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# Assignment Model - Introduction | Linear Programming Problem (LPP)

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## What is Assignment Model?

→ Assignment model is a special application of Linear Programming Problem (LPP), in which the main objective is to assign the work or task to a group of individuals such that;

i) There is only one assignment.

ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

→ In assignment problem, the cost of performing each task by each individual is known.
→ It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.

#### For example:

• Suppose there are 'n' tasks, which are required to be performed using 'n' resources.

• The cost of performing each task by each resource is also known (shown in cells of matrix) • In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.

→ Assignment Model is a special application of Linear Programming (LP).

→ The mathematical formulation for Assignment Model is given below: → Let, $\text {C}_{ij}$ denotes the cost of resources 'i' to the task 'j'; such that

\begin{aligned} x_{ij} &= 1 \ ; \text{if} \ \ i^{th} \text{resource is original to } j^{th} \text{task}\\ x_{ij} &= 0 \ ; \text{if} \ \ i^{th} \text{resource is not original to } j^{th} \text{task} \end{aligned}  → Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.

$\therefore \text{Minimize Z}= \displaystyle\sum_{j=1}^n \ \sdot \ \displaystyle\sum_{i=1}^n \ C_{ij} \ \sdot \ x_{ij}$

→ Subjected to constraint;

(i) For all $j^{th}$ task, only one $i^{th}$ resource is possible:

$\therefore \displaystyle\sum_{j=1}^n \ x_{ij} = 1 \quad \text{(i=1,2,3, ...,n)}$

(ii) For all $i^{th}$ resource, there is only one $j^{th}$ task possible;

$\therefore \displaystyle\sum_{i=1}^n \ x_{ij} = 1 \quad \text{(j=1,2,3, ...,n)}$

(iii) $x_{ij}$ is '0' or '1'.

## Types of Assignment Problem:

#### (i) Balanced Assignment Problem

• It consist of a suqare matrix (n x n).
• Number of rows = Number of columns

#### (ii) Unbalanced Assignment Problem

• It consist of a Non-square matrix.
• Number of rows $\not=$ Number of columns

## Methods to solve Assignment Model:

#### (i) Integer Programming Method:

• In assignment problem, either allocation is done to the cell or not.

• So this can be formulated using 0 or 1 integer.

• While using this method, we will have n x n decision varables, and n+n equalities.

• So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.

• So this method becomes very lengthy and difficult to solve.

#### (ii) Transportation Methods:

• As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.

• In transportation methods (NWCM, LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)

• But, here in assignment problems, the matrix is a square matrix (m=n).

• So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5

• But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.

• So, if are we will use transportation methoods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.

• So, the method becomes lengthy and time consuming.

#### (iii) Enumeration Method:

• It is a simple trail and error type method.

• Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.

• For 3 x 3 matrix, the total possible trails are 3!
So total 3! = 3 x 2 x 1 = 6 trails are possible.

• The assignments which gives minimum cost is selected as optimal solution.

• But, such trail and error becomes very difficult and lengthy.

• If there are more number of rows and columns,
( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible)
then such methods can't be applied for solving assignments problems.

#### (iv) Hungarian Method:

• It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.

• It is also know as Reduced matrix method or Flood's technique.

• There are two main conditions for applying Hungarian Method:

(1) Square Matrix (n x n).
(2) Problem should be of minimization type.

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