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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.
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
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.
The formal definition of the quadratic assignment problem is as follows: Given two sets, P ("facilities") and L ("locations"), of equal size, together with a weight function w : P × P → R and a distance function d : L × L → R. Find the bijection f : P → L ("assignment") such that the cost function:
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 ...
It is related to the quadratic assignment problem in the same way as the linear bottleneck assignment problem is related to the linear assignment problem, the "sum" is replaced with "max" in the objective function. The problem models the following real-life problem: There are a set of n facilities and a set of n locations.
While it is easy to verify whether a given assignment renders the formula true, [1] no essentially faster method to find a satisfying assignment is known than to try all assignments in succession. Cook and Levin proved that each easy-to-verify problem can be solved as fast as SAT, which is hence NP-complete.
It is required to perform all tasks by assigning exactly one agent to each task in such a way that the maximum cost among the individual assignments is minimized. The term "bottleneck" is explained by a common type of application of the problem, where the cost is the duration of the task performed by an agent. In this setting the "maximum cost ...