PERFORMANCE OF THE BEST SOLUTION FOR THE PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN IMPROVED
VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

PERFORMANCE OF THE BEST SOLUTION FOR THE
PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN
IMPROVED VOGEL'S APPROXIMATION METHOD
E.M.D.B. Ekanayake1, E. M. U. S. B. Ekanayake1
Volume 3 Issue 3
(December 2022)
e-ISSN 2722-6395
doi: 10.30997/ijar.v3i3.241
ARTICLE INFO
Article history:
Received: 11-11-2022
Revised version received: 16-11-2022
Accepted: 19-12-2022
Available online: 26-12-2022
Keywords:
Balance and Unbalance Transportation
Problem, Initial Basic Feasible Solution,
Optimal Solution, Prohibited Route,
“VAM" method
How to Cite:
Ekanayake, E., M., D., B., & Ekanayake,
E. M. U. S. B. (2022). PERFORMANCE OF
THE BEST SOLUTION FOR THE
PROHIBITED ROUTE TRANSPORTATION
PROBLEM BY AN IMPROVED VOGEL'S

1

Department of Physical Sciences, Faculty of Applied
Sciences, Rajarata University of Sri Lanka,
Mihinthale, Sri Lanka.
ABSTRACT

The transportation problem (TP) is a significant factor in
operational research. Numerous researchers have put forth
various solutions to these problems. The goal is to reduce
the overall cost of distributing resources from multiple
sources to numerous destinations. If there are road risks
(snow, flood, etc.), traffic limitations, etc., it might not be
feasible to transport products from one place to another. In
these circumstances, the appropriate route(s) can be given
an extremely high unit cost, such as M (or ∞). Following
that, a specific case of the prohibited transportation problem
is introduced. Therefore, the focus of this study is to provide
a novel algorithm that will reduce the cost of restricted
transportation problems. With a few modifications, the
traditional Vogel approach has been enhanced. The
proposed method would perform better than the other
approaches now in use. The numerical problem is resolved
to demonstrate the effectiveness of the proposed approach
and make comparisons with different approaches already in
use.

APPROXIMATION METHOD. Indonesian
Journal of Applied Research
(IJAR), 3(3), 190-206.
https://doi.org/10.30997/ijar.v3i3.241
Corresponding Author:
E.M.D.B. Ekanayake
dananjayaekanayake96@gmail.com

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1. INTRODUCTION
One of the most colorful and demanding problems in the history of operations research
is the transportation problem (TP). The fundamental goal of these problems is to get an initial
basic feasible solution (IBFS) product from the source to the destination while keeping the
supply limit and demand in check. Therefore, many researchers are researching the initial
basic, feasible solution and the optimal solution to transportation problems. Hitchcock
(1941) developed the first method for calculating the IBFS. Subsequently, various methods
of calculating the IBFS were proposed. The most commonly used methods are the Northwest
corner method by Charmes and Cooper (1954–1955) (Mhlanga et al., 2014), the Least cost
method, and the Vogels approximation method, introduced by Reinfeld and Vogel (Mhlanga
et al., 2014) in 1958. In addition, various novel methods for calculating the IBFS using
various research methodologies have been established in recent years. Including "An
effective alternative new approach to solving transportation problems" and a “Modified ant
colony optimization algorithm for solving transportation problems” (Ekanayake et al.,
2020). an improved algorithm to solve transportation problems for an optimal solution, a
new approach to finding the initial solution to the unbalanced transportation problem (Shaikh
et al., 2018). In all the optimization mentioned above, the focus is on cost minimization.
However, in a particular case of transportation problems, various algorithms have
been created based on time. For example, Małachowski et al. (2019) published "Application
of the transport problem from the criterion of time to optimize supply network with
production fast running" 2019. "Algorithmic approach to calculate the minimum time
shipment of a transportation problem" published in 2013 (Ullah et al., 2013) and "Problem
of modeling road transport" (Ziółkowski & Lęgas, 2019) can be pointed out. Meanwhile, the
multi-objective transportation problem can be mentioned as a problem that is solved by
considering several things like time and cost. Related articles have been published by Kaur
et al. (2018), Nomani et al. (2017), and Bharathi and Vijayalakshmi (2016).
All the articles mentioned above are exceptional cases of transportation problems.
Here we will study transportation problems with restricted roads, a particular type of
problem that could be more popular among researchers. it is called "prohibited route
transportation problems." Some specific routes may not be accessible due to issues such as
construction projects, poor road conditions, strikes, unforeseen disasters, and traffic laws.
Such problems belong to this category, and in the case of the transportation problem,
transportation on forbidden routes incurs a substantial cost (M or [infinity (∞)]). Therefore,
this new approach tries to derive IBFS so that these paths are not included in the optimal
solution to drive these problems. Examples of such reports are "problems with prohibited
routes." published by Ekanayake et al. (2022), "an improved ant colony algorithm to solve
prohibited transportation problems (Ekanayake et al., 2022)" published by Prah et al. (2022);
and "A 2-phase method for solving transportation" published (Prah et al., 2022).
Also, Currin (1986) has made similar observations. In this way, this research paper
aims to present a new algorithm to obtain a basic feasible solution for transportation
problems with restricted roads. The study aims to provide a preliminary solution so that the
most expensive route is not included in the calculation of the relevant answer, and whether
that solution is optimal or close to it is the aim of the study. For this, an improved Vogel's
approximation method is used, and satisfactory solutions are expected. It is not appropriate
to use standard methods such as the "Northwest corner method," "least cost method," and

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"Vogel's approximation method," used in solving general transportation problems, to solve
transportation problems with restricted routes.
In some cases, the costs of prohibited roads may be included in the essential solutions
obtained. Then the corresponding total transportation cost will be high. However, the
primary objective of a transportation problem is to construct an algorithm that minimizes
transportation costs. Where the value obtained as the initial solution is its minimum cost, it
is known as the optimal solution. We presented a new algorithm to find the immediate
solution to the prohibited road transport problem. Since the traditional Vogel method is
unsuitable, it has been improved, and the related problems have been adjusted accordingly.
Mathematical problems indicate the primary solutions obtained thereby. It has been shown
by the transportation problems that the cost values obtained by this method do not include
the cost of the prohibited route and provide the optimal solution or a solution close to it.

2. METHODS
Generally, express the transportation problem in a mathematical table as shown
below since the problem is simple to solve (Jude et al., 2016).
Table 1 General representation of transportation tableau
To
destination→

D1

…

D2

Dn

Supply
𝑎𝑖

↓From source
S1

C11

C12

…

C1n

X12
C22

…

X1n
C2n

𝑎1

X11
S2

C21
X21

⁞
Sm

⁞
Cm1
Xm1

Demand
𝑏𝑗

X22
⁞
Cm2

⁞
…

Xm2
𝑏1

𝑎2
X2n

⁞
Cmn
Xmn

𝑏2

…

𝑏𝑛

⁞
𝑎𝑚
m

i=1

•
•
•
•

n

∑ a i = ∑ bj
j=1

Where; "m" implies the quantity of sources S1, S2,.. Sm) with supply capabilities (a1,
a2,.. am).
Source - It is the location where commodities are placed.
"n" implies the quantity of destinations (D1, D2,.., Dn) with the capabilities of
demand (b1, b2,.., bn.)
Destinations - It is the location where commodities are transported.

Take into account that Cij (here particular case in transportation problem, some Cij
values are infinite.) is the cost of transporting a unit from the ith source to the jth destination,
and Xij indicates the number of units transported from the source (i) to the destination (j).
Given the following formula, the transportation problem can be represented mathematically
as a linear programming problem.

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Objective function;
𝑛
Minimize∑𝑚
𝑖=1 ∑𝑗=1 𝑋𝑖𝑗 𝐶𝑖𝑗

S.t; ∑𝑛𝑗=1 𝑋𝑖𝑗 = 𝑎𝑖 ,𝑖 = 1,2, … , 𝑚
∑𝑚
𝑖=1 𝑋𝑖𝑗 = 𝑏𝑗 , 𝑗 = 1,2, … , 𝑛 and
𝑋𝑖𝑗 ≥ 0 for all 𝑖 = 1,2, … , 𝑚 , 𝑗 = 1,2, … , 𝑛
One of the main requirements is to be a balanced transportation problem. i.e.,
m

n

∑ a i = ∑ bj
i=1

j=1

When a problem is unbalanced (total demand and total supply are not equal), the problem
should be balanced by properly adding a dummy column or row before determining the
solution. After solving IBFS and determining whether it is the optimum solution: 1,) IBFS
are those that satisfy the transportation problem and also have a total number of allocations
equal to (m + n-1); 2) When the total cost of transportation is the lowest, it is an "optimum
solution." We use the modified Vogel’s approximation approach in this study to find the
most basic, feasible solutions to the problems. Before introducing the method, we need to
study some assumptions and theoretical information; 3) Vogel's approximation method
(VAM) (Balakrishnan, 1990).
One of the most popular methods is Vogel's approximation method, which can identify
the best initial feasible solution to the transportation problem. The most important steps to
solving problems by that method can be listed as follows (Priya & Maheswari, 2022): Step.1)
First, the penalties for each row and column in the transportation table will be determined
by subtracting the most negligible unit cost from the smallest unit cost; Step.2) Next, select
the column or row with the greatest penalty value. Select the cell with the lowest unit cost
that is a part of that column or row. After completing as much of the supply or demand for
this cell as possible, remove the row or column. If both the row and the column are satisfied,
step cross over the line; Step.3) Repeat the previous process until all supplies and demands
are met. Complete the procedure, determine the shipping cost, and ensure that all supply and
demand are satisfied.
Vogel's method mentioned above has been modified to provide solutions for these
particular problems. The related numerical problems were obtained through various research
papers and reference books. The aim is to test this improved Vogel method. Also, the steps
of the improved Vogel's method can be shown as follows:
2.1. Proposed Algorithm
The steps consist of: Step.1) First, whether the transport problem is balanced or
unbalanced should be checked. If the problem is unbalanced, it could be balanced by using
a suitable dummy column or row; Step.2) Then calculate the reciprocal of the values in all
the cells of the transportation problem table. (Do not calculate the reciprocal values in the
dummy column or dummy row; Step.3) Then, for each column and row, subtract the highest
value in that row or column from the next highest value to get the penalty value. Select the
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row or column with the highest value among those penalty values. After assigning
assignments to the highest common value in the respective column or row, cross the row or
column that satisfies supply or demand; Step. 4) Follow the above steps to satisfy all demand
and supply in the system. In an unbalanced transportation problem, after checking all suitable
cells to give an assignment in the considered column or row, if the corresponding supply or
demand values are not satisfied, finally select a dummy cell and give the assignment; Step
5) Finally, each assigned value is assigned to the original cost values of the corresponding
transportation table, and the problem's initial solution is computed.
In this research, the relevant calculations are made subject to the following
assumptions:
1) The maximum unit cost of the cell with the Prohibited path is said to be infinite (∞).
2) Dividing by one assumes that the value of infinity is equal to zero (Ufuoma, 2020).
How to obtain solutions using the above-mentioned improved Vogel's method is
presented in detail in the following section. It is shown in each step. It is also essential to
compare the solutions obtained by the proposed method with those obtained by the existing
methods. Accordingly, the analytical comparison is mentioned in Tables 27, 28, and 47. Also,
analytical graphs related to it are shown in this section.

3. RESULTS AND DISCUSSION
This section shows some of the transportation problems solved through the aftermentioned algorithms in detail. It is intended to show the effectiveness of this newly introduced
method according to the result obtained.
Problem 1 (Prah et al., 2022)
Table 2 Balance prohibited route transportation problem table
D1
∞
11
12
180

S1
S2
S3
demand

D2
14
10
8
320

D3
12
6
15
120

D4
17
10
7
380

supply
250
350
400

Table 3 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0
0.091
0.083
180

S1
S2
S3
demand

D2
0.071
0.100
0.125
320

D3
0.083
0.167
0.067
120

D4
0.059
0.100
0.143
380

supply
250
350
400

Table 4 Step 03 & step 04 – Table showing the steps used in the improved Vogel’s method
D1

D2

D3

D4

supply

S1

0

0.071 (250)3

0.083

0.059

S2

0.091 (180)6

0.100 (50)5

0.167 (120)1

0.100

S3

0.083

0.125 (20)4

0.067

0.143 (380)2

250
(0)
350
(230)
(180)
400
(20)
(0)

Penalty
value
0.012 0.012 0.071 ----0.067 0

------ -------

0.009 0.009 0.009 0.091

0.018 0.018 0.042

0.042 ------

-----

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demand

180

Penalty
value

0.008
0.008
0.008
0.008
0.091
0.091

320 (70) (50)
(0)
0.025
0.025
0.025
0.025
0.100
------

120 (0)

380 (0)

0.084
-----------------------------

0.043
0.043
-------------------------

Table 5 Step 05 - The corresponding assignment values were assigned to the appropriate cells
of the transportation problem, and the corresponding total cost was calculated.
D1
∞
11 (180)
12
180

S1
S2
S3
demand

D2
14 (250)
10 (50)
8 (20)
320

D3
12
6 (120)
15
120

D4
17
10
7 (180)
380

supply
250
350
400

Transportation cost =(11 × 180) + (14 × 250) + (10 × 50) + (8 × 20) + (6 × 120) +
(7 × 380) = 9520
Problem 2 (Prah et al., 2022)
Table 6 Balance prohibited route transportation problem Table.
D1
9
11
12
180

S1
S2
S3
demand

D2
14
10
8
320

D3
12
6
15
120

D4
17
∞
7
380

supply
250
350
400

Table 7 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0.111
0.091
0.083
180

S1
S2
S3
demand

D2
0.071
0.100
0.125
320

D3
0.083
0.167
0.067
120

D4
0.059
0
0.143
380

supply
250
350
400

Table 8 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method
D1

D2

D3

D4

supply

S1

0.111 (180)5

0.071 (70)6

0.083

0.059

S2

0.091

0.100 (230)4

0.167 (120)2

0

S3

0.083

0.125 (20)3

0.067

0.143 (380)1

250 (70)
(0)
350 (230)
(0)
400 (20)
(0)

demand

180 (0)

120 (0)

380 (0)

Penalty
value

0.020
0.020
0.020
0.020
0.111
------

320 (300)
(70) (0)
0.025
0.025
0.025
0.029
0.071
0.071

0.084
0.084
------------------

0.084
-----------------

Penalty
value
0.028 0.028 0.040 0.04 0.04 0.071
0.067 0.067 0.009 0.09 ----- ----0.018 0.042 0.042 ----

----- -----

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Table 9 Step 05 - The corresponding assignment values were assigned to the appropriate cells
of the transportation problem, and the corresponding total cost was calculated.
D1
9 (180)
11
12
180

S1
S2
S3
demand

D2
14 (70)
10 (230)
8 (20)
320

D3
12
6 (120)
15
120

D4
17
∞
7 (380)
380

supply
250
350
400

Transportation cost =(9 × 180) + (14 × 70) + (10 × 230) + (8 × 20) + (6 × 120) +
(7 × 380) = 8440
Problem 3 (Prah et al., 2022)
Table 10 Balance prohibited route transportation problem table
D1
9
11
12
180

S1
S2
S3
demand

D2
14
10
∞
320

D3
12
6
15
120

D4
17
10
7
380

supply
250
350
400

Table 11 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0.111
0.091
0.083
180

S1
S2
S3
demand

D2
0.071
0.100
0
320

D3
0.083
0.167
0.067
120

D4
0.059
0.1
0.143
380

supply
250
350
400

Table 12 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method

S1
S2
S3
demand
Penalty
value

D1

D2

D3

D4

supply

0.111
(160)4
0.091

0.071
(90)6
0.100
(230)5
0

0.083

0.059

0.167
(120)1
0.067

0.1

250 (90)
(0)
350 (230)
(0)
400 (20)
(0)

320 (90)
(0)
0.025
0.029
0.029
0.029
0.029
0.071

120 (0)

380 (0)

0.084
--------------------------

0.043
0.043
-------------------

0.083
(20)3
180 (160)
(0)
0.020
0.020
0.020
0.020
-----------

0.143 (380)2

Penalty
value
0.028 0.040 0.040 0.040 0.071 0.071
0.067 0

0.009 0.009 0.1

0.060 0.060 0.083 -----

----

-----------

Table 13 Step 05 - The corresponding assignment values were assigned to the appropriate cells
of the transportation problem, and the corresponding total cost was calculated.
S1
S2
S3
demand

D1
9 (160)
11
12 (20)
180

D2
14 (90)
10 (230)
∞
320

D3
12
6 (120)
15
120

D4
17
10
7 (380)
380

supply
250
350
400

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Transportation cost =(9 × 160) + (14 × 90) + (10 × 230) + (12 × 20) + (6 × 120) +
(7 × 380) = 8620
Problem 4 (Prah et al., 2022)
Table 14 Balance prohibited route transportation problem table
D1
9
11
12
180

S1
S2
S3
demand

D2
∞
10
8
320

D3
12
∞
15
120

D4
17
10
7
380

supply
250
350
400

Table 15 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0.111
0.091
0.083
180

S1
S2
S3
demand

D2
0
0.100
0.125
320

D3
0.083
0
0.067
120

D4
0.059
0.1
0.143
380

supply
250
350
400

Table 16 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method
D1

D2

D3

D4

supply

0

0.083 (120)4

0.059

S2

0.111
(130)5
0.091 (50)

0.100 (300)3

0

0.1

S3

0.083

0.125 (20)2

0.067

demand

180 (50) (0)

320 (300) (0)

120 (0)

0.143
(380)11
380 (0)

250 (130)
(0)
350 (50)
(0)
400 (20)
(0)

Penalty
value

0.020
0.020
0.020
0.020
0.020
0.091

0.025
0.025
0.1
-----------

0.016
0.016
0.083
0.083
------

0.043
---------------------

S1

Penalty
value
0.028 0.028 0.028 0.028
0

0.111 -----

0.009 0.009 0.009 0.091 0.091

0.018 0.042 -----

-----

------ ------

Table 17 Step 05 - The corresponding assignment values were assigned to the appropriate cells
of the transportation problem, and the corresponding total cost was calculated.
D1
9 (130)
11 (50)
12
180

S1
S2
S3
demand

D2
∞
10 (300)
8 (20)
320

D3
12 (120)
∞
15
120

D4
17
10
7 (380)
380

supply
250
350
400

Transportation cost =(9 × 130) + (11 × 50) + (10 × 300) + (8 × 20) + (12 × 120) +
(7 × 380) = 8980
Problem 05 (Ekanayake et al., 2022)
Table 18 Unbalance prohibited route transportation problem table
S1
S2
S3
S4
demand

D1
14
∞
∞
∞
20

D2
15
16
∞
∞
30

D3
16
17
15
∞
50

D4
17
18
16
17
40

Supply
40
50
30
50

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Table 19 Step 01 - Since the transportation problem is unbalanced, balance the problem using
a dummy column.

S1
S2
S3
S4
demand

D1

D2

D3

D4

14
∞
∞
∞
20

15
16
∞
∞
30

16
17
15
∞
50

17
18
16
17
40

Dummy
column
0
0
0
0
30

Supply
40
50
30
50

Table 20 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell

S1
S2
S3
S4
demand

D1

D2

D3

D4

0.071
0
0
0
20

0.067
0.063
0
0
30

0.063
0.059
0.067

0.059
0.056
0.063
0.059
40

50

Dummy
column
0
0
0
0
30

Supply
40
50
30
50

Table 21 Step 03 & 04 - Table showing the steps used in the improved Vogel’s method.
D1

D2
0.067 (20)5

S2

0.071
(20)1
0

S3
S4

0
0

demand

20
(0)
0.071
--------------------------

S1

Penalty
value

D3
0.063

Dummy
column
0

0.059
7

0.056

0 (20)

0
0

0.067 (30)4

0
0 (10)3

30 (10) (0)

50 (20) (0)

0.063
0.059
(40)2
40 (0)

0.004
0.004
0.004
0.004
0.004
0.063
------

0.004
0.004
0.004
0.004
0.004
0.059
0.059

0.004
0.004
-----------------------

0.063 (10)

6

D4

0.059 (20)

8

Supply
40 (20)
(0)
50 (40)
(20)
30 (0)
50 (10)
(0)

Penalty
value
0.004 0.004 0.004 0.004 0.004 ----- ----0.004 0.004 0.004 0.004 0.004 0.004 0.059
0.004 0.004 0.004 0.067 ----0.059 0.059 0.059 ------ -----

------ ---------- -----

30 (20)

Table 22 Step 05 - The corresponding assignment values were assigned to the appropriate
cells of the transportation problem, and the corresponding total cost was calculated.

S1
S2
S3
S4
demand

D1

D2

D3

D4

14 (20)
∞
∞
∞
20

15 (20)
16 (10)
∞
∞
30

16
17 (20)
15 (30)
∞
50

17
18
16
17 (40)
40

Dummy
column
0
0 (20)
0
0 (10)
30

Supply
40
50
30
50

Transportation cost =(14 × 20) + (15 × 20) + (16 × 10) + (17 × 20) + (15 × 30) +
(17 × 40) + (0 × 20) + (0 × 10) = 2210
Problem 06 (Ekanayake., 2022)
Table 23 Unbalance prohibited route transportation problem table
S1

D1
10

D2
13

D3
16

D4
19

supply
700

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PERFORMANCE OF THE BEST SOLUTION FOR THE PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN IMPROVED
VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake
∞
∞
∞
∞
∞
300

S2
S3
S4
S5
S6
demand

10
15
∞
∞
∞
700

13
18
15
20
∞
900

16
21
18
23
15
800

700
200
700
200
700

Table 24 Step 01 - Since the transportation problem is unbalanced, balance the problem using
a dummy column.

S1
S2
S3
S4
S5
S6
demand

D1

D2

D3

D4

10
∞
∞
∞
∞
∞
300

13
10
15
∞
∞
∞
700

16
13
18
15
20
∞
900

19
16
21
18
23
15
800

Dummy
column
0
0
0
0
0
0
500

supply
700
700
200
700
200
700

Table 25 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell

S1
S2
S3
S4
S5
S6
demand

D1

D2

D3

D4

0.100
0
0
0
0
0
300

0.077
0.100
0.067
0
0
0
700

0.063
0.077
0.056
0.067
0.050
0
900

0.053
0.063
0.048
0.056
0.043
0.067
800

Dummy
column
0
0
0
0
0
0
500

supply
700
700
200
700
200
700

Table 26 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method
D1

D2

D3

D4

S1

0.100
(300)1

0.077

0.063
(200)5

0.053
(100)6

S2

0

0.077

0.063

0

S3
S4

0
0

0.100
(700)3
0.067
0

700
(400)
(200)
(100) (0)
700 (0)

0.048
0.056

0 (200)
0

200 (0)
700 (0)

0.011 0.011 0.011 0.008 0.008 0.048
0.011 0.011 0.011 0.011 ------

S5
S6

0
0

0
0

0.056
0.067
(700)4
0.050
0

0 (200)
0

200 (0)
700 (0)

0.007 0.007 0.007 0.007 0.007 0.043
0.067 0.067 ------ ------ ------

demand

300 (0)

700 (0)

Penalty
value

0.100
---------------------

0.023
0.023
0.023
--------------

0.043
0.067
(700)2
800 (100)
(0)
0.004
0.004
0.007
0.003
0.005
0.005

900 (200)
(0)
0.010
0.010
0.010
0.004
0.007
------

Dummy
column
0 (100)

supply

Penalty
value
0.023 0.014 0.014 0.01 0.010 0.053

0.023 0.023 0.023 -----

------

500 (0)

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PERFORMANCE OF THE BEST SOLUTION FOR THE PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN IMPROVED
VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

Table 27 Step 05 - The corresponding assignment values were assigned to the appropriate cells
of the transportation problem, and the corresponding total cost was calculated.

S1
S2
S3
S4
S5
S6
demand

D1

D2

D3

D4

10 (300)
∞
∞
∞
∞
∞
300

13
10 (700)
15
∞
∞
∞
700

16 (200)
13
18
15 (700)
20
∞
900

19 (100)
16
21
18
23
15 (700)
800

Dummy
column
0 (100)
0
0 (200)
0
0 (200)
0
500

supply
700
700
200
700
200
700

Transportation cost =(10 × 300) + (10 × 700) + (16 × 200) + (15 × 700) +
(15 × 700) + (0 × 100) + (0 × 200) + (0 × 200) = 36100
The following table is intended to evaluate the performance of the proposed method. It
is compared with the new algorithm created to find the basic solution to this special
transportation problem. The proposed method has been checked with the initial solution as well
as the optimal solution. The proposed method has been tested for its suitability in finding the
initial solution to transportation problems.

Table 28 Comparison of the solution proposed method with the other research's algorithm
Problem no.
A 2-phase method for solving transportation
(Ackora-Prah et al., 2022)
Optimum solution (Ackora-Prah et al., 2022)
Proposed algorithm

01
9520

02
8440

03
8620

04
8980

9520
9520

8440
8440

8620
8620

8980
8980

Table 29 Comparison of the solution proposed method with the other research's algorithm
Problem
an improved ant colony algorithm to solve prohibited transportation
problems (Ekanayake., 2022)
Optimal solution
Proposed method

05
2210

06
36100

2210
2210

36100
36100

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PERFORMANCE OF THE BEST SOLUTION FOR THE PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN IMPROVED
VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

4

2

3

1

2
0

1
7500

8000

8500

9000

9500

10000

20000

30000

40000

10000

Proposed algorithm

Proposed method
Optimal solution

Optimum solution (Ackora-Prah et al., 2022)

an improved ant colony algorithm to solve prohibited
transportation problems (Ekanayake., 2022)

A 2-phase method for solving transportation
(Ackora-Prah et al., 2022)

Figure 1 Comparison graph exiting method
Figure 2 Comparison graph exiting
With proposed method with the proposed method
Problem 07 (Taha, 2011)
Table 30 Balance prohibited route transportation problem table
D1
∞
7
1
5

S1
S2
S3
demand

D2
3
4
8
6

D3
5
9
6
19

supply
4
7
19

Table 31 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0
0.143
1
5

S1
S2
S3
demand

D2
0.333
0.25
0.125
6

D3
0.2
0.111
0.167
19

supply
4
7
19

Table 32 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method
D1

D2

D3

supply

S1
S2
S3

0
0.143
1 (5)1

0.333
0.25 (6)2
0.125

0.2 (4)3
0.111 (1)5
0.167 (14)4

4 (0)
7 (1) (0)
19 (14)
(0)

demand
Penalty
value

5 (0)
0.857
---------------------

6 (0)
0.083
0.083
----------------

19 (15) (1) (0)
0.033
0.033
0.033
0.056
0.111

Penalty
value
0.133 0.133 0.200 ------ -----0.107 0.139 0.111 0.111 0.111
0.833 0.042 0.167 0.167 ------

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Table 33 Step 05 - The corresponding assignment values were assigned to the appropriate cells
of the transportation problem, and the corresponding total cost was calculated.
D1
∞
7
1 (5)
5

S1
S2
S3
demand

D2
3
4 (6)
8
6

D3
5 (4)
9 (1)
6 (14)
19

supply
4
7
19

Transportation cost =(1 × 5) + (4 × 6) + (5 × 4) + (9 × 1) + (6 × 14) = 142
Problem 8
Table 34 Balance prohibited route transportation problem table
D1
10
5
15
20
30

S1
S2
S3
S4
demand

D2
2
10
5
15
10

D3
3
15
14
13
20

D4
15
2
7
∞
25

D5
9
4
15
8
20

supply
25
30
20
30

Table 35 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0.100
0.2
0.067
0.050
30

S1
S2
S3
S4
demand

D2
0.500
0.100
0.200
0.067
10

D3
0.333
0.067
0.071
0.077
20

D4
0.067
0.500
0.143
0
25

D5
0.111
0.250
0.067
0.125
20

supply
25
30
20
30

Table 36 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method
D1

D2

D3

D4

D5

supply

S1

0.100

0.111

0.2 (5)4

0.333
(15)3
0.067

0.067

S2

0.500
(10)2
0.100

0.250

S3

0.067
(20)7
0.050
(5)8
30 (25)
(5) (0)
0.100
0.100
0.100
0.133
0.017
0.017
0.017
0.050

0.200

0.071

0.500
(25)1
0.143

25 (15)
(0)
30 (5)
(0)
20 (0)

0.067

0

10 (0)

0.077
(5)6
20 (0)

25 (0)

0.125
(20)5
20 (0)

0.300
0.300
-------------------------

0.256
0.250
0.250
0.006
0.006
0.006
------

0.357
-------------------------------

0.125
0.125
0.125
0.125
0.058
-----------

S4
demand
Penalty
value

0.067

30 (10)
(5) (0)

Penalty
value
0.167 0.167 0.222 ----- ------ ------ ----0.250 0.050 0.050 0.050 ------ ----- ----0.057 0.129 0.004 0.004 0.004 0.004 0.067
0.048 0.048 0.048 0.048 0.048 0.027 0.050
0.050

---------

Table 37 Step 05 - The corresponding assignment values were assigned to the appropriate
cells of the transportation problem, and the corresponding total cost was calculated.
S1
S2
S3
S4
demand

D1
10
5 (5)
15 (20)
20 (5)
30

D2
2 (10)
10
5
15
10

D3
3 (15)
15
14
13 (5)
20

D4
15
2 (25)
7
∞
25

D5
9
4
15
8 (20)
20

supply
25
30
20
30

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VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

Transportation cost =(5 × 5) + (15 × 20) + (20 × 5) + (2 × 10) + (3 × 15) +
(13 × 5) + (2 × 25) + (8 × 20) = 765

Problem 09 (Srirangacharyulu & Srinivasan, 2010)
Table 38 Unbalance prohibited route transportation problem table
D1
30
30
∞
∞
3

S1
S2
S3
S4
demand

D2
40
∞
40
∞
4

D3
25
30
40
25
5

supply
3
5
4
12

Table 39 Step 01- Since the transportation problem is unbalanced, balance the problem
using a dummy column.
D1
30
30
∞
∞
3

S1
S2
S3
S4
demand

D2
40
∞
40
∞
4

D3
25
30
40
25
5

Dummy column
0
0
0
0
12

supply
3
5
4
12

Table 40 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0.033
0.033
0
0
3

S1
S2
S3
S4
demand

D2
0.025
0
0.025
0
4

D3
0.040
0.033
0.025
0.040
5

Dummy column
0
0
0
0
12

supply
3
5
4
12

Table 41 Step 03 & step 04 - Table showing the steps used in the improved Vogel’s method

S1
S2
S3
S4
demand
Penalty
value

D1

D2

D3

0.033
0.033 (3)2
0
0
3 (0)
0.000
0.000
---------

0.025 (3)3
0
0.025 (1)4
0
4 (1)
0.000
0.000
0.025
0.025

0.040
0.033
0.025
0.040 (5)1
5 (0)
0.000
-------------

Dummy
column
0
0 (2)
0 (3)
0 (7)
12

supply
3 (0)
5 (2)
4 (3)
12 (7)

Penalty
value
0.007 0.008 0.025 ---0.000 0.033 0.000 ---0.000 0.025 0.025 0.025
0.040 0.000 0.000 0.000

Table 42 Step 05 - The corresponding assignment values were assigned to the appropriate
cells of the transportation problem, and the corresponding total cost was calculated.
S1
S2
S3
S4
demand

D1
30
30 (3)
∞
∞
3

D2
40 (3)
∞
40 (1)
∞
4

D3
25
30
40
25 (5)
5

Dummy column
0
0 (2)
0 (3)
0 (7)
12

supply
3
5
4
12

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PERFORMANCE OF THE BEST SOLUTION FOR THE PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN IMPROVED
VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

Transportation cost =(30 × 3) + (40 × 3) + (40 × 1) + (25 × 5) + (0 × 2) + (0 × 3) +
(0 × 7) = 375
Problem 10 (Srirangacharyulu & Srinivasan, 2010)
Table 43 Unbalance prohibited route transportation problem table
D1
3
4
3
50

S1
S2
S3
demand

D2
6
∞
4
30

D3
5
5
4
40

D4
2
5
4
20

supply
40
50
30

Table 44 Step 01 - Since the transportation problem is unbalanced, balance the problem
using a dummy row.
D1
3
4
3
0
50

S1
S2
S3
Dummy row
demand

D2
6
∞
4
0
30

D3
5
5
4
0
40

D4
2
5
4
0
20

supply
40
50
30
20

Table 45 Step 02 - Table of Calculating the reciprocal value of unit costs for each cell
D1
0.333
0.250
0.333
0
50

S1
S2
S3
Dummy row
demand

D2
0.167
0
0.250
0
30

D3
0.200
0.200
0.250
0
40

D4
0.500
0.200
0.250
0
20

supply
40
50
30
20

Table 46 Step 03 & 04 - Table showing the steps used in the improved Vogel’s method.

S1
S2
S3
Dummy
row
demand
Penalty
value

D1

D2

D3

D4

supply

0.333
(20)2
0.250
(30)4
0.333

0.167

0.200

0.500 (20)1

0

0.200 (20)5

0.200

0.250
(30)3
0

0.250

0.250

40 (20)
(0)
50 (20)
(0)
30 (0)

0 (20)

0

20 (0)

30 (0)

40 (20) (0)

20 (0)

0.083
0.083
0.250
---------

0.050
0.050
0.050
0.200
0.200

0.250
----------------

0
50 (30)
(0)
0.000
0.000
0.083
0.250
------

Penalty
value
0.167 0.133 -----

------

-----

0.050 0.050 0.050 0.050 0.200
0.083 0.083 0.083 ----- ------

Table 47 Step 05 - The corresponding assignment values were assigned to the appropriate
cells of the transportation problem, and the corresponding total cost was calculated.
S1
S2
S3
Dummy row
demand

D1
3 (20)
4 (30)
3
0
50

D2
6
∞
4 (30)
0
30

D3
5
5 (20)
4
0 (20)
40

D4
2 (20)
5
4
0
20

supply
40
50
30
20

Indonesian Journal of Applied Research (IJAR), volume 3 issue 3 – December 2022

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PERFORMANCE OF THE BEST SOLUTION FOR THE PROHIBITED ROUTE TRANSPORTATION PROBLEM BY AN IMPROVED
VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

Transportation cost =(3 × 20) + (4 × 30) + (5 × 20) + (2 × 20) + (0 × 20) = 440

The following table shows the compression of the initial solutions obtained by the
traditional "VAM method" and the newly proposed "improved VAM method" used in
constructing this new algorithm. Also, the initial solution obtained is compared with the
optimal solution.
Table 48 Comparison of the solution proposed method with the other research's algorithm
Problem
VAM method
improved VAM
method
Optimal
solution

07
142
142

08
∞
765

09
∞
375

10
440
440

142

665

375

440

()1, ()2, ()3 represent – the order of the allocations in assignment values for each cell.

4. CONCLUSION
This paper proposes a method to find the basic solution through a new approach to the
prohibited route transportation problem, a special type of transportation problem. The
traditional Vogels method can be pointed out as a method that can successfully obtain
essential solutions to transportation problems. However, instead of the traditional Vogel
method, this research aims to improve it differently and present it to suit the particular
transportation problem. A new method has been proposed to improve the method and is
suitable for the prohibited road transport problem. This study found basic solutions for
balanced and unbalanced transportation problems using the improved Vogels method. We
compared the primary solution to the transportation problem obtained by the proposed
method with the essential solutions obtained by other research methods and checked its
efficiency. Consequently, the solutions obtained from the problems solved by the proposed
method were compared with those obtained from the existing methods. It was concluded that
the solutions of the proposed method give similar or more accurate answers than those of
other methods. Moreover, many of those solutions can be considered the best. Furthermore,
through this proposed method, it is possible to check the basic solution of special
transportation problem models. It can confirm the accuracy of this proposed method.
Accordingly, this proposed method can be concluded as an approach that can obtain highbasic solutions efficiently to the prohibited route transportation problem.

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VOGEL'S APPROXIMATION METHOD – Ekanayake & Ekanayake

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