






























Admasu Tadesse *1 , Sirkumar Acharya 2, Berhanu Belay3
1 Department of Mathematics, Hawassa university, Ethiopia
2 Department of Mathematics, KIIT University, Bhubaneswar, India
3Department of Mathematics, Debretabor University, Ethiopia

KEYWORDS:

Multi-Objective Programming;
Triangular Fuzzy Number;
Fuzzy Transportation Problem;
Fuzzy Decision Variables;
Ranking Function;
Fuzzy Programming Method.

ABSTRACT

The aim this study is presenting the solution methodology of multi-
objective fuzzy transportation problem with fuzzy decision vari-
ables, where all the input parameters and decisions variables of the
programming problems are assumed to be triangular fuzzy num-
ber and triangular fuzzy decision variables respectively. More-
over the objectives under considerations are minimization of cost
of transportation and minimization of shipping time under fuzzy
environment. The fuzziness of the objective functions and the
fuzzy constraints of the programming problem are defuzzified us-
ing the ranking function and the equality property between two
fuzzy numbers, respectively. The consequent crisp multi-objective
fuzzy transportation problem is tackled by employing fuzzy math-
ematical programming approach. Finally fuzzy decision is made
after solving the resultant mathematical programming problem us-
ing LINGO(Schrage and LINDO Systems (1997)) software. Illus-
trative numerical example is presented in support of the proposed
methodology.

the transportation of a product man-
ufactured at different plants ( supply origins)
to a number of different warehouses (demand
destinations). Transportation problems were
well known as a basic network problem in its
classical category. The formulation and dis-
cussion of transportation model was intro-
duced by Hitchcock (1941).

Classical TP models and techniques have
been effectively applied to problems with a
well-defined or accurately known parameters
for many years. The coefficient parameters
of the majority of TP models are considered

*Corresponding author:

East African Journal of Biophysical and Computational Sciences 

Journal homepage : https://journals.hu.edu.et/hu-journals/index.php/eajbcs 
College of Natural & Com

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tat
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ion
sa U

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Sc
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nc
sit

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s

Year 2021

Volume xx No xx

Email: admasut@hu.edu.et

Transportation problem(TP) is a particular
class of linear programming, which is associ-
ated with day-to-day activities in our real life
and mainly deals with logistics. It helps in
solving problems on distribution and trans-
portation of resources from one place to an-
other. The goods are transported from a set
of sources (e.g., factory) to a set of desti-
nations (e.g., warehouse) to meet the spe-
cific r equirements, Das et al. (2016). Inother
words, transportation problems deal with
  _______________________________ 

 https://dx.doi.org/10.4314/eajbcs.v4i2.4S

East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53

43

INTRODUCTION

Fuzzy Programming Approach to Solve Multi-Objective Fully Fuzzy 
Transportation Problem

 
Research article



to be single real number which is accurately
known. However, such assumptions are not
suitable for dealing with a number of issues
that occur in real life because many of the
parameters are imprecise and vague as a re-
sult of both natural and anthropogenic ef-
fects. This motivates to formulate the 
TP models in uncertain environment.
One of the uncertain environments is fuzzy
environ-ment. Moreover in most of the
literatures, authors assumed the decision
parameters as fuzzy numbers while the
decision variables as crisp ones. Since the
variables are crisp, the solution obtained as
the crisp, which is a real number. The
solution is exact value in the fuzzy
programming problems with fuzzy
parameters. The fuzzy aspect of the decision
is partly lost in this case so, it is reasonable
and important to consider fuzzy mathemat-
ical programming problem with fuzzy deci-
sion variables. Fuzzy transportation prob-
lems involving fuzzy decision variables are
considered in this study. Kaur and Kumar
(2012) studied a special type of fuzzy TP by
assuming that a decision maker is uncertain
about the precise values of transportation
cost only, where the transportation cost is
represented by generalized trapezoidal fuzzy
numbers. In their proposed work all the sup-
ply and demand of products are crisp pa-
rameters that means there is no uncertainty
about the supply and demand of the prod-
uct. According to the explanation of Ku-
mar and Kaur (2011), there may be factors
which imposes the occurrences of fuzziness
in TP. Some of these are, the decision maker
has not enough information about the unit
transportation cost of transportation opera-
tion and thus the transportation cost uncer-
tain, there may be some sort of vagueness
with respect to the demand of a newly intro-
duced product to the market and there exists
uncertainty about the product availability at
a source or supplier because of time factor.
Many authors introduced tools to solve TP.
Since transportation problem (TP) is spe-
cial case of linear programming (LP) prob-
lem, one straightforward approach is to ap-
ply the existing LP techniques to the fuzzy
TP. These techniques are discussed by many

researchers namely; Buckley (1988),
Buckley (1990), Mitlif (2016),
Ebrahimnejad (2013), Ebrahimnejad (2015),
etc. However, some these techniques only
give crisp solutions, which represent a
compromise in terms of fuzzy data.

The traditional view on TP is mainly con-
cerned with distributing any homogeneous
product from a group of supply centers,
called sources, to any group of receiving cen-
ters, called destinations, in such a way as
to minimize the single objective total trans-
portation cost, where the transportation cost
per unit product is constant regardless of
the amount transported, but most of the
time in real-life situation, the TPs are not
designed as single objective function. The
TP that deals with multiple-objective func-
tions is called a multi-objective transporta-
tion problem (MOTP). The MOTP is a
special type of multi-objective linear pro-
gramming problem in which objective func-
tions conflict with each other. Further-
more, objective functions are frequently in
conflict, thus there is no one best (global
optimum) solution, but rather a group of
equally good (non-dominated) alternatives
known as pareto optimal (PO) solutions.
In the framework of multi-objective pro-
gramming problems, numerous scholars from
a wide variety of academic disciplines dis-
cussed their work. Recently, multi-objective
problems have been proposed by researchers
such as Sayyah et al. (2019); Sahih et al.
(2021); Sosa and Dhodiya (2021); Geshni-
ani et al. (2020), and others. Researchers
namely; Acharya et al. (2014) and Dutta
et al. (2016) introduced MOT problem in
stochastic environment. Chakraborty and
Chakraborty (2010) discussed cost-time min-
imization TP, where the demand, supply and
transportation cost per unit of the quantities
are fuzzy. Nomani et al. (2017) introduced
a weighted goal programming to solve multi-
objective transportation problems with crisp
parameters. They used weighted approach
based on goal programming to obtain com-
promise solutions. Roy et al. (2018) pro-

East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53

44



posed multi-objective transportation prob-
lem (MOTP) under intuitionistic fuzzy en-
vironment. They have assumed transporta-
tion cost, the supply and the demand param-
eters as a intuitionistic fuzzy numbers. Jalil
et al. (2017) proposed a solution approach
for obtaining compromise optimal solution
of fully fuzzy(all the parameters and deci-
sion variables are fuzzy) multi-objective solid
transportation problems. In their proposed
problem they used ranking function for the
defuzzification of fuzzy objective function
and the property of equality between fuzzy
numbers for the defuzzification of fuzzy con-
straints. El Sayed and Abo-Sinna (2021);
Moges et al. (2023); Malik and Gupta (2022);
Niksirat (2022), etc. are the works done un-
der fuzzy environment.
As is mentioned above (Paragraph 2), in
most of the literature, authors regarded the
decision variables as being crisp while the de-
cision parameters are assumed to be fuzzy.
This assumptions leads to crisp decisions
which is illogical. A fuzzy decision multi-
objective fully fuzzy transportation prob-
lem proposed in this study. Ranking func-
tion and equality between two triangular
fuzzy numbers are employed for defuzzifica-
tion purpose and finally the equivalent crisp
multi-objective model is solved fuzzy pro-
gramming method.
The paper is organized as follows: following
the introduction, basic preliminaries are pre-
sented in Sect. 2. The mathematical model
of multi-objective fuzzy TP is presented in
Sect. 3. Solution procedures are provided
in 4. Numerical examples are provided in
support of the proposed method in Sect. 5.
Finally, Conclusion is provided in Sect. 6
followed by supportive references.

Definition: [Roy et al. (2018)]: A tri-angular 
fuzzy number ã is denoted by (ap, a, ao), 
where ap, a, ao are real numbers. The 
membership function (µã(x)) of ã is given
below:

µã(x) =


0, x ≤ ap
x−ap

a−ap
, ap ≤ x ≤ a

ao−x
ao−a

, a ≤ x ≤ ao

0, otherwise

Note: The point ’a’ is the core value of tri-
angular fuzzy number Ã, where µã(a) =1 
ap and ao are the lower and upper bounds 
of support of triangular fuzzy number Ã re-
spectively.

Definition : [Roy et al. (2018)]: Let ã = (ap, 
a, ao) and b̃ = (bp, b, bo) be two triangular 
fuzzy numbers ,then
(i) (ap, a, ao)⊕(bp, b, bo) = (ap+ap, a+b, ao+
bo)
(ii) k(ap, a, ao) = (kap, ka, kao), k ≥ 0
(iii) (ap, a, ao)⊗ (bp, b, bo) = (apbp, ab, aobo, if
ap ≥ 0 and bp ≥ 0

Definition : [Ebrahimnejad (2017)]: Let ã = 
(ap, a, ao) and b̃ = (bp, b, bo) be two tri-
angular fuzzy numbers ,then
(i) ã = b̃ iff ap = bp, a=b and ao = bo

(ii) ã = (ap, a, ao) ≥ 0 iff ap ≥ 0

Definition :Ebrahimnejad (2017)]:
Fuzzy TP is said to be balanced transporta-
tion problem when total supply from all the
sources is equal to the total demand in all
destinations.

Definition : [Kumar et al. (2011)]: A

ranking function is a function R:F (R) → R, 
where F(R) is a set of fuzzy numbers defined
on set of real numbers, which maps each
fuzzy number into the real line, where a nat-
ural order exists. Let ã = (ap, a, ao) be a tri-
angular fuzzy number, then R(ã) = ap+2a+ao

4

Definition : [Hasan et al. (2015)]:
Multi-Objective Optimization Problem
(MOOP) involves more than one objective
function that are to be minimized or maxi-
mized. Answer of MOOP is the set of solu-
tions that define t he b est t radeoff between
competing objectives. The following is gen-
eral mathematical Form of MOOP:

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45

BASIC PRELIMINARIES



max /min Zm(x),m = 1, 2, 3, ...,M (2.1)

s.t. gj(x) ≥ 0, j = 1, 2, ..., J (2.2)

hk(x) ≥ 0, k = 1, 2, ..., K (2.3)

xL
≤xi ≤ xU (2.4)

The mathematical model for MOFTP with fuzzy decision variables is represented as:

min : Z̃k ≈
m∑
i=1

n∑
j=1

((ckij)
pxp

ij, c
k
ijxij, (c

k
ij)

oxo
ij), k ∈ {1, 2...K} (3.1)

subject to
n∑

j=1

(xp
ij, xij, x

o
ij) ≈ (api , ai, a

o
i ), i ∈ {1, 2, 3, ...,m} (3.2)

m∑
i=1

(xp
ij, xij, x

o
ij) ≈ (bpj , bj, b

o
j), j ∈ {1, 2, 3, ..., n} (3.3)

x̃ij ⪰ 0, i ∈ {1, 2, 3, ...,m}; j ∈ {1, 2, 3, ..., n} (3.4)

where,
i. the fuzzy total availability and fuzzy

total demand are assumed to be
equal(balanced fuzzy transportation
problem), ′m′ is total number of sup-
ply points and ’n’ is total number of
destination points,

ii. ãi=(api , ai, a
o
i ) is the fuzzy availability

of the commodity at ith origin and as-
sumed to be triangular fuzzy number,

iii. b̃j=(bpj , bj, b
o
j) is the fuzzy requirement

of the commodity at jth destination
and assumed to be triangular fuzzy
number,

iv. c̃ij=(cpi j, cij, c
o
i j) is the fuzzy cost coef-

ficient involved with fuzzy variables in
the objective function from ith origin to
jth destination, which is also assumed
to be triangular fuzzy number,

v. x̃ij=(xp
ij, xij, x

o
ij) is the fuzzy quantity

that should be transported from ith ori-
gin to jth destination and assumed to
be triangular fuzzy decision variables.

Crisp equivalent of multi-
objective fuzzy transporta-
tion problem

Since FMP model from (3.1) to (3.4) can-
not be solved directly, so ranking function
and the property of equality between the
fuzzy numbers are respectively applied on
the fuzzy objective functions and fuzzy con-
straints models of MOFTP. The resultant
crisp equivalent is obtained as:

min : Zk =
1

4

m∑
i=1

n∑
j=1

((cpij)
kxp

ij+2ckijxij+(coij)
kxo

ij), k = 1, 2.....K

(3.5)
subject to

n∑
j=1

xp
ij = api , i ∈ {1, 2, 3, ...,m}; (3.6)

n∑
j=1

xij = ai, i ∈ {1, 2, 3, ...,m} (3.7)

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46

MATHEMATICAL MODEL



n∑
j=1

xo
ij = aoi , i ∈ {1, 2, 3, ...,m} (3.8)

m∑
i=1

xp
ij = bpj , j ∈ {1, 2, 3, ..., n} (3.9)

m∑
i=1

xij = bj, j ∈ {1, 2, 3, ..., n} (3.10)

m∑
i=1

xo
ij = boj , j ∈ {1, 2, 3, ..., n} (3.11)

xo
ij−xij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n}

(3.12)
xij−xp

ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n}
(3.13)

xp
ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n}

(3.14)

Solution procedure

Now we use the fuzzy programming pro-
gramming technique to solve the crisp multi-
objective programming problem of 3.5 to
3.14. The solution procedures based on the
fuzzy programming method is detailed be-
low.

Step 1: Find the ideal solutions x1, x2, ..., xk

by picking one objective function at a
time and leaving the other objective
functions.

Step 2: Construct a pay-matrix with the
help of individual best solutions found
by the above step. Using table 1 es-
timate the bounds of zk (k=1,2,3...,K)
from the Payoff matrix.

Step 3: For every objective function zk
(k=1,2,3...,K), we formulate member-
ship function using any one of the fol-
lowing techniques of maximization or
minimization:

Table 1: Payoff matrix

z1(x) z2(x) . . . zK(x)

x(1) z1(x
(1)) z2(x

(1)) . . . zK(x
(1))

x(2) z1(x
(2)) z2(x

(2)) . . . zK(x
(2))

. . . . . . .

. . . . . . .

. . . . . . .
x(K) z1(x

(K)) z2(x
(K)) . . . .zK(x

(K))

Step 4

Case 1: Membership function is formulated
in the case of maximization problem as:

µzk(x) =


0, if zk ≤ lb−k
zk−lb−k
ub∗−lb−k

, if lb−k ≤ zk ≤ ub∗k

1, if zk ≥ ub∗k

where lb−k denotes the worst lower
bound of zk and ub∗k denotes the best
upper bound of zk

Case 2: For minimization problem the mem-
bership function is formulated as:

µzk(x) =


0, if zk ≥ ub−k
ub−−zk
ub−−lb∗k

, if lb∗k ≤ zk ≤ ub−k

1, if zk ≤ lb∗k

Where ub−k denotes the worst upper
bound of zk and lb∗k denotes best lower
bound of zk.

Case 1 : Apply the augmented variable, λ
with max-min operator to formulate
a crisp single objective mixed integer
programming problem as:

max : λ (3.15)

subject to

µzk(x) ≥ λ, k = 1, 2, ...K (3.16)

n∑
j=1

xp
ij = api , i ∈ {1, 2, 3, ...,m} (3.17)

East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53

47



n∑
j=1

xij = ai, i ∈ {1, 2, 3, ...,m} (3.18)

n∑
j=1

xo
ij = aoi , i ∈ {1, 2, 3, ...,m} (3.19)

m∑
i=1

xp
ij = bpj , j ∈ {1, 2, 3, ..., n} (3.20)

m∑
i=1

xij = bj, j ∈ {1, 2, 3, ..., n} (3.21)

m∑
i=1

xo
ij = boj , j ∈ {1, 2, 3, ..., n} (3.22)

xo
ij−xij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n}

(3.23)
xij−xp

ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n}
(3.24)

xp
ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n}

(3.25)
0 ≤ λ ≤ 1 (3.26)

Case 2 : Apply the augmented variable, λ
with min-max operator to formulate
a crisp single objective mixed integer
programming problem as:

min : λ (3.27)

subject to

µzk(x) ≤ λ, k ∈ {1, 2, 3, ....K} (3.28)

n∑
j=1

xp
ij = api , i ∈ {1, 2, 3...m} (3.29)

n∑
j=1

xij = ai, i ∈ {1, 2, 3...m} (3.30)

n∑
j=1

xo
ij = aoi , i ∈ {1, 2, 3...m} (3.31)

m∑
i=1

xp
ij = bpj , j ∈ {1, 2, 3...n} (3.32)

m∑
i=1

xij = bj, j ∈ {1, 2, 3...n} (3.33)

m∑
i=1

xo
ij = boj , j ∈ {1, 2, 3...n} (3.34)

xo
ij−xij ≥ 0, i ∈ {1, 2, 3...m}; j ∈ {1, 2, 3...n}

(3.35)
xij−xp

ij ≥ 0, i ∈ {1, 2, 3...m}; j ∈ {1, 2, 3...n}
(3.36)

xp
ij ≥ 0, i ∈ {1, 2, 3...m}; j ∈ {1, 2, 3...n}

(3.37)
0 ≤ λ ≤ 1 (3.38)

Step 5 At the end, the equivalent single ob-
jective MP model is solved by using
appropriate techniques or existing soft-
ware. The obtained PO solutions sub-
stituted back to original fuzzy objec-
tive function, as the result we can find
fuzzy optimal values of each fuzzy ob-
jective functions.

Numerical example

In this part, application numerical example
is solved using the provided approach, and
the conclusions drawn from the results are
discussed in further detail.
A firm has two sources O 1 and O 2 and three
destinations D1 ,D2 and D3. The fuzzy sup-
ply of the commodity from O1 and O2 are
(75, 95, 125) and (45, 65, 95), respectively.
Request of fuzzy demanded product at D1 
,D2 and D3 are (35, 45, 55), (25, 35, 45)
and (60, 80, 110), respectively. The com-
pany wants to determine the fuzzy quan-
tity of the commodity that should be trans-
ported from each origin to each destination
so that the total fuzzy transportation cost is
minimum with minimum transfer time. For
i=1,2, j=1,2,3., let the fuzzy transportation
cost for unit quantity of the commodity from
ith source to jth destinations be c̃ij , the 
fuzzy transportation time is also considered
as t̃ij and x̃ij represents the allocations(or 
amounts), which is non negative triangular
fuzzy real variable. The theoretical data on
fuzzy cost of transportation and fuzzy deliv-
ery time, is given in the table 2 below.

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48



Table 2: Fuzzy transportation cost per
unit(in Rupees) and fuzzy transportation
time per unit(in minute)

D1 D2 D3

c̃1j(O1) (15,25,35) (55, 65,85) (85,95,105)
c̃2j(O2) (65,75,85) (80,90,110) (30,40,50)
t̃1j(O1) (4, 6, 8) (6, 8,10) (7,9,11)
t̃2j(O2) (3,5,7) (5,7,9) (11,13,15)

From the table 2 mathematical model for MOFTP with fuzzy decision variables becomes:

min : z̃1 ≈ (15, 25, 35)⊗ x̃11 ⊕ (55, 65, 85)⊗ x̃12 ⊕ (85, 95, 105)⊗ x̃13 ⊕ (65, 75, 85)⊗ x̃21

⊕(65, 75, 85)⊗ x̃21 ⊕ (80, 90, 110)⊗ x̃22 ⊕ (30, 40, 50)⊗ x̃23 (3.39)

min : z̃2 ≈ (4, 6, 8)⊗ x̃11 ⊕ (6, 8, 10)⊗ x̃12 ⊕ (7, 9, 11)⊗ x̃13 ⊕ (3, 5, 7)⊗ x̃21 ⊕ (5, 7, 9)⊗ x̃22

⊕(11, 13, 15)⊗ x̃23 (3.40)

Subject to

x̃11 ⊕ x̃12 ⊕ x̃13 ≈ (75, 95, 125) (3.41)

x̃21 ⊕ x̃22 ⊕ x̃23 ≈ (45, 65, 95) (3.42)

x̃11 ⊕ x̃21 ≈ (35, 45, 65) (3.43)

x̃12 ⊕ x̃22 ≈ (25, 35, 45) (3.44)

x̃13 ⊕ x̃23 ≈ (60, 80, 110) (3.45)

x11, x̃12, x̃13 ⪰ 0 (3.46)

x̃21, x̃22, x̃23 ⪰ 0 (3.47)

defuzzification is done using the concept
discussed in the subsection 3.1. Then the
case(3.27-3.37) of evaluation of membership
functions for the minimization (Max-min op-
erator) is applied on the crisp MOTP along

with all the procedures discussed above. The
fuzzy programming method is applied on the
aforementioned crisp equivalent MOTP to
find the ideal solutions as detailed below

x(1) = (xp
11, x11, x

o
11, x

p
12, x12, x

o
12, x

p
13, x13, x

o
13, x

p
21, x21, x

o
21, x

p
22, x22, x

o
22x

p
23, x23, x

o
23)=

(35,35,55,10,10,10,30,50,60,0,10,10,15,25,35,30,30,50)
x(2) = (xp

11, x11, x
o
11, x

p
12, x12, x

o
12, x

p
13, x13, x

o
13, x

p
21, x21, x

o
21, x

p
22, x22, x

o
22, x

p
23, x23, x

o
23)=

(0,10,30,25,35,35,50,50,60,35,35,35,0,0,10,10,30,50)

At the corresponding ideal points the val-
ues of crisp objective functions are obtained
and given as: z1=10737.5, z2=1440.
After constructing a pay-of matrix 2, the
bounds of the two objective functions are
given by:

10737.5 ≤ z1(x) ≤ 11825

1440 ≤ z2(x) ≤ 1465.

The the single objective crisp problem is ob-
tained by formulating membership function.
Thus the programming problem becomes:

max : λ (3.48)

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49



subject to

z1 + 1387.5λ ≤ 11825 (3.49)

z2 + 25λ ≤ 1465 (3.50)

xp
11 + xp

12 + xp
13 = 75 (3.51)

x11 + x12 + x13 = 95 (3.52)

xo
11 + xo

12 + xo
13 = 125 (3.53)

xp
21 + xp

22 + xp
23 = 45 (3.54)

x21 + x22 + x23 = 65 (3.55)

xo
21 + xo

22 + xo
23 = 95 (3.56)

xp
11 + xp

21 = 35 (3.57)

x11 + x21 = 45 (3.58)

xo
11 + xo

21 = 65 (3.59)

xp
12 + xp

22 = 25 (3.60)

x12 + x22 = 35 (3.61)

xo
12 + xo

22 = 45 (3.62)

xp
13 + xp

23 = 60 (3.63)

x13 + x23 = 80 (3.64)

xo
13 + xo

23 = 110 (3.65)

xo
ij − xij ≥ 0 (3.66)

xij − xp
ij ≥ 0 (3.67)

xp
ij ≥ 0 (3.68)

0 ≤ λ ≤ 1 (3.69)

By employing the LINGO software, the resultant equivalent crisp programming problem
(3.48)-(3.69) is solved. The PO solutions and the agumated variable λ obtained are given
consecutively as: x∗=(31.66375,41.66375,61.66375,0,3.336248,3.33666248,43.33675,50,60,
3.336248 ,3.336248 ,3.336248 ,25,31.66375,41.66375,16.66375,30,50), λ=0.6668124. The com-
promising fuzzy objective functions values are z̃1=(6875.31, 10341.77, 16075.04) and z̃2=(748.32,
1355.13, 2335).
Figurative description of fuzzy optimal triangular cost and triangular transfer time are detailed
in the figure below.

Figure 1: Minimum fuzzy transportation cost

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50



Figure 2: Minimum fuzzy transfer time

The defuzzified crisp MOTP is solved by
fuzzy programming method. Using Max-min
operator, the resultant single objective crisp
integer MP problem is coded into LINGO
optimization package version 19.0 software
and fuzzy PO solutions are obtained. The
minimum total fuzzy cost of transportation
and the minimum fuzzy transfer time are ob-
tained and interpreted as follows:
Fuzzy transportation cost and fuzzy transfer
time z̃1=(6875.31, 10341.77, 16075.04) and 
z̃2=(748.32, 1355.13, 2335) respectively are
calculated by back substitutions of PO solu-
tions into fuzzy objective functions (3.1) of
the the programming problem.
It is found that the least amounts of min-
imum total transportation cost and trans-
fer time are 6875.31 and 748.32 units re-
spectively. The the most possible amounts
of minimum total transportation cost and
transfer time are 10341.77 and 1355.13 units
respectively. Furthermore, the greatest
amounts of minimum total transportation
cost and total transfer time are 16075.04 and
2335 units respectively.

The classical transportation problem (TP)
is primarily concerned with distributing any
homogeneous product from a group of sup-
ply centers, known as sources, to any group
of receiving centers, known as destinations,
in such a way that the single objective to-
tal transportation cost is minimized, where
all parameters are crisp (precisely defined).
However, in many circumstances, the deci-
sion maker lacks precise knowledge of the
TP parameters. and the nature of the TPs
are not designed as single objective func-
tion. If the nature of the information is vague
and the decision maker objectives preference
are conflicting, t he c orresponding program-
ming problem is fuzzy multi-objective pro-
gramming problem, and thus fuzzy MOTP
arises. In this paper, we have discussed a
solution approach for solving TP, with more
than one objective function by considering
the presence of vagueness in the real life data
of transportation problems, where all the pa-
rameters and decision variables are consid-
ered as triangular fuzzy numbers. Initially,
the fuzzy objective functions are defuzzified
by applying the ranking function for trian-

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51

DISCUSSION
CONCLUSION



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We, the authors, would like to express our
gratitude to editors and the anonymous re-
viewers for their valuable suggestions which
improved the quality of this work. We also
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study.

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Acknowledgments

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East Afr. J. Biophys. Comput. Sci.
Hawassa University


	Introduction
	Mathematical Model
	Crisp equivalent of multi-objective fuzzy transportation problem
	Solution procedure
	Numerical example
	Discussion of the results

	Conclusion
	Basic Preliminaries
	Untitled

