The Assignment Problem is a special case of linear programming where the objective is to assign a set of tasks to a set of agents in a way that minimizes the total cost or maximizes the total profit. Each task must be assigned to exactly one agent, and each agent can only be assigned to one task. The Assignment Problem can be formulated as follows:

Formulation:

• Let $n$

  $m$ be the number of agents.

• Let cij​ represent the cost or profit of assigning task $i$$j$.
• Let xij​ be a binary decision variable, where
  $=1$

   

  xij​=1 if task i is assigned to agent j, and xij​=0 otherwise.

The objective is to minimize or maximize the total cost or profit, subject to the following constraints:

1. Assignment Constraints:
   • Each task must be assigned to exactly one agent:
     
     ${\sum }_{\mathrm{<br>}=1}^{\mathrm{<br>}}{\mathrm{<br>}}_{}=1\text{for }\mathrm{<br>}=1,2,\dots ,\mathrm{<br>}$

      

      

      

     ∑j=1m​xij​=1for i=1,2,…,n

   • Each agent can only be assigned to one task:
     
     ${\sum }_{\mathrm{<br>}=1}^{\mathrm{<br>}}{\mathrm{<br>}}_{}=1\text{for }\mathrm{<br>}=1,2,\dots ,\mathrm{<br>}$

      

      

      

     ∑i=1n​xij​=1for j=1,2,…,m

2. Non-negativity Constraints:
   • 
     
     ${\mathrm{<br>}}_{}\ge 0$

      

      

      

     xij​≥0 for all

     
     $,\mathrm{<br>}$

      

      

      

     i,j.

3. Binary Constraints:
   • 
     
     ${\mathrm{<br>}}_{}$

      

      

      

     xij​ must be binary:

     
     $\in \{0,1\}$

      

      

      

     xij​∈{0,1} for all i,j.

Solution Methods:

1. Hungarian Algorithm:
   • The Hungarian Algorithm is a widely used method for solving the Assignment Problem efficiently. It uses a combination of assignment, augmentation, and alternation steps to find the optimal assignment.
2. Linear Programming Formulation:
   • The Assignment Problem can also be solved using linear programming techniques. By formulating the problem as a linear program and applying methods such as the simplex method or interior point methods, the optimal assignment can be determined.
3. Branch and Bound:
   • Branch and Bound is a general-purpose algorithmic technique that can be adapted to solve the Assignment Problem optimally by systematically exploring the solution space and pruning branches that cannot lead to the optimal solution.
4. Heuristic Approaches:
   • Various heuristic approaches, such as the nearest neighbor method, greedy algorithms, and genetic algorithms, can be used to find suboptimal solutions to the Assignment Problem in a reasonable amount of time, especially for large problem instances.

The Assignment Problem has numerous applications in various fields, including operations research, logistics, scheduling, and resource allocation. Its efficient solution methods make it a valuable tool for optimizing assignment decisions in practical scenarios.