operations research transportation problems and solutions

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Operations Research/Transportation and Assignment Problem

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

The whole model is:

subject to,

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

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Transportation Problem: Definition, Formulation, and Types

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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

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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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:

( c 11 x 11   + c 12 x 12 + c 13 x 13 + …… + c 1n x 1n ) +  ( c 21 x 21   + c 22 x 22 + c 23 x 23 + …… + c 2n x 2n ) + …….. + ( c m 1 x m 1   + c m 2 x m 2 + c m 3 x m 3 + …… + c mn x mn )

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

Min Z  = ( c 11 x 11   + c 12 x 12 + c 13 x 13 + …… + c 1n x 1n ) +  ( c 21 x 21   + c 22 x 22 + c 23 x 23 + …… + c 2n x 2n ) + …….. + ( c m 1 x m 1   + c m 2 x m 2 + c m 3 x m 3 + …… + c mn x mn )

Subject to:

x i 1 + x i 2 + ……. + x in = a i & x 1 j + x 2 j + …….. + 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.

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.

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

Must check: difference between transporation problem and assignment problem.

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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What is Transportation Problem?

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

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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