A New Method to Solving Generalized Fuzzy Transportation Problem-Harmonic Mean Method

International journal of Chemistry, Mathematics and Physics (IJCMP)
[Vol-4, Issue-3, May-Jun, 2020]
https://dx.doi.org/10.22161/ijcmp.4.3.3
ISSN: 2456-866X
http://www.aipublications.com/ijcmp/ Page | 51
Open Access
A New Method to Solving Generalized Fuzzy
Transportation Problem-Harmonic Mean Method
S. Senthil Kumar1
, P. Raja2
, P. Shanmugasundram3
, Srinivasarao Thota4
1
Department of Mathematics, Parks College, Tirupur-5, Tamil Nadu, India.
2
P.G. & Research Department of Mathematics, PSA College of Arts and Science, Dharmapuri, Tamil Nadu, India.
3
Department of Mathematics, Mizan Tepi University, Ethiopia.
4
Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No. 1888,
Adama, Ethiopia
Abstract— Transportation Problem is one of the models in the Linear Programming problem. The objective
of this paper is to transport the item from the origin to the destination such that the transport cost should be
minimized, and we should minimize the time of transportation. To achieve this, a new approach using
harmonic mean method is proposed in this paper. In this proposed method transportation costs are
represented by generalized trapezoidal fuzzy numbers. Further comparative studies of the new technique with
other existing algorithms are established by means of sample problems.
Keywords— Fuzzy Transportation Problem (FTP); Generalized Trapezoidal Fuzzy Number (GTrFN);
Ranking function; Harmonic Mean Method (HMM).
I. INTRODUCTION
In transportation problem, different sources supply to
different destinations of demand in such a way that the
transportation cost should be minimized. We can obtain
basic feasible solution by three methods. They are
1. North West Corner method
2. Least Cost method
3. Vogel’s Approximation method (VAM)
In these three methods, VAM method is best according to
the literature. We check the optimality of the transportation
problem by MODI method. The transportation problem is
classified into two types. They are balanced transportation
problem and unbalanced transportation problem. If the
number of sources is equal to number of demands, then it is
called balanced transportation problem. If not, it is called
unbalanced transportation problem. If the source of item is
greater than the demand, then we should add dummy
column to make the problem as balanced one. If the demand
is greater than the source, then we should add the dummy
row to convert the given unbalanced problem to balanced
transportation problem.
Transportation problem is an important network structured
in linear programming problem that arises in several
contexts and has deservedly received a great deal of
attention in the literature. The central concept in the problem
is to find the least total transportation cost of a commodity in
order to satisfy demands at destinations using available
supplies at origins. Transportation problem can be used for a
wide variety of situations such as scheduling, production,
investment, plant location, inventory control, employment
scheduling, and many others. In general, transportation
problems are solved with the assumptions that the
transportation costs and values of supplies and demands are
specified in a precise way i.e., in crisp environment.
However, in many cases the decision makers has no crisp
information about the coefficients belonging to the
transportation problem. In these cases, the corresponding
coefficients or elements defining the problem can be
formulated by means of fuzzy sets, and the fuzzy
transportation problem (FTP) appears in natural way. The
basic transportation problem was originally developed by
Hitchcock [14].The transportation problems can be modeled
as a standard linear programming problem, which can then

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be solved by the simplex method. However, because of its
very special mathematical structure, it was recognized early
that the simplex method applied to the transportation
problem can be made quite efficient in terms of how to
evaluate the necessary simplex method information
(Variable to enter the basis, variable to leave the basis and
optimality conditions). Charnes and Cooper [4] developed a
stepping stone method which provides an alternative way of
determining the simplex method information. Dantzig and
Thapa [8] used simplex method to the transportation
problem as the primal simplex transportation method. An
Initial Basic Feasible Solution (IBFS) for the transportation
problem can be obtained by using the North- West Corner
rule, Row Minima Method, Column Minima Method,
Matrix Minima Method or Vogel’s Approximation Method
(VAM) [21]. The Modified Distribution Method (MODI) [5]
is useful for finding the optimal solution of the
transportation problem. In general, the transportation
problems are solved with the assumptions that the
coefficients or cost parameters are specified in a precise way
i.e., in crisp environment. In real life, there are many diverse
situations due to uncertainty in judgments, lack of evidence
etc. Sometimes it is not possible to get relevant precise data
for the cost parameter. This type of imprecise data is not
always well represented by random variable selected from a
probability distribution. Fuzzy number [23] may represent
the data. Hence fuzzy decision making method is used here.
Zimmermann [24] showed that solutions obtained by fuzzy
linear programming method and are always efficient.
Subsequently, Zimmermann’s fuzzy linear programming has
developed into several fuzzy optimization methods for
solving the transportation problems. Chanas et al. [2]
presented a fuzzy linear programming model for solving
transportation problems with crisp cost coefficient, fuzzy
supply and demand values. Chanas and Kuchta [3] proposed
the concept of the optimal solution for the transportation
problem with fuzzy coefficients expressed as fuzzy
numbers, and developed an algorithm for obtaining the
optimal solution. Saad and Abbas [22] discussed the solution
algorithm for solving the transportation problem in fuzzy
environment. Liu and Kao [18] described a method for
solving fuzzy transportation problem based on extension
principle. Gani and Razak [13] presented a two stage cost
minimizing fuzzy transportation problem (FTP) in which
supplies and demands are trapezoidal fuzzy numbers. A
parametric approach is used to obtain a fuzzy solution and
the aim is to minimize the sum of the transportation costs in
two stages. Lin [166] introduced a genetic algorithm to solve
transportation problem with fuzzy objective functions.
Dinagar and Palanivel [9] investigated FTP, with the aid of
trapezoidal fuzzy numbers. Fuzzy modified distribution
method is proposed to find the optimal solution in terms of
fuzzy numbers. Pandian and Natarajan [20] proposed a new
algorithm namely, fuzzy zero point method for finding a
fuzzy optimal solution for a FTP, where the transportation
cost, demand and supply are represented by trapezoidal
fuzzy numbers. Edward Samuel [10-12] a solving
generalized trapezoidal fuzzy transportation problems,
where precise values of the transportation costs only, but
there is no uncertain about the demand and supply. In this
paper, a proposed method, namely, Harmonic Mean Method
(HMM) is used for solving a special type of fuzzy
transportation problem by assuming that a decision maker is
uncertain about the precise values of transportation costs
only. In the proposed method transportation costs are
represented by generalized trapezoidal fuzzy numbers. To
illustrate the proposed method HMM a numerical example is
solved. The proposed method HMM is easy to understand
and to apply in real life transportation problems for the
decision makers.
II. PRELIMINARIES
In this section, some basic definitions, arithmetic operations
and an existing method for comparing generalized fuzzy
numbers are presented.
2.1. Definition [15] A fuzzy set ,A defined on the universal
set of real numbers  is said to be fuzzy number if it is
membership function has the following characteristics:
(i) )(~ xA
 :   [0,1] is continuous.
(ii) 0)(~ xA
 for all ( , ] [ , ).x a d   
(iii) )(~ xA
 Strictly increasing on [a, b] and strictly
decreasing on [c, d]
(iv) )(~ xA
 =1 for all [ , ], .x b c where a b c d   
2.2. Definition [15] A fuzzy number ( , , , )A a b c d is
said to be trapezoidal fuzzy number if its membership
function is given by

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( )
, ,
( )
1 , ,
( )
( )
, ,
( )
, .
A
x a
a x b
b a
b x c
x
x d
c x d
c d
o otherwise


  

  
 
  
 


2.3. Definition [6] A fuzzy set ,A defined on the universal
set of real numbers , is said to be generalized fuzzy
number if its membership function has the following
characteristics:
(i) )(~ xA
 :   [0, ] is continuous.
(ii) 0)(~ xA
 for all ( , ] [ , ).x a d   
(iii) )(~ xA
 Strictly increasing on [a, b] and strictly
decreasing on [c, d]
(iv) )(~ xA
 = , for all [ , ], 0 1.x b c where   
2.4. Definition [6] A fuzzy number ( , , , ; )A a b c d  is
said to be generalized trapezoidal fuzzy number if its
membership function is given by
( )
, ,
( )
, ,
( )
( )
, ,
( )
, .
A
x a
a x b
b a
b x c
x
x d
c x d
c d
o otherwise





  

  
 
  
 


Arithmetic operations: In this section, arithmetic
operations between two generalized trapezoidal fuzzy
numbers, defined on the universal set of real numbers ,
are presented [6,7].
Let 1 1 1 1 1 1( , , , ; )A a b c d  and 2 2 2 2 2 2( , , , ; )A a b c d 
are two generalized trapezoidal fuzzy numbers, then the
following is obtained.
(i) 1 2 1 2 1 2 1 2 1 2 1 2( , , , ; min( , )),A A a a b b c c d d       
(ii) 1 2 1 2 1 2 1 2 1 2 1 2( , , , ; min( , )),A A a d b c c b d a       
(iii) 1 2 1 2( , , , ;min( , )),A A a b c d    where
1 2 1 2 2 1 1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 2 1 2 2 1 1 2
min( , , , ),
min( , , , ),
max( , , , ),
max( , , , ).
a a a a d a d d d
b b b b c c b c c
c b b b c c b c c
d a a a d a d d d




(iv)
1 1 1 1 1
1
1 1 1 1 1
( , , , ; ), 0,
( , , , ; ), 0.
a b c d
A
d c b a
     

     

 

Ranking function: An efficient approach for
comparing the fuzzy numbers is by the use of ranking
function [7,17,19],  )(: F , where F( ) is a set of
fuzzy numbers defined on the set of real numbers, which
maps each fuzzy number into the real line, where a natural
order exists, i.e.,
(i) BA
~~
 if and only if )
~
()
~
( BA 
(ii) BA
~~
 if and only if )
~
()
~
( BA 
(iii) BA
~~
 if and only if )
~
()
~
( BA 
Let 1 1 1 1 1 1( , , , ; )A a b c d  and 2 2 2 2 2 2( , , , ; )A a b c d  be
two generalized trapezoidal fuzzy numbers and
1 2min( , )   . Then
1 1 1 1
1
( )
( )
4
a b c d
A
   
  and
2 2 2 2
2
( )
( )
4
a b c d
A
   
  .
III. PROPOSED METHOD
The methodology of harmonic mean method is presented as
follows.
Step 1: Check whether the given fuzzy transportation
problem is balanced or not. If not, balance or by
adding dummy row or column with costs are fuzzy
zero. Then go to step2.
Step 2: Find the fuzzy harmonic mean for each row and
each column. Then find the maximum value among
that.

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Step 3: Allocate the minimum supply or demand at the
place of minimum value of the corresponding row
or column.
Step 4: Repeat the step 2 and 3 until all the demands are
satisfied and all the supplies are exhausted.
Step 5: Total minimum fuzzy cost = sum of the product of
the fuzzy cost and its corresponding allocated
values of supply or demand.
IV. NUMERICAL EXAMPLE
To illustrate the proposed method namely, Harmonic Mean
Method (HMM) the following Fuzzy Transportation
Problem is solved
Example 1: Table 1 gives the availability of the product
available at three sources and their demand at three
destinations, and the approximate unit transportation cost of
the product from each source to each destination is
represented by generalized trapezoidal fuzzy number.
Determine the fuzzy optimal transportation of the products
such that the total transportation cost is minimum.
Table 1
D1 D2 D3 Supply (ai)
S1 (11,13,14,18;.5) (20,21,24,27;.7) (14,15,16,17;.4) 13
S2 (6,7,8,11;.2) (9,11,12,13;.2) (20,21,24,27;.7) 20
S3 (14,15,17,18;.4) (15,16,18,19;.5) (10,11,12,13;.6) 5
Demand (bj) 12 15 11
Using step1,
3 3
1 1
38i j
i j
a b
 
   , so the chosen problem is a balanced FTP. Using Step2 to Step4 we get:
D1 D2 D3 Supply (ai)
S1
(11,13,14,18;.5)
7
(20,21,24,27;.7)
(14,15,16,17;.4)
6
*
S2
(6,7,8,11;.2)
5
(9,11,12,13;.2)
15
(20,21,24,27;.7) *
S3 (14,15,17,18;.4) (15,16,18,19;.5)
(10,11,12,13;.6)
5
*
Demand
(bj)
* * *
The minimum fuzzy transportation cost is equivalent to
7(11,13,14,18;.5) + 6(14,15,16,17;.4) + 5(6,7,8,11;.2) +
15(9,11,12,13;.2) + 5(10,11,12,13;.6) = (376,436,474,543;
.2). Therefore, the ranking function R(A) = 91.45
Results with normalization process: If all the values of the
parameters used in problem.1 are first normalized and then
the Problem is solved by using the HMM, then the fuzzy
optimal value is 0 (376,436,474,543;1).x 
Results without normalization process: If all the values of
the parameters of the same problem.1 are not normalized
and then the Problem is solved by using the HMM, then the
fuzzy optimal value is 0 (376,436,474,543;.2).x 
Remark: Results with normalization process represent the
overall level of satisfaction of decision maker about the
statement that minimum transportation cost will lie between
436 and 474 units as 100% while without normalization
process, the overall level of satisfaction of the decision
maker for the same range is 20%. Hence, it is better to use
generalized fuzzy numbers instead of normal fuzzy
numbers, obtained by using normalization process.

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Example 2:
D1 D2 D3 D4 (ai)
S1 (11,12,13,15;.5) (16,17,19,21;.6) (28,30,34,35;.7) (4,5,8,9;.2) 8
S2 (49,53,55,60;.8) (18,20,21,23;.4) (18,22,25,27;.6) (25,30,35,42;.7) 10
S3 (28,30,34,35;.7) (2,4,6,8;.2) (36,42,48,52;.8) (6,7,9,11;.3) 11
(bj) 4 7 6 12
Example 3: [12]
D1 D2 D3 (ai)
S1 (1,4,9,19;.5) (1,2,5,9;.4) (2,5,8,18;.5) 10
S2 (8,9,12,26;.5) (3,5,8,12;.2) (7,9,13,28;.4) 14
S3 (11,12,20,27;.5) (0,5,10,15;.8) (4,5,8,11;.6) 15
(bj) 15 14 10
V. COMPARATIVE STUDY AND RESULT ANALYSIS
From the investigations and the results given in Table 2 it clear that HMM is better than NWCR [1], MMM [1] and VAM [21]
for solving fuzzy transportation problem and also, the solution of the fuzzy transportation problem is given by HMM is an
optimal solution.
Table 2
S.No ROW COLUMN NWCR MMM VAM MODI HMM
1. 3 3 108.80 99.50 97.50 91.45 91.45
2. 3 4 134.18 95.00 83.00 75.60 75.60
3. 3 3 64.35 73.10 67.60 64.35 64.35
Table 2 represents the solution obtained by NWCR [1], MMM [1], VAM [21], MODI [5] and HMM. This data speaks the
better performance of the proposed method. The graphical representation of solution obtained by varies methods of this
performance, displayed in graph.
0
20
40
60
80
100
120
140
160
NWCR
MMM
VAM
MODI
HMM

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VI. CONCLUSION
From the comparison table 2, we can observe that the
optimum solution obtained by the proposed method is less
than that of other methods and same that of MODI Method.
But, the proposed method is very easy since we have less
computation works. So, we can conclude that if we use
harmonic mean method to solve transportation problem, we
can get global optimum solution in a lesser step.
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