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A DIFFERENT INITIATIVE TO FIND AN OPTIMAL SOLUTION TO THE TRIANGULAR FUZZY TRANSPORTATION PROBLEM BY IMPLEMENTING THE ROW-COLUMN MAXIMA METHOD
A. Kokila
Research Scholar, Department of Mathematics, SAS, VIT, Vellore, Tamil Nadu, India
kokila.a@vit.ac.in
G. Deepa
Assistant Professor, Department of Mathematics, SAS, VIT, Vellore, Tamil Nadu, India
deepa.g@vit.ac.in
Abstract
In this paper, we discussed an issue in fuzzy transportation problem, which involves fuzzy costs, fuzzy supply, and fuzzy product needs. The goal of this article is to convey the item from point of origin to point of destination at the least possible cost. For fuzzy transportation problems with balance and unbalance types, the proposed technique provides a superior optimal. Transportation costs, supply, and demand are represented by generalized triangular fuzzy numbers using this proposed named Row - Column Maxima Method (RCMM). A numerical example of a fuzzy transportation problem is illustrated and the solution is compared with the outcomes of other approaches. This method reduces iterations and which help to understand and implement easily in real life applications.
Keywords: Fuzzy set, Fuzzy Number, Triangular fuzzy number, Fuzzy Transportation problem, RCM- Method, Fuzzy optimal solution.
1. Introduction
In 1941, Hitchcock had his initial idea regarding the transportation problem. In 1965, L.A. Zadeh [17] created fuzzy set theory and successfully applied it to a variety of fields. There is a need to send products from various origins (Factories) to various destinations in a variety of real-world situations (warehouses). The decision maker's goal is to figure out how much product to order. Many distribution challenges in today's actual world, such as in business or industrial settings are imprecise in nature due to parameter variances. However, due to some unavoidable circumstances, all of these elements of the transportation problem may not be precisely understood in real time. In 1978, the fuzzy decision-making method was introduced. Zimmermann developed a variety of fuzzy optimization algorithms for TP and FTP [18]. Hitchcock [5] was the first to come up with the basic transportation problem. To handle the totally fuzzy transportation problem,
Dhanaseker et al. [4] presented the Hungarian-Modi technique. Muthuperumal et al. [9] offered an algorithmic solution to the problem of unbalanced triangular fuzzy transportation. Senthil Kumar et al. [13] suggested the Harmonic Mean Way as a new method for solving the Generalized Fuzzy Transportation problem. A new strategy for finding an optimal solution to Generalized Fuzzy Transportation Problems was proposed by Srinivasaro Thota and Raja [16]. Fuzzy Transportation Problem By Using Triangular Fuzzy Numbers With Ranking Using Area Of Trapezium, Rectangle, And Centroid At Different Levels Of -Cut was discussed by Ambadas Deshmukh et al[1]. Balasubramanian et al. [2], [3] explored utilizing a ranking function to solve the Fuzzy Transportation Problem. Srinivasan et al. [14] established a method for handling fully fuzzy transportation problems in which the materials are transformed, and this method is straightforward to evaluate and can rank many forms of triangular fuzzy numbers. This study by Ladji Kane et al. [8] addressed a Simplified approach for Solving Transportation Problems with Triangular Fuzzy Numbers in Fuzzy Environments. Purushoth kumar et al. [10] proposed employing the diagonal optimum method to address fully fuzzy transportation problems. Indira Singuluri et al. [6] proposed their strategies to address a novel transportation approach to solving type-2 triangular intuitionistic fuzzy transportation problems.
In this study, we offer a new method for solving the fuzzy transportation problem called the RCM method, which assumes supply, demand, and unit transportation cost as triangular fuzzy integers. It provides a minimal value when compared to other approaches such as the NWCM [North-West Corner Method], LCM [Least Cost Method], VAM [Vogel's Approximation Method], and RMM [Row Minima Method]. Finally, an example is provided to aid in the comprehension of the method.
The remainder of this work is arranged in the following manner. Present the fundamental definitions and mathematical constructions of transportation problems in section 2. Present a new algorithm to handle the fully fuzzy transportation problem in section 3. The proposed approach is illustrated numerically in Section 4. The conclusion and future study is presented in section 5.
2. Preliminaries
Definition 2.1[17]
Let U is a collection of elements indicated by u then a fuzzy set T is a set of ordered pairs in U: T = %(u, (u))| u £ U/, where the membership function or grade of membership of u in J1 is ^(u):U > [0,1].
Definition 2.2[4]
P is a fuzzy set of real numbers that is defined on the universal set of real numbers. If R's membership function satisfies the following properties, R is said to be a fuzzy number.
1. y.p(u) is a piecewise continuous
2. T is convex. + (1 — S)u#) > min(^y(U"), ^(u#)), Vur, u# £ K & VS £ [0, 1].
3. T is Normal.
Definition 2.3[4]
If the membership function [0,1] of a fuzzy number P on R satisfies the following
characteristics, it is said to be a triangular fuzzy number (TFN) or linear fuzzy number.
P(u) =
( u — Pi
Pi — Pi 1
, for pi <u< p2
, U = P" , for p2 <u< p#
p# — P"
0 , elsewhere
0 Pi p2 P3
Fig 1 .Triangular fuzzy number
2.4 Arithmetic Operation on Fuzzy Numbers [4]:
The operations that can be performed on triangular fuzzy numbers are as follows: Then, if T = (P1-P2-P3) and Q = (i?!,^,^).
(i) Addition: T + Q = (px + q1,p2 + q2,p3 + q3).
(ii) Subtraction: T -Q = (Pl- q3,p2 - q2,p3 - qi).
(iii) Multiplication: T x Q = (p1q1,p2,p3q3)-
2.5 MATHEMATICAL CONSTRUCTION [9]:
A fuzzy transportation problem can be expressed mathematically as follows:
Minimize (Total cost) Z = c%&
Subject to the constraints
£"=1 Xi& = , i = 1,2,........., m(Fuzzy Supply constraints)
X™" = d&, j = 1,2,.........,n(Fuzzy Demand constraints)
Xij >0, i = 1,2,........., m and j = 1,2,........., n
Where m: Total number of sources, n: Total number of destinations
Notations:
: The product's fuzzy availability at ith the source. d&: The product's fuzzy demand at jth destination.
Cy: The fuzzy transportation cost of transporting one unit of commodity from ith source to jth destinations.
Xij: To minimize total fuzzy transportation, a fuzzy quantity is delivered from i*+source to jth destination (or fuzzy decision variables).
E'=1 : The fuzzy cost of transporting one unit of the product from ith source to the jth destination.
: The product's total fuzzy availability X'=1 d&: The product's total fuzzy demand
Destination 1 Destination 2 .. Destination n Supply
Source 1 C12^12 ■■ &1
Source 2 C22X22 ■■ C2n^2n *>2
Source m Cm2^m2 ■ ■ ^mn^mn ^m
mn
Demand d1 ^2 ■■ d' = Rdj
%(1 j=1
2.6 Balanced and unbalanced FTP [11]:
Balanced fuzzy transportation problem: The total fuzzy supply is equal to total fuzzy demand
) n
i.e. = R&
i=i &=i
Unbalanced fuzzy transportation problem:
The total fuzzy supply is not equal to total fuzzy demand
) n
i=l j=1
2.7 To Modify Unbalanced FTP to Balanced FTP: An Unbalanced FTP may occur in two different forms: (i) Excess of availability, (ii) Shortage in availability.
We now discuss these two cases by considering the usual m- sources, n- destinations FTP with the condition that 2)=1^ ^ ^j Case1: (Excess of Availability, i.e. 2 >2 dy) The general FTP may be stated as follows: Minimize (Total cost) £ = 2)=12n=1 ^y cy Subject to the constraints
2'=" ^ , i = 1,2,........., m(Fuzzy Supply constraints)
2)(1 = ^j, j = 1,2,.........,n(Fuzzy Demand constraints)
and Xij >0, i = 1,2,........., m and j = 1,2,........., n
The problem will possess a fuzzy feasible solution if 2^% > . In the first constraints, the introduction of slack variable #i,n+1 (i = 1,2,........., m) gives
R R ^ij + ^i,n+1 = ¿i, i = 1,2, ■
j=1
) / n \ )
R I R + *i,n+1 I = R ^
,,m
i=1 v=1
n / ) \ ) )
j=1 i=1 i=1 i=1 ^ 2n = 1 + 2i=1 ^"i,n+1 = 2i=1 ¿i (" 2i=1 ^ij = ^j
^ 2)=1 #i,n+1 = 2)=1 - 2n(1 = Excess of Availability If this excess availability is denoted by ¿n+1, the modified FTP, can be reformulated as: Minimize Z = 2)=12n=i *y cy , Subject to the constraints
R R ^ij + ^i,n+1 = ¿i, ^ = 1,2, ■
,,m
j=1
2)=1 = ^j, j = 1,2,.........,n + 1 and > 0, for all i and j
and ci n+1 = 0, for i = 1,2,........., m and 2)=1 = 2'=+! ^j
This is clearly the balanced FTP and thus can be easily solved by fuzzy transportation algorithm.
Working Rule:If2 >2 ^j, avoid using a fake row or column when converting to balance. Let see M = 2)=1 — 2n(1 ^j. The difference w added to the demand (^1,^2,^3) minimum. Reconstruct the provided Fuzzy transportation table using + w1, + oi2, + oi3).
Case2: (Shortage in Availability, i.e. £.6; < £ dy) In this case, the general FTP becomes: Minimize Z = £'=" Cy Subject to the constraints
R R ^y = ,
i = 1,2,.........,m
&=1
,,n
< , ; = 1,2,.„
%=i
and >0, i = 1,2,........., m; j = 1,2,........., n
Now, introducing the slack variable #m+1j (/ = 1,2,........., n) in the second constraint, we get
R R ^y + ^m+1j = , 7 = 1,2,
i=1
n / m \ n
R [ R *y + *m+1j ) = R \i=1 / & = 1 ) / n \ n n
,,n
d&
i = 1 \& = 1 f & = 1 &=1 ^ Si=1 ^i + Zy=1 ^"m+1J = Sj = 1 (v Zy = 1 ^-y = ^i ^ Sn=1 ^m+1,& = Sn=1 - = Shortage in availability say
Thus the modified FTP, in this case becomes:
Minimize Z = D+1 Sn=1 *y cy,
Subject to the constraints
Tl
R Xij = i = 1,2,.........,m+1
&=1
+ ^m+1,& = , j = 1,2,........., n and Xi& > 0, for all i and j
where cm+1,& = 0, for j = 1,2,.........,n and = Sn=1
This is clearly the balanced FTP and thus can be easily solved by fuzzy transportation algorithm.
Working Rule: IfS)=1 ^ Sn=1 , avoid using a fake row or column when converting to balance. Let see m = — Sn=1 The difference w added to the supply minimum.
Reconstruct the provided Fuzzy transportation table using + + w2,^3 + oi3).
2.8 Fuzzy Feasible Solution [9]:
A fuzzy feasible solution is any set of fuzzy non negative allocations xi& (xi& > 0) that fulfills (in the sense equivalent) the row and column requirements.
2.9 Fuzzy Basic Feasible Solution [9]:
If the number of positive allocations is exactly equal to (m + n — 1), a fuzzy feasible solution to a fuzzy transportation problem with m origins and n destinations is said to be fuzzy basic feasible solution.
2.10 Fuzzy Optimal Solution [9]:
If the entire fuzzy transportation cost is minimized, a fuzzy feasible solution is said to be fuzzy optimum.
Theorem 2.11 [11]: (Existence of Fuzzy feasible solution)
A necessary and sufficient condition for the existence of feasible solution of a fuzzy
transportation problem is £J=1= Yj=1 dj (i = 1,2,.........,m; j = 1,2,.........,n).
Proof: The condition is necessary: Let there exist a feasible solution to the fuzzy transportation problem. Then,
Hi = 1 Hj = i xij = Hi = i , (1)
Z'=1 ZE1 xi} =X j=1 dj, (2)
From equation (1) and (2), we get
«ZE1 = Zj=1 dj.
The condition is sufficient: Let £J=1 ¿i = Sj=1 dj = K (say).
If toi ^ 0 be any real number such that x%& = toidjV i,j, then toiis given by
j j j
R xij = R ^ d> = to R dJ = Kto j=1 j=1 j=1 ^ toi = ^ Hj = 1 xij = ft (v Hj=1 xij =
Thus, xtj = to^j = > 0, since > 0,dj> 0 V i,j. Hence a Fuzzy feasible solution exists.
3. Proposed algorithm
In this paper, we proposed Row-Column maxima method [RCMM] to find optimum solution and
this result compared with NWCM, LCM, RMM, VAM methods.
Step 1: Check to see if the given FTP is balanced or not.
Casel: If £J=1 ¿i = S'=1 dj. then go to step 3.
Case2: IfYH=1 ¿i ^ S'=1 dj, possible, avoid using a fake row or column when converting to balanced. Let see = £J=1 - S'=1 dj ifTj=1 dj < £J=1 ¿i or (ii)u = ZK1 dj - X'=1 ¿i ifzv=1 Si < X'=1 dj.
Step 2: The difference m will be divided into three parts (m1,m2,m3) such that m = X$=1 ^ and added to the supply (s1,s2,s3) or demand (d1,d2,d3) minimum. Reconstruct the provided Fuzzy transportation table using (s1 + m1,s2 + m2,s3 + m3)/(d1 + M1,d2 +
u>2,d$ +
Step 3: For each row, find the difference between the first and second maximum values and use that value instead of the first maximum value.
Step 4: After completing step 3, calculate the difference between the 1st and 2nd maximum values and use that value to replace the 1st maximum value in each column.
Step 5: Choose the fuzzy cost's minimum value in either a row or a column. Then determine the minimum supply and demand value and assign it.
Step 6: After step 5, delete the row/column in which supply/demand has reached its limit.
Step 7: Steps 5 - 6 should be repeated until (m + n-1) cells have been allotted.
Step 8: Calculate the minimum Fuzzy Transportation Cost. That is, Total Cost= X)=1 cij S'=1 ^j.
4. Numerical Example
A manufacturing company produces diesel engines in 10 cities C1,C2, C3,C4, C5, C6, C7, C8, C9, C10 and they are purchased by ten trucking companies T1, T2, T3, T4, T5, T6, T7, T8, T8, T10. The table below indicates how many engines are required by T1, T2, T3, T4, T5, T6, T7, T8, T8, T10. It also displays the cost of transportation per engine from origin to destination. The corporation wants to maintain the total transportation cost to a minimum.
Table 1: Triangular Fuzzy Transportation Problem
T1 T¿ T3 Tt T¡ T„ T7 T„ T, Tw Supply
Cr (3,5,7) (4,6,8) (3,6,9) (11,12,13) (2,3,4) (7,8,9) (5,6,7) (12,14,16) (1,4,7) (7,8,9) (15,20,25)
C-L (2,5,8) (5,7,9) (4,5,6) (7,8,9) (14,16,18) (12,13,14) (5,7,9) (0,1,2) (2,4,6) (6,7,8) (5,10,15)
c3 (5,6,7) (4,6,8) (11,13,15) (3,6,9) (14,15,16) (2,3,4) (8,9,10) (4,8,16) (9,10,11) (14,16,18) (25,30,35)
c4 (9,10,11) (2,5,7) (2,3,4) (3,5,7) (10,15,20) (5,6,7) (7,8,9) (1,3,5) (3,6,9) (10,11,12) (40,45,50)
Cs (8,9,10) (2,4,6) (8,10,12) (6,8,10) (3,6,7) (10,12,14) (1,4,7) (14,16,18) (1,2,3) (8,9,10) (90,95,100)
Ce (6,7,8) (12,13,14) (14,16,18) (1,2,3) (1,3,5) (3,5,7) (6,8,10) (2,4,6) (7,8,9) (13,15,17) (70,75,80)
c7 (5,6,7) (12,14,16) (13,15,17) (5,6,7) (0,1,2) (11,13,15) (14,16,18) (2,4,6) (7,9,11) (3,5,7) (50,55,60)
C„ (16,18,20) (1,3,5) (7,8,9) (8,10,12) (3,6,9) (4,5,6) (10,11,12) (3,6,9) (14,15,16) (4,6,8) (65,70,75)
c, (4,6,8) (1,2,3) (2,4,6) (11,12,13) (1,2,3) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6) (85,90,95)
(7,8,9) (5,7,9) (6,8,10) (9,11,13) (4,6,8) (14,15,16) (11,12,13) (14,16,18) (0,2,4) (2,3,4) (55,60,65)
Demand (15,20,25) (40,45,50) (25,30,35) (5,10,15) (50,55,60) (70,75,80) (90,95,100) (85,90,95) (65,70,75) (55,60,65)
Applying the proposed algorithm [RCMM]: Step 1:
^ = (500,550,600) and Sn=1 d& = (500,550,600). ^ = Sn=1 (Total supply = Total demand).
Since the given Fuzzy Transportation Problem is balanced. So go to step 3,
Step 3:
In first row, First maximum value = (12,14,16) Second maximum value = (11,12,13) The difference between 1st and 2nd maximum value i.e., (12,14,16) — (11,12,13) = (—1,2,5)
Then replace the subtracted value instead of the first maximum value i.e., (12,14,16) = (—1,2,5)
Similarly, apply step 3 other 2nd, 3rd up to 10th row, then we get table 2.
Table 2: Row-wise Difference Table
Tr T2 T3 Ts Te T7 T, T, Tw Supply
Cr (3,5,7) (4,6,8) (3,6,9) (11,12,13) (2,3,4) (7,8,9) (5,6,7) (-1,2,5) (1,4,7) (7,8,9) (15,20,25)
C-L (2,5,8) (5,7,9) (4,5,6) (7,8,9) (0,3,6) (12,13,14) (5,7,9) (0,1,2) (2,4,6) (6,7,8) (5,10,15)
c3 (5,6,7) (4,6,8) (11,13,15) (3,6,9) (14,15,16) (2,3,4) (8,9,10) (4,8,16) (9,10,11) (-2,1,4) (25,30,35)
Ci (9,10,11) (2,5,7) (2,3,4) (3,5,7) (-2,4,10) (5,6,7) (7,8,9) (1,3,5) (3,6,9) (10,11,12) (40,45,50)
Ca (8,9,10) (2,4,6) (8,10,12) (6,8,10) (3,6,7) (10,12,14) (1,4,7) (0,4,8) (1,2,3) (8,9,10) (90,95,100)
Ce (6,7,8) (12,13,14) (-3,1,5) (1,2,3) (1,3,5) (3,5,7) (6,8,10) (2,4,6) (7,8,9) (13,15,17) (70,75,80)
c-, (5,6,7) (12,14,16) (13,15,17) (5,6,7) (0,1,2) (11,13,15) (-3,1,5) (2,4,6) (7,9,11) (3,5,7) (50,55,60)
C„ (0,3,6) (1,3,5) (7,8,9) (8,10,12) (3,6,9) (4,5,6) (10,11,12) (3,6,9) (14,15,16) (4,6,8) (65,70,75)
c, (4,6,8) (1,2,3) (2,4,6) (1,3,5) (1,2,3) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6) (85,90,95)
(7,8,9) (5,7,9) (6,8,10) (9,11,13) (4,6,8) (14,15,16) (11,12,13) (-2,1,4) (0,2,4) (2,3,4) (55,60,65)
Demand (15,20,25) (40,45,50) (25,30,35) (5,10,15) (50,55,60) (70,75,80) (90,95,100) (85,90,95) (65,70,75) (55,60,65)
Step 4: In table 2, apply the step 4 of proposed algorithm
In first column, First maximum value = (9,10,11), Second maximum value = (8,9,10) The difference between ^ and 2nd maximum value = (9,10,11) — (8,9,10) = (—1,1,3)
Then replace the subtracted value instead of the first maximum value. [ (9,10,11) = (-1,1,3)] Similarly, apply step 4 other 2nd, 3rd up to 10th column, then we get table 3.
_Table 3: Column-wise difference table_
T, T2 T3 n Ts Te T7 T„ T, Tw Supply
C, (3,5,7) (4,6,8) (3,6,9) (-2,1,4) (2,3,4) (7,8,9) (5,6,7) (-1,2,5) (1,4,7) (7,8,9) (15,20,25) - (5,10,15) = (0,10,20)
C2 (2,5,8) (5,7,9) (4,5,6) (7,8,9) (0,3,6) (12,13,14) (5,7,9) (0,1,2) (2,4,6) (6,7,8) (5,10,15)
c3 (5,6,7) (4,6,8) (11,13,15) (3,6,9) (6,9,12) (2,3,4) (8,9,10) (-5,2,13) (9,10,11) (-2,1,4) (25,30,35)
c4 (-1,1,3) (2,5,7) (2,3,4) (3,5,7) (-2,4,10) (5,6,7) (7,8,9) (1,3,5) (3,6,9) (10,11,12) (40,45,50)
Cs (8,9,10) (2,4,6) (8,10,12) (6,8,10) (3,6,7) (10,12,14) (1,4,7) (0,4,8) (1,2,3) (8,9,10) (90,95,100)
Ce (6,7,8) (12,13,14) (-3,1,5) (1,2,3) (1,3,5) (3,5,7) (6,8,10) (2,4,6) (7,8,9) (1,4,7) (70,75,80)
c7 (5,6,7) (-2,1,4) (-2,2,6) (5,6,7) (0,1,2) (11,13,15) (-3,1,5) (2,4,6) (7,9,11) (3,5,7) (50,55,60)
C„ (0,3,6) (1,3,5) (7,8,9) (8,10,12) (3,6,9) (4,5,6) (10,11,12) (3,6,9) (3,5,7) (4,6,8) (65,70,75)
c, (4,6,8) (1,2,3) (2,4,6) (1,3,5) (1,2,3) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6) (85,90,95)
(7,8,9) (5,7,9) (6,8,10) (9,11,13) (4,6,8) (-1,2,5) (-1,1,3) (-2,1,4) (0,2,4) (2,3,4) (55,60,65)
Demand (15,20,25) (40,45,50) (25,30,35) (5,10,15) (50,55,60) (70,75,80) (90,95,100) (85,90,95) (65,70,75) (55,60,65)
Follow step 5 of the outlined procedure in table 4 to assign the initial allocation.
Table 4: First allocation table
T, T2 T3 Tb Te T7 T„ T, Tw Supply
c, (3,5,7) (4,6,8) (3,6,9) (5,10,15) (-2,1,4) (2,3,4) (7,8,9) (5,6,7) (-1,2,5) (1,4,7) (7,8,9) (15,20,25) - (5,10,15) = (0,10,20)
c2 (2,5,8) (5,7,9) (4,5,6) (7,8,9) (0,3,6) (12,13,14) (5,7,9) (0,1,2) (2,4,6) (6,7,8) (5,10,15)
C3 (5,6,7) (4,6,8) (11,13,15) (3,6,9) (6,9,12) (2,3,4) (8,9,10) (-5,2,13) (9,10,11) (-2,1,4) (25,30,35)
c4 (-1,1,3) (2,5,7) (2,3,4) (3,5,7) (-2,4,10) (5,6,7) (7,8,9) (1,3,5) (3,6,9) (10,11,12) (40,45,50)
Cs (8,9,10) (2,4,6) (8,10,12) (6,8,10) (3,6,7) (10,12,14) (1,4,7) (0,4,8) (1,2,3) (8,9,10) (90,95,100)
Ce (6,7,8) (12,13,14) (-3,1,5) (1,2,3) (1,3,5) (3,5,7) (6,8,10) (2,4,6) (7,8,9) (1,4,7) (70,75,80)
C7 (5,6,7) (-2,1,4) (-2,2,6) (5,6,7) (0,1,2) (11,13,15) (-3,1,5) (2,4,6) (7,9,11) (3,5,7) (50,55,60)
C„ (0,3,6) (1,3,5) (7,8,9) (8,10,12) (3,6,9) (4,5,6) (10,11,12) (3,6,9) (3,5,7) (4,6,8) (65,70,75)
c, (4,6,8) (1,2,3) (2,4,6) (1,3,5) (1,2,3) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6) (85,90,95)
(7,8,9) (5,7,9) (6,8,10) (9,11,13) (4,6,8) (-1,2,5) (-1,1,3) (-2,1,4) (0,2,4) (2,3,4) (55,60,65)
Demand (15,20,25) (40,45,50) (25,30,35) (5,10,15) (50,55,60) (70,75,80) (90,95,100) (85,90,95) (65,70,75) (55,60,65)
Step 6: Using step 6 of the proposed method, remove T3 from table 4, and then the new reduction indicated in table 5, and again execute steps 5 to 6 for the second allocation shown in table 6.
Table 5: New Reduced Table
Ti T2 T3 r5 T6 Tj Ts r9 Tio Supply
Cl (3,5,7) (4,6,8) (3,6,9) (2,3,4) (7,8,9) (5,6,7) (-1,2,5) (1,4,7) (7,8,9) (0,10,20)
C2 (2,5,8) (5,7,9) (4,5,6) (0,3,6) (12,13,14) (5,7,9) (0,1,2) (2,4,6) (6,7,8) (5,10,15)
C3 (5,6,7) (4,6,8) (11,13,15) (6,9,12) (2,3,4) (8,9,10) (-5,2,13) (9,10,11) (-2,1,4) (25,30,35)
c4 (-1,1,3) (2,5,7) (2,3,4) (-2,4,10) (5,6,7) (7,8,9) (1,3,5) (3,6,9) (10,11,12) (40,45,50)
C5 (8,9,10) (2,4,6) (8,10,12) (3,6,7) (10,12,14) (1,4,7) (0,4,8) (1,2,3) (8,9,10) (90,95,100)
C6 (6,7,8) (12,13,14) (-3,1,5) (1,3,5) (3,5,7) (6,8,10) (2,4,6) (7,8,9) (1,4,7) (70,75,80)
C7 (5,6,7) (-2,1,4) (-2,2,6) (0,1,2) (11,13,15) (-3,1,5) (2,4,6) (7,9,11) (3,5,7) (50,55,60)
C8 (0,3,6) (1,3,5) (7,8,9) (3,6,9) (4,5,6) (10,11,12) (3,6,9) (3,5,7) (4,6,8) (65,70,75)
c9 (4,6,8) (1,2,3) (2,4,6) (1,2,3) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6) (85,90,95)
^10 (7,8,9) (5,7,9) (6,8,10) (4,6,8) (-1,2,5) (-1,1,3) (-2,1,4) (0,2,4) (2,3,4) (55,60,65)
Demand (15,20,25) (40,45,50) (25,30,35) (50,55,60) (70,75,80) (90,95,100) (85,90,95) (65,70,75) (55,60,65)
Table 6: Second allocation table
Ti T2 T3 r5 T6 Tj Ts r9 Tio Supply
Ci (3,5,7) (4,6,8) (3,6,9) (2,3,4) (7,8,9) (5,6,7) (-1,2,5) (1,4,7) (7,8,9) (0,10,20)
C2 (2,5,8) (5,7,9) (4,5,6) (0,3,6) (12,13,14) (5,7,9) (5,10,15) (0,1,2) (2,4,6) (6,7,8) (5,10,15)
C3 (5,6,7) (4,6,8) (11,13,15) (6,9,12) (2,3,4) (8,9,10) (-5,2,13) (9,10,11) (-2,1,4) (25,30,35)
c4 (-1,1,3) (2,5,7) (2,3,4) (-2,4,10) (5,6,7) (7,8,9) (1,3,5) (3,6,9) (10,11,12) (40,45,50)
C5 (8,9,10) (2,4,6) (8,10,12) (3,6,7) (10,12,14) (1,4,7) (0,4,8) (1,2,3) (8,9,10) (90,95,100)
C6 (6,7,8) (12,13,14) (-3,1,5) (1,3,5) (3,5,7) (6,8,10) (2,4,6) (7,8,9) (1,4,7) (70,75,80)
C7 (5,6,7) (-2,1,4) (-2,2,6) (0,1,2) (11,13,15) (-3,1,5) (2,4,6) (7,9,11) (3,5,7) (50,55,60)
C8 (0,3,6) (1,3,5) (7,8,9) (3,6,9) (4,5,6) (10,11,12) (3,6,9) (3,5,7) (4,6,8) (65,70,75)
c9 (4,6,8) (1,2,3) (2,4,6) (1,2,3) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6) (85,90,95)
(7,8,9) (5,7,9) (6,8,10) (4,6,8) (-1,2,5) (-1,1,3) (-2,1,4) (0,2,4) (2,3,4) (55,60,65)
Demand (15,20,25) (40,45,50) (25,30,35) (50,55,60) (70,75,80) (90,95,100) (85,90,95) -(5,10,15) =(70,80,90) (65,70,75) (55,60,65)
Step 7: Using Steps 5 to 6 of the proposed technique once again, all allocations are made as indicated in Table 7.
Table 7: Final allocations of fuzzy transportation table
Ti T2 T3 Ti r5 r6 Tj Ts r9 T10
Cl (3,5,7) (4,6,8) (3,6,9) (5,10,15) (-2,1,4) (2,3,4) (7,8,9) (5,6,7) (0,10,20) (-1,2,5) (1,4,7) (7,8,9)
C2 (2,5,8) (5,7,9) (4,5,6) (7,8,9) (0,3,6) (12,13,14) (5,7,9) (5,10,15) (0,1,2) (2,4,6) (6,7,8)
C3 (5,6,7) (4,6,8) (11,13,15) (3,6,9) (6,9,12) (2,3,4) (8,9,10) (-5,2,13) (9,10,11) (25,30,35) (-2,1,4)
c4 (15,20,25) (-1,1,3) (2,5,7) (2,3,4) (3,5,7) (-2,4,10) (5,6,7) (7,8,9) (15,25,35) (1,3,5) (3,6,9) (10,11,12)
C5 (8,9,10) (2,4,6) (8,10,12) (6,8,10) (3,6,7) (10,12,14) (15,25,35) (1,4,7) (0,4,8) (65,70,75) (1,2,3) (8,9,10)
C6 (6,7,8) (12,13,14) (25,30,35) (-3,1,5) (1,2,3) (1,3,5) (3,5,7) (6,8,10) (35,45,55) (2,4,6) (7,8,9) (1,4,7)
C7 (5,6,7) (40,45,50) (-2,1,4) (-2,2,6) (5,6,7) (0,10,20) (0,1,2) (11,13,15) (-3,1,5) (2,4,6) (7,9,11) (3,5,7)
C8 (0,3,6) (1,3,5) (7,8,9) (8,10,12) (3,6,9) (5,30,55) (4,5,6) (-10,10,30) (10,11,12) (-40,0,40) (3,6,9) (3,5,7) (20,30,40) (4,6,8)
c9 (4,6,8) (1,2,3) (2,4,6) (1,3,5) (30,45,60) (1,2,3) (25,45,65) (2,4,6) (3,5,7) (5,6,7) (8,9,10) (4,5,6)
^10 (7,8,9) (5,7,9) (6,8,10) (9,11,13) (4,6,8) (-1,2,5) (55,60,65) (-1,1,3) (-2,1,4) (0,2,4) (2,3,4)
As a result, (m + n — 1) = (10 + 10 — 1 = 19, cells are assigned and we have a feasible solution. Then find the minimum fuzzy transportation cost.
Step 8: Calculate the minimum Fuzzy Transportation Cost. Total cost Z = YL" Cy Y'=1 -^y.
= (5,10,15) (-2,1,4) + (0,10,20) (-1,2,5) + (5,10,15) (0,1,2) + (25,30,35) (-2,1,4) + (15,20,25) (-1,1,3) + (15,25,35) (1,3,5) + (15,25,35) (1,4,7) + (65,70,75) (1,2,3) + (25,30,35) (-3,1,5) + (35,45,55) (2,4,6) + (40,45,50) (-2,1,4) + (0,10,20) (0,1,2) + (5,30,55) (4,5,6) + (-10,10,30) (10,11,12) + (40,0,40) (3,6,9) + (20,30,40) (4,6,8) + (30,45,60) (1,2,3) + (25,45,65) (2,4,6) + (55,60,65) (-1,1,3) Z = (-160, 1440, 3930)
4.1 Result and discussion:
The fuzzy transportation cost Z of the given FTP is a TFN as given below:
Z = (-160, 1440, 3930). The result can be explained (Refer to Fig. 2) as follows:
□
l
0 -160 1440 3930
Fig. 2 fuzzy transportation cost
The least amount of the minimum total transportation cost is -160.
The most possible amount of the minimum total transportation cost is 1440.
The greatest amount of the minimum total transportation cost is 3930. i.e., the minimum total transportation cost will always be greater than -160 and less than 1440, and highest chances are that the minimum total transportation cost will be 3930. The above result was verified by MATLAB.
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Given Fuzzy Transportation problem is balanced TC =
-160 1440 3930
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Fig 3, Result of the Fuzzy Transportation problem
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500 1000 1500 2000 2500 3000 3500 4000 Triangular Fuzzy Number
Fig. 4 Triangular Fuzzy Number
Table 8: Comparative results ofNWCM, LCM, RMM, VAM and proposed method (RCMM) for example 1
Proposed
um erica NWCM LCM RMM VAM method
eXampk (RCMM)
1
(-570,4050,11080) (445,2160,4755) (455,2110,4775) (-70,1760,4260) (-160,1440,3930)
The comparative results in table 8 are also depicted using bar graphs and the results are given in the Figure 5.
4.2 Comparison of results: The numerical examples 2, 3, 4, 5, 6 are taken from the referred journals 1, 8, 9, 10, 14 respectively, and it is verified with our proposed method and the existing methods NWCM, LCM, RMM, VAM.
Table 9: Comparative results of NWCM, LCM, RMM, VAM and proposed method (RCMM) for example 2 to 6
Numerical examples NWCM LCM RMM VAM Proposed method RCMM
2[Ref:1] (68,176,316) (62,150,258) (62,150,258) (52,149,274) (62, 150, 258) (-2,83,208)
3[Ref:8] (1850,6609,12882) (1790,6609,12998) (1900,8292,19080) (1850,6609,12882) (3532, 6609, 9852) (-462,3249,8798)
4[Ref:9] (40,1230.3560) (-120,1210,3860) (140,1250,3220) (120,1210,3140) (270,1210,2750) (-1140,500,3760)
5[Ref:10] (125,1000,2950) (-175,850,2925) (-275,950,3450) (-25,850,2625) (-75,850,2750) (-175,350,2425)
6[Ref:14] (-270,4285,10470) (160,2455,5470) (-330,2290,6500) (-25,2220,5455) (1825,2455,3085) (-340,1025,4180)
5. Conclusion and future study
Our proposed method uses the comparison table to find the best initial feasible solution to the balanced and unbalanced fuzzy transportation problems. We compared our strategy to others and discovered that ours is the most effective. This technique considers the entire fuzzy cost of each origin and destination for allotment, allowing for a reduction in iterations to provide the best basic feasible solution to FTP. In addition, the proposed method is used to achieve the best solution for an unbalanced TP by converting it to a balanced TP without the need of a dummy source/destination, saving time and space. The proposed method is simple to implement and can be used to solve a variety of fuzzy transportation problems, including minimizing the total transportation costs. In the future, this technique might be expanded to fuzzy multiple objective transportation problems and used to solve real-world transportation problems using fuzzy numbers.
Conflict of interest: There are no conflicts of interest declared by the authors. References:
[1] Ambadas Deshmukh, Dr. Arun Jadhav, Ashok S. Mhaske, K. L. Bondar, " Fuzzy Transportation Problem By Using TriangularFuzzy Numbers With Ranking Using Area Of Trapezium, Rectangle And Centroid At Different Level Of a-Cut", Turkish Journal of Computer and Mathematics Education, Vol.12 No.12 (2021), 3941-3951, 2021.
[2] Balasubramanian K, Subramanian S, "Optimal Solution of Fuzzy transportation problem using Ranking Function", International journal of Mechanical and Production Engineering Research and development (IJMPERD), ISSN (P): 2249 - 6890; ISSN (E): 2249 - 8001, Vol.8, Issue 4, 551-558, 2018.
[3] Balasubramanian K, Subramanian S, "Solving Fuzzy transportation problem using Ranking Function", International journal of Mechanical and Production Engineering Research and development (IJMPERD), ISSN (P): 2249 - 6890; ISSN (E): 2249 - 8001, Vol.9, Issue 4, 93-98, 2019.
[4] Dhansekar S, Hariaran S, Seker, "Fuzzy Hungarian MODI Algorithem to solve Fully Fuzzy Transportation Problems", International journal of fuzzy systems, 19(5); 1479-149, 2017.
[5] Hitchcock FL, "The distribution of a product from several sources to numerous localities", J. Math. Phys., 20, 224-230, 1941.
[6] Indira Singuluri, N. Ravishankar, "A Novel Transportation Approach To Solving Type-2 Triangular Intuitionistic Fuzzy Transportation Problems", Reliability: Theory and Applications, No 4 (65) Volume 16, December 2021.
[7] Klir G.J., Yuan B, "Fuzzy Sets and Fuzzy Logic-Theory and Applications", Prentice Hall, New York, 1995.
[8] Ladji Kane, Hamala sidbe, Soulelmane kane, Hawa Bado, Moussa Konate, Daouda Diawara and Lassina Diabate, "A Simplified Method for solving Transportation problem with Triangular Fuzzy Numbers under Fuzzy Circumstances", Journal of Fuzzy Extension & Applications, Vol. 2, No.1, 89-105, 2021.
[9] Muthuperumal S, Titus P, and Venkatachalapathy M, "An Algorithmic approach to Solve Unbalanced Triangular Fuzzy Transportation Problems", Soft computing 24: 18689 - 18689, https://doi:org/10.10007/s000500-020-05103-3, 2020.
[10] Purushoth kumar M.K, Ananathanarayanan M and Dhansekar S, "Fuzzy Diagonal Optimal Algorithm to solve Fully Fuzzy Transportation problems", ARPN Journal of Engineering and Applied Sciences, Vol.14, No.19, ISSN: 1819-6608, 2019.
[11] S.D Sharma: Operation Research, (Kedar nath, Ram nath publication 2015).
[12] Senthil Kumar R, Jahir Hussain, "computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems", Int J sys Assur 7(Suppl. 2):S90-S101 DOI 10.1007/s13198-014-0334-2, 2016.
[13] Senthil Kumar S, Raja P, Shanmugasundram P, Srinivas Thota, "A New method to solving Generalized Fuzzy Transportation problem - Harmonic Mean method". International journal of Chemistry, Mathematics and Physics (IJCMP) [Vol -4, Issue -3, ISSN: 2456-866X], 2020.
[14] Srinivasan R, Karthikeyan N, Renganathan K and Vijan, "Method for solving fully fuzzy transportation problem to transform the materials", Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.05.423, 2020.
[15] Srinivasan R, Muruganandam R.S and Vijan V, "A New Algorithm for Solving Fuzzy Transportation Problem with Triangular Fuzzy Number", Asian Journal of information Technology, 3501-3805, 2016.
[16] Srinivasarao Thaota, Raja P, "A New Method for finding an optimal solution of Generalized Fuzzy Transportation Problems", Asian Journal of Mathematical Sciences (AJMS), ISSN: 2581 -3463, 2020.
[17] Zadeh L.A., "Fuzzy sets", Information and Control Vol. 8, pp. 338-353, 1965.
[18] Zimmermann HJ, "Fuzzy programming and linear programming with several objective functions", Fuzzy sets System, 1:45-55, 1978.
