unbalanced transportation problem means

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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UNBALANCED TRANSPORTATION PROBLEM

Unbalanced transportation problem is a transportation problem where the total availability at the origins is not 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 900 units or 1,100 units, the transportation problem is said to be an unbalanced one. To make an unbalanced transportation problem, a balanced one, a dummy origin(s) or a dummy destination (s) (as the case may be) is introduced with zero transportation cost per unit.

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Encyclopedia of Operations Research and Management Science pp 855 Cite as

UNBALANCED Transportation problem

• Saul I. Gass 3 &
• Carl M. Harris 4  
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A transportation problem in which the total amount to be shipped is not equal to the total demand. The unbalanced problem can be stated as a standard transportation problem by the addition of a fictitious destination when the supply is greater than the demand, or by adding a fictitious origin if the demand is greater than the supply. In the first case, the demand at the fictitious destination is the difference between the total supply and total demand, while in the second case, the supply at the fictitious origin is the difference between the total demand and total supply. Transportation problem .

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Carl M. Harris

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Gass, S.I., Harris, C.M. (2001). UNBALANCED Transportation problem . In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_1080

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Balanced and Unbalanced Transportation Problem | Operations Research

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Balanced Transportation Problem:

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Unbalanced Transportation Problem

So far we have assumed that the total supply at the origins is equal to the total requirement at the destinations.

Specifically,

But in certain situations, the total supply is not equal to the total demand. Thus, the transportation problem with unequal supply and demand is said to be unbalanced transportation problem .

How to solve?

If the total supply is more than the total demand, we introduce an additional column, which will indicate the surplus supply with transportation cost zero. Similarly, if the total demand is more than the total supply, an additional row is introduced in the table, which represents unsatisfied demand with transportation cost zero. The balancing of an unbalanced transportation problem is illustrated in the following example.

Example of Unbalanced Transportation Problem

The total demand is 1000, whereas the total supply is 800.

Total supply < total demand.

To solve the problem, we introduce an additional row with transportation cost zero indicating the unsatisfied demand.

Using matrix minimum method , we get the following allocations.

Use Horizontal Scrollbar to View Full Table Calculation.

Initial basic feasible solution

50 X 28 + 450 X 26 + 250 X 12 + 50 X 16 + 200 X 0 = 16900.

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Solving the Transportation Problem

Minimizing the cost of transporting products from production and storage locations to demand centers is an essential part of maintaining profitability for companies who deal with product distribution. Since transportation costs are generally not controllable, minimizing total cost requires making the best product routing decisions. This essential problem was first formulated as a linear programming problem in the early 1940’s and is popularly known as the transportation problem.

In this blog, we give an overview of the transportation problem and how the transportation algorithm — as implemented in IMSL — can be used to solve different variations of the transportation problem.

What Is the Transportation Problem?

Balanced transportation problem example, unbalanced transportation problem example.

The transportation problem is a type of linear programming problem designed to minimize the cost of distributing a product from \(M\) sources to \(N\) 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. Another is the optimal assignment of agents or workers to different jobs or positions. By far the most common application is of moving goods from multiple factories to multiple warehouse locations, or from warehouses to storefronts.

Transportation Problem Example

Consider the following example:

XYZ Inc. has two factories in different locations around the country where they produce widgets. Their sales partner has three central warehouses where they ship these widgets to their various customers. The factories can produce a given number of widgets per week each and the expected demand for each warehouse is also known. There is a shipping cost from each factory to each warehouse. Which factory should produce and ship how many widgets to which warehouses to meet the demand at each location with minimal cost?

This problem statement has all the components of a typical transportation problem. The sources and destinations are generic — they could be logging sites and sawmills, factories and warehouses, warehouses and stores, bases, and battlefields, and so on.

In each case, there is some demand or need \(D\) at each of \(N\) locations, some supply \(S\) at each of \(M\) locations, and a cost, \(c\), associated with transporting (or using) one unit from a particular \(M\) location to a particular \(N\) location. There will be a total of \(M\) x \(N\) such costs.

The cost, \(c\), can be a calculation involving factors such as time, distance, material costs, and so on, but it may be any quantity that is relevant to the problem.

Once supplies, demands, and costs are known, the problem is to determine the number of units, \(x\), that should be produced and sent from each of the \(M\) supply centers to each of the \(N\) demand locations.

The total cost is the sum of all individual costs times the individual units to be produced and shipped from each supply center to each demand center. Framing this as an optimization problem, the goal is to minimize the total cost:

$$ \sum_{i=1}^M \sum_{j=1}^N c_{ij}x_{ij} $$

Simultaneously, there are some rules (constraints) that must be satisfied:

• The number of units shipped must be less than or equal to the total supply.
• The number shipped must match, or meet, the demand at each location.
• The number of units to ship must be greater than or equal to zero (no negative values).

In the case of a balanced transportation problem, for which the total demand is equal to the total supply, the constraints have the following mathematical representation:

$$ \sum_{j=1}^N x_{ij} = S_{i}, i=1, …, M $$

$$ \sum_{i=1}^M x_{ij} = D_{j}, j=1, …, N $$

$$ x_{ij} \geqslant 0, i=1 …,M, j=1 …N $$

Note that there may be cases of excess supply or excess demand leading to an unbalanced problem. This case is addressed below and can be solved similarly using dummy variables and possibly penalties for unmet demand or storage costs for excess supply.

The total cost function along with the three constraints define a well-formed linear programming (LP) optimization problem with linear constraints. It is known specifically as the balanced transportation problem.

Unlike many LP problems, the transportation problem is feasible to solve by hand using a series of tables and well-documented strategies such as the Northwest-Corner Method to find an initial basic feasible solution and then using techniques like the Least-Cost Method or the Stepping Stone Method .

Working through a problem by hand simply requires repeatedly drawing tables and manual refinement starting with an initial basic feasible solution. While it can be instructive, solving by hand is not practical or scalable for real-world problems.

For larger problems or to realize any level of scalability, a computer-based method is preferred. Spreadsheet solutions are possible as well, but often require significant rework whenever the number of supply or demand centers changes.

The IMSL Library algorithm allows users to solve problems of nearly any size with a simple programming interface. The IMSL C function is imsl_f_transport() with a short list of required arguments and a few optional arguments. Using the terminology introduced above, the calling sequence using only the required arguments is:

where each variable is defined as:

• x = a pointer to the solution matrix of size M x N containing the solution (the optimal routing of supply)
• M = the number of sources (supply locations)
• N = the number of sinks (demand locations)
• S = an array of size \(M\) containing the supply capacities
• D = an array of size \(N\) containing the demand requirements
• c = an array of size \(M\) x \(N\) containing the per-unit cost matrix

Want to try this for yourself? Request a free trial of IMSL C now. >>

Optional arguments to the function provide additional capabilities such as limiting the number of iterations (by default there is no limit) and output of the dual solution and total cost of the optimal routing.

The algorithm used is a revised simplex method to solve a sparse LP problem with \(M\) + \(N\) constraints and \(M\) x \(N\) variables. Further details of the algorithm can be found in chapter 5 of Sewell (2005).

Consider the word problem stated above again, but this time with some actual values. XYZ Inc. has two factories with weekly production rates of 40 and 20 widgets. These widgets must be shipped to three warehouses that have a demand of 25, 10, and 25 units. The cost to ship between each location is known (see the grid below). Which factory should produce and ship how many widgets to which warehouses to meet the demand at each location with minimal cost?

The standard table format of this problem is:

This table format is used throughout this paper for both the problem statement and solutions. When the upper leftmost cell is “(solution)” the contents are the solution matrix, otherwise it’s a cost matrix of specified units. The central values in the table represent the point-to-point per unit shipping cost, such that it costs $550 to ship from Factory A to Warehouse 1. Note that this is a balanced problem where the total supply of 60 units equals the total demand.

Balanced Transportation Problem Solution

The C code to solve this problem definition using the IMSL algorithm is as follows, including the optional output parameter to display the total cost:

Executing the program produces the following output:

Using the previous table format, replacing unit costs with the number of units gives:

This table view allows easy confirmation that the supply for each factory and the demand at each warehouse is fully satisfied.

Not every transportation is necessarily balanced. When supply exceeds demand, an alternate technique is not required, especially if there are no storage costs associated with not sending units to the destination. However, when there is excess demand, the problem statement is infeasible as there is no solution that can satisfy the demand at each destination. The IMSL algorithm will issue a warning of IMSL_INSUFFICIENT_CAPACITY if this condition is encountered, although it will still return the best solution possible. When attempting to solve an unbalanced problem, best practices dictate the use of dummy variables to take up the slack.

Consider an example of allocating rental cars. For simplicity, generic locations will be used and the cars are assumed to be fungible (any car can be substituted for any other car). There is also a storage cost to keep an unwanted car at an origin location which must be accounted for. The base problem given is: Sites A, B, and C have 8, 12, and 10 cars on site, while Destinations X, Y, and Z require 9, 7, and 11 cars respectively. The storage cost for each site is 100, 100, and 80. The cost to move the cars between sites is shown in the table:

Unbalanced Transportation Problem Solution

From this problem statement, it is clear that demand (27 cars) is less than the supply (30 cars), so three cars will need to be kept at one or more source locations. To account for this, one would add a dummy destination site with the appropriate storage cost:

In this form, the problem is balanced. Except for now including the storage cost and location of the extra supply, the solution returned from the IMSL transport function is the same in either case, thus there is no strict requirement in the algorithm for the problem to be balanced. The source code for this problem is included in Appendix A, with the solution matrix as follows, and a total cost of $12,880:

Alternatively, if a problem indicates an excess demand, a similar dummy column can be utilized, especially if there is a known cost for missing the demand (like lost revenue from dissatisfied customers). Sometimes there is no cost associated with missing demand or excess supply, and in those cases the value in the cost matrix for the dummy row or column would simply be zero.

There may also be special cases where one or more destinations cannot be reached from one or more source locations. In this case, the user should use an arbitrarily large number in the cost matrix to encourage the algorithm to allocate zero units for this unreachable condition. A value an order of magnitude or two larger than any other cost in the matrix usually suffices.

Sewell, Granville (2005), Computational Methods of Linear Algebra, second edition, John Wiley & Sons, New York.

Final Thoughts

In this blog, we looked at the transportation problem and demonstrated the use of the IMSL C algorithm imsl_f_transport() to solve both balanced and unbalanced transportation problems.

In our next blog, we’ll look at two special cases of the transportation problem, the assignment problem, and the transshipment problem.

Easily Solve the Transportation Problem With IMSL

Solving the transportation problem, regardless of the complexity, is much more surmountable with transportation algorithms included in IMSL.

Whether you’re working in C/C++, Fortran, Java, or Python, you can evaluate the IMSL library for your application free. Try it today via the link below.

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Unbalanced Problem - Meaning & Definition

What is unbalanced problem.

Unbalanced problems are typically encountered in transportation problems in operations research where the total supply does not equal the total demand. The main objective of a transportation problem is to determine the transportation schedule that minimizes the overall transportation cost, while meeting the supply and demand requirements. But in reality, the problems that we encounter involve unbalanced conditions where the supply and demand are not equal.

Let us consider a problem where a company has three warehouses – warehouse 1, warehouse 2, and warehouse 3. The company provides supplies to retailers Retailer A and Retailer B. The supply and demand for the warehouses and retailers is shown below. The table shows the supply from each warehouse, the demand of individual retailer and the distance between the individual warehouses and the retailers. The cost of transportation can be taken proportionate to the distance between the warehouse and retailer.

The problem can be simplified by introducing dummy dealers or dummy suppliers, who will either supply or consume the excess. So the above problem boils down to

The cost of transportation from the individual warehouses to the retailers is taken as zero, because the dummy supplier or dealer does not exist in reality. Now, algorithms or linear programming equations can be used to find out find out a solution.

Hence, this concludes the definition of Unbalanced Problem along with its overview.

This article has been researched & authored by the Business Concepts Team . It has been reviewed & published by the MBA Skool Team. The content on MBA Skool has been created for educational & academic purpose only.

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Transportation Problem Explained and how to solve it?

• Introduction
• Transportation Problem
• Balanced Problem
• Unbalanced Problem

Contributed by: Patrick

Operations Research (OR) is a state of art approach used for problem-solving and decision making. OR helps any organization to achieve their best performance under the given constraints or circumstances. The prominent OR techniques are,

• Linear programming
• Goal programming
• Integer programming
• Dynamic programming
• Network programming

One of the problems the organizations face is the transportation problem. It originally means the problem of transporting/shipping the commodities from the industry to the destinations with the least possible cost while satisfying the supply and demand limits.  It is a special class of linear programming technique that was designed for models with linear objective and constraint functions. Their application can be extended to other areas of operation, including

• Scheduling and Time management
• Network optimization
• Inventory management
• Enterprise resource planning
• Process planning
• Routing optimization  

The notations of the representation are:

m sources and n destinations

(i , j) joining source (i) and destination (j)

c ij 🡪  transportation cost per unit

x ij 🡪  amount shipped

a i   🡪 the amount of supply at source (i)

b j   🡪 the amount of demand at destination (j)

Transportation problem works in a way of minimizing the cost function. Here, the cost function is the amount of money spent to the logistics provider for transporting the commodities from production or supplier place to the demand place. Many factors decide the cost of transport. It includes the distance between the two locations, the path followed, mode of transport, the number of units that are transported, the speed of transport, etc. So, the focus here is to transport the commodities with minimum transportation cost without any compromise in supply and demand. The transportation problem is an extension of linear programming technique because the transportation costs are formulated as a linear function to the supply capacity and demand. Check out the course on transportation analytics .

Transportation problem exists in two forms. 

• Balanced 

It is the case where the total supply equals the total demand.

It is the case where either the demand is greater than the supply, or vice versa.

In most cases, the problems take a balanced form. It is because usually, the production units work, taking the inventory and the demand into consideration. Overproduction increases the inventory cost whereas under production is challenged by the demand. Hence the trade-off should be carefully examined. Whereas, the unbalanced form exists in a situation where there is an unprecedented increase or decrease in demand.

Let us understand this in a much simpler way with the help of a basic example. 

Let us assume that there is a leading global automotive supplier company named JIM. JIM has it’s production plants in many countries and supplies products to all the top automotive makers in the world. For instance, let’s consider that there are three plants in India at places M, N, and O. The capacity of the plants is 700, 300, 550 per day. The plant supplies four customers A, B, C, and D, whose demand is 650, 200, 450, 250 per day. The cost of transport per unit per km in INR and the distance between each source and destination in Kms are given in the tables below.

Here, the objective is to determine the unknown while satisfying all the supply and demand restrictions. The cost of shipping from a source to a destination is directly proportional to the number of units shipped.

Many sophisticated programming languages have evolved to solve OR problems in a much simpler and easier way. But the significance of Microsoft Excel cannot be compromised and devalued at any time. It also provides us with a greater understanding of the problem than others. Hence we will use Excel to solve the problem.

It is always better to formulate the working procedure in steps that it helps in better understanding and prevents from committing any error.

Steps to be followed to solve the problem:

• Create a transportation matrix (define decision variables)
• Define the objective function
• Formulate the constraints
• Solve using LP method 

Creating a transportation matrix:

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.

Define the objective function: 

An objective function is our target variable. It is the cost function, that is, the total cost incurred for transporting. It is known as an objective function because our interest here is to minimize the cost of transporting while satisfying all the supply and demand restrictions.

The objective function is the total cost. It is obtained by the sum product of the cost per unit per km and the decision variables (highlighted in red), as the total cost is directly proportional to the sum product of the number of units shipped and cost of transport per unit per Km.

The column “Total shipped” is the sum of the columns A, B, C, and D for respective rows and the row “Total Demand” is the sum of rows M, N, and O for the respective columns. These two columns are introduced to satisfy the constraints of the amount of supply and demand while solving the cost function. 

Formulate the constraints:

The constraints are formulated concerning the demand and supply for respective rows and columns. The importance of these constraints is to ensure they satisfy all the supply and demand restrictions.

For example, the fourth constraint, x ma + x na + x oa = 650 is used to ensure that the number of units coming from plants M, N, and O to customer A should not go below or above the demand that A has. Similarly the first constraint x ma + x mb + x mc + x md  = 700 will ensure that the capacity of the plant M will not go below or above the given capacity hence, the plant can be utilized to its fullest potential without compromising the inventory. 

Solve using LP method:

The simplest and most effective method to solve is using solver. The input parameters are fed as stated below and proceed to solve. 

This is the best-optimized cost function, and there is no possibility to achieve lesser cost than this having the same constraints.

From the solved solution, it is seen that plant M ships 100 units to customer A, 350 units to C and 250 units to D. But why nothing to customer B? And a similar trend can be seen for other plants as well. 

What could be the reason for this? Yes, you guessed it right! It is because some other plants ship at a profitable rate to a customer than others and as a result, you can find few plants supplying zero units to certain customers. 

So, when will these zero unit suppliers get profitable and can supply to those customers? Wait! Don’t panic. Excel has got away for it too. After proceeding to solve, there appears a dialogue box in which select the sensitivity report and click OK. You will get a wonderful sensitivity report which gives details of the opportunity cost or worthiness of the resource.

Basic explanation for the report variables,

Cell: The cell ID in the excel

Name: The supplier customer pairing

Final value: Number of units shipped (after solving)

Reduced cost: How much should the transportation cost per unit per km should be reduced to make the zero supplying plant profitable and start supplying

Objective coefficient: Current transportation cost per unit per Km for each supplier customer pair

Allowable Increase: It tells us the maximum cost of the current transportation cost per unit per Km can be increased which doesn’t make any changes to the solution

Allowable Decrease: It tells how much maximum the current transportation cost per unit per Km can be lowered which doesn’t make any changes to the solution

Here, look into the first row of the sensitivity report. Plant M supplies to customer A. Here, the transportation cost per unit per Km is ₹14 and 100 units are shipped to customer A. In this case, the transportation cost can increase a maximum of ₹6, and can lower to a maximum of ₹1. For any value within this range, there will not be any change in the final solution. 

Now, something interesting. Look at the second row. Between MB, there is not a single unit supplied to customer B from plant M. The current shipping cost is ₹22 and to make this pair profitable and start a business, the cost should come down by ₹6 per unit per Km. Whereas, there is no possibility of increasing the cost by even a rupee. If the shipping cost for this pair comes down to ₹16, we can expect a business to begin between them, and the final solution changes accordingly.

The above example is a balanced type problem where the supply equals the demand. In case of an unbalanced type, a dummy variable is added with either a supplier or a customer based on how the imbalance occurs.

Thus, the transportation problem in Excel not only solves the problem but also helps us to understand how the model works and what can be changed, and to what extent to modify the solution which in turn helps to determine the cost and an optimal supplier. 

If you found this helpful, and wish to learn more such concepts, head over to Great Learning Academy and enroll in the free online courses today.

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