This paper is built on a study in relation to transportation problem as it affects most organisational decision in a decomposed setting. The case study used in this work is Dangote cement factory (in Ibese, Nigeria) with three sources and four destinationscentres. The factory is supported by increasing number of cement delivery trucks. Some models for solving balanced transportation problems (TPs) are considered in order to determine the optimal and initial basic feasible solutions (IBFS). From the analysis, it is observed that Modified Vogel Approximation Method (MVAM) is a better method. This is partly because MVAM considers each unit cost in its solution algorithm and minimises total cost comparatively with Vogel Approximation Method (VAM). The results arefurther justified and validated using windows version 2.00 Tora package.

1. http://www.iaeme.com/IJCIET/index.asp 358 editor@iaeme.com
International Journal of Civil Engineering and Technology (IJCIET)
Volume 9, Issue 12, December 2018, pp. 358–366, Article ID: IJCIET_09_12_039
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=12
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
©IAEME Publication Scopus Indexed
MODIFIED VOGEL APPROXIMATION
METHOD FOR BALANCED TRANSPORTATION
MODELS TOWARDS OPTIMAL OPTION
SETTINGS
I. D. Ezekiel
Department of Mathematics & Statistics, Federal Polytechnic, Ilaro, Nigeria
Department of Mathematics, Covenant University, Ota, Nigeria
S. O. Edeki
Department of Mathematics, Covenant University, Ota, Nigeria
Corresponding Author’s Email: soedeki@yahoo.com
ABSTRACT
This paper is built on a study in relation to transportation problem as it affects
most organisational decision in a decomposed setting. The case study used in this
work is Dangote cement factory (in Ibese, Nigeria) with three sources and four
destinations centres. The factory is supported by increasing number of cement delivery
trucks. Some models for solving balanced transportation problems (TPs) are
considered in order to determine the optimal and initial basic feasible solutions
(IBFS). From the analysis, it is observed that Modified Vogel Approximation Method
(MVAM) is a better method. This is partly because MVAM considers each unit cost in
its solution algorithm and minimises total cost comparatively with Vogel
Approximation Method (VAM). The results are further justified and validated using
windows version 2.00 Tora package.
Key words: Balanced transportation problem, Option pricing, Modified Vogel
approximation method, Vogel approximation method, Decomposition method
Cite this Article: I. D. Ezekiel and S. O. Edeki, Modified Vogel Approximation
Method for Balanced Transportation Models towards Optimal Option Settings,
International Journal of Civil Engineering and Technology (IJCIET) 9(12), 2018, pp.
358–366.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=12
1. INTRODUCTION
This study contains the data from Dangote cement factory located in Ibese, Nigeria. Dangote
Cement operates three manufacturing plants in Obajana, Ibese and Gboko, with longer-term
plans to expand its capacity across two sites at Itori and Okpella. However, the Ibese plant is
considered herein. The factory supplies cement product to South-South states, Lagos state,

2. I. D. Ezekiel and S. O. Edeki
http://www.iaeme.com/IJCIET/index.asp 359 editor@iaeme.com
Ogun state and other South west states in Nigerian markets with production capacity of 6.0
MT cement produced from lines 1, 2 and 3 respectively. The factory is supported by
increasing number of cement delivery trucks. Hence, the need for enhanced means to resolve
transportation problem (TP). This data article therefore aims at making informed decision on
effective techniques for finding initial basic feasible solution and applying Modified Vogel
Approximation Method (MVAM) to determine the optimal solution to the corresponding
balanced TP. Table 1 contains the sources alongside the destinations of the trunk lines, Table
2 contains the General Transportation Problem Tableau (GTPT), while the analysed data are
presented in Tables 3-5 and Figures 1-4. Related work on Transportation, economic, and
financial problems of relevance to this study can be found in [1-15].
The simplest transportation problem was first presented in 1941 and developed in 1949
and 1951. Since then several extensions of transportation models and methods have emerged.
Among early pioneers of transportation problems (TPs) are Hitchcock [16] through his
paper “the Distribution of a Product from Several sources to numerous Localities”. Closely
related to Hitchcock is Koopman [17] who with his paper “Optimum Utilization of
Transportation System” simplified better understanding of transportation methods involving
a number of shipping sources and destinations. Gass [5] in his own contribution explains
practical issues for solving transportation problems and offered comments on various aspects
of transportation problem (TP) methodologies along with discussions on the computational
results by various researchers. Tzeng, Teodorovic, and Hwang, [18] applied time TP to
formulate a LP that minimizes cost of transportation to distribute and transport imported Coal
to power plants by determining the quantity and quality required amounts under stable supply
with least delay. Their model yield optimum results and serves as decision support system to
manage coal allocation, voyage scheduling and dynamic fleet assignment problems. Since
then, a number of researchers such as Sonia [13] contributed to Transportation Problem in
area of decision support. However, their work was restricted on time transportation problem.
A number of researchers including Wahead and Lee [19], and Joshi [20], in “Optimization
Techniques for Transportation Problems of Three Variables” proposed for the procedure for
full allocation of available total quantity required in each cell.
The existing techniques or methods are the minimum cost rule method, Northwest corner
rule method, Vogel approximation method, Russell approximation method and Roland
Larson’s method among others. Resolved TPs in differential model forms can be handled with
approximate and numerical methods [21-25].
This paper therefore aims at making comparison of existing techniques for finding initial
basic feasible solution and apply modified Vogel approximation method in determining initial
basic feasible solutions so as to determine the optimal solution to any given balanced TP.
2. MATERIALS AND METHODS
For the purpose of this paper, the table below gives simplified data structure obtained from
the cement factory with three sources and four destination centres without service level
requirements. The following are considered.
 There are three sources (lines 1, 2 and 3) and four destinations centres (South-South, Lagos,
Ogun, other South west states) represented by A, B, C and D respectively.
 All the mentioned routes in this work are always accessible.
 Movement of cement products are always from the sources to the specified destinations.
 The same means of transportation of the products and constant speed are allowed.

3. Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal
Option Settings
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Table1 specifies sources ( ), destination centres ( ), where ;
and the unit transportation cost from to .
Table 1 Case study of Dangote Cement (Ibese-Station)
3. GENERAL TRANSPORTATION PROBLEM AND APPLICATIONS
Mathematically, we state TP compactly as given below:
∑ ∑
subject to:
∑
∑
, and
where is the capacity of supply at source , is capacity of demand at destination ,
is the amount shipped from source to destination , and ∑ ∑ is the total
shipping cost and it is non-negative. ∑ ∑ is the condition for existence of
feasible solution for standard TP. In other words, total units demanded must match total units
supplied. A balanced transportation problem having sources and destinations is usually
represented in tabular form as shown below:
Table 2 General Transportation Problem Tableau
S/d Destination
Source 1 2 3 . . .
1 . . .
2 . . .
3 . . .
. . . . . . . . .
. . . . . . . . .
. . .
. . .
Here, the methods applied include: Northwest-Corner Rule (NCR), Least-Cost Method
(LCM), Vogel Approximation Method (VAM), and Modified Vogel Approximation Method
(MVAM).
S/D Destination
Source A B C D
1 8 6 3 9 120
2 2 6 1 4 140
3 7 8 6 3 100
100 60 80 120 360

4. I. D. Ezekiel and S. O. Edeki
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Table 3 Northwest Corner Rule
S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6
Source A B C D
Supply 1 8 6 3 9 120 20 0 0 0 0 0
2 2 6 1 4 140 140 140 100 20 0 0
3 7 8 6 3 100 100 100 100 100 100 0
100 60 80 120 360
Stage 1 0 60 80 120 260
Stage 2 0 40 80 120 240
Stage 3 0 0 80 120 200
Stage 4 0 0 0 120 120
Stage 5 0 0 0 100 100
Stage 6 0 0 0 0 0
Using Northwest-corner rule, we have the following initial basic feasible solution for
as: , , , , , .
The optimal minimization cost is given by
( ) ( ) ( ) ( ) ( ) ( )
Figure 1 Northwest Corner Rule
Table 4 Least Cost Rule
0
50
100
150
200
250
300
350
400
A B C D SS S1 S2 S3 S4 S5 S6
Sourcesandsupply-stages
Destinations and stages
Supply:1
Supply:2
Supply:3
dj
Stage:1
Stage: 2
Stage: 3
Stage: 4
Stage: 5
Stage: 6
S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6
Origin A B C D
Supply 1 8 6 3 9 120 120 120 120 60 20 0
2 2 6 1 4 140 60 0 0 0 0 0
3 7 8 6 3 100 100 100 0 0 0 0
100 60 80 120 360
Stage 1 100 60 0 120 280
Stage 2 40 60 0 120 220
Stage 3 40 60 0 20 120
Stage 4 40 0 0 20 60
Stage 5 0 0 0 20 20
Stage 6 0 0 0 0 0

5. Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal
Option Settings
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For , we have: , , , , ,
The optimal minimization cost is given by
( ) ( ) ( ) ( ) ( ) ( )
Figure 2 Least Cost Rule
Table 5 Vogel Approximation Method
S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6
Source A B C D
Supply 1 8 6 3 9 120 3 120 3 40 3 40 3 40 6 0 0 0
2 2 6 1 4 140 1 40 3 40 2 40 2 20 6 20 6 0
3 7 8 6 3 100 3 100 3 100 5 0 0 0 0 0 0 0
100 60 80 120 360
Stage 1 5 0 2 1
0 60 80 120 260
Stage 2 0 0 2 1
0 60 0 120 180
Stage 3 0 0 0 1
0 60 0 20 80
Stage 4 0 0 0 5
0 60 0 0 60
Stage 5 0 0 0 0
0 20 0 0 20
Stage 6 0 6 0 0
0 0 0 0 0
Using Vogel approximation rule, where is the row difference at stage
and is the column difference at stage . Thus the following initial basic
feasible solution for becomes: , , , ,
,
The optimal minimization cost is given by
( ) ( ) ( ) ( ) ( ) ( )
0
50
100
150
200
250
300
350
400
A B C D SS S1 S2 S3 S4 S5 S6
Sourcesandsupply-stages
Destinations and stages
Supply: 1
Supply: 2
Supply: 3
dj
Stage: 1
Stage: 2
Stage: 3
Stage: 4
Stage: 5
Stage: 6

6. I. D. Ezekiel and S. O. Edeki
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Figure 3 Vogel Approximation Method
Table 6 Modified Vogel Approximation Method (MVAM)
Using modified Vogel approximation rule method, where is the row
mean deviation at stage and is the column mean deviation at stage .
Therefore, the initial basic feasible solution for becomes:
, , , , ,
The optimal minimization cost is given by:
( ) ( ) ( ) ( ) ( ) ( )
0
50
100
150
200
250
300
350
400
A
B
C
D
SS
S1A
S1B
S2A
S2B
S3A
S3B
S4A
S4B
S5A
S5B
S6A
S6B
Sourcesandsupply-stages
Destinations and stages
Supply: 1
Supply: 2
Supply: 3
dj
Stage 1
Stage: 1
Stage 2
Stage: 2
Stage 3
Stage: 3
Stage 4
Stage: 4
Stage 5
Stage: 5
Stage 6
Stage: 6
S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6
Sour. A B C D
Sup. 1 8 6 3 9 120 2.3 120 2.3 120 2.4 120 1.5 120 1.5 60 1.7 0
2 2 6 1 4 140 1.9 140 1.9 40 2.0 20 2.5 0 0 0 0 0
3 7 8 6 3 100 1.9 0 0 0 0 0 0 0 0 0 0 0
100 60 80 120 360
Stage
1
2.6 0.9 2.1 2.6
100 60 80 20 260
Stage 2 3 0 1 2.5
0 60 80 20 160
Stage 3 0 0 1 2.5
0 60 80 0 140
Stage 4 0 0 1 0
0 60 60 0 120
Stage 5 0 2.4 1.7 0
0 0 60 0 60
Stage 6 0 0 1.7 0
0 0 0 0 0

7. Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal
Option Settings
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Figure 4 Modified Vogel Approximation Method (MVAM)
Table 7 Comparison of the Methods
S/N Methods IBFS
1 Northwest corner rule
2 Least cost rule
3 Vogel Approximation
4 Modified Vogel Approximation
4. DISCUSSION OF RESULTS, CONCLUDING REMARKS, AND
RECOMMENDATIONS
Table 7 shows the result for selected existing approaches available for solving balanced
transportation problem and their order of efficiency. MVAM is applied as well as other
existing methods for solving balanced transportation problem and compared the efficiency of
the methods at reducing total cost using the same case study. From the analysed data in Table
7, it is obvious that the MVAM gives better improvement than any other selected techniques
in determining the initial basic feasible solution (IBFS) in terms of reduction in total cost of
production in the cement factory. The results of the solution obtained in Table 7 was further
justified and validated using windows version 2.00 Tora package. The use of the modified
VAM for solving transportation problem gives a systematic and transparent solution when
compared with other existing methods. Thus for more scientific transportation problem, the
modified VAM gives a better result. Hence the modified VAM guarantees optimal solution.
The researcher, therefore, recommends that the modified VAM should be adopted and
encouraged by companies, producers and other agencies involved in transportation business.
The researcher also recommends further research in this area
ACKNOWLEDGEMENTS
The authors are indeed grateful to Covenant University for the provision of resources, and
enabling working environment.
0
50
100
150
200
250
300
350
400
A
B
C
D
SS
S1A
S1B
S2A
S2B
S3A
S3B
S4A
S4B
S5A
S5B
S6A
S6B
Sourcesandsupply-stages
Destinations and stages
Supply: 1
Supply: 2
Supply: 3
dj
Stage 1
Stage 1
Stage 2
Stage 2
Stage 3
Stage 3
Stage 4
Stage 4
Stage 5
Stage 5
Stage 6
Stage 6

8. I. D. Ezekiel and S. O. Edeki
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