Multiplicative determination of priority weights in a Fuzzy Analytic Hierarchy Process
T. Van Hecke
Ghent University, Faculty of Engineering and Architecture Voskenslaan 270, Ghent, Belgium
Tanja.VanHecke@ugent.be
Abstract:Many application fields need evaluation tools and ranking techniques of several alternatives. The multi-criteria decision-making approach, known as the Analytic Hierarchy Process, has yet been extended by a variant based on fuzzy logic, which is useful when dealing with uncertain judgements. This paper describes an important alternative method to determine the fuzzy priority weights of pairwise comparison matrices based on logarithmic calculus as a natural companion of the geometric mean. A significant advantage of our approach is the linear character of the resulting mathematical problem. The linearity allows the use of well-known linear programming techniques. The functioning of the method is illustrated by a numerical example which exposes the consistent way of priority weight determination.
Keywords:AHP; analytic hierarchy process; MCDM; multi-criteria decision- making; simplex method; priority weights; fuzzy logic; evaluation criteria;
ranking.
Biographical notes: Tanja Van Hecke obtained a master degree in applied mathematics in 1995 at the Ghent University. In 1998 she finished her doctoral thesis in numerical analysis in the field of solving ordinary differential equations.
For several years she worked at the faculty of applied engineering sciences at the University College Ghent where she was part of the department of mathematics and statistics. In 2013 this faculty became part of the faculty of Engineering and Architecture at the Ghent University.
1 Introduction
The Analytic Hierarchy Process (AHP) is a multi-criteria decision-making approach that was introduced by Saaty (1980). The AHP has caught the attention of many researchers (among them Sitorus, Cilliers and Brito-Parada (2019) and Bhaskar and Narayan Kudal (2019)), mainly due to the nice mathematical properties of the method and the fact that the required input data are rather easy to obtain. The AHP is a decision support tool which can be used to solve complex decision problems. It uses a multi-level hierarchical structure of a finite number of criteria and alternatives. The data is obtained by means of a set of pairwise reciprocal comparisons. These comparisons are used to obtain the weights of importance of the decision criteria, as well as the relative performance measures of the alternatives in terms of the decision criteria. If the comparisons are not perfectly consistent, then AHP provides a mechanism for improving consistency. Industrial engineering applications of the AHP are plentiful: in the evaluation of technology investment decisions as described by Boucher (1991), in flexible manufacturing systems as described by Wabalickis (1988), in layout design as described by Cambron and Evans (1991) and also in other engineering problems as described by Wang and Raz (1991).
A natural way to cope with uncertain judgments is to express the comparison ratios as fuzzy numbers, as it incorporates the vagueness of human thinking. Wagenknecht and Hartmann (1983) employs the least squares method to calculate fuzzy priorities weights wi. The obtained priorities are represented as fuzzy numbers. In Groselj and Zadnik Stirn (2018) a comparison is given of several methods to determine these weights.
The approach proposed by Buckley (1985) is based on trapezoidal fuzzy numbers with a prioritization process, which is also based on the geometric mean. The obtained priorities are combined in the Saaty hierarchy to compute the final fuzzy scores, which are then compared by fuzzy ranking. Chang (1996) gives another approach for fuzzy prioritisation, called synthetic extent analysis. He applies a simple arithmetic mean (AM) algorithm to find fuzzy priorities from comparison matrices, whose elements are represented by triangular fuzzy numbers (TFN). However, the arithmetic mean can only be used if the comparison matrices are consistent.
Our work focusses on the consistent use of the multiplicative approach throughout the whole procedure of the determination of the priority weights in the Fuzzy Analytic Hierarchy Process (FAHP). Applications are described in Haldar et. al. (2018), Nitkratoke and Aengchuan (2019), Mahtani and Garg (2018). This multiplicative approach is already included by all authors using the geometric mean to summarize different comparisons.
However, the geometric mean (1) can be rewritten as the arithmetic mean of the logarithm- transformed values, which explains the synonym ’log-average’ for the geometric mean.
GM(a1, a2, . . . , an) = ( n
∏
i=1
ai
)1/n
= exp (
1 n
∑n i=1
ln(ai) )
(1)
Our contribution is mainly based on the extension of this principle of using logarithm to the priority weights calculations (see Zhang and Guo (2017) and Wu et. al. (2019)). It implies that the optimization problem turns out into a linear programming problem.
2 Fuzzy numbers
Following Zadeh (1965), each triangular fuzzy number can be expressed as
˜
n= (l, m, u) (2)
where the uncertainty is expressed by means of the degree of membership function
f(x) =
0, x < l
x−l
m−l, l≤x≤m
u−x
u−m, m≤x≤u 0, x > u
(3)
which can be seen in Figure 2 supplemented by theα-cut[lα, uα] ={x:f(x)≥α}.
Figure 1Degree of membership function andα-cut
The arithmetic operations on the fuzzy numbers n˜i= (li, mi, ui) (1≤i≤a) are defined as follows:
addition⊕:
˜
n1⊕˜n2⊕. . .⊕n˜a = (
∑a i=1
li,
∑a i=1
mi,
∑a i=1
ui) (4)
multiplication⊙:
˜
n1⊙˜n2⊙. . .⊙n˜a = (
∏a i=1
li,
∏a i=1
mi,
∏a i=1
ui) (5)
division⊘:
˜
n1⊘˜n2= (l1 u2
, m1 m2
, u1 l2
) (6)
Geometric mean GM:
GM(˜n1,n˜2, . . . ,˜na) = ((
∏a i=1
li)1/a,(
∏a i=1
mi)1/a,(
∏a i=1
ui)1/a) (7)
Additively normalization of a set{n˜1,n˜2, . . . ,˜na}: (
li
∑a
j=1lj, ∑ami
j=1mj, ∑aui j=1uj
)
(1≤i≤a) (8)
Defuzzification:
The transformation of a TFN (2) into one single number is called defuzzification and can be done by means of the best non-fuzzy performance (BNP) value as defined by Tzeng and Huang (2011).
BNP(˜n) = l+m+u
3 . (9)
3 AHP
Steps in the AHP process:
• Construct the hierarchical framework of pairwise comparisons of criteria and alternatives for each criterion
• Calculate the weights of the alternatives for each evaluation criterion
• Calculate the weights of the evaluation criteria
• Calculate the global ranking for the alternatives
Suppose n criteria are used to evaluate m alternatives. A (n×n) matrix C contains the pairwise comparisons of the criteria, while the m×m matrices A[k] (1≤k≤n) represent the pairwise comparisons of the alternatives concerning criterionkwherea[k]ij = (l[k]ij, m[k]ij, u[k]ij )is a TFN. The elementa[k]ij shows the preference weight of alternative iobtained by comparing it with alternativej for criterionk. The elementsa[k]ij estimate the ratiosw[k]i /w[k]j wherew[k]= (w[k]1 , w[k]2 , . . . , wn[k])is the vector of current weights of the alternatives for criterionk. Chang (1996) calculates the weights as the normalized eigenvectorP[k] of the dominant or largest eigenvalue of the pairwise comparison matrix A[k](see Saaty (1980)). Then×nmatrixPis made up of the vectorsP[k]as its columns.
The total scores for the alternatives can be calculated asP⊙P C, whereP Cis the additively normalized eigenvector of the largest eigenvalueλmaxof the pairwise comparison matrix C.
Table 1 Random consistency index for different dimensions
d 3 4 5 6
RI 0.5245 0.8815 1.1086 1.2479
Table 2 Definition of triangular fuzzy numbers fuzzy number definition
˜9=(7,9, 9) Extremely important
˜8=(6,8, 9) Intermediate value between extremely and very strongly important
˜7=(5,7, 9) Very strongly important
˜6=(4,6, 8) Intermediate value between very strongly and strongly important
˜5=(3,5, 7) Strongly important
˜4=(2,4, 6) Intermediate value between strongly and moderately important
˜3=(1,3, 5) Moderately important
˜2=(1,2, 4) Intermediate value between moderately and equally important
˜1=(1,1, 3) Equally important
Saaty (1980) elaborated a technique to check the consistency of comparison matrices.
The consistency index of a comparison matrix of dimension (d×d) was defined by him as the ratio of the CI and the Random consistency index (RI) as in (10).
CR=CI
RI, with CI=λmax−d
d−1 (10)
RI is the average value of CI for random matrices using the Saaty scale obtained by Forman (1990) as in Table 1.
If the value of CI is smaller than 10%, the inconsistency is acceptable.
4 FAHP
When taking uncertainty into account, we can fuzzify the crisp comparison numbers as in Table 2 describing the intensities of the different TFNs.
4.1 Fuzzy priority weights by the fuzzy synthetic extent method
Following Buckley (1985) and Chang (1996), the determination process of the weights of matrixA∈ {A[1], A[2], . . . , A[n], C}of dimensiond×dwill be obtained by the following steps:
• Sum up each row of the fuzzy judgment matrixAto get the fuzzy number vectorRS.
RS=
∑d j=1a1j
∑d j=1a2j
...
∑d j=1adj
=
∑d
j=1(l1j, m1j, u1j)
∑d
j=1(l2j, m2j, u2j) ...
∑d
j=1(ldj, mdj, udj)
(11)
• Normalize the row fuzzy number vectorRS to get the fuzzy synthetic extent value vectorS.
S=
RS1⊙N RS2⊙N
... RSd⊙N
(12)
with N=
(
∑d 1
i=1
∑d
j=1uij, 1
∑d i=1
∑d
j=1mij, 1
∑d i=1
∑d j=1lij
)
. (13)
• Compute the degree of possibility to get the non-fuzzy weight vectorV.
V =
mink̸=1V(S1≥Sk) mink̸=2V(S2≥Sk)
...
mink̸=dV(Sd≥Sk)
(14)
with
V(Sj≥Si) =
1, ifmj ≥mi
0, ifli≥uj li−uj
(mj−uj)−(mi−li),else
(15)
• Define the final weight vectorW after normalization ofV.
The priority vectorsP[k] andP C can be obtained from the comparison matrices by normalization of the geometric mean of the column vectors ofA[k]andCrespectively (see Chang (1996)). A defuzzification of these results resumes all information into single total scores for each alternative.
The geometric mean GM is used because property (16) of the pairwise comparison matrices is preserved, which is not the case for the arithmetic mean. Figure 4.1 visualizes (for one example) the difference between combining two TFNs by GM and combining two TFNs by AM.
aij = 1 aji
, ∀i,∀j (16)
Figure 2Degree of membership function of the geometric mean and the arithmetic mean
4.2 Fuzzy priority weights by linear optimization with the simplex method We consider a general (d×d) fuzzy pairwise comparison matrix A. The goal is to determine the best choice of weightswi (i= 1,2, . . . , d) to ensure thataij ≈ wwji. Our method logFAHP is based on the logarithms of the comparison values, which is a better representation of the multiplicative approach. We define matrix B by bij = ln(aij) = (ln(lij),ln(mij), ln(uij)). This makes that the following sets of properties are equivalent.
aii = 1 aij = 1/aji
aij = wwi
j
⇔
bii = 0 bij =−bji
bij = ln(wi)−ln(wj) =vi−vj
(17)
Theα-cut of the TFN (2) can be described as the interval
[lij+α(mij−lij), uij−α(uij−mij)]. (18) After the translation v∗i =vi−li1−α(mi1−li1), the (d−1) conditions vi ≥li1+ α(mi1−li1)coincide with the natural positivity conditionsvi∗≥0of a linear programming problem. The feasible region is delineated by the linear inequality constraints (19) for (2≤j < i≤d).
v∗i −v∗j ≤uij−α(uij−mij)−li1−α(mi1−li1) +lj1+α(mj1−lj1) v∗i −v∗j ≥lij+α(mij−lij)−li1−α(mi1−li1) +lj1+α(mj1−lj1) v∗i ≤ui1−li1−α(ui1−li1)
(19)
Without losing generality, we choosev1= 0to reduce the number of unknowns to(d−1).
The optimalv∗-values are obtained by means of the simplex method (see Murty (1983)) as we are dealing with a linear programming problem whereαis to maximized.
At last the original weightswi are obtained by using the expression exp(ui+li1+ α(mi1−li1))(2≤i≤d) and after normalization.
Table 3 Criteria in case of metal roofing selection ready for ((log)F)AHP Criterion 1: wind Is a roof with an inclination of30◦susceptible
to blow-off?
Criterion 2: water Is a roof with an inclination of30◦watertight?
Criterion 3: reflection Is the surface reflective?
Criterion 4: cost Does the roof require a large investment?
5 Numerical example
Table 3 explains the criteria that are relevant to evaluate metal roofing. These criteria are incorporated in an AHP hierarchy as in Figure 5.
Figure 3AHP hierarchy for choosing the best metal roofing
In case of pairwise comparison matrices as in Table 4, we get priority vectors for the three methods: AHP, FAHP and logFAHP. For the first criterion there is some disagreement on the objects among the three specialists:A[1]12 is evaluated as 12, 13 and 14 respectively.
AnalogousA[1]13is evaluated as 4, 5 and 7. The geometric mean resumes the global opinion of the specialists.
When applying the different multi-criteria decision-making methods, we get the total scores as in Table 5.
It is clear that the results depend on the approach. The results are basically showing that a margin of fuzziness affects the overall result.
6 Conclusion
Our logFAHP method determines the priority weights of pairwise comparison matrices on the basis of the geometric mean in an advantageous way. The linear programming problem
Table 4 Consistency rate and priority vectors for pairwise comparison matrices with the methods AHP, FAHP and logFAHP
CR AHP FAHP logFAHP
A[1]=
1 1
2√3 3
√3
140 2√3
3 1 9
1
√3
140 1
9 1
2.8%
0.273 0.665 0.062
0.332 0.650 0.018
0.342 0.592 0.066
A[2]=
1 3 1
1 31 17 1 7 1
7.7%
0.388 0.097 0.515
0.286 0.120 0.594
0.467 0.067 0.466
A[3]=
1 5 9
1 51 4
1 9 1 41
6.8%
0.743 0.194 0.063
0.791 0.209 0.000
0.800 0.160 0.040
A[4]=
1 135 3 1 9
1 5 1 91
2.8%
0.265 0.672 0.063
0.323 0.656 0.021
0.333 0.600 0.067
C=
1 4 3 7
1 4 1 133
1 3 3 1 5
1 7 1 3
1 51
4.5%
0.548 0.127 0.270 0.055
0.404 0.188 0.295 0.113
0.500 0.137 0.262 0.101
Table 5 Overall scores for the different objects with multiple weight defining methods.
object AHP FAHP logFAHP
sheet metal tiles 0.414 0.458 0.478 sheet metal panels 0.466 0.421 0.408 standing-seam system 0.120 0.121 0.114
that arises through the logarithmic approach, is easy to solve and delivers evaluation scores for the multi-criteria decision problem. These scores reflect the opinion of multiple experts in a more consistent way.
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