Introduction
Transportation problems involve optimally distributing goods from supply locations or sources to demand locations or destinations while minimizing total transportation costs. They belong to the class of logistical problems and have wide applications in manufacturing, distribution and supply chain management. This research paper aims to provide an in-depth review of the transportation problem, its mathematical formulation, solution methods and applications through examples.
Mathematical formulation
The transportation problem can be mathematically formulated as a linear programming problem. Consider a transportation problem with m supply locations (sources) and n demand locations (destinations). Let:
a1, a2, …, am = supplies available at each of the m supply locations
b1, b2, …, bn = demands at each of the n demand locations
cij = unit transportation cost from supply location i to demand location j
xij = amount of goods to be transported from supply location i to demand location j
The objective is to determine the values of xij so as to minimize the total transportation cost, given by:
Minimize Z = ΣiΣj cijxij (1)
Subject to the following supply and demand constraints:
Σj xij ≤ ai for all i = 1,2,…,m (supply constraints) (2)
Σi xij ≥ bj for all j = 1,2,…,n (demand constraints) (3)
xij ≥ 0 for all i, j (non-negativity constraints) (4)
The supply constraints state that the total quantity dispatched from a supply location cannot exceed its available supply. The demand constraints state that the total quantity received at a demand location must be at least equal to its demand. The non-negativity constraints restrict the transportation quantities xij to be non-negative. This linear programming formulation converts the transportation problem into a form that can be solved using algorithms for linear programming.
Solution methods
Some commonly used methods to solve transportation problems include the North-West Corner Rule, Least Cost Method and Vogel’s Approximation Method.
North-West Corner Rule
This is one of the simplest methods to obtain an initial basic feasible solution. It works as follows:
Assign quantities in the order of rows and columns, starting from the top-left (north-west) cell.
Fully satisfy the demand/supply of the nearest cell before moving to the next cell in the matrix.
Stop when all supplies are dispatched and demands are met.
The solution obtained may not be optimal. This method provides a good starting point for improvement methods.
Least Cost Method
This iterative method aims to improve the basic feasible solution obtained by the North-West Corner Rule and reach an optimal solution. The steps are:
Obtain an initial basic feasible solution using NWCR.
Identify the shortest side – row or column with negative slack.
Find the least cost cell in the shortest side that is not already at its upper limit.
Increase the quantity in the least cost cell and decrease another quantity along the same row/column to create space while keeping feasibility.
Terminate if optimality conditions are met, else repeat steps 2-4.
Vogel’s Approximation Method (VAM)
VAM is more efficient and accurate than the above two methods. It works as follows:
Find the transport cost coefficients c*ij – normalized cost coefficients by dividing each cij by the smallest cost.
Assign initial values to cells starting from minimum c*ij and proceeding to larger values, like NWCR but without fully utilizing the closest cell.
Check for optimality conditions. If not satisfied, formulate a modified transportation problem using a penalty approach and solve it.
Repeat step 3 iteratively until optimal solution is obtained.
The underlying principle is that the minimum normalized cost cell has the highest probability of being in the optimal solution. This method is faster and more accurate compared to the others.
Applications
Some real-world applications of transportation problems include:
Production and distribution planning: Optimally allocating production quantities across plants to distribution centers and ending demands.
Freight transportation: Finding least cost routes for shipping goods via rail, road, ship from suppliers to warehouses and retailers.
Air traffic route planning: Scheduling optimal routes for aircraft movement with considerations of passenger demand and fuel costs.
Energy distribution: Optimal power flow problems involve minimizing transmission losses when distributing electricity across a network.
Logistics network design: Determining location, capacity and transportation routes in supply chain networks.
Facility location: Selecting optimal locations of plants, distribution centers, storage facilities based on transportation costs from suppliers to customers.
With advances in algorithms and computing power, large-scale and complex transportation problems involving millions of variables can now be solved to obtain near-optimal solutions in reasonable time. This has enabled their wide application in supply chain optimization.
Conclusion
This research paper discussed the mathematical formulation and modeling of transportation problems as linear programs. Commonly used classical solution methods like NWCR, LCM and VAM were explained along with their working mechanisms. Real-world applications of transportation problems across different domains were also highlighted. Transportation problem is a fundamental problem in operations research and continues to be relevant with increasing supply chain complexities in the current digital era. Advances in solution techniques have expanded its solvability to large industry-size problems.
Andrew Stroud