The North-West Corner Method is a powerful technique for solving transportation problems in further mathematics. Its simplicity and effectiveness make it a popular choice for decision-making and resource allocation tasks. This article will provide an in-depth understanding of the method, its applications in decision mathematics, and a step-by-step guide to solving transportation problems using this approach. In addition, it will cover examples of real-life problems and offer tips and techniques for effectively dealing with unbalanced transportation situations. So, fasten your seatbelts as you embark on a journey to master the North-West Corner Method in further mathematics.
Understanding the North-West Corner Method
The North-West Corner Method is a simple and efficient technique used primarily for solving transportation problems in operations research and linear programming. The objective of these problems is typically to allocate resources optimally from a set of origins or sources to a set of destinations or sinks while minimizing overall transportation costs.
A transportation problem can be mathematically represented as finding the optimal solution to the following linear programming problem:
\[ \text{Minimise } Z = \sum_{i=1}^{m}\sum_{j=1}^{n} c_{ij}x_{ij} \]
Subject to the following constraints:
- \(\sum_{j=1}^{n} x_{ij} = a_i\) for \(i = 1, 2, ..., m\)
- \(\sum_{i=1}^{m} x_{ij} = b_j\) for \(j = 1, 2, ..., n\)
- \(x_{ij} \ge 0\) for all \(i = 1, 2, ..., m\) and \(j = 1, 2, ..., n\)
Where:
- \(m\) is the number of origins (sources)
- \(n\) is the number of destinations (sinks)
- \(c_{ij}\) is the unit cost of transporting resources from Origin \(i\) to Destination \(j\)
- \(x_{ij}\) represents the quantity of resources transported from Origin \(i\) to Destination \(j\)
- \(a_i\) and \(b_j\) represent the supply and demand for each Origin \(i\) and Destination \(j\), respectively
Note that a transportation problem is considered balanced if the total supply equals the total demand, i.e., \(\sum_{i=1}^{m} a_i = \sum_{j=1}^{n} b_j\). If the problem is unbalanced, a dummy source or destination is added to make the problem balanced before applying the North-West Corner Method.
The North-West Corner Method involves the following steps:
- Choose the first (north-west) cell of the transportation table as the starting point.
- Allocate as many resources as possible to the chosen cell, without exceeding the supply or demand constraints for that cell.
- If the supply or demand of a row or column is completely satisfied, move to the next available cell in the remaining rows or columns, respectively.
- Repeat steps 2 and 3 until all supplies and demands are satisfied.
Example: Suppose there are two origins (O1 and O2) with supplies of 40 and 60 units, respectively, and three destinations (D1, D2, and D3) with demands of 50, 30, and 20 units. The unit costs of transportation are given in the following table:
Applying the North-West Corner Method for this example, the optimal allocation results in the following transportation table:
The total cost of transportation is 330 units.
Applications of the North-West Corner Method in Decision Mathematics
The North-West Corner Method is an essential tool in decision mathematics, as it provides an easy-to-follow approach to solve a wide range of real-world resource allocation problems. Some common applications include:
- Transportation and logistics: determining the most efficient route for transporting goods from factories/warehouses to retailers/customers
- Supply chain management: finding the optimal distribution of raw materials or finished goods from suppliers to manufacturers or distribution centers
- Economics: understanding the effects of different subsidy systems on resource allocation in various market sectors
- Network routing: efficiently routing information packets in telecommunication networks or internet data exchange
Although the North-West Corner Method may not always provide the optimal solution to a transportation problem, it offers a quick and straightforward initial solution that can serve as the starting point for more advanced optimization techniques, such as the stepping-stone method or the modified distribution (MODI) method.
How to Solve the Transportation Problem Using North-West Corner Method
The North-West Corner Method provides a simple and systematic approach to solving transportation problems by allocating resources from a set of origins to a set of destinations while minimizing overall transportation costs. In this section, we will delve deeper into the process and explore examples and various scenarios for using this method.
North-West Corner Method example
Let us start with a detailed example to understand how the North-West Corner Method works in practice. Suppose we have a transportation problem involving three origins (O1, O2, and O3) with supplies of 40, 50, and 60 units, respectively, and four destinations (D1, D2, D3, and D4) with demands of 30, 35, 45, and 40 units. The unit costs of transportation are given in the table below:
| D1 | D2 | D3 | D4 |
| O1 | 3 | 1 | 7 | 4 |
| O2 | 2 | 6 | 5 | 2 |
| O3 | 8 | 3 | 3 | 2 |
Following the steps outlined previously for the North-West Corner Method, we proceed as below:
- Select the top-left cell (O1-D1) as the starting point.
- Allocate the maximum possible amount (30 units) to the selected cell without violating the supply and demand constraints.
- Eliminate the row or column if its supply or demand is satisfied entirely, and move to the next available cell in the remaining rows or columns. In this case, both O1's supply is satisfied, and D1's demand is satisfied, so we move to the next available cell O2-D2.
- Repeat steps 2 and 3 until all supplies and demands are satisfied.
By applying the method, the resulting transportation table is as follows:
| D1 | D2 | D3 | D4 |
| O1 | 30 | 0 | 0 | 10 |
| O2 | 0 | 35 | 5 | 10 |
| O3 | 0 | 0 | 40 | 20 |
The total cost of transportation, in this case, is 600 units.
Transportation model North-West Corner Method
The transportation model is a mathematical representation of the transportation problem. It consists of origins, destinations, supplies, demands, and costs associated with transporting resources from each origin to each destination. When using the North-West Corner Method to solve a transportation model, we can summarise the process into the following steps:
- Check if the transportation problem is balanced (i.e., total supply equals total demand). If not, add a dummy origin or destination to balance the problem.
- Construct the transportation table, listing origins on the vertical axis and destinations on the horizontal axis. Fill in the associated unit transportation costs for each cell.
- Apply the North-West Corner Method as previously described, starting with the north-west corner of the table and moving through the cells to allocate resources optimally.
- Calculate the total transportation cost by summing the cost of each allocated resource multiplied by its corresponding unit transportation cost.
The transportation model North-West Corner Method provides a robust framework for solving transportation problems, though it might not guarantee the optimal solution. However, it offers an efficient starting point for more advanced optimisation techniques.
Unbalanced transportation Problem Using North-West Corner Method Example
An unbalanced transportation problem occurs when the total supply does not equal the total demand. Before applying the North-West Corner Method to an unbalanced problem, we must first balance the problem by introducing a dummy origin or destination with a supply or demand equal to the imbalance. Let's consider an example of an unbalanced transportation problem.
Suppose there are two origins (O1 and O2) with supplies of 40 and 60 units, respectively, and three destinations (D1, D2, and D3) with demands of 30, 40, and 20 units. Notice that the total supply (100 units) is greater than the total demand (90 units). To balance the problem, we introduce a dummy destination (D4) with a demand equal to the imbalance (10 units).
After adding the dummy destination, the unit costs of transportation are given in the table below:
Now, we can apply the North-West Corner Method to the balanced transportation problem to find the optimal resource allocation. Once the problem is solved, you can exclude the costs and allocations associated with the dummy destination to get the final solution for the original unbalanced problem.
Step-by-Step Guide to the North-West Corner Method
The North-West Corner Method consists of several essential steps, which we will now discuss in greater detail. It is essential to understand and follow these steps to apply the method accurately in solving transportation problems:
- Check for balance: Ensure the transportation problem is balanced by verifying that the total supply equals the total demand. If the problem is unbalanced, add a dummy origin or destination to balance the problem.
- Construct the transportation table: Create a table with origins along the rows and destinations along the columns. Fill in the corresponding unit transportation costs for each cell. If applicable, include the costs associated with the dummy origin or destination.
- Initialise the starting point: Begin at the top-left corner (north-west) of the transportation table.
- Allocate resources: Allocate the maximum possible amount to the selected cell without exceeding the supply or demand constraints for that cell.
- Update the table: If the supply or demand of a row or column is satisfied entirely, mark the row or column as exhausted and move to the next available cell in the remaining rows or columns, respectively.
- Repeat step 5: Continue allocating resources and updating the table until all supplies and demands have been satisfied.
- Calculate total cost: Determine the total transportation cost by summing the product of each allocated resource and the corresponding unit transportation cost.
Dealing with unbalanced problems in the North-West Corner Method
Unbalanced problems in the North-West Corner Method require special attention. It is crucial to balance these problems correctly to avoid any inaccuracies in the final solution. To handle unbalanced transportation problems, follow these steps:
- Identify the imbalance: Calculate the difference between the total supply and total demand. If the supply is greater than the demand, add a dummy destination. If demand is greater than supply, add a dummy origin.
- Update the transportation table: Add the dummy origin or destination to the table and assign an appropriate supply or demand to balance the problem. Update the unit transportation costs for the dummy row or column, typically using zero costs.
- Proceed with the North-West Corner Method: Apply the method as previously explained, including the dummy row or column in the resource allocation process.
- Remove the dummy row or column: Once the problem is solved, disregard the dummy row or column and its associated costs to arrive at the final solution.
Tips for efficient solving with the North-West Corner Method
To ensure that the North-West Corner Method is applied efficiently and accurately, consider the following tips:
- Double-check your inputs: Carefully entering the supply, demand, and cost data will help to avoid errors that could lead to incorrect results.
- Visualise allocation steps: Tracking the allocation process step-by-step can help to identify any potential mistakes and refine your understanding of the method.
- Confirm balance and constraints: Routinely ensure that the supply and demand constraints are met during resource allocation.
- Compare with alternative methods: Although the North-West Corner Method provides an efficient initial solution, verifying the results with other techniques, such as the stepping-stone method or the MODI method, can help confirm the optimality of the solution.
- Practice with a variety of problems: Solving a range of transportation problems with different complexities will help to build your expertise and familiarity with the North-West Corner Method.
By applying these tips in conjunction with the essential steps and procedures for dealing with unbalanced problems, you can efficiently and effectively solve transportation problems using the North-West Corner Method.
The North-West Corner Method - Key takeaways
- The North-West Corner Method: a simple and efficient technique for solving transportation problems in operations research and linear programming.
- Objective: allocate resources optimally from a set of origins or sources to a set of destinations or sinks while minimizing overall transportation costs.
- Steps: select the first cell, allocate maximum possible resources to the cell without exceeding supply or demand constraints, and move to the next cell until all supplies and demands are satisfied.
- Dealing with unbalanced problems: balance the problem by adding a dummy origin or destination to equalize supply and demand before applying the method.
- Applications: transportation and logistics, supply chain management, economics, and network routing.
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