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Author's personal copy Applied Soft Computing 12 (2012) 3260–3275
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A balanced solution of a fuzzy soft set based decision making problem in medical science Tanushree Mitra Basu a , Nirmal Kumar Mahapatra b,∗ , Shyamal Kumar Mondal a a b
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, WB, India Department of Mathematics, Panskura Banamali College, Panskura RS 721152, WB, India
a r t i c l e
i n f o
Article history: Received 21 September 2010 Received in revised form 16 May 2012 Accepted 17 May 2012 Available online 2 June 2012 Keywords: Soft set Fuzzy soft set (FSS) Choice value Balanced solution Mean potentiality approach (MPA) Parameter reduction
a b s t r a c t The purpose of this paper is two folded. Firstly, the concept of mean potentiality approach (MPA) has been developed and an algorithm based on this new approach has been proposed to get a balanced solution of a fuzzy soft set based decision making problem. Secondly, a parameter reduction procedure based on relational algebra with the help of the balanced algorithm of mean potentiality approach has been used to reduce the choice parameter set in the parlance of fuzzy soft set theory and it is justified to the problems of diagnosis of a disease from the myriad of symptoms from medical science. Moreover the feasibility of this proposed method is demonstrated by comparing with Analytical Hierarchy Process (AHP), Naive Bayes classification method and Feng’s method. © 2012 Elsevier B.V. All rights reserved.
1. Introduction For tackling real life ambiguous situations we require methodologies which provide some form or other flexible information processing capacity. Generally the soft set theory is used to solve such problems. In the initial phase Molodtsov [3] proposed soft set as a completely generic mathematical tool for modeling uncertainties in the year 1999. As there is no limitation for the description of the objects; as a result researchers can select the form of parameters they require, which immensely simplifies the decision making process and make it more efficient in the absence of partial information. Maji et al. have done further research on soft set theory [4,6] and on fuzzy soft set theory [1,2,5]. To cope with fuzzy soft set based decision making problems, Maji et al. [5] presented a novel method for finding an optimal choice object. But Kong et al. [12] argued that the Roy-Maji method was incorrect and they presented a revised algorithm. Then Feng et al. [7] have proposed an adjustable approach to FSS based decision making by means of level soft sets. On the other hand Aktas and Cagman [13] initiated soft groups and Jun [14] applied soft sets to the theory of BCK/BCI-algebras. Then Feng et al. [15] defined soft semirings and established a connection between soft sets and semirings. Cagman et al [16,17] introduced a
∗ Corresponding author. Tel: +91 9434321347; fax: +91 3228252816. E-mail addresses:
[email protected] (T. Mitra Basu), nirmal
[email protected] (N.K. Mahapatra), shyamal
[email protected] (S.K. Mondal). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.05.006
new soft set based decision making method (uni-int decision making method) [17] which selects a set of optimum elements from the alternatives. They also gave the definition of soft matrices [16] and also proposed an algorithm for solving soft set based decision making problems using several operations of soft matrices defined by them. Moreover Maji et al. [4] utilize the thinking of attributes reduction in rough set theory to reduce parameters set of a soft set. But unreal optimal choice object may be obtained through this way. So Chen et al. [9] showed that the method of attributes reduction in rough set theory cannot simply transplant to parameters reduction in soft set theory, but the elaborate process of parameters reduction in soft set theory is not described by him. Zou et al [8] have given the detailed process of parameter reduction by help of SQL (Structural Quarry Language) introduced by Chambelin and Boyce [10]. According to Zou each parameter in soft set can be regarded as an attribute of relation and the values of each object in parameter set constitute a record of relation and therefore it is possible to utilize corresponding theories and thoughts of relational algebra to research several problems in soft set theory. Moreover Zhi Kong et al. [11] have given a new definition of normal parameter reduction of fuzzy soft sets and proposed an algorithm for it. But according to Feng’s method [7], the decision maker can select any level to form the level soft set. There does not exist any unique or uniform criterion for the selection of the level. So by this method the decision maker cannot decide which level is suitable to select the optimal choice object.
Author's personal copy T. Mitra Basu et al. / Applied Soft Computing 12 (2012) 3260–3275 Table 1(a) Tabular representation of the soft set (F, A).
d1 d2 d3 d4
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Table 1(b) Tabular representation of the fuzzy soft set (F, A).
e1
e2
e3
e4
0 1 0 1
1 0 1 0
0 1 1 0
0 0 0 1
To overcome such difficulties, in this paper we have introduced an efficient solution procedure named as mean potentiality approach and the corresponding algorithm, to obtain a balanced solution of a fuzzy soft set based decision making problem. Feng et al [7] shown that their method of solution of a fuzzy soft set based decision making problem is more efficient than Maji et al. [5] and Kong et al. [12]. In our presentation we have shown that our approach is more deterministic and so accurate than Feng’s method. Moreover we have compared our method with the well known Analytical Hierarchy Process (AHP) [18], Naive Bayes classification method [25] and showed that our result gives better output than those methods also. Again, parameter reduction is very important in decision making problem as it helps us to get the key parameters from a set of choice parameters. So we have proposed a new method based on relational algebra for the parameter reduction of fuzzy soft sets with the help of the balanced algorithm of mean potentiality approach. At last we have justified this method to a problem for diagnosis of a disease from the myriad of symptoms from medical science [19–24] and compared this method with the normal parameter reduction procedure introduced by Kong et al [11]. 2. Soft set based decision making in fuzzy environment
C1 C2 C3 C4
e2
.2 .9 .4 .6
0 1 .3 .4
e3 .3 .4 .8 0
e4 .9 .1 .5 .3
called a fuzzy soft set (FSS) over U, where F is a mapping given by, F : A → P(U). Example 2.2. Let U be the set of four cities, given by, U = {C1 , C2 , C3 , C4 } . Let E be the set of parameters (each parameter is a fuzzy word), given by, E = { highly, immensely, moderately, average, less}. Let A ⊂ E, given by, A = {highly, immensely, moderately, less} = {e1 , e2 , e3 , e4 }, where e1 e2 e3 e4
stands for the parameter ‘highly’, stands for the parameter ‘immensely’, stands for the parameter ‘moderately’, stands for the parameter ‘less’.
Now suppose that, F(e1 ) = {C1 /.2, C2 /.9, C3 /.4, C4 /.6}, F(e2 ) = {C2 /1, C3 /.3, C4 /.4}, F(e3 ) = {C1 /.3, C2 /.4, C3 /.8}, F(e4 ) = {C1 /.9, C2 /.1, C3 /.5, C4 /.3} Then the fuzzy soft set is given by (F, A)
We recall the definitions and properties of Soft Sets, Fuzzy Soft Sets over the common universe with some examples.
e1
= {highly polluted city = {C1 /.2, C2 /.9, C3 /.4, C4 /.6}, immensely polluted city = {C2 /1, C3 /.3, C4 /.4}, moderately polluted city = {C1 /.3, C2 /.4, C3 /.8},
2.1. Soft sets: [6] Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Let A ⊂ E. A pair (F, A) is called a soft set over U, where F is a mapping given by, F: A → P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. Example 2.1. Let U be the set of four dresses, given by, U = {d1 , d2 , d3 , d4 } . Let E be the set of parameters, given by, E = {costly, cheap, comfortable, beautiful, gorgeous }. Let A ⊂ E, given by, A = { costly, cheap, comfortable, beautiful } = {e1 , e2 , e3 , e4 } where e1 e2 e3 e4
stands for the parameter ‘costly’, stands for the parameter ‘cheap’, stands for the parameter ‘comfortable’, stands for the parameter ‘beautiful’.
Now suppose that, F is a mapping, defined as “dresses (·)” and given by, F(e1 ) = {d2 , d4 }, F(e2 ) = {d1 , d3 }, F(e3 ) = {d2 , d3 }, F(e4 ) = {d4 } . Then the Soft Set (F, A) = { costly dresses = {d2 , d4 }, cheap dresses = {d1 , d3 }, comfortable dresses = {d2 , d3 }, beautiful dresses = {d4 }}. The tabular representation of this soft set (F, A) is given in Table 1(a).
less polluted city = {C1 /.9, C2 /.1, C3 /.5, C4 /.3}} The tabular representation of (F, A) is given in Table 1(b).
2.3. Choice parameter and choice value [1,6] According to a decision making problem the parameters of a decision maker’s choice or requirement which forms a subset of the whole parameter set of that problem are known as choice parameters. Choice value of an object is the sum of the membership values of that object corresponding to all the choice parameters associated with a decision making problem. Example 2.3. Consider the Example 2.1. Now suppose Mr. X wants to buy a dress which is cheap, comfortable and beautiful. Then the choice parameters of Mr. X are e2 , e3 , e4 . Therefore the choice value of d1 the choice value of d2 the choice value of d3 the choice value of d4
is 1 + 0 +0 = 1 is 0 + 1 +0 = 1 is 1 + 1 +0 = 2 is 0 + 0 +1 = 1
2.4. Level soft set of a fuzzy soft set [7] 2.2. Fuzzy soft sets [1] Let U be an initial universe set and E be a set of parameters (which are fuzzy words or sentences involving fuzzy words). Let P(U) denotes the set of all fuzzy sets of U. Let A ⊂ E. A pair (F, A) is
Let ˝ = (F, A) be a fuzzy soft set over a finite universe U, where A ⊆ E and E is the parameter set. Let : A → [0, 1] be a membership function defined on A, then the fuzzy set in A, i.e., A is called a threshold fuzzy set. The level soft set of the fuzzy soft set ˝ with
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respect to the fuzzy set A is a crisp soft set L(˝ ; ) = (F , A) defined by F (a) = L(F(a); (a)) = {x ∈ U; F(a)(x) ≥ (a), ∀a ∈ A}
(1)
In order to better understand the above idea, Feng et al have given the following implementations. 2.4.1. The top-level soft set of a fuzzy soft set [7] Let ˝ = (F, A) be a fuzzy soft set over a finite universe U, where A ⊆ E and E is the parameter set. Based on the fuzzy soft set ˝ = (F, ˜ ˝ : A → [0, 1] by A), we can define a fuzzy set Amax ˜ ˝ where max ˜ ˝ (a) = maxF(a)(x), ∀a ∈ A max
(2)
x∈U
2.6.3. Selection operation of a relation The selection operation of a relation R is a new relation S composed of records selected from R which satisfy the given condition and denoted by S = F (R) = {t/t R ∧ F(t) = true }
where F is a logic expression which represents a selecting condition and the result of F is a logic value “true” or “false”. Therefore, actually selection operation selects records that can make logic expression F to be true. It is an operation from the angle of rows. 2.6.4. Projection operation of a relation The projection operation of a relation R is a new relation S only containing appointed attributes from R and denoted by
The fuzzy set Amax ˜ ˝ is called the max-threshold of fuzzy soft set ˝. The level soft set of ˝ with respect to the max-threshold Amax ˜ ˝, ˜ ˝ ) is called the top-level soft set of ˝ and simply namely L(˝; max denoted by L(˝ ; max).
S=
2.4.2. The mid-level soft set of a fuzzy soft set [7] Let ˝ = (F, A) be a fuzzy soft set over a finite universe U, where A ⊆ E and E is the parameter set. Based on the fuzzy soft set ˝ = (F, ˜ ˝ : A → [0, 1] by A), we can define a fuzzy set Amid where mid ˜
2.7. Analytical hierarchy process (AHP) [18]
˝
˜ ˝ (a) = 1 mid F(a)(x), ∀a ∈ A |U|
(3)
x∈U
The fuzzy set Amid ˜
˝
is called the mid-threshold of fuzzy soft set ˝
and simply denoted by L(˝ ; mid). 2.5. Normal parameter reduction [11] For a fuzzy soft set (F, P); P = {e1 , e2 , . . ., em } if there exists a subset A = {e1 , e2 , . . . , ep } of P satisfying d = d = ··· = e ∈A 1k e ∈A 2k
d , ek ∈A nk
k
k
(4)
(R) = {t[A]/t R}
(5)
A
where A represents a subset of the attributes set of R. Therefore, the projection operation is an operation from the angle of columns.
Analytical hierarchy process (AHP) is one type of approach for discriminating between competing options in the light of a range of criteria to be met in a decision making problem. It involves with (i) structuring the multiple choice of criteria into a set of hierarchy, (ii) assessing the relative importance of these criteria, (iii) comparing alternatives for each criterion, and (iv) determining an overall ranking of the alternatives. In this process the decision problem is decomposed into a hierarchy that consists of the most important elements of the decision problem. In developing an AHP, the top level is the ultimate goal of the decision at hand. The hierarchical structure consists of an overall goal, a group of options or alternatives for reaching the goal and a group of factors or criteria that relate the alternatives to the goal. The stepwise procedure of AHP is described in Fig. 1.
[where dik ; i = 1, 2, . . ., n ; k = 1, 2, . . ., p be the entries of
the tabular representation of (F, A)], A is dispensable, otherwise, A is indispensable. B ⊆ P is a normal parameter reduction of P if B is indispensable and d = d = ··· = d e ∈P−B 1k e ∈P−B 2k e ∈P−B nk
2.8. Naive Bayes classification method [25]
[here dik be the entries of the tabular representation of (F, P − B)], that is to say P − B is the maximal subset of P that the value of fP−B (·) keeps constant.
A Naive Bayes classifier is a very useful probabilistic classifier based on applying Bayes’ theorem (from Bayesian statistics) with strong (naive) independence assumptions. Using Bayes’ theorem, it can be written as
2.6. The elementary operations in relational algebra [8]
P(C/F1 , F2 , . . . , Fn ) = (P(C)P(F1 , F2 , . . . , Fn /C))/P(F1 , F2 , . . . , Fn )
k
k
k
The four elementary operations in relational algebra are union, difference, selection and projection. In a relation, a column describes an attribute and a row represents a record. 2.6.1. Union of two relations Let R and S be two relations with n common attributes and the values of the same attributes come from the same domain. The union of R and S is a new relation H with n attributes composed of records which belong to R or S and denoted by H = R ∪ S = {t/tR ∨ tS}. 2.6.2. Difference of two relations Let R and S be two relations with n common attributes and the values of the same attributes come from the same domain. The difference of R and S is a new relation H with n attributes composed of records which belong to R and not belong to S and denoted by H = R − S = {t/tR ∨ ¬ tS}.
= (P(C)ni=1 P(Fi /C))/Z
(6)
where C is the class variable, F1 , F2 , . . ., Fn are the feature variables and Z (the evidence) is a scaling factor dependent only on F1 , F2 , . . ., Fn , i.e., a constant if the values of the feature variables are known. In plain English the above equation can be written as Posterior =
(prior × likelihood) evidence
(7)
This is the Naive Bayes probabilistic model. The Naive Bayes classifier combines this model with a decision rule. One common rule is to pick the hypothesis that is most probable; this is known as the maximum posterior decision rule. The corresponding classifier is a function denoted by Classify and it is defined as Classify(f1 , f2 , . . . , fn ) = argmaxP(C = c)ni=1 P(Fi = fi /C = c)
(8)
c
where f1 , f2 , . . ., fn are the values of n features F1 , F2 , . . ., Fn respectively.
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Table 2 Tabular representation of (F, P) with choice values.
O1 O2
e1
e2
e3
Choice value
0.9 0.6
0.8 0.7
0.4 0.8
2.1 2.1
3.2. Optimality criteria To get a balanced solution of a fuzzy soft set based decision making problem with equally weighted choice parameters the following criteria must be satisfied: (i) At least one object satisfy all the choice parameters mostly. In other words, the choice value of at least one object be maximum. (ii) At least in one object with maximum choice value, the satisfaction (i.e., membership value) for every choice parameters, are almost same. There is not a huge difference of satisfaction from one choice parameter to another. They should be close to each other. The criterion − (ii) is illustrated through the following example. Example 3.1. Consider the tabular representation of a fuzzy soft set (F, P) with choice values. Table 2 shows that the choice values for the objects O1 , O2 are i ) between the membersame, but the non-negative differences (jk ship values of each object (Oi ) associated with the parameters ej i = | (O ) − (O )| and ek is given by jk ej ek i i 1 = 0.1, 1 = 0.5, 1 = 0.4 for O and 2 = So in this case, 12 1 13 23 12 2 = 0.2, 2 = 0.1 for O 0.1,13 2 23 Therefore the sum of these differences for O1 and O2 are given by
1 =
3 3
ij1 = 0.1 + 0.5 + 0.4 = 1.0
i=1 (j=1)(i = / j)
2 =
3 3
ij2 = 0.1 + 0.2 + 0.1 = 0.4
i=1 (j=1)(i = / j)
Fig. 1. Flow chart of AHP.
3. Optimality criteria for a balanced solution of a fuzzy soft set based decision making problem 3.1. Balanced solution Till now some researchers [5,7,12] have worked to get solution of the fuzzy soft set based decision making problems with equally weighted choice parameters. According to their methods the selected object may have considerable difference between the membership values for the choice parameters though they are equally weighted. In real life there are many problems in which selection is expected in such a way that all criteria, i.e., choice parameters associated with the selected object will be more or less of same importance, i.e., there will not be any significant difference between the membership values of the selected object for the choice parameters. In this parlance this paper focus a balanced solution in which all the choice parameters are satisfied mostly and the satisfactions (membership values) for every choice parameters are closed to each other as much as possible.
Now 1 is very much larger than 2 , which implies that there are far differences between the membership values of O1 for the choice parameters e1 , e2 , e3 . On the other hand, the small value of 2 indicates that satisfaction (membership values) of O2 associated with these same choice parameters are very closed to each other. Hence one of the most important optimality criteria for a balanced solution is to minimize this -value. 3.3. Measure of performance The measure of performance of a method (M) which satisfies the optimality criteria to solve a fuzzy soft set based decision making problem is defined as, the sum of the inverse of the summation of the non-negative differences between the membership values of the optimal object for the choice parameters and the choice value of the optimal object, i.e., it is mathematically defined as, M =
1 m m i=1 (j=1)(i = / j)
|ei (Op ) − ej (Op )|
+
m i=1
ei (Op )
(9)
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Table 3 Tabular representation of (F, P).
h1 h2 h3
Table 5 Tabular representation of L((F, P) ; mid) with choice values.
e3
e6
e7
0.8 0.6 0.7
0.7 0.9 0.5
0.2 0.7 0.8
where m is the number of choice parameters and ei (Op ) is the membership value of the optimal object (Op ) for the choice parameter ei . Suppose there are two methods M1 , M2 which satisfy the optimality criteria and their measure of performances are respectively M1 and M2 . Now if M1 > M2 then M1 be a better method than M2 , if M1 < M2 then M2 be a better method than M1 and if M1 = M2 then the performance of the both methods be the same.
h1 h2 h3
Example 3.2. Let U be the set of three houses, given by, U = {h1 , h2 , h3 }. Let E be the set of parameters (each parameter is a fuzzy word), given by, E = { beautiful, wooden, modern, well furnished, in the green surroundings, well ventilated, well situated } = {e1 , e2 , e3 , e4 , e5 , e6 , e7 } (say). Let the fuzzy soft set (F, E) describes the “attractiveness of the houses” and (F, E) is given by, (F, E)
= {beautiful houses = {h1 /.2, h2 /.9, h3 /.4}, wooden houses = {h2 /1, h3 /.3}, modern houses = {h1 /.8, h2 /.6, h3 /.7}, well furnished houses = {h1 /.9, h2 /.1, h3 /.5} houses with the green surroundings = {h2 /.8, h3 /.3}, well ventilated houses = {h1 /.7, h2 /.9, h3 /.5}, well situated houses = {h1 /.2, h2 /.7, h3 /.8}}
The set of choice parameters of Mr. X is, P = {modern, well ventilated, well situated } = {e3 , e6 , e7 }. Now the tabular representation of (F, P) is given in Table 3. Using top level soft set: Using this method, the tabular representation of the corresponding top-level soft set of (F, P) with the choice values of the houses is given in Table 4. In Table 4 the choice values of all the houses are equal. In this situation the decision maker may select any one of the houses (according to Feng’s algorithm) as his optimal choice. Suppose that the decision maker select the first house (h1 ). But by observation from Table 3, it should not be the best compared to other two houses (h2 and h3 ). Since for this house (h1 ), the third parameter is satisfied with membership value 0.2 which is distinctively less than that of other two houses h2 and h3 . Moreover all the choice parameters in this problem have the same weight. So it is necessary to balance among the membership values of the choice parameters of Table 4 Tabular representation of L((F, P) ; max) with choice values.
h1 h2 h3
e6
e7
Choice value
1 0 1
1 1 0
0 1 1
2 2 2
Table 6 Tabular representation of the fuzzy soft set (S1 , P1 ) with choice values.
o1 o2 o3
e1
e2
e3
e4
Choice value
0.9 0.8 0.7
0.7 0.6 0.5
0.3 0.4 0.5
0.2 0.3 0.4
2.1 2.1 2.1
Table 7 Tabular representation of L((S1 , P1 ) ; max) with choice values.
3.4. Limitations of Feng’s method [7] The Feng’s method is not very efficient in selecting the optimal object of a fuzzy soft set based decision making problem with equally weighted parameters. It posses some inherent drawbacks. We can realize this by the following examples.
e3
o1 o2 o3
e1
e2
e3
e4
Choice value
1 0 0
1 0 0
0 0 1
0 0 1
2 0 2
the optimal choice house. But the previous algorithms not necessarily striking the balance among the same. So, top level soft set approach is not suitable for this problem. Therefore, we will apply the mid level soft set approach to choose the best one (i.e., a balanced solution). Using mid level soft set: Using this method, the tabular representation of the corresponding mid-level soft set of (F, P) with the choice values of the houses is given in Table 5. Here also the choice values of all the houses are equal. In this situation the decision maker may select (according to Feng’s algorithm) any one of the houses as his optimal choice. Hence the same problem arises in this case also. In this situation, according to Feng’s method, the decision maker can select any level for further investigation for the optimal choice object. There does not exist any unique or uniform criterion for the selection of the level. So we have to choose the appropriate level by mathematical inspection which is laborious. There is no clear indication to choose the level. So by this method the decision maker cannot decide which level is suitable to select a balanced solution of the problem. Now we consider another example of a fuzzy soft set based decision making problem cited in Table 7 [7] to find out the draw back of Fengs’ Method. Here the tabular representation of this example is shown in Table 6. Example 3.3. Using top level soft set of Fengs’ Method, the following is the tabular representation of the soft set L((S1 , P1 ) ; max) with choice values. According to Feng’s algorithm, Table 7 gives o1 or o3 as the optimal choice object. Again the tabular representation of the soft set using mid level soft set, we have the following. Now Table 8 gives o2 as the optimal choice object (according to Feng’s method). But we can see that there is a huge difference in between the membership values of o1 as well as of o2 for the choice parameters. Such as for o1 ; e1 /0.9, e2 /0.7, e3 /0.3, e4 /0.2. Whereas we can see that there is a balance among the membership values Table 8 Tabular representation of the soft set L((S1 , P1 ) ; mid) with choice values.
e3
e6
e7
Choice value
1 0 0
0 1 0
0 0 1
1 1 1
o1 o2 o3
e1
e2
e3
e4
Choice value
1 1 0
1 1 0
0 1 1
0 1 1
2 4 2
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for the choice parameters of o3 . As all the choice parameters are equally weighted, o3 should be the optimal choice object. So we try to overcome such problems by proposing a new approach for decision making problem of fuzzy soft set as follows. 4. Mean potentiality approach To overcome the difficulties of Feng’s method, we introduce a new approach to obtain a synchronized solution of a fuzzy soft set based decision making problem with equally weighted choice parameters which comprises of some new notions as follows. Definition. The potentiality of a fuzzy soft set (pfs ) is defined as the sum of all membership values of all objects with respect to all parameters, i.e., mathematically it is defined as, pfs =
n m
ij
(10)
i=1 j=1
where ij is the membership value of the ith object with respect to the jth parameter, m is the number of objects and n is the number of parameters. Definition. The mean potentiality (mp ) of a fuzzy soft set is defined as it’s average weight among the total potentiality, i.e., mathematically it is defined as, mp =
pfs m×n
(11)
This is an approach by which we can find the appropriate level (for obtaining the optimal choice object maintaining a balance in between the choice parameters) to get the level soft of a fuzzy soft set. The detailed step-wise procedure as an algorithm is given in the following subsection. 4.1. Balanced algorithm for finding a balanced solution of a fuzzy soft set based decision making problem with equally weighted choice parameters In many decision making problems there is a selection of one object from a list of objects on the basis of some parameters chosen by a decision maker. In such problems when choice parameters are equally weighted and the decision maker likes to choose an object maintaining an equal balance between the membership values of all the choice parameters then the following algorithm is very useful to get the balanced solution of these decision making problems. Step 1
Step 2 Step 3
Step 4
Step 5
Step 6
Find a normal parameter reduction Q of the choice parameter set P. If it exists construct the tabular representation of (F, Q). Otherwise construct the tabular representation of the FSS (F, P) with the choice values of each object. Compute the potentiality (pfs ) of the FSS according to our definition. Then find out the Mean Potentiality (mp ) of the fuzzy soft set up to significant figures (where is the maximum number of significant figures among all the membership values of the objects concerned with the problem). Now form a mp -level soft set of the FSS and represent this in tabular form, then compute the choice value ci for each object Oi ∀i. If ck is maximum and unique among c1 , c2 ,. . .,cm where m is the number of objects (rows) then the optimal choice object is Ok and then the process will be stopped. If ck is not unique, then go to Step 6. Determine the non-negative difference between the largest and the smallest membership value in each
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Table 9 The tabular representation of (F, P).
h1 h2 h3
e3
e6
e7
Choice value
0.8 0.6 0.7
0.7 0.9 0.5
0.2 0.7 0.8
1.7 2.2 2.0
column and exhibit it as ˛i , i = 1, 2, . . ., n where n be the number of choice parameters. The same procedure is followed for each row (object) and Step 7 denote the difference values as ˇj ; j = 1, 2, . . ., m. Now take the average (˛) of the ˛i ’s up to significant Step 8 figures and named it as . Step 9 Then construct -level soft set and then compute the choice value ci for each object Oi , ∀i = 1, 2, . . ., m from its tabular representation. where Step 10 If cl is maximum and unique among c1 , c2 , . . . , cm m is the number of objects (rows) then the optimal choice object is Ol and then the process will be stopped. If cl is not unique, then go to Step 11. Step 11 If l has more than one value then we have to consider the object corresponding to the minimum value of ˇj (which may or may not be unique),j = 1, 2, . . ., m, as the optimal choice of the decision maker (Fig. 2). Example 4.1. To illustrate the basic idea of this algorithm, we apply it to some FSS based decision making problems. First, let us consider the decision making problem of Example 3.2. (1) Since P is indispensable, there does not exist any normal parameter reduction of P. Now the tabular representation of (F, P) with choice values is given in Table 9. (2) The potentiality of (F, P) is, pfs = 5.9. (3) The mean potentiality of (F, P) is, mp = 5.9/(3 × 3) = 5.9/9 =0.6 (since all membership values of Table 9 have one significant figure, therefore the maximum number of significant figures, i.e., = 1. So mp is taken up to one significant figure). (4) mp -level soft set of (F, P) is given in Table 10. (5) The optimal choice house is h2 , since max{ci , i = 1, 2, 3} = c2 . Example 4.2. Now consider a same type of decision making problem whose associated FSS is given in Table 11. Now let us apply the balanced algorithm on (F, P). (1) Let Q = {e1 , e2 , e3 } Then Q ⊂ P be a normal parameter reduction of P, h = h = since Q is indispensable and 1k e ∈P−Q e ∈P−Q 2k
h ek ∈P−Q 3k
= 10.7 = constant
k
k
Now the tabular representation of (F, Q) with choice values is given in Table 12. (2) So the potentiality of (F, Q) is, pfs = 6.12. mean potentiality of (F, Q) is, (3) The mp = 6.12/(3 × 3) = 6.12/9 =0.68 (since all the membership values of Table 12 have two significant figures, therefore here
= 2. So the value of mp is taken up to two significant figures). (4) Now the tabular representation of the mp -level soft set of (F, Q) is given in Table 13. Table 10 The tabular representation of L((F, P) ; 0.6).
h1 h2 h3
e3
e6
e7
Choice value
1 1 1
1 1 0
0 1 1
2 3 2
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Fig. 2. Flow chart of synchronized algorithm.
Table 11 Tabular representation of (F, P).
h1 h2 h3
Table 13 Tabular representation of L((F, Q) ; 0.6) with choice values.
e1
e2
e3
e4
e5
0.85 0.47 0.75
0.73 0.92 0.57
0.26 0.75 0.82
0.32 0.64 0.56
0.75 0.43 0.51
(5) Since all the houses have the same choice value (2), calculate the ˛i and ˇj values of (F, Q). (6, 7) See Table 14. 3 (8) Now ˛ = ˛ /3 = (0.38 + 0.35 + 0.56)/3 = 0.43 i=1 i Therefore = 0.43 (Since here = 2, the value of is taken up to two significant figures.) (9) So, the tabular representation of the -level soft set of (F, Q) is given in Table 15. (10) Here max{ci ; i = 1, 2, 3} = 3 = {c2 , c3 }. So we have to consider the ˇj values for j = 2 and 3. (11) Since in between ˇ2 (=0.45) and ˇ3 (=0.25); ˇ3 is minimum, we have h3 as the optimal choice house. Example 4.3. Consider the decision making problem of Example 3.3 (Table 16).
e1
e2
e3
Choice value
1 0 1
1 1 0
0 1 1
2 2 2
Table 14 Tabular representation of (F, Q) with ˛i and ˇj values. e1
e2
e3
ˇj
h1 h2 h3
0.85 0.47 0.75
0.73 0.92 0.57
0.26 0.75 0.82
0.59 0.45 0.25
˛i
0.38
0.35
0.56
(1) Since P is indispensable, there does not exist any normal parameter reduction of P. (2) The potentiality of (F, P) is, pfs = 6.3 (3) The mean potentiality of (F, P) is, mp = 6.3/(3 × 4) = 6.3/12 = 0.5. (4) mp -level soft set of (F, P) is given in Table 17. (5) The optimal choice object is o3 , since max{ci , i = 1, 2, 3} = c3 .
Table 15 Tabular representation of L((F, Q) ; 0.43) with choice values.
Table 12 The tabular representation of (F, Q) with choice values.
h1 h2 h3
h1 h2 h3
e1
e2
e3
Choice value
0.85 0.47 0.75
0.73 0.92 0.57
0.26 0.75 0.82
1.84 2.14 2.14
h1 h2 h3
e1
e2
e3
Choice value
1 1 1
1 1 1
0 1 1
2 3 3
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Table 16 Tabular representation of the fuzzy soft set with choice values.
o1 o2 o3
e1
e2
e3
e4
Choice value
0.9 0.8 0.7
0.7 0.6 0.5
0.3 0.4 0.5
0.2 0.3 0.4
2.1 2.1 2.1
5. A new approach for parameter reduction procedure of fuzzy soft set Parameter reduction is very important in decision making problem. By this process the number of parameters in a problem can be efficiently minimized. So in a decision making problem, the parameter reduction helps us to present the key parameters. Here we are proposing the following algorithm to reduce the number of parameters in the set of choice parameters in a fuzzy soft set based decision making problem. (i) At first apply the Balanced Algorithm to get the tabular representation (R) of the level soft set corresponding to the optimal choice object (Omax ). (ii) Secondly apply the relational algebra based reduction algorithm which consists of the following steps. (a) First construct a subset E1 of the choice parameter set P such that, E1 = {e : e ∈ E
and
Omax ∈ F(e)}
(12)
(b) Then compute RE1 which is the result of projection operation on E1 of R, i.e., RE1 is a relation comprising after selecting the columns corresponding to all parameters of E1 from R. (c) Compute each Rei ∀ei E1 where Rei is a relation obtained by selection operation on R which satisfies the selecting condition ei = 0 ; i = 1, 2, . . ., p where p be the cardinality of the set E1 . (d) Finally apply union operation on the relations Rei , ei E1 in such a way that the difference operation of the resulting relations from the relation RE1 , i.e., the combinations RE1 − ∪k Rei gives a relation comprising a row corresponding the optimal choice object Omax , where ∪k Rei is a relation resulting from union set of k- no of Rei (1 ≤ k ≤ p). Hence the set {ei : ∪k Rei } be the reduced parameter set of P (Fig. 3).
Fig. 3. Flow chart of parameter reduction procedure in a fuzzy soft set.
6. The diagnosis of a disease from the myriad of symptoms Generally in medical science a patient suffering from a disease may have multiple symptoms. Again it is also observed that there are certain symptoms which may be common to more than one diseases leading to diagnostic dilemma. Example 6.1 ([19,20,22]). Now we consider from medical science seven symptoms such as abdominal pain, fever, headache, weight loss, muscle pain, nausea vomiting, diarrhea which have more or less contribution in four diseases such as typhoid, peptic ulcer, food poisoning, acute viral hepatitis. Now, from medical statistics, the degree of availability of these seven symptoms in these four diseases are observed as follows. The degree of belongingness of all the symptoms abdominal pain, fever, headache, weight loss, muscle pain, nausea vomiting and diarrhea for the
diseases typhoid, peptic ulcer, food poisoning and acute viral hepatitis are {0.3,0.8,0.2,0.2,0.2,0.1,0.2},{0.9,0.2,0.1,0.1,0.1,0.1,0.1}, {0.6,0.3,0.1,0.6,0.2,0.6,0.7}, and {0.2,0.6,0.4,0.2,0.2,0.5,0.1} respectively. Suppose a patient who is suffering from a disease, have the symptoms P (abdominal pain, fever, headache, nausea vomiting and diarrhea). Now the problem is how a doctor detects the actual disease with effective symptoms among these four diseases for that patient. To solve this problem first we detect the disease which is most suited with the observed symptoms of the patient and then secondly we find the actual symptoms which is optimal for that disease. These are solved by Fuzzy Soft Set Decision making Technique. For solving these the following notations are used:
Table 17 The tabular representation of L((F, P) ; 0.5) with choice values.
o1 o2 o3
e1
e2
e3
e4
Choice value
1 1 1
1 1 1
0 0 1
0 0 0
2 2 3
(i) {abdominal pain, fever, headache, weight loss, muscle pain, nausea vomiting, diarrhea} = {e1 , e2 , e3 , e4 , e5 , e6 , e7 } (ii) {typhoid, peptic ulcer, food poisoning, acute viral hepatitis} = {d1 , d2 , d3 , d4 }
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T. Mitra Basu et al. / Applied Soft Computing 12 (2012) 3260–3275 Table 20 Tabular representation of L((F, P) ; max) with choice values.
Table 18 Tabular representation of (F, P) with choice values.
typhoid(d1 ) peptic ulcer(d2 ) food poisoning(d3 ) a . v . hepatitis(d4 )
e1
e2
e3
e6
e7
Choice value
0.3 0.9 0.6 0.2
0.8 0.2 0.3 0.6
0.2 0.1 0.1 0.4
0.1 0.1 0.6 0.6
0.2 0.1 0.7 0.1
1.6 1.4 2.3 1.9
e2
e3
e6
e7
Choice value
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 1
0 0 1 0
1 1 2 2
Table 21 Tabular representation of L((F, P) ; mid) with choice values.
Table 19 The tabular representation of L((F, P) ; 0.3) with choice values.
d1 d2 d3 d4
d1 d2 d3 d4
e1
e1
e2
e3
e6
e7
Choice value
1 1 1 0
1 0 1 1
0 0 0 1
0 0 1 1
0 0 1 0
2 1 4 3
d1 d2 d3 d4
e1
e2
e3
e6
e7
Choice value
0 1 1 0
1 0 0 1
1 0 0 1
0 0 1 1
0 0 1 0
2 1 3 3
Table 22 Pairwise comparison values.
Therefore, in the parlance of fuzzy soft set, the finite universe, U = {d1 , d2 , d3 , d4 } and the set of parameters, E = {e1 , e2 , e3 , e4 , e5 , e6 , e7 }. Now the fuzzy soft set (F, E) is defined as (F, E)
=
{the disease having abdominal pain = {d1 /.3, d2 /.9, d3 /.6, d4 /.2}, the disease having fever = {d1 /.8, d2 /.2, d3 /.3, d4 /.6}, the disease having headache = {d1 /.2, d2 /.1, d3 /.1, d4 /.4}, the disease having weight loss = {d1 /.2, d2 /.3, d3 /.1, d4 /.2}, the disease having nausea vomiting = {d1 /.1, d2 /.1, d3 /.6, d4 /.5}, the disease having diarrhea = {d1 /.2, d2 /.1, d3 /.7, d4 /.1}}
Value aij
Comparison description
1 3 5 7 9
Criteria i and j are of equal importance Criteria i is weakly more important than j Criteria i is strongly more important than j Criteria i is very strongly more important than j Criteria i is absolutely more important than j
6.2. By Feng’s method
Then the tabular representation of this fuzzy soft set (F, P) is given in Table 18. Now out of those diseases in U, the doctor has to detect the disease which qualifies with all of the symptoms of the patient as much as possible. So to solve this problem first we apply our developed method, i.e., (A) mean potentiality approach, then to compare the result obtained by this method subsequently we apply other three existing methods – (B) Feng’s method, (C) analytical hierarchy process and (D) Naive Baye’s classification method.
Using top level soft set (Table 20): Now from this table we observe that either d3 or, d4 be the optimal choice disease corresponding to the maximum choice value 2. Using mid level soft set (Table 21): Again from this table we also observe that either d3 or, d4 be the optimal choice disease corresponding to the maximum choice value 3. Hence according to Feng’s method the doctor may detect that the patient is suffering from either food poisoning (d3 ) or, acute viral hepatitis (d4 ). In this situation, the doctor will be puzzled regarding to which disease he treat the patient (Fig. 5).
6.1. By mean potentiality approach
6.3. By analytical hierarchy process
Since P is indispensable, there does not exist any normal parameter reduction of P. Now according to our mean potentiality approach, the potentiality of this fuzzy soft set (F, P) is, pfs = 7.2. The mean potentiality of (F, P) is, mp = 7.2/(4 × 5) = 7.2/20 = 0.3. Therefore, the tabular representation of the mp -level soft set of (F, P) is given in Table 19. Here the choice value of the disease d3 is maximum among the four diseases and therefore the disease most suitable with the symptoms is food poisoning (d3 ) (Fig. 4).
There are four alternatives d1 , d2 , d3 , d4 and five criteria e1 , e2 , e3 , e6 , e7 . Now the first step in AHP is to ignore the alternatives and just decide the relative importance of the criteria by comparing each pair of criteria and ranking them on the following scale: Comparing criterion i and criterion j, give a value aij (Table 22). Furthermore, if we set aij = k, then we set aji = 1/k. Now since in our problem all the criteria have been given equal importance so the preference on criteria is given in Table 23. After normalizing the preference matrix ( ij )5×5 (by dividing each entry of the matrix by the sum of the entries of the corresponding column), the overall weight (Wi ) assigned to ith criterion is given by this method as, Wi = 1/5 for i = 1, 2, 3, 4, 5 corresponding to the criteria e1 , e2 , e3 , e6 , e7 respectively. According to AHP, our next step is to evaluate all the alternatives (diseases) for each criterion (symptom), i.e., in this step the Table 23 The preference matrix ( ij )5×5 on criteria.
Fig. 4. Block diagram of MPA in medical diagnosis.
e1 e2 e3 e6 e7
e1
e2
e3
e6
e7
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
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Fig. 5. Block diagram of Feng’s method in medical diagnosis.
Table 24 The preference matrix (aij )4×4 for e1 .
d1 d2 d3 d4
d1
d2
d3
d4
1 5 3
1 5
1 3
1
3 1
2 7 4 1
1 3 1 7
1 2
1 4
Table 25 Relative scores for e1 .
d1 d2 d3 d4
d1
d2
d3
d4
Average
0.10526 0.52631 0.31578 0.052631
0.11931 0.59658 0.19886 0.08522
0.07272 0.65454 0.21818 0.05454
0.14285 0.5 0.2857 0.071428
0.11004 0.56936 0.25463 0.06595
preference matrix (aij )4×4 on alternatives for each criterion is determined by the same ranking scale described in Table 22. For instance, the preference matrix (aij )4×4 on alternatives for the 1st criterion e1 is given in Table 24. ) assigned to Then after normalizing Table 24, the weight (W1j jth alternatives (d1 , d2 , d3 , d4 ) for the criterion e1 will be obtained as the average value in the last column of Table 25. Similarly, we can calculate the weights Wij assigned to j-he alternatives (d1 , d2 , d3 , d4 ) for the ith criteria (e2 , e3 , e6 , e7 ). Now the table presenting the overall weights (Wij ) of all the alternatives for each criteria will be as in Table 26. Now the value (Vj ) for the alternative, i.e., disease (dj ) is given by, Vj =
5
Wi Wij ,
j = 1, 2, 3, 4
(13)
i=1
Therefore in this problem V1 = 0.22826 (the value for typhoid), V2 = 0.2799 (the value for peptic ulcer), V3 = 0.393195 (the value for food poisoning) and V4 = 0.267948 (the value for a.v. hepatitis) Table 26 Overall weights (Wij ) of alternatives.
e1 e2 e3 e6 e7
d1
d2
d3
d4
0.11004 0.50683 0.263049 0.08333 0.17807
0.56936 0.0777 0.56456 0.08333 0.10466
0.25463 0.11755 0.56456 0.41666 0.612599
0.06595 0.297837 0.45467 0.41666 0.10466
Since the value for food poisoning is maximum (0.393195), therefore this method (AHP) indicates that the patient is suffering from food poisoning (Fig. 6). 6.4. By Naive Baye’s classification method By this method it will be classified that a set of symptoms of a patient will be most suited with a disease based on the measured features. In this problem the features are E. Now a patient is affected with the symptoms P which is a sample to be classified. Now training set associated to the problem is the fuzzy soft set (F, E), which is given in Table 27. From the training set we have that the degree of the jth symptom ej in the ith disease di is the membership value (ij ) of di for ej . So ij can be treated as the mean value of the occurrence of ej in di . So the classifier created from the training set using the Gaussian distribution assumption as shown in Table 28. Now it is not possible to find out the variance (ij2 ) of the occurrence of the symptoms in the diseases from the training set. So due to unavailability of ij2 in such type of problems Naive Baye’s method cannot be applied for classification. Inspite of that if we apply this method for classification using Gaussian distribution, then the value of ij is assumed as a very small non-zero positive value (0.00001), since in such type of problems the variance ij2 will be 0 ∀ i, j as only one value (mean) of occurrence of the symptoms in the diseases is given. Since we have equiprobable classes so, the probability of the diseases are P(d1 ) = P(d2 ) = P(d3 ) = P(d4 ) =
1 4
(14)
Since the symptoms of the patient are equally weighted then the sample to be classified as a disease is given in Table 29. According to Naive Baye’s classification method we first find out the posterior of d1 , d2 , d3 , d4 and then choose the maximum posterior value to predict the disease. Now using Gaussian distribution for the class d1 the likelihoods are given by √ −(1−0.3)2 /2(0.00001)2 = P(e √ 1 = 1/d1 ) = (1/( 2 × 0.00001))e (1/( 2 × 0.00001)) × 1,
Table 27 Tabular representation of (F, E).
typhoid(d1 ) peptic ulcer(d2 ) food poisoning(d3 ) a . v . hepatitis(d4 )
e1
e2
e3
e4
e5
e6
e7
0.3 0.9 0.6 0.2
0.8 0.2 0.3 0.6
0.2 0.1 0.1 0.4
0.2 0.3 0.1 0.2
0.1 0.1 0.1 0.5
0.1 0.1 0.6 0.6
0.2 0.1 0.7 0.1
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Fig. 6. Block diagram of AHP in medical diagnosis.
Table 28 Mean values of the occurrence of symptoms. Class (disease)
Mean(e1 )
Mean(e2 )
Mean(e3 )
Mean(e4 )
Mean(e5 )
Mean(e6 )
Mean(e7 )
d1 d2 d3 d4
0.3 0.9 0.6 0.2
0.8 0.2 0.3 0.6
0.2 0.1 0.1 0.4
0.2 0.3 0.1 0.2
0.1 0.1 0.1 0.5
0.1 0.1 0.6 0.6
0.2 0.1 0.7 0.1
Table 29 The sample. Disease
e1
e2
e3
e6
e7
Sample
1
1
1
1
1
√ P(e √ 2 = 1/d1 ) = (1/( 2 × 0.00001)) × 1, P(e √ 3 = 1/d1 ) = (1/( 2 × 0.00001))√× 1, P(e6 = 1/d1 ) = (1/( 2 × 0.00001)) × 1 P(e7 = 1/d1 ) = (1/( 2 × 0.00001)) × 1. Therefore Posterior(d1 ) =
P(d1 )P(e1 =1/d1 )P(e2 =1/d1 )P(e3 = 1/d1 )P(e6 = 1/d1 )P(e7 = 1/d1 ) evidence 5/2
(1/2) = ×1=C evidence
5/2
(1/2) whereC = = constant evidence
(15)
Similarly, Posterior(d2 ) = Posterior(d3 ) = Posterior(d4 ) = C
(16)
Now since Posterior of d1 , d2 , d3 and d4 are all same, then by this method classification is not possible in such type of fuzzy soft set based decision making problems.
Hence this table shows that the methods mean potentiality approach and analytical hierarchy process give the same unique diagnosis and their performances are better than Feng’s method. So the doctor treats the patient according the disease food poisoning (d3 ). Now the effective symptoms of this disease will be found out according to our proposed algorithm of parameter reduction. Let d3 will be denoted by dmax . Now construct a subset E1 of the choice parameter set P such that, E1 = {e : e ∈ E and dmax ∈ F(e)}
(17)
Therefore, in this case E1 = {e1 , e2 , e6 , e7 }. Now let the relation, i.e., the tabular representation of the mp -level soft set of (F, P) be denoted by R. Then compute RE1 which is the result of projection operation on E1 of R, i.e., RE1 is a relation comprising after selecting the columns corresponding to e1 , e2 , e6 , e7 from R. So, the tabular representation of the relation RE1 is shown in Table 31. Now applying selection operation on the relation RE1 to get the another four new relations Rei ; i = 1, 2, 6, 7 such that e1 = 0, e2 = 0, e6 = 0 and e7 = 0 respectively in the table RE1 , i.e., in the terms of SQL, these four relations can be written as Rei = select ∗ from RE1 where ei = 0; i = 1, 2, 6, 7
6.5. Comparison of the above three methods Using above three methods to this problem we get the results given in Table 30. Table 30 Comparison table . Name of the methods
Solution of the problem
Measure of performance ( )
Analytical hierarchy process Feng’s method
Food poisoning (d3 ) any one of the diseases d3 , d4 Food poisoning (d3 )
2.63
Mean potentiality approach
2.63 or, 2.26 2.63
(18)
and using these selections we get the relations given in Tables 32–35. Finally we apply union operation on the relations Re1 , Re2 , Re6 and Re7 in such a way that the difference operation of the resulting relations from the relation RE1 gives the following relation comprising a row corresponding the optimal choice object dmax (Table 36). Table 31 The tabular representation of the relation RE1 .
d1 d2 d3 d4
e1
e2
e6
e7
1 1 1 0
1 0 1 1
0 0 1 1
0 0 1 0
Author's personal copy T. Mitra Basu et al. / Applied Soft Computing 12 (2012) 3260–3275 Table 32 The tabular representation of the relation Re1 .
d4
Table 37 Tabular representation of (F, A).
e1
e2
e6
e7
0
1
1
0
Table 33 The tabular representation of the relation Re2 .
d2
e1
e2
e6
e7
1
0
0
0
Table 34 The tabular representation of the relation Re6 .
d1 d2
e1
e2
e6
e7
1 1
1 0
0 0
0 0
Table 35 The tabular representation of the relation Re7 .
d1 d2 d4
act . dntl . abscess(d1 ) migraine(d2 ) act . sinusities(d3 ) ptslr . abscess(d4 )
e1
e2
e3
e4
e5
e6
e7
0.6 0.2 0.3 0.4
0 0 0.7 0
0.6 0.1 0.3 0.2
0.9 0.9 0.8 0.7
0 0.8 0.3 0.1
0.7 0 0.4 0.6
0.8 0 0 0.5
how a doctor reaches to the most suitable diagnosis based on the symptoms, history, physical examination and laboratory investigation of the patient. To solve this problem first we have to take the “AND Product” of the two fuzzy soft sets (F, P) and (G, B) and then we will apply (A) Mean Potentiality Approach to detect the disease which is most suited with the symptoms and other investigative procedures. Then to solve the same problem we will go through the method introducing by (B) Feng and the well known (C) analytical hierarchy process and at last we will compare all these three methods with respect to this example. 6.6. Solution of the problem
e1
e2
e6
e7
1 1 0
1 0 1
0 0 1
0 0 0
Now this relation can be obtained by each of the ways {RE1 − {Re6 ∪ Re1 }}, {RE1 − Re7 }, {RE1 − {Re1 ∪ Re7 }}, {RE1 − {Re2 ∪ Re7 }}, {RE1 − {Re6 ∪ Re7 }}. Therefore {e6 , e1 }, {e7 }, {e1 , e7 }, {e2 , e7 } and {e6 , e7 } are the parameter reductions of (F, P). This result indicates that when the doctor identifies the most suitable disease with the symptoms (P) of the patient, then the symptom list (P) can be reduced into either {e6 , e1 } or, {e7 } or, {e1 , e7 } or, {e2 , e7 } or, {e6 , e7 } which give the same result as the parameter set P in this decision making problem. Hence the key parameters, i.e., the effective symptoms of the patient are either {e6 , e1 } or, {e7 } or, {e1 , e7 } or, {e2 , e7 } or, {e6 , e7 }, i.e., {nausea vomiting, abdominal pain} or, {diarrhea} or, {abdominal pain, diarrhea} or, {fever, diarrhea} or, {nausea vomiting, diarrhea} which confirm the diagnosis of the patient suffering from food poisoning (d3 ). Example 6.2 ([21–24]). Suppose that the set of universe U contains four diseases, given by, U = {acute dental abscess, migraine, acute sinusities, peritonsillar abscess} = {d1 , d2 , d3 , d4 }(say) The set of parameters E is given by, E = {fever, runningnose, weakness, oro − facial pain, nausea vomiting, swelling, trismus (inability to open the mouth), history, physical examination, laboratory investigation } = {e1 , e2 , e3 , e4 , e5 , e6 , e7 , s1 , s2 , s3 }(say) Let A, B be two subsets of E given by, A = {e1 , e2 , e3 , e4 , e5 , e6 , e7 } and B = {s1 , s2 , s3 } Now suppose that (F, A) and (G, B) be two fuzzy soft sets describing respectively “symptoms of the diseases” and “decision making tools of the diseases”. Let the tabular representation of (F, A) and (G, B) are respectively given in Tables 37 and 38. Now suppose a patient who is suffering from a disease, have the symptoms P (fever, running nose, oro-facial pain). The problem is Table 36 The relation giving optimal object.
d3
3271
e1
e2
e6
e7
1
1
1
1
The tabular representation of the “AND Product” of the two fuzzy soft sets (F, P) and (G, B)is given in Table 39. 6.7. By mean potentiality approach Since P is indispensable, there does not exist any normal parameter reduction of P. Now the potentiality of (F, P) ∧ (G, B) is, pfs = 13. The mean potentiality of (F, P) is, mp = 13/(4 × 9) = 13/36 = 0.3(taking the first significant figure). Therefore, the tabular representation of the mp -level soft set of (F, P) ∧ (G, B) is given in Table 40. Here the choice value of the disease d3 is maximum among the four diseases and therefore the disease most suitable with the symptoms and the other investigative procedures is, acute sinusitis (d3 ). Hence the patient is suffering from acute sinusitis. 6.8. By Feng’s method Using top level soft set (Table 41): Here also the choice value of the disease d3 is maximum among the four diseases and hence this method also indicates the same disease acute sinusitis. 6.9. By analytical hierarchy process There are four alternatives d1 , d2 , d3 , d4 and nine pairs of criteria (e1 , s1 ), (e1 , s2 ), (e1 , s3 ), (e2 , s1 ), (e2 , s2 ), (e2 , s3 ), (e4 , s1 ), (e4 , s2 ), (e4 , s3 ) where each criterion is a pair of one symptom and one decision making tool. Now since in our problem all the criteria have been given equal importance, so by normalizing (by dividing each entry of the matrix by the sum of the entries of the corresponding column) the matrix of preference on criteria ( ij )9×9 , the overall weight (Wi ) assigned to ith criterion is given by this method as, Wi = 1/9 for i = 1, 2, 3, 4, 5, 6, 7, 8, 9 corresponding to the criteria (e1 , s1 ), (e1 , s2 ), (e1 , s3 ), (e2 , s1 ), (e2 , s2 ), (e2 , s3 ), (e4 , s1 ), (e4 , s2 ), (e4 , s3 ) respectively. According to AHP, our next step is to evaluate all the alternatives (diseases) for each criterion ((symptom, decision making tool)), i.e., Table 38 Tabular representation of (G, B).
act . dntl . abscess(d1 ) migraine(d2 ) act . sinusitis(d3 ) ptslr . abscess(d4 )
s1
s2
s3
0.6 0.8 0.8 0.6
0.8 0.3 0.4 0.8
0.4 0.6 0.7 0.3
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Table 39 Tabular representation of (F, P) ∧ (G, B).
d1 d2 d3 d4
(e1 , s1 )
(e1 , s2 )
(e1 , s3 )
(e2 , s1 )
(e2 , s2 )
(e2 , s3 )
(e4 , s1 )
(e4 , s2 )
(e4 , s3 )
Choice value
0.6 0.2 0.3 0.4
0.6 0.2 0.3 0.4
0.4 0.2 0.3 0.3
0 0 0.7 0
0 0 0.4 0
0 0 0.7 0
0.6 0.8 0.8 0.6
0.8 0.3 0.4 0.7
0.4 0.6 0.7 0.3
3.4 2.3 4.6 2.7
Table 40 Tabular representation of the mp -level soft set of (F, P) ∧ (G, B).
d1 d2 d3 d4
(e1 , s1 )
(e1 , s2 )
(e1 , s3 )
(e2 , s1 )
(e2 , s2 )
(e2 , s3 )
(e4 , s1 )
(e4 , s2 )
(e4 , s3 )
Choice value
1 0 1 1
1 0 1 1
1 0 1 1
0 0 1 0
0 0 1 0
0 0 1 0
1 1 1 1
1 1 1 1
1 1 1 1
6 3 9 6
Table 41 Tabular representation of the top-level soft set of (F, P) ∧ (G, B).
d1 d2 d3 d4
(e1 , s1 )
(e1 , s2 )
(e1 , s3 )
(e2 , s1 )
(e2 , s2 )
(e2 , s3 )
(e4 , s1 )
(e4 , s2 )
(e4 , s3 )
Choice value
1 0 0 0
1 0 0 0
1 0 0 0
0 0 1 0
0 0 1 0
0 0 1 0
0 1 1 0
1 0 0 0
0 0 1 0
4 1 5 0
Table 42 The tabular representation of (F, P).
d1 d2 d3 d4
(e1 , s1 )
(e1 , s2 )
(e1 , s3 )
(e2 , s1 )
(e2 , s2 )
(e2 , s3 )
(e4 , s1 )
(e4 , s2 )
(e4 , s3 )
0.6 0.2 0.3 0.4
0.6 0.2 0.3 0.4
0.4 0.2 0.3 0.3
0 0 0.7 0
0 0 0.4 0
0 0 0.7 0
0.6 0.8 0.8 0.6
0.8 0.3 0.4 0.7
0.4 0.6 0.7 0.3
Table 43 Overall weights (Wij ) of alternatives.
d1 d2 d3 d4
(e1 , s1 )
(e1 , s2 )
(e1 , s3 )
(e2 , s1 )
(e2 , s2 )
(e2 , s3 )
(e4 , s1 )
(e4 , s2 )
(e4 , s3 )
0.4 0.133 0.2 0.26
0.4 0.133 0.2 0.26
0.33 0.166 0.25 0.25
0 0 1 0
0 0 1 0
0 0 1 0
0.2143 0.286 0.286 0.2143
0.36 0.136 0.182 0.32
0.2 0.3 0.35 0.15
in this step the preference matrix (aij )4×4 on alternatives for each criterion is determined. Then after normalizing (aij )4×4 the overall weight (Wij ) assigned to ith alternative (i = 1, 2, 3, 4) for jth criterion (j = 1, 2, 3, 4, 5, 6, 7, 8, 9) is to be evaluated. But in such type of fuzzy soft set based decision making problems the overall weights (Wij ) of the alternatives (di ) for jth criterion is directly obtained by normalizing the membership values in the tabular representation of the fuzzy soft set (F, P). Now the fuzzy soft set associated with this problem is given in Table 42. After normalizing Table 40, Wij is obtained in Table 43. Now the value (Vi ) for the alternative, i.e., disease (di ) is given by, Vi =
5
Wj Wij ,
i = 1, 2, 3, 4
6.10. Comparison of the above three methods Using the above three methods to this problem we get the results which are given in Table 44. Hence this table shows that the methods MPA, AHP and Feng’s method give the same unique and confirmed diagnosis. So the doctor treats the patient according the disease acute sinusitis (d3 ). Example 6.3 ([19,23,24]). Suppose that the set of universe U contains three diseases, given by, U = {mumps, acute dental abscess, allergy} = {d1 , d2 , d3 }(say) The set of parameters E is given by, E = {running nose, weakness, fever, nausea vomiting, diarrhea, oro − facial pain, swelling }
(19)
j=1
Therefore in this problem V1 = 0.2116 (the value for acute dental abscess), V2 = 0.1282 (the value for migraine), V3 = 0.496 (the value for acute sinusitis) and V4 = 0.1616 (the value for peritonsillar abscess) Since the value for acute sinusitis is maximum (0.496), therefore this method (AHP) indicates that the patient is suffering from acute sinusitis.
Table 44 Comparison table. Name of the methods
Solution of the problem
Measure of performance ( )
Analytical hierarchy process Feng’s method Mean potentiality approach
Acute sinusitis (d3 )
4.71627
acute sinusitis (d3 ) Acute sinusitis (d3 )
4.71627 4.71627
Author's personal copy T. Mitra Basu et al. / Applied Soft Computing 12 (2012) 3260–3275 Table 45 The tabular representation of (F, P).
d1 d2 d3
e3
e6
e7
Choice value
0.8 0.6 0.7
0.7 0.9 0.6
0.2 0.6 0.8
1.7 2.1 2.1
= {e1 , e2 , e3 , e4 , e5 , e6 , e7 }(say) Let the fuzzy soft set (F, E) describes “the symptoms of the diseases” and given by, (F, E)
= {the disease having running nose = {d1 /.3, d2 /0, d3 /.6}, the disease having weakness = {d1 /.5, d2 /.6, d3 /.2}, the disease having fever = {d1 /.8, d2 /.6, d3 /.7}, the disease having nausea vomiting = {d1 /0, d2 /.1, d3 /.1}, the disease having diarrhea = {d1 /0, d2 /.1, d3 /0}, the disease having oro − facial pain = {d1 /.7, d2 /.9, d3 /.6}, the disease having swelling = {d1 /.2, d2 /.6, d3 /.8}}
Now suppose a patient who is suffering from a disease, have the symptoms P (fever, oro-facial pain, swelling). The problem is how a doctor reaches to the most suitable diagnosis based on these symptoms.
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Table 47 Column wise absolute differences and row wise absolute differences (˛i ˇj ) between the largest and smallest membership values in (F, P). e3
e6
e7
d1 d2 d3
0.8 0.6 0.7
0.7 0.9 0.6
0.2 0.6 0.8
˛i
0.2
0.3
0.6
ˇj 0.6 0.3 0.2
Table 48 Tabular representation of -level soft set of (F, P).
d1 d2 d3
e3
e6
e7
Choice value
1 1 1
1 1 1
0 1 1
2 3 3
Table 49 Tabular representation of L((F, P) ; max) with choice values.
d1 d2 d3
e3
e6
e7
Choice value
1 0 0
0 1 0
0 0 1
1 1 1
Now out of those diseases in U, the doctor has to detect the disease which qualifies with all of the symptoms of the patient as much as possible. So to solve this problem first we apply our developed method, i.e., (A) mean potentiality approach, then to compare the result obtained by this method subsequently we apply other two existing methods – (B) Feng’s method and (C) Analytical Hierarchy Process.
Using mid level soft set (Table 50): Here also the choice values of all the disease be the same. Hence according to Feng’s method the doctor may detect either d1 disease or, d2 disease or, d3 disease. In this situation, the doctor will be puzzled regarding to which disease he treat the patient.
6.11. By mean potentiality approach
6.13. By analytical hierarchy process
(1) Tabular representation of (F, P) is given in Table 45. Since P is indispensable, there does not exist any normal parameter reduction of P. (2) The potentiality of (F, P) is, pfs = 5.9 (3) The mean potentiality of (F, P) is, mp = 5.9/(3 × 3) = 5.9/9 =0.6 (4) mp -level soft set of (F, P) with choice values is given in Table 46. (5) Since max{ci , i = 1, 2, 3} = {c2 , c3 }, i.e., not unique, we have to go to the next step. (6, 7) The tabular representation of (F, P) with ˛i ˇj values is given in Table 47. (8) ˛ = (0.2 + 0.3 + 0.6)/3 =1.1/3 =0.366 Therefore = 0.3 (9) The -level soft set of (F, P) with choice values is given in Table 48. (10) Since max{ci , i = 1, 2, 3} = {c2 , c3 }, i.e., not unique, we have to go to the next step. (11) Since min{ˇ2 , ˇ3 } = ˇ3 , the optimal choice disease is d3 , i.e., Allergy.
There are three alternatives d1 , d2 , d3 and three criteria e3 , e6 , e7 . Now according to AHP, the value (Vj ) for the alternative, i.e., disease (dj ) is given by, Vj =
3
Wi Wij ,
j = 1, 2, 3
(20)
j=1
where Wi is the weight of ith criterion and Wij is the overall weight of jth alternative (d1 , d2 , d3 ) for ith criterion (e3 , e6 , e7 ). Therefore in this problem V1 = 0.311054 (the value for mumps), V2 = 0.349238 (the value for acute dental abscess) and V3 = 0.33966 (the value for allergy) Since the value for acute dental abscess is maximum (0.349238), therefore this method (AHP) indicates that the patient is suffering from acute dental abscess. 6.14. Comparison of the above three methods
Using top level soft set (Table 49): As the choice values of all the diseases are same, according to Feng’s method, the doctor may detect any one of the three diseases d1 , d2 , d3 .
Using above three methods to this problem we get the results which are given in Table 51. Hence this table shows that the performance of Mean Potentiality Approach is better than AHP and Feng’s method and it gives the unique and confirmed diagnosis compared to AHP and Feng’s method. So the doctor treats the patient according the disease allergy (d3 ).
Table 46 Tabular representation of L((F, P) ; 0.6) with choice values.
Table 50 Tabular representation of L((F, P) ; mid) with choice values.
6.12. By Feng’s method
d1 d2 d3
e3
e6
e7
Choice value
1 1 1
1 1 1
0 1 1
2 3 3
d1 d2 d3
e3
e6
e7
Choice value
1 0 1
1 1 0
0 1 1
2 2 2
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Table 51 Comparison table.
satisfying
Name of the method
Solution of the problem
Measure of performance ( )
Analytical hierarchy process Feng’s method
Acute dental abscess (d2 )
3.766
Any one of the three diseases d1 , d2 , d3 Allergy (d3 )
2.533 or, 3.766 or, 4.6
Mean potentiality approach
d ek ∈A 1k
d ek ∈A 2k
= ··· =
d , ek ∈A nk
[where
dik ; i = 1, 2, . . ., n ; k = 1, 2, . . ., p be the entries of the tabular representation of (F, A)], A is dispensable, otherwise, A is indispensable. B ⊆ P is a reduction of normal parameter P if B is indispensable and d = d = · · · = d 1k 2k e ∈P−B e ∈P−B e ∈P−B nk k
4.6
=
k
k
[here dik be the entries of the tabular representation of (F, P − B)], that is to say P − B is the maximal subset of P that the value of fP−B (·) keeps constant. Now in case of the Example 6.1, there does not exist any such dispensable subset A of P. Therefore here P is indispensable which implies that, it is not possible to have any normal parameter reduction of the choice parameter set P. Though we have found the parameter reductions {e6 , e1 }, {e7 }, {e1 , e7 }, {e2 , e7 }, {e6 , e7 } of the choice parameter set P by the parameter reduction method introduced by us. So our method of parameter reduction seems to be more efficient than Kong’s method. 9. Conclusion
Fig. 7. Performance graph of the methods.
7. Discussion In Example 6.1, Table 30 shows that measure of performance ( ) of mean potentiality approach (MPA) and Analytical hierarchy process (AHP) have the same and unique value (2.63). In one of the approaches in Feng’s method the is also 2.63 which is same as the other methods but in contrast applying another approach of Feng’s method the value is 2.26 which is less than that of the other methods and so there is no unique way by which we can get the better result. Moreover Naive Bayes classification method cannot be successfully applied in dealing with such type of decision making problems because standard deviations cannot be ascertained from the training set of such problems. In another Example 6.2 the measure of performance of MPA, AHP and Feng’s method are same and unique (4.71627). Furthermore in the Example 6.3 Feng’s method gives three different results by two different ways and the measure of performances are 2.533, 3.76, 4.6. Despite 4.6 is the highest value in this case there is no clear indication of obtaining this one. In other two methods AHP and MPA have unique result and MPA gives the maximum measure of performance (4.6) than AHP. The following graph represents the measure of performance of these three methods additionally involving six more problems of which three (4.1, 4.2, 4.3) have been cited (Fig. 7). 8. Comparison of Kong’s normal parameter reduction procedure with our newly proposed parameter reduction method At first we will try to apply the normal parameter reduction procedure introduced by Kong et al. [11] in Example 6.1 to obtain a normal parameter reduction of the choice parameter set P. In general, by Kong’s method, for the fuzzy soft set (F, P); P = {e1 , e2 , . . ., em } if there exists a subset A = {e1 , e2 , . . . , ep } of P
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