The transportation problem is a classical linear programming problem that involves shipping goods from several sources to several destinations while minimizing the total shipping cost. The least-cost method is one of the popular methods used to solve transportation problems.

In this article, we will discuss the least cost method and its application to solve transportation problems.

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**Introduction to Transportation Problem**

A transportation problem is a linear programming problem that involves determining the optimal way to transport goods from several sources to several destinations. The objective of the problem is to minimize the total cost of transportation while satisfying the supply and demand constraints.

**The Least Cost Method**

The least cost method is one of the popular methods used to solve transportation problems. In this method, we start by allocating the minimum cost cell (i.e., cell with the smallest transportation cost) to the respective supply and demand values.

The following steps are involved in the least cost method:

**Step 1**: Identify the minimum cost cell in the transportation matrix.**Step 2**: Allocate as much as possible to the respective supply and demand values.**Step 3**: Update the remaining supply and demand values.**Step 4**: Repeat steps 1 to 3 until all the supply and demand values are satisfied.

**An Example of Solving a Transportation Problem Using the Least Cost Method**

Let us consider a transportation problem with three sources (S1, S2, and S3) and three destinations (D1, D2, and D3). The transportation costs and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 3 | 5 | 4 | 150 |

S2 | 2 | 6 | 3 | 200 |

S3 | 4 | 2 | 5 | 100 |

Demand | 120 | 180 | 150 |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S3, D2) with a transportation cost of 2.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 100, to the cell (S3, D2).

**Step 3**: Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 3 | 5 | 4 | 150 |

S2 | 2 | 6 | 3 | 200 |

S3 | 4 | 5 | ||

Demand | 120 | 80 | 150 |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S1, D1) with a transportation cost of 3.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 120, to the cell (S1, D1).

**Step 3**: Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 5 | 4 | ||

S2 | 2 | 6 | 3 | 200 |

S3 | 4 | 5 | ||

Demand | 80 | 150 |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S1, D3) with a transportation cost of 4.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 150, to the cell (S1, D3).

**Step 3:** Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 5 | |||

S2 | 2 | 6 | 3 | 200 |

S3 | 4 | 1 | 50 | |

Demand | 80 |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S2, D1) with a transportation cost of 2.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 80, to the cell (S2, D1).

**Step 3**: Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 5 | |||

S2 | 6 | 3 | 120 | |

S3 | 4 | 1 | 50 | |

Demand |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S3, D3) with a transportation cost of 1.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 50, to the cell (S3, D3).

**Step 3**: Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 5 | |||

S2 | 6 | 2 | 120 | |

S3 | 4 | |||

Demand |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S2, D3) with a transportation cost of 2.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 2, to the cell (S2, D3).

**Step 3**: Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 5 | |||

S2 | 4 | 118 | ||

S3 | 4 | |||

Demand |

**Step 1**: Identify the minimum cost cell in the transportation matrix.

The minimum cost cell in the transportation matrix is (S2, D2) with a transportation cost of 4.

**Step 2**: Allocate as much as possible to the respective supply and demand values.

We allocate the maximum possible value, which is 118, to the cell (S2, D2).

**Step 3**: Update the remaining supply and demand values.

The updated transportation matrix and supply and demand values are given in the following table:

Source/Destination | D1 | D2 | D3 | Supply |
---|---|---|---|---|

S1 | 5 | |||

S2 | 0 | |||

S3 | 4 | |||

Demand |

As we have satisfied all the demand values, we can stop the algorithm. The total cost of transportation is given by the sum of the products of the allocated quantities and their corresponding transportation costs, which is:

**Total Cost** = (5 x 6) + (150 x 4) + (50 x 1) + (80 x 2) + (2 x 3) + (50 x 1) + (2 x 4) + (118 x 4) = 1466

In this way, we can solve transportation problems using the least cost method. However, it should be noted that this method may not always give the optimal solution, and other methods such as Vogel’s approximation method or the North-West corner method may need to be used. It is also possible to use software packages such as Excel Solver or OpenSolver to solve transportation problems quickly and efficiently.