Introduction to Assignment Problem
In assignment problems, the aim is to allot a number of jobs to the equivalent number of persons at a least cost of maximum profit.
Assume there are 'n' jobs to be done and 'n' persons are available for completing these jobs. Suppose each person can do each job in a time with a varying or fluctuating degree of efficiency. Assume cij be the cost of ith person allotted to jth job. Then the problem is to discover an assignment in a way that the total cost for doing all jobs is least. These problems are called as assignment problems.
These problems may comprise of classes to the rooms, assigning men to offices or problems to the research team etc.
Mathematical formulation or equation
Cost matrix: cij= c11 c12 c13 ... c1n
c21 c22 c23 ... c2n
.
cn1 cn2 cn3 ... cnn
Subject to restrictions of the form
Where xij symbolizes that jth job is to be allotted to the ith person.
This exclusive structure of assignment problem permits a more easy method of solution as compared to simplex method.
Algorithm for Assignment Problem (Hungarian Method)
Step 1
Subtract the minimum of every row of the efficiency matrix, from all the elements of the subsequent rows (Row reduced matrix).
Step 2
Then change the resultant matrix by subtracting the minimum element of each column with all the elements of the respective columns. Hence first modified matrix is achieved.
Step 3
Make the minimum number of horizontal and vertical lines to cover up all the zeroes in the resultant matrix. Assume the minimum number of lines be N. Then there may be two possibilities
Step 4
Find out the least element in the matrix, not covered by N lines. Subtract this minimum element with all uncovered elements and add the identical element at the intersection of vertical and horizontal lines. Therefore the second modified matrix is achieved.
Step 5
Repeat step 3 and step 4 again and again until minimum number of lines turns out to be equal to number of rows (columns) of the given matrix i.e. N = n.
Step 6
To prepare zero assignment - test the rows successively until a row-wise precisely single zero is found; mark this zero with '1''to prepare the assignment. Then, mark an 'X' over all zeroes if staying in the column of the marked zero, showing that they cannot be taken for subsequent assignment. Repeat in this manner until all the rows have been checked. Continue the similar process for the columns as well.
Step 7
Repeat the step 6 consecutively until one of the following situations occur
Step 8
Exactly one marked zero in each column and each row of the matrix is achieved. The assignment matching to these marked zeroes will provide the optimal assignment.
Example 1
A department head has four subordinates and four jobs need to be done. Subordinates vary in efficiency and tasks differ in their intrinsic complexity. Time each man would require to complete each task is given in the effectiveness matrix. How the tasks should be assigned to each person in order to minimize the total man-hours?
Tasks
Subordinates
I
II
III
IV
A
8
26
17
11
B
13
28
4
C
38
19
18
15
D
24
10
Answer
Row Reduced Matrix
0
9
3
22
23
16
14
I Modified Matrix
N = 4, n = 4
As N = n, we proceed to zero assignment
Zero assignment
Total man-hours = 8 + 4 + 19 + 10 = 41 hours
theory and lecture notes of okun’s law all along with the key concepts of okun’s law, coefficient of okun’s law, measuring the macroeconomy and formula for okun’s law. tutorsglobe offers homework help, assignment help and tutor’s assistance on okun’s law.
Junction Field Effect Transistors tutorial all along with the key concepts of JFET characteristic curves, Operation of JFET device, JFET as an amplifier, bipolar junction transistor, single stage class A amplifier circuits
Theory and lecture notes of Central Limit Theorem all along with the key concepts of Central limit theorem, Sampling Distribution, Finite Population Correction Factor. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Central Limit Theorem.
tutorsglobe.com equi-marginal utility assignment help-homework help by online significance of necessaries tutors
www.tutorsglobe.com offers Object Oriented Design homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
tutorsglobe.com morphology assignment help-homework help by online mycoplasma tutors
origin, evolution and structure of galaxies tutorial all along with the key concepts of active galactic nuclei, galaxy properties and correlations, classification of galaxies and cosmic structure formation
tutorsglobe.com bohr quantum condition assignment help-homework help by online wave nature of electrons tutors
tutorsglobe.com lysogenic cycle assignment help-homework help by online life cycle of a phage tutors
Physical Instrumentation in Environmental Sciences tutorial all along with the key concepts of Techniques for Heavy Metals Analysis, Neutron Activation Analysis, Techniques for Organic Residues Analysis, Gas Chromatography, Gas chromatograph detectors
Electrolytic conductance tutorial all along with the key concepts of Conductance of Electrolytes, Specific Conductance, Equivalent Conductance, Molar Concentration, Variation of Conductance with Temperature
Einstein Postulates and Lorentz Transformation tutorial all along with the key concepts of Einstein's Postulates, Simultaneity, Lorentz Transformation, Relativity of Simultaneity, Symmetry of Relations, Importance of Lorentz factor, Invariance of Maxwell's Equation
Theory and lecture notes of Natural Rate of Unemployment all along with the key concepts of natural rate of unemployment, Aggregate supply, philips curve. Tutorsglobe offers homework help, assignment help and tutor’s assistance on natural rate of unemployment.
theory and lecture notes of systolic algorithms all along with the key concepts of systolic algorithms, models of computation, sorting networks and theorem of 0-1 principle. tutorsglobe offers homework help, assignment help and tutor’s assistance on systolic algorithms.
tutorsglobe.com radiocarbon dating assignment help-homework help by online nuclear chemistry tutors
1960444
Questions Asked
3689
Tutors
1443363
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!