A NOVEL TRANSPORTATION APPROACH TO SOLVING TYPE 2 TRIANGULAR INTUITIONISTIC FUZZY TRANSPORTATION PROBLEMS Текст научной статьи по специальности «Математика»

A NOVEL TRANSPORTATION APPROACH TO SOLVING TYPE - 2 TRIANGULAR INTUITIONISTIC FUZZY TRANSPORTATION PROBLEMS

Indira Singuluri

Vignan's Institute of Information Technology (A), Duvvada, Visakhapatnam.

[email protected]

N. Ravishankar

Gitam Deemed to be University, GIS, Visakhapatnam. [email protected]

Abstract

In this article we propose a new transportation strategy to achieve an ideal answer for triangular intuitionistic fuzzy transportation problem of type - 2 i.e., limits and requests are considered as real numbers and the transportation cost from cause to objective is considered as triangular intuitionistic fuzzy numbers as product cost per unit. The proposed method is solving by using ranking function. The appropriate response system is delineated with a numerical model.

Keywords: IFN, TIFN, IF Optimum solution, TIFTP of type-2.

I. Introduction

In genuine world, there are general complex circumstances in each field, in which specialists and chiefs battle with uncertainty and hesitation. In useful circumstances, assortment of fresh information of different boundaries is troublesome because of absence of precise interchanges, mistake in information, market information and consumer loyalties. The data accessible is some of the time ambiguous and inadequate. The real-life problems, when defined by the decision maker with uncertainty leads to the notion of fuzzy sets. Due to imprecise information, the exact evaluation of participation values is not possible. Moreover, the evaluation of non-participation esteems is consistently impossible. This prompts an in deterministic climate where dithering endures. Managing estimated data while deciding, idea of fuzziness was presented by Bellman and Zadeh [6]. K. T, Atanassov [4] presented idea of Intuitionistic fuzzy set hypothesis, which is more able to manage such issues. B. Chetia and P. K. Das [1] demonstrated a few outcomes on intuitionistic fuzzy delicate network. Intuitionistic fuzzy sets [5], [7], [8] discovered to be exceptionally powerful in managing ambiguity, among a few higher request fuzzy sets. S.K. Singh, S.P. Yadav [9] proposed their strategies to address case 2 sort of intuitionistic fuzzy transportation problem (IFTP) for example IFTP of type-2. G. Gupta and A. Kumara [3] a capable technique was introduced in which limit and request factors are taken as TIFN's utilized in this article to tackle mathematical model. This paper proposes another transportation strategy for tackling TIFTP of type - 2 by applying ranking function found in [2].

RT&A, No 4 (65) Volume 16, December 2021

The association of this article is as per the following: In Section 2, a review on essentials IFS and IFN's. Segment 3, presents the Ranking function and Comparison of TIFN's. Area 4, briefs the numerical detailing and proposed TP technique. Delineates the mathematical model in Section 5. At long last, Section 6 exposes the conclusion.

II. Preliminaries

In this part a couple of essential definitions and math tasks are examined.

Intuitionistic Fuzzy Set (IFS): An IFS A,FS in X an IFS is described as an object of following design

~A1FS

— { (x, №%ifs

(x) ,v%ifs(x)) '-x e x}

where, functions^%IFS : X ^ [0, 1] and v%ifs : X ^ [0, 1] defines degree of Enrollment work and non-participation elementx E X, respectively and 0 < p%ifs(x),v%ifs(x) < 1, for everyx E X. Intuitionistic Fuzzy Numbers (IFN's): A subset of IFS,^,fs — { (x,^%ifs(x),v%ifs(x))\x E X}, of real line ^ is called an IFN if the following holds:

(i) 3m E %v~ifs (m) — 1 and v~i#s (m) — 0

(ii)

: : ^ ^ [0, 1] is continuous and for every xE^,0< ifs (x),v~ifs(x) < 1 holds.

^ifs (X) — ? h1(x),' x E (m,m + P&] and ^ifs (x) — } 0,

Enrollment work and non-participation capacity ofA,FS is as follows,

f1(x), xE[m-a1,m) ( 1, xE(-m,m-a2)

1, x — m _ _ J f2(x), xE[m- a2,m)

x — m,x E [m + p2,m) 0, otherwise V h2(x), xE(m,m + p2]

Where, f((x)and hi(x); i — 1,2 are strictly increasing and decreasing functions in [m — ai,m) and (m, m — respectively. ai and b are left and right spreads of ifs(x)and v~ifs(x) respectively.

Triangular Intuitionistic Fuzzy Number (TIFN): A TIFN A,FNis an IFS in ^ with the following

and non-participation capacity v~ IFN defined by

a1 <x<a2

Enrollment function ^ IFN

"FN

(X) —

x—a&

a2-a& ' a&—x

a3 — a2 '

0,

a2 <x<a3 andVAIFN(x) — otherwise

x—a2 f i

0,

& <x<a2

a2 < x < a'3 otherwise

Wherea'& < a& < a2 < a3 < a'3. This TIFN is denoted by A,FN — (a&, a2, a3; a'&, a2, a'3) in Fig 1.

«i «2 «3 «3

Figure 1: Participation and non-enrollment elements of TIFN

Arithmetic operations of TIFN:

Forany two TIFN's^,FN — (a1,a2,a3-,a'1,a2,a'3)andBIFN — (b1,b2,b3;b'1,b2,b'3), arithmetic operations are as follows, (i) Addition:

Indira Singuluri, N. Ravishankar

A NOVEL TRANSPORTATION APPROACH TO SOLVING TYPE-2 RT&A, No 4 (65)

TRIANGULAR INTUTIONISTIC FUZZY TRANSPORTATION PROBLEMS Volume 16, December 2021 aifn q bifn = (a& + b&, a' + ba- + b-. a,& + v^ a' + ba,3 + b,3)

(ii) Subtraction:

AIFN - BIFN = (a& - b-i - a- - b&. a'& - b,- ba,3 - b,&)

(iii) Multiplication:

Aifn ®Bifn = (a1b1,a2b2,a3b3ma'&b'&,a2b2,a'3b'3)

(iv) Scalar multiplication:

^ ~IFN ((kal, ka-> &, 3), k ^ 0

k <0

III. Ranking Function

Ranking function is taken from [2], i.e., the ranking function is defined for Trapezoidal and triangular Intuitionistic fuzzy number as

a + b + 2 (a 2 + b3) + 5 (a3 + b2) + (a4 + b4 ) V 4w1 + 5 w2

R ( AIFN ) =

R ( A ) =

18 J0 18

( ai + bl ) + 14a 2 +( a 4 +b4 ) V Aw1 + 5w2

18 0 18 Consider w& = w2 = 1, we get ranking function is

R ( A) =

( al + ¿j) + 14a 2 +( a4 +b 4 )N 36

Comparison of TIFN's: To contrast TIFN's and one another, we need to rank them. A function such as R: F(^) ^ which maps each TIFN's into real line, is called ranking function. Here, F(^)means the arrangement of all TIFN's. By using ranking function" R", TIFN's can be compared.

Let Aifn = (a&, a2, a3; a'&, a2, a'3) and Bifn = (b&, b2, b3; b'&, b2, b'3)are two TIFN's then R (>n) = ,&0&1,'02+a'l+a'3 and R (bifn) = b&+14b'+b(+b'&+b'( then the orders are defined as follows Aifn > Bifn if R (>n) > R (bifn) ,

(i)

AIFN = BIFN if r(Àifn) = R(BIFN)

AIFN < BIFN if R (>N) < R (BIFN) , and

(iii)

Ranking function R also holds the following properties:

(i) R (ÀIFN) + R (BIFN) = R (ÀIFN + BIFN), (ii) R (kÀIFN) =k R (ÀIFN) Vk E R

IV. Mathematical Formulation of Triangular Intuitionistic Fuzzy transportation problem (TIFTP) and proposed method I. TIFTP of type - 2:

Consider a transportation with 'm' Intuitionistic Fuzzy (IF) origins and 'n' IF destination.

Let C(j(i = 1,2, ...,mmj = 1,2, ...,n) be the cost of transporting one unit of the product form ithorigin

to ythdestination.

~]IFS = (a[, a'2, a'3; a[ ', a'2, a'3 ) be IF extent at ith vendor. bjPS = (b, b'2, b ; b(, b'2, bl ) be IF abundant at jth insistent.

~ijFS = (xi , X2> X3 X , x'ij, x'3 ) be IF quantity transformed fromi56vendor to jth insistent Then balanced triangular IFTP of type - 2 is given by

MinZIFN xxT

i=i j=i

Xj=i! =a! , i = i,2,..., m XhX = ,j^...n

XN >0; i = 1,2,...., m; j = 1,1,...., n II. Proposed Transportation strategy

Stage 1: Utilizing separation formula, considered in "Comparison of IFTN's" segment, adopt least and greatest IFN from each archive and segmentof intuitionistic fuzzy price matrix of TIFTP of type - 2 and deduct it from each IFN's of their relating line and segment.

Stage 2: Find sum of row difference and column difference and denote row sum by R and column sum by C. Identify Maximum sum of row and column. Select maximum difference in row and column.

Stage 3: Choose the cell having most minimal expense in row and column identified in stage 2. Stage 4: Make a feasible assignment to the cell picked in stage 5. Delete fulfilled row/column. Stage 5: Repeat the technique until all the designations has been made.

Stage 6: The Optimum solution and triangular intuitionistic optimum value is attained in step 5, is optimum solution {x(j} and triangular intuitionistic fuzzy optimum value is YHL& Yj=i cij®xij.

V. Numerical Example In this part, an existing mathematical model ([2]) is solved to illustrate the proposed transportation strategy.

Table 1: TIFTP of type - 2

D d2 D3 D4 Suppl

y

(st)

Sl (2,45; (2,5,7; (4,6,8; (4,7,8; 11

1,4,6) 1,5,8) 3,6,9) 3,7,9)

(4,6,8; (3,7,12; (10,15,20; (11,12,13; 11

3,6,9) 2,7,13) 8,15,22) 10,12,14)

(3,4,6; (8,10,13; (2,3,5; (6,10,14; 11

1,4,8) 5,10,16) 1,3,6) 5,10,15)

(2,4,6; (3,9,10; (3,6,10; (3,4,5; 12

1,4,7) 2,9,12) 2,6,12) 2,4,8)

Dema 16 10 8 11 45

nd

( dj)

Example 1: An existing TIFTP of type - 2, with four suppliers i.e., S1,S2,S3,S1 and four destinations i.e.,D1, D2, D3,D4, respectively by Table 1, is solved using the proposed method. This problem is solved in the following steps. Select maximum and minimum TIFN in each row and column take the difference as given in table 2.

Table 2: Roio and Column Difference Table

Di D2 D3 d4 Supply Oi) Row diff

Si (2,4,5; 1,4,6) (2,5,7; 1,5,8) (4,6,8; 3,6,9) (4,7,8; 3,7,9) 11 1.4444

S2 (4,6,8; 3,6,9) (3,7,12; 2,7,13) (10,15,20; 8,15,22) (11,12,13; 10,12,14) 11 4.5

S3 (3,4,6; 1,4,8) (8,10,13; 5,10,16) (2,3,5; 1,3,6) (6,10,14; 5,10,15) 11 3.5

S4 (2,4,6; 1,4,7) (3,9,10; 2,9,12) (3,6,10; 2,6,12) (3,4,5; 2,4,8) 12 2.125

Demand 16 10 8 11 45 R=11.56

(d ) Column diff 1.0555 2.6111 5.9444 3.9444 C=13.55

The problem given in Table 2, transformed in Table 3 by using the Stage 2 and assign first allocation using stage 4 of proposed method.

Table 3: First allocation Table

oi d2 D3 d4 Supply Row difference

Si (2,4,5; (2,5,7; (4,6,8; (4,7,8; 11 1.4444

1,4,6) 1,5,8) 3,6,9) 3,7,9)

s2 (4,6,8; (3,7,12; (10,15,20; (11,12,13; 11 4.5

3,6,9) 2,7,13) 8,15,22) 10,12,14)

S3 (3,4,6; (8,10,13; (2,3,5; (6,10,14; 3.5

1,4,8) 5,10,16) 1,3,6) 5,10,15) 3

[8]

(2,4,6; (3,9,10; (3,6,10; (3,4,5; 12 2.125

1,4,7) 2,9,12) 2,6,12) 2,4,8)

Demand 16 10 11 45 R =

11.

5

416

Column 1.0555 2.6111 5.9444 3.9444 C =

difference 13.5

554

Using Stage 4 of proposed method remove D3 from Table 3. New reduced shown in Table 4 again apply the procedure.

Table 4: Neio Reduced Table

Di D2 Ü4 Supply Oi) Row difference

Si (2,4,5; 1,4,6) (2,5,7; 1,5,8) (4,7,8; 3,7,9) 11 1.4444

S2 (4,6,8; 3,6,9) (3,7,12; 2,7,13) (11,12,13; 10,12,14) 11 3

S3 (3,4,6; 1,4,8) (8,10,13; 5,10,16) (6,10,14; 5,10,15) 3 3

S4 (2,4,6; 1,4,7) (3,9,10; 2,9,12) (3,4,5; 2,4,8) 12 2.125

Demand 16 10 11 45 R = 9.5694

(d.) Column dlfferen 1.0555 2.6111 3.9444 C = 7.6110

Table 5: Second Allocation table

D i D2 Ü4 Supply (Si) Row diff

Si (2,4,5; 1,4,6) (2,5,7; 1,5,8) (4,7,8; 3,7,9) 11 1.4444

s2 (4,6,8; 3,6,9) (3,7,12; 2,7,13) (11,12,13; 10,12,14) 11 3

S3 (3,4,6; 1,4,8) [3] (8,10,13; 5,10,16) (6,10,14; 5,10,15) 3 3

S4 (2,4,6; 1,4,7) (3,9,10; 2,9,12) (3,4,5; 2,4,8) 12 2.125

Demand (d ) Column difj 13 1.0555 10 2.6111 11 3.9444 45 R = 9.5694 C = 7.6110

Again, applying the Stage 5 of the proposed method, all the allocations are made as shown in Table 6.

Table 6: Final allocation table

01 d2 d3 d4

S1 (2,4,5; (2,5,7; (4,6,8; (4,7,8;

1,4,6) 1,5,8) 3,6,9) 3,7,9)

[1] [10]

(4,6,8; (3,7,12; (10,15,20; (11,12,13;

3,6,9) 2,7,13) 8,15,22) 10,12,14)

[11]

(3,4,6; (8,10,13; (2,3,5; (6,10,14;

1,4,8) 5,10,1 6) 1,3,6) 5,10,15)

[3] [8]

(2,4,6; (3,9,10; (3,6,10; (3,4,5;

1,4,7) 2,9,12) 2,6,12) 2,4,8)

[1] [11]

Step 6: Optimum solution and IF optimum value

The optimum solution, obtained in Step 5, is xn = 1, x12 = 10, x21 = 11, x31 = 3, x33 = 8, x41 = 1 and x44 = 11. The IF optimum value of IFTP of type - 2, given in Table 1, is

1 ® (2,4,5;1,4,6) 010 ® (2,5,7;1,5,8) 011 ® (4,6,8;3,6,9) 0 3 ® (3,4,6;1,4,8) 0 8 ® (2,3,5;1,3,6) 01® (2,4,6;1,4,7) 011® (3,4,5; 2,4,8) = (126,204,282;78,204,3 52).

VI. Conclusion

Numerical Formulation for IFTP of type - 2 and system for acquiring an IF ideal arrangement is examined with relevant numerical example. The proposed transportation strategy is utilized to get the ideal arrangement of TIFTP of type - 2. The proposed transportation technique gives same outcome, as found by G. Gupta, A. Kumara [3], in single emphasis. Consequently, this might be favored over the current strategies.

References

[1] Chetia. B and Das. P. K., (2012). Some results of intuitionistic fuzzy soft matrix theory, Advances in Applied Science Research, 3, 412-413.

[2] Pardhasaradhi . B, Madhuri.M.V. and Ravi Shankar. N, (2017). Ordering of Intuitionistic fuzzy numbers using centroid of centroids of Intuitionistic fuzzy numbers", International Journal of Mathematics Trends and Technology, Vol. 52, No.5, 276-285.

[3] Gupta.G, Kumara. A (2017). An efficient method for solving intuitionistic fuzzy transportation problem of type-2, International Journal of Applied and Computational Mathematics, 3, 37959804.

[4] Atanassov. K. T., (1986). Intuitionstic fuzzy sets", Fuzzy Sets and Systems, 20, 87-96.

[5] Atanassov. K. T., (1989). More on intuitionstic fuzzy sets", Fuzzy Sets and Systems, 33, 37-46.

[6] Bellman.R.E. and Zadeh.L. A., (1970). Decision making in a fuzzy environment", Management Sciences, vol.17, 141-164.

Indira Singuluri, N. Ravishankar

A NOVEL TRANSPORTATION APPROACH TO SOLVING TYPE-2 RT&A, No 4 (65)

TRIANGULAR INTUTIONISTIC FUZZY TRANSPORTATION PROBLEMS Volume 16, December 2021

[7] Jahir Hussain. R and Senthil Kumar. P., (2012). Algorithm approach for solving intuitionistic fuzzy transportation problem, Applied mathematical sciences, vol. 6(80), 3981-3989.

[8] SagayaRoseline and Henry Amirth raj, (2015). New Approaches to Find the Solution for the Intuitionistic Fuzzy Transportation Problem with Ranking of Intuitionistic Fuzzy Numbers, International Journal of Innovative Research in Science Engineering and Technology, vol. 4(10), 10222-10230.

[9] Singh.S.K., Yadav.S.P., (2016). A new approach for solving intuitionistic fuzzy transportation problem of type-2, Ann. Oper. Res. 243, 349-363.