CC BY
18
A.M. Dar, K. Selvakumar, S. Ramki, K. M. Karuppasamy, J. A. Ansari, . __
rr j RT&A, No 4 (76)
Aafaq A. Rather y■ '
OPTIMIZING MULTI-OBJECTIVE MULTI-INDEX ..._Volume 18, December 2023
OPTIMIZING MULTI-OBJECTIVE MULTI-INDEX TRANSPORTATION PROBLEMS: A SMART ALGORITHMIC SOLUTION WITH LINDO SOFTWARE
Ajjaz Maqbool Dar1, K. Selvakumar2, S. Ramki3, K. M. Karuppasamy4, Jameel A.
Ansari5, Aafaq A Rather6'*
1Jammu and Kashmir Institue of Mathematical Sciences, Srinagar-190008, J&K, India ^Department of Mathematics, University College of Engineering-Nagercoil-629004, Tamil Nadu, India 34Department of Mathematics & Statistics, Bharath Institute of Higher Education and Research,
Selaiyur, Chennai - 73
^Department of Engineering Sciences, Faculty of Science and Technology, Vishwakarma University,
Pune, India
6Symbiosis statistical Institute, Symbiosis International (Deemed University), Pune-411004, India [email protected], [email protected], [email protected], [email protected], [email protected], 6*[email protected]
Abstract
In the present paper, we create an algorithm to address the transportation problem with numerous objectives and indexes. The transportation problem exists when there are more supply points, more demand points, and various means of transportation are used to meet demand or when moving certain types of goods. The transportation problem may frequently be more complex than the typical form of transportation problem. We create a model that blends fuzzy multi-objective programming and the multiindex transportation problem' by using LINDO software to resolve all related problems. Additionally, the decision-maker may present a variety of data and it may be further improved. The new algorithmfor addressing transport problems in fuzzy environments is demonstrated numerically.
Key Words: Fuzzy transportation problem (FTP), Linear Programming Problem(LP P), ¡Multi-index transportation problem (MITP)
1. Introduction
The transportations prolem has a extensive range of real-world applications and can be seen as a specific example of the LPP. It is one of the best optimization techniques and has a wide range of real-world applications. a combination of various goods from any of the m origins to any of the n destination places. In order to reduce the overall cost of a transportation issue, we control the amount to be transported from all origins to all destinations. We may not have focused on a single objective function in this situation, which is multi-objective. All of the objectives of MOTP are in competition with one another, and all of the restrictions are of the equality kind. The technique for multi-objective fuzzy linear programming with uncertain goals. iThe multi-objective transportation problem has the best compromise solutionlwhen applied to fuzzy linear programming.
Numerous academics have evaluated the use of the MITP to optimization, mathematical modelling and industry. Wang et al. [10] established a decomposition technique for handling the standard three-index transportation problem that is entirely dependent on the successive adjustment of the optimality criterion. They looked at the transportation problem's solution using a linear and quadratic objective function. In addition to recommending the adoption of such transportation efficiency, Rautman et al. [8] find a solution to the shipment scheduling conflict by utilising a multiindex transportation problem method to optimize the integral system Bit et al. [4] used a fuzzy programming method with a hyperbolic membership function to solve the Multi-Objective Capacitated Transportation Problem, in which the targets are non-commensurable and incompatible and the deliver and demand constraints are all of the same kind. In order to identify the best and most effective compromise solution to a multi-objective capacitated transportation problem, fuzzy programming with hyperbolic membership function was applied. For the first time, all parameters are taken into consideration when using fuzzy multi index bi-criteria constant fee bottleneck transportation (FMIBCFCBTP) by Sungeeta et al. [9]. An algorithm was created to detect FMIBCFCBTP fuzzy time-value change-off pairings. A numerical example was provided to explain the said algorithm. An exponential membership function was employed in the fuzzy programming method by Kaur et al. [5] to resolve a multi-objective and multi-index transportation problem. The main emphasis is on reducing the prices, decreasing rates, and underutilized capacity of transporting raw materials via various modes of transportation from various points of origin to various destination sites. Each target function is given a unique form of non-linear membership function by employing the fuzzy programming technique to solve actual transport problems using an exponential function and creating a non-dominated compromise solution. To tackle the multi-index fixed charge bi-criterion transportation problem , Archana and Veena [2] provided a method for determining the ideal trade-off pair amongst efficient cost-time trade-off pairs.
The linear-multi-objective-solid transportation problem was approached from a fuzzy-linear programming perspective by Bit et al. [3]. The outcome is a compromise approach that is both cost-effective and ideal. The fuzzy linear programming approach was used to develop the FORTRAN programme. With profit maximisation and time minimization as the objectives, Anjana et al. [1] developed a multi-objective multi-object strong transportation problem (MMSTP) with fuzzy inequality constraints. Fuzzy chance programming and the additional specificity opportunity-necessity principle are used to create a method for changing the ambiguous MMSTP to the equivalent -deterministic shape. The fuzzy interactive satisfied method was used to develop optimal compromise solutions for the MMSTP through a generalised reduced gradient strategy. The finest non-dominating solution was created using the technique for order preference by similar to perfect solution. An algorithm was created by Porchelvi and Anitha [7] to address the multi-objective transportation problem. The source and destination parameters , along with the cost coefficients of the goal function, are expressed as interval costs. To solve MOTP, they used fuzzy programming techniques with linear membership functions for various costs.
The MOMITP,we assume pyi be a multidimensional arrayl <i <m,1 < j <n and 1 <l <k and let P = p;j, Q = q ji and R = ra be multi-matrices then the MITP is as fallows
2. Mathematical Model
Min Z = Yll PijiXiji
i j l
Such that
£ Xiji = Pj for alli, j
2 Xjji = ra for alii, I j
2 Xj = qjj forall j, l, & Xjjj > 0, for alii, j,l j
it further, implies that
2 Pj = 2 qji ; 2 Pj =2 rl ; 2 qji =2 rl i l j 1 j 1
All the 3-conditions are necessary but not sufficient. MOMITP is defined as fallows
m n
Minimize Zk = II j xf +1 I j x(2)
' j ' j
Such that
(2)
(3)
(4)
î] = pu »Vi (5)
îj = P2i » V i j (6)
î xf = qij , V j i (7)
î j = q2 j , V j i j + j = rt] , Vi, j xj > 0, j > 0 The following set of conditions are necessary for the existance solution. n îril = Pli + P2i» Vi» j (8) (9) (10) (11)
n î rii = qi j + q2 j » V j j (12)
m n î Pii =î qi j » V i»j i i (13)
m n î P2i = î q2 j »V »»j ' j (14)
n î % ^ min (Pi, + qi j ) + min ^ + q2 j )» V /'» j j (15)
3. Proposed Algorithm
Step 1: Formulate a FTP.
Step 2: Solving the MOTP, k times, taking, one at a time, we first develop a matrix form inorder to get corresponding values for each objectives for each solution.
Zi Z 2 . . Zk
zii Zi2 ... Z1k
* 2 Z 21 Z 22 ... Z 2k
_Zk1 Zk2 ... Zkk
Where, each X',i = l,2,..k represent the isolated optimal solutions to the K distinct transportation
problems for k distict objective functions Zj = Zj (X') Vi, j, where i, j = 1,2,3........k, respectavelly the
ith row and jth column members of the matrix.
Step 3: Using step 2, we set upper and lower bounds for each objective and defining the range of values for the membership function that represents the degree of acceptance and rejection for a perticular solution. The values of such functions can be calculated as.
U£ = Max(Zr (Xr))
L£ = Min (Zr (Xr)), 0< r < k,
Where U£ and L£ are rtespectivelly the upper and lowerbound for the (kth objective functionZ^)
k = 1,2,3........K, dk = U£ — L£ ,the degradation allowance for objective k .
Step 4: We define the membership function as:
£k {Z (X )k} =
L£ > Zk (X)
1 —(Zl(X)— L£) , L£< Zk(X)<U£ , wheredk = U£—L£
(16)
Zk (X) > U£
Step 5: We use a LMF for the initial fuzzy model, the crisp model can be simplified as: Minimize (X Subj ect to
Zk (X)+xdk <U£,
¿40 = pu, ^ j
j
Zj = P2i, V J
(17)
j
Z xj = qi j , Vi
m n\
Z xf = q2 j , VJ i
j + j = rj, Vi, J xj > 0, j > 0
Above system of LP can be solved by using LONDO statistical software.
Step 6: Using the precise mathematical Programming approach, we are able to solve the crisp model.
Min (
...K, (18)
Cjxj +xdk < U£, k = 1,2,
iZxy1) = Pu, Vh j
J
Zj = p2l, Vi,
J
m
Z xj = qy , V j
i
m (2)
Z x(- = q2 j, V j
i
xj + xf = rj, Vi, j
1
k
0
m
j > 0, J > 0
Another membership function, like the Hyperbolic Tangent function, is one that we utlise. Min X
1 1 U£ + L£
x> 1 + — tanh{—k-^ + Zk } zk,
2 2 2 k
Where, rt = -
U£ - K
-, where S is number of constraints.
x^ >0,Vi, J and x > 0. Step 7: An intuitionistic fuzzy optimization for MOLP is defined as
Lk > Zk (X)
4 Z (X)} =
1 I Zk -Lk
2 I d
where k = 1,2,.
.K.
(19)
Lk < Zk (X) < U£ where dk = U£ - Lk (20) Zk (X) > U£
S
4. Illustrations
Example 4.1: Consider the fuzzy MOMITP discussed by Lohgaonkar et al. [6]
Minimize Z1 = 4Y1(1) + 3Y1(1) + 5Y1(1) + 8Y;(1) + 6Y:(1) + 2Y;(1) + 7Y(1) + 4Y^1)
+ Y^ + 9Y(1) + 10Y(1) + 12Y(1) + 8Y1(2) + 6Y1(2) + 3Y1(2) + 5Y:(2) +4Y^ + Y^2 + 9Y<2) + 2Y42 + 6Y-(|) + 4Y|2) + 9Y12? + 3Y22)
Subj ect to
Y® + Y® + Y« = 9 Y« + Y2(2) + Y« = 14 Y3« + Y31) + Y31) = 6
y(1) , y(1) , y(1) _ 7 Y41 ^ y42 ^ y43 = 7
Y(2) , y(2) , y(2) = 6
Y11 + Y12 + Y13 = 6
(2) (2) (2) Y21 ^ y22 ^ y23 = 7 (2) (2) (2)
Y31 ^ y32 ^ Y33 = 5
Y(2) , Y(2) , Y(2) = 6
Y41 ^ y42 ^ y43 = 6
Y (1) , Y (1) , Y (1) , Y (1) = 14 Y11 ^ Y 21 ^ Y31 ^ Y 41 = 14
p(1Ky(1L7(1L7(1) _19
Y12 ^Y 22 ^ Y32 ^ Y 42 = 12
Y1(31) + Y® + Y,® + = 10
y(2Ky(2K7(2K7(2) Y11 ^ Y 21 ^ Y31 ^ Y 41 = 5
y(2) y(2) y(2) y(2) _o
Y 12 ^ Y 22 ^ Y 32 ^ Y 42 = 8
Y1(32) + Y23) + 42) + Y4(? = 11 Y« + Y1(12) = 5
y(1) , y(2) _ 7 Y12 ^ Y12 = 7
A.M. Dar, K. Selvakumar, S. Ramki, K. M. Karuppasamy, J. A. Ansari, . __
rr j RT&A, No 4 (76)
Aafaq A. Rather ,T , „„ ^ '
q Volume 18, December 2023
OPTIMIZING MULTI-OBJECTIVE MULTI-INDEX ...
Y(1) + Y(2) = 3 Y13 + Y13 = 3
Y(1) + Y^ = 8
Y(2) + Y2(22) = 4 Y(1) + Y(2) = 9
y23 + y23 = 9
Y« + Y3(2) = 4
Y^ + Y3(22) = 1 (21)
^3(3) + ^3(3 ) — 6
Y(1) + Y(2) = 2 y41 + y41 = 2
Y(2) + Y4(22) = 8 Y(1) + Y(2) = 3 y43 + y43 = 3
j, Yj > 0, i = 1,2,3,4 and j = 1,2,3
Example 4.2:
Minimize Z2 = 5Y1(1) + 6Y1(1) + 7Y1(1) + 4Y^ + + 2Y(1) + 1Y:(1) + 3Y(1) + 4 Y^1) + 4Y^1) + 2Y|1) + 3Y^1) + 10Y1(2) + 9Y1(2) + 9Y1(2) + 7Y(2) +9Y22) + 2Y2(2) + 8Y3(2) + 7Y3(2) + 9Y3(2) + 8Y42> + 4Y^2) + 5Y442)
Subj ect to
Y1« + Y1« + Y1« = 9 + Y« + = 14 Y« + Y® + = 6
yO) , y(1) _ 7
y41 + Y42 + Y43 = 7 Y(2) + Y(2) + Y(2) = 6
133 + y12 + Y13 = 6
(2) (2) (2) 123 + y22 + y23 = 7
(2) (2) (2)
133 + y32 + Y33 =5 Y(2) + Y(2) + Y(2) = 6
y41 + Y42 + y43 = 6
Y (1) + Y (1) + Y (1) + Y (1) = 14 111 + y21 + Y31 + y41 = 14
Y« + Y« + Y« + Y® = 12
rj? + Y2(3) + + y4(3) = 10
Y(2) + Y2(12) + Y312) + Y^2 = 5
+ Y™ + 73(2) + Y™ = 8
Y[(32) + Y(2) + Y3(32) + Y4(2) = 11 ^ + C = 5
y(1) , y(2) _7 132 + 132 = 7
Y a) + Y (2) = 3
Y13 + Y13 =3
Y (1) + Y (2) = 8
Y 21 + Y 21 = 8
Y fl) + Y (2) = 4
1 22 + 1 22 = 4
Y (1) + Y (2) = 9 Y 23 + Y 23 = 9
Y(1 Y (1
Y 32
Y(1
Y33
Y(1
Y 41 Y(1
Y 42 Y(1 y43
Y(1
+ Y3(2} = 4
+ Y3<2) = 1
+ Y3(32) = 6
+ Y4(2} = 2
+ Y4(^} = 8
+ Y4(32) = 3
,Yj >0, i = 1,2,3,4 and j = 1,2,3
If no errors are found, then the LINGO Solver status window appears of the illustration 4.1 is given below (by changing variable Y to X and also, taking X (1) and X(2) respectively X11M
and X11N so on. In LINGO solver)
Figure 1: Illustration of model4.1 on LINGO window
By closing above window, we can view the solution of the Model 4.1 is given (Global optimal solution found).
Objective value: 298.0000
Infeasibilities: 0.000000
Total solver iterations: 4
Model Class: LP
Total variables: 24
Nonlinear variables: 0
Integer variables: 0
Total constraints: 27
Nonlinear constraints: 0
Total nonzeros 96
Nonlinear nonzeros: 0
A.M. Dar, K. Selvakumar, S. Ramki, K. M. Karuppasamy, J. A. Ansari, . __
rr j RT&A, No 4 (76)
Aafaq A. Rather y■ '
OPTIMIZING MULTI-OBJECTIVE MULTI-INDEX ..._V°lume 18, December 2023
Table 1: Optimal Solution of model 4.1
Variable Value Reduced Cost
X11M 5.000000 0.000000
X12M 4.000000 0.000000
X13M 0.000000 6.000000
X21M 8.000000 0.000000
X22M 1.000000 0.000000
X23M 5.000000 0.000000
X31M 1.000000 0.000000
X32M 0.000000 6.000000
X33M 5.000000 0.000000
X41M 0.000000 2.000000
X42M 7.000000 0.000000
X43M 0.000000 9.000000
X11N 0.000000 3.000000
X12N 3.000000 0.000000
X13N 3.000000 0.000000
X21N 0.000000 1.000000
X22N 3.000000 0.000000
X23N 4.000000 0.000000
X31N 3.000000 0.000000
X32N 1.000000 0.000000
X33N 1.000000 0.000000
X41N 2.000000 0.000000
X42N 1.000000 0.000000
X43N 3.000000 0.000000
If no errors are found, then the LINGO Solver status window appears for the illustration 4.2 is given below
Figure 2: Illustration of model4.2 on LINGO window
By closing above window, we can view the solution of the Model 4.2 is given (Global optimal solution found).
Objective value: 283.0000
Infeasibilities: 0.000000
Total solver iterations: 3
Model Class: LP
Total variables 24
Nonlinear variables: 0
Integer variables: 0
Total constraints: 27
Nonlinear constraints: 0
Total nonzeros: 96
Nonlinear nonzeros: 0
Table 2: Optimal Solution of model 4.2
Variable Value Reduced Cost
X11M 3.000000 0.000000
X12M 6.000000 0.000000
X13M 0.000000 1.000000
X21M 8.000000 0.000000
X22M 4.000000 0.000000
X23M 2.000000 0.000000
X31M 1.000000 0.000000
X32M 0.000000 1.000000
X33M 5.000000 0.000000
X41M 2.000000 0.000000
X42M 2.000000 0.000000
X43M 3.000000 0.000000
X11N 2.000000 0.000000
X12N 1.000000 0.000000
X13N 3.000000 0.000000
X21N 0.000000 1.000000
X22N 0.000000 4.000000
X23N 7.000000 0.000000
X31N 3.000000 0.000000
X32N 1.000000 0.000000
X33N 1.000000 0.000000
X41N 0.000000 0.000000
X42N 6.000000 0.000000
X43N 0.000000 0.000000
Now, we can determine Z1 and Z2, for (X (2) , X (1)) respectively as given below Z2 (X (1)) =295 and Z1(X (2)) =335, and written in the form of matrix.
X
(1) r
X(2) L
Zi 298
335
Z2 295"
283
From the above, we have Uf = 335 Uf = 295 We define Z1 and Z2 respectively
M( X ) =
1,
L = 298
= 283
if Z1(X) < 298
1 _ Zi(X) - 298 , if 298 < z1(X) < 335
0,
335 _ 298
dkl = 37
if Z1(X) > 335
X ) =
if Z1( X ) < 283 1 _ Z2(X) _ 283, f 283 < z1(X) < 295
0,
295 _ 283
dkl = 12
fZ1( X ) > 295
We find Minimize a
Z1(X) + 37a < 335 Z 2(X) + 12a < 295
We Solve the above crisp model and the Solver window appears given below
(22) (23)
Lingo 19.0 Solver Status ILingol] 5 olvei S KOA №
X
Model Llass: State: Objective I nfeaî iti il ihy: I teraticms:
LP
Globai Opt O.7SÍ098 O
7
■ Extended Solver Status S olver Type. Best Ob-j: O hi 1 EtoLjrpd.
Steps:: Active:
Variable
T ot^l
M r il ii i ^
I ntegeiç:
25 O O
Update I nteível: | 2
1 ii^ll.=inLs T otal: Nonlinear: 2 9 O
T Otâl tJ onlinear: 123 O
Generator Memorji Used 1K| 3 a
Elapsed Rijritpnrpe (lit« r» rp OO:OO:OO
I I ej j i lj j j t Solver
Close
Figure 3: Illustration of model 22 and 23 on LINGO window
By closing above window, we can view the solution of the Model 22 and 23 is given (Global optimal solution found). Objective value: 0.7540984
Infeasibilities: 0.000000
Total solver iterations: 7
Elapsed runtime seconds: 0.11
Model Class: LP
Total variables: 25
Nonlinear variables: 0
Integer variables: 0
Total constraints: 29
Nonlinear constraints: 0
Total nonzeros: 123
Nonlinear nonzeros: 0
Table 3: Optimal Solution of crisp model 22 and 23
Variable Value Reduced Cost
A 0.7540984 0.000000
X11M 5.000000 0.000000
X12M 3.316940 0.000000
X13M 0.6830601 1.000000
X21M 7.000000 0.000000
X22M 3.683060 0.000000
X23M 3.316940 0.000000
X31M 0.000000 0.4918033E-01
X32M 0.000000 0.2622951
X33M 6.000000 0.000000
X41M 2.000000 0.000000
X42M 5.000000 0.000000
X43M 0.000000 0.1639344E-01
X11N 0.000000 0.1311475
X12N 3.683060 0.000000
X13N 2.316940 0.000000
X21N 1.000000 0.000000
X22N 0.3169399 0.000000
X23N 5.683060 0.000000
X31N 4.000000 0.000000
X32N 1.000000 0.000000
X33N 0.000000 0.000000
X41N 0.000000 0.4918033E-01
X42N 3.000000 0.000000
X43N 3.000000 0.000000
, * ~„ . ~ *
Z = 307.04 Z2 = 285.92 a = 0.754
Maximize Ct
a> 1 + 1tanh{ ^^ - Zk }rk
Further implies that
U?+ Lt
Zkrk + tanh-1(2«-1)< k ^ k rk (24)
Maximize w
A.M. Dar, K. Selvakumar, S. Ramki, K. M. Karuppasamy, J. A. Ansari, . __ .
rr j RT&A, No 4 (76)
Aa fa q A. Ra the r
OPTIMIZING MULTI-OBJECTIVE MULTI-INDEX ..._V°lume 18, December 2023
6(4 • X^ + 3 • X1(j) + 5 • X1(13) + 8 • X(\} + 6 • X(22} + 2 • X^ + 7 • X(\) + 4 • X(1} +1 • X(j) + 9 • X
+10 • x(3) +12 • X^ + 8 • X((2) + 6 • X((2) + 3 • X((2) + 5 • X(2)+4 • X(2) +1 • X(23) + 9 • X(21) + 2 • X(2)
+ 6 • X23) + 4 • X(3) + 9 • X(42) + 3 • X(433) ) + 37 • a < 1899
6(5 • X(1) + 6 • X(12) + 7 • X(13) + 4 • X(1) + 5 • X(¿) + 2 • X(¿) +1 • X(\) + 3 • X(1) + 4 • X(1) + 4 • X(4\) + 2 • X(1) + 3 • X41) +10 • X® + 9 • X® + 9 • X^ + 7 • X® + 9 • x42) + 2 • X42) + 8 • X(21) + 7 • X(2)
+ 9 • X(23) + 8 ■ X^ + 4 • X((2) + 5 • X(2)) +12 - a <1734 Subject to the condition (21) Solver window of model (24) appears below
Figure 4: Illustration of model 24 on LINGO window
Solution of above model (24) is given by (Global optimal solution found).
Objective value: 1.525591
Infeasibilities: 0.000000
Total solver iterations: 7
Elapsed runtime seconds: 0.17
Model Class: LP
Total variables: 25
Nonlinear variables: 0
Integer variables: 0
Total constraints: 29
Nonlinear constraints: 0
Total nonzeros: 123
Nonlinear nonzeros: 0
A.M. Dar, K. Selvakumar, S. Ramki, K. M. Karuppasamy, J. A. Ansari, . __
rr j RT&A, No 4 (76)
Aafaq A. Rather y■ '
OPTIMIZING MULTI-OBJECTIVE MULTI-INDEX ..._V°lume 18, December 2023
Table 4 : Optimal solution of model 24
Variable Value Reduced Cost
A 1.525591 0.000000
X11M 5.000000 0.000000
X12M 3.316273 0.000000
X13M 0.6837270 0.000000
X21M 7.000000 0.000000
X22M 3.683727 0.000000
X23M 3.316273 0.000000
X31M 0.000000 0.4918033E-01
X32M 0.000000 1.574803
X33M 6.000000 0.000000
X41M 2.000000 0.000000
X42M 5.000000 0.000000
X43M 0.000000 0.9842520E-0
X11N 0.000000 0.7874016
X12N 3.683727 0.000000
X13N 2.316273 0.000000
X21N 1.000000 0.000000
X22N 0.3162730 0.000000
X23N 5.683727 0.000000
X31N 4.000000 0.000000
X32N 1.000000 0.000000
X33N 0.000000 0.000000
X41N 0.000000 0.2952756
X42N 3.000000 0.000000
X43N 3.000000 0.000000
Z* = 307.04 Z2 = 285.92
Where w =tanh-i (2a - 1) and w=1.52, a= 0.92
ße1{Z1(X)}=
1, 298 < Z1 (X)
Zj-298"
: e ^ {o, Z1(x) > 335
298 < Z1(X) < 335 ,
dki = 37
Maximize (X Such that
ße2{Z2(X)}=
1, 283 < Zi (X)
_lf Z2 -283'
{o, Z2(X) > 295
283 < Z2(X) < 295,
a < e
_lf 307.04-298^ _ 1| 285.92-283
37 J and a < e 12
dk7 = 12
The solution of the problem is given by a = 0.89
5. Conclusion
A fuzzy MOMIT algorithm is constructed in this paper, and with the help of numerical examples, a solution is demonstrated using three various kinds of membership functions, including linear, hyperbolic, and exponential membership functions. The numerous modes of transporting goods between points of origin and destination are represented by the multi-index transportation problem. The crisp model becomes a linear one when the hyperbolic membership functionis used. When compared to the linear membership function and hyperbolic membership function, the optimum compromise solution is drastically different. However, there is no significant different between the linear membership function's solution and the exponential membership function's ideal compromise solution.
There are numerous methods that future research in the field of fuzzy programming might be carried out. For problems involving many scales and several objectives in linear programming, employing fuzzy programming to design decision support systems will be particularly beneficial in real-world scenarios. Future research may take into account the use of fuzzy programming to solve MOTP's when the supply and demand factors are simply made up of fuzzy integers. There is still room for more research into duality theory and post optimality analysis in multi-objective two- and three-dimensional transportation problems. The demand parameters, agency capacity, and mode of transportation capacity can be anticipated as random variables that follow specific probability distributions in multi-index transportation problems, in addition to the two index transportation problem.
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