Transportation Problem: Definition, Formulation, and Types

# Transportation Problem: Definition, Formulation, and Types

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Vikram Singh
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Updated on Jan 29, 2024 10:44 IST

Transportation problems are used to find the minimum cost of transportation of goods from m source to n destination. In this article, we will learn about transportation problems, formulation, types and how they differ from assignment problems.

A transportation problem in operation research is a special type of Linear Programming Problem used to optimize (minimize) the transportation cost and allocate resources from M source to N destination. This article will briefly discuss transportation problems, types of transportation problems, and how to solve them.

So, let’s dive into learning all about Transportation Problems.

## What is the Transportation Problem?

A transportation problem is a Linear Programming Problem that deals with identifying an optimal solution for transportation and allocating resources to various destinations and from one site to another while keeping the expenditure to a minimum.

In simple words, the main objective of the Transportation problem is to deliver (from the source to the destination) the resources at the minimum cost.

• It is also referred to as the Hitchcock Problem.
• It involves transporting a single product from ‘m’ source (origin) to ‘n’ destinations.
• Assumptions: The supply level of each source and the demand at each destination are known.
• Objective: To minimize the total Transportation Cost.

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## Formulation of Transportation Problem

Let you are supplying the resources from m source (Si) to n destinations (Dj) such that:

ai: the quantity available at the source Si

bj: the quantity required at the destination Dj

cij: cost of transportation of one unit resource from Si to Dj

xij: units of resources transported from Si to Dj

1<= i <= m, 1 <= j <= n

So, the Total Cost of Transposition is:

(c11x11  + c12x12 + c13x13 + …… + c1nx1n) +  (c21x21  + c22x22 + c23x23 + …… + c2nx2n) + …….. + (cm1xm1  + cm2xm2 + cm3xm3 + …… + cmnxmn)

As we already mentioned, our objective is to minimize the Total Cost:

Min Z = (c11x11  + c12x12 + c13x13 + …… + c1nx1n) +  (c21x21  + c22x22 + c23x23 + …… + c2nx2n) + …….. + (cm1xm1  + cm2xm2 + cm3xm3 + …… + cmnxmn)

Subject to:

xi1 + xi2 + ……. + xin = ai & x1j + x2j + …….. + xmn = bj

xij >= 0,

i = 1, 2, 3, ……, m, j = 1, 2, 3, ……., n

The matrix below can also represent the above diagram.

## Types of Transportation Problems

Transportation problems are broadly classified into balanced and unbalanced, depending on the source’s supply and the requirement at the destination.

### Unbalanced Transportation Problem

Example – 1: Check which types of Transportation Problem it is.

Answer – 1: From the above, we have

Total Supply = 5 + 8 + 7 + 14 = 34

Total Demand = 7 + 9 + 18 = 34

Hence, Total Supply = Total Demand

Therefore, it is a Balanced Transportation Problem.

Example – 2: Check whether the given problem is Balanced or Unbalanced.

Answer – 2: From the above matrix, we have:

Total Supply = 10 + 13 + 12 = 35

Total Demand = 8 + 5 + 4 = 17

Hence, Total Supply != Total Demand; therefore, the given transportation problem is an Unbalanced Transportation Problem.

Now, let’s see the difference between a transportation problem and an assignment problem.

## Conclusion

Transportation Problem in operational research is a special kind of linear programming problem, having an objective to find the minimum cost of transportation of goods from m source to n destination.

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## FAQs

What is Transportation Problem?

A transportation problem is a Linear Programming Problem that deals with identifying an optimal solution for transportation and allocating resources to various destinations and from one site to another while keeping the expenditure to a minimum.

What are the key elements of Transportation Problem?

The key elements of the Transporation problems are: 1. Source 2. Destination 3. Supply 4. Demand and 5. Transportation Cost

What is the objective of transportation problem?

The main of the transportation problem is to minimize the cost while meeting the demand requirements of each destination and supply cost requirements.

What are the different methods for solving transportation problem?

There are different methods to solve the transportations problem, such as: 1. NorthWest Corner Method, 2. Least Cost Method, 3. Vogel's Approximation Method, and 4. Steppingstone Method

What are the assumptions made in transportation problem?

The assumption for solving transportation problems are: 1. Transportation costs are linear and constant. 2. Supply and demand are fixed 3. Goods are homogeneous. 4. Sources and Destinations are mutually exclusive