Fuzzy Assignment Problems Using Normalised Heptagonal ...acadpubl.eu/jsi/2017-117-11-14/articles/11/46.pdf · points of heptagonal fuzzy numbers and the new operations addition, sub-traction

Fuzzy Assignment Problems Using NormalisedHeptagonal Fuzzy Numbers

P.Selvam1 , A.Rajkumar2 and J.Sudha Easwari3

1Research scholar,1,2 Hindustan Institute of Technology and Science, Chennai -603 103,India

and Assistant Professor,Adithya Institute of Technology,Coimbatore-641 107,India

[email protected] professor, [email protected]

3B.T Assistant,Government Higher Secondary School, Tirupur- 638 701,India.

[email protected]

Abstract

Fuzzy numbers are based on membership function have been classifiedinto shape of triangle , trapezoidal, bell etc., in various different points inreal numbers. In many real life cases, the decision data of human judgmentwith preference are often vague. So that the traditional ways of using crispvalues are inadequate and using fuzzy numbers such as triangular ,trapezoidalare not apt in some case. Even though seven points is been used where theuncertainties arises. In such case heptagonal fuzzy number can be used tosolve the problems. In this research paper has expressed the seven differentpoints of heptagonal fuzzy numbers and the new operations addition, sub-traction have been defined. It is a simple method of solving fuzzy assignmentproblem using heptagonal fuzzy number. It gives the optimum cost which ismuch lower than the non trapezoidal shape of heptagonal fuzzy numbers andtrapezoidal fuzzy numbers.

Key Words :fuzzy numbers, triangular fuzzy number, trapezoidal fuzzynumber[TpFN], heptagonal fuzzy number [HFN], fuzzy arithmetic opera-tions, alpha cut, normal fuzzy numbers, fuzzy assignment problem[FAP],ranking of fuzzy numbers.

International Journal of Pure and Applied MathematicsVolume 117 No. 11 2017, 405-415ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

405

1 Introduction

L.A.Zadeh[1] was introduced by the concept of fuzzy numbers and fuzzy arithmetic.Masaharu Mizumoto and Kokichi Tnaka [3][7] have investigated the algebraic prop-erties of fuzzy sets and fuzzy numbers with some operations. J.G. Dijkman, H. VanHaeringen and S. J. De Lange [2] have investigated nine operations of fuzzy num-bers for addition, subtraction, multiplication, division, joint and meet operations.Triangular fuzzy numbers are frequently used in many application. In some casestriangular fuzzy numbers is not apt where the uncertainties arise in more than threeand four points. So important contributes to the theory of fuzzy numbers have sevendifferent points by numerous researchers with triangular shape fuzzy numbers. Inthis paper the two different operations like addition, subtraction which have beenintroduced using alpha cut principle and as new approach for a simple method ofsolving FAP using HFN.

2 Preliminaries and Notations

2.1 Definition A fuzzy set A is defined on the set of real line, R is said to be afuzzy number if its membership function µA : < → [0, 1] satisfies

• Convex and Normal of fuzzy set.

• A is piecewise continuous.

3 New HFNs

3.1 Definition: Fuzzy numberAHFN is a HFN denoted byAHFN = (λ1, λ2, λ3, λ4, λ5, λ6, λ7)whereas λ1, λ2, λ3, λ4, λ5, λ6, λ7 are real numbers and its membership function

µAHFN(x) =

13( x−λ1λ2−λ1 ) for λ1 ≤ x ≤ λ2

13

+ 13( x−λ2λ3−λ2 ) for λ2 ≤ x ≤ λ3

23

+ 13( x−λ3λ4−λ3 ) for λ3 ≤ x ≤ λ4

1− 13( x−λ4λ5−λ4 ) for λ4 ≤ x ≤ λ5

23− 1

3( x−λ5λ6−λ5 ) for λ5 ≤ x ≤ λ6

13( λ7−xλ7−λ6 ) for λ6 ≤ x ≤ λ7

0 for x < λ1 and x > λ7

4 Operations of HFNs

Let AHFN = (λ1, λ2, λ3, λ4, λ5, λ6, λ7) and BHFN = (β1, β2, β3, β4, β5, β6, β7) be theircorresponding HFN then

1. Addition:AHFN(+) BHFN = (λ1 + β1, λ2 + β2, λ3 + β3, λ4 + β4, λ5 + β5, λ6 +β6, λ7 + β7)

International Journal of Pure and Applied Mathematics Special Issue

406

Figure 1: Graphical repesentation of a normal HFNs for x ∈ [0, 1]

2. Subtraction:AHFN(−) BHFN = (λ1− β7, λ2− β6, λ3− β5, λ4− β4, λ5− β3, λ6− β2, λ7− β1)

4.1 Example:Let AHFN = (2, 4, 6, 8, 10, 12, 14) andBHFN = (1, 2, 3, 4, 5, 6, 7) be two HFNs

then

1. Addition:AHFN(+) BHFN = (3, 6, 9, 12, 15, 18, 21)

2. Subtraction: AHFN(−) BHFN = (−5,−2, 1, 4, 7, 10, 13)

4.2 Definition:A HFN can be defined as HFN= Pl(t), Ql(u), Rl(v), Pu(t), Qu(u), Ru(v), t ∈

[0, 0.3333),u ∈ [0.3333, 0.6666), v ∈ [0.6666, 1.0] whereas

Pl(t) = 13( x−λ1λ2−λ1 ) Pu(t) = 1

3( λ7−xλ7−λ6 )

Ql(u) = 13

+ 13( x−λ2λ3−λ2 ) Qu(u) = 2

3− 1

3( x−λ5λ6−λ5 )

Rl(v) = 23

+ 13( x−λ3λ4−λ3 ) RU(v) = 1− 1

3( x−λ4λ5−λ4 )

Pl(t), Ql(u), Rl(v) is monotonic ascending with bounded under [0, 0.3333), [0.3333, 0.6666)and [0.6666,1.0] .Pu(t), Qu(u), Ru(v) is monotonic descending with bounded under [0, 0.3333), [0.3333, 0.6666)and [0.6666,1.0] .

4.3Definition :The alpha cut of the HFN in the set of elements in X is defined as

HFNα = {x∈X/µA(x) ≥ α} =

[Pl(α), Pu(α)] for α ∈ [0, 0.3333)

[Ql(α), Qu(α)] for α ∈ [0.3333, 0.6666)

[Rl(α), Ru(α)] for α ∈ [0.6666, 1.0]

where as α ∈ [0, 1]


407

4.4 Definition:If Pl(x) = α and Pu(x) = α, then α- cut operations interval HFNα is obtained as

• [Pl(α), Pu(α)] = [3α(λ2 − λ1) + λ1,−3α(λ7 − λ6) + λ7] similarly

• [Ql(α), Qu(α)] = [3(α− 13)(λ3 − λ2) + λ2,−3(α− 2

3)(λ6 − λ5) + λ5]

• [Rl(α), Ru(α)] = [3(α− 23)(λ4 − λ3) + λ3,−3(α− 1)(λ5 − λ4) + λ4]

Hence α cut of HFN

HFNα =

[3 α(λ2 − λ1) + λ1,−3α(λ7 − λ6) + λ7] for α ∈ [0, 0.3333)

{ [3 (α− 13)(λ3 − λ2) + λ2,−3(α− 2

3)(λ6 − λ5) + λ5] for α ∈ [0.3333, 0.6666)

{ [3(α− 23)(λ4 − λ3) + λ3,−3(α− 1)(λ5 − λ4) + λ4] for α ∈ [0.6666, 1]

4.5 Note: The triangular fuzzy numbers and HFNs are same for the points havethe equal intervals and distinct for the points have the unequal intervals.

5 A new operation for addition and subtraction on HFN:

5.1 Definition: LetAHFN = (λ1, λ2, λ3, λ4, λ5, λ6, λ7) and BHFN = (β1, β2, β3, β4, β5, β6, β7),for all x, λ1, λ2, . . . , λ7, β1, β2, . . . , β7 ∈ <, λ1 ≤ λ2 ≤ . . . ≤ λ7,β1 ≤ β2 ≤ . . . ≤ β7 betheir corresponding HFN then for all α ∈ [0,1]. let us take the membership functionon the basis of α -cut αA and αB of AHFN and BHFN by using interval arithmetic.

5.2 Addition of two HFNs:

AHFN(+) BHFN =

[3 α((λ2 + β2)− (λ1 + β1)) + (λ1 + β1),

−3α((λ7 + β7)− (λ6 + β6)) + (λ7 + β7)] for α ∈ [0, 0.3333)

[3(α− 13)((λ3 + β3)− (λ2 + β2)) + (λ2 + β2),

−3(α− 23)((λ6 + β6)− (λ5 + β5)) + (λ5 + β5)] for α ∈ [0.3333, 0.6666)

[3(α− 23)((λ4 + β4)− (λ3 + β3)) + (λ3 + β3),

−3(α− 1)((λ5 + β5)− (λ4 + β4)) + (λ4 + β4] for α ∈ [0.6666, 1.0]5.3 Subtraction of two HFNs:

AHFN(−) BHFN =

[3α((λ2 + β7)− (λ1 + β6)) + (λ1 − β7),−3α((λ7 + β2)− (λ6 + β1)) + (λ6 + β1)] for α ∈ [0, 0.3333)

[(3(α− 13)(λ3 − λ2) + λ2)− (−3(α− 2

3)(β6 − β5) + β5,

(-3(α− 23)(λ6 − λ5) + λ5)− 3(α− 1

3)(β3 − β2) + β2] for α ∈ [0.3333, 0.6666)

[(3(α− 23)(λ4 − λ3) + λ3)− ((−3(α− 1)(β5 − β4)) + β4),

(−3(α− 1)(λ5 − λ4) + λ4)− (3(α− 23)(β4 − β3)) + β3) for α ∈ [0.6666, 1.0]


408

5.4 Numerical example in arithmetic operation using α cut:

we take the arithmetic example 4.2, To validate this new operation,AHFN = (2, 4, 6, 8, 10, 12, 14)andBHFN=(1,2,3,4,5,6,7) be two HFN then

α(A+B)HFN α = 0 α = 0.3333 α = 0.6666 α = 1 HFN(9α + 3,−9α + 21) [3, 21] [6, 18] [9, 15] [12, 12] (3, 6, 9, 12, 15, 18, 21)

Figure 2: Summation of two normal HFNs A=(2,4,6,8,10,12,14) andB=(1,2,3,4,5,6,7)


409

α(A−B)HFN α = 0 α = 0.3333 α = 0.6666 α = 1 HFN(9α− 5,−9α + 13) [−5, 13] [−2, 10] [1, 7] [4, 4] (−5,−2, 1, 4, 7, 10, 13)

Figure 3: Subtraction of two normal HFNs A=(2,4,6,8,10,12,14) andB=(1,2,3,4,5,6,7)

6 Ranking of HFNs

6.1 Definition [5]:If(1-ω)y2 +ωy1, where ω ∈ [0, 1], I = [y1, y2], y1, y2 ∈ <, y1 ≤ y2is defined by ντ (F ; t) = (1− ω)V∗(F ;R) + ωV ∗(F ;R) whereas V∗(F ; t) =

∫ 1

0FLαd(αs)

and V∗(F ; t) =∫ 1

0FUαd(αs) where F is a normal fuzzy number.

6.2 Note: Frequently used function r is r(α) = αs, s > 0;in this case, α = s/(s+ 1)can easily depend on s; when r varies from 0 to ∞,which have convex function of afuzzy number (r >1).

6.3 Remark: The value of the HFN AHFN = (λ1, λ2, λ3, λ4, λ5, λ6, λ7 : ρ1, ρ2, 1) isgiven by ντ (AHFN ; r) = τV ∗(AHFN ; r) + (1− τ)V∗(AHFN ; r) where the upper andlower value of AHFN are given by V∗(AHFN ; r) = (γ(λ1+rλ2

r+1) + ψ(λ2+rλ3

r+1) + η(λ3+rλ4

r+1))

andV ∗(AHFN ; r) = (γ(λ7+rλ6

r+1) + ψ(λ6+rλ5

r+1) + η(λ5+rλ4

r+1)) whereγ = ρr1, β = (ρ2−ρ1)r and

η = (1− ρ2)r.

6.4 Remark: The value of the ATHFN = (λ1, λ2, λ3, λ4; 1) is given byντ (ATHFN ; 1) = τV ∗(ATHFN ; 1) + (1− τ)V∗(ATHFN ; 1)where the upper and lowervalue of ATHFN are given by V∗(ATHFN ; 1) = (λ1+λ2

2) and V ∗(ATHFN ; 1) = (λ3+λ4

2)

7 Fuzzy Assignment Problem[FAP]

7.1 Definition:The matrix representation of FAP can be stated in the form of mx n effectiveness matrix is

The mathematically stated by the FAP ismin =

∑ni=1

∑nj=1CijXij

Subject to constraints∑ni=1Xij = 1, i = 1, 2, . . . n,


410

- Machine 1 Machine 2 . . . . . . Machine n

Job 1 C11 C12 . . . . . . C1n

Job 2 C21 C22 . . . . . . C2n

. . . . . . . . . . . . . . . . . .

Job m Cm1 Cm2 . . . . . . Cmn

∑nj=1Xij = 1, j = 1, 2, . . . n, Xij ∈ 0, 1

Xij =

{1 if the ith machine is assigned to jth job

0 otherwise

8 Fuzzy Hungarian Algorithm

Create the fuzzy cost table for the given problem and denote the vagueness in costby HFNs. Let us take the matrix representation of the FAP. If the fuzzy cost table isnot a square matrix, then we convert the cost table into a square matrix by addingfuzzy zero element rows or fuzzy zero element columns , then we use the Hungarianalgorithm to acquire the fuzzy optimal solution.

8.1 Numerical example

The following numerical example is taken from the paper,’Representation and Rank-ing of Fuzzy Numbers with Heptagonal Membership Function Using value and Am-biguity Index’ by K. Rathi, S. Balamohan [4]

- Machine1 Machine2 Machine3

Job 1 (1,4,6,8,11,14,16 (8,9,10,13,15,17,18 (6,9,10,12,13,14,15;0.15,0.65,1) ;0.22,0.5,1) ;0.3,0.42,1)

Job 2 (11,14,16,18,22,24,26 (9,13,15,18,20,22,23 (15,18,20,22,24,27 ,28;0.25,0.55,1) ;0.11,0.44,1) ;0.2,0.61,1)

Job 3 (7,10,12,14,16,18,20 (11,12,15,17,19,20,21 (10,12,14,15,18,20,22;0.31,0.45,1) ;0.32,0.6,1) ;0.1,0.53,1)

Table 1: Heptagonal Fuzzy Cost of a FAP

Let λ = 0.5 and r =1 ,then we calculate the each fuzzy cost using ranking valueas follows

Machine 1 Machine 2 Machine 3

Job 1 MC11=8.38 MC12=11.37 MC13=8.71

Job 2 MC21=16.80 MC22=14.74 MC23=22.05

Job 3 MC31=10.90 MC32=15.91 MC33=14.24

Table 2: The parameters = 0.5 and r =1 in the ranking value of heptagonal fuzzycost


411

The following reduced matrix is the row and column reduction

0 (−8,−5,−1, 5, 9, 13, 17, ; 0.15, 0.5, 1) 0(−12,−8,−4, 0, 7, 11, 17, ; 0.11, 0.44, 1) 0 (−22,−14,−7, 0, 10, 19, 29; 0.15, 0.42, 1)

0 (−9,−6,−1, 3, 7, 10, 14; 0.31, 0.4, 1) (−24,−16,−9,−3, 7, 15, 25; 0.1, 0.4, 1)

The minimum assignment is done by making the assignment and checking J1 →M3, J2 →M2, J3 →M1 with the optimal assignment costC=(22,32,37,44,49,54,58;0.11,0.4,1) and its crisp value is 34.35.

In the same problem if we put the values of k1 = 0.3333 and k2 = 0.6666 we getthe minimum assignment as J1 → M1, J2 → M2, J3 → M3 with the optimal assign-ment cost C=(20,29,35,41,49,56,61;0.1,0.44,1) and its crisp value is 40.16.

8.2 Example: Trapezoidal Fuzzy cost for example 8.1 [5]

The following trapezoidal fuzzy cost is reduced from Table 1 is

- Machine1 Machine2 Machine3

Job 1 (1, 4, 14, 16; 1)) (8, 9, 17, 18; 1)) (6, 9, 14, 15; 1))Job 2 (11, 14, 24, 26; 1) (9, 13, 22, 23; 1) (15, 18, 24, 27, 28; 1)Job 3 (7, 10, 18, 20; 1) (11, 12, 20, 21; 1) (10, 12, 18, 20, 22; 1)

Table 3: Trapezoidal Fuzzy cost of a FAP(Converted)

Let λ = 0.5 and r =1 ,then we calculate the each fuzzy cost using ranking valueas follows.

Machine 1 Machine 2 Machine 3

Job 1 MC11=8.76 MC12=13.00 MC13=11.02

Job 2 MC21=18.76 MC22=16.79 MC23=22.02

Job 3 MC31=13.76 MC32=16.00 MC33=16.00

Table 4: The parameters = 0.5 and r =1in the ranking value of Trapezoidal Fuzzycost

The optimality assignment is done by making the assignment and checkingJ1 →M1, J2 →M2, J3 →M3 with the minimum assignment cost C=(20,29,56,61;1)and its crisp value is 41.5.


412

Authors Fuzzy number Assignment cost Crisp value

K. Rathi Non-Normal HFN C =(20,29,35,41,49 41.5Balamohan.S[5] ,56,61:0.3,0.47,1)

Normal HFN k1and C =(22,32,37,44,49 34.35Proposed and k2 are distinct ,54,58:0.11,0.4,1)approach Normal HFN k1=0.3333 C =(30,29,35,41,49,56, 40.16

and k2 =0.6666 61:0.1,0.44,1)Normal Trapezoidal FN C =(20,29,56,61:1) 41.5

Table 5: Proposed approach in FAP using heptagonal Fuzzy Cost

9 Conclusion

In this paper, the triangle shape of seven various points of HFN are introduced andsome alpha cut arithmetic operations like addition, subtraction is illustrated withnumerical examples. Using HFNs solve a FAP (Job and Machine are normalizedHFN) in simple way.It gives the optimum cost which is much lower than the nonnormal with non triangle shape of HFN and trapezoidal fuzzy numbers.

References

[1] L.A.Zadeh, Inform. Sci., 8 (1975), 199-249, 301-357;

[2] J.G. Dijkman, H. Van Haeringen,And S. J. De Lange, Fuzzy Numbers, CJournalOf Mathematical Analysis And Applications, 92 (1983), 301-341.

[3] Mizumoto, M., and Tanaka, K.,The four Operations of Arithmetic on FuzzyNumbers, Systems Comput. Controls 7(5),(1977), 73-81.

[4] K. Rathi, S. Balamohan, Representation and Ranking of Fuzzy Numbers withHeptagonal Membership Function Using value and Ambiguity Index, AppliedMathematical Sciences, 8(87) (2014), 4309 4321.

[5] George J.Klir, and Tina.A.Folger,Fuzzy sets, Uncertainty and Information,,Prentice Hall of India Pvt. Ltd., New Delhi (2005).

[6] H.W.Kuhn,The Hungarian method for the assignment problem, Naval ResearchLogistics Quartely, 2,(1955),83-97.

[7] Mizumoto, M., and Tanaka, K.,Some properties of Fuzzy Numbers, Advances infuzzy set theory and applications,North-Holland Publishing Company,(1979),153-164.

[8] P. Selvam,A.Rajkumar,J.Sudha Easwari,A New Method To Find OctagonalFuzzy Number, International Journal of Control Theory and Applications, In-ternational Science Press,9(28) (2016), 447-461.


413

[9] P. Selvam,A.Rajkumar,J.Sudha Easwari,Dodecagonal Fuzzy Number, Interna-tional Journal of Control Theory and Applications, International SciencePress,9(28) (2016), 499-515.


414

415

416

Documents

Fuzzy Assignment Problems Using Normalised Heptagonal ...acadpubl.eu/jsi/2017-117-11-14/articles/11/46.pdf · points of heptagonal fuzzy numbers and the new operations addition, sub-traction