CC BY
44
DOI 10.18551/rjoas.2019-03.42
NEW METHOD OF CRITERIA WEIGHTING FOR SUPPLIER SELECTION
Ristono Agus
UPN Veteran Yogyakarta, Indonesia E-mail: agus.ristono@upnyk.ac.id
ABSTRACT
In the future, researchers focusing on supplier selection are likely to use a combination of multi-criteria decision-making (MCDM) methods. The analytical hierarchy process (AHP) is often used in such combinations. The function of the AHP method in MCDM is criteria weighting. When there are a relatively large number of participants involved in an evaluation judgment, it is difficult to obtain consistent opinions. In such cases, the AHP is a useful method to obtain consistent opinions over time by repeatedly conducting pairwise comparison matrices. This study proposes a new methodology to resolve such problems. In the proposed method, the decision maker assesses the level of contribution of each criterion to the selection of suppliers. Using the proposed method, comparing the contributions of these criteria to supplier selection will always produce a consistent value. The advantage of the proposed method is that decision makers do not have to assess the degree of importance of each individual criterion. So, if there are n criteria, the decision maker has to access as much as n times. The results of this study indicate that the proposed method consistently produces a solution, without the need for repeated human judgements and without consideration of the number of criteria.
KEY WORDS
Analytic hierarchy process, comparison matrix, criteria, consistency ratio, AHP weighting.
In the selection of suppliers, companies generally have various criteria to consider. Usually, if one criterion is considered more important than other criteria, then this criterion is given greater weight. Problems arise when a supplier has to be selected based on a number of criteria [1]. In the future, researchers in the supplier selection field are likely to use a combination of methods of MCDM [2, 3]. One MCDM method that is often used in such combinations is the analytic hierarchy process (AHP) [4, 5]. Therefore, the success of the combination method is determined by AHP weighting [6].
AHP weighting can be considered valid (i.e., to have produced a consistent solution) if the consistency ratio is less than 0.1 [7, 8]. The validity of AHP weighting is determined based on the consistency of the resulting pairwise comparison matrix [9]. In such cases, the ranking or weighting of criteria is based on the judgement of the decision maker [10]. The decision maker have to do (repeat) until the pairwise comparison matrix is consistent Therefore, the role of human judgment of decision makers is very important in supplier selection using the AHP method.
As the number of criteria increases, human judgments become increasingly sensitive and may become inconsistent [7]. It may be difficult to obtain consistent results when the size of the matrix is relatively large [11] or when there are more then seven criteria [12]. According to the literature, the optimum number of criteria is seven or fewer [13]. Thus, most supplier selection studies use the hierarchical method when there are multiple criteria [14]. However, the latter cannot guarantee consistent results, mainly due to inconsistency in human judgment. To address the aforementioned issues, this study proposes a new method to aid human judgment and ensure consistent decision making. The basic idea underpinning the proposed method is that decision makers are not required to draw comparisons between criteria. As noted above, the higher the number of criteria, the greater the risk of confusion among decision makers.
MATERIALS AND METHODS OF RESEARCH
Much research has focused on overcoming inconsistencies of pairwise comparison matrices using the AHP method. The simplest and most widely performed method involves the use of hierarchical criteria (i.e., AHP criteria weighting), in which criteria are separated into different groups [15-17]. The weakness of the hierarchical method is the extended calculation time the absence of any guarantee of consistency if the number of major criteria or sub-criteria exceeds seven.
Besides the hierarchical method, researchers have described other methods to overcome inconsistencies of pairwise comparison matrices [18, 19]. The crisp value of each criterion is included in the pairwise comparison matrix using Eq (1). In Eq (1), if the Lp and Lq values are equal, then the element aj in the pairwise comparison matrix is 1. Lq and Lp represent the value of the importance of criteria q and p. If the value of Lp is greater than that of Lq, then the element aj in the pairwise comparison matrix is Lp - Lq + 1. If the value of Lp is smaller than that that of Lq, fhen the element a¡j in the pairwise comparison matrix is 1/(Lp -Lq + 1). Inconsistency often occurs because the interval of the comparison value between the criteria is very large. Using equation (Lp - Lq + 1), the pairwise comparison matrix will be consistent.
1, if Lp = Lq I Lp-Lq +1, if(Lp -Lq) > 0 ' -, if (Lp-Lq) <0
(1)
Lv-L„ +1
Table 1 shows a comparison of the scales of Saaty [8] and Li et al. [19]. Replacing the crisp values (1, 3, 5, 7, and 9) with decimal numbers and the reverse comparison, in which Aj=1/ nj with ry=1 -rj. is expected to minimize inconsistencies. Disadvantages of the modified scale of Li et al. [19] is not produces accurate results.
Table 1 - Importance scale of Saaty [8] and Li et al. [19]
Scale [8] Scale [19] Value
1 0.5 Equally important
3 0.6 Moderately more
important
5 0.7 Strongly more
important
7 0.8 Very strongly more
important
9 0.9 Extremely more
important
1/3; 1/5; 0.1; Reverse
1/7; 1/9 0.2; comparison
0.3;
0.4
Definition
Criterion i and criterion j are equally important Criterion i is moderately more important than criterion j
Criterion i is much more important than than criterion j
Criterion i is very much more important than than criterion j
Criterion i is extremely more important than criterion j
If criterion a, is compared with criteria aj and a judgment matrix rj is obtained, then the judgement matrix of aj and ai is as follows: • rji=1/rj ([8])
_• j=1 - rs ([19])_
The consistency ratio in AHP is obtained first by calculating the eigenvalue maximum [20]. The consistency index is obtained by nine stages in Figure 1. Figure 1 shows the stages of the AHP. More details on the process can be found elsewhere [8, 21].
In Figure 1, a¡j denote the importance assigned to different criteria (i and j), n is the number of criteria, W-, indicates the relative weight of criterion i. CI is the consistency index, and CR is the consistency ratio. The CR is a probability measure that the matrix is filled randomly. Thus, the CR value is the ratio between the current matrix and question and answer matrix [22]. For example, a12 denotes the importance of criteria C1 as compared with that of C2. This matrix aims to determine the relative importance levels of supplier selection criteria. Matrix N contains normalized aj values. The data from this matrix is as an input into a relative weight matrix. The content of matrix Wj is the result of calculating the relative
aij =
weights of each criterion. In terms of the value W, the greater the value assigned to the weight of a criterion, the more this criterion is prioritized by the decision maker. The evaluation matrix Eg and sum matrix V are the first two stages in obtaining the CI. To generate an evaluation matrix, Eg, each element in matrix ag is divided by the weight of the criteria w. To obtain the sum matrix V, the elements of the evaluation matrix Eg that are in the same row are summed. The consistency index can be obtained after calculating the eigen vector. The eigen vector is the weight of each element used for prioritizing elements at the lowest hierarchy level. After determining the consistency of the index, the results are compared using a random consistency index for each n criterion. To ensure the validity of decision making, the consistency ratio should be < 10% [22].
Pair-wise comparison matrix (a,j)
1 ai2 ■■■ ain
n aTT ■■■ aT«
h:
Nj =
Eigenvalue matrix (/,)
W
W
Normalized matrix (Nj)
V =
Sum matrix (V})
Vi
a11 a12 a1n
Z aü n Z ai2 z a. / J in
a21 a22 a2n
Z aü n Z ai2 n Za in >1
i=1 i=1 i=1 W
W =
an1 an2 a nn
n Z aü i=1 n Z ai2 i=1 n Za in i=1
an + a12 + . . + a1n
W1 W2 Wn
a21 + a22 + . . + a2n
W1 W2 Wn
an1 + an2 + . . + a nn
W1 W2 Wn
Relative weight matrix ( W)
- +... + -
Êaii Za
+... +
Z<
+... +
Y,aii Za
Z<
Evaluation matrix (Ej
Ej =
an a12 a1n
W1 W2 . . Wn
a21 a22 a2n
W1 W2 . . Wn
an1 an2 a nn
W1 W2 . . Wn
H
Eigenvalue max (/max)
^max
V V
VI , V 2
+ + ... + -
W w2
W
4
Consistency index (CI)
CI = ^max—n = numberof criteria
Consistency ratio (CR)
CI
CR = — ; RI = randomindex RI
Figure 1 - The stages of the AHP
If a decision maker considers criterion C2 to be highly important than criterion C1 and criterion C3 to be highly important than criterion C2 then criterion C3 to be highly important than criterion C1. If C2 > C1 and C3 > C2, then C3 > C1. Thus, C3 < C1 is not possible. As shown in Appendix 1, using only three criteria, AHP will always yield an inconsistent value if pair-wise comparison matrix is inconsistent in the comparison of supplier selection criteria. Essentially, the higher the number of criteria, the higher the inconsistency. This problem can be resolved by assigning importance value to the determination of criteria. If each criterion has a fixed value, then the value assigned to the supplier selection criteria will always be fixed and consistent. If these conditions are met, then the results will always be consistent, regardless of the number of criteria.
In the proposed method, decision makers are asked to assess the level of importance (contribution) of particular criteria to supplier selection. Based on the assessment of the decision maker, supplier selection is adjusted to the level of the contribution of each criterion.
a
a
a
ii
12
+
aij =
i2
i=1
i=1
i=1
a
a
a
n2
nn
a
a
a
21
22
+
i2
i=1
i=1
i=1
a
a
a
n2
nn
+
i2
i=1
i=1
i=1
2
2
n
Table 2 shows the contribution values and importance assigned to various criteria in supplier selection. Comparison of the criterion values of the decision maker results in a pairwise comparison matrix. The difference between the original AHP and the proposed method is illustrated in Figure 2. The stages of the generation of the pairwise comparison matrix using the proposed method are shown in Figure 3.
Table 2 - Values and importance assigned to various supplier selection criteria
Contribution level Definition
1 Weakly or slightly important
2 Important
3 Moderately important
4 Very moderately important
5 Highly important
6 Very highly important
7 Very very highly important
8 Extremely important
9 Very extremely important
: Criteria Ci is compared with criteria Cj which it' s value is aij, where aij e {1, 2, 3, . . . , 9 }. If criteria Cjis compared with criteria Ci so it's value is 1/aij.
= 1, if Ci is equal importance with Cj = 2, if Ci is more slight or weak importance than Cj = 3, if Ci is more importance than Cj = 4, if Ci is more moderate importance than Cj = 5, if Ci is more strong importance than Cj = 6, if Ci is more strong plus importance than Cj = 7, if Ci is more very strong importance than Cj = 8, if Ci is more very very strong importance than Cj = 9, if Ci is more extreme importance than Cj
Wi : weight of criteria Ci.
: Criteria Ci has contribution level Xi if it' s used in the supplier selection, where Xi e {1, 2, 3, . . . , 9 } in scale 9 or Xi e {0, 1, 2, 3, . . . , 1 00 } in scale 100
In the scale 9, so
Xi = 1, if Ci is slight important to supplier selection Xi = 2, if Ci is important to supplier selection Xi = 3, if Ci is moderate important to supplier selection Xi = 4, if Ci is moderate plus important to supplier selection Xi = 5, if Ci is strong important to supplier selection Xi = 6, if Ci is strong plus important to supplier selection Xi = 7, if Ci is very strong important to supplier selection Xi = 8, if Ci is veiy very strong important to supplier selection Xi = 9, if Ci is extreme important to supplier selection
In the scale 100, so
Xi = 0, if Ci has no contribution level to supplier selection
Xi = 100, if Ci has the highest contribution level to supplier selection
: contribution level of criteria Ci to supplier selection
Original AHP Proposed method
Figure 2 - Basic idea of the proposed method
Stagel:
Decision maker provides an assessment of the criteria.
Purpose of this stage: level of contribution of each criteria
Usage for the next stage: as a basis for comparative value scoring among criteria
Stage2:
The value of comparison between criteria
Purpose of this stage: Importance of each criteria when compared to other criteria
Usage for the next stage: as input for pairwise comparison matrix
Stage 3: Pairwise comparison matrix
Purpose of this stage: To make it easy to use in AHP
Usage for the next stage: as an input for the calculation of the weight of criteria
Figure 3 - Stages of the proposed method
Ci
xi
The difference between the proposed model and the original AHP lies in the process of generating the pairwise comparison matrix. In the original AHP, decision makers are required to compare one criterion against another. The results of each comparison are then included in the pairwise comparison matrix. This matrix is not necessarily consistent. In the proposed model, the decision maker assigns the level of importance (contribution value) of each criterion in supplier selection. Based on these values, a matrix pairwise comparison is generated.
DISCUSSION OF RESULTS
The performance of the proposed method. We assumed that there were nine criteria (C1, C2, C3, ., C9), where the level of contribution was x1, x2, x3, ., x9 and Xj e {1, 2, 3, ., 9 Based on a comparison of the criteria, C1 and C2 were assigned a value of 1 and 2, respectively. Criteria C2 and C1 were assigned a value of 2. The values of the comparisons among the other criteria were obtained using the same calculation. Furthermore, we calculated the weight of each criterion and its consistency ratio. The results of the weight calculation and the consistency ratio, as well as the pairwise comparison matrix, are presented in Table 3.
Table 3 - Performance of the proposed method using successive levels
Criteria Ci C2 C3 C4 C5 C6 C7 C8 Cg
C 1.00 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11
C2 2.00 1.00 0.67 0.50 0.40 0.33 0.29 0.25 0.22
C3 3.00 1.50 1.00 0.75 0.60 0.50 0.43 0.38 0.33
C4 4.00 2.00 1.33 1.00 0.80 0.67 0.57 0.50 0.44
C5 5.00 2.50 1.67 1.25 1.00 0.83 0.71 0.63 0.56
C6 6.00 3.00 2.00 1.50 1.20 1.00 0.86 0.75 0.67
C7 7.00 3.50 2.33 1.75 1.40 1.17 1.00 0.88 0.78
C8 8.00 4.00 2.67 2.00 1.60 1.33 1.14 1.00 0.89
C9 9.00 3.50 2.33 1.75 1.80 1.50 1.29 1.13 1.00
Contr. level 1 2 3 4 5 6 7 8 9
Weight 0.023 0.045 0.068 0.090 0.113 0.136 0.158 0.181 0.186
CR 0.008 (consistent)
If x1 < x2 < x3 < x4 < x5 < x6 < x7 < x8 < x9, then x1 = 1, x2 = 2, x3 = 3, ., and x9 = 9. If criteria C9 are compared with the other criterion, the consistency ratio will always resultin a value greater than 1, as shown in Table 3 (row C9). Thus, the pairwise comparison matrix of C9 criteria with other criteria gives consistent results. As the decision maker assigned C1 the highest value (i.e., weakly or slightly important), the paired comparison value of criteria C1 versus that of other criteria will always be < 1. Therefore, C9 will have the greatest weight. Likewise, the reverse is true for criteria C1. Thus, if criteria C1 is compared with the other criteria, it will always result in a value less than 1 and never more than 1, as shown in Table 3 in the second row (row C1). Thus, the pairwise comparison matrix of criteria C1 with other criteria gives consistent results. As the decision maker assigned C1 the lowest value (i.e., lowest importance), the paired comparison value of criteria C9 versus that of other criteria will always be > 1. As a result, C1 will have the lowest weight. In terms of the other criteria, their weights will be in accordance with the order of the contribution value. Thus, it is logical that the weight of each criterion is determined by its contribution to supplier selection. Therefore, if each criterion makes the same contribution to supplier selection, it will have the same weight. Although all the supplier selection criteria have same value, the weights of all the criteria have the same value, as depicted in Table 4.
The contribution level of a particular criterion can be further evaluated by assigning a value of 0 to 100 (integer number), where 0 indicates no contribution to supplier selection, and 100 indicates the highest contribution (importance) of a criterion to supplier selection. The advantage of using contribution levels between 0 and 100 is the broad scope it gives decision makers to input the value of contributions of various criteria to supplier selection. In addition, if we use only an integer value range between 1 and 9, and there are more than
nine criteria, then some criteria will have the same contribution level. However, if we use an integer value range between 0 and 100, this will minimize the chances of multiple criteria being assigned the same contribution value (level of importance).
Table 4 - Performance of the proposed method using same contribution value
Criteria_Ci_C_C3_C_C_C_C7_C_Cg
1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4
Contribution level 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9
Weight 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111
CR 0.000 (consistent)
Testing the proposed method using data from the literature. The proposed method was tested using previous data in which there were nine selection criteria [23, 24]. The pairwise comparison matrix in one study was inconsistent [23], whereas that in the other was consistent [24]. Tables 5 and 6 show the results of assessing the different contribution levels of decision makers for different criteria. As apparent in these tables, the proposed method yielded a consistent pairwise comparison matrix in the presence of more than nine criteria. Thus, the resulting supplier selection criteria will always be valid, regardless of the number of criteria. The proposed method was also capable of making inconsistent pairwise comparison matrices consistent and providing a definitive solution to supplier selection.
Table 5 - Example 1
n/n [23] Proposed method
Criteria Weight Contribution level (1-9) Weight Contribution level (0-100) Weight
Ci 0.291 9 0.181 60 0.327
C2 0.229 8 0.161 25 0.136
C3 0.114 7 0.141 22 0.120
C4 0.114 7 0.141 20 0.109
C5 0.036 3 0.060 12 0.065
Ce 0.037 3 0.060 8 0.044
C7 0.036 3 0.060 7 0.038
C8 0.068 5 0.100 15 0.082
Cg 0.052 4 0.076 10 0.052
C10 0.023 1 0.020 5 0.027
CR 0.184 (inconsistent) 0.022 (consistent) 0.003 (consistent)
Table 6 - Example 2
[24] Proposed method
Criteria Weight Contribution level (1-9) Weight Contribution level (0-100) Weight
Ci 0.165 9 0.176 60 0.191
C2 0.135 8 0.156 45 0.144
C3 0.111 6 0.117 32 0.102
C4 0.092 4 0.078 28 0.089
C5 0.080 4 0.078 25 0.080
Ce 0.078 4 0.078 24 0.077
C7 0.052 2 0.039 15 0.048
C8 0.059 2 0.039 18 0.057
Cg 0.076 4 0.063 23 0.065
C10 0.104 6 0.117 30 0.096
C11 0.048 3 0.059 16 0.051
CR 0.008 (consistent) 0.012 (consistent) 0.003 (consistent)
Comparison of the proposed method with that of Li et al. [19]. We compared the performance of the proposed method with that of Li et al. [19] using data from previous studies [23, [24]. In the pairwise comparison matrix of Hruska et al. [23], the method by Li et al. [19] does not accommodate numbers other than 1, 3, 5, 7, and 9. Thus, number 8 is placed between numbers 7 and 9, number 6 is placed between numbers 5 and 7, and number 4 is placed between numbers 3 and 5. Figure 4 show the results of the pairwise comparison conversion using the method of Hruska et al. [23] and that of Li et al. [9]. As shown in Figure 4, the pairwise comparison matrix based on the method of Li et al. [19] is inconsistent. Although the method used by Li et al. [19] can minimize inconsistencies, when there are more than seven criteria. This is one of the weaknesses of the method [19].
Hruska et al. [23] C, C,
C ? C a
C
C6 C 7 C8 C9 C 1t
Li et al. [19] C1
C2 C3 C 4 C 5 C6 C 7 C 8 C 9 C11
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
Weight CR
1.00 3.00 0.33 1.00 0.20 0.20 0.20 0.20 0.14 0.14 0.13 0.13 0.13 0.13 0.17 0.17 0.17 0.13 0.20 0.20
5.00 5.00 7.00 5.00 5.00 7.00
1.00 0.33 5.00 3.00 1.00 4.00
0.20 0.25 1.00
0.17 0.33 0.25 0.14 0.14 5.00 0.20 0.20 7.00 0.14 0.14 3.00 0.25 0.20 0.33
8.00 8.00 8.00 8.00 6.00 7.00 3.00 7.00 4.00 0.20 1.00 5.00 0.20 1.00 5.00 3.00 3.00 3.00 0.33 0.33
6.00 5.00 5.00 6.00 5.00 5.00 5.00 3.00 4.00 5.00 1.00 5.00
0.14 0.33 3.00
0.20 0.33 3.00
0.33 0.33 3.00 1.00 1.00 3.00 1.00 1.00 3.00
0.33 0.33 1.00
0.29 0.23 0.1845
0.11 0.11 0.04 (inconsistent)
0.04 0.04 0.07 0.05 0.02
Weight CR
0.50 0.60 0.70 0.70 0.40 0.50 0.70 0.70 0.30 0.30 0.50 0.40 0.30 0.30 0.60 0.50 0.20 0.20 0.30 0.35
0.15 0.15 0.25 0.40
0.15 0.15 0.20 0.20
0.25 0.25 0.30 0.30
0.25 0.15 0.20 0.20 0.30 0.30 0.35 0.30
0.80 0.80 0.70 0.65 0.50 0.35 0.70 0.80 0.60 0.40
0.85 0.85
0.85 0.85
0.75 0.80 0.60 0.80
0.65 0.30 0.50 0.70 0.30 0.50 0.70 0.60 0.60 0.60 0.40 0.40
0.75 0.70 0.70 0.75 0.70 0.70 0.70 0.60 0.65 0.70 0.50 0.70 0.20 0.40 0.60 0.30 0.40 0.60 0.40 0.40 0.60 0.50 0.50 0.60 0.50 0.50 0.60 0.40 0.40 0.50
0.15 0.14 0.11 0.11 0.07 0.4056 (inconsistent)
0.07 0.07 0.09 0.08 0.08
Figure 4 - Li et al. [19] for example 1
Example 2 provides additional evidence illustrating the disadvantages of the method of Li et al. [19]. Thus, the matrix data of by Polat et al. [24] must be rounded first, and then the matrix must be converted using the Li et al.'s method [19]. Figure 5 show the results of the pairwise comparison conversion using the method of Polat et al. [24] and that of Li et al. [9]. As can be seen in this figure, in a matrix that contains more than seven criteria, Li et al.'s method [19] produces an inconsistent solution. Table 7 provides a comparison of the results obtained using the method of Li et al. [19] with those obtained using the proposed method. Based on this table, it can be seen that the proposed method is better than that of Li et al. [19] because it always produces a consistent value. The proposed method is capable of giving a CI value close to zero and a value of zero if the matrix is perfectly consistent [25].
Polat [24]
Li et al [19] C,
C, 1.00 1.26 1.44 1.82 2.29 2.15 3.30 2.88 1.65 1.59 3.17
C 2 0.79 1.00 1.44 1.38 1.65 1.59 2.62 2.52 1.44 1.26 2.71
C3 0.69 0.69 1.00 1.31 1.36 1.55 2.15 1.82 1.26 1.10 2.29
C4 0.55 0.72 0.76 1.00 1.18 1.14 1.74 1.59 1.05 0.94 1.82
C5 0.44 0.61 0.74 0.85 1.00 1.10 1.58 1.31 0.87 0.79 1.71
C6 0. 47 0.63 0.65 0.88 0.91 1.00 1.65 1.26 0.85 0.69 1.70
C7 0.30 0.38 0.47 0.57 0.63 0.61 1.00 0.94 0.72 0.55 0.91
C 8 0.35 0.40 0.55 0.63 0.76 0.79 1.06 1.00 0.63 0.60 1.31
C9 0.35 0.38 0.47 0.57 1.15 1.18 1.39 1.59 1.00 0.87 1.96
C10 0.63 0.79 0.91 1.06 1.27 1.45 1.82 1.67 1.15 1.00 2.52
C11 0.32 0.37 0.44 0.55 0.58 0.59 1.10 0.76 0.51 0.40 1.00
Weight 0.165 0.135 0.111 0.092 0.080 0.078 0.052 0.059 0.076 0.104 0.048
CR 0.0088 (consistent)
C1 0.50 0.50 0.50 0.55 0.55 0.55 0.60 0.60 0.55 0.55 0.60
C2 0.50 0.50 0.50 0.50 0.55 0.55 0.60 0.60 0.50 0.50 0.60
C3 0.50 0.50 0.50 0.50 0.50 0.55 0.55 0.55 0.50 0.50 0.55
C4 0.45 0.50 0.50 0.50 0.50 0.50 0.55 0.55 0.50 0.50 0.55
Cs 0.45 0.45 0.50 0.50 0.50 0.50 0.55 0.50 0.50 0.50 0.55
Cs 0.45 0.45 0.45 0.50 0.50 0.50 0.55 0.50 0.50 0.50 0.55
C7 0.40 0.40 0.45 0.45 0.45 0.45 0.50 0.50 0.50 0.50 0.50
C78 0.40 0.40 0.45 0.45 0.50 0.50 0.50 0.50 0.50 0.50 0.50
C9 0.40 0.40 0.45 0.45 0.50 0.50 0.50 0.50 0.50 0.50 0.55
C10 0.45 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.60
C11 0.40 0.40 0.45 0.45 0.45 0.45 0.50 0.50 0.45 0.40 0.50
Weight 0.100 0.098 0.095 0.093 0.091 0.090 0.085 0.086 0.087 0.092 0.082
CR 0.3622 (inconsistent)
Figure 5 - Li et al. [19] for example 2
Comparison of the proposed method with that of Chandavarkar and Guddeti [18]. The performance of the proposed method was compared with that of Chandavarkar and Guddeti [18]. Chandavarkar and Guddeti [18] to construct a pairwise comparison matrix. In the pairwise comparison matrices of Hruska et al. [23] and Polat et al. [24], Lp = Lq. In equation (1), the value assigned to aj is infinity. Thus, in the method used by Chandavarkar
C
C
2
C
3
C
4
C
5
C
C
7
C
8
C
9
C
10
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
and Guddeti [18], the a¡j value (infinity) is replaced by zero. This is one of the weaknesses of their method [18]. A comparison of the results obtained using their method [18] and those generated using the proposed method is presented in Tables 8 and 9. Based on these tables, it can be seen that the proposed method is better than that of Chandavarkar and Guddeti [18] because it always produces a consistent value. The proposed method is capable of giving a CI value close to zero and a CI value of zero if the matrix is perfectly consistent [25].
Table 7 - Summary of the results obtained using Li et al. [19] and those obtained
using the proposed method
Consistency ratio (CR)
Matrix data Size -
Original AHP Li et al. [19] Proposed method
Hruska et al. [23] 0.1845 0.4056 0.022
10 x 10
(inconsistent) (inconsistent) (consistent)
Polat [24] 0.0088 0.3622 0.012
11 x 11
(consistent) (inconsistent) (consistent)
Table 8 - Summary of the results obtained using the method Chandavarkar and Guddeti [18] and those obtained using the proposed method for example 1
n/n [231 Chandavarkar and Guddeti method [18] Proposed method
Criteria Weight Weight Contribution level (1-9) Weight
Ci 0.291 0.309 9 0.181
C2 0.229 0.223 8 0.161
C3 0.114 0.158 7 0.141
C4 0.114 0.158 7 0.141
C5 0.036 0.023 3 0.060
Ce 0.037 0.023 3 0.060
Cj 0.036 0.023 3 0.060
C8 0.068 0.034 5 0.100
Cg 0.052 0.031 4 0.076
Cio 0.023 0.018 1 0.020
CR 0.184 (inconsistent) 0.066 (consistent) 0.022 (consistent)
Table 9 - Summary of the results obtained using the Chandavarkar and Guddeti method [18] and
those of the proposed method for example 2
[24] Chandavarkar and Guddeti method [18] Proposed method
Criteria Weight Weight Contribution level (1-9) Weight
Ci 0.165 0.382 9 0.181
C2 0.135 0.289 8 0.161
C3 0.111 0.151 7 0.141
C4 0.092 0.039 7 0.141
C5 0.080 0.039 3 0.060
Ce 0.078 0.039 3 0.060
Cj 0.052 0.007 3 0.060
C8 0.059 0.007 5 0.100
Cg 0.076 0.005 4 0.076
Cio 0.104 0.013 1 0.020
Cii 0.048 0.028
CR 0.008 (consistent) 0.299 (inconsistent) 0.022 (consistent)
Test of the effect of criteria weight on supplier selection using the proposed method. We examined the effect of criteria weight on supplier selection using the proposed method as compared with that using the original AHP. The data used in the test are shown in Tables 10 and 11. These data are performance data from each supplier for each criterion. As shown in the tables, there are six suppliers (SC1, SC2, SC3, SC4, SC5, and SC6).
Table 10 - Supplier data for example 1
Supplier Ci C2 C3 C4 C5 Ce Cj Cs Cg Cio
SC1 8 5 3 1 8 7 8 3 5 3
SC2 10 6 5 2 7 10 5 1 8 1
SC3 10 6 3 3 5 8 6 4 5 5
SC4 9 7 4 2 4 11 2 3 7 0
SC5 12 8 4 2 6 9 4 0 8 2
SC6 10 6 8 4 5 6 3 2 7 1
Table 11 - Supplier data for example 2
Supplier Ci C2 C3 C4 C5 Ce Cj Cs Cg Cio Cii
SC1 8 5 3 1 8 7 8 3 5 3 8
SC2 10 6 5 2 7 10 5 1 8 1 6
SC3 10 6 3 3 5 8 6 4 5 5 10
SC4 9 7 4 2 4 11 2 3 7 0 4
SC5 12 8 4 2 6 9 4 0 8 2 5
SC5 10 6 8 4 5 6 3 2 7 1 9
Ranking of supplier using a contribution level of 0 to 100
for example 1 Ranking of supplier using Chandavarkar and Guddeti [18]
for example 1
0700
SCI SC2 SCS SKA SCS SC»
Ranking of supplier using Li etal. [19] for example 1
Figure 6 - Supplier selection solution (example 1)
Figure 6 is the result of the proposed method using supplier data in Table 10. Figures 7 is the result of the proposed method using supplier data in Table 11. The results of the sequence of suppliers are the same in Figure 6. Test supplier is SC 5, and the worst supplier is Sc 1. It was inconsistent in the pairwise comparison matrices of Hruska et al. [23] but consistent when using the proposed method Thus, the results obtained by Hruska et al. [23] and those obtained using the method of Li et al. [19] are invalid, although they yield the same solution as that obtained using the proposed method.
Using the pairwise comparison matrix of by Polat [24] gives the solution shown in Figure 7. Figure 7 shows that the solution of the proposed method is the same as that generated using the original AHP. The results obtained using the method of Chandavarkar and Guddeti [18] are invalid, although the method yields the same solution as those generated using the proposed method. Proposed method produce the same solution as that
obtained using a consistent pairwise comparison matrix. An inconsistent matrix results in a different supplier selection solution.
Figure 7 - Supplier selection solution (example 2) CONCLUSION
The solution obtained using the proposed method is better than that achieved using the method of Li et al. [19] and that of Chandavarkar and Guddeti [18], as the proposed method is capable of generating a valid solution, regardless of the number of criteria and without having to revise the pairwise comparison matrix. The proposed method is also easier to use because decision makers have only to assign a contribution level to each criterion rather than drawing comparisons between criteria. Furthermore, the proposed method is simpler than the original AHP, as it does not require a consistency test. In addition, using the proposed method, the pairwise comparison matrix does not have to be complete.
This research was financially supported by the PDD Program of the Ristekdikti Republic of Indonesia.
Limitations of the study. The proposed method has not been tested using real data.
Conflict of interest. The authors declare that there are no conflicts of interest.
ACKNOWLEDGEMENTS
The authors thank the Ministry of Research, Technology and Higher Education of the
Republic of Indonesia for its financial support.
REFERENCES
1. Rezaei, J., Nispeling, T., Sarkis, J., and Tavasszy, L., "A supplier selection life cycle approach integrating traditional and environmental criteria using the best worst method," Journal of Cleaner Production, Vol. 135, pp. 577-588, 2016.
2. Chai, J., Liu, J. N. K., and Ngai, E. W. T., "Application of decision-making techniques in supplier selection: A systematic review of literature," Expert Systems with Applications, Vol. 40, no. 10, pp. 3872-3885, 2013.
3. Banaeian, N., Mobli, H., Fahimnia, B., Nielsen, I. E., and Omid, M., "Green supplier selection using fuzzy group decision making methods: A case study from the agri-food industry," Computers & Operations research, Vol. 89, no. 1, pp. 337-347, 2017.
4. Govindan, K., Rajendran, S., Sarkis, J., and Murugesan, P., "Multi criteria decision making approaches for green supplier evaluation and selection: a literature review," Journal of Cleaner Production, Vol. 98, pp. 66 - 83, 2015.
5. Zimmer, K., Frohling, M., and Schultmann, F., "Sustainable supplier management - a review of models supporting sustainable supplier selection, monitoring and development," International Journal of Production Research, Vol. 54, no. 5, pp. 1412-1442, 2017.
6. Rosyidi, C. N., Murtisari, R., and Jauhari, W. A., "A concurrent optimization model for suppliers selection, tolerance and component allocation with fuzzy quality loss," Cogent Engineering, Vol. 3, no. 1, pp. 1-10, 2016.
7. Saaty, T. L., "Decision-making with the AHP: Why is the principal eigenvector necessary," European Journal of Operational Research, Vol. 145, no. 1, pp. 85-91, 2003.
8. Saaty, T. L., "How to make a decision: The Analytic Hierarchy Process," European Journal of Operational Research, Vol. 48, no. 1, pp. 9-26, 1990.
9. Darabi, S. and Heydari, J., "An interval-valued hesitant fuzzy ranking method based on group decision analysis for green supplier selection," IFAC-paper On Line, Vol. 49, no. 2, pp. 12-17, 2016.
10. Tavana, M., Fallahpour, A., Di-Caprio, D. and Santos-Arteaga, F. J., "A hybrid intelligent fuzzy predictive model with simulation for supplier evaluation and selection," Expert Systems With Applications, Vol. 61, pp. 129-144, 2016.
11. Hamdan, S. and Cheaitou, A., "Supplier selection and order allocation with green criteria: An MCDM and multi-objective optimization approach", Computers and Operation Research, Vol. 81, pp. 282 - 304, 2017.
12. Saaty, T. L. and Kearns, K. P., "Analytical planning: The organization systems," Oxford: Pergamon press, pp. 34-36, 1985.
13. Saaty, T. L. and Ozdemir, M. S., "Why the magic number seven plus or minus two," Mathematical and Computer Modelling, Vol. 38, no. 3-4, pp. 233-44, 2003.
14. Luzon, B. and El-Sayegh, S. M., "Evaluating supplier selection criteria for oil and gas projects in the UAE using AHP and Delphi," International Journal of Construction Management, Vol. 140, no. 3, pp. 1686-1698, 2016.
15. Dweiri, F., Kumar, S., Khan, S. A., and Jain, V., "Designing an integrated AHP based decision support system for supplier selection in automotive industry," Expert Systems With Applications, Vol. 62, pp. 273-283, 2016.
16. Pandey, P., Shah, B. J., and Gajjar, H., "A fuzzy goal programming approach for selecting sustainable suppliers," Benchmarking: An International Journal, Vol. 24, no. 5, 2017.
17. Luthra, S., Govindan, K., Kannan, D., Mangla, S. K., and Garg, C. P., "An integrated
framework for sustainable supplier selection and evaluation in supply chains," Journal of Cleaner Production, Vol. 140, no. 3, pp. 1686-1698, 2017.
18. Chandavarkar, B. R. and Guddeti, R. M. R., "Simplified and Improved Analytical Hierarchy Process Aid for Selecting Candidate Network in an Overlay Heterogeneous Networks," Wireless Personal Communications, Vol. 83, no. 4, pp. 2593-2606, 2015.
19. Li, Z., Li, J., Liang, D., and Lee, T., "Building multilevel governance and partnerships: an evaluation approach," Journal of Entrepreneurship in Emerging Economies, Vol. 8, no. 2, pp. 263 - 278, 2016.
20. Galankashi, M. R., Helmi, S. A., and Hashemzahi, P., "Supplier selection in automobile industry: A mixed balanced scorecard-fuzzy AHP approach," Alexandria Engineering Journal, Vol. 55, pp. 93 - 100, 2016.
21. Saaty, T. L., "Rank from comparisons and from ratings in the analytic hierarchy/network processes," European Journal of Operational Research, Vol. 168, no. 2, pp. 557-570, 2006.
22. Lin, Y. and Xiaohong, D., "The Computation and Application of random consistency index," 2nd International Conference on Information Science and Engineering (ICISE) IEEE, 4-6 Dec 2010, pp. 693 - 695, 2010.
23. Hruska, R., Prùsa, P. and Babic, D., "The use of AHP method for selection of supplier," Transport, Vol. 29, no. 2, pp. 195 - 203, 2014.
24. Polat, G., "Subcontractor selection using the integration of the AHP and PROMETHEE methods," Journal of Civil Engineering and Management, Vol. 22, no. 8, pp. 1042-1054, 2015.
25. Franek, J. And Kresta, A., "Judgment scales and consistency measure in AHP," Procedia Economics and Finance, Vol. 12, pp. 164 - 173, 2014.
APPENDIX
Criteria C1 is compared with criteria C2 Criteria C2 is compared with criteria C3 Criteria C1 is compared with criteria C3 Consistency ratio (CR) Conclusion
2 / 0.209 Inconsistent
3 / 0.356 Inconsistent
4 / 0.490 Inconsistent
1 5 / 0.613 Inconsistent
6 / 0.730 Inconsistent
7 / 0.843 Inconsistent
8 / 0.951 Inconsistent
9 / 1.057 Inconsistent
1 / 0.209 Inconsistent
2 / 0.481 Inconsistent
3 / 0.700 Inconsistent
4 / 0.890 Inconsistent
2 5 / 1.063 Inconsistent
6 / 1.224 Inconsistent
7 / 1.378 Inconsistent
8 / 1.526 Inconsistent
9 / 1.670 Inconsistent
1 / 0.356 Inconsistent
2 / 0.700 Inconsistent
3 / 0.967 Inconsistent
4 / 1.193 Inconsistent
3 5 / 1.397 Inconsistent
6 / 1.586 Inconsistent
7 / 1.764 Inconsistent
8 / 1.936 Inconsistent
9 / 2.103 Inconsistent
1 / 0.906 Inconsistent
2 / 1.450 Inconsistent
3 / 1.841 Inconsistent
4 / 2.163 Inconsistent
4 5 / 2.448 Inconsistent
6 / 2.711 Inconsistent
7 / 2.958 Inconsistent
8 / 3.194 Inconsistent
9 / 3.424 Inconsistent
1 / 0.608 Inconsistent
2 / 1.063 Inconsistent
3 / 1.400 Inconsistent
4 / 1.680 Inconsistent
5 5 / 1.926 Inconsistent
6 / 2.152 Inconsistent
7 / 2.364 Inconsistent
8 / 2.567 Inconsistent
9 / 2.762 Inconsistent
1 / 0.721 Inconsistent
2 / 1.224 Inconsistent
3 / 1.592 Inconsistent
4 / 1.893 Inconsistent
6 5 / 2.157 Inconsistent
6 / 2.397 Inconsistent
7 / 2.622 Inconsistent
8 / 2.836 Inconsistent
9 / 3.042 Inconsistent
1 / 0.829 Inconsistent
2 / 1.378 Inconsistent
3 / 1.774 Inconsistent
4 / 2.096 Inconsistent
7 5 / 2.375 Inconsistent
6 / 2.629 Inconsistent
7 / 2.865 Inconsistent
8 / 3.089 Inconsistent
9 / 3.304 Inconsistent
1 / 0.933 Inconsistent
2 / 1.526 Inconsistent
3 / 1.949 Inconsistent
4 / 2.291 Inconsistent
8 5 / 2.585 Inconsistent
6 / 2.851 Inconsistent
7 / 3.097 Inconsistent
8 / 3.330 Inconsistent
9 / 3.553 Inconsistent
1 / 1.033 Inconsistent
2 / 1.670 Inconsistent
3 / 2.120 Inconsistent
4 / 2.480 Inconsistent
9 5 / 2.789 Inconsistent
6 / 3.066 Inconsistent
7 / 3.322 Inconsistent
8 / 3.563 Inconsistent
9 / 3.793 Inconsistent
1 1/3 0.131 Inconsistent
2 1/3 0.356 Inconsistent
3 1/3 0.546 Inconsistent
4 1/3 0.714 Inconsistent
1 5 1/3 0.869 Inconsistent
6 1/3 1.014 Inconsistent
7 1/3 1.152 Inconsistent
8 1/3 1.286 Inconsistent
9 1/3 1.417 Inconsistent
1 1/3 0.356 Inconsistent
2 1/3 0.700 Inconsistent
3 1/3 0.967 Inconsistent
4 1/3 1.195 Inconsistent
2 5 1/3 1.400 Inconsistent
6 1/3 1.592 Inconsistent
7 1/3 1.774 Inconsistent
8 1/3 1.949 Inconsistent
9 1/3 2.120 Inconsistent
1 1/3 0.546 Inconsistent
2 1/3 0.967 Inconsistent
3 1/3 1.282 Inconsistent
4 1/3 1.547 Inconsistent
3 5 1/3 1.784 Inconsistent
6 1/3 2.004 Inconsistent
7 1/3 2.212 Inconsistent
8 1/3 2.412 Inconsistent
9 1/3 2.606 Inconsistent
1 1/3 0.711 Inconsistent
2 1/3 1.193 Inconsistent
3 1/3 1.547 Inconsistent
4 1/3 1.841 Inconsistent
4 б 1/3 2.102 Inconsistent
б 1/3 2.342 Inconsistent
l 1/3 2.5б8 Inconsistent
S 1/3 2.1S6 Inconsistent
9 1/3 2.99б Inconsistent
1 1/3 0.8б1 Inconsistent
2 1/3 1.39l Inconsistent
3 1/3 1.lS4 Inconsistent
4 1/3 2.102 Inconsistent
б б 1/3 2.3S3 Inconsistent
б 1/3 2.64O Inconsistent
l 1/3 2.SS1 Inconsistent
S 1/3 3.112 Inconsistent
9 1/3 3.335 Inconsistent
1 1/3 0.999 Inconsistent
2 1/3 1.58б Inconsistent
3 1/3 2.004 Inconsistent
4 1/3 2.344 Inconsistent
б б 1/3 2.б42 Inconsistent
б 1/3 2.913 Inconsistent
l 1/3 3.161 Inconsistent
S 1/3 3.409 Inconsistent
9 1/3 3.643 Inconsistent
1 1/3 1.130 Inconsistent
2 1/3 1.164 Inconsistent
3 1/3 2.212 Inconsistent
4 1/3 2.513 Inconsistent
l б 1/3 2.886 Inconsistent
б 1/3 3.110 Inconsistent
l 1/3 3.436 Inconsistent
S 1/3 3.688 Inconsistent
9 1/3 3.930 Inconsistent
1 1/3 1.254 Inconsistent
2 1/3 1.936 Inconsistent
3 1/3 2.412 Inconsistent
4 1/3 2.192 Inconsistent
S б 1/3 3.120 Inconsistent
б 1/3 3.411 Inconsistent
l 1/3 3.692 Inconsistent
S 1/3 3.953 Inconsistent
9 1/3 4.203 Inconsistent
1 1/3 1.315 Inconsistent
2 1/3 2.103 Inconsistent
3 1/3 2.606 Inconsistent
4 1/3 3.005 Inconsistent
9 б 1/3 3.341 Inconsistent
б 1/3 З.ббб Inconsistent
l 1/3 3.940 Inconsistent
S 1/3 4.209 Inconsistent
9 1/3 4.466 Inconsistent
1 У 0.211 Inconsistent
2 У 0.4S1 Inconsistent
3 У 0.111 Inconsistent
4 У 0.906 Inconsistent
1 б У 1.084 Inconsistent
б У 1.250 Inconsistent
l У 1.408 Inconsistent
S У 1.561 Inconsistent
9 У 1.109 Inconsistent
1 У 0.490 Inconsistent
2 У 0.S90 Inconsistent
3 У 1.193 Inconsistent
4 У 1.450 Inconsistent
2 б У 1.680 Inconsistent
б У 1.893 Inconsistent
l У 2.096 Inconsistent
S У 2.291 Inconsistent
9 У 2.4S0 Inconsistent
1 У 0.714 Inconsistent
2 У 1.195 Inconsistent
3 У 1.547 Inconsistent
4 У 1.841 Inconsistent
3 5 У 2.102 Inconsistent
6 У 2.344 Inconsistent
7 У 2.573 Inconsistent
8 У 2.792 Inconsistent
9 У 3.005 Inconsistent
1 У 0.906 Inconsistent
2 У 1.450 Inconsistent
3 У 1.841 Inconsistent
4 У 2.163 Inconsistent
4 5 У 2.448 Inconsistent
6 У 2.711 Inconsistent
7 У 2.958 Inconsistent
8 У 3.194 Inconsistent
9 У 3.424 Inconsistent
1 У 1.078 Inconsistent
2 У 1.677 Inconsistent
3 У 2.102 Inconsistent
4 У 2.448 Inconsistent
5 5 У 2.753 Inconsistent
6 У 3.032 Inconsistent
7 У 3.294 Inconsistent
8 У 3.544 Inconsistent
9 У 3.786 Inconsistent
1 У 1.235 Inconsistent
2 У 1.886 Inconsistent
3 У 2.342 Inconsistent
4 У 2.711 Inconsistent
6 5 У 3.032 Inconsistent
6 У 3.325 Inconsistent
7 У 3.600 Inconsistent
8 У 3.861 Inconsistent
9 У 4.113 Inconsistent
1 У 1.383 Inconsistent
2 У 2.083 Inconsistent
3 У 2.568 Inconsistent
4 У 2.958 Inconsistent
7 5 У 3.295 Inconsistent
6 У 3.601 Inconsistent
7 У 3.886 Inconsistent
8 У 4.158 Inconsistent
9 У 4.418 Inconsistent
1 У 1.524 Inconsistent
2 У 2.272 Inconsistent
3 У 2.786 Inconsistent
4 У 3.194 Inconsistent
8 5 У 3.547 Inconsistent
6 У 3.864 Inconsistent
7 У 4.160 Inconsistent
8 У 4.439 Inconsistent
9 У 4.708 Inconsistent
1 У 1.659 Inconsistent
2 У 2.455 Inconsistent
3 У 2.996 Inconsistent
4 У 3.424 Inconsistent
9 5 У 3.790 Inconsistent
6 У 4.119 Inconsistent
7 У 4.423 Inconsistent
8 У 4.711 Inconsistent
9 У 4.987 Inconsistent
1 1/5 0.289 Inconsistent
2 1/5 0.608 Inconsistent
3 1/5 0.861 Inconsistent
4 1/5 1.078 Inconsistent
1 5 1/5 1.274 Inconsistent
6 1/5 1.457 Inconsistent
7 1/5 1.631 Inconsistent
8 1/5 1.798 Inconsistent
9 1/5 1.960 Inconsistent
1 1/5 0.613 Inconsistent
2 1/5 1.063 Inconsistent
3 1/5 1.391 Inconsistent
4 1/5 1.611 Inconsistent
2 б 1/5 1.926 Inconsistent
6 1/5 2.151 Inconsistent
1 1/5 2.315 Inconsistent
S 1/5 2.5S5 Inconsistent
9 1/5 2.1S9 Inconsistent
1 1/5 0.869 Inconsistent
2 1/5 1.400 Inconsistent
3 1/5 1.184 Inconsistent
4 1/5 2.102 Inconsistent
3 б 1/5 2.3S3 Inconsistent
6 1/5 2.642 Inconsistent
1 1/5 2.886 Inconsistent
S 1/5 3.120 Inconsistent
9 1/5 3.341 Inconsistent
1 1/5 1.084 Inconsistent
2 1/5 1.680 Inconsistent
3 1/5 2.102 Inconsistent
4 1/5 2.44S Inconsistent
4 б 1/5 2.153 Inconsistent
6 1/5 3.032 Inconsistent
1 1/5 3.295 Inconsistent
S 1/5 3.541 Inconsistent
9 1/5 3.190 Inconsistent
1 1/5 1.214 Inconsistent
2 1/5 1.926 Inconsistent
3 1/5 2.3S3 Inconsistent
4 1/5 2.153 Inconsistent
б б 1/5 3.011 Inconsistent
6 1/5 3.313 Inconsistent
1 1/5 3.650 Inconsistent
S 1/5 3.915 Inconsistent
9 1/5 4.110 Inconsistent
1 1/5 1.448 Inconsistent
2 1/5 2.152 Inconsistent
3 1/5 2.640 Inconsistent
4 1/5 3.032 Inconsistent
6 б 1/5 3.313 Inconsistent
6 1/5 3.683 Inconsistent
1 1/5 3.912 Inconsistent
S 1/5 4.24S Inconsistent
9 1/5 4.514 Inconsistent
1 1/5 1.610 Inconsistent
2 1/5 2.364 Inconsistent
3 1/5 2.SS1 Inconsistent
4 1/5 3.294 Inconsistent
1 б 1/5 3.650 Inconsistent
6 1/5 3.913 Inconsistent
1 1/5 4.213 Inconsistent
S 1/5 4.559 Inconsistent
9 1/5 4.S33 Inconsistent
1 1/5 1.163 Inconsistent
2 1/5 2.561 Inconsistent
3 1/5 3.112 Inconsistent
4 1/5 3.544 Inconsistent
S б 1/5 3.915 Inconsistent
6 1/5 4.249 Inconsistent
1 1/5 4.559 Inconsistent
S 1/5 4.S53 Inconsistent
9 1/5 5.136 Inconsistent
1 1/5 1.910 Inconsistent
2 1/5 2.162 Inconsistent
3 1/5 3.335 Inconsistent
4 1/5 3.186 Inconsistent
9 б 1/5 4.110 Inconsistent
6 1/5 4.516 Inconsistent
1 1/5 4.S35 Inconsistent
S 1/5 5.131 Inconsistent
9 1/5 5.426 Inconsistent
1 1/6 0.364 Inconsistent
2 1/6 0.721 Inconsistent
3 1/6 0.999 Inconsistent
4 1/6 1.235 Inconsistent
1 5 1/6 1.448 Inconsistent
6 1/6 1.645 Inconsistent
7 1/6 1.831 Inconsistent
8 1/6 2.010 Inconsistent
9 1/6 2.184 Inconsistent
1 1/6 0.730 Inconsistent
2 1/6 1.224 Inconsistent
3 1/6 1.586 Inconsistent
4 1/6 1.886 Inconsistent
2 5 1/6 2.152 Inconsistent
6 1/6 2.397 Inconsistent
7 1/6 2.629 Inconsistent
8 1/6 2.851 Inconsistent
9 1/6 3.066 Inconsistent
1 1/6 1.014 Inconsistent
2 1/6 1.592 Inconsistent
3 1/6 2.004 Inconsistent
4 1/6 2.342 Inconsistent
3 5 1/6 2.640 Inconsistent
6 1/6 2.913 Inconsistent
7 1/6 3.170 Inconsistent
8 1/6 3.417 Inconsistent
9 1/6 3.655 Inconsistent
1 1/6 1.250 Inconsistent
2 1/6 1.893 Inconsistent
3 1/6 2.344 Inconsistent
4 1/6 2.711 Inconsistent
4 5 1/6 3.032 Inconsistent
6 1/6 3.325 Inconsistent
7 1/6 3.601 Inconsistent
8 1/6 3.864 Inconsistent
9 1/6 4.119 Inconsistent
1 1/6 1.457 Inconsistent
2 1/6 2.157 Inconsistent
3 1/6 2.642 Inconsistent
4 1/6 3.032 Inconsistent
5 5 1/6 3.373 Inconsistent
6 1/6 3.683 Inconsistent
7 1/6 3.973 Inconsistent
8 1/6 4.249 Inconsistent
9 1/6 4.516 Inconsistent
1 1/6 1.645 Inconsistent
2 1/6 2.397 Inconsistent
3 1/6 2.913 Inconsistent
4 1/6 3.325 Inconsistent
6 5 1/6 3.683 Inconsistent
6 1/6 4.006 Inconsistent
7 1/6 4.309 Inconsistent
8 1/6 4.596 Inconsistent
9 1/6 4.873 Inconsistent
1 1/6 1.610 Inconsistent
2 1/6 2.364 Inconsistent
3 1/6 2.881 Inconsistent
4 1/6 3.294 Inconsistent
7 5 1/6 3.650 Inconsistent
6 1/6 3.973 Inconsistent
7 1/6 4.273 Inconsistent
8 1/6 4.559 Inconsistent
9 1/6 4.833 Inconsistent
1 1/6 1.983 Inconsistent
2 1/6 2.836 Inconsistent
3 1/6 3.409 Inconsistent
4 1/6 3.861 Inconsistent
8 5 1/6 4.248 Inconsistent
6 1/6 4.596 Inconsistent
7 1/6 4.919 Inconsistent
8 1/6 5.224 Inconsistent
9 1/6 5.517 Inconsistent
1 1/6 2.141 Inconsistent
2 1/6 3.042 Inconsistent
3 1/6 3.643 Inconsistent
4 1/6 4.113 Inconsistent
9 5 1/6 4.514 Inconsistent
6 1/6 4.S13 Inconsistent
1 1/6 5.204 Inconsistent
S 1/6 5.511 Inconsistent
9 1/6 5.S11 Inconsistent
1 1/1 0.436 Inconsistent
2 1/1 0.S29 Inconsistent
3 1/1 1.130 Inconsistent
4 1/1 1.3S3 Inconsistent
1 5 1/1 1.610 Inconsistent
6 1/1 1.S19 Inconsistent
1 1/1 2.011 Inconsistent
S 1/1 2.206 Inconsistent
9 1/1 2.3S9 Inconsistent
1 1/1 0.S43 Inconsistent
2 1/1 1.31S Inconsistent
3 1/1 1.164 Inconsistent
4 1/1 2.0S3 Inconsistent
2 5 1/1 2.364 Inconsistent
6 1/1 2.622 Inconsistent
1 1/1 2.865 Inconsistent
S 1/1 3.091 Inconsistent
9 1/1 3.322 Inconsistent
1 1/1 1.152 Inconsistent
2 1/1 1.774 Inconsistent
3 1/1 2.212 Inconsistent
4 1/1 2.568 Inconsistent
3 5 1/1 2.SS1 Inconsistent
6 1/1 3.161 Inconsistent
1 1/1 3.436 Inconsistent
S 1/1 3.692 Inconsistent
9 1/1 3.940 Inconsistent
1 1/1 1.408 Inconsistent
2 1/1 2.096 Inconsistent
3 1/1 2.513 Inconsistent
4 1/1 2.95S Inconsistent
4 5 1/1 3.294 Inconsistent
6 1/1 3.600 Inconsistent
1 1/1 3.886 Inconsistent
S 1/1 4.160 Inconsistent
9 1/1 4.423 Inconsistent
1 1/1 1.631 Inconsistent
2 1/1 2.315 Inconsistent
3 1/1 2.886 Inconsistent
4 1/1 3.295 Inconsistent
5 5 1/1 3.650 Inconsistent
6 1/1 3.912 Inconsistent
1 1/1 4.213 Inconsistent
S 1/1 4.559 Inconsistent
9 1/1 4.S35 Inconsistent
1 1/1 1.S31 Inconsistent
2 1/1 2.629 Inconsistent
3 1/1 3.110 Inconsistent
4 1/1 3.601 Inconsistent
6 5 1/1 3.913 Inconsistent
6 1/1 4.309 Inconsistent
1 1/1 4.622 Inconsistent
S 1/1 4.919 Inconsistent
9 1/1 5.204 Inconsistent
1 1/1 2.011 Inconsistent
2 1/1 2.865 Inconsistent
3 1/1 3.436 Inconsistent
4 1/1 3.886 Inconsistent
1 5 1/1 4.213 Inconsistent
6 1/1 4.622 Inconsistent
1 1/1 4.945 Inconsistent
S 1/1 5.251 Inconsistent
9 1/1 5.546 Inconsistent
1 1/7 2.191 Inconsistent
2 1/7 3.089 Inconsistent
3 1/7 3.688 Inconsistent
4 1/7 4.158 Inconsistent
8 5 1/7 4.559 Inconsistent
6 1/7 4.918 Inconsistent
7 1/7 5.251 Inconsistent
8 1/7 5.566 Inconsistent
9 1/7 5.868 Inconsistent
1 1/7 2.357 Inconsistent
2 1/7 3.304 Inconsistent
3 1/7 3.930 Inconsistent
4 1/7 4.418 Inconsistent
9 5 1/7 4.833 Inconsistent
6 1/7 5.204 Inconsistent
7 1/7 5.546 Inconsistent
8 1/7 5.868 Inconsistent
9 1/7 6.177 Inconsistent
1 1/8 0.507 Inconsistent
2 1/8 0.933 Inconsistent
3 1/8 1.254 Inconsistent
4 1/8 1.524 Inconsistent
1 5 1/8 1,763 Inconsistent
6 1/8 1.983 Inconsistent
7 1/8 2.191 Inconsistent
8 1/8 2.389 Inconsistent
9 1/8 2.581 Inconsistent
1 1/8 0.951 Inconsistent
2 1/8 1.526 Inconsistent
3 1/8 1.936 Inconsistent
4 1/8 2.272 Inconsistent
2 5 1/8 2.567 Inconsistent
6 1/8 2.836 Inconsistent
7 1/8 3.089 Inconsistent
8 1/8 3.330 Inconsistent
9 1/8 3.563 Inconsistent
1 1/8 1.286 Inconsistent
2 1/8 1.949 Inconsistent
3 1/8 2.412 Inconsistent
4 1/8 2.786 Inconsistent
3 5 1/8 3.112 Inconsistent
6 1/8 3.409 Inconsistent
7 1/8 3.688 Inconsistent
8 1/8 3.953 Inconsistent
9 1/8 4.209 Inconsistent
1 1/8 1.561 Inconsistent
2 1/8 2.291 Inconsistent
3 1/8 2.792 Inconsistent
4 1/8 3.194 Inconsistent
4 5 1/8 3.544 Inconsistent
6 1/8 3.861 Inconsistent
7 1/8 4.158 Inconsistent
8 1/8 4.439 Inconsistent
9 1/8 4.711 Inconsistent
1 1/8 1.798 Inconsistent
2 1/8 2.585 Inconsistent
3 1/8 3.120 Inconsistent
4 1/8 3.547 Inconsistent
5 5 1/8 3.915 Inconsistent
6 1/8 4.248 Inconsistent
7 1/8 4.559 Inconsistent
8 1/8 4.853 Inconsistent
9 1/8 5.137 Inconsistent
1 1/8 2.010 Inconsistent
2 1/8 2.851 Inconsistent
3 1/8 3.417 Inconsistent
4 1/8 3.864 Inconsistent
6 5 1/8 4.249 Inconsistent
6 1/8 4.596 Inconsistent
7 1/8 4.918 Inconsistent
8 1/8 5.224 Inconsistent
9 1/8 5.517 Inconsistent
1 1/8 2.206 Inconsistent
2 1/8 3.091 Inconsistent
3 1/8 3.692 Inconsistent
4 1/8 4.160 Inconsistent
1 5 1/8 4.559 Inconsistent
6 1/8 4.919 Inconsistent
1 1/8 5.251 Inconsistent
S 1/8 5.566 Inconsistent
9 1/8 5.868 Inconsistent
1 1/8 2.389 Inconsistent
2 1/8 3.330 Inconsistent
3 1/8 3.953 Inconsistent
4 1/8 4.439 Inconsistent
s 5 1/8 4.853 Inconsistent
6 1/8 5.224 Inconsistent
1 1/8 5.566 Inconsistent
S 1/8 5.889 Inconsistent
9 1/8 6.199 Inconsistent
1 1/8 2.563 Inconsistent
2 1/8 3.553 Inconsistent
3 1/8 4.203 Inconsistent
4 1/8 4.108 Inconsistent
9 5 1/8 5.136 Inconsistent
6 1/8 5.511 Inconsistent
1 1/8 5.868 Inconsistent
S 1/8 6.199 Inconsistent
9 1/8 6.515 Inconsistent
1 1/9 0.516 Inconsistent
2 1/9 1.033 Inconsistent
3 1/9 1.315 Inconsistent
4 1/9 1.659 Inconsistent
1 5 1/9 1.910 Inconsistent
6 1/9 2.141 Inconsistent
1 1/9 2.351 Inconsistent
S 1/9 2.563 Inconsistent
9 1/9 2.162 Inconsistent
1 1/9 1.051 Inconsistent
2 1/9 1.610 Inconsistent
3 1/9 2.103 Inconsistent
4 1/9 2.455 Inconsistent
2 5 1/9 2.162 Inconsistent
6 1/9 3.042 Inconsistent
1 1/9 3.304 Inconsistent
S 1/9 3.553 Inconsistent
9 1/9 3.193 Inconsistent
1 1/9 1.411 Inconsistent
2 1/9 2.120 Inconsistent
3 1/9 2.606 Inconsistent
4 1/9 2.996 Inconsistent
3 5 1/9 3.335 Inconsistent
6 1/9 3.643 Inconsistent
1 1/9 3.930 Inconsistent
S 1/9 4.203 Inconsistent
9 1/9 4.466 Inconsistent
1 1/9 1.109 Inconsistent
2 1/9 2.480 Inconsistent
3 1/9 3.005 Inconsistent
4 1/9 3.424 Inconsistent
4 5 1/9 3.186 Inconsistent
6 1/9 4.113 Inconsistent
1 1/9 4.418 Inconsistent
S 1/9 4.108 Inconsistent
9 1/9 4.981 Inconsistent
1 1/9 1.960 Inconsistent
2 1/9 2.189 Inconsistent
3 1/9 3.341 Inconsistent
4 1/9 3.190 Inconsistent
5 5 1/9 4.110 Inconsistent
6 1/9 4.514 Inconsistent
1 1/9 4.833 Inconsistent
S 1/9 5.136 Inconsistent
9 1/9 5.426 Inconsistent
1 1/9 2.184 Inconsistent
2 1/9 3.066 Inconsistent
3 1/9 3.655 Inconsistent
4 1/9 4.119 Inconsistent
6 5 1/9 4.516 Inconsistent
6 1/9 4.873 Inconsistent
7 1/9 5.204 Inconsistent
8 1/9 5.517 Inconsistent
9 1/9 5.817 Inconsistent
1 1/9 2.389 Inconsistent
2 1/9 3.322 Inconsistent
3 1/9 3.940 Inconsistent
4 1/9 4.423 Inconsistent
7 5 1/9 4.835 Inconsistent
6 1/9 5.204 Inconsistent
7 1/9 5.546 Inconsistent
8 1/9 5.868 Inconsistent
9 1/9 6.177 Inconsistent
1 1/9 2.581 Inconsistent
2 1/9 3.563 Inconsistent
3 1/9 4.209 Inconsistent
4 1/9 4.711 Inconsistent
8 5 1/9 5.137 Inconsistent
6 1/9 5.517 Inconsistent
7 1/9 5.868 Inconsistent
8 1/9 6.199 Inconsistent
9 1/9 6.515 Inconsistent
1 1/9 2.762 Inconsistent
2 1/9 3.793 Inconsistent
3 1/9 4.466 Inconsistent
4 1/9 4.987 Inconsistent
9 5 1/9 5.426 Inconsistent
6 1/9 5.817 Inconsistent
7 1/9 6.177 Inconsistent
8 1/9 6.515 Inconsistent
9 1/9 6.838 Inconsistent
