ISSN: 3030-3931, Impact factor: 7,241
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Original article
760
TRANSPORT MASALASINI BOSHLANG‘ICH REJALASHTIRISH VA OPTIMAL
YECHIMGA KELTIRISH
Mamatova Zilolaxon Xabibulloxonovna
Farg‘ona davlat universiteti dotsenti,
pedagogika fanlari bo‘yicha falsafa doktori (PhD)
E-mail: mamatova.zilolakhon@gmail.com
ORCID ID
Raimova Shohida Ravshanbek qizi
Farg‘ona davlat universiteti talabasi
E-mail :raimovashohida2@gmail.com
Annotatsiya:
Ushbu maqolada transport masalasi tushunchasi, uning matematik modeli
hamda optimal yechimlarni topish usullari — Shimol-gʻarb burchagi usuli, Vogel taxminiy usuli
(VAM), va MODI (Modified Distribution) usullari yoritilgan. Har bir usulning algoritmi misollar
bilan ko‘rsatilib, amaliy ahamiyati baholanadi.
Аннотация:
В статье рассматривается концепция транспортной задачи, ее
математическая модель и методы поиска оптимальных решений — метод северо-
западного угла, метод аппроксимации Фогеля (VAM) и методы MODI
(модифицированного распределения). Алгоритм каждого метода проиллюстрирован
примерами и оценена его практическая значимость.
Annotation:
This article discusses the concept of the transportation problem, its
mathematical model, and methods for finding optimal solutions — the Northwest Corner
Method, the Vogel Approximation Method (VAM), and the MODI (Modified Distribution)
methods. The algorithm of each method is illustrated with examples and its practical
significance is assessed.
Kalit so‘zlar:
Transport masalasini matematik modeli, Boshlang‘ich yechim topish
usullari,optimal yechim topish.
Ключевые слова:
Математическая модель транспортной задачи, Методы
нахождения начального решения, Нахождение оптимального решения.
Keywords:
Mathematical model of the transportation problem, Methods for finding an
initial solution, finding an optimal solution.
Kirish
. Transport masalasi – bu chiziqli dasturlashning keng qo‘llaniladigan masalalaridan
biri bo‘lib, maqsad – berilgan manbalar va talab nuqtalari o‘rtasida tashish xarajatlarini
ISSN: 3030-3931, Impact factor: 7,241
Volume 7, issue 1, Aprel 2025
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Original article
761
minimallashtirishdan iborat. Ushbu masala ishlab chiqarish, logistika va resurslar taqsimoti
sohalarida muhim ahamiyatga ega.
Transport masalasining matematik modeli
Berilgan:
- m ta yetkazib beruvchi (manbalar), har biri
�
�
birlik mahsulotga ega.
- n ta iste’molchi (talab nuqtalari), har biri
�
�
birlik mahsulot talab qiladi.
- Har bir i-manbadan j-iste’molchiga yetkazish xarajati
�
��
.
Maqsad:
Tashish xarajatlari yig‘indisini minimallashtiruvchi reja
�
��
ni topish:
Min Z =
�=1
�
�=1
�
�
��
× �
��
cheklovlar:
�
��
= �
�
�
��
= �
�
�
��
≥
0
Yechish usullari: Boshlang‘ich yechim topish usullari
a)Shimol-gʻarb burchak usuli:
Bu usulda jadvalning chap yuqori burchagidan boshlab talab va taklif bo‘yicha ketma-ket
taqsimlash amalga oshiriladi.
b) Vogel taxminiy usuli (VAM):
Har bir satr va ustundagi ikki eng kichik qiymat orasidagi farq hisoblanadi. Eng katta farqli
yo‘nalish tanlanadi. Optimalga yaqin boshlang‘ich yechim beradi.
Transport masalasining optimal usullarini topish
Optimal yechim topish: MODI usuli
MODI usuli boshlang‘ich asosiy yechimni optimallashtirish uchun qo‘llaniladi:
1. Potensiallar
�
�
,
�
�
ni topish.
2. Bo‘sh kataklar uchun
∆
��
=
�
��
- (
�
�
+
�
�
) hisoblash.
ISSN: 3030-3931, Impact factor: 7,241
Volume 7, issue 1, Aprel 2025
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Original article
762
3. Agar barcha
∆
��
≥
0 bo‘lsa – yechim optimal. Aks holda, eng salbiy
∆
��
bo‘yicha yangi bazisga
o‘tish.
Amaliy misol
: 6 ta ta’minotchi va 7 ta iste’molchi mavjud bo‘lgan quyidagi transport
masalasini ko‘rib chiqamiz. Berilgan transport jadvali (qiymatlar – xarajatlar).
a)
Shimoliy-g‘arb burchak usulida yechilishini ko‘ramiz:
1-jadval
Bu yerdan biz
�
��
ni qiymatlarini va Z ni topamiz.
�
��
=
min(a,b) deb olamiz.
ISSN: 3030-3931, Impact factor: 7,241
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Original article
763
2-jadval
Shimol-g‘arb usuli asosida boshlang‘ich taqsimot
:
�
��
=
min(40,100)=40
�
11
= 40 �
12
= 0
�
13
=0
�
14
= 0 �
15
= 0
�
16
= 0
�
17
= 0
�
21
= 60 �
22
= 10 �
23
=0
�
24
= 0 �
25
= 0
�
26
= 0
�
27
= 0
�
31
= 0
�
32
= 70 �
33
=30
�
34
= 0
�
35
= 0 �
36
= 0
�
37
= 0
�
41
= 0
�
42
= 0
�
43
=40
�
44
= 50 �
45
= 0
�
46
= 0
�
47
= 0
�
51
= 0
�
52
= 0
�
53
=0
�
54
= 50 �
55
= 0 �
56
= 0
�
57
= 0
�
61
= 0
�
62
= 0
�
63
=0
�
64
= 20 �
65
= 50 �
66
= 80 �
77
= 70
Xarajat
ni topish uchun:
Min Z =
�=1
�
�=1
�
�
��
× �
��
formuladan foydalanamiz.
�
��
����
barcha elementlarini
�
��
ning mos elementlariga ko‘paytirib qo‘shamiz.
�
1
=16
×
40+17
×
60+20
×
10+22
×
70+18
×
30+13
×
40+22
×
50+20
×
50+23
×
20+10
×
50+22
×
80+15
×
70=
10330
$
ISSN: 3030-3931, Impact factor: 7,241
Volume 7, issue 1, Aprel 2025
https://worldlyjournals.com/index.php/Yangiizlanuvchi
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Original article
764
Amaliy misol
: 6 ta ta’minotchi va 7 ta iste’molchi mavjud bo‘lgan quyidagi transport
masalasini ko‘rib chiqamiz. Berilgan transport jadvali (qiymatlar – xarajatlar).
b)
Vogel taxminiy usuli (VAM) da yechilishini ko‘ramiz:
1-jadval
Minimal element usulini qo‘llash tartibi
:
1.
Xarajatlarni tekshirib chiqish:
Har bir manba uchun har bir manzilga transport xarajatlarini
taqqoslash.
2.
Eng kichik xarajatni topish:
Har bir manba va manzil juftligi bo‘yicha eng kichik transport
xarajatini aniqlash.
3.
Manba va manzillarni to‘ldirish:
Eng kichik xarajat bo‘lgan elementni tanlash va unga
to‘g‘ri keladigan miqdorni belgilab, resurs va talabni yangilash.
4.
Yangi qiymatlarni hisoblash:
Har bir to‘ldirilgan elementdan keyin resurslar va talablarni
yangilab borish, va yana eng kichik xarajatni tanlash.
ISSN: 3030-3931, Impact factor: 7,241
Volume 7, issue 1, Aprel 2025
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Original article
765
2-jadval
Bu jadvalda
�
61
xarajat tanlab olindi.
Xarajat
ni topish uchun:
Min Z =
�=1
�
�=1
�
�
��
× �
��
formuladan foydalanamiz.
�
��
����
barcha elementlarini
�
��
ning mos elementlariga ko‘paytirib qo‘shamiz.
�
2
=9
×
40+4
×
70+18
×
10+14
×
10+21
×
10+9
×
70+4
×
80+3
×
10+15
×
50+2
×
100+7
×
60+10
×
40+22
×
20=4360$
Minimal element usuli transport masalalarida boshlang‘ich yechimni topishda samarali
yondashuvdir. Bu usulda xarajatlar bo‘yicha eng kichik qiymatlar tanlanib, resurslar va talablar
to‘ldiriladi. Bu metodni ishlatish, masalani tezda hal qilish imkonini beradi, ammo bu faqat
boshlang‘ich yechimni beradi, ya'ni u optimal yechimni ta'minlamasligi mumkin.
Umumiy xarajat
ni topish uchun:
�
���
=
�
1
− �
2
=10330
$ −
4360$=5970$
Minimal element usuli orqali xarajat miqdori 5970$ ga kamaytirildi.
Boshqa usullar orqali umumiy xarajatni yanada kamaytirish mumkin.
Xulosa
: Transport masalasi — iqtisodiy samaradorlikka katta ta’sir qiluvchi masala bo‘lib,
uni optimal hal qilish tashkilot xarajatlarini kamaytiradi. VAM boshlang‘ich reja uchun samarali
ISSN: 3030-3931, Impact factor: 7,241
Volume 7, issue 1, Aprel 2025
https://worldlyjournals.com/index.php/Yangiizlanuvchi
worldly knowledge
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Original article
766
bo‘lsa, MODI usuli esa optimal yechimni topishda asosiy vositadir. Ushbu usullarni
integratsiyalash transport va logistika tizimlarini samarali boshqarishga xizmat qiladi.
Foydalanilgan adabiyotlar
1.Taha H. A. Operations Research: An Introduction, 9th Edition, Pearson.
2.Hillier, F. S., Lieberman, G. J. Introduction to Operations Research, McGraw-Hill.
3.Mirobodiy I. Operatsion tadqiqotlar, Toshkent, 2020.
4.Chopra, S., & Meindl, P. (2013). Supply Chain Management: Strategy, Planning, and
Operation. Pearson.
