Q.To convert an assignment problem into a minimization problem, the smallest element in the matrix is deducted from all other elements.

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.

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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. …

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.

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