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673
MATHEMATICAL MODEL OF THE TRANSPORTATION PROBLEM AND FINDING
THE OPTIMAL SOLUTION
Mamatova Zilolaxon Xabibulloxonovna
Associal Professor at Fergana State University,
Doctor of Philosophy (PhD) in Pedagogical Sciences
E-mail:
Xafizova Muhlisa Rahmatjon kizi
Student of Fergana State University
E-mail:
Annotation:
This article is dedicated to the "Transportation Problem"
,
one of the key areas of
operations research and optimal control. It explores the theoretical foundations, mathematical
models, and practical applications of this problem. The transportation problem is considered an
optimal resource allocation problem and is widely applied in fields such as economics, logistics,
and supply chain management. The article provides a detailed explanation of the main solution
methods for the transportation problem, including the “Northwest Corner Method”, “Minimum
Cost Method”, and “Potential Method”
.
The principles for determining the optimal solution
using these methods are analyzed. Additionally, practical examples of the transportation
problem’s application in real-world economics and logistics are presented, illustrating the
efficiency and applicability of these methods. The research findings demonstrate that
understanding and solving the transportation problem can help optimize logistics systems,
minimize transportation costs, and improve resource utilization efficiency. This article serves as
an important theoretical and practical guide for researchers, economists, and logistics specialists
working with transportation problems.
Keywords:
transportation problem, operations research, optimal control, optimal allocation,
logistics, efficient resource allocation, Northwest Corner Method, Minimum Cost Method,
Potential Method, mathematical modeling, freight transportation problem, optimal solution,
transportation cost minimization.
The general formulation of the transportation problem is as follows: A
1
, A
2
, A
3
, … A
m
Am are supply points dealing with the same type of product, where A
i
– represents the amount of
product at point a
i
and the quantity is denoted as ai units. The task is to distribute these products
to the consumption pointsB
1
, B
2
, B
3
, … B
n.
At each consumption point B
j
– the required amount
of product to be delivered is b
j
units.
Let c
ij
be the cost in soums to transport one unit of product from supply point A
i
to
consumption point B
j
. The task is to distribute the products from the supply points to the
consumers with the minimum total cost. To solve this problem, the amount of product to be
transported from A
i
to B
j
, denoted as x
ij
, is determined. This leads to the construction of the
mathematical model for the problem,
1
1
min
m
n
ij ij
i
j
c x
=
=
®
(1)
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674
1
n
ij
i
j
x
a
=
=
,
1,2,..., ,
i
m
=
(2)
1
m
ij
j
i
x
b
=
=
,
1,2,..., ,
j
n
=
(3)
0
ij
x
,
1,2,..., ,
i
m
=
1,2,..., ,
j
n
=
(4)
Here, (2) represents the constraint on the amount of product to be taken by each supplier,
and (3) represents the constraint on the amount of product to be delivered to each consumer. (1)
is called the objective function, which determines the total transportation cost. Therefore, (1)
indicates that we need to minimize this total cost.
(2), (3), and (4) represent the set of constraints that define the feasible solutions for the
transportation problem. The set of plans corresponding to this feasible solution is called the plan
set. Each xij vector in the plan set is referred to as a transportation plan for the corresponding
transportation problem.
Thus, the formulation of the transportation problem is as follows: a plan must be selected from
the plan set such that it minimizes the value of the objective function (1).
Definition 1:
If equality
1
1
m
n
i
j
i
j
a
b
=
=
=
is satisfied, the corresponding transportation problem is
called a closed transportation problem. Otherwise, it is called an open transportation problem.
Theorem
1:
Any
closed
transportation
problem
has
a
solution.
If the transportation problem is open, it can be transformed into a closed transportation problem
and solved. There are several methods for solving transportation problems, and we can gain
detailed information about these methods by solving the problem presented below.
Problem:
A large logistics company provides product delivery services in Uzbekistan. The
company has three warehouses located in the cities of Tashkent, Samarkand, and Bukhara.
Products need to be delivered from these warehouses to three cities – Fergana, Karshi, and
Nukus. The objective is to minimize the total transportation costs.
The shipping capabilities of the warehouses (in units):
Tashkent: 200 units
Samarkand: 300 units
Bukhara: 500 units
Demand quantities (in units):
Fergana: 250 units
Karshi: 350 units
Nukus: 400 units
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Solution of the problem:
Consumer
Supplier
Andijan
Bukhara
Karshi
Nukus
Available
product
quantites
B
1
B
2
B
3
B
4
Tashkent
A
1
12$
18$
25$
20$
200
Samarkand
A
2
10$
14$
15$
22$
150
Fergana
A
3
8$
16$
24$
30$
250
Demands
180
120
170
130
600
North-West Corner Method:
This method starts by satisfying the demand of consumer b
1
with
the available product from supplier A
1
. If the demand is satisfied (for this to happen
1
1
a b
the
quantity should be sufficient), then the remaining product from A
1
is used to satisfy the demand
of consumer B
2
, and so on. If supplier A
1
cannot fully satisfy the demand of consumer B
1
, then
the product from supplier A
2
is used, and with its help, the demand of B
1
is either fully or partially
satisfied. Since a closed transportation problem is being considered, this process continues until
all the available products from the suppliers are fully distributed to the consumers. In each step
of this process, either the product of the corresponding supplier is fully distributed, or the
demand of the corresponding consumer is fully satisfied.
This method starts by satisfying the demand of consumer b
1
with the available product from
supplier A
1
. If the demand is satisfied (for this to happen,
1
1
a b
must hold), then the remaining
product from A
1
is used to satisfy the demand of consumer B
2
, and so on. If supplier A
1
cannot
fully satisfy the demand of consumer B
1
, then the product from supplier A
2
is used, and with its
help, the demand of B
1
is either fully or partially satisfied. Since a closed transportation problem
is being considered, this process continues until all the available products from the suppliers are
fully distributed to the consumers. In each step of this process, either the product of the
corresponding supplier is fully distributed, or the demand of the corresponding consumer is fully
satisfied.
Consummer
Supplier
Andijan
Bukhara
Karshi
Nukus
Available
product
quantites
B
1
B
2
B
3
B
4
Tashkent
A
1
12
180
18
20
25
0
20
0
200
Samarkand
A
2
10
0
14
100
15
50
22
0
150
Fergana
A
3
8
0
16
0
24
120
30
130
250
Demands
180
120
170
130
600
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676
If both situations occur simultaneously — that is, if the available products of several
suppliers are fully distributed while the corresponding consumers' demands are also fully
satisfied — then a zero is written in the corresponding right or lower cell. By assigning such
zeros in some cells, a special initial feasible solution is constructed. This method will be
explained through the following example: Let A
1
, A
2,
A
3
be the suppliers with available product
quantities of a
1
=200, a
2
=150, a
3
=250 units, respectively. The consumers B
1
, B
2
, B
3
, B
4
have
product demands of b
1
=180, b
2
=120, b
3
=170, b
4
=130 units, respectively.
The cost cij of transporting one unit of product from supplier A
i
to consumer B
j
is given by the
following values: c
11
=12$, c
12
=18$, c
13
=25$, c
14
=20$, c
21
=10$, c
22
=14$, c
23
=15$, c
24
=22$,
c
31
=8$, c
32
=16$, c
33
=24$, c
34
=30$
This problem was placed into our first table. Now, in the second table, we will explain how to
work
using
the
"North-West
Corner
Method."
The available product quantity at supplier A
1
is 200 units. We begin by satisfying the demand of
the
consumer
with
the
smallest
index,
B
1
,
using
these
products.
Out of the 200 units, 180 units are allocated to B
1
, and the remaining 20 units are given to B
2
.
Then, the unmet 100 units of demand for B
2
are fulfilled using the product from supplier A2. The
remaining quantity of supplier A
2
, which is 150 – 100 = 50 units, is allocated to satisfy part of the
demand
of
B
3
.
The remaining demand of B
3
, 170 – 50 = 120 units, is satisfied by supplier A
3
.
The remaining product from A
3
, 250 – 120 = 130 units, is sufficient to meet the demand of B
4
.
These values are recorded in the second table. From the table, it is clear that, based on this
process,
we
have
obtained
an
initial
basic
feasible
solution:
x
11
=180, x
12
=20, x
13
=0, x
14
=0, x
21
=0, x
22
=100, x
23
=50, x
24
=0, x
31
=0, x
32
=0, x
33
=120, x
34
=130. To
determine the total transportation cost corresponding to this initial solution, we use the following
formula:
11
11
12
12
33
33
34
34
1
1
...
m
n
ij ij
i
j
c x
c x
c x
c x
c x
=
=
=
+
+ +
+
g
g
g
g
If we denote this expression by
Z
, then the total transportation cost for our transportation
problem will be equal to the following value:
Z=180∙12+20∙18+0∙25+0∙20+0∙10+100∙14+50∙15+0∙22+0∙8+0∙16+120∙24+130∙30=2160+360+0
+0+0+1400+750+0+0+0+2880+3900=11450$
Thus, for the given example, the total transportation cost corresponding to the obtained initial
solution amounts to
$11,450
.
Conclusion
This article discusses the transportation problem, where the transportation costs were
calculated
using
the
North-West
Corner
Method.
The main focus was not on delivering products with minimal cost, but rather on supplying
consumers sequentially according to their demands, without prioritizing cost minimization. In
this approach, suppliers deliver products to consumers step-by-step, satisfying the demands in
order.
References
1. M. To‘xtasinov,
Process Research
, Tashkent, 2017 – "Transportation Problem".
https://ijmri.de/index.php/jmsi
volume 4, issue 3, 2025
677
2. Toshpulatov Sh.,
Process Research
, Tashkent, 2019 – "Classical Models of the
Transportation Problem, Vogel’s Method, MODI Method, and Algorithm Explanations".
3. Eshchanov R., Ergashev M.,
Modeling of Economic Processes
, Tashkent, 2020 – "Economic
Analysis of Transportation and Logistics Issues".
4. Ismoilov A.K.,
Process Research and Operations Modeling
, 2021 – "Practical Problems and
Solutions Using Excel".
