# Assignment Problem

```Assignment Problem
Concepts
Concepts
What is the Assignment Problem?
The assignment problem deals with assigning machines to tasks,
workers to jobs, soccer players to positions, and so on.
The goal is to determine the optimum assignment that,
minimizes the total cost or maximizes the team effectiveness.
Concepts
What is the Assignment Problem?
Suppose we have “n” resources to which we want to assign to
“n” tasks on a one-to-one basis.
Suppose also that we know the cost of assigning a given resource
to a given task (i.e. cost matrix).
We wish to find an optimal assignment - one which minimizes
total cost.
Assignment Example
Assignment Example
We have 3 workers and 3 tasks
Decision: Which worker to assign to which task?
Objective: Minimize cost in wages to get all 3 tasks done
Estimated Wages Cost of Possible Assignments
Worker
Dish Washing
Car Cleaning
Laundry
John
Peter
Robert
Rs. 11
Rs. 8
Rs. 9
Rs. 14
Rs. 10
Rs. 12
Rs. 6
Rs. 11
Rs. 7
Assignment Example
Workers (Supply)
John
Dish Washing
Peter
Car Cleaning
Robert
Laundry
Possible
Assignments
The “flow” on each arc is either “0” (false) or “1” (true)
Assignment Example
Decision Variables:
Xij = “1” if worker “i” is assigned to task “j” , else it is “0”
Objective Function:
Min 11Xjd + 14Xjc + 6Xjl
+ 8Xpd + 10Xpc + 11Xpl
+ 9Xrd + 12Xrc + 7Xrl
Constraints:
(supply nodes)
- (11Xjd + 14Xjc + 6Xjl) = - 1
- (8Xpd + 10Xpc + 11Xpl) = - 1
- (9Xrd + 12Xrc + 7Xrl) = - 1
Estimated Wages Cost of Possible Assignments
Worker
Dish
Washing
Car
Cleaning
Laundry
John
Rs. 11
Rs. 14
Rs. 6
Peter
Rs. 8
Rs. 10
Rs. 11
Robert
Rs. 9
Rs. 12
Rs. 7
Constraints:
(demand nodes)
11Xjd + 8Xpd + 9Xrd = 1
14Xjc + 10Xpc + 12Xrc = 1
6Xjl + 11Xpl + 7Xrl = 1
Solution to the Assignment Problem
There are 4 methods of solving the assignment problem:
1. Complete Enumeration Method
2. Transportation Method
3. Simplex Method
4. Hungarian Method
Hungarian Method
Hungarian Method
Hungarian Method handles assignment method in the most
efficient way.
It is based on the concept of opportunity cost.
Theorem: If a number is added to or subtracted from all of the
entries of any one row or column of a cost matrix, then on
optimal assignment for the resulting cost matrix is also an
optimal assignment for the original cost matrix.
Hungarian Formulation of Model
The Mathematical Model: Let cij be the cost of assigning the
ith resource to the jth task.
the n &times; n matrix
We define the cost matrix to be
- An assignment is a set of n entry positions in the cost matrix,
no two of which lie in the same row or column.
- An assignment with the smallest possible cost is called an
optimal assignment.
Maximization / Minimization Problems
• In some situations, the assignment problem may call for
maximization of profit, revenue, etc. as the objective.
• Examples:
– Minimization:
Find the optimum assignment of products to machine
setups resulting in minimum cost/time.
– Maximization:
Find the optimum assignment of salesmen to markets
resulting in maximum revenue/profits.
Balanced / Unbalanced Problems
• In some situations the number of supply nodes do not
identically match the number of demand nodes resulting in
unbalanced problems.
• Examples:
– Balanced:
5 workers and 5 machines
– Unbalanced:
4 workers and 6 machines
In unbalanced situations, either of the nodes will remain idle.
Prohibited Assignment Problems
• In some situations some specific supply nodes may not be
able to match some specific demand nodes resulting in
constrained assignment problems.
• Examples:
– Constrained: Worker ‘A’ cannot perform Task ‘Y’
To cope with constrained situations, the cost of performing
that job by such a person is taken extremely large (which is
written as “M”)
Hungarian Method of Solution
The Hungarian Method:
The following algorithm
applies the theorem to
a given n &times; n cost
matrix to find an
optimal assignment.
Numerical
Minimization, Balanced Type
Maximization, Unbalanced Problem
Multiple Assignment, One Optimal Solution
Constrained Assignment, Multiple Optimal Solutions
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