Assignment Model | Linear Programming Problem (LPP) | Introduction

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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)

Fig 1-assigment model intro

  • 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:

Fig 1-assigment model intro

→ Let, Cij\text {C}_{ij} denotes the cost of resources 'i' to the task 'j'; such that

xij=1 ;if  ithresource is original to jthtaskxij=0 ;if  ithresource is not original to jthtask\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.

Minimize Z=j=1n  i=1n Cij  xij\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 jthj^{th} task, only one ithi^{th} resource is possible:

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

(ii) For all ithi^{th} resource, there is only one jthj^{th} task possible;

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

(iii) xijx_{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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