Question

For a belanced transportation problem with 3 rows and 3 evlurnns, the number of basie variables will be ___ .

Short answer

The number of basic variables is 5.

Step-by-step solution

Understand the concept of basic variables

In a balanced transportation problem, each route or path between suppliers (rows) and demand points (columns) is potentially a variable. The number of variables that can be 'basic' or non-zero in the optimal solution is determined by the mathematical formulation of the problem.

Apply the formula for the number of basic variables

The number of basic variables in a balanced transportation problem is calculated using the formula: \[ m + n - 1 \] where \( m \) is the number of rows and \( n \) is the number of columns.

Substitute the number of rows and columns

Given that there are 3 rows and 3 columns, substitute these values into the formula:\[ 3 + 3 - 1 = 5 \]

Calculate the result

Perform the calculations:\[ 3 + 3 - 1 = 5 \]Therefore, the number of basic variables in the solution is 5.

Key concepts

Basic Variables

In a balanced transportation problem, basic variables are essential for determining solutions that satisfy both supply and demand constraints. Think of them as the active components or 'routes' that carry goods between suppliers and demand points. In simpler terms, when dealing with a matrix structure of a transportation problem, each cell or route can either carry some load (be basic) or carry no load (be non-basic). Basic variables are the minimum set of non-zero variables required to maintain a feasible solution. They are usually determined by subtracting one from the total sum of rows and columns, expressed mathematically as: \[ m + n - 1 \]This means that in a transportation agreement where there are 3 sources (rows) and 3 destinations (columns), exactly 5 of these routes will be used to satisfy supply and demand initially.

Mathematical Formulation

The mathematical formulation of a balanced transportation problem is designed to find the most efficient way of transporting goods from multiple suppliers to multiple buyers. Each transportation 'problem' is a mathematical model that involves minimizing the total transportation cost while respecting supply at each source and demand at each destination. Here's how it generally works:

• Each supplier has a maximum supply capacity.
• Each demand point has a specific demand requirement.
• Transportation costs are associated with moving goods from each supplier to each demand point.

The task is to solve for the variables that indicate how much goods should move via each route, while staying within the constraints. These are the basic variables we outlined earlier. The balanced problem guarantees that total supply equals total demand, making the problem more straightforward to solve.

Optimal Solution

An optimal solution in the context of a balanced transportation problem refers to the allocation of shipments that results in the least possible transportation cost. After determining the number of basic variables, the next task is to use them to reach the optimal solution. In practice, this can involve:

• Setting up a transportation tableau.
• Using methods like the Northwest Corner, Minimum Cost, or Vogel’s Approximation Method to find a starting feasible solution.
• Enhancing the initial solution to reach optimality using techniques like the Stepping Stone Method or the MODI Method.

Each approach aims to adjust the quantities shipped along the routes to meet supply and demand perfectly, ensuring that no adjustments would lower the cost further. Once optimal, all shipments will fulfill the basic variables while minimizing expenses.

Supply and Demand in Transportation Models

Supply and demand in transportation models function as the core balancing factors that drive the problem-solving process. Any balanced transportation model ensures that the total supply equals the total demand, which simplifies finding solutions. Here's how they work conceptually:

• Supply refers to what the suppliers are offering. It's finite and needs efficient allocation to various demands.
• Demand indicates what the end points need to receive from suppliers. It's predetermined and must be met for a solution to be valid.
• Both supply and demand are used to create constraints that the transportation problem must satisfy.

This balancing act between supply and demand defines the use of basic variables, leading to efficiently optimized transportation plans that align perfectly with operational costs and logistical constraints.