Transportation & Transshipment Models: Initial Solution Methods (NWCM, MMM & VAM)

Transshipment Model or Transportation Model

The transportation problem is one of the subclasses of linear programming problem where the objective is to transport various quantities of a single homogeneous product that are initially stored at various origins, to different destinations in such a way that the total transportation is minimum. Transportation models or problems are primarily concerned with the optimal (best possible) way in which a product produced at different factories or plants (called supply origins) can be transported to a number of warehouses (called demand destinations)

The objective in a transportation problem is to fully satisfy the destination requirements within the operating production capacity constraints at the minimum possible cost. Whenever there is a physical movement of goods from the point of manufacture to the final consumers through a variety of channels of distribution (wholesalers, retailers, distributors etc.), there is a need to minimize the cost of transportation so as to increase the profit on sales. It is a transportation model with intermediate destination between the source and the destination. For example – goods are often transported for manufacturing plants to distribution centers or warehouse, then finally to stores constraint involving source & destination are similar –

1. Everything leaving source must not exceed supplies.
2. Everything entering destination must not exceed demand.

New constraint – Everything entering an intermediate point must equal everything leaving. 

Initial solution for a transportation problem

1. North – west corner method 

The North West corner rule is a method for computing a basic feasible solution of a transportation problem where the basic variables are selected from the North – West corner (i.e., top left corner). 

Steps

• Select the north west (upper left-hand) corner cell of the transportation table and allocate as many units as possible equal to the minimum between available supply and demand requirements, i.e., min (s1, d1).
• Adjust the supply and demand numbers in the respective rows and columns allocation.
• If the supply for the first row is exhausted then move down to the first cell in the second row.
• If the demand for the first cell is satisfied then move horizontally to the next cell in the second column.
• If for any cell supply equals demand then the next allocation can be made in cell either in the next row or column.
• Continue the procedure until the total available quantity is fully allocated to the cells as required.

2. Minimum Matrix Method (MMM) 

Matrix minimum method is a method for computing a basic feasible solution of a transportation problem where the basic variables are chosen according to the unit cost of transportation. 

Steps

• Identify the box having minimum unit transportation cost (cij).
• If there are two or more minimum costs, select the row and the column corresponding to the lower numbered row.
• If they appear in the same row, select the lower numbered column.
• Choose the value of the corresponding xij as much as possible subject to the capacity and requirement constraints.
• If demand is satisfied, delete the column.
• If supply is exhausted, delete the row.
• Repeat steps 1-6 until all restrictions are satisfied.

3. Vogel’s Approximation Method (VAM) 

The Vogel approximation method is an iterative procedure for computing a basic feasible solution of the transportation problem. 

Steps

• Identify the boxes having minimum and next to minimum transportation cost in each row and write the difference (penalty) along the side of the table against the corresponding row.
• Identify the boxes having minimum and next to minimum transportation cost in each column and write the difference (penalty) against the corresponding column
• Identify the maximum penalty. If it is along the side of the table, make maximum allotment to the box having minimum cost of transportation in that row. If it is below the table, make maximum allotment to the box having minimum cost of transportation in that column.
• If the penalties corresponding to two or more rows or columns are equal, select the top most rows and the extreme left column.

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Sandeep Ghatuary

Finance & Accounting blogger simplifying complex topics.

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