New Methodology to Find Initial Solution for Transportation ...

Applied Mathematical Sciences, Vol. 9, 2015, no. 19, 915 - 927

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121018

New Methodology to Find Initial Solution for

Transportation Problems:

a Case Study with Fuzzy Parameters

Giancarlo de França Aguiar

Federal Institute of Paraná and Positivo University, Street João Negrão

Post box 1285, Curitiba- Paraná, 80230-150, Brazil

Bárbara de Cássia Xavier Cassins Aguiar

Federal University of Paraná - DEGRAF, Jardim das Américas


Volmir Eugênio Wilhelm

Federal University of Paraná - DEGRAF, Jardim das Américas


Copyright © 2014 Giancarlo de França Aguiar et al. This is an open access article distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we propose a new algorithm for finding a feasible initial solution to

transportation problems. The MOMC Method (Maximum Supply with Minimum

Cost) as it was called, was compared with three classical methods: Corner

Northwest, Minimum Cost and Vogel, obtaining significant results (better than the

methods shown). A numerical example was presented to a rigorous understanding

of the algorithm, and the results in a case study with a transportation problem with

fuzzy parameters after the application of a classification technique showed great

computational advantage (higher processing speed and less use of memory).

Keywords: Fuzzy Transportation Problem, MOMC Method, Feasible Initial

Solution

916 Giancarlo de França Aguiar et al.

1 Introduction

The transportation problems are a very broad study class within the Linear

Programming. In mid 1941 the Professor Hitchcock [8] formalized originally one

of the first basic problems of transportation. Appa in 1973 [1] discussed several

variations of the transportation problem. The solution to many of these problems

can be found using the simplex method, for example with teachers Charnes and

Cooper [4] in 1954, and Dantzig Thapa [5] in 1963 and Arsham and Khan [2] in

1989, but because of its very particular mathematical structure, new approaches

have been proposed and studied.

The simplex method is an iterative algorithm, it requires an initial feasible

solution to the problem. The four main classical methods, very widespread in the

literature for finding an initial solution are: the north-west corner method (see

Zionts [20] in 1974), Vogel method (see Murty [10] in 1983), the Cost Minimum

method (see Puccinni and Pizzolato [11] in 1987) and Russell's method (see Ruiz

and Landín [14] in 2003) each with its own advantages, disadvantages and

particularities. However researchers as Rodrigues [12] has proposed and studied

new approaches to find an initial solution to transportation problems.

Armed with an initial solution, the new goal becomes: find out if this solution

is optimal or not. Several heuristics as in Kaur and Kumar [9] in 2011, Silva [16]

in 2012, Samuel and Venkatachalapathy [15] in 2013 and Ebrahimnejad [6] in

2014, have been developed to find an optimal solution to transportation problems,

from an initial feasible solution. These new heuristics have gotten increasingly

satisfactory results (more efficient algorithms) and with various practical

applications. Another branch of very exploited Transportation problems, and that

has become increasingly important in scientific circles, are the problems involving

inaccuracies in the collection and treatment of measures of cost factors, supply

and demand, and that many of these measures are uncontrollable, requiring then

the use of fuzzy logic in dealing and development of these new techniques. The

pioneers in the study of fuzzy environment in decision-making were the

researchers Zadeh [19] and Bellmann and Zadeh [3] in 1965 and 1970

respectively. Following this work, new authors as Tanaka and Asai [18] in 1984,

Rommelfanger, Wolf and Hanuscheck [13] in 1989, Fang Hu, Wang and Wu [7]

in 1999, Sudha, Margaret and Yuvarani [17] in 2014 have developed fuzzy linear

programming techniques to transportation problems.

However, the initial solution has been little exploited, probably because

efficiency of the traditional methods and also by extensive computational power

of modern computers, which easily rotate the simplex method to a problem with a

large number of variables and constraints.

This paper proposes a new methodology to find a feasible initial solution to

transportation problems. The Algorithm of MOMC Method (Maximum Supply

with Minimum Cost) will be illustrated in the following sections, a case study

with a trapezoidal fuzzy transportation problem after applying a ranking technique

will be explored and a comparison of this method with the classical methods will

be outlined.

New methodology to find initial solution for transportation problems 917

2 Fuzzy Transportation Problem

Following the basics concepts of a fuzzy number are illustrated, the general

model of a fuzzy transportation problem in their tabular form (in a frame) and its

linear programming model.

2.1 Number Fuzzy

According Narayanamoorthy, Saranya and Maheswari [11] in 2013, a real

fuzzy number is a fuzzy subset of the real number R with membership function

satisfying the following conditions:

1. is continuous from R to the closed interval [0, 1];

2. is strictly increasing and continuous on [a1, a2];

3. is strictly decreasing and continuous on [a3, a4].

2.2 Trapezoidal Fuzzy Number

According to yet Narayanamoorthy, Saranya and Maheswari [11] in 2013, a fuzzy number is said to be a trapezoidal fuzzy number if its membership function is given by:

𝜇𝐴 =

0 𝑥 < 𝑚𝑥 − 𝑎

𝑏 − 𝑎 𝑎 ≤ 𝑥 ≤ 𝑏

𝑑 − 𝑥

𝑑 − 𝑐 𝑐 ≤ 𝑥 ≤ 𝑑

0 𝑐 > 𝑑

Being represented graphically as in the following Figure 2.1:

𝜇𝐴 𝑋

𝑋 0 𝑎 𝑏 𝑐 𝑑

FIGURE 2.1: Trapezoidal Number Fuzzy


The following Table 2.1 illustrates the first general model of a fuzzy

transportation problem in a tabular form.

TABLE 2.1: Tabulated Form for to Fuzzy Transportation Problem

Destinations

1 … 𝑛 Supply

Sources 1 𝑐 11 … 𝑐 1𝑛 𝑎 1

⋮ ⋮ ⋮ ⋮ 𝑚 𝑐 𝑚1 … 𝑐 𝑚𝑛 𝑎 𝑚

Demand 𝑏 1 … 𝑏 𝑛

Inserted in linear programming, the fuzzy transportation model deserves

special attention, having its own characteristics and a general model:

n,...,jem,...,i,x

ba

)demanda(n,...,j,bx

)oferta(m,...,i,ax:.a.s

x.cCMin

~

ij

~

n

j

j

~m

i

i

~

j

~m

i

ij

~

i

~n

j

ij

~

m

i

n

j

ij

~

ij

~~

110

1

1

11

1

1

1 1

In this model, 𝑐 𝑖𝑗 represent transportation costs from place of origin i to

destination j, 𝑎 𝑖 is the offer of origin i, 𝑏 𝑗 is the demand j , e 𝑥 𝑖𝑗 is the quantity to

be transported from origin i to destination j.

3 Initial Solution for Transportation Problems

There are three classic models very worked in the literature to find an initial

solution to transportation problems. The North-west Corner Method, the

Minimum Cost Method and Vogel Method.

3.1 The North-west corner method

The North-west corner method (can be evaluated in Zionts [20]) as it is known

does not consider transport costs in decision-making, depending only on the supply and demand of the origins and destinations respectively and having relatively


simple computational development. In general your solution is not as efficient

(near optimal), but its application is a feasible solution.

3.2 Minimum Cost Method

The Minimum Cost Method as the name suggests is based on an analysis of

transport costs and also the examination of the values of supply and demand,

aiming a closer initial solution of the optimal than that provided by the Northwest

Corner Method. A detailed study of the method can be seen in Puccini and

Pizzolato [11] in 1987.

3.3 Vogel Method

In Vogel method does not interest the transportation of lower cost, but the

transportation that has the greater penalty if you do not choose the lowest cost. In

this method, each transmission frame must be calculated for each row and each

column, the difference between the two lower cost, the result is called penalty. In

the column or row where the biggest difference occurs choose the cell, called

basic cell, which has the lowest value. The algorithmic process can be seen in

Murty [10] in 1983.

3.4 New Methodology

This paper proposes a new methodology to find a feasible initial solution to

transportation problems. The Maximum Supply Method with Minimum Cost or

MOMC will be named as described in the following algorithm.

3.4.1 The MOMC Algorithm

The following steps are illustrated to find a feasible initial solution for a

balanced transportation problem, that is, where the supply of origins is equal to

demand of destinations.

Step 1: Select i line with the highest supply ( ); If a tie, choose the line that has

the lowest cost;

Step 2: In line with the increased supply of step 1, choose the column with the

lowest cost ( ); If a tie, choose the column j with the highest demand;

Step 3: The cell intersection of the line (step 1) with the column (step 2) is

selected;


Step 4: Allocate the maximum supply (quantity) of step 1 line to the cell ( )

selected, up to the amount of demand j column;

Step 5: Decrease of the supply (line i) and Demand (column j) the amount

allocated to the cell ( ) from step 4;

If demand < supply, then the demand is annulled;

If demand > supply, then the supply will be annulled;

If demand = supply, then the demand and the supply is annulled;

Step 6: Eliminate of the transportation model (Table), the column or line with

demand or supply annulled after step 5; If demand (column) and supply (line) are

zero simultaneously, then eliminate them;

Step 7: If there no more supply, end. Otherwise, return to step 1, starting the next

iteration.

4 Numerical Example

Right now we use the numerical example worked for Poonam, Abbas and

Gupta [12] in 2012. It is a fuzzy transportation problem with three sources and

four destinations as illustrated in the following Table 4.1.

TABLE 4.1: Fuzzy Transportation Problem with 3 Sources and 4 Destinations

FD1 FD2 FD3 FD4 Supply

FS1 (1,2,3,4) (1,3,4,6) (9,11,12,14) (5,7,8,11) (1,6,7,12)

FS2 (0,1,2,4) (-1,0,1,2) (5,6,7,8) (0,1,2,3) (0,1,2,3)

FS3 (3,5,6,8) (5,8,9,12) (12,15,16,19) (7,9,10,12) (5,10,12,17)

Demand (5,7,8,10) (1,5,6,10) (1,3,4,6) (1,2,3,4)

Poonam, Abbas and Gupta [12] in 2012 used the previous problem a Robust

Ranking Technique for the parameters (transport costs, supply and demand) fuzzy,

satisfying the properties: compensation, linearity and additivity. After ranking, the

transportation problem with fuzzy numbers has become a transportation problem

with crisp numbers as shown in the following Table 4.2:


TABLE 4.2: Crisp Transport Problem with 3 Sources and 4 Destinations


FS1 2,5 3,5 11,5 7,7 6,5

FS2 1,7 0,5 6,5 1,5 1,5

FS3 5,5 8,5 15,5 9,5 11

Demand 7,5 5,5 3,5 2,5

From Table 4.2 above will be illustrated as finding an initial solution to the

transportation problem using MOMC algorithm. Let’s consider how line 1, the

corresponding line source 1 (FS1), and consider how one column, the column

corresponding to the destination 1 (FD1). Below is illustrated the development of

the steps 7 of the first iteration.

Step 1: Select I line with the highest supply; If a tie, choose the line that has the

lowest cost:

The line with the highest supply (11) is the line 3.

Step 2: In line with the increased supply of step 1, choose the column with the

lowest cost ( ); If a tie, choose the column j with the highest demand:

The column of line 3 with lower cost (5.5) is the column 1.

Step 3: The cell intersection of the line (step 1) with the column (step 2) is

selected:

The corresponding cell is , as shown in Table 4.3 below.

TABLE 4.3: Choice of cell


FS1 2,5 3,5 11,5 7,7 6,5

FS2 1,7 0,5 6,5 1,5 1,5

FS3 5,5 8,5 15,5 9,5 11

Demand 7,5 5,5 3,5 2,5

Step 4: Allocate the maximum supply (quantity) of step 1 line to the cell selected,

up to the amount of demand j column:

The source 3 can offer 11 units, however the destination 1 requires 7.5 units.

So will be allocated 7.5 units from the origin to the destination 3. .

Step 5: Decrease of the supply and Demand the amount allocated to the cell from

step 4:


The following Table 4.4 illustrates step 5.

TABLE 4.4: Step 5


FS1 2,5 3,5 11,5 7,7 6,5

FS2 1,7 0,5 6,5 1,5 1,5

FS3 5,5 8,5 15,5 9,5 11 – 7,5 = 3,5

Demand 7,5 – 7,5 = 0 5,5 3,5 2,5

Step 6: Eliminate of the transportation model (Table), the column or line with

demand or supply annulled after step 5; If demand (column) and supply (line) are

zero simultaneously, then eliminate them:

As the demand of the destination 1 was annulled, then leaves the table column

1, as shown in Table 4.5 below.

TABLE 4.5: Reduced Numerical Example

FD2 FD3 FD4 Supply

FS1 3,5 11,5 7,7 6,5

FS2 0,5 6,5 1,5 1,5

FS3 8,5 15,5 9,5 3,5

Demand 5,5 3,5 2,5

Step 7: If there no more supply, end. Otherwise, return to step 1, starting the next

iteration.

As there are still supply, return to step 1. The following Table 4.6 illustrates the

results of the application of MOMC Algorithm.

TABLE 4.6: Reduced Numerical Example

Iteration Cell

1 x31 = 7,5

2 x12 = 5,5

3 x34 = 2,5

4 x23 = 1,5

5 x13 = 1

6 x33 = 1

5 Results

Will be presented at this time the results of the application of the North-west

Corner Method, of Minimum Cost Method, Vogel method and MOMC method

for the numerical example of the previous section. Table 5.1 shows the results.


TABLE 5.1: Results of Numerical Example

Variáveis

Método do

Canto

Noroeste

Método do

Custo

Mínimo

Método

de Vogel

Método

MOMC

𝒙𝟏𝟏 6,5 6,5 1 0

𝒙𝟏𝟐 0 0 5,5 5,5

𝒙𝟏𝟑 0 0 0 1

𝒙𝟏𝟒 0 0 0 0

𝒙𝟐𝟏 1 0 0 0

𝒙𝟐𝟐 0,5 1,5 0 0

𝒙𝟐𝟑 0 0 0 1,5

𝒙𝟐𝟒 0 0 1,5 0

𝒙𝟑𝟏 0 1 6,5 7,5

𝒙𝟑𝟐 5 4 0 0

𝒙𝟑𝟑 3,5 3,5 3,5 1

𝒙𝟑𝟒 2,5 2,5 1 2,5

Min C 138,7 134,5 123,5 121

The application of the Northwest Corner method shows the following results:

𝑥11 = 6,5, 𝑥21 = 1, 𝑥22 = 0,5, 𝑥32 = 5, 𝑥33 = 3,5 and 𝑥34 = 2,5. The other

variables are null. Soon the minimum cost function returns 𝑀𝑖𝑛 𝐶 = 2,5 6,5 +

1,7 1 + 0,5 0,5 + 8,5 5 + 15,5 3,5 + 9,5 2,5 = 138,7.

The use of the Minimum Cost method presents the results: 𝑥11 = 6,5, 𝑥22 =

1,5, 𝑥31 = 1, 𝑥32 = 4, 𝑥33 = 3,5 and 𝑥34 = 2,5. The other variables have zero

value, and the minimum cost function returns 𝑀𝑖𝑛 𝐶 = 2,5 6,5 + 0,5 1,5 +

5,5 1 + 8,5 4 + 15,5 3,5 + 9,5 2,5 = 134,5. This method was a better result

(lower cost) than the Northwest Corner Method, however, is still not the best.

The use of Vogel method yielded the following results: 𝑥11 = 1, 𝑥12 =

5,5, 𝑥24 = 1,5, 𝑥31 = 6,5, 𝑥33 = 3,5 e 𝑥34 = 1. The other variables are null. The

minimum cost function obtained as a result 𝑀𝑖𝑛 𝐶 = 2,5 1 + 3,5 5,5 +

1,5 1,5 + 5,5 6,5 + 15,5 3,5 + 9,5 1 = 123,5. This method was a better

result than the Northwest Corner Method and method of Min Cost.

The MOMC method showed the following results: 𝑥12 = 5,5, 𝑥13 = 1, 𝑥23 =

1,5, 𝑥31 = 7,5, 𝑥33 = 1 e 𝑥34 = 2,5. The other variables are null. Thus, the

minimum cost function resulted in 𝑀𝑖𝑛 𝐶 = 3,5 5,5 + 11,5 1 + 6,5 1,5 +

5,5 7,5 + 15,5 1 + 9,5 2,5 = 121, and the best feasible initial solution

between the four methods evaluated. This solution is also the optimal solution of

the problem.


The great advantage of using MOMC method is not only a good approximation

(in the previous example has found the optimal solution) of the optimal solution,

but the computational time to solve problems. The following Table 5.2 shows a

comparison between the Vogel method and of MOMC method.

TABLE 5.2: Comparison of the Vogel method with the MOMC method

Pro

ble

m

So

urc

es

x

Des

tin

ati

on

s

Vogel

(1a iteration)

MOMC

(1a iteration)

Ga

in %

M

OM

C

Scan

cells

Lo

gic

al

Op

era

tio

ns

(Su

btr

act

ion

)

Total

Scan

cells

Lo

gic

al

Op

era

tio

ns

Total

1 3 x 4 24 7 31 7 0 7 77,42

2 10 x 4 80 14 94 14 0 14 85,11

3 100 x 90 18.000 190 18.190 190 0 190 98,96

4 4.000 x 300 2.400.000 4.300 2.404.300 4.300 0 4.300 99,82

In this case study with three sources and four destinations, the Vogel method

Makes a reading in 12 horizontal cells, more 12 vertical cells to determine the two

smaller each row and each column of costs (a total of 24 cells). It performs three

logical operations (difference between the lowest costs) horizontal and 4 vertical

operations (a total of 7 operations). Performing in total 31 decisions, only the first

iteration.

The MOMC method done to the same problem a total of 7 decisions (3 vertical

scans 4 more horizontal and no logic operation). In this problem, analyzing the

first iteration MOMC method achieved a gain of approximately 77.42% in making

supply allocation decision. In a problem with 100 sources and 90 destinations, the

MOMC method achieved a gain of 98.96% in making supply allocation decision.

Computationally this is greater processing speed and less memory usage.

6 Conclusions

This paper presented a new algorithm called MOMC method. This method has

shown significant advantages in simplifying calculations and decision making to

find an initial solution to transportation problems.

In many worked examples by author so far, the MOMC method has proven

effective finding the optimal solution without the need for optimality algorithm. From the computational point of view (higher processing speed and less memory


usage), the example text could illustrate the efficiency of capacity MOMC method

with up to 99.82% gains even in the first iteration with respect to

Vogel method (one the most implemented according to the literature).

The next work will be the implementation (in language with graphical

programming interface), computer simulation of classical problems and compared

to most methods discussed in the literature.

References

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79-99, (1973). http://dx.doi.org/10.1057/jors.1973.10

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http://dx.doi.org/10.1057/jors.1989.95

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Intelligent Systems and Applications, no.2, 71-75, 2013.

http://dx.doi.org/10.5815/ijisa.2013.02.08

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Apr. 1975.

[15] Rommelfanger, H., Wolf, J., Hanuscheck, R. Linear programming with fuzzy

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[17] Samuel, A. E., Venkatachalapathy, M. A simple heuristic for solving

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Applied Mathematics, vol. 83, n. 1, 91-100, (2013).

http://dx.doi.org/10.12732/ijpam.v83i1.8

[18] Silva, T. C. L. New methods for solving transportation problems in sparse

cases. Doctoral Thesis. PPGMNE, (2012).

[19] Sudha, S., Margaret, A. M., Yuvarani, R. A New Approach for Fuzzy

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April, 2014, Vol3, Issue 4, 251-260.

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http://dx.doi.org/10.1016/s0019-9958(65)90241-x

http://dx.doi.org/10.5815/ijisa.2013.02.08

http://dx.doi.org/10.12732/ijpam.v83i1.8

http://dx.doi.org/10.1016/s0019-9958%2865%2990241-x


[22] Zionts, S. Linear and integer programming. Englewood Cliffs, New Jersey,

Prentice Hall, 1974.

Received: December 23, 2014; Published: January 30, 2015

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