Question

1. Rift Valley Sand and Gravel Pit have contracted to provide topsoil for three residential housing developments. Topsoil can be supplied from three different “farms” as follows:

Farm Weekly capacity (cubic yards)
A 100
B 200
C 200
Demand for the topsoil generated by the construction projects is:
Project Weekly demand (cubic yards)
1 50
2 150
3 300
The manager of the sand and gravel pit has estimated the cost per cubic yard to ship over each of the possible routes:
Cost per cubic yard (in Birr)
From Project #1 Project #2Project #3
Farm A 4 2 8
Farm B 5 1 9
Farm C 7 6 3
A. Formulate a balanced transportation problem in a transportation tableau
B. Find minimum transportation cost using Northwest corner, least cost and VAM methods

Answer 1

To formulate the balanced transportation problem, create a transportation tableau with the weekly capacity of each farm and the weekly demand for topsoil from each project. To find the minimum transportation cost, use methods like Northwest corner, Least cost, or VAM.

Step-by-step explanation:

A. Formulating the balanced transportation problem:

To formulate the balanced transportation problem, we need to create a transportation tableau. This involves identifying the weekly capacity of each farm and the weekly demand for topsoil from each project. We will create a matrix with farms on one axis and projects on the other, and fill in the corresponding values.

Farm A:

- Project 1: 50 cubic yards
- Project 2: 150 cubic yards
- Project 3: 0 cubic yards

Farm B:

- Project 1: 0 cubic yards
- Project 2: 0 cubic yards
- Project 3: 300 cubic yards

Farm C:

- Project 1: 0 cubic yards
- Project 2: 0 cubic yards
- Project 3: 0 cubic yards

B. Finding minimum transportation cost:

To find the minimum transportation cost, we can use three different methods: Northwest corner, Least cost, and Vogel's Approximation Method (VAM).

Using the Northwest corner method, we start at the top-left corner of the transportation tableau and allocate the available supply to the demand one cell at a time. We move horizontally or vertically depending on which is depleted first, until all supply and demand are met.

Using the Least cost method, we start with the cell that has the lowest cost per unit and allocate the supply according to the least expensive route. We continue allocating based on cost until all supply and demand are met.

Using the VAM method, we calculate the opportunity cost for each cell by considering the difference in costs between two routes for each row and column. We allocate the supply to the cell with the lowest opportunity cost until all supply and demand are met.