North West Corner Rule

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North-West Corner Method for Transportation Problems

The North-West Corner Method is a technique used to find an initial feasible solution for transportation problems. This method is particularly useful for solving transportation problems in operations research where you need to minimize the cost of transporting goods from sources to destinations.

Understanding the North-West Corner Method

Concept Overview:

  • The North-West Corner Method is named for the "north-west" (or top-left) corner of the transportation matrix where the solution process begins.
  • It focuses on filling the transportation matrix by allocating as much of the supply to the demand as possible starting from the top-left corner of the matrix.

Steps to Implement the North-West Corner Method:

  • Prepare the Transportation Table:
    • Sources (supply points) are listed along the rows.
    • Destinations (demand points) are listed along the columns.
    • Each cell in the matrix contains the cost of transporting one unit from a source to a destination.
  • Check Supply and Demand Balance:
    • Ensure that total supply equals total demand.
    • If not balanced, add dummy rows or columns with zero costs to balance the matrix. A dummy row is added if total supply exceeds total demand, and a dummy column is added if total demand exceeds total supply.
  • Allocate Supplies:
    • Start at the North-West Corner:
      • Begin at the top-left cell of the matrix.
      • Allocate the minimum of supply and demand to this cell.
      • Adjust the supply and demand values accordingly.
    • Move Horizontally or Vertically:
      • If the supply at the current cell is exhausted, move horizontally to the next cell in the same row.
      • If the demand at the current cell is satisfied, move vertically to the next cell in the same column.
    • Repeat Allocation:
      • Continue the process until all supply and demand are met.
  • Calculate Total Cost:
    • Compute the total transportation cost by multiplying the allocated units in each cell by the corresponding cost per unit and summing them up.

Example Problem and Solution:

Problem Statement:

  • Three sources: A, B, C with supplies of 50, 40, and 60 units respectively.
  • Three destinations: D, E, F with demands of 20, 95, and 35 units respectively.
  • The transportation costs are given in the matrix.

Transportation Cost Matrix:

D E F
A 5 8 6
B 9 6 8
C 6 5 7

Solution Steps:

  • Initial Allocation:
    • Cell AD (North-West Corner):
      • Allocate 20 units (min of supply at A and demand at D).
      • Update supply at A to 30 units and demand at D to 0 units.
      • Move to the next cell in the row (AE).
    • Cell AE:
      • Allocate 30 units (supply at A exhausted).
      • Update demand at E to 65 units (95 - 30).
      • Move to the next cell in the column (BE).
    • Cell BE:
      • Allocate 40 units (min of supply at B and demand at E).
      • Update supply at B to 0 units and demand at E to 25 units (65 - 40).
      • Move to the next cell in the row (CE).
    • Cell CE:
      • Allocate 25 units (min of supply at C and demand at E).
      • Update supply at C to 35 units and demand at E to 0 units.
      • Move to the next cell in the column (CF).
    • Cell CF:
      • Allocate 35 units (supply at C exhausted, demand at F met).
      • Supply and demand fully satisfied.
  • Matrix After Allocation:
D E F
A 20 30 0
B 0 40 0
C 0 25 35
  • Calculate Total Cost:
  • Total Cost = (20×5)+(30×8)+(40×6)+(25×9)+(35×6) 
Total Cost=100+240+240+225+210=1015

Key Points:

  • Feasibility:
    • The North-West Corner Method ensures an initial feasible solution but may not always be optimal. Further optimization techniques (like the Stepping Stone or MODI Method) are often required to find the optimal solution.
  • Dummy Rows/Columns:
    • Used to balance the supply and demand. They have zero transportation costs and do not affect the total cost calculation.
  • Allocation Strategy:
    • Allocation starts from the top-left corner and proceeds either horizontally or vertically. This systematic approach helps in achieving a feasible solution efficiently.
  • Cost Calculation:
    • The total transportation cost is the sum of the products of the allocated units and their respective transportation costs.

The North-West Corner Method provides a simple and effective way to begin solving transportation problems by offering an initial feasible solution that can be refined further using other optimization methods.