“The different assignment problems usually demand you to assign n number of objects to m other spaces in an injective manner”.
Most of the assignment problems usually contain two components, the basic structure and the function which is to be optimized. By optimization, t is implied that the costs of performing certain tasks will be minimized and total profit will be maximized.
There are lots of methods for solving these problems. In real life too, you have to know how to solve assignment problem step by step. This method is especially useful in assigning routes to vehicles, assigning different workers in different sectors of a company etc.
Tables in use
Different assignment problems have tables or matrices associated with them. The row of these tables, usually contain people be assigned. The columns in turn contain tasks that these people would be doing. You have to take into consideration different activities as well as costs involved in these operations.
You should realize that a cost matrix is very similar to a T.P. A slight difference existing here is that requirements present in these destinations are unity.
Methods to be used
By use of Flood’s technique or Hungarian technique of minimization, we can get a really efficient means of getting our hands on optimal solution. You do not really require making comparisons of all available options. This method works by means of matrix reduction. It means that by addition or subtraction of required numbers, you will be getting yourself a matrix containing opportunity costs. These opportunity costs are really useful for solving real life scenarios.
Hence you have to know “how to solve assignment problem step by step”. By using opportunity costs, you will know the difference between assigning any job to any worker in contrast to making the best assignment.
Steps involved in solving problem
- Firstly, you will require finding opportunity cost by performing subtraction of smallest number present in each row of a table from each number in that row. After this, you require performing similar subtractions in the columns of your table.
- You require creating opportunity costs by beginning your examination of rows until you find one where there is an unmarked zero. This zero needs to be enclosed within a box, according to which assignment will be made. You have to cross off the other zeroes as they will not be considered further. All rows need to be examined in this manner.
- After examination of rows, the columns need to be examined in a similar manner. The zeroes in columns would be marked by putting squares around them. All the different columns would have to be examined for getting desired results.
- You have to keep on repeating these operations until all zeroes present in the columns are crossed off. Finally, you will have one assignment in the rows as well as columns. This means that you have got optimal assignment.
Revising opportunity cost
The table that you have got after following the above steps is also called the reduced table. You have to now draw vertical as well as horizontal lines to cover zeroes in this given table. All rows that do not have assignments are marked now.
These steps are repeated until you do not have any more rows or columns to mark. To exactly know how to solve assignment problem step by step, you have to draw lines carefully. If these lines are equal in number to rows or columns, then you have an optimal solution in your hands. Sometimes you have to make adjustments in your table and shift zeroes from one location to a new one.
This new location is usually uncovered in nature. It is expected that it will bring forward a zero based opportunity cost. This can be achieved by you when you subtract the smallest valued number not covered by a line from all numbers not covered by a line.
This smallest number is thereafter added to all numbers found at intersection of drawn lines. As you can see, lots of complex operations are involved in solving these problems. Therefore, online help for solving them is very much required. You can get online tutors for these problems quite easily too.
Algorithm for better problem solving
You begin solving your problem by firstly setting up a cost table. You have to then decide whether it is minimization problem or maximization problem. If it is a maximization problem, then to needs to be converted to a minimization problem by doing subtractions. All the different elements are required to be subtracted from the largest value element of your table.
If it is not a maximization problem, you have to check whether it is balanced in nature or not. If the answer is no, then balance will be brought by adding dummy rows or columns. Having an idea of how to solve assignment problem step by step, is further enhanced by use of these algorithms.
Variations in problems
As a part of your assignments, you will be getting many types of assignment problems. Not only will the type of data will vary in these problems, but the method of solving can also vary sometimes. In unbalanced assignment problems, for example number of facilities is never equal to number of available jobs.
For using Hungarian method for solving, you require a square matrix. Therefore, some imaginary coats or facilities need to be added to the table for continuing with this method. While you are solving this table, the imaginary values are also treated like real ones.
In some problems, restrictions creep into framing of problems. Therefore, a specific facility cannot be assign to a particular job. These problems are usually solved by putting in infinite cost or very heavy cost in a particular cell.
This problem would thereafter be undergoing further considerations. You must always remember to change maximization problems to minimization based one. This is done by subtraction of all different elements from the one having highest value. You may also proceed by multiplying all matrix elements by -1. Knowing about how to solve assignment problem step by step, is really useful.
Special cases in assignment problems
The problems associated with travelling salesman, requires you to pay particular attention and solve accordingly. You can begin solving these problems like regular assignment problems. If the solution is cyclic in nature, then that is your answer.
However noncyclic solution requires you to select a west side entry having values other than zero. This row and column needs to be deleted and assignment needs to commence throughout the matrix again. If solution is still not cyclic, then you have to aim for a higher value, till cyclicity is achieved.
Step by step MOA method for solving assignment problem
The MOA method for solving assignment problem is a bit different from the Hungarian method. It is also known as matrix one’s method as assignments are made here in terms of 1s. Some 1s are placed in the assignment matrix and then a complete assignment created in terms of these 1s. It is essential to understand how to solve assignment problem step by step, in order to be ahead of your peers. When you have more than one way of solving a problem, you can use them as and when required.
- You will firstly require finding the minimum element in each row. And note it down on the right side of your matrix.
- After this, you require diving each element present in the i-th row by i. this operation will create at least one 1s in each row.
- The minimum element present in each column is then found out and similar operations are repeated. Hence 1s are now created in columns too.
- You have to now cover all 1s you see in your matrix using minimum number of lines. If you find that the number of lines drawn equals to order of your matrix, then you can get complete assignment.
- If complete assignment is not possible, then you should select that element which is smallest in value and do not lie on these lines. You need to keep repeating the above two steps till you get complete assignment.
Through this blog, you have come to know about the most efficient Hungarian approach to solving assignment problem. Other than that, MOA method can also be used for solving both maximization and minimization problems. In this method you have to solve the matrix with respect to 1s. Everything is got here in terms of these 1s. Both methods, are more interesting than all known previous methods of assignment solving.
With all these methods used for solving assignment problems, it is natural for students to get confused regarding which method to follow. Hence we provide you in depth details regarding these methods. If you have any doubts regarding “how to solve assignment problem step by step”, then we provide best services. You can really count on us to give you the best in business.