The transportation problem represents a particular type of linear programming problem used for allocating resources in an optimal way; it is a highly useful tool for managers and supply chain engineers for optimizing costs.
For clearly understand you can watch this video on my youtube channel
https://www.youtube.com/watch?v=5Ssnew58Yfc&t=2s
Transportation Problem In Linear ProgrammingMirza Tanzida
This work is an assignment on the course of 'Mathematics for Decision Making'. I think, it will provide some basic concept about transportation problem in linear programming.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
Transportation Problem In Linear ProgrammingMirza Tanzida
This work is an assignment on the course of 'Mathematics for Decision Making'. I think, it will provide some basic concept about transportation problem in linear programming.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
OPERATIONS RESEARCH
TRANSPORTATION PROBLEM
LEAST COST METHOD
method used to obtain the initial feasible solution for the transportation problem
Method: Least Cost Method (LCM)
The least cost method is more economical than north-west corner rule,since it starts with a lower beginning cost. Various steps involved in this method are summarized as under.
Step 1: Find the cell with the least(minimum) cost in the transportation table.
Step 2: Allocate the maximum feasible quantity to this cell.
Step:3: Eliminate the row or column where an allocation is made.
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources.A transportation matrix is a way of understanding the maximum possibilities the shipment can be done. It is also known as decision variables because these are the variables of interest that we will change to achieve the objective, that is, minimizing the cost function.
This presentation is made to represent the basic transportation model. The aim of this presentation is to implement the transportation model in solving transportation problem.
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
Transportation
Quantitative techniques for business decisions
Operations Research
Objective,Definition,Perks,Outlook
Procedure with Illustration for Northwest, Least cost, vogel's approximation, MODI method, Comparision of the total cost
Constructing a network
1 Introduction and definitions:
-Activity and Project
-Project Management Process
-Network
2 Situations in network diagram
-Concurrent activities
-Predecessors and Successors Activities
-Dummy Activity
3 Errors to be Avoided in constructing a network
4 Rules in constructing a network
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Why we use sensitivity analysis? and why we use it?
For clearly understand you can watch this video on my youtube channel
https://www.youtube.com/watch?v=R7g3KO_wroo&t=14s
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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unwillingness to rectify this violation through action requires accountability.
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students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...
Transportation problem
1. Faculty of Economics and Business Administration
Lebanese University
Chapter 4: Transportation problem
Dr. Kamel ATTAR
attar.kamel@gmail.com
! 2020 !
2. 1
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
1 Definition and basic notation
2 Transportation Model
Balanced transportation problem
Un-balanced transportation problem
3 Methods for obtaining initial feasible basic solution
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
4 Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
3. 2
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Definition
The transportation problem is a special type of linear programming problem where the objective
consists in minimizing transportation cost of a given commodity from a number of sources or
origins to a number of destinations.
In the transportation problem, we have
• m sources (warehouses, factories) producing
items, and
• n destinations (shops, businesses) requiring these
items.
The items need to be transported from sources to desti-
nations which has associated cost.
• The i-th source denotes by ai has available items
to transfer, and
• The j-th destination denotes by bj has demands
items to be delivered.
It costs cij to deliver one item from i-th source to j-th
destination.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
4. 3
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Assumption: destinations do not care from which source the items come, and sources do not care
to which destinations they deliver.
Decision variables: xij = the number of items transported from the i-th source to j-th destination.
Objective: The goal is to minimize the total cost of transportation.
Minimize
m
i=1
n
j=1
cij xij
Subject to
n
j=1
xij = ui for i = 1, · · · , m
m
i=1
xij = vi for j = 1, · · · , n
with xij ≥ 0 for i = 1, · · · , m and j = 1, · · · , n
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
5. 4
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
TABLE
b1 b2 b3 Supply
a1
c11
x11
c12
x12
c13
x13 u1
a2
c21
x21
c22
x22
c23
x23 u2
a3
c31
x31
c32
x32
c33
x33 u3
Demand v1 v2 v3 z =?
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
6. 5
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Balanced transportation problem
Un-balanced transportation problem
# Transportation Model #
There are two different types of transportation problems based on the initial
given information balanced and un-balanced transportation problems.
# Balanced transportation problems #
Cases where the total supply is equal to the total amount demanded.
Factory Transportation Supply
X Y
A 6 8 24
B 4 3 40
Demand 40 24
64
64
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
7. 6
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Balanced transportation problem
Un-balanced transportation problem
# Un-balanced transportation problems #
Cases where the total supply is less than or greater than to the total amount demanded.
• When the supply or availability is higher than the demand, a dummy destination is introduced in
the equation to make it equal to the supply (with shipping costs of 0$); the excess supply is
assumed to go to inventory.
Factory Transportation Supply
X Y
A 6 8 20
B 4 3 50
Demand 30 60
90
70
=⇒
Factory Transportation Supply
X Y
A 6 8 20
B 4 3 50
dummy 0 0 20
Demand 30 60
90
90
• On the other hand, when the demand is higher than the supply, a dummy source is introduced
in the equation to make it equal to the demand (in these cases there is usually a penalty cost
associated for not fulfilling the demand).
Factory Transportation Supply
X Y
A 6 8 30
B 4 3 60
Demand 20 50
70
90
=⇒
Factory Transportation Supply
X Y dummy
A 6 8 0 30
B 4 3 0 60
Demand 20 50 20
90
90
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
8. 7
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
# Methods for obtaining initial feasible basic solution #
To find the optimal solution we use two steps.
• First we need to find the initial basic feasible solution by using one of the
following three methods
North-west corner , Least cost entry or Vogel’s Approximation Method
• Then we obtain an optimal solution by making successive improvements in
the initial basic feasible solution until no further decrease in transportation
cost is possible. We can use on of the following two methods:
Stepping Stone or Modified Distribution Method
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
9. 8
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
# North-West Corner Method #
This method is the most systematic and easiest method for obtaining initial basic feasible solution.
To understand this method more clearly, let us take an example and discuss the rationale of
transportation problem.
Example
Transportation Availability in tons
(Cost per ton in $)
Factory W X Y sugar
A 4 16 8 72
B 8 24 16 102
C 8 16 24 41
Demand in Tons 56 82 77 215
Step‚ This problem is balanced since
m
i=1
ui
total supply
=
n
i=1
vi
total demand
= 215. If the problem is not
balanced, then we open a dummy column or row as the case may be and balance the problem.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
10. 9
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Stepƒ Construct an empty m × n matrix, completed with rows & columns.
Step„ Indicate the rows and column totals at the end.
W X Y Quantity Supplied
A
4 16 8
72
B
8 24 16
102
C
8 16 24
41
Demand 56 82 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
11. 9
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step… Starting with the first cell at the north west corner of the matrix :
• If u1 < v1 , then set x11 = u1 , remove the first row, and decrease v1 to v1 − u1.
• If u1 > v1 , then set x11 = v1, remove the first column, and decrease u1 to u1 − v1.
W X Y Quantity Supplied
A
4
56
16 8
72 16
B
8
−
24 16
102
C
8
−
16 24
41
Demand 56 0 82 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
12. 9
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step† Repeat:
• If ui < vj , then set xij = ui , remove the i-th row, and decrease vj to vj − ui .
• If ui > vj , then set xij = vj , remove the j-th column, and decrease ui to ui − vj .
W X Y Quantity Supplied
A
4
56
16
16
8
− 72 16 0
B
8
−
24 16
102
C
8
−
16 24
41
Demand 56 0 8266 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
13. 9
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step† Repeat:
• If ui < vj , then set xij = ui , remove the i-th row, and decrease vj to vj − ui .
• If ui > vj , then set xij = vj , remove the j-th column, and decrease ui to ui − vj .
W X Y Quantity Supplied
A
4
56
16
16
8
− 72 16 0
B
8
−
24
66
16
102 36
C
8
−
16
−
24
41
Demand 56 0 8266 0 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
14. 9
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step† Repeat:
• If ui < vj , then set xij = ui , remove the i-th row, and decrease vj to vj − ui .
• If ui > vj , then set xij = vj , remove the j-th column, and decrease ui to ui − vj .
W X Y Quantity Supplied
A
4
56
16
16
8
− 72 16 0
B
8
−
24
66
16
36 102 36 0
C
8
−
16
−
24
41
Demand 56 0 8266 0 77 41 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
15. 9
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step‡ Once all the allocations are over, i.e., both rim requirement (column and row i.e., availability
and requirement constraints) are satisfied, write allocations and calculate the cost of transportation.
W X Y Quantity Supplied
A
4
56
16
16
8
− 72 16 0
B
8
−
24
66
16
36 102 36 0
C
8
−
16
−
24
41 41 0
Demand 56 0 8266 0 77 41 0 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
16. 10
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step‡ Once all the allocations are over, i.e., both rim requirement (column and row i.e., availability
and requirement constraints) are satisfied, write allocations and calculate the cost of transportation
Rim Requirements
Stone squares must be equal to the nb. of columns+nb. of rows−1. In this
table we have
3columns + 3rows − 1 = 6 − 1 = 5Stones
Transportation cost
From To Units in tons Cost in $
A W 56 56 × 4 = 224
A X 16 16 × 16 = 256
B X 66 66 × 24 = 1584
B Y 36 36 × 16 = 576
C Y 41 41 × 24 = 984
Total 3624
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
17. 11
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
# Least Cost entry Method #
This method takes into consideration the lowest cost and therefore takes less time to solve problem.
To understand this method more clearly, let us resolve the previous example.
Example
Transportation Availability in tons
(Cost per ton in $)
Factory W X Y sugar
A 4 16 8 72
B 8 24 16 102
C 8 16 24 41
Demand in Tons 56 82 77 215
Step‚ This problem is balanced since
m
i=1
ui
total supply
=
n
i=1
vi
total demand
= 215.
If the problem is not balanced, then we open a dummy column or dummy row as the case may be
and balance the problem.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
18. 12
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Stepƒ Construct an empty m × n matrix, completed with rows & columns.
Step„ Indicate the rows and column totals at the end.
W X Y Quantity Supplied
A
4 16 8
72
B
8 24 16
102
C
8 16 24
41
Demand 56 82 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
19. 12
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step… Select the cell with the lowest transportation cost cij among all the rows and columns of the
transportation table. If the minimum cost is not unique then select arbitrarily any cell with the lowest
cost. Then allocate as many units as possible to the cell determined and eliminate that row in which
either capacity or requirement is exhausted.
W X Y Quantity Supplied
A
4
56
16 8
72 16
B
8
−
24 16
102
C
8
−
16 24
41
Demand 56 0 82 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
20. 12
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step† Repeat the steps… for the reduced table until the entire capacities are exhausted to fill the
requirements at the different destination.
W X Y Quantity Supplied
A
4
56
16
−
8
16 72 16 0
B
8
−
24 16
102
C
8
−
16 24
41
Demand 56 0 82 77 61 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
21. 12
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step† Repeat the steps… for the reduced table until the entire capacities are exhausted to fill the
requirements at the different destination.
W X Y Quantity Supplied
A
4
56
16
−
8
16 72 16 0
B
8
−
24 16
102
C
8
−
16
41
24
− 41 0
Demand 56 0 8241 77 61 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
22. 12
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step† Repeat the steps… for the reduced table until the entire capacities are exhausted to fill the
requirements at the different destination.
W X Y Quantity Supplied
A
4
56
16
−
8
16 72 16 0
B
8
−
24 16
61 102 41
C
8
−
16
41
24
− 41 0
Demand 56 0 8241 0 77 61 0 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
23. 12
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step‡ Once all the allocations are over, i.e., both rim requirement (column and row i.e., availability
and requirement constraints) are satisfied, write allocations and calculate the cost of transportation.
W X Y Quantity Supplied
A
4
56
16
−
8
16 72 16 0
B
8
−
24
41
16
61 102 41 0
C
8
−
16
41
24
− 41 0
Demand 56 0 8241 0 77 61 0 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
24. 13
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step‡ Once all the allocations are over, i.e., both rim requirement (column and row i.e., availability
and requirement constraints) are satisfied, write allocations and calculate the cost of transportation
Rim Requirements
Stone squares must be equal to the nb. of columns+nb. of rows−1. In this
table we have
3columns + 3rows − 1 = 6 − 1 = 5Stones
Transportation cost
From To Units in tons Cost in $
A W 56 56 × 4 = 224
A Y 16 16 × 8 = 128
B X 41 41 × 24 = 984
B Y 61 61 × 16 = 976
C X 41 41 × 16 = 656
Total 2968
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
25. 14
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
# Vogel’s Approximation Method #
This method is preferred over the (NWCM) and (LCM), because the initial basic feasible solution
obtained by this method is either optimal solution or very nearer to the optimal solution. So the
amount of time required to calculate the optimum solution is reduced.
To understand this method more clearly, let us resolve the previous example.
Example
Transportation Availability in tons
(Cost per ton in $)
Factory W X Y sugar
A 4 16 8 72
B 8 24 16 102
C 8 16 24 41
Demand in Tons 56 82 77 215
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
26. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step‚ Find the cells having smallest and next to smallest cost in each row (resp. column) and write
the difference (called penalty) along the side of the table in row penalty (resp. column penalty).
W X Y Supply Penalty
A
4 16 8
72 4 = 8 − 4
B
8 24 16
102 8 = 16 − 8
C
8 16 24
41 8 = 16 − 8
Demand 56 82 77 215
Penalty 4 = 8 − 4 0 = 16 − 16 8 = 16 − 8
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
27. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Stepƒ Select the cell with the lowest transportation cost cij among all the rows and columns of the
transportation table. If the minimum cost is not unique then select arbitrarily any cell with the lowest
cost. Then allocate as many units as possible to the cell determined and eliminate that row in which
either capacity or requirement is exhausted. If there is a tie in the values of penalties then select
the cell where maximum allocation can be possible
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0 4 = 8 − 4
B
8 24 16
102 8 = 16 − 8
C
8 16 24
41 8 = 16 − 8
Demand 56 82 77 5 215
Penalty 4 = 8 − 4 0 = 16 − 16 8 = 16 − 8
The maximum penalty is 8, occurs in column Y. The minimum cij in this column is c13 = 8.
The maximum allocation in this cell is min(72, 77) = 72.
It satisfy supply of A and adjust the demand of Y from 77 to 5 (77 − 72 = 5).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
28. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step„ Repeat this steps until all supply and demand values are 0.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8 24 16
102 8 = 16 − 8
C
8 16 24
41 8 = 16 − 8
Demand 56 82 77 5 215
Penalty 4 = 8 − 4 8 = 24 − 16 8 = 24 − 16
The maximum penalty is 8, occurs in row B.
The minimum cij in this row is c21 = 8.
The maximum allocation in this cell is min(102, 56) = 56.
It satisfy demand of W and adjust the supply of B from 102 to 46 (102 − 56 = 46).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
29. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step„ Repeat this steps until all supply and demand values are 0.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8
56
24 16
102 46 8 = 16 − 8
C
8
−
16 24
41 8 = 16 − 8
Demand 56 0 82 77 5 215
Penalty 4 = 8 − 4 8 = 24 − 16 8 = 24 − 16
The maximum penalty is 8, occurs in row B.
The minimum cij in this row is c21 = 8.
The maximum allocation in this cell is min(102, 56) = 56.
It satisfy demand of W and adjust the supply of B from 102 to 46 (102 − 56 = 46).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
30. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step„ Repeat this steps until all supply and demand values are 0.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8
56
24 16
102 46 8 = 24 − 16
C
8
−
16 24
41 8 = 24 − 16
Demand 56 82 77 5 215
Penalty 8 = 24 − 16 8 = 24 − 16
The maximum penalty is 8, occurs in row C.
The minimum cij in this row is c32 = 16.
The maximum allocation in this cell is min(41, 82) = 41.
It satisfy supply of C and adjust the demand of X from 82 to 41 (82 − 41 = 41).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
31. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step„ Repeat this steps until all supply and demand values are 0.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8
56
24 16
102 46 8 = 24 − 16
C
8
−
16
41
24
− 41 0 8 = 24 − 16
Demand 56 82 41 77 5 215
Penalty 8 = 24 − 16 8 = 24 − 16
The maximum penalty is 8, occurs in row C.
The minimum cij in this row is c32 = 16.
The maximum allocation in this cell is min(41, 82) = 41.
It satisfy supply of C and adjust the demand of X from 82 to 41 (82 − 41 = 41).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
32. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step„ Repeat this steps until all supply and demand values are 0.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8
56
24 16
102 46 8 = 24 − 16
C
8
−
16
41
24
− 41 0
Demand 56 82 41 77 5 215
Penalty 24 16
The maximum penalty is 24, occurs in column X.
The minimum cij in this column is c22 = 24.
The maximum allocation in this cell is min(46, 41) = 41.
It satisfy demand of X and adjust the supply of B from 46 to 5 (46 − 41 = 5).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
33. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step„ Repeat this steps until all supply and demand values are 0.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8
56
24
41
16
102 46 8 = 24 − 16
C
8
−
16
41
24
− 41 0
Demand 56 82 41 77 5 0 215
Penalty 24 16
The maximum penalty is 24, occurs in column X.
The minimum cij in this column is c22 = 24.
The maximum allocation in this cell is min(46, 41) = 41.
It satisfy demand of X and adjust the supply of B from 46 to 5 (46 − 41 = 5).
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
34. 15
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step… Once all the allocations are over, i.e., both rim requirement (column and row i.e., availability
and requirement constraints) are satisfied, write allocations and calculate the cost of transportation.
W X Y Supply Penalty
A
4
−
16
−
8
72 72 0
B
8
56
24
41
16
5 102 46
C
8
−
16
41
24
− 41 0
Demand 56 82 41 0 77 5 0 215
Penalty
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
35. 16
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
North-west corner method
Least cost entry method
Vogel’s Approximation Method (VAM) or penalty method
Step… Once all the allocations are over, i.e., both rim requirement (column and row i.e., availability
and requirement constraints) are satisfied, write allocations and calculate the cost of transportation.
Rim Requirements
Stone squares must be equal to the nb. of columns+nb. of rows−1. In this
table we have
3columns + 3rows − 1 = 6 − 1 = 5Stones
Transportation cost
From To Units in tons Cost in $
A Y 72 72 × 8 = 576
B W 56 56 × 8 = 448
B X 41 41 × 24 = 984
B Y 5 5 × 16 = 80
C X 41 41 × 16 = 656
Total 2744
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
36. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
# Stepping stone method of optimality test #
Once, we get the basic feasible solution for a transportation problem, the next duty is to test
whether the solution got is an optimal one or not? This can be done by using Stepping Stone
methods.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
37. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Step‚ Starting from the empty cell draw a loop moving horizontally and vertically from loaded cell
to loaded cell. We have to take turn only at loaded cells and move to vertically downward or
upward or horizontally to reach another loaded cell. In between, if we have a loaded cell, where we
cannot take a turn, ignore that and proceed to next loaded cell in that row or column. After
completing the loop, mark minus (−) and plus (+) signs alternatively, but begin with the + mark in
the empty cell, and calculate the cost change.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
38. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Step‚ Starting from the empty cell draw a loop moving horizontally and vertically from loaded cell
to loaded cell. We have to take turn only at loaded cells and move to vertically downward or
upward or horizontally to reach another loaded cell. In between, if we have a loaded cell, where we
cannot take a turn, ignore that and proceed to next loaded cell in that row or column. After
completing the loop, mark minus (−) and plus (+) signs alternatively, but begin with the + mark in
the empty cell, and calculate the cost change.
A −→ Y : +AY − BY + BX − AX = +8 − 16 + 24 − 16 = 0
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
39. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Stepƒ Repeat these steps again until all the empty cells get evaluated.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
40. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Stepƒ Repeat these steps again until all the empty cells get evaluated.
B −→ W : +BW − BX + AX − AW = +8 − 24 + 16 − 4 = −4
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
41. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Stepƒ Repeat these steps again until all the empty cells get evaluated.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
42. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Stepƒ Repeat these steps again until all the empty cells get evaluated.
C → W : +CW−CY+BY−BX+AX−AW = +8−24+16−24+16−4 = −12
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
43. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Stepƒ Repeat these steps again until all the empty cells get evaluated.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
44. 17
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Stepƒ Repeat these steps again until all the empty cells get evaluated.
C → X : +CX − CY + BY − BX = +16 − 24 + 16 − 24 = −16
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
45. 18
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Step„ Evaluate the cost change. If the cost change is positive, it means that if we include the
evaluated cell in the program, the cost will increase. If the cost change is negative, it means that
the total cost will decrease, by including the evaluated cell in the program.
Now, if all the cost changes are positive or are equal to or greater than zero, then the optimal
solution has been reached. But in case, if any, value comes to be negative, then there is a scope to
reduce the transportation cost further. Then, select that loop which has the most negative of cost
change.
S.No. Empty Cell Evalution Loop formation C. change
B → W +BW − BX + AX − AW +8 − 24 + 16 − 4 −4
C → W +CW − CY + BY − BX + AX − AW +8 − 24 + 16 − 24 + 16 − 4 −12
C → X +CX − CY + BY − BX +16 − 24 + 16 − 24 -16
A → Y +AY − BY + BX − AX +8 − 16 + 24 − 16 0
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
46. 18
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Step„ Evaluate the cost change. If the cost change is positive, it means that if we include the
evaluated cell in the program, the cost will increase. If the cost change is negative, it means that
the total cost will decrease, by including the evaluated cell in the program.
Now, if all the cost changes are positive or are equal to or greater than zero, then the optimal
solution has been reached. But in case, if any, value comes to be negative, then there is a scope to
reduce the transportation cost further. Then, select that loop which has the most negative of cost
change.
C −→ X
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
47. 18
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Step… Identify the lowest load in the cells marked with negative sign. This number is to be added to
the cells where plus sign is marked and subtract from the load of the cell where negative sign is
marked. Do not alter the loaded cells, which are not in the loop. The process of adding and
subtracting at each turn or corner is necessary to balance the demand and supply requirements.
-
66
+
36
+
−
-
41
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
48. 18
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Step† Construct a table of empty cells and work out the cost change for a shift of load from loaded
cell to loaded cell.
-
66
+
36
+
−
-
41
=⇒
41 − 66
25
41 + 36
77
41 + 0
41
41 − 41
−
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
49. 19
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
New table
W X Y Demand
A
4
56
16
16
8
− 72
B
8
−
24
25
16
77 102
C
8
−
16
41
24
− 41
Supply 56 82 77 215
Rim Requirements
3columns + 3rows − 1 = 6 − 1 = 5Stones
Transportation cost
From To Units in tons Cost in $
A W 56 56 × 4 = 224
A X 16 16 × 16 = 256
B X 25 25 × 24 = 600
B Y 77 77 × 16 = 1232
C X 41 41 × 16 = 656
Total 2968
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
50. 20
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Second iteration
A −→ Y : +AY − BY + BX − AX = +8 − 16 + 24 − 16 = 0
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
51. 20
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Second iteration
B −→ W : +BW − AW + AX − BX = +8 − 4 + 16 − 24 = −4
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
52. 20
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Second iteration
C −→ W : +CW − CX + AX − AW = +8 − 16 + 16− = 4
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
53. 20
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Second iteration
C −→ Y : +CY − BY + BX − CX = +24 − 16 + 24 − 16 = 16
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
54. 21
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Empty boxes
S.No. Empty Cell Evalution Loop formation Cost change in $
A → Y +AY − BY + BX − AX +8 − 16 + 24 − 16 0
B → W +BW − AW + AX − BX +8 − 4 + 16 − 24 -4
C → W +CW − CX + AX − AW +8 − 16 + 16 − 4 4
C → Y +CY − BY + BX − CX +24 − 16 + 24 − 16 16
-
56
+
16
+
−
-
25
=⇒
56 − 25
31
16 + 25
41
25 + 0
25
25 − 25
−
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
55. 22
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
New table
W X Y Supply
A
4
31
16
41
8
− 72
B
8
25
24
−
16
77 102
C
8
−
16
41
24
− 41
Demand 56 82 77 215
Rim Requirements
3columns + 3rows − 1 = 6 − 1 = 5Stones
Transportation cost
From To Units in tons Cost in $
A W 31 31 × 4 = 124
A X 41 41 × 16 = 656
B X 25 25 × 8 = 200
B Y 77 77 × 16 = 1232
C X 41 41 × 16 = 656
Total 2868
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
56. 23
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Third iteration
S.No. Empty Cell Evalution Loop formation Cost change in $
A → Y +AY − BY + BW − AW +8 − 16 + 8 − 4 -4
B → X +BX − AX + AW − BW +24 − 16 + 4 − 8 4
C → W +CW − AW + AX − CX +8 − 4 + 16 − 16 4
C → Y +CY − BY + BW − AW + AX − CX 24 − 16 + 8 − 4 + 16 − 16 12
-
31
+
−
+
25
-
77
=⇒
− 31
56 46
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
57. 24
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
New table
W X Y Supply
A
4
−
16
41
8
31 72
B
8
56
24
−
16
46 102
C
8
−
16
41
24
− 41
Demand 56 82 77 215
Rim Requirements
3columns + 3rows − 1 = 6 − 1 = 5Stones
Transportation cost
From To Units in tons Cost in $
A Y 31 31 × 8 = 248
A X 41 41 × 16 = 656
B W 56 56 × 8 = 448
B Y 46 46 × 16 = 736
C X 41 41 × 16 = 656
Total 2744
Dr. Kamel ATTAR | Chapter 4: Transportation problem |
58. 25
Definition and basic notation
Transportation Model
Methods for obtaining initial feasible basic solution
Optimality Test: (Method for optimal solution)
Stepping stone method of optimality test
Final iteration
Empty boxes
S.No. Empty Cell Evalution Loop formation Cost change in $
B → X +BX − AX + AY − BY +24 − 16 + 8 − 16 0
C → W +CW − BW + BY − AY + AX − CX +8 − 8 + 16 − 8 + 16 − 16 8
C → Y +CY − AY + AX − CX +24 − 8 + 16 − 16 16
A → W +AW − BW + BY − AY +4 − 8 + 16 − 8 4
Then the optimal solution of the transportation cost is 2744$.
Dr. Kamel ATTAR | Chapter 4: Transportation problem |