE
IN ASCII] 1987), AMP quality criteria weight indices [Q.sub.j] are
identified. For this purpose, Eqs (10)-(12) are used, when in the
beginning [[bar.q].sub.j] and [d.sub.j], and finally [Q.sub.j] are
computed from them. Calculation data are presented in Table 5 and Fig.
2.
[FIGURE 2 OMITTED]
Correlation of the 1st criterion (C) average rank and the sum of
ranks of all quality criteria is computed from the Eq 10):
[[bar.q].sub.1] = [[bar.R].sub.1]/[m.summation over (j=1)]
[R.sub.j] = 1.395/45.000 = 0.031.
Reciprocal quantity is obtained when the calculated correlation of
ranks [[bar.q].sub.1] of this criterion is subtracted from one according
to Eq (11):
[d.sub.1] = 1 - [[bar.q].sub.1] = 1-0.031 = 0.969,
which, according to formula (12) is divided by the sum of all
quality criteria [d.sub.j], which is equal to m - 1, weight index is
obtained:
[Q.sub.1] = [d.sub.1]/[m.summation over (j=1)] = [d.sub.1]/m - 1 =
0.969/8.000 = 0.1211.
Weight index of the 2nd quality criterion (T) is [Q.sub.2] =
0.1175, the 3rd quality criterion (H) - [Q.sub.3] = 0.1183, the 4th
quality criterion (E) - [Q.sub.4] = 0.1110, the 5th quality criterion
(P) - [Q.sub.5] = 0.108, the 6th quality criterion (W) - [Q.sub.6] =
0.1069, the 7th quality criterion (R) - [Q.sub.7] = 0.1040, the 8th
quality criterion (B) - [Q.sub.8] = 0.1032 and the 9th quality criterion
(U) - [Q.sub.9] = 0.1099
The sum of AMP quality criteria weight indices calculated according
to the methodology of Zavadskas ([TEXT NOT REPRODUCIBLE IN ASCII] 1987)
is equal to 1.0000. The difference between the max weight index
[Q.sub.1] = 0.1211 and the min [Q.sub.8] = 0.1032 is equal to 0.0179.
Their ratio 1.17 shows a slight (approx 15%) change of max and min
values.
AMP quality criteria significances (weights) are determined
according to the methodology developed by the author. For this purpose,
the new Eq (13) developed by the author is used, from which the
significance of AMP's 1st quality criterion C (produced HMA mixture
composition compliance with JMF) is computed:
[Q.sub.1] = (m + 1) - [[bar.R].sub.1]/[9.summation over (j=1)]
[[bar.R].sub.j] = (9 + 1) - 1.395/45.000 = 0.1912.
Weight index of the 2nd (T) quality criterion is [Q.sub.2] =
0.1622, the 3rd (H) - quality criterion [Q.sub.3] = 0.1685, the 4th (E)
quality criterion [Q.sub.4] = 0.1101, the 5th (P) - quality criterion
[Q.sub.5] = 0.0868, the 6th (W)--quality criterion [Q.sub.6] = 0.0770,
the 7th (R) - quality criterion [Q.sub.7] = 0.0543, the 8th (B) -
quality criterion [Q.sub.8] = 0.0486 and the 9th (U)--quality criterion
[Q.sub.9] = 0.1013. The sum of all quality criteria coefficients is
equal to 1. The difference between the max [Q.sub.1] = 0.1912 and the
min [Q.sub.8] = 0.0486 weight indices is equal to 0.1426, and ratio 3.93
shows a significant change (approx 75%) of max and min values (Table 5,
Fig. 2.). A dotted horizontal line shows average weight index
[[bar.Q].sub.j] = 1:9 = 0.1111 of all AMP quality criteria.
Having used weight indices [Q.sub.j] of independent quality
criteria, AMP quality complex index is obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Normalized values of variable quantities [x.sub.1], [x.sub.2], ...,
[x.sub.9] of each AMP differ and vary from 0 to 1. The higher they are,
the better quality of the operating AMP is.
4. Conclusions
AMP quality is evaluated according to 9 independent quality
criteria showing the compliance of HMA mixture properties produced in it
with JMF (C, T and H); its capabilities to meet the environmental
protection requirements (E) according to the pollutants'
concentration emitted from its equipment; its economy, expressed by
production manufacturing costs per one ton of the produced HMA (P); its
technical condition (degree of depreciation) and investments allocated
to its improvement (W and R); the use of its capacity (technical
productivity) when producing HMA mixture, required to lay transport
roads, streets, airfields pavement structure courses in the servicing
rational zone (B) and its technological versatility, defined by a
possibility to produce in it all groups, kinds and types of asphalt
mixtures (U) presented in "TRA ASFALTAS 08". The importance of
these quality criteria to the quality were evaluated by 43 experts
according to a 9point scale.
Average ranks [[bar.R].sub.j] obtained by replacing points given to
criteria by ranks enabled to identify their hierarchy showing that the
most important quality criteria for AMP quality are influenced by the
properties of HMA mixture produced in it (C = 1.395, H = 2.419, T =
2.698). The less important quality criteria are influenced by
environmental protection and technological versatility (E = 5.046, U =
5.442); the next less important quality criteria are those showing the
mixture production cost price and AMP depreciation degree (P = 6.093, W
= 6.353), and the least important quality criteria are those reflecting
the scope of investments allocated to AMP repair and reconstruction and
the degree of their capacity to produce HMA mixture (R = 7.558, B =
7.814). It is logical that indices of the produced HMA mixture
properties mostly influence on the quality of the technological
equipment complex. Experts almost do not care that most of the time AMP
will not produce HMA mixture (will not generate income, benefit and
added value). It may be considered that due to long idle time the
company will not incur huge losses and generate sufficient benefit
during the HMA mixture production period.
The opinion of all 43 experts are concordant as the calculated
concordance coefficient is W = 0.719, Pearson's chi-kvadrat
statistics [chi square] = 247.5 is much higher than critical value [[chi
square].sub.[upsilon],[alpha]], which corresponds to the number of the
degree of freedom v = 8 and significance level [alpha] = 0.01 ([[chi
square].sub.v,[alpha]] = 20.0902). Min concordance coefficient is
[W.sub.min] = 0.0584, at which it could be still stated (when v = 8 and
[alpha] = 0.01) that the opinions of all experts are concordant.
AMP quality criteria normalized significance (weight) indices
calculated according to Zavadskas methodology max value [Q.sub.max] =
0.1211 (C quality criterion) and min [Q.sub.min] = 0.1032 (B quality
criterion) show their slight difference equal to 0.0179, i.e. it differs
1.17 times (approx 15%). The max value of these quality criteria weight
indices calculated according to the author's methodology is
[Q.sub.max] = 0.1912, and the min value is [Q.sub.min] = 0.0486: the
difference is 0.1426, i.e. 3.93 times (approx 75%). Quality criteria
weight indices calculated according to the 2nd method are more
different; therefore they are more "sensitive" and have
greater impact on the weight of quality criteria when used to calculate
the values of AMP complex quantity index K additive model components.
doi: 10.3846/bjrbe.2011.07
Received 8 September 2010; accepted 20 January 2011
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Henrikas Sivilevicius
Dept of Transport Technological Equipment, Vilnius Gediminas
Technical University, Plytines g. 27, 10105 Vilnius, Lithuania E-mail:
[email protected]
Table 1. Weight values given by experts to AMP quality indices
in points [B.sub.ij]
Expert's Quality criterion (index) notation
(respondent's) and its point (j = 1, 2, ..., m)
code
[X.sub.1] [X.sub.2] ... [X.sub.m]
i=1,2, ..., n
[E.sub.1] [B.sub.11] [B.sub.12] ... [B.sub.1m]
[E.sub.2] [B.sub.21] [B.sub.21] ... [B.sub.2m]
[E.sub.3] [B.sub.31] [B.sub.31] ... [B.sub.3m]
... ... ... ...
[E.sub.n] [B.sub.n1] [B.sub.n2] ... [B.sub.nm]
Total
[n.summation [B.sub.1] [B.sub.2] ... [B.sub.m]
over (i=1)
[B.sub.ij] =
[B.sub.J]
Table 2. Experts' opinion ranks and their use in determining
an average rank and value W of Kendall's coefficient of
concordance
Quality criterion (index) notation
Expert's and its rank (j = 1, 2, m)
(respondent's)
code [X.sub.1] [X.sub.2]
i=1,2, ..., n
[E.sub.1] [R.sub.11] [R.sub.12]
[E.sub.2] [R.sub.21] [R.sub.22]
[E.sub.3] [R.sub.31] [R.sub.32]
... ... ...
[E.sub.n] [E.sub.n1] [E.sub.n2]
Sum of ranks
[n.summation [R.sub.1] [R.sub.2]
over (i=1)]
[R.sub.ij]
Average rank
[bar.[R.sub.j]]= [bar.[R.sub.1]] [bar.[R.sub.2]]
[n.summation
over (i=1)]
[R.sub.ij]/n
[n.summation over (i=1)] [R.sub.ij] - 1/2 n (m+1)
[[[n.summation over (i=1)] [R.sub.ij] - 1/2 n (m+1)].sup.2]
Quality criterion (index) notation
Expert's and its rank (j = 1, 2, m)
(respondent's)
code ... [X.sub.m]
i=1,2, ..., n
[E.sub.1] ... [E.sub.1m]
[E.sub.2] ... [E.sub.2m]
[E.sub.3] ... [E.sub.3m]
... ... ...
[E.sub.n] [??] [E.sub.nm]
Sum of ranks
[n.summation ... [R.sub.m]
over (i=1)]
[R.sub.ij]
Average rank
[bar.[R.sub.j]]= ... [bar.[R.sub.m]]
[n.summation
over (i=1)]
[R.sub.ij]/n
[n.summation over (i=1)] [R.sub.ij] - 1/2 n (m+1)
[[[n.summation over (i=1)] [R.sub.ij] - 1/2 n (m+1)].sup.2]
Table 3. Questionnaire of determining the weight of the operating
AMP quality complex evaluation quality criteria (points By given by
the 1st expert [E.sub.1] and calculated ranks [R.sub.1j])
Number of Indices (quality criteria) Weight Ranks
criterion showing AMP quality values in [R.sub.1j]
points
[B.sub.1j]
Quality of HMA mixture
production
1 Compliance of the produced 9 1
HMA mixture composition
(amount of components in it)
with the job mix formula
(JMF) requirement (C)
2 Compliance of the produced 7 3
HMA mixture temperature with
the temperature specified in
TRA ASFALTAS 08 (T)
3 Homogeneity of the produced 8 2
HMA mixture (in mix batch),
expressed by the quality of
its mixing (H)
4 Environmental protection 1 9
when polluting atmospheric
air with the pollutants
emitted from AMP (E)
5 Costs of HMA mixture 6 4
production per 1 ton (cost
price of producing 1 ton of
HMA mixture) (P)
6 The degree of physical and 4 6
moral wear (depreciation) of
the operating AMP (W)
7 AMP repair and 5 5
reconstruction costs (R)
8 The use (exploitation) of 2 8
AMP capacity to produce HMA
or other asphalt mixtures
(B)
9 AMP technological 3 7
versatility (capability to
produce mixtures of various
kinds, types and marks) (U)
Table 4. Ranks [R.sub.ij] of the operating (not newly purchased) AMP
quality criterion (indices) and values calculated from them
Expert's code, AMP quality criterion (index)
i = 1, 2, ..., n notation; j = 1, 2, ..., m
C T H E P
[E.sub.1] 1 3 2 9 4
[E.sub.2] 1 3 2 4 9
[E.sub.3] 2 3 1 7 8
[E.sub.4] 1 2 4 3 9
[E.sub.5] 1 2 3 4 5
[E.sub.6] 1 2 3 4 5
[E.sub.7] 1 3 2 4 7
[E.sub.8] 1 3 4 2 7
[E.sub.9] 1 5 2 4 7
[E.sub.10] 1 3 2 4 7
[E.sub.11] 1 3 2 5 7
[E.sub.12] 3 1 5 4 8
[E.sub.13] 1 3 2 6 8
[E.sub.14] 2 1 3 5 6
[E.sub.15] 1 3 2 4 6
[E.sub.16] 1 4 3 6 8
[E.sub.17] 1 3 2 9 6
[E.sub.18] 1 4 2 5 6
[E.sub.19] 3 1 2 5 9
[E.sub.20] 1 2 3 5 6
[E.sub.21] 1 2 3 5 4
[E.sub.22] 1 2 3 6 5
[E.sub.23] 1 3 4 5 8
[E.sub.24] 2 1 4 5
[E.sub.25] 1 3 2 6 5
[E.sub.26] 1 3 2 4 7
[E.sub.27] 2 1 6 5
[E.sub.28] 1 3 2 4 5
[E.sub.29] 1 3 2 5 6
[E.sub.30] 1 2 3 4 6
[E.sub.31] 1 3 2 6 5
[E.sub.32] 1 3 8 5 6
[E.sub.33] 1 2 3 6 8
[E.sub.34] 1 2 3 5 4
[E.sub.35] 1 2 3 5 4
[E.sub.36] 1 3 2 9 4
[E.sub.37] 1 3 2 5 6
[E.sub.38] 1 2 3 4 7
[E.sub.39] 7 2 4 1
[E.sub.40] 1 3 2 4 6
[E.sub.41] 2 1 4 6
[E.sub.42] 1 2 3 5 6
[E.sub.43] 1 3 2 7 5
Sum of ranks
[n.summation 60 116 104 217 262
over (i=1)]
[R.sub.ij] =
[R.sub.j]
Average rank n
[bar.[R.sub.j]]= 1.395 2.698 2.419 5.046 6.093
[n.summation
over (i=1)]
[R.sub.ij]/n
Difference
[n.summation -155 -99 -111 2 47
over (i=1)]
[R.sub.ij] -
n(m+1)/2
[[[n.summation 24 025 9801 12 321 4 2209
over (i=1)]
[R.sub.ij] -
1/2 n
(m+1)].sup.2]
Expert's code, AMP quality criterion (index)
i = 1, 2, ..., n notation; j = 1, 2, ..., m
W R B U
[E.sub.1] 6 5 8 7
[E.sub.2] 7 8 5 6
[E.sub.3] 5 6 9 4
[E.sub.4] 5 6 8 7
[E.sub.5] 6 7 8 9
[E.sub.6] 6 7 8 9
[E.sub.7] 5 8 9 6
[E.sub.8] 8 9 6 5
[E.sub.9] 6 8 9 3
[E.sub.10] 9 5 6 8
[E.sub.11] 9 8 6 4
[E.sub.12] 7 9 6 2
[E.sub.13] 5 9 7 4
[E.sub.14] 8 7 9 4
[E.sub.15] 7 9 8 5
[E.sub.16] 7 9 5 2
[E.sub.17] 4 5 8 7
[E.sub.18] 7 8 9 3
[E.sub.19] 8 7 6 4
[E.sub.20] 4 8 9 7
[E.sub.21] 8 6 7 9
[E.sub.22] 7 8 9 4
[E.sub.23] 7 6 9 2
[E.sub.24] 6 8 7 9
[E.sub.25] 9 8 7 4
[E.sub.26] 6 8 9 5
[E.sub.27] 9 7 8 4
[E.sub.28] 8 9 7 6
[E.sub.29] 7 8 9 4
[E.sub.30] 5 7 8 9
[E.sub.31] 7 9 8 4
[E.sub.32] 7 8 9 4
[E.sub.33] 4 9 7 5
[E.sub.34] 7 6 9 8
[E.sub.35] 7 9 8 6
[E.sub.36] 6 8 7 5
[E.sub.37] 8 9 7 4
[E.sub.38] 6 8 9 5
[E.sub.39] 8 5 9 3
[E.sub.40] 5 7 9 8
[E.sub.41] 7 8 9 5
[E.sub.42] 4 7 8 9
[E.sub.43] 4 9 8 6
Sum of ranks
[n.summation 281 325 336 234
over (i=1)]
[R.sub.ij] =
[R.sub.j]
Average rank n
[bar.[R.sub.j]]= 6.535 7.558 7.814 5.442
[n.summation
over (i=1)]
[R.sub.ij]/n
Difference
[n.summation 66 110 121 19
over (i=1)]
[R.sub.ij] -
n(m+1)/2
[[[n.summation 4356 12 100 14 641 361
over (i=1)]
[R.sub.ij] -
1/2 n
(m+1)].sup.2]
Table 5. The results of AMP quality criteria (indices) significance
(weight) and priority calculation, obtained when applying different
methodologies
Quantity AMP quality criterion (index) notation
(Eq)
C T H E P
[bar.[q.sub.j]] (10) 0.031 0.060 0.054 0.112 0.135
[d.sub.j] (11) 0.969 0.940 0.946 0.888 0.865
[Q.sub.j] (12) 0.1211 0.1175 0.1183 0.1110 0.1081
[Q.sub.j] (13) and 0.1912 0.1622 0.1685 0.1101 0.0868
[[??].sub.j] (14)
Priority 1 3 2 4 6
Quantity AMP quality criterion (index) notation Sum
(Eq)
W R B U
[bar.[q.sub.j]] (10) 0.145 0.168 0.174 0.121 1.000
[d.sub.j] (11) 0.855 0.832 0.826 0.879 8.000
[Q.sub.j] (12) 0.1069 0.1040 0.1032 0.1099 1.0000
[Q.sub.j] (13) and 0.0770 0.0543 0.0486 0.1013 1.0000
[[??].sub.j] (14)
Priority 7 8 9 5
Fig. 1. Average ranks of operating AMP quality criteria (indices)
C 1.395
T 2.698
H 2.419
E 5.046
P 6.093
W 6.535
R 7.558
B 7.814
U 5.442
Note: Table made from bar graph.
