In an Unbalanced Assignment Problem, the number of tasks (or agents) is not equal to the number of agents (or tasks), resulting in an imbalance between supply and demand. One set (tasks or agents) has more elements than the other. This scenario often arises in real-world applications where the requirements or capacities of tasks and agents differ. Here’s how the Unbalanced Assignment Problem can be formulated:

Formulation:

• Let
  $$n
  

   

  m be the number of agents. Without loss of generality, assume
  $\mathrm{<br>}$

   

  n>m.

• Let
  
  ${\mathrm{<br>}}_{}$

   

  cij​ represent the cost or profit of assigning task
  $\mathrm{<br>}$

   

  i to agent
  $�$

   

  j.

• Let
  
  ${}_{}$

  xij​ be a binary decision variable, where
  ${\mathrm{<br>}}_{}$

   

  xij​=1 if task
  $\mathrm{<br>}$

  i is assigned to agent
  $\mathrm{<br>}$

   

  j, and
  ${\mathrm{<br>}}_{}=0$

   

  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 at most one agent:
     
     ${\sum }_{\mathrm{<br>}=1}^{\mathrm{<br>}}{\mathrm{<br>}}_{}\le 1\text{for }\mathrm{<br>}=1,2,\dots ,\mathrm{<br>}$

      

     ∑j=1m​xij​≤1for i=1,2,…,n

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

      

     ∑i=1n​xij​≤1for j=1,2,…,m

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

     xij​≥0 for all
     $$

     i,j.

3. Binary Constraints:

   • xij​ must be binary: ∈{0,1}

      

     xij​∈{0,1} for all
     $\mathrm{<br>}$

      

     $i,j$.

4. Balancing Constraints:
   • In case of an excess of tasks, add dummy agents with zero cost/profit. Similarly, in case of an excess of agents, add dummy tasks with zero cost/profit.

Solution Methods:

The Unbalanced Assignment Problem can be solved using various methods, including adaptations of those used for the balanced assignment problem:

1. Adapted Hungarian Algorithm:
   • Adapt the Hungarian Algorithm to handle unbalanced assignment problems by adding dummy tasks or agents to balance the problem. These dummy elements ensure that there is a one-to-one assignment between tasks and agents.
2. Linear Programming Formulation:
   • Formulate the unbalanced assignment problem as a linear program and solve it using linear programming techniques such as the simplex method or interior point methods.
3. Heuristic Approaches:
   • Use heuristic approaches such as greedy algorithms, genetic algorithms, or simulated annealing to find suboptimal solutions for unbalanced assignment problems in a reasonable amount of time.
4. Modified Algorithms:
   • Modify existing algorithms for the balanced assignment problem to accommodate unbalanced scenarios by adjusting the allocation rules and constraints to handle the excess tasks or agents.

Addressing unbalanced assignment problems requires careful consideration of the excess tasks or agents and the introduction of dummy elements to ensure a feasible solution. The choice of solution method depends on factors such as problem size, complexity, and desired solution quality