Q.What is an assignment problem? State any three assumptions on which the standard assignment model is based, and explain why enumerating all possible assignments is impractical.

To turn a maximization assignment problem into a minimization one, subtract each element from the largest element in the matrix; the statement (deduct the smallest element from all others) is False.

The Hungarian method minimizes total cost. If the given assignment problem is a maximization problem (e.g. maximizing profit or output), it is first converted into an equivalent minimization problem by forming the loss/regret matrix:

$\text{new element} = (\text{largest element in the matrix}) - (\text{each element})$

Subtracting every entry from the maximum turns the highest payoff into the lowest cost ($0$), so minimizing this new matrix is equivalent to maximizing the original one.

The statement claims we "deduct the smallest element from all other elements," which is not the conversion rule (that operation resembles a row/column reduction step, and it does not switch a maximization objective to minimization). Hence the statement is incorrect.

The objective of a standard assignment problem is to assign an equal number of jobs to persons (one job per person) so as to minimise the total cost (or, equivalently in a maximisation version, maximise total profit).

In a balanced assignment problem there are $n$ jobs and $n$ persons, and each person is given exactly one job and each job is done by exactly one person. Among all such one-to-one assignments, the aim is to choose the one whose total cost is the least.

Examining the options:

• "...equal number of persons at maximum cost" — wrong objective (we do not maximise cost).
• "...equal number of persons at minimum cost" — correct: equal numbers, one-to-one, least total cost.
• "Only to maximize cost" / "Only to minimize cost" — incomplete; they omit the essential one-to-one assignment of equal numbers of jobs and persons.

Hence the objective is to assign the equal number of jobs to persons at minimum cost.

A balanced assignment problem has equal rows and columns, i.e. a square cost matrix.

In an assignment problem, rows represent jobs (or workers) and columns represent machines (or tasks). The problem is said to be balanced when the number of rows equals the number of columns — that is, the number of jobs equals the number of facilities to which they are assigned.

A matrix with an equal number of rows and columns is a square matrix. If the numbers differ (a rectangular matrix), the problem is unbalanced and must first be balanced by adding a dummy row or column with zero costs.