Difference Between Transportation and Assignment Problems

Both the transportation and the assignment problems are parts of linear programming. While the transportation problem is concerned with the optimal distribution of resources and goods from multiple sources to the destinations, assignment problems deal with the allocation of tasks, resources, and jobs on a one-on-one basis. Both these methods are of inherent use in resource allocation, minimization of cost, planning workforce, management of supply chain, management of time, and decision making.

Transportation Problem

A transportation problem is a Linear Programming problem that involves determining the best solution for transportation and allocating resources to various destinations and from one location to another while keeping costs to a minimal. The primary goal of the Transportation problem is to deliver resources (from source to destination) at the lowest possible cost.

1. Components of the transportation problem

The transportation problem is made up of numerous important components that define its structure and influence how it might be modeled and solved.

• Supply nodes or sources are the points from which products are shipped. These could be factories, warehouses, or distribution centers.
• Demand nodes or destinations are the locations where commodities are delivered. These could include retail stores, consumers, and other warehouses.
• The cost matrix shows the cost of transporting one unit of products from each supply node to each demand node.
• The objective function calculates the overall transportation cost and seeks to minimize it while meeting supply and demand limitations.
• Feasibility Conditions under which a workable solution can be found. These include ensuring that overall supply equals or exceeds total demand.

2. Types of transportation problem

Transportation challenges can be classified depending on a variety of characteristics, including the nature of supply and demand, the structure of the cost matrix, and the constraints involved. It is classified as follows,

• The total supply will be equal to the total demand in a balanced transportation problem.
• In the unbalanced transportation problem, the total supply will not equal the total demand.
• In the symmetric transportation problem, the cost of transporting the goods from a supply node to the demand node is the same in both directions.
• In the asymmetric transportation problem, the transportation cost will be different in each direction.
• It is also categorized as a single and multi-commodity transportation problem depending on the type of goods transported.

3. Solution for the transportation problem

There are four different solution approaches to find the initial and the most feasible solution for the transportation problem.

• North-West Corner Method: The solution is obtained by starting from the top-left corner and working the way down or right to find an initial workable solution.
• Least Cost Method: Selecting the lowest cost cell yields an initial workable answer.
• Vogel’s Approximation Method (VAM): The penalty costs are taken into account to arrive at an initial solution.
• Modified Distribution Method (MODI): This is used to ensure that the initial solution is optimal and, if necessary, improve it.

=> Read Also: Difference Between Logistics and Transportation

Assignment Problem

An Assignment Problem is a form of Transportation Problem in Operations Research that involves assigning workers or instances to jobs or machines. Each origin and destination must be assigned to a single origin. The Hungarian procedure can be used to find the solution for the assignment procedure.

1. Components of Assignment Problem

The components of the assignment problems are the agents, tasks, cost matrix, decision variables, and objective function. Agents are the entities that perform the tasks, and it includes the machines, workers and delivery vehicles. The next component is the jobs or tasks that are to be completed and could be destinations, projects, or activities. The third component is the cost matrix, which depicts the cost associated with each agent completing the activity. The decision variables are binary variables that indicate which agent is assigned to each task.

2. Solutions to solve assignment problems

The Hungarian method is the best approach to solve the assignment problem. The assignment problem can alternatively be handled using linear programming techniques, notably by transforming it into a binary integer programming problem and solving it with methods such as the simple method or specialized algorithms like branch and bound. Heuristic and metaheuristic procedures can be utilized to solve big and complicated assignment issues where accurate methods may be computationally expensive.

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Differences between Transportation and Assignment Problems

Transportation Problem	Assignment Problem
Transportation problem deals with the distribution of goods and resources from multiple sources, such as warehouses and factories, to multiple destinations, which could be either the customers or the retail stores.	This is a special type of transportation problem that assigns the workers to destinations or jobs.
By addressing the transportation problem, we can easily minimize the total transportation cost.	By addressing the assignment problem, we can minimize the total cost or the time associated with completing the tasks.
Constraints on transportation problems include constraints on supply and demand.	Constraints with the assignment problem are that every single task will be assigned only to one resource, and each resource will be assigned to exactly one task only.
Transportation problems can be used in real-world scenarios like supply chain optimization, traffic flow optimization, distribution network design, etc.	Assignment problems can be applied to scheduling personnel for jobs, allocation of machines etc.
It can be solved using Vogel’s approximation or Modi method, Least Cost method, or Northwest method.	It can be solved using the Hungarian assignment method.
The number of sources and number of demands need not be equal	The number of sources should be equal to the number of demand.