Post on 12-Sep-2021
FUZZY HIERARCHICAL ANALYSIS AND APPLICATION
V. S. MATHU SURESH
Assistant Professor, Department of Mathematics, Udaya School of Engineering, Ammandi Villai, Kanyakumari District, INDIA
Abstract: An evolutionary algorithm is used to directly fuzzify the hierarchies. While finding the fuzzy weights
original saaty method of estimation is compared with this. As an illustration the choice for the best college for admission based on five criteria is illustrated.
Keywords: Fuzzification , Evolutionary algorithm , eigenvectors.
1. Introduction
Our present complex environment calls for a new logic – a new way to cope with myriad factors that affect the achievement of goals and the consistency of the Judgments we use to draw valid conclusions. This approach should be Justifiable and appeal to our wisdom and good sense. It should not be so complex that only the educated can use it, but should serve as a unifying tool for thought in general. People are now finding it hard to take decisions when qualitative and quantitative parameters are involve in Judgement making. The analytical hierarchy process (AHP) offer a method for organizing and making Judgments in such situations. It contributes to solving complex problems by structuring a hierarchy of criteria, stake holders and outcomes and by eliciting Judgements to develop priorities. It provides an effective structure for group decisions making by imposing a discipline on the group’s thought process. We now present the Computational details of finding weights in the original AHP, then fuzzify the hierarchical analysis (HA) by using the fuzzy numbers for the pair wise comparisons. Direct Computation of fuzzy eigen values and fuzzy eigen vectors (the fuzzy weight) from positive, reciprocal matrix is very complicated hence we fuzzify with an equivalent method to get the fuzzy weight. A supplementary evolutionary algorithm to estimate the fuzzy weight is also reviewed.
We use a bar over a letter to denote a fuzzy set. All the fuzzy sets will fuzzy subsets of the real numbers. If is a fuzzy set, is the value of the membership function at . An cut of , is
written as , the support of is , is the closure of the union of , .
A trapezoidal fuzzy number is defined by four numbers . Assuming
, the graph of (x) is a trapezoid with base on the interval
. If we get a triangular fuzzy. We write as
. Here we fuzzify using saaty’s original method of computing the weights.
The geometric mean method
is used to get weights if is a positive reciprocal matrix.
2. Hierarchical analysis:
We review the basic concepts needed to find weights in HA. In HA each person is asked to give ratios for each pair wise Comparison between issues for each criterion in a hierarchy, and also
between the criteria. For any criteria , if a person consider to be more important than ,then might
be or or . The numbers for the ratios will be taken from the set so could
be with , S and . The ratio indicate, for this expert, the strength in which
V. S. Mathu Suresh et al. / International Journal of Engineering Science and Technology (IJEST)
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dominates If then . That is , for all i,j with
. Let H be the matrix whose entries are the ratios . H is called
a positive reciprocal matrix. Since H is for criterion we will write for this matrix.
Assume that there are K criteria with a positive reciprocal matrix for each ,
. Also, the Judge must give pair wise comparisons of the criteria producing a positive reciprocal
matrix E. This structure is as in fig 1:
Fig. 1 Structure of the Hierarchy
Now we have to compute the weighs for each and
for E . Given any positive reciprocal matrix H, let the eigen values, counting a root of
multiplicity n n-times, be .The dominant (highest real positive) eigen value, is denoted by ,
so that for all , is a root of multiplicity 1. Corresponding to there is a
unique eigen vector so that
Where for all i and . This positive normalized vector gives the weights for
H. Then is positive , normalized, eigen vector corresponding to for , is the eigen
vector for E. The objective of HA is to rank the alternatives across all the criteria. Assuming that the reciprocal matrices and E are reasonably consistent, the final ranking of the alternatives is determined
by the vector where
. The weight for alternative is , . The alternatives are ranked according to the
numbers , . The hierarchial structure in fig 1 can be extended to any number of levels.
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Let , a vector of length m of all ones, let H be any positive
reciprocal matrix with sum = sum of all the elements in , = 1,2,…… . Now define
Then if
then ω is unique, positive , normalized eigen vector of H corresponding to (T.L. Satty, 1997). This
gives a method of computing the weights for and of E. We compute (4) for = 2,4,….. until the
vector stabilizes. Now eqn.(5) produces the approximation to ω.
3. Fuzzy hierarchical analysis:
Fuzzy ratios can now replace the ratios by experts as follows: The , , is now a fuzzy
number. , for all . The fuzzy numbers can be formed as follows
where As shown in Fig 2, there are eight types of fuzzy numbers,
with and other seven.
(a) Trapezoidal (b) Triangle (c) More than to (d) Less than to
(e) Between and (f) At (g) At most .
Fig. 2 Fuzzy numbers in FHA
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If then, is the reciprocal of the fuzzy
number . The reciprocals are shown in fig.3. Here the Judges are allowed to express uncertainty using the
fuzzy equivalents in fig 2.
(a) Reciprocal of fig 2 (a) , (b) Reciprocal of fig 2(b) , (c) Reciprocal of fig 2 (c) , (d) Reciprocal of fig 2 (d) , (e) Reciprocal of fig 2 (e) , (f) Reciprocal of fig 2 (f) , (g) Reciprocal of fig 2 (g)
Fig . 3 Reciprocals of fuzzy numbers in fig 2:
Now we need the α-cuts in Fig 2(a),(b) and 3(a),(b) it is obvious and simple in 2(c) and (d). In fig 2(c) the cut is , . For 3(c) it is
, . For fig 2(e) all α-cuts are . In fig 2(g) an cut is
and its reciprocal gives an α-cut for fig 3(g).
Now we assume that the elements in the fuzzy positive reciprocal matrices and are
, and . Some of the can be real numbers
. Now we have to compute fuzzy weights vectors and .
For this we now fuzzyfy equations (4) and (5). Let be a fuzzy positive, reciprocal matrix.
Choose . Let and we write
. Define positive , reciprocal , matrix as follows
(1) (2)
and
(3) for let
And define . Set .
The above is a continuous mapping for each in .
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As in J.J.Buckley , let
for all .
Now becomes cuts of fuzzy numbers which
produce the fuzzy weight vector .
To compute and we used the Evolutionary algorithm(EA) proposed by James J.
Buckly et.al (2001). Computing is a complicated, non-linear optimization problem and EAs are very
good tools for optimization. The search space is , so members of the population will be vectors in
. We have to estimate for selected values of , say and
1. Then for each and each We run EA to approximate the minimum
Then, for each and each we apply the EA to approximate the
maximum of This produces the approximations to
and the fuzzy weight vector for .
The general procedure used to solve (4) and (5) is : (1) Find the Crisp solutions: (2) Fuzzify the Crisp solution (3) The fuzzified Crisp solution gives the answer to the fuzzy problem
To verify the validity of the procedure we adopted the following with and
(i) when m = 3 Let
be the positive, reciprocal matrix. If is the unique positive, normalized, eigen vector
corresponding to , then according to Satty(1980)
Where,
We now fuzzify by substituting fuzzy numbers for , for and for ,
and use the extension principle to find fuzzy weights and ( which are obtained as
follows ) Since the are continuous, according to J.J.Buckley(1990):
For , where is an cut of . We see that
; .
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However may be increasing for some and decreasing for some other . Similarly for a function of
and a function of . Hence
For in , and
For in , and
For in , where
In this way we find the .
Let us denoted by method , the procedure of calculating the weights using the expression in this
section. Method I is the one which uses EA and equation and .
To test Method I we use the both methods on the following reciprocal matrix :
From J.J.Buckjey (1985), for and , the results obtained by both the
methods are presented in table 1. The result shows that EA computes good approximation to the fuzzy weights obtained using method II.
Table 1: Testing the evolutionary algorithm method ( method I ) on the ( ) problem.
Method I Method II
0 0.2 0.4 0.6 0.8 1.0
[ 0.5198 , 0.6788 ] [ 0.5416 , 0.6683 ] [ 0.5575 , 0.6701 ] [ 0.5757 , 0.6561 ] [ 0.6049 , 0.6339 ] [ 0.6261 , 0.6219 ]
[ 0.5193 , 0.6819 ] [0.5398 , 0.6612 ] [ 0.5512 , 0.6693 ] [ 0.5710 , 0.6522 ] [ 0.6013 , 0.6299 ] [ 0.6210 , 0.6196 ]
0
0.2 0.4 0.6 0.8 1.0
[ 0.1863 , 0.2483 ] [ 0.1916 , 0.2458 ] [ 0.1990 , 0.2385 ] [ 0.2024 , 0.2311 ] [ 0.2100 , 0.2230 ] [ 0.2173 , 0.2179 ]
[ 0.1857 , 0.2459 ] [ 0.1916 , 0.2488 ] [ 0.1972 , 0.2399 ] [ 0.2015 , 0.2313 ] [ 0.2099 , 0.2243 ] [ 0.2172 , 0.2172 ]
0.2 0.4 0.6 0.8 1.0
[ 0.1216 , 0.2146 ] [ 0.1266 , 0.1964 ] [ 0.1320 , 0.1827 ] [ 0.1450 , 0.1600 ] [ 0.1520 , 0.1520 ]
[ 0.1214 , 0.2143 ] [ 0.1263 , 0.1970 ] [ 0.1312 , 0.1835 ] [ 0.1440 , 0.1615 ] [ 0.1520 , 0.1520 ]
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Testing the EA with the regular method on a 3×3 example problem: (ii) when m = 4 :
In this case let
be a positive, reciprocal matrix. If is the positive, normalized, eigen
vector of H corresponding to , then from satty(1980) we have the formulas for
. We fuzzify these expressions producing fuzzy weights
. We use the same extension principle of Saaty to find for all ,
so that
For . As given in Satty (1980) the expressions are very complicated and hence
the difficulties in the computation of , , hence, used
another EA to estimate the , in eqns (24) and (25).
Method I is tested in the 4 × 4 matrix given in (26)
With .
The results are displayed in Table 2. In this also method I is a good approximation to the fuzzy weights , assuming that method II produced good estimate. We thus conclude that the new
method I can obtain good estimate of the fuzzy weights.
4. Application Selection of the best Professional College for Engineering Degree
We consider the problem of selecting the best Engineering College in the state among the top three for getting admission. The aspirants ( a group of 10 students who expect top ranks in their school ) wanted to evaluate the choice based on five criteria (i) Quality of teachers (ii) Infrastructure (iii) Hostel facilities (iv) Fees structure (v) Distance from the home town. Using F H A they constructed the following fuzzy reciprocal matrices.
For Quality of teachers,
For infrastructure ,
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For Hostel facility
For Fee structure ,
For Distance from home town.
For the criteria, where Q = Quality of teachers
I = Infrastructure H = Hostel facilities F = Fees structure D = Distance from home town. In the matrices the : (1) the first row / column corresponds to alternative ;(2) Second row /
column is ; (3) the third row / column corresponds to . Using our EA we compute the fuzzy weight
vectors for , , and for . Then from (3)
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Table 2. Testing method I on a example problem
Method I Method II
0 0.2 0.4 0.6 0.8 1.0
[ 0.4532 , 0.5271 ] [ 0.4599 , 0.5200 ] [ 0.4667 , 0.5147 ] [ 0.4715 , 0.5999 ] [ 0.4800 , 0.5031 ] [0.4889 , 0.5000 ]
[ 0.4514 , 0.5289 ] [ 0.4590 , 0.5215 ] [ 0.4661 , 0.5155 ] [ 0.4728 , 0.5102 ] [ 0.4805 , 0.5044 ] [ 0.4868 , 0.4978 ]
0
0.2 0.4 0.6 0.8 1.0
[ 0.1658 , 0.2568 ] [ 0.1665 , 0.2528 ] [ 0.1679 , 0.2499 ] [ 0.1715 , 0.2468] [ 0.1753 ,0.2438] [ 0.1791 , 0.2378 ]
[ 0.1609 , 0.2581 ] [ 0.1649 , 0.2538 ] [0.1685 , 0.2481 ] [ 0.1721 , 0.2461 ] [ 0.1758 , 0.2403 ] [ 0.1788 , 0.2365 ]
0
0.2 0.4 0.6 0.8 1.0
[ 0.1249 , 0.2299 ] [ 0.1285 , 0.2247 ] [ 0.1347 , 0.2199 ] [ 0.1400 , 0.2120 ] [ 0.1458 , 0.2099 ] [ 0.1495 , 0.2020 ]
[ 0.1231 , 0.2301 ] [ 0.1284 , 0.2249 ] [ 0.1338 , 0.2210 ] [ 0.1399 , 0.2128 ] [ 0.1445 , 0.2089 ] [ 0.1500 , 0.2039 ]
0 0.2 0.4 0.6 0.8 1.0
[ 0.1118 , 0.1428 ] [ 0.1124 , 0.1420 ] [ 0.1132 , 0.1409 ] [ 0.1148 , 0.1401 ] [ 0.1153 , 0.1387 ] [ 0.1159 , 0.1360 ]
[ 0.1118 , 0.1428 ] [ 0.1125 , 0.1412 ] [ 0.1138 , 0.1411 ] [ 0.1147 , 0.1385 ] [ 0.1159 , 0.1367 ] [ 0.1165 , 0.1351 ]
we get
For all j. The fuzzy weight for admission . Before showing the results, we discuss the
consistency and the ranking of fuzzy numbers.
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(33)
4.1 Consistency
A positive reciprocal matrix is said to be Consistent if for all . This implies that if
the Judge states for versus and gives for against , then to be
logically consistent this Judge must state for versus . If is consistent then
and in general Hence a measure of consistency is built around the difference
. is said to be the ‘ reasonably ’ consistency when is .
Now we need the meaning of and for two fuzzy numbers and
. Now we define
Now we can write if and where
.Let us use . Next , we write when is not greater than and is not
greater than . or , if
Then . We also say if or .
Now A fuzzy positive reciprocal matrix is said to be consistent if
for all . We shall now use the following theorem of J.J.Buckley ( 1985
).
Theorem : Let be a fuzzy, positive , reciprocal matrix with
.
Choose and from .
If is consistent , then is also consistent.
Under this we see that for and all are reasonally consistent. In fact
and are exactly consistent.
4.2. Ranking Fuzzy Numbers
In equation (32) we need to rank in order to get the final ranking of alternatives.
Let be all the un dominated fuzzy numbers . We say is un dominated if no .
Again be all un dominated after deleting all the fuzzy numbers in . Similarly, construct
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. Then all the corresponding to a in have the highest ranking , all the
having in have the second ranking etc.
5. Results
Here we estimated the fuzzy weight vectors for , . The EA method was
applied to get the fuzzy weight vector for . Fuzzy numbers are calculated for cuts for the
values of . Instead of displaying all we display in Table 3 the
for and . From the analysis we have and . We find
that the students most preformed criteria is the Infrastructure and then comes the quality of teachers and Hostel facilities.
The result reveals that the Fee structure and the distance from home town are not binding the selection provided the institution has good Infrastructure with Quality Teachers and a good Hostel.
Table 3. Fuzzy weights in the application
For
[ 0.1273 , 0.1793 ] [ 0.1340 , 0.1704 ]
[ 0.5941 , 0.7534 ] [ 0.6570 , 0.7135 ]
[ 0.2189 , 0.3263 ] [ 0.2526 , 0.2726 ]
For
[ 0.1122 , 0.2016 ] [ 0.1443 , 0.1443 ]
[ 0.3814 , 0.4758 ] [ 0.4716 , 0.4321 ]
[ 0.3852 , 0.4812 ] [ 0.4326 , 0.4319 ]
For
[ 0.4502 , 0.5001 ] [ 0.4558 , 0.4773 ]
[ 0.4472 , 0.5011 ] [ 0.4748 , 0.4956 ]
[ 0.0521 , 0.0692 ] [ 0.0615 , 0.0662 ]
For
[ 0.2212 , 0.4456 ] [ 0.2178 , 0.2179 ]
[ 0.4773 , 0.6992 ] [ 0.6847 , 0.6847 ]
[ 0.0678 , 0.1211 ] [ 0.1072 , 0.1072 ]
For
[ 0.0872 , 0.1269 ] [ 0.1042 , 0.1156 ]
[ 0.4361 , 0.5062 ] [ 0.4352 , 0.4481 ]
[ 0.3992 , 0.4695 ] [ 0.4474 , 0.4676 ]
For
[0.0762,0.1179] [0.0855,0.1075]
[0.0411,0.0712] [0.0462,0.0575]
[0.3211,0.3689] [0.3272,0.3695]
[0.0899,0.1356] [0.0999,0.1255]
[0.3847,0.4292] [0.3888,0.4292]
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