Bipartite Matching Assignment Problem And Transportation
Solve the linear sum assignment problem.
The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a “worker”) and vertex j of the second set (a “job”). The goal is to find a complete assignment of workers to jobs of minimal cost.
Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost
\[\min \sum_i \sum_j C_{i,j} X_{i,j}\]
s.t. each row is assignment to at most one column, and each column to at most one row.
This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.
The method used is the Hungarian algorithm, also known as the Munkres or KuhnMunkres algorithm.
Parameters:  cost_matrix : array


Returns:  row_ind, col_ind : array

Notes
References
 http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html
 Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly, 2:8397, 1955.
 Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly, 3: 253258, 1956.
 Munkres, J. Algorithms for the Assignment and Transportation Problems. J. SIAM, 5(1):3238, March, 1957.
 https://en.wikipedia.org/wiki/Hungarian_algorithm
Examples
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. It consists of finding a maximum weight matching (or minimum weight perfect matching) in a weightedbipartite graph.
In its most general form, the problem is as follows:
 The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agenttask assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized.
If the numbers of agents and tasks are equal and the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called the linear assignment problem. Commonly, when speaking of the assignment problem without any additional qualification, then the linear assignment problem is meant.
Algorithms and generalizations[edit]
The Hungarian algorithm is one of many algorithms that have been devised that solve the linear assignment problem within time bounded by a polynomial expression of the number of agents. Other algorithms include adaptations of the primal simplex algorithm, and the auction algorithm.
The assignment problem is a special case of the transportation problem, which is a special case of the minimum cost flow problem, which in turn is a special case of a linear program. While it is possible to solve any of these problems using the simplex algorithm, each specialization has more efficient algorithms designed to take advantage of its special structure.
When a number of agents and tasks is very large, a parallel algorithm with randomization can be applied. The problem of finding minimum weight maximum matching can be converted to finding a minimum weight perfect matching. A bipartite graph can be extended to a complete bipartite graph by adding artificial edges with large weights. These weights should exceed the weights of all existing matchings to prevent appearance of artificial edges in the possible solution. As shown by Mulmuley, Vazirani & Vazirani (1987), the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph. Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least ½. For a graph with n vertices, it requires time.
Example[edit]
Suppose that a taxi firm has three taxis (the agents) available, and three customers (the tasks) wishing to be picked up as soon as possible. The firm prides itself on speedy pickups, so for each taxi the "cost" of picking up a particular customer will depend on the time taken for the taxi to reach the pickup point. The solution to the assignment problem will be whichever combination of taxis and customers results in the least total cost.
However, the assignment problem can be made rather more flexible than it first appears. In the above example, suppose that there are four taxis available, but still only three customers. Then a fourth dummy task can be invented, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it. The assignment problem can then be solved in the usual way and still give the best solution to the problem.
Similar adjustments can be done in order to allow more tasks than agents, tasks to which multiple agents must be assigned (for instance, a group of more customers than will fit in one taxi), or maximizing profit rather than minimizing cost.
Formal mathematical definition[edit]
The formal definition of the assignment problem (or linear assignment problem) is
 Given two sets, A and T, of equal size, together with a weight functionC : A × T → R. Find a bijectionf : A → T such that the cost function:
is minimized.
Usually the weight function is viewed as a square realvalued matrixC, so that the cost function is written down as:
The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms.
The problem can be expressed as a standard linear program with the objective function
subject to the constraints
The variable represents the assignment of agent to task , taking value 1 if the assignment is done and 0 otherwise. This formulation allows also fractional variable values, but there is always an optimal solution where the variables take integer values. This is because the constraint matrix is totally unimodular. The first constraint requires that every agent is assigned to exactly one task, and the second constraint requires that every task is assigned exactly one agent.
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