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Solving Transportation Problem using Linear Programming in Python
Learn how to use Python PuLP to solve transportation problems using Linear Programming.
In this tutorial, we will broaden the horizon of linear programming problems. We will discuss the Transportation problem. It offers various applications involving the optimal transportation of goods. The transportation model is basically a minimization model.
The transportation problem is a type of Linear Programming problem. In this type of problem, the main objective is to transport goods from source warehouses to various destination locations at minimum cost. In order to solve such problems, we should have demand quantities, supply quantities, and the cost of shipping from source and destination. There are m sources or origin and n destinations, each represented by a node. The edges represent the routes linking the sources and the destinations.
In this tutorial, we are going to cover the following topics:
Transportation Problem
The transportation models deal with a special type of linear programming problem in which the objective is to minimize the cost. Here, we have a homogeneous commodity that needs to be transferred from various origins or factories to different destinations or warehouses.
Types of Transportation problems
- Balanced Transportation Problem : In such type of problem, total supplies and demands are equal.
- Unbalanced Transportation Problem : In such type of problem, total supplies and demands are not equal.
Methods for Solving Transportation Problem:
- NorthWest Corner Method
- Least Cost Method
- Vogel’s Approximation Method (VAM)
Let’s see one example below. A company contacted the three warehouses to provide the raw material for their 3 projects.
This constitutes the information needed to solve the problem. The next step is to organize the information into a solvable transportation problem.
Formulate Problem
Let’s first formulate the problem. first, we define the warehouse and its supplies, the project and its demands, and the cost matrix.
Initialize LP Model
In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.
Define Decision Variable
In this step, we will define the decision variables. In our problem, we have various Route variables. Let’s create them using LpVariable.dicts() class. LpVariable.dicts() used with Python’s list comprehension. LpVariable.dicts() will take the following four values:
- First, prefix name of what this variable represents.
- Second is the list of all the variables.
- Third is the lower bound on this variable.
- Fourth variable is the upper bound.
- Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are LpContinuous or LpInteger .
Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.
Define Objective Function
In this step, we will define the minimum objective function by adding it to the LpProblem object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.
In this code, we have summed up the two variables(full-time and part-time) list values in an additive fashion.
Define the Constraints
Here, we are adding two types of constraints: supply maximum constraints and demand minimum constraints. We have added the 4 constraints defined in the problem by adding them to the LpProblem object.
Solve Model
In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.
From the above results, we can infer that Warehouse-A supplies the 300 units to Project -2. Warehouse-B supplies 150, 150, and 300 to respective project sites. And finally, Warehouse-C supplies 600 units to Project-3.
In this article, we have learned about Transportation problems, Problem Formulation, and implementation using the python PuLp library. We have solved the transportation problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. In upcoming articles, we will write more on different optimization problems such as transshipment problem, assignment problem, balanced diet problem. You can revise the basics of mathematical concepts in this article and learn about Linear Programming in this article .
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Transshipment Problem in Python Using PuLP
Python Decorators
- There are three supply nodes: IE Junction, Centerville and Wayover. Let s1 = 4 be the supply at IE Junction, s2 = 1 be the supply at Centerville and s3 =2 be the supply at Wayover City.
- Let A-Station be demand node 1, Fine Place demand node 2, Goodville be demand node 3 and Somewhere Street demand node 4. Since a single locomotive is needed at each demand point, di = 1 for i = 1,2,3 and 4.
- The cost cij of shipping one unit from supply node i to demand node j is the distance from the supply point to the demand point given in the table above.
A new approach for solving cost minimization balanced transportation problem under uncertainty
- Published: 18 September 2014
- Volume 7 , pages 339–345, ( 2014 )
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- Sandeep Singh 1 &
- Gourav Gupta 1
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In the literature, there are several methods for solving fuzzy transportation problems and finding the fuzzy optimal values. In this paper a new method is proposed for finding an optimal solution for fuzzy transportation problem. The proposed method always gives a fuzzy optimal value without disturbance of degeneracy condition. This requires least computational work to reach optimality as compared to the existing methods available in the literature.
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Singh, S., Gupta, G. A new approach for solving cost minimization balanced transportation problem under uncertainty. J Transp Secur 7 , 339–345 (2014). https://doi.org/10.1007/s12198-014-0147-1
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Published : 18 September 2014
Issue Date : December 2014
DOI : https://doi.org/10.1007/s12198-014-0147-1
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Transportation Problem | Set 6 (MODI Method – UV Method)
There are two phases to solve the transportation problem. In the first phase, the initial basic feasible solution has to be found and the second phase involves optimization of the initial basic feasible solution that was obtained in the first phase. There are three methods for finding an initial basic feasible solution,
- NorthWest Corner Method
- Least Cost Cell Method
- Vogel’s Approximation Method
This article will discuss how to optimize the initial basic feasible solution through an explained example. Consider the below transportation problem.
Check whether the problem is balanced or not. If the total sum of all the supply from sources
is equal to the total sum of all the demands for destinations
then the transportation problem is a balanced transportation problem.
If the problem is not unbalanced then the concept of a dummy row or a dummy column to transform the unbalanced problem to balanced can be followed as discussed in
Finding the initial basic feasible solution. Any of the three aforementioned methods can be used to find the initial basic feasible solution. Here,
will be used. And according to the NorthWest Corner Method this is the final initial basic feasible solution:
Now, the total cost of transportation will be
(200 * 3) + (50 * 1) + (250 * 6) + (100 * 5) + (250 * 3) + (150 * 2) = 3700
U-V method to optimize the initial basic feasible solution. The following is the initial basic feasible solution:
– For U-V method the values
have to be found for the rows and the columns respectively. As there are three rows so three
values have to be found i.e.
for the first row,
for the second row and
for the third row. Similarly, for four columns four
. Check the image below:
There is a separate formula to find
is the cost value only for the allocated cell. Read more about it
. Before applying the above formula we need to check whether
m + n – 1 is equal to the total number of allocated cells
or not where
is the total number of rows and
is the total number of columns. In this case m = 3, n = 4 and total number of allocated cells is 6 so m + n – 1 = 6. The case when m + n – 1 is not equal to the total number of allocated cells will be discussed in the later posts. Now to find the value for u and v we assign any of the three u or any of the four v as 0. Let we assign
in this case. Then using the above formula we will get
). Similarly, we have got the value for
so we get the value for
which implies
. From the value of
. See the image below:
Now, compute penalties using the formula
only for unallocated cells. We have two unallocated cells in the first row, two in the second row and two in the third row. Lets compute this one by one.
- For C 13 , P 13 = 0 + 0 – 7 = -7 (here C 13 = 7 , u 1 = 0 and v 3 = 0 )
- For C 14 , P 14 = 0 + (-1) -4 = -5
- For C 21 , P 21 = 5 + 3 – 2 = 6
- For C 24 , P 24 = 5 + (-1) – 9 = -5
- For C 31 , P 31 = 3 + 3 – 8 = -2
- For C 32 , P 32 = 3 + 1 – 3 = 1
If we get all the penalties value as zero or negative values that mean the optimality is reached and this answer is the final answer. But if we get any positive value means we need to proceed with the sum in the next step. Now find the maximum positive penalty. Here the maximum value is 6 which corresponds to
cell. Now this cell is new basic cell. This cell will also be included in the solution.
The rule for drawing closed-path or loop.
Starting from the new basic cell draw a closed-path in such a way that the right angle turn is done only at the allocated cell or at the new basic cell. See the below images:
Assign alternate plus-minus sign to all the cells with right angle turn (or the corner) in the loop with plus sign assigned at the new basic cell.
Consider the cells with a negative sign. Compare the allocated value (i.e. 200 and 250 in this case) and select the minimum (i.e. select 200 in this case). Now subtract 200 from the cells with a minus sign and add 200 to the cells with a plus sign. And draw a new iteration. The work of the loop is over and the new solution looks as shown below.
Check the total number of allocated cells is equal to (m + n – 1). Again find u values and v values using the formula
is the cost value only for allocated cell. Assign
then we get
. Similarly, we will get following values for
Find the penalties for all the unallocated cells using the formula
- For C 11 , P 11 = 0 + (-3) – 3 = -6
- For C 13 , P 13 = 0 + 0 – 7 = -7
- For C 14 , P 14 = 0 + (-1) – 4 = -5
- For C 31 , P 31 = 0 + (-3) – 8 = -11
There is one positive value i.e. 1 for
. Now this cell becomes new basic cell.
Now draw a loop starting from the new basic cell. Assign alternate plus and minus sign with new basic cell assigned as a plus sign.
Select the minimum value from allocated values to the cell with a minus sign. Subtract this value from the cell with a minus sign and add to the cell with a plus sign. Now the solution looks as shown in the image below:
Check if the total number of allocated cells is equal to (m + n – 1). Find u and v values as above.
Now again find the penalties for the unallocated cells as above.
- For P 11 = 0 + (-2) – 3 = -5
- For P 13 = 0 + 1 – 7 = -6
- For P 14 = 0 + 0 – 4 = -4
- For P 22 = 4 + 1 – 6 = -1
- For P 24 = 4 + 0 – 9 = -5
- For P 31 = 2 + (-2) – 8 = -8
All the penalty values are negative values. So the optimality is reached. Now, find the total cost i.e.
(250 * 1) + (200 * 2) + (150 * 5) + (50 * 3) + (200 * 3) + (150 * 2) = 2450
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Balanced and Unbalanced Transportation Problems
The two categories of transportation problems are balanced and unbalanced transportation problems . As we all know, a transportation problem is a type of Linear Programming Problem (LPP) in which items are carried from a set of sources to a set of destinations based on the supply and demand of the sources and destinations, with the goal of minimizing the total transportation cost. It is also known as the Hitchcock problem.
Introduction to Balanced and Unbalanced Transportation Problems
Balanced transportation problem.
The problem is considered to be a balanced transportation problem when both supplies and demands are equal.
Unbalanced Transportation Problem
Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem.
Methods of Solving Transportation Problems
There are three ways for determining the initial basic feasible solution. They are
1. NorthWest Corner Cell Method.
2. Vogel’s Approximation Method (VAM).
3. Least Call Cell Method.
The following is the basic framework of the balanced transportation problem:
The destinations D1, D2, D3, and D4 in the above table are where the products/goods will be transported from various sources O1, O2, O3, and O4. The supply from the source Oi is represented by S i . The demand for the destination Dj is d j . If a product is delivered from source Si to destination Dj, then the cost is called C ij .
Let us now explore the process of solving the balanced transportation problem using one of the ways known as the NorthWest Corner Method in this article.
Solving Balanced Transportation problem by Northwest Corner Method
Consider this scenario:
With three sources (O1, O2, and O3) and four destinations (D1, D2, D3, and D4), what is the best way to solve this problem? The supply for the sources O1, O2, and O3 are 300, 400, and 500, respectively. Demands for the destination D1, D2, D3, and D4 are 250, 350, 400, and 200, respectively.
The starting point for the North West Corner technique is (O1, D1), which is the table’s northwest corner. The cost of transportation is calculated for each value in the cell. As indicated in the diagram, compare the demand for column D1 with the supply from source O1 and assign a minimum of two to the cell (O1, D1).
Column D1’s demand has been met, hence the entire column will be canceled. The supply from the source O1 is still 300 – 250 = 50.
Analyze the northwest corner, i.e. (O1, D2), of the remaining table, excluding column D1, and assign the lowest among the supply for the appropriate column and rows. Because the supply from O1 is 50 and the demand for D2 is 350, allocate 50 to the cell (O1, D2).
Now, row O1 is canceled because the supply from row O1 has been completed. Hence, the demand for Column D2 has become 350 – 50 = 50.
The northwest corner cell in the remaining table is (O2, D2). The shortest supply from source O2 (400) and the demand for column D2 (300) is 300, thus putting 300 in the cell (O2, D2). Because the demand for column D2 has been met, the column can be deleted, and the remaining supply from source O2 is 400 – 300 = 100.
Again, find the northwest corner of the table, i.e. (O2, D3), and compare the O2 supply (i.e. 100) to the D2 demand (i.e. 400) and assign the smaller (i.e. 100) to the cell (O2, D2). Row O2 has been canceled because the supply from O2 has been completed. Column D3 has a leftover demand of 400 – 100 = 300.
Continuing in the same manner, the final cell values will be:
It should be observed that the demand for the relevant columns and rows is equal in the last remaining cell, which was cell (O3, D4). In this situation, the supply from O3 was 200, and the demand for D4 was 200, therefore this cell was assigned to it. Nothing was left for any row or column at the end.
To achieve the basic solution, multiply the allotted value by the respective cell value (i.e. the cost) and add them all together.
I.e., (250 × 3) + (50 × 1) + (300 × 6) + (100 × 5) + (300 × 3) + (200 × 2) = 4400.
Solving Unbalanced Transportation Problem
An unbalanced transportation problem is provided below. Because the sum of all the supplies, O1, O2, O3, and O4, does not equal the sum of all the demands, D1, D2, D3, D4, and D5, the situation is unbalanced.
The idea of a dummy row or dummy column will be applied in this type of scenario. Because the supply is more than the demand in this situation, a fake demand column will be inserted, with a demand of (total supply – total demand), i.e. 117 – 95 = 22, as seen in the image below. A fake supply row would have been introduced if demand was greater than supply.
Now this problem has been changed to a balanced transportation problem, and it can be addressed using any of the ways listed below to solve a balanced transportation problem, such as the northwest corner method mentioned earlier.
Frequently Asked Questions on Balanced and Unbalanced Transportation Problems
What is meant by balanced and unbalanced transportation problems.
The problem is referred to as a balanced transportation problem when both supplies and demands are equal. Unbalanced transportation is defined as a situation where supply and demand are not equal.
What is called a transportation problem?
The transportation problem is a type of Linear Programming Problem in which commodities are carried from a set of sources to a set of destinations while taking into account the supply and demand of the sources and destinations, respectively, in order to reduce the total cost of transportation.
What are the different methods to solve transportation problems?
The following are three approaches to solve the transportation issue:
- NorthWest Corner Cell Method.
- Least Call Cell Method.
- Vogel’s Approximation Method (VAM).
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PRACTICAL STEPS INVOLVED IN SOLVING TRANSPORTATION PROBLEMS OF MINIMIZATION TYPE
The practical steps involved in solving transporation problems of minimization type are given below:
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Maximization Transportation Problem
There are certain types of transportation problems where the objective function is to be maximized instead of being minimized.
These problems can be solved by converting the maximization problem into a minimization problem.
"Profit maximization is the single universal objective for most commercial organizations." -Vinay Chhabra & Manish Dewan
Example: Maximization Problem in Transportation
Surya Roshni Ltd. has three factories - X, Y, and Z. It supplies goods to four dealers spread all over the country. The production capacities of these factories are 200, 500 and 300 per month respectively.
Determine a suitable allocation to maximize the total net return.
Maximization transportation problem can be converted into minimization transportation problem by subtracting each transportation cost from maximum transportation cost.
Here, the maximum transportation cost is 25. So subtract each value from 25. The revised transportation problem is shown below.
An initial basic feasible solution is obtained by matrix minimum method and is shown in the final table.
Final table
Use Horizontal Scrollbar to View Full Table Calculation.
The maximum net return is
25 X 200 + 8 X 80 + 7 X 320 + 10 X 100 + 14 X 100 + 20 X 200 = 14280.
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Step-by-step guide on how to solve a balanced minimization transportation problem. We reimagined cable. Try it free.* Live TV from 100+ channels. No cable box or long-term contract required....
To solve a transportation problem, the following information must be given: m= The number of sources. n= The number of destinations. The total quantity available at each source. The total quantity required at each destination. The cost of transportation of one unit of the commodity from each source to each destination. ASSUMPTIONS
Methods for Solving Transportation Problem: NorthWest Corner Method Least Cost Method Vogel's Approximation Method (VAM) Let's see one example below. A company contacted the three warehouses to provide the raw material for their 3 projects. Supply Table Demand Table Cost Matrix This constitutes the information needed to solve the problem.
The transportation problem is a type of linear programming problem designed to minimize the cost of distributing a product from M M sources to N N destinations. The transportation problem can be described using examples from many fields. One application is the problem of efficiently moving troops from bases to battleground locations.
If the total demand is equal to the total supply, the problem is said to be a balanced transportation problem. If the total supply is greater than the total demand, we can add a dummy demand node to create a balanced problem. The constraint matrix for a balanced transportation problem has a special structure for which researchers have developed ...
The Transportation Problem (TP) is a characteristic optimization technique that aims to find the most effective way to deliver items from suppliers to customers while keeping transportation...
To solve a crisp transportation problem Taha (Taha 2006) uses tabular methods such as Northwest Corner Rule, Least Cost Method and Vogel's Approximation Method (Reinfeld and Vogel, (Reinfeld and Vogel 1958 )). These methods use techniques of a Linear programming problem. They differ only in steps of carrying out optimality conditions.
The objective of the fuzzy transportation problem is to determine the transportation schedule that minimizes the total fuzzy transportation cost while satisfying the availability and...
Balanced Transportation Problem is a transportation problem where the total availability at the origins is equal to the total requirements at the destinations. For example, in case the total production of 4 factories is 1000 units and total requirements of 4 warehouses is also 1000 units, the transportation problem is said to be a balanced one ...
A new approach for solving cost minimization balanced transportation problem under uncertainty Authors: Sandeep Singh Chahal Akal University Gourav Gupta Thapar University Abstract and...
A square matrix A is triangular i it is nonsingular & both Ax = b,&ˇA= c, can be solved by backsubstitution. Theorem: Every basis for the balanced transportation prob- lem is triangular. Theorem: The determinant of every basis for the balanced transportation problem is 1. As a consequence, every basic solution is integral if the RHS constants vector is integral.
An introduction to the basic transportation problem and its linear programming formulation:Transshipment Problem video: https://youtu.be/ABMPgSApdUw Solve Tr...
Solution: Step 1: Check whether the problem is balanced or not. If the total sum of all the supply from sources O1 , O2 , and O3 is equal to the total sum of all the demands for destinations D1 , D2 , D3 and D4 then the transportation problem is a balanced transportation problem. Note:
Solving Balanced Transportation problem by Northwest Corner Method Consider this scenario: With three sources (O1, O2, and O3) and four destinations (D1, D2, D3, and D4), what is the best way to solve this problem? The supply for the sources O1, O2, and O3 are 300, 400, and 500, respectively.
Mixed Constraints Cost Minimization Transportation Problem: An Effective Algorithmic Approach Farhana Rashid1,*, Aminur Rahman Khan2, Md. Sharif Uddin2 ... Then balanced it by using VAM and solve it with any transportation algorithm. Place the load(s) of the dummy row(s)/ column(s) of the balanced LBP at the ...
Comparison of balanced fuzzy transportation problem (BFTP) and unbalanced fuzzy Transportation problem (UFTP) shows that the optimal transportation cost of UFTP is less than BFTP. A fuzzy transportation problem (FTP) includes cost, supply and demand of transportation problems. Its numbers are fuzzy numbers. Fuzzy transportation problem works to reduce transportation cost of some commodities ...
problem is termed as a Balanced Transportation problem. Here ∑ 2.5 2.2 Unbalanced Transportation Problem If the sum of the supplies of all the sources is not equal to the sum of the demands of all the destinations then the problem is termed as unbalanced transportation problems. Here X ij = a i for i = 1,2,…,m X j = bj for j = 1,2,…,n X
In this paper, we proposed an efficient algorithm for solving transportation problems in which the origin and destination constraints consist not only of equality but also inequality. The proposed ...
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Transportation, Transshipment, and Assignment Problems Learning Objectives After completing this chapter, you should be able to: Describe the nature of transportation transshipment and assignment problems. Formulate a transportation problem as a linear programming model. Use the transportation method to solve problems with Excel.
The practical steps involved in solving transporation problems of minimization type are given below: Step 1 → See whether Total Requirements are equal to Total Availability; if yes, go to Step 2; if not, introduce a Dummy Origin/Destination, as the case may be, to make the problem a balanced one, taking Transportation Cost per unit as zero ...
400. Determine a suitable allocation to maximize the total net return. Solution. Maximization transportation problem can be converted into minimization transportation problem by subtracting each transportation cost from maximum transportation cost. Here, the maximum transportation cost is 25. So subtract each value from 25.
This problem is unbalanced fuzzy transportation problem then the problem convert to balanced fuzzy transportation problem defined as follows. The modified minimization problem is unbalanced. To make it balance we introduce a dummy destination FO4 with demand 2.6 units with zero costs c ij. Hence the balanced minimization transportation problem ...