Various Method of Finding Initial Basic Feasible Solution-North West Corner Method

The North-West Corner method belongs to one of the popular methods of identifying the initial basic feasible solution of transportation problems under linear programming. It offers a simple and fast solution to getting started with the iterative simplex method to find the optimal solution.

Under the large-scale transportation and logistics problems, among the first steps is to find an Initial Basic Feasible Solution (IBFS). This is just like placing the goal of a race – you need to be strategically placed before working towards the finish line, or, in this case, the ideal solution.

IBFS allows all supply and demand constraints to be satisfied (without violating the constraints). There are various ways to achieve an IBFS, however the North West Corner Method (NWCM) is probably the easiest one. The present article will delve into this procedure.

Overview of Transportation Problems

The transportation problem in operations research determines the best way to get goods from several sources (factories) to several destinations (warehouses) at minimum cost.

Components of a Transportation Problem

Supply: Quantity available at each source
Demand: Quantity required at each destination
Cost Matrix: Transportation cost from each source to each destination

These problems are typically represented using a matrix or table that shows costs and constraints.

Methods to Find IBFS

Before solving a transportation problem optimally, we need a good starting solution. There are three main methods:

North West Corner Method (NWCM) – Our main focus
Least Cost Method (LCM) – Focuses on lowest cost entries
Vogel’s Approximation Method (VAM) – Considers cost penalty

Deep Dive into North West Corner Method

What is the North West Corner Method?

The North West Corner Method is a method for identifying an initial feasible solution for a transportation problem. It is easy and it does not take into account transportation costs first.

Why is it called “North West”?

It derives its name in that it begins in the top left (north-west) corner of the transportation table. From there the allocation is either to the right or downward depending on remaining supply and demand.

Assumptions for NWCM

The problem must be balanced (total supply = total demand)
Transportation costs are not considered at this stage
Allocations must satisfy both row and column constraints

Step-by-Step Explanation of NWCM

Let’s break down how the NWCM works:

Step 1: Start at the Top-Left Cell

Go to the cell at the top-left (north-west) corner of the matrix.

Step 2: Allocate Minimum of Supply or Demand

Compare supply for the row and demand for the column. Allocate the minimum of these two.

Step 3: Adjust Supply and Demand

If supply is used up, cross out the row and move down
If demand is met, cross out the column and move right

Step 4: Move to the Next Cell

Repeat the allocation process in the new cell.

Step 5: Repeat Until Table is Filled

Keep moving until all supplies and demands are exhausted.

Solved Example of NWCM

Let’s consider a small transportation problem:

	D1	D2	D3	Supply
S1	4	6	8	20
S2	5	3	7	30
S3	6	5	4	25
	15	25	35

Step-by-Step Allocation:

Start at S1-D1: min(20,15) = 15 → allocate 15 → S1 has 5 left, D1 fulfilled
Move to S1-D2: min(5,25) = 5 → allocate 5 → S1 done, D2 has 20 left
Move to S2-D2: min(30,20) = 20 → allocate 20 → S2 has 10 left, D2 done
Move to S2-D3: min(10,35) = 10 → allocate 10 → S2 done, D3 has 25 left
Move to S3-D3: min(25,25) = 25 → allocate 25 → done

This allocation satisfies all constraints and gives us an IBFS.

Characteristics of NWCM

Simple and quick to implement
Does not consider cost in allocation
Provides a basic feasible solution, but not always optimal

Here’s how the North-West Corner Method works:

Start at the North-West Corner: Begin at the top-left cell of the transportation table, which represents the intersection of the first supply location and the first demand destination.
Allocate Minimum: Allocate the minimum value between the supply available at the current row and the demand required at the current column. This minimum value is assigned to the cell you’re currently on.
Update Supply and Demand:
- Supply: Subtract the allocated value from the available supply at the current row. If the supply becomes zero at this location, move down to the next row (south) and repeat steps 2 and 3.
- Demand: Subtract the allocated value from the demand required at the current column. If the demand becomes zero at this location, move right to the next column (east) and repeat steps 2 and 3.
Continue Until Completion: Keep iterating through the steps above, allocating the minimum value, updating supply and demand, and moving to the next logical cell (south if supply is zero, east if demand is zero) until all supply and demand requirements are fulfilled.

Advantages of North-West Corner Method:

Simple and Easy to Apply: The method follows a clear set of steps, making it easy to understand and implement for beginners.
Provides a Feasible Solution: It guarantees to find an initial solution that satisfies all the supply and demand constraints.

Disadvantages of North-West Corner Method:

May Not Be Optimal: The solution obtained might not be the most cost-efficient (minimum total transportation cost) solution. It often leads to a higher initial cost compared to other methods.
Tendency to Overlook Other Options: By focusing on the top-left corner first, the method might miss out on potentially better allocation possibilities in other parts of the table.

Other Methods for Finding Initial Basic Feasible Solution:

Least Cost Method: This method prioritizes assigning shipments to the cells with the lowest transportation cost. While it can provide a more cost-effective initial solution, it can be slightly more complex to implement compared to the North-West Corner Method.
Vogels Approximation Method (VAM): This method takes into account both the transportation cost and the penalty associated with leaving a supply point unsatisfied or a demand point unmet. It aims to find a good balance between these factors to arrive at a near-optimal initial solution.

Applications of NWCM

Supply Chain Logistics: Planning initial shipment plans
Warehouse Management: Determining basic inventory distribution
Transport Planning: Drafting shipment routes

Common Mistakes in NWCM

Not updating supply/demand correctly after allocation
Skipping cells or jumping to wrong cell
Unbalanced problems being handled without dummy rows/columns

Tips for Mastering NWCM

Double-check totals before starting
Practice with varied sizes of problems
Try solving unbalanced problems using dummy rows/columns
Compare with LCM/VAM to see difference in cost

Conclusion

North West Corner Method is one of the simplest ways to begin the solution plan in the shuttle problems. Though not guaranteeing cost-effective result, it ensures constraints on supply and demand are being met – a basis to the optimization. After you have mastered this method, you’ll be well prepared to tackle tougher techniques such as MODI or Vogel’s Approximation. Practice is the key and great start is a firm grasp of NWCM!

FAQs

Q1: Is the North West Corner Method always accurate?
No, it gives a feasible solution, but not necessarily an optimal one.

Q2: Can NWCM handle unbalanced problems?
Yes, but you need to add dummy supply or demand to balance it first.

Q3: What’s the main benefit of NWCM?
Simplicity and speed in getting a starting solution.

Q4: How does NWCM differ from Vogel’s method?
VAM considers penalties and costs; NWCM doesn’t.

Q5: Is NWCM useful in real-life logistics?
Yes, as a starting point for distribution planning.