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This is an unbalanced assignment problem. One way to solve it is to invent a fourth dummy task, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it. This reduces the problem to a balanced assignment problem, which can then be solved in the usual way and still give the best solution to the problem.
This problem can be seen as a generalization of the linear assignment problem. [2] In words, the problem can be described as follows: An instance of the problem has a number of agents (i.e., cardinality parameter) and a number of job characteristics (i.e., dimensionality parameter) such as task, machine, time interval, etc. For example, an ...
The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods.It was developed and published in 1955 by Harold Kuhn, who gave it the name "Hungarian method" because the algorithm was largely based on the earlier works of two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry.
The formal definition of the bottleneck assignment problem is Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function: (, ()) is minimized.
For example, i = arr[i] = f() is equivalent to arr[i] = f(); i = arr[i]. In C++ they are also available for values of class types by declaring the appropriate return type for the assignment operator. In Python, assignment statements are not expressions and thus do not have a value. Instead, chained assignments are a series of statements with ...
In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the assignment problem. When the costs and profits of all tasks do not vary between different agents, this problem reduces to the multiple knapsack problem. If there is a single agent, then, this problem reduces to the knapsack problem.
Flow Shop Scheduling Problem; Generalized assignment problem; Integer programming. The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling
One example is the travelling salesman problem mentioned above: for each number of cities, there is an assignment of distances between the cities for which the nearest-neighbour heuristic produces the unique worst possible tour. [4] For other possible examples, see horizon effect.