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Manufacturing Engineering & Management The Proceedings
A Fuzzy Based Decision Support Model for Non-traditional Machining Process Selection Tolga Temuçin1 ‐ Hakan Tozan1 ‐ Jan Valíček2 ‐ Marta Harničárová3 1
Turkish Naval Academy, Department of Industrial Engineering, Tuzla, Istanbul/TURKEY, E‐mail:
[email protected];
[email protected] Mining and Geological Faculty, Institute of Physics, VŠB‐Technical University of Ostrava, Czech Republic, E‐mail:
[email protected] Nanotechnology Centre, VŠB‐Technical University of Ostrava, Czech Republic, E‐mail:
[email protected]
2
3
ARTICLE INFO: Category : Original Scientific Paper Received : 21 February 2011 / Revised: 24 October 2011 / Accepted: 24 October 2011
Keywords: (in causal order)
Abstract:
Multi‐criteria decision making Fuzzy logic TOPSIS
Usage of non-traditional machining (NTM) processes has increased recently since demand for materials like high strength and temperature resistant alloys has expanded proportionally to the improvements in technologically advanced industries such as aeronautics, nuclear reactors, automobiles, and etc. Such developments in the field of material science points out them as indispensable processes due to some benefits such as economic cutting speed, production of complex shapes. In this respect; the selection process for the proper NTM process requires the usage of multi criteria decision making (MCDM) methods due to conflicting criterions such as initial cost of technology, quality of surface finished, environmental impact, time of process, and etc. This study provides distinct systematic approaches both in fuzzy and crisp environments to deal with the selection problem of appropriate NTM process and proposes a decision support model helping decision makers to assess potentials of distinct NTM processes. The required data for decision matrixes is obtained via a questionnaire to specialists as well as deep discussions with experts, and making use of past studies. An application of the proposed model is also performed to show the applicability of the model.
Citation: T. Temuçin et al.: A Fuzzy Based Decision Support Model for Non‐traditional Machining Process Selection, 2nd International Conference Manufacturing Engineering & Management 2012, (2012), p. 170‐175, ISBN 978‐80‐553‐1216‐3
INTRODUCTION methods to be widely used [4‐7]. The purpose in this study is to propose a decision support model which is used to select the best non‐traditional machining process option for cutting operations of a specific material. Criterions for the proposed model and weights that represent the rate of importance for those criterions were identified via questionnaires to specialists, deep discussions with experts, and making use of past studies. The remainder of this study is structured as follows. In the second section research methods are introduced briefly. In the third section firstly the proposed decision support model is introduced and then a case study is performed. Finally, conclusions and further recommendations are highlighted in the last section.
The well‐known philosophical statement denoted by René Descartes “I think, therefore I exist” is perhaps the best comment that discloses the importance of thinking in human life. Humans used this vital ability in decision making process which can be defined as the procedure to find the best alternative among a set of feasible ones [1]. However; complexity of decision making processes changed in the course of time proportional to the improvements in science. Consequently, decision making process wouldn’t be easy as formerly because of the increased number of inputs such as conflicting criterions, importance weights, and etc. According to the concept of “restricted rationalism”, it is claimed that the capacity of human is limited in the solution of complex problems. Thus, evolution of MCDM methods has started. They help to improve the quality of decisions by making the process more explicit, rational, and efficient. These methods are powerful tools which are widely used for complex problems featuring high uncertainty, conflicting objectives, multi interests, and perspectives [2]. MCDM has been one of the fastest growing problem areas during at least the last two decades [3]. Advances in industries like nuclear reactors, automobiles, missiles, and turbines require high strength and temperature resistant alloys. This forced scientists in the field of material science to develop higher strength materials. However in the course of time traditional machining processes wouldn’t be sufficient to produce complex shapes in the strengthened materials such as titanium and stainless steel. Consequently, proportional to the increase in the strength of work material, cutting tool materials also required to be harder which caused the evolution of NTM processes. NTM processes are characterized by the presence of a large number of viable alternatives, uncertainties concerning the process capabilities, and shortage of the experienced planners [4]. In this context, the ill‐structured and multi criteria nature of NTM process selection problems involving many uncertainties concluded MCDM
Research Methods: TOPSIS and Fuzzy TOPSIS As a decision support tool, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is used both in crisp and fuzzy environments. In this section, TOPSIS, fuzzy set theory, and Fuzzy TOPSIS methods are explained briefly.
TOPSIS The first method used in this study is TOPSIS which is developed by Yoon and Hwang in 1980. This method is employed for [8‐11]: Computation procedure is simple; Method’s logic is understandable; and Importance weights are incorporated into the procedure. Ideal Solution is the choice with best performances in every criterion which is indeed impossible to come true. In this case, the choice nearest to the ideal solution must be preferred. The method uses the concepts of Positive Ideal Solution and Negative Ideal Solution to determine the best choice. The positive ideal solution is one maximizing the benefit criterion and minimizing the cost one while for the negative ideal solution the opposite is true [8]. According to this method the best alternative is the one which is
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A Fuzzy Based Decision Support Model for Non-traditional Machining Process Selection
nearest to the positive ideal solution and farthest to the negative ideal solution [12]. The algorithm of this method is described as follows: Step 1: Construct Decision () and Weight (ࢃ) Matrixes: The decision matrix, A consisting three components has to be determined. These components are viable alternatives
T. Temuçin, et al.
Step 7: Calculate Relative Closeness to the Ideal Solution: The relative closeness to the ideal solution, C i* , for each alternative can be calculated with Eq. (7).
Ci*
defined by a1 , a 2 ,…, ai , ak ,…, am ; criterions defined
c1 , c2 ,…, c j ,…, cn
by
alternatives with respect to each criterion defined by a ij i 1, 2,..., m j 1, 2,..., n . In this step the
Step 8: Rank Preference Order: The alternative with the highest C i* is the best choice.
w1 ,
Fuzzy sets theory in MCDM
n
w2 ,…, w j ,…, wn for each criterion satisfying w j 1 j 1
a11 a1n A a m1 amn
Fuzzy logic (FL) notion was first introduced by L.A. Zadeh in 1965. It is a precise logic of imprecision and approximate reasoning [13]. It provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information [14]. “On the contrary to many cases that involves human judgment, crisp sets divide the given universe of discourse into basic two groups; members, which are certainly belonging the set and non‐members, which certainly are not. This delimitation which arises from their mutually exclusive structure enforces the decision maker to set a clear‐cut boundary between the decision variables and alternatives. The basic difference of FL is its capability of data processing using partial set membership functions” [15]. Very often in MCDM problems data is imprecise and fuzzy [3]. On the contrary of crisp set theory, fuzzy set theory is developed to cope with these kinds of indefiniteness.
is also need to be constituted.
Step 2: Convert Criterions to the Same Type: Conversion is done by dividing “1” with the evaluation values in the column of criterion which is desired to be converted. Step 3: Construct Normalized Decision Matrix (ࢄ): The normalized decision matrix can be constructed with Eq. (1) to make the data dimensionless.
xij
aij
Fuzzy TOPSIS
(1)
m
Principal steps of fuzzy TOPSIS method can be described as follows:
a
2 kj
k 1
(7)
where 0 C i* 1 and i 1, 2, 3, ..., m .
and the performances of
weight matrix, W , having the weights defined by
Si Si Si*
Step 1: Construct Decision () and Weight (ࢃ) Matrixes Step 4: Construct Weighted Normalized Decision Matrix (ࢅ): The weighted normalized decision matrix can be constructed with Eq. (2).
Step 2: Convert Criterions to the Same Type
Step 5: Determine Positive and Negative Ideal Solutions:
෩ ) and Weight (ࢃ ෪) Step 3: Construct Fuzzy Decision ( Matrixes: Criterions can be grouped as objective and subjective ones [16]. Error rate defined for decision and weight matrixes helps to obtain a triangular fuzzy number (TFN) for each crisp number. Considering a TFN formed by
If c1 and c2 are benefit and cost criterions respectively,
a triplet
positive and negative ideal solutions can be determined with Eqs. (3) and (4).
and the middle one can be computed with Eq. (8).
y ij xij * w j
j
P
j
P
(2)
max p ,J c ;m inp ,J c m inp ,J c ; max p ,J c ij
1
ij
1
ij
ij
2
(3)
2
(4)
Si
n
(y j 1
ij
S i* and S i
(y
ij
y j ) 2
(5)
(6)
Determine the highest value of a3 ' s in that column. Equalize the highest value of a3 ' s in that *
column to x .
171
3
෩ ): Step 4: Construct Normalized Fuzzy Decision Matrix (ࢄ Normalized fuzzy decision matrix is constructed as follows: For benefit criterions:
n
j 1
y *j ) 2
2
error rate a3 crisp data crisp data * 100
respectively, can be calculated with Eqs. (5) and (6).
Si*
1
error rate a1 = crisp data crisp data * 100 a2 (crisp data) (8)
Step 6: Calculate Separation Measures: The positive and negative ideal separation measures,
a a , a , a , the most extreme values
The normalized value of a a1 , a2 , a3 . * * *
For cost criterions:
x
x
x
T. Temuçin, et al.
A Fuzzy Based Decision Support Model for Non-traditional Machining Process Selection
are benefit criterions where higher values are desired; while OC, IC, TSU, DTE, W, SR, V, N, AP, and R are cost criterions where lower values are always preferred.
Determine the smallest value of a1 ' s in that column. Equalize the smallest value of a1 ' s in that *
column to x .
x* x* x* , , . a3 a2 a1
The normalized value of a
Step 5: Construct Weighted Normalized Fuzzy Decision ෩ ) Matrix (ࢅ Each fuzzy evaluation value, aij i 1, 2,3,..., m j 1, 2,3,..., n , has to be multiplied
with fuzzy weight,
w j 1, 2,3,..., n , j
to obtain
weighted normalized fuzzy decision matrix. Step 6: Determine Fuzzy Positive and Fuzzy Negative Ideal Solution Sets: Fuzzy positive ideal reference point FPIRP, A and fuzzy negative ideal reference point
FNIRP, A are defined with Eqs. (9) and (10).
A v1 , v2 ,..., vn
(9)
A v , v ,..., v
(10)
where v j 1,1,1 and v j 0,0,0 , j 1, 2,..., n.
Fig.1 Structure of the machining process selection decision support model
Step 7: Calculate Separation Measures: Vertex method can be used to compute the distance between fuzzy
m1 , m2 , m3
A case study for machining process selection AISI 309 stainless steel is considered during the evaluation phase of each alternative machining process in terms of criterions. AISI 309 is a heat resistant alloy with oxidation resistance to 19000F. The high chromium and relatively low nickel content of it provide good resistance to high temperature sulphur bearing atmosphere. Some features of this stainless steel are moderate strength at high temperature, ease of fabrication, and good weldability. The chemical composition of AISI 309 is given in Table 1 [20].
are two TFNs then the distance between them is calculated with Eq. (11). 1 2 2 2 n1 m1 n2 m2 n3 m3 3
(11)
Step 8: Calculate Relative Closeness to the Ideal Solution Step 9: Rank Preference Order
Tab.1 Chemical composition of AISI 309 (%) [20]
Proposed Decision Support Model and a Case Study for Machining Process Selection Proposed decision support model Oxy‐fuel, laser, and plasma machining processes are the most common ones [18]. Additionally, water jet and abrasive water jet are the most rapidly improving technological methods of machining materials [19]. Therefore, in this study concern is focused on these five machining process alternatives. Determination of the criteria for the proposed decision support model was done via questionnaires filled in by specialists as well as deep discussions with experts studying in Faculty of Manufacturing Technologies of The Technical University of Kosice, and making use of the past studies [4,6]. Surface finish, cost, safey, toxicity are some criterions that should be used in the comparison of some distinct non‐traditional machining technologies [4,6]. Fig. 1 illustrates the skeleton of the proposed model including criterions and alternatives. Among these criterions, DTE (mm), SR (µm), and CS (m/min) are objective criterions that have absolute numerical values; while OC, IC, TSU, W, V, N, AP, R, S, HH, SO, ESCAS, PC, and UF are subjective criterions which are evaluated on a scale of 1‐10 by specialists and experts of this field. Additionally; S, HH, CS, SO, ESCAS, PC, and UF
MIN MAX
22.00 24.00
12.00 15.00
‐ 0.20
‐ 0.045
‐ 0.030
‐ 2.00
‐ 1.00
Determination of the weights concerning each criterion, error rate (10%), and performances of alternatives in terms of each criterion was done via a questionnaire filled in by specialists as well as deep discussions with experts studying in Faculty of Manufacturing Technologies of The Technical University of Kosice. The developed decision matrix is illustrated in Table 2. Tab.2 Decision matrix
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Silicon
d n ,m
Manganese
numbers [17]. If n n1 , n2 , n3 and m
Sulphur
Phosphorus
n
Carbon
2
Nickel
1
Chromium
A Fuzzy Based Decision Support Model for Non-traditional Machining Process Selection
T. Temuçin, et al.
to the ideal solution for each alternative shown in Table 10.
Machining Process Selection with TOPSIS: Cost criterions in the decision matrix shown in Table 2 are converted to benefit criterions. The reconstructed decision matrix is presented in Table 3. Determined weights shown in Table 2 are normalized and presented in Table 4 within the normalized decision matrix which is constructed via Eq. (1). The weighted normalized decision matrix shown in Table 5 is constructed via Eq. (2). Equations (5), (6) and (7) are used to determine positive ideal solution, negative ideal solution, and relative closeness’s to the ideal solution for each alternative shown in Table 6. The Machining Process Selection with Fuzzy TOPSIS : decision matrix shown in Table 2 and the weight matrix shown in Table 4 (last row of the table) are fuzzified via Eq. (8). The fuzzy decision matrix and fuzzy weight matrix are presented in Table 7. The fuzzy normalized decision matrix and the fuzzy weighted normalized decision matrix are presented in Table 8 and Table 9, respectively. Finally, Equations (11) and (7) are used to determine positive ideal solution, negative ideal solution, and relative closeness’s
CONCLUSION In this study, a comprehensive decision support model is proposed to assist decision makers in the selection of the right machining process for a specific material. A case study is also performed. The required data for the study is obtained via questionnaires given to experts and making use of past studies. The results reached by TOPSIS and fuzzy TOPSIS methods showed that WJM is found to be the best alternative while AWJM is the second and LBM is the third alternatives in the rank order. On the other hand; PAM and Oxy‐Fuel Machining seem to be on the fourth and sometimes on the fifth rank in the sequence depending on the chosen method for application. Further researches can be performed using other fuzzy MCDM methods such as fuzzy ELECTRE, fuzzy PROMETHEE or the ones which take into consideration the influences between alternatives and criterions such as fuzzy Analytic Network Process (FANP).
Tab.3 Reconstructed Decision matrix
S
HH
CS
SO ESCAS PC
UF
LBM
3
4
0.375
9
9
9
2
WJM
5
7
1.500
9
9
9
3
OC
IC
DTE
W
SR
0.143 0.200 0.125
1.667
0.250
8
0.333 0.500 0.333
10
1
TSU
V
N
AP
R
0.333 0.500 0.143 0.143
5
0.333 0.154 0.250 0.125 0.250
1
5
0.250
1
2
0.250 0.167 0.500
WJM
6
8
0.050
9
9
10
7
0.500
0.500
10
PAM
2
2
5
8
9
7
1
0.500 0.500 0.250
2
0.143 0.003 0.167 0.111 0.111 0.167
O‐FUEL 5
5
1
8
7
7
7
0.200 0.250 0.200
0.200
0.111 0.002 0.250 0.143 0.167 0.250
ESCAS
PC
Tab.4 Normalized Decision matrix
S
HH
CS
SO
UF
OC
IC
TSU
DTE
W
SR
V
N
AP
R
‐5
LBM
0.302 0.318 0.070 0.467 0.466 0.474 0.189 0.174 0.158 0.181 1.2*10 0.482 0.970 0.583 0.876 0.234 0.098
AWJM
0.503 0.557 0.282 0.467 0.466 0.474 0.283 0.407 0.395 0.482
0.707
0.643 0.019 0.438 0.219 0.409 0.688
WJM
0.603 0.636 0.009 0.467 0.466 0.527 0.661 0.610 0.790 0.722
0.707
0.482 0.242 0.438 0.292 0.819 0.688
‐5
‐4
PAM
0.201 0.159 0.938 0.415 0.466 0.369 0.094 0.610 0.395 0.361 1.4*10 0.276 4*10 0.292 0.195 0.182 0.115
O‐FUEL
0.503 0.398 0.188 0.415 0.362 0.369 0.661 0.244 0.197 0.289
1*10
0.214 2*10
WEIGHTS 0.065 0.065 0.065 0.065 0.052 0.052 0.065 0.065 0.052 0.045
0.065
0.065 0.065 0.052 0.052 0.058 0.058
DTE
W
SR
‐6
‐4
0.438 0.250 0.273 0.172
Tab.5 Weighted Normalized Decision matrix
S
HH
CS
SO
ESCAS
PC
UF
OC
IC
TSU
‐6
N
AP
R
LBM
0.019 0.021 0.005 0.030 0.024 0.024 0.012 0.011 0.008 0.008
0.031
0.063
0.030 0.045 0.014 0.006
AWJM
0.032 0.036 0.018 0.030 0.024 0.024 0.018 0.026 0.020 0.022 0.046 0.041
0.001
0.023 0.011 0.024 0.040
0.016
0.023 0.015 0.048 0.040
WJM
0.039 0.041 6*10
PAM
‐4
10
V
0.030 0.024 0.027 0.043 0.039 0.041 0.033 0.046 0.031
0.013 0.010 0.061 0.027 0.024 0.019 0.006 0.039 0.020 0.016
O‐FUEL 0.032 0.026 0.012 0.027 0.019 0.019 0.043 0.016 0.010 0.013
‐6
‐5
10
0.018 2.6*10 0.015 0.010 0.011 0.007
0
0.014 1.6*10 0.023 0.013 0.016 0.010
‐5
Tab.6 Positive and Negative Ideal Solutions, Relative Closeness’s to the Ideal Solution and Preference Orders Positive Ideal Solution
Negative Ideal Solution
Relative Closeness’s to the Ideal Solution
Preference Orders
LBM
0.109
0.077
0.415
3
AWJM
0.093
0.080
0.461
2
WJM
0.083
0.104
0.556
1
PAM
0.119
0.068
0.364
4
OXY‐FUEL
0.120
0.047
0.283
5
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T. Temuçin, et al.
A Fuzzy Based Decision Support Model for Non-traditional Machining Process Selection
Tab.7 Fuzzy Decision Matrix and Fuzzy Weight Matrix CRI. /ALT. S H CS SO ESCAS PC UF OC IC TSU DTE W SR V N AP R
LBM
AWJM
WJM
PAM
OXY‐FUEL
WEIGHTS
(2.700;3.000;3.300) (3.600;4.000;4.400) (0.338;0.375;0.413) (8.100;9.000;9.900) (8.100;9.000;9.900) (8.100;9.000;9.900) (1.800;2.000;2.200) (0.129;0.143;0.157) (0.180;0.200;0.220) (0.113;0.125;0.138) (1.500;1.667;1.833) (0.225;0.250;0.275) (7.200;8.000;8.800) (0.299;0.333;0.367) (0.450;0.500;0.550) (0.129;0.143;0.157) (0.129;0.143;0.157)
(4.500;5.000;5.500) (6.300;7.000;7.700) (1.350;1.500;1.650) (8.100;9.000;9.900) (8.100;9.000;9.900) (8.100;9.000;9.900) (2.700;3.000;3.300) (0.299;0.333;0.367) (0.450;0.500;0.550) (0.299;0.333;0.367) (9*104 ; 105;11*104) (0,299;0.333;0.367) (0.138;0.154;0.169) (0.225;0.250;0.275) (0.113;0.125;0.138) (0.225;0.250;0.275) (0.900;1.000;1.100)
(5.400;6.000;6.600) (7.200;8.000;8.800) (0.045;0.050;0.055) (8.100;9.000;9.900) (8.100;9.000;9.900) (9.000;10.00;11.00) (6.300;7.000;7.700) (0.450;0.500;0.550) (0.900;1.000;1.100) (0.450;0.500;0.550) (9*104; 105 ;11*104) (0.225;0.250;0.275) (1.800;2.000;2.200) (0.225;0.250;0.275) (0.150;0.167;0.183) (0.450;0.500;0.550) (0.900;1.000;1.100)
(1.800;2.000;2.200) (1.800;2.000;2.200) (4.500;5.000;5.500) (7.200;8.000;8.800) (8.100;9.000;9.900) (6.300;7.000;7.700) (0.900;1.000;1.100) (0.450;0.500;0.550) (0.450;0.500;0.550) (0.225;0.250;0.275) (1.800;2.000;2.200) (0.129;0.143;0.157) (0.003;0.003;0.003) (0.150;0.167;0.183) (0.099;0.111;0.122) (0.099;0.111;0.122) (0.150;0.167;0.183)
(4.500;5.000;5.500) (4.500;5.000;5.500) (0.900;1.000;1.100) (7.200;8.000;8.800) (6.300;7.000;7.700) (6.300;7.000;7.700) (6.300;7.000;7.700) (0.180;0.200;0.220) (0.225;0.250;0.275) (0.180;0.200;0.220) (0.180;0.200;0.220) (0.099;0.111;0.122) (0.002;0.002;0.002) (0.225;0.250;0.275) (0.129;0.143;0.157) (0.150;0.167;0.183) (0.225;0.250;0.275)
(0.058;0.065;0.071) (0.058;0.065;0.071) (0.058;0.065;0.071) (0.058;0.065;0.071) (0.046;0.052;0.057) (0.046;0.052;0.057) (0.058;0.065;0.071) (0.058;0.065;0.071) (0.046;0.052;0.057) (0.041;0.045;0.050) (0.058;0.065;0.071) (0.058;0.065;0.071) (0.058;0.065;0.071) (0.046;0.052;0.057) (0.046;0.052;0.057) (0.052;0.058;0.064) (0.052;0.058;0.064)
Tab.8 Fuzzy Normalized Decision Matrix CRI./ALT. S HH CS SO ESCAS PC UF OC IC TSU DTE W SR V N AP R
LBM (0.409;0.455;0.500) (0.409;0.455;0.500) (0.061;0.068;0.075) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.736;0.818;0.900) (0.234;0.260;0.286) (0.234;0.260;0.286) (0.164;0.182;0.200) (0.205;0.227;0.250) (14*10‐6; 15*10‐6;17*10‐6) (0.614;0.682;0.750) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.234;0.260;0.286) (0.117;0.130;0.143)
AWJM (0.682;0.758;0.833) (0.716;0.795;0.875) (0.245;0.273;0.300) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.736;0.818;0.900) (0.351;0.390;0.429) (0.545;0.606;0.667) (0.409;0.455;0.500) (0.545;0.606;0.667) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.016;0.017;0.019) (0.614;0.682;0.750) (0.205;0.227;0.250) (0.409;0.455;0.500) (0.818;0.909;1.000)
WJM (0.818;0.909;1.000) (0.818;0.909;1.000) (0.008;0.009;0.010) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.818;0.909;1.000) (0.614;0.682;0.750) (0.205;0.227;0.250) (0.614;0.682;0.750) (0.273;0.303;0.333) (0.818;0.909;1.000) (0.818;0.909;1.000)
PAM (0.273;0.303;0.333) (0.205;0.227;0.250) (0.818;0.909;1.000) (0.727;0.808;0.889) (0.818;0.909;1.000) (0.573;0.636;0.700) (0.117;0.130;0.143) (0.818;0.909;1.000) (0.409;0.455;0.500) (0.409;0.455;0.500) (2*10‐5; 2*10‐5; 2*10‐5) (0.351;0.390;0.429) (0.0003;0.0004;0.0004) (0.409;0.455;0.500) (0.182;0.202;0.222) (0.182;0.202;0.222) (0.136;0.152;0.167)
OXY‐FUEL (0.682;0.758;0.833) (0.511;0.568;0.625) (0.164;0.182;0.200) (0.727;0.808;0.889) (0.636;0.707;0.778) (0.573;0.636;0.700) (0.818;0.909;1.000) (0.327;0.364;0.400) (0.205;0.227;0.250) (0.327;0.364;0.400) (16*10‐7;18*10‐7;2*10‐6) (0.273;0.303;0.333) (0.0002;0.0002;0.0003) (0.614;0.682;0.750) (0.234;0.260;0.286) (0.273;0.303;0.333) (0.205;0.227;0.250)
Tab.9 Fuzzy Weighted Normalized Decision Matrix CRI./ALT. LBM S (0.024;0.029;0.035) HH (0.024;0.029;0.035) CS (0.004;0.004;0.005) SO (0.048;0.059;0.071) ESCAS (0.038;0.047;0.057) PC (0.034;0.042;0.051) UF (0.014;0.017;0.020) OC (0.014;0.017;0.020) IC (0.008;0.009;0.011) TSU (0.008;0.010;0.012) DTE (8*10‐6;1*10‐6;12*10‐7) W (0.036;0.044;0.053) SR (0.048;0.059;0.071) V (0.038;0.047;0.057) N (0.038;0.047;0.057) AP (0.012;0.015;0.018) R (0.006;0.008;0.009)
AWJM WJM (0.040;0.049;0.059) (0.048;0.059;0.071) (0.042;0.051;0.062) (0.048;0.059;0.071) (0.014;0.018;0.021) (0.00048;0.00059;0.00071) (0.048;0.059;0.071) (0.048;0.059;0.071) (0.038;0.047;0.057) (0.038;0.047;0.057) (0.034;0.042;0.051) (0.038;0.047;0.057) (0.020;0.025;0.030) (0.048;0.059;0.071) (0.032;0.039;0.047) (0.048;0.059;0.071) (0.019;0.023;0.028) (0.038;0.047;0.057) (0.033;0.041;0.050) (0.022;0.027;0.033) (0.048;0.059;0.071) (0.048;0.059;0.071) (0.048;0.059;0.071) (0.036;0.044;0.053) (0.0009;0.0011;0.0014) (0.012;0.015;0.018) (0.029;0.035;0.043) (0.029;0.035;0.043) (0.009;0.012;0.014) (0.013;0.016;0.019) (0.021;0.026;0.032) (0.043;0.053;0.064) (0.043;0.053;0.064) (0.043;0.053;0.064)
PAM (0.016;0.020;0.024) (0.012;0.015;0.018) (0.048;0.059;0.071) (0.042;0.052;0.063) (0.038;0.047;0.057) (0.027;0.033;0.040) (0.007;0.008;0.010) (0.048;0.059;0.071) (0.019;0.023;0.028) (0.017;0.021;0.025) (1*10‐5;12*10‐7;14*10‐7) (0.020;0.025;0.030) (2*10‐5;24*10‐6;3*10‐5) (0.019;0.023;0.028) (0.008;0.010;0.013) (0.010;0.012;0.014) (0.007;0.009;0.011)
OXY‐FUEL (0.040;0.049;0.059) (0.030;0.037;0.044) (0.010;0.012;0.014) (0.042;0.052;0.063) (0.030;0.036;0.044) (0.027;0.033;0.040) (0.048;0.059;0.071) (0.019;0.023;0.028) (0.010;0.012;0.014) (0.013;0.016;0.020) (1*10‐6;12*10‐7;14*10‐7) (0.016;0.020;0.024) (12*10‐6;15*10‐6;18*10‐6) (0.029;0.035;0.043) (0.011;0.013;0.016) (0.014;0.018;0.021) (0.011;0.013;0.016)
Tab.10 Positive and Negative Ideal Solutions, Relative Closeness’s to the Ideal Solution and Preference Orders Positive Ideal Solution
Negative Ideal Solution
Relative Closeness’s to the Ideal Solution
Preference Orders
LBM
16.514
0.493
0.029
3
AWJM
16.371
0.638
0.037
2
WJM
16.246
0.764
0.045
1
PAM
16.582
0.424
0.025
5
OXY‐FUEL
16.569
0.436
0.026
4
Acknowledgements
funded by Structural Funds of the European Union and state budget of the Czech Republic and the project RMTVC No. CZ.1.05/2.1.00/01.0040
This research has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme 'Research and Development for Innovations'
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