The document discusses applying linear programming to solve employee assignment problems. It describes the assignment problem as assigning personnel to tasks to optimize resources like time and cost. It provides an example of assigning 5 candidates to 4 work jobs to minimize the total time. It defines decision variables and constraints, formulates the objective function to minimize total time, and explains how to solve it using the Hungarian method algorithm. The algorithm involves subtracting row/column minimums, drawing lines to cover all zeros, checking for optimality, and adjusting uneovered elements iteratively until optimal. It also discusses converting a maximization assignment problem to minimization to use the Hungarian method.

1. Application of Linear Programming
in Employee Assignment

2. Assignment problem
• Assignment Problem is the fundamental optimization
problem which deals with assigning most suitable
personnel's to the tasks at hand so as to optimize the
resources like time , cost , Profit ,sales etc.
• In its most general form, the problem is as follows:
There are a number of agents and a number of tasks. Any
agent can be assigned to perform any task, incurring some
cost that may vary depending on the agent-task
assignment. It is required to perform all tasks by assigning
exactly one agent to each task in such a way that the total
cost of the assignment is minimized.

3. Assigment Problem to minimize Time
Tamsung India Pvt Ltd. has reserved 5 candidates to occupy 4
work jobs. The work jobs consist in driving 4 different tasks
related to mobile hardwares (one worker for each hardware).
It tested 5 workers at 4 machines, every worker doing the
same work at every machine, Obtaining the following times:
The HR manager needs to decide which candidates are to be
selected and what machines are to be assigned to them.

4. Determining decision variables and formulating
constraints
• Xij: action that the i worker is assigned to j machine. Ex:
X34 refers to the assignment of machine no. 4 to
candidate no. 3
• Each worker must be assigned only to one machine or
none if is not chosen.
• In each machine must be assigned to strictly one worker.
• Xij ≥ 0 & Xij are Booleans (0 or 1)

5. Determining objective function
Minimize
Z = 10X11 + 8X21 + 8X31 + 9X41 + 8X51 + 6X12 + 7X22
+ 6X32 + 7X42 + 7X52 + 6X13 + 6X23 + 5X33 + 7X43
+ 6X53 + 5X14 + 6X24 + 6X34 + 6X44 + 5X54
Assignment Problem is generally solved by
Hungarian Method algorithm . Simplex
Method algorithm becomes tedious for the
solving.

6. Hungarian Method Algorithm
• 1. Subtract the entries of each row by the row
minimum.
• 2. Subtract the entries of each column by the column
minimum.
• 3. Select rows and columns across which you draw
lines, in such a way that all the zeros are covered in
Minimum number of lines.

7. Hungarian Method Algorithm
• 4. A test for optimality.
• (i) If the number of the lines is n (number of rows
or columns) , choose a combination from the
modified cost matrix in such a way that the sum is
zero.
• (ii) If the number of the lines is < n, go to 5.
• 5. Find the smallest element which is not covered by
any of the lines. Then subtract it from each entry
which is not covered by the lines and add it to each
entry which is covered by a vertical and a horizontal
line. Go back to 3.

8. Hungarian method Algorithm
Special Case for a non square Matrix Problem
and Maximization Problem
• If we have, instead of a minimization problem,
a maximization problem, multiply the matrix C
by -1 and proceed as above. If C is not a
square matrix (there are more tasks than
workers or conversely), we have to augment C
into a square matrix by adding zero rows or
columns ( Dummy worker or Dummy m/c).

9. Maximization Case in an Assignment
Problem
• A ABC company has 4 Sales Representatives who are
to be assigned to 4 sales territories. The Monthly Sales
increased estimates for each sales representative for
different territories are as follows.
Sales Rep North South West East
Pratap 200 150 170 220
Prasad 160 120 150 140
Virat 190 195 190 200
Kumar 180 175 160 190
• HR needs to Assign Different Territories to different
Sales Representatives so as to maximise monthly sales.

10. Converting The maximization Problem
to minimization Problem
The maximization Assignment Problem needs to be converted to
Minimization Assignment Problem So that Hungarian Method
algorithm can be used to solve it. This is accomplished by multiply
every element of the matrix by by -1 .
Sales Rep North South West East
Pratap -200 -150 -170 -220
Prasad -160 -120 -150 -140
Virat -190 -195 -190 -200
Kumar -180 -175 -160 -190