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# 数学代写|运筹学代写Operations Research代考|MTH360 ASSUMPTIONS

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## 数学代写|运筹学代写Operations Research代考|ASSUMPTIONS

The solution of the above assignment problem, formulated as shown in Table $6.2$, follows certain assumptions which are:

The number of employees to be assigned a job is equal to the number of jobs available. Thus, there would be $\mathrm{n}$ assignments.

Each employee, though qualified to do all jobs, will be assigned only one job.

Similarly, though each job can be performed by any of the employees, each task has to be performed by only one employee.

Performing a job incurs cost. Cell elements indicate cost associated with each possible assignment. As each employee can perform each task, the matrix indicates cost for each possible combination of employee and task.

As pointed by 2 and 3, any employee can be assigned to any job incurring a certain cost. Thus, the purpose of the assignment model is to identify that combination which results in minimization of total cost.

However, assumptions 1,2 and 3 are mostly not applicable in realistic situations. For instance, fewer people are available to work on more machines; much more applications are received for a limited number of positions; limited classrooms are available for assigning a large number of periods, etc. Thus, certain variations are required in the assignment model, which will be discussed in later sections.

## 数学代写|运筹学代写Operations Research代考|COMPARISON WITH TRANSPORTATION MODEL

The assignment problem is a special case of the transportation model. This is better understood by comparing the assignment model shown in Table $6.1$ with transportation model shown in Table $5.1$ of the chapter on Transportation model. This is reproduced here as Table 6.3.
First, let’s discuss similarities between the two models.
Similarities:

First, both models have a limited and fixed number of suppliers or employees and a limited and known number of demand centres or jobs.

Second, both models deals with aspect of combining resources with demand and tasks. In the case of transportation, the number of units from different supply centres is allocated to fulfil demand of various destinations in a manner to minimize transportation cost. Whereas, in the assignment model, the same kind of allocation is done based on certain criteria of either minimizing cost or minimizing time to do a specific job.
Difference:

But both models have one major difference. In the case of transportation problem, different supply and demand centres can have different capacities of output and requirements respectively, shown by $b_i$ and $c_j$ in Table 6.3. Whereas in the case of assignment problem, all supplies and demands equal to one. This is because of fundamental of assignment problem, i.e. one worker can be assigned to do only job at a time, one geographic area can be assigned only one store or department and so on. Thus, the cost incurred in transporting ‘ $x$ ‘ units from supply centre 1 to demand centre 1 would be $c_{11} x_{11}$. For the assignment model, it would be the same but the value of $x_{11}$ would be either 1 or 0 .The optimality of the transportation model is checked by equating total available units with total requirement. Whereas in the case of assignment, it is checked by equating the number of persons with the number of tasks. Thus, the condition of suppliers/employees to be equal to demand centres/ tasks in assignment is essential, though not a necessary condition.

Finally, feasibility of transportation is checked by equating $m+n-1$ with the number of basic variables (allocated cells). If it is not fulfilled then deficit variables are termed as degenerate. Thus in the case of assignment where $m=n$ number of allocations/assignments should be equal to $2 n-1$. However, because the assignment problem involves assigning $n$ employees to $\mathrm{n}$ jobs so, there would be only $\mathrm{n}$ assignments. So deficit, i.e. $\mathrm{n}-1$ variables would degenerate. This is depicted by $\mathrm{x}_{\mathrm{ij}}=0$, implying no assignment.

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