The Assignment Model
The assignment model is used to solve the traditional one to one assignment problem of assigning employees to jobs, employees to machines, machines to jobs, etc. The model is a special case of the transportation method. In order to generate an assignment problem it is necessary to provide the number of jobs and machines and indicate whether the problem is a minimization or maximization problem. The number of jobs and machines do not have to be equal but usually they are.
Objective function. The objective can be to minimize or to maximize. This is set at the creation screen but can be changed in the data screen.
The table below shows data for a 7 by 7 assignment problem. Our goal is to assign each salesperson to a territory at minimum total cost. There must be exactly one salesperson per territory and exactly one territory per salesperson.
The data structure is nearly identical to the structure for the transportation model. The basic difference is that the assignment model does not display supplies and demands since they are all equal to one.
The results are very straightforward.
Assignments. The 'Assigns's in the main body of the table are the assignments which are to be made. For example, Mort is to be assigned to Pennsylvania, Chorine is to be assigned to Florida, Bruce is to be sent to Canada, Beth is to work the streets of New York, Lauren is across the river in New Jersey, Eddie works Europe and Brian will work in Mexico.
Total cost. The total cost appears in the upper left cell. In this example the total cost is given by $191.
The assignments can also be given in list form as shown below.
The marginal costs can be displayed also. For example, if we want to assign Chorine to Pennsylvania then the total will increase by $6 to $197.
NOTE: To preclude an assignment from being made (in a minimization problem) you should enter a very large cost. If you enter an 'x' then the program will place a high cost in that cell.
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 agent-task 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
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.
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
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:
Usually the weight function is viewed as a square real-valued 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.