Applied Soft Computing 24 (2014) 559–571
Contents lists available at ScienceDirect
Applied Soft Computing
journal homepage: www.elsevier.com/locate/asoc
A new method for solving dual hesitant fuzzy assignment problems
with restrictions based on similarity measure
Pushpinder Singh ∗
Department of Computer Science, Palacky University, 17. listopadu 12, CZ-77146 Olomouc, Czech Republic
a r t i c l e
i n f o
Article history:
Received 3 February 2014
Received in revised form 20 May 2014
Accepted 1 August 2014
Available online 12 August 2014
Keywords:
Fuzzy sets
Hesitant fuzzy sets
Dual hesitant fuzzy sets
Similarity measures
Assignment problems
Bidirectional approximate reasoning
systems
a b s t r a c t
Zhu et al. (2012) proposed dual hesitant fuzzy set as an extension of hesitant fuzzy sets which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual
hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are
represented by two sets of possible values. Therefore, in accordance with the practical demand these sets
are more flexible, and provides much more information about the situation. In this paper, the axiom definition of a similarity measure between dual hesitant fuzzy sets is introduced. A new similarity measure
considering membership and nonmembership degrees of dual hesitant fuzzy sets has been presented
and also it is shown that the corresponding distance measures can be obtained from the proposed similarity measures. To check the effectiveness, the proposed similarity measure is applied in a bidirectional
approximate reasoning systems. Mathematical formulation of dual hesitant fuzzy assignment problem
with restrictions is presented. Two algorithms based on the proposed similarity measure, are developed
to finds the optimal solution of dual hesitant fuzzy assignment problem with restrictions. Finally, the
proposed method is illustrated by numerical examples.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
In 1952, Votaw and Orden [28] first proposed the assignment
problems. It marked the beginning of the development of the classical assignment problem. Assignment problems are widely applied
in manufacturing and service systems. Assignment is a special type
of linear programming problem where the objective is to assign n
number of jobs to n number of persons at a minimum cost (time).
In classical assignment problems all the parameters are taken as
crisp or precise, but in real life situations the parameters of the
assignment problems are often imprecise rather than a fixed real
number. The time/cost taken by a machine/person may vary due
to various reasons eg., the job performance of a worker may correlate to the time taken to finish the task. Every worker needs a
minimum time to perform an assigned task. Therefore, the time
parameter behaves as an imprecise number for the decision maker.
To deal with these type of situations Zadeh [35] in 1965, proposed
the theory of fuzzy sets. Fuzzy set theory proved to be a useful tool
to handel the uncertainty in real life problems.
∗ Tel.: +420 608828914.
E-mail addresses: [email protected], [email protected]
http://dx.doi.org/10.1016/j.asoc.2014.08.008
1568-4946/© 2014 Elsevier B.V. All rights reserved.
In recent years, fuzzy assignment and fuzzy transportation problems have received much attention. For instance, Lin and Wen [19]
concentrate on the assignment problem where costs are not deterministic numbers but imprecise ones. The elements of the cost
matrix of the assignment problem are subnormal fuzzy intervals
with increasing linear membership functions, whereas the membership function of the total cost is a fuzzy interval with decreasing
linear membership function. By the max–min criterion suggested
by Bellman and Zadeh, the fuzzy assignment problem can be treated
as a mixed integer nonlinear programming problem and also they
showed that this problem can usually be simplified into either a
linear fractional programming problem or a bottleneck assignment
problem. Huang and Zhang [8] proposed a mathematical model for
the fuzzy assignment problem with restriction of qualification. The
transforming model as certain assignment problem with restriction
of qualification is set methods for judging the existence of solution for this problem is given by transforming the beneficial matrix
into the decision matrix; furthermore, they transform the beneficial matrix into solution matrix when the problem has a solution
and then the assignment problem with restriction of qualification is
transformed into the traditional assignment problem. Jana and Roy
[11] presented a new intuitionistic fuzzy optimization approach
to solve the a multi-objective linear programming problem under
uncertainty. The idea was based on extension of fuzzy optimization.
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P. Singh / Applied Soft Computing 24 (2014) 559–571
They considered a multi-objective linear programming with
equality and inequality constraints with intuitionistic fuzzy goals.
Their fuzzy non-linear membership and non-membership function
have been taken for the degree of rejection of objectives and constraints together with the degree of satisfaction. Then it converts
the said problem into a conventional linear programming problem. Kumar and Gupta [13] chose some fuzzy assignment problems
and fuzzy travelling salesman problems which cannot be solved by
using the fore-mentioned method. Two methods were proposed for
solving fuzzy assignment problems and fuzzy travelling salesman
problems. Kaur and Kumar [12] proposed a new algorithm for solving fuzzy transportation problems using generalized trapezoidal
fuzzy numbers.
Furthermore, Li et al. [16] investigated that numerous papers
based on various search methods across a wide variety of applications have appeared in the literature over recent years. Most of
these methods apply the following same approach to address the
problems at hand: at each iteration of the search, they first apply
their search methods to generate new solutions, then they calculate the objective values (or costs) by taking some constraints
into account, and finally they use some strategies to determine the
acceptance or rejection of these solutions based upon the calculated objective values. However, they pointed out that calculating
the exact objective value of every resulting solution is not a must,
particularly for highly constrained problems where such a calculation is costly and the feasible regions are small and disconnected.
Moreover, in many combinatorial problems there are poor-cost
solutions where possibly just one component is misplaced and
all others work well. Although these poor-cost solutions can be
the intermediate states towards the search of a high quality solution, any cost-oriented criteria for solution acceptance would deem
them as inferior and consequently probably suggest a rejection.
To address the above issues, they proposed a pattern recognitionbased framework with the target of designing more intelligent
and more flexible search systems for two real world cases of the
assignment problem, i.e. the hospital personnel scheduling and
educational timetabling. Mukherjee and Basu [23] set a mathematical models of the assignment problem with restriction on person
cost depending on efficiency/qualification and restriction on job
cost where both the costs are considered as intuitionistic fuzzy
numbers. Restriction of qualification is in the form of the maximum intuitionistic fuzzy cost which can be offered to a person
depending on his/her efficiency/qualification. Also, Mukherjee and
Basu believe that restrictions on the intuitionistic fuzzy cost which
can be spent for doing a particular job makes the problem of intuitionistic fuzzy assignment problem with restrictions more realistic
than the problems found in the literature so far. A heuristic method
was constructed for showing the existence of the solution so that
both the constraints are satisfied. Lin [18] constructed an algorithm
to solve the fractional assignment problem based on parametric
approaches. The algorithm performs optimization once and overcomes degeneracy and also the main features of the algorithm are
an effective initial heuristic approach, a simple labelling procedure and an implicit primal-dual schema. Mehlawat and Kumar
[22] studied a multi-objective multi-choice assignment problem
considering cost and time objectives subject to some realistic constraints including multi-job assignment. They assume that the
decision-maker provides multiple aspiration levels regarding both
cost and time objectives using discrete choices as well as interval
values. To obtain efficient allocation plans, they use multi-choice
goal programming methodology to solve the assignment problem.
On the other hand, the measure of similarity between two fuzzy
concepts, as an important content in fuzzy mathematics, has gained
attention from researcher for their wide applications in some areas
such as pattern recognition, machine learning, decision making,
real and market prediction, etc. In literature very large number
of distance and similarity measures for fuzzy sets and intuitionistic fuzzy set have been proposed, for example Chen and Tan [4]
proposed two similarity measures for measuring the degree of similarity between vague sets. De et al. [5] defined some operations
on intuitionistic fuzzy sets. Szmidt and Kacprzyk [25] introduced
the Hamming distance between intuitionistic fuzzy sets and proposed a similarity measure between intuitionistic fuzzy sets based
on the distance. Dengfeng and Chuntian [6] also proposed similarity
measures of intuitionistic fuzzy sets and applied these similarity measures to pattern recognition. Liang and Shi [17] proposed
several similarity measures to differentiate different intuitionistic
fuzzy sets and discussed the relationships between these measures.
Mitchell [21] interpreted intuitionistic fuzzy sets as ensembles of
ordered fuzzy sets from a statistical viewpoint to modify Dengfeng
and Chuntian measures [6]. Hung and Yang [9] proposed another
method to calculate the distance between intuitionistic fuzzy sets
based on the Hausdorff distance and used it to propose several
similarity measures between intuitionistic fuzzy sets. Liu [20] proposed some similarity measures between intuitionistic fuzzy sets
and applied these measure methods in pattern recognition. Hung
and Yang [10] proposed similarity measures by inducing Lp metric. Xu [30] pointed out the the drawbacks of existing methods
on measures of similarity between vague sets and proposed a
new method on measures of similarity between vague sets. Lee
[14] proposed a novel score function by taking into account the
expectation of the hesitancy degree of interval-valued intuitionistic fuzzy sets and identified the best alternative in muticriteria
decision-making problems by proposing a multicriteria fuzzy decision making method which deals with interval-valued intuitionistic
fuzzy sets. Xu [32] introduced some relations and operations of
interval-valued intuitionistic fuzzy numbers and define some types
of matrices, including interval-valued intuitionistic fuzzy matrix,
interval-valued intuitionistic fuzzy similarity matrix and intervalvalued intuitionistic fuzzy equivalence matrix and also proposed
a method, based on distance measure, for group decision making with interval-valued intuitionistic fuzzy matrices. Guha and
Chakraborty [7] introduced a distance measure for intuitionistic
fuzzy numbers and studied the metric properties of the proposed
measure. Singh [24] proposed a similarity measure for intervalvalued intuitionistic fuzzy sets by studying some properties of
similarity measure and applied in pattern recognition and in bidirectional approximate reasoning systems.
In 2012, Zhu et al. [36] proposed a dual hesitant fuzzy sets
(DHFSs) as a extension of HFSs which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets [34]
as special cases, and investigated the basic operations and properties of DHFSs. Also, they gave the application of DHFSs in group
forecasting. DHFSs consist of two parts, one is the membership
hesitancy function and another is the nonmembership hesitancy
function. The existing sets, including fuzzy sets, intuitionistic fuzzy
set, HFSs, and fuzzy multi sets, can be regarded as special cases
of DHFSs. The membership degrees and nonmembership degrees
of DHFSs are represented by two sets of possible values. Therefore, it has the desirable characteristics and advantages of its own
and appears to be a more flexible method to be valued in multifold ways according to the practical demands, taking into account
much more information given by decision makers. However, the
existing measures cannot be used to deal with distance and similarity measure between DHFSs. Due to the fact that membership
hesitancy and nonmembership hesitancy are very common in decision making, therefore in order to find the optimal solutions of dual
hesitant fuzzy assignment problem with restrictions, it is necessary to develop similarity measure for DHFSs. In this paper, a new
similarity measure for DHFSs based on the membership degree
and nonmembership degree has been proposed. Also, it is verified that the proposed measure satisfy the axiom definition of a
P. Singh / Applied Soft Computing 24 (2014) 559–571
similarity measure between DHFSs. Effectiveness of the proposed
similarity measure has been tested in bidirectional approximate
reasoning systems. Mathematical formulation of dual hesitant
fuzzy assignment problem with restrictions has been introduced.
Two algorithms, based on the proposed similarity measure, are
developed to finds the optimal solution of dual hesitant fuzzy
assignment problem with restrictions. The rest of this paper is organized as follows: Section ‘Preliminaries’ discusses basic definitions
of HFSs, DHFSs, ranking of DHFSs and some the existing distance
and similarity measures for HFSs. Section ‘Similarity measure for
dual hesitant fuzzy sets’ proposes new similarity measures based
membership degree and nonmembership degree and, the relationship between distance and similarity measure is analyzed. In
Section ‘Dual hesitant fuzzy assignment problem with restrictions’,
mathematical formulation of dual hesitant fuzzy assignment problem with restrictions is presented and also a procedure to find the
optimal solution of dual hesitant fuzzy assignment problem with
restrictions is developed. In Section ‘Illustrative example’, a numerical example is taken to illustrate the proposed method. Finally, we
conclude this paper in “Conclusions” section.
2. Preliminaries
In this section, we review the basic definitions of HFSs, DHFs and
some existing distance measures for DHFSs.
pair d(x) = {h(x), g(x)} is called a dual hesitant fuzzy element (DHFE) denoted by d = {h, g}, with the conditions:
0 ≤ hD , gD ≤ 1, 0 ≤ hD + gD ≤ 1 and hD ∈ h(x), gD ∈ g(x), hD ∈
h+ (x) = ∪hD ∈h(x) max{hD }, gD ∈ g + (x) = ∪gD ∈g(x) max{gD }
From Definition 4, we can see that it consists of two parts, that is,
the membership hesitancy function and the nonmembership hesitancy function, supporting a more exemplary and flexible access
to assign values for each element in the domain, and can handle
two kinds of hesitancy in this situation. The existing sets, including
fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy
multisets, can be regarded as special cases of DHFSs [36].
Given a DHFE d, then some special DHFEs are defined as follows
[36]:
(1)
(2)
(3)
(4)
Complete uncertainty: d = {{0}, {1}};
Complete certainty: d = {{1}, {0}};
Complete ill-known (all is possible): d = [0, 1];
Nonsensical element: d = ∅ (h = ∅ , g = ∅).
2.3. Ranking of dual hesitant fuzzy sets
Let d = {hd , gd } be a DHFSs. Zhu et al. [36] introduced a score function sd , and accuracy function pd , of an DHFSs, which is represented
as follows:
sd =
2.1. Hesitant fuzzy sets
1 1 hd −
gd
#g
#h
(2)
1 1 hd +
gd
#g
#h
(3)
hd ∈h
Torra and Narukawa [27] and Torra [26] firstly proposed the
concept of a hesitant fuzzy set, which is defined as follows:
and
Definition 1. [26] Let X be a fixed set, a HFS on X is in terms of a
function that when applied to X returns a subset of [0,1], which can
be represented as the following mathematical symbol:
pd =
A = {x, hA (x)|x ∈ X}
,where hA (x) is a set of values in [0,1], denoting the possible membership degrees of the elements x ∈ X to the set M. For convince,
hA (x) is called a hesitant fuzzy element denoted by h [33].
Definition 2. [26,27] Given a hesitant fuzzy element h, its lower
and upper bounds are defined as h− (x) = min h(x) and h+ (x) = max
h(x), respectively.
Definition 3. [26,27] Given a hesitant fuzzy element h, Aenv(h) is
called the envelope of h which is represented by (h− , 1 − h+ ), with
the lower bound h− and upper bound h+ .
From this definition, we can obtain the relation between a
hesitant fuzzy set and an intuitionistic fuzzy set, i.e., Aenv(h) is
defined as {x, (x), (x)}, with and defined by (x) = h− (x),
(x) = 1 − h+ (x), x ∈ X.
561
hd ∈h
gd ∈g
gd ∈g
where # h and # g be the number of elements in h and g, respectively.
Let d1 and d2 be any two DHFSs. Based on the score function s
and the accuracy function p, Zhu et al. [36] defined the following
order relation:
(i) if sd1 > sd2 , then d1 is superior to d2 , denoted by d1 d2 ;
(ii) if sd1 < sd2 , then d2 is superior to d1 , denoted by d1 ≺ d2 ;
(ii) if sd1 = sd2 , then
(a) if pd1 = pd2 , then d1 is equivalent to d2 , denoted by d1 ∼ d2 ;
(b) if pd1 > pd2 , then d1 is superior than to d2 , denoted by
d1 d2 .
(c) if pd1 < pd2 , then d2 is superior than to d1 , denoted by
d1 ≺ d2 .
2.4. Distance measures for hesitant fuzzy sets
2.2. Dual hesitant fuzzy sets
Xu and Xia [33] introduced the following axioms for distance
and similarity measure between HFSs.
Definition 4. [36] Let X be the fixed set, then dual hesitant fuzzy
set (DHFS) D on X is defined as:
Definition 5. [33] Let M and N be two HFSs on X = {x1 , x2 , . . . , xn },
then the distance measure between M and N is defined as d(M, N),
which satisfies the following properties:
D = {x, h(x), g(x)}
in which h(x) and g(x) are two sets of some vales in [0,1], denoting
the possible membership degrees and nonmembership degrees of
the element x ∈ X to the set D, respectively, with the conditions:
0 ≤ hD , gD ≤ 1, 0 ≤ hD + gD ≤ 1
(1)
where hD ∈ h(x), gD ∈ g(x), hD ∈ h+ (x) = ∪hD ∈h(x) max{hD }, and
gD ∈ g + (x) = ∪gD ∈g(x) max{gD } for all x ∈ X. For convenience, the
(1) 0≤ d(M, N) ≤ 1 ;
(2) d(M, N) = 0 if and only if M = N ;
(3) d(M, N) = d(N, M).
Definition 6. [33] Let M and N be two HFSs on X = {x1 , x2 , . . . , xn },
then the similarity measure between M and N is defined as S(M, N),
which satisfies the following properties:
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P. Singh / Applied Soft Computing 24 (2014) 559–571
(i) 0≤ S(M, N) ≤ 1 ;
(ii) S(M, N) = 1 if and only if M = N ;
(iii) S(M, N) = S(N, M).
(6) If If = 2, a hesitant normalized Euclidean-Hausdorff distance
obtained as:
By analyzing Definitions 5 and 6, it is noted that S(M,
N) = 1 − d(M, N), accordingly, Xu and Xia [33] mainly discussed the
distance measures for HFSs.
Also, they pointed out that in most cases, l(hM (xi )) =
/ l(hN (xi )),
and for convenience, let lxi = max{l(hM (xi )), l(hN (xi ))} for each xi ∈ X.
To operate correctly, we should extend the shorter one until both
of them have the same length when we compare them. To extend
the shorter one, the best way is to add the same value several times
in it. In fact, we can extend the shorter one by adding any value
in it. The selection of this value mainly depends on the decision
makers’ risk preferences. Optimists anticipate desirable outcomes
and may add the maximum value, while pessimists expect unfavorable outcomes and may add the minimum value. For example,
let hM (xi ) = {0.1, 0.2, 0.3}, hN (xi ) = {0.4, 0.5}, and l(hM (xi )) > l(hN (xi )).
To operate correctly, we should extend hN (xi ) to hN (xi ) = {0.4, 0.4,
0.5} until it has the same length of hM (xi ), the optimist may extend
hN (xi ) as hN (xi ) = {0.4, 0.5, 0.5} and the pessimist may extend it as
hN (xi ) = {0.4, 0.4, 0.5}. Although the results may be different if we
extend the shorter one by adding different values, this is reasonable
because the decision makers’ risk preferences can directly influence
the final decision.
Based on the well known Hamming distance and Euclidean distance, Xu and Xia [33] proposed the following distance measures:
(1) Hesitant normalized Hamming distance:
⎡
⎤
lx
⎡
⎢1 1 i
(
n
2
lx
n
i=1
lxi
(j)
(j)
|hM (xi ) − hN (xi )|)
⎤1/2
⎥
⎦
j=1
⎡
⎢1 1 i
(
n
lx
n
i=1
(j)
lxi
(j)
(j)
|hM (xi ) − hN (xi )|)
⎤1/
⎥
⎦
, > 0
j=1
(j)
where hM (xi ) and hN (xi ) are the largest values in hM (xi ) and
hN (xi ), respectively.
On applying the Hausdroff metric to the distance measure the
following distance measures were proposed:
(4) Generalized hesitant normalized Hausdroff distance
dghnh (M, N) =
1
(j)
(j)
max|hM (xi ) − hN (xi )|
n
j
n
1/
i=1
(5) Specially, if = 1 the hesitant normalized HammingHausdorff distance obtained as:
1
(j)
(j)
max|hM (xi ) − hN (xi )|
n
j
n
dhnhh (M, N) =
Combining the above equations, Xu and Xia [33] proposed a
hybrid hesitant normalized Hamming distance, a hybrid hesitant
normalized Euclidean distance and a generalized hybrid hesitant
normalized distance:
(7)
1
dhhne (M, N) =
2n
n
1
lxi
lx
i
i=1
(j)
|hM ((xi )
−
(j)
hN (xi )|
+
(j)
max|hM (xi )
j
+
2
(j)
(j)
max|hM (xi )−hN (xi )|
j
+
2
(j)
(j)
max|hM (xi )−hN (xi )|
j
−
(j)
hN (xi )|
j=1
(8)
dghne (M, N) =
1
2n
n
1
lxi
lx
i
1/2
2
(j)
(j)
|hM (xi )−hN (xi )|
j=1
i=1
(9)
dhhnh (M, N) =
1
2n
n
i=1
1
lxi
lx
i
1/
(j)
(j)
|hM (xi )−hN (xi )|
j=1
3.1. Motivation
(3) Generalized hesitant normalized distance:
i=1
j=1
(2) Hesitant normalized Euclidean distance:
dghn (M, N) = ⎣
1/2
i
i=1
dhne (M, N) = ⎣
n
3. Similarity measure for dual hesitant fuzzy sets
1 1 (j)
(j)
⎣
dhnh (M, N) =
|hM (xi ) − hN (xi )|⎦
n
lxi
n
dhneh (M, N) =
2
1
(j)
(j)
max|hM (xi ) − hN (xi )|
n
j
i=1
Dual hesitant fuzzy sets are extension of hesitant fuzzy sets,
unlike hesitant fuzzy sets, it permits membership value and nonmembership value of an elements to a set of several possible
values between 0 and 1. DHFSs consist of two parts, that is, the
membership hesitancy function and the nonmembership hesitancy
function, supporting a more exemplary and flexible access to assign
values for each element in the domain, and we have to handle two
kinds of hesitancy in this situation. In real life cases, we do not confront an interval of possibilities (as in interval valued fuzzy sets
or interval valued intuitionistic fuzzy sets [1]), or some possibility
distributions (as in Type-2 Fuzzy sets [3,29]) on the possible values, or multiple occurrences of an element (as in Fuzzy multisets
[34]), but several different possible values indicate the epistemic
degrees whether certainty or uncertainty. For example, in a multicriteria decision-making problem, some decision makers consider
as possible values for the membership degree of x into the set A a
few different values 0.1, 0.2, and 0.3, and for the nonmembership
degrees 0.4, 0.5 and 0.6 replacing just one number or a tuple. So,
the certainty and uncertainty on the possible values are somehow
limited, respectively, which can reflect the original information
given by the decision makers as much as possible. DHFSs can take
much more information into account, the more values we obtain
from the decision makers, the greater epistemic certainty we have,
and thus, compared to the existing sets, DHFS can be regarded as a
more comprehensive set, which supports a more flexible approach
when the decision maker provide their judgments. The similarity
measure for hesitant fuzzy sets, proposed by Xu and Xia [36] and
solution of the fuzzy assignment problem with restrictions using
similarity measures, proposed by Mukherjee and Basu [23], motivate us to propose the similarity measures for DHFSs as a new
extension. Also in the next section we will give a procedure to find
the optimal solution of dual hesitant fuzzy assignment problem
with restrictions.
P. Singh / Applied Soft Computing 24 (2014) 559–571
3.2. Proposed similarity measure
The values in a DHFE are usually given as disorders, so we
arrange them in a decreasing order. For a DHFE d = {h, g}, let : (1,
2, . . . , n) → (1, 2, . . . , n) be a permutation satisfying h(s) ≥ h(s+1) for
s = 1, 2, . . . , n − 1, and h(s) be the sth largest value in h; let ı : (1, 2,
. . . , m) → (1, 2, . . . , m) be a permutation satisfying gı(t) ≥ gı(t+1) for
t = 1, 2, . . . , m − 1, and gı(t) be the tth largest value in g.
Now, first we define the axioms definition of similarity, then we
propose the similarity measures for DHFSs.
Definition 7. Let A and B be two DHFSs on a unverse of discourse X = {x1 , x2 , . . . , xn } denoted as A = {xi , hA (xi ), gB (xi )|xi ∈ X}
and B = {xi , hB (xi ), gB (xi )|xi ∈ X}, respectively. then the similarity
measure between A and B is defined as s(A, B), which satisfies the
following properties:
(P1)
(P2)
(P3)
(P4)
0≤ S(A, B)) ≤ 1 ;
S(A, B) = 1 if and only if A = B ;
S(A, B) = S(B, A);
Let C be any DHFS, if A ⊆ B ⊆ C, then S(A, B) ≥ S(A, C) and S(B,
C) ≥ S(A, C).
Practically, in most of the cases, the values of number of
elements in membership degree and non membership degree
may not be equal i.e., l(hA (xi )) =
/ l(hB (xi )) and k(gA (xi )) =
/ k(gB (xi )),
where l(hA (xi )) and l(hB (xi )) represents the number of values in
hA (xi ) and hB (xi ), respectively and k(gA (xi )) and k(gB (xi )) represents the number of values in gA (xi ) and gB (xi ), respectively. Let
lxi = max{l(hA (xi )), l(hB (xi ))} and kxi = max{k(gA (xi )), k(gB (xi ))} for
each xi ∈ X. One can make them have the same number of elements through adding some elements to the DHFS which has less
number of elements. According to the pessimistic principle, the
smallest element will be added. Therefore, if l(hA (xi )) < l(hB (xi )) or
k(hA (xi )) < k(hB (xi )), hA (xi ) of hB (xi ) should be extended by adding
the minimum value in it until it has the same length as gA (xi ) of
gB (xi ).
We define the similarity measure the similarity measure
between A and B as follows:
Let hAB (i) =
Definition 8.
gAB (i) =
1
2kx
kx
i
1
2lx
lx
i
i
s=1
|hA(s) (i) − hB(s) (i)| and
|g
(i) − gB(s) (i)|. Then we define
s=1 A(s)
i
n
1 p
p
S(A, B) = 1 − √
(hAB (i) + gAB (i))
p
n
(4)
i=1
Here, 0< p < ∞, lxi = max{l(hA ), l(hB )}, kxi = max{k(hA ), k(hB )} for
each xi ∈ X, where l(hA ) and l(hB ) represents the number of values in
hA (xi ) and hB (xi ), respectively, and, k(hA ) and k(hB ) represents the
number of values in gA (xi ) and gB (xi ), respectively.
Theorem 1. Let A, B and C be any DHFSs, then S(A, B) is the similaity
measure.
Proof. We prove S(A, B) satisfy axioms (P1)−(P4). It is seen that
S(A, B) satisfies (P1)−(P3). For (P4), we gave the following proof.
Let
AC (i) = hAC (i) + gAC (i)
and
AB (i) = hAB (i) + gAB (i).
Then
AB (i) − AC (i) = (hAB (i) + gAB (i)) − (hAC (i) + gAC (i))
lx
kx
i |h
i |g
= ( 2l1
(i) − hB(s) (i)| + 2k1
(i) − gB(s) (i)|) −
s=1 A(s)
s=1 A(s)
( 2l1
xi
xi
lx
|h
(i) − hC(s) (i)| +
s=1 A(s)
i
xi
kx
1
2kx
i
i
s=1
|gA(s) (i) − gC(s) (i)|) =
BC (i) ≥ 0.
Therefore S(A, B) ≥ S(A, C) .
In similar way, we can prove S(B, C) ≥ S(A, C) . Theorem 2.
S(A, B) = 1 if and only if A is the same as B
563
Proof.
Let A is the same as B⇒ hA(s) (i) = hB(s) (i) and
gA(s) (i) = gB(s) (i), therefore from Definition 9, we have S(A, B) = 1.
Conversely,
let
S(A,
lx
B)kx= 1⇒ hAB (i) = − gAB (i)) ⇒
1
1
i |h
i |g
(i)
−
h
(i)|
=
−
(i) − gB(s) (i)|⇒
B(s)
s=1 A(s)
s=1 A(s)
2l
2k
1
lx
xi
i
hA(s) (i) +
1
kx
i
gA(s) (i) =
1
lx
i
xi
hB(s) (i) +
1
kx
i
gB(s) (i).
Therefore
from Section ‘Ranking of dual hesitant fuzzy sets’, we have
p(A) = p(B) ⇒ A = B
It is well known that distance measure and similarity measure are dual concept. Therefore we may use similarity measure
to define a distance measure. Let f be monotonic decreasing function. Since 0 ≤ S(A, B) ≤ 1. On applying f, we have f (1) ≤ f (S(A, B)) ≤
(1)
≤ 1. f (0) ⇒ 0 ≤ f (S(A,B))−f
f (0)−f (1)
Thus we may define the distance measure between DHFSs A and
B as follows:
d(A, B) =
f (S(A, B)) − f (1)
f (0) − f (1)
(5)
According to the Theorem 1, d(A, B) satisfies the properties
P(1)−P(4).
If we select f(x) = 1 − x. Then a distance measure between A and
B is denoted as follows:
d(A, B) = 1 − S(A, B)
(6)
If f(x) = e−x . Then a distance measure between A and B is denoted
as follows:
d(A, B) =
e−S(A,B) − e−1
1 − e−1
Also if f (x) =
d(A, B) =
1
.
1+x
(7)
Then,
1 − S(A, B)
1 + S(A, B)
(8)
3.3. Effectiveness of proposed similarity measure in a
bidirectional approximate reasoning systems
Here, we apply the proposed similarity measure for DHFSs
in a bidirectional approximate reasoning systems to check the
validity of proposed similarity measure. First, let us consider the following single-input-single-output forward approximate reasoning
scheme:
1
2
3
4
5
6
R1 : If Ỹ is Ã1 , then Z̃ is B̃1 .
R2 : If Ỹ is Ã2 , then Z̃ is B̃2 .
...
Rp : If Ỹ is Ãp , then Z̃ is B̃p .
Fact: Ỹ is Ã∗
Consequence: Z̃ is B̃∗
In this scheme, Rj (1 ≤ j ≤ t) is the jth production rule, Ãj and Ã∗
are DHFSs of the universe of discourse O, where O = {o1 , o2 , . . . , on },
and B̃j and B̃∗ are DHFSs of the universe of discourse P, where P = {p1 ,
p2 , . . . , pm }.
Applying Chen et al. method [2], we can derive the following
results:
S(Ã1 , Ã∗ ) = k1 ⇒ the deduced consequence of rule R1 is “Z̃ is B̃1 ,
∗
where S(Ã1 , Ã ) is the similarity measurement between DHFSs Ã1
and Ã∗ ,
k
B̃1∗ = k1 ∗ B̃1 = [p1 {{1 − (1 − h11 ) 1 }, {(g11 )k1 }}, p2 {{1 −
k1
k
k1
(1 − h12 ) }, {(g12 ) }}, ..., pm {{1 − (1 − h1m ) 1 }, {(g1m )k1 }}].
S(Ã2 , Ã∗ ) = k2 ⇒ the deduced consequence of rule R2 is “Z̃ is
k
B̃2 , where B̃2∗ = k2 ∗ B̃2 = [p1 {{1 − (1 − h21 ) 2 }, {(g21 )k2 }}, p2 {{1 −
k2
k
k2
(1 − h22 ) }, {(g22 ) }}, ..., pm {{1 − (1 − h2m ) 2 }, {(g2m )k2 }}].
...
564
P. Singh / Applied Soft Computing 24 (2014) 559–571
S(Ãt , Ã∗ ) = kt ⇒ the deduced consequence of rule Rt is “Z̃
k
is B̃t , where B̃t∗ = kt ∗ B̃t = [p1 {{1 − (1 − ht1 ) t }, {(gt1 )kt }}, p2 {{1 −
kt
kt
kt
(1 − ht2 ) }, {(gt2 ) }}, ..., pm {{1 − (1 − htm ) }, {(gtm )kt }}].
Thus, the deduced consequence of the approximate reasoning
scheme is “Z̃ is B̃∗ ”, where
= [p1 {h∗j1 ∈ (h11 ∪ h21 ∪ ... ∪ hj1 )|h∗j1 ≥
B̃∗ = B̃1∗ ∪ B̃2∗ ∪, ..., ∪B̃t∗
k
∗ ∈ (g
∗
max(∪min[1 − (1 − hj1 ) j ])}, {gj1
11 ∪ g21 ∪ ... ∪ gj1 )|gj1 ≥
j
max(∪min[(gj1 )kj ])}}, p2 {h∗j2 ∈ (h12 ∪ h22 ∪ ... ∪ hj2 )|h∗j2 ≥
j
k
∗ ∈ (g
∗
max(∪min[1 − (1 − hj2 ) j ])}, {gj2
12 ∪ g22 ∪ ... ∪ gj2 )|gj2 ≥
j
max(∪min[(gj2 )kj ])}}, ..., pm {h∗jm ∈ (h1m ∪ h2m ∪ ... ∪ hjm )|h∗jm ≥
j
k
∗ ∈ (g
∗
max(∪min[1 − (1 − hjm ) j ])}, {gjm
1m ∪ g2m ∪ ... ∪ gjm )|gjm ≥
j
max(∪min[(gjm )kj ])}}] and “∪” is union operator of DHFSs,
j
(1 ≤ j ≤ t).
Example 1. Let us consider the following forward approximate
reasoning system based on DHFSs:
In this scheme, Rj (1 ≤ j ≤ t) is the jth production rule, Ãj and Ã∗
are DHFSs of the universe of discourse O, where O = {o1 , o2 , . . . , on },
and B̃j and B̃∗ are DHFSs of the universe of discourse P, where P = {p1 ,
p2 , . . . , pm }.
Again, on applying Chen et al. method [2], we can derive the
following results:
S(Ã1 , Ã∗ ) = l1 ⇒ the deduced consequence of rule R1 is “Z̃
l
is Ã1 , where Ã∗1 = l1 ∗ Ã1 = [o1 {{1 − (1 − h11 ) 1 }, {(g11 )l1 }}, o2 {{1 −
l
l
(1 − h12 ) 1 }, {(g12 )l1 }}, ..., om {{1 − (1 − h1m ) 1 }, {(g1m )l1 }}]
S(Ã2 , Ã∗ ) = l2 ⇒ the deduced consequence of rule R2 is “Z̃
l
is Ã2 , where Ã∗2 = l2 ∗ Ã2 = [o1 {{1 − (1 − h21 ) 2 }, {(g21 )l2 }}, o2 {{1 −
l
l
(1 − h22 ) 2 }, {(g22 )l2 }}, ..., om {{1 − (1 − h2m ) 2 }, {(g2m )l2 }}].
...
S(Ãt , Ã∗ ) = lt ⇒ the deduced consequence of rule Rt is “Z̃
l
is Ãt , where Ã∗t = lt ∗ Ãt = [o1 {{1 − (1 − ht1 ) t }, {(gt1 )l1 }}, o2 {{1 −
lt
lt
l1
(1 − ht2 ) }, {(gt2 ) }}, ..., om {{1 − (1 − htm ) }, {(gtm )l1 }}].
Thus, the deduced consequence of the approximate reasoning
scheme is “Z̃ is Ã∗ ”, where
Ã∗ = Ã∗1 ∪ Ã∗2 ∪, ..., ∪Ã∗t
= [p1 {h∗j1 ∈ (h11 ∪ h21 ∪ ... ∪ hj1 )|
l
1
2
3
4
5
R1 : If Ỹ is Ã1 , then Z̃ is B̃1 .
R2 : If Ỹ is Ã2 , then Z̃ is B̃2 .
R3 : If Ỹ is Ã3 , then Z̃ is B̃3 .
Fact: Ỹ is Ã∗
Consequence: Z̃ is B̃∗
∗ ∈ (g
∗
h∗j1 ≥ max(∪min[1 − (1 − hj1 ) j ])}, {gj1
11 ∪ g21 ∪ ... ∪ gj1 )|gj1 ≥
j
max(∪min[(gj1 )lj ])}}, p2 {h∗j2 ∈ (h12 ∪ h22 ∪ ... ∪ hj2 )|h∗j2 ≥
j
j
max(∪min[(gj2 )lj ])}}, ..., pm {h∗jm ∈ (h1m ∪ h2m ∪ ... ∪ hjm )|h∗jm ≥
j
In this system, DHFSs, Ã1 = [o1 {0.5, 0.4, 0.3}, {0.4, 0.3}, o2
{0.6, 0.4}, {0.4, 0.2}, o3 {0.3, 0.2, 0.1}, {0.6, 0.5}] Ã2 = [o1 {0.7,
0.6, 0.4}, {0.3, 0.2}, o2 {0.7, 0.6}, {0.3, 0.2}, o3 {0.7, 0.6, 0.4},
{0.2, 0.1}] Ã3 = [o1 {0.6, 0.4, 0.3}, {0.3}, o2 {0.6, 0.5}, {0.3}, o3
{0.6, 0.5}, {0.3, 0.1}] Ã∗ = [o1 {0.8, 0.7, 0.6}, {0.2, 0.1}, o2 {0.7,
0.6}, {0.2}, o3 {0.4, 0.3}, {0.2, 0.1}] B̃1 = [p1 {0.3, 0.2, 0.1}, {0.6,
0.4}, p2 {0.3, 0.2}, {0.6, 0.1}, p3 {0.8, 0.4}, {0.2, 0.1}] B̃2 = [p1
{0.2, 0.1}, {0.4, 0.3}, p2 {0.7, 0.5}, {0.2, 0.1}, p3 {0.6, 0.3}, {0.3,
0.2}] B̃3 = [p1 {0.6, 0.3, 0.1}, {0.4, 0.3}, p2 {0.3, 0.2}, {0.6, 0.1},
p3 {0.5, 0.3}, {0.4, 0.3}] On applying the proposed similarity measure, we get S(Ã1 , Ã∗ ) = 0.72, S(Ã2 , Ã∗ ) = 0.84, and
S(Ã3 , Ã∗ ) = 0.78, respectively.
Then, we obtain B̃1∗ = [p1 {0.236, 0.12, 0.007}, {0.69, 0.52},
p2 {0.23, 0.15}, {0.69, 0.19}, p3 {0.686, 0.307}, {0.314, 0.191}]
B̃2∗ = [p1 {0.171, 0.085}, {0.363, 0.259}, p2 {0.636, 0.441}, {0.259,
0.145}, p3 {0.537, 0.259}, {0.363, 0.259}] B̃3∗ = [p1 {0.511,
0.243, 0.079}, {0.489, 0.391}, p2 {0.243, 0.160}, {0.671, 0.166},
p3 {0.418, 0.243}, {0.489, 0.391}].
Thus, B̃∗ = B̃1∗ ∪ B̃2∗ ∪ B̃3∗ = [p1 {0.511, 0.243, 0.079, 0.85.0.171,
0.15}, {0.52, 0.489, 0.391, 0.363, 0.259}, p2 {0.636.0.441, 0.243,
0.23, 0.16}, {0.671, 0.259, 0.166, 0.145, 0.19}, p3 {0.686, 0.537,
0.418, 0.307, 0.259}, {0.391, 0.363, 0.314, 0.259, 0.191}].
Again, on applying the proposed similarity measure, we obtain
S(B̃1 , B̃∗ ) = 0.763, S(B̃2 , B̃∗ ) = 0.749, and S(B̃3 , B̃∗ ) = 0.7954. The
results show that S(Ã1 , Ã∗ ) < S(Ã3 , Ã∗ ) < S(Ã2 , Ã∗ ) and S(B̃2 , B̃∗ ) <
S(B̃1 , B̃∗ ) < S(B̃3 , B̃∗ ) i.e., S(Ã1 , Ã∗ )) and S(Ã2 , Ã∗ ) have the smallest and largest values, respectively, among the values of
S(Ã1 , Ã∗ ), S(Ã2 , Ã∗ ) and S(Ã3 , Ã∗ ), whereas B̃3 and B̃2 are located at
the minimum and maximum distances from B̃∗ , respectively.
Conversely, let us consider the following backward approximate
reasoning scheme:
l
∗ ∈ (g
∗
max(∪min[1 − (1 − hj2 ) j ])}, {gj2
12 ∪ g22 ∪ ... ∪ gj2 )|gj2 ≥
l
∗ ∈ (g
∗
max(∪min[1 − (1 − hjm ) j ])}, {gjm
1m ∪ g2m ∪ ... ∪ gjm )|gjm ≥
j
max(∪min[(gjm )lj ])}}] and “∪” is union operator of DHFSs,
j
(1 ≤ j ≤ t).
Example 2. Let us consider the following backward approximate
reasoning system based on DHFSs:
1
2
3
4
5
R1 : If Ỹ is Ã1 , then Z̃ is B̃1 .
R2 : If Ỹ is Ã2 , then Z̃ is B̃2 .
R3 : If Ỹ is Ã3 , then Z̃ is B̃3 .
Fact: Z̃ is B̃∗
Consequence: Ỹ is Ã∗
In this system, DHFSs,
Ã1 = [o1 {0.6, 0.4, 0.2}, {0.4, 0.1}, o2 {0.3, 0.2}, {0.5, 0.2},
o3 {0.2, 0.1}, {0.7, 0.5}]
Ã2 = [o1 {0.7, 0.3, 0.2}, {0.2, 0.1}, o2 {0.5, 0.3, 0.2}, {0.3, 0.2},
o3 {0.4, 0.1}, {0.6, 0.2}]
Ã3 = [o1 {0.7, 0.4, 0.2}, {0.3, 0.2},
o2 {0.8, 0.4}, {0.2}, o3 {0.6, 0.5}, {0.3, 0.2}] B̃1 = [p1 {0.5, 0.1},
{0.3}, p2 {0.8, 0.6}, {0.1}, p3 {0.3, 0.2}, {0.5, 0.4}] B̃2 = [p1 {0.6,
0.1}, {0.3}, p2 {0.7, 0.6}, {0.2, .1}, p3 {0.5, 0.3, 0.2}, {0.5, 0.4}]
B̃3 = [p1 {0.5, 0.1}, {0.3, 0.1}, p2 {0.8, 0.7, 0.6}, {0.2, 0.1}, p3
{0.4, 0.3, 0.2}, {0.6, 0.4}] B̃∗ = [p1 {0.5, 0.4, 0.1}, {0.3, 0.2}, p2
{0.4, 0.1}, {0.5, 0.4}, p3 {0.7, 0.3, 0.2}, {0.1}]
Similarity as in Example 1, on applying the proposed similarity measure, we get backward approximate reasoning scheme for
above system
By considering the results of Examples 1 and 2 jointly, we can
conclude that the proposed method is effective for approximate
reasoning. In addition, the results of Examples 1 and 2 can be
obtained only with the proposed distance measure.
3.4. Rationality of proposed similarity measure
1
2
3
4
5
6
R1 : If Ỹ is Ã1 , then Z̃ is B̃1 .
R2 : If Ỹ is Ã2 , then Z̃ is B̃2 .
...
Rp : If Ỹ is Ãp , then Z̃ is B̃p .
Fact: Ỹ is B̃∗
Consequence: Z̃ is Ã∗
Li et al. [15] pointed out that some existing similarity measures between intuitionistic fuzzy sets have shortcomings and
also they found counter-intuitive cases. From Table 1, we can
see that most of the existing similarity measures give counterintuitive results. It is worth noticing that all the existing similarity
P. Singh / Applied Soft Computing 24 (2014) 559–571
565
Table 1
Shortcoming in existing similarity measures [15].
Expression
(1)
Counter examples
n
|SÃ (xi )−SB̃ (xi )|
n
(2)
SH (Ã, B̃) = 1 −
(3)
SL (Ã, B̃) = 1 −
i=1
n
−
i=1
(|Ã (xi )−B̃ (xi )|+|Ã (xi )−B̃ (xi )|)
à = {(x, 0.3, 0.3)}, B̃ = {(x, 0.4, 0.4)},
C̃ = {(x, 0.3, 0.4)}, D = {(x, 0.4, 0.3)},
SH (Ã, B̃) = SH (C̃, D̃) = 0.9
2n
n
i=1
|SÃ (xi )−SB̃ (xi )|
à = {(x, 0.4, 0.2)}, B̃ = {(x, 0.5, 0.3)},
4n
(|Ã (xi )−B̃ (xi )|−|Ã (xi )−B̃ (xi )|)
C̃ = {(x, 0.5, 0.2)},
SL (Ã, B̃) = SL (Ã, C̃) = 0.95
4n
n
(4)
à = {(x, 0, 0)}, B̃ = {(x, 0.5, 0.5)},
SC (Ã, B̃) = 1
i=1
SC (Ã, B̃) = 1 −
2n
SÃ (xi ) = Ã (xi ) − Ã (xi ), SB̃ (xi ) = B̃ (xi ) − Ã (xi )
SO (Ã, B̃) = 1 −
n
i=1
(Ã (xi )−B̃ (xi ))2 +(Ã (xi )−B̃ (xi ))2
Same as that of SH (Ã, B̃)
2n
|
(xi )− B̃ (xi )|p
i=1 Ã
=1−
n
(Ã (xi )+1−Ã (xi ))
( (x )+1− (x ))
, B̃ (xi ) = B̃ i 2 B̃ i
2
p
(5)
SDC (Ã, B̃)
(6)
n
(7)
Same as that of SC (Ã, B̃)
=
SHB (Ã, B̃) = ( (Ã, B̃) + (Ã, B̃))/2
(Ã, B̃) = SDC (Ã , B̃ )
(Ã, B̃) = SDC (1 − Ã , 1 − B̃ )
à (xi )
p
Se (Ã, B̃)
p
=1−
(xi ) =
˜ (xi ) = |
p
i=1
|Ã (xi )−B̃ (xi )|
2
1−Ã (xi )
2
( (xi )− (xi ))p
p
|
(s1 (xi )−s2 (xi ))p
(8)
i=1
Ss (Ã, B̃) = 1 −
n
s1 (xi ) = (|mÃ1 (xi ) − mB̃1 (xi )|)/2,
s2 (xi ) = (|mÃ2 (xi ) − mB̃2 (xi )|)/2,
mÃ1 (xi ) = (à (xi ) + mà (xi ))/2,
mÃ2 (xi ) = (mà (xi ) + 1 − à (xi )+)/2,
mB̃1 (xi ) = (B̃ (xi ) + mB̃ (xi ))/2,
mB̃2 (xi ) = (mB̃ (xi ) + 1 − B̃ (xi ))/2,
mà (xi ) = (|à (xi ) + 1 − à (xi )|)/2,
mB̃ (xi ) = (|B̃ (xi ) + 1 − B̃ (xi )|)/2
(9)
i=1
Sh (Ã, B̃) = 1 −
3n
1 (i) = (xi ) + (xi )
2 (i) = mà (xi ) − mB̃ (xi )
3 (i) = max(là (i), lB̃ (i)) − min(là (i), lB̃ (i)),
là (i) = (1 − à (xi ) − à (xi ))/2,
lB̃ (i) = (1 − B̃ (xi ) − B̃ (xi ))/2.
n
p
p
Same as that of SH (Ã, B̃)
n
(1−B̃ (xi ))
−n 2
Same as that of SH (Ã, B̃)
Same as that of SL (Ã, B̃)
(1 (i)+2 (i)+3 (i))p
measure between HFSs [33] and intuitionistic fuzzy sets were constructed to satisfy the following conditions: (i) S(Ã, B̃) ∈ [0, 1], (ii)
S(Ã, B̃) = 1, (iii) S(Ã, B̃) = S(B̃, Ã), (iv) if à ⊆ B̃ ⊆ C̃, then S(Ã, C̃) ≤
S(Ã, B̃) and S(Ã, C̃) ≤ S(B̃, C̃) Unfortunately none of the above
similarity measure is able to satisfy all the properties. On the
other hand proposed similarity measures satisfies all the properties which are required for the similarity measures. Hence the
proposed similarity measure is more reasonable and don’t give any
counter-intuitive result.
Same as that of SL (Ã, B̃)
cost of doing the jth job by the ith person is satisfied. This problem
is more realistic in the sense that instead of cost we have used the
membership degree and non-membership degree.
Let us now formulate the problem: Let the dual fuzzy cost matrix
be (cij )n×n . Let xij be 0 − 1 variable, where
xij =
⎧
⎨ 1, iftheperson i isassignedthejob j; i, j = 1, 2, ..., n;
⎩
0,
otherwise.
4. Dual hesitant fuzzy assignment problem with
restrictions
In this section first we formulate the mathematical model of dual
hesitant fuzzy assignment problem with restrictions (DHFAPR)
than we give the procedure to find the optimal solution of DHFAPR.
4.1. Formulation of DHFAPR
Let there be n persons and n jobs. Each job must be done by
exactly one person and one person can do, at most, one job the
problem is to assign the persons to the jobs so that the total cost of
completing all jobs becomes minimum. The cost of person i doing
the job j is considered as an DHFS, cij = {hij , gij } i, j = 1, 2, . . . , n. Here
hij denotes the membership degree that cost of doing the jth job by
the ith person is satisfied and gij the non-membership degree that
Corresponding
n to the (ij)th event of assigning person i to job j, the
x = 1,
j = 1, 2, ..., n means each person must
constraint
i=1 ij
n
x = 1,
i=
be assigned exactly one job, and the constraint
j=1 ij
1, 2, ..., n means that each job must be done by exactly one person.
Also we have the restriction on the maximum dual hesitant
fuzzy cost cj that can be spent for the job j. This gives us an additional
constraint cij xij ≤ cj , i, j = 1, 2, . . . , n.
On the other hand, we have the data for the restriction on the
maximum fuzzy cost cpi , i = 1, 2, . . . , n, which can be offered to the
ith person depending on his/her efficiency/qualification. This gives
us another additional constraint cij xij ≤ cpi , i, j = 1, 2, . . . , n. Thus we
can formulate this kind of dual hesitant fuzzy assignment problem
with two restrictions on the dual hesitant fuzzy job cost and the
566
P. Singh / Applied Soft Computing 24 (2014) 559–571
dual hesitant fuzzy cost which can be offered to the ith person for
doing any job based on their qualification and efficiency, as follows:
Min z =
n
n cij xij
(9)
i=1 j=1
Subjectto
n
xij = 1, j = 1, 2, ..., n
(10)
i=1
n
xij = 1, i = 1, 2, ..., n
(11)
j=1
cij xij ≤ cj ,
i, j = 1, 2, ..., n
cij xij ≤ cpi ,
i, j = 1, 2, ..., n
xij = 0 or 1,
i, j = 1, 2, ..., n
(12)
(13)
(14)
This cost cij is usually deterministic in nature. But in real situations, it may not be practicable to know the precise values of these
costs. In that case we can replace the cost with dual hesutant fuzzy
cost, cij = {hij , gij } then Eq. (9) becomes
Min z =
n
n
{hij , gij }xij
(15)
i=1 j=1
Our objective is to maximize acceptance degree hij and to minimize the rejection degree gij . So the objective function (15) can be
written as
Max z =
n
n hij xij
(16)
gij xij
(17)
i=1 j=1
Min z =
n
n 4.2. Procedure for finding the optimal solution of DHFAPR
In this section we gave the solution procedure, adapted from
Mukherjee and Basu [23], for finding the optimal solution of assignment problem taking cost values as dual hesitant fuzzy sets.
Definition 9. A judgment matrix A = (aij )n×n corresponding to
one of the constraints (23) and (24) is such that aij = 0, if the corresponding constraint is not satisfied for the (ij)th position; otherwise
aij = 1.
We form two judgment matrices A and B respectively corresponding to the constraints (23) and (24) using the score function
and the accuracy function (using the Eqs. (2) and (3)). Then we form
the composite judging matrix Comp(AB) = (aij bij )n×n , where aij bij is
the product of the corresponding elements of the matrices A and B.
In matrix A = (aij )n×n , we call the element 1 at different row
and different column as the independent element 1 of the judging matrix A. Moreover the existence of the solution is shown by
a heuristic method which is used to find the number of independent 1’s in the judging matrix. The method of finding the number
of independent 1’s in a judging matrix is as follows:
(i) Select a 1 from the row (column) which has the least number of
1’s, and mark it as 1* . Cover the corresponding row and column
containing the 1* with lines.
(ii) Ignoring rows and columns containing 1* (i.e. ignoring the covered rows and columns), repeat the step (i) until there are no
uncovered 1 left out
Suppose the number of 1* in the judging matrix be k. If k = n, then
the problem has optimal solution; if k < n, then the problem has no
solution.
If the problem has solution, then we form the decision matrix
R = (rij )n×n , where
i=1 j=1
Hence the dual hesitant fuzzy assignment problem with
restrictions with two restrictions become a multi-objective linear
programming problem in the form,
Max z =
n
n hij xij
(18)
gij xij
(19)
i=1 j=1
Min z =
n
n
i=1 j=1
subject to
(hij + gij − 1)xij ≤ 0
(20)
hij xij ≥gij xij
(21)
gij xij ≥0
(22)
n
rij =
⎧
c ,
⎪
⎨ ij
⎪
⎩
“− ,
aij = 1;
aij = 0.
(28)
Here “ − represents the situation that the jth job cannot be
assigned to the ith person if aij = 0.
The procedure for judging the existence of the solution as well
as for finding the decision matrix of the assignment problem with
restriction of job cost and person-cost based on their efficiency/
qualification has been summarized in the form of the Algorithm 1.
Algorithm 1
Step 1. Construct the judgment matrix A = (aij )n×n by using Eqs. (2)
and (3), considering the rescription on jobs such that aij =
1, cij ≤ cj ;
0, cij > cj . Also construct the judgment matrix B =
(bij )n×n by using Eqs. (2) and (3), considering the rescription
xij = 1, j = 1, 2, ..., n
(23)
xij = 1, i = 1, 2, ..., n
(24)
on person-cost such that bij =
1,
0,
cij ≤ cpi ;
cij > cpi .
i=1
n
j=1
cij xij ≤ cj ,
i, j = 1, 2, ..., n
cij xij ≤ cpi ,
xij = 0 or 1,
i, j = 1, 2, ..., n
i, j = 1, 2, ..., n
(25)
(26)
(27)
Step 2. Then form the composite judging matrix Comp(AB) = (aij bij ),
where aij bij is the product of the corresponding elements of
the matrices A and B.
Step 3. Then count the number of independent 1’s in judging
matrix Comp(AB), and denote it as k. If k < n, the problem
has no solution, stop; if k = n, go to step 4.
Step 4. Construct the decision matrix R = (rij )n×n by using
Comp(AB), cost matrix (cij )n×n and Eq. (28).
P. Singh / Applied Soft Computing 24 (2014) 559–571
Step 5. Find the optimal solution of dual hesitant fuzzy assignment
problem with cost/profit, matrix R = (rij )n×n by using Algorithm 2.
Above DHFAPR cannot be solved by the traditional Hungarian
method, since the elements of this matrix are in the form of dual
hesitant fuzzy sets. So, the concept of composite relative degree
of similarity to positive ideal dual hesitant fuzzy solution can be
applied and this DHFAPR with the cost matrix as R = (rij )n×n can be
solved by using Algorithm 2.
For an DHFAPR, let A = {A1 , A2 , A3 , . . . , Am } be a set of alternatives
for a row or column in the assignment (cost) matrix, and let C be
an attribute (like cost or time or profit etc.) describing the selection
alternative. Assume that the characteristics of the alternative Ai are
represented by the DHFSs as:
Ai = {C, hAi (C), gAi (C)}, where hAi (C) indicates the degree that
the alternative Ai satisfies the attribute C, hAi (C) indicates the
degree that the alternative Ai does not satisfies the attribute C, and
0 ≤ hAi (C) + gAi (C) ≤ 1.
Algorithm 2
Step 1. Determine the positive ideal dual hesitant fuzzy
and negative ideal dual hesitant fuzzy solution as
follows,
respectively:
A+ = {C, hA+ (C), gA+ (C)}
and
where
hA+ (C) = max{hAi (C)},
A− = {C, hA− (C), gA− (C)},
i
gA+ (C) = min{gAi (C)} and hA− (C) = min{hAi (C)}, gA− (C) =
i
i
max{gAi (C)}
i
Step 2. Using Eq. (4), calculate the degree of similarity between
positive ideal dual hesitant fuzzy sets A+ and alternative
Ai , and the degree of similarity between negative ideal dual
hesitant fuzzy sets A− and alternative Ai . i.e., for p = 1, calculate
1 S(Ai , A+ ) = 1 − √ (hAi A+ (i) + gAi A+ (i))
n
n
(29)
567
Step 4. Repeat Step 1 to Step 3 for the rest of the columns of the
cost matrix and find the relative similarity measure di corresponding to the alternative Ai for these columns i.e. for
the jobs with respect to the persons.
Step 5. Form the matrix R1 where [R1 ] = [pij ]n×n , pij is the relative
similarity measure representing how much the jth person
prefers the ith job considering all the dual hesitant fuzzy
attributes, by using relative similarity measure dj of the jobs
with respect to the persons. Put > 0, a very small number
(degree of similarity) in the positions of the matrix R1 to
denote the situation that the jth person cannot be assigned
to the ith job for these positions, if the data in the original
problem considers that option.
Step 6. Calculate the relative similarity measure di for the persons
with respect to each job, considering the data of the first
row of the cost matrix. Repeat Step 1 to Step 3 for this row
and also the rest of the rows of the cost matrix and find the
relative similarity measure for these rows.
Step 7. With these relative similarity measure di of the persons
with respect to the jobs, form the matrix R2 where [R2 ] =
[qij ]n×n , qij is the relative similarity measure representing
how much the ith job is suitable for the jth person considering all the dual hesitant fuzzy atributes attributes. If jth
person cannot be assigned to the ith job, put > 0, a very
small number (degree of similarity) for these positions of
R2 .
Step 8. Construct the composite matrix Comp(R1 R2 ) = (pij qij )n×n =
(dij )n×n whose elements are the composite relative degree
of similarity representing the preference or suitability to
offer the ith job to the jth person or that the jth person is
chosen for performing the ith job.
Step 9. Then considering this matrix Comp(R1 R2 ) as the initial table
for an assignment problem in the maximization form, it is
solved by Hungarian method or by any standard software
to find the optimal assignment which maximizes the total
composite relative degree of similarity.
i=1
hAi A+ (i) =
where,
gAi A+ (i)) =
kx
1
2kx
i
s=1
i
lx
1
2lx
i
i
s=1
|hAi (s) (i) − hA+ (s) (i)|,
|gAi (s) (i) − gA+ (s) (i)| and,
1 S(Ai , A− ) = 1 − √ (hAi A− (i) + gAi A− (i))
n
n
(30)
5. Illustrative example
i=1
where, hAi A− (i) =
gAi A− (i)) =
1
lx
1
2kx
i
s=1
2lx
i
kx
i
i
s=1
|hAi (s) (i) − hA− (s) (i)|,
|gAi (s) (i) − gA− (s) (i)|
Step 3. Based on (29) and (30) calculate the relative similarity measure di corresponding to the alternative Ai as:
di =
Using algorithm 1 and 2, we can solve the dual hesitant fuzzy
assignment problem with restriction on the maximum cost that
can be spent on job and restriction on the maximum cost that can
be offered to each person for doing a job based on their efficiency/
qualification.
S(Ai , A+ )
, i = 1, 2, ..., n.
S(Ai , A+ )S (Ai , A− )
(31)
Bigger the value of di , the more similar is Ai to the positive
ideal dual hesitant fuzzy sets A+ and hence better is the
alternative Ai .
In this section, we illustrate the proposed method by solving the
optimal assignment of projects to teams based on certain attributes
which are represented by dual hesitant fuzzy sets.
Let us consider an DHFAPR with rows representing 3 alternative
projects P1 , P2 , P3 and columns representing the three teams T1 ,
T2 , T3 . The cost matrix [cij ] is given whose elements are DHFSs.
The problem is to find the optimal assignment, subject to certain
restrictions so that the total cost becomes minimum for project
assignment.
The data is shown in Table 2. Suppose the maximum cost that
can be spent for doing the jth project be cj j = 1, 2, 3 given by
Table 2
Data for the assignment problem with restrictions on jobs and persons.
Project
P1
P2
P3
Restriction on team
Team
T1
T2
T3
Restrictions on project
{{0.5,0.1},{0.5,0.4,0.2}}
{{0.4,0.2},{0.6,0.4}}
{{0.3,0.2,0.1},{0.6,0.5}}
{{0.6,0.4,0.3},{0.3,0.1}}
{{0.7,0.6,0.5},{0.3,0.2}}
{{0.7,0.6},{0.3,0.2}}
{{0.2,0.1},{0.4,0.3,0.2}}
{{0.6,0.3,0.1},{0.4,0.3,0.2}}
{{0.6,0.4,0.3},{0.3}}
{{0.6,0.5},{0.3}}
{{0.6,0.5},{0.3,0.1}}
{{0.5,0.3,0.1},{0.3,0.2,0.1}}
{{0.8,0.7,0.6},{0.2,0.1}}
{{0.7,0.6},{0.2}}
{{0.4,0.3},{0.2,0.1}}
{{0.8,0.6},{0.2}}
568
P. Singh / Applied Soft Computing 24 (2014) 559–571
Table 3
Score matrix of Table 2.
Table 8
Decision matrix.
Project
Team
s
T1
P1
P2
P3
Score value of restriction
on team
−0.067
−0.2
0.35
0.23
Project Team
T2
0.35
0.4
−0.15
0.033
T3
Score value of
restrictions on project
0.134
0.25
0.35
0.1
0.55
0.45
0.20
0.3
Project
Project
Team
p
T1
T2
T3
Accuracy value of
restrictions on project
P1
P2
P3
Accuracy value
restriction on team
0.667
0.8
0.75
0.63
0.85
0.9
0.45
0.63
0.73
0.85
0.75
0.5
0.85
0.85
0.5
0.9
Project
P1
P2
P3
Team
T1
T2
T3
1
1
0
1
1
1
1
1
0
Table 6
Judging matrix [B] = [bij ]n×n for the restriction on the cost of teams.
P1
P2
P3
Team
T1
T2
T3
1
1
0
1
1
1
1
1
0
Step 1. Using Eq. (2) and (3), construct the judgment matrix A =
(aij )n×n and B = (bij )n×n , as as shown in Tables 5 and 6.
Step 2. Form the composite judging matrix Comp(AB) = (aij bij ),
where aij bij is the product of the corresponding elements
of the matrices A and B as shown in Table 7.
Step 3. The number of independent 1’s in judging matrix Comp(AB)
are 3, so problem has solution.
Table 7
Composite judging matrix Comp[AB] = [aij bij ]n×n .
P1
P2
P3
T1
T2
T3
0.571
0.452
–
–
–
0.726
0.712
0.841
–
Team
T1
T2
T3
1*
1
0
0
0
1*
1
1*
0
Team
T1
T2
T3
0.534
0.612
–
–
–
0.726
0.743
0.646
–
Table 11
Matrix R1 containing values of dj for projects with respect to the teams (column
wise).
P1
P2
P3
c1 = {{0.8, 0.7, 0.6}, {0.2, 0.1}}, c2 = {{0.7, 0.6}, {0.2}}, c3 = {{0.4, 0.3},
{0.2, 0.1}} The maximum cost, denoted by cpi which can be offered
to the ith team depending on its efficiency/qualification, i = 1, 2, 3,
is given that cp1 = {{0.6, 0.4, 0.3}, {0.3, 0.1}}, cp2 = {{0.6, 0.3, 0.1},
{0.4, 0.3, 0.2}}, cp3 = {{0.5, 0.3, 0.1}, {0.3, 0.2, 0.1}}.
Using Eq. (2) and (3) find score and accuracy matrix, as shown
in Tables 3 and 4.
Algorithm 1
Project
Team
P1
P2
P3
Project
Project
T3
Table 10
Values of S(A− , Pj ) for projects with respect to the teams (column wise).
Table 5
Judging matrix [A] = [aij ]n×n for the restriction on the project cost.
P1
P2
P3
T2
{{0.5,0.1},{0.5,0.4,0.2}} –
{{0.6,0.4,0.3},{0.3}}
{{0.4,0.2},{0.6,0.4}}
–
{{0.7,0.6},{0.2}}
–
{{0.2,0.1},{0.4,0.3,0.2}} –
Table 9
Values of S(A+ , Pj ) for projects with respect to the teams (column wise).
Table 4
Accuracy matrix of Table 2.
Project
T1
P1
P2
P3
Team
T1
T2
T3
0.516
0.427
0.489
0.565
0.5
Step 4. Using Comp(AB) and Eq. (28) construct the decision matrix
as shown in Table 8.
Step 5. Now considering the decision matrix R = (rij )n×n as the cost
matrix the optimal solution of assignment problem can be
obtained by using Algorithm 2.
Algorithm 2
Step 1. The positive ideal dual hesitant fuzzy and negative ideal
dual hesitant fuzzy solution for the first column are given
as follows, respectively: A+ = {{0.5}, {0.2}} and A− = {{0.1},
{0.6}}.
Step 2. Using Eqs. (29) and (30) calculate the degree of similarity
between positive ideal dual hesitant fuzzy set A+ and alternative Pi , and the degree of similarity between negative
ideal dual hesitant fuzzy set A− and alternative Ai i.e.,
S(A+ , P1 ) = 0.571, S(A+ , P2 ) = 0.452 and S(A− , P1 ) = 0.534,
S(A− , P2 ) = 0.612.
Vales of S(A+ , Pj ) and S(A− , Pj ) for all the column are shown
in Tables 9 and 10, respectively.
Step 3. Using Eq. (31), calculate the relative similarity measure dj
corresponding to the alternative Pi :
d1 = 0.516 and d2 = 0.427.
Step 4. Repeat Step 1 to Step 3 for the rest of the columns of the
cost matrix and find the relative similarity measure for all
the column.
Step 5. With these relative similarity measures dj , construct the
matrix R1 , as shown in Table 11. Put > 0, a very small number (degree of similarity) in the positions of the matrix R1 to
P. Singh / Applied Soft Computing 24 (2014) 559–571
569
Table 12
Values of S(A+ , Ji ) for projects with respect to the teams (row wise).
Project
P1
P2
P3
Team
T1
T2
T3
0.516
0.5
–
–
–
0.721
0.635
0.841
–
Table 13
Values of S(A− , Ji ) for projects with respect to the teams (row wise).
Project
P1
P2
P3
Team
T1
T1
T1
0.591
0.683
–
–
–
0.726
0.591
0.429
–
Table 14
Matrix R2 containing values of di for teams with respect to the projects (row wise).
Project
P1
P2
P3
Team
T1
T2
T3
0.466
0.422
0.517
0.661
0.5
Table 15
Composite judging matrix Comp(R1 R2 ) = (pij qij )n×n .
Project
P1
P2
P3
Step 6.
Step 7.
Step 8.
Step 9.
Team
Fig. 1. Flow chart for DHFAPR.
T1
T2
T3
0.240
0.180
0.252
0.294
5.1. Advantages of the proposed method
0.25
(i) As mentioned earlier, the dual hesitant fuzzy set is a further
generalization of the hesitant fuzzy set and intuitionistic fuzzy
set. So the dual hesitant fuzzy set contains more information (both membership hesitant degrees and nonmembership
hesitant degrees) than the hesitant fuzzy set (only membership hesitant degrees) and the intuitionistic fuzzy set (both
membership degree and nonmembership degree). Thus, the
proposed similarity measures of DHFSs can be considered as a
further generalization of the distance and similarity measures
of hesitant fuzzy sets [33] and intuitionistic fuzzy sets [31].
(ii) Since Mukherjee and Basu [23] proposed an intuitionistic fuzzy
assignment problem by using similarity measures between
intuitionistic fuzzy sets. Also from literature we know that
there are some limitations in the similarity measures between
intuitionistic fuzzy sets. So Mukherjee and Basu’s [23] method
does not give appropriate results.
(iii) The distance and similarity measures of hesitant fuzzy sets
[33] and intuitionistic fuzzy sets [31] are special cases of the
distance and similarity measures of DHFSs proposed in this
paper. Therefore, DHFAPR proposed in this paper can be used
to solve not only assignment problems with DHFSs but also
the assignment problems of hesitant fuzzy sets and intuitionistic fuzzy sets, whereas the method in [23] is only suitable to
find the optimal solution of intuitionistic fuzzy fuzzy assignment problem with restrictions. The procedure, through flow
chart, for finding the optimal solution of DHFAPR is depicted
in Figs. 1 and 2
denote the situation that the jth person cannot be assigned
to the ith job for these positions, if the data in the original problem considers that option. Where “ − represents,
aij = 0 in that place.
Repeat Step 1 to Step 3 for the first rows and also for the rest
of the rows of the cost matrix and find the relative similarity
measure for these rows. Values of S(A+ , Ji ) and S(A− , Ji ) for
all the rows are shown in Tables 12 and 13, respectively
With these relative similarity vales construct matrix R2 , as
shown in Table 14. If jth person cannot be assigned to the
ith job, put > 0, a very small number (degree of similarity)
for these positions of R2 .
Construct the composite matrix Comp(R1 R2 ) as shown in
Table 15.
Considering this matrix Comp(R1 R2 ) as the initial table for
an assignment problem in the maximization form. We can
solve this problem by Hungarian method or by any standard
software. The optimal solution for this dual hesitant fuzzy
assignment problem with restrictions is given as follows:
Project P1 is assigned to Team T1
Project P2 is assigned to Team T3
Project P3 is assigned to Team T2
570
P. Singh / Applied Soft Computing 24 (2014) 559–571
Fig. 2. Flow chart for DHFAPR (continued).
6. Conclusions
In this paper, we proposed the axiomatic definition of a similarity measure between dual hesitant fuzzy sets. A new similarity
measure between dual hesitant fuzzy sets has been introduced.
The relationship between similarity measure and distance measure of dual hesitant fuzzy sets is analyzed. Limitations of some
existing similarity measures have been studied. Effectiveness of
the proposed similarity measure has been tested in bidirectional
approximate reasoning systems. Further, we formulate the dual
hesitant fuzzy assignment problem with restrictions and a procedure is presented to find the optimal solution of dual hesitant fuzzy
assignment problem with restrictions, based on proposed similarity measure. To illustrated the proposed method, the problem is
to find the optimal assignment, subject to certain restrictions so
that the total cost becomes minimum for project assignment, is
presented.
Acknowledgements
The author is very grateful to the Editor-in-Chief, and the
anonymous referees, for their constructive comments and suggestions that led to an improved version of this paper. The author
was supported by the Education for Competitiveness Operational
Programme project “Encourage the creation of excellent research
teams and intersectoral mobility at Palacky University in Olomouc”
reg. no. CZ.1.07/2.3.00/30.0004, which is co-financed by the European Social Fund and the state budget of the Czech Republic.
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