Transportation Problem: Definition, Formulation, and Types

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?
• Formulation of Transportation Problem
• Types of Transportation Problems
• Difference between Transportation and Assignment Problem

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 (S_i) to n destinations (D_j) such that:

a_i: the quantity available at the source S_i

b_j: the quantity required at the destination D_j

c_ij: cost of transportation of one unit resource from S_i to D_j

x_ij: units of resources transported from S_i to D_j

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

So, the Total Cost of Transposition is:

(c₁₁x₁₁  + c₁₂x₁₂ + c₁₃x₁₃ + …… + c_1nx_1n) +  (c₂₁x₂₁  + c₂₂x₂₂ + c₂₃x₂₃ + …… + c_2nx_2n) + …….. + (c_m1x_m1  + c_m2x_m2 + c_m3x_m3 + …… + c_mnx_mn)

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

Min Z = (c₁₁x₁₁  + c₁₂x₁₂ + c₁₃x₁₃ + …… + c_1nx_1n) +  (c₂₁x₂₁  + c₂₂x₂₂ + c₂₃x₂₃ + …… + c_2nx_2n) + …….. + (c_m1x_m1  + c_m2x_m2 + c_m3x_m3 + …… + c_mnx_mn)

Subject to:

x_i1 + x_i2 + ……. + x_in = a_i & x_1j + x_2j + …….. + x_mn = b_j

x_ij >= 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.

Balanced Transportation Problem

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.

	D1	D2	D3	Availability
S1	4	3	2	10
S2	2	5	0	13
S3	3	8	6	12
Required	8	5	4	–

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.

Transportation Problem vs. Assignment Problem

Transportation Problem	Assignment Problem
It is used to optimize the transportation cost.	It is about assigning a finite source to a finite destination (one source is alloted to one destination).
A number of Sources and demands may or may not be equal.	The number of sources and the number of destinations must be equal.
If demand and supply are not equal, then the transportation problem is known as the Unbalanced Transportation Problem.	If the number of rows and the number of columns are not equal, then the assignment problem is known as the Unbalanced Assignment Problem.
It requires two steps to solve:Find the Initial Solution using North West, Least Cost or Vogel ApproximationFind Optimal Solution using the MODI method.	It requires only one step to solve.The Hungarian Method is sufficient to find the optimal solutions.

Must Check: Difference Between Transporation Problem and 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.

I hope this article helps you learn more about transportation problems in operational research.

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