The assignment problem is concerned with the linear programming problem. It aims to assign the resources to an equal number of activities for cost minimization and profit maximization. In general, the assignment problem is:
Cost minimization or profit maximization by dividing a certain number of jobs by a certain number of employees. Here, we take into account the effectiveness of each employee for a job as well.
Further, assignment problems occur when human/machine resources have different capabilities to perform different tasks. Thus, in such scenarios, variables like profit, loss and cost regarding different tasks are different. There are many problems in management sciences that resemble the assignment problem. Like:
Suppose a manager has four employees to assign four different jobs. if the manager knows the cost of each assignment and has to divide the task among four employees so that the total cost of the assignment is low. This management assignment resembles the balance assignment problem task.
Let’s consider another one:
A textile industry operator has four different types of sewing machines in four different warehouses. And, he has to allot each machine to a separate employee after knowing the speed of each machine. Then, the balance assignment problem or task management approach demands a manager to assign the task to separate employees to minimize the time required.
All these examples are well-suited to the balance assignment problems. The balance assignment problem is a type of general assignment problem. Through, it has a few other examples as well. Before describing the types of assignment problems, let’s first introduce the Hungarian method.
The Hungarian method is a popular technique to solve assignment problems. It is one of the most efficient methods that find the optimal solutions without comparing different situations. In the Hungarian method, if we write the time or cost for completion of a task in matrix form, then it will be the cost matrix. The working principle of this method explains: if we add a constant number to all elements present in a matrix, then a solution of the assignment problem will remain the same as the original problem, and the reverse is also possible.
Hungarian method suggests that we can make another cost matrix by adding any constant to all elements to the original matrix, only if the total cost and time is zero. In this way, the optimum solutions will remain unchanged. Likewise, if the purpose is to maximize the effectiveness, another aspect of the Hungarian method can be used. Here, the Hungarian methods are applicable to the final cost matric: instead of the original one.
Type Of Assignment Problems:
Unbalanced Assignment Problem:
The unbalanced assignment problems are opposite to the balanced assignment problems. In such assignments, the number of jobs and the facilities are not equal. In some cases, the number of jobs exceeds the number of faculties. While in others, the number of the facilities exceeds the number of jobs. In either case, to develop an unbalanced assignment problem, the dummy facilities, a balanced problem or a dummy job (with zero time and cost) helps.
Balanced Assignment Problem:
Balanced Assignment Problems are the problems where the number of jobs/tasks is equal to the number of resources. All the above examples fall into the balanced assignment problems.
Unlike the balanced assignment problem, a dummy job is a supposed job/facility. Dummy jobs or facilities (with zero time or cost) help us to introduce the unbalanced assignment problem.
Infeasible assignments problems arise in the assignment cost matrix. We call this assignment infeasible because it occurs in a cell where a certain facility is not capable of doing a specific task. Another condition is when a certain machine has no build-in configuration to perform a task. Here, the solution of the balanced assignment problems must consider the restrictions as well. Such a problem helps us to sort the limitation along with cost and effectiveness. However, if still someone faces any difficulty, he/she can hire best assignment writing services in UK.
Steps To Use The Hungarian Method For Solving A Balance Assignment Problem:
The first task in finding the balance assignment problem solution is to count the number of facilities or jobs. For this, you must make a cost table from the problem under study. If they are equal, then you can move to step 2. If not, you have to add a dummy job/facility.
If the problem under study is unbalanced, add a dummy row or column to turn it into a balanced assignment problem. To do so, you should allot the zero to all cells in the dummy column or row.
Step 2 is to find the opportunity cost table. For this, you need to find the smallest element in rows. After locating the smallest number, subtract that number from the other elements of that row. This step will give you a modified matrix from where you again have to locate the smallest number (from column). And subtract the number from all elements of the column. Now, you will have at least one zero value in rows and columns. At this point, you will have an opportunity cost matrix.
Step three is to make an assignment in the opportunity cost matrix. For this, observe rows to find one unmarked zero. Take its square and eliminate all other zeros from the same rows or columns. Repeat this step for all columns until all zeros are assigned or stuck to get answer.
If the member of the allotted cells is equal to the number of columns in the row, this will be the optimal solution. The total cost of this solution is obtained by adding the original cost figures to the assigned cells.
If you got an optimal solution, then stop here. If not, then you have to repeat the procedure with a little modification. Though, these are some general steps that help you to find a correct solution of balanced assignment problems. Further, these steps may differ depending upon the purpose- minimization goals or maximization goals.