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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 following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
Various branch-and-bound algorithms, which can be used to process TSPs containing thousands of cities. Solution of a TSP with 7 cities using a simple Branch and bound algorithm. Note: The number of permutations is much less than Brute force search. Progressive improvement algorithms, which use techniques reminiscent of linear programming. This ...
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.
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Branch and price is a branch and bound method in which at each node of the search tree, columns may be added to the linear programming relaxation (LP relaxation). At the start of the algorithm, sets of columns are excluded from the LP relaxation in order to reduce the computational and memory requirements and then columns are added back to the LP relaxation as needed.
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