conditions for assignment problem

Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Lesson 1. assignment problems - introduction.

Assignment problems – introduction – mathematical representation - comparison with transportation problems  - theorems of assignment problems. 

1. Introduction

The assignment problem is defined as  assigning each facility to one and only one job so as to optimize the given measures of effectiveness, when n facilities and n jobs are available and given the effectiveness of each facility for each job.

Let there be  n facilities (machines) to be assigned to n jobs. Let c ij is cost of assigning i th facility to j th job and x ij   represents the assignment of i th facility to j th job. If i th facility can be assigned to j th job, x ij =1, otherwise zero. The objective is to make assignments that minimize the total assignment cost or maximize the total associated gain.

Thus an assignment problem can be represented by n x n matrix which continues n ! possible ways of making assignments. One obvious way to find the optimal solution is to write all the n ! possible arrangements, evaluate the cost of each and select the one involving the minimum cost.  

However, this enumeration method is extremely slow and time consuming even for small values of n. For example, for n = 10, a common situation, the number of possible arrangements is 10! = 3,628,800. Evaluation of so large a number of arrangements will take a probability large time. This confirms the need for an efficient computational technique for solving such problems.

  2. Mathematical Representation of Assignment Model

  Mathematically, the assignment model can be expressed as follows:  

Let x ij   denote the assignment of facility i to job j such that

                        x ij = 0, if the i th facility is not assigned to j th job,

                        x ij = l, if the i th facility is assigned to j th job.

Then, the model is given by

            subject to constraints   

and                  x ij  = 0  or 1 (or x ij = x ij 2 ).

  If the last condition is replaced by x ij ≥0, this will be a transportation model with all requirements and available resources equal to 1.

3. Comparison with the transportation model

An assignment model may be regarded as a special case of the transportation model. Here, facilities represent the ‘sources’ and jobs represent the ‘destinations’. Number of sources is equal to the number of destinations, supply at each source is unity ( a i = 1 for all i ) and demand at each destination is also unity ( b j = 1, for all j ). The cost of ‘transporting’ (assigning) facility i to job j is c ij and the number of units allocated to a cell can be either one or zero. i.e., they are non-negative quantities.

However the transportation algorithm is not very useful to solve this model, when an assignment is made, the row as well as column requirements are satisfied simultaneously (rim conditions being always unity) resulting in degeneracy. Thus the assignment problem is a completely degenerate form of the transportation problem. In n x n problem, there will be n assignments instead of n + n –1 or 2 n –1– n = n –1 epsilons which will make the computations quite cumbersome. However, the special structure of the assignment model allows a more convenient and simple method of solution.

Difference between the transportation problem and the assignment problem

4. Theorems

The technique used for solving assignment model makes use of the following two theorems:

4.1. Theorem I

It states that in an assignment problem, if we add or subtract a constant to every element of a row (or column) in the cost matrix, then an assignment which minimizes the total cost on one matrix also minimizes the total cost on the other matrix”.

Let, c ij represent the original cost elements of the matrix. If constants u i and v j are subtracted from the i th row j th column respectively, the new cost elements will be

If Z is the original objective function, the new objective function will be

    Now with reference,                             

          or z' is minimum when z is minimum. This proves the theorem.

  Likewise, if in an assignment problem some cost elements are negative, we may convent them into an equivalent assignment problem where all the cost elements are non-negative by adding a suitably large constant to the elements of the relevant row.

  4.2. Theorem II

  It states “If all c ij ≥ 0 and we can find a set x ij = x’ ij ,  such that

  then this solution is optimal.

  The result follows automatically since as neither of c ij is negative, the value of

Hence, its minimum value is zero which is attained when x ij =x ij ' .  

Thus the present solution is optimal solution.

The above two theorems indicate that if one can create a new c ij ' matrix with zero entries, and if these zero elements, or a subset thereof, constitute feasible solution, then this feasible solution is the optimal solution.

  Thus the method of solution consists of adding and subtracting constant from rows and columns until sufficient number of c ij ' s become zero to yield a solution with a value of zero. 

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Generalized Assignment Problem

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• O. Erhun Kundakcioglu 3 &
• Saed Alizamir 3  

15 Citations

Article Outline

Introduction

  Multiple-Resource Generalized Assignment Problem

  Multilevel Generalized Assignment Problem

  Dynamic Generalized Assignment Problem

  Bottleneck Generalized Assignment Problem

  Generalized Assignment Problem with Special Ordered Set

  Stochastic Generalized Assignment Problem

  Bi-Objective Generalized Assignment Problem

  Generalized Multi-Assignment Problem

  Exact Algorithms

  Heuristics

Conclusions

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Combinatorial clustering: literature review, methods, examples.

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Kundakcioglu, O.E., Alizamir, S. (2008). Generalized Assignment Problem . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_200

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