In a Maximization Assignment Problem, the objective is to assign tasks to agents in such a way that the total profit or benefit is maximized. Each task must be assigned to exactly one agent, and each agent can handle only one task. Here’s how the Maximization Assignment Problem can be formulated:

Formulation:

• Let
  
  $n$

   

   

  be the number of tasks and

  $\mathrm{<br>}$

   

   

  m be the number of agents.

• Let pij​ represent the profit or benefit of assigning task
  
  $\mathrm{<br>}$

   

   

  i to agent

  $\mathrm{<br>}$

   

   

  j.

• Let
  
  ${\mathrm{<br>}}_{\mathrm{<br>}}$

   

   

  xij​ be a binary decision variable, where

  $=1$

   

   

  xij​=1 if task
  i is assigned to agent

  $\mathrm{<br>}$

   

   

  j, and

  $=0$

   

   

  xij​=0 otherwise.

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

1. Assignment Constraints:
   • Each task must be assigned to exactly one agent:
     
     ${\sum }_{=1}^{\mathrm{<br>}}{\mathrm{<br>}}_{}=1\text{for }\mathrm{<br>}=1,2,\dots ,\mathrm{<br><span\; class="base"><span\; class="mop"><span\; class="mop\; op-symbol\; small-op"> </span></span></span>}$

      

      

   • 
     
     $∑j=1m​xij​=1for i=1,2,…,n$

      

      

   • 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:
   • xij​ must be binary:
     
     ${\mathrm{<br>}}_{\mathrm{<br>}}\in \{0,1\}$

      

      

     xij​∈{0,1} for all i,j.

Solution Methods:

1. Hungarian Algorithm:
   • The Hungarian Algorithm can be adapted to solve Maximization Assignment Problems by maximizing the total profit instead of minimizing the total cost. It efficiently finds the optimal assignment by iteratively finding augmenting paths in the assignment graph.
2. Linear Programming Formulation:
   • The Maximization Assignment Problem can also be formulated as a linear program, where the objective is to maximize the total profit subject to the assignment constraints. Linear programming techniques, such as the simplex method or interior point methods, can then be applied to find the optimal solution.
3. Branch and Bound:
   • Branch and Bound can be used to solve Maximization Assignment Problems optimally by systematically exploring the solution space and pruning branches that cannot lead to the optimal solution. It guarantees finding the optimal solution but may be computationally intensive for large problem instances.
4. Heuristic Approaches:
   • Heuristic approaches, such as greedy algorithms, genetic algorithms, or simulated annealing, can be used to find suboptimal solutions to Maximization Assignment Problems in a reasonable amount of time, especially for large problem instances where exact methods may be impractical.

The Maximization Assignment Problem has various applications in resource allocation, project assignment, job scheduling, and task optimization, where the goal is to maximize overall benefit or profit while efficiently assigning tasks to agents.