Using an integrated model for shaft sinking method selection/Kompleksinio modelio naudojimas greziniu irengimo metodui parinkti.
Lashgari, Ali ; Fouladgar, Mohamad Majid ; Yazdani-Chamzini, Abdolreza 等
1. Introduction
Shaft is one of the most important capital openings of underground
deep mines, which is used to have access to the ore body, as well as
providing all services for underground operations including water
supply, drainage, ventilation, personnel and ore transportation,
communications and power. Shaft sinking operation may consume up to 60%
of time of the underground mine development stage (Unrug 1992). This
time depends on the selected sinking method and the depth of the
underground mine (Hustrulid 1982). Therefore, selection of a proper
method to sink the shafts is an important issue to minimize the
development time and cost and assure success of the stage of development
openings.
A number of technical issues should be concerned for the design of
the shaft such as the approximate shaft location and underground space
outline including a description of the characteristics of the shaft and
its functions; the shaft capacity, diameter, hoisting depth and gear,
shaft lining type, number of shaft insets, the main pipelines and
cables, the quantity of airflow through the shaft and the depth of the
shaft sump along with cost specifications (Read, Napierf 1994; Lin
2010). A preliminary evaluation of the hoisting depths and shaft
diameter are needed beyond the initial phase of the project. Taking a
view of over-designing is a good idea in this phase, to prevent facing a
bottleneck and requiring the sinking of another shaft in case of
potential increase of production (Hustrulid 1982).
In addition to the technical parameters, the safety and economic
issues are also important to make the accurate decisions regarding the
design and sinking the shafts (Bhulose 2004; Medineckiene et al. 2010).
The costs associated with the shaft sinking operation can divided into
two different categories; capital and operating costs. Capital costs are
those costs that have accrued or accrue just to have the potential of
using the required equipment and facilities and the use of piece of
these equipment and facilities generate a constant stream of operating
costs (Vorster 1980).
Many different methods can be applied to sink the shafts of the
underground mines. To select the best alternative, different issues
affecting the selection of the shaft sinking method should be considered
and all possible options should be evaluated. Some of these criteria are
quantitative and some are qualitative which need to be made quantified.
Apart from selecting the most efficient shaft sinking method, decision
maker should have enough knowledge and expertise of use it. As a result,
it is complicated to consider all associated parameters simultaneously.
In traditional approaches of shaft sinking method selection, some
critical factors such as safety are not taken into account. Moreover,
the importance weights of different criteria are considered as equal.
Moreover in these approaches the merits of mechanized boring systems
such as rapid excavation, safety and performance of operations and
simultaneous installation of rock supports are not considered.
In this paper, an applicable approach based on Multi Criteria
Decision Making (MCDM) techniques including Fuzzy analytical hierarchy
process
(FAHP) and Fuzzy TOPSIS (FTOPSIS) for selection of shaft sinking
method is introduced. TOPSIS method is utilized because of being
rational, simple computations, and results are obtained in shorter time
than other methods (Per?in 2009). However, TOPSIS is often criticized
for its inability to deal with vague and uncertain problems (Yu et al.
2011). On the other hand, fuzzy sets are able to model the uncertainty.
Moreover, Fuzzy AHP is widely used for solving MCDM problems in real
issues (Karimi et al. 2011). Thus, FAHP, and FTOPSIS are combined to
rank shaft sinking methods which applies FAHP to obtain criteria weights
and FTOPSIS to acquire the final ranking order of shaft sinking methods.
As a field study, this approach is applied to select the best
method for shaft sinking operation in Parvadeh Coal Mine, located in
coal zone of Tabas which is the major collieries in central part of Iran
and the largest Iranian coal mine. This is a semi-mechanized coal mine
which both traditional and new shaft sinking methods are applicable
there.
2. Shaft sinking methods
Shaft sinking methods can be divided into mechanized and
conventional methods. Nowadays, the mechanization of underground mining
and construction development is becoming increasingly significant with
increased stress on efficient and safe operation (Douglas, Pfutzenruter
1989). Mechanized excavation is one of the alternatives to improve
overall mine performance; since more process phases can take place
simultaneously, e.g. excavation and muck removal. In some cases, the
installation of rock support can also be performed simultaneously
(Puhakka 1997).
On one hand, the conventional methods such as drilling and blasting
and Alimak have been broadly utilized, so far have been used to drive
more shafts and raises than any other system; in all kinds of rock,
pilot and full-face, vertical and inclined, and even for raise and vein
mining (Hustrulid 1982). On the other hand, the new mechanized systems
such as rotary drilling method of boring large diameter holes by using
Raise Boring Machine (RBM) and Shaft Boring Machine (SBM) for the mining
and construction industries has proven to be extremely safe and
economical (Robbins 2000).
Raise Boring Machine (RBM) is an automated boring machine which has
been generally used in the underground mining and construction
development since 1968 for boring shafts up to 1260 m in length and
ranging from 0.7 m in diameter to 7.1 m. RBMs can be used in various
types of operations including conventional raise boring, down boring,
blind hole boring, pilot hole boring and horizontal boring (Breeds,
Conway 1992). The RBM installs on top of the planned raise and bores a
pilot hole, breaking through at the target point at the level below. The
pilot bit is then replaced by a reamer head, with the diameter of the
planned raise. The RBM pulls the reamer head upward, with strong force,
while rotating, to break a circular hole in rock (Ozdemir 1986). Fig. 1
shows these various types of operations.
[FIGURE 1 OMITTED]
The Shaft Boring Machine (SBM) is a development for the mechanized
excavation of deep vertical blind shafts in hard rock conditions which
can be used to sink deep vertical shafts with a diameter of up to 8.5 m
(Ozdemir 1986). An overview of the system is shown in Fig. 2.
[FIGURE 2 OMITTED]
The Alimak has been around since 1948. The Alimak Method consists
of five steps which make up a cycle: drilling, loading, blasting,
ventilation and scaling. It is better if the rock structure is
continuous over several hundred feet vertically. The Alimak is by no
means a new technology (Hustrulid 1982). It is a relatively fast method
and can be used in excavations in excess of 200 m in length. Experience
has shown that raises from 75 m to 150 m length are the most economical.
A further advantage is that support components can be installed as one
develops. The Alimak method offers solutions in the development of reef
raises, boxholes, ventilation passes, shafts, etc. at practical
diameters ranging from [+ or -] 1.8 m to 6 m (Ferreira 2005). Drilling
and blasting is a conventional method with no mechanization in which the
operations handled by manpower (Hustrulid 1982). Table 1 shows
advantages and disadvantages of these methods.
3. Decision making
Decision making is the study of identifying and alternatives based
on the values and preferences of the decision maker. Making a decision
implies that there are alternative choices to be considered, and in such
a case we want not only to identify as many of these alternatives as
possible but to choose the one that best fits with our goals,
objectives, desires, values, and so on (Harris 1998). Decision making
process can be divided into eight following steps (Fiilop 2005):
--Define the problem.
--Determine requirements.
--Establish goals.
--Identify alternatives.
--Define criteria.
--Select a decision making tool.
--Evaluate alternatives against criteria.
--Validate solutions against problem statement.
Various tools can be used to select the best alternative, including
expert systems, Delphi decision making process, paired comparison, grid
analysis, influence diagram, pro/con approach, decision tree, game
theory, cost/benefit analysis, multi-voting technique, linear
programming, trial and error approach, affinity diagrams and multiple
criteria decision analysis.
Expert systems try to make decisions base on some rules and the
knowledge of experts (Kreider et al. 1992). This method is based on
personal judgment and provides no guarantee about the quality of the
rules on which it operates. Moreover, these systems are not optimal for
all problems, and significant knowledge is needed to obtain accurate
consequences (Denby, Schofield 1990).
The Delphi decision making process was developed in the early
1950s. In this method, a series of surveys, questionnaires, etc. are
sent to selected respondents who are selected because they are experts
or they have significant knowledge (the Delphi group). The group does
not converse in person (Yang, Hsieh 2009). All exchange of information
or idea is normally in letters. The responses are collected and
evaluated to determine conflicting opinions on each point. The process
goes on in order to work towards synthesis and building consensus. In
this method, the success depends upon the respondents' proficiency
and communication skill. Also, each response requires enough time for
reflection and analysis (Clayton 1997).
Paired comparison analysis helps the user to evaluate the
importance of a number of alternatives relative to each other. It is
particularly useful where decision maker do not have objective data to
base this on (Katz et al. 2001).
Grid analysis is a useful technique to make a decision particularly
where the decision maker has a number of alternatives to choose from,
and several factors to take into consideration. Grid analysis is a great
technique to use in almost any decision where there isn't a clear
and obvious preferred alternative (Pike 2004).
Influence diagram is beneficial where impacts are graphically
represented for a decision situation. Influence diagrams provide an
alternative to decision trees which grow exponentially with more
parameters (Cobb, Shenoy 2008).
Pro/Con and the similar or related techniques (such as pro/con/fix,
T-chart, weighted pro/con, force field analysis and
plus/minus/interesting,) are the age old method of considering the pros
and cons of two alternatives. A key restriction of these methods is that
only two alternatives are considered simultaneously (Ullman 2006).
Decision tree is useful to visualize multi-stage decision problems
while dealing with uncertain outcomes. It can be beneficial in making
decisions between investment opportunities or strategies with
constrained resources (Choi, Lee 2010).
Game theory is useful for making complex strategic decision where
it is beneficial to consider the likely response of outside participants
(e.g. government, competitors or customers). This method can be regarded
as an extension to influence diagrams. Game theory needs some
simplifying assumptions to restrict a decision to a solvable game
problem which is the most important limitation of this method (Tsoukias
2008).
Cost/benefit analysis is bounded to making decision about financial
problems or can be considered as an extension for evaluation of
financial criteria to other decision making methods (Almansa,
Martinez-Paz 2011).
Multi-voting technique is beneficial for group decisions to select
fairly between a large numbers of alternatives. It is much more useful
to omit lower priority options before using a more precise method to
finalize a decision on a smaller number of alternatives (Ou et al.
2005).
Linear Programming is commonly used for optimization of limited
resources. This is a mathematical method in which the objectives and
constraints are presented in form of linear equations (Huang et al.
2010).
Trial and error approach is another method for decision making. The
main restrictions of this technique are that impacts for decision
failure should be small and suitable reaction shoud be implicated after
the failure to ensure that acceptable cause/effect relationships are
recognized in the learning procedure. As an instance, heuristic
techniques are trial and error decision making approaches which start
with a model that is refined with ongoing experimentation (Whitehead,
Ballard 1991).
Affinity diagrams and the similar or related methods (such as KJ
method) address information overload by classifying a number of ideas
and large amounts of data using this qpproach. Affinity diagram is
generally used as part of a brainstorming exercise (Ho et al. 1999).
The decision making problems in which the number of the
alternatives and criteria is finite and the alternatives are specified
explicitly are named multi-attribute decision making (MADM) problems.
Multiple criteria decision analysis and the same or related techniques
(such as grid analysis, Kepner-Tregoe matrix) are techniques provide a
good compromise between intuition and analysis by using a systematic
framework that evaluates options against a defined set of success
criteria (Chang, Wang 2009; Zavadskas, Turskis 2010; Ulubeyli, Kazaz
2009; Ginevi?ius, Podvezko 2009). Analytical Hierarchy Process (AHP) is
an enhanced multiple criteria technique that uses paired comparison with
additional mathematics to help address the subjectivity and intuition
that is inherent in a human decision making technique (Kahraman 2008).
In some cases decision criteria are not rigid, where the boundary
between a value and its inverse is gradual and there is an inexact
boundary or class overlap. Boolean logic is in binary form in which an
element is false or true, an object fit in a set or it doesn't
(Goetcherian 1980). Fuzzy logic began with the 1965 proposal of fuzzy
set theory by Zadeh and permits the concept of nuance. Based on this
theory, a proposition may be anything from hardly to approximately true.
A fuzzy set does not have strictly defined borders. The notion of a
fuzzy set is beneficial to dealing with imprecision problems with
uncertain criteria and conditions (Zadeh 1965). A brief review of fuzzy
sets, Fuzzy AHP and Fuzzy TOPSIS are presented in appendix A, B and C,
respectively.
4. Case study
4.1. Parvadeh mine
The coal region of Tabas is divided into three sections; Parvadeh,
Nayband and Mezino areas. These areas are shown in Fig. 3.
[FIGURE 3 OMITTED]
Parvadeh underground coal mine (Tabas coal mine No.1.) is a
semi-mechanized coal mine, with a seam thickness of 1.8 m and dip angle
of 29.5[degrees], located in a remote rugged desert environment some 85
km south of Tabas city in mid east Iran, in an area of 1200 [km.sup.2,]
and production rate of this mine is about 4000 t of coal per day.
Because of the suitable geometry of the coal seams and large extent of
the deposit, mechanized longwall mining is applied. The face length
(panel width) varies from 200 m to 220 m. The panel length is about 1000
m (Hosseini 2007). An international tender for Parvadeh coal mine was
issued by the National Iranian Steel Company and a joint venture between
IRITEC and IRASCO was selected as the preferred bidder. This project
consists of preparation of infrastructures, carrying out engineering of
one of the mine and supply of suitable technology and machineries for
the development of the mine as well as training, technical assistance
and commissioning of the longwall, coal handling and the coal
preparation plants. Fig. 4 shows Parvadeh mine and other districts of
the coal region of Tabas with location of the exploration shafts
(IRITECH 1992; NISCOIR 1996).
[FIGURE 4 OMITTED]
4.2. Results
In this paper, FAHP is used to analyze the structure of the problem
of shaft sinking method selection and to determine weights of the
criteria, and FTOPSIS is used to obtain final ranking. The steps are
summarized as follows:
Step 1. Forming a board of 13 academic and industrial experts are
involved in mining and construction and explain the shaft sinking method
selection problem.
Step 2. Decomposing the problem into a hierarchical structure in
which the overall goal, at the top level of the hierarchy, can be
separated into several criteria at a lower level of the hierarchy. The
bottom level of the hierarchy represents potential alternatives. The aim
of the hierarchy is to determine the importance rating of different
methods based on the criterion that decision maker would like to attain
in implication of the project, including water inflow rate ([C.sub.1]),
mechanization and advance rate ([C.sub.2]), rock properties ([C.sub.3]),
hoisting depth (C4), shaft diameter ([C.sub.5]), safety ([C.sub.6]),
operating cost ([C.sub.7]) and capital cost ([C.sub.8]) and potential
sub-criterions. Water inflow rate, capital and operating costs have
negative impact on the selection and the rest of criteria have positive
impact.
This purpose is done through pairwise comparison of the importance
of different shaft sinking methods towards each criterion and pairwise
comparison of the importance of different criteria towards the target.
Step 3. Developing a questionnaire to gather the expert knowledge
regarding the subject. The experts will be asked to compare each of the
paired factors in the matrices through questionnaires, regarding the
technical parameters of the project. In this case, shaft diameter and
hoisting depth will be 5.5 m and 580 m, respectively. At the first
level, they need to state decisions about the relative importance of
each criterion in terms of how it contributes to attaining the overall
goal. Then a preference for each potential alternative in terms of its
contribution to each criterion must be made.
A nine-point scale is suggested to state preferences between
alternatives as extremely preferred, very strongly, strongly, moderately
or equally, with pairwise weights of 9, 7, 5, 3 or 1, respectively. The
values between mentioned points are the intermediate values for the
preference scale. For the inverse comparisons, reciprocal values can be
used. The matrix of paired comparison is constructed, after each factor
has been compared.
Step 4. calculating the fuzzy pair-wise comparison matrix as
follows (Jaskowski et al. 2010):
[[??].sub.ij] = ([l.sub.ij], [m.sub.ij], [u.sub.ij]); (1)
[l.sub.ij] = min {[x.sup.k.sub.ij]}, [m.sub.ij] = 1/k [k.summation
over (k=1)][x.sup.k.sub.ij],[u.sub.ij]}, (2)
where [[??].sub.ij] indicates the fuzzy importance weights of each
criterion which are calculated by experts, k is the number of expert and
[x.sub.ij] is the crisp weight of each criterion (Table 2).
Thereafter, obtained weights of all criteria are compared by Eq.
(B-6) and are presented in Table 3.
Step 5. Priority weights are determined by using Eq. (B-8) and are
presented in Table 4.
Step 6. By comparing the alternatives under each of the criteria, a
decision matrix based on the experts' opinion is established and
the performance ratings of the alternatives are determined by Eq. (3)
(Torfi et al. 2010):
[[??].sub.ij] = ([[??].sub.ij.sup.1] [cross product] ... [cross
product] [[??].sub.ij.sup.k]); k = 13. (3)
The membership functions of fuzzy numbers which is shown in Table
A. 1 are used to quantify the linguistic values. Table 5 shows fuzzy
decision matrix.
Then normalized fuzzy decision matrix is determined by Eqs (4) and
(5).
[r.sub.ij] = [x.sub.ij] - min {[x.sub.ij]}/[max {[x.sub.ij]} - min
{[x.sub.ij]}], (4)
the larger, the better type
[r.sub.ij] = min {[x.sub.ij]} - [x.sub.ij]/[max {[x.sub.ij]} - min
{[x.sub.ij]}], (5)
Step 7. The weighted normalized decision matrix is established
using the criteria weights calculated by FAHP in step 5 by Eq. (1).
Table 6 shows weighted normalized fuzzy decision matrix.
Step 8. The distance of each alternative from [D.sup.+] and
[D.sup.-] can be currently determined using Eq. (C-7) and Eq. (C-8). At
last, FTOPSIS solves the similarities to an ideal solution by Eq. (C-9).
The results of the analyses are summarized in Table 7. According to
[CL.sub.i] values, the ranking of the alternatives in descending order
are RBM, drilling and blasting, alimak and SBM. Fig. 5 presented a
schematic view of the rank of alternatives.
It can be inferred from Table 6 that shaft diameter ([C.sub.4]),
hoisting depth ([C.sub.5]) and capital cost ([C.sub.8]) are the main
reasons to select SMB as the worst alternative. From technical and
economical point of view, shaft sinking by the SMB impose a great amount
of capital to the project. Moreover, this method is appropriate for
large diameter deep shafts. Therefore this method is appropriate for
large scale shaft.
[FIGURE 5 OMITTED]
5. Conclusions
Shaft sinking is a critical part of underground construction
operation and selection of an appropriate method to minimize sinking
time and cost along with assure uninterrupted operation is of great
importance. A number of techniques are available for shaft sinking
operation. Each method has several inherent advantages and entails some
limitations and problem. Consequently, selection of an appropriate
method for shaft sinking operation requires consideration of many
technical and economical criteria. In this study, a decision support
system for shaft sinking method selection is presented to facilitate
consideration of many effective parameters simultaneously in the shaft
sinking method selection process. The present study explored the use of
a hybrid method of Fuzzy Analytical Hierarchy Process (FAHP) and Fuzzy
TOPSIS (FTOPSIS) in solving this multi criteria decision making issue.
For this purpose, the existing criteria have been weighted by FAHP and
then FTOPSIS is used to prioritize the alternatives. A real world case
study of Parvadeh Coal Mine located in coal zone of Tabas in selecting
the most appropriate shaft sinking method is presented to examine the
practicality of the proposed model. This hybrid method considers both
quantitative and qualitative effective parameters along with existing
uncertainty, simultaneously and solves the problems of traditional shaft
sinking method selection approaches. By applying the model, using of
Raise Boring Machine (RBM) is selected as optimal method for shaft
sinking in this mine.
Appendix A. Fuzzy sets
This theory can change concepts, variables and systems which are
vague and imprecise to mathematical forms and this can provide
background for reasoning, inference, control and decision making in
uncertainty conditions (Ross 2004).
If [??] = ([a.sub.1], [a.sub.2], [a.sub.3]) is considered as a
Triangular Fuzzy Number (TFN), where ab a2, a3 are crisp numbers and
[a.sub.1] < [a.sub.2] < [a.sub.3], then membership function
[f.sub.([??])] is as Eq. (A-1):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-1)
A TFN is shown in Fig. A. 1. The factors L, M and U represent the
smallest possible value, the most promising and the largest possible
value that describe a fuzzy event, respectively (Antuchevi?ien? 2005; Xu
et al. 2010).
[FIGURE A.1. OMITTED]
If [??], [??] are two triangle fuzzy numbers as A = ([a.sub.1],
[a.sub.2], [a.sub.3]), [??] = ([b.sub.1], [b.sub.2], [b.sub.3]) the
mathematical relationship between [??] and [??] will be as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (A-2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (A-3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (A-4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (A-5)
U = {[u.sub.1], [u.sub.2], ..., [u.sub.m]}. (A-6)
A fuzzy set is characterized by a characteristics (membership)
function, which allocates to each object a grade of membership belongs
to [0-1] (Bardossy, Fodor 2004; Kala 2008). The characteristics function
of fuzzy numbers, is applied to expert's questionnaire results to
establish fuzzy weights, is defined in Table A.1. Fig. A.2. shows an
interview of the membership function of fuzzy numbers.
[FIGURE A.2. OMITTED]
http://dx.doi.org/ 10.3846/13923730.2011.628687
Table A.1. Characteristic function of the fuzzy numbers
Linguistic Corresponding
terms Fuzzy Number
Very bad (0, 0, 1)
Bad (0, 1, 3)
Medium bad (1, 3, 5)
Medium (3, 5, 7)
Medium good (5, 7, 9)
Good (7, 9, 10)
Very good (9, 10, 10)
Appendix B. Fuzzy AHP
Analytical hierarchy process (AHP) was introduced by Saaty (1980).
This method is based on three fundamental concepts; structure of the
model, comparative judgment of the options and the criteria and
synthesis of the priorities. In order to develop a methodology for
selection of the best alternatives in case of imprecision problems with
uncertain criteria, AHP method has been combined with fuzzy theory by
miscellaneous approaches (Buckley 1985; Chang 1996; Cheng 1997;
Sivilevi?ius, Maskeli?nait? 2010; Pan 2008). This method not only
effectively handles the imprecision and uncertainty of the decision
making but also supplies the flexibility and robustness required for the
decision maker to realize the decision problem (Nang-Fei 2008).
Assume that X = {[x.sub.1], [x.sub.2] ..., [x.sub.n]} and U =
{[u.sub.1], [u.sub.2] ..., [u.sub.m]} are object and goal sets,
respectively. Based on the extent FAHP methodology which was introduced
by Chang (Chang 1996) each object is considered and extent analysis for
each goal, [g.sub.i], is applied, respectively. Therefore, m extent
analysis values for each object can be given as follows:
[M.sup.1.sub.gi], [M.sup.2.sub.gi], ..., [M.sup.m.sub.gi], i = 1,
2, ..., n. (B-1)
In the above [M.sup.j.sub.gi](j = 1, 2, ..., m) are TFNs.
The procedure of implication of Chang's FAHP methodology can
be divided into three following steps:
Step. 1. Determination of the value offuzzy synthetic extent: the
following equation is applied to determine the value of fuzzy synthetic
extent with respect to ith object:
[S.sub.i] = [m.summation over (j=1)][M.sup.j.sub.gi][cross
product][[[n.summation over (i=l)] [m.summation over (j=l)]
[M.sup.j.sub.gi]].sup.-1]. (B-2)
Performing the fuzzy addition operation of m extent analysis values
for a particular matrix, the term of
[[summation].sup.m.sub.j=i][M.sup.j.sub.gi] will be determined as:
[m.summation over (j=l)][M.sup.j.sub.gi] = ([m.summation over
(j=l)][l.sub.i], [m.summation over (j=l)] [m.sub.i], [m.summation over
(j=l)] [u.sub.i]). (B-3)
Performing the fuzzy addition operation of [M.sup.j.sub.gi](j = 1,
2, ..., m) values, the term of [[[summation].sup.n.sub.i=l]
[summation].sup.m.sub.j=l][M.sup.j.sub.gi]].sup.-1] will be obtained as:
[summation].sup.n.sub.i=l]
[summation].sup.m.sub.j=l][M.sup.j.sub.gi] = ([n.summation over
(i=1)][l.sub.i], [n.summation over (i=1)][m.sub.i], [n.summation over
(i=1)] [u.sub.i]). (B-4)
Thereafter, the inverse of the vector the above equation can be
computed such that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B-5)
Step. 2. Determining the degree of possibility : the degree of
possibility of [M.sub.2] = ([l.sub.2], [m.sub.2], [u.sub.2]) [greater
than or equal to] [M.sub.1] = ([l.sub.1], [m.sub.1], [u.sub.1]) can
mathematically expressed by Eq. (B-6):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B-6)
Assume that d is the ordinate of highest intersection point, D,
between [[mu].sub.M1] and [[mu].sub.M2] (Fig. B.1).
[FIGURE B.1. OMITTED]
Eq. (B-7) can be also defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-7)
The values of ([M.sub.1] [greater than or equal to] [M.sub.2]) and
V([M.sub.2] [greater than or equal to] [M.sub.1]) are needed to compare
[M.sub.1] and [M.sub.2].
Step. 3. Determining the weight vector: Eq. (B-8) can be implicated
to determine the degree of possibility for a convex fuzzy number to be
greater than k convex fuzzy numbers M. (i = 1, 2,k) :
V(M [greater than or equal to] [M.sub.1],[M.sub.2], ...,
[M.sub.k])= V(M [greater than or equal to] [M.sub.1]) and (M [greater
than or equal to] [M.sub.2]) ... and (M [greater than or equal to]
[M.sub.k])] = min V(M [greater than or equal to] [M.sub.i]), i = 1,2 ...
k. (B-8)
For d'([A.sub.i]) = min V ([S.sub.i], [S.sub.k]), k = 1,2 ...,
n; k [not equal to] i and [A.sub.i](i = 1,2 ..., n) are n elements, the
weight vector is defined as:
W' = (d'[([A.sub.i]),d'([A.sub.2]),
...,d'([A.sub.n])).sup.T]. (B-9)
Step. 4. Normalizing the weight vectors: the normalized weight
vectors can be defined as:
W = (d[([A.sub.i]),d([A.sub.2]), ..., d([A.sub.n])).sup.T], (B-10)
where W is a non-fuzzy number.
Appendix C. Fuzzy TOPSIS
Technique for Order Performance by Similarity to Ideal Solution
(TOPSIS) approach was introduced by Hwang and Yoon (Hwang, Yoon 1981).
This method is based on the concept that the separation of the best
alternative from the positive and negative ideal solution should have
the shortest and the farthest, respectively which seems rational (Lin et
al. 2008; Tupenaite et al. 2010; Zavadskas et al. 2010; Zavadskas,
Antucheviciene 2006). TOPSIS is an easy-to-apply method and the
computations involved are uncomplicated.
For the situation of incomplete information and nonobtainable
information, TOPSIS technique has been combined with fuzzy theory which
uses the fuzzy numbers to allocate the relative importance of the
criteria instead of crisp numbers (Ning et al. 2011). The approach to
extend the FTOPSIS method can be summarized as follows (Chen 2000;
Braglia et al. 2003; Wang, Chang 2007):
Assume that [[??].sub.ij], [[??].sub.j],; i = 1,2 ...,m; j = 1,2
..., n are linguistic triangular Fuzzy numbers which are defined as
[[??].sub.ij] = ([a.sub.ij], [b.sub.ij], [c.sub.ij]) and [[??].sub.j] =
([a.sub.j1], [b.sub.j2], [c.sub.j3]). To express the fuzzy MCDM in the
form of matrix, Eq. (C-1) and Eq. (C-2) can be developed:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (C-1)
W = [[??].sub.1], [[??].sub.2] ..., [[??].sub.n]]. (C-2)
where [[??].sub.ij] donates the performance rating of the ith
alternative ([A.sub.i]) concerning the jth criterion ([C.sub.j]). Also
the weight of [C.sub.j] is represented by [[??].sub.1].
The normalized Fuzzy decision matrix ([??]) and the weighted Fuzzy
normalized decision matrix can be expressed as Eq. (C-3) and Eq. (C-4),
respectively:
[??] = [[[[??].sub.ij]].sub.mxn]. (C-3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (C-4)
The procedure of implication of FTOPSIS can be divided into 6
different steps.
Step. 1. Choosing the linguistic ratings for alternatives: the
linguistic ratings for alternatives ([[??].sub.ij]; i = 1,2 ..., m; j =
1,2 ..., n) concerning criteria and the appropriate linguistic variables
([[??].sub.ij] j = 1,2, ..., n) for the weights of the criteria are
selected.
Step. 2. Developing the weighted normalized fuzzy decision matrix.
Step. 3. Determining the positive and negative ideal solutions: The
positive ([A.sup.+]) and negative ([A.sup.-]) ideal solutions are
calculated as Eqs (C-5) and (C-6), respectively:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (C-5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (C-6)
Step. 4. Measuring the separation of alternatives from the positive
and negative ideals: the separation of each alternative from positive
ideal ([d.sup.+.sub.i]) and from negative ideal ([d.sup.-.sub.i]) can be
calculated as follows:
[d.sup.+.sub.i] = [n.summation over (j=1)]d ([[??].sub.ij],
[[??].sup.+.sub.j]), i = 1,2, ..., m; (C-7)
[d.sup.-.sub.i] = [n.summation over (j=1)]d([[??].sub.ij],
[[??].sup.-.sub.j]), i = 1,2, ..., m. (C-8)
Step. 5. Calculating the relative closeness of each alternative to
the idea solution: the relative closeness of each alternative to the
idea solution is determined according to Eq. (C-9):
[CL.sup.*.sub.i] = [d.sup.-.sub.i]/[d.sup.-.sub.i] +
[d.sup.+.sub.i]. (C-9)
Step. 6. Final ranking: the alternative with maximum value of
relative closeness ([CL.sup.*.sub.i]) will be selected as the best
option.
References
Almansa, C.; Martinez-Paz, J. M. 2011. What weight should be
assigned to future environmental impacts? A probabilistic cost benefit
analysis using recent advances on discounting, Science of the Total
Environment 409(7): 1305-1314. doi:10.1016/j.scitotenv.2010.12.004
Antucheviciene, J. 2005. Alternatyvu vertinimo budai TOPSIS metodu,
esant neapibrePtumui, Technological and Economic Development of Economy
11(4): 242-247.
Asadi, A.; Shahriar, K.; Goshtasbi, K.; Najm, K. 2005. Development
of a new mathematical model for prediction of surface subsidence due to
inclined coal-seam mining, The Journal of the South African Institute of
Mining and Metallurgy 105(1): 15-20.
Bardossy, G.; Fodor, J. 2004. Evaluation of uncertainties and risks
in geology: new mathematical approaches for their handling. New York:
Springer. 234 p.
Bhulose, M. C. 2004. An investigation into the physical constraints
of the current sub-decline at Turffontein Shaft, The Journal of the
South African Institute of Mining and Metallurgy 104(3): 153-162.
Braglia, M.; Frosolini, M.; Montanari, R. 2003. Fuzzy TOPSIS
approach for failure mode, effects and criticality analysis, Quality and
Reliability Engineering International 19(5): 425-143.
doi:10.1002/qre.528
Breeds, C. D.; Conway, J. J. 1992. SME mining engineering handbook.
Chapter 22.1. 2nd Ed., Vol. 2, SME, Colorado, USA, 1871-1907.
Buckley, J. J. 1985. Fuzzy hierarchical analysis, Fuzzy Sets and
Systems 17(3): 233-247. doi:10.1016/0165-0114(85)90090-9
Chang, D.-Y. 1996. Application of the extent analysis method on
fuzzy AHP, European Journal of Operational Research 95(3): 649-655.
doi:10.1016/0377-2217(95)00300-2
Chang, T.-H.; Wang, T.-C. 2009. Using the fuzzy multi-criteria
decision making approach for measuring the possibility of successful
knowledge management, Information Sciences 179(4): 355-370.
doi:10.1016/j.ins.2008.10.012
Chen, C.-T. 2000. Extensions of the TOPSIS for group
decision-making under fuzzy environment, Fuzzy Sets and Systems 114(1):
1--9. doi10.1016/S0165-0114(97)00377-1
Cheng, C.-H. 1997. Evaluating naval tactical missile systems by
fuzzy AHP based on the grade value of membership function, European
Journal of Operational Research 96(2): 343--350.
doi:10.1016/S0377-2217(96)00026-4
Choi, M.; Lee, G. 2010. Decision tree for selecting retaining wall
systems based on logistic regression analysis, Automation in
Construction 19(7): 917-928. doi:10.1016/j.autcon.2010.06.005
Clayton, M. J. 1997. Delphi: a technique to harness expert opinion
for critical decision-making tasks in education, Educational Psychology
17 (4): 373-386. doi:10.1080/0144341970170401
Cobb, B. R.; Shenoy, P. P. 2008. Decision making with hybrid
influence diagrams using mixtures of truncated exponentials, European
Journal of Operational Research 186(1): 261-275.
doi:10.1016/j.ejor.2007.01.036
Denby, B.; Schofield, D. 1990. Applications of expert systems in
equipment selection of surface mine design, International Journal of
Surface Mining, Reclamation and Environment 4(4): 165-171.
doi:10.1080/09208119008944184
Douglas, A. A. B.; Pfutzenreuter, F. R. B. 1989. Overview of
current South African vertical circular shaft construction practice, in
The Shaft Engineering Conference, Harrogate, England, 5-7 June, 1989,
140-158.
Ferreira, P. H. 2005. Mechanised mine development utilising rock
cutting and boring through raise and blind boring techniques, in The
3>Td Southern African Conference on Base Metals, Kitwe, Zambia, 26-29
June, 2005, 297-314.
Fulop, J. 2005. Introduction to decision making methods. Working
paper of the Laboratory of Operations Research and Decision Systems
(LORDS), WP05-6. Hungary. 15 p.
Ginevicius, R.; Podvezko, V. 2009. Evaluating the changes in
economic and social development of Lithuanian countries by multiple
criteria methods, Technological and Economic Development of Economy
15(3): 418-436. doi:10.3846/1392-8619.2009.15.418-436
Goetcherian, V. 1980. From binary to grey tone image processing
using fuzzy logic concepts, Pattern Recognition 12(1): 7-15.
doi:10.1016/0031-3203(80)90049-7
Harris, R. 1998. Introduction to Decision Making [online].
VirtualSalt. [accessed 10 May 2011]. Available from Internet:
<http://www.virtualsalt.com/crebook5.htm>.
Herrenknecht Copmany's Brochure [online], [accessed 10 May
2011]. Available from Internet: <http://www.herrenknecht.
com/products/vertical-drilling/shaft-sinkingmachines.html>.
Ho, E. S. S. A.; Lai, Y.-J.; Chang, S. I. 1999. An integrated group
decision-making approach to quality function deployment, IIE
Transactions 31(6): 553-567. doi:10.1080/07408179908969858
Hosseini, N. 2007. Modelling of pillars in longwall method using
advanced numerical techniques. MSc. Thesis. Islamic Azad University,
South of Tehran branch. 235 p. (in Persian).
Huang, C.; Wong, C. K.; Tam, C. M. 2010. Optimization of tower
crane and material supply locations in a high-rise building site by
mixed-integer linear programming, Automation in Construction 20(5):
571-580. doi:10.1016/j.autcon.2010.11.023
Hustrulid, W. A. 1982. Underground mining methods handbook, SME
mining engineering handbook. SME, Colorado. 1754 p.
Hwang, C. L.; Yoon, K. 1981. Multiple attribute decision making
methods and applications. New York: Springer-Verlag. 259 p.
IRITECH. 1992. Internal reports of Tabas coal mine, 110-135.
Jaskowski, P.; Biruk, S.; Bucon, R. 2010. Assessing contractor
selection criteria weights with fuzzy AHP method application in group
decision environment, Automation in Construction 19(2): 120--126.
doi:10.1016/j.autcon.2009.12.014
Kahraman, C. 2008. Fuzzy multi-criteria decision making: theory and
applications with recent development. Istanbul: Springer. 590 p.
doi:10.1007/978-0-387-76813-7 Kala, Z. 2008. Fuzzy probability analysis
of the fatigue resistance of steel structural members under bending,
Journal of Civil Engineering and Management 14(1): 67-72.
doi:10.3846/1392-3730.2008.14.67-72
Karimi, A. R.; Mehrdadi, N.; Hashemian, S. J.; Nabi Bidhendi, G.
R.; Tavakkoli Moghaddam, R. 2011. Selection of wastewater treatment
process based on the analytical hierarchy process and fuzzy analytical
hierarchy process methods, International Journal of Environmental
Science and Technology 8(2): 267-280.
Katz, B. E.; Bruck, M. C.; Coleman III, W. P. 2001. The Benefits of
Powered Liposuction Versus Traditional Liposuction: A Paired Comparison
Analysis, Dermatologic Surgery 27(10): 863-867.
doi:10.1046/j.1524-4725.2001.01077.x
Kreider, J. F.; Wang, X. A.; Anderson, D.; Dow, J. 1992. Expert
systems, neural networks and artificial intelligence applications in
commercial building HVAC operations, Automation in Construction 1(3):
225-238. doi:10.1016/0926-5805(92)90015-C
Lin, K.-L. 2010. Determining key ecological indicators for urban
land consolidation, International Journal of Strategic Property
Management 14(2): 89-103. doi:10.3846/ijspm.2010.08
Lin, Y.-H.; Lee, P.-C.; Chang, T.-P.; Ting, H.-I. 2008.
Multi-attribute group decision making model under the condition of
uncertain information, Automation in Construction 17(6): 792--797.
doi:10.1016/j.autcon.2008.02.011
Medineckiene, M.; Turskis, Z.; Zavadskas, E. K. 2010. Sustainable
construction, taking into account the building impact on the
environment, Journal of Environmental Engineering and Landscape
Management 18(2): 118-127. doi:10.3846/jeelm.2010.14
Nang-Fei, P. 2008. Fuzzy AHP approach for selecting the suitable
bridge construction method, Automation in Construction 17(8): 958--965.
doi:10.1016/j.autcon.2008.03.005
Ning, X.; Lam, K.-C.; Lam, M. C.-K. 2011. A decision-making system
for construction site layout planning, Automation in Construction 20(4):
459-473. doi:10.1016/j.autcon.2010.11.014
NISCOIR (National Iranian Steel Company). 1996. Exploration Plan
Report of Tabas Coal Region, Vol. 2, Parvadeh Mine. 20 p.
Ou, M.; Chen, Y.; Orady, E. 2005. Genetic algorithm and fuzzy
C-means based multi-voting classification scheme in data mining, in The
Annual Meeting of the North American Fuzzy Information Processing
Society, 2005, 222-227.
Ozdemir, L. 1986. Improving the performance of tunnel, raise, and
shaft boring machines for coal mine applications. Technical Report.
Bureau of Mines, U.S. Department of the Interior, Colorado, USA. 342 p.
Pan, N.-F. 2008. Fuzzy AHP approach for selecting the suitable
bridge construction method, Automation in Construction 17(8): 958-965.
doi:10.1016/j.autcon.2008.03.005
Percin, S. 2009. Evaluation of third-party logistics (3PL)
providers by using a two-phase AHP and TOPSIS methodology, Benchmarking:
An International Journal 16(5): 588-604.
Pike, S. 2004. The use of repertory grid analysis and
importance-performance analysis of identify determinant attributes of
universities, Journal of Marketing for Higher Education 14(2): 1-18.
doi:10.1300/J050v14n02_01
Puhakka, T. (Ed.). 1997. Underground Drilling and Loading Handbook.
Finland: Tamrock Corporation. 271 p.
Read, H. W.; Napierf, L. G. D. 1994. Project management, and the
design of shaft-sinking projects, The Journal of the South African
Institute of Mining and Metallurgy 94(7): 147-171.
Robbins, R. J. 2000. Mechanization of underground mining: a quick
look backward and forward, International Journal of Rock Mechanics and
Mining Sciences 37(1-2): 413-421.
Ross, T. J. 2004. Fuzzy Logic with Engineering Applications. 2nd
ed. Wiley, USA. 640 p.
Saaty, T. 1980. The Analytic Hierarchy Process: Planning, Priority
Setting, Resource Allocation (Decision Making Series). New York: McGraw
Hill. 287 p.
Sivilevicius, H.; Maskeliunaite, L. 2010. The criteria for
identifying the quality of passengers' transportation by railway
and their ranking using AHP method, Transport 25(4): 368-381.
doi:10.3846/transport.2010.46
Terratech Copmany's Brochure [online], [accessed 10 May 2011].
Available from Internet: <http://www.terratec. com.au/pro ductsdetail
.php?ci d=2>.
Torfi, F.; Farahani, R. Z.; Rezapour, S. 2010. Fuzzy AHP to
determine the relative weights of evaluation criteria and Fuzzy TOPSIS
to rank the alternatives, Applied Soft Computing 10(2): 520-528.
doi:10.1016/j.asoc.2009.08.021
Tsoukias, A. 2008. From decision theory to decision aiding
methodology, European Journal of Operational Research 187(1): 138-161.
doi:10.1016/j.ejor.2007.02.039
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
Ullman, D. G. 2006. Making robust decisions: decision management
for technical, business and service teams. USA: Trafford Publishing. 349
p.
Ulubeyli, S.; Kazaz, A. 2009. A multiple criteria decision-making
approach to the selection of concrete pumps, Journal of Civil
Engineering and Management 15(4): 369-376.
doi:10.3846/1392-3730.2009.15.369-376
Unrug, K. 1992. Construction of Development Openings, in H. L.
Hartman (Ed.). SME mining engineering handbook. 2nd ed., vol. 2. SME,
Colorado, USA, 1580-1643.
Vorster, M. C. 1980. A systems approach to the management of civil
engineering construction equipment. PhD dissertation. University of
Stellenbosch, Stellenbosch, South Africa.
Wang, T. C.; Chang, T. 2007. Application of TOPSIS in evaluating
initial training aircraft under a fuzzy environment, Expert Systems with
Applications: An International Journal 33(4): 870-880.
doi:10.1016/j.eswa.2006.07.003
Whitehead, S. D.; Ballard, D. H. 1991. Learning to perceive and act
by trial and error, Machine Learning 7(1): 45-83. doi:10.1007/BF00058926
Xu, Y.; Yeung, J. F. Y.; Chan, A. P. C.; Chan, D. W. M.; Wang, S.
Q.; Ke, Y. 2010. Developing a risk assessment model for PPP projects in
China-A fuzzy synthetic evaluation approach, Automation in Construction
19(7): 929-943. doi:10.1016/j.autcon.2010.06.006
Yang, T.; Hsieh, C. H. 2009. Six-Sigma project selection using
national quality award criteria and Delphi fuzzy multiple criteria
decision-making method, Expert Systems with Applications: An
International Journal 36(4): 7594-7603. doi:10.1016/j.eswa.2008.09.045
Yu, X.; Guo, S.; Guo, J.; Huang, X. 2011. Rank B2C e-commerce
websites in e-alliance based on AHP and fuzzy PSIS, Expert Systems with
Applications: An International Journal 38(4): 3550-3557.
doi:10.1016/j.eswa.2010.08.143
Zadeh, L. A. 1965. Fuzzy sets, Information and Control 8(3):
338-353. doi:10.1016/S0019-9958(65)90241-X
Zavadskas, E. K.; Antucheviciene, J. 2006. Development of an
indicator model and ranking of sustainable revitalization alternatives
of derelict property: a Lithuanian case study, Sustainable Development
14(5): 287-299. doi:10.1002/sd.285
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.; Turskis, Z.; Tamosaitiene, J. 2010. Risk
assessment of construction projects, Journal of Civil Engineering and
Management 16(1): 33-46. doi:10.3846/jcem.2010.03
Ali Lashgari (1), Mohamad Majid Fouladgar (2), Abdolreza
Yazdani-Chamzini (3), Miroslaw J. Skibniewski (4)
(1) Tarbiat Modares University, Tehran, Iran
(2,3) Fateh Research Group, No. 5 Block. 3 Milad Complex Artesh
Blvd. Aghdasieh Tehran, Iran
(4) Visiting Professor, Department of Management, Bialystok
University of Technology, 16-001 Kleosin, Poland
E-mails: (1) ali.lashgari@gmail.com; (2) manager@fatehidea.com; (3)
a.yazdani@fatehidea.com; (4) mirek@umd.edu (corresponding author)
Received 21 Jun. 2011; accepted 30 Aug. 2011
Ali LASHGARI. Master of Science at the Department of Mining
Engineering, Tarbiat Modares University, Tehran, Iran, member of Fateh
Research Group. He is author of more than 15 journal and conference
papers in the last couple of years. His interests include construction
economics and management, heavy equipment management, feasibility
studies, cost estimation and equipment maintenance and operation.
Mohammad Majid FOULADGAR. Master of Science in the Dept of
Strategic Management, Manager of Fateh Reaserch Group, Tehran-Iran.
Author of 10 research papers. In 2007 he graduated from the Science and
Engineering Faculty at Tarbiat Modares University, Tehran-Iran. His
interests include decision support system, water resource, and
forecasting.
Abdolreza YAZDANI-CHAMZINI. Master of Science in the Dept of
Strategic Management, research assistant of Fateh Reaserch Group,
Tehran-Iran. Author of more than 20 research papers. In 2011 he
graduated from the Science and Engineering Faculty at Tarbiat Modares
University, Tehran-Iran. His research interests include decision making,
forecasting, modeling, and optimization.
Miroslaw J. SKIBNIEWSKI. A. J. Clark Chair Professor of
Construction Engineering and Project Management at the University of
Maryland, College Park, USA. His research interests include construction
equipment operations, construction information technologies and
construction automation. He is past President of the International
Association for Automation and Robotics in Construction, and author or
coauthor of over 200 publications on a wide range of topics in
construction engineering and management.
Table 1. Advantages and disadvantages of shaft sinking methods
Method Usage Advantages Disadvantages
Drilling Applicable for --Applicable in --Low performance
and all sinking small shafts rate
Blasting operations --Low capital
cost --Unsafe
Alimak Excavations in --No need to operation
method excess of 200 m mechanization environment
in length, with --Applicable in
no restriction on all sizes and --Ventilation
raise angles and angles system is
sizes required
Raise Sinking shafts up --Mechanized --Straight line
Boring to 1260 m in method drilling makes
Machine length and --High speed it a relatively
ranging from 0.7 excavation inflexible
m in diameter to --Applicable in method
7.1 m various types
of boring --Expensive on
Shaft Boring deep operations cost per meter
Boring vertical shafts --Very safe as
Machine with a diameter few people --Limited to
of up to 8.5 m involved certain sizes
--No in-hole and lengths
ventilation
system required --Fast drilling
--Accurate requires high
drilling to tonnage chip
accuracies of removal
0.035%
deviation --Requires
reasonably
--No blasting and stable ground
thus no conditions
blasting
related
fractures
--Cost effective,
especially
where time is
of the essence
--The drilling of
long holes has
now become the
norm
Table 2. Fuzzy pair-wise comparison matrix
[C.sub.1] [C.sub.2] [C.sub.3]
[C.sub.1] (1.00 1.00 1.00) (0.50 1.07 2.00) (0.14 0.26 1.25)
[C.sub.2] (0.50 0.93 2.00) (1.00 1.00 1.00) (0.20 0.31 1.33)
[C.sub.3] (0.80 3.87 7.00) (0.75 3.21 5.00) (1.00 1.00 1.00)
[C.sub.4] (0.60 1.32 3.00) (0.50 1.33 3.00) (0.33 0.56 1.67)
[C.sub.5] (0.80 2.77 4.00) (0.50 2.33 4.00) (0.50 0.89 2.00)
[C.sub.6] (0.50 1.36 2.50) (0.40 1.67 3.00) (0.33 0.41 1.67)
[C.sub.7] (0.80 3.47 5.00) (0.75 3.21 4.00) (0.50 0.94 2.50)
[C.sub.8] (0.75 2.65 7.00) (0.60 3.67 5.00) (0.40 0.80 2.50)
[C.sub.4] [C.sub.5] [C.sub.6]
[C.sub.1] (0.33 0.76 1.67) (0.25 0.36 1.25) (0.40 0.70 2.00)
[C.sub.2] (0.33 0.73 2.00) (0.25 0.43 2.00) (0.33 0.60 2.50)
[C.sub.3] 0.60( 1.89 3.00) (0.50 1.12 2.00) (0.60 2.40 3.00)
[C.sub.4] 1.00( 1.00 1.00) (0.40 0.81 1.67) (0.40 1.60 2.50)
[C.sub.5] 0.60( 1.23 2.50) (1.00 1.00 1.00) (0.80 2.30 4.00)
[C.sub.6] 0.40( 0.61 2.50) (0.25 0.43 1.25) (1.00 1.00 1.00)
[C.sub.7] 0.60( 2.63 4.00) (0.60 2.13 3.00) (0.75 2.30 5.00)
[C.sub.8] 0.80( 2.09 4.00) (0.60 2.52 4.00) (0.80 2.70 4.00)
[C.sub.7] [C.sub.8]
[C.sub.1] (0.20 0.29 1.25) (0.14 0.38 1.33)
[C.sub.2] (0.25 0.31 1.33) (0.20 0.27 1.67)
[C.sub.3] (0.40 1.06 2.00) (0.40 1.25 2.50)
[C.sub.4] (0.25 0.38 1.67) (0.25 0.48 1.25)
[C.sub.5] (0.33 0.47 1.67) (0.25 0.40 1.67)
[C.sub.6] (0.20 0.42 1.33) (0.25 0.37 1.25)
[C.sub.7] (1.00 1.00 1.00) (0.67 1.02 1.50)
[C.sub.8] (0.67 0.98 1.49) 1.00( 1.00 1.00)
Table 3. The comparison of fuzzy weights
V(S1>S2)=1 V(S2>S1)=0.997 V(S3>S1)=1
V(S1>S3)=0.687 V(S2>S3)=0.72 V(S3>S2)=1
V(S1>S4)=0.899 V(S2>S4)=0.91 V(S3>S4)=1
V(S1>S5)=0.776 V(S2>S5)=0.801 V(S3>S5)=1
V(S1>S6)=0.938 V(S2>S6)=0.945 V(S3>S6)=1
V(S1>S7)=0.664 V(S2>S7)=0.698 V(S3>S7)=0.997
V(S1>S8)=0.662 V(S2>S8)=0.696 V(S3>S8)=0.975
V(S1>S2)=1 V(S4>S1)=1 V(S5>S1)=1
V(S1>S3)=0.687 V(S4>S2)=1 V(S5>S2)=1
V(S1>S4)=0.899 V(S4>S3)=0.81 V(S5>S3)=0.9
V(S1>S5)=0.776 V(S4>S5)=0.893 V(S5>S4)=1
V(S1>S6)=0.938 V(S4>S6)=1 V(S5>S6)=1
V(S1>S7)=0.664 V(S4>S7)=0.786 V(S5>S7)=0.898
V(S1>S8)=0.662 V(S4>S8)=0.785 V(S5>S8)=0.896
V(S1>S2)=1 V(S6>S1)=1 V(S7>S1)=1 V(S8>S1)=1
V(S1>S3)=0.687 V(S6>S2)=1 V(S7>S2)=1 V(S8>S2)=1
V(S1>S4)=0.899 V(S6>S3)=0.771 V(S7>S3)=1 V(S8>S3)=1
V(S1>S5)=0.776 V(S6>S4)=0.965 V(S7>S4)=1 V(S8>S4)=1
V(S1>S6)=0.938 V(S6>S5)=0.855 V(S7>S5)=1 V(S8>S5)=1
V(S1>S7)=0.664 V(S6>S7)=0.748 V(S7>S6)=1 V(S8>S6)=1
V(S1>S8)=0.662 V(S6>S8)=0.746 V(S7>S8)=0.999 V(S8>S7)=1
Table 4. Final weight obtained of FAHP
Local Global
Weight Weight
V(S1>S2,S3,S4,S5,S6,S7,S8)= 0.662 0.0979
V(S2>S1,S3,S4,S5,S6,S7,S8)= 0.696 0.1030
V(S3>S2,S1,S4,S5,S6,S7,S8)= 0.975 0.1443
V(S4>S2,S3,S1,S5,S6,S7,S8)= 0.785 0.1161
V(S5>S2,S3,S4,S1,S6,S7,S8)= 0.896 0.1326
V(S6>S2,S3,S4,S5,S1,S7,S8)= 0.746 0.1104
V(S7>S2,S3,S4,S5,S6,S1,S8)= 0.999 0.1478
V(S8>S2,S3,S4,S5,S6,S7,S1)= 1.000 0.1480
Table 5. Fuzzy decision matrix
[A.sub.1] [A.sub.2]
[C.sub.1] (1.87 3.11 4.22) (2.33 3.88 5.51)
[C.sub.2] (6.94 7.8 8.56) (6.22 7.21 7.89)
[C.sub.3] (3.12 4.02 4.97) (3.27 3.68 5.06)
[C.sub.4] (0.41 1.04 1.37) (2.17 3.11 3.82)
[C.sub.5] (3.09 4.15 4.85) (5.28 6.12 7.06)
[C.sub.6] (7.22 7.98 8.54) (6.58 7.16 7.44)
[C.sub.7] (2.36 3.93 4.69) (3.16 4.06 4.86)
[C.sub.8] (7.83 8.36 8.76) (6.38 6.88 7.71)
[A.sub.3] [A.sub.4]
[C.sub.1] (4.65 5.96 6.97) (7.20 8.18 8.79)
[C.sub.2] (3.96 4.65 5.69) (3.33 3.82 4.36)
[C.sub.3] (4.86 4.78 5.95) (5.20 5.96 6.87)
[C.sub.4] (5.33 6.13 6.91) (6.88 7.65 8.68)
[C.sub.5] (6.25 6.96 7.66) (6.89 7.21 8.19)
[C.sub.6] (4.14 4.87 5.66) (2.23 3.15 3.77)
[C.sub.7] (4.78 5.22 6.90) (7.19 8.02 8.66)
[C.sub.8] (4.43 5.10 5.64) (0.29 1.21 1.94)
Table 6. Weighed normalized fuzzy matrix
[A.sub.1] [A.sub.2]
[C.sub.1] (0.065 0.081 0.098) (0.047 0.070 0.092)
[C.sub.2] (0.067 0.083 0.098) (0.054 0.072 0.085)
[C.sub.3] (0.000 0.348 0.072) (0.006 0.022 0.075)
[C.sub.4] (0.000 0.009 0.014) (0.025 0.038 0.048)
[C.sub.5] (0.000 0.028 0.046) (0.057 0.079 0.104)
[C.sub.6] (0.088 0.010 0.111) (0.765 0.087 0.092)
[C.sub.7] (0.094 0.112 0.149) (0.090 0.109 0.130)
[C.sub.8] (0.000 0.007 0.016) (0.018 0.033 0.042)
[A.sub.3] [A.sub.4]
[C.sub.1] (0.026 0.040 0.059) (0.000 0.009 0.023)
[C.sub.2] (0.012 0.025 0.044) (0.000 0.009 0.019)
[C.sub.3] (0.067 0.064 0.109) (0.081 0.110 0.145)
[C.sub.4] (0.069 0.081 0.092) (0.091 0.102 0.012)
[C.sub.5] (0.083 0.101 0.119) (0.099 0.108 0.133)
[C.sub.6] (0.034 0.047 0.060) (0.000 0.016 0.027)
[C.sub.7] (0.042 0.081 0.092) (0.000 0.015 0.035)
[C.sub.8] (0.055 0.064 0.076) (0.120 0.138 0.149)
Table 7. Final ranking of alternatives
Alternative Ranking [CL.sub.i] [d.sup.- [d.sup.+
.sub.i] .sub.i]
SBM 4 0.0603 0.484 7.544
RBM 1 0.0671 0.538 7.484
Alimak 3 0.0659 0.528 7.487
Drilling- Blasting 2 0.0666 0.534 7.487
